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Warren Platts
2007-Nov-13, 04:56 PM
That Bode's Law is an ATM concept is evidenced by the fact that the main planetary science journal Icarus does not accept manuscripts that discuss Bode's Law.

I take it that the essence of Bode's Law is that planetary spacing is approximately logarithmic in scale.

One problem is that there's not a lot of data sets to test it on. However, five exoplanets have been discovered around 55 Cancri A. Amazingly, the semimajor axes are well predicted by the following simple relation:




a = 0.039en-1

where a is the semimajor axis, e is the natural logarithm constant (2.7 ...), n is the number of the planet starting from it's sun, and the 0.039 is the semimajor axis of the closest planet in AU's.

The tables show the data for 55 Cancri and the Sol system (1st column = observed value; 2nd column = predicted value; 3rd column = the square of the difference between the first two columns; data from the Wikipedia):



55 Cancri



0.038 --- 0.039 --- 0.0000010
0.115 --- 0.106 --- 0.0000808
0.240 --- 0.288 --- 0.0023207
0.781 --- 0.783 --- 0.0000055
????? --- 2.129 --- ??????????
5.770 --- 5.788 --- 0.0003281
????? --- 15.73 --- ??????????



Solar system (planet #5 is Ceres)



00.400 --- 0.390 --- 0.0001
00.700 --- 0.720 --- 0.0004
01.000 --- 1.000 --- 0.0000
01.600 --- 1.520 --- 0.0064
02.800 --- 2.770 --- 0.0009
05.200 --- 5.200 --- 0.0000
10.000 --- 9.540 --- 0.2116
19.600 --- 19.20 --- 0.1600


Thus the average for the squared differences for 55 Cancri is 0.0005472, whereas that for the Sol system is 0.0474250. Even if we just take the average of the squared differences from Mercury through Jupiter, the average is 0.00130--still higher than that predicted for 55 Cancri.

It is striking that the very first extrasolar test of Bode's Law-like relations agrees with the predictions better than our own solar system does.

If Bode's Law like spacings are ubiquitous, that suggests that logarithmic spacings are not in fact merely coincidental, but result from a deep, underlying physical explanation.

The best reference so far I've found is Graner and Dubulle (1994). (http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994A%26A...282..262G&data_type=PDF _HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf)

Edit: this thread is a continuation of a discussion started in the Is there a pattern to how our solar system is laid out? (http://www.bautforum.com/astronomy/66784-there-pattern-how-our-solar-system-laid-out.html) thread in the Astronomy section.

Warren Platts
2007-Nov-13, 06:19 PM
This is interesting: earlier I wrote:


I think there is something to Bode's law other than mere coincidence, but its more like a law of "biology", than a law of physics. Protoplanets "compete" for mass. As larger and larger protoplanets "grow" by "ingesting" matter out of the primordial disk, there is a limit to the "search space" they can cover governed by the average orbital eccentricity. In other words, Bode's Law is the result of "niche partitioning" among the major planets.

Then this from Graner and Dubrulle (1994):


Accretion produces a Titius-Bode law via the "feeding zone" effect (Vityazev et al. 1977): diring its rotation around the central object, a large planetesimal at a mean distance r sweeps and annular area of radial extent 2er where e is the eccentricity of the orbit.

Yet another example of teleological reasoning in astronomy.

Celestial Mechanic
2007-Nov-13, 06:26 PM
[Snip!] Yet another example of teleological reasoning in astronomy.
You're on a roll, Mr. Platts -- why spoil it by dragging in that teleology nonsense?

antoniseb
2007-Nov-13, 06:48 PM
Seeing the TB relation in 55 Cancri is pretty cool. I agree that it will be interesting to see if this holds up for broad classes of stars.

Warren Platts
2007-Nov-13, 08:15 PM
You're on a roll, Mr. Platts -- why spoil it by dragging in that teleology nonsense?I'm not trying to make too big of a deal out of it. Of course we all know that mere physical systems don't have souls or spirits, and are not alive. Nevertheless, it is a fact of the history of science that physical scientists occasionally employ biological language and reasoning when trying to come up with a working hypothesis that can explain a given phenomenon.

korjik
2007-Nov-13, 08:38 PM
Seeing the TB relation in 55 Cancri is pretty cool. I agree that it will be interesting to see if this holds up for broad classes of stars.

It will be interesting to see if it holds up for 55 Cancri when we actually get images of the planets in question.

Warren Platts
2007-Nov-13, 09:06 PM
It will be interesting to see if it holds up for 55 Cancri when we actually get images of the planets in question.That's right, the values I used aren't written in stone; therefore, any conclusions reached on those numbers must be regarded as provisional.

stutefish
2007-Nov-13, 10:58 PM
Heh. I found out about Bode's law about a month ago, and immediately started wondering if it would be found to apply in other planetary systems.

tusenfem
2007-Nov-14, 02:13 PM
That Bode's Law is an ATM concept is evidenced by the fact that the main planetary science journal Icarus does not accept manuscripts that discuss Bode's Law.


Icarus maybe, but as I wrote in the other thread in "Astronomy" papers have been published, at least in 1999 by Graner & Dubrelle, who investigated Titus-Bode like laws and where they come from in great detail and they published it in "Astronomy & Astrophysics". I do not know where you get the information that Icarus would not publish a well written scientific paper on the appearance of TB-laws. Naturally, only fitting stuff will not get you published, you will have to give a reasonable explanation why this "law" exists, just like Graner & Dubrelle did.

Disinfo Agent
2007-Nov-14, 03:47 PM
Seeing the TB relation in 55 Cancri is pretty cool. I agree that it will be interesting to see if this holds up for broad classes of stars.Although I've been very critical of Warren's idea, I can't help agreeing. It is true that many natural phenomena, in a wide variety of fields, seem to follow power laws* at least approximately (for an old example, google "Zipf's Law"; for a slightly mind-boggling one, google "Benford's Law"). Unfortunately, I don't think anyone has ever managed to clearly explain why this happens in general, though there are plenty of partial attempts in the mathematical and probabilistic literature.

*I should add that the Titius-Bode law is not a power law. It's a logarithmic/exponential law. But it still made me think of power laws. :)

Warren Platts
2007-Nov-14, 04:49 PM
Icarus maybe, but as I wrote in the other thread in "Astronomy" papers have been published, at least in 1999 by Graner & Dubrelle, who investigated Titus-Bode like laws and where they come from in great detail and they published it in "Astronomy & Astrophysics". I do not know where you get the information that Icarus would not publish a well written scientific paper on the appearance of TB-laws. Naturally, only fitting stuff will not get you published, you will have to give a reasonable explanation why this "law" exists, just like Graner & Dubrelle did.


Icarus (http://icarus.cornell.edu/information/authors.html) does not publish papers that provide "improved" versions of Bode’s law, or other numerical relations, unless they are accompanied by some detailed physical/chemical arguments to explain why the new relation is to be preferred.

And thanks for the Graner & Dubrulle link, but I beat you to it in the first post of this thread. Graner and Dubrulle were able to get away with their paper because they framed it as a paper about papers about Titius-Bode laws. I found the following quote to be insightful regarding the historical discussion:


The general attitude towards the Titius-Bode laws (2) is twofold: skepticism or faith. Skeptic people argue that the "laws" are pure numerical coincidences and were produced by chance alone. . . . Faithful people use the fact that Titius-Bode laws are observed in the solar system and in satellite systems of giant planets to justify its possible physical significance. The law is then used as a constraint of theories of solar and satellite system formation and explanations are sought. A difficulty is to find a mechanism working in both solar and planetary systems, which are a priori rather different in nature (temperature, density, etc.). However, this difficulty does not seem to be a major limitation to the imagination of the theoreticians since there are over 15 explanations to the various forms of the Titius-Bode law.

Skeptical theories can be divided into two camps: (1) those that say the observed pattern is real but that it's a mere coincidence--there's no underlying physical explanation for the pattern; and (2) those that say the pattern is actually indistinguishable from random expectations and that anybody can "curve-fit" some sort of relation to any set of points.

Theory (1) is plausible, and one can always say that a sample-size of one solar system is too small to go off half-cocked inventing physical explanations for a pattern that may not be general. On the other hand, there are the solar system's gas giants, and now we have tantalizing results from 55 Cancri, and soon data will be flooding in from many other systems, so (1) is increasingly harder to maintain. Theory (2), however, is false on the face of it. All you have to do is look at the pattern in this solar system--the observed pattern is not similar to that generated by a blindfolded dart thrower. So proponents of (2) have to make assumptions like the planets cannot be too close together, and so the distribution is random within those constraints. But this is just sour grapes. By imposing excluded volume constraints they're invoking a physical explanation in order to deny the physical explanation. Moreover, even the true believers don't deny that there's going to be random variation within fairly wide error bars for any real world solar system.

Of the physical explanations for Titius-Bode relations, Graner and Dubrulle identify two broad classes: (1) those that are the result of celestial mechanics operating after the planets form; and (2) those that say the observed pattern is a fossil relict of conditions at the time of planetary formation. I don't see how theories of type (1) can work, just based on my own fooling around with tony873004's GravitySimulator computer program: when orbits get in a stable configuration, they just stay there pretty much, and there are lots of stable patterns that aren't necessarily Bode law-type patterns.

So that leaves the "dynamical" theories having to do with the formation of planets and moons out of primordial disks. Graner and Dubrulle discuss several. The two I find most plausible are (1) the pattern is based on the average eccentricity of particles making up the primordial disk; and (2) the observed pattern is the result the scale of vorticity in the primordial disk that's assumed to be fully turbulent. Although, my original thinking was more along the lines of (1), I'm now leaning towards (2) mainly because a large vortex is a better design for matter-eating than are single planitesimals, and it's clear that the gas giants formed out of minidisks within the main primoridial disk.

Warren Platts
2007-Nov-14, 05:46 PM
*I should add that the Titius-Bode law is not a power law. It's a logarithmic/exponential law. But it still made me think of power laws. :)What's the difference between a power law and a logarithmic/exponential law?

Disinfo Agent
2007-Nov-14, 05:59 PM
In a power law, the base is variable and the exponent (i.e. the power) is a constant. In a logarithmic/exponential law, the base is constant and the exponent is variable. Their graphs are different curves.

Warren Platts
2007-Nov-14, 07:12 PM
I have a confession to make: in the first post I said that the planets for 55 Cancri obey Bode's Law better than the planets of our own solar system. That's not exactly true. I took the straight errors, and then squared them, and because most of the planets at 55 Cancri are very close in compared to our solar system, the absolute errors are smaller, but if the errors are expressed percentage-wise, the errors at 55 Cancri are larger than the classic formulation of Bode's Law for our solar system. To wit: under the formulation I proposed, there average of the absolute value of the percentage errors is about 6% for 55 Cancri and about 2% for the classic formulation for the Sol system (excluding Neptune). The biggest source of error is for 55 Cancri III (to use the old Star Trek classification), which is 20%.

Graner and Dubrulle list several forms of mathematical structures a Titius-Bode law can take, but there are two that are in common currency: (1) what I call the classic formulation; and (2) what I call the "natural" formulation. The general form of the classic formulation is:



an = a1 + (a2 - a1)Km

where m is a number from the following series: (-∞, 0, 1, 2, 3, ...). The "-∞" makes the second term go to zero (whoever said 1/0 wasn't useful! :D) so you get a1, and when m = 0, Km = 1, so you get a2. In other words, you get your first two planets for free, so the classic formulation of Bode's law is more open to the charge of "curve-fitting" than the natural formulation. In our solar system, the values usually used are a1 = 0.4 and a2 = 0.3, with K set to 2. So the formula becomes:



an = 0.4 + (0.3)2m

One can attempt a classic formulation for 55 Cancri, but one has to set K to equal 3 instead of 2:



an = 0.039 + (0.077)3m

in which case the average percentage error can be reduced to about 5%.

The natural formulation, according to Graner and Dubrulle is:



an = a1Kn-1

which is the formulation I first used for 55 Cancri. Which also apparently entails that my value for K (i.e., e) must be a total coincidence. But what an utterly, freaky coincidence!!! The only number that works better than 2.71828182845904 is 2.7180. So, naturally, I went with e for K. But apparently that's a pure fluke since the K for our solar system, according to Graner and Dubrulle (which I subsequently confirmed myself) is 1.7. On this model, the average error for our solar system is 10% (but it has the virtue of bringing Neptune back into the fold at least). So, on the natural formulation of Bode's Law, 55 Cancri does in fact do better than our own solar system. If, however, you divide our solar system into inner and outer planets, and assume a K of 1.57 for the inner and a K of 1.855 for the outer, the errors can be reduced to 4%.

Now, the main point of the Graner and Dubrulle paper is that any attempt to physically model the layout of a solar system that assumes initial conditions are radially independent (i.e., there is no variation that depends on the distance r from the center of the system), a Titius-Bode's like relation will fall out, no matter which of the 15 underlying physical theories you use. They then use this result of theirs to trivialize attempts to more deeply understand the Titius-Bode relation: hence the subtitle of their paper: "Scale indepence explains [away] everything".

But I don't think that's fair. If it's really the case that a system obeys the Titius-Bode relation, that says something important about that system: namely, that the process that caused the pattern was radially independent, and that that is going to place huge contraints on what a theory to explain the layout of the solar system must be like. Conversely, if a solar system does not obey the Bode relation (as our solar system apparently does not under the natural formulation), that also says something: that conditions changed as one progressed toward the outer edge of the primoridial disk.

Also, the scale factor K says something important. I suggest that the K factor says something about the rate of formation of planets and solar systems: the higher the K factor, the faster the rate of formation (you heard it here first folks! :D).

On both the classic and natural formulations for 55 Cancri and the Sol system, 55 Cancri has a larger K factor. So why would I say that the 55 Cancri system formed faster than our own system?


55 Cancri A is more enriched than our sun in elements heavier than helium, with 186% the solar abundance of iron; it is therefore classified as a rare "super metal-rich" (SMR) star. (Wikipedia)

So what happened is that at 55 Cancri--because of the superabundance of iron and other heavy elements--heavy, iron-rich, powerful, hungry cores capable of holding mass-sucking atmospheres were able to rapidly form at the center of large vortices; and so planetary and orbital evolution at 55 Cancri was accelerated, compared to the Earth's solar system. Hence, the wider spacing of planets at 55 Cancri--which, paradoxically, allowed planets to form much closer to the sun than at our solar system.

Meanwhile, back here, there was less iron, core's formed less readily, evolution was slower, and so the primordial disk had time to become more organized (less eccentric), and that furthermore, since K, the Bode scale factor, for the outer planets is 1.86, but only 1.57 for the inner planets, then the evolution must have progressed inward starting with the outer planets first, with the inner planets forming somewhat later.

:cool:

Warren Platts
2007-Nov-14, 07:13 PM
In a power law, the base is variable and the exponent (i.e. the power) is a constant. In a logarithmic/exponential law, the base is constant and the exponent is variable. Their graphs are different curves.Thank you. But then what do you call it when both the base and the exponent are variables?

Disinfo Agent
2007-Nov-14, 07:56 PM
If it has a name, I don't know what it is. The simplest case is y=xx, but this is just one curve. There's no free shape parameter there we can adjust to define a family of similar curves (like your K).

Jim
2007-Nov-15, 04:00 PM
Let's be very clear on one important point. The Titus-Bode (or Titius-Bode) relationship is not a law in the scientific sense. At best, it is an hyposthesis. I say "at best" because, while it claims a relationship of secondary to orbital distance from its primary, it offers no explanation for why this might hold.

Warren Platts
2007-Nov-15, 04:24 PM
Let's be very clear on one important point. The Titus-Bode (or Titius-Bode) relationship is not a law in the scientific sense. At best, it is an hyposthesis. .

Couldn't we call it an empirical law, as in Galileo's laws of motion? We can say that heavy and light objects fall at the same rate, and call that a law without having to come up with a deep physical explanation for how that happens. Be that as it may, I've been calling it Bode's Law just because that's how most encyclopedia entries refer to it--but it may not be a true scientific law in the strict sense.


I say "at best" because, while it claims a relationship of secondary to orbital distance from its primary, it offers no explanation for why this might hold.The BT relation does not entail any particular physical explanation; it does, however, impose constraints on what such physical explanations must be like. As Graner and Dubrulle point out, all physical explanations for the BT relation share one thing in common: that whatever causal factors are at work do not vary with the distance from the primary within the zone that the BT relation holds.

Jim
2007-Nov-15, 06:18 PM
Couldn't we call it an empirical law, as in Galileo's laws of motion? We can say that heavy and light objects fall at the same rate, and call that a law without having to come up with a deep physical explanation for how that happens. Be that as it may, I've been calling it Bode's Law just because that's how most encyclopedia entries refer to it--but it may not be a true scientific law in the strict sense.

You could call them laws of motion because they have been tested many times, in many ways, in many venues and always - always - give the same results. T-B has been "tested" only on one example, our solar system, and has not held up that well. (The asteroid belt and the outer planets don't fit all that closely.) 55Cancri is an incomplete test as we don't have all the information on that system. Test T-B a bit more and we'll see if it holds up; then you can begin to think of calling it a true scientific law.

Also, the laws of motion can be explained scientifically; T-B cannot.

T-B is an empirical observation and an unexplained relationship, not a law.

Warren Platts
2007-Nov-16, 12:49 AM
Also, the laws of motion can be explained scientifically; T-B cannot.
I respectfully beg to differ. The T-B relation can be explained scientifically: Graner and Dubrulle claim to have identified at least 15 different scientific explanations for the T-B relation. Personally, I find the explanations based on eccentricity and vorticity scale to be the most compelling. Such explanations can explain why it is that apparently 55 Cancri has a larger scale factor than does the Sol system.

Jim
2007-Nov-16, 02:36 AM
Graner and Dubrulle claim to have identified at least 15 different scientific explanations ...

Shotgun science.

I can come up with at least 15 "scientific" explanations for astrology. You make an observation and you toss out an idea or three as to why that observation happened, but that doesn't make it a law.

If you want to discuss T-B as an observation, a coincidence, or even an hypothesis, fine. Just don't call it a law.

Ari Jokimaki
2007-Nov-16, 09:33 AM
If you want to discuss T-B as an observation, a coincidence, or even an hypothesis, fine. Just don't call it a law.
Calling it a law is just common terminology (http://en.wikipedia.org/wiki/Titius-Bode_law). There is no need to confuse the issue with new terminology, so I say keep calling it a law.

hhEb09'1
2007-Nov-16, 10:47 AM
so I say keep calling it a law.Agreed. It's been "Bode's Law" for ever :)

Warren Platts
2007-Nov-16, 11:07 AM
Graner and Dubrulle claim to have identified at least 15 different scientific explanations ...

Shotgun science.

I can come up with at least 15 "scientific" explanations for astrology. You make an observation and you toss out an idea or three as to why that observation happened, but that doesn't make it a law.
Shotgun science?!? Well, judging from the threads in this ATM section, Hubble red shifts, gravity, and the BBT itself must all be shotgun science as well.


If you want to discuss T-B as an observation, a coincidence, or even an hypothesis, fine. Just don't call it a law.


Calling it a law is just common terminology. There is no need to confuse the issue with new terminology, so I say keep calling it a law.
Results from a Google search of "Titius-Bode"

Law: 50
Rule: 6
Relation: 4
Number sequence: 1
Series: 1

It doesn't really matter to me what it's called, but common parlance definitely prefers "Law".

Van Rijn
2007-Nov-16, 11:31 AM
Calling it a law is just common terminology (http://en.wikipedia.org/wiki/Titius-Bode_law). There is no need to confuse the issue with new terminology, so I say keep calling it a law.

The issue is that the terminology is confusing, since it certainly doesn't qualify as what is usually called a scientific law. As said here (http://en.wikipedia.org/wiki/List_of_laws_in_science#Other_laws):

The laws of science are various established scientific laws, or physical laws as they are sometimes called, that are considered universal and invariable facts of the physical world. Laws of science may, however, be disproved if new facts or evidence arise to contradict them. A "law" differs from hypotheses, theories, postulates,principles, etc., in that a law is an analytic statement, usually with an empirically determined constant. A theory may contain a set of laws, or a theory may be implied from an empirically determined law.

Personally, I don't care - much - if it is called a law as long as everyone in the discussion recognizes that the word "law" is meaningless in this case.

hhEb09'1
2007-Nov-16, 02:14 PM
Personally, I don't care - much - if it is called a law as long as everyone in the discussion recognizes that the word "law" is meaningless in this case.I don't recognize that. :)

As your wiki quote says, "Laws of science may, however, be disproved if new facts or evidence arise to contradict them." In that sense, Bode's Law has been contradicted. It still makes sense to talk about it as a law. Surely there are other laws that have fallen by the wayside, but we still refer to them as laws. I just can't think of any right now...

Warren Platts
2007-Nov-16, 04:27 PM
I don't recognize that. :)

As your wiki quote says, "Laws of science may, however, be disproved if new facts or evidence arise to contradict them." In that sense, Bode's Law has been contradicted. It still makes sense to talk about it as a law. Surely there are other laws that have fallen by the wayside, but we still refer to them as laws. I just can't think of any right now...Well, Newton's Law of Gravity (F = m1m2Gr2) yields false predictions under relativistic conditions, and when the objects of mass m1 and m2 also have net electrical charges; yet we still refer to Newton's Law of Gravity. Sure, I suppose you could say that Newton's Law is in the same category as Bode's Law--a conventional, meaningless law, as opposed to truly genuine scientific laws. But if we're going to say that only strict, proviso-free generalizations describing exceptionless regularities count as genuine laws of nature, then I'm afraid we'll have to get rid of all such laws. If you then say it's OK to incorporate ceteris-paribus conditions, you'll have a hard time nailing down the truth conditions unless you simply say law L is true whenever law L is true.

So quibbling over whether T-B is a genuine law of nature opens a philosophical can of worms we best not get into.

The real question is whether solar systems with T-B layouts are accidental or not. Obviously, within our solar system there are numerous, but trivial, counterexamples to the T-B relation because every single asteroid and Kuiper Belt object does not obey the relation. Neptune, however, is not so trivial. But this just goes to ceteris-paribus clause that must be incorporated. Rephrased--other things being equal--we can say that the Titius-Bode Law is:


Under normal conditions, major planets forming out of a primordial disk will have an approximately logarithmic spacing.

That's the general formulation of the T-B relation, which if true is worthy of law status, because if T-B is true, that says something interesting that places nontrivial constraints on models of planetary formation.

Van Rijn
2007-Nov-16, 08:34 PM
I don't recognize that. :)

As your wiki quote says, "Laws of science may, however, be disproved if new facts or evidence arise to contradict them." In that sense, Bode's Law has been contradicted. It still makes sense to talk about it as a law. Surely there are other laws that have fallen by the wayside, but we still refer to them as laws. I just can't think of any right now...

If the idea had ever risen above the level of a hypothesis, I might agree.

Van Rijn
2007-Nov-16, 08:38 PM
Well, Newton's Law of Gravity (F = m1m2Gr2) yields false predictions under relativistic conditions, and when the objects of mass m1 and m2 also have net electrical charges; yet we still refer to Newton's Law of Gravity.


I generally avoid that. Instead, I might talk about Newtonian physics, or Newton's theory of gravity. However, there are at least a wide variety of conditions where Newtonian physics has been shown to be quite useful, unlike Titius-Bode.

Warren Platts
2007-Nov-16, 09:02 PM
If the idea had ever risen above the level of a hypothesis, I might agree.

Thanks for the substantive critique. It makes all the effort worth it. :rolleyes:

Van Rijn
2007-Nov-16, 09:15 PM
Thanks for the substantive critique. It makes all the effort worth it. :rolleyes:

What's your point? As has already been said in this thread, TB has problems in the solar system, we certainly don't have enough information on exoplanets to draw a conclusion, and there are rather huge questions on the mechanism. Now, when we have detailed maps of, say, fifty systems and if then we see a consistent pattern, then it might rise above the level of hypothesis.

Warren Platts
2007-Nov-17, 12:06 AM
What's your point? . . .My point is that I'm sorry I didn't include a poll in the first thread: it would have saved you a few keystrokes to register your vote. Thanks for voting! :)


Now, when we have detailed maps of, say, fifty systems and if then we see a consistent pattern, then it might rise above the level of hypothesis.
What's your point here? That it's pointless to speculate that extrasolar planetary spacing is often logarithmic before we have a sample size of 50 solar systems with at least 5 planets each, or is it that BT is so fouled-up that it cannot predict where some of those planets will be found before they are found as we build the sample size of 50 systems? :confused:

Van Rijn
2007-Nov-17, 12:56 AM
The point is that it would need a lot more supporting data before it could go beyond a hypothesis. There are problems with the idea already, based on our own solar system, and we don't have enough data for any other solar system, including 55 Cancri, to take it very far.

Jim
2007-Nov-17, 02:15 AM
In the interests of brevity, what Van Rijn said (in his last several posts).

Warren Platts
2007-Nov-17, 02:31 AM
I have a quick question for you guys: What would be the curve that better fits the data: (1) a curve that minimized the average of the percentage errors; or (2) a curve that minimized the standard deviation of the average of the percentage errors?

Warren Platts
2007-Nov-17, 03:16 AM
The point is that it would need a lot more supporting data before it could go beyond a hypothesis.We were having enough philosophical trouble with 'law'; now you want to bring in 'hypothesis'. . . . Very well then. What do you mean by 'hypothesis' and what would it be like to "go beyond" a hypothesis? :whistle:

And which hypothesis are you referring to?

H1: Major planets that formed out of a primordial disk under normal conditions have an approximately logarithmic spacing; or
H2: The semimajor axis of 55 Cancri g is greater than 1.9 AU and less than 2.3 AU.

How much more supporting data will it take for H2 to go beyond a hypothesis?


There are problems with the idea already, based on our own solar system.
I seriously hope you aren't referring to asteroids and Kuiper Belt objects. . . .

And what about the fact that the major satellites of the gas giants in this solar system obey the TBL?


[A]nd we don't have enough data for any other solar system, including 55 Cancri, to take it very far.
How can you possibly say that being able to predict the location of a new planet doesn't count as "taking it very far"? Is it that extrasolar planetary science is an utterly trivial and boring pursuit?

Warren Platts
2007-Nov-17, 08:14 AM
Here's a chart showing the position of where Planet V is predicted to be found.

Also, I revised the TBL formula for 55 Cancri by minimizing the sum of the square of the (percentage) errors:




ai = 0.0374(2.722)n-1

To still insist we don't have enough data to make useful predictions for 55 Cancri is just pure FUD.

Van Rijn
2007-Nov-17, 08:50 AM
Here's a chart showing the position of where Planet V is predicted to be found.

Also, I revised the TBL formula for 55 Cancri by minimizing the sum of the square of the errors:




ai = 0.0374(2.722)n-1

Eh? The TBL formula is:


a = 0.4 + 0.3 · 2m

where "a" is the distance from the star in AU, and "m" starts with negative infinity (that to fit Mercury), then 0, 1, 2, 3 etc. Your equation no longer is "the TBL formula" but rather a formula of your own devising, designed to fit different data.

Van Rijn
2007-Nov-17, 09:03 AM
Here's a chart showing the position of where Planet V is predicted to be found.

Also, I revised the TBL formula for 55 Cancri by minimizing the sum of the square of the (percentage) errors:




ai = 0.0374(2.722)n-1

To still insist we don't have enough data to make useful predictions for 55 Cancri is just pure FUD.

So, to be clear, you're proposing to make a prediction of another planet for 55 Cancri based on your custom calculation, but this calculation would not be applicable to any other star system?

Van Rijn
2007-Nov-17, 09:21 AM
We were having enough philosophical trouble with 'law'; now you want to bring in 'hypothesis'. . . .


I thought that both Jim and I had covered this. The "philosophical trouble" is because it does not conform to what is usually referred to as a physical law. It is an observation based on limited data, and there is a hypothesis that there is an underlying, simple rule to the observation. I didn't "bring in" the hypothesis aspect: That's always been there.



Very well then. What do you mean by 'hypothesis' and what would it be like to "go beyond" a hypothesis? :whistle:


Definition of hypothesis:

A hypothesis (from Greek ὑπόθεσις) consists either of a suggested explanation for a phenomenon or of a reasoned proposal suggesting a possible correlation between multiple phenomena.

Normally, you go beyond a hypothesis by showing validity through supporting data. That generally requires a decent sample size.



And which hypothesis are you referring to?

H1: Major planets that formed out of a primordial disk under normal conditions have an approximately logarithmic spacing; or
H2: The semimajor axis of 55 Cancri g is greater than 1.9 AU and less than 2.3 AU.

How much more supporting data will it take for H2 to go beyond a hypothesis?


I was actually picturing something like:

Spacing of planets follow the pattern described by the Titius-Bode formula:

a = 0.4 + 0.3 · 2m

where "a" is the distance from the star in AU and "m" starts with negative infinity, then 0, 1, 2, 3, etc.




I seriously hope you aren't referring to asteroids and Kuiper Belt objects. . . .


Those, the Mercury "fix" and especially Neptune. Why would I ignore that data? Mercury is a special case in the equation. Neptune doesn't follow the rule. Pluto seemed to follow the rule, except that it turns out it is just one rather low mass object among many, so that's meaningless. There was supposed to be a planet between Mars and Jupiter according to the rule, but again, we just find a large number of small objects (and Ceres is like Pluto). In fact, of course, Jupiter would prevent formation of a large planet. To me, that suggests that the situation is too complex for a blanket rule, and a generally applicable method would need to consider (among other things) the mass of the individual planets.



And what about the fact that the major satellites of the gas giants in this solar system obey the TBL?


I wasn't aware that was a "fact." I am aware that apparent patterns show up in systems of satellites, but I'm not aware that systems of satellites follow the TB formula. References, please?

Warren Platts
2007-Nov-17, 05:15 PM
This isn't your average "Titus [sic]-Bode Law--ooh shiny!" thread. I'm trying to take it to the next level--to "go beyond", as some might say. Before, I merely suspected that you did not read thread--now I know for sure. So allow me to bring you up to speed.


[ai = 0.0374(2.722)n-1]? The TBL formula is:


a = 0.4 + 0.3 · 2m

where "a" is the distance from the star in AU, and "m" starts with negative infinity (that to fit Mercury), then 0, 1, 2, 3 etc. Your equation no longer is "the TBL formula" but rather a formula of your own devising, designed to fit different data.

In post #1, I first posted this formula:




a = 0.039en-1

where a is the semimajor axis, e is the natural logarithm constant (2.7 ...), n is the number of the planet starting from it's sun, and the 0.039 is the semimajor axis of the closest planet in AU's.


Then in post #14, I basically parrot Graner and Dubrulle's (1994) paper (http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994A%26A...282..262G&data_type=PDF _HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf), where they identify two basic forms of the TBL (and there are more):


Graner and Dubrulle list several forms of mathematical structures a Titius-Bode law can take, but there are two that are in common currency: (1) what I call the classic formulation; and (2) what I call the "natural" formulation. The general form of the classic formulation is:



an = a1 + (a2 - a1)Km

where m is a number from the following series: (-∞, 0, 1, 2, 3, ...). The "-∞" makes the second term go to zero (whoever said 1/0 wasn't useful! :D) so you get a1, and when m = 0, Km = 1, so you get a2. In other words, you get your first two planets for free, so the classic formulation of Bode's law is more open to the charge of "curve-fitting" than the natural formulation. In our solar system, the values usually used are a1 = 0.4 and a2 = 0.3, with K set to 2. So the formula becomes:



an = 0.4 + (0.3)2m

One can attempt a classic formulation for 55 Cancri, but one has to set K to equal 3 instead of 2:



an = 0.039 + (0.077)3m

in which case the average percentage error can be reduced to about 5%.

The natural formulation, according to Graner and Dubrulle is:



an = a1Kn-1

which is the formulation I first used for 55 Cancri. Which also apparently entails that my value for K (i.e., e) must be a total coincidence. But what an utterly, freaky coincidence!!!

Actually, the classical formulation is easily shown to be a special case of the natural formulation where a solar system is broken into two zones, each with its own scaling factor K.

And to say that TBL doesn't apply to 55 Cancri because the parameters of the model for 55 Cancri are different from our solar system is such a total red herring: you might as well say that someone who applies Newton's Laws to an extrasolar system is using a formula "of their own devising designed to fit different data" because they adjusted the mass parameter of the primary to something different from the Sun. :lol:


So, to be clear, you're proposing to make a prediction of another planet for 55 Cancri based on your custom calculation, but this calculation would not be applicable to any other star system?
Re-read the above.


I thought that both Jim and I had covered this. The "philosophical trouble" is because it does not conform to what is usually referred to as a physical law.
That's just false on the face of it. The TBL is usually referred to as a law (and I guess it would have to be a physical law, because it sure isn't a biological or economic law) at least 5 to 1 (as I point out in post #24) compared to other descriptions like 'rule' or 'relation' (you're the first to speak of the TB "hypothesis").

BTW, the wiki article on scientific laws you cite is junk. If you want an understanding of the issues involved, read the Stanford Encyclopedia of Philosophy article (http://plato.stanford.edu/entries/laws-of-nature/) instead. But it's going to take more than a couple of minutes. . . .


Normally, you go beyond a hypothesis by showing validity through supporting data. That generally requires a decent sample size.
I've got 5 data points for 55 Cancri. I've done the least squares statistical reduction. The standard deviation for the results is rather wide (about 6%), but that's still not bad, and it generates good results, and at least one concrete prediction, as you can see from the attached graph. If you're going to say that's statistically insignificant, you'd better be prepared to back up that assertion with something more than IIRC handwaving about sample sizes.


Those [asteroids and Kuiper Belt objects], the Mercury "fix" and especially Neptune. Why would I ignore that data? Mercury is a special case in the equation. Neptune doesn't follow the rule. Pluto seemed to follow the rule, except that it turns out it is just one rather low mass object among many, so that's meaningless. There was supposed to be a planet between Mars and Jupiter according to the rule, but again, we just find a large number of small objects (and Ceres is like Pluto). In fact, of course, Jupiter would prevent formation of a large planet. To me, that suggests that the situation is too complex for a blanket rule, and a generally applicable method would need to consider (among other things) the mass of the individual planets.

First of all, you contradict yourself within one paragraph when you say that we shouldn't ignore asteroids and KBO's, and then you say that Pluto fits the rule but doesn't count because it's a KBO.

Secondly, the TBL is, and never was, intended to be a "blanket rule" covering every object in the night sky. The proper domain of the TBL is major planets in normal solar systems. TBL--like Newton's Law of Gravity--incorporates a common-sense ceteris paribus proviso. Thus, in situations where other things are not equal--as when Jupiter prevents the formation of a planet--you're not going to find a planet there. Nevertheless, the presence of the asteroid belt where TBL says a planet should be is usually considered a confirmation of TBL--you're the first to suggest otherwise.

Thirdly, as for Neptune and Mercury--they both do fit the TBL. The formula for the inner solar system is:




ai = 0.413(1.56)n-1

The standard deviation for the average of the (percent) errors above (using the s.d. for a sample of a population [=STDEVA(...) in Excel]) is 4.4%. The formula for the outer system (Jupiter through Neptune) is (s.d. 3.6%):




ai = 5.32(1.81)n-1

The above formula predicts Neputune's orbit to within 5%. Now before you go off saying I am engaging in ad hoc curve-fitting, reread what I said before about ceteris paribus clauses. In other words, by giving two formulas, I am saying that conditions during the origin of the solar system were not the same for the inner and outer zones. My evidence? As an old professor of mine used to say, just look at it! The inner planets are all rocky, and the outer planets are all gas giants--so other things are obviously not equal. If there's any conclusion to be drawn from the ~256+ extrasolar planets discovered so far, it is that the situation here in our solar is not normal--gas giants regularly form elsewhere at distances much less than 2 AU. Therefore, it's not surprising that the scaling factor K for the two zones is different. In post #14, I speculate on the physical significance for the K factor for planetary formation:


Also, the scale factor K says something important. I suggest that the K factor says something about the rate of formation of planets and solar systems: the higher the K factor, the faster the rate of formation (you heard it here first folks! :D).

And your suggestion that planetary mass has something to do with planetary spacing is belied by the evidence.


I wasn't aware that [TB spacing of the major satellites of gas giants] was a "fact." I am aware that apparent patterns show up in systems of satellites, but I'm not aware that systems of satellites follow the TB formula. References, please?This fact has been noted several times in other threads on the TBL in this forum and Graner and Dubrulle (1994) (http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994A%26A...282..262G&data_type=PDF _HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf) list the K factors for Jupiter, Saturn, and Uranus as 1.6, 1.5, and 1.4 respectively (which according to my theory implies that the Jupiter system formed first).

grant hutchison
2007-Nov-17, 07:24 PM
And what about the fact that the major satellites of the gas giants in this solar system obey the TBL?I wasn't aware that was a "fact." I am aware that apparent patterns show up in systems of satellites, but I'm not aware that systems of satellites follow the TB formula. References, please?There is a tendency towards increased spacing with increased orbital radius for the regular moons of the gas giants, which is of course what allows exponential fits to be carried out.
Good fits in this case need ad hoc rejections of uncomfortable data points.
Here (http://www.ghutchison.pwp.blueyonder.co.uk/moons.jpg) is a log plot of the orbital radii of all the regular moons of the four gas giants. I've suppressed only the Saturnian ring shepherds, and have treated each coorbital family around Saturn as a single data point. If an exponential fit were going to be good, we would expect to see a family of straight lines.

Grant Hutchison

Warren Platts
2007-Nov-17, 07:48 PM
Here (http://www.ghutchison.pwp.blueyonder.co.uk/moons.jpg) is a log plot of the orbital radii of all the regular moons of the four gas giants. I've suppressed only the Saturnian ring shepherds, and have treated each coorbital family around Saturn as a single data point. I don't know what you mean by "regular moon". You're falling into the same fallacy that says that the BTL has to apply to every asteroid and KBO out there. If you restrict your analysis to the five largest moons in each, I think you'll find a good correlation.

The physical basis of the BTL has to do with the formation of planets. Celestial mechanical events that happen subsequently to formation would be expected to cause deviations from the primitive logarithmic spacing.

grant hutchison
2007-Nov-17, 08:39 PM
I don't know what you mean by "regular moon".The inner moons of the gas giants have near circular prograde orbits in approximately the equatorial planes of the their parent planets. The are called "regular" for that reason.


You're falling into the same fallacy that says that the BTL has to apply to every asteroid and KBO out there.Please let us know which logical fallacy we are commiting by asking you to not to select only those data that best fit your curves.


Celestial mechanical events that happen subsequently to formation would be expected to cause deviations from the primitive logarithmic spacing.In which case, in what way and to what extent can we expect your "law" to be predictive of orbital spacings?

Grant Hutchison

Jim
2007-Nov-17, 09:30 PM
And which hypothesis are you referring to?

H1: Major planets that formed out of a primordial disk under normal conditions have an approximately logarithmic spacing; or
H2: The semimajor axis of 55 Cancri g is greater than 1.9 AU and less than 2.3 AU.

How much more supporting data will it take for H2 to go beyond a hypothesis? ...

And what about the fact that the major satellites of the gas giants in this solar system obey the TBL?

Let's take H1 (if that is what you are proposing).

Define "major planet."
Define "normal conditions."
Define "approximately" logarithmic.

Oh, and since you mentioned it,
Define "major satellites."
Explain why this applies only to the gas giants.

Jim
2007-Nov-17, 09:51 PM
That's just false on the face of it. The TBL is usually referred to as a law ... at least 5 to 1 (as I point out in post #24) compared to other descriptions like 'rule' or 'relation' (you're the first to speak of the TB "hypothesis").

I frankly don't give a flip. Just because most people use a word erroneously does not change its scientific meaning. Some people try to denigrate evolution by saying it is "only a theory," failing to understand the scientific implications of the word; don't try to elevate T-B by calling it a law, especially when you do seem to understand what that means.

(Idle thought... If we "usually" - 5-1, say - referred to you as a woman, would you protest, or give in to the majority?)


... The proper domain of the TBL is major planets in normal solar systems. TBL--like Newton's Law of Gravity--incorporates a common-sense ceteris paribus proviso. ...

Thirdly, as for Neptune and Mercury--they both do fit the TBL. The formula for the inner solar system is:




ai = 0.413(1.56)n-1

... The formula for the outer system (Jupiter through Neptune) is (s.d. 3.6%):




ai = 5.32(1.81)n-1

The Newtonian equation for gravitational attraction (which, I'm happy to see, you admit works within proper context... inded, I doubt you can show any example where it does not work within context) includes an "explanation" for the result; it has the masses of the two bodies.

Your formulae have no such "explanation" beyond order from the primary. Is there any other explanation?

Also, you now say the formula changes from inner to outer planets.

So, define "inner" and "outer" planets.


The above formula predicts Neputune's orbit to within 5%. ...

No, it "predicts" nothing. You have generated a formula which fits the observed data. Congratulations.


... If there's any conclusion to be drawn from the ~256+ extrasolar planets discovered so far, it is that the situation here in our solar is not normal...

But, didn't you say something about T-B working when a system forms under "normal conditions?" So, why does it apply here, to a "not normal" system?


And your suggestion that planetary mass has something to do with planetary spacing is belied by the evidence.

So what does determine spacing?

Warren Platts
2007-Nov-17, 11:00 PM
Please let us know which logical fallacy we are commiting by asking you to not to select only those data that best fit your curves.It's called a category mistake.

Let me take it from the top again: there are two broad classes of explanations for explanations of the BTL:
those who say the explanations are much ado about nothing;
those who say the explanations describe a real process.


Of the theories that purport to describe a real underlying process that caused the TBL pattern, there are two broad classes:
what G^D call "kinematic" theories, where the TBL layout is the result of celestial mechanical resonances;
what G^D call "dynamical" theories, where the TBL layout is a fossil relict resulting from the formation of the planets.

Kinematic theories are evolutionary; the planets gradually evolved to their present arrangement because of Newtonian mechanics.

Dynamical theories of the TBL assert that the TBL pattern was formed when the planets were formed, and that the pattern hasn't changed much in billions of years in the solar system's case, or several millions of years in the case of 55 Cancri.

Here, I've been favoring dynamical theories because of my teleological predilections, and the fact that I've fooled around with GravitySimulator enough to know that all kinds of crazy orbits are mostly stable over millions of years, so it's just not intuitively plausible that kinematic theories could be true.

However, the recent findings with regard to 55 Cancri can at last shed some empirical light on what was previously a dark mystery: according to the kinematic theories, it's highly unlikely that the TBL layout could be preserved over billions of years, unless that was a somehow favored orbital layout, because even though orbits can look stable, miniscule variations add up to substantial evolution over 100's of millions of years. Thus kinematic theories predict that systems less than 10 million years should be randomly organized because they haven't had time to evolve to a TBL layout yet.

Now that the data from 55 Cancri is in, however, it shows that a TBL layout can be formed quite early a solar system's evolution, suggesting that TBL layout formed when the planets formed; it also showed that TBL layouts can be stable for billions of years, but not because the TBL layout itself is especially stable, but because lots of classes of orbital layouts are stable, and the TBL is just one of many possible orbital layouts.


In which case, in what way and to what extent can we expect your "law" to be predictive of orbital spacings?
The primordial disk was fully turbulent, and therefore, largely well-mixed. Then large vortices form whose diameter is proportional to a characteristic scale-factor K that doesn't ordinarily vary much with distance from the primary. The result is that as one progresses outward from the center of the solar system, the diameter of the large vortices increase. Each vortex then stakes out its own "territory" or "feeding zone". The result is that each major planet sweeps out a zone, and the center of each zone is represented by the orbit of each major planet, and that is why the orbits of the major planets exhibit the TBL layout.

grant hutchison
2007-Nov-17, 11:32 PM
It's called a category mistake.
Because the rejected points fall into the category you classify as "points I would reject"?



In which case, in what way and to what extent can we expect your "law" to be predictive of orbital spacings?The primordial disk was fully turbulent, and therefore, largely well-mixed. ...No, I didn't ask you to repeat a post hoc justification for how your "law" might work if it actually existed. I asked you to tell us how useful you think your "law" is going to be, in making predictions of how the real world is laid out.
So stop dodging around and tell us:
On what physical basis you choose to reject data points, other than that they don't conform to your pet hypothesis.
On what physical basis you will decide whether your curve, plotted from these selected data points, is capable of predicting new data points (given that you've already offered "stuff moving around" as an excuse for a bad fit).
To what extent your hypothesis will have failed if it does not correctly predict new data points (since at present it seems to be infinitely adaptable by the simple expedient of rejecting the inconvenient).

Grant Hutchison

Van Rijn
2007-Nov-17, 11:42 PM
To what extent your hypothesis will have failed if it does not correctly predict new data points (since at present it seems to be infinitely adaptable by the simple expedient of rejecting the inconvenient).

Grant Hutchison

. . . or modifying the formula to fit.

grant hutchison
2007-Nov-17, 11:46 PM
. . . or modifying the formula to fit.Ah yes. :)
So how many new data points do we have to receive before deciding that this is the predictive graph, and no further data will be required to fix it?

It does seem that five is the magic number we want, and that more points just cause confusion. Warren Platts has asked us (http://www.bautforum.com/1115121-post43.html) to reject all but five of the moons of each gas giant, and has now (http://www.bautforum.com/1115045-post41.html) divided the solar system in two for more convenient curve fitting.

Grant Hutchison

Van Rijn
2007-Nov-18, 03:14 AM
Actually, the classical formulation is easily shown to be a special case of the natural formulation where a solar system is broken into two zones, each with its own scaling factor K.


Actually, it appears every example is a special case, and there aren't many examples.



I've got 5 data points for 55 Cancri. I've done the least squares statistical reduction. The standard deviation for the results is rather wide (about 6%), but that's still not bad, and it generates good results, and at least one concrete prediction, as you can see from the attached graph. If you're going to say that's statistically insignificant, you'd better be prepared to back up that assertion with something more than IIRC handwaving about sample sizes.


And you fit a custom equation to the data. So what?



First of all, you contradict yourself within one paragraph when you say that we shouldn't ignore asteroids and KBO's, and then you say that Pluto fits the rule but doesn't count because it's a KBO.


This is getting silly. I was responding to your statement:


I seriously hope you aren't referring to asteroids and Kuiper Belt objects. . . .

And I was pointing out that of course we shouldn't ignore that data. In both cases, there is supposed to be a planet, according to the formula. Picking a match out of many objects is pointless.



Secondly, the TBL is, and never was, intended to be a "blanket rule" covering every object in the night sky. The proper domain of the TBL is major planets in normal solar systems. TBL--like Newton's Law of Gravity--incorporates a common-sense ceteris paribus proviso. Thus, in situations where other things are not equal--as when Jupiter prevents the formation of a planet--you're not going to find a planet there. Nevertheless, the presence of the asteroid belt where TBL says a planet should be is usually considered a confirmation of TBL--you're the first to suggest otherwise.


So, the mass of Jupiter means that the rule can't work, but at the same time, asteroids confirm the rule?



Thirdly, as for Neptune and Mercury--they both do fit the TBL. The formula for the inner solar system is:




ai = 0.413(1.56)n-1

The standard deviation for the average of the (percent) errors above (using the s.d. for a sample of a population [=STDEVA(...) in Excel]) is 4.4%. The formula for the outer system (Jupiter through Neptune) is (s.d. 3.6%):




ai = 5.32(1.81)n-1

The above formula predicts Neputune's orbit to within 5%. Now before you go off saying I am engaging in ad hoc curve-fitting, reread what I said before about ceteris paribus clauses. In other words, by giving two formulas, I am saying that conditions during the origin of the solar system were not the same for the inner and outer zones.


By giving two formulas, you are doing even more customization to fit the data, which makes it even less interesting.



My evidence? As an old professor of mine used to say, just look at it! The inner planets are all rocky, and the outer planets are all gas giants--so other things are obviously not equal.


Exactly, which should give you a hint that simple rules, without consideration of underlying physics, probably aren't going to work.



If there's any conclusion to be drawn from the ~256+ extrasolar planets discovered so far, it is that the situation here in our solar is not normal--gas giants regularly form elsewhere at distances much less than 2 AU. Therefore, it's not surprising that the scaling factor K for the two zones is different.


Yes, it is likely you will need to do a lot of customization.



And your suggestion that planetary mass has something to do with planetary spacing is belied by the evidence.


Eh? You said it yourself, already quoted above:



Thus, in situations where other things are not equal--as when Jupiter prevents the formation of a planet--you're not going to find a planet there.

Here's what it boils down to: You're going to have to start nailing down the specifics (as per the questions Jim and Grant have asked) if you are going to have any hope of showing that this isn't any more than selecting data and adjusting formula to suit you. It already appears you're admitting that the actual underlying physics is more complex, and that conditions change over time.

Ari Jokimaki
2007-Nov-18, 06:52 AM
I frankly don't give a flip. Just because most people use a word erroneously does not change its scientific meaning. Some people try to denigrate evolution by saying it is "only a theory," failing to understand the scientific implications of the word; don't try to elevate T-B by calling it a law, especially when you do seem to understand what that means.
Warren is not using it erroneously. It is exactly correct to use the term that practically everyone else uses, including the scientists (http://adsabs.harvard.edu/cgi-bin/nph-abs_connect?db_key=AST&db_key=PRE&qform=AST&arxiv_sel=astro-ph&arxiv_sel=cond-mat&arxiv_sel=cs&arxiv_sel=gr-qc&arxiv_sel=hep-ex&arxiv_sel=hep-lat&arxiv_sel=hep-ph&arxiv_sel=hep-th&arxiv_sel=math&arxiv_sel=math-ph&arxiv_sel=nlin&arxiv_sel=nucl-ex&arxiv_sel=nucl-th&arxiv_sel=physics&arxiv_sel=quant-ph&arxiv_sel=q-bio&sim_query=YES&ned_query=YES&aut_logic=OR&obj_logic=OR&author=&object=&start_mon=&start_year=&end_mon=&end_year=&ttl_logic=AND&title=titius+bode&txt_logic=OR&text=&nr_to_return=100&start_nr=1&jou_pick=ALL&ref_stems=&data_and=ALL&group_and=ALL&start_entry_day=&start_entry_mon=&start_entry_year=&end_entry_day=&end_entry_mon=&end_entry_year=&min_score=&sort=SCORE&data_type=SHORT&aut_syn=YES&ttl_syn=YES&txt_syn=YES&aut_wt=1.0&obj_wt=1.0&ttl_wt=0.3&txt_wt=3.0&aut_wgt=YES&obj_wgt=YES&ttl_wgt=YES&txt_wgt=YES&ttl_sco=YES&txt_sco=YES&version=1) (linked page shows all occurances of "Titius" and "Bode" in all scientific papers in NASA Astrophysics Data System, you can see that most of them refer to Titius Bode law, few call it rule and few something else. Word "hypothesis" is not being used.) I still say keep calling it a law, and if one shouldn't give a flip about someone's sayings (about law or not) here, it's not Warren's sayings in my opinion.

Warren Platts
2007-Nov-18, 09:20 AM
H1: Major planets that formed out of a primordial disk under normal conditions have an approximately logarithmic spacing; or
H2: The semimajor axis of 55 Cancri g is greater than 1.9 AU and less than 2.3 AU.
Let's take H1 (if that is what you are proposing).

Define "major planet."
I define major planet according to the standard definition as a body large enough to gravitationally achieve hydrostatic equilibrium (it's got to be big enough to be round), and it has to have cleared out its orbit of most debris--as opposed to dwarf planets like Pluto and Ceres. Thus, Mercury counts on my definition as a major planet. Moreover, Mercury's high density, and extensive simulations using GravitySimulator suggest that Mercury formed where it is, and is not an escaped moon from Venus or somewhere else.


Define "normal conditions."I am referring specifically to the conditions of the primordial disk, as well as to the subsequent history of a solar system. Different mixtures of dust and gas large enough to form a solar system out of will exhibit common patterns of solar system formation; and once a solar system forms, it will pretty much stay the same, unless historically contingent events perturb the primitive pattern.

Define "approximately" logarithmic.The TBL doesn't claim to have the accuracy of QED. Deviations as much as 15% from the predicted orbit are typical.


Oh, and since you mentioned it,
Define "major satellites." Explain why this applies only to the gas giants.If the TBL extends to some planet-moon systems, it is because they share a similar causal history compared to the solar system as a whole. In other words, in order for the TBL to apply, the moons have to form out of a primordial minidisk that forms around the planet, and then the moons have to remain in their primordial orbits for billions of years in order for us to observe the TBL pattern.

So we only observe TBL patterns around gas giants because only gas giants have enough moons for there to be enough data points to make a curve for; and only the "major" satellites count, because only the fairly larger ones have enough inertia to maintain their orbits for billions of years.

So, I have to retract my rebuttle of Van Rijn's claim that mass has something to do with TBL patterns. Lots of mass is a good design for planets and satellites not only because it helps maintain identity and creates a positive feedback so that the planet or satellite can "ingest" ever more matter, but lots of mass also enhances orbital stability. An asteroid can pass close by the Earth and be thrown into a crazy orbit, whereas the effect on the Earth will be lucky to be measurable at all. And so that's why every little rock, whether "regular" or not, in orbit around a gas giant doesn't figure into calculations for determining the presense of a TBL pattern. Incidentally, the printouts kindly provided by grant hutchison show that other orbital patterns different from TBL layouts do in fact commonly occur in nature, but that just demonstrates that TBL layouts themselves require a special explanation.

So thanks for the contribution to TBL theory, Van Rijn and grant hutchison! :clap: I appreciate your input!


(Idle thought... If we "usually" - 5-1, say - referred to you as a woman, would you protest, or give in to the majority?)Since I'm a transgendered republican, I'm already referred to as a woman 5 to 1. :lol:


The Newtonian equation for gravitational attraction ... includes an "explanation" for the result; it has the masses of the two bodies.

Your formulae have no such "explanation" beyond order from the primary. Is there any other explanation?
As I've said before, the TBL pattern is a "fossil". Thus the paradigm science for understanding the TBL is not Newtonian mechanics--it is paleontology. What's the explanation for the pattern of dinosaur bones beneath, but not above, the K-T boundary, beyond the pattern itself? The TBL formula itself is the pattern--unlike Newton's equations, it does not purport to explain the process that caused the pattern. Just as one must reach beyond the fossil record itself for an explanation of the causal process that caused the mass extinction pattern, one must reach beyond the TBL pattern, as described by the TBL formula, in order to arrive at a causal explanation for the TBL pattern.


Also, you now say the formula changes from inner to outer planets.

So, define "inner" and "outer" planets.
The inner planets are the rocky planets (Mercury through Mars), and the outer planets are the gas giants (Jupiter through Neptune). Obviously, there were different causal regimes under which the inner and outer planets formed. Thus, it is appropriate to calculate different TBL formulas for the inner and outer planets.



Originally Posted by Warren Platts
The above formula predicts Neputune's orbit to within 5%. ... No, it "predicts" nothing. You have generated a formula which fits the observed data. Congratulations.I think my use of 'predict' in the above context is in accord with common scientific practice ("Scientific sense of 'to have as a deducible consequence' is recorded from 1961." Dictionary.com (http://dictionary.reference.com/browse/predict))


So what does determine spacing?The primordial disk is a big swirl, then swirls within the swirl form based on a characteristic vorticity scale factor that in turn is determined primarily by the level of heavy elements (metals, rocks, and ices, i.e., the Z mass fraction, defined as total mass less hydrogen (X) and helium (Y)). The higher the Z, the faster vortices form, and the bigger the scale factor.

Warren Platts
2007-Nov-18, 10:00 AM
I didn't ask you to repeat a post hoc justification for how your "law" might work if it actually existed. I asked you to tell us how useful you think your "law" is going to be, in making predictions of how the real world is laid out.I guess I'm having a hard time answering your question despite my best efforts because your questions are so vague. E.g., you want me to tell you how the "real world is laid out". :confused: Where should I start? With the Big Bang or with God?

So stop dodging around and tell us:
On what physical basis you choose to reject data points, other than that they don't conform to your pet hypothesis.The physical basis is the evolutionary origin of the item in question. Asteroids, KBO's, major planets, and major moons all form differently--dont they? The TBL only applies to major planets and moons for the reasons I stated above.


On what physical basis you will decide whether your curve, plotted from these selected data points, is capable of predicting new data points (given that you've already offered "stuff moving around" as an excuse for a bad fit).
See, I don't understand this question. I think I've answered it before, but my previous answers aren't apparently what you want. My H2 (that a new planet will be found at around 2 AU from 55 Cancri A) is a specific, concrete hypothesis. That would be a new data point. I said before that the physical basis is the vorticity scale factor at the time of planetary formation. But you apparently want the "physical basis" for predicting new data points. So I have absolutely not the foggiest idea what the heck you are asking. I want to help, but I can't. Could you please be a little more specific?


To what extent your hypothesis will have failed if it does not correctly predict new data points.
IIRC, three standard deviations is about a 95% confidence factor, so if a planet were found between 55 Cancri d and 55 Cancri f that wasn't at 2 ± 0.36 AU from 55 Cancri A, then H2 could be rejected with 95% confidence.


(since at present it seems to be infinitely adaptable by the simple expedient of rejecting the inconvenient)This is really unfair. I don't reject data points because they're inconvenient. If a major planet at 55 Cancri were found at 4 AU, I would be the first to admit that my H2 was falsified.

Warren Platts
2007-Nov-18, 10:29 AM
I think part of the problem here is that you guys just don't get what the TBL is all about.

It's not about some magic formula that provides esoteric insight into the mysteries of orbital mechanics.

It's not even about predicting the orbits of unknown planets--though it has done so in the past, and may very well do so again in the near future.

It's about what happens during planetary formation. It preserves information regarding the conditions at the time of the origin of the solar system. Thus, the TBL scaling factor K is of intrinsic interest in itself. It is a fundamental, measurable parameter essential for a complete understanding of a given solar system.

So, when I divide the solar system into two zones, and calculate the K for both zones, I'm not engaging in curve fitting for the sake of curve fitting. Rather, I'm trying to understand the history of the solar system.

So here's an exam question--if you have been following this thread, and understand what I've been saying (and are therefore qualified to criticize what I've been saying), then you should be able to readily answer it:


I said before that planetary spacing depends on the size of the vorticity scaling factor, and that that in turn depends on the heavy mass fraction Z (higher Z entails higher K and that entails greater spacing). Yet Kin < Kout. How is this possible, given that rocky planets have an extremely high Z? Hint: Jupiter has a higher Z than the Sun.

:D

grant hutchison
2007-Nov-18, 01:58 PM
I guess I'm having a hard time answering your question despite my best efforts because your questions are so vague. E.g., you want me to tell you how the "real world is laid out". :confused: Where should I start? With the Big Bang or with God?Bluff. I am attempting to get detail out of your vague formulation, despite your best efforts to dodge. You are using data to make predictions about orbital spacing. It is not unreasonable to ask you to reveal how you reject data (the majority of data, in the case of planetary moons), and what you would consider to be a failure of prediction.


The physical basis is the evolutionary origin of the item in question. Asteroids, KBO's, major planets, and major moons all form differently--dont they? The TBL only applies to major planets and moons for the reasons I stated above.So the "triumph" of TBL, predicting Ceres, was in fact a failure? Because there's a gap in the TBL progression at Ceres.
So how do you decide what consitutes a "major" moon? The regular satellites of the gas giants are all hypothesised to have formed in the planetary nebula, and much migration has occurred since, according to theory.


See, I don't understand this question. I think I've answered it before, but my previous answers aren't apparently what you want. My H2 (that a new planet will be found at around 2 AU from 55 Cancri A) is a specific, concrete hypothesis. That would be a new data point.We all know that you have made that "prediction". But how are you sure that the rather selected data available from exoplanet studies is the "right" data, given that you feel free to reject "wrong" data, as in the case of the regular moon systems, and have told us that deviations from your "law" will occur if stuff moves about after it forms, which appears to be the case at 55 Cnc. Given the failure of TBL at Ceres, which we've just found out about, how do you know that it is reasonable or appropriate to predict a planet at 55 Cancri?


IIRC, three standard deviations is about a 95% confidence factor, so if a planet were found between 55 Cancri d and 55 Cancri f that wasn't at 2 ± 0.36 AU from 55 Cancri A, then H2 could be rejected with 95% confidence.Two standard deviations gives you approximate 95% confidence. This would appear to be just the confidence with which you can place a point on your existing curve, however. It doesn't say how well your "law" would tolerate a missing or deviant value. Would one empty space break it? How about two? Two in different systems? If a body appeared outside your current 95% confidence interval, you might well be able to recompute the whole curve to reaccommodate it. Would that be acceptable to you?


This is really unfair. I don't reject data points because they're inconvenient. If a major planet at 55 Cancri were found at 4 AU, I would be the first to admit that my H2 was falsified.You cheerily rejected more than half the regular moons of the gas giants without blinking. What are we supposed to deduce from that? What happened there was that you were prepared to fit a curve to a minority of the moons, which thereafter could predict nothing about the remainder of the moons.

Grant Hutchison

grant hutchison
2007-Nov-18, 02:10 PM
I think part of the problem here is that you guys just don't get what the TBL is all about.

It's not about some magic formula that provides esoteric insight into the mysteries of orbital mechanics.We all get this very well, I suspect.
Our problem is that you (and TBL) are starting with a magic formula, which seems to lack justification. Only once we have some sense that the magic formula has any relevance to the real world do we need to start looking for a specific hypothesis to explain it. Graner & Dubrulle provide a selection.

So your "exam question" is a cart placed well ahead of the horse.

Grant Hutchison

R.A.F.
2007-Nov-18, 02:38 PM
You cheerily rejected more than half the regular moons of the gas giants without blinking. What are we supposed to deduce from that?

I would deduce that Warren is quite capable of "messaging" any data until it "fits" into his already decided upon conclusion.

Bode's "law" is numerological coincidence...and not even a close coincidence at that.

grant hutchison
2007-Nov-18, 02:50 PM
Warren is not using it erroneously. It is exactly correct to use the term that practically everyone else uses, including the scientists (http://adsabs.harvard.edu/cgi-bin/nph-abs_connect?db_key=AST&db_key=PRE&qform=AST&arxiv_sel=astro-ph&arxiv_sel=cond-mat&arxiv_sel=cs&arxiv_sel=gr-qc&arxiv_sel=hep-ex&arxiv_sel=hep-lat&arxiv_sel=hep-ph&arxiv_sel=hep-th&arxiv_sel=math&arxiv_sel=math-ph&arxiv_sel=nlin&arxiv_sel=nucl-ex&arxiv_sel=nucl-th&arxiv_sel=physics&arxiv_sel=quant-ph&arxiv_sel=q-bio&sim_query=YES&ned_query=YES&aut_logic=OR&obj_logic=OR&author=&object=&start_mon=&start_year=&end_mon=&end_year=&ttl_logic=AND&title=titius+bode&txt_logic=OR&text=&nr_to_return=100&start_nr=1&jou_pick=ALL&ref_stems=&data_and=ALL&group_and=ALL&start_entry_day=&start_entry_mon=&start_entry_year=&end_entry_day=&end_entry_mon=&end_entry_year=&min_score=&sort=SCORE&data_type=SHORT&aut_syn=YES&ttl_syn=YES&txt_syn=YES&aut_wt=1.0&obj_wt=1.0&ttl_wt=0.3&txt_wt=3.0&aut_wgt=YES&obj_wgt=YES&ttl_wgt=YES&txt_wgt=YES&ttl_sco=YES&txt_sco=YES&version=1) (linked page shows all occurances of "Titius" and "Bode" in all scientific papers in NASA Astrophysics Data System, you can see that most of them refer to Titius Bode law, few call it rule and few something else. Word "hypothesis" is not being used.) I still say keep calling it a law, and if one shouldn't give a flip about someone's sayings (about law or not) here, it's not Warren's sayings in my opinion.I think we're talking two different forms of "erroneous" here.
It's not an error to use what has become standard terminology, as in "Titius-Bode Law", any more than it's an error to use the expression "black hole". But it is an error to assume that, because everyone uses the expression, the expression must be accurate. So the phrase "black hole" is erroneous to the extent that some black holes may emit visible light, and the expression "Titius-Bode Law" is erroneous to the extent that it (so far) describes only a perceived relationship in a small dataset.

Grant Hutchison

R.A.F.
2007-Nov-18, 02:52 PM
It's not even about predicting the orbits of unknown planets--though it has done so in the past, and may very well do so again in the near future.

This is simply NOT TRUE. I know this is ATM, but Please try to stick to the facts of the matter. It's coincidence plus bending the evidence to suit a conclusion, and unless/until you can explain the mechanism behind bodes "law", it will remain that.


...I'm trying to understand the history of the solar system.

How is rehashing the debunked "bodes law" going to help you understand the history of the Solar System?

grant hutchison
2007-Nov-18, 02:59 PM
IIncidentally, the printouts kindly provided by grant hutchison show that other orbital patterns different from TBL layouts do in fact commonly occur in nature, but that just demonstrates that TBL layouts themselves require a special explanation.I just noticed this. So "TBL layouts" are what is left when "other orbital patterns" are removed? That's as clear an admission of data dredging as one could hope to encounter.
You are picking a pattern out of the data by suppressing what doesn't conform to the pattern, and then claiming that the pattern requires a special explanation. Of course it does: but the explanation is that you created the pattern, by sifting the data.

Grant Hutchison

Warren Platts
2007-Nov-18, 03:44 PM
So your "exam question" is a cart placed well ahead of the horse.

If you can't answer the question, it shows you're criticizing something you don't understand. You might as well try and criticize the new Garrett Lisi paper.

grant hutchison
2007-Nov-18, 03:54 PM
If you can't answer the question, it shows you're criticizing something you don't understand.Your question is about the cart. I'm criticizing the rather glaring absence of a horse.

Grant Hutchison

orionjim
2007-Nov-18, 04:33 PM
IMHO Warren is laying the ground work for a model of how a star system is laid out; right or wrong. He is doing the obvious thing any real scientist would do and that is use all of the available data; in this case it’s only two, our solar system and the 55 Cancri system. He has a rough model and he is making a prediction of a fifth planet in the 55 Cancri system. Warren differs from the Titus-Bode Law because he is indirectly trying to define a mechanism.


We all get this very well, I suspect.
Our problem is that you (and TBL) are starting with a magic formula, which seems to lack justification. Only once we have some sense that the magic formula has any relevance to the real world do we need to start looking for a specific hypothesis to explain it. Graner & Dubrulle provide a selection.

So your "exam question" is a cart placed well ahead of the horse.

Grant Hutchison

Anytime you develop a model you have to put the cart before the horse/ the horse before the cart/ the horse on the cart/ the cart on the horse/ just the horse and just the cart. When Warren is finished with his model odds are that it will be wrong. But as George Box once said “How wrong does a model have to be not to be useful”.


This is simply NOT TRUE. I know this is ATM, but Please try to stick to the facts of the matter. It's coincidence plus bending the evidence to suit a conclusion, and unless/until you can explain the mechanism behind bodes "law", it will remain that.

How is rehashing the debunked "bodes law" going to help you understand the history of the Solar System?

The mechanism is the key. If Warren was to define a mechanism (or model) that fits TBL the first thing you would all do is go back to TBL and Warren’s beginning post and try to find different flaws.

It’s kind of funny how science works.

Jim

R.A.F.
2007-Nov-18, 05:06 PM
If Warren was to define a mechanism (or model) that fits TBL...

Since bodes law is based on coincidence and data "messaging", I don't see just how it would be possible for Warren to ever define a "mechanism".


...the first thing you would all do is go back to TBL and Warren’s beginning post and try to find different flaws.

The flaws in Warren's reasoning are clear and (thanks to this thread) abundent. Implying that those who disagree with him are being "closed-minded" is an obvious attempt to dodge the issue...bodes "law" doesn't work.


It’s kind of funny how science works.

Even funnier is how some people "think" science works.

grant hutchison
2007-Nov-18, 05:26 PM
The mechanism is the key. If Warren was to define a mechanism (or model) that fits TBL the first thing you would all do is go back to TBL and Warren’s beginning post and try to find different flaws. We have lots of models that "fit" TBL: Graner & Dubrulle have reviewed them. What we don't have is any apparent need for those models, until we get some compelling evidence that TBL is any more than a figment of selective data analysis. If some varient of exponential scaling actually turned out to be the case, in a statistically robust way, then we would have an embarrassment of potential explanations immediately to hand. So Warren is departing from the mainstream view not by coming up with a model (we've already got plenty of those) but with his claim that the data are strong enough to require (or justify) such a model in the first place.
This is why we're keen for him to defend his curve-fitting.

In addition, Warren's model relies on preservation of the original spacing that pertained when the planets formed. This requires some quite dramatic assumptions when applied to the 55 Cnc system. Current theory requires that b, c, and f have migrated inwards from their site of formation. Planet d could have formed in situ; planet e may have formed locally (if it's some sort of superterrestrial) or may also have migrated (if its some kind of overcooked gas giant). It's also part of current theory that the solar system giants migrated (spreading out from their primeval spacing), and that the moons of the gas giants migrated after formation, passing through a variety of resonance relationships. So I'm not disposed to fret about the detail of Warren's metallicity question when his model already requires the willing suspension of disbelief.

Warren's little "exam question" is therefore a classic example of misdirection in argumentation: a recurring favourite in my limited experience of ATM. When your argument begins to fall apart on obvious matters of gross structure, point urgently off into the distance and claim that a peripheral detail is absolutely key to everyone's understanding, and then refuse to proceed until that detail has been addressed.

Grant Hutchison

Warren Platts
2007-Nov-18, 07:21 PM
We have lots of models that "fit" TBL: Graner & Dubrulle have reviewed them. What we don't have is any apparent need for those models, until we get some compelling evidence that TBL is any more than a figment of selective data analysis. If some varient of exponential scaling actually turned out to be the case, in a statistically robust way, then we would have an embarrassment of potential explanations immediately to hand. So Warren is departing from the mainstream view not by coming up with a model (we've already got plenty of those) but with his claim that the data are strong enough to require (or justify) such a model in the first place.
This is why we're keen for him to defend his curve-fitting.
I've said this before, but let me say it again in a slightly different way. TBL only applies to major planets. Why? Because of the physical model Kuiper himself originally proposed. Kuiper's model says that the spacing has to do with how the planets were formed. Therefore, objects that formed through different processes are going to have different spacing. Therefore, every rock in the sky will not exhibit TBL spacing. Therefore, such objects aren't included in the statistical analysis that generates the mathematical model.

Oh, that's right! I almost forgot! I'm not allowed to talk about the physical mechanism until I explain why I didn't include every rock in the sky in my statistical analysis.

OK, let me try again. I can't include every rock in my statistical analysis mainly because most rocks in the sky don't exhibit the TBL spacing that the major planets exhibit. But maybe that's because most rocks in the sky didn't form the same way the major planets formed, and Kuiper himself said that the major planets formed from these giant vortices, and that the planetary spacing is because of the primitive vorticity scale.

Oh, that's right. I almost forgot. I'm not allowed to talk about the physical mechanism until I explain why I didn't include every rock in the sky in my statistical analysis.

OK, let me try again. Major planets are special. They exhibit an approximately logarithmic spacing pattern that's not typical of every rock in the sky. I wonder why that is? Well, Kuiper says that there were these vortices that have to do with how the major planets were formed, and that's what gave rise to the TBL pattern, but since every rock in the sky did not form the same way that the major planets formed, it's not surprising that every rock in the sky doesn't exhibit the TBL spacing pattern.

Oh, that's right. I almost forgot. I'm not allowed to talk about the physical mechanism until I explain why I didn't include every rock in the sky in my statistical analysis.

OK, so let's try again. . . .

You've stuck me in a classic, crazy-making double bind, a box, a catch-22, there's no way out of. I can't talk about the physical mechanism for TBL until I explain why I didn't include every rock in the sky in my statistical analysis; but I can't explain why I didn't include every rock in the sky without talking about the physical mechanism.

So it's a no-win situation, a sucker's bet, a fool's game. But I'll continue to play for the sake of the lurkers, and because I guarantee you, when this thread is finally locked, I'm not going to be the one who looks like the fool.


In addition, Warren's model relies on preservation of the original spacing that pertained when the planets formed. This requires some quite dramatic assumptions when applied to the 55 Cnc system. Current theory requires that b, c, and f have migrated inwards from their site of formation. Planet d could have formed in situ; planet e may have formed locally (if it's some sort of superterrestrial) or may also have migrated (if its some kind of overcooked gas giant). It's also part of current theory that the solar system giants migrated (spreading out from their primeval spacing), and that the moons of the gas giants migrated after formation, passing through a variety of resonance relationships. So I'm not disposed to fret about the detail of Warren's metallicity question when his model already requires the willing suspension of disbelief.
Dr. Hutchison, now you're coming dangerously close to agreeing that there is a TBL pattern to be explained, but you apparently prefer Velikovskyian, kinematic explanations. But yeah, it's kind of hard to see how the gas giants in this solar system would migrate outward to form a TBL pattern, whereas the planets at 55 Cancri migrated inward to form a TBL pattern.

Besides that, if every rock in the sky is subject to gravitational resonances, then there should be TBL patterns everywhere, but there isn't.

And besides that, it's hard to see how a Jupiter sized planet is going to go anywhere but where it wants to go. That the major planets in this solar system have moved around a lot since they were formed is news to me--Velikovsky notwithstanding. Sure, I know they exchange angular momentum and have their resonances, like the way Venus is in rotational resonance with the Earth, and their eccentricities vary slightly over time. But my experience with GravitySimulator is that I can't get the big planets to go much anywhere unless I crank up the time-step, and then things go crazy, but it's hard to tell if the craziness is inherent to the universe or just the result of sampling errors that build up in numerical simulations that are necessarily approximations. Tony, if you're reading this, now would be a good time to chime in and explain to everyone the difficulties inherent in accurately simulating orbital evolution for billions of years.

As for gas giants forming close to their primaries, I am aware that one theory is that gas giants formed in outer regions and then--somehow--migrated into the interior reaches of their solar systems. . . . Drag because of interplanetary dust? Or maybe the Electric Universe people have got it right, and powerful electromagnetic fields act as giant electric brakes. :lol:

However, I am not aware that this is a settled question and that in situ formation of gas giants within a couple of AU's of their primaries is theoretically impossible. My understanding of the evolutionary theory is we don't have gas giants closer than Jupiter in this solar system, so, reasoning by analogy, it is assumed that gas giants can't form close-in in other solar systems.

But maybe the reason we don't have gas giants in the inner solar system is because the biggest rocky planet here is only one Earth-mass--too low to grab onto and hold a hydrogen atmosphere. Jupiter, after all, probably has a rocky-metallic core on the order of 8-14 Earth-masses. So, in a solar system like 55 Cancri where the metalliticity of the sun there is 186% of our Sun, maybe it's the case that large metallic cores with masses on the order of tens of Earth-masses might form and quickly grab a hydrogen atmosphere before all the hydrogen got sucked into the parent star or blown away by the solar wind. Then powerful magnetic fields within such inner gas giants would generate a force field protecting their atmosphere from erosion by strong solar winds.

Or maybe it's the case that 55 Cancri's inner gas giants will eventually lose their hydrogen, but since the 55 Cancri system is < 10 million years old, there just hasn't been time enough for this to occur yet.


Warren's little "exam question" is therefore a classic example of misdirection in argumentation: a recurring favourite in my limited experience of ATM. When your argument begins to fall apart on obvious matters of gross structure, point urgently off into the distance and claim that a peripheral detail is absolutely key to everyone's understanding, and then refuse to proceed until that detail has been addressed.
I'm not misdirecting anything; I'm trying to move the discussion forward. And my question is a test for anyone interested in the theory here to test themselves and see how well they understand the model presented. If you don't want to type out the answer here, that's fine, and I understand why. But if you can't figure it out in your own head at least, you really don't understand my theory. (It should be easier now, however, as I just gave some more clues.) :)

grant hutchison
2007-Nov-18, 08:09 PM
OK, so let try again. . . .

You've stuck me in a classic, crazy-making double bind, a box, a catch-22, there's no way out of.I haven't. We've all read your physical explanation with various levels of interest. My objection is to "exam questions" on topics of minor importance while the glaring problems lie all around unaddressed. But your little fit there demonstrates rather nicely where the circularity lies: all of your offered explanations are for data that you have extracted as "requiring explanation" because they appear to conform to a spacing which is the basis of your explanation. The process by which you levitate your own "law" into existence is very clear.


That the major planets in this solar system have moved around a lot since they were formed is news to me--Velikovsky notwithstanding.Oh dear. Well, I kind of figured it would be news to you. And yet it is so.
The migration of the gas giants has been part of the standard model for more than twenty years. There are entire textbooks and learned conferences devoted to the topic.
The first paper off my pile, Planetary migration in a planetesimal disk: why did Neptune stop at 30AU (http://www.boulder.swri.edu/~hal/PDF/migration.pdf), by Gomes et al, seems like it will provide you with an introduction and a starter's reference list, and a broad hint on why Gravity Simulator isn't going to help you with this one.
You might also care to search on "Type-I migration" and "Type-II migration", the latter of which is the current preferred mechanism for the creation of "hot Jupiters".


And my question is a test for anyone interested in the theory here to test themselves and see how well they understand the model presented. If you don't want to type out the answer here, that's fine, and I understand why. But if you can't figure it out in your own head at least, you really don't understand my theory.See above for why I'm devoting so little effort to understanding your hypothesis.

Grant Hutchison

R.A.F.
2007-Nov-18, 08:38 PM
I'm trying to move the discussion forward.

What is there to "discuss"??...Bode's "law" doesn't work.

grant hutchison
2007-Nov-18, 09:07 PM
Or maybe it's the case that 55 Cancri's inner gas giants will eventually lose their hydrogen, but since the 55 Cancri system is < 10 million years old, there just hasn't been time enough for this to occur yet.BTW, what makes you think that the system is less than 10 million years old?
The star is generally described as being chromospherically quiet, and Baliunas et al (http://www.journals.uchicago.edu/ApJ/journal/issues/ApJL/v474n2/5583/5583.html) assigned a stellar age of 5 billion years. which more or less matches the reference given in the Extrasolar Planets Encyclopedia (http://exoplanet.eu/star.php?st=55+Cnc).
Fischer, Marcy et al, in their recent announcement of the fifth planet, describe the star as having "... a modest age of 2-8Gyr, where 2Gyr is a strong lower limit on age" based on their most recent seven years of observation.

It would seem a bit surprising if the planetary system was so much younger than the star.

Grant Hutchison

Warren Platts
2007-Nov-18, 09:56 PM
The first paper off my pile, Planetary migration in a planetesimal disk: why did Neptune stop at 30AU (http://www.boulder.swri.edu/~hal/PDF/migration.pdf), by Gomes et al, seems like it will provide you with an introduction and a starter's reference list,
Grant, thank you for the link--that is very interesting. They provide a mechanism that would explain why the scaling factor K is greater than in the inner solar system: the primitive K factor was the same as the current inner system, but then Jupiter moved inward, while Neptune moved outward; thus the K factor increased.

However, a quick glance at my Bode's model shows that the main deviation from theoretical expectations of the primitive condition is that Uranus and Neptune might have migrated up to 3 AU towards each other (that's the main source of error in my model).

And the overall movements described by Levison and his colleagues are hardly Velikovskyian in scale, the are on the order of a few AU at most, which is in within the expected error bars anyway.
EDIT: I take back what I said about the movements describe by Levison and his colleagues not being Velikovskyian in scale:


In all integrations that we have made, Neptune reverses its migration at ~ 110-120 AU. We note that Neptune is not more likely to eject planetesimals from the Solar System when it is further from the Sun because for a particle encountering the planet the probability to be ejected to hyperbolic orbit depends exclusively on the Tisserand parameter, and the latter is independent of the semi major axis units. Therefore the reversion of Neptune's migration requires a more subtle explanation.

This all goes to show that dynamic and kinematic theories aren't mutually exclusive. Dynamical theories say what the primitive condition was, and kinematic theories say how conditions evolved until we arrived at the present layout. Thus, kinematic and dynamical theories of the TBL are in fact complementary. Indeed kinematic theorests should be grateful for an independent criterion against which to constrain their own theories.

And this goes to show a fundamental weakness of all you TBL naysayers, because you can't answer this question: What do you think was the primitive planetary spacing?


And I know what you're going to say:


WE DON'T KNOW! :hand:Therefore, you should show a little humility because you couldn't possibly have the answer either!:wall:
This is a council of FUD. The TBL, however, has a plausible answer: it was logarithmic, as evidenced by the present spacing, and as explained by Kuiper's theory.

Warren Platts
2007-Nov-18, 10:08 PM
BTW, what makes you think that the system is less than 10 million years old?
You're quite right about the age. I must have confused 55 Cancri with another star I was reading about.

grant hutchison
2007-Nov-18, 10:14 PM
By the way, what is FUD?
When I was a boy, here in Scotland, it was a taboo word of anatomical specificity. It's quite unnerving to see it repeated in large bold capital letters so frequently.

Grant Hutchison

grant hutchison
2007-Nov-18, 10:36 PM
And this goes to show a fundamental weakness of all you TBL naysayers, because you can't answer this question: What do you think was the primitive planetary spacing?Honestly, Warren, and I ask out of genuine curiosity fueled by bafflement: Did you get any training at all in logical argumentation during your philosophy course?

This is not a weakness in our argument, which has always been that you are speculating beyond the data. Pointing out that the available data are not the data you need merely strengthens that argument: you are now seen to be speculating beyond data that you don't even have access to.
Pack up your tent and go home.

Grant Hutchison

Warren Platts
2007-Nov-18, 11:06 PM
By the way, what is FUD?
When I was a boy, here in Scotland, it was a taboo word of anatomical specificity. It's quite unnerving to see it repeated in large bold capital letters so frequently.

Grant Hutchison

What, "2. Woolen waste, for mixing with mungo and shoddy."? Because that's the only other meaning for FUD in the dictionary I use. FUD is an acronym for Fear, Uncertainty, and Doubt. :)

Van Rijn
2007-Nov-18, 11:15 PM
By the way, what is FUD?
When I was a boy, here in Scotland, it was a taboo word of anatomical specificity. It's quite unnerving to see it repeated in large bold capital letters so frequently.

Grant Hutchison

Here, it's an acronym: Fear, Uncertainty, and Doubt. It's a very well known term in the computer industry. The old line was, "Nobody was ever fired for buying IBM," usually said by an IBM salesman. Perhaps the modern version would put "Microsoft" in place of "IBM." You hear it outside the computer industry these days. Here's the wikipedia entry on it:

http://en.wikipedia.org/wiki/Fear%2C_uncertainty_and_doubt

Now that I've heard of this other definition of the term, I'll be a bit more careful using it.

Van Rijn
2007-Nov-18, 11:18 PM
What, "2. Woolen waste, for mixing with mungo and shoddy."? Because that's the only other meaning for FUD in the dictionary I use. FUD is an acronym for Fear, Uncertainty, and Doubt. :)

Try google. You'll find it quickly enough.

grant hutchison
2007-Nov-18, 11:21 PM
What, "2. Woolen waste, for mixing with mungo and shoddy."? Because that's the only other meaning for FUD in the dictionary I use.So you don't keep a dictionary of Scottish slang expressions? How negligent of you.
You might try typing "fud" and "Scottish" into the search engine of your choice. Suffice it to say, typing the word in big bold letters here gives you a passing resemblance to an overexcited, attention-seeking wee boy. :)


FUD is an acronym for Fear, Uncertainty, and Doubt.Ah, thanks. So it's one of those accusations people who think themselves innovative use against people who think themselves sensible. I see.

Grant Hutchison

Edit: No, now I've looked at the post and link from Van Rijn (thanks!), it appears to be something of a more sinister nature. Warren, if it was your intention to accuse us here of spreading deliberate misinformation, I do most heartily resent it.

Van Rijn
2007-Nov-18, 11:25 PM
[Titius Bode is] about what happens during planetary formation. It preserves information regarding the conditions at the time of the origin of the solar system. Thus, the TBL scaling factor K is of intrinsic interest in itself. It is a fundamental, measurable parameter essential for a complete understanding of a given solar system.


That is what we call a "hypothesis." First, you have to find that there actually is a recognizable TB pattern, and as I said early in the thread, you're going to need a lot more data than we currently have to determine that, especially given that even you now realize that systems evolve over time.

Van Rijn
2007-Nov-18, 11:39 PM
Ah, thanks. So it's one of those accusations people who think themselves innovative use against people who think themselves sensible. I see.

Grant Hutchison

Sometimes, but in the industry it usually refers to deliberate manipulation. IBM had a habit of waiting until another company had a successful product, then announcing they would have their own version "soon." Often, that led to sales for the competition to drop. By the way, that worked well in the mainframe years, but not so well in the later PC years. I remember when they announced they would be coming out with laptops, and my astonishment at their sales demonstration when they finally got around to it: Their early laptops were insanely heavy and at least a year behind everyone else's technology (they finally did become innovative, but that's another story).

grant hutchison
2007-Nov-18, 11:51 PM
Sometimes, but in the industry it usually refers to deliberate manipulation.Yes, thanks, I took that meaning from your post and link (which had appeared while I was preparing my reply to Warren) and I have now been back to edit my post accordingly while (I think) you've been preparing your most recent. :doh:
I'm hoping Warren intended my original understanding of the term, rather than the one you report.

Grant Hutchison

Warren Platts
2007-Nov-19, 12:05 AM
I would deduce that Warren is quite capable of "messaging" any data until it "fits" into his already decided upon conclusion.

Bode's "law" is numerological coincidence...and not even a close coincidence at that.
I said I would be the first to reject my H2 hypothesis, that there is a planet at about 2 AU, if 55 Cancri g was found at 4 AU. You all imply that I would just curve-fit my way out of such an inconvenience.

So I plugged 4 AU into the observed slot in my spreadsheet model, and then applied my curve fitting skills to first come up with a new least squares model that best fit the new data. Then I compared the results with the previous model. Here is the data for the new (hypothesis rejected) model (H2'):



ai = 0.0368(2.798)n-1


name observed predicted error abs(%err)
e 0.038 0.037 -0.001 0.03158
b 0.115 0.103 -0.012 0.10464
c 0.240 0.288 0.048 0.20042
f 0.781 0.806 0.025 0.03214
V 4.000 2.255 -1.745 0.43613
d 5.770 6.311 0.541 0.09373
average abs(error) = 0.14977
stdeva(abs(errors)) = .15333

compared to for the original model (H2)
e 0.038 0.037 -0.001 0.01579
b 0.115 0.102 -0.013 0.11476
c 0.240 0.277 0.037 0.15461
f 0.781 0.754 -0.027 0.03421
d 5.770 5.589 -0.181 0.03142
average abs(error) = 0.07016
stdeva(abs(errors)) = 0.06097

So, despite my best curve-fitting skills, the new model incorporating Planet V at 4 AU instead of 2 AU has twice the size average error. The error for Planet V itself remains 44% despite my best efforts. The situation is made even more clear by the accompanying chart. You can see by tweaking the curve as much as possible and doubling the error bar, a Planet V distance at 4 AU woud still result in rejection of H2'. So even the tweaked model makes defeasible predictions. So to keep on insisting that I'll massage the curves to fit any data is just false, and misleading.

So I'm sticking to my guns. The 5 data points in hand fit a well behaved exponential curve remarkably well to a statistically significant degree. The first extrasolar system for which we have data for five planets exhibits the TBL pattern.

grant hutchison
2007-Nov-19, 12:09 AM
So to keep on insisting that I'll massage the curves to fit any data is just false, and misleading, and borderline ad hom.Have you considered splitting the system into two sections and curve-fitting separately? You've done that in the solar system already.

Grant Hutchison

Warren Platts
2007-Nov-19, 12:15 AM
Have you considered splitting the system into two sections and curve-fitting separately? You've done that in the solar system already.

Grant Hutchison

Because the inner and outer planets formed under different conditions.

grant hutchison
2007-Nov-19, 12:30 AM
Because the inner and outer planets formed under different conditions.Ah.

Grant Hutchison

Warren Platts
2007-Nov-19, 07:03 PM
Here is the statistical analysis for the 55 Cancri data.

y = ln(observed semimajor axis of planets at 55 Cancri in AU)

x y (x-avg(x))2 (y-avg(y))2 x-avg(x)*y-avg(y)

1 -3.270 4.8400 4.8367 4.8383
2 -2.163 1.4400 1.1922 1.3103
3 -1.427 0.0400 0.1269 0.0712
4 -0.247 0.6400 0.6786 0.6590
6 1.753 7.8400 7.9727 7.9061

avg(x) = 3.2
avg(y) = -1.071
sum((x-avg(x))2) = 14.800
sum((y-avg(y))2) = 14.807
sum(x-avg(x)*y-avg(y)) = 14.785
slope (b) = 0.99898
y intercept = -4.267
SSResid = 0.03719
estimated standard deviation se of variation about the estimated regression line = 0.111337
r2 coeffient of determination = 0.9975
estimated standard deviation of the slope sb = 0.02894
degrees of freedom (n-2) = 3
t value for 80% confidence level (d.f.=3) = 1.64
confidence interval for β = ± 0.04746

Now take a look at the attached log graph. The blue lines in the graph show the upper and lower 80% confidence level for the slope β.

Clearly, it is obvious to visual inspection that the observations are linearly arranged and well-described by a straight-line regression. However, the coefficient of determination r2 shows that 99.75% of the total variation is explained by the model--that is, there is very little scatter about the regression line.

Now to calculate a few more statistics in order to get a prediction confidence interval for predicting the location of the apparently missing Planet V.

s2a+b(x=5) = 0.00519
t value for 95% (d.f.=3) confidence interval = 3.18
prediction interval = ± 0.422
ln(Planet V) = 0.727
95% confidence interval = e0.727+0.422, e0.727-0.422 = 3.155, 1.357 AU

Thus, if the planetary system at 55 Cancri is arranged in a Titius-Bode pattern, then the statistical regression predicts at a 95% confidence level that 55 Cancri V will be found somewhere between 1.357 AU to 3.155 AU. The 90% confidence interval is from 2.515 to 2.826 AU. The 80% confidence interval is 1.665 to 2.572 AU.

The confidence interval can be seen in context in the next graph. The error bar shows the 95% confidence interval. The blue lines show the 80% confidence interval in the slope of the regression line, and the 80% confidence level error bar for the Planet V prediction would be very slightly to the inside of the blue lines.

So there!

R.A.F.
2007-Nov-19, 07:36 PM
So there!

:lol:

Warren Platts
2007-Nov-19, 07:53 PM
"There is a single light of science, and to brighten it anywhere is to brighten it everywhere."
:lol:

R.A.F.
2007-Nov-19, 08:46 PM
I don't care if you laugh at me, but laugh at Asimov and we're gonna have TROUBLE.

Kapish??

Warren Platts
2007-Nov-19, 10:38 PM
I don't care if you laugh at me, but laugh at Asimov and we're gonna have TROUBLE.

Kapish??Don't worry, it wasn't Asimov; it was the irony!


So you don't keep a dictionary of Scottish slang expressions? How negligent of you.

You might try typing "fud" and "Scottish" into the search engine of your choice. Suffice it to say, typing the word in big bold letters here gives you a passing resemblance to an overexcited, attention-seeking wee boy. :)

Ah, thanks. So it's one of those accusations people who think themselves innovative use against people who think themselves sensible. I see.

Grant Hutchison

Edit: No, now I've looked at the post and link from Van Rijn (thanks!), it appears to be something of a more sinister nature. Warren, if it was your intention to accuse us here of spreading deliberate misinformation, I do most heartily resent it.

Dr. Hutchison, you're one to complain about rudeness--your tone has been awful throughout this thread!

Why don't we let bygones be bygones and settle this like gentlemen, shall we? Thus, I propose a bet:



If 55 Cancri g is not found where I say it will be, I will send you a 1-liter bottle of America's finest bourbon--a.k.a. Booker's. However, if 55 Cancri g turns up where the Titius-Bode Law says it should be, then you shall send me a 1-liter bottle of 18-year-old Glen Morangie. To make it fair, let's go by the 80% confidence level prediction interval. The bet becomes decided when the Encyclopedia of Extrasolar Planets (http://exoplanet.eu/catalog-all.php) first posts a status "R" measure of the semimajor axis for 55 Cancri g. I win if the semimajor axis posted by the Encyclopedia is equal to or greater than 1.665 AU or less than or equal to 2.572 AU. That's way more than fair, since there's the Jupiter-asteroid belt factor in your favor.

How about it?

:D

grant hutchison
2007-Nov-19, 11:28 PM
Dr. Hutchison, you're one to complain about rudeness--your tone has been awful throughout this thread!I believe I've been admirably restrained, occasionally verging on the solicitous.


Why don't we let bygones be bygones and settle this like gentlemen, shall we? Thus, I propose a bet:I don't bet, Warren.
But now I know you're viewing this as some sort of competition, perhaps I understand the thread a little better.

Grant Hutchison

grant hutchison
2007-Nov-19, 11:50 PM
I win if the semimajor axis posted by the Encyclopedia is equal to or greater than 1.665 AU or less than or equal to 2.572 AU.BTW: Just in case anyone else wants to jump in and take up Warren's bet, please note that his choice of wording means he wins no matter what the reported semimajor axis is.

Grant Hutchison

Warren Platts
2007-Nov-20, 12:45 AM
BTW: Just in case anyone else wants to jump in and take up Warren's bet, please note that his choice of wording means he wins no matter what the reported semimajor axis is.

Grant Hutchison

GARGH! :razz: They really do teach good logic up there in Scotland. But believe me, I've had my share of logic here. Once I even developed a completely expressible propositional caluculus (PC) based solely on unary truth-functions. Which I'm sure you'd put into the same class as the Titius-Bode relation. :)


Oh, by the way, "Titius-Bode" is kind of reminiscent of a certain parts of the female anatomy in certain foreign languages. Sorry if I offended anyone by starting this thread.

As for this thread being a competition, you should remember this is a science forum, and as the cover of this philosophy of science book attached demonstrates, science consists of a bunch of naked men battling with swords, Braveheart-style.

I wish you'd reconsider the bet; really it's a sucker's bet, because if 55 Cancri is like our solar system, there's an asteroid belt in slot V. Something about slot V. . . . Probably the solar system bifurcated more than once. So the asteroid belt is a sort of phase transition zone, where original symmetry was broken. The outer protoplanetary vortices started doing their own thing, while the central zone still thought it was all one big empire. After a while, though, the close quarters became untenable, so the parent retracted to enjoy its dotage, leaving the younger kids a meagre inheritance.

So, perhaps the same thing happened on 55 Cancri--which is billions of years old.

So what do you say. I know you're not a gambling man. But hopefully that's because of your principles and not bitter experience. So why not make an exception. It's really not a bet, it's more of a duel. But dueling with guns and swords is outlawed these days, and even a good old-fashioned fist-fight is liable to wind yourself up in jail no matter what country your in these days, so forcing the other to buy a superexpensive bottle of booze is the closest either one of us can hope to inflicting any real pain. So what do you say there, mate? (I win if 55 Cangri g semimajor axis is greater than or equal to 1.665 AU AND less than or equal to 2.572 AU.)

grant hutchison
2007-Nov-20, 01:13 AM
I don't bet, because a bet on the outcome either glorifies the trivial or demeans the significant.
In this case, the former applies, since it won't matter a whit whether or not there's a planet in the appointed slot: you will still have speculated beyond the data available to you today.
There could be a planet in the slot, but no general exponential relationship to be found when 55 Cnc is included in the analysis of good data from multiple systems. There could be no planet in the slot, but widespread exponential scaling detectable across multiple systems, sufficient to require explanation.

We don't know. You don't know. And that's the point people have been trying to make.

Grant Hutchison

R.A.F.
2007-Nov-20, 01:29 AM
I wish you'd reconsider the bet.

Wagering is frowned upon around here, so if would be best for you to discontinue this and simply present your evidence.

Warren Platts
2007-Nov-20, 01:39 AM
No, the point of this thread, as stated in the OP is that 55 Cancri is already a data point to be chalked up on the TBL side of the board, as we build up that database of 50 systems that Van Rijn said would satisfy him. Right? 55 Cancri certainly can't be counted as evidence against the TBL. Can it? Sure, it's only one data point; but data points have to be added together. Don't they?

55 Cancri is Iowa, baby, and she's voting for TBL!!! :clap::dance::clap:

That cannot be denied.

That just cannot be denied.

I did the statistics myself.

To deny that--as I'm sure none of you can--though you will anyway--is the very essence of pure




FUD.

And that's a nice line about gambling demeaning the admirable--I'll have to remember that one. But like I said, this isn't about gambling for the sake of gambling anymore. This is between you and me buddy. I'm supposed to pack up my tent and leave?!? How bout we just put our money where our mouths are instead?

Warren Platts
2007-Nov-20, 02:05 AM
Wagering is frowned upon around here, so if would be best for you to discontinue this and simply present your evidence.
R.A.F., don't be such a control freak. You know as well as I do that there's been a tradition among scientists and philosophers to place long term bets on the outcome of some future crucial experiment or observation. Wasn't Fermi taking bets on whether the atmosphere would ignite when they lit off the first atomic bomb?

And I did present my evidence as simply as standard statistics allows. It's there. In the post above with the two charts attached. Read it. Read it again. The post where it says that the planets at 55 Cancri have a logarithmic spacing pattern. The one where the regression line had an r2 of 99.7%.

FUD will not work anymore.

R.A.F.
2007-Nov-20, 02:15 AM
R.A.F., don't be such a control freak.

Hey...just friendly advice.


You know as well as I do that there's been a tradition...

NOT ON THIS BOARD.


And I did present my evidence as simply as standard statistics allows. It's there. In the post above with the two charts attached. Read it. Read it again.

Done and done, yet the "evidence" you present is simply not convincing.

Bodes-Titus doesn't work...not without "twisting" the data all to hell.

Warren Platts
2007-Nov-20, 02:42 AM
Hey...just friendly advice.



NOT ON THIS BOARD.
Maybe they should get around to finally passing rule # 37.1.412.III(a): "No bets on future scientific observations allowed."


Done and done, yet the "evidence" you present is simply not convincing. What? 99.75% of the variance explained by the logarithmic model is not good enough for you?


Bodes-Titus doesn't work...not without "twisting" the data all to hell.Just where did I twist the data? Was it when I computed the logarithms of the semimajor axes, or when I summed the squares of the residuals; was it when I looked up the t-value in the table at the back of the book, or was it when I plotted that suspicious-looking error bar on that one chart?

Van Rijn
2007-Nov-20, 03:03 AM
I don't bet, because a bet on the outcome either glorifies the trivial or demeans the significant.
In this case, the former applies, since it won't matter a whit whether or not there's a planet in the appointed slot: you will still have speculated beyond the data available to you today.
There could be a planet in the slot, but no general exponential relationship to be found when 55 Cnc is included in the analysis of good data from multiple systems. There could be no planet in the slot, but widespread exponential scaling detectable across multiple systems, sufficient to require explanation.

We don't know. You don't know. And that's the point people have been trying to make.

Grant Hutchison

That pretty much sums it up. Fitting equations to the data, then yelling "FUD" at anyone that points out you can't leap to conclusions, is pretty sad behavior. I'm open to the possibility that a common, simple, relationship may fall out with more data, but at this point, it's speculation. I don't have any problem discussing speculation, but I do have a problem when someone insists it is more than that.

R.A.F.
2007-Nov-20, 03:21 AM
Just where did I twist the data?

What makes you think that I would have the slightest interest in proving you wrong?

You're the proponent of this idea...prove yourself right.

...and repeating "this can not be denied" is not proof, it's handwaving.

publius
2007-Nov-20, 04:00 AM
Could someone shed some light on Graner & Dubrulle for me, please. My understanding was, that if you start out with a gas cloud with axial symmetry and scale invariance and let it collapse to a star with planetary system, you're going to get some sort of "regular relation" in the spacing of the final bodies.

That relation can be anything, but once the collpase gets started and the chaotic processes fall a certain way, you will get "some sort of relation". :)

-Richard

grant hutchison
2007-Nov-20, 08:50 AM
55 Cancri is Iowa, baby, and she's voting for TBL!!! :clap::dance::clap:

That cannot be denied.

That just cannot be denied.

I did the statistics myself.You might just pause, during your little dance, to recall that these data are necessarily correlated, just because of the way you've generated them. You numbered the semimajor axes in ascending order of magnitude. You've sorted them: it's pretty much impossible for them to show no correlation.
This sort of thing is a common enough error in data analysis, when people accidentally put in "pre-correlated" data (because of an unnoticed mathematical linkage between the dependent and independent variable, or because the two variables turn out to be measuring the same thing in two different ways), and are then impressed by the results of a correlation test. But it's unusual, I think, for it to be quite so flagrant as correlating a dataset with a simple measure of the relative magnitude of its data points.

And then you inserted a gap in order to meet your specific expectation of an exponential. :doh:

Grant Hutchison

grant hutchison
2007-Nov-20, 08:53 AM
Could someone shed some light on Graner & Dubrulle for me, please. My understanding was, that if you start out with a gas cloud with axial symmetry and scale invariance and let it collapse to a star with planetary system, you're going to get some sort of "regular relation" in the spacing of the final bodies.Yes, I think that's what they're saying.

Grant Hutchison

Warren Platts
2007-Nov-20, 01:56 PM
Originally Posted by publius
Could someone shed some light on Graner & Dubrulle for me, please. My understanding was, that if you start out with a gas cloud with axial symmetry and scale invariance and let it collapse to a star with planetary system, you're going to get some sort of "regular relation" in the spacing of the final bodies.

Yes, I think that's what they're saying.

Grant Hutchison

No, what they're saying is that if whatever physical mechanism responsible for planetary spacing is scale invariant with respect to distance from the primary, then a logarithmic planetary spacing pattern will result.

grant hutchison
2007-Nov-20, 02:14 PM
No, what they're saying is that if whatever physical mechanism responsible for planetary spacing is scale invariant with respect to distance from the primary, then a logarithmic planetary spacing pattern will result.You're suggesting that a "logarithmic planetary spacing" is not a "'regular relation' in the spacing of the final bodies"?

Grant Hutchison

publius
2007-Nov-20, 04:45 PM
Here's a good one on Arxiv:

http://arxiv.org/abs/astro-ph/9710116




Simple ``solar systems'' are generated with planetary orbital radii r distributed uniformly random in log(r) between 0.2 and 50 AU. A conservative stability criterion is imposed by requiring that adjacent planets are separated by a minimum distance of k Hill radii, for values of k ranging from 1 to 8. Least-squares fits of these systems to generalized Bode laws are performed, and compared to the fit of our own Solar System. We find that this stability criterion, and other ``radius-exclusion'' laws, generally produce approximately geometrically spaced planets that fit a Titius-Bode law about as well as our own Solar System. We then allow the random systems the same exceptions that have historically been applied to our own Solar System. Namely, one gap may be inserted, similar to the gap between Mars and Jupiter, and up to 3 planets may be ``ignored'', similar to how some forms of Bode's law ignore Mercury, Neptune, and Pluto. With these particular exceptions, we find that our Solar System fits significantly better than the random ones. However, we believe that this choice of exceptions, designed specifically to give our own Solar System a better fit, gives it an unfair advantage that would be lost if other exception rules were used. We conclude that the significance of Bode's law is simply that stable planetary systems tend to be regularly spaced; this conclusion could be strengthened by the use of more stringent methods of rejecting unstable solar systems, such as long-term orbit integrations.



IOW, generate a random solar system, apply stability criteria, a voila!, you get regular spacing that follows a pattern, roughly, especially if you allow yourself some "peremptory challenges" so to speak of data points you don't like.

-Richard

R.A.F.
2007-Nov-20, 05:13 PM
IOW, generate a random solar system, apply stability criteria, a voila!, you get regular spacing that follows a pattern, roughly, especially if you allow yourself some "peremptory challenges" so to speak of data points you don't like.

Hmm...ignoring data based solely on a predetermined outcome sounds like bad science to me. :)

Disinfo Agent
2007-Nov-20, 05:40 PM
IOW, generate a random solar system, apply stability criteria, a voila!, you get regular spacing that follows a pattern, roughly, especially if you allow yourself some "peremptory challenges" so to speak of data points you don't like.I'm not sure what they did was as simple as generating a "random" solar system. By assigning a uniform distribution to log(r), aren't they basically assuming Bode's Law?

grant hutchison
2007-Nov-20, 06:13 PM
I'm not sure what they did was as simple as generating a "random" solar system. By assigning a uniform distribution to log(r), aren't they basically assuming Bode's Law?It certainly seems so, and I don't see any discussion or defence of that particular random distribution in the paper, unless I'm missing something. They also use a Hill radius exclusion, which will help to weed out systems that deviate strongly, by chance, from the log distribution they've created. And then they get to "weed" the result! Very odd.

Grant Hutchison

Warren Platts
2007-Nov-20, 07:13 PM
You might just pause, during your little dance, to recall that these data are necessarily correlated, just because of the way you've generated them. You numbered the semimajor axes in ascending order of magnitude. You've sorted them: it's pretty much impossible for them to show no correlation.
This sort of thing is a common enough error in data analysis, when people accidentally put in "pre-correlated" data (because of an unnoticed mathematical linkage between the dependent and independent variable, or because the two variables turn out to be measuring the same thing in two different ways), and are then impressed by the results of a correlation test. But it's unusual, I think, for it to be quite so flagrant as correlating a dataset with a simple measure of the relative magnitude of its data points.

And then you inserted a gap in order to meet your specific expectation of an exponential. :doh:

Grant Hutchison
It still holds. I just did a Monte Carlo simulation where I generated 31 random 55 Cancri systems with 5 planets each. I then looked at the log plot for each one, and gave myself one unknown planet to fool around with. I manipulated the arrangement by hand, to give the best r2. The r2's ranged from .7722 to .99479. The mean was 0.91840. So, that's pretty darn close to being in the 95% confidence area.

I'd like to do more simulations, but I need to automate the process, and I don't have programming capability right now.

But even with the sample size I did generate, 55 Cancri is obviously exceptional.

There can be no doubt about it.

publius
2007-Nov-20, 07:50 PM
I'm not sure what they did was as simple as generating a "random" solar system. By assigning a uniform distribution to log(r), aren't they basically assuming Bode's Law?

That's true -- random in log(r) vs r just didn't hit me. But here's what they say about it:




It is easy to see why radius-exclusion laws tend to produce planetary distances that approximately follow
a geometric progression. If a fixed fractional radius exclusion V is used for every planet, and planets are
packed as tightly as possible according the radius-exclusion law, then the physical extent of radius exclusion
at distance r is rV , and the resulting planetary separations would follow an exact geometric progression with
semi-major axis ratio (1 + V )/(1 − V ). If the planets are packed less tightly, then the progression will be
only approximately geometric.



So their argument seems to be that available stable slots have to follow such a rule anyway, and so let's fill that in at random for some number of planets, letting no two planets be in each other's exclusion zone.


-Richard

grant hutchison
2007-Nov-20, 07:59 PM
But even with the sample size I did generate, 55 Cancri is obviously exceptional.

There can be no doubt about it.Were your randomly generated systems demonstrably stable (and demonstrably detectible)?
You have three problems I can think of with your claim that 55 Cnc is exceptional. (Ummm. At the risk of sounding like a Monty Python sketch, you have four problems.)
1) The data are already strongly correlated (as previously described).
2) The population from which the data are drawn is not random. Considerations of stability require some minimal spacing, which will depend on the masses of the planets and the possibility of resonant interactions. So each successive "random" member of a planetary system constrains the probability distribution for the next, in a way which will depend upon mass and eccentricity.
3) The population is also not random by reason of detectibility: close and massive sends a more detectible signal than far and low-mass.
4) Titius-Bode allows, indeed encourages, the "adjustment" of data that don't match the required relationship. This further "de-randomizes" the dataset on which a correlation is performed.

So any sensible hypothesis test on the 55 Cnc data would require us to allow for the strong correlation going in, the non-random spacing considerations arising from stability (and detectibility), and the rules by which we decide to "adjust" poorly matched data. Only then could we decide whether there was any statistical significance to your correlation. I suspect the prospect of framing a sampling distribution for that lot would make a strong statistician blench, but Disinfo Agent may be able to cast more light.

So "obviously exceptional" seems just a tad overstated. :)

In the absence of a sampling distribution, doesn't just looking at the data give you the tiniest frisson of misgivings? You've got a rough line of four objects and then a clear distant outlier, the location of which you have adjusted to catch the upstroke of a fitted exponential. Looks to me as if your exponential fit pivots on a single datum, which happens to be the one you've messed with.

Grant Hutchison

Warren Platts
2007-Nov-20, 09:14 PM
Were your randomly generated systems demonstrably stable?
I didn't run GravitySimulator on them--yet! But see below.


You have three problems I can think of with your claim that 55 Cnc is exceptional.
1) The data are already strongly correlated (as previously described).

True, and I was surprised with my simulation just how strongly correlated an exponential plot was with several of the random systems.


2) The population from which the data are drawn is not random. Considerations of stability require some minimal spacing, which will depend on the masses of the planets and the possibility of resonant interactions. So each successive "random" member of a planetary system constrains the probability distribution for the next, in a way which will depend upon mass and eccentricity.
True again. For my first attempt at a Monte Carlo simulation, I kept it as simple as possible. But taking into account the Hill radius, for example, will mainly affect the left-ended, lower-valued outliers. Just by looking at them, the worst-performing simulations were the ones with one planet relatively close-in with four other planets planets that were farther out that were clustered together.

A planetary spacing system for a half dozen (and we should always be very cautious about extending TBL type relations much beyond that) planets that perfectly followed a logarithmic pattern would be spaced apart far enough so that Hill radii wouldn't be a causal factor. The Hill radii would simply spread those four clustered planets, increasing the r2 for the best linear model of the log plot for that trial. But such trials would not affect the right handed tail of the distribution curve for r2, which is what we're interested in.



3) Titius-Bode allows, indeed encourages, the "adjustment" of data that don't match the required relationship. This further "de-randomizes" the dataset on which a correlation is performed.I must admit that as I started through the random trials, I was thinking those guys at BAUT will say that I would just add unknown planets to fudge the model. So I started allowing two unknown planets, but then I got an additional two r2's in the 99%+ category.

However, the highest 7-slot trial [99.6%] was still lower than the 6-slot model I proposed for 55 Cancri (r2 = 0.9975). I then redid, and expanded the data set to only include the 6-slot trials. Nevertheless, the exercise clearly demonstrates your point that adding slots for sure makes it easier to fit a curve to.



So any sensible hypothesis test on the 55 Cnc data would require us to allow for the strong correlation going in,Which is only ~ 0.9


the non-random spacing considerations arising from stability,
Which are too small to affect TB models

and the rules by which we decide to "adjust" poorly matched data. We do that by minimizing the number of missing planets.


Only then could we decide whether there was any statistical significance to your correlation.
ALL RIGHT!!! :dance::clap::dance:


I suspect the prospect of framing a sampling distribution for that lot would make a strong statistician blench, but Disinfo Agent may be able to cast more light.It wouldn't be that hard. Just run 10,000 random trials incorporating your hill radius or whatever, and then see if the observed, apparent TBL pattern still falls in the right-handed tail.

Is anyone aware of a good, simple, free BASIC programming language out there?


So "obviously exceptional" seems just a tad overstated.:)Not to me!!! :D


In the absence of a sampling distribution, doesn't just looking at the data give you the tiniest frisson of misgivings? You've got a rough line of four objects and then a clear outlier, the location of which you have adjusted to catch the upstroke of a fitted exponential. Looks to me as if your exponential fit pivots on a single datum, which happens to be the one you've messed with.I must say I did feel a little shiver as I started making random models. But that's the thing, the line of four objects are not so rough, and they point right to the outlier. It's almost like a road sign to the missing planet.

grant hutchison
2007-Nov-20, 09:43 PM
But that's the thing, the line of four objects are not so rough, and they point right to the outlier. It's almost like a road sign to the missing planet.And one which, by pointing at the planet, seems to say "We're not part of an exponential relationship!" :)

I edited in a fourth consideration for your delectation, presumably while you were posting. Any sampling distribution must also simulate what we can detect, not what's actually out there.

With reference to your suggestion of "just" knocking off 10,000 simulations of the dataset, i have to wonder how you might do that, since no-one on Earth knows how planetary systems are actually constructed, or how much of them we currently detect. This, once again, refers back to our tediously repetitive suggestion that you are speculating beyond the data. And publius's paper is a rather timely and lovely illustration of the whole principle that you get out exactly what you put in, when you build simulations.

Anyway. You seem to be rather more chipper and less confrontational today. Have you banished all thoughts of duels and combat and inflicting pain? Or is there more of that stuff to come?

Grant Hutchison

Disinfo Agent
2007-Nov-20, 09:45 PM
I must say I did feel a little shiver as I started making random models. But that's the thing, the line of four objects are not so rough, and they point right to the outlier. It's almost like a road sign to the missing planet.But you have to agree that 5 points is very little. Given that we know they'll fall along some increasing curve, it's almost fatal that an exponential model will provide a good fit. The thing is, so would literally infinitinely many other curves.

Grant raised another caveat which I thought was pertinent: correlation. It's quite possible that the data for the five planets are not independent (remember that it all started out as one estimated planet). If not, this invalidades all the t-tests and confidence intervals you made. The oddly large correlation coefficients you got between x and y (though, as Grant noted, a positive correlation was predictable) could also be due to correlations between different data points.

Warren Platts
2007-Nov-20, 10:25 PM
Here's a graph to illustrate what I was talking about regarding Hill radii.

The four planets are way too close to be realistic, but imposing minimum spacing won't improve the r2 enough to rival the correlation at 55 Cancri.

grant hutchison
2007-Nov-20, 10:29 PM
It's quite possible that the data for the four planets are not independent (remember that it all started out as one estimated planet).Oh yes; isn't that a good point!
We're not looking at individual little sparkly points in the sky, after all, which we can guarantee are unique entities. (Warren, this links back to our previous exchange about "following an exoplanet for a complete orbit to be sure it's real" as compared to "following Pluto for a just bit of an orbit to refine the data".) The original datset for 55 Cnc is just a single, wobbly line connecting individual Doppler observations and their accompanying error bars, which has been deconvoluted into discrete frequencies to produce the best fit. Tweak just one of those frequencies, and the others will shift to compensate.
So none of these "planets" (temporary scare quotes) is actually independent of the data describing all the others. I'm remote from my filing cabinet, but I think I correctly recall the paper from Macy's team, announcing the fifth planet. They took the two best-represented frequencies, across all observations, as "givens", and then fitted the residuals. Three additional frequencies did the trick, four did no better.

The mutual dependence of the data is pretty clear.

Grant Hutchison

grant hutchison
2007-Nov-20, 10:35 PM
The four planets are way too close to be realistic, but imposing minimum spacing won't improve the r2 enough to rival the correlation at 55 Cancri.How do you know? It's an impossible system. Eliminate and roll the dice again; It's not acceptable just to nudge excluded data towards reality. (Although medical research would work so much better if that were the case. :))

Grant Hutchison

Warren Platts
2007-Nov-20, 10:41 PM
But you have to agree that [5] points is very, very little. Given that we know they'll fall along some increasing curve, it's almost fatal that an exponential model will provide a good fit. The thing is, so would literally infinitinely many other curves. Some models fit better than others. The proportion of variation in spacing explained by the linear version of the logarithmic model (r2) for randomly generated models is about 0.92 (1.0 is perfect)--so there's a grain of truth to your charge.

However, the r2 for 55 Cancri is 0.9975; I handcrafted 31 models for describing 31 randomly generated solar systems, and not one of them had a higher r2.

No matter how you slice it, the data for 55 Cancri are exceptional.


Grant raised another caveat which I thought was pertinent: correlation. It's quite possible that the data for the [five] planets are not independent (remember that it all started out as one estimated planet). If not, this invalidades all the t-tests and confidence intervals you made. The oddly large correlation coefficients you got between x and y (though, as Grant noted, a positive correlation was predictable) could also be due to correlations between different data points.Assuming the astronomers are correct for their empirical estimates for the semimajor axes of the alledged planets at 55 Cancri, they are independent variables. It is trivially true that as one progresses from the center to the outer edge, the absolute distance must necessarily increase.

However, the spacing between planets need not always increase. It's the spacing we're after, and that is an independent variable.

Warren Platts
2007-Nov-20, 10:48 PM
How do you know? It's an impossible system. I don't know that it's an impossible system. I didn't run GravitySimulator on it. But it is radically different from the system observed at 55 Cancri.

But the closer it is theoretically possible to pack planets together, the more exceptional 55 Cancri looks.

Eliminate and roll the dice again; It's not acceptable just to nudge excluded data towards reality. (Although medical research would work so much better if that were the case. :)) There was that one outlier with the r2 of 0.99497--it made me wonder if I made a mistake entering in the data or something--but I know I've got to leave it in. When I get the program up and running, then I can start over and do the 10,000 trial run.

grant hutchison
2007-Nov-20, 11:09 PM
Assuming the astronomers are correct for their empirical estimates for the semimajor axes of the alledged planets at 55 Cancri, they are independent variables.How do we know they're correct? They weren't correct the last three times. If they re-estimate again, everything changes.


I don't know that it's an impossible system. I didn't run GravitySimulator on it.I'll take Tony's guidance on that, but I doubt you'll have assessed the billion-year stability of a system on a PC in the last 24 hours.


When I get the program up and running, then I can start over and do the 10,000 trial run.Be sure to correctly reproduce an appropriate mass and eccentricity distribution for both the observable and unobservable planets in each system, and then to make a billion-year stability assessment for each. Then automate your "tweaking" rule, so you have no control over it during the run. Otherwise you're not reproducing the population from which your 55 Cnc data were sampled, and we can make no judgement about how exceptional or otherwise 55 Cnc will prove to be.

Grant Hutchison

Warren Platts
2007-Nov-21, 12:31 AM
How do we know they're correct? They weren't correct the last three times. If they re-estimate again, everything changes.

I'll take Tony's guidance on that, but I doubt you'll have assessed the billion-year stability of a system on a PC in the last 24 hours.

Be sure to correctly reproduce an appropriate mass and eccentricity distribution for both the observable and unobservable planets in each system, and then to make a billion-year stability assessment for each. Then automate your "tweaking" rule, so you have no control over it during the run. Otherwise you're not reproducing the population from which your 55 Cnc data were sampled, and we can make no judgement about how exceptional or otherwise 55 Cnc will prove to be.

Grant Hutchison
If the TBL theory is correct, then TBL spacings are not exceptional at all. They are what we see, and what we expect. In fact, we have never yet encountered a solar system that has not exhibited the TBL spacing. But let's forget about the Sol system for now and focus on 55 Cancri. The data is clearly exceptional with respect to random expectations. Theoretical expectations, according to TBL, are that 55 Cancri is just what we should expect to find. So, it is definately not my claim that 55 Cancri is exceptional relative to other observed star systems. It is only exceptional relative to randomly generated models.

Let's be clear. The data for 55 Cancri--as it has been reported--is very highly correlated with a logarithmic model that postulates a missing planet at slot V.

We can debate everything else, once we all agree on that.

I don't doubt that it's not unlikely that the 55 Cancri "planets" will be radically revised in the future. So what? The point of this thread is that the data for 55 Cancri--as it has been reported--is very highly correlated with a logarithmic model that postulates a missing planet at slot V.

What we are looking for now is an objective, scientific, statistical method for pronouncing that planetary spacing for x1, x2, x3, x4, x5, x6 exhibits a TBL pattern or not.

You cannot state that there is no TBL pattern without a mechanism for determining what a true TBL pattern would be like.

The TBL says that ideally, solar systems will be modeled by exponential models with an r2 of 1.000. That's the ideal r2. So, the question is how much slack are we willing to grant before we say "that's not a TBL pattern"?

I say we go by randomly generated solar systems. Incorporating a bunch of Hill radii, etc. should only happen after it can be demonstrated that 55 Cancri is different from random expecations at a 95% confidence level.

Are you willing to concede that 55 Cancri--as it has been reported--exhibit a TBL pattern at a 95% confidence level or better?

Disinfo Agent
2007-Nov-21, 12:38 AM
But there are all sorts of problems, caveats and questions marks in the data you have to work with...

Have you performed the same kind of data analysis for the planets in our solar system? I'm curious about what kind of correlations you get.

Warren Platts
2007-Nov-21, 01:08 AM
But there are all sorts of problems, caveats and questions marks in the data you have to work with...Undoubtedly! But that's not the point. Is the data--as it is--suggestive of a TBL pattern or not in a way that randomly generated solar system don't ordinarly exhibit?


Have you performed the same kind of data analysis for the planets in our solar system? I'm curious about what kind of correlations you get.Not to the extent that I have for 55 Cancri. However, my preliminary results agree with those reported by others, namely, that the best fitted models divide the solar system into two zones, outer gas giants surrounding inner rocky planets. Each zone has its own characteristic spacing pattern.

However, as I said above, let's not keep going in circles, and let's focus on the one, fundamental, empirical claim that 55 Cancri (for now, anyway)--as the data has been reported--exhibits a remarkable--that is to say, statistically significant--TBL pattern.

grant hutchison
2007-Nov-21, 09:05 AM
However, as I said above, let's not keep going in circles, and let's focus on the one, fundamental, empirical claim that 55 Cancri (for now, anyway)--as the data has been reported--exhibits a remarkable--that is to say, statistically significant--TBL pattern.But circles are all you've got.
If you generate truly random "solar systems", you are not generating solar systems that obey stability laws (appropriate spacing), nor are you generating the sort of partially sensed solar system we have in 55 Cnc.
You're setting yourself up in advance to generate unrealistic data to compare to your single dataset. If you discover a statistically significant deviation, we have no idea if that's not just because you have generated impossible solar systems, or solar systems that don't obey the sensing bias of our current knowledge. Given that your data are strongly correlated from the start, small deviations from correct simulation can only give misleading results.
So to perform your trial, you have to put in knowledge we have yet to glean from observation.
We need observation, not speculation from small datasets.

Grant Hutchison

Disinfo Agent
2007-Nov-21, 09:11 AM
However, as I said above, let's not keep going in circles, and let's focus on the one, fundamental, empirical claim that 55 Cancri (for now, anyway)--as the data has been reported--exhibits a remarkable--that is to say, statistically significant--TBL pattern.How do you know that it's statistically significant? From the correlations and the significance tests? But they may not be valid for your data!

Van Rijn
2007-Nov-21, 09:18 AM
If the TBL theory is correct, then TBL spacings are not exceptional at all.


Terminology issues again: You wanted to call it a law, you appeared to propose a hypothesis, but somehow it has morphed into "the Titius-Bode Law theory"? :confused: Care to explain that?

Warren Platts
2007-Nov-21, 09:43 AM
Oh yes; isn't that a good point!
We're not looking at individual little sparkly points in the sky, after all, which we can guarantee are unique entities. . . . The original datset for 55 Cnc is just a single, wobbly line connecting individual Doppler observations and their accompanying error bars, which has been deconvoluted into discrete frequencies to produce the best fit. Tweak just one of those frequencies, and the others will shift to compensate.

. . .

I'm remote from my filing cabinet, but I think I correctly recall the paper from Macy's team, announcing the fifth planet. They took the two best-represented frequencies, across all observations, as "givens", and then fitted the residuals. Three additional frequencies did the trick, four did no better.

One thing I've been waiting for you all to point out is that not only is the spacing factor for 55 Cancri much larger than for the Sol system, but that the spacing factor K is so darn close to the natural logarithm base 2.178.....

It just makes me wonder whether there is something seriously and horribly wrong at the very heart of the calculations used to disentangle the observations that has to do with natural logarithms.

And that the "planets" aren't real at all.
:think:

Warren Platts
2007-Nov-21, 10:14 AM
But circles are all you've got.

If you generate truly random "solar systems", you are not generating solar systems that obey stability laws (appropriate spacing), Think about what "appropriate spacing" really means. If you're saying the minimum spacing for stable orbits just is the ideal TBL spacing, then you've offered a kinematic, physical theory to explain the TBL pattern itself--you've just solved the problem for me--and I doubt you want to do that. :razz:

However, if the minimum spacing is less much less than the TBL spacing, that will not affect the r2 distribution for the purpose of figuring out what percentile the observed pattern at 55 Cancri fits into.

What I'm doing (I found a sweet little, no-frills freeware BASIC program) is creating a data base--a histogram really--of 10,000 r2. Each trial will figure out the best fit exponential model to explain the randomly generated data. The best fit model will have an r2 associated with it that says how well the data fit the best-fit model.

A few of those data will fit the exponential model better than 55 Cancri itself, most will be worse with an r2 ranging from ~ 0.88 to 0.98. And a few will have horrible fits (r2 < 0.800).

The distribution will have an "F-scale" shape--that is, since r2 cannot be greater than one, the distribution runs into a wall on the right side, while the left tail will be long and asymptotic.

So, small effects from minimum spacing requirements will only affect the shape of the left handed tail of the distribution.

Large effects that would affect the right-handed tail are physical explanations for the TBL itself.

Let's just focus on the pattern itself for now. There's no point in worrying about physical processes if the pattern itself is not a real pattern. Once we all agree that there really is a TBL spacing pattern at 55 Cancri, then we can debate whether the responsible causal process is celestial mechanics or planetary formation.

tusenfem
2007-Nov-21, 02:05 PM
And what exactly happens when it turns out that there is no planet V, as is the case until now?

Secondly, you cannot just use randomly created solar systems with some program you found on the web. In order to see whether there is really something to TB (TB or not TB that is the question) you need to build a realistic solar system creation code, that starts at the collapse of the primordial cloud, creation of the sun, collapsing of the rest material into a disk around the sun and coalescence of the planets in that disk.

You can fit whatever you like to the data available. If the planets are created in a scale free environment, just like G&D proposed, it is probably the most logical thing that there will be some power law behaviour, as scale free processes, e.g. turbulence tend to create power laws.

Warren Platts
2007-Nov-21, 03:05 PM
And what exactly happens when it turns out that there is no planet V, as is the case until now? [quote]
Well, in our solar system, the slot for Planet V is filled by the asteroid belt. So, it's possible the same situation could hold for 55 Cancri.

[QUOTE]Secondly, you cannot just use randomly created solar systems with some program you found on the web. In order to see whether there is really something to TB (TB or not TB that is the question [nice one! :lol:]) you need to build a realistic solar system creation code, that starts at the collapse of the primordial cloud, creation of the sun, collapsing of the rest material into a disk around the sun and coalescence of the planets in that disk. There is no program on the web for doing this. I had to do it myself. Sure it would be nice to do a numerical simulation of planetary formation. But the first step is to make sure the apparent TBL pattern really deviates from random expectations.


You can fit whatever you like to the data available. If the planets are created in a scale free environment, just like G&D proposed, it is probably the most logical thing that there will be some power law behaviour, as scale free processes, e.g. turbulence tend to create power laws.That is indeed the question my friend.

Warren Platts
2007-Nov-21, 03:09 PM
Attached is the BASIC code I used for the Monte Carlo simulation. It may not be the most elegant program ever written, but it works. I downloaded a freeware version called "Decimal BASIC (http://hp.vector.co.jp/authors/VA008683/english/)". It's great, no frills, just like when you were back in (old-school) high school--only better. The help function is in Japanese, but you can still figure it out.

EDIT: I replaced the program I originally posted with the most recent one that does not look at every possible way to arrange 5 planets in 6 orbits, but only at sequences of the type {I, II, III, IV, ___, VI}

REM *** generates random solar systems using
REM *** the Monte Carlo technique.
REM *** Model assumes 5 planets and 6 available
REM *** orbital slots, and calculates the r-squared for
REM *** each sequence of the form {I, II, III, IV, __, VI}
REM *** for CYCLE times TRIAL total trials.
REM *** If you have a dual processor, it's best to run 2
REM *** programs at once and then combine the data with Excel.
REM *** The output is data for a histogram with a bin size of
REM *** 0.001 where each bin has the number of trials with
REM *** an r-squared from the listed r-squared to the next
REM *** highest r-squared.

DIM CANCRI (5)
DIM X(5)
DIM HIST(1000)
LET X(1) = 1
LET X(2) = 2
LET X(3) = 3
LET X(4) = 4
REM *** note that X(5) is set to 6 because we are assuming a
REM *** missing planet in slot number 5 {I, II, III, IV, __, VI}
LET X(5) = 6

FOR CYCLE = 1 TO 100
RANDOMIZE
FOR TRIAL = 1 TO 500000

REM ***** sort the planets according to natural log(a)
LET CANCRI(1) = LOG(RND)
LET CANCRI(2) = LOG(RND)
IF CANCRI(2) < CANCRI(1) THEN
LET A = CANCRI(2)
LET CANCRI(2) = CANCRI(1)
LET CANCRI(1) = A
END IF
LET CANCRI(3) = LOG(RND)
IF CANCRI(3) < CANCRI(2) THEN
IF CANCRI(3) < CANCRI(1) THEN
LET A = CANCRI(3)
LET CANCRI(3) = CANCRI(2)
LET CANCRI(2) = CANCRI(1)
LET CANCRI(1) = A
ELSE
LET A = CANCRI(3)
LET CANCRI(3) = CANCRI(2)
LET CANCRI(2) = A
END IF
END IF
LET A = LOG(RND)
IF A < CANCRI(3) THEN
LET CANCRI(4) = CANCRI(3)
IF A < CANCRI(2) THEN
LET CANCRI(3) = CANCRI(2)
IF A < CANCRI(1) THEN
LET CANCRI(2) = CANCRI(1)
LET CANCRI(1) = A
ELSE
LET CANCRI(2) = A
END IF
ELSE
LET CANCRI(3) = A
END IF
ELSE
LET CANCRI(4) = A
END IF
LET A = LOG(RND)
IF A < CANCRI(4) THEN
LET CANCRI(5) = CANCRI(4)
IF A < CANCRI(3) THEN
LET CANCRI(4) = CANCRI(3)
IF A < CANCRI(2) THEN
LET CANCRI(3) = CANCRI(2)
IF A < CANCRI(1) THEN
LET CANCRI(2) = CANCRI(1)
LET CANCRI(1) = A
ELSE
LET CANCRI(2) = A
END IF
ELSE
LET CANCRI(3) = A
END IF
ELSE
LET CANCRI(4) = A
END IF
ELSE
LET CANCRI(5) = A
END IF
REM *** END OF SORT ALGORITHM ******
REM *** BEGIN CALCULATING R-SQUARED ****
LET RSQ = 0
LET SUMX = 0
LET SUMY = 0
FOR J = 1 TO 5
LET SUMX = SUMX + X(J)
LET SUMY = SUMY + CANCRI(J)
NEXT J
LET AVGX = SUMX / 5
LET AVGY = SUMY / 5
LET SUMXSQ = 0
LET SUMYSQ = 0
LET SUMXY = 0
FOR J = 1 TO 5
LET SUMXSQ = SUMXSQ + ((X(J) - AVGX) * (X(J) - AVGX))
LET SUMYSQ = SUMYSQ + ((CANCRI(J) - AVGY) * (CANCRI(J) - AVGY))
LET SUMXY = SUMXY + ((X(J) - AVGX) * (CANCRI(J) - AVGY))
NEXT J
LET B = SUMXY / SUMXSQ
LET SSRESID = SUMYSQ - (B * SUMXY)
LET RSQ = 1 - (SSRESID / SUMYSQ)
REM *** Load the r-squared data into the histogram array
LET HIST(INT(RSQ * 1000)) = HIST(INT(RSQ * 1000)) + 1
NEXT TRIAL
PRINT CYCLE; "% DONE"
NEXT CYCLE
REM *** end of r-squared algorithm
REM *** now printout histogram data
FOR I = 1 TO 999
REM *** print the r-squared score
PRINT USING "%.###": (I / 1000);
REM *** print the number of trials in each r-squared bin
PRINT USING "########": HIST(I);
REM *** print a running total
LET COUNTER = COUNTER + HIST(I)
PRINT USING "##########": COUNTER
NEXT I
REM *** this final number should equal the total number of trials
PRINT "TOTAL COUNT = "; COUNTER
PRINT "{I, II, III, IV, ___, VI}"
END

Warren Platts
2007-Nov-21, 03:18 PM
I set the program to generate one million random solar systems:

NUMBER OF TRIALS EXCEEDING 55 CANCRI R-SQUARED: 9440
NUMBER OF TRIALS: 1000000

In other words, the observed TBL pattern has a higher r2 than 99.06% of the random trials.

So, I have to up my confidence level from 95% to 99%.

:cool:

Now we can finally lay to rest the repeated claim any old jumble of points can generate a TBL pattern.
:hand:

So now we can finally move the discussion forward to possible physical mechanisms that generated this very remarkable Titius-Bode planetary spacing for the very first extrasolar system for which we have data.

:clap::dance::clap:

I vote for the vorticity model!

KaiYeves
2007-Nov-21, 03:27 PM
I thought that Bode's Law didn't even work for all of the planets in this solar system.

R.A.F.
2007-Nov-21, 04:04 PM
I thought that Bode's Law didn't even work for all of the planets in this solar system.

Sure it does...you just have to "message" the data until it says what you want it to say. :)

Warren knew from the onset what he "wanted" his conclusion to be...if it took a bit of "hammering" to make the evidence "fit" that preconceived notion, well, that doesn't seem to bothering Warren at all...

Although it should.

Warren Platts
2007-Nov-21, 04:14 PM
Sure it does...you just have to "message" the data until it says what you want it to say. :)

Warren knew from the onset what he "wanted" his conclusion to be...if it took a bit of "hammering" to make the evidence "fit" that preconceived notion, well, that doesn't seem to bothering Warren at all...

Although it should.

R.A.F. rude abbreviation requesting a cessation of discourse from one party removed by moderator


I didn't massage anything. Quit spreading lies.

tusenfem
2007-Nov-21, 05:06 PM
Well, I had some time to kill before I go and see Beowulf, and so I took the 55 Cancri planets (only the 5) and fitted log10(semi major axis) versus location number (i.e. 1 through 5). I found the following fit values

intercept: -2.0235
slope: 0.5195
variance in slope: 0.0020
variance in intercept: 0.0118
regression coefficient: 0.983

which was done with an ordinary least squares fit with all error in y (so F1 in the function below)

so my "law" would be: 0.009 × 100.5195 n

############################################

for people interested in a Matlab function doing 5 different variations of fitting, based on the paper by Isobe et al. look below (might as well do something useful for a change)

function [F1, F2, F3, F4, F5, r] = all_fits( x, y );
% all_fits.m
% this program performs the 5 different OLS described in:
% Isobe et al., ApJ 364, 104-113, 1990
% the fits are numbered 1 to 5 and describe the following:
% 1. OLS( x | y ) ordinary least squares fit, all errors in y
% 2. OLS( y | x ) ordinary least squares fit, all errors in x
% 3. OLS bisector the "average" between 1 and 2
% 4. Othogonal regression, minizing the error perpendicular to the fit
% 5. Reduced major axis
% note:
% In the paper the conclusion is that #3 is the best to use in astrophysics
% The orthogonal regression can only be used if both axis have the same
% scale
% input:
% x an array of length n
% y an array of length n
% output:
% F1 to F5 are (4,1) arrays containing the following
% 1: a1 to a5: the intercepts of the 5 different fits
% 2: b1 to b5: the slopes of the 5 different fits
% 3: e1 to e5: the variance estimates in the slopes
% 4: v1 to v5: the variance estimates in the intercept
% the fits are then in F-terms written as:
% y = F1(1) + x * F1(2)

% creating the starting variables for the fit, cf. Eqs (2) to (6)
n = length( x ); % number of data points
xbar = sum( x ) / n; % Eq. (2a)
ybar = sum( y ) / n; % Eq. (2b)
x0 = x - xbar; % Eq. (5)
y0 = y - ybar; % Eq. (6)
sxx = sum( x0.^2 ); % Eq. (3a)
syy = sum( y0.^2 ); % Eq. (3b)
sxy = sum( x0 .* y0 ); % Eq. (4)

% calculating the slopes for all 5 fits, cf. Table 1
b1 = sxy / sxx;
b2 = syy / sxy;
b3 = ( b1 * b2 - 1 + sqrt( ( 1 + b1^2 ) * ( 1 + b2^2 ) ) ) / ( b1 + b2 );
b4 = 0.5 * ( ( b2 - 1 / b1 ) + sign( sxy ) * sqrt( 4 + ( b2 - 1 / b1 )^2 ) );
b5 = sign( sxy ) * sqrt( b1 * b2 );

% calculating the intercept defined by: ai = ybar - bi xbar, Eq. (8)
a1 = ybar - b1 * xbar;
a2 = ybar - b2 * xbar;
a3 = ybar - b3 * xbar;
a4 = ybar - b4 * xbar;
a5 = ybar - b5 * xbar;

% calculating the variance of the slope, cf. Table 1
% this is the Var(b1) = sum( ( y - ybar ) - b1 * ( x - xbar ) ) / sxx. Eq. (7)
% Bevington, Data reduction and error analysis for the physical sciences, 1989, p 114
e1 = sum( x0.^2 .* ( y - b1 * x - ybar + b1 * xbar ).^2 ) / sxx^2;
e2 = sum( y0.^2 .* ( y - b2 * x - ybar + b2 * xbar ).^2 ) / sxy^2;
% now we need the Cov(b1, b2), given in the footnote of Table 1
cov = sum( x0 .* y0 .* ( y0 - b1 * x0 ) .* ( y0 - b2 * x0 ) ) / (b1 * sxx^2 );
e3 = b3 * ( ( 1 + b2^2 )^2 * e1 + ( 1 + b1^2 ) * ( 1 + b2^2 ) * cov + ( 1 + b1^2 ) * e2 ) / ...
( ( b1 + b2 )^2 * ( 1 + b1^2 ) * ( 1 + b2^2 ) );
e4 = b4 * ( e1 / b1^2 + 2 * cov + b1^2 * e2 ) / ( 4 * b1^2 + ( b1 * b2 - 1 )^2 );
e5 = ( b2 * e1 / b1 + 2 * cov + b1 * e2 / b2 ) / 4;

% the variance in the intercept, using Eq. (9)
% gam1 and gam2 are the only differences for each fit
gam1 = 1; % Eq. (10)
gam2 = 0; % Eq. (15)
v1 = sum( ( y0 - b1 * x0 - n * xbar * ...
( ( gam1 / sxx ) * x0 .* ( y0 - b1 * x0 ) + ( gam2 / sxy ) * y0 .* ( y0 - b2 * x0 ) ) ).^2 ) / n^2 ;

% two extra variables g1 and g2, cf. Eqs. (20) and (21)
g1 = b3 / ( ( b1 + b2 ) * sqrt( ( 1 + b1^2 ) * ( 1 + b2^2 ) ) ); % Eq. (20)
g2 = b4 / sqrt( 4 * b1^2 + ( b1 * b2 - 1 )^2); % Eq. (21)

gam1 = 0; % Eq. (11)
gam2 = 1; % Eq. (16)
v2 = sum( ( y0 - b1 * x0 - n * xbar * ...
( ( gam1 / sxx ) * x0 .* ( y0 - b1 * x0 ) + ( gam2 / sxy ) * y0 .* ( y0 - b2 * x0 ) ) ).^2 ) / n^2 ;

gam1 = g1 * ( 1 + b2^2 ); % Eq. (12)
gam2 = g1 * ( 1 + b1^2 ); % Eq. (17)
v3 = sum( ( y0 - b1 * x0 - n * xbar * ...
( ( gam1 / sxx ) * x0 .* ( y0 - b1 * x0 ) + ( gam2 / sxy ) * y0 .* ( y0 - b2 * x0 ) ) ).^2 ) / n^2 ;

gam1 = g2 / abs( b1 ); % Eq. (13)
gam2 = g2 * abs( b1 ); % Eq. (18)
v4 = sum( ( y0 - b1 * x0 - n * xbar * ...
( ( gam1 / sxx ) * x0 .* ( y0 - b1 * x0 ) + ( gam2 / sxy ) * y0 .* ( y0 - b2 * x0 ) ) ).^2 ) / n^2 ;

gam1 = 0.5 * sqrt( b2 / b1 ); % Eq. (14)
gam2 = 0.5 * sqrt( b1 / b2 ); % Eq. (19)
v5 = sum( ( y0 - b1 * x0 - n * xbar * ...
( ( gam1 / sxx ) * x0 .* ( y0 - b1 * x0 ) + ( gam2 / sxy ) * y0 .* ( y0 - b2 * x0 ) ) ).^2 ) / n^2 ;

% creating the output arrays
F1 = [a1, b1, e1, v1];
F2 = [a2, b2, e2, v2];
F3 = [a3, b3, e3, v3];
F4 = [a4, b4, e4, v4];
F5 = [a5, b5, e5, v5];

% regression coefficient of the data from xlsqfit by Wolfgang Baumjohann
nsx = sum( x );
nsy = sum( y );
nsx2 = sum( x .* x );
nsy2 = sum( y .* y );
nsxy = sum( x .* y );
r = ( n * nsxy - nsx * nsy ) / sqrt( ( n * nsx2 - nsx^2 ) * ( n * nsy2 - nsy^2 ) );

return;

grant hutchison
2007-Nov-21, 05:32 PM
Think about what "appropriate spacing" really means. If you're saying the minimum spacing for stable orbits just is the ideal TBL spacing, then you've offered a kinematic, physical theory to explain the TBL pattern itself--you've just solved the problem for me--and I doubt you want to do that. :razz:No, we already know there are many plausible explanations for an exponential distribution if one is ever observed to prevail generally. One of those plausible explanations involves the Hill sphere. I'm doing nothing that isn't already out there and well-acknowledged.
But what you've just done above is to acknowledge that spacing "rules" (in which I would include resonant interactions as well as perturbations) can signicantly alter the probability distribution for "random solar systems": some are more probable than others. In addition, some of these systems will be more detectible than others; and some will be more readily deconvoluted than others. So at the end of the day we will have a very complex probability distribution underlying any potential "family" of random, stable, detectible, deconvolutable five-member systems.
We have to imagine a fifteen-dimensional probability space (semi-major axis, mass and eccentricity for each of five objects): some regions of that space will have higher probabilities than others; some very much lower.
Treating that space as if it were uniform in probability (as you've done) invalidates any possible conclusion you can draw.

As a dumb example, suppose I am 1.85m tall. I wish to know if I am taller than most people. Armed only with the datum that few humans are ever taller than 2.5m, I simulate a flat probability space for all heights between 0m and 2.5m, and check how many members of the resulting dataset exceed my height.
It's readily apparent that I have done a dumb thing, and my conclusions have no validity. Likewise, you have simplified the probability distribution to the point of making your conclusion meaningless.

What did I need to do to make my conclusion about my height more valid? I needed a real-world sample with which to assess the probability distribution of heights in my population of interest. (At which point I could dispense with a simulation and make the comparison directly.)
What do you need to make your conclusion valid? You need a real-world sample, or at least a robust simulation involving many parameters.

This is why people keep telling you that you are (persistently, now) speculating beyond the data. And this is why there is still debate about the reality or otherwise of an exponential spacing in the solar system. If it were as simple as knocking together a dumb simulation one morning (by which I wish you to understand that the simulation is naive and uncomplicated), then we'd know, with some level of confidence, the truth or falsity of that claim. But we don't. Because answering the question is a lot harder than what you've just done.

Grant Hutchison

orionjim
2007-Nov-21, 06:02 PM
I just read Grant Hutchison's response and I am echoing some of what he is saying:

Warren, one thing you didn’t do was to consider the Hill Radius. I used a value of .005 and between 5 and 6 percent of the runs were within this radius. (I just picked .005 out of the air because it seemed like a small value). Also I was getting minus values, but I wasn’t using the language you were using.

All and all I wasn’t impressed. You need to consider the gravitational effect of the planets, and when you do that you will see your .997 number isn’t that spectacular.

My suggestion Warren is to forget the statistics and work on the mechanism.

Jim

Warren Platts
2007-Nov-22, 02:56 PM
No, we already know there are many plausible explanations for an exponential distribution if one is ever observed to prevail generally. One of those plausible explanations involves the Hill sphere. I'm doing nothing that isn't already out there and well-acknowledged.
But what you've just done above is to acknowledge that spacing "rules" (in which I would include resonant interactions as well as perturbations) can signicantly alter the probability distribution for "random solar systems": some are more probable than others. In addition, some of these systems will be more detectible than others; and some will be more readily deconvoluted than others. So at the end of the day we will have a very complex probability distribution underlying any potential "family" of random, stable, detectible, deconvolutable five-member systems.
We have to imagine a fifteen-dimensional probability space (semi-major axis, mass and eccentricity for each of five objects): some regions of that space will have higher probabilities than others; some very much lower.
Treating that space as if it were uniform in probability (as you've done) invalidates any possible conclusion you can draw.

As a dumb example, suppose I am 1.85m tall. I wish to know if I am taller than most people. Armed only with the datum that few humans are ever taller than 2.5m, I simulate a flat probability space for all heights between 0m and 2.5m, and check how many members of the resulting dataset exceed my height.
It's readily apparent that I have done a dumb thing, and my conclusions have no validity. Likewise, you have simplified the probability distribution to the point of making your conclusion meaningless.

What did I need to do to make my conclusion about my height more valid? I needed a real-world sample with which to assess the probability distribution of heights in my population of interest. (At which point I could dispense with a simulation and make the comparison directly.)
What do you need to make your conclusion valid? You need a real-world sample, or at least a robust simulation involving many parameters.

This is why people keep telling you that you are (persistently, now) speculating beyond the data. And this is why there is still debate about the reality or otherwise of an exponential spacing in the solar system. If it were as simple as knocking together a dumb simulation one morning (by which I wish you to understand that the simulation is naive and uncomplicated), then we'd know, with some level of confidence, the truth or falsity of that claim. But we don't. Because answering the question is a lot harder than what you've just done.

Grant Hutchison

Debating you is like playing Wack-a-Mole. I'm trying to focus on one thing: what you have repeatedly brought up on this thread but now seem to be ignoring: Is the planetary spacing at 55 Cancri logarithmic or not? This thread isn't about the Sol system. It isn't about Hill radii or resonance--those are possible explanations for a logarithmic spacing patterns. It's about statistics and identifying the pattern itself. You have repeatedly suggested that the logarithmic pattern is a figment of my imagination. For example:


Our problem is that you (and TBL) are starting with a magic formula, which seems to lack justification. Only once we have some sense that the magic formula has any relevance to the real world do we need to start looking for a specific hypothesis to explain it.


You are picking a pattern out of the data by suppressing what doesn't conform to the pattern, and then claiming that the pattern requires a special explanation. Of course it does: but the explanation is that you created the pattern, by sifting the data.


We have lots of models that "fit" TBL: Graner & Dubrulle have reviewed them. What we don't have is any apparent need for those models, until we get some compelling evidence that TBL is any more than a figment of selective data analysis. If some varient of exponential scaling actually turned out to be the case, in a statistically robust way, then we would have an embarrassment of potential explanations immediately to hand. So Warren is departing from the mainstream view not by coming up with a model (we've already got plenty of those) but with his claim that the data are strong enough to require (or justify) such a model in the first place.
This is why we're keen for him to defend his curve-fitting.



all of your offered explanations are for data that you have extracted as "requiring explanation" because they appear to conform to a spacing which is the basis of your explanation. The process by which you levitate your own "law" into existence is very clear.



You might just pause, during your little dance, to recall that these data are necessarily correlated, just because of the way you've generated them. You numbered the semimajor axes in ascending order of magnitude. You've sorted them: it's pretty much impossible for them to show no correlation.

This sort of thing is a common enough error in data analysis, when people accidentally put in "pre-correlated" data (because of an unnoticed mathematical linkage between the dependent and independent variable, or because the two variables turn out to be measuring the same thing in two different ways), and are then impressed by the results of a correlation test. But it's unusual, I think, for it to be quite so flagrant as correlating a dataset with a simple measure of the relative magnitude of its data points.

And then you inserted a gap in order to meet your specific expectation of an exponential. :doh:


In the absence of a sampling distribution, doesn't just looking at the data give you the tiniest frisson of misgivings? You've got a rough line of four objects and then a clear distant outlier, the location of which you have adjusted to catch the upstroke of a fitted exponential. Looks to me as if your exponential fit pivots on a single datum, which happens to be the one you've messed with.

So I've been doing my best to address your concern that the 55 Cancri spacing pattern is properly described as "logarithmic". But now you're telling me I need to create 15-dimensional physical models to see if 55 Cancri is exceptional. Maybe your just confused by what I mean by "exceptional". I don't mean exceptional with regard to other star systems--if TBL is correct, then 55 Cancri is a typical planetary system. What I mean is that 55 Cancri is exceptional with respect to solar systems whose planetary spacing is generated randomly. I don't doubt that planetary systems are not randomly generated. Of course they're not! The first step, however, is to come up with a spacing classification system. We can then classify the few systems for which we have empirical data, and then use the classified empirical data to constrain the umpteen-dimensional physical models for solar system formation and evolution.

In proceeding as I have by doing a Monte Carlo simulation, I have self-consciously followed in the footsteps of my old professors at the University of Chicago, Raup and Sepkowski, who were interested in mass extinctions. But how can you tell if a purported mass extinction event is real or not. Well, you do a random Monte Carlo simulation, where lineages are randomly modeled with characteristic speciation and extinction probabilities, and then observe the diversity changes through time. Under random expectations, the diversity will dip from time to time, and so a true mass extinction event is defined as diversity dips that are greater than those typically found with the randomly generated lineages.

So that's what I've attempted to do here. The question is whether 55 Cancri has a logarithmic spacing-pattern. The very definition of any real pattern is that it deviate from random expectations. Moreover, my model wasn't completely random: it was constrained by 55 Cancri itself. Orbital distances were limited to 6 AU, and to 5 planets with up to 6 orbital slots.

First I was told that my data is pre-correlated--and that's true. All the random systems I generated were also pre-correlated.

Then I was told that exponential curves to a fine job in describing 5 randomly selected data points. And you know what? That's quite true as well. The average r2 for the million randomly generated is 0.93. The median r2 is 0.95. The peak of the r2 is 0.97. (See the attached Figure 1.)

But the r2 for 55 Cancri is 0.9975. Which puts it in the 99th percentile (See attached Figure 2.)

So your analogy with human height isn't very analogous at all. A better analogy would be a high school math test. A student is given the following series: {1, 2, 4, 8, ____, 32}, and is asked to fill in the blank and come up with a formula for describing the series. Clearly, there's only one answer: the blank is 16, and the formula is x = 2n-1.

But what if the series is {0.943, 2.168, 3.949, 8.234, _____, 31.145}? The answer is still x = 2n-1. But now the question is, is this series properly termed "logarithmic". And the only way to answer that is compare the r2's for randomly generated series, and see if the logarithmic description is exceptionally better than logarithmic descriptions of randomly generated, ordered series.

Warren Platts
2007-Nov-22, 03:44 PM
One thing you all might have pointed out as a flaw in my analysis is that there is a standard of deviation associated with the r2 estimate itself for the TBL equation for 55 Cancri. The point estimate I derived (0.9975) falls within the 99th percentile. So I said that my confidence level was now 99%; but really, I need a confidence interval for the r2 estimate, and the level of signifcance for the curve as a whole would then depend on where the left handed tail of the r2 distribution resulting from measurement error would fall.

So that's the next step. Once that's completed, I will have generated a general statistical technique for identifying TBL patterns as future data from other planetary systems comes in.

Warren Platts
2007-Nov-22, 04:21 PM
I just read Grant Hutchison's response and I am echoing some of what he is saying:

Warren, one thing you didn’t do was to consider the Hill Radius. I used a value of .005 and between 5 and 6 percent of the runs were within this radius. (I just picked .005 out of the air because it seemed like a small value). Also I was getting minus values, but I wasn’t using the language you were using.If you're using the code I wrote, negative values are generated by taking the logarithm of the randomly generated distances--short distances generate minus values. The r2 values should all be positive, and none should be lower than about 0.6 (the six valued array I used stores the natural logarithms of the five distances, and the sixth value stores the r2 value. If the Hill radius is small relative to TBL spacing, then it won't have much of an effect--or so I would guess.


All and all I wasn’t impressed. You need to consider the gravitational effect of the planets, and when you do that you will see your .997 number isn’t that spectacular.

My suggestion Warren is to forget the statistics and work on the mechanism.The first step is statistics; there's no point in identifying special mechanisms if we can't identify special patterns first.

R.A.F.
2007-Nov-22, 04:25 PM
Quit spreading lies.

It is NOT a lie to point out that you are playing "fast and loose" with the evidence. I would appreciate it if you wouldn't make groundless accusations.


This thread isn't about the Sol system.

Do you really think that you have "control" of where a discussion will "go" simply because you are the one who started this thread?

...anyhow, once you mentioned Titus-Bode, the burden became yours to demonstrate the TB "works" in our Solar System...you have not fulfilled that obligation.

Warren Platts
2007-Nov-22, 07:18 PM
It is NOT a lie to point out that you are playing "fast and loose" with the evidence. I would appreciate it if you wouldn't make groundless accusations. How dare you impugn my scientific integrity! Absolutely everything I have presented here is utterly transparent. I even posted the code that I used to analyze the data. In fact, I have pointed out more flaws with my own position than you have--which is zero. If you really believe I'm fudging the evidence, now is the time to present your evidence.


...anyhow, once you mentioned Titus-Bode, the burden became yours to demonstrate the TB "works" in our Solar System...you have not fulfilled that obligation.
There have been several threads in this forum regarding the Titius-Bode Law in this solar system. They all seem to end with, "until we get more data from other solar systems, we're pretty much barking at the moon." Well, two weeks ago, data from another solar system finally became available. And guess what, it apparently exhibits the TBL pattern. If there's something wrong with my statistical method then say so. But to accuse me of "messaging" the data is false. I have posted every single number and calculation I have used so far. Therefore, for you to insist that I am playing fast and loose with the evidence is in fact a lie--unless, that is, you haven't bothered to read the thread you are commenting on--in which case, I apologize for calling you a liar. . . .

R.A.F.
2007-Nov-22, 09:22 PM
How dare you impugn my scientific integrity!

Oh, please get over yourself. It's not like you're presenting a scientific paper.


If you really believe I'm fudging the evidence, now is the time to present your evidence.

I see I need to clarify myself. Coincidences found in other systems matter little unless/until you can reconcile the "problems" TB has describing THIS system. For instance, care to explain why there is no planet at 2.8 AU's from the Sun??


...to accuse me of "messaging" the data is false. I have posted every single number and calculation I have used so far. Therefore, for you to insist that I am playing fast and loose with the evidence is in fact a lie--unless, that is, you haven't bothered to read the thread you are commenting on--in which case, I apologize for calling you a liar. . . .

So I'm either a liar, or I'm to lazy to read this thread?

If you believe that insults will help convince others that your ideas are valid, then good luck to you.

grant hutchison
2007-Nov-22, 09:55 PM
Debating you is like playing Wack-a-Mole. I'm trying to focus on one thing: what you have repeatedly brought up on this thread but now seem to be ignoring: Is the planetary spacing at 55 Cancri logarithmic or not?No. I've repeatedly wondered whether it is significantly logarithmic or not. And I've repeatedly pointed out, with examples and explanation, that that question is pretty much impossible to answer with the tiny dataset you have. So you needn't be surprised if I keep pointing out problems to you.
Finding that 55 Cnc is significantly different from a "random solar system", generated by a random number generator, is not nearly enough to answer that question.

I am very happily ignoring everything that might influence the formation of planets: so you may imagine that some benevolent deity initially simply drops planets into the locations prescribed by your program.
But we have no direct knowledge of such systems. First of all, they must be stable, otherwise we will never observe that particular spacing in a mature system: hence, spacing rules must be understood and applied. Secondly, they must be observable: if planets are too far out, or too small in mass, we can't see them. Thirdly, they must deconvolute reliably: what signal will we see, when we break down the velocity waveform from the resultant signal?
That is the sampling distribution that must be understood and simulated, because that represents all the possible "systems" which can appear in our data. We are asking ourselves (or at least, I am asking myself) if we should be surprised by the 55 Cnc system, given that its data have passed through a pretty rigorous selection process just to end up impinging on our consciousness.


In proceeding as I have by doing a Monte Carlo simulation, I have self-consciously followed in the footsteps of my old professors at the University of Chicago, Raup and Sepkowski, who were interested in mass extinctions. But how can you tell if a purported mass extinction event is real or not. Well, you do a random Monte Carlo simulation, where lineages are randomly modeled with characteristic speciation and extinction probabilities, and then observe the diversity changes through time.Excellent example. All such studies need to address the different sampling distributions for particular animal groups. If a species that fossilizes poorly (a bird, for instance, or a species that prevailed in a time and place with poor fossilizing conditions) drops out of the fossil record at some point, is it the case that it has become extinct, or has it simply become so rare that we are unlikely to have encountered one of its fossils for the relevant time period?
That's the problem which causes the Signor-Lipps effect, in which any mass extinction appears to have a more gradual onset for rarely fossilized species than for commonly fossilized species. So all sensible Monte Carlo simulations of this phenomenon take into account the probabilistic nature of the "signal" we receive from the different species, through the fossil record, as well as the underlying random process of speciation and extinction.
Do you see the analogy with observing distant planetary systems?


First I was told that my data is pre-correlated--and that's true.And that's what makes your simulation so exquisitely sensitive to the sampling conditions, which is why I made a fuss about it. If random data were completely uncorrelated going in and the signal was well correlated coming out, there would be less reason to fret about the detail of the sampling distribution. But when random data are pretty well correlated going in and the signal is just rather better correlated coming out, we know that statistical significance will be strongly influenced by the detail of the sampling distribution.

Grant Hutchison

The Albatross Slayer
2007-Nov-23, 01:42 AM
If random data were completely uncorrelated going in and the signal was well correlated coming out, there would be less reason to fret about the detail of the sampling distribution. But when random data are pretty well correlated going in and the signal is just rather better correlated coming out, we know that statistical significance will be strongly influenced by the detail of the sampling distribution.

I do not see how this is a problem here. The equation Warren is looking at is a linear regression equation:

y=a+b*x+eps

Usually when we talk about statistical significance, we are talking about whether the b coefficient is significantly different than zero. In that case, if we had serial correlation in the eps (serial correlation in the x is ok) then t-statistics and p-values would have to be estimated by methods that take serial correlation into account.

However, that is not what Warren is doing. He is using the r-squared as a test statistic. That is a little bit unconventional, but so what? He calculates a test statistic from the observed data, and compares it to the same test statistic calculated from random data, over a large number of simulations. And the test statistic is quite unusual. Serial correlation and other violations of the standard linear regression assumptions are irrelevant, because the Monte Carlo experiment takes all of this into account. His result sure looks statistically significant to me; the problem I see is not that, but that it does not mean what he thinks it means. More on that soon.

Another statistical irrelevance, in my view, is this:


One thing you all might have pointed out as a flaw in my analysis is that there is a standard of deviation associated with the r2 estimate itself for the TBL equation for 55 Cancri. The point estimate I derived (0.9975) falls within the 99th percentile. So I said that my confidence level was now 99%; but really, I need a confidence interval for the r2 estimate, and the level of signifcance for the curve as a whole would then depend on where the left handed tail of the r2 distribution resulting from measurement error would fall.

Warren, I would recommend that you not do this - what you have is done all the time in asymptotic statistics. You have already found the distribution of the r-squared statistic (which you are using as a test-statistic) empirically. What are you going to do now? Run a million simulations a million times to see how accurate your estimated distribution of the r-squared statistic is? I would say if you have got that kind of computing power, just run a million times a million simulations. This is not your problem - that lies elsewhere.


Now we can finally lay to rest the repeated claim any old jumble of points can generate a TBL pattern.

I am not so sure that is the conclusion I would draw here. I can generate lots of random data sets by rules that do not match TBL very well, and I can also generate some that match it really well. Does your statistically significant result tell us that TBL works really well, or does it mean that the random planetary system generator you wrote conforms to TBL very poorly?

I think what you really want to do is to test the observed data against various alternatives. For example, you could try:

y=a+b1*x1+b2*x2+b3*x3+eps

where x1 are positions predicted by TBL (logarithmic), x2 are positions predicted by some other simple rule, x3 are positions predicted by yet another simple rule. But then what people are saying to you about limited data points is really going to bite; you just do not have enough observations to come to any reasonable conclusion there. If all you do is test the statistical significance of b in:

y=a+b*x+eps

(NB-not what you have done so far!) where x are positions by TBL, and find it is statistically significant, then what have you done? You have shown that TBL explains planetary positions better than:

y=a+eps

In other words, a rule that says the second planet is closer to the star than the first planet half the time. Not a very high hurdle to jump over. You really want to test TBL against reasonable alternatives, and the limited availability of data is going to bite you on that one.

Another statistical issue - Warren, it sounds to me from reading this thread that you have been comparing r-squared statistics across different models when the y variable has been modified. You can not do that. The r-squared in a regression of y on some explanatory variables is not comparable to the r-squared in a regression of the log of y on some explanatory variables, because the variable whose fit you are measuring has changed.


the one, fundamental, empirical claim that 55 Cancri (for now, anyway)--as the data has been reported--exhibits a remarkable--that is to say, statistically significant--TBL pattern.

Comments on statistical significance above - the statistical significance you have found is that the distribution of the planets was not generated by the rule you used in the random simulations. That is not quite the same as saying it exhibits a TBL pattern. But, if you think it is remarkable, wait until you see what I have. Try this model:

y=a+b*(x^5)

where y is the distance (not logarithmic), and x is the planet number, skipping 5, just as you did. What r-squared do you get now? I get 0.9999. Once I got a spreadsheet rigged up to do the calculations, it took me less than one minute to find this model.

Jerry
2007-Nov-23, 02:53 AM
Also, the scale factor K says something important. I suggest that the K factor says something about the rate of formation of planets and solar systems: the higher the K factor, the faster the rate of formation (you heard it here first folks! :D).

As soon as you fix orbits according to a rule that is not a part of current planet formation theory, other assumptions fall by the wayside, especially the assumption that the planets all started from a dusty cloud. If there is a set of rules that puts discrete spaces in naturally defined orbits, planets may be captured rather than evolved from a dusty orbital plane.

You have to be really careful when looking at logrythmic curves. There is an expectancy in mature systems for planet-cleared orbits to define discrete distances at which moons should be expected, no unusual physics required. It is not unlike looking at a Talus slope - which is logrythmic; and concluding there is an underlying physical mechanism that defines all debris fields in nature.

Warren Platts
2007-Nov-23, 03:22 AM
No. I've repeatedly wondered whether it is significantly logarithmic or not. And I've repeatedly pointed out, with examples and explanation, that that question is pretty much impossible to answer with the tiny dataset you have.
Which data set are you referring to? I have 1,000,000 data points in my Monte Carlo simulation. I can crank it up to a billion if you want, but it's going to take a couple of days.

Then there's 55 Cancri itself, consisting of 5 observed semimajor axes. Sure it's small, but that doesn't entail that nothing useful can be said about the arrangement. Consider the sequence {1, 2, 4, 8, _____, 32} and just forget everything you ever knew about astronomy. Is the sequence significantly logarithmic or not? Of course it is. It has a perfect r2 (1.00).

What about the sequence {1, _____, 29, 30, 31, 32}? Obviously, it is not well-described by an exponenential formula--although it's easy enough to construct one anyway. But if you do, the r2 is only 0.6853.

What about the sequence {0.038, 0.103, 0.281, 0.762, _____, 5.619}? You can't tell just by looking at it, but it also has an r2 of 1.0000.

What about {1.01, 1.99, 4.01, 7.99, 32.01}? It's not perfect, but it's pretty darn good at r2 = 0.9996. Can we call that significantly logarithmic?

What about {0.031, _____, 0.654, 1.325, 3.303, 4.716}? Well, just looking at the numbers, they seem intriguing perhaps. Do the math and the r2 is 0.9457. Not too shabby, right? But wait: now that I've done the Monte Carlo simulation, we can put 0.9457 into context. If you look at Figure 1 again you'll see that 0.9457 is less than the median r2 score for the million star system simulation. In other words, despite the seemingly high r2 (0.9457), more than half of the randomly generated solar systems did better--indeed, I can look at the table I generated and tell you that {0.031, _____, 0.654, 1.325, 3.303, 4.716} is in the 45th percentile, clearly not significantly logarithmic.

If you'll look at Figure 2 again, you'll see a fairly steep drop off at about the 95th percentile. But looking at my table, the 95th percentile corresponds to an r2 of about 0.9925. The 90th percentile corresponds to an r2 of about 0.988. So which percentile shall we decide counts as "significantly" logarithmic? It doesn't matter to me--that's a conventional human decision. But let's just go with the top 5%.

Thus, we can say a sequence of five numbers with one empty slot allowed is significantly logarithmic if the r2 is greater than or equal to 0.9925.

So if we apply that criteria to 55 Cancri, we see that 55 Cancri's r2 is 0.9975--so apparently, 55 Cancri counts as significantly logarithmic.



Finding that 55 Cnc is significantly different from a "random solar system", generated by a random number generator, is not nearly enough to answer that question. You're confusing me again with all the pronouns. I take "that question" to refer to the same "that question" above that asked "whether it is significantly logarithmic or not". What other kind of significance is there besides statistical significance? Any attempt at classifying planetary spacing as "regular" or not is going to have to go through an analysis pretty much like I did. And it first has to proceed as a mathematical excersize with all astronomical inputs abstracted out.

Spacing rules and stability criteria absolutely must not be taken into account, because otherwise you'll be just assuming what you're trying to detect.


Excellent example. All such studies need to address the different sampling distributions for particular animal groups. If a species that fossilizes poorly (a bird, for instance, or a species that prevailed in a time and place with poor fossilizing conditions) drops out of the fossil record at some point, is it the case that it has become extinct, or has it simply become so rare that we are unlikely to have encountered one of its fossils for the relevant time period?
That's the problem which causes the Signor-Lipps effect, in which any mass extinction appears to have a more gradual onset for rarely fossilized species than for commonly fossilized species. So all sensible Monte Carlo simulations of this phenomenon take into account the probabilistic nature of the "signal" we receive from the different species, through the fossil record, as well as the underlying random process of speciation and extinction.
Do you see the analogy with observing distant planetary systems? Absolutely! And I'll tell you if Raup and Sepkowski had listened to all the reasons why you say I can't do this analysis, they would have never got off the ground either. And don't get me wrong: I appreciate your input, because it has spurred me to take my analysis to the next level. :) But the fact is, they dealt with birds and other hard to detect fossils by ignoring them completely. Moreover, they ignored the lower level categories of species and genera, and focused instead on families of shelly, marine invertebrates. So that's the analogy when it comes to extrasolar planetary spacing. We are of necessity forced to look at the only good data there is, which are the shelly marine invertebrates--or in this case, the massive, close-in planets. But that's OK. For purposes of TBL, we are only interested in major planets anyway, and we know from Earth and now 55 Cancri that we can get good data on several planets within 5-6 AU of the primary.


If random data were completely uncorrelated going in and the signal was well correlated coming out, there would be less reason to fret about the detail of the sampling distribution. But when random data are pretty well correlated going in and the signal is just rather better correlated coming out, we know that statistical significance will be strongly influenced by the detail of the sampling distribution. The histogram produced by the Monte Carlo program is the sampling distribution. And you're right, the statistical significance is strongly influenced by the shape of the histogram. But now we have the histogram. We can look at it in detail, and now that we have, we know that to count as significantly logarithmic requires insanely high r2's. And it just so happens that 55 Cancri has an insanely high r2.

Warren Platts
2007-Nov-23, 04:00 AM
I But, if you think it is remarkable, wait until you see what I have. Try this model:

y=a+b*(x^5)

where y is the distance (not logarithmic), and x is the planet number, skipping 5, just as you did. What r-squared do you get now? I get 0.9999. Once I got a spreadsheet rigged up to do the calculations, it took me less than one minute to find this model.Thanks for the comments, I'm still trying to digest most of them, but one quick question for now: Is the above formula intended to model the numbers for 55 Cancri? If so, what values for a and b did you use? I plugged in 0.038 for a, and 0.000737 for b, and that works fairly well (except for being 46% off for planet II). But when I plug it into the spreadsheet that calculates r2[sup], (which converts the numbers to logarithms before doing the least-squared thing), I get an r[sup]2 of 0.9876216. I think your formula is what Disinfo Agent called a power law, I think, and therefore, it's only approximately logarithmic.

The Albatross Slayer
2007-Nov-23, 05:29 AM
Thanks for the comments, I'm still trying to digest most of them,

Ok, I hope they are helpful.


but one quick question for now: Is the above formula intended to model the numbers for 55 Cancri?

Yes, or at least I think so. I pulled the distances for the planets from an earlier post of yours, and was able to replicate your r-squared in the logarithmic model to at least four decimal places, so I think I am using the right data.


If so, what values for a and b did you use?

I didn't save the results, but the r-squared part of the BASIC program you wrote would pretty much do it for you. If you change your y variable to the distance of the planet (instead of the logarithmic distance), and the x variable to the planet number to the fifth power (beginning with one, and skipping five as you did), then I think you should get the same r-squared I did from your program (skip the simulations and use the actual data instead of the simulated data for the y variable). The program explicitly calculates b, and if you change x and y to be what I used (x is the planet number to the fifth power, and y is the planetary distance, no logarithm taken), you should get the b I used. My a would be the average value of y, minus b times the average value of x. These are the coefficients that minimize the sum of squared deviations of the y from the predicted values, and there is nothing specific to this application here. I calculated the best fit coefficients for your model, replicating the r-squared you found, exactly the same way, just using the appropriate y and x variables (for your model, y being the logarithm of distance, and x being the planet number).

I get a very high r-squared this way. It is possible I made a mistake, because I did it quickly, but I did replicate your r-squared very closely, and all I did was change the definition of the y and x variables to get mine, so I think the number is correct.

I agree that some of the statistical complaints being thrown at you (including some you threw at yourself :)) are not valid, because the Monte Carlo simulation takes care of them. Correlation in the residuals, non-normality, whatever, all the usual problem that make t-statistics and the like incorrect, are taken care of because you simulated the actual distribution of the r-squared test statistic. And with a million simulations, I think the distribution is probably pretty accurate. If you are so inclined and run another million and get pretty much the same result, that would tend to suggest that you are ok on that front. I would be real careful about interpretation though.

In the last post, where you are responding to Grant, I would have to argue that when your confidence level is high, you have a statistically significant result, but the result is not that the logarithmic (or whatever) model is correct. The result is that the model used to generate the simulated data is wrong. The deck is intrinsically stacked against you here, that is just the way statistics is. You can never prove your model correct; the best you can hope for is that it will not be proved incorrect. By example, I would say, suppose you came up with some model that did not really explain the positions of the planets all that well. Let us say the r-squared is 0.3. You then simulate random data according to some algorithm that is absolutely terrible. You then find that only 2% of the simulations have a higher r-squared. We can not really conclude that your model is correct; we can only conclude that the actual data does not conform to the algorithm used to generate the random data. Maybe someone will come up with a better algorithm that is different than yours, but which generates r-squared statistics routinely much greater than 0.3. Until it happens, we do not really know.

The other thing I would mention is, assuming I calculated my r-squared correctly, the fact that it is higher than yours does not mean the model I cooked up has a better fit than yours, because the y variable is different. In mine, the r-squared measures how much of the variation in the planetary distances is explained by the regression variable (in my case, planet number to the fifth power), whereas in yours, the r-squared measures how much of the variation in the logarithm of distance is explained. The two are not directly comparable. If we converted my numbers to logarithms, and calculated a comparable statistic, maybe it would be worse than yours. Or maybe it would be higher still; I did not do it, so I am not sure.



I plugged in 0.038 for a, and 0.000737 for b, and that works fairly well (except for being 46% off for planet II). But when I plug it into the spreadsheet that calculates r2[sup], (which converts the numbers to logarithms before doing the least-squared thing), I get an r[sup]2 of 0.9876216. I think your formula is what Disinfo Agent called a power law, I think, and therefore, it's only approximately logarithmic.

Power law is not terminology I use a lot, so I am not sure of the exact definition. But it could be made logarithmic in the following way:

ln(y-a)=ln(b)+ln(x)

If that is a power law or not, I am not sure.

I just think one has to be really careful in interpretation here. We are fitting five observations with two-parameter curves, which can nail two of the five observations perfectly. Then, we get to choose the shape of the curve (logarithmic, power, whatever), which is almost like having another free parameter. There is not a whole lot left to fit. I did not do a direct comparison, but I think our two models probably make pretty similar predictions over a wide range, then deviate somewhat afterwards. There are almost certainly a lot of curves you can draw through five data points. The other thing that is a little iffy here is that the last planet looks like an extreme outlier. If the last planet had not yet been discovered, I think the fitted models would make very different predictions.

grant hutchison
2007-Nov-23, 12:05 PM
I do not see how this is a problem here. The equation Warren is looking at is a linear regression equation:

y=a+b*x+eps

Usually when we talk about statistical significance, we are talking about whether the b coefficient is significantly different than zero. In that case, if we had serial correlation in the eps (serial correlation in the x is ok) then t-statistics and p-values would have to be estimated by methods that take serial correlation into account.

However, that is not what Warren is doing. He is using the r-squared as a test statistic. That is a little bit unconventional, but so what? He calculates a test statistic from the observed data, and compares it to the same test statistic calculated from random data, over a large number of simulations.Yeah, I'm comfortable with r2 as the test statistic, and I'm happy with Monte Carlo as a way of deciding if that test statistic is unusual. My problem is that Warren is using the wrong Monte Carlo model to make that decision. There is no way that our received data on distant solar systems corresponds to a flat probability distribution: we're well aware of many deviations from that, which I've tried to summarize. But see my question list below.
Edit: Reading through again what you've been saying to Warren about deducing that the model is incorrect, I wonder if we're perhaps not saying similar things in different ways. I guess I may have inadvertently misused the phrase "statistical significance"; I was thinking of this as arising only when the correct Monte Carlo model is used, but it seems that you're using it whenever the test statistic deviates from the model currently being used. So (if I understand you) you're saying that Warren has achieved statistical significance with the wrong model, and I'm saying Warren has failed to establish useful statistical significance because he is using the wrong model (and because the right model is unknown to us).


Which data set are you referring to? I have 1,000,000 data points in my Monte Carlo simulation. I can crank it up to a billion if you want, but it's going to take a couple of days.All these points are simply part of the statistical test you're doing on your real-world dataset: they're standing in for a sampling distribution. Your dataset, from which are attempting to make deductions about the real world, is confined to the five points of the original 55 Cnc observations.


Absolutely! And I'll tell you if Raup and Sepkowski had listened to all the reasons why you say I can't do this analysis, they would have never got off the ground either. And don't get me wrong: I appreciate your input, because it has spurred me to take my analysis to the next level. But the fact is, they dealt with birds and other hard to detect fossils by ignoring them completely. Moreover, they ignored the lower level categories of species and genera, and focused instead on families of shelly, marine invertebrates.And their statistical analysis was strongly criticized, by Tony Hallam and others. And several of their smaller extinction events, together with the periodicity that they thought they had identified, are now considered to have arisen from mere background noise, as a result of more detailed analyses of the sort I described.

There's a whole raft of stuff been posted, and it strikes me that there are several different problems which are now in danger of all getting mixed together. I've actually been trying to stay focussed on a single question here, despite Warren's protestations to the contrary, but now that others have joined the discussion it seems to me that the sub-questions perhaps need to be more formally stated.
Here's the list, as I see it, together with the answers as I see them; I've been concentrating on my question 5) since I first pointed out the "pre-correlation" problem to Warren.

1) Is it acceptable to interpolate a point on the 55 Cnc graph?Of course it is. We can do what we like.

2) Does that interpolation tell us anything about the probability of a planet actually occupying that slot in the real world? At present, no. Even assuming the exponential pattern exists in the real world, we have no data at all on the frequency of "drop-outs" from the sequence, and we cannot expect to have complete data on a large number of solar systems any time soon.

3) Is the r2 statistic of 55 Cnc unusual when compared to a completely random distribution of radii, treated in the same way as 55 Cnc? Yes, it is. Warren's Monte Carlo simulation addresses that.

4) Does the r2 statistic of 55 Cnc give us cause to believe, with reasonable confidence, that similar, future observations of other systems will also show exponential relationships? No it doesn't, because the data accessible to us through observation are highly selected: they demonstrably cannot correspond to the complete randomness of Warren's Monte Carlo simulation. To answer this question, we would have to perform a Monte Carlo simulation which includes the selectivity of our observations.

5) Does the r2 statistic of 55 Cnc give us cause to believe, with reasonable confidence, that real, complete solar systems (as distinct from what we can currently observe of those systems) follow exponential relationships?No, again, for all the reasons in 4) plus the consideration that some components of distant solar systems (including 55 Cnc) will be unobservable with our current observation techniques, and that stability considerations further limit the randomness of real solar systems.

Grant Hutchison

KaiYeves
2007-Nov-23, 03:34 PM
Welcome to BAUT, The Albatross Slayer!
Great handle, by the way!

The Albatross Slayer
2007-Nov-23, 05:13 PM
Edit: Reading through again what you've been saying to Warren about deducing that the model is incorrect, I wonder if we're perhaps not saying similar things in different ways. I guess I may have inadvertently misused the phrase "statistical significance"; I was thinking of this as arising only when the correct Monte Carlo model is used, but it seems that you're using it whenever the test statistic deviates from the model currently being used. So (if I understand you) you're saying that Warren has achieved statistical significance with the wrong model, and I'm saying Warren has failed to establish useful statistical significance because he is using the wrong model (and because the right model is unknown to us).

I don't think we have any serious disagreement. I am arguing that there is a statistically significant result, but we have to interpret that result of rejection of the model used to generate the random data, not confirmation of the TBL model. Correctness of TBL is possible given the results, but not mandatory. I could, for example, generate the random data using a really bad model, find that the r-squared for the observed data is much higher than the r-squared in almost all of the simulations, and find a statistically significant result. But that tells me the simulation model is wrong; it doesn't tell me the TBL model is right.


All these points are simply part of the statistical test you're doing on your real-world dataset: they're standing in for a sampling distribution. Your dataset, from which are attempting to make deductions about the real world, is confined to the five points of the original 55 Cnc observations.

I would add that one of these observations (the farther planet) seems to be strongly influential, and probably drives most of the result. Most of the variation in planetary distances comes from this one observation, so a model that nails that one will have a good fit. Just on lark, I decided to delete the last planet, and still use the one-extra-slot method, and found the r-squared from the logarithmic model actually went down a bit.


Welcome to BAUT, The Albatross Slayer!
Great handle, by the way!

Thanks, you make these decisions impulsively, and then live with the consequences a long time :) Glad you like it though.

grant hutchison
2007-Nov-23, 05:29 PM
I don't think we have any serious disagreement. I am arguing that there is a statistically significant result, but we have to interpret that result of rejection of the model used to generate the random data, not confirmation of the TBL model. Correctness of TBL is possible given the results, but not mandatory. I could, for example, generate the random data using a really bad model, find that the r-squared for the observed data is much higher than the r-squared in almost all of the simulations, and find a statistically significant result. But that tells me the simulation model is wrong; it doesn't tell me the TBL model is right.OK, good.
With regard to my concern about the modelling when the data are "pre-correlated": I'm saying that if the model r2s were distributed around zero (which is what we're used to when looking at correlation coefficients), then Warren's very high r2 for 55 Cnc would be very impressive in many different possible models. But in the "pre-correlated" situation that prevails, the model r2s are necessarily going to be centred somewhere up beside Warren's 55 Cnc r2: a small change in the model can therefore more readily make the difference between significance and insignificance.

Grant Hutchison

Warren Platts
2007-Nov-23, 06:41 PM
I have time for some quick responses to these questions.


Originally posted by grant hutchison

Here's the list, as I see it, together with the answers as I see them; I've been concentrating on my question 5) since I first pointed out the "pre-correlation" problem to Warren.

1) Is it acceptable to interpolate a point on the 55 Cnc graph?Of course it is. We can do what we like. Agreed.


2) Does that interpolation tell us anything about the probability of a planet actually occupying that slot in the real world? At present, no.
Even assuming the exponential pattern exists in the real world, we have no data at all on the frequency of "drop-outs" from the sequence.I agree here as well. There is the asteroid belt, which wouldn't show up on extrasolar surveys. So empty slots should be expected even according to TBL. Nevertheless, the empty slot is intriguing; I'm sure astronomers will be keeping a close eye on 55 Cancri in the future, as they have been for the past 18 years (Hubble was recently granted a crack at 55 Cancri).


3) Is the r2 statistic of 55 Cnc unusual when compared to a completely random distribution of radii, treated in the same way as 55 Cnc? Yes, it is. Warren's Monte Carlo simulation addresses that.Thank you.


4) Does the r2 statistic of 55 Cnc give us cause to believe, with reasonable confidence, that similar, future observations of other systems will also show exponential relationships? No it doesn't, because the data accessible to us through observation are highly selected: they demonstrably cannot correspond to the complete randomness of Warren's Monte Carlo simulation.I agree here, but for different reasons. The TBL is a purely empirical relation. It implies no particular causality at all. Indeed, the random Monte Carlo simulation suggests that even if planetary spacing in the universe were totally random, 1% of those random systems would have a more pronounced logarithmic spacing than 55 Cancri. To see if the TBL is a general pattern will require data from several star systems. But to do that, we need a method and a criterion for scoring planetary systems into TBL versus non-TBL spacings.


5) Does the r2 statistic of 55 Cnc give us cause to believe, with reasonable confidence, that real, complete solar systems (as distinct from what we can currently observe of those systems) follow exponential relationships?No, again, for all the reasons in 4) Even if we could construct full-blown numerical simulations that took everything into account (planetary formation and a few billion years of celestial mechanics) were available, and they reliably generated TBL systems or not, our confidence in such models would still depend on our ability to gather empirical data from several different star systems.


plus the consideration that some components of distant solar systems (including 55 Cnc) will be unobservable with our current observation techniques, and that stability considerations further limit the randomness of real solar systems. I'm still having trouble following your logic here. We're not going to be able to detect every extrasolar asteroid out there, no doubt, but that's not a problem, because the TBL only applies to major planets (including perhaps asteroid belts), and major moons of gas giant planets (which will probably remain unresolved). And I agree that stability considerations limit the randomness of real solar systems. But if I'm not mistaken, typical TBL patterns are much further apart than are minimum spacings that result from orbital stability considerations. But maybe that question is up in the air too.

grant hutchison
2007-Nov-23, 07:34 PM
To see if the TBL is a general pattern will require data from several star systems. But to do that, we need a method and a criterion for scoring planetary systems into TBL versus non-TBL spacings.No, that's never going to work. You seem to be treating this as if exponential spacing is an all-or-nothing phenomenon. But it's not going to be like that: it isn't in the solar system, it isn't in 55 Cnc. What we find are patterns that look a bit exponential, but not perfectly and with many exceptions. The more data of this kind we collect, the more an exponential pattern will emerge from the noise, or disappear into the noise.
I think everyone here is pointing out, in different ways, that the five data points of 55 Cnc are insufficient to make a statement either way with any confidence.


Even if we could construct full-blown numerical simulations that took everything into account (planetary formation and a few billion years of celestial mechanics) were available, and they reliably generated TBL systems or not, our confidence in such models would still depend on our ability to gather empirical data from several different star systems.Exactly. So how much less confidence should we have in our incomplete understanding of solar system formation, coupled with incomplete data from only a few systems? A great deal less, is the answer.


I'm still having trouble following your logic here. We're not going to be able to detect every extrasolar asteroid out there, no doubt, but that's not a problem, because the TBL only applies to major planets (including perhaps asteroid belts), and major moons of gas giant planets (which will probably remain unresolved).Well ... We're also missing the outer reaches of every single planetary system out there, at present, as well as almost anything the size of Earth or smaller. Most variants of TBL at least pretend to treat the terrestrial planets and the ice giants. So those are show-stopping gaps, which when filled could either strengthen or destroy any perceived tendency to exponential spacing.


And I agree that stability considerations limit the randomness of real solar systems. But if I'm not mistaken, typical TBL patterns are much further apart than are minimum spacings that result from orbital stability considerations.Stability covers many more aspects than just stuff being very close together: we already know that resonances are reasonably common in extrasolar planetary systems, and resonances could either support or destroy exponential spacing. I think you're also neglecting the known prevalence of high eccentricity orbits in extrasolar planetary systems. A massive planet on an eccentric orbit (and we already know of quite a few of these) will enforce a large spacing around its semi-major axis.

Grant Hutchison

Warren Platts
2007-Nov-23, 10:08 PM
6TH PLANET DISCOVERED AT 55 CANCRI BY WARREN PLATTS!

A 6th planet was recently reported according to a new analysis of data produced by Fischer et al. (2007) (http://exoplanets.org/55cnc5th.pdf). According to the Titius-Bode relation, there should be a 6th planet at 55 Cancri at a distance of about 2 AU. According to Kepler's laws, the orbital period squared is proportional to the cube of the semimajor axis. For 55 Cancri:




P2 / D3 ~ 140,000

For a 2 AU semimajor axis, this works out to a three Earth-year orbital period. Thus, there should be a planet with a period of about 1,000 to 1,200 days at 55 Cancri. Not surprisingly, a new peak on the periodograms published by Fisher et al. (2007) show a recalcitrant peak on several figures (Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 7) within the predicted range. Also their Fig. 1 shows a three year periodicity.

When asked, Mr. Platts denied receiving any phone calls from the Nobel Prize Committee in Stockholm.

grant hutchison
2007-Nov-23, 10:58 PM
I guess I'm just unlucky when I download the linked version of the paper to find that the text is corrupted while the graphs are visible.

Checking a copy in which the graph labels are legible (http://exoplanet.eu/papers/debra.pdf) reveals the following: (Fig 5, residuals of four-planet model) "The tallest peak is at a period of 265.6 d ... No other period is compelling between periods of 1 and 3000 d"; (Fig 7, residuals of five planet model) "No other compelling periods are apparent."
Earlier in the paper (Section 5) is a discussion of the unsuccessful search for a 6th signal, and an estimate of the (significant) planetary masses which might so far have evaded detection. Also, handily, reference to the "signal deconvolution" problem I've been wittering on about, reporting how neighbouring planets might produce fictitious periodogram spikes, and resonant orbits might elude detection.

Grant Hutchison

Warren Platts
2007-Nov-23, 11:15 PM
I love the word "compelling".

grant hutchison
2007-Nov-23, 11:35 PM
What was the point of that?
You must have known that someone on this thread would be familiar enough with the paper to call your bluff immediately.

Grant Hutchison

Warren Platts
2007-Nov-23, 11:45 PM
Stability covers many more aspects than just stuff being very close together: we already know that resonances are reasonably common in extrasolar planetary systems, and resonances could either support or destroy exponential spacing. I think you're also neglecting the known prevalence of high eccentricity orbits in extrasolar planetary systems. A massive planet on an eccentric orbit (and we already know of quite a few of these) will enforce a large spacing around its semi-major axis.

Grant Hutchison
Quite true, sir. That's why I sought average eccentricity of the protoplanentary planitesimals as an explanation for TBL. Indeed, planetary spacing could give information on what the primitive eccentricity was. But how do you break the symmentry? That's why I turned to vorticity--the disk breaks into vortices whose diameter depend on the average eccentricity. And probably, the average eccentricity depends on the average metallicity.

Warren Platts
2007-Nov-23, 11:57 PM
What was the point of that?
You must have known that someone on this thread would be familiar enough with the paper to call your bluff immediately.

Grant Hutchison

Well, if you'll look at the figures, you'll see the peak I'm talking about. There's no mistaking it. And yeah sure, there's multiple peaks in some of the figures; but if you read the paper, you'll see that there are "aliases"--false signals for closely spaced peaks. The peaks for 260 day and 460 day orbits are only two hundred days apart, but on a logarithmic scale, that's about the same as the time between 1,000 and 1,100 days.

To me it's just kind of striking that there are in fact medium-sized peaks where I predicted. So, I'm no doubt biased. And then there is my genetic predisposition to inordinate strange pattern detection.

So what we need is access to the raw data, and then we can do a real statistical analysis ourselves.

grant hutchison
2007-Nov-24, 12:05 AM
So what we need is access to the raw data, and then we can do a real statistical analysis ourselves.I'll leave you to it.
That last little effort overdrew at my tolerance bank.

Grant Hutchison

Warren Platts
2007-Nov-24, 01:46 AM
Seriously: you've got to look close at Fig. 4 and Fig. 7: those two charts have the same scale, but one shows the profile with the 260-day signal taken out.

To do this, you have to load 2 different PDF windows, one with Fig. 4, and one with Fig. 7. Then you have to horizontally minimize the screens and double check to see if the PDF magnification is the same for both windows. Then you have to line them up on top of each other real close. Jack up the magnification in both windows to 200% at least. Then compare the three peaks that surround the 1,000 day period predicted by the TBL.

Fig. 4 shows the "four-planet" model, where the effects of the first four known planets were sucked out of the periodogram. Then, Fig. 7 shows the "five-planet" model, where the effects of the fifth planet (55 Cancri f) are sucked out.

At around 1,000 days, there are three peaks that are relatively prominant. When the effects of 55 Cancri f are sucked out, however, the first two peaks on the left decline, whereas the third peak remains pretty much the same.

These generate two data sets of the form Fig x = {(Pi, di), . . . } where Pi and di) are the measurements for the power and the period in days for each of the three peaks from left to right:



Fig. 4. = {(7.0, 900), (3.0, 1020), (3.75, 1080)}
Fig. 7. = {(3.5, 800), (2.5, 1030), (3.75, 1090)}

The recalcitrant peak is at ~1085, about where the TBL predicts a recalcitrant peak should be.

And this was a prediction, mind you. I figured out the 1,100 day period first, and then looked at the spectra, and there was a recalcitrant peak at about where there should have been one.

Granted, the effect is small--but it's real nevertheless. Look for yourself. Becareful, the vertical scale on Fig 4 is slightly larger than Fig 7. You have to use an envelope to make accurate measurements.

55 Cancri is the star system with the longest running continuous observation that we have. The peak at 1085 days is obvious now, and as more, better observations pile on top of the previous 18 years of observation, the peak will only become more undeniable.

Lucky for you that you didn't take my bet! :)

grant hutchison
2007-Nov-24, 01:58 AM
Let me just pause for a moment to consider whom I trust to analyse these data:
1) Marcy & Butler's team, who have some small experience and track record in these matters.
2) Warren "Award Myself A Nobel Prize Every Week" Platts, and his measurements with an envelope.

OK. I've decided.
'Bye, Warren. :)

Grant Hutchison

Warren Platts
2007-Nov-24, 02:09 AM
Let me just pause for a moment to consider whom I trust to analyse these data:
1) Marcy & Butler's team, who have some small experience and track record in these matters.
2) Warren "Award Myself A Nobel Prize Every Week" Platts, and his measurements with an envelope.

OK. I've decided.
'Bye, Warren. :)

Grant Hutchison

How do you think Marcy & Butler really do it? They take a printout and a ruler like I did, and then just look at it. Really, that is their very first observation. Everything else, the telescopes, the computers, and and all the math that went into those printouts, are all lens and filters, and its the printout that the human retina actually sees. And they pick out peaks with their eye and take that as a working hypothesis, and then they see where that leads them. Maybe they've already taken a close look at the 1,085 peak. But maybe they've got so much on their plates they haven't really had a chance to. Or they did, and they're intrigued just because it fits the TBL pattern, but chose to wait before going public because of the harsh reception that TBL claims are guaranteed to receive.

Warren Platts
2007-Nov-24, 02:29 AM
With regard to my concern about the modelling when the data are "pre-correlated": [B]I'm saying that if the model r2s were distributed around zero (which is what we're used to when looking at correlation coefficients), then Warren's very high r2 for 55 Cnc would be very impressive in many different possible models.

There seems to be a bit of confusion regarding r2, the coefficent of determination ("It is the proportion of variation in y that has been explained by the simple linear regression model, and 100r2 is the percentage of total variation explained by the model", according to the old statistics textbook I dug up, as opposed to Pearson's sample correlation coefficient r.

The correlation coefficient is calculated of a scatter plot to decide if it's worth it to fit a line to the data. It is related to the slope of the least-squares line b. If r = 0, then b = 0 (flat), if r < 0, b < 0 (downslope), if r > 0, b > 0 (upslope). Since the data for the Monte Carlo simulation was precorrelated necessarily, then r is necessarily positive for our purposes.

r2 on the other hand, measures how closely the data fits a fitted line. It is always positive, with 1.00 indicating a perfect fit, no matter what the slope is.

As far as I can tell, there is no direct relationship between r and r2--one certainly can't take the square root of r2 and expect to get r.

Warren Platts
2007-Nov-24, 03:03 AM
Attached is the chart from post #84 that shows the 95% confidence level prediction interval, with the new data point taken from 1085 days orbital periods from Fig. 4 and 7.

As you can see it falls pretty much where it was expected to.

The Albatross Slayer
2007-Nov-24, 03:57 AM
As far as I can tell, there is no direct relationship between r and r2--one certainly can't take the square root of r2 and expect to get r.

In a single variable regression, there is a pretty direct relationship. The regression equation is:

y=a+b*x+eps

where x and eps are independent. Then you can find the variance of y:

var(y)=(b^2)*var(x)+var(eps)

The r-squared statistic is the ratio of the amount of variance explained by x to the amount of variance in y, so:

r^2=(b^2)*var(x)/var(y).

You can also find the covariance between x and y:

cov(x,y)=b*var(x)

The correlation (the r you refer to) is:

cov(x,y)/sq(var(x))/sq(var(y))=b*sq(var(x))/sq(var(y))

Square the r, and you get r^2. Pretty direct relationship :)

Things are more complicated in a multiple variable regression, but we're talking about single variables here.

I think the statistics of the Monte Carlo are fine, the problem is the interpretation. The statistical significance does not prove the TBL; fit to the TBL is just the test statistic. The correct interpretation is rejection of the algorithm you used to simulate the random data; if this algorithm were the true algorithm generating planetary systems, then the observed system would be extremely unusual. But this tells us the simulation generation algorithm is bad; it doesn't tell us the TBL is good.

I didn't completely understand Grant's comment myself, but I understood clustering "around" zero to mean clustering at small positive values. But perhaps I should let Grant speak for himself.

Warren Platts
2007-Nov-24, 12:57 PM
In a single variable regression, there is a pretty direct relationship. The regression equation is:

y=a+b*x+eps

where x and eps are independent. Then you can find the variance of y:

var(y)=(b^2)*var(x)+var(eps)

The r-squared statistic is the ratio of the amount of variance explained by x to the amount of variance in y, so:

r^2=(b^2)*var(x)/var(y).

You can also find the covariance between x and y:

cov(x,y)=b*var(x)

The correlation (the r you refer to) is:

cov(x,y)/sq(var(x))/sq(var(y))=b*sq(var(x))/sq(var(y))

Square the r, and you get r^2. Pretty direct relationship :)

Things are more complicated in a multiple variable regression, but we're talking about single variables here.
Looks like you're right--I stand corrected. :)


I think the statistics of the Monte Carlo are fine, the problem is the interpretation. The statistical significance does not prove the TBL; fit to the TBL is just the test statistic. The correct interpretation is rejection of the algorithm you used to simulate the random data; if this algorithm were the true algorithm generating planetary systems, then the observed system would be extremely unusual. But this tells us the simulation generation algorithm is bad; it doesn't tell us the TBL is good.I'm with you here. We're not proving the truth of a TBL layout, we're rejecting the hypothesis that the 55 Cancri layout was randomly generated.

I didn't completely understand Grant's comment myself, but I understood clustering "around" zero to mean clustering at small positive values. But perhaps I should let Grant speak for himself.[/QUOTE]

Warren Platts
2007-Nov-24, 01:40 PM
What was the point of that?
You must have known that someone on this thread would be familiar enough with the paper to call your bluff immediately.


I'll leave you to it.
That last little effort overdrew at my tolerance bank.


Let me just pause for a moment to consider whom I trust to analyse these data:
1) Marcy & Butler's team, who have some small experience and track record in these matters.
2) Warren "Award Myself A Nobel Prize Every Week" Platts, and his measurements with an envelope.

OK. I've decided.
'Bye, Warren.


First of all, I'm sorry if I got a little overexcited. I get that way sometimes. Hey, I'm just trying to have some fun here. So please lighten up. Besides, I don't care about the Nobel. I'm going for the Templeton--it pays better! :D

Second, you say I'm bluffing. I only bluff when I play poker. A bluff is an intentional misrepresentation, and while I am fully capable of making huge mistakes, there is nothing on this thread that I have intentionally misrepresented.

I was doing my duty by actually checking to see what evidence there is regarding a planet at 2 AU. I calculated what period that corresponds to (3 years), and then looked at the periodogram for myself. That's what scientists, whether amateur or professional, are supposed to do. Ideally, if there was nothing at 2 AU, then the line would be flat at around 1,100 days; ideally, if there was a major planet there, there would be an unmistakable peak compelling to everyone and not just me. Instead, the situation is ambiguous. There are peaks at the predicted locations, but they aren't towering over everything else.

On the other hand, there is an overall trend toward declining peaks as one moves to the right. And I'm fully aware about the point you were "wittering" on about regarding "aliases"--neighboring, spurious peaks generated by the actual planets. Those can be corrected by removing the effects generated by those planets, however.

As one proceeds from Fig. 3 to Fig. 4 to Fig. 7, the effects of planets are progressively taken out. Fig. 7 has all the known planets taken out. Of the three peaks I talked about, the one on the right (@ 1085 days) is not affected by removing the effects of 55 Cancri f--it remains recalicitrant.

Granted, it may not look particularly compelling (especially to people emotionally invested in making sure Warren's teleological theories are not vindicated). However, part of the reason the peak doesn't stand out is an optical illusion resulting from the fact that the periodogram is printed out on a log scale. If Fig. 7 were converted to a linear scale,the peak at 1085 would indeed stand out by itself, and tower over it's immediate neighbors.

It took 18 years of continous observation to tease out the 55 Cancri f signal. So it may take some more years of observation before the peak at 1085 days becomes statistically significant--or not. But if the TBL prediction of a Planet V at 2 AU is eventually "observed", it's going to happen by taking a very close look at the 1085 day peak on the periodograms generated in the future.

grant hutchison
2007-Nov-24, 02:15 PM
I didn't completely understand Grant's comment myself, but I understood clustering "around" zero to mean clustering at small positive values. But perhaps I should let Grant speak for himself.I was thinking about "r" and typing "r2", with the inevitable result.:sad:
So, yes, fold and squish the symmetrical distribution of "r" around zero (for independent, random x and y) to see the corresponding distribution of "r2" close to zero.

Grant Hutchison

tusenfem
2007-Nov-26, 09:26 AM
6TH PLANET DISCOVERED AT 55 CANCRI BY WARREN PLATTS!


There is no need for "planet V" which is only an artifact of your specific modelling technique, which I have shown by expecting that the planets there are spaced by A × 10n B.

However, my "law" is just as useless as yours, because both are just numerology, come up with a real explanation and then maybe one of the umpteen models that you can fit may have some relevance to the creation of particularly spaced planets.

Warren Platts
2007-Nov-26, 11:13 AM
There is no need for "planet V" which is only an artifact of your specific modelling technique, which I have shown by expecting that the planets there are spaced by A × 10n B.

Huh? How do you figure? In Post #138 you gave your model as:


0.009 × 100.5195 n

I didn't comment on it earlier because it gives the following horrible results:

{(1, 0.0298), (2, 0.098), (3, 0.326), (4, 1.077), (5, 3.562), (6, 11.78)}

The observed pattern is:

{(1, 0.038), (2, 0.115), (3, 0.227), (4, 0.781), (5, 2.023*), (6, 5.77)}
(*observation based on periodogram peak at 1085 days.)

My model calculates the following distances:

{(1, 0.038), (2, 0.102), (3, 0.281), (4, 0.762), (5, 2.07), (6, 5.62)}




However, my "law" is just as useless as yours, because both are just numerology, come up with a real explanation and then maybe one of the umpteen models that you can fit may have some relevance to the creation of particularly spaced planets.To the extent that you could get your model to come close to the observed distances, it would be mathematically trivial to show that the models were equivalent.

Moreover, it is not my claim that the mathematical models are explanations for planetary spacing. Such models are mere descriptions intended to show that the pattern is real, and not the figment of an overactive imagination.

The truly useful result is that showing that TBL spacings are common provides a welcome constraint on models of solar system evolution. However, one decides to construct such evolutionary models, such models must be able to reproduce the TBL spacing.

That is why "kinematic" models based solely on celestial mechanical considerations can't possibly provide a complete picture of solar system evolution: they can't reproduce the TBL spacing pattern.

Hence, it's much more likely that TBL spacing patterns are the fossil remains of ancient "feeding zones" whose width is ultimately a function of the solar system's metallicity.

tusenfem
2007-Nov-26, 01:27 PM
guess there must have been a typo in my "law"
secondly I did no use your imaginary non-existent "planet V" as I clearly stated in my post



That is why "kinematic" models based solely on celestial mechanical considerations can't possibly provide a complete picture of solar system evolution: they can't reproduce the TBL spacing pattern.


Yahyah, blah blah, why don't you then start to develop a numerical model that takes the primordial cloud and lets it collapse and creates the planets. It's a bit of work, but it would, at some point, get you the recognition that you are so yearning for (although I don't think it will be the Nobel prize).

Warren Platts
2007-Nov-26, 04:45 PM
guess there must have been a typo in my "law"
secondly I did no use your imaginary non-existent "planet V" as I clearly stated in my postHow do you know that Planet V isn't where I say it is? Theortical calculations suggest there should be a planet at 2 AU, and the periodograms of Fischer et al. (2007, Fig. 3-7) show a recalcitrant peak in the predicted location (did you bother to look at their printouts?). Both the combination of theory and empirical evidence show there is a Planet V at 2 AU.


Yahyah, blah blah, why don't you then start to develop a numerical model that takes the primordial cloud and lets it collapse and creates the planets. It's a bit of work, but it would, at some point, get you the recognition that you are so yearning for (although I don't think it will be the Nobel prize).To make my point that kinematic models can't explain TBL layouts, all I have to do is do numerical simulations of planetary orbits. To that end, I've been using GravitySimulator to investigate the stability of linearly spaced orbits. The first run had 6 Jupiters evenly separated by from 1 to 6 AU (1 AU between each semimajor axis). I was surprised when the simulation turned chaotic. The trouble started from the outside and worked inward.

So I redid the simulation, this time using 6 Earths instead of 6 Jupiters. This time the simulation worked perfectly--I let it run for over 10,000 years before I got bored looking at the perfect circles.

So I redid it again, this time with only 4 Jupiters spaced 1 AU starting from the Sun, as before. Results: stability.

Just from fooling around for an hour this morning, I have already learned a lot. Minimum spacing apparently is a function of distance from the primary. However, minimum spacing is also a function of the mass of the planets.

Moreover, there is nothing inherently unstable with linearly arranged layouts, as long as the spacing is greater than the minimum spacing. That is, there is no tendency for stable linear arrangements to evolve over time to more logarithmic arrangements.

Therefore, it is likely that the logarithmic arrangements that we do observe represent the primitive arrangement.

tusenfem
2007-Nov-26, 05:15 PM
To make my point that kinematic models can't explain TBL layouts, all I have to do is do numerical simulations of planetary orbits. To that end, I've been using GravitySimulator to investigate the stability of linearly spaced orbits. The first run had 6 Jupiters evenly separated by from 1 to 6 AU (1 AU between each semimajor axis). I was surprised when the simulation turned chaotic. The trouble started from the outside and worked inward.


That gravity simulator does not show anything. Inherently, numerical modeling of multibody systems is very very difficult (the so called 3-body problem), and you cannot do anything serious with a programme just downloaded from the web. It is to be expected that it turns chaotic. I am not surprised.

You just keep on playing with toys, and don't have the foggiest what you are doing with them. You can have fun with them I suppose.

tusenfem
2007-Nov-26, 05:35 PM
How do you know that Planet V isn't where I say it is? Theortical calculations suggest there should be a planet at 2 AU, and the periodograms of Fischer et al. (2007, Fig. 3-7) show a recalcitrant peak in the predicted location (did you bother to look at their printouts?). Both the combination of theory and empirical evidence show there is a Planet V at 2 AU.


The calculations say that there could be a planet not should (see page 11 on all the specifics that a possible planet will have to have in order not to be detected. They do not say that there is one.

They also say that:
Several explanations for the modest 6.74 m s−1 scatter in the residuals are possible. Perhaps we are underestimating our internal errors. Perhaps the jitter for this metal-rich star is somewhat higher than the average for G8 main-sequence stars. Or perhaps there are other planets that cause a sufficiently low signal that they are not apparent in the periodograms but nonetheless add a few m s−1 of “noise” to the velocities.

So there are at least 3 explanations for the modest scatter they observe.

And I guess you are going to call TB in our solar system the "emperical evidence"?

and then they also say at the end of section 5
For orbital periods of 300 - 850 d, a 6th planet with Msini below 50 MEarth would have eluded detection as the periodogram peaks would not have loomed even 50% above the noise. For periods 850 d -1500 d, a 6th planet could avoid detection by having M sin i below 100 MEarth. For periods
1750 - 4000 d, planets below 250 MEarth would elude detection.

So, there are various period windows in which the 6th planet could be, nothing there about "there has to be a planet at 2 AU".

Warren Platts
2007-Nov-26, 05:59 PM
That gravity simulator does not show anything. Inherently, numerical modeling of multibody systems is very very difficult (the so called 3-body problem), and you cannot do anything serious with a programme just downloaded from the web. It is to be expected that it turns chaotic. I am not surprised.
Numerical modeling of multibody systems is (relatively) easy. It's the analytic solutions to such problems that are impossible to do. Only one of the three simulations I ran turned chaotic. The other two were perfectly stable.


You just keep on playing with toys, and don't have the foggiest what you are doing with them. You can have fun with them I suppose.

GravitySimulator (http://www.orbitsimulator.com/gravity/articles/what.html) is a great program. Although it is fun and easy to use, it is no mere toy. Tony Dunn, the designer, regularly posts in this forum. The math he uses has been discussed here. It is free, and it is a must-have for any amateur astronomy buff. You should check it out. :)

Warren Platts
2007-Nov-26, 06:43 PM
From page 11:


Thus such planets could exist around 55 Cnc and yet have avoided detection by our current 18 years of Doppler measurements. Indeed several such planets could exist in the large gap between periods of 260 d and 13 yr and probably maintain dynamical stability.

So, it's quite clear that Fischer, and her 11 other colleagues view the presence of more planets to be quite plausible. Thus, it's striking that when you do look at the peaks between 260 d and 13 yr, the best one is at 1085 days--right where the TBL says it should be.

Warren Platts
2007-Nov-27, 04:05 PM
The least-squares method I've been using lately to calculate predicted planetary positions is useful because it simplifies the statistical analysis used to determine statistical significance. However, it generates formulas of the form:




ai = e(a+bi)

Where ai is the semimajor axis of the ith planet, and a and b are the y-intercept and slope of the least-squares regresstion line. For 55 Cancri, a = -4.2677, and b = 0.99898.
This equation generates good results, but just looking at it, it's hard to judge the physical significance of the numbers. Thus it would be desirable to convert the formula back into the "natural" formulation that I posted a while back:




ai = a1K(i-1)

where K is the TBL scaling factor characteristic of that solar system and a1 is the semimajor axis of the first planet. Thus, one can compare the scaling factor of different solar systems at a glance, and one can tell the distance to the first planet.

So, we can do a little algebra:




e(a+bi) = ea(eb)i = eaeb(eb)(i-1) = e(a+b)(eb)(i-1) = a1K(i-1)

In other words, a1 = e(a+b), and K = eb. So, the complete equation for 55 Cancri is as follows:




ai = 0.038(2.1755)(i-1)

Warren Platts
2007-Nov-27, 07:43 PM
The first step, of course, was to see if the predicted Planet V is in a stable orbit--and unsurprisingly, it was. The obvious logarirthmic layout on the screen is too beautiful to be false.

I then investigated the minimum orbital spacings for 55 Cancri. Fisher et al. (2007) (http://exoplanets.org/55cnc5th.pdf), (p. 11) stated


For orbital periods of 300 - 850 d, a 6th planet with Msini below 50 MEarth would have eluded detection as the periodogram peaks would not have loomed even 50% above the noise. For periods 850 d - 1500 d, a 6th planet could avoid detection by having M sin i below 100 MEarth. For periods 1750 - 4000 d, planets below 250 MEarth would elude detection. Thus such planets could exist around 55 Cnc and yet have avoided detection by our current 18 years of Doppler measurements. Indeed several such planets could exist in the large gap between periods of 260 d and 13 yr and probably maintain dynamical stability.

So I calculated the semimajor axes associated with periods of 850 days and 4000 days (0.859 AU, and 4.828 AU, respectively). I found that orbits at both 0.859, 4.828, 4.5, and 4.25 AU were unstable, but that orbits at 1.0 and 4.0 AU were stable. Thus, the maximal minimum spacing are about 0.22 AU for 55 Cnc f (IV), and -1.8 AU for 55 Cnc d (VI), so that an alleyway from 1 to 4 AU is available for extra planets.

Warren Platts
2007-Nov-27, 09:47 PM
A comment on a point that got lost in the excitement.



Originally Posted by Warren Platts
To see if the TBL is a general pattern will require data from several star systems. But to do that, we need a method and a criterion for scoring planetary systems into TBL versus non-TBL spacings.

No, that's never going to work. You seem to be treating this as if exponential spacing is an all-or-nothing phenomenon. But it's not going to be like that: it isn't in the solar system, it isn't in 55 Cnc. What we find are patterns that look a bit exponential, but not perfectly and with many exceptions. The more data of this kind we collect, the more an exponential pattern will emerge from the noise, or disappear into the noise.

I think everyone here is pointing out, in different ways, that the five data points of 55 Cnc are insufficient to make a statement either way with any confidence.
Is the TBL an all-or-nothing phenomenon?

Well, that depends on how "all-or-nothing" is taken. There are several ways to take it.


1. An individual solar system is either logarthmically spaced or not. Is this my position? No. No solar system will ever exhibit a spacing with an r2 of 1.000000000. Nevertheless, there are those systems that approach the logarithmic, platonic ideal to a statistically surprising degree. So we decide on a "level of significance", say 95%, and then say that if the spacing pattern has an r2 > 0.9930, then that solar system exhibits the TBL pattern at the 95% confidence level. Thus, when I say "55 Cancri has a TBL pattern", this must be understood as shorthand for "the major planets at 55 Cancri exhibit the TBL spacing pattern at a 95% confidence level." Thus, the TBL classification is a human construct--there is no magic difference between an r2 = 0.9930 (95th percentile) and an r2 = 0.9920 (94th percentile); on the other hand, making classifications on the basis of statistical criteria are routine in science--that's how things get done. It would be unfair to hold the TBL theory to some higher absolute criterion.

2. All solar systems either have the TBL spacing, or none of them do. Again, this is not my position. Thus, what would it mean to say that "Solar systems in general tend to exhibit a TBL pattern." Well, that just means the same as "Most solar systems have a TBL pattern", or more strictly speaking, "Most solar systems exhibit the TBL pattern at the 95% confidence level". So how would one go about establishing or disestablishing such a statement? Well, one looks at the 50 different systems for which we have good data on at least 5 major planets, then we figure the r2 of the best logarithmic model for each system, and count the number of systems with r2 scores in the 95th percentile, and then divide by the total number of systems. This will give us a percentage of systems with a TBL pattern (strictly speaking this will give us a percentage of systems that have a TBL-like spacing at the 95% confidence level). Then all we have to do is argue over the percentage that will count as "most". Is 51% good enough, or should we make in an even 2/3 majority. In practice, people will just write down the percentage, and leave it at that.

So, I'm not taking the TBL relation to be an all-or-nothing phenomenon. Indeed, to say that "The more data of this kind we collect, the more an exponential pattern will emerge from the noise, or disappear into the noise"--that is an all-or-nothing statement. The fact is that what will emerge--indeed what we now have--is a percentage of systems for which we have data that have the TBL pattern to a certain level of confidence. That percentage will be either high or low. If the percentage turns out to be low, does that mean that something disappeared into the noise? Absolutely not. The percentage is what it is. It's a real pattern unto itself. Just because it's low doesn't entail that anything was lost to the noise; on the contrary, the low percentage emerged from the noise. Moreover, even if the next 99 systems did not have the TBL pattern, that would leave the TBL pattern of 55 Cancri itself untouched. 55 Cancri has its own spacing pattern, it's real, and it does not depend on other solar systems.

Do I believe that most extrasolar systems have the TBL spacing pattern? No. Probably, most solar system do not have the TBL spacing. There are all kinds a crazy systems out there. Systems with low metallicity may not have planets at all, and systems that have undergone major gravitational perturbations would not be expected to have the TBL pattern. For example, a solar system with a Jupiter-sized planet that has a comet-sized eccentricity would not be a good candidate for having the TBL spacing pattern.

Do I believe that most systems that formed from well-behaved primordial disks with relatively high metallicity, that have more-or-less circular orbits, and no history of major gravitation perturbations have the TBL spacing pattern? Absolutely! On one hand, there are the a priori theoretical considerations that planets form within logarithmically spaced "feeding zones" based on characteristic scale of vorticity. On the other hand, there is the empirical evidence for the two such well-behaved systems for which we have good data--the Sol system and 55 Cancri--and both systems exhibit the TBL pattern. Thus, it's likely that the TBL is the general pattern for systems meeting such rigid criteria.

tusenfem
2007-Nov-28, 09:30 AM
So, it's quite clear that Fischer, and her 11 other colleagues view the presence of more planets to be quite plausible. Thus, it's striking that when you do look at the peaks between 260 d and 13 yr, the best one is at 1085 days--right where the TBL says it should be.

Yes, they say it is a possibility, but your special planet is only 1 of 5 possibilities, which are:

1. underestimating our internal errors
2. jitter for this metal-rich star is somewhat higher than the average for G8 main-sequence stars
3. other planets that cause a sufficiently low signal that they are not apparent in the periodograms but nonetheless add a few m s−1 of “noise” to the velocities

And for number 3 they give 3 possibilities:
1. orbital periods of 300 - 850 d, a 6th planet with Msini below 50 MEarth
2. periods 850 d - 1500 d, a 6th planet could avoid detection by having M sin i below 100 MEarth
3. periods 1750 - 4000 d, planets below 250 MEarth would elude detection

So the probability of your planet is 20%

Warren Platts
2007-Nov-28, 11:51 AM
Yes, they say it is a possibility, but your special planet is only 1 of 5 possibilities, which are:

1. underestimating our internal errors
2. jitter for this metal-rich star is somewhat higher than the average for G8 main-sequence stars
3. other planets that cause a sufficiently low signal that they are not apparent in the periodograms but nonetheless add a few m s−1 of “noise” to the velocities

And for number 3 they give 3 possibilities:
1. orbital periods of 300 - 850 d, a 6th planet with Msini below 50 MEarth
2. periods 850 d - 1500 d, a 6th planet could avoid detection by having M sin i below 100 MEarth
3. periods 1750 - 4000 d, planets below 250 MEarth would elude detection

So the probability of your planet is 20%
Actually, the range where possible new planets might be found is probably less than that suggested by Fischer et al. (2007) (http://exoplanets.org/55cnc5th.pdf) According to my numerical simulations, the range for stable orbits runs from about 375 to about 3,000 days, rather 300 to 4,000 days. Note that my simulation was done using the minimum mass estimates not taking into account that the "derived orbital inclination is i = 53±6.8 deg (37 deg from edge-on) for that outer planet, implying that its actual mass is 4.9 MJup." (p. 18) If these heavier masses were taken into account, that might narrow the range a little more.

But still, that leaves plenty of room. Fischer et al. state that "Indeed several such planets could exist in the large gap between periods of 260 d and 13 yr and probably maintain dynamical stability." (p. 11) Thus, I take it from comments such as these that Fischer and her 11 colleagues consider the possibility of a sixth planet to be likely.

Indeed, I should note that Fischer et al.'s search for the fifth planet was motivated by reasoning as did old Titius himself, that the Author of the Universe would not leave such empty slots unfilled.

Warren Platts
2007-Nov-28, 04:38 PM
I reran the program this time to create 100,000,000 random solar systems. The data that follows is from an algorithm that only explored the {I, II, III, IV, ___, VI} model, rather than choosing the best r2 out of all possible 5-planet/6-orbital slot models

The minimum r2 was 0.409.
The median r2 was 0.8154

The peak r2 of the negatively skewed distribution was 0.861 (64th percentile)

Confidence levels:

95% r2 ≥ 0.9661
99% r2 ≥ 0.9873
99.9% r2 ≥ 0.9965

(The r2 for 55 Cancri was 0.9975.)

_________________________________________________
0.408 0 0
0.409 5 5
0.410 33 38
0.411 104 142
0.412 181 323
0.413 301 624
0.414 449 1,073
0.415 611 1,684
0.416 735 2,419
0.417 935 3,354
0.418 1149 4,503
0.419 1424 5,927
0.420 1686 7,613
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0.998 45185 99,975,400
0.999 24600 100,000,000

Warren Platts
2007-Nov-29, 07:06 PM
In an earlier post, grant hutchison had mentioned that planetary resonances might be a confounding factor for TBL theory, in that resonances might either cause or disrupt TBL patterns. However, I don't think that such resonances are important for the following reasons.

In our own solar systems, there are no important resonances that could dramatically cause evolution of semimajor axes. There is the rotational resonance of Mercury with its orbit around the Sun, and there is the possible rotational resonance of Venus with the Earth; but there is little reason to think that either resonance has had much effect on their semimajor axes.

Then there is the orbital resonance of Neptune and Pluto. However, since Pluto is not a major planet, it is irrelevant for TBL; and there is in any case no reason to think that Pluto has had a major influence on the orbital distance of Neptune.

Then there are all sorts of resonances involving asteroids and moons all over the solar system. But again, since TBL is concerned with major planets such resonances are irrelevant in themselves, and there is little reason to think that such asteroids and moons have had substantive effects on the orbits of the major planets.

As for 55 Cancri, the second and third planets are almost in an orbital resonance. 55 Cnc b and 55 Cnc c have orbital periods of 14.65 and 44.34 days respectively, with a ratio of 3.026. Fischer et al. (2007, 12) (http://exoplanets.org/55cnc5th.pdf) looked this potential resonance, but concluded it was not significant:


It is interesting to note that during the course of the numerical integration, the 3:1 resonant arguments for planets b and c are all circulating. This indicates that planets ”b” and ”c” do not currently participate in a low-order mean motion resonance, despite the near commensurability of their orbital periods.

However, it is also interesting to note that both "b" and "c" have the largest deviations from the primitive orbits predicted by the TBL; to wit, they have apparently moved closer together; so, one is tempted to wonder whether the potential 3:1 resonance might have had something to do with it. If "b" and "c" were in their predicted orbits, however, their orbital periods would be 12.5 and 56 days, respectively, having a ratio of 4.5:1. Thus, as the planets moved towards each other, they would have had to pass through a 4:1 resonance, and if such resonances are stable and are able to control orbital distances, then "b" and "c" should have stopped migrating once the 4:1 resonance was achieved. Therefore, if "b" and "c" have in fact migrated since they were first formed, it is probably not due to orbital resonances.

Fischer et al. also note that mean-motion resonances could effectively hide a sixth planet. Happily, however, the orbital period for the predicted location for Planet V at about 2 AU (1,122 days) is in a 4.6:1 ratio with its outer neighbor and a 4.3:1 ratio with its inner neighbor. Therefore, resonances will not hinder the eventual unequivocal detection of Planet V.

snowflakeuniverse
2007-Nov-30, 06:24 PM
Bodes Law.

First, Compliments

Warren Platts you have posted an interesting post and you are doing well. The criticisms of what you have posted fall into either of two categories, meaningless comments or wonderfully helpful with insight.

Second, Order

I believe that Bodes Law reveals an underlying order to which the formation of planetary systems conforms. The argument that the law is simply the result of coincidence is becoming weaker with additional discoveries of extra planetary systems.

Third, Space around Atoms

I believe that the underlying explanation for planetary order is similar to the same reason the space around an atom results in the patterns to which electrons probabilistically reside. Since there is a pattern around atoms, it would seem logical to expect some kind of pattern exhibited around gravitationally bound systems, especially if one believes in a Unified Field Theory.

Forth, Predictions of Bodes Law
http://en.wikipedia.org/wiki/Ceres_%28dwarf_planet%29
Bodes law made predictions as to the locations of two planets, Uranus and the “dwarf” planet Ceres, before they were discovered.

Fifth, Criticisms and Justifications of the Bode Titus Law

a. Empirically or observationally established.

One of the criticisms, or reasons used to dismiss the Bode – Titus law is that the formulas are adjusted to fit what is observed and it is thus an empirical law, rather than a “pure” law based on theoretical constructs. As pointed out in previous postings, Newton’s Law of gravity is technically an empirical law and we have accepted as permissible to refer to Newton’s Law of Gravity, despite the lack of theoretical construct; an issue that is addressed in General Relativity.

b. Ignoring is ignorance

Ignoring the potential significance of the empirical Bode –Titus Law may short circuit any advancement in understanding the evolutionary development of planetary systems, just as ignoring Newtons empirical law of gravity would have failed to produce General Relativity.

c. Lack of Accuracy

Another criticism of the Bode – Titus Law is that it is not entirely accurate or consistent; variations in the empirical values vary for the “inner” planets, and the “outer” planets, with the gaseous giant planet of Jupiter establishing the inner most “outer” planet. Also Uranus falls off the curve a bit, and the outer most dwarf planets are not even close. Ceres is also way too small, a fraction of the size of our own moon.

d. Chaotic beginning

These deviations are, I believe, due to the chaotic early evolution of the solar system. Given the premise of the Nebular Cloud hypothesis describing the beginning of the solar system, something would have to “whack” or dramatically stir up the constituent matter existing in our solar system before the planets reached any significant size. This kind of disturbance could explain how Uranus has such a tipped axis of rotation, and an impact on what would have become the planet Ceres could explain its diminutive size due to the interruption and ensuing delay in the gathering of sufficient mass to become a full fledged planet.

This explosive era in the early evolution of our solar system is also evidenced by other observations. I have proposed an explanation for this but I do not want to digress to my own work too much.

This chaotic beginning of our planetary system is the reason deviations from the Bode –Titus Law is observed.


Sixth, Conclusion

The Bode Titus Law is an Empirical Law, and it provides valuable information about the early formation of our Solar System.

Snowflake aka John Kulick

Hornblower
2007-Nov-30, 10:46 PM
Bodes law made predictions as to the locations of two planets, Uranus and the “dwarf” planet Ceres, before they were discovered.
And it fell flat on its face for Neptune.

R.A.F.
2007-Nov-30, 10:53 PM
The Bode Titus Law is an Empirical Law, and it provides valuable information about the early formation of our Solar System.

Then you should have little difficulty discribing the mechanism involved.

Van Rijn
2007-Nov-30, 11:06 PM
And it fell flat on its face for Neptune.

Oh, but he has a fix for that - a "two piece" equation. :)

snowflakeuniverse
2007-Dec-01, 01:56 AM
Hi Hornblower and Van Rijn

You both point out the nonconformance of Neptune to the Bode Titus Law. I also mentioned that the dwarf planets found outside of Neptune do not conform to the B-T Law.

However, this criticism does not invalidate the model. It indicates that the early formation of the solar system was chaotic. (see note d in post 190).

Snowflake

snowflakeuniverse
2007-Dec-01, 01:57 AM
Hi R.A.F

When I stated
“The Bode Titus Law is an Empirical Law, and it provides valuable information about the early formation of our Solar System.”

You responded with.

“Then you should have little difficulty describing the mechanism involved.”

Why do you assume I would have little difficulty describing the mechanism?

Newton saw order in his gravitational law, but he did not know that mechanism that connected the Earth and Moon together. I see order revealed in the BT Law, don’t you?

(I could hypothesize mechanisms, but that would result in proposing my ideas on another person’s posting.)

Snowflake

Van Rijn
2007-Dec-01, 02:04 AM
Hi Hornblower and Van Rijn

You both point out the nonconformance of Neptune to the Bode Titus Law. I also mentioned that the dwarf planets found outside of Neptune do not conform to the B-T Law.

However, this criticism does not invalidate the model. It indicates that the early formation of the solar system was chaotic. (see note d in post 190).

Snowflake

I wasn't aware there was a model. I just see an equation that is fit to the data.

jkmccrann
2007-Dec-01, 07:42 AM
This is simply NOT TRUE. I know this is ATM, but Please try to stick to the facts of the matter. It's coincidence plus bending the evidence to suit a conclusion, and unless/until you can explain the mechanism behind bodes "law", it will remain that.



How is rehashing the debunked "bodes law" going to help you understand the history of the Solar System?

Just on this last point - how can "bodes law" have been comprehensively debunked when we only have one reference Solar System to test the theory against?

That seems to me to be jumping to a hasty conclusion! Wouldn't we need multiple data sets to reach a scientific consensus on the merits or otherwise of "Bodes Law?"

jkmccrann
2007-Dec-01, 09:11 AM
And it fell flat on its face for Neptune.

Are you arguing that our Solar System is static?

That the orbits we observe today will be the orbits in place for the next 3 billion years? (Or whatever number you would like to plug in there)
(Absent the desturctive perturbations that would occur if a rogue star were to career through our neck of the woods and throw all these sorts of discussions into complete irrelevance)

I would ask a few questions of you (and anyone else) to back up your statement that TB fell flat on it face in regards to Neptune.

For how many years has Neptune been in its current orbit?
(Scientific Proof thankyou)

If it is accepted that Neptune has in the past experienced periods of planetary migration.

When did Neptune's planetary migration stop?
(Scientific Proof thankyou)

Can you prove that in fact Neptune's planetary migration has stopped?
(Scientific Proof thankyou)

Can you provide proof that Neptune will not in fact experience planetary migration in future.
(Scientific Proof thankyou)

If there is a prospect for Neptune to experience further planetary migration in future then that completely and utterly refutes your claim that the current snapshot location of Neptune debunks TB.

I would like to see reference to scientific proof on these matters because simply saying, Neptune's current location disproves TB implicitly suggests that the orbits of the major planets is static.

Is not TB supposed to be describing a static relationship between the major planets?

Thus, if we actually live in a star system that is still experiencing levels of dynamism in terms of the orbits of the major planets - then it is impossible to use the current locations as proof or disproof of TB's validity.


Please provide links to Scientific articles regarding the status of Neptune's planetary migration. I am keen to read the proof of your argument rather than just taking at face value your implicit assertion that planetary migration in our Solar System has well and truly stopped.

jkmccrann
2007-Dec-01, 09:27 AM
You could call them laws of motion because they have been tested many times, in many ways, in many venues and always - always - give the same results. T-B has been "tested" only on one example, our solar system, and has not held up that well. (The asteroid belt and the outer planets don't fit all that closely.) 55Cancri is an incomplete test as we don't have all the information on that system. Test T-B a bit more and we'll see if it holds up; then you can begin to think of calling it a true scientific law.

Also, the laws of motion can be explained scientifically; T-B cannot.

T-B is an empirical observation and an unexplained relationship, not a law.

Using the asteroid belt to disprove TB is a red herring. No one has ever suggested that TB should be used to explain the minutiae of any given star system.

R.A.F.
2007-Dec-01, 01:06 PM
I see order revealed in the BT Law, don’t you?

If you had bothered to read ANY of my previous posts on the subject you would KNOW what my opinion is...


Can you prove that...snip...Can you provide proof...

Why in the heck would you think that the burden of proof is on anyone to prove T-B wrong??? Why don't you show us the evidence that proves T-B
right???

The best thing about this thread is that it will be locked in less than 2 weeks. :)

jkmccrann
2007-Dec-01, 03:01 PM
If you had bothered to read ANY of my previous posts on the subject you would KNOW what my opinion is...



Why in the heck would you think that the burden of proof is on anyone to prove T-B wrong??? Why don't you show us the evidence that proves T-B
right???

The best thing about this thread is that it will be locked in less than 2 weeks. :)

The burden of proof is on everyone in relation to TB. But it can't conclusively be proved either way at the moment. Isn't that obvious?!?

If TB is such a bad theory, and I grant that it may well be, can you please show me the theory that explains how planetary systems form, accrete matter to various bodies throughout the dust cloud, migrate to various positions in relation to the central star (forgetting about binary systems which are obviously inherently more unstable than our own in terms of planetary formation) and then settle into whatever positions they eventually settle into?

I doubt you can provide this scientific proof - because I don't believe it exists in its entirety! I realise much of those processes has been worked out and are generally accepted, and I definitely accept this consensus - but I don't believe there is a consensus viewpoint on whether bodies in a system will end up in a particular configuration once you start factoring in all the variables a system will have. I believe this is a contested area at present - mainly owing to a lack of information to feed into possible models!

Sure, proponents of the theory need to clearly enunciate where they're coming from, and just FYI, I am not a proponent of TB, but I do find it intriguing, I also find planetary formation and subsequent accretion and migration very interesting - and particularly how they relate to our own Solar System.

The number of variables that could and would impact on planetary formation is mind-boggling - which I believe presents a huge problem in classifying any particular star system as "normal," and as such with regard to TB - it could be that it may apply to only a very small sample of star systems - effectively making it a useless kind of theory/law - but what if systems that turned out to fit into some sort of TB model had an enhanced chance of living organisms appearing in these systems?

Pure speculation, but its impossible to refute at the moment.

What I do wonder about all these extra-solar systems that we've observed is - I think to properly classify them in terms of planetary formation - one has to really mention how old the system is believed to be - and whether it is thought the system is still undergoing substantial planetary migration (if not accretion!) - I tend to think that when I hear about how many star systems have these large gas giants much closer to their stars - and how unusual that makes our own Solar System - well does it really?

How old are these systems? Why does that make our Solar System the least bit unusual given that in the past the gas giants were themselves much closer to the Sun many millions and billions of years ago. It really is quite annoying that when talking of extra-solar systems they are rarely time-stamped in terms of their age - because it makes comparisons with our own pretty useless if the time scales are incomparable.

Which gets me back to TB in relation to our own system. I have the understanding that the TB theory was put out there with the belief of Titius and Bode etc. that our Solar System was static at the time, and not evolving - and planetary migration was not something they factored into their theory.

If that is not the case, can someone please let me know - because it is important given what we have learnt of planetary migration in recent years.

If there is a chance that over the next 50 million years, or 100 million years or 250 million years that there will be further planetary migration within our own Solar System (particularly in relation to Neptune (when discussing TB) and absent external factors such as passing stars) - then I don't believe you can invalidate TB's application in relation to our own Solar System - because they're "measuring" 2 different things!

Some more speculation, but when they came up with this theory - Neptune was an undiscovered body - but had they come up with this theory at some stage after Neptune's discovery - do you think it is possible that the theory of planetary migration might have been advanced by TB proponents at the time to explain the fact that it was in the "wrong place"?

Could this have led to theorising on planetary migration fully a century before it was extensively theorised about? I believe it is more than possible that could have happened.

The bottom line is - we do not have enough information at the moment for scientific rigour to be applied in this area with certainty - and until we do it is enjoyable to speculate about this sort of thing, I don't see anything wrong with doing that really - its one way to test our boundaries and come up with new explanations for things - and in that sense there is nothing wrong with refining a theory as more information becomes available.

Warren Platts
2007-Dec-01, 03:23 PM
The best thing about this thread is that it will be locked in less than 2 weeks. :)

I may have to ask for an extension on this one; there is still a lot to figure out.

Warren Platts
2007-Dec-01, 06:05 PM
I was thinking that maybe I did the test statistic wrongly. With 5 planets and 6 potential orbital slots, there are 6 different ways to arrange the 5 planets:


{_____, II, III, IV, V, VI}
{I, _____, III, IV, V, VI}
{I, II, _____, IV, V, VI}
{I, II, III, _____, V, VI}
{I, II, III, IV, _____, VI}
{I, II, III, IV, V, _____}


So I had my program go through each randomly spaced solar system and generate an r2 for each possible arrangement, and then return the best r2 for addition to the histogram against which the test statistic is tested. However, for 55 Cancri, we are only interested in the fifth possibility: {I, II, III, IV, _____, VI}. By working through all 6 possibilities, that generates a bias by inflating the r2's to higher values than they would be if only the fifth possibility were checked.

So I redid the analysis. This time it only took 58 minutes. Taking out the loop that calculated the r2 for all 6 possibilities really speeded it up; and since my computer has two processors, I set up two programs that each generated 50,000,000 random solar systems, and so each processor could be dedicated to each separate program and finish the task in half the time. The data were then combined in an Xcel spreadsheet.

Results (figures are the r2):

Total random solar system: 100,000,000

Minimum: 0.413
Median: 0.9093
Peak: 0.965 (85th %-tile)
Confidence levels:

80%: 0.9577
90%: 0.9722
95%: 0.9816
99%: 0.9929

Note that the r2 for 55 Cancri (0.9975) isn't affected: it's still in the 99th %-tile. However, I calculated the r2 for the analogous situation in the Sol system (Mercury through Jupiter, skipping the Asteroid Belt) and got an r2 of 0.9883, which is at the 92nd %-tile under the first test statistic, but at the 97.5th %-tile under the one I just figured up.

I think I should go with the latter statistic.

Any thoughts?
____________________________________
EDIT:
I just looked at my program, and it was set to check models of the form {I, _____, III, IV, V, VI}. So the above data are for that model. I reran the programs to generate the correct {I, II, III, IV, _____, VI} histogram. Here are the results:

Total random solar system: 100,000,000

Minimum: 0.409
Median: 0.8158
Peak: 0.864 (65th %-tile)
Confidence levels:

80%: 0.9089
90%: 0.9459
95%: 0.9670
99%: 0.9880

Note that this (the correct formulation) puts the Mercury through Jupiter model in the 99th %-tile.

:o

R.A.F.
2007-Dec-01, 06:17 PM
If TB is such a bad theory...

TB isn't a theory at all or else there would be some discernable mechanism involved. All that has been presented here is an "interesting" coincidence, nothing more.


I may have to ask for an extension on this one; there is still a lot to figure out.

That's really too bad, and demonstrates the need to "iron" the difficulities out of ATM ideas before starting discussions.

Warren Platts
2007-Dec-01, 06:27 PM
Which gets me back to TB in relation to our own system. I have the understanding that the TB theory was put out there with the belief of Titius and Bode etc. that our Solar System was static at the time, and not evolving - and planetary migration was not something they factored into their theory.

. . .

Some more speculation, but when they came up with this theory - Neptune was an undiscovered body - but had they come up with this theory at some stage after Neptune's discovery - do you think it is possible that the theory of planetary migration might have been advanced by TB proponents at the time to explain the fact that it was in the "wrong place"?

Could this have led to theorising on planetary migration fully a century before it was extensively theorised about? I believe it is more than possible that could have happened.
That's a very good point. TBL generated expectations about where Neptune should have been, and when it wasn't there, that motivated research into possible reasons why.

BTW, 55 Cancri is about the same age as our solar system (http://en.wikipedia.org/wiki/55_Cancri): 4.5 billion years old.

Warren Platts
2007-Dec-01, 06:31 PM
That's really too bad, and demonstrates the need to "iron" the difficulities out of ATM ideas before starting discussions.

Sorry, I didn't realize you were the new moderator. You might want to update your personal profile to reflect that fact. :)

R.A.F.
2007-Dec-01, 06:40 PM
Sorry, I didn't realize you were the new moderator. You might want to update your personal profile to reflect that fact. :)

Instead of wasting time being "snippy", you should be working on your "theory".

I'm glad I'm not a mod...it's much easier just to use the little red button. :)

tusenfem
2007-Dec-01, 09:02 PM
Are you arguing that our Solar System is static?

That the orbits we observe today will be the orbits in place for the next 3 billion years? (Or whatever number you would like to plug in there)
(Absent the desturctive perturbations that would occur if a rogue star were to career through our neck of the woods and throw all these sorts of discussions into complete irrelevance)

I would ask a few questions of you (and anyone else) to back up your statement that TB fell flat on it face in regards to Neptune.

For how many years has Neptune been in its current orbit?
(Scientific Proof thankyou)


In that case, could you please prove that Mercury - ... - Uranus are in stable orbits and if they have not shifted and maybe still are shifting and that the whole TB thingamabob is just a chance occurrence that will be gone in 1 million years.

Thank you.

Warren Platts
2007-Dec-01, 09:50 PM
In that case, could you please prove that Mercury - Uranus are in stable orbits and [that] they have not shifted maybe [they] still are shifting and that the whole TB thingamabob is just a chance occurrence that will be gone in 1 million years.

Thank you.
The fact that planets don't move around much is proven by good old-fashioned anthropic reasoning:


[B]If planets moved around much, then we wouldn't be here to talk about it.

You're welcome.

Hornblower
2007-Dec-01, 11:25 PM
Are you arguing that our Solar System is static?

That the orbits we observe today will be the orbits in place for the next 3 billion years? (Or whatever number you would like to plug in there)
(Absent the desturctive perturbations that would occur if a rogue star were to career through our neck of the woods and throw all these sorts of discussions into complete irrelevance)It appears that you read between the lines of my one-liner and jumped to some amazing conclusions about what I might be thinking.


I would ask a few questions of you (and anyone else) to back up your statement that TB fell flat on it face in regards to Neptune.I will back up my statement with simple arithmetic. Neptune is about 22% short of the radius predicted by the TB sequence, while the others are within 5%. I never expressed an opinion one way or the other about possible physical significance of these facts.


For how many years has Neptune been in its current orbit?
(Scientific Proof thankyou)

If it is accepted that Neptune has in the past experienced periods of planetary migration.

When did Neptune's planetary migration stop?
(Scientific Proof thankyou)

Can you prove that in fact Neptune's planetary migration has stopped?
(Scientific Proof thankyou)

Can you provide proof that Neptune will not in fact experience planetary migration in future.
(Scientific Proof thankyou)

If there is a prospect for Neptune to experience further planetary migration in future then that completely and utterly refutes your claim that the current snapshot location of Neptune debunks TB.

I would like to see reference to scientific proof on these matters because simply saying, Neptune's current location disproves TB implicitly suggests that the orbits of the major planets is static.

Is not TB supposed to be describing a static relationship between the major planets?

Thus, if we actually live in a star system that is still experiencing levels of dynamism in terms of the orbits of the major planets - then it is impossible to use the current locations as proof or disproof of TB's validity.


Please provide links to Scientific articles regarding the status of Neptune's planetary migration. I am keen to read the proof of your argument rather than just taking at face value your implicit assertion that planetary migration in our Solar System has well and truly stopped.
If you know of any peer-reviewed publications of computer simulations that indicate ongoing orbital migration, please show us a link or otherwise identify them. I have never heard of any such thing in the absence of massive swarms of leftover planetesimals with which the planets could exchange orbital energy. The survival of Earth's biosphere for upwards of the past billion years appears to be to be strong evidence against such migration, at least for the four inner planets. I think the burden of proof is on you to convince us that Neptune might be behaving otherwise.

In closing, let me recommend that we stick to discussion of Warren's line of thought. If I am not mistaken he has moved beyond adherence to the original TB formula and is analyzing generalized sequences of which the power series component of TB is a special case.

tusenfem
2007-Dec-02, 11:21 AM
The fact that planets don't move around much is proven by good old-fashioned anthropic reasoning:


If planets moved around much, then we wouldn't be here to talk about it.

You're welcome.

Maybe the Earth is special because we are there and the other planets just move around in millions of years, you never know .... Venus was shot out of Jupiter.

Just to make my case clearer I forgot to put on the answer to jkmccrann on.

This thread goes from bad to worse. Ah well, it is the BadAstronomy board.

tony873004
2007-Dec-03, 02:24 AM
...Tony, if you're reading this, now would be a good time to chime in and explain to everyone the difficulties inherent in accurately simulating orbital evolution for billions of years.
Sorry, this is the first time I've visited this thread. I'm not a big fan of TBL, so I never ventured in here. But my web server log told me I was getting hits from this thread, so I figured I'd have a look.


...just based on my own fooling around with tony873004's GravitySimulator computer program: when orbits get in a stable configuration, they just stay there pretty much...

That's what I used to think. But last year I took Astro 405, "Astrobiology" from Debra Fisher (author of the Cancri paper). She thought Gravity Simulator was neat so she let me demonstrate it in class to show how the solar system barycenter moves. She then gave me a challenge. She wanted to know what would happen if Venus suddenly had Jupiter's mass. I tried it, and after many hours of letting the simulation run, everything looked completely undisturbed. So I pumped Venus' mass up to 2 Jupiter masses. Again, after many hours of letting the simulation run, everything looked completely undisturbed. I concluded it was stable. But I went to bed without turning off the computer, and when I woke up in the morning, Mars was missing. This surprised me. For 10s of thousands of years, Mars had no noticable change in its orbit, and then suddenly it is missing. Why Mars? Why not Earth? It's closer to Heavy Venus.

So I repeated the simulation several times, this time using Auto Save, so I could witness the ejection event. Mars was happily orbiting the Sun for 10s of thousands of years, and one day, its orbit was just slightly more eccentric than normal, and orbit after orbit it grew. The ejection was fast. If I remember correctly, only a few hundred years seperated Mars from being in an orbit virtually unchanged from its starting condition to being sent by Jupiter on a hyperbolic trajectory.

But different starting conditions, as well as different time steps yield different results. Simply increasing Venus' mass in a different part of its orbit affects when Mars gets ejected. So this is where it's important to realize what part of the results of the simulation are useful, and what parts are not. Although the simulator will give you an exact date (i.e. Mars was ejected on April 22, 138582), this changes from run to run when the initial conditions are tinkered with, or when the time step is changed. But in each simulation there was a trend. Mars lasted anywhere from 50,000 years to 150,000 years before getting ejected. So the new conclusion that I drew was that Mars is unstable in this configuration, and every year Mars is rolling the dice, taking a 1 out of ~100,000 chance that this is the year that a certain alignment will quickly put it on an ejection trajectory.


...As for 55 Cancri, the second and third planets are almost in an orbital resonance.
It's interesting to note that the Sun's inner solar system is filled with near resonances too: Venus:Earth = ~7:5, Mercury:Venus=~2:5 Mercury: Earth~4:1, Earth:Mars ~15:8, Mercury:Mars ~8:1. Although the 3:1 resonance you're referring to in the Cancri55 system is a bit closer to a true 3:1 than the examples from our inner solar system, it makes me wonder if these near-resonances are common in the universe. And what causes them? Were they once perfect mean motion resonances that let go after the orbits circularized, allowing the ratio to slightly drift? (If so, why did the orbits circularize?) Or could it just be a statistical thing, that any two periods are going to coincidently have a near-integer ratio?


...I'll take Tony's guidance on that, but I doubt you'll have assessed the billion-year stability of a system on a PC in the last 24 hours...
Billions of years are difficult to do without a very high time step. And high timesteps are bad for accuracy. But it does make me wonder. What is more likely? A stable system falls apart due to a high timestep, or an unstable system stays together?




...The first paper off my pile, Planetary migration in a planetesimal disk: why did Neptune stop at 30AU (http://www.boulder.swri.edu/~hal/PDF/migration.pdf), by Gomes et al, seems like it will provide you with an introduction and a starter's reference list, and a broad hint on why Gravity Simulator isn't going to help you with this one...
You can make planets migrate with the Beta version of Gravity Simulator. But you have to tell the planet how to migrate, so like you said, Gravity Simulator won't be of any help here. Technically I guess you could do it by setting up thousands of planetesimals and letting the gas giants perturb them around, but realistically it would probably take you hundreds of years of real time to simulate the few million years you'd need for results.

I haven't read that paper, but I would guess that Neptune stopped at 30 AU because the solar system ran out of significant amounts of small bodies for Neptune to perturb when Neptune was at about 30AU.

This is a little off topic, but Resonant Capture Theory is a fascinating subject. Why are so many TNOs in orbital resonance with Neptune? It's theorized that as Neptune migrated outward, as its exterior resonances swept through the Kuiper Belt, that it captured these objects into resonances. Motovited by Eugene Chaing's paper on Resonant Capture Theory, I performed a simulation where 30 test particles were placed in circular orbits at 20AU +-40%, and Neptune was placed at 10 AU and told to migrate out 30 AU. By 7 million years, every single object was captured into either a 1:2 or a 2:3 resonance with Neptune. As Neptune continued to migrate, the eccentricity of the captured objects grew. Neptune took 12 million years to complete its migration in this simulation (which took nearly 24 hours of computer time, which shows why billions of years is difficult to do). Using a faster migration, Neptune only captured particles in a 2:3 resonance, and faster still, it captured nothing. I would guess that with a slower migration, it may capture objects into some weaker resonances such as 3:5, which are observed in the real solar system, especially if the objects start in slightly eccentric orbits (pre-heated according to Chiang, et. al).

So since the observed populations of resonant objects may be a signature of Neptune's migration, it is possible to use Gravity Simulator to learn something about planetary migration. Although you need to perform simulation after simulation after simulation and spot trends, rather than just rather than draw your conclusions from one simulation.


...and you cannot do anything serious with a programme just downloaded from the web...
There are some serious things you can do with it, but it is very important to understand what it can and can't do. I get very similar results to those of Chiang, et. al. There's another paper by Alessandro Morbidelli and Harold F. Levison where they perform a simulation showing how Sedna could have been captured from a passing brown dwarf. My results were virtually identical to the results they got using the Swift rmvs3 orbit integrator. And the list goes on. There's more simulations and discussion in the forum on my website describing the useful things Gravity Simulator can do. And Gravity Simulator's results for planet stability are very much in agreement with the results from two other methods.

Warren, you might find these methods useful, rather than performing long-term simulations:
Method 1:
There's a website called oklo.org , which is run by Greg Laughlin's team from UC Santa Cruz. Sky & Telescope magazine had a feature article on this website last year. Using its downloadable console, users can play with both hypothetical and real velocity data to attempt to find planets hidden in the data. When a user reports a find, he uploads his new system to their website where they perform a stability check. It's a really quick and dirty method too. They simulate the planet for 100 years, and if its semi-major axis drifts by less than 1%, they consider the planet to be stable. 100 years is fine for the tight orbits they're discovering, but for more distant planets you'd want to simulate longer. This is where Gravity Simulator can help you. In fact, on their old forum board, Greg even endorsed the idea of using Gravity Simulator before their stability checker was in place.

To use Gravity Simulator for this, use the menu File> Output File, to create an Excel-readable data file so you can examine the semi-major axes of your objects. Just use (min-max)/mean on your SMA column to see if your planet violates the 1% rule. Using this method, it exposed Mars as unstable in the "Heavy Venus" simulation I described earlier. But it only took a few minutes to perform the simulation, rather than 24 hours, as I only needed to simulate for about 100 years. It also exposed Earth as unstable in this configuration. Earth's SMA deviation was smaller than Mars' (presumably because Mars is closer to the real Jupiter), which is probably why Earth was never ejected in my simulations. I'm guessing if I ran the "Heavy Venus" simulation longer, that Earth too would be ejected.

Method 2:
On my "formulas" page ( http://www.orbitsimulator.com/formulas/ ) there are two formulas called "interior reach" and "exterior reach", as well as a link to the paper where these formulas came from. These compute analytically the unstable regions caused by a planet. For example, using Jupiter in these formulas shows that anything orbiting between 4.13 AU from the Sun up to Jupiter's position is unstable. And anything orbiting from Jupiter's position out to 6.3 AU is also unstable. I've ran a few simulations that suggest that Gravity Simulator's predictions and these formulas are also in agreement with each other. But these formulas don't account for multiple massive planets. So using them in the "Heavy Venus" simulation shows that Earth and Mars are stable, simply because you can only account for either "Heavy Venus" or "Real Jupiter", while it is their combination that makes Earth and Mars unstable.

Jim
2007-Dec-03, 03:44 AM
Sorry, I didn't realize you were the new moderator. ...

He doesn't need to be; he only needs to be familiar with the BAUT Rules for ATM.

All threads close automatically after 30 days. This is a software function and cannot be overridden for any individual thread.

Jim
2007-Dec-03, 03:59 AM
Using the asteroid belt to disprove TB is a red herring. No one has ever suggested that TB should be used to explain the minutiae of any given star system.

Yet the proponents of the "classical" T-B formula claim that it puts a fifth planet where the asteroid belt is. I don't see how pointing out that the formula isn't terribly precise (geographic center, mass center, something else?) is a red herring.

Jim
2007-Dec-03, 04:06 AM
... I would like to see reference to scientific proof on these matters because simply saying, Neptune's current location disproves TB implicitly suggests that the orbits of the major planets is static.

...

Thus, if we actually live in a star system that is still experiencing levels of dynamism in terms of the orbits of the major planets - then it is impossible to use the current locations as proof or disproof of TB's validity. ...

Well, that's bad news for Warren, then. He has presented two formulae for our solar system, one for the inner four planets and another for the outer four... which includes Neptune.


The burden of proof is on everyone in relation to TB. ...

Nonsense. Anyone making an extraordinary claim has the burden of proof.

tusenfem
2007-Dec-03, 01:30 PM
There are some serious things you can do with it, but it is very important to understand what it can and can't do. I get very similar results to those of Chiang, et. al. There's another paper by Alessandro Morbidelli and Harold F. Levison where they perform a simulation showing how Sedna could have been captured from a passing brown dwarf. My results were virtually identical to the results they got using the Swift rmvs3 orbit integrator. And the list goes on. There's more simulations and discussion in the forum on my website describing the useful things Gravity Simulator can do. And Gravity Simulator's results for planet stability are very much in agreement with the results from two other methods.


Hi tony. I did not want to degrade your program, in which you probably have invested a lot of time and effort. I just meant that using a program as a black box does limit its use for Warren Platts, as your example with Mars suddenly being kicked out overnight clearly shows.

I am just baffeled by the "statistics" that WP seems to be doing with your program and then drawing significances from it.

R.A.F.
2007-Dec-03, 02:19 PM
Something I don't understand...

If we use the Earths distance from the Sun (1 AU) as the Primary measuring unit (sorry, but I don't know how else to put it) for T-B in our solar System, then what (primary measuring unit) should we be using for other solar systems? I can "almost" understand using 1AU for our system, Earth is "important" because we all live on it. :), but does it really make sense to use our (primary measuring unit) for other solar systems?

antoniseb
2007-Dec-03, 02:36 PM
does it really make sense to use our (primary measuring unit) for other solar systems?

I don't think anyone is claiming that 1AU (of the original TB-Relation) would be the basis for all planetary systems. The idea that there is an individual basis for each planetary system is what is being talked about here.

R.A.F.
2007-Dec-03, 02:54 PM
I don't think anyone is claiming that 1AU (of the original TB-Relation) would be the basis for all planetary systems.

My error...

What is being used instead of 1AU?...and by what criteria was that number decided upon?

...or is it just the spacing between planets??

...but then how would you find (the spacings) without a "baseline" to work from?...

tusenfem
2007-Dec-03, 04:07 PM
I guess they just measured the distances of the five planets and then converted them to AUs, just because most planetary physicists will have a feeling about that kind of distance. But we could just as well use CUs, but then one would have to define what a CU is.

But basically you can work with any system, you can normalize on the smallest orbit on the middle or the largest, as long as you normalize all distances in the same way.

R.A.F.
2007-Dec-03, 04:49 PM
Nope...still doesn't work for me...

Warren Platts
2007-Dec-03, 11:28 PM
The AU is just a convenient, intuitive linear measure that has nothing directly to do with TBL. The TBL works just as well in miles or centimeters.

Warren Platts
2007-Dec-04, 11:25 AM
Sorry, this is the first time I've visited this thread. I'm not a big fan of TBL, so I never ventured in here. But my web server log told me I was getting hits from this thread, so I figured I'd have a look.
Hi Tony, thanks for the thoughtful post. It's very refreshing.


That's what I used to think. But last year I took Astro 405, "Astrobiology" from Debra Fisher (author of the Cancri paper). She thought Gravity Simulator was neat so she let me demonstrate it in class to show how the solar system barycenter moves. Schwheat! :cool:


She then gave me a challenge. She wanted to know what would happen if Venus suddenly had Jupiter's mass. I tried it, and after many hours of letting the simulation run, everything looked completely undisturbed. So I pumped Venus' mass up to 2 Jupiter masses. Again, after many hours of letting the simulation run, everything looked completely undisturbed. I concluded it was stable. But I went to bed without turning off the computer, and when I woke up in the morning, Mars was missing. This surprised me. For 10s of thousands of years, Mars had no noticable change in its orbit, and then suddenly it is missing. Why Mars? Why not Earth? It's closer to Heavy Venus.

So I repeated the simulation several times, this time using Auto Save, so I could witness the ejection event. Mars was happily orbiting the Sun for 10s of thousands of years, and one day, its orbit was just slightly more eccentric than normal, and orbit after orbit it grew. The ejection was fast. If I remember correctly, only a few hundred years seperated Mars from being in an orbit virtually unchanged from its starting condition to being sent by Jupiter on a hyperbolic trajectory.

But different starting conditions, as well as different time steps yield different results. Simply increasing Venus' mass in a different part of its orbit affects when Mars gets ejected. So this is where it's important to realize what part of the results of the simulation are useful, and what parts are not. Although the simulator will give you an exact date (i.e. Mars was ejected on April 22, 138582), this changes from run to run when the initial conditions are tinkered with, or when the time step is changed. But in each simulation there was a trend. Mars lasted anywhere from 50,000 years to 150,000 years before getting ejected. So the new conclusion that I drew was that Mars is unstable in this configuration, and every year Mars is rolling the dice, taking a 1 out of ~100,000 chance that this is the year that a certain alignment will quickly put it on an ejection trajectory.I know why she asked you about Venus: the second planet from 55 Cancri A is Jupiter sized. You've got an interesting result. But there's always that nagging question, "if only I reduced the time-step by an order of magnitude (and let it run for a week or a month), maybe that would make a difference." 55 Cancri has four pretty massive planets packed in close (less than 1 AU, the smallest is roughly Uranus massed), and they've obviously been stable. I created a .gsim model for 55 Cancri using the masses published in the Encyclopedia of Exoplanets (http://vo.obspm.fr/exoplanetes/encyclo/catalog-RV.php), and it ran just fine, even when I placed a 100 Earth-mass planet at 2 AU (for the 10,000 years or so that I ran it). Maybe I'll try it again and let it run for a longer period of time. However, one surprising and worthwhile result of the present investigation is that one big difference between the 55 Cancri system and the Sol system is the TBL scaling factor itself (K): KCnc = 2.17; KSol ~ 1.7. So, even though you've got more planets packed into a tighter space at 55 Cancri, the scaling factor is actually greater there, so orbits should be more stable there than here.


It's interesting to note that the Sun's inner solar system is filled with near resonances too: Venus:Earth = ~7:5, Mercury:Venus=~2:5 Mercury: Earth~4:1, Earth:Mars ~15:8, Mercury:Mars ~8:1. Although the 3:1 resonance you're referring to in the Cancri55 system is a bit closer to a true 3:1 than the examples from our inner solar system, it makes me wonder if these near-resonances are common in the universe. And what causes them? Were they once perfect mean motion resonances that let go after the orbits circularized, allowing the ratio to slightly drift? (If so, why did the orbits circularize?) Or could it just be a statistical thing, that any two periods are going to coincidently have a near-integer ratio?There's probably a mathematical theorem that says that the ratio any two periods can be represented by a whole number fraction to within a 1% error!



Billions of years are difficult to do without a very high time step. And high timesteps are bad for accuracy. But it does make me wonder. What is more likely? A stable system falls apart due to a high timestep, or an unstable system stays together?Nice one! :D


I haven't read that paper, but I would guess that Neptune stopped at 30 AU because the solar system ran out of significant amounts of small bodies for Neptune to perturb when Neptune was at about 30AU.What about Neptune? That's the question everyone's been asking. I guess a priori theory suggests that the outermost planet of a system is most likely to migrate significantly--angular momentum is traded and handed off from planet to planet and get concetrated in the outermost planet. The paper cited by grant hutchison suggested that Neptune might have migrated over 100 AU--and then came back! The empirical difficulty with such a scenario, however, is that one would expect Neptune to clear out Kuiper belt objects out to that distance, yet they are still there. However, as you mention below, perhaps the large numbers of TNO's locked in orbital resonance with Neptune is evidence of earlier migrations. In any case, for purposes of the TBL, if it is the case that TBL spacing patterns are the fossil remains of in situ planet formation, then position of the outermost planet of a system must be viewed with the most caution, as it is the most likely to have been moved around much.


This is a little off topic, but Resonant Capture Theory is a fascinating subject. Why are so many TNOs in orbital resonance with Neptune? It's theorized that as Neptune migrated outward, as its exterior resonances swept through the Kuiper Belt, that it captured these objects into resonances. Motivated by Eugene Chaing's paper on Resonant Capture Theory, I performed a simulation where 30 test particles were placed in circular orbits at 20AU +-40%, and Neptune was placed at 10 AU and told to migrate out 30 AU. By 7 million years, every single object was captured into either a 1:2 or a 2:3 resonance with Neptune. As Neptune continued to migrate, the eccentricity of the captured objects grew. Neptune took 12 million years to complete its migration in this simulation (which took nearly 24 hours of computer time, which shows why billions of years is difficult to do). Using a faster migration, Neptune only captured particles in a 2:3 resonance, and faster still, it captured nothing. I would guess that with a slower migration, it may capture objects into some weaker resonances such as 3:5, which are observed in the real solar system, especially if the objects start in slightly eccentric orbits (pre-heated according to Chiang, et. al).

So since the observed populations of resonant objects may be a signature of Neptune's migration, it is possible to use Gravity Simulator to learn something about planetary migration. Although you need to perform simulation after simulation after simulation and spot trends, rather than just rather than draw your conclusions from one simulation.Good job! :cool:



Warren, you might find these methods useful, rather than performing long-term simulations:
Method 1:
There's a website called oklo.org (http://www.oklo.org/), which is run by Greg Laughlin's team from UC Santa Cruz. Sky & Telescope magazine had a feature article on this website last year. Using its downloadable console, users can play with both hypothetical and real velocity data to attempt to find planets hidden in the data. When a user reports a find, he uploads his new system to their website where they perform a stability check. It's a really quick and dirty method too. They simulate the planet for 100 years, and if its semi-major axis drifts by less than 1%, they consider the planet to be stable. 100 years is fine for the tight orbits they're discovering, but for more distant planets you'd want to simulate longer. This is where Gravity Simulator can help you. In fact, on their old forum board, Greg even endorsed the idea of using Gravity Simulator before their stability checker was in place.Thanks for the link; there's a lot of stuff there!


To use Gravity Simulator for this, use the menu File> Output File, to create an Excel-readable data file so you can examine the semi-major axes of your objects. Just use (min-max)/mean on your SMA column to see if your planet violates the 1% rule. Using this method, it exposed Mars as unstable in the "Heavy Venus" simulation I described earlier. But it only took a few minutes to perform the simulation, rather than 24 hours, as I only needed to simulate for about 100 years. It also exposed Earth as unstable in this configuration. Earth's SMA deviation was smaller than Mars' (presumably because Mars is closer to the real Jupiter), which is probably why Earth was never ejected in my simulations. I'm guessing if I ran the "Heavy Venus" simulation longer, that Earth too would be ejected.It probably wouldn't be too hard to do this for the tentative "Planet V". The "Jupiter" in this simulation has a mass from 3-5 Jupiters, and the "Venus" is about 1 MJup, but it's at 0.115 AU instead of 0.7 AU.


Method 2:
On my "formulas" page ( http://www.orbitsimulator.com/formulas/ ) there are two formulas called "interior reach" and "exterior reach", as well as a link to the paper where these formulas came from. These compute analytically the unstable regions caused by a planet. For example, using Jupiter in these formulas shows that anything orbiting between 4.13 AU from the Sun up to Jupiter's position is unstable. And anything orbiting from Jupiter's position out to 6.3 AU is also unstable. I've ran a few simulations that suggest that Gravity Simulator's predictions and these formulas are also in agreement with each other.
A consilience between two different models is always a good thing! Actually, using Gravity simulator, I found that for 55 Cancri d (3-5 MJup, a = 5.77 AU), orbits at 4.25 AU were unstable, whereas orbits at 4.0 AU were stable; so that's in pretty close agrement too.

But these formulas don't account for multiple massive planets. So using them in the "Heavy Venus" simulation shows that Earth and Mars are stable, simply because you can only account for either "Heavy Venus" or "Real Jupiter", while it is their combination that makes Earth and Mars unstable.That stands to reason; a "Light Venus" doesn't have much gravitational power. I wonder what the minimum mass of Jupiter would be required such that there are no stable orbits between Jupiter and the Sun?

Warren Platts
2007-Dec-04, 05:32 PM
I've been running my program, changing the parameters to incorporate other scenarios. It turns out that a new r-squared distribution has to be constructed for just about every combination of numbers of planets and empty slots.

The attached graph shows the r2 for Monte Carlo simultions for systems having 4 - 9 planets with no empty slots.

The lines for the 4-5 planet scenarios are kind of weird looking, whereas the ones with more planets take on the classic F-scale distribution.

The next graph shows the distribution for 3 planets. As you can see, it is completely different. I checked the math, and I'm pretty sure I did it right. I had to give it a separate picture, because the right hand scale goes off the charts otherwise.

With these extra distribution, I'll be able to take a closer look at the Sol system.

tony873004
2007-Dec-05, 02:28 AM
...I wonder what the minimum mass of Jupiter would be required such that there are no stable orbits between Jupiter and the Sun?

Just use this formula: http://orbitsimulator.com/formulas/reachinterior.html
Using n=3, e=0, M=1 Sun Mass, a=5.2AU, and adjusting m until d was close to 0 shows that if Jupiter had 115 Jupiter masses (which would make it a star), there would be no stable orbits inbetween the Sun and Jupiter.

Your best bet for trying to figure out whether or not a planet exists in the gap inbetween planets d & f in the Cancri 55 system, is to download the console from oklo.org. The radial velocity data for Cancri 55 is publically available there as well. Then you can put additional planets anywhere you like, and see if you can reduce the chi2 of the existing fits.

As far as stability, I ran a simulation for you. I placed 7 massless test particles inbetween planets d and f. The distances and corresponding percentage change in semi-major axes are as follows:

Planet f: 138 million km
particle 1: 200 million km, 1.6%
particle 2: 307 million km, 1.9% (I made it 307 instead of 300 to match your predicted location)
particle 3: 400 million km, 2.3%
particle 4: 500 million km, 3.3%
particle 5: 600 million km, 6.6%
particle 6: 700 million km, ejected
particle 7: 800 million km, 2.2%
Planet d: 840 million km

They all failed the 1% test. The conclusion is that there are no stable orbits in the large gap between Planets d and f.

Warren Platts
2007-Dec-05, 03:41 PM
Your best bet for trying to figure out whether or not a planet exists in the gap inbetween planets d & f in the Cancri 55 system, is to download the console from oklo.org. The radial velocity data for Cancri 55 is publically available there as well. Then you can put additional planets anywhere you like, and see if you can reduce the chi2 of the existing fits.This is a sweet package. I downloaded it. It's just what I was looking for. I'm still on the learning curve for the next few days though. This package is mandatory for anyone interested finding a planet for themselves.

The rush is on!


As far as stability, I ran a simulation for you. I placed 7 massless test particles inbetween planets d and f. The distances and corresponding percentage change in semi-major axes are as follows:

Planet f: 138 million km
particle 1: 200 million km, 1.6%
particle 2: 307 million km, 1.9% (I made it 307 instead of 300 to match your predicted location)
particle 3: 400 million km, 2.3%
particle 4: 500 million km, 3.3%
particle 5: 600 million km, 6.6%
particle 6: 700 million km, ejected
particle 7: 800 million km, 2.2%
Planet d: 840 million km

They all failed the 1% test. The conclusion is that there are no stable orbits in the large gap between Planets d and f.
I've been running two GravitySimulator simulations for the 55 Cancri system with a 50 Earth-mass planet at the periodogram peak at 1085 days (semimajor axis = 2.033 AU). The timestep was set at 16384 (which I think means that one second of computer time = 16,384 seconds (~4.5 hours) of simulated time).

So far after well over one million years of simulation both simulations show all six planets merrily spinning in their original orbits. The timestep I set is insanely high--it's one step below the 32K time step which ejects the innermost planet right away. So if any planets are ejected in my simulations, that in itself can't be taken as strong evidence for orbital instability between planets d(VI) and f(IV). However, I figure that if the simulation can run for several million years with a fast but inaccurate time-step, that would be evidence for the stability of the orbit at 2 AU.

At the rate it's going (1,000,000 years per 7 hours of run time), it will take about 3 days to do 10,000,000 years, a month to do 100,000,000 years, 10 months for 109 and about 4 years of run time for 4.5 billion years of simulated time.

EDIT: I must say that as I'm running these simulations, the Planet IV (f) has developed a large eccentricity that's visible to the eye. But in Fischer et al. (2007, Tab. 2) (http://exoplanets.org/55cnc5th.pdf), they say that IV(f) has an eccentricity of 0.2--roughly an order of magnitude higher than the other planets. So, I'm thinking that maybe the high eccentricity noted by Fischer et al. might in itself be evidence of the existence of Planet V! :D

Warren Platts
2007-Dec-05, 07:18 PM
You know, it might be possible to estimate the mass of Planet V just by varying the mass of Planet V in the simulation such that it caused 55 Cnc f to have the observed eccentricity of 0.20.

BTW: 1,500,000+ simulated years so far. All conditions normal.

tony873004
2007-Dec-05, 11:01 PM
You know, it might be possible to estimate the mass of Planet V just by varying the mass of Planet V in the simulation such that it caused 55 Cnc f to have the observed eccentricity of 0.20.

BTW: 1,500,000+ simulated years so far. All conditions normal.

The problem is that if this planet existed, it would already be exposed in the radial velocity data unless it were not very massive, in which case it would not be able to perturb anything to 0.2.

And this 0.2 eccentricity is likely to be an oscillating orbital element. It may grow or shrink with time. The methods of the planet hunters account for this already.

If one additional increase of time steps starts ejecting planets, then you're running way too fast. You'll be adding energy to the orbits and causing them to precess unnaturally.

A quick and dirty way of testing your time step is to let it run for a few orbits, tracing paths. Then use menu Time > Time Backwards and see if it follows its own footprints.

Another method is to delete all but one planet and the star. This way you've turned it into a 2 body problem and your orbit should hold steady indefinately. Therefore any changes can be attributed to numerical error.

Warren Platts
2007-Dec-06, 01:45 AM
The problem is that if this planet existed, it would already be exposed in the radial velocity data unless it were not very massive, in which case it would not be able to perturb anything to 0.2.

And this 0.2 eccentricity is likely to be an oscillating orbital element. It may grow or shrink with time. The methods of the planet hunters account for this already.

If one additional increase of time steps starts ejecting planets, then you're running way too fast. You'll be adding energy to the orbits and causing them to precess unnaturally.
Granted, they may be precessing unnaturally, but that should only increase the instability, right? The orbits themselves do not seem to be changing appreciably in semimajor axis. After nearly 2.3 million years, Planet V in the left screen has a semimajor axis of 2.03322, which I calculated by taking the square root of X2 + Y2. (I originally entered 2.033 AU as the semimajor axis to start the simulation.)

My thinking is that if I increase the instability (by jacking up the time step) I can probe deeper into time, and if the system remains stable anyway, we are warranted in asserting that the system in question would in fact be stable in real life.


A quick and dirty way of testing your time step is to let it run for a few orbits, tracing paths. Then use menu Time > Time Backwards and see if it follows its own footprints.

Another method is to delete all but one planet and the star. This way you've turned it into a 2 body problem and your orbit should hold steady indefinately. Therefore any changes can be attributed to numerical error.
Thanks for tips. I'm afraid to fool around with the simulations too much at this point--I don't want to mess them up until they've run at least 10 million years. If the computer GravitySimulator (http://www.orbitsimulator.com/gravity/articles/what.html) is running on crashes, will the arrangement be saved so it can be reloaded?

snowflakeuniverse
2007-Dec-07, 01:46 AM
1. Three or more objects in orbit defined by gravitational interaction are inherently unstable.

2. Over time all systems will collapse to a one or two body system. Either one of the “extra” bodies will collide with another body, or the “extra” body will be ejected from the system, (which some may argue is still part of the large cosmological system).

3. The more powerful the gravitational interaction between the objects in the system, either due to proximity or mass, the quicker the destruction of the three body system.

4. The weaker the gravitational interaction, the longer the system is preserved.

5. Resonate orbital periods of much smaller objects, orbiting a much larger mass, are inherently more stable.

Given the above physical characteristics of orbiting systems, the only "stable" , or longest lasting orbital relationships will be those that result in resonate orbital relationships. If planetary systems are built from constituent pieces over time, it is only those locations that are at resonate relationships that will be preserved for the longest periods of time.

Snowflake.

snowflakeuniverse
2007-Dec-07, 01:47 AM
The gravity simulator is a great tool, but some mention as to the role of Chaos theory needs to be considered in using the application over extended periods of time. Tony 873004 alludes to this somewhat in his study of a planetary system with a large planet located in the orbital position of Venus, and the subsequent ejection of the equivalent to Planet Mars. (Post number 212). Slight changes in the elliptical orientation significantly altered when Mars was ejected. This poses a problem in the accuracy of the initial conditions.

(The ejection of Mars also indicates the significance of the importance of resonate relationships, one would initially think planets closer to a large Venus would be ejected instead of Mars. Mars with its more elliptical orbit and least resonate relationship to the other planets made it prone to instability).

Also important in any kind of iterative analysis is the accumulation of small rounding errors. Calculations carried out to 5 places will produce different results from those carried out to 10 places, and even calculations carried out to 100 places will be different from calculations carried out to 1,000 places. Even more surprising is that there is not a convergence to one result with increasing accuracy.

Snowflake.

tony873004
2007-Dec-07, 05:22 AM
Granted, they may be precessing unnaturally, but that should only increase the instability, right?
I don't know the answer to this. High time steps are bad for close encounters. An object that might otherwise make a close passage to a massive body can simply jump over it instead as something moving at 10 km/s at a time step of 65536 seconds moves 655,360 km per iteration.


...After nearly 2.3 million years, Planet V in the left screen has a semimajor axis of 2.03322, which I calculated by taking the square root of X2 + Y2. (I originally entered 2.033 AU as the semimajor axis to start the simulation.)
That's how you compute distance, but not semi-major axis. You also need to include Z2 if your simulation has any Z values. If you're working from the Cancri 55 simulation from my web site, everything is co-planar in the XY plane, so you can ignore Z.

By simply comparing its current distance to its initial distance, you have no guarantee that it didn't wander out to 3 AU and back to 2.0233.

To compute semi-major axis is more difficult. But Gravity Simulator will do it for you. Just use menu View > Orbital Elements Box.

Since semi-major axis is an oscillating orbital element, you can't simply take two points and conclude anything. For example if you sampled a sine wave at 0 and 2*pi, you may conclude you had a linear function. The best way to examine semi-major axis is to have Gravity Simulator output this data for you, then you can open it in Excel and examine it. From the menu, File > Output File. (If you're not using the Beta version, I suggest you upgrade. It's got nicer data output features). Then choose to output semi-major axis for your desired object, and choose a sampling frequency that is much smaller than 1 orbital period. I'd suggest choosing 10 days and 10,000 data points. Excel can easily handle this. This will give you about 274 years of data. Your data file will have the same name as your simulation, except with a .txt extension. It will be in the same folder as your simulation. Open it in Excel as a comma-delimited txt file. Under your 10000th entry, enter the formula =(min(C2..C10001) / max(C2..C10001)) / average(C2..C10001) , replacing C with whatever column your data is in. This will tell you how much your SMA changes over the 274 year period. More than 0.01 is bad for stability. It is the threshold the people on oklo.org use to catagorize a system as stable or unstable.


...My thinking is that if I increase the instability (by jacking up the time step) I can probe deeper into time, and if the system remains stable anyway, we are warranted in asserting that the system in question would in fact be stable in real life...
You're assuming that the higher time step will amplify the instability, and I don't know that this is the case. In my earlier post when I asked "What is more likely? A stable system falls apart due to a high timestep, or an unstable system stays together?" I truly meant that as a question. If you read scientific papers, you'll notice that a lot of detail go into describing mathmatically the errors and assumptions. So you can't just simply dismiss this with a single sentence. It would take a lot of simulations, running both known stable systems and known unstable systems at different time steps to find a trend that might hint at an answer to my question.

You might already know this, to make your simulation run much faster, turn off the plotting. It's the "P" button on the Graphics Options interface. The program will appear paused, but its still crunching the numbers in the background, much faster that it would if it had to plot the objects too. Or if you want to watch it draw the trails, and still have it run nearly as fast as if the plotting was turned off, use the Preferences menu to tell it to update the graphics every 100th iteration.


If the computer GravitySimulator (http://www.orbitsimulator.com/gravity/articles/what.html) is running on crashes, will the arrangement be saved so it can be reloaded?

No.

Not by default at least. You can simply hit menu File > Save As... and give it a new name, then you have your simulation to date backed up.

Or you can use menu > File > Auto Save to tell the program to save a copy for you at an interval specified by you. This is your best protection against losing your simulation if your computer crashes or the power fails.

Warren Platts
2007-Dec-07, 07:18 AM
Given the above physical characteristics of orbiting systems, the only "stable" , or longest lasting orbital relationships will be those that result in resonate orbital relationships. If planetary systems are built from constituent pieces over time, it is only those locations that are at resonate relationships that will be preserved for the longest periods of time.

Snowflake.But that's the thing. The TBL spacing doesn't seem to be correlated with any particular resonances. 55 Cnc "b" and "c" are sort of in one (3:1), but they are the furthest from the TBL "predictions" (and they would have had to move through a 4:1 resonance.

Warren Platts
2007-Dec-07, 07:21 AM
Gargh! Three days of simulations have to be thrown away because I had 55 Cnc d (Planet VI) set at 3.835 Earth masses instead of Jupiter masses! :cry:

Thanatos
2007-Dec-07, 07:29 AM
Warren, reexamine your model. It is deficient. Science includes both that which is observed, and that which is not observed [hint].

tony873004
2007-Dec-07, 07:55 AM
Gargh! Three days of simulations have to be thrown away because I had 55 Cnc d (Planet VI) set at 3.835 Earth masses instead of Jupiter masses! :cry:
That might help explain why your system has been stable for this long.

There's a Cancri 55 simulation on my web forum that you can use as a starting point: http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1194407194

Warren Platts
2007-Dec-07, 06:03 PM
That might help explain why your system has been stable for this long.
I'm rerunning one version of it with the correct Fischer et al. (2007, Tab. 2) data, Planet V at 2.033 AU (e=0) and 50 Earth masses, same time step (16384).

In the other window I'm running the same version but with Planet V weighing 1 kg. After ~1.3 million years both systems are still stable. However, this time I set the eccentricity for planet "f" (IV) at 0, in order to see what effect Planet V might have on "f"'s eccentricity. So far, the eccentricity hasn't moved much for either simulation. Intriguingly, however, the eccentricity for the Planet V = 50 earth masses scenario is at ~0.02, whereas that for the Planet V = 1 kg scenario the eccentricity is ~0.01. So I'll let 'em crank for a while more to see what happens. Since the eccentricity for "f" reported by Fischer et al. (http://exoplanets.org/55cnc5th.pdf)is 0.2, then if a missing Planet V were responsible for the high eccentricity, it might have to be much bigger than 50 or 100 Earth masses, and therefore must be located in one of the mean-motion resonance orbits (either 3:1 or 4:1 in order for the TBL pattern to hold).


There's a Cancri 55 simulation on my web forum that you can use as a starting point: http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1194407194Thanks for the link. I'm checking it out along with the new beta version (http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1176774875).

Cheers,
Warren

John Mendenhall
2007-Dec-07, 06:50 PM
And it fell flat on its face for Neptune.

Reminds one of the notorious prime number formulas, doesn't it?

Nevertheless, Warren, I've enjoyed your thread. A few more extrasolar planetary systems in detail, and we'll know if the T-B description fits well elsewhere. And maybe whether it's correlation and causation, or not.

Warren Platts
2007-Dec-08, 03:23 AM
Reminds one of the notorious prime number formulas, doesn't it?

Nevertheless, Warren, I've enjoyed your thread. A few more extrasolar planetary systems in detail, and we'll know if the T-B description fits well elsewhere. And maybe whether it's correlation and causation, or not.
This is a common misconception--that we'll know if the TBL fits or not elsewhere after we get a big enough sample size. The reality is that a few special systems will have the TBL spacing pattern and probably the majority of solar systems will not.

The Holy Grail is of course a roughly Earth-like mass in a stable orbit within the habitable zone. So far its turning out that the special sorts of systems likely to produce another temperate Earth might also reliably produce systems with exponential orbital spacing.

Why? Who knows. But the TBL is a pattern that planet hunters would be stupid to not take into account.

Warren Platts
2007-Dec-08, 04:45 AM
OK, I've ran the simulation to ~2.3 million years. Everything seems normal. Yes I don't know for sure that the orbits didn't rapidly change back and forth when I wasn't looking. However, I've been keeping the simulations as basically my desktop wallpaper (plot off of course), so I check it quite often at random intervals, and I haven't seen anything obvious to the human eye.

So at least GravitySimulator simulations out to ~106 years indicate stable orbits.

Eccentricity of planet "f" (IV) did not increase as expected. Instead it declined somewhat. The simulation with the 50 Earth mass Planet V seemed to have a higher volatility in eccentricity to my eye, but I have no statistics for that.

Fischer et al. (2007) said that planets within ~2AU zone predicted by TBL should not be more than 50-100 Earth masses, or else they would have noticed the doppler signature--with the possible exception of planets located at mean motion resonances. As noted earlier, planets "b" and "c" might have apparently moved from their predicted primitive orbits "in order to" form a mean motion resonance.

The simulation so far says that a 50 Earth mass planet might have a small effect on the eccentricity of "f". However, the eccentricity of "f" is 0.2! What could cause such an eccentricity?

There is always the deus ex machina of Velikovskyian collisions. Thank me for saving you the keystrokes.

Then there might be other explanations. Like the gravitational effect of a neighboring planet. . . This would suggest that perhaps Planet V is a massive Jupiter-plus sized planet in a 3:1 or 4:1 orbital resonance with "f". The 3:1 period is 782.4 days. The 4:1 period is 1043 days. These correspond to semimajor axes of 1.626 and 1.97 AU respectively.

So the plan is to run GravitySimulator on two simulations. One with a Jupiter massed planet at the 3:1 resonance, and one simulation with a Jupiter massed planet at the 4:1 resonance. The hope is to measure an effect on the eccentricity of "f".

Van Rijn
2007-Dec-08, 06:02 AM
This is a common misconception--that we'll know if the TBL fits or not elsewhere after we get a big enough sample size. The reality is that a few special systems will have the TBL spacing pattern and probably the majority of solar systems will not.


If TB doesn't apply above the chance level, it is meaningless. Still, I find it fascinating how much you can say with so little data.



Why? Who knows. But the TBL is a pattern that planet hunters would be stupid to not take into account.

I think planet hunters would be wise to look for planets without making too many assumptions.

Warren Platts
2007-Dec-08, 01:57 PM
If TB doesn't apply above the chance level, it is meaningless. Still, I find it fascinating how much you can say with so little data.I'm finally coming around to your view that the TB "Law" appellation is unfortunate because it leads to the sort of verbal traps that you've fallen into. This thread is so not about whether the entire universe exhibits the TBL pattern or not.

It's about:

what the TBL pattern is;
how you mathematically characterize a TBL pattern;
how you can judge whether a purported TBL pattern is real (i.e., statistically significant).


The TBL pattern is not a property of galaxies; it is a property of solar systems.

Pop quiz question to see if you understand the thread so far: If it turned out that 55 Cancri and the Sol system are the only solar systems in the galaxy with TBL spacing, would that entail that 55 Cancri and the Sol system do not have TBL spacing?


I think planet hunters would be wise to look for planets without making too many assumptions.
Then you've never done it yourself. There are so many parameters and the search space is so large that you will never get anywhere if you don't start making assumptions that can constrain the search space. Check out how they do it at oklo.org. How they do it in principle is not that hard.

Warren Platts
2007-Dec-08, 02:35 PM
Then there might be other explanations. Like the gravitational effect of a neighboring planet. . . . This would suggest that perhaps Planet V is a massive Jupiter-plus sized planet in a 3:1 or 4:1 orbital resonance with "f". The 3:1 period is 782.4 days. The 4:1 period is 1043 days. These correspond to semimajor axes of 1.626 and 1.97 AU respectively.

So the plan is to run GravitySimulator on two simulations. One with a Jupiter massed planet at the 3:1 resonance, and one simulation with a Jupiter massed planet at the 4:1 resonance. The hope is to measure an effect on the eccentricity of "f".

I ran the simulations for about 1,000,000 years. Systems appear perfectly stable. And the eccentricity for planet "f" (IV) if anything was less than in the non-resonant simulations with the low mass Planet V.

So I'm jacking up the Planet V masses to 5 Jupiter masses and will see what happens. Already after a few thousand years, the 4:1 resonant (5 MJup in both simulations) doesn't show much of anything happening (ef < ~0.01) but the 3:1 resonance system the ef seems to vary from 0.07 to 0.17. So maybe we're getting somewhere. Will let her rip some more to see what happens.

Van Rijn
2007-Dec-09, 09:07 AM
Then you've never done it yourself. There are so many parameters and the search space is so large that you will never get anywhere if you don't start making assumptions that can constrain the search space. Check out how they do it at oklo.org. How they do it in principle is not that hard.

Assumptions based on known physics are one thing. Assumptions based on numerology (like you are doing) are another.

snowflakeuniverse
2007-Dec-10, 03:43 AM
Hi Van Rijn

You said
“Assumptions based on known physics are one thing. Assumptions based on numerology (like you are doing) are another.”

Your criticism is a bit extreme. Warren Platts is testing his theory or hypothesis by looking for planets in the predicted locations, according to the Bode Titus Law. This is good physics worthy of encouragement.

Newton’s Law of Gravity was used to predict the location of planets due to the perturbations observed of other planets. Most would consider these historical discoveries “good physics”, even though the structural relationship of spacetime and mass was not yet understood. (And I would argue is still not completely understood, even with General Relativity).The discoveries of the reliability of Newton’s Law of Gravity helped spur the continued development, and search for the “string” that holds planetary systems together.

Snowflake

Van Rijn
2007-Dec-10, 05:01 AM
Hi Van Rijn

You said
“Assumptions based on known physics are one thing. Assumptions based on numerology (like you are doing) are another.”

Your criticism is a bit extreme. Warren Platts is testing his theory or hypothesis by looking for planets in the predicted locations, according to the Bode Titus Law. This is good physics worthy of encouragement.


You apparently are reading something other than what I'm reading. My criticism was for his recommendation that planet hunters make assumptions not based on physics, but on a questionable, untested idea. As I've said repeatedly, I'll get interested if a consistent pattern falls out after we have good data for a number of solar systems. We aren't anywhere near that yet, and TB has big problems even in the solar system.

By the way, is TB a "law," "theory" or "hypothesis" in your view?

jkmccrann
2007-Dec-10, 09:25 AM
Yet the proponents of the "classical" T-B formula claim that it puts a fifth planet where the asteroid belt is. I don't see how pointing out that the formula isn't terribly precise (geographic center, mass center, something else?) is a red herring.


It is a "red herring" in that instance for exactly the reason you cite. Proponents will say that it proves a 5th body would have formed there but for the obvious gravitational pull of Jupiter.

That may be so, which also points out an obvious variable in play in any given Solar System - clumping of material may follow a pattern that resembles TB, possibly, but because of the distribution of material, and the varying elements in any given system that leads to a tipping point being reached with the accretion of the material at a given point which gravitationally alter nearby clumping points - as apparently occured with Jupiter in relation to the Asteroid Belt - which is why I refer to it as a red herring.

It neither proves nor disproves TB in the basic sense, it merely illustrates the fact that there are other variables that will effect any TB formation and will necessarily alter it to some degree in any given system.

Having given TB a little thought recently, I can probably sum up my views on it just so you know where I stand.

I think in a given Solar System that has exactly the same components, mass distribution, physical properties etc. as our own, then you are as likely as not to see the planets come about into what we know as TB (barring passing stars, black holes etc - but how common are they to come through the neighbourhood over the 4-6-7billion year life cycle of a given star system - I have no idea)

So, if you change these physical property variables in a given star system, then the fact is you are extremely unlikely to see TB occur as it has in ours, and the evidence is that our star system is really not the run-of-the-mill type of system many of us living here would naturally think it is (not you folks, but the layman in the street), the evidence would seem to point to the fact that there are all types of star systems out there and we lie somewhere on this great continuum.

So in looking to apply TB to other star systems, you are probably wasting your time Mr. Platts, because it is very unlikely you'll be able to convince anyone here that you can find evidence of TB like relationships elsewhere. But, you should keep trying because it is highly amusing.

As for Neptune and its orbital migration - I don't pretend to know the ins and outs of that - which is precisely why I ask for documented scientific papers on the subject so I might learn more of it.

All I know is that over the past 10-20 perhaps more years, there has been a wide discussion about Neptune's orbital migration.

If so, when did this occur and where is the proof, or where is the theory, that it has indeed stopped. Given our lack of knowledge about the Kuiper Belt and the properties of bodies even beyond the Kuiper Belt - are the stability of these orbits indicative of a long-term stasis in their relationship with Neptune? Or is there perhaps evidence that Neptune continues to exert changing gravitational effects on these bodies - perhaps because it has yet to truly find its natural equilibrium in our system.

That question appears particularly pertinent when you consider that although many of the Kuiper Belt and other objects in that region have resonant orbital relationships with Neptune - what about the many hundreds, and thousands of objects in the region that don't have resonant orbital relationships with Neptune?

Why is there such a disparity? What is the breakdown for the region as a whole? Maybe someone can tell me, and how do the percentages for resonant/non-resonant in the Kuiper Belt stack up against those in the Asteroid Belt?

Is there a difference? If there is, is that difference possibly explained by orbital migration? And then we get back to the original question I asked in regards to Neptune - if we're talking about Neptune's orbital migration having occurred at some point in the past - where is the real proof that it has stopped?

I'm sure with enough study of the Kuiper Belt it can likely be discerned, but that is unfortunately not something I'm currently capable of carrying out - which is why I ask for the direction to the scientific papers!!! I'm not trying to prove anything, I'm trying to learn myself!

Finally back on point, if indeed there is still planetary migration going on in this system, with regards to Neptune, then using Neptune as proof of TBs failing in its own backyard is an incorrect assumption to make - because in that case it would fail to consider the slow equilibrium processes that are carrying out to reach a so-called "TB equilibrium" point.

Warren Platts
2007-Dec-10, 03:13 PM
So in looking to apply TB to other star systems, you are probably wasting your time Mr. Platts, because it is very unlikely you'll be able to convince anyone here that you can find evidence of TB like relationships elsewhere. But, you should keep trying because it is highly amusing.
I believe I have already found a TBL pattern at 55 Cancri that is statistically significant at the 99% confidence level. If 99% is not good enough for you, then what level of siginificance would convince you? Or is there some glaring flaw in my statistical analysis? If so, why won't you share it with us?

snowflakeuniverse
2007-Dec-11, 04:10 AM
Hi Van Rijn

You asked
“By the way, is TB a "law," "theory" or "hypothesis" in your view?.”

If it were up to me, General Relativity would be a “Law”, not a theory of gravity since general relativity is based on geometrical relationships of distance and time, which are experimentally or observationally verified (locally). Newton’s “Law” of gravity would be a theory since it is observationally established and is not based upon a theoretical framework that predicts the relationships. A hypothesis would be an untested relationship that may yield an insight on relationships.

However, physicists have screwed up the English language. Newton’s “Law” of gravity is called a “law” and will always be. By allowing this convention, physical “laws” are established by observation followed by discovering the mathematical relationships that conform to the observation. The Bode Titus “Law” falls into this process, so just as Newton’s “Law” of Gravity is called a “Law”, the Bode-Titus relationship would be called a “Law”. A “Theory”, by common usage in the science community, is ambiguous in usage. The Theory of General Relativity is hardly a theory, and the Theory of Evolution/natural selection is hardly a theory.

Snowflake

snowflakeuniverse
2007-Dec-11, 04:22 AM
Hi jkmccrann

You said
“So in looking to apply TB to other star systems, you are probably wasting your time Mr. Platts, because it is very unlikely you'll be able to convince anyone here that you can find evidence of TB like relationships elsewhere. But, you should keep trying because it is highly amusing.”

As unlikely as it may be, he has convinced me.

I would estimate that at least 10 percent of the planetary systems would conform to the Bode Titus law and that deviations from the BT Law could be attributed to, and evidence of, disturbances in the early evolution of the planetary system. .

Snowflake