View Full Version : Moment of Inertia of the Earth

trinitree88

2016-Aug-04, 12:28 AM

Ok. Question. Taking a Physics 1 training Course @ Bridgewater State to accommodate Supervisors and to learn some more neat stuff...and this came up. Exactly what is the methodology used to determine the moment of inertia of the Earth, and it's accompanying inference to the models of it's interior?....earthquake propagation times? Thanks. pete

it's not my homework.

Solfe

2016-Aug-04, 02:56 AM

Interesting post, but I have no answer. I believe it is mass vs. radius vs. density, and they aren't all uniform. Exactly how it is done is beyond me, but I am interested myself.

I did get my hands on some data about earthquakes for a contest to develop a predictive model. I think it was hosted or sponsored by Google. After about 15 seconds of reviewing the data, I determined it was way more than I could process. I think it came in GB chunks and was pretty seriously detailed.

lemming

2016-Aug-04, 03:12 AM

The answer is two-fifths times mass times radius squared for a solid spherical cow of uniform density.

Are you looking for the method to derive this idealised, approximate result, or something more exact that takes into account things like the non-uniform density of the earth?

ShinAce

2016-Aug-04, 10:21 AM

If you knew the size of the Earth's tidal bulges and the rate at which the rotation is slowing, couldn't that give you the torque/moment of inertia?

StupendousMan

2016-Aug-04, 02:48 PM

There are a number of satellites orbiting the Earth, the primary goal of which is to provide evidence for the distribution of mass within the Earth's body. Very careful observations of the satellites' positions and velocities reveal tiny little perturbations from the ideal ellipses that simple theory predicts. If one has many measurements over many years, one can gradually modify a model for the interior mass (or density) distribution of the Earth -- away from the simple uniform model -- in order to explain these small perturbations.

The result is a very detailed model of the Earth's density, which can then be used to compute the Earth's moment of inertia.

This field of study is known as "satellite geodesy." You might read a basic introduction at

https://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003D.HTM

Ken G

2016-Aug-05, 06:32 AM

If you knew the size of the Earth's tidal bulges and the rate at which the rotation is slowing, couldn't that give you the torque/moment of inertia?

Yes, that might work, I don't know if that's how they do it though. It would also work to see how the rotation changes in response to known changes in the moment of inertia, that could give you the unperturbed moment of inertia.

Ken G

2016-Aug-05, 06:33 AM

There are a number of satellites orbiting the Earth, the primary goal of which is to provide evidence for the distribution of mass within the Earth's body. Very careful observations of the satellites' positions and velocities reveal tiny little perturbations from the ideal ellipses that simple theory predicts. It's not obvious this can give you the moment of inertia, however, For example, a spherical mass distribution would give you the same orbits as a point mass, but they have hugely different moments of inertia. You might have to actually see how the planet responds to torques, not the gravity it produces.

profloater

2016-Aug-05, 09:01 AM

If there is a rotating core, as is thought, it might be a different value in opposed rotational directions, although I would imagine the same for small perturbations. cf the observed effects of a spinning bucket of liquid. the vortex produced by acceleration or deceleration are different patterns. The M of I will also be different for different forcing frequencies for the same reason

grant hutchison

2016-Aug-05, 12:20 PM

The equilibrium oblateness of the Earth for its know rotation rate, overall density and mass gives you some information about the moment of inertia - the equilibrium shape of the Earth under rotation would be different if most of its mass was near the surface, or if its mass was strongly concentrated at the core.

The rate at which non-equatorial orbits precess around the Earth [because of its oblateness] depends on a parameter J2, which is determined by the three principal moments of inertia.

Grant Hutchison

Ken G

2016-Aug-05, 01:13 PM

The rate at which non-equatorial orbits precess around the Earth [because of its oblateness] depends on a parameter J2, which is determined by the three principal moments of inertia.Yet notice the problem of inverting the direction of the logic. You have shown that if you know the details of the mass distribution, you can calculate the precession of orbits. But this does not imply the converse, there is often an extreme uniqueness issue. For example, if you find no precession, you only know the mass distribution is spherical, so the three principal moments are equal to each other-- but you still get no idea what they actually are in that case.

Ken G

2016-Aug-05, 01:15 PM

If there is a rotating core, as is thought, it might be a different value in opposed rotational directions, although I would imagine the same for small perturbations.Good point, the entire notion of a "moment of inertia" involves the idealization that the object is rigid-- which the Earth is not. Presumably it is close enough to rigid that this is not your main source of uncertainty, but maybe that is indeed the largest source of error.

profloater

2016-Aug-05, 03:58 PM

Good point, the entire notion of a "moment of inertia" involves the idealization that the object is rigid-- which the Earth is not. Presumably it is close enough to rigid that this is not your main source of uncertainty, but maybe that is indeed the largest source of error.

Indeed for large objects the concept of rigidity is complex for any use of the moment of inertia, the progress of sound and shock waves through the solid must be taken into account. Even tidal gravity couples will cause internal vibrations with energy loss so that the small object useful calculations based on a moment of inertia will not hold. It would be at best an approximation for back of envelope calculations, much though I love those.

grant hutchison

2016-Aug-05, 04:19 PM

Yet notice the problem of inverting the direction of the logic. You have shown that if you know the details of the mass distribution, you can calculate the precession of orbits. But this does not imply the converse, there is often an extreme uniqueness issue. For example, if you find no precession, you only know the mass distribution is spherical, so the three principal moments are equal to each other-- but you still get no idea what they actually are in that case.But in the Earth's case there is precession. And the relevant equations are actually easily reversible in terms of J2. The ratios of the principle moments of inertia are also constrained by the fact that we're dealing with an equilibrium ellipsoid, at least in terms of J2.

Grant Hutchison

Ken G

2016-Aug-05, 05:02 PM

But in the Earth's case there is precession. And the relevant equations are actually easily reversible in terms of J2. The ratios of the principle moments of inertia are also constrained by the fact that we're dealing with an equilibrium ellipsoid, at least in terms of J2.

I'm still skeptical they are reversible. The basic problem is, for a nearly spherical Earth, you get very nearly the same orbits even if the (nearly equal) moments of inertia around the principal axes are drastically different in the two cases. This is a classic example of what is known as "ill conditioning" in the inverse problem-- you are starting with two perfectly spherical planets with totally different moments of inertia, giving exactly the same orbits, and now you perturb those planets to get precession. The perturbations will not need to be large, because the precession effect is small. It does not sound like a recipe for uniqueness to me!

StupendousMan

2016-Aug-05, 05:23 PM

It's not obvious this can give you the moment of inertia, however, For example, a spherical mass distribution would give you the same orbits as a point mass, but they have hugely different moments of inertia. You might have to actually see how the planet responds to torques, not the gravity it produces.

I guess this guy (and others in the business) ought to give up, then.

http://adsabs.harvard.edu/abs/2013GeoJI.195..260T

profloater

2016-Aug-05, 05:51 PM

Monte Carlo methods? I used to use those for problem solving, it's basically the average guess of a group, or it was then, and it was surprisingly powerful even when individual guesses were wide spread. It might be related to the jury theory, or something. When I tested it against various esoteric facts, I was amazed how close the average guess could be to the fact, even when no single person knew the correct answer.

grant hutchison

2016-Aug-05, 06:25 PM

I'm still skeptical they are reversible. The basic problem is, for a nearly spherical Earth, you get very nearly the same orbits even if the (nearly equal) moments of inertia around the principal axes are drastically different in the two cases. This is a classic example of what is known as "ill conditioning" in the inverse problem-- you are starting with two perfectly spherical planets with totally different moments of inertia, giving exactly the same orbits, and now you perturb those planets to get precession. The perturbations will not need to be large, because the precession effect is small. It does not sound like a recipe for uniqueness to me!

This (https://farside.ph.utexas.edu/teaching/celestial/Celestial/node93.html) from Richard Fitzpatrick at the University of Texas Department of Physics:

It should be noted that the dimensionless quadrupole moment of the Earth's gravitational field, J2, (as well as higher-order coefficients such as J3 ; see Exercise 3 (https://farside.ph.utexas.edu/teaching/celestial/Celestial/node96.html#exj3)) is most accurately determined via observations of the precession rates of the perigees and ascending nodes of orbiting satellites.

Knowing J2 effectively gives you a difference in principal moments of inertia. Measuring some aspect of spin evolution (such as precession) then gives you a different constraint on the moments of inertia, which lets you parse J2 into its components.

Grant Hutchison

Ken G

2016-Aug-08, 01:45 AM

I guess this guy (and others in the business) ought to give up, then.

http://adsabs.harvard.edu/abs/2013GeoJI.195..260TPlease note the first sentence from the abstract:

Gravity inversion allows us to constrain the interior mass distribution of a planetary body using the observed shape, rotation and gravity.

My bold. So the point here is, there is a very large difference between constraining an already well determined mass distribution by using orbital data, versus solving for the moment of inertia based only on orbital data. This is completely routine, we face difficult inversion problems all the time. We use the data to constrain solutions, but we always have to know a great deal about the nature of the solution already, or else we face drastic non-uniqueness problems. This is nothing new, there are whole books about the difficulties involved in doing inversions.

Ken G

2016-Aug-08, 01:47 AM

Knowing J2 effectively gives you a difference in principal moments of inertia. My point exactly. You already have to know an enormous amount about the moment of inertia or else this data would be useless to you. Hence, you cannot measure the moment of inertia this way, but you can fine tune a large set of pre-existing assumptions. A very typical problem in inversion techniques, which has led to more than one surprise over the years.

lemming

2016-Aug-08, 03:55 AM

Monte Carlo methods? I used to use those for problem solving, it's basically the average guess of a group, or it was then, and it was surprisingly powerful even when individual guesses were wide spread. It might be related to the jury theory, or something. When I tested it against various esoteric facts, I was amazed how close the average guess could be to the fact, even when no single person knew the correct answer.

I won't claim to know every application, but whenever I have seen (or used) Monte Carlo methods, it has had nothing to do with averaging different people's subjective opinions on something. Rather, it involved analysing a random system which was too complicated for closed-form analysis.

grant hutchison

2016-Aug-08, 10:11 AM

My point exactly.Well ... um ... that's good.

Grant Hutchison

profloater

2016-Aug-08, 03:48 PM

I won't claim to know every application, but whenever I have seen (or used) Monte Carlo methods, it has had nothing to do with averaging different people's subjective opinions on something. Rather, it involved analysing a random system which was too complicated for closed-form analysis.

Well I am recalling from the 70 s 80s when we used Monte Carlo for brainstorming on new product ideas and problem solving. It was published as an interesting phenomena and I remember it became useful is risk assessment and risk identification. With the rush of computer applications the word became used to avoid saying random when adding some randomness into simulations. I do wonder if there is a more complex derivation than just gambling.

glappkaeft

2016-Aug-08, 05:11 PM

That sounds more like the Delphi method.

profloater

2016-Aug-08, 05:25 PM

That sounds more like the Delphi method.

You are right. Now I am wondering about how I mixed them up in that I remember using the name Monte Carlo in my induction training in 1972. I may have to see if I have notes from then. The main subject was brainstorming which evolved out of Value Engineering discipline. It became devalued by undisciplined attempts, IMO. Anyway it's tangential to the OP so not much help in calculating moment of Inertia.

Ken G

2016-Aug-09, 09:26 PM

Well ... um ... that's good.

In case there remains any uncertainty in what I am saying:

One cannot invert orbital information into the moment of inertia of the Earth, or its values along principal axes. Period. Trying to do so would face drastic non-uniqueness problems, stemming from the fact that a wide array of hugely different moments of inertia give very nearly the same orbits to the smallest detail. What you can do is apply a set of pre-existing information about how the moment of inertia needs to behave, and tinker with small corrections, within those assumptions, by looking at orbital data. This makes the orbital data very useful, but certainly not a measurement of the moment of inertia of the Earth, because if you are using wrong assumptions, your corrections will settle on a completely wrong answer. This is important to realize whenever you apply techniques like that.

grant hutchison

2016-Aug-09, 10:35 PM

In case there remains any uncertainty in what I am saying:

One cannot invert orbital information into the moment of inertia of the Earth, or its values along principal axes. Period. I don't believe I ever said anything different.

You, however, seemed to go off on a tack in which you denied that J2 was derivable from orbital precession. Which is wrong, and I'm sure now it's not what you intended.

Grant Hutchison

Ken G

2016-Aug-10, 12:19 AM

I don't believe I ever said anything different.

You, however, seemed to go off on a tack in which you denied that J2 was derivable from orbital precession. Which is wrong, and I'm sure now it's not what you intended.

I never said anything other than that you cannot infer the Earth's moment of inertia, or its values along principal axes, by looking at orbital precession. The problem is simply not invertible in that manner. The thread was about how to infer the Earth's moment of inertia. Orbital precession is not nearly enough for that.

grant hutchison

2016-Aug-10, 12:46 AM

So when I wrote:

... And the relevant equations are actually easily reversible in terms of J2. ...

And you replied:

... I'm still skeptical they are reversible. ...

You weren't actually expressing scepticism about the reversibility in terms of J2. But I think you'll agree it's easy to see how someone might think otherwise.

Grant Hutchison

Ken G

2016-Aug-10, 01:52 AM

The reversibility I was talking about is different from the reversibility you mentioned. I was saying you cannot reverse orbital observations into an inference of the moment of inertia. Differences between moments of inertia along different principal axes is a second order question to the structural issues that actually set the moment of inertia of the Earth, the latter of which is how I interpreted the OP question. Perhaps I should have read the point you were making more clearly when I used your word "reversible", but what I was saying was always this: you cannot infer a moment of inertia from an orbit. The OP was not about J_2, because J_2 by itself is a weak constraint on structure models, but that doesn't make your comment wrong. So take my posts as saying, in effect, you are right that we can learn about J_2 that way, but we cannot learn about the moment of inertia that way. We would have to ask the OPer if they were interested in different models that produce different moments of inertia, or just different J_2. If my posts sounded like "you are wrong, you can't get the J_2 parameter that way" then I should have framed them more carefully to say "although it is interesting that you can get J_2 that way, it is important to recognize the extreme nonuniqueness problem you still face if you want to go from J_2 to a moment of inertia."

grant hutchison

2016-Aug-10, 09:33 AM

Deriving J2 in the way I described is, however, part of the standard way of deriving moments of inertia for planets and satellites. It constrains the possible values of the three principal moments in one way.

Symmetry arguments from the equilibrium ellipsoid constrain in another way.

Information from precession or libration constrain in a third way.

Solve.

Grant Hutchison

Ken G

2016-Aug-14, 11:50 PM

Yet my point is one can never "solve" for the moment of inertia that way, because even when you put all those together, you will still face a drastic nonuniqueness problem. Hence, there is no substitute for modeling, unless you can observe the rotational response to known changes in the moment of inertia, or known torques, which is more or less the same thing. That is the sole way to measure a moment of inertia. And even then, modeling is needed, in the sense of idealizations pointed out above.

grant hutchison

2016-Aug-15, 12:40 AM

Yet my point is one can never "solve" for the moment of inertia that way, because even when you put all those together, you will still face a drastic nonuniqueness problem. Hence, there is no substitute for modeling, unless you can observe the rotational response to known changes in the moment of inertia, or known torques, which is more or less the same thing. That is the sole way to measure a moment of inertia. And even then, modeling is needed, in the sense of idealizations pointed out above.

Doesn't observing precession or libration correspond to "observ[ing] the rotational response to known changes in the moment of inertia, or known torques"?

Grant Hutchison

Ken G

2016-Aug-15, 03:50 AM

Doesn't observing precession or libration correspond to "observ[ing] the rotational response to known changes in the moment of inertia, or known torques"?

No, those torques are not on the Earth, so you don't get the Earth's moment of inertia from those things. However, you can constrain an already well-constrained set of assumptions about the Earth's moment of inertia, and that is the tack taken there, as I'm sure those papers will show. The basic problem that brings this into focus is that you can start from vastly different mass distributions (and moments of inertia for the Earth) that produce precisely the same orbits to the smallest detail (spherical mass distributions), and then you perturb them to get the actual orbits you see. Clearly, that will matter a whole lot what mass distribution you start with when you do that. Something similar happens when you go to calculate the oblateness of the Earth-- the "first cut" at any oblateness calculation will use a spherical mass distribution! Later iterations will then perturb that in ways that depend quite a lot on how you distribute the mass initially.

grant hutchison

2016-Aug-15, 09:26 AM

No, those torques are not on the Earth, so you don't get the Earth's moment of inertia from those things.But I'm talking specifically about torques on the Earth (or, in general terms, on the body being examined). What I called "some aspect of spin evolution" earlier in the thread. Precession (or libration) of the spin axis induced by tidal forces from other bodies.

There's precession and there's satellite orbital precession. Two entirely different processes, constraining the moments of inertia in two different ways.

Grant Hutchison

Ken G

2016-Aug-15, 10:38 AM

But I'm talking specifically about torques on the Earth (or, in general terms, on the body being examined). What I called "some aspect of spin evolution" earlier in the thread. Precession (or libration) of the spin axis induced by tidal forces from other bodies.Everything that I have been saying about nonuniqueness of inversion strategies relates to the impossibility of measuring the Earth's moment of inertia by looking at things orbiting the Earth, which was a topic raised several times and which I restricted my comments in response to. Look at posts #5 and #7 in this thread, for example, it was a very early topic of discussion. Also consider post #10, which was also about objects orbiting the Earth. In a related vein, we had the citation http://adsabs.harvard.edu/abs/2013GeoJI.195..260T offered, which is a perfectly classic example of everything that I am talking about, even though it was originally offered as some kind of counterexample, which it is certainly not.

That is what I have been pointing out, since I already presumed it was obvious that we can learn about the moment of inertia of the Earth by looking at how the Earth itself responds to torques, see posts #4 and #6, and the times you were talking about the Earth's precession and libration, rather than the precession and libration of satellite orbits. The consistent picture that I think we can all agree on is this: you cannot learn much about the moment of inertia of the Earth strictly by looking at things orbiting the Earth, you have to look at the various responses of the Earth's own rotation, and you have to include models of the Earth's interior that involve a host of assumptions. However, you can use objects orbiting the Earth to winnow down an already highly constrained parameter space dealing with the Earth's mass distribution, which is what that cited paper was talking about.

grant hutchison

2016-Aug-15, 02:28 PM

Everything that I have been saying about nonuniqueness of inversion strategies relates to the impossibility of measuring the Earth's moment of inertia by looking at things orbiting the Earth ...Yes, I know.

The consistent picture that I think we can all agree on is this: you cannot learn much about the moment of inertia of the Earth strictly by looking at things orbiting the Earth, you have to look at the various responses of the Earth's own rotation, and you have to include models of the Earth's interior that involve a host of assumptions. However, you can use objects orbiting the Earth to winnow down an already highly constrained parameter space dealing with the Earth's mass distribution, which is what that cited paper was talking about.You just seemed so intent on repeatedly kicking the concept of satellite orbital precession raised by StupendousMan, there was a danger that we might lose track of the fact that satellite orbital precession actually is a standard method used in estimating the moment of inertia of planets and satellites. It's part of the standard toolkit, and it's therefore part of the answer to the OP.

Grant Hutchison

Ken G

2016-Aug-15, 05:37 PM

You just seemed so intent on repeatedly kicking the concept of satellite orbital precession raised by StupendousMan, there was a danger that we might lose track of the fact that satellite orbital precession actually is a standard method used in estimating the moment of inertia of planets and satellites. Well that is certainly a cartoon version of everything that I have actually said (and yes, I realize I am borrowing your turn of phrase there, it's a good one). For example, I'm sure your sophisticated mind understands the difference between "kicking" a claim, versus pointing out its crucial limitations! I mean seriously, are we children kicking toys, or scientists trying to understand the limitations in our knowledge?

Of course I know that satellite orbits are used to winnow down the parameter space needed to model planetary moments of inertia, you can tell because I said exactly that right from the start. But we must not pretend that you can actually measure moments of inertia that way, because that would represent a complete lack of understanding of how such methods work. All I have been trying to do is explain how these methods actually do work, and you can read that article to get more sophisticated detail. But the bottom line is this: you face a drastic nonuniqueness problem if you attempt to use satellite orbits to deduce the moment of inertia of a planet. I've said that many times, in many ways, and it's still just as true now as it was when I first said it. To see what I mean, read that paper, and notice the extreme mathematical backflips they are forced to do to try to keep that bugbear in its cage. This is nothing new-- it is the common nature of the whole process of using precise astronomical observations to work backward to the nature of the systems that gave rise to them. We see it all over the place in astronomy, moments of inertia is just a nice example of a far deeper and much more important principle here.

It's part of the standard toolkit, and it's therefore part of the answer to the OP.As I said above, it is a part of the toolkit, indeed a small part, a part that only works if you already know most of the answer, which is the point here. So yes, it is relevant to the OP, and nobody ever said it wasn't relevant. But what I did ask is, should we not notice the difference between such a limited tool, and an actual measurement? Should we pretend we cannot be bothered to understand that difference? That "part of the toolkit" is a whole lot different from a means of measuring a moment of inertia. And then I mentioned all the problems with those "tools", problems that are not going to go away by pretending they aren't there. Instead, one requires the incredibly painstaking analysis that is in that cited paper, because of the severity of the problem of nonuniqueness in inversion methods just like that. That paper is exactly what I am talking about.

Now, I wouldn't make anything of it, if it were not such a hugely and centrally important issue pervading most of astronomy. Ergo, far from "losing track" of the issue of understanding moments of inertia (or dark matter, or dark energy, or much of the rest of the unknowns of astronomy), it is absolutely central to the entire issue. The scientist who loses track of the difference between making a measurement versus merely winnowing a set of preconceived assumptions is on the verge of having egg in his/her face. Not like it doesn't happen constantly, by the way-- the Greeks fell victim to it when they rejected Aristarchus' heliocentric model, Newton and Einstein fell victim to it when they rejected a dynamical universe. The list goes on and on-- in astronomy, we very rarely get to measure what we want to know, so we are instead forced to winnow a set of preconceived assumptions. Not always with desired consequences, but we have little choice-- though we can at least notice we are doing it. Or we can just gloss over the distinction between a measurement and a modeling assumption, and keep making the same mistakes over and over.

grant hutchison

2016-Aug-15, 05:44 PM

Well that is certainly a cartoon version of everything that I have actually said. Of course I know that satellite orbits are used to winnow down the parameter space needed to model planetary moments of inertia, you can tell because I said exactly that right from the start. But we must not pretend that you can actually measure moments of inertia that way, because that would represent a complete lack of understanding of how such methods work.Given that I agree with you, I'm not sure what you're characterizing as hooey.

[Oh, I see you deleted the "hooey" bit by the time I hit the "Quote" button.]

Grant Hutchison

Ken G

2016-Aug-15, 08:05 PM

What I originally called "hooey", then replaced with your more eloquent "cartoon" metaphor, is the idea that I am being overly hard on methods for inferring moments of inertia that involve orbiting satellites. What I said from the start is this: orbiting satellites cannot tell you the moment of inertia of a planet until you have already made a large number of modeling assumptions, which could of course be wrong. Hence, they don't measure the moment of inertia, an important fact to bear in mind as we press forward in our understanding of planetary interiors-- and everything else in astronomy. Why you would characterize this simple and perfectly correct observation as "kicking" the wrong idea that you can infer the moment of inertia by looking at orbiting satellites, I just could not say. Of course I don't really care about moments of inertia, but I do care about the widespread and important nonuniqueness problem in astronomy, which I am using this rather insignificant example of moments of inertia to explain. That's the great thing about physics and astronomy-- the most insignificant example can be used as a door to understanding deeper principles that underpin not only the history of astronomy, but also its future course. Rest assured that nonuniqueness, such as you encounter when you try to invert orbital motions into an understanding of interior planetary structure, is the central problem of astronomy, and has been for thousands of years. It dogged Copernicus, Kepler, Newton, and Einstein, and even today is the forefront issue in the large-scale dynamics of galaxies and the universe.

grant hutchison

2016-Aug-15, 08:49 PM

What I originally called "hooey", then replaced with your more eloquent "cartoon" metaphor, is the idea that I am being overly hard on methods for inferring moments of inertia that involve orbiting satellites.Not overly hard, just repetitive in your criticism. When people agree with you, it's often taken as a sign to stop arguing.

Grant Hutchison

Ken G

2016-Aug-15, 10:03 PM

So you are now prolonging the issue to help underscore how important my point was? Excellent, we agree. Before, I was not convinced the key point was coming across, for pretty obvious reasons (do complaints about "losing track" of how to infer a moment of inertia ring a bell? I even had a paper that is talking about precisely the same problem I was, cited at me as if it refuted what I was saying!). So I clarified. Hopefully some have benefited from thinking about the nonuniqueness issues surrounding inferring what is invisible and interior from what is visible and exterior in situations where we know we have vastly different possible solutions that yield indistinguishable observations-- a key problem in much of science. I felt it was worth clarifying the point. Still do.

grant hutchison

2016-Aug-15, 10:28 PM

So you are now prolonging the issue to help underscore how important my point was?I thought I was spending my time trying to unprolong the issue, to be honest. <Sigh.>

Grant Hutchison

Ken G

2016-Aug-16, 03:01 AM

Here was always the way to unprolong the issue: simply let my point stand, it's just plain true that you cannot measure a moment of inertia by looking at satellite orbits. The reason the point matters is that this is nothing less than the central problem of astronomy: how to navigate all the nonuniquenesses of all the possible solutions, based on highly sparse and incomplete datasets, bolstered by a host of modeling assumptions that we often cannot avoid but also must not forget when we are invoking them. And here is the best weapon for doing it: notice that this is indeed our issue, as many times as it needs to be pointed out. I believe this was at the heart of the OP question-- how do we really know the moment of inertia of a planet? Like, if we have not actually measured it, maybe we are missing something. The only way to know is to keep careful track of the difference between an actual measurement, versus merely the winnowing of assumptions. That is I believe what the OP is actually asking, what is that difference as it applies to the concept of a moment of inertia. And by extension, what can happen when we fail to recognize that difference? Is that what is happening with dark matter? With dark energy? With inflationary models? With tests of general relativity, and unification with quantum mechanics? When do we know we have measured something, and when have we merely constrained a set of preconceived ideas we may someday need to cut loose.

grant hutchison

2016-Aug-16, 02:28 PM

Here was always the way to unprolong the issue: simply let my point stand, it's just plain true that you cannot measure a moment of inertia by looking at satellite orbits.Aaaaaaargh.

Agreement! Is! Futile!

Grant Hutchison

Ken G

2016-Aug-17, 12:19 AM

Then I guess all we don't agree on is the futility, as what I see is the opposite of futility. Complete agreement is always impossible, and partial agreement is the basis of human communication. Communication is what has happened here, and I see it as valuable, and possibly not just to the two of us. If there was any single lesson I would try to convey to any astronomy student, it would be the difference between a measurement and a winnowing of model assumptions (for example, the difference between determining the value of the cosmological constant, versus measuring the history of universal expansion). There is no more profound or important topic anywhere in science. So if you agree with that statement, I say the exchange is just the opposite of futile!

grant hutchison

2016-Aug-17, 12:56 AM

I don't see what is futile about it at all. We agree, all is good, hardly futile. I guess all we don't agree on is the futility, as what I see is the opposite of futility.That was a joke - a reference to the Borg phrase "Resistance is futile!" from Star Trek.

Now that we've established that we've agreed that J2 of itself does not measure planetary moment of inertia, maybe I can just (again) point out how it is used to estimate the planetary moments of inertia that are provided in reference tables. Here's a little more detail.

From satellite precession rates we get J2, which is proportional to (C-A), the difference between the polar moment of inertia (C) and equatorial moment of inertia (A), approximating the planet as an ellipsoid of rotation.

The precession rate of the object itself under known tidal forces is proportional to (C-A)/C.

Knowing the value of (C-A) and (C-A)/C lets us solve for C and A.

If we can't measure precession (and we often can't) we can extract an estimate of C from J2 by measuring the polar-equatorial flattening and using the Darwin-Radau approximation, which assumes hydrostatic equilibrium. Darwin-Radau turns out to give a pretty good agreement for the Earth's moment of inertia as derived from its J2 and its precession rate, for instance - to within one part in a thousand.

There are assumptions. And there are refinements. But what I've written is fundamentally where the numbers in the books come from.

Grant Hutchison

Ken G

2016-Aug-17, 03:38 AM

I am well aware that you can measure J_2 that way. This is the key difference between what you can measure, and what winnows a parameter set. The latter comes up when you want to use J_2 to infer something about the moment of inertia, which this thread is about.

Ken G

2016-Aug-17, 04:54 AM

For anyone who thinks we are simply having a personal and pointless argument, here is the deeper point here. This kind of thing happens all the time in astronomy. We want to know one thing, say the moment of inertia of the Earth. Maybe there's a way to measure that, but we don't have access to it, we can't actually do it. But we can make assumptions, and model it, and test that the model is working well. Then we can improve the model, that's where J_2 comes in-- we say that the Earth is spinning, and our model predicts J_2. Then we find we can measure J_2 much more easily than we can measure the moment of inertia, because vastly different moments of inertia can lead to the same orbits of satellites, but different J_2 will leave its mark on those orbits. So we use J_2 to improve our model, indirectly. But if we were way off to begin with, because our model was wrong, we are using J_2 to make small improvements on a totally wrong model. So we need to understand that this is what might be happening-- we need to understand how measuring J_2 is different from measuring the moment of inertia.

Now for the real point. Consider how exactly this kind of thing has happened, over and over, in the history of astronomy. It's the reason we don't ever get to do anything more than make models that work-- for what they work for. Then we measure something we could never measure before, and we get knocked on our keester. We always think that can't happen, and then it does. We keep forgetting what we have actually measured, and what is dependent on our models.

grant hutchison

2016-Aug-17, 08:11 AM

I am well aware that you can measure J_2 that way.The post wasn't addressed to you, of course.

It's a response to the OP question, "Exactly what is the methodology used to determine the moment of inertia of the Earth ...?"

We find that the precession rate of a planet varies with the applied torque and its polar moment of inertia. But we can't know the torque without knowing the moment of inertia already, because the torque depends on the difference between the polar and equatorial moments - on J2, in other words. So we can only get at the moment of inertia embedded in the rate of planetary precession by finding another way to measure J2. Which is where satellites come in, because the precession of their orbits gives us an independent measure of J2. Once we know that, we can figure out the torque that's driving the precession, and so dig out the polar moment of inertia.

Grant Hutchison

Ken G

2016-Aug-17, 03:13 PM

That is true if the torque on Earth only depends on J_2. Maybe it does work out that way-- that would be a useful coincidence.

Ken G

2016-Aug-17, 05:22 PM

Indeed, on further thought, I cannot see that the torque would indeed depend only on J_2. It should depend in a more complicated way on the mass distribution, and that's the problem. Simplification is an important tool of science, but we must beware of oversimplification. Is it not an oversimplification to say that the torque is determined by J_2? So if it is true that it is not determined by J_2, we are right back to being forced to treat J_2 as nothing but a constraint on a set of modeling assumptions-- assumptions that cannot be checked in that manner. I believe that is precisely why the paper cited in this thread needed to be so mathematically sophisticated.

grant hutchison

2016-Aug-17, 07:58 PM

Indeed, on further thought, I cannot see that the torque would indeed depend only on J_2. It should depend in a more complicated way on the mass distribution, and that's the problem.You'll notice I wrote "approximating the planet as an ellipsoid of revolution". So J2 does the job for that approximation.

The torque also depends on even-numbered higher zonal harmonics - their contribution is often small, both because they're often small relative to J2 (ie, the ellipsoid is a good approximation), and because there's a power relationship as you go up the series, which drives the contributions small quickly.

The even-numbered higher zonal harmonics are also recoverable from satellite orbits, so if their contribution is large, they can be detected and measured, and ploughed into the precession calculation. For instance, we got J2 and J4 from the orbits of the Galilean satellites of Jupiter, back in (IIRC) the 60s. We're up to J20 from Galileo and Juno.

And if we're dealing with a tri-axial ellipsoid (like a tidally locked moon), then we have another set of harmonics to deal with. for instance, our estimates of the moments of inertia of the Galilean satellites themselves come from J2 and C22, derived from satellite flybys, and Darwin-Radau.

Hence the "assumptions" and "refinements" caveat I made at the end of my post. But the principle's the same. It's why studying satellite orbital perturbation is key to estimating moments of inertia.

Grant Hutchison

Ken G

2016-Aug-17, 10:12 PM

You'll notice I wrote "approximating the planet as an ellipsoid of revolution". So J2 does the job for that approximation.I don't think so, that still doesn't sound like enough information. Are you not also assuming it has a constant density (as in MacCullagh's formula from those notes you cited)? Or you might even go a step farther and make some kind of assumption about the composition, and get compressibility models and so on, but that's what I mean by modeling assumptions as opposed to measurements of the mass distribution. Whenever you start with a simplification, it's very important not to forget the simplification and pretend that you have a measurement instead of an assumption.

Here's the basic problem. If you have a spherical mass distribution, you can have a million orbits with different eccentricities and radii and planes of orbit, and all you will ever get for that entire series of Js is a bunch of zeroes. So that's your starting point, knowing exactly nothing about the moment of inertia of the planet. Then you can give the planet some rotation, treat its equipotentials like ellipsoids of revolution but still with unknown mass distribution, and somehow you have to recover the mass distribution well enough to know the gravitational torque from a list of tiny Js? I don't think that's going to work, you are going to need to model that mass distribution and winnow the set of parameters using those Js. Certainly that's what you will do, you might even start (for schematic purposes only) with an assumption of constant density, just to get the ball rolling. But it's not a measurement, and it's going to encounter nonuniqueness as soon as you start relaxing those assumptions. When that happens, there is the potential for another headline starting with "Astronomers are shocked to discover...". And there's me reading that headline thinking, "either astronomers are incredibly slow learners, or they are not that easy to shock any more."

grant hutchison

2016-Aug-18, 12:12 PM

I don't think so, that still doesn't sound like enough information. Are you not also assuming it has a constant density (as in MacCullagh's formula from those notes you cited)?I don't think so. The assumption is hydrostatic equilibrium under a pure gravitational+centrifugal potential. J3 and higher measure deviations from that equilibrium ellipsoid.

Here's the basic problem. If you have a spherical mass distribution, you can have a million orbits with different eccentricities and radii and planes of orbit, and all you will ever get for that entire series of Js is a bunch of zeroes. So that's your starting point, knowing exactly nothing about the moment of inertia of the planet. Then you can give the planet some rotation, treat its equipotentials like ellipsoids of revolution but still with unknown mass distribution, and somehow you have to recover the mass distribution well enough to know the gravitational torque from a list of tiny Js? I don't think that's going to work, you are going to need to model that mass distribution and winnow the set of parameters using those Js. Certainly that's what you will do, you might even start (for schematic purposes only) with an assumption of constant density, just to get the ball rolling. But it's not a measurement, and it's going to encounter nonuniqueness as soon as you start relaxing those assumptions.Okay, so it seems to me there are several possibilities:

1) Actually, the planetary moments of inertia in the textbooks are derived in some other, more satisfactory, way that you know about but have so far resisted telling us about.

2) The planetary moments of inertia in the textbooks are derived in general terms as I describe, but you believe the method has so many assumptions as to be useless.

3) The planetary moments of inertia in the textbooks are derived in general terms as I describe, and you're OK with that, but still feel I need a lesson in the particular assumptions being made, and the general problem of untracked assumptions in science.

4) You don't know how the planetary moments of inertia in the textbooks are derived and don't intend to check, but are just poking away at my description for entertainment and education.

5) You think I've just made all this stuff up.

I'm so far failing to tease out which of these applies to our conversation.

Grant Hutchison

Ken G

2016-Aug-18, 05:29 PM

I don't think so. The assumption is hydrostatic equilibrium under a pure gravitational+centrifugal potential. J3 and higher measure deviations from that equilibrium ellipsoid.

Let's stick for now with ellipsoidal equipotentials, which means, I suspect, we are treating the gravity as purely that of a spherical source, coupled with the centrifugal potential of a rigid rotation (another assumption), which then generates all the deviations from a spherical gravitational potential in which to satisfy hydrostatic equlibrium in our interior model. In other words, we only iterate the centrifugal corrections once-- we don't go back and update the gravitational sources to get the new equipotentials. That's fine, such an iteration would converge quickly anyway, and we are not interested in tiny corrections, we are wondering if we could have the answer completely wrong, say at the level of tens of percent. Even so, you are far from done making assumptions if you only take first-order centrifugal corrections to the mass distribution, because all that gives you is the equipotentials of the effective gravity. That's not the mass distribution, so it's not a moment of inertia!

Now you need an equation of state, say an ideal gas law for a star, or some model of the compressibility of matter for a planet. That's what I'm talking about-- the winnowing of model parameters within a set of assumptions that could be completely wrong. Jupiter, for example, does not obey the ideal gas law, so you need to understand hydrogen under high pressure. The Earth's composition is not known, we have to use models of the compressibility of iron and various rocks, along with models of convection of magma, to get a working equation of state for the Earth's interior. You need all that to understand the mass distribution-- the equipotentials tell you very little about that, expressly because they come from a spherical (i.e., highly nonunique) mass assumption, coupled with a centrifugal correction that also does not care about the mass distribution. Yet you have a J_2 at this point-- and you haven't said anything at all yet about the mass distribution, other than the shape of the outer surface!

Okay, so it seems to me there are several possibilities:

1) Actually, the planetary moments of inertia in the textbooks are derived in some other, more satisfactory, way that you know about but have so far resisted telling us about.Why am I supposed to tell you what assumptions you are making? I'm only telling you to find your own assumptions when you make claims about what satellite orbits tell us about the Earth's moment of inertia. For example, you linked to some lecture notes that assume a constant density, when you track the assumptions there. This is my entire message here: the assumptions we make distinguish what we can call a measurement from what we should actually call a winnowing of parameter space within some set of assumptions that could turn out to be wrong in ways that "shock the astronomers."

I'm so far failing to tease out which of these applies to our conversation.

Then you are not reading my posts very carefully, because it's none of your possibilities, it is what I have been saying: notice your own assumptions, they are what distinguish your conclusions from things that could accurately be described as measurements, and they are entry points for reaching wrong conclusions that you can later be "shocked" about, like all those headlines we constantly get.

grant hutchison

2016-Aug-18, 06:33 PM

Well, this sort of "Socratic Dialogue with unseen textbooks through the interpretation of an intermediary" approach rarely works, even face-to-face. I speak from prior experience of both ends of such dialogues.

Here are two sets of introductory lecture notes on the topic:

Cornell: Astronomy 6570 (http://astro.cornell.edu/academics/courses/astro6570/Precession_Free_and_Forced.pdf)

UCSC: EART162 - Planetary Interiors (http://www.es.ucsc.edu/~fnimmo/eart162_10/Week2.pdf)

I believe they illustrate that I've described the accepted way of finding a value for the moment of inertia of a planet.

If you want to use them as a starting point to describe the disabling assumptions they involve, then that'll be fine. At least we'll both be looking at the same information.

Grant Hutchison

George

2016-Aug-18, 07:03 PM

With too little time to enjoy the nuances, it seems to me that, in georgeeze, Ken is addressing the possible hiccups that are latent within an indirect method (orbital variations) to achieve a practical result (torque). It works fine for blackholes and, very likely, planetary torques, but Ken, true to his cuidado commentary, is adding the cautionary touch. Is this close to right?

grant hutchison

2016-Aug-18, 08:01 PM

With too little time to enjoy the nuances, it seems to me that, in georgeeze, Ken is addressing the possible hiccups that are latent within an indirect method (orbital variations) to achieve a practical result (torque). It works fine for blackholes and, very likely, planetary torques, but Ken, true to his cuidado commentary, is adding the cautionary touch. Is this close to right?Ken will no doubt describe his motivation shortly, but to me he's making a very important point about the problem of unrecognized or unexamined assumptions in science. When such assumptions intrude between the things we can measure and the things we want to know, they can be a time-bomb.

It's a problem I recognize from my own profession.

And there are certainly assumptions involved in calculating planetary moments of inertia, though it's not clear to me that all of them are the assumptions Ken thinks, or that they are necessarily unexamined and/or dangerous.

Despite these assumptions that Ken is concerned about, there are a bunch of standard equations sitting in the astronomy textbooks, being provided in lecture slides, and appearing in scientific publications. So these equations are being used, educationally and practically.

What are we too make of that?

Grant Hutchison

Ken G

2016-Aug-18, 10:28 PM

Here are two sets of introductory lecture notes on the topic:

Cornell: Astronomy 6570 (http://astro.cornell.edu/academics/courses/astro6570/Precession_Free_and_Forced.pdf)

UCSC: EART162 - Planetary Interiors (http://www.es.ucsc.edu/~fnimmo/eart162_10/Week2.pdf)

I believe they illustrate that I've described the accepted way of finding a value for the moment of inertia of a planet.Ah, thank you for that, the notes make it clear what is going on. Although the notes don't seem to realize this, it turns out that all reference to J2 is a red herring, none of it matters in determining the moment of inertia of the Earth. What is actually happening is that we are doing what we all agreed we need to do to get the moment of inertia of the Earth-- we need to look at how the Earth responds to torques. And as Grant said, that means we need to know what torques the Earth experiences, which come mostly from the Moon but depend on the Earth's mass distribution, and that's what we are trying to figure out. But there is a lovely trick here, which doesn't really depend on J2 at all (so is not subject to the assumptions that go into letting the gravitational perturbation be a rapidly converging expansion that starts with a J2 term). The trick is to realize that a system of the Earth and something in orbit with the Earth (whether the Moon or a small satellite) is a closed system that conserves angular momentum. So if you want to know the torque the Earth feels, find the torque it produces, and take the negative of that torque. The problem is that the Earth produces too small a torque on the Moon to notice (apparently, because they don't just look at changes in the orbital angular momentum of the Earth/Moon system), so we instead have to look at satellites orbiting the Earth to see what torque the Earth produces, and simply scale that to the torque the Earth produces on the Moon (the scaling is straightforward because they are both changes in orbital angular momentums due to nonsphericity in the Earth, so the torques will scale with M/r2 where M is the mass of the object and r is its distance from Earth, so Earth-produced torque rescalings are trivial, they just scale with the strength of the gravitational force). Then say the negative of that torque is the torque the Moon produces on the Earth's rotation, causing precession that obeys *dw/dt = torque (where [I] is the moment of inertia tensor that we want, and w the omega vector we can measure).

So this is why no assumptions are needed about the Earth's mass distribution, the torque the Earth feels from the Moon (or Sun, to be even more accurate) is exactly a scaled version of the torque the Earth produces on a satellite. J2 is irrelevant, the equations are simpler without ever mentioning it, and all these results are much more elegant in the form of simple conservation equations. That's a criticism of the cited notes, but it certainly doesn't make them wrong, nor does it make what Grant said above wrong. He is correct that the technique for measuring the moment of inertia of the Earth is looking at the torque the Earth produces on satellites (which depends on nonsphericity of an unspecified nature in the Earth's structure), scale that to the torque the Earth applies to the Moon (which shows up in the orbital angular momentum of the Earth-Moon system, but we can't measure that very easily apparently), and notice that this must be the negative of the torque the Earth experiences from the Moon (which shows up as precession in the Earth's spin, which we can measure). Measuring that precession then is a measurement of the Earth's moment of inertia, with relatively few assumptions (indeed, no need to assume anything about the Earth being an ellipsoid, for example, or even that it be anywhere close to a sphere).

But here's the key point-- it is not the Earth's effect on the satellites that comes from the Earth's moment of inertia, indeed there is no need for any direct connection between those things at all (by which I mean, the Earth could have an arbitrary internal shape and this whole approach still works fine, though the mathematics is much easier if the Earth has axial symmetry so that it has a diagonalizable [I] with two equal principal moments of inertia). What really matters is that we can get the torque on the Earth by noticing its torque on something else, that's the guts of it. It's pretty clever, I'll admit, but all the business about ellipsoids and J2 is just a kind of distractor. All you need for the simple precession measurement to work is that the Earth have an axially symmetric mass distribution [I]of any kind, and even that only simplifies the measurement you need to do. That measurement is just the rate of change of orbital angular momentum of an Earth satellite, and the rate of change of the spin vector of the Earth. The connection always gives you [I].

However, we are still not completely free from modeling assumptions, as was pointed out above-- we are assuming rigid rotation to use [I]*dw/dt = torque. That turns out to be wrong, which is why that equation doesn't work well for the "free precession" of the Earth (a roughly yearly kind of effect due to the Earth not being spherical and not having a spin angular momentum that lines up with its polar axis, but it is a wobble that plays out over a scale of just a few meters), except as a kind of schematic approximation. This is mentioned in those notes rather blithely-- they just say it took a long time to figure out how a non-rigid mantle messes with the results, but they never mention how it messes with "forced precession"! They seem to imply it doesn't affect the forced precession at all, but that can only be justified by assuming the non-rigidity doesn't manifest on the different timescales of the forced precession, and we have no evidence cited to justify that ( the forced precession takes waay longer, but is manifest over much larger distance scales, and the speeds are of roughly similar order, so I cannot see how we can know that magma motions won't affect things). It would seem that additional modeling assumptions are hidden in there, a parameter space about the elasticity of the Earth that has been winnowed and could present more places for errors at the few-percent level in the moment of inertia.

slang

2016-Aug-18, 11:11 PM

Moving to Astronomy as this thread seems to have evolved beyond what is proper for Q&A

Ken G

2016-Aug-19, 02:54 AM

I should correct one thing from post #55, the rescaling of the torque to different satellites has an additional factor of 1/r that I left out. I forgot that as r changes, the force not only scales like 1/r2, but the component of that force that is perpendicular to the radius (and that's what goes into the torque) has an additional 1/r dependence (because the force from any dm mass element inside the Earth has its force component perpendicular to the radius fall off like 1/r). This effect can also be seen in the equations in those notes Grant just cited, though those notes seem unnecessarily restricted to ellipsoids and J2 type gravitational perturbations, but they are actually not so limited. The rescaling of the torque works fine for arbitrary mass distributions of the Earth, and so does the conservation of angular momentum. The only subtle point is that the torque must be averaged over an orbit, so the simplest rescaling of the torque is for satellites with the same inclination as the Moon's orbit. To use other satellites in the torque rescaling, one needs a correction for the inclination i, which the notes say depends on sin(2i). That sin(2i) correction may require an ellipsoidal mass distribution, it's not given in detail, but in any event it would be unnecessary if we used a satellite with the same i as the Moon to do the torque rescaling, and then the mass distribution of the Earth would once again be irrelevant.

So for those who like the equations, let's call T the (vector) torque of the Earth on some satellite with the same inclination i as the Moon's orbit, and then the torque on the Moon would be T*M/m*(a/A)3, where M is the Moon mass, m is the satellite mass, A is the distance to the Moon, and a is the distance to the satellite. The torque on the Earth from the Moon is then -T*M/m*(a/A)3. The value of T is found from looking at the rate of change of the angular momentum of the satellite orbit, and the moment of inertia tensor [I] for a rigid Earth is then found from:

-T*M/m*(a/A)3 = [I]*dw/dt, where dw/dt is the measurable (vector) rate of change of the angular velocity of the Earth's spin.

This equation only admits a unique solution for [I] in special cases, but it is particularly easy to solve if [I] is diagonal, i.e., if the Earth's mass distribution can be regarded as axially symmetric. No other assumption about the shape of the Earth or its mass distribution is needed, except that the Earth rotates rigidly, which of course isn't even true.

chornedsnorkack

2016-Aug-19, 05:12 AM

Hm. Letīs say... how would this operate regardless of mass distribution?

Imagine that Earth were of such shape that Moon produces no torque. (It would be odd for Earth to have such a shape, and we have observations that Earth does receive torque from Moon). For example, suppose that the surface of Earth is indeed orange shaped, but Earth possesses a dense core which is lemon shaped, such that the gravitational torques of Moon on Earth surface and Earth core would sum to exactly zero torque.

Would artificial satellites of Earth then experience any torque?

grapes

2016-Aug-19, 01:01 PM

Hm. Letīs say... how would this operate regardless of mass distribution?

Imagine that Earth were of such shape that Moon produces no torque. (It would be odd for Earth to have such a shape, and we have observations that Earth does receive torque from Moon). For example, suppose that the surface of Earth is indeed orange shaped, but Earth possesses a dense core which is lemon shaped, such that the gravitational torques of Moon on Earth surface and Earth core would sum to exactly zero torque.

Would artificial satellites of Earth then experience any torque?

The moon torque (that causes tidal slowing) depends upon the dynamic bulge created by the moon being drawn off-line by "friction"

The moon is also much farther away than the satellites, so is much less affected by the differences, mostly because of the exponent-dependent terms mentioned earlier.

grant hutchison

2016-Aug-19, 01:16 PM

The moon torque (that causes tidal slowing) depends upon the dynamic bulge created by the moon being drawn off-line by "friction"But the precession torque is caused by the gravitational interaction of the Earth's equatorial bulge and the moon (+sun+planets).

Grant Hutchison

grant hutchison

2016-Aug-19, 01:19 PM

Hm. Letīs say... how would this operate regardless of mass distribution?

Imagine that Earth were of such shape that Moon produces no torque. (It would be odd for Earth to have such a shape, and we have observations that Earth does receive torque from Moon). For example, suppose that the surface of Earth is indeed orange shaped, but Earth possesses a dense core which is lemon shaped, such that the gravitational torques of Moon on Earth surface and Earth core would sum to exactly zero torque.

Would artificial satellites of Earth then experience any torque?That sort of mass distribution would show up in the higher even-numbered values of J, so would certainly show up in satellite orbits.

A perfectly spherical Earth would experience no lunisolar precession and no satellite orbit precessions, because all values of J (and other harmonics) would equal zero.

Grant Hutchison

George

2016-Aug-19, 02:52 PM

A perfectly spherical Earth would experience no lunisolar precession and no satellite orbit precessions, because all values of J (and other harmonics) would equal zero. But, I assume, you are also saying it would have to remain spherical (i.e. inelastic) in order to be true, which would never happen (not that you're suggesting otherwise, of course).

[Added: I am finishing "Lucky Planet" and it is interesting that, according to the author's calculations, a slightly larger Moon would have moved into the 40,000 year planetary resonance by now and our axis would have gone nuts, with serious impacts to life.]

grant hutchison

2016-Aug-19, 03:30 PM

But, I assume, you are also saying it would have to remain spherical (i.e. inelastic) in order to be true, which would never happen (not that you're suggesting otherwise, of course).That's right - it would have to infinitely rigid and (as I should have made clearer) have a spherically symmetrical imass distribution inside.

I think another issue with chornedsnorkack's scenario is that there's no reason gravitational torques that cancel down at one distance should cancel at another distance. A satellite in low orbit "sees" the various mass concentrations within the Earth at significantly different distances, whereas from the vantage point of the moon they're at similar distances.

Grant Hutchison

George

2016-Aug-19, 05:17 PM

That's right - it would have to infinitely rigid and (as I should have made clearer) have a spherically symmetrical imass distribution inside. Yep, and it is intuitive.

I think another issue with chornedsnorkack's scenario is that there's no reason gravitational torques that cancel down at one distance should cancel at another distance. A satellite in low orbit "sees" the various mass concentrations within the Earth at significantly different distances, whereas from the vantage point of the moon they're at similar distances. That's a good point. I would guess two satellites in different orbits would be needed in any hope to determine internal lumpiness in 3D. Then there might be headaches when considering that elasticity will be an issue for moon-like bodies, but not little artificial satellites. [Is "artificial" still used anymore?]

chornedsnorkack

2016-Aug-19, 05:58 PM

But the precession torque is caused by the gravitational interaction of the Earth's equatorial bulge and the moon (+sun+planets).

Is it possible to find out what are the precession torques of Moon and Sun respectively?

grant hutchison

2016-Aug-19, 06:10 PM

Is it possible to find out what are the precession torques of Moon and Sun respectively?Yes, it's calculated in one of my earlier links. The lunar torque accounts for 69% of the lunisolar precession rate.

Grant Hutchison

Ken G

2016-Aug-19, 10:38 PM

My point is, if the assumption that the Earth is an ellipsoid is correct, then you can use the mathematics offered in those notes, and the results can be expressed in terms of J2. But this assumption is not necessary, a more elegant solution appears by never invoking that assumption, and simply rescaling the torque on some satellite to the torque on the Moon. I admit that how to do this rescaling is not always obvious, and the J2 approach does that for you, but it can also distract from the key fact that the torque rescaling from the satellite to the Moon is the guts of the approach. So one must look closely at the assumptions that go into that rescaling. But the point I'm making is this: the rescaling step is never checked, anywhere in that calculation. It is always a model assumption, it is never itself measured. As such, we are winnowing the parameters within a set of model assumptions, and if those model assumptions prove wrong, then our answer is wrong. The torque rescaling is the guts of the whole procedure, and that rescaling assumes the deviation from sphericity is handled by the J2 parameter. The second key assumption is that the Earth responds rigidly to external torques. Neither of those assumptions are checked, so the result is only as good as they are. Measurements, on the other hand, don't require assumptions like that.

Worse, the only thing that is checked in that calculation is the unforced precession rate, and it comes out wrong to several tens of percent! Not exactly a rousing endorsement of the approach, it asks us to believe that the same elasticity that messes up the unforced precession rate does not mess up the forced precession. Also, mass distributions that deviate from the axisymmetric approach that leads to the J2 could alter the correct torque rescaling, or what's even more interesting, there could be mass distributions that behave similarly in the torque rescaling (especially if the inclination angle i is not important) as the J2 type does, and if the torque rescaling is the same, the result for the moment of inertia is the same, even if J2 is the wrong way to describe the deviation of the mass distribution from spherical. The answer those notes get will be just the same any time the torque rescaling is the same as they use-- the value of J2 never matters in that analysis, its numerical value can be removed completely in the way I showed above.

grant hutchison

2016-Aug-19, 11:23 PM

I think the "J2 assumption" is tested outside the model discussed in my links, because low satellite orbits are a sensitive probe for harmonics other than J2. So you check the model before you use the model.

Grant Hutchison

Ken G

2016-Aug-20, 01:15 AM

Deriving J2 in the way I described is, however, part of the standard way of deriving moments of inertia for planets and satellites.But you do understand that the only place J2 appears in the derivation of the moment of inertia of the Earth is in the torque rescaling from the satellite to the Moon, right? The business end of the torque rescaling is the sin(2i)/r3 factor, that's the whole story there, the sole requirement for the determination of the Earths moment of inertia tensor using that method. If one simply uses that torque rescaling, in the equations I gave above, one gets precisely the same result without ever mentioning J2. Indeed, if one uses a satellite with the same i as the Moon, one only needs the 1/r3 part, and that's simply the next term in the potential after the spherically symmetric part and the necessary absence of a dipole moment about the center of mass.

chornedsnorkack

2016-Aug-20, 07:25 AM

We have a three body system. Earthīs bulge, Moon and Sun. And therefore three torques. Torque between Earth bulge and Moon, torque between Earth bulge and Sun, and torque between Moon and Sun.

How can these three bulges be measured separately?

Ken G

2016-Aug-20, 03:59 PM

An important question, but there is actually a way to do it. It's the usual thing in astronomy that makes these kinds of separations possible-- the extreme scale separations, in time and distance, that we have the luxury of (consider for example how the Sun's and Moon's gravity affect the Earth's orbit), but sometimes two effects are competitive (like the tidal effects of the Sun and Moon on Earth). But even when the effects are competitive, there are ways to separate them.

So here, it turns out that the Sun's gravity and Moon's gravity do indeed produce torques on Earth that are quite similar (indeed the scale of that effect is on the same order as the tidal effects, the notes Grant gave showed both types of effect scale like M/r3), so the effects of both those torques on the Earth's rotation (they both produce mostly precession of the Earth's axis) are similar and add together. That would make them hard to separate, but it's OK because each of those torques would be hard to measure anyway-- as they both depend on the Earth's interior mass distribution. So the trick is to recognize that the torque of the Sun on Earth is the negative of the torque of the Earth on the Sun, and the torque of the Moon on Earth is the negative of the torque of the Earth on the Moon. Those can be separated (in principle) by looking at the motions of the Sun and Moon separately.

Now the plot thickens. First of all, we must recognize that the main way the torques on the Earth affect the Earth is by causing precession of the Earth's spin (it also slows the Earth's spin, but that's a much weaker effect that takes much longer, again a scale separation), but the main way the torques by the Earth on the Sun and Moon affect things is by changing the orbits of the Sun/Earth system and the Moon/Earth system (so not by changing the spin angular momentum of the Sun or Moon). These facts are part of the assumptions that go into those cited notes, and although they are not justified there or even mentioned, they must be in place for the analysis to work. Presumably those assumptions are pretty solid, so our skepticism must focus on the most questionable of the assumptions (which I think would be the assumption of rigid rotation).

So now it seems we have an easy way to distinguish torques from the Sun from torques from the Moon, we simply look at the opposite torques on the orbits of those objects relative to Earth. However, this is apparently hard to do-- the easy thing to measure is what is happening to the Earth's rotation, not what is happening to those orbits. Enter torque rescaling.

The crux of the torque rescaling is the idea that with relatively few assumptions about the mass distribution of the Earth, we can know how the torque on Earth from any one object will scale to the torque from any other object that has a different distance r and orbital inclination i. So if you have that torque rescaling factor, you can immediately separate the torques from the Sun from the torques from the Moon, because you know the factor by which the two differ. Then you would only need to observe either the orbit of the Moon around the Earth, or the orbit of the Earth around the Sun, to understand one of the torques, and you could easily scale to the other torque, thereby separating them.

However, it seems we cannot even do that-- neither of those orbits are easy to observe precisely enough, the perturbations on them are too weak. So instead we pick a third object whose orbit we can observe very precisely and is affectly strongly enough to observe, and that is the torque on an orbiting satellite. Then we simply apply the assumed torque rescaling from that satellite to the Moon and to the Sun, and we have the torques from the Moon and the Sun.

So that's the game that is played in those notes to get the torques. The only necessary assumption is that we can know the torque rescaling factor, which in the case of ellipsoidal equipotentials due to oblateness scales like sin(2i)/r3. By looking at satellites with different i and r, we can check our torque rescaling, so even that isn't really an assumption, and it is indeed the torque rescaling that we get from an ellipsoidal Earth. The notes don't say that factor is checked by comparing satellite orbits, but it seems likely to be true.

At this point, what we can know is the torque rescaling factor, and thus the torques on the Earth from all these various objects, including the Moon and the Sun. That will only tell us the moment of inertia of the Earth if we can further assume the Earth is rigid, such that torque = [I]*dw/dt. This actually doesn't work very well for the free precession of the Earth, which is the reason I am skeptical it should work to three decimal places of precision as it is reported in those notes.

Ken G

2016-Aug-20, 04:17 PM

By the way, it would probably help to mention that there is another type of torque in the Earth/Moon system which comes from the bulges the Moon induces on the Earth's oceans. You can't get that torque by rescaling from a satellite because it's too weak to observe, so that's just not the kind of torque these notes are about. That torque takes a very long time to change the Earth's spin, and it works to slow the spin, not create precession-- it's a torque in a different direction and on a different scale. We always have to make assumptions about what to include and what to neglect, so I'm not saying we shouldn't make assumptions, I'm just saying that the one thing you rarely see connected with a nice analysis like the one given in those notes is a reasonably complete list of the assumptions made, along with special emphasis on the assumption regarded as most suspect. When we don't do that, that's when we get "shocked" when we discover a wrong assumption was being ignored.

chornedsnorkack

2016-Aug-22, 06:46 AM

Three obvious problems with "satellite" orbits:

1) Earth has no natural satellites other than Moon

2) A satellite near Earth would be affected by higher unevennesses of Earth mass distribution, not only equatorial bulges

3) A small satellite has no notable effect on Earth rotation.

Whereas, with Moon and Sun, it can be assumed that both Moon and Sun are very far from Earth and are therefore mainly affected by ellipticity of Earth, rather than higher order unevennesses.

However, the orbit of Moon is affected by both Earth ellipticity and by Sun.

Is it possible to predict the effect of Sun on the orbit of Moon with such precision as to measure the effect of Earth ellipticity on Moon orbit?

Ken G

2016-Aug-26, 12:38 AM

Looks like a few posts were lost. This is what I said before, I think:

Three obvious problems with "satellite" orbits:

1) Earth has no natural satellites other than MoonThey don't use natural satellites, they use artificial ones. The closer to Earth the better-- greater torque effects.

2) A satellite near Earth would be affected by higher unevennesses of Earth mass distribution, not only equatorial bulgesThis is why the torque rescaling is important to do right. Some assumptions must be made to rescale the torque on a nearby satellite to the Moon. The trickiest one is the sin(2i) scaling with inclination i, that one requires an axisymmetric mass perturbation (which at least we can see goes to zero torque for i=0 and i=90). I presume they've checked their scaling by looking at multiple satellites, though they don't say they do. If the mass isn't axisymmetric, that won't be right. But it's different from my point that you can't go from the orbits to the mass distribution, because spherical mass components all produce the same orbits. Here you actually want there to be ambiguity-- you want lots of different mass distributions to yield the same torque rescaling, because we're not getting the mass distribution by looking at the orbits, we are getting the torque rescaling that way. We still get the mass distribution by looking at how the Earth's spin responds to torques.

3) A small satellite has no notable effect on Earth rotation.This doesn't matter, because we never care about how the artificial satellite affects the Earth, we care about how the Earth affects the satellite. We want to know the torque of Earth on other things, since we will rescale that to its torque on the Moon. That gives us the Moon's torque on Earth, which does matter.

Whereas, with Moon and Sun, it can be assumed that both Moon and Sun are very far from Earth and are therefore mainly affected by ellipticity of Earth, rather than higher order unevennesses.The torque rescaling must connect those distant objects to nearby orbits, that's true. That torque rescaling is the key assumption being made, it's the guts of everything in those notes, but I think it works as long as the mass distribution is axisymmetric. Of course, it might not be, strange things happen. The bigger problem is probably the assumption that the Earth responds to the Moon's torque like a rigid object would respond-- that apparently doesn't work at all well for the small "unforced" precession of the Earth's spin (the "Chandler wobble"), so I'm not sure how they can know it should work better for the forced precession from the Moon and Sun.

However, the orbit of Moon is affected by both Earth ellipticity and by Sun.We never look at the orbit of the Moon, we don't need it. Presumably it's hard to measure, or there are other effects making it hard. That's why we don't get the torque from the Moon by looking directly at the torque on the Moon.

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