Trojan

[Dave Lee]

I would like to get better understanding on trojan phenominun.

Trojan:

https://en.wikipedia.org/wiki/Trojan_(astronomy)

Please look at Jupiter_trojan

https://en.wikipedia.org/wiki/Jupiter_trojan

"The Jupiter trojans are divided into two groups: The Greek camp in front of and the Trojan camp trailing behind Jupiter in their orbit."

What kind of force holds all the Asteroids of "Trojan" camp or "Greeks" camp together at the same spot?
Is it just a simple gravity by Newton power?
How do they do it?

[glappkaeft]

What kind of force holds all the Asteroids of "Trojan" camp or "Greeks" camp together at the same spot?
Is it just a simple gravity by Newton power?
How do they do it?

Yes.

https://en.wikipedia.org/wiki/Lagran...atical_details (you really should read the wiki pages more carefully - that article was linked in the first paragraph of both the pages you linked)

[Dave Lee]

Yes.

Thanks

It's amazing!
Trojan - Hundred of thousands of Asteroids are forced to be together at the "Trojan" camp or "Greeks" camp while they orbit the Sun and keep their position - all of that, just by gravity power.

[Dave Lee]

However, if Torjan works for Asteroids why it can't work for stars?

[swampyankee]

However, if Torjan works for Asteroids why it can't work for stars?

Gravity is universal; nonetheless a stellar triplet with a star in one of the stable Lagrange points is unlikely exist, as the mass at the Lagrange point has to be very small relative to either of the major bodies and the larger major body must be about 25 times the mass of the lesser. I don’t know what the limiting mass of the object at the Trojan points is, but the derivation of the Lagrange points is likely invalid if there is non-negligible mass at the 5 Lagrange points.

See http://math.ucr.edu/home/baez/lagrange.html for more information

[Dave Lee]

Gravity is universal; a stellar triplet with a star in one of the stable Lagrange points is unlikely exist, asthe mass at the Lagrange point has to be very small relative to either of the major bodies and the larger major body must be about 25 times the mass of the lesser. See http://math.ucr.edu/home/baez/lagrange.html for more people information

If gravity is Universal, why it can't work also in a spiral galaxy?

https://en.wikipedia.org/wiki/Milky_...y_Way_Arms.svg

Why we are so sure that there is no gravity effect in spiral arms?
Why we do not try to understand how Newton power had set the shape of spiral galaxy?

Let's focus on Orion Arm as it is quite Unique.
There is no significant difference in the distance between its head or tail to the center of the galaxy.
So it really looks similar to Trojan camp in Jupiter.

Hence, why Orion Arm couldn't be considered as direct product of trojan phenomenon?
Why the Sun couldn't be considered as part of the "Orion camp"?

If we verify carefully the trojan phenomenon in Jupiter, try to understand how Hundred of thousands of Asteroids are forced to be together at the "Trojan" camp due to gravity power, we will surly solve the Spiral galaxy enigma.

[Strange]

If gravity is Universal, why it can't work also in a spiral galaxy?

Why do you think it doesn't?

[Hornblower]

To have the Trojan phenomenon we need a primary central body, a satellite body with not more than about 1/25 the primary's mass, and some small stuff of vanishingly small mass relative to the secondary. In addition we must not have too much perturbation from other massive satellites. Then the small stuff can be maintained in stable Trojan orbits. This sort of mass distribution simply does not occur in a galaxy.

[Dave Lee]

To have the Trojan phenomenon we need a primary central body, a satellite body with not more than about 1/25 the primary's mass, and some small stuff of vanishingly small mass relative to the secondary. In addition we must not have too much perturbation from other massive satellites. Then the small stuff can be maintained in stable Trojan orbits. This sort of mass distribution simply does not occur in a galaxy.

Spiral galaxy is not a simple orbital system as the Solar system.
If it was so simple, than by definition it had to be a simple orbital system.

It seems that we do not understand how spiral system really works.
We are missing the key functionality of the spiral structure.
It has several key elements including the primary central body (SMBH), ring, spiral arms and more.
If you focus only on one segment you are missing the main point of the spiral galaxy activity.
Therefore, it is a severe mistake to take a decision just by looking on the primary central body of the galaxy or any other key element which is needed to proper functionality of simple orbital system.

Again - if our expectation for spiral galaxy is based on a simple orbital system as solar system then by definition the spiral galaxy has to be a simple orbital system.
As the spiral galaxy is completely different from simple system - the outcome must be different
Why we do not try to understand the unique structure of spiral galaxy and look how Newton power and Trojan phenomenon effects the spiral galaxy?
How can we neglect The MOST important power in the nature?

Why do you think it doesn't?

Do you mean that gravity and Trojan phenomenon works in spiral galaxy?
If so, how?

[grant hutchison]

To have the Trojan phenomenon we need a primary central body, a satellite body with not more than about 1/25 the primary's mass, and some small stuff of vanishingly small mass relative to the secondary. In addition we must not have too much perturbation from other massive satellites. Then the small stuff can be maintained in stable Trojan orbits. This sort of mass distribution simply does not occur in a galaxy.

Yes, the operative phrase is "three body problem" - once you introducing perturbers, those stable points may become unstable. We haven't found any Trojans associated with Saturn, for instance (the last time I looked), because Jupiter's gravity means that trojan orbits in that location are unstable in the short (astronomical) term.

Grant Hutchison

[grant hutchison]

How can we neglect the gravity effects and Trojan phenomenon in spiral galaxy?

There's a huge body of literature on the effect of gravity in spiral galaxies. Why would you think it was being ignored?
The "Trojan phenomenon" is a restricted solution to the three body problem, and simply isn't relevant to the n-body dynamics of galaxies.

Grant Hutchison

[Dave Lee]

Why do you claim that :

The "Trojan phenomenon" is a restricted solution to the three body problem, and simply isn't relevant to the n-body dynamics of galaxies.

Grant Hutchison

Have you looked at "Trojan" camp or "Greeks" camp in Jupiter?
Have you noticed how many orbital objects are forced together due to "Trojan phenomenon"?

[swampyankee]

Why do you claim that :



Have you looked at "Trojan" camp or "Greeks" camp in Jupiter?
Have you noticed how many orbital objects are forced together due to "Trojan phenomenon"?


Why do you claim that :

The Langrage points are a solution to the restricted three-body problem, where essentially all the mass is in two bodies. The generalized three-body problem, where all objects have significant mass, has no analytical solution. There may be numerous bodies in the L4 and L5 points of the Sun-Jupiter system, but their combined mass is still miniscule compared to that of either Jupiter or the Sun.

[grant hutchison]

Why do you claim that :



Have you looked at "Trojan" camp or "Greeks" camp in Jupiter?
Have you noticed how many orbital objects are forced together due to "Trojan phenomenon"?

Yes, I have looked. In fact, back in the day, I used to run simulations of the trajectories of these bodies.
Hornblower gave you the answer. The pure Trojan situation, mathematically, involves a zero-mass test object at the Trojan position. In the rotating reference frame, such objects follow a looping trajectory around the Trojan point. Given that they have zero mass, you can put a whole bunch of them in slightly different orbits at that point, and the situation will remain stable. In reality, you can place a whole bunch of objects of very low mass (compared to the primary and secondary) in orbits at that location, and they will interact minimally - but you'll lose some as time goes by, because they can exchange momentum during close encounters with each other.
If you ramp up the mass of the Trojan objects(s), the situation becomes progressively less stable.

Grant Hutchison

[Dave Lee]

The pure Trojan situation, mathematically, involves a zero-mass test object at the Trojan position. In the rotating reference frame, such objects follow a looping trajectory around the Trojan point. Given that they have zero mass, you can put a whole bunch of them in slightly different orbits at that point, and the situation will remain stable.
Grant Hutchison

Wow!
Thanks for your great answer!!!
I couldn't expect for better reply.

First - as Usual, it is relativity.
In the "Trojan" camp or "Greeks" camp in Jupiter the total orbital mass is relativity small compare to the primary.
In the same token, the total mass in Orion arm is small compare to the relevant primary mass in the spiral galaxy (which is not the SMBH).

Second - You have mentioned the "Trojan point".
and this is the MOST important issue in spiral galaxy.
If we could understand this key issue we could also understand the Unique path of the Sun which is going up and down from the disc plan.
It is a simple gravity path.
If you understand how "Trojan point" works in spiral galaxy - you have solved the spiral galaxy structure enigma

[grant hutchison]

First - as Usual, it is relativity.
In the "Trojan" camp or "Greeks" camp in Jupiter the total orbital mass is relativity small compare to the primary.
In the same token, the total mass in Orion arm is small compare to the relevant primary mass in the spiral galaxy (which is not the SMBH).

The fact that Trojan orbits require low relative mass does not imply that low relative mass causes Trojan orbits.

Second - You have mentioned the "Trojan point".
and this is the MOST important issue in spiral galaxy.
If we could understand this key issue we could also understand the Unique path of the Sun which is going up and down from the disc plan.
It is a simple gravity path.

It is indeed a simple gravity path - the sort you get when you place a test mass in orbit within a gravitating disc. There's nothing surprising or unique about it. It has nothing to do with Trojan points.

Grant Hutchison

[Strange]

Do you mean that gravity and Trojan phenomenon works in spiral galaxy?

I thought you were saying that gravity didn't work in spiral galaxies. Maybe I misunderstood.

[tony873004]

It is not necessary that the trojan have insignificant mass compared to the planet. It is only necessary that the ratio of the (mass of the planet + mass of trojan) to the Sun's mass be in one of 2 ranges: ratio is less than 2.85% or the ratio is between 3.1% and 4%.

[George]

It is not necessary that the trojan have insignificant mass compared to the planet. It is only necessary that the ratio of the (mass of the planet + mass of trojan) to the Sun's mass be in one of 2 ranges: ratio is less than 2.85% or the ratio is between 3.1% and 4%.

That's a little surprising. Do you have a recommended reference on this?

[grant hutchison]

That's a little surprising. Do you have a recommended reference on this?

Laughlin and Chambers (2002) did some work on the stability of the Trojan solution for bodies with non-zero mass.

Grant Hutchison

[tony873004]

That's a little surprising. Do you have a recommended reference on this?

This was discussed here a few years ago: https://forum.cosmoquest.org/showthr...-Earth-changes
There are some references in this thread.

[George]

Laughlin and Chambers (2002) did some work on the stability of the Trojan solution for bodies with non-zero mass.

Yes, but that seems to be for equal mass planets in 1:1 resonance where the mass of each would be < 1.98% (3.96% combined), which is mu/2, slightly greater than the actual mass ratio. I don't understand why a mass gets tossed between 2.85% and 3.1%. Is there a georgeeze answer here?

[George]

This was discussed here a few years ago: https://forum.cosmoquest.org/showthr...-Earth-changes
There are some references in this thread.

Oops, I just missed this post. So I will read it when I can.

[tony873004]

I don't understand why a mass gets tossed between 2.85% and 3.1%. Is there a georgeeze answer here?

I don't understand either. I only know that's what I read, and when I simulated it, the results were consistent with those ranges.

Here's a new version of the simulation. This will run in your web browser. It lets you choose the masses of 2 objects spaced 60 degrees apart. It seems to confirm that strange gap between 2.85% and 3.1%. It only takes a few hundred years for the unstable ones to get ejected, so the simulations should only take a few minutes to run.

http://orbitsimulator.com/gravitySim...nMassTest.html

In this simulation, Earth is held stationary by rotating the frame.

To use:
Enter the mass of each object as a percentage of Sun Mass. Initially, M1 is an Earth mass, and M2 is massless.
Below your inputs, it will add them for you.
Press Apply.
Press [>] play on the time step interface. Don't increase the time step or you may introduce numerical errors.

[grant hutchison]

With reference to StupendousMan's post in that other thread:

Tony's excellent work prompted me to do what I should have done a few days ago: get up and walk to the library. I scanned through a book called "The Three-Body Problem" by Christian Marchal, published by Elsevier in 1990. Chapter 8 in that book discusses "Simple Solutions of the Three-Body Problem", one of which is the system Tony proposes: a very massive star (the Sun) and two much less massive bodies (Earth-1 and Earth-2), with the two small bodies orbiting the massive one in circular orbits of the same radius, separated by 60 degrees.

Marchal discusses the stability of these systems at some length, using first-order and higher-order analysis. He provides references to papers by C. Richa (thesis from University of Pierre and Marie Curie, 15 Oct 1980) and himself (seminar at the Bureau of Longitudes, 7 Mar 1968). In his discussion, he defines a couple of parameters which involves the relative masses of the three bodies:

N = sqrt( m1^2 + m2^2 + m3^3 - m1m2 - m1m3 - m2m3) / (m1 + m2 + m3)

which, in our case of m2 = m3, simplifies to (please check my algebra here!)

N = (m1 - m2) / (m1 + 2m2)

and the parameter

R = ( 3 - sqrt(12 N^2 - 3) ) / 6

Now, in the case Tony has suggested, in which m1 = Sun's mass and m2 = m3 = Earth's mass, the parameter N is a fraction very close to 1 (I find N=0.999994), and the parameter R is a number very close to zero (I find R=4 x 10^(-6)). Again, check my arithmetic!

The payoff for computing these values is that Marchal presents a nice graph (his Figure 13 on page 49) showing zones of stability for this "Trojan" arrangement. For circular orbits, there are two zones in which the orbit can be proved stable:

0 <= R < 0.02860...
and
0.02860... < R <= 0.03852....

Note that Tony's system falls well within this first zone of stability. In fact, this suggests a test: the arrangement should still be stable until the two "Earth-like" planets grow in mass so that R = 0.03852, which means N = 0.9428 or so. And that, in turn, if I can do algebra again, means that the system should be stable until each of the two "Earth-like" planets has a mass of m1*(0.01982). In other words, until the two Earth-like planets added together have a mass of about 4 percent of Sun.

Sorry for being so forceful in my incorrectness earlier :-(

Tony, thanks for sticking to your guns and helping me to learn something new today ....

Marchal actually gives a precise value for the parameter R at which circular motion is unstable - it's (3-sqrt(8)/6). That corresponds to his parameter N = sqrt(11/12). So in Marchal's analysis there's no range of unstable conditions in this vicinity - just a single point. Of course in the real world there are always perturbations, so we can expect to see unstable behaviour arising in the vicinity of this point in parameter space, but I'm not clear why Tony's simulation should show a range of unstable values.

For reference, here's the relevant page (click to enlarge):


Grant Hutchison

[Dave Lee]

It is not necessary that the trojan have insignificant mass compared to the planet. It is only necessary that the ratio of the (mass of the planet + mass of trojan) to the Sun's mass be in one of 2 ranges: ratio is less than 2.85% or the ratio is between 3.1% and 4%.

Thanks Tony.

[Dave Lee]

So the main question should be:
How can we explain the structure of spiral galaxy by using "Trojan point"?

However, before we start, we need to understand the basic ideas of "Trojan point":

1. "Trojan point" works locally. (As the needed mass ratio is quite low - Please see the above message from Tony)
2. Locally is relativity
3. Each star in the galaxy might have its own "Trojan point".
4. However, "Trojan points" works as branches in a tree.

After that, lets look again on the following image of the Milky way:

https://en.wikipedia.org/wiki/Milky_...y_Way_Arms.svg

Let me start by the broken chain of the arms.
Please focus on the presues tail.
There are about 12 parts for this chain.

So what kind of glue is needed to hold this chain?
Why the parts do not disconnect from each other?

Please look carefully, it seems that each part of the chain is a single Trojan camp.
I assume that in each Trojan camp there are hundreds of thousands stars.

So, where is the primary point for each Trojan camp?
Is it the center of the galaxy?
No, as the center is too far away and gravity works locally.
So where is it?

[glappkaeft]

So the main question should be:
How can we explain the structure of spiral galaxy by using "Trojan point"?

We don't. The spiral structure is explained by density waves.

[tusenfem]

Okay, this thread has passed well beyond the Q&A stage, but interesting answers!
I will move this to S&T, however, if Dave Lee keeps insisting on "trojan spiral galaxy" he needs to take this to ATM.

[Dave Lee]

Clear.

Would you kindly move this tread to ATM so I can continue the discussion?