How to calculate the force of gravitational attraction in co-orbiting planets?

[Robittybob1]

I help out many who know less than me, and appreciate help too from others, but I was getting excited by the topic and the two articles at the same time appearing that read like I thought it did. I'll re-read in light of what you've said.

 

So in your simulation did you start Theia off from zero (or virtually zero)?

Another article about the Moon formation in a similar vein to yesterday's article.

http://www.irishexaminer.com/examviral/science-world/is-this-the-final-piece-to-the-puzzle-of-how-the-moon-was-formed-323087.html

If it was formed from the same stuff where was it positioned? Was Theia at L3 Lagrangian Point or in one of these horseshoe orbits?

Janus - in your simulation did you start Theia off at the L3 point from stationary?

I didn't say I agree what was written is this article other than it was "in a similar vein" to the one quoted the day before.

It seems the writer is a bit confused as to when the mixing occurred. I will have to try and read the original work by the scientists from Maryland.

 

A team from the University of Maryland has managed to reconcile the accepted theory with the bits that don’t quite fit.

 

“The problem is that Earth and the moon are very similar with respect to their isotopic fingerprints, suggesting that they are both ultimately formed from the same material that gathered early in the solar system’s history,” said Richard Walker, a professor of geology at the University of Maryland and co-author of the study.

Richard Walker maryland university.

https://cmns.umd.edu/news-events/features/2944

https://www.geol.umd.edu/directory.php?id=21

 

 

 

“The small, but significant, difference in the Tungsten isotopic composition between Earth and the moon perfectly corresponds to the different amounts of material gathered by Earth and the moon post-impact,” Walker said. “This means that, right after the moon formed, it had exactly the same isotopic composition as Earth’s mantle.”

This finding supports the idea that the mass of material created by the impact, which later formed the moon, must have mixed together thoroughly before the moon coalesced and cooled. This would explain both the overall similarities in isotopic fingerprints and the slight differences in Tungsten-182.

That still does not eliminate mixing prior to the Earth and Theia forming in co-orbits. Has he considered that yet?

Unless it is hidden in these words, the "Earth and the moon are very similar with respect to their isotopic fingerprints, suggesting that they are both ultimately formed from the same material that gathered early in the solar system’s history”.

"Early in the solar systems history" would then be before the planet Theia collided with the Moon.

I have just re-read the post above that I wrote yesterday and I find the word "isotopically" in my post that I don't know what it means and it doesn't appear to be a spelling mistake. That seems to be a miracle if that is the right word used in the right context. Where and how did that word appear in my post?

In fact that part sentence "they don't have to be on the same orbit but to be very close isotopically," is not what I understand or intended to write, so not only a word is inserted but a whole thought! What is going on?

 

What does isotopically mean?

I think I do remember writing that now. I was trying to clarify that a planet at the L3 position is not necessarily at exactly the same distance from the Sun (not on the same orbit) but close to it and hence in a region that while the material was in the protoplanetary dust disc stage it would have mixed sufficiently to be isotopically similar.

Edited April 10, 2015 by Robittybob1

[Robittybob1]

.....That still does not eliminate mixing prior to the Earth and Theia forming in co-orbits. Has he considered that yet?

Unless it is hidden in these words, the "Earth and the moon are very similar with respect to their isotopic fingerprints, suggesting that they are both ultimately formed from the same material that gathered early in the solar system’s history”.

"Early in the solar systems history" would then be before the planet Theia collided with the Moon.

I think I do remember writing that now. I was trying to clarify that a planet at the L3 position is not necessarily at exactly the same distance from the Sun (not on the same orbit) but close to it and hence in a region that while the material was in the protoplanetary dust disc stage it would have mixed sufficiently to be isotopically similar.

"close isotopically" must be more than just some other spot in the Solar System with close to the exact same ratio of isotopes. Could there be a similar place other than within a co-orbital region?

Edited April 12, 2015 by Robittybob1

[Robittybob1]

Janus if you are around - in the simulation did you start Theia off at the L3 point from stationary?

[imatfaal]

Janus if you are around - in the simulation did you start Theia off at the L3 point from stationary?

 

A planet at another planet's L3 which is stationary with respect to the central mass will fall almost straight inwards.

[pavelcherepan]

 

A planet at another planet's L3 which is stationary with respect to the central mass will fall almost straight inwards.

 

From a lot of arguing with Robittybob1 I think what he meant was "stationary with respect to the Earth", not with respect to the Sun

[swansont]

 

From a lot of arguing with Robittybob1 I think what he meant was "stationary with respect to the Earth", not with respect to the Sun

 

That doesn't work, either. The L3 point orbits the earth, in the earth's frame.

[Robittybob1]

 

That doesn't work, either. The L3 point orbits the earth, in the earth's frame.

Have you seen that written down somewhere? The L3 point can be a point hidden behind the Sun from the Earth's frame. I can't see how anyone can say it orbits the Earth sorry.

 

From a lot of arguing with Robittybob1 I think what he meant was "stationary with respect to the Earth", not with respect to the Sun

Yes, like those horseshoe orbits demonstrated by Janus they were all wrt the Earth. Both the Earth and the L3 planet orbit the Sun on a yearly basis as well as doing the toing and froing wrt to each other. Well that is the picture in my head at the moment.

 

A planet at another planet's L3 which is stationary with respect to the central mass will fall almost straight inwards.

The planets set themselves up in these positions because they have a degree of stability at these points. The orbital speeds are increased (shortened radius and period) sufficient to overcome the additional apparent central mass due to the associated L3 planet.

 

Each astronomical body has it's own Lagrangian points and I think the ones we a tlking about are the Sun's Lagrangian points.

So there are the Earth's Lagrangian points as well as shown in this animation.

Here the L3 orbits the Earth but is is a completely different L3 than I'm speaking of in the thread.

"lagrange points animation" should really be "Earth's lagrange points animation"

 

The better mathematical approach to the Sun's Lagrangian points is in "The Lagrangian Points are Awesome... Tutorial"

Edited April 12, 2015 by Robittybob1

[imatfaal]

For someone who clearly does not have a grounding in the subject you really do jump into lecturing others very quickly

 

Have you seen that written down somewhere? The L3 point can be a point hidden behind the Sun from the Earth's frame. I can't see how anyone can say it orbits the Earth sorry.

 

Plot the displacement of planet X with respect to planet Y on a piece of paper. It stays the same distance away - but does that define rest (hint the earth stays the same distance from the sun)

 

 

Yes, like those horseshoe orbits demonstrated by Janus they were all wrt the Earth. Both the Earth and the L3 planet orbit the Sun on a yearly basis as well as doing the toing and froing wrt to each other. Well that is the picture in my head at the moment.

 

I am not sure what the period of the variation in the orbit was - hopefully Janus will be able to enlighten

 

 

Each astronomical body has it's own Lagrangian points and I think the ones we a tlking about are the Sun's Lagrangian points.

So there are the Earth's Lagrangian points as well as shown in this animation.

 

How can you have got so far into this thread thinking that a single body has Lagranian points. The Lagrangian Point is the point at which TWO bodies allow a third smaller body to orbit in a stable position with respect to both of them.

[Robittybob1]

....

How can you have got so far into this thread thinking that a single body has Lagranian points. The Lagrangian Point is the point at which TWO bodies allow a third smaller body to orbit in a stable position with respect to both of them.

What I was wondering is how do you name a Lagrangian Point? In your above statement the "TWO bodies" are orbiting a barycenter eg the Sun - Earth barycenter or the Earth - Moon barycenter, so when you introduce the third object do we say "at the Sun's Lagrange point" or "at the Earth's Lagrange point" or combination "Sun - Earth system Lagrange point" or the "Earth - Moon system Lagrange point". Everyone up to now just says "at the Lagrange point" but not what system the Lagrange point relates to.

So I getting a bit confused if Swansont says the L3 orbits the Earth.

http://www.scienceforums.net/topic/88515-how-to-calculate-the-force-of-gravitational-attraction-in-co-orbiting-planets/page-2#entry863348

Which system's L3 orbits the Earth?

Edited April 12, 2015 by Robittybob1

[imatfaal]

1. Take a read of the Wikipedia page - you wil note the terminology as Sun-Earth L₃.

2. You will note I said two large masses and one small - this means you try to simplify away the fact that it is a three body by treating the small mass at the lagrange point as not affecting the others. You don't get objects of comparative mass at a steady orbit at Lagrange points - the maths only works when you can almost ignore the mass of the third object.

3. I don't think anyone is confused apart from yourself - you are talking about a planet in orbit around the sun interacting with Earth. It makes sense that the two large masses are the Earth and the Sun - this would be the Sun-Earth L₃. Which btw is an inherently unstable equilibrium (ie small perturbations will disrupt it) - I don't believe there is much there. L₄ and L₅ are much more stable

 

I must admit I haven't be paying great attention - other lagrange points could have been mentioned or alluded to. As I, studiot, SwansonT, Janus and others have all mentioned - this is not a simple matter. We can make some approximations in some circumstances but for any precision in real world scenarios you need awesome maths and then heavy duty simulation

[Robittybob1]

Plot the displacement of planet X with respect to planet Y on a piece of paper. It stays the same distance away - but does that define rest (hint the earth stays the same distance from the sun)

from http://www.physicsclassroom.com/class/1DKin/Lesson-1/Scalars-and-Vectors

Scalars are quantities that are fully described by a magnitude (or numerical value) alone.

Vectors are quantities that are fully described by both a magnitude and a direction.

 

So the distance can be constant but the distance vector is continually changing.

What was the point of that question?

"Rest" defined by Wikipedia "Rest, in physics, refers to an object being stationary relative to a particular frame of reference or another object."

1. Take a read of the Wikipedia page - you wil note the terminology as Sun-Earth L₃.

2. You will note I said two large masses and one small - this means you try to simplify away the fact that it is a three body by treating the small mass at the lagrange point as not affecting the others. You don't get objects of comparative mass at a steady orbit at Lagrange points - the maths only works when you can almost ignore the mass of the third object.

3. I don't think anyone is confused apart from yourself - you are talking about a planet in orbit around the sun interacting with Earth. It makes sense that the two large masses are the Earth and the Sun - this would be the Sun-Earth L₃. Which btw is an inherently unstable equilibrium (ie small perturbations will disrupt it) - I don't believe there is much there. L₄ and L₅ are much more stable

 

I must admit I haven't be paying great attention - other lagrange points could have been mentioned or alluded to. As I, studiot, SwansonT, Janus and others have all mentioned - this is not a simple matter. We can make some approximations in some circumstances but for any precision in real world scenarios you need awesome maths and then heavy duty simulation

The instability of L3 is not an issue for if Theia started from there it is able to escape the L3 position and move around (in those horseshoe orbits) for some millions of years picking up mass until the orbits allowed for Earth and Theia to collide or for Theia (the Moon) to be captured into an orbit around the Earth.

I'll see if I can get someone just to run the Three Body program but I'm not in any hurry. I think the questions from the OP have been answered as best as possible at the moment.

Thanks to those who have contributed.

[Robittybob1]

 

2. You will note I said two large masses and one small - this means you try to simplify away the fact that it is a three body by treating the small mass at the lagrange point as not affecting the others. You don't get objects of comparative mass at a steady orbit at Lagrange points - the maths only works when you can almost ignore the mass of the third object.

http://en.wikipedia.org/wiki/Lagrangian_point#History

In 1772, Joseph-Louis Lagrange published an "Essay on the three-body problem". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits.[3]

 

 

Does the phrase "for any three masses" mean that the third one has to be much smaller than the rest?

 

That doesn't work, either. The L3 point orbits the earth, in the earth's frame.

http://en.wikipedia.org/wiki/Lagrangian_point#/media/File:Lagrange_points_simple.svg

How does the Sun - Earth L3 point orbit the Earth?

"In the Earth's frame" is that like saying "the Sun orbits the Earth in the Earth's frame"? If that is so then the Sun - Earth L3 orbits the Earth too but what a way of looking at the Solar System!

Edited April 13, 2015 by Robittybob1

[xyzt]

 

Strange - agree completely. BTW - last time I was telling RBob this I tried looking for a list of the three body problems that were amenable to an analytic rather than numerical/simulation solution. I know that there is a decent number of situations which have been "solved" - anybody ever seen a list?

See here

[Robittybob1]

http://en.wikipedia.org/wiki/Lagrangian_point#History

 

Does the phrase "for any three masses" mean that the third one has to be much smaller than the rest?

http://en.wikipedia.org/wiki/Lagrangian_point#/media/File:Lagrange_points_simple.svg

How does the Sun - Earth L3 point orbit the Earth?

"In the Earth's frame" is that like saying "the Sun orbits the Earth in the Earth's frame"? If that is so then the Sun - Earth L3 orbits the Earth too but what a way of looking at the Solar System!

I keep on thinking about the dynamics of that situation. It will work as long as you don't calculate that the centripetal force has to equal the gravitational force. The Sun has just too much inertia to ever be made to orbit the Earth.

See here

I can't imagine ever doing the math, but they don't seem to work with the centrifugal forces as much as I think they should. So I am tempted.

Thanks xyzt.

Here is a paper that talks of gravitational and centrifugal forces solving the Three-body Problem and is just about clear enough for me to give the math a go.

Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem

http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec18.pdf

 

From a lot of arguing with Robittybob1 I think what he meant was "stationary with respect to the Earth", not with respect to the Sun

Janus did come up with a very interesting number http://www.scienceforums.net/topic/88515-how-to-calculate-the-force-of-gravitational-attraction-in-co-orbiting-planets/#entry862603

 

I've [Janus] been running a simulation since last night of this situation, it has been running for 25,000 simulator years, repeating this pattern roughly every 550 yrs with no collision yet. How long this will continue is anyone's guess and there is no good way to estimate it.

But I always wanted to know if the first loop of the horseshoe orbit was started off at zero km/sec.

At least tell us how you got the L3 planet to start moving around the orbital perimeter?

Edited April 13, 2015 by Robittybob1

[swansont]

How does the Sun - Earth L3 point orbit the Earth?

"In the Earth's frame" is that like saying "the Sun orbits the Earth in the Earth's frame"? If that is so then the Sun - Earth L3 orbits the Earth too but what a way of looking at the Solar System!

Yes, precisely. Everything orbits the earth in the earth's frame. I can't think of a way that something at L3 would be considered stationary, and was pointing out that using the earth as a frame does not do that.

[Robittybob1]

Yes, precisely. Everything orbits the earth in the earth's frame. I can't think of a way that something at L3 would be considered stationary, and was pointing out that using the earth as a frame does not do that.

What I was calling "stationary" was that line on which "the first three Lagrangian points are on the line connecting the two large bodies". If it was still on that line it was "stationary" and therefore I was interested in how fast did it move off that line on an orbital path around the Sun? That would have been its initial speed.

[swansont]

What I was calling "stationary" was that line on which "the first three Lagrangian points are on the line connecting the two large bodies". If it was still on that line it was "stationary" and therefore I was interested in how fast did it move off that line on an orbital path around the Sun? That would have been its initial speed.

No, that's not at all clear from your post. It's not proper use of the terminology.

[Robittybob1]

No, that's not at all clear from your post. It's not proper use of the terminology.

OK so what would you call that? All those bodies and points are on a line, and while they orbit and stay on the line we'd call them stable, but they are prone to be unstable at these 3 Lagrangian Points (L1, L2 and L3), so how do you describe the bodies moving away from these points? They move at a rate wrt that line.

Edit:

The other points that are on that line are the two - five barycenters depending on the number of objects at the Lagrangian Points along that line.

Edited April 13, 2015 by Robittybob1

[Robittybob1]

No, that's not at all clear from your post. It's not proper use of the terminology.

I'd like you to tell me what it should be called then.

[swansont]

I'd like you to tell me what it should be called then.

The simplest reference would be "at rest with respect to L3"

 

Motion is always in reference to something.

[Robittybob1]

The simplest reference would be "at rest with respect to L3"

 

Motion is always in reference to something.

Good answer, but not that much different as the L3 point is on the same line anyway.

[swansont]

Good answer, but not that much different as the L3 point is on the same line anyway.

Which isn't how you phrased it, so that's moot.

[Robittybob1]

Which isn't how you phrased it, so that's moot.

My first question to Janus (regarding this) used the L3 point as my reference point "Janus - in your simulation did you start Theia off at the L3 point from stationary?" http://www.scienceforums.net/topic/88515-how-to-calculate-the-force-of-gravitational-attraction-in-co-orbiting-planets/page-2#entry862799

The speed through or away/toward the L3 would make a big difference to the simulation. How did Janus assign an initial value to that?

[swansont]

My first question to Janus (regarding this) used the L3 point as my reference point "Janus - in your simulation did you start Theia off at the L3 point from stationary?" http://www.scienceforums.net/topic/88515-how-to-calculate-the-force-of-gravitational-attraction-in-co-orbiting-planets/page-2#entry862799

The speed through or away/toward the L3 would make a big difference to the simulation. How did Janus assign an initial value to that?

The L3 point isn't stationary. That's what started this triviality.

[Robittybob1]

The L3 point isn't stationary. That's what started this triviality.

We must be talking relative motion here. Of course the L3 stays in the same alignment as ever but the unstable planet there moves off with some speed. What speed did Janus give it?