Three body problem solution

[Sigma88]

I have found a "PHYSICAL REVIEW LETTERS" paper that describes this solution to the three body problem

(C. Moore, Phys. Rev. Lett. 70, 3675 (1993).)

Now, I'm not a Physicist, so I need some help to understand the relationship between mass, distance and orbital period in this solution.

anyone here studies this kind of things?

[K^2]

I have found a "PHYSICAL REVIEW LETTERS" paper that describes this solution to the three body problem

(C. Moore, Phys. Rev. Lett. 70, 3675 (1993).)

Now, I'm not a Physicist, so I need some help to understand the relationship between mass, distance and orbital period in this solution.

anyone here studies this kind of things?

I haven't looked at the full article, but from the abstract it follows that these are numerical solutions based on desired topology. Therefore, a formula for period does not exist. It has to be numerically evaluated for a particular trajectory. That said, once you have the trajectory, it's a trivial thing to compute. Given the DoF, if you know positions of all 3 bodies, you know their velocities from relevant conservation laws.

I also wouldn't jump to conclusions about stability, unless article specifically states that they have found this configuration to be stable.

P.S. Actually, if I was solving this, I'd make period fixed, and let it expand/contract during optimization. So it'd be a way of computing distances from masses and period instead. But the point stands, there isn't a formula. It's a numerical problem.

Edited July 14, 2015 by K^2

[Sigma88]

I've managed to mod this using kopernicus (I have obtained the shape of the orbit).

Now I can tweak distance/mass/period to suit whatever ratio I want.

I don't need the numbers to be 100% accurate, I just need them to be not clearly wrong.

I know the moment I release this people will go

Is there some simulator I could use?

[Sigma88]

A lot of sites that talk about this kind of orbits say that

They use carlos simo's data.

But noone puts a reference to which data they are talking about

[Weywot8]

Is there some simulator I could use?

There'd only be a handful of people around the forum really qualified go "Actually..." that like this guy http://forum.kerbalspaceprogram.com/threads/68502-WIP-Principia-N-Body-Gravitation-and-Better-Integrators-for-Kerbal-Space-Program. (Hint on who to ask next:wink:. Or bug Scott Manley? Pretty sure he has an astro-computer PhD background from the occasional lingo he uses)

Came to the same conclusion as K^2 but figured you'd want to replicate and scale this. Data is locked in the file located here http://www.maia.ub.es/dsg/3body , assumed you've found it. However, scaling (heavier bodies, trying for longer orbital periods) would be computational as well because there isn't an underlying formula unlike Kepler's Law. The figure 8 is actually more stable than Laplace's equilateral triangle orbit. Also more robust cause m1=m2=m3 is not a strict requirement (orbit changes from ideal though). So less accurate numerical methods can be used and "actually..." KSP isn't "real" either.

My two pence worth of help would be directions to Section 5 of this website: http://www.artcompsci.org/msa/web/vol_1/v1_web/node6.html and bumping across http://www.control.aau.dk/~jan/undervisning/MechanicsI/mechsys/chapter8 which is very followable.

Hope it helps and leave the other 350 3-body orbits alone:sticktongue:

[Sigma88]

leave the other 350 3-body orbits alone:sticktongue:

Sorry, I can't promise anything

Jokes aside

it looks like it would be a lot of effort. I think I'll look for a gravity simulator and try some combination of mass/distance/velocity to see if I can get three stars to orbit themselves for like 2-3 orbits (that will be enough for me)

as you said KSP is not really "real", and anyway my solution is already very gimmicky so it's not like it'll be much realistic anyways.

Scott Manley was of course the first person I thought of, but if you've seen his video on installing RSS you may have noticed how many notifications he has.

I'll try writing to him, but I won't be surprised if he don't even see the message.

[Weywot8]

http://www.emis.de/journals/Annals/152_3/chencine.pdf initial velocity vectors and position at bottom of first page. middle of figure 8 is origin coordinates "0, 0i". Mass and gravitational constant set at "1" I think so μ is 1. Try accounting for that if things crash in cause the real "G" ain't 1. Note relationship of positions and magnitude of velocity.

Good Luck!

[SciMan]

To me, this looks related to a Klemperer rosette. They use an even number of bodies with 2 different mass types. (I'll refer to them as "True Klemperer rosettes")

However, the article also mentions a configuration that has 5 bodies of equal mass, and this seems to be a 3 body version of a similar system. (I'll refer to them as "Niven rosettes" because of their use by that Sci-Fi author)

The article states that True Klemperer rosettes can be created with any even number of bodies.

Because of this, I think that Niven rosettes can be created with any prime number of bodies, and possibly any odd number of bodies.

As far as needing a good gravity simulator, the only affordable one that I can think of that would even remotely fit this problem's requirements would be Universe Sandbox, but I hesitate to suggest it as it's not exactly known for stability AFAIK.

[Sigma88]

To me, this looks related to a Klemperer rosette. They use an even number of bodies with 2 different mass types. (I'll refer to them as "True Klemperer rosettes")

However, the article also mentions a configuration that has 5 bodies of equal mass, and this seems to be a 3 body version of a similar system. (I'll refer to them as "Niven rosettes" because of their use by that Sci-Fi author)

The article states that True Klemperer rosettes can be created with any even number of bodies.

Because of this, I think that Niven rosettes can be created with any prime number of bodies, and possibly any odd number of bodies.

As far as needing a good gravity simulator, the only affordable one that I can think of that would even remotely fit this problem's requirements would be Universe Sandbox, but I hesitate to suggest it as it's not exactly known for stability AFAIK.

I knew I was going to regret passing on Universe Sandbox at 2.5$ when it was on sale last weekend, damn!

EDIT:

I think I've found a solution

Edited July 14, 2015 by Sigma88

[Sigma88]

ok, so to me it looks like I can use the standard formula for orbital period,

T = 2À sqrt ( a^3 / ( G M ) )

where in this example:

a is the maximum distance from the barycenter

and

M is the mass of one star

thank you everybody for your tips

[kiwi1960]

From a non technical standpoint... and looking at the example, the three objects are always exactly far away from each other at all times.

That sort of makes it simpler for me to understand, however, in reality they wouldn't be the exact same mass. Eventually, something will have to give.

[Sigma88]

Eventually, something will have to give.

Actually going through the various papers I found some quotes saying that this system is more stable if the masses are a little different.

it has something to do with the precession of orbits in a sphere or something like that.

anyways, this is KSP so I just needed a "rule of thumb"

[SAI Peregrinus]

http://www.orbitsimulator.com

Free n-body gravitational system simulator.

[Weywot8]

ok, so to me it looks like I can use the standard formula for orbital period,

T = 2À sqrt ( a^3 / ( G M ) )

where in this example:

a is the maximum distance from the barycenter

and

M is the mass of one star

thank you everybody for your tips

didn't notice the "elliptical orbit with torsion" proved numerically. Good catch!