Gravity Simulator - Bound trajectories check

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Bound trajectories check (Read 2146 times)

frankuitaalst

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Bound trajectories check
09/21/14 at 11:14:10

It is known that in a two body problem a planet is bound if the total energy is smaller than zero
( elliptical orbit ) .
In this case we have : E= Ep+Ek <0 : mv²/2 - GmM/r <0 . This means the velocity should not be too big .

I now have a system of a planet orbiting a binary.
And want to have a quick check if the planet is bound to the binary.
Any idea's how I should proceed ?

mv²/2 -GmMtot <0 (vrel to center of mass Mtot )
or :
m Vrel1²/2+ mVrel2²/2 - GmM1/r1 -GmM2/r2 <0 ( two bodies considered )

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Tony YaBB Administrator Posts: 1054 Gender:	Re: Bound trajectories check Reply #1 - 09/22/14 at 06:38:58 Compute the center of mass of the binary. Pretend all the mass was there, and treat it as a single object. Now see if energy is negative, or check if ecc < 1 or sma is positive.
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frankuitaalst

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Re: Bound trajectories check
Reply #2 - 09/23/14 at 13:21:43

Quote from Tony on 09/22/14 at 06:38:58:

Compute the center of mass of the binary. Pretend all the mass was there, and treat it as a single object. Now see if energy is negative, or check if ecc < 1 or sma is positive.

Thanks Tony , you think treating the binary at its COG gives better results than taking account of the two seperate masses ?
I think in general yes it is but :

If COG is used one does not take into account a change in energy due to a close encounter of one part of the binary .
I'm not very sure in this case .
Is there a way to test in GravSim the result if I would test the second approach ?

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frankuitaalst

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Re: Bound trajectories check
Reply #3 - 09/24/14 at 22:41:33

I gave it another thought .
Of couse you are right Tony , at long distances this is indeed the criterium .
I just wondered If at close distances the separate approach could be appropriate . I had the impression that although a body which is in eliptical orbit around a binary can be trown out after a "close " approach to one part of the binary .

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Tony

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Re: Bound trajectories check
Reply #4 - 09/25/14 at 18:26:47

I think the secondary bodies have to be about 5x as far as the aphelion of the primary stars for stable orbits.
Gravity Simulator can do this.

File > New
Create another star around the Sun.

File > New
Create a low-mass body around the Sun and choose "barycenter" in the focus object.

There's also a View > View Orbital Energy option, but I think I coded it wrong. It's not doing what I expect. You could always output data to view in Excel. Energy of the system is the sum of U + KE between every body in the system (-GMm/d^2 + 0.5 * m * v^2). It should be constant.

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