Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Atomic-S on 15/10/2009 05:28:10
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Given 2 bodies each of mass M, orbiting each other in a common circular orbit of radius
R, how close (in terms of average radius from this center) can another much smaller
body orbit this pair stabily, that is, without being drawn into the pair's orbit and
risking a collision, and without being ejected from the system? (Let us assume masses
and speeds such that non-relativistic mechanics can be used).
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I believe that what we refer to as 'The Pole Star' is such a system but I do not know its parameters and long time stability.
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Algol is a ternary star system where Algol A (3.59 solar masses) and B (0.79 solar masses) are separated by 0.062 AU, compared with Algol C (1.67 solar masses), which averages 2.62 AU (numbers from wikipedia).
Algol A and B are very close, of course, but in this case Algol C is orbiting at roughly 42.25 times the A-B separation. At such relative distances, I don't think that the relative mass of the distant object is very important - it'll affect the close pairs almost equally.
I think the main risk may not be with the distant object being drawn in or being thrown out, but rather of the distant object perturbing the close pair and upsetting their orbits.
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I have taken a quick look to Google on the mathmatics of three body systems and find them rather daunting
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There does not exist a solution for which the setup remains stable. Nor is it easy to calculate the trajectories of each body to see whether they end up smashing together or flying apart.
This is because the system is non-linear and sensitive to initial configurations, i.e. it's chaotic. The same is true for solar system(s); however, as the masses of the planets are so small compared to the Sun and their separations so large, it appears as if each is in a stable orbit independant of the others.
If we say the binary stars exist for some time T and you ask whether a sufficiently light body sufficiently far away can remain in a roughly-fixed orbit around it for the duration of T, the answer would depend on how 'fixed' you want the orbit to be. There will also be no limits on the 3rd body's distance or mass as wacky-trajectory orbital solutions will exist for small distance/high mass pairings: they'd just be highly irregular and impossible to compute (or set up). The smallest of changes will send them flying out.
EDIT: When I say 'stable', I'm allowing for tiny fluctuations in each of the body's properties e.g. the orbit should remain unchanged if a teenie grain of space-dust lands on one body. The sensitivity to initial conditions means that this definition of stability is violated.
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This is worse than I thought.
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This is because the system is non-linear and sensitive to initial configurations, i.e. it's chaotic.
This almost sounds like such a system resembles that of randomly colliding particles. Now concerning randomly colliding particles,there is some kind of a theorem regarding the equipartition of energy, which goes something like, each possible energy state of the system will, at equilibrium, have the same energy. "Equilibrium" here refers to thermodynamic equibrium, not mechanical equlibrium. There is no lasting mechanical equilibrium in such a system. Raising the question: in a system of 3 orbiting bodies, how does energy flow in the long term? Is it true or false that given enough time, the distribution of energy between the bodies will assume every possible percentage distribution scheme among them (assuming, of course, that they do not smash up first; we could postulate that the bodies are small and dense and separated by distances large enough that we can ignore the chance of that happening)? And related to this question, what is the long term average energy value for each body?
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There is a pair of stable locations close to a pair of orbiting bodies of a similar mass. That is around the point in the orbit of the major bodies where the bodies form an isosceles triangle with the smaller body this is is a Lagrange point and a body can "orbit" there in a complex figure indefinitely.
The body can be quite large right up to the mass of the other bodies and the system remain stable. three bodies orbiting in an equilateral triangle is the only stable solution to the three body problem (except for some very peculiar solutions found recently for bodies with very precisely constrained conditions)