For a given system consisting of two bodies, when will one body orbit another body as given by Kepler's law? Sometimes the body just gets attracted linearly and sometimes it orbits the other body instead. What conditions need to be met for either scenario to happen? And in linear movement both bodies move towards each other whereas in the latter case only one body orbits whereas the first body is at one of the focii, how does that work?
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3Does this answer your question? Conservative central force and stable orbits – Farcher Feb 08 '23 at 13:50
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Kepler's laws of planetary motion are only correct if one of the two bodies has a much larger mass than the other, and so can be treated as if it were stationary.
If the two bodies have similar masses then each will orbit their mutual centre of mass in an orbit that is a conic section with the COM at a focus - this is known as the two-body problem. As long as they are not going too fast relative to one another, these orbits will both be ellipses.
The case of linear motion occurs if the initial velocity of each body lies along the line between them, so that this then reduces to a one-dimensional problem. You can think of this as a limiting case, where the conic section orbits become thinner and thinner and are eventually reduced to straight lines.
Note that all of this assumes that we can treat this as a classical mechanics problem with Newtonian gravity. If relativistic effects are significant then the problem becomes much more complex - see this Wikipedia article for an overview.
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What are the conditions for an ellipse vs a hyperbola for 2 body problem? – John Alexiou Feb 08 '23 at 14:12
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@Gert I don't think I have used the word "retardation" in my answer - but "retarded" appears in the other answer to this question from ScienceAJ. Is it possible you meant to attach your comment to the other answer ? – gandalf61 Feb 08 '23 at 17:37
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This makes it sound like Kepler's third law couldn't be applied to a binary star system? – ProfRob Feb 09 '23 at 09:04
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@ProfRob No, Kepler's third law does not apply to binary systems because the orbits of the two objects about their COM have the same period but if their masses are different then their orbits have different sizes. – gandalf61 Feb 09 '23 at 09:52
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I suggest you look at that again. Kepler's third law is often applied to binary systems and is universal, regardless of the mass ratio of the binary. If you want to say "Kepler's laws of motions for planets orbiting the Sun can only be applied to planets orbiting the Sun" well fair enough. But $G(M_1+M_2) = \omega^2 a^3$, usually called "Kepler's third law" is universal. – ProfRob Feb 09 '23 at 11:25
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@ProfRob Kepler's third law of planetary motion is usually given as "The square of the orbital period is proportional to the cube of the length of the semi-major axis of the orbit". This obviously applies to planetary systems outside of the solar system (to a good approximation). But in this form it does not apply to binary star systems for the reason I gave above. I agree that the more general form that you gave, which is the binary mass function, does apply to binary star systems. However, since you are clearly not commenting in good faith I am done here. – gandalf61 Feb 09 '23 at 12:09