For example if something was orbiting the sun only slightly further out than the earth, it would risk capture by the earth when the earth caught up to it in it's orbit. If it was orbiting at a great distance, then it would have an almost circular orbit around the barycenter of the sun-earth system (ignoring other planets for now). I understand the Lagrange points, but I'm wondering what happens when things are orbiting slightly further out.

  • $\begingroup$ It's kind of like using the dipole moment approximation of the complete E-M field created with two charged particles, i.e. if you get reasonably far away, then to first order the two bodies can be treated as a single item at the CM $\endgroup$ Oct 14, 2021 at 13:18

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This is called a "restricted three-body" problem. It is slightly simpler than a general three-body problem since one of the bodies can be assumed to be "light". However, even this simplified problem doesn't have an analytical solution.

The orbit of the satellite doesn't have a particular shape, and can be chaotic. In some situations it can be approximated usefully as perturbed ellipse: that is it can be approximated as a Keplerian orbit but with non-constant orbital parameters, which nevertheless vary in a fairly regular way: for example, the orientation of the ellipse might rotate, or precess, or the values of the inclination and eccentricity might oscillate.

  • $\begingroup$ Thanks. Just knowing what to call this situation is a help to researching it. But Mars sorta fits the description of my problem and it surely is in a stable orbit, or at least the variations you describe will not become chaotic, yes? So even tho absolute regularity might not obtain, the system might yet be stable? And I'm thinking that if the satellite gets too close, then indeed it goes chaotic, perhaps being flung by Earth's gravity out to God know where. $\endgroup$ Oct 6, 2021 at 16:05
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    $\begingroup$ Mars's orbit is actually chaotic (like the rest of the solar system) on a long enough time scale, But not chaotic enough to actually cause it to be thrown out. You can usefully model mars as "Keplarian +perturbation" $\endgroup$
    – James K
    Oct 6, 2021 at 16:55
  • $\begingroup$ By 'chaotic' I meant, liable to be thrown out. But I guess that's not the correct usage. Do we have a term to distinguish 'Mars style' chaos (variations on a theme, but nothing drastic) from an ejection style situation? Or, I guess a capture situation? I do presume that if the orbit came too close to the earth or the sun, then the orbit of the satellite could terminate in ejection or capture. $\endgroup$ Oct 6, 2021 at 20:45
  • $\begingroup$ @RayAndrews you are correct -- "chaotic" implies a StrangeAttractor, and a nonpredictable exact path, but if the level of strangeness is small, you can bound the path inside a toroidal-ish region of space. $\endgroup$ Oct 14, 2021 at 13:20

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