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So here's a strange question that came up in a class tonight. It is axiomatic that satellites in the same orbit travel at the same speed.

But is there some size of satellite whose mass is so great that it distorts that axiom. For example, would a satellite 1/2 the mass of Earth also travel at the same speed as a co-orbital paint chip? 3/4 the mass? If so, where is the tipping point. Any help on the math of this is welcome!

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    $\begingroup$ In that case, the paint chip can't be "co-orbital". The two large bodies are nicely orbiting their combined center of mass, but now you have a three-body problem, where the paint chip will have a complicated trajectory. $\endgroup$ – Mark Adler Nov 1 '18 at 4:27
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    $\begingroup$ Sounds like a moon to me $\endgroup$ – Antzi Nov 1 '18 at 5:28
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    $\begingroup$ That's no moon... that's a space station! $\endgroup$ – GdD Nov 1 '18 at 10:37
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    $\begingroup$ @Antzi: Moons are, in most of the cases, satellites. Hence the name natural satellite. $\endgroup$ – AtmosphericPrisonEscape Nov 1 '18 at 11:27
  • $\begingroup$ Its size would make very little difference. Its mass otoh... and if it was an earth made satellite, then the earth's mass would be considerably reduced. $\endgroup$ – Strawberry Nov 1 '18 at 14:15
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It is axiomatic that satellites in the same orbit travel at the same speed.

Your axiom is not axiomatic.

The idea that there is something called an "orbit" that a satellite can be "in" is a simplification, just like cars are in lanes. They are, roughly, most of the time, but in many countries they are all over the place, without discrete "lanes" evident nor consistent speeds. That's a much better model for thinking about satellite trajectories.

Once you start discussing a satellite's mass, then any mass at all is going to change the motion of everything else in the solar system a little. There is no minimum mass that starts to affect things.

See answers to Does launching a device into orbit change earth's orbit? for example.

As you increase the mass of the satellite, the orbit of the Earth and the mass around each other continuously changes, different paint chips move the Earth different amounts. Likewise the effect of the two paint chips on each other. There is no minimum. Certainly if one gets very massive it will start to noticeably perturb other objects in orbit, but that threshold is only related to how carefully you look and how little you can notice.

If so, where is the tipping point.

There is no spoon tipping point: YouTube

Your question really just asks about how long an approximation or simplification is valid, and that's purely up to you and how much error you can tolerate by using your approximation.


Also:

While you would like to imagine that there can be two satellites in the same orbit but one behind the other, the problem is that the Earth's gravitational field is not uniform, so it will not follow in the same path. Orbits are not perfectly closed or repeatable because of this.

It's a hard task to get out of the idea that there are really such things as fixed orbits, it's a jump of intuition, but that's the reality; there isn't. All you have is a gravity field with it's lumpy deviations, plus drag, solar pressure, and the Sun's and Moon's gravity (plus more), and a sophisticated numerical propagator that calculates step-by-step how a body might move in this complex and ever-changing field of accelerations.

Welcome to the edge of space!

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I'm going to add to the pre-existing answer, from the perspective of classical mechanics and astronomy:

Any two masses that you can take, with any initial conditions in position and velocity, will pose a so-called two-body problem. This 'problem' is a, for all times, exactly solvable mathematical description and solution of the setup you've been asking about.

Now what you propose, to increase the mass of the satellite to the point where it becomes 'relevant' would simply be a binary system (of stars, planets, blackholes, whatever). Both bodies then orbit the common barycenter, which is a constant of motion.
This is still true for the special case when one mass is much smaller than the other, but then the barycenter happens to coincide with the center of the more massive body, so it looks as if there is an orbit around the more massive body.

In real-life things are more complicated than just a two-body problem. Then all masses tug on all the others, and binary barycenters (for example the barycenter for the Sun-Jupiter system) move as well and are not a constant of motion anymore. For earthly satellites in geostationary orbit, perturbations from the other solar system bodies play a role already, and/or station-keeping in L2 requires firing of thrusters from time to time.

In order to say what is binary and what's a satellite-like system one would use the position of the barycenter of the more massive body as a guideline. Therefore the Pluto-Charon system is a binary, where the Jupiter-Sun system is a satellite-like orbit, although their barycenter sometimes leaves the solar radius by about 5%. Mind you that even this binary barycenter's position changes over time as Jupiter exchanges angular momentum with the other gas giants over time, that's why the barycenter lies mostly inside the sun.

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    $\begingroup$ From the perspective of classical mechanics both bodies always orbit around a common barycenter, and that point is coincident with the more massive body's center exactly never, because if one body has no mass, it will not orbit. (spherically symmetric bodies, etc...) From the perspective of Astronomy, or practical problem solving, that's a different matter. $\endgroup$ – uhoh Nov 1 '18 at 11:56
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    $\begingroup$ @AtmosphericPrisonEscape The issue is probably the exact wording of "coincide with the center of" which is factually wrong, and you even provide a counterexample in the last paragraph. The center of an (idealized) celestial body is a point, and any object with some mass will shift the barycenter away from there. The solution could be to change the wording to "happens to lie within". $\endgroup$ – MauganRa Nov 1 '18 at 13:10
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    $\begingroup$ Additionally, in "from the perspective of classical mechanics and astronomy", the word "perspective" is singular, implying there is one perspective shared by both. If you meant to draw a contrast between their perspectives (mathematical versus practical) then plural, plus an indication of when one stops and the other begins would be great. $\endgroup$ – uhoh Nov 1 '18 at 13:40
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    $\begingroup$ I just recommended small, clarifying edits, for the benefit of future readers who may be less familiar with absolutes and approximations. It's always good to try to write as accurately as possible when using absolutes, and when approximations are involved, indicate accordingly. You wrote a nice answer from your perspective, not from the perspective of classical mechanics but from a practical perspective. $\endgroup$ – uhoh Nov 1 '18 at 15:11
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    $\begingroup$ Usually when this happens a small edit is made, and the comment disappears. Answers are generally held to a higher standard than questions, so community involvement in post-refinement can be more detailed for answers than for questions. It shouldn't be about who can sound the most knowledgable, but instead about great answers for the OP and for future readers, or for the AI bots who will one day eat all of the Stack Exchange ecosystem for breakfast and then take over the world. $\endgroup$ – uhoh Nov 1 '18 at 15:19

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