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Can satellites moving in orbits that lie in different planes remain at the same distance from each other at all times? How can I prove that they cannot?

Satellites can revolve at different angular velocities. Both orbits have the planet's centre of mass at their centres, so their planes must intersect.

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  • $\begingroup$ One trivial way they can: parallel planes and constant difference vector. $\endgroup$ – aschepler Jul 1 '18 at 1:25
  • $\begingroup$ @aschepler Yes, but these aren't valid orbits, as both orbits must have the planet's centre of mass at their centres. $\endgroup$ – GingerBadger Jul 1 '18 at 1:26
  • $\begingroup$ And you're probably also assuming ideal adherence to Kepler's laws? $\endgroup$ – aschepler Jul 1 '18 at 1:48
  • $\begingroup$ Not necessarily, just that fact above. $\endgroup$ – GingerBadger Jul 1 '18 at 1:51
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Can satellites moving in orbits that lie in different planes remain at the same distance from each other at all times? How can I prove that they cannot?

They can in at least once scenario, thus proving it false isn't possible.

90° Planes

One orbit circular, the other elliptical, at 90° to each other.

When the polar orbiting satellite crosses the equator have the equitorial orbiting satellite positioned opposite (so they don't collide). When the polar orbiting satellite is passing over a pole have the equitorial orbiting satellite at 90° from where it was, as described in the prior sentence. An elliptical orbit maintains the distance.

It is possible for more than two satellites to do this simultaneously, so there's more than one solution.

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  • $\begingroup$ Could you please develop the explanation? I could not understand why the distance is constant. Which orbit is elliptic, the polar or the equatorial? Is Earth on the center or on the focus of the ellipse? $\endgroup$ – rafa11111 Jul 1 '18 at 2:51
  • $\begingroup$ @rafa11111: why did you accept the answer if you don't understand it? I think it is incorrect. The ellipse of a satellite has the earth at a focus, not the center. $\endgroup$ – Ross Millikan Jul 1 '18 at 4:57
  • $\begingroup$ If both orbits are circular with the same radius we can let the equatorial one be in the $xy$ plane and the polar one in the $yz$ plane. Let the radius be $1$. At one time the equatorial satellite is at $(1,0,0)$ and the polar satellite at $(0,1,0)$ for a distance of $\sqrt 2$. One eighth orbit later they are at $(\frac {\sqrt 2}2,\frac {\sqrt 2}2,0)$ and $(0,\frac{\sqrt 2}2, \frac{\sqrt 2}2)$ for a distance of $1$. $\endgroup$ – Ross Millikan Jul 1 '18 at 5:03
  • $\begingroup$ Thanks for explaining @RossMillikan. Please notice that I am not the OP. I thought that the construction in the answer would only work if the earth was at the center, which is nonsense considering the celestial mechanics. Furthermore, thanks for presenting this simpler construction with two circular orbits. $\endgroup$ – rafa11111 Jul 1 '18 at 5:54

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