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Is it possible that two toroidal, rigid, homogeneous planets are forming a system in which the toroids are chained through each other, and during their mutual motion, the planets are never touching each other? How should I go about studying the systems of the above kind, and classify them according to their geometrical parameters, masses and initial dynamical parameters.

Are there systems which are stable? (I considered the fact that there are a sizeable number of dynamical degrees of freedom.) What happens in the case for example, if the ‘orbit’ frequency reduces a bit, or if the plane of one of the toroids tilts slightly?

(I'm aware that Analytic calculations are favorable. For numerical calculations, the results should be cross-checked in some semi-analytic way, what would that be?)

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    $\begingroup$ The orbits would have to be perfectly circular and the model could not allow for any perturbations. So my knee-jerk guess is no, it would not be stable and the planets in the toroid would eventually collide and ultimately merge or be thrown out. $\endgroup$ – honeste_vivere Oct 13 '16 at 13:53
  • $\begingroup$ @honeste_vivere But can that be proven mathematically? $\endgroup$ – Naveen Balaji Oct 13 '16 at 14:12
  • $\begingroup$ Oh I am sure it can but I am not a sufficient master of symbol gymnastics to help on that front. I just know from spacecraft orbits that when elliptical, the inter-spacecraft separation changes throughout the orbit. So there really would be no way to keep a bunch of oblate spheroids very close together unless they were in a perfectly circular orbit. This would require no perturbations from pressure or gravity to maintain, which is unrealistic/unphysical. Thus, why I feel fairly confident in my comment. $\endgroup$ – honeste_vivere Oct 13 '16 at 14:41
  • $\begingroup$ @honeste_vivere Well yeah that makes sense, but, again why can't I assume the motion of a system to be rotational and/or vibrational; because when such a system is in place there would be immense gravity acting from one toroid's centre to the other and also the gravitational effects on the edges of the toroids. Again there can be a sizeable number of dynamical degrees of freedom, so why can't the system translate and/or rotate and/or vibrate within an imaginary sphere. What I mean is like a ball moving within a sphere, constrained to the geometry of the sphere and never retracing its path. $\endgroup$ – Naveen Balaji Oct 13 '16 at 14:59

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