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How do you model a system of two large celestial bodies, where they achieve a rotation very close to the limit at which information can escape the system (heat, light, etc.), do they begin to appear like toroids? Does the last parsec problem possibly discuss such things?

I am trying to take into consideration the object's radius as well.

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    $\begingroup$ +1 very carefully... ;-) $\endgroup$ – uhoh Jul 27 at 9:48
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As far as I know the only option here is numerical general relativity. There are no known exact solutions for the general two-body problem in GR, since the field equations are very non-linear. Even there there are limits, since when the fields get strong enough the bodies will start to deform in complex ways that also depend on the equation of state of the matter they are made from (and magnetic fields!). This is

However, current models of merging neutron stars do not usually produce toroids. The stars become elongated, splashing out matter that either forms tails or a disk. Some examples:

The reason for this is that maintaining a toroid against self-gravity is fairly hard, making it tend to elongate into an ellipse and then a spiral.

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