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Black hole spacetimes, like the Kerr spacetime, admit both stable andplunging orbits, separated in parameter space by the separatrix. Determiningthe location of the separatrix is of fundamental interest in understandingblack holes, and is of crucial importance for modeling extreme mass-ratioinspirals. Previous numerical approaches to locating the Kerr separatrix werenot always efficient or stable across all of parameter space. In this paper weshow that the Kerr separatrix is the zero set of a single polynomial inparameter space. This gives two main results. First, we thoroughly analyzespecial cases (extreme Kerr, polar orbits, etc.), finding strict bounds on thelimits of roots, and unifying a number of results in the literature. Second, wepose a stable numerical method which is guaranteed to quickly and robustlyconverge to the separatrix. This new approach is implemented in the Black HolePerturbation Toolkit, and results in a ~45x speedup over the prior robustapproach.

The location of the last stable orbit in Kerr spacetime