Parameterizing black hole orbits for adiabatic inspiral

Adiabatic binary inspiral in the small mass ratio limit treats the small body as moving along a geodesic of a large Kerr black hole, with the geodesic slowly evolving due to radiative backreaction. Up to initial conditions, geodesics are typically parameterized in two ways: using the integrals of motion energy E, axial angular momentum Lₙ, and Carter constant Q; or, using orbit geometry parameters semilatus rectum p, eccentricity e, and (cosine of) inclination x₈. The community has long known how to compute orbit integrals as functions of the orbit geometry parameters, i. e. , as functions expressing E (p, e, x₈), and likewise for Lₙ and Q. Mappings in the other direction---functions p (E, Lₙ, Q), and likewise for e and x₈---have not yet been developed in general. In this note, we develop generic mappings from (E, Lₙ, Q) to (p, e, x₈). The mappings are particularly simple for equatorial orbits (Q=0, x₈=1), and can be evaluated efficiently for generic cases. These results make it possible to more accurately compute adiabatic inspirals by eliminating the need to use a Jacobian which becomes singular as inspiral approaches the last stable orbit.