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Adiabatic binary inspiral in the small mass ratio limit treats the small bodyas moving along a geodesic of a large Kerr black hole, with the geodesic slowlyevolving due to radiative backreaction. Up to initial conditions, geodesics aretypically parameterized in two ways: using the integrals of motion energy $E$,axial angular momentum $L_z$, and Carter constant $Q$; or, using orbit geometryparameters semi-latus rectum $p$, eccentricity $e$, and (cosine of )inclination $x_I \equiv \cos I$. The community has long known how to computeorbit integrals as functions of the orbit geometry parameters, i.e., asfunctions expressing $E(p, e, x_I)$, and likewise for $L_z$ and $Q$. Mappingsin the other direction -- functions $p(E, L_z, Q)$, and likewise for $e$ and$x_I$ -- have not yet been developed in general. In this note, we developgeneric mappings from ($E$, $L_z$, $Q$) to ($p$, $e$, $x_I$). The mappings areparticularly simple for equatorial orbits ($Q = 0$, $x_I = \pm1$), and can beevaluated efficiently for generic cases. These results make it possible to moreaccurately compute adiabatic inspirals by eliminating the need to use aJacobian which becomes singular as inspiral approaches the last stable orbit.

Parameterizing black hole orbits for adiabatic inspiral