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 $L_z$, and Carter constant $Q$; or, using orbit geometry

parameters semi-latus rectum $p$, eccentricity $e$, and (cosine of )

inclination $x_I equiv cos I$. The community has long known how to compute

orbit integrals as functions of the orbit geometry parameters, i.e., as

functions expressing $E(p, e, x_I)$, and likewise for $L_z$ and $Q$. Mappings

in 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 develop

generic mappings from ($E$, $L_z$, $Q$) to ($p$, $e$, $x_I$). The mappings are

particularly simple for equatorial orbits ($Q = 0$, $x_I = pm1$), 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.

## Mapping Orbit Integrals to Orbit Geometry Parameters

This article discusses the development of mappings from orbit integrals (energy E, axial angular momentum L_z, Carter constant Q) to orbit geometry parameters (semi-latus rectum p, eccentricity e, and inclination x_I). These mappings are essential for accurately computing adiabatic inspirals where a small body moves along the geodesic of a large Kerr black hole.

### Current Understanding

The community has long been able to compute orbit integrals as functions of the orbit geometry parameters. However, the reverse mappings, i.e., functions that express p, e, and x_I in terms of E, L_z, and Q, have not yet been developed in general.

### New Developments

In this article, the authors present generic mappings that translate E, L_z, and Q into p, e, and x_I. These mappings are particularly simple for equatorial orbits (Q = 0, x_I = ±1) and can be efficiently evaluated for generic cases.

### Potential Opportunities

- Accurate Computation: The developed mappings provide a more accurate method to compute adiabatic inspirals, eliminating the need for a singular Jacobian as the inspiral approaches the last stable orbit.
- Improved Understanding: By bridging the gap between orbit integrals and orbit geometry parameters, researchers can gain a deeper understanding of the dynamics of small bodies moving in the vicinity of Kerr black holes.

### Potential Challenges

- Validation: The newly developed mappings need to be validated through further research and comparison with existing methods. This will ensure their reliability and accuracy in various scenarios.
- Complex Scenarios: While the mappings are efficient for generic cases, there may be complex scenarios or extreme conditions where their applicability needs to be further studied.

### Roadmap for Readers

- Understand the current state of knowledge regarding the computation of orbit integrals and their dependence on orbit geometry parameters.
- Explore the limitations and challenges faced in the absence of reverse mappings from orbit integrals to orbit geometry parameters.
- Examine the new developments presented in this article, focusing on the generic mappings that allow for accurate computation of adiabatic inspirals.
- Consider the potential opportunities stemming from these developments, such as improved accuracy and a deeper understanding of small body dynamics around Kerr black holes.
- Recognize the potential challenges in validating the mappings and their applicability in complex scenarios.
- Stay updated on further research in this field to gain insights into the refinement and expansion of the developed mappings.

**In conclusion**, this article presents a significant advancement in understanding adiabatic inspirals by developing mappings from orbit integrals to orbit geometry parameters. While offering opportunities for accurate computation and improved understanding, these mappings need validation and careful consideration of their applicability in various scenarios.