Analytical solutions of bound timelike geodesic orbits in effective-one-body frame

TL;DR

This paper derives approximate analytical solutions for bound timelike geodesic orbits in the effective-one-body frame, expressed via elliptic integrals, facilitating gravitational wave calculations for extreme mass-ratio inspirals.

Contribution

It provides explicit analytical solutions and orbital frequencies in the EOB frame using Mino time, enabling efficient Fourier analysis of geodesic orbits for gravitational wave modeling.

Findings

Analytical solutions expressed with elliptic integrals.

Explicit formulas for orbital frequencies using Fourier series.

Application to gravitational wave calculations from EMRIs.

Abstract

We derive the approximate analytical solutions of the bound timelike geodesic orbits in the effective-one-body (EOB) frame with extreme-mass ratio limit. The analytical solutions are expressed in terms of the elliptic integrals using Mino time $\lambda$ as the independent variable. Since Mino time decouples the $r$ and $\theta$-motion, we also give explicit expressions for three orbital frequencies $\Omega_r, ~\Omega_\theta, ~\Omega_\phi$ using the Fourier series expansion. With these analytical expressions at hand, we can perform Fourier expansions in Mino time $\lambda$ for any function expressed in terms of the coordinates $(r,\theta,\phi)$. In particular, the observer's time t is decomposed into Mino time $\lambda$, and the frequency-domain description is constructed from the $\lambda$-Fourier expansion and the expansion of t. These analytical expressions are quite simple to implement, and can be applicable for calculating gravitational waves (GWs) from extreme mass-ratio inspirals (EMRIs) with the frequency-domain Teukolsky equation.