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

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 $λ$ as the independent variable. Since Mino time decouples the $r$ and $θ$-motion, we also give explicit expressions for three orbital frequencies $Ω_r, ~Ω_θ, ~Ω_ϕ$ using the Fourier series expansion. With these analytical expressions at hand, we can perform Fourier expansions in Mino time $λ$ for any function expressed in terms of the coordinates $(r,θ,ϕ)$. In particular, the observer's time t is decomposed into Mino time $λ$, and the frequency-domain description is constructed from the $λ$-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.