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arXiv · 2511.06684

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

Chen Zhang·Wen-Biao Han
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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.

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Chen Zhang, Wen-Biao Han. 2025-11-10. Analytical solutions of bound timelike geodesic orbits in effective-one-body frame. https://doi.org/10.1103/7g4q-5crv

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