We derive the approximate analytical solutions of the bound timelike geodesic orbits in the effective-one-body frame with an 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 ${\mathrm{\Omega }}_r,{\mathrm{\Omega }}_{\theta },{\mathrm{\Omega }}_{\varphi }$ 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 $(r,\theta ,\varphi )$. 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 from extreme mass-ratio inspirals with the frequency-domain Teukolsky equation.

Close

PrevNext

View figure in article

• Gravitational waves