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

Chen Zhang, Wen-Biao Han

TL;DR

This paper develops analytical solutions for bound timelike geodesics in an extreme-mass-ratio binary within a deformed Kerr background used by the EOB framework. By employing Mino time $\lambda$ to decouple radial and polar motion and expressing the geodesics with Legendre elliptic integrals, it derives mass-ratio–corrected fundamental frequencies $\Omega_r$, $\Omega_\theta$, and $\Omega_\phi$ via Fourier expansions and constructs $t$-domain representations for gravitational-wave calculations. The work validates the analytical frequencies against numerical integrations, showing good agreement and substantial speedups, and demonstrates the practicality of using these results in the frequency-domain Teukolsky formalism for EMRI waveform generation. Overall, the results enable efficient EMRI waveform modeling with mass-ratio corrections within the EOB framework and set the stage for incorporating these analytical solutions into Teukolsky-based GW calculations.

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.

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

TL;DR

This paper develops analytical solutions for bound timelike geodesics in an extreme-mass-ratio binary within a deformed Kerr background used by the EOB framework. By employing Mino time to decouple radial and polar motion and expressing the geodesics with Legendre elliptic integrals, it derives mass-ratio–corrected fundamental frequencies , , and via Fourier expansions and constructs -domain representations for gravitational-wave calculations. The work validates the analytical frequencies against numerical integrations, showing good agreement and substantial speedups, and demonstrates the practicality of using these results in the frequency-domain Teukolsky formalism for EMRI waveform generation. Overall, the results enable efficient EMRI waveform modeling with mass-ratio corrections within the EOB framework and set the stage for incorporating these analytical solutions into Teukolsky-based GW calculations.

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

Paper Structure

This paper contains 8 sections, 42 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Effective-one-body Hamiltonian
  3. The fundamental frequencies
  4. Geodesics in Mino time
  5. Frequencies of $r$ and $\theta$-motion
  6. Frequencies of $t$ and $\phi$-motion
  7. Consistency check of the fundamental frequencies
  8. Conclusions and Outlook

Figures (1)

  • Figure 1: Fitting error of the integral $\int_{\rm r_2}^{\rm r}\frac{dr}{\sqrt{R(r)}}$. Replacing ${R(r)}$ with $\mathcal{R}(r)$ provides an highly precise approximation. Even if we choose extreme orbital parameters that bring the periapsis $r_2$ near LSO(orange line), the error is less than $0.16\nu$.