Secular evolution of orbital parameters for general bound orbits in Kerr spacetime

Abstract

We analytically derive the secular changes of the orbital parameters, i.e., energy, angular momentum, and Carter constant, for general bound orbits in Kerr spacetime, at leading order in the mass ratio, through the 6th post-Newtonian (6PN) order and the 16th order in orbital eccentricity. We validate the formulas against high-precision numerical Teukolsky results and quantify how eccentricity affects both the achievable accuracy and the PN convergence. We then construct and test a simple ``hybrid'' approximation that combines different PN and eccentricity truncations to retain accuracy at reduced computational cost. We also assess the performance of exponential resummation at higher PN orders. These results provide building blocks for fast, (analytic) adiabatic inspiral and waveform models for extreme mass ratio inspirals relevant to space-based detectors such as the Laser Interferometer Space Antenna (LISA).

Figures (9)

• Figure 1: Relative errors in the analytic PN formulas for the secular change of the particle's energy $E$ as a function of the (dimensionless) semi-latus rectum $p$ for the dimensionless spin parameter $q=0.90$ (see Eq. \ref{['eq:relative_error']}). We truncate the plots at $p=6$. These plots correspond to Fig. 1 of Ref. Sago:2015rpa. The dotted line in each plot is reference proportional to $1/p^{13/2}=v^{13}$. The cases with the eccentricity $e=0.10$, $0.40$, and $0.70$ are shown from the top to the bottom panels, and those with the inclination $\theta_\textrm{inc}=20^\circ$, $50^\circ$, and $80^\circ$ are shown from the left to the right panels.
• Figure 2: Relative contributions of the 5.5PN and 6PN terms in the analytic formula for $\langle dE/dt \rangle_t$ in the case of $q=0.90$, $e=0.10$, and $\theta_\mathrm{inc}=80^\circ$, corresponding the top right panel of Fig. \ref{['fig:DelE_q09']}. $\delta_E^{(j)}$ is defined by Eq. \ref{['eq:delta_I_j']}.
• Figure 3: Relative errors in the 6PN $O(e^{16})$ formulas for $\langle dI/dt \rangle_t$ ($I=\{E,\, L,\, C\}$) as a function of $p$ for $\theta_\textrm{inc}=50^\circ$ (see Eq. \ref{['eq:relative_error']}). The cases with the dimensionless spin parameter $q=0.90$, $0.50$, $0.10$ and $-0.90$ are shown from the top to the bottom panels, and those with the eccentricity $e=0.10$, $0.40$, and $0.70$ are shown from the left to the right panels. This corresponds to Fig. 2 in Ref. Sago:2015rpa.
• Figure 4: Relative errors in the analytic 6PN formula for $\langle dE/dt \rangle_t$ with different order of expansion with respect to eccentricity. The cases with the eccentricity $e=0.10$, $0.40$, and $0.70$ are shown from the top to the bottom panels, and those with the inclination angle $\theta_\mathrm{inc}=20^\circ$, $50^\circ$, and $80^\circ$ are shown from the left to the right panels. We fix $q=0.90$ for all plots.
• Figure 5: Relative contribution of the $O(e^j)$ term in the PN formula for $\langle dE/dt \rangle_t$ as a function of $p$ (see Eq. \ref{['eq:Edot_e_exp']}). The cases with the eccentricity $e=0.10$, $0.40$, and $0.70$ are shown from the top to the bottom panels, and those with the inclination angle $\theta_\mathrm{inc}=20^\circ$, $50^\circ$, and $80^\circ$ are shown from the left to the right panels. We fix $q=0.90$ in all plots.