Captured are circularized: A relativistic treatment of extreme mass ratio inspirals crossing accretion disks

TL;DR

This work develops a fully relativistic treatment of extreme mass-ratio inspirals that repeatedly cross a geometrically thin accretion disk around a supermassive black hole. By modeling the SMO as a forced geodesic in Schwarzschild spacetime and applying an adiabatic, phase-averaged formalism, the authors derive secular evolution equations for the orbital elements (p, e, i) under two matter-interaction mechanisms: aero-drag for stars and dynamical friction for stellar-mass black holes. The study shows that disk crossings tend to align orbits with the disk and typically damp eccentricity for stars, while sBHs can experience eccentricity excitation at large inclinations but still circularize upon capture; importantly, only a small subset of objects near the SMBH and disk can be captured within typical AGN lifetimes, highlighting the role of two-body scattering in boosting capture rates. The results yield simple scaling laws for capture timescales, t_cap ∝ p^{3/2} (aero-drag) and t_cap ∝ p^{-1/2} (dynamical friction), and it places the findings in the context of previous Newtonian studies, offering a relativistic benchmark for modeling wet EMRIs relevant to LISA and related observables.

Abstract

A small body orbiting around an accreting massive object and periodically crossing its accretion disk is a common configuration in astrophysics. In this work, we investigate the secular evolution of extreme mass-ratio inspirals (EMRIs), in which a stellar-mass object (SMO), e.g., a star or a stellar-mass black hole (sBH), collides with the accretion disk of a central supermassive black hole (SMBH), within a fully relativistic framework. We find (1) the disk always tends to align the SMO no matter what the initial orbital inclination $ι$ relative to the disk is, (2) the final orbital eccentricity of the SMO captured by the disk is always low though the orbital eccentricity may temporarily grow when the orbital inclination $ι$ is large and the SMO is an sBH, and (3) via collisions with the accretion disk only, only a small fraction of sBHs that are initially close to the SMBH and close to the disk can be captured by the disk within typical disk lifetime of active galactic nuclei. Two-body scatterings between SMOs in the nuclear stellar cluster play an essential role in randomly kicking sBHs towards the disk and boosting the capture rate.

Figures (14)

• Figure 1: Schematic illustration of the wet EMRIs with SMO-disk collision scenario. The relations between the labeled parameters are $z_{1}\equiv\cos\theta_{\rm min}=\sin\iota$, where $\Vec{L}_{\rm disk}$ is the disk angular momentum, and $\Vec{L}_{\rm orbit}$ is the orbital angular momentum of the SMO, the angle between these two vectors is equivalent to the orbital inclination, $\cos^{-1}(\hat{L}_{\rm disk}\vdot\hat{L}_{\rm orbit})=\iota$. $0<\iota<90^{\circ}$ corresponds to the prograde cases, while $90^{\circ}<\iota<180^{\circ}$ corresponds to the retrograde cases. In both cases $\iota$ will decrease, which will be verified in this work.
• Figure 2: Evolution of the semi-major axis $a$, eccentricity $e$, and orbital inclination angle $\iota$ a of star. Solid lines labeled "full" are obtained from full E.O.M. \ref{['eq: forced geodesic']}, while dashed lines labeled "adiabatic" show the secular evolution results obtained by evolving Eq. \ref{['eq:secular ode']}. The zoomed-in panels display the variations of $a,\ e,\ \iota$ over the relatively short initial time interval $(0,\ 6\times 10^5\ M_{\bullet})$. In the bottom panel, the intersection point of the two red dotted lines indicates the critical inclination $\iota_{\rm crit}=3H_{\rm disk}/p_{\rm ini}$ and the corresponding time $t_{\rm cap}=2.7\times 10^{8}\ M_\bullet$.
• Figure 3: The comparison of $\expval{\mathrm{d}e/\mathrm{d}\tau}$ variations with orbital inclination angle $\iota$ for different $e$, with $p_{\rm ini}=300\ M_{\bullet}$ assuming the aero-drag force model. The vertical black dotted line marks $\iota=\pi\ (180^{\circ})$.
• Figure 4: The comparison of estimated shrink timescale with $\abs{a_{\rm ini}/\expval{\dv{a}{\tau}}_{\rm ini}}$ (solid dotted line) and capture timescale with $\abs{\iota_{\rm ini}/\expval{\dv{\iota}{\tau}}_{\rm ini}}$ (solid line) with orbital inclination angle $\iota$ for different $e$, fixing $p_{\rm ini}=300\ M_{\bullet}$, using the aero-drag force model for stars. The vertical black dotted line marks $\iota=\pi\ (180^{\circ})$.
• Figure 5: Evolution of the semi-major axis $a$, eccentricity $e$, and orbital inclination angle $\iota$ using the aero-drag force model with a constant damping coefficient $\gamma_{0}=2\times 10^{-6}\ M_{\bullet}^{-1}$. The initial conditions are $p_{\rm ini}=300\ M_{\bullet}$, and $\ e_{\rm ini}=0.3$. The prograde cases $(\iota_{\rm ini}<90^{\circ})$ are shown with solid lines, while the retrograde cases $(\iota_{\rm ini}>90^{\rm \circ})$ are shown with dashed lines. The blue dotted horizontal line at $a=10\ M_{\bullet}$ in the top panel marks the threshold below which the orbit is considered to have shrunk in this work. The black dotted horizontal line in the bottom panel marks $\iota=\pi/2$.