The Transition from Inspiral to Plunge for a Compact Body in a Circular Equatorial Orbit Around a Massive, Spinning Black Hole

TL;DR

The paper addresses how a compact body with mass ratio $\eta = \mu/M \ll 1$ transitions from adiabatic inspiral to plunge when in a circular equatorial orbit around a Kerr black hole. It develops a near-ISCO, non-geodesic radial equation of motion by expanding the effective potential and derives a universal dimensionless form $\frac{d^2 X}{dT^2} = -X^2 - T$ that interpolates between inspiral and plunge, with scaling laws leading to final energy and angular momentum deficits of order $\eta^{4/5}$. It shows the transition lasts for a proper time $\Delta\tau \sim M \eta^{-1/5}$ and that gravitational waves during this regime are dominated by the second harmonic at frequency $f \sim \tfrac{\tilde\Omega_{\rm isco}}{\pi M}$, providing estimates for the RMS strain amplitude and LISA signal-to-noise under favorable orientation. The work highlights the potential observability of the transition by LISA for circular-equatorial orbits and outlines the need to extend the analysis to generic eccentric and inclined orbits using the Teukolsky formalism, aligning with parallel work by Buonanno and Damour. Overall, it lays analytic groundwork for including the transition regime in EMRI waveform modeling and informs the feasibility of extracting transition dynamics from LISA data.

Abstract

There are three regimes of gravitational-radiation-reaction-induced inspiral for a compact body with mass mu, in a circular, equatorial orbit around a Kerr black hole with mass M>>mu: (i) The "adiabatic inspiral regime", in which the body gradually descends through a sequence of circular, geodesic orbits. (ii) A "transition regime", near the innermost stable circular orbit (isco). (iii) The "plunge regime", in which the body travels on a geodesic from slightly below the isco into the hole's horizon. This paper gives an analytic treatment of the transition regime and shows that, with some luck, gravitational waves from the transition might be measurable by the space-based LISA mission.

Figures (3)

• Figure 1: The gradually changing effective potential $V(\tilde{r},\xi)$ for radial geodesic motion. Each curve is for a particular value of $\xi \equiv \tilde{L} - \tilde{L}_{\rm isco}$. As $\xi$ decreases due to radiation reaction, the body, depicted by the large dot, at first remains at the minimum of the effective potential ($\xi_1$; "adiabatic regime"). However, as $\xi$ nears zero (at $\xi\simeq\xi_2$), the body cannot keep up with the rapid inward motion of the minimum; it lags behind in a manner described by the transition-regime analysis of Sec. \ref{['sec:Transition']}. At $\xi\simeq\xi_5$ the effective potential has become so steep that radiation reaction is no longer important, the transition regime ends, and the body plunges toward the black hole with nearly constant energy and angular momentum.
• Figure 2: Dimensionless orbital radius $X$ as a function of dimensionless proper time $T$ for an orbit near the isco. Adiabatic Inspiral: The analytic solution (\ref{['XTAdiabatic']}) for adiabatic inspiral outside the isco. Transition: The numerical solution to the dimensionless equation of motion (\ref{['eomdimensionless']}) for the transition regime in the vicinity of the isco. Plunge: The analytic solution (\ref{['XTPlunge']}) for the orbital plunge inside the isco.
• Figure 3: Same as Fig. \ref{['figtransition1']}, but drawn on a different scale.