Spin the black circle II: tidal heating and torquing of a rotating black hole by a test mass on generic orbits

Abstract

Horizon fluxes of energy and angular momentum are a key strong-field effect in the dynamics of black holes, encoding direct information about their nature. In this work, we present a numerical study of these fluxes for a test particle orbiting a Kerr black hole on equatorial geodesics, covering circular, eccentric, and hyperbolic trajectories across a wide range of orbital parameters and black hole spins. We reproduce known results for circular orbits and uncover a richer phenomenology for eccentric and hyperbolic ones: the instantaneous fluxes can exhibit multiple peaks and sign changes, indicating a complex interplay between superradiant and non-superradiant regimes. We then compare these results against existing analytical post-Newtonian expressions, exploring resummation strategies to improve their performance against numerical data. In particular, we propose a factorized and resummed representation of the horizon fluxes that predicts the onset frequency of the superradiant regime to within $10\%$ for $\gtrsim 73\%$ of configurations for both the energy and angular momentum fluxes. This representation exactly reduces to the circular limit by construction, independently of the perturbative order of the remaining analytical terms. For peak and orbit-averaged fluxes, the analytical models achieve acceptable accuracy -- with relative errors at the $10\%$ level or below -- at large separations and low eccentricities. However, they can exhibit deviations of $\sim \mathcal{O}(100\%)$ in the strong-field regime, motivating the need for improved flux prescriptions and further investigations.

Figures (11)

• Figure 1: Summary corner plot of the initial parameters and maximum horizon fluxes for all simulated systems, divided by orbital configuration: circular (pink), eccentric (green) and hyperbolic (blue). Our entire dataset comprises 257 simulations, with 8 circular, 140 eccentric, and 109 hyperbolic ones. The maximum fluxes are obtained for the lowest approach distances $r_{\rm min}$, as expected.
• Figure 2: Three representative configurations considered in this work. Left panel: trajectories of the test particle around the Kerr black hole for circular (pink), eccentric (green), and hyperbolic (blue) orbits. Right panel: corresponding horizon fluxes of energy (top panel) and angular momentum (bottom panel) as functions of (normalized) time. The eccentric and hyperbolic configurations exhibit peaks in the fluxes at closest approach, while the circular orbit shows constant fluxes, as expected.
• Figure 3: Left: orbit-averaged energy (top) and angular momentum (bottom) fluxes as a function of the eccentricity $e_0$ of the orbit at fixed semilatus rectum $p_0$ and spin $a$ for a representative sample of simulations. The orbit-average procedure simplifies the complex behavior of the instantaneous fluxes for positive $\hat{a}$. In this case, we observe that for fixed values of ($p_0, \hat{a}$) there exists a critical value of eccentricity beyond which the fluxes change sign from negative to positive. Notably, this eccentricity value is not the same for energy and momentum fluxes. Right: peak energy (top) and angular momentum (bottom) fluxes during the close encounter as a function of the orbital energy $E_0$ at fixed angular momentum $p_{\varphi}$ and spin $\hat{a}$. Similar to the eccentric case, for positive spins there exists a critical value of $E_0$ beyond which the peak fluxes change sign from negative to positive. Again, this value is not the same for energy and angular momentum fluxes.
• Figure 4: Examples of energy (top) and angular momentum (bottom) fluxes for an eccentric (green) and a hyperbolic (blue) orbit, showcasing their sign changes around the time of periastron passage/closest approach. Vertical lines mark times when the orbital frequency $\Omega = \Omega_{\rm H}$. Remarkably, $\dot{S}_{m=1}$ changes sign at the same time as $\dot{S}_{m=2}$.
• Figure 5: Comparing how well several models (see Sec. \ref{['subsec:mode_by_mode_superradiance']}) for the tidal frequency entering the superradiance prefactors predict the sign change in $\dot{S}$ (top) and $\dot{M}$ (bottom), for the global fluxes (left), $m=1$ (center) and $m=2$ (right). Triangles and crosses represent hyperbolic and eccentric orbits, respectively, while filled and unfilled markers differentiate between sign changes occurring before and after the minimum approach separation. Globally, the $\Omega_{\rm T}^{\rm{NS, dt}}$ models (red) perform best for both $\dot{S}$ and $\dot{M}$. This is especially evident by looking at the $m=1$ modes: $\Omega_{21}^{\rm dt}$ models are necessary to capture the sign change in $\dot{S}_{21}$, while the nonspinning corrections are crucial for the energy flux. Note that for the former, no $\dot{S}_{21}$ "NS" corrections are available, so the red and light blue models coincide, as do the dark blue and green ones.