Accretion of a Vlasov gas by a Kerr black hole

TL;DR

The authors address relativistic Bondi-type accretion of a collisionless Vlasov gas onto a Kerr black hole by solving the Vlasov equation with asymptotic distributions depending on energy. They formulate the problem in terms of Kerr geodesics, introduce a novel $(E,Q, obracket heta)$ parametrization to handle phase-space regions, and derive closed integral expressions for the particle current and energy–momentum observables, decoupled into absorbed and scattered contributions. They compute accretion rates for monoenergetic and Maxwell–Jüttner distributions, showing that rotation reduces mass and energy accretion while angular momentum accretion anti-aligns with spin, thus slowing the black hole. A slow-rotation expansion up to cubic order in the spin parameter $oldsymbol{\alpha}$ provides analytic approximations that agree with full Kerr results at the percent level, validating a practical approach for studying accretion morphologies and flow properties in moderately to rapidly spinning black holes.

Abstract

We investigate the accretion of a collisionless, relativistic kinetic gas by a rotating Kerr black hole, assuming that at infinity the state of the gas is described by a distribution function depending only on the energy of the particles. Neglecting the self-gravity of the gas, we show that relevant physical observables, including the particle current density and the accretion rates associated with the mass, the energy, and the angular momentum, can be expressed in the form of closed integrals that can be evaluated numerically or approximated analytically in the slow-rotation limit. The accretion rates are computed in this manner for both monoenergetic particles and the Maxwell-Jüttner distribution and compared with the corresponding results in the non-rotating case. We show that the angular momentum accretion rate decreases the absolute value of the black hole spin parameter. It is also found that the rotation of the black hole has a small but non-vanishing effect on the mass and the energy accretion rates, which is remarkably well described by an analytic calculation in the slow-rotation approximation to quadratic order in the rotation parameter. The effects of rotation on the morphology of the accretion flow are also analyzed.

Figures (22)

• Figure 1: Left plot (case A): Function $K(\vartheta)$ for the parameter values $a = 0.95M$, $L_z = 0.4Mm$, and $E = m$. There is a global minimum at $\vartheta = \pi/2$. Right plot (case B): Same as in the previous case except that the energy is $E = 2.5m$. The global minima are located at $\vartheta=\vartheta^*$ and $\vartheta = \pi - \vartheta^*$ with $\vartheta^* \approx 0.443$.
• Figure 2: Left plot: Function $W_+(r)$ for the parameter values $a = 0.95M$, $\beta = \hat{L}_z/L = -0.6$, and $L = 4.83 Mm$. The maximum of the potential well approximately coincides with the asymptotic value $m$ and is located at $r\approx r_\mathrm{mb} \approx 4.505 M$. Right plot: Same as in the previous case except that $L = 6Mm$.
• Figure 3: Profiles of critical values of $Q$: $Q_c$ (left panel) and $Q_\mathrm{max}$ (right panel) versus the energy $\varepsilon=E/m$ for various values of the rotation parameter $\alpha=a/M$ and fixed $\vartheta=\pi/2$, $\chi=\pi/2$, and $r=10M$.
• Figure 4: Time components of the particle current density $J_t$ for a monoenergetic model with $\varepsilon_0 = 2$. Left: $\alpha = 1/2$. Right: $\alpha = 0.95$. Both graphs are plotted for $\vartheta = \pi/4$. Dashed vertical lines mark locations of the outer horizon.
• Figure 5: Angular components of the particle current density $J_\varphi$ for a monoenergetic model with $\varepsilon_0 = 2$. Left: $\alpha = 1/2$. Right: $\alpha = 0.95$. Both graphs are plotted for $\vartheta = \pi/4$. Dashed vertical lines mark locations of the outer horizon.