Stable Collisionless Tori Around Kerr Black Holes

TL;DR

This work establishes analytic, fully kinetic equilibria for collisionless tori with finite angular momentum around Kerr black holes and implements them in the GPU-based GRPIC framework Aperture. By enforcing Jeans's theorem with a distribution f$(x^j,p_j)=f_0\exp((E_{\max}-E)/T)\,\delta(L_z-L_0)$ and bounding energies via $E_{\min}(r,\theta)$, the authors produce steady tori that are stable in 2D axisymmetric simulations without seed magnetic fields. They detail the practical steps to translate the analytic torus into GRPIC initial conditions, including momentum-space sampling, current-source construction, and BL-to-KS coordinate transformations, enabling controlled kinetic studies of collisionless accretion and jet launching. The results highlight structured bulk motion, a position-dependent effective temperature, and a highly anisotropic pressure tensor, laying the groundwork for future 3D investigations of Weibel instability and MRI-driven accretion in low-luminosity AGN. Overall, the analytic tori provide robust, physically motivated starting points for first-principles kinetic simulations of collisionless black-hole accretion and outflows.

Abstract

In low-luminosity active galactic nuclei like M87$^\ast$ and Sgr A$^\ast$, the accretion flow in the vicinity of the black hole is in the collisionless regime, meaning that the collisional mean free path of charged particles is much larger than the dynamical length scales. To properly model the particle energization and emission from the collisionless accretion flow, a promising approach is to employ the global general-relativistic particle-in-cell simulations$\unicode{x2014}$a newly developed, fully kinetic, first-principles method. However, it has been challenging to set up an initial condition that involves collisionless gas with finite angular momentum. We present, for the first time, a class of analytic kinetic equilibria of collisionless tori around a Kerr black hole. We have successfully implemented the collisionless tori in our GPU-based GRPIC code framework Aperture, and found them to be stable for hundreds to thousands of dynamical times in 2D axisymmetric simulations when there is no initial seed magnetic field. These kinetic equilibria serve as ideal starting points for future studies of the physics of collisionless accretion and jet launching.

Figures (3)

• Figure 1: Cross sections of the density distribution for three axisymmetric tori. Panels I, II, and III correspond to the initial tori with inner radii $r_{\mathrm{in}} = 11\,r_g$, $10\,r_g$, and $9\,r_g$, respectively, each normalized to unity. Panels IV, V, and VI show these same tori after evolving for $800\,r_g/c$. We include electromagnetic effects but there is no initial background field. All cases have temperature $T=10^{-3}m$ and constant angular momentum $L_0 \approx 3.986\,m$, corresponding to $r_0=14\,r_g$ around a black hole with spin $a = 0.999$. The fluctuations seen in the right column are consistent with particle shot noise for 20 particles per cell. In each panel, the black circle shows the black hole event horizon, $r_h = r_g(1 + \sqrt{1 - a^2})$.
• Figure 2: Angular velocity on the equatorial plane as a function of radius. The blue line shows the angular velocity of the collisionless torus, defined as $\Omega = S^{\phi} / S^0$, while the orange line shows the angular velocity of stable circular geodesic orbits, $\Omega_C = \left( a + r^{3/2}/\sqrt{M} \right)^{-1}$. The vertical line marks the location of $r_0 = 14$, where the lower bound of the particle energy reaches its minimum. The plotted range spans the radial extent of the torus from $r_{\rm in}$ to $r_{\rm out}$.
• Figure 3: Panel I: Cross section of the effective temperature, defined as the average squared particle velocity in the fluid rest frame, see Equation \ref{['eq:Teff']}. The equilibrium corresponds to the middle row of Figure \ref{['fig:density']}. Four labeled regions (A–D) are outlined, each containing approximately $3\times10^4$ particles. The box shapes follow the local grid structure to preserve consistent particle count. Panel II: Velocity distribution functions $P(v_{\mathrm{FRF}})$ for particles within each region, plotted as probability density (normalized to unity) versus fluid-frame speed $v_{\mathrm{FRF}}$. Each distribution resembles a truncated Maxwellian, with truncation occurring at lower velocities for regions closer to the outer boundary. This illustrates how the energy cutoff in the distribution function varies with position in the torus.