Accretion of the relativistic Vlasov gas in the equatorial plane of the Kerr black hole

We investigate stationary accretion of the collisionless Vlasov gas onto the Kerr black hole, occurring in the equatorial plane. The solution is specified by imposing asymptotic boundary conditions: at infinity the gas obeys the Maxwell-Jüttner distribution, restricted to the equatorial plane (both in positions and momenta). In the vicinity of the black hole, the motion of the gas is governed by the spacetime geometry. We compute accretion rates of the rest-mass, the energy, and the angular momentum, as well as the particle number surface density, focusing on the dependence of these quantities on the asymptotic temperature of the gas and the black hole spin. The rest-mass and energy-accretion rates, normalized by the black hole mass and appropriate asymptotic surface densities of the gas, increase with increasing asymptotic temperature. The accretion slows down the rotation of the black hole.

Figure 1

Sample solutions of Eq. (59) for $\alpha =2/3$, ${\epsilon }_{\sigma }=+1$ (continuous line), ${\epsilon }_{\sigma }=-1$ (dashed line). Physically relevant branches, given by the parametric formula (53), are depicted with red lines. Note the existence of mirrored branches in the remaining three quadrants of the full $ϵ-\lambda$ plane (not shown).

Figure 2

Sample solutions of Eq. (59) for $\alpha =1$, ${\epsilon }_{\sigma }=+1$ (continuous line), ${\epsilon }_{\sigma }=-1$ (dashed line). Physically relevant branches are depicted with red lines. For $\alpha =1$ and ${\epsilon }_{\sigma }=+1$, ${\lambda }_c(ϵ,{\epsilon }_{\sigma }=+1)=ϵ$.

Figure 3

Sample graphs of ${ϵ}_{\min }(\xi ,{\epsilon }_{\sigma })$ for $\alpha =1/10$ and ${\epsilon }_{\sigma }=+1$ (blue continuous line) and ${\epsilon }_{\sigma }=-1$ (orange dashed line). Vertical lines denote locations of circular photon orbits ${\xi }_{\mathrm{ph}}$ for ${\epsilon }_{\sigma }=\pm 1$. Note that ${ϵ}_{\min }(\xi ,+1)$ is regular at ${\xi }_{\mathrm{ph}}$ corresponding to ${\epsilon }_{\sigma }=-1$.

Figure 4

Components of the particle current surface density $J_{\phi }$, $J_{\phi }^{(\mathrm{abs})}$, and $J_{\phi }^{(\mathrm{scat})}$ corresponding to ${\epsilon }_{\sigma }=+1$. The spin parameter $\alpha =7/8$; $\beta =0.1$. The vertical line marks the location of the circular photon orbit ${\xi }_{\mathrm{ph}}$.

Figure 5

Same as in Fig. 4, but for ${\epsilon }_{\sigma }=-1$.

Figure 6

Components of the particle current surface density $J_{\phi }$, $J_{\phi }^{(\mathrm{abs})}$, and $J_{\phi }^{(\mathrm{scat})}$ obtained by summing the parts corresponding to ${\epsilon }_{\sigma }=\pm 1$. The spin parameter $\alpha =7/8$; $\beta =0.1$.

Figure 7

Components of the particle current surface density $J_t$, $J_t^{(\mathrm{abs})}$, and $J_t^{(\mathrm{scat})}$ corresponding to ${\epsilon }_{\sigma }=+1$. The spin parameter $\alpha =7/8$; $\beta =0.1$. The vertical line marks the location of the circular photon orbit ${\xi }_{\mathrm{ph}}$.

Figure 8

Same as in Fig. 7, but for ${\epsilon }_{\sigma }=-1$.

Figure 9

Components of the particle current surface density $J_t$, $J_t^{(\mathrm{abs})}$, and $J_t^{(\mathrm{scat})}$ obtained by summing the parts corresponding to ${\epsilon }_{\sigma }=\pm 1$. The spin parameter $\alpha =7/8$; $\beta =0.1$.

Figure 10

Components of the particle current surface density $J_{r+}^{(\mathrm{abs})}$ (for ${\epsilon }_{\sigma }=1$), $J_{r-}^{(\mathrm{abs})}$ (for ${\epsilon }_{\sigma }=-1$) and $J_r$ obtained by summing the parts corresponding to ${\epsilon }_{\sigma }=\pm 1$. The spin parameter $\alpha =7/8$; $\beta =0.1$.

Figure 11

Streamlines of the vector field $(J^x,J^y)$. Circular photon orbits are shown in red and blue colors. The region inside the black hole horizon is marked in black. The black hole rotates counterclockwise. The parameters used to create this plot are $\alpha =0.5$ and $\beta =1$. The units on the axes are mass units $M$.

Figure 12

The invariant particle surface density $n_s$ vs the dimensionless radius $\xi$. The left panel shows the dependence of $n_s$ on the value of the parameter $\beta$ for $\alpha =7/8$. In the right panel we set $\beta =1$ and plot the graphs of $n_s$ for three different values of $\alpha$. Vertical lines in the right plot correspond to inner Kerr horizons.

Figure 13

Accretion rate $\dot{M}$ vs black hole spin parameter $\alpha$. The abscissa in the left panel shows ${\lim }_{\beta \to \infty }\dot{M}/(M{\rho }_{s,\infty })={\sum }_{{\epsilon }_{\sigma }=\pm 1}{\lambda }_{*}({\epsilon }_{\sigma })$ (solid blue line), ${\lambda }_{*}({\epsilon }_{\sigma }=+1)$ (dotted orange line), and ${\lambda }_{*}({\epsilon }_{\sigma }=-1)$ (dashed green line). The right panel depicts accretion rates $\dot{M}$, rescaled according to Eq. (130). Blue and orange graphs correspond to the limits $\beta \to 0$ and $\beta \to \infty$, respectively. A graph of the half circle $\sqrt{1-{\alpha }^2}$ (dashed green line) is shown for comparison.

Figure 14

$\dot{M}/(M{\rho }_{s,\infty })$ versus $\alpha$. The dots represent numerically computed values. Solid lines depict the limits $\beta \to 0$ (blue line) and $\beta \to \infty$ (orange line), given by Eqs. (124) and (92), respectively. Numerical data are computed for $\beta =1/4000$, $1/10$, 1, 10, 100, and 10000.

Figure 15

$\dot{M}/(M{\rho }_{s,\infty })$ versus $\beta$. Blue dots correspond to $\alpha =0$. Orange dots correspond to $\alpha =\pm 1$. For $\alpha =0$ the data interpolate between $6\sqrt{3}$ and 8. For $\alpha =\pm 1$, between 9 and $2(2+\sqrt{2})$. The inflection point is located around $\beta \simeq 2.8$.

Figure 16

$\dot{\mathcal{E}}/(M{ϵ}_{s,\infty })$ versus $\alpha$. The dots represent numerically computed values. Solid lines depict the limits $\beta \to 0$ (blue line) and $\beta \to \infty$ (orange line), given by Eqs. (125) and (95), respectively. Numerical data are computed for $\beta =1/1000$, $1/10$, 1, 10, 100, and 10000.

Figure 17

$\dot{\mathcal{L}}/(M^2{ϵ}_{s,\infty })$ versus $\alpha$. The dots represent numerically computed values. Solid lines depict the limits $\beta \to 0$ (blue line) and $\beta \to \infty$ (orange line), given by Eqs. (128) and (97), respectively. Numerical data are computed for $\beta =1/100$, $1/10$, 1, 10, 100, and 1000. Note a nearly linear behavior for moderate values of $\alpha$; the deviation from a first order series expansion around $\alpha =0$ is of the order of 3.5% at $\alpha =\pm 0.5$.