Choked accretion onto a Kerr black hole

The choked accretion model consists of a purely hydrodynamical mechanism in which, by setting an equatorial to polar density contrast, a spherically symmetric accretion flow transitions to an inflow-outflow configuration. This scenario has been studied in the case of a (nonrotating) Schwarzschild black hole as central accretor, as well as in the nonrelativistic limit. In this article, we generalize these previous works by studying the accretion of a perfect fluid onto a (rotating) Kerr black hole. We first describe the mechanism by using a steady-state, irrotational analytic solution of an ultrarelativistic perfect fluid, obeying a stiff equation of state. We then use hydrodynamical numerical simulations in order to explore a more general equation of state. Analyzing the effects of the black hole’s rotation on the flow, we find in particular that the choked accretion inflow-outflow morphology prevails for all possible values of the black hole’s spin parameter, showing the robustness of the model.

Figure 1

Example of the axisymmetric quadrupolar flow with $AM=0.01$ and a central Kerr black hole with $a=0.5$. The stagnation points in this case are located along the polar axis at a coordinate distance $r=\mathcal{S}\simeq 4.2M$. The figure shows isocontours of the three-velocity’s magnitude $V$. Note that $V$ becomes luminal at the event horizon ($r=r_{+}$) and at the outer ellipsoid indicated by a black, thick line. Fluid streamlines are indicated by thick, solid lines with an arrow. The axes correspond to the cylindrical-like coordinates $R=\sqrt{r^2+a^2}\sin \theta$, $z=r\cos \theta$.

Figure 2

Same as in Fig. 1, except that $AM=-0.01$ is negative. The stagnation points in this case are located on an equatorial ring at a coordinate distance $r=\mathcal{S}\simeq 5M$.

Figure 3

Outermost boundary $\mathcal{E}$ of the axisymmetric, quadrupolar flow for several values of the coefficient $A$. Solid, colored lines represent cases with $A>0$, while dashed colored lines correspond to $A<0$. The central black hole has a spin parameter $a=0.5M$. The event horizon is indicated by a solid black line. Note that the curves $\mathcal{E}$ corresponding to $A=A_{\max }\simeq 0.41$ and $A=A_{\min }\simeq -0.21$ touch the event horizon at the equator and pole, respectively. Curves with $A>A_{\max }$ or $A<A_{\min }$ actually pierce through the horizon. The axes correspond to the cylindrical-like coordinates $R=\sqrt{r^2+a^2}\sin \theta$, $z=r\cos \theta$.

Figure 4

Same as in Fig. 1, except that now we consider an inclination angle ${\theta }_0=30°$. In order to show the stagnation points, the plot corresponds to the plane $\varphi =\varphi (\epsilon )=-0.15882$ (see Table 2 and accompanying discussion).

Figure 5

Schematic representation of the choked accretion model in the polar plane. Shown are the black hole region ($r<r_{+}$), the location of the injection sphere ($r=\mathcal{R})$, the location of the reference point $(r=\mathcal{R},\theta =\pi /2)$ where the data $({\rho }_0,h_0,V_0)$ characterizing the solution are specified, the location of the stagnation point at $r=\mathcal{S}$ and the critical angle ${\theta }_c$ which separates the inflow from the outflow regions on the injection sphere.

Figure 6

Analytic model of choked accretion for a Kerr black hole with $a=0.99M$ and flow parameters $\mathcal{R}=10M$ and $V_0=0.2$. The figure shows isocontours of the fluid’s normalized rest-mass density (left panel) as well as the magnitude of the three-velocity (right panel). The stagnation points are located on the symmetry axis with radius $\mathcal{S}\simeq 4.6011M$. Fluid streamlines are indicated by thick, solid lines with an arrow. The axes correspond to the cylindrical-like coordinates $R=\sqrt{r^2+a^2}\sin \theta$ and $z=r\cos \theta$.

Figure 7

Mass accretion rate as a function of the model parameters ($a$, $V_0$) in units of ${\dot{M}}_0=8\pi M^2{\rho }_0$. The value of the injection radius $\mathcal{R}$ in each case is indicated by a central label on each panel. The solid, black lines in each panel indicate the range of validity of the model parameters according to $V_0\in (V_{\min },V_{\max })$, with the lower boundary corresponding to $V_{\min }$ and the upper one to $V_{\max }$.

Figure 8

Dependence of different properties of the choked accretion model on the spin parameter $a/M$ and the injection velocity $V_0$, for an injection sphere at $\mathcal{R}=10M$. From top to bottom, each panel shows: the mass accretion rate $\dot{M}$ in units of ${\dot{M}}_0=8\pi M^2{\rho }_0$, the location of the stagnation points $\mathcal{S}/M$, and the ejection-to-injection mass rate ratio $\eta ={\dot{M}}_{\mathrm{ej}}/{\dot{M}}_{\mathrm{in}}$. The black line in the first panel corresponds to the “spherical” case $(\ell ,m)=(0,0)$, for which $V_0=2M{r}_{+}/\sqrt{{\mathrm{\Sigma }}_0}\simeq 0.0003$.

Figure 9

Validation test of the aztekas code. In this figure we show the steady state of numerical simulation for the benchmark test, which corresponds to the analytic solution presented in Sec. 4, with parameters $\mathcal{R}=10M$, $V_0=0.2$, and $a=0.99M$ (compare with Fig 6). The figure shows the isocontour levels of the normalized rest-mass density $\rho /{\rho }_0$ (left panel) and the magnitude of the three-velocity (right panel) $V$, as measured by a ZAMO at this location. The fluid streamlines are indicated with black solid arrows. The simulation reached the stationary state at $t=180M$, showing a good agreement with the analytic solution.

Figure 10

Validation test for the aztekas code. In this figure we show the evolution in time of the relative error between the numerical mass accretion rate $\dot{M}$ and analytic value ${\dot{M}}_A$ for the solution with parameters $\mathcal{R}=10M$, $V_0=0.2$, and $a=0.99M$. The four different resolutions used for this benchmark test are represented with different dashed lines, showing a diminishing of the error as the resolution increases which, as shown in Appendix pp3, is consistent with second order convergence.

Figure 11

Stationary state of the numerical simulations of the choked accretion mechanism for a rotating black hole with $a=0.99M$, an injection radius of $\mathcal{R}=10M$, a dimensionless temperature of ${\mathrm{\Theta }}_0=1$ at the equator of the injection sphere, and a polytropic fluid with $\gamma =4/3$ (top panels) and $5/3$ (bottom panels). On the left panels of this figure we show the isocontour levels of the normalized rest-mass density $\rho /{\rho }_0$ while, on the right panels, the magnitude of the three-velocity $V$ as measured by the ZAMO. The fluid streamlines are indicated with black solid arrows, while a white solid line shows the location of the event horizon $r_{+}$.

Figure 12

Mass accretion rate as a function of the spin parameter. The blue and red dotted lines correspond to $\gamma =4/3$ and $5/3$, respectively. The black dashed lines represent their respective “spherical” accretion values, i.e., $\delta =0$.

Figure 13

Dependence of different properties of the choked accretion simulations on the spin parameter $a/M$ and $\gamma$. From top to bottom, each panel shows: the mass accretion rate $\dot{M}$, the location of the stagnation point $\mathcal{S}/M$, and the ejection-to-injection mass rate ratio $\eta ={\dot{M}}_{\mathrm{ej}}/{\dot{M}}_{\mathrm{in}}$. Each panel is normalized by its corresponding nonrotating case value, which is denoted with the subscript $\mathrm{S}$. The black solid line represents the analytic solution presented in Sec. 4 for an ultrarelativistic stiff fluid.

Figure 14

$L^1$-norm of the error in the mass accretion rate, for the benchmark test presented in Sec. 5a. The test is performed using the radial resolutions $N_r=64$, 128, 256, 512. The black dashed lines represent the expected tendency for a first (top) and second order (bottom) convergence.

Figure 15

Self-convergence tests of the polytropic fluid simulations, for both values of $\gamma$: $4/3$ (top panel) and $5/3$ (bottom panel). In this figure we show the evolution in time of the convergence rate $Q$ for three different values of the spin parameter $a$. The gray stripe shows the expected convergence zone (given the numerical methods used in aztekas).