Bondi accretion

Overview

Spherically symmetric, radial accretion onto a Schwarzschild black hole. The relativistic Bondi solution (Michel 1972, Hawley et al. 1984) provides an exact steady-state for density, internal energy, and radial velocity as functions of radius, fully determined by the mass accretion rate $\dot{M}$ and the sonic radius $r_s$. The code is initialized with this analytic solution and evolved forward in time; because the solution is stationary, any deviation at the final dump is purely numerical error. This test uses a curved spacetime and exercises the GR sector of the code.

Setup

The domain spans $[R_{\rm hor}, R_{\rm out}]$ radially and $[0, \pi]$ in polar angle, using Modified Kerr-Schild (MKS) coordinates with $a = 0$ (Schwarzschild). The analytic solution is characterized by two conserved constants,

$$ C_4 = r^2T^{N/2}u^r, \qquad C_3 = \left[1 + \bigg(1 + \frac{N}{2}\bigg)T\right]^2!\left[1 - \frac{2}{r} + (u^r)^2\right], $$ where $N = 2/(\Gamma-1)$ is the number of degrees of freedom, and $T$ is the dimensionless temperature ($\propto kT/mc^2$). These are fixed by the sonic conditions at $r = r_s$,

$$ u^r_s = \sqrt{\frac{1}{2r_s}}, \quad T_c = \frac{2N c_{s,c}^2}{(N+2)(2-Nc_{s,c}^2)}, \quad c_{s,c}=\frac{(u^r_c)^2}{1-3(u^r_c)^2}, $$ where $c_{s,c}$ is the sound speed at $r_s$. At each radius, $T(r)$ is obtained by Newton-Raphson root-finding of $C_3(r, T) - C_3 = 0$, then

\[ u^r = -\frac{C_4}{r^2 T^{N/2}}, \qquad \rho = K^{-N/2} T^{N/2}, \qquad u = \frac{\rho T}{\Gamma - 1}, \]

where $K = (4\pi C_4/\dot{m})^{1/n}$. The outer boundary applies the analytic solution at every timestep (X1R_BOUND USER); the inner boundary is outflow with no inflow enforced (X1L_BOUND OUTFLOW, X1L_INFLOW 0). There is no magnetic field.

Parameters

Problem-specific runtime parameters are:

Parameter	Meaning
`mdot`	Mass accretion rate $\dot{M}$
`rs`	Sonic radius
`Rhor`	Inner radial boundary (set slightly outside $r = 2M$)

Relevant compile-time parameters are:

Parameter	Default	Notes
`N1TOT`, `N2TOT`	`64`	Grid resolution; change for convergence study
`METRIC`	`MKS`	Modified Kerr-Schild
`RECONSTRUCTION`	`WENO`
`X1L_BOUND`	`OUTFLOW`	Inner radial boundary
`X1R_BOUND`	`USER`	Outer radial boundary — analytic Bondi solution
`X2L/R_BOUND`	`POLAR`
`X1L_INFLOW`	`0`	No inflow at inner boundary

Convergence

Because the solution is stationary, the L1 error is computed by comparing the final dump to the initial dump for $\rho$, $u$, and $u^r$, excluding zones inside the event horizon,

\[ L_1(q) = \frac{1}{N_{\rm active}}\sum_{r_i > R_{\rm hor}}\left|q_{ij}(t_f) - q_{ij}(0)\right|. \]

The errors in $\rho$, $u$, and $\tilde{u}^1$ exhibit the expected second-order convergence, $L_1 \propto N^{-2}$.

Convergence

References

Michel (1972) — relativistic Bondi solution.
Hawley et al. (1974) - relativistic Bondi solution formulated in terms of thermodynamic quantities at sonic point instead of asymptotic values.

Parameter	Meaning
`mdot`	Mass accretion rate \(\dot{M}\)
`rs`	Sonic radius
`Rhor`	Inner radial boundary (set slightly outside \(r = 2M\))