Bondi Accretion

Am working on a 3D simulation of Bondi Flow onto a central sink particle and studying the accretion properties using the Krumholz accretion algorithm.

The following are the fiducial IC's used in the simulations (in computational units), unless otherwise noted:

Lx=Ly=Lz	20
Mx=My=Mz	64
MaxLevel (AMR)	2
Effective resolution (dx)	.15625
Sink dx = 4dx	.625

From Bondi (1952) we have the following quantities (in order: Bondi radius, accretion rate, sonic radius, nondimensional density and velocity):

R_{BH} = \frac{GM_*}{C_\infty^2}

\dot{M}_{BH} = 4 \pi \lambda_{crit}(G M_*)^2c_\infty^{-3}\rho_{\infty}

R_s=\frac{5-3\gamma}{4}\frac{GM_*}{C_{\infty}^2}

z\_=\frac{\rho}{\rho_{\infty}}

y\_=\frac{v_r}{C_\infty}

In the above, \lambda is a numerical coefficient which controls the strength of the accretion flow. At its critical value, \lambda=\lambda_{crit}, \dot{M} is strongest. In the code, \lambda_{crit} is solved for using Eqn. 18 in Bondi's paper. With respect to these 'Bondi' variables, the sims used the following (in computational units), unless otherwise noted:

C_{\infty}	1.291
T_{\infty}	~1
\rho_{\infty}	~1
M_*	.14318E+13
R_{BH}	25.63
\dot{M}_{BH}	2692.17

with the computational domain set up using:

Lx=Ly=Lz	20
Mx=My=Mz	64
MaxLevel (AMR)	2
Effective resolution (dx)	.15625
Sink dx = 4dx	.625

Note, the sink dx is equivalent to the accretion volume radius (r_{acc}).

Research Log:

3/18/18, \gamma=7/5, dynamical cases

Resolving sonic point

Not-resolving sonic point

3/18/18, \gamma=1.66, steady-state case

The case of \gamma=5/3 flow is an extreme limit for the solution space. This value of \gamma has a few interesting features:

In other words, when \gamma=5/3 exactly, we have no accretion flow.

In addition to the Bondi/flow params cited in the top section of this page, the sim was set up using the following (in CU):

\gamma	1.66
\lambda_{crit}	.2526
R_{s}	.0128

Note, the sonic radius is not resolved by ~ factor of 20 in this simulation, as seen when checking dx quoted above. The problem module for this sim sets of the Bondi solution everywhere in the grid at t=0 (except within a small inner radius, described below). In subsequent timesteps, the Bondi solution is re-pasted into the ghost zones and a small inner spherical region at the origin. A sink particle is initialized at the origin with mass given above (M_{sink}=M_*). The particle can accrete gas following the Krumholz prescription. The only source of gravity in the sims is the point gravity object associated with the particle (self-gravity is turned off).

The simulation setup reproduces the correct nondimensional profiles. Here's a comparison of the t=0 nondimensional density and velocity profiles astrobear calculates for the bondi module (left), compared to the Bondi solutions for \gamma=5/3 (curves II and III for the case of \lambda=\lambda_{crit}).

The profiles are cut-off within an inner radius of r=.875 to avoid extremely high speeds there (am going to get rid of this in the next round of sims). Sampling the mass flux across a spherical shell less than this radius doesn't match up with the theoretical prediction of \dot{M}=2692 since the solution is getting stepped on there. Sampling the mass flux across a shell larger than this, however, produces agreement (2689 compared to 2692).

Since the solution is getting stepped on within some small inner radius, can't meaningfully check the behavior of the accretion algorithm and any spurious waves it might be generating there. Instead, will be removing this inner boundary in the next run. These next sims will then be able to test the effect of the sonic radius being inside the accretion radius as opposed to outside. To test this will do a resolution study on \gamma=7/5 flow, where the sonic point is \sim .2R_{BH}.

As the following images show, this set of ICs produces a steady-state solution.. This is due to the solution being stepped on within the accretion volume, which would likely be the first place any deviation from the similarity solutions would occur due to spurious pressure waves that the accretion algorithm might produce in such strongly subsonic flow as this. The attached module files that produce these files are thus called "*_steadystate*"

Mesh
Radial velocity
Radial mach
Velocity field
Isosurface (r=rsink)
Mass flux (r=rsink)

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