Sink Particles

The Sink Particle algorithm is more or less taken from ​Federrath et al.

Before each step after overlaps have been applied - each processor searches through each info structure looking for new particles. This requires data out to 4 ghost cells. The processors then gather all of the new particles and the moments of the surrounding cells. Then each processor updates all of its grids by accreting material onto the sink particles. Then the change in particle mass from each processor is accumulated and the particles new properties (mass, mom, etc…) are then updated. The sink particles are then advanced redundantly using a leap-frog integration on every processor.

There are several criteria for creating sink particles. The Truelove criteria states the following must be satisfied:

\lambda_J=\left( \frac{\pi c_s^2}{G\rho} \right )^{1/2} > ~4\Delta x $

else artificial fragmentation can occur. Thus, the first criteria is that the density and sound speed in a cell must violate the Truelove criteria:

begin{equation} \lambda_J=\left( \frac{\pi c_s^2}{G\rho} \right )^{1/2} < ~4\Delta x \end{equation}

Regions that satisfy (1) are subject to potential artificial fragmentation.

A cell that violates the truelove criteria must also have converging flow and must be located at the minimum of a potential well within a radius of 4\Delta x

In addition to the central cell violating the truelove criteria - the surrounding region must be Jeans unstable or from the virial equation 2T+U < 0 where we have ignored external pressure. This implies that:

begin{equation} 2 \int{(c_V nKT) dV} - \frac{1}{2}G\int{\rho(r') \int{\frac{\rho(r)}{(r-r')}}} < 0 \end{equation}

For an ideal gas we have c_V nKT = \frac{P}{\gamma-1} = \frac{\rho c_s^2}{(\gamma-1) \gamma} or c_V nKT = \frac{3}{2} \rho c_s^2 for an isothermal gas.

If we assume a uniform sphere of radius R=4 \Delta x we get

begin{equation} 3\rho c_s^2 V - \frac{3 G \rho^2 V^2}{5R} < 0 \rightarrow c_s^2 - \frac{G\rho V}{5R} < 0 \end{equation}

or that

begin{equation} R^2 > \frac{15 c_s^2}{4 \pi G \rho} \rightarrow R > \left (\frac{15}{4 \pi^2} \right )^{1/2} \lambda_J \sim .6164 \lambda_J < \lambda_J < 4\Delta x \equiv R \end{equation}

So if a cell satisfies the Truelove criteria - then assuming that it is surrounded by cells of similar density within a radius of 4\Delta x it will also be Jeans unstable.

Magnetic fields modify the Jeans stability / virial equation to be 2T+U+M < 0 where M=\int{\frac{B^2}{2}}

Finally the region must be gravitationally bound

begin{equation} T+U+M+K<0 \end{equation}

For reference on another way astrobear has implemented sinks in the past, see SinkParticle