Radiation feedback from sink particles

The amount of thermal radiation produced in the grid is a function of temperature. Since sinks are a subgrid model they themselves do not have temperature (we are not sure how big the forming star is, how fast it is growing by contraction, etc., so there isn't an easy way of assigning the sub-grid object a 'temperature'). Thus, we are left to estimate the amount of radiation produced by the sink. We envision the sink particle as a protostar, and not yet producing radiation through fusion. Thus, the radiation our sinks produce comes from accretion.

Accretion Luminosity

For spherical symmetry, a gas parcel starting from rest and freely falling to the star from infinity will have its kinetic and gravitational energy balance at the stellar surface:

\frac{1}{2} m v_{ff}^2 = \frac{GmM_{*}}{R_{*}}

As material passes through the accretion shocks at the surface of the star, its kinetic energy is converted into heat that is then radiated away. For an accretion rate \dot{m}, the rate at which this heat is produced, or the luminosity L, is given by:

L = \frac{1}{2}\dot{m} v_{ff}^2 = \frac{G\dot{m}M_{*}}{R_{*}}

Since we do not track energy accretion onto the sink, we are left to assume gas that is accreted from the surrounding zones contribute to this accretion luminosity directly. Thus, the best we can do for tracking the energy release from infall is to calculate the RHS of this equation in the code and use it as an estimate of the true accretion luminosity. (By the way, this form of the accretion luminosity was shown to be a good approximation for our purposes ​here).

The accretion energy (L*dt) will be distributed smoothly in a kernel surrounding the sink every time step. It will then diffuse away from the sink via FLD radiative transfer. In this way, sinks will act as additional sources of radiation within the grid.

Tracking accretion luminosity in the code

At each time-step i, the luminosity is computed as:

L_i=\frac{G M_i dm_i}{R ~dt}

where M_i is the mass of the sink particle at i, dm_i is the total accreted mass for that time-step, G is the gravitational constant in computational units, R is the radius of the star, taken to be 1 solar radius by default, but modifiable by the user at run-time, and dt is the hydro time-step. This luminosity is only computed after the sink particle has grown to have non-zero mass, and then will only be a source of luminosity after surrounding material has been deemed unstable and collapsing inward to the sink particle (i.e. is accreted).

Kernel

Work arrays and feeding L into the source

Tests

2D, fixed grid - radiating sink

Energy output

Different opacities

Marshak Waves

2D, fixed grid - collapsing clump