Outflow Feedback

Adding a precessing, bipolar, 2-component outflow to the sink particle routine, following ​Federrath, et al. 2014

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Density injection

-Define a spherical cone centered on the sink particle of height rsink
-Prescribe some function that distributes accreted mass over the cone, with some radial and angular dependance
-Given this is a discretized operation, and the sink is not necessarily positioned at the center of the conical region, the masses in the 2 lobes may be slightly different
-Thus, compute a correction factor for each of the cells to multiply the mass by when applying it to the cones
-Rinse and repeat

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References: ​https://astrobear.pas.rochester.edu/trac/blog/johannjc05052015 and ​https://astrobear.pas.rochester.edu/trac/wiki/u/johannjc/scratchpad

Radiative Feedback

The amount of thermal radiation produced in the grid is a function of temperature. Since sinks are a subgrid model they 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'). However, we track the amount of energy that falls onto the sink. We can imagine that as material hits the surface of the star (i.e. as it is accreted by sinks), it contributes to the energy that is re-radiated back into the grid. That is because young stars emit energy through many means: mechanical (e.g. outflows) and various radiation processes. Since we are not modeling stellar evolution on the sub-grid scale we are not following how much energy is being released due to thermonuclear reactions in the core of our invisible star. Instead we can just imagine that as the material hits the surface of the star (i.e. passes through the sink particle) it is slowed and compressed, thereby producing thermal radiation, which we then distribute to the zones surrounding the star.

To mock this up, we will prescribe some fraction of infalling energy to be recycled back into the grid. We will have this radiative energy distributed smoothly in a kernel surrounding the sink, so that it diffuses away back into the grid through the solution of the radiative transfer equations. In this way, the sinks will act as an additional source of radiation. The kernel of cells surrounding the sinks will be stepped on each radiative time step with the values of Erad from the star.

Accretion Luminosity

The amount of energy deposited into the kernel around a sink is intuitively given by the accretion energy. As infalling material hits the surface of the star, its kinetic energy is converted to heat. For spherical symmetry, a gas parcel starting from rest and freely falling to the star from infinity will have:

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

at the surface of the star. As the material strikes the surface of the star (i.e. is accreted) the kinetic energy is converted to heat. 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_{*}}

However, in our simulations the gas parcel isn't falling into the sink from infinity, but rather from some distance r away from the sink. At this radius it has some initial kinetic and gravitational potential energy. For this situation, energy conservation gives:

\frac{1}{2}m v^2(r) - \frac{GmM_{r}}{r} = \frac{1}{2}m v^2(R_{*}) - \frac{GmM_{*}}{R_{*}}

Rearranging for the accretion luminosity, we have:

L=\frac{1}{2} \dot{m}v^2(r)+G(\frac{M_{*}}{R_{*}}-\frac{M_{r}}{r})\dot{m}

To first order, we make the following approximations. 1. Only mass within the accretion volume will contribute to the accretion luminosity. This is reasonable because only mass within this volume is accreted. 2. We take v(r) to be the velocity of the ith cell at a distance r away from the sink (for r within the accretion volume). 3. We take \dot m to be the 'Bondi Accretion Rate' already calculated in the code. 4. M_{*} is the mass of the sink particle. Thus, radiative feedback is only possible after sinks have performed their first accretion. 5. R_{*}=\frac{1}{4}\Delta X_{min}. 6. M_r is the total mass within the accretion volume (sum of all q_i(1) + mass of sink).