Changes between Version 19 and Version 20 of u/erica/RadFeedback

Timestamp:

02/16/16 20:28:56 (9 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/RadFeedback

@@ -1 @@
-== Radiation feedback from sink particles ==
+[[PageOutline]]
 
-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.
+= Radiation feedback from sink particles =
 
-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. 
+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 must come from accretion.
+
 
 == 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:
+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:
 
 [[latex($\frac{1}{2} m v_{ff}^2 = \frac{GmM_{*}}{R_{*}}$)]]
 
-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 [[latex($\dot{m}$)]], the rate at which this heat is produced, or the luminosity L, is given by:
+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 [[latex($\dot{m}$)]], the rate at which this heat is produced, or the luminosity L, is given by:
 
 [[latex($ 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:
+Since we do not track energy accretion onto the sink, we are left to assume that the gas that is accreted from the surrounding zones contributes to this accretion luminosity directly. Thus, the best we can do for tracking the energy released 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 [https://astrobear.pas.rochester.edu/trac/blog/erica01262016 here]). 
 
-[[latex($\frac{1}{2}m v^2(r) - \frac{GmM_{r}}{r} = \frac{1}{2}m v^2(R_{*}) - \frac{GmM_{*}}{R_{*}}$)]] 
+The accretion energy ([[latex($L*dt$)]]) will then be distributed smoothly in a kernel surrounding the sink every time step. From there it will 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 energy in the code = 
 
-Rearranging for the accretion luminosity, we have:
+At each time step i, the accretion energy is computed as:
 
-[[latex($L=\frac{1}{2} \dot{m}v^2(r)+G(\frac{M_{*}}{R_{*}}-\frac{M_{r}}{r})\dot{m}$)]]
+[[latex($E=\frac{G M dm}{R }$)]]
 
-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 [[latex($v(r)$)]] 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 [[latex($\dot m$)]] to be the 'Bondi Accretion Rate' already calculated in the code. 4. [[latex($M_{*}$)]] is the mass of the sink particle. Thus, radiative feedback is only possible after sinks have performed their first accretion. 5. [[latex($R_{*}=\frac{1}{4}\Delta X_{min}$)]]. 6. [[latex($M_r$)]] is the total mass within the accretion volume (sum of all [[latex($q_i(1)$)]] + mass of sink).
+where [[latex($M$)]] is the mass of the sink particle at time i, [[latex($dm$)]] is the total accreted mass for that time step, [[latex($G$)]] is the gravitational constant in computational units, [[latex($R$)]] is the radius of the protostar, taken to be 1 solar radius by default, but modifiable by the user at run-time. 
+
+== Kernel ==
+
+After computing the accretion energy for the particle, we then smooth this energy over a kernel of cells to be fed into the radiative transfer module as a source function. To ensure that the sum of the differential energies over the kernel equals the total accretion energy calculated, we have the following equation:
+
+[[latex($\sum \Delta E_i*dV_i=E ~~~~~~(1)$)]]
+
+where [[latex($E$)]] is the accretion energy, [[latex($\Delta E_i$)]] is the differential amount of E to be distributed in the ith cell, and [[latex($dV_i$)]] is the volume of the ith cell. As of now, the units don't balance in this equation. We then need to find a normalization constant that has units of 1/volume, which we will do next.
+
+We want the amount of E in each cell to drop off smoothly with radius away from the sink. For this we choose a decaying exponential. Let, 
+
+[[latex($\Delta E_i= k E e^{-r_i/\sigma} ~~~~~~~(2)$)]]
+
+where [[latex($\sigma$)]] is a scaling factor. In the code the exponential function used is such that it falls smoothly to zero at the boundary of the kernel, and covers 4 e-foldings. Now, to solve for the normalization constant, we insert (2) into (1):
+
+[[latex($\sum k E e^{-r_i/\sigma}dV_i=E$)]]
+
+
+and solve for k:
+
+[[latex($k= \frac{1}{\sum e^{-r_i/\sigma} dV_i}$)]]
+
+
+While the set of equations for the kernel is arbitrary, as we will see in the next section, this normalization constant allows us to easily feed into the source function a ''specific'' accretion energy (i.e. E/V), which is necessary for the code's solvers. 
+
+
+
+== Work arrays and feeding E into the radiative source function ==
+
+There is a subroutine in the code that is called by the radiative transfer module, 'apply kernel to work array'. This goes through and populates the work array for each cell's accretion energy as:
+
+
+[[latex($workarray(i,j,k,iaccretion\_energy)= k*E*e^{-r/\sigma}/dt $)]]
+
+This is actually an average luminosity over the hydro step. This is then fed into the source function as:
+
+[[latex($ source(i,j,k,iaccretion\_energy)= workarray(i,j,k,iaccretion\_energy)*dt_{rad}$)]]
+
+Note, that the source is being populated by a ''specific'' accretion energy (recall k has units of 1/volume), averaged over a radiative time step.
+
+== Physical meaning of the accretion energy source term == 
+
+-how it couples to the gas, how it couples to the radiation, some equations go here.
+
+= Tests =
+
+== 2D, fixed grid - radiating sink ==
+
+=== Energy output === 
+
+=== Different opacities ===
+
+=== Marshak Waves ===
+
+== 2D, fixed grid - collapsing clump ==