Changes between Version 14 and Version 15 of u/erica/scratch4

Timestamp:

02/16/16 18:43:31 (9 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/scratch4

@@ -33 +33 @@
 [[latex($\sum \Delta E_i*dV_i=E ~~~~~~(1)$)]]
 
-where [[latex($E$)]] is the accretion energy from the time step, [[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. When we normalize the kernel, the normalization constant will then have units of 1/volume to satisfy this equation as we will do next.
+where [[latex($E$)]] is the accretion energy from the time step, [[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.
 
-Now, 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, 
+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($\Delta E_i= k E e^{-r_i/scale}$)]]
+[[latex($\Delta E_i= k E e^{-r_i/\sigma} ~~~~~~~(2)$)]]
 
-To solve for the normalization constant, we insert this into (1):
+where [[latex($\sigma$)]] is a scaling factor. In the code this scaling factor lets the exponential fall to zero at the boundary of the kernel, and a width that gives ... how to describe this. Now, to solve for the normalization constant, we insert (2) into (1):
 
-[[latex($\sum k E e^{-r_i/scale}=E$)]]
+[[latex($\sum k E e^{-r_i/\sigma}dV_i=E$)]]
 
 
 and solve for k:
 
-[[latex($k= \frac{1}{\sum e^{-r_i/scale} dV_i}$)]]
+[[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.