| 31 | | To calculate the kernel (i.e. volume of cells that we will be injecting the accretion energy into), we have the following equations: |
| | 31 | To ensure that the sum of the differential energies over the kernel (i.e. volume of cells that we will be injecting the accretion energy into) equals the total accretion energy calculated, we have the following equation: |
| | 32 | |
| | 33 | [[latex($\sum \Delta E_i*dV_i=E ~~~~~~(1)$)]] |
| | 34 | |
| | 35 | 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. |
| | 36 | |
| | 37 | 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, |
| | 38 | |
| | 39 | [[latex($\Delta E_i= k E e^{-r_i/scale}$)]] |
| | 40 | |
| | 41 | To solve for the normalization constant, we insert this into (1): |
| | 42 | |
| | 43 | [[latex($\sum k E e^{-r_i/scale}=E$)]] |
| | 44 | |
| | 45 | |
| | 46 | and solve for k: |
| | 47 | |
| | 48 | [[latex($k= \frac{1}{\sum e^{-r_i/scale} dV_i}$)]] |
| | 49 | |
| | 50 | |
| | 51 | |
