Changes between Version 17 and Version 18 of u/erica/scratch


Ignore:
Timestamp:
02/08/16 12:15:09 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/scratch

    v17 v18  
    11The accretion luminosity is summed up for all cells surrounding the sink particle, and the total is given by 'E_acc'. This total accretion energy is then smoothed over a kernel of cells surrounding the sink, proportionally to a decreasing exponential, such that,
    22
    3 Sum dE_acc | cells = E_acc
     3[[latex($\Sigma ~dE_{acc} = E_{acc}$)]]
    44
    5 In other words, the exponential function is 'normalized'.
     5In other words, the exponential function is ''normalized''.
    66
    77We also want the exponential function to go to zero at the boundary of the kernel.
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    9 So given these 2 constraints, we have the following set of equations,
     9So given these 2 constraints, we have the following equation,
    1010
     11[[latex($E_{acc} = \sum_{i=0}^{4} k(~e^{-dx*i} - e^{-4dx} )~E_{acc}  $)]]
     12
     13which gives the normalization constant,
     14
     15[[latex($ \boxed{k = \frac{1}{~\Sigma ~e^{-dx*i}-e^{-4dx}}} $)]]
     16
     17Note that the sum runs over cells i=0 to 1, where the '0th' cell is the cell the sink is in, and the '4th' cell is the furthest cell from the sink in the kernel, that the exponential goes to zero at this boundary, and dx*i gives a position (which here assumes the sink is at the cell center, but will be more general in the code).
     18