Changes between Version 1 and Version 2 of u/johannjc/scratchpad8


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Timestamp:
11/20/15 16:33:26 (9 years ago)
Author:
Jonathan
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  • u/johannjc/scratchpad8

    v1 v2  
    22http://www.aanda.org/articles/aa/pdf/2013/09/aa21551-13.pdf
    33
    4 1.) Self-shielding
    54
    6 delta v = 20 km/s
     5The prescription of Bourrier et al.. is to calculate (at each pixel) the average optical depth $\tau_i$ for range of velocity bins with spacing $\Delta v$ (or frequency bins with size $\Delta \nu$, so that
    76
    8 velocity is projected along star/earth los
     7$\sum \tau_i \Delta \nu = \tau$
    98
    10 $\tau \left(x,y,v \right ) = \sigma_{\nu 0} \int \int n \left (x,y,z,v' \right ) \phi (v'-v ) dv' dz $
     9So we have
    1110
    12 They don't explicitly say what $\phi$ is - but they imply that it is Lorentzian, and based on the parameterization I am assuming that
     11$\tau_i \Delta \nu = \tau_i \frac{\Delta v}{\lambda_0} = \displaystyle \int_{v_i-\frac{1}{2}\Delta v}^{v_i + \frac{1}{2}\Delta v} \tau (v) dv$
    1312
    14 $\phi(\nu) = \frac{1}{\pi} \frac{\frac{1}{2}\Gamma}{\left (\nu-\nu_0 \right )^2 + \left ( \frac{1}{2} \Gamma \right ) ^2 }$
     13and
    1514
    16 where they specify
     15$\tau (v) = \sigma_{v0} \int N(v') \phi ( v-v' ) dv'$
    1716
    18 || $\lambda_0$  ||   $1215.6702 \mbox{ Angstroms}$  ||
    19 || $\sigma_{\nu0}$  ||   $1.102 \times 10^{-6} m^2 s^{-1}$  ||
    20 || $\Gamma$  ||   $6.27 \times 10^8 s^{-1}$  ||
    2117
    22 and I am assuming $\nu_0 = \frac{c}{\lambda_0}$
     18Now to get the line profile in terms of velocities, we can start with the line profile in terms of frequencies:
     19
     20$\phi ( \nu - \nu' ) = \frac{\Gamma}{4 \pi^2 (\nu - \nu')^2 + \left ( \Gamma / 2 \right )^2}$
     21
     22and realizing that this is a distribution, we can use
     23
     24$\phi ( v - v') dv' = \phi ( \nu - \nu') d \nu$
     25
     26where
     27
     28$\nu = \nu_0 + \nu_0 \frac{v}{c}$
     29
     30so
     31
     32$ \nu - \nu' = \frac{v - v'}{\lambda_0}$
     33
     34and
     35
     36$d \nu = \frac{dv}{\lambda_0}$
     37
     38to get
     39
     40$ \phi ( v-v' ) =\frac{\Gamma \lambda_0}{4 \pi^2 (v - v')^2 + \left ( \Gamma \lambda_0 / 2 \right )^2}$
     41
     42
     43Putting this all together we have...
     44
     45$\tau_i  = \frac{\lambda_0}{\Delta v}\displaystyle \int_{v_i-\frac{1}{2}\Delta v}^{v_i + \frac{1}{2}\Delta v} \sigma_{v0} \int N(v') \frac{\Gamma \lambda_0}{4 \pi^2 (v - v')^2 + \left ( \Gamma \lambda_0 / 2 \right )^2} dv' dv$
     46
     47Now if we bin the column densities into velocity bins, we have
     48
     49$N_j  = \displaystyle \int_{v_j-\frac{1}{2}\Delta v}^{v_j + \frac{1}{2}\Delta v} N (v') dv'$
     50
     51and
     52
     53$ \int N(v') \frac{\Gamma \lambda_0}{4 \pi^2 (v - v')^2 + \left ( \Gamma \lambda_0 / 2 \right )^2} dv' = \sum N_j \frac{\Gamma \lambda_0}{4 \pi^2 (v - v_j)^2 + \left ( \Gamma \lambda_0 / 2 \right )^2}$
     54
     55and we have (wave arms in air for a bit)
     56
     57$\tau_i  =  \frac{\lambda_0}{\Delta v} \sigma_{v0} \sum N_j \displaystyle \int_{v_i-\frac{1}{2}\Delta v}^{v_i + \frac{1}{2}\Delta v} \sigma_{v0}  \frac{\Gamma \lambda_0}{4 \pi^2 (v - v_j)^2 + \left ( \Gamma \lambda_0 / 2 \right )^2} dv$
     58
     59or
     60
     61$\tau_i =  \frac{\lambda_0}{\Delta v} \sigma_{v0} \sum N_j C_{ij}$
     62
     63where
     64
     65$C_{ij}=\displaystyle \int_{v_i-\frac{1}{2}\Delta v}^{v_i + \frac{1}{2}\Delta v} \sigma_{v0}  \frac{\Gamma \lambda_0}{4 \pi^2 (v - v_j)^2 + \left ( \Gamma \lambda_0 / 2 \right )^2} dv$
     66
     67
     68Now $\Delta v$ is of order 20 km/s and $\Gamma \lambda_0$ is 76 m/s...  so if $i == j$, then the integral may as well go to infinity and we have
     69
     70$C_{ii} = 1$
     71
     72and for $i \ne j$
     73
     74$C_{ij} = \frac{\Gamma \lambda_0}{4 \pi^2 (v_i-v_j)^2} \Delta v$
     75
     76so we have
     77
     78$\tau_i =  \frac{\sigma_{v0} N_i \lambda_0}{\Delta v} + \displaystyle \sum_{i \ne j} N_j \sigma_{v0} \frac{\Gamma \lambda_0^2}{4 \pi^2 (v_i-v_j)^2} $
     79
     80
     81Or at least that's how I get agreement with their equation 11...
     82
     83
     84So to implement this approach, I plan on creating projections (integrations along LOS) of $\kappa_i$ where
     85
     86$\kappa_i(n,v) = \delta_{ij}\frac{\sigma_{v0} n \lambda_0}{\Delta v} + \left ( 1 - \delta_{ij} \right ) n \sigma_{v0} \frac{\Gamma \lambda_0^2}{4 \pi^2 (v_i-v_j)^2}$
     87
     88where $j$ corresponds to the bin that the LOS cell velocity $v$ lies within
    2389
    2490
    2591
    26 
    27 
    28 http://mnras.oxfordjournals.org/content/436/3/2179.full.pdf