wiki:u/johannjc/scratchpad8

Version 2 (modified by Jonathan, 9 years ago) ( diff )

http://www.aanda.org/articles/aa/pdf/2013/09/aa21551-13.pdf

The prescription of Bourrier et al.. is to calculate (at each pixel) the average optical depth for range of velocity bins with spacing (or frequency bins with size , so that

So we have

and

Now to get the line profile in terms of velocities, we can start with the line profile in terms of frequencies:

and realizing that this is a distribution, we can use

where

so

and

to get

Putting this all together we have…

Now if we bin the column densities into velocity bins, we have

and

and we have (wave arms in air for a bit)

or

where

Now is of order 20 km/s and is 76 m/s… so if , then the integral may as well go to infinity and we have

and for

so we have

Or at least that's how I get agreement with their equation 11…

So to implement this approach, I plan on creating projections (integrations along LOS) of where

where corresponds to the bin that the LOS cell velocity lies within

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