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Conversion between density and column density 
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#1
Jul1814, 05:27 AM

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What's the way to convert n(r) to [itex] \rho(r) [/itex] in case of a spherical cloud.
n(r) is the column density, [itex] \rho(r) [/itex] is the density. I tried (but didn't manage) to get it from their relation to the total mass: [itex] M = 4 \pi \int r^2 \rho(r)dr = 2 \pi \int r n(r)dr [/itex] (is it correct anyway?) Thank you, 


#2
Jul1814, 07:45 AM

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P: 13,067

In general, no there isn't.
The reason being that the column density calculation hides information. But if you know the density distribution has a special symmetry  i.e. it is radial, then you may be able to. In this case it appears you have a cloud whose density function depends on on the distance from the center. \rho(r) is the density of the cloud a distance r from the center, so the mass of the spherical shell between r and r+dr is dm = \rho (4\pi r^2 dr) so the total mass is the integral over all radii. See how that works? This is r in spherical coordinates. To get to the column density, you need to convert to cylindrical coordinates. This is because the column at a horizontal distance r from the center, area dxdy (say) will have a different density for different z values (since different z values will have different spherical r values) It should be clearer if you use a different symbol for r in the two equations. 


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