Conversion between density and column density

[bagherihan]

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

 

[Simon Bridge]

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.