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Conversion between density and column density

  1. Jul 18, 2014 #1
    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. jcsd
  3. Jul 18, 2014 #2

    Simon Bridge

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    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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