Conversion between density and column density

  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. Simon Bridge

    Simon Bridge 15,053
    Science Advisor
    Homework Helper
    Gold Member

    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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