Slide 9 of 48
Notes:
n_H is the interstellar hydrogen number density, and N_H is the interstellar hydrogen ‘column’ density and has units of m^-2.
These derivations assume that the ratio of n_Z to n_H is NOT a function of distance, while n_H MAY BE a function of distance.
For values of the effective cross-section, see eg. Morrison and McCammon, 1983, ApJ, 270, 119.
The column density of hydrogen in any given direction is measured from the flux in the 21cm atomic hydrogen line using radio telescopes. Unfortunately though, there is a factor of 2 uncertainty in making these measurements - the 21cm line is emitted during a transition in which spins of the electron and proton change from parallel to anti-parallel. There are two possible spins of the electron and proton, thus a factor of 2 uncertainty arises.
There are also problems due to the very wide beams used (2x3 degrees for the Stark et al all-sky survey) which give poor spatial resolution… and there is an unknown contribution to the column from molecular hydrogen (although the distribution of molecular hydrogen is generally found to follow the atomic hydrogen distribution).
