The fact that different frequency modes of an electromagnetic wave traveling through a plasma medium, propagate at different group velocities, creates a dispersion that enables us to calculate the distance of different astrophysical sources. Imagine, for instance, a pulsar at a distance from earth. The time it takes for a pulse with frequency to get to earth is:
where is a path element along the line of sight and is the group velocity for a wave with frequency . The plasma frequency in the ISM is very low (about Hz) so we can safely assume for practical purposes. We can then use the plasma dispersion relation
which leads to:
and plugging into :
The first term is just the time it would take the signal to travel the same distance in a vacum. The second term is the correction for plasma, which is frequency dependent. In practice, one measures the difference in arrival time as a function of frequency . Plugging this is:
where is known as the dispersion measure of the plasma. Finally, if the density of the plasma is known, one can estimate the distance to the pulsar by measuring .