Dispersion measure

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 $d$ from earth. The time it takes for a pulse with frequency $\omega$ to get to earth is:

$t_p=\int_0^d \frac{ds}{v_g(\omega)}$

where $ds$ is a path element along the line of sight and $v_g(\omega)$ is the group velocity for a wave with frequency $\omega$. The plasma frequency in the ISM is very low (about $10^3$Hz) so we can safely assume $\omega \gg \omega_p$ for practical purposes. We can then use the plasma dispersion relation

$\omega^2=\omega_p^2+c^2k^2$ and:

$v_g(\omega)=\frac{\partial \omega}{\partial k}=c \sqrt{1-\frac{\omega_p^2}{\omega^2}}$

which leads to:

$\frac{1}{v_g}=\frac{1}{c} (1-\frac{\omega_p^2}{\omega^2})^{-1/2} \approx \frac{1}{c} (1+\frac{1}{2} \frac{\omega_p^2}{\omega^2})$

and plugging into $t_p$:

$t_p=\int_0^d \frac{1}{c}(1+\frac{1}{2} \frac{\omega_p^2}{\omega^2})ds=\frac{d}{c} +\frac{1}{2c\omega^2} \int_0^d \omega_p^2 ds$

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 $\frac{dt_p}{d\omega}$. Plugging $\omega_p$ this is:

$\frac{dt_p}{d\omega}=-\frac{4 \pi e^2}{c m_e \omega^3} DM \approx - \frac{c r_e}{\omega^3} DM$

where $DM \equiv \int_0^d n ds$ is known as the dispersion measure of the plasma and $r_e$ is the classical electron radius. Finally, if the density of the plasma is known, one can estimate the distance to the pulsar by measuring $\frac{dt_p}{d\omega}$.