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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} D

where  D\equiv \int_0^d n ds 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  \frac{dt_p}{d\omega} .

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