Pulsars and neutron stars/The interstellar medium

Introduction
The nature of the pulsed radiation observed on Earth is affected by charged particles in the interstellar medium (ISM).

Dispersion
The radiation travels through the ionized gas of the ISM with group velocity (Shapiro & Teukolsky 1983):

$$v(\nu) = c\left(1-\frac{\nu_e^2}{\nu^2}\right)^{1/2}$$

where $$c$$ is the vacuum speed of light and $$\nu_e$$ is the plasma frequency. The time difference, $$\Delta T$$ between two frequencies $$\nu_1$$ and $$\nu_2$$ after travelling a distance $$d$$ equals:

$$\Delta T = \int_0^d \left( \frac{1}{v_1(l)} - \frac{1}{v_2(l)}\right) dl$$

where $$v_1$$ and $$v_2$$ are the group velocities corresponding to the two frequencies. Writing the plasma frequency in terms of fundamental constants and $$n_e(l)$$ the charged particle density we get:

$$\Delta T \approx \frac{e^2}{2c\pi m_e}\left(\frac{1}{\nu_1^2}-\frac{1}{\nu_2^2}\right)\int_0^d n_e(l)dl$$

where $$e$$ is the electronic charge and $$m_e$$ is the electron rest mass. We define the pulsar's dispersion measure (DM) as

$${\rm DM} = \int_0^d n_e(l)dl$$

Hence, the time delay, $$t$$ between an observed pulse at observing frequency $$\nu$$ and a pulse of infinite frequency (or travelling through a vacuum) is given by:

$$t [s] \approx 4.15 \times 10^3 \frac{(\nu [{\rm MHz}])^2}$$

When a pulsar is observed with a frequency channel resolution of ($$\Delta \nu$$), the dispersion will lead to a smearing of the profile:

$$\Delta t_{\rm DM} \approx 8.30 \times 10^6 {\rm DM} \Delta \nu \nu^{-3}$$ ms

Modelling the interstellar medium
Cordes & Lazio (2002) presented the most commonly used model for the Galactic distribution of free electrons.