# 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:

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