# 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.