# 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):

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

where ${\displaystyle c}$ is the vacuum speed of light and ${\displaystyle \nu _{e}}$ is the plasma frequency. The time difference, ${\displaystyle \Delta T}$ between two frequencies ${\displaystyle \nu _{1}}$ and ${\displaystyle \nu _{2}}$ after travelling a distance ${\displaystyle d}$ equals:

${\displaystyle \Delta T=\int _{0}^{d}\left({\frac {1}{v_{1}(l)}}-{\frac {1}{v_{2}(l)}}\right)dl}$

where ${\displaystyle v_{1}}$ and ${\displaystyle v_{2}}$ are the group velocities corresponding to the two frequencies. Writing the plasma frequency in terms of fundamental constants and ${\displaystyle n_{e}(l)}$ the charged particle density we get:

${\displaystyle \Delta T\approx {\frac {e^{2}}{2c\pi m_{e}}}\left({\frac {1}{\nu _{1}^{2}}}-{\frac {1}{\nu _{2}^{2}}}\right)\int _{0}^{d}n_{e}(l)dl}$

where ${\displaystyle e}$ is the electronic charge and ${\displaystyle m_{e}}$ is the electron rest mass. We define the pulsar's dispersion measure (DM) as

${\displaystyle {\rm {DM}}=\int _{0}^{d}n_{e}(l)dl}$

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

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

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

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

## Modelling the interstellar medium

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