# Dispersion measure

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\section{Overview} For an electromagnetic wave of frequency $\nu$ emitted at a distance $d$ propagating through an electron plasma with uniform number density $n_e$, the pulse travel time $t_p$ to the observer is

\begin{align} t_p = \frac{d}{c}+\frac{e^2}{2\pi m_ec}\frac{\int_0^d n_e\,dl}{\nu^2} \end{align} Therefore, the \textit{delay} time due to dispersion is \begin{align} t_d &= \frac{e^2}{2\pi m_ec}\frac{\mathcal{DM}}{\nu^2}\\ &= 4140.\,\,\left(\frac{\mathcal{DM}}{{\rm cm}^{-3}\,\,{\rm pc}}\right)\,\left(\frac{\nu}{1\,{\rm MHz}}\right)^{-2}\,\,{\rm sec.} \end{align} where \begin{align} \boxed{\mathcal{DM}\equiv \int_0^d n_e\,dl} \end{align} is the \textbf{dispersion measure}.

\section{Description}

The speed at which an electromagnetic wave propagates through a plasma depends on its frequency due to dispersive effects (see Plasma Frequency). Suppose a source (typically a pulsar) emits an electromagnetic pulse composed of several different frequencies. The intervening interstellar medium (ISM) causes lower frequencies to travel more slowly. Thus, if we are observing a pulsar signal, we will see the higher frequencies arrive at us first, followed by the lower frequencies. The light is \textit{dispersed} by the ISM. \par The \textit{dispersion measure}, DM, is simply a constant of proportionality relating the frequency of the light to the extra amount of time (relative to vacuum) required to reach the observer due to dispersion. It depends on two quantities: the (electron) number density $n_e$ and the pathlength through the plasma $d$. For large values of DM, either the source is relatively nearby but is traveling through a dense plasma, or it is far away, and traveling through a relatively less dense plasma. \par Shown at the right is a signal measured from a pulsar (Source: Lorimer & Kramer). The top parts show the power measured per channel as a function of time, showing clear frequency-dependent arrival times, or dispersion. The bottom shows the dispersion-corrected pulse, which is found to be quite sharp.

\section{Measuring the Dispersion Measure}
What is typically measured is the rate of change of delay time with respect to frequency,
\begin{align}
\frac{dt_d}{d\nu}=-\frac{e^2}{\pi m_e c}\frac{\mathcal{DM}}{\nu^3}
\end{align}
which is obtained simply by taking the derivative of the expression for the time delay. Thus, the dispersion measure is
\begin{align}
\mathcal{DM}=-\frac{\pi m_e c}{e^2}\left(\frac{dt_d}{d\nu}\right)\nu^3
\end{align}
Recall that
\begin{align}
\mathcal{DM}\equiv \int_0^d n_e\,dl
\end{align}
Suppose, as a rough first approximation, that the number density $n_e$ is uniform. In the ISM, $n_e\sim 0.05\,{\rm cm}^{-3}$. Then we can estimate the distance to a source as
\begin{align}
d\sim -\frac{\pi m_e c}{n_ee^2}\left(\frac{dt_d}{d\nu}\right)\nu^3
\end{align}
Alternatively, if we know the distance to a source (e.g. through trigonometric parallax), we can estimate the ISM number density
\begin{align}
n_e\sim -\frac{\pi m_e c}{e^2d}\left(\frac{dt_d}{d\nu}\right)\nu^3
\end{align}
Note that this is a rough approximation, since we know that the electron number density is more enhanced along sightlines passing through the spiral arms due to the presence of HII regions.

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