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\title{Faraday Rotation in the Interstellar Medium}
\author{ Paul Thorman Jr.\\
     \\
         Department of Physics\\
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}

\date{11/14/2001}
\maketitle
\begin{abstract}

   Although wavelength and intensity of radiation are the most frequently used measures in astrophysics, some distant sources of light also have polarization properties that can be used to learn about their constitution. This polarized light is also affected by the interstellar medium, which can be treated as a rarefied electron gas, and the effect of the medium is dependent on the presence and intensity of magnetic fields, as well as upon the density of the medium itself. The presence of magnetic fields is indicated by Faraday rotation, a change in the polarization plane of a signal as it passes through a medium in the presence of a magnetic field. These measurements can easily be applied to determine the gross magnetic field of our galaxy. \\[1em]
\end{abstract}

\pagebreak

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    The light from a majority of stellar sources is not polarized
to any measurable degree. There are a few exceptions: light from
"reddened stars," which has been scattered by grains of dust,
often shows signs of polarization due to such scattering, and
pulsar light, while not constant in its polarization, can be
averaged over a sample of perhaps 100 pulses to give a stable mean
polarization \cite{refA}. Pulsar light has the added advantage
that each pulse contains light of many different frequencies which
all originated at the same instant, so the spread of the pulse is
due to the frequency-dependent conductivity of the interstellar
medium, and thus can be used to determine its path length while in
the plasma.
    For a plasma made up of ions and electrons, we can
assume that only the electrons contribute to the conductivity, the
ions being so massive as to be practically immobile. The result
for an isotropic plasma is well-known,
\[
\epsilon=1-(\frac{\omega^{2}}{\omega_{p}^{2}})
\]
where $\omega_{p}$ is the plasma frequency, defined by
\[\omega_{p}^{2}\equiv\frac{4\pi n e^{2}}{m}\]


    Clearly if $\omega<\omega_{p}$ then $\epsilon$ is negative and the
amplitude of the wave falls off exponentially, so $\omega_{p}$ is
also called the \textit{plasma cutoff frequency}, below which
there is no propagation of the wave through the plasma.\cite{refB}
    Now consider propagation through such a plasma in the presence
of an external magnetic field $(B_{0})$. The system is no longer
isotropic, but has azimuthal symmetry around the direction of the
magnetic field lines. Any plane-polarized wave can be decomposed
into a component parallel to the magnetic field and a component
perpendicular to the field. Assuming $(B_{0})$ is large compared
to the magnetic field of the wave, the equation of motion for an
electron in the plasma will be

\begin{equation}
m (d{\bf{v}}/dt) = -e{\bf{E}}-(e/c)
{\bf{v}}\times{\bf{B_{0}}}

\end{equation}

    For a circularly polarized incoming wave, we know
\[{\bf{E}}(t)=Ee^{-i \omega t}(\epsilon_{1}\pmi\epsilon_{2})\]
\\
Defining the cyclotron frequency $\omega_{B}\equiv\frac{e
B_{0,parallel}}{mc}$ and solving for the steady-state velocity of
the electron, we find
\begin{equation}
{\bf{v}}(t)=\frac{-ie}{m(\omega\pm\omega_{B})}{\bf{E}}(t)
\end{equation}
which gives
\begin{equation}
\epsilon_{R}=1-\frac{\omega_{p}^{2}}{\omega(\omega+\omega_{B})}
\end{equation}

for a right-handed polarization, and\\
\begin{equation}
\epsilon_{L}=1-\frac{\omega_{p}^{2}}{\omega(\omega-\omega_{B})}
\end{equation}

for a left-handed polarization. \pagebreak

    A plane-polarized wave can be treated as the superposition of a
left-hand and a right-hand circularly polarized wave of equal
weight. As a result of the differential in speed between the
different left- and right-hand modes, the plane of polarization
will rotate as it passes through the plasma. This is called
\textsc{Faraday rotation}.
    In general, the angle through which the electric vector of a
circularly polarized wave proceeds is given by
\begin{equation}
\phi=\int_{0}^{d}k ds
\end{equation}

    where k is the wave number,\\

\[k=\frac{\omega}{c} \root{\epsilon}\]

    Assuming that $\omega\gg\omega_{p}$ and $\omega\gg\omega_{B}$, we
can expand k as
\begin{equation}
k\approx\frac{\omega}{c}[1-\frac{\omega_{p}^{2}}{2\omega^{2}}(1+\frac{\omega_{B}}{\omega})]
\end{equation}

    Substituting the known expressions for $\omega_{p}^{2}$ and
$\omega_{b},$ we obtain for the Faraday rotation
\begin{equation}
\Delta\theta=\frac{2\pi
    e^{3}}{m^{2}c^{2}\omega^{2}}\int_{0}^{d} nB_{parallel}ds
\end{equation}

    Since $\Delta\theta$ varies with frequency in a known manner,
measurements made at different frequencies can be combined to give
a value for the integral.\cite{refB}

    One useful astrophysical application of Faraday rotation is
the determination of the magnetic field of the galaxy using
extragalactic pulsars. The dispersion relation given in equation
(1) leads to an expression for the travel time for a pulse of a
given frequency through a path length L of plasma
\[t(\nu)=\frac{L}{c}+\frac{e^{2}}{2\pi
    mc}\times\frac{D_{m}}{\nu^{2}}\]

    where \(D_{m}\) is the "dispersion measure",

    \[D_{m}=\int_{0}^{L} n ds\]

    $D_{m}$ can be determined for pulsars based on the frequency
spread of what was originally a simultaneous pulse. Then
$B_{parallel}$ is directly proportional to the ratio of
$\Delta\theta$ to $D_{m}$, and the constant of proportionality is
known.

\begin{figure}[htb]

\centerline{\psfig{figure=galxymag.ps,height=2.5in}}

\caption[]{\em{Galactic Magnetic field as a function of Galactic
Longitude }}
\end{figure}
\cite{refA}

    The following plot was constructed from a study of 26 pulsars; the
solid line is the least squares fit, which yields a net value of
$(2.2\pm0.4)\times 10^{-6}$ gauss for the galactic magnetic field.

\pagebreak
\begin{thebibliography}{99}
\bibitem{refA} L. Spitzer, "Physical Processes in the Interstellar
Medium", John Wiley and Sons, NY, 1978
\bibitem{refB} G. Rybicki and A. Lightman, "Radiative Processes in
Astrophysics", John Wiley and Sons, NY, 1979
\bibitem{refc} J.E. Dyson and D.A. Williams, "The Physics of the
Interstellar Medium", IOP Publishing, Philadelphia, 1997
\bibitem{refc} F. Shu, "The Physics of Astrophysics: Vol. I,
Radiation", University Science Books, Mill Valley, CA, 1991



\end{thebibliography}
\end{document}
