# Appendix D Wave Propagation in a Plasma

The propagation of radio waves is governed by Maxwell’s equations: $\displaystyle\vec{\nabla}\cdot\vec{D}=4\pi\rho,$ (D.1) $\displaystyle\vec{\nabla}\cdot\vec{B}=0,$ (D.2) $\displaystyle\vec{\nabla}\times\vec{E}=-\frac{1}{c}\frac{\partial\vec{B}}{% \partial t},$ (D.3) $\displaystyle\vec{\nabla}\times\vec{H}=\frac{4\pi\vec{J}}{c}+\frac{1}{c}\frac{% \partial\vec{D}}{\partial t}.$ (D.4)

# D.1 Dispersion and Reflection in a Low-Density Plasma

The interstellar medium (ISM) of our Galaxy is a nearly perfect vacuum, so $\vec{D}\approx\vec{E}$ and $\vec{B}\approx\vec{H}$. However, the current density (charge flow rate per unit area) $\vec{J}$ need not be zero because there are free electrons in the ISM, with mean density $n_{\mathrm{e}}\sim 0.03\mathrm{~{}cm}^{-3}$. The transverse electric field $E=E_{0}\exp(ikx-i\omega t)$ of a monochromatic plane wave with angular frequency $\omega$ traveling in the $x$-direction past a free electron at $x=0$ causes the electron to oscillate with frequency $\omega$ and generate an oscillating electric current. The ISM density is low enough that damping by electron collisions can be ignored, so the electron’s equation of motion is

 $F=m_{\mathrm{e}}\dot{v}=eE=eE_{0}\exp(-i\omega t),$ (D.5)

where $\dot{v}$ is the electron’s acceleration. Integrating Equation D.5 yields the electron velocity

 $v=\int_{0}^{t}\dot{v}\,dt=-\frac{eE_{0}}{i\omega m_{\mathrm{e}}}\exp(-i\omega t).$ (D.6)

Since the resulting current density is proportional to the applied electric field strength, the ISM obeys Ohm’s law

 $\vec{J}=\sigma\vec{E},$ (D.7)

where the constant of proportionality $\sigma$ is the called the conductivity of the medium. The current density is the rate at which charge flows through a unit area, so $J=en_{\mathrm{e}}v$ and the conductivity of the ISM is

 $\sigma=\frac{J}{E}=en_{\mathrm{e}}\biggl{(}-\frac{e}{i\omega m_{\mathrm{e}}}% \biggr{)}=i\biggl{(}\frac{e^{2}n_{\mathrm{e}}}{\omega m_{\mathrm{e}}}\biggr{)}.$ (D.8)

The conductivity of a collisionless plasma is purely imaginary. That means that the current density and electric field are 90 degrees out of phase, and radio waves can propagate without suffering resistive power losses.

For a plane wave traveling in the $x$-direction,

 $\vec{\nabla}\rightarrow ik\qquad\mathrm{and}\qquad\frac{\partial}{\partial t}% \rightarrow-i\omega,$ (D.9)

and Maxwell’s two curl equations become

 $\displaystyle ikE$ $\displaystyle=\frac{i\omega H}{c},$ (D.10) $\displaystyle-ikH$ $\displaystyle=\frac{4\pi\sigma E}{c}-\frac{i\omega E}{c}.$ (D.11)

Using the first to eliminate $H$ from the second gives

 $-ikH=-ik\biggl{(}\frac{ckE}{\omega}\biggr{)}=\frac{4\pi\sigma E}{c}-\frac{i% \omega E}{c}.$ (D.12)

Thus

 $k^{2}=\biggl{(}\frac{\omega}{c}\biggr{)}^{2}\biggl{(}1+i\frac{4\pi\sigma}{% \omega}\biggr{)}.$ (D.13)

The ISM conductivity (Equation D.8) can be expressed in terms of the plasma frequency defined by

 $\boxed{\nu_{\mathrm{p}}\equiv\biggl{(}\frac{e^{2}n_{\mathrm{e}}}{\pi m_{% \mathrm{e}}}\biggr{)}^{1/2}\approx 8.97\mathrm{~{}kHz}\,\biggl{(}\frac{n_{% \mathrm{e}}}{\mathrm{cm}^{-3}}\biggr{)}^{1/2}.}$ (D.14)

The plasma frequency in the ISM where $n_{\mathrm{e}}\sim 0.03\mathrm{~{}cm}^{-3}$ is only $\nu_{\mathrm{p}}\sim 0.3\mathrm{~{}kHz}$.

In terms of $\nu_{\mathrm{p}}$ the magnitude of the wave vector $k$ obeys

 $k^{2}=\biggl{(}\frac{\omega}{c}\biggr{)}^{2}\biggl{(}1-\frac{\nu_{\mathrm{p}}^% {2}}{\nu^{2}}\biggr{)},$ (D.15)

where $\nu=\omega/(2\pi)$. The wave vector is imaginary for radio waves whose frequency $\nu$ is less than the plasma frequency $\nu_{\mathrm{p}}$. Low-frequency radiation cannot propagate through the plasma, and radio waves with $\nu<\nu_{\mathrm{p}}$ incident on a plasma are reflected. For example, the maximum electron density in the F layer of the Earth’s ionosphere is $n_{\mathrm{e}}\approx 10^{6}\mathrm{~{}cm}^{-3}$, so celestial radio radiation at frequencies lower than $\nu\sim 10\mathrm{~{}MHz}$ is reflected back into space by the ionosphere and cannot be observed from the ground.

The plasma index of refraction is

 $\boxed{\mu=\frac{ck}{\omega}=\biggl{(}1-\frac{\nu_{\mathrm{p}}^{2}}{\nu^{2}}% \biggr{)}^{1/2}.}$ (D.16)

In the limit $\nu_{\mathrm{p}}\ll\nu$ appropriate for radio waves in the ISM,

 $\mu\approx 1-\frac{\nu_{\mathrm{p}}^{2}}{2\nu^{2}},$ (D.17)

and the group velocity $v_{\mathrm{g}}$ of radio pulses traveling through the ISM depends on frequency:

 $\boxed{v_{\mathrm{g}}\approx\mu c\approx c\biggl{(}1-\frac{\nu_{\mathrm{p}}^{2% }}{2\nu^{2}}\biggr{)}.}$ (D.18)

# D.2 Faraday Rotation in a Magnetized Plasma

If the ISM contains a magnetic field of strength $B$, all nonrelativistic electrons orbit around field lines with angular frequency

 $\omega_{\mathrm{G}}=\frac{eB}{m_{\mathrm{e}}c},$ (D.19)

where $\nu_{\mathrm{G}}=\omega_{\mathrm{G}}/(2\pi)$ is called the gyro frequency (Section 5.1.1). In the (noninertial) coordinate frame rotating with the electron, circularly polarized radiation at frequency $\omega$ will drive the electron at frequency $\omega+\omega_{\mathrm{G}}$ or $\omega-\omega_{\mathrm{G}}$ depending on the sense (left handed or right handed) of the circular polarization. Because the index of refraction in a collisionless plasma depends on frequency (Equation D.16), there are two indices of refraction in a magnetized plasma, which is said to be birefringent. A linearly polarized wave is the sum of left- and right-handed circularly polarized waves, so the position angle of the linearly polarized wave will rotate as the wave travels parallel to the magnetic field. This rotation is called Faraday rotation, and it is a useful tool for measuring the line-of-sight component $B_{\parallel}$ of the magnetic field in a radio source. Equations describing Faraday rotation are easily derived by extending the derivation in Section D.1.

Equation D.5 can be expanded to include the force on an electron from a steady ambient magnetic field of strength $B$:

 $F=m_{\mathrm{e}}\dot{v}=eE_{0}\exp(-i\omega t)+\frac{e}{c}{vB_{\parallel}},$ (D.20)

where $B_{\parallel}$ is the component of the magnetic field parallel to the direction of the electromagnetic field $E=E_{0}\exp(ikx-i\omega t)$. If the incident wave is circularly polarized:

 $\vec{E}=E_{y}\hat{y}\pm iE_{z}\hat{z},$ (D.21)

then the electron velocity is

 $\vec{v}=v(\hat{y}\pm i\hat{z})\exp({-i\omega t)}$ (D.22)

and

 $v=i\biggl{[}\frac{eE}{m_{\mathrm{e}}(\omega\pm\omega_{\mathrm{G}})}\biggr{]}.$ (D.23)

The resulting conductivity of the magnetized plasma is

 $\sigma=i\biggl{[}\frac{e^{2}n_{\mathrm{e}}}{m_{\mathrm{e}}(\omega\pm\omega_{% \mathrm{G}})}\biggr{]},$ (D.24)

and the refractive index takes on two values

 $\mu=\biggl{[}1-\frac{\nu_{\mathrm{p}}^{2}}{\nu(\nu\pm\nu_{\mathrm{G}})}\biggr{% ]}^{1/2}.$ (D.25)

Faraday rotation changes the polarization position angle by

 $\Delta\chi\approx\lambda^{2}\biggl{[}\frac{e^{3}}{2\pi(m_{\mathrm{e}}c^{2})^{2% }}\int_{\mathrm{los}}n_{\mathrm{e}}(l)B_{\parallel}(l)dl\biggr{]}\equiv\lambda% ^{2}\,\mathrm{RM},$ (D.26)

where the quantity in square brackets defines the rotation measure RM. The sign convention is that RM is positive for a magnetic field pointing from the source to the observer. In astronomically convenient units,

 $\biggl{(}\frac{\mathrm{RM}}{\mathrm{rad~{}m}^{-2}}\biggr{)}\approx 8.1\times 1% 0^{5}\int_{\mathrm{los}}\biggl{(}\frac{n_{\mathrm{e}}}{\mathrm{cm}^{-3}}\biggr% {)}\biggl{(}\frac{B_{\parallel}}{\mathrm{gauss}}\biggr{)}\biggl{(}\frac{dl}{% \mathrm{pc}}\biggr{)}.$ (D.27)

The phenomenon of Faraday rotation can also lead to Faraday depolarization when the rotation occurs within the emission region.