Introduction

At the low densities encountered in the further reaches of the earth's atmosphere and in outer space, collisions between particles are very rare. Hence, unlike in a terrestrial laboratory, it is possible for gas to remain in an ionized state for long periods of time. Such plasmas are ubiquitous in astrophysics, and have been extensively studied for their own sake. In this chapter however, we focus on the effects of this plasma on radio waves propagating through them, and will find astrophysical plasmas to be largely of nuisance value.

The refractive index of a cold neutral plasma is given by

$$\mu(\nu) = \sqrt{1 - {\nu_p^2 \over \nu^2}},$$ (16.1.1)

where $\nu_p$ the ``plasma frequency is given by

$$\nu_p = \sqrt{{ n_e e^2\over \pi m_e}} \simeq 9 \sqrt{n_e}~~{\rm kHz}$$ (16.1.2)

where $e$ is the charge on the electron, $m_e$ is the mass of the electron and $n_e$ is the electron number density (in cm$^{-3}$). At frequencies below the plasma frequency $\nu_p$ the refractive index becomes imaginary, i.e. the wave is exponentially attenuated and does not propagate through the medium. The earth's ionosphere has electron densities $\sim 10^4 - 10^5$ cm$^{-3}$, which means that the plasma frequency is $\sim 1 - 10$ MHz. Radio waves with such low frequencies do not reach the earth's surface and can be studied only by space based telescopes. The plasma between the planets is called the Interplanetary Medium (IPM) and has electron densities $\sim 1$ cm$^{-3}$ (at the earth's location); the corresponding cut off frequency is $\sim 9$ kHz. The typical density in the Interstellar Medium (ISM) is $\sim 0.03$ cm$^{-3}$ for which the cut off frequency is $\sim 1$ kHz. Waves of such low frequency from extra solar system objects cannot be observed even by spacecraft since the IPM and ISM will attenuate them severely.

The dispersion relationship in a cold plasma is given by $c^2 k^2 = \omega^2 - \omega_p^2$. Since this is a non linear relation there are two characteristic velocities of propagation, the phase velocity given by

$$v_p = {\omega \over k} = {c \over \mu } \simeq c\ (1 + {1 \over 2 } {\nu_p^2 \over \nu^2})$$ (16.1.3)

and the group velocity which is given by

$$v_g = {d\omega \over dk } = c\mu \simeq c\ (1 - {1 \over 2 } {\nu_p^2 \over \nu^2}).$$ (16.1.4)

Where for the last expression we have assumed that $\nu >> \nu_p$ (which is usually the regime of interest).