Radiation in a Plasma. I. Čerenkov Effect

We start with the linearized plasma equations containing an isotropic pressure term, plus extra source terms ${\mathrm{J}}^s$ and ${\rho }^s$ in the Maxwell equations. The fields of (${\mathrm{J}}^s, {\rho }^s$) can be decomposed into two modes. The electromagnetic (EM) mode has all the magnetic field and no charge accumulation; it is the ordinary EM field of (${\mathrm{J}}^s, {\rho }^s$) in a dispersive medium of relative dielectric constant ${\epsilon }_r=1-{\left(\frac{{\omega }_p}{\omega }\right)}^2$. The plasma (P) mode has all the charge accumulation and no magnetic field; at great distances from the source, it becomes a longitudinal (radial) plasma wave with the usual dispersion relation for plane plasma waves. Various potentials for the EM and P modes are given by the inhomogeneous Klein-Gordon equation. The fields of a uniformly moving charged particle are found by a Lorentz transformation. When $\left(\frac{u}{v_0}\right)<\mathrm{}1\mathrm{}$ ($u=\mathrm{particle}\mathrm{velocity}$, $v_0=\mathrm{rms}\mathrm{thermal}\mathrm{velocity}$), the EM and P fields are exponentially screened outside oblate spheroids foreshortened in the direction of motion. When $\left(\frac{u}{v_0}\right)>\mathrm{}1\mathrm{}$, the $P$ field exists only within the Mach (Čerenkov) cone trailing the particle. The frequency and angular spectra of the Čerenkov radiation are found, and the total radiated energy is found by assuming an arbitrary high-frequency cutoff due to Landau damping. The expression for total radiated energy agrees with that given by Pines and Bohm, except for the logarithmic terms.