General Analytical Solution for the Dispersion Equation of an Ionized Gas in a Constant Background Magnetic Field

Understanding how radio waves propagate in the Earth's upper atmosphere requires a model for the conductivity of an ionized gas. A basic linear model requires the solution of three simultaneous vector equations for v
_i
, v
_e
, and v
_n
, respectively the ion, electron, and neutral gas velocities. Of these, v
_i
and v
_e
alone determine the current density. When a uniform background magnetic field is present, some of the coefficients in the equation are cross products, rather than just simple scalars. This article demonstrates that in contrast to the more obvious route of finding an approximate solution or one computed via 9×9 matrices, an algebraic approach leads to a systematic analytical solution of this reasonably complex problem. For example, the dispersion equations for plane electromagnetic waves may be found exactly, and using the effective resistivity requires less effort than the conductivity. In addition, an interesting method of finding v
_i
/v
_e
develops out of an equation of the form av
_i
+ bΩ × v
_i
= a'v
_e
+ bΩ × v
_e
.