It is shown that by choosing an ellipsoid of revolution to describe the angular dependence of the velocity distribution function, the Boltzmann equation can be reduced to a set of two equations that have validity over a wide range of conditions. These equations reduce to the common two-term spherical harmonic expansion for nearly isotropic cases, but also properly describe highly anisotropic conditions. An example is given of the application of this approach to the Townsend discharge in helium over a very wide range of $E/N.$

The Boltzmann equation can be reduced to two equations that are valid for nearly isotropic cases and highly anisotropic conditions, with applications to the Townsend discharge in helium.

Elliptic representation of the Boltzmann equation with validity for all degrees of anisotropy

Key Takeaway: The Boltzmann equation can be reduced to two equations that are valid for nearly isotropic cases and highly anisotropic conditions, with applications to the Townsend discharge in helium.