Oblate electron holes are not attributable to anisotropic shielding

Shielding mechanisms' influence on the ratio of perpendicular to parallel scale lengths of multidimensional plasma electron hole equilibria are analyzed theoretically and computationally. It is shown that the ``gyrokinetic'' model, invoking perpendicular polarization, is based on a misunderstanding and cannot explain the observational trend that greater transverse extent accompanies lower magnetic field. Instead, the potential in the wings of the hole, outside the region of trapped-electron depletion, has isotropic shielding giving $\phi\propto {\rm e}^{-r/L}/r$, with the shielding length $L$ equal to the Debye length for holes much slower than the electron thermal speed. Particle in cell simulations confirm the analysis. Trapped electron charge distribution anisotropy must therefore instead underlie the oblate shape of electron holes.