  Finite electron gyro-radius influences on the trapping and charge density
distribution of electron holes of limited transverse extent are calculated
analytically and explored by numerical orbit integration in low to moderate
magnetic fields. Parallel trapping is shown to depend upon the gyro-averaged
potential energy and to give rise to gyro-averaged charge deficit. Both types
of average are expressible as convolutions with perpendicular Gaussians of
width equal to the thermal gyro-radius. Orbit-following confirms these
phenomena but also confirms for the first time in self-consistent potential
profiles the importance of gyro-bounce-resonance detrapping and consequent
velocity diffusion on stochastic orbits. The averaging strongly reduces the
trapped electron deficit that can be sustained by any potential profile whose
transverse width is comparable to the gyro-radius $r_g$. It effectively
prevents equilibrium widths smaller than $\sim r_g$ for times longer than a
quarter parallel-bounce-period. Avoiding gyro-bounce resonance detrapping is
even more restrictive, except for very small potential amplitudes, but it takes
multiple bounce-periods to act. Quantitative criteria are given for both types
of orbit loss.
