  We consider an ideal gas of active Brownian particles that undergo
self-propelled motion and both translational and rotational diffusion under the
influence of gravity. We solve analytically the corresponding Smoluchowski
equation in two space dimensions for steady states. The resulting one-body
density is given as a series, where each term is a product of an
orientation-dependent Mathieu function and a height-dependent exponential. A
lower hard wall is implemented as a no-flux boundary condition. Numerical
evaluation of the suitably truncated analytical solution shows the formation of
two different spatial regimes upon increasing Peclet number. These regimes
differ in their mean particle orientation and in their variation of the
orientation-averaged density with height.
