The flow resulting from the steady rotation of a gravitating sphere in a monatomic gas is studied in two special cases. If the rotation is slow, then the motion consists of a swirling flow whose streamlines are concentric circles around the rotation axis, and a slower meridional flow whose structure depends on the strength of the gravitational field. This meridional flow is investigated in detail for small and large field strength. In the latter case, the scale height is small compared to the sphere radius, and most of the fluid is in a thin recirculating layer on the surface. However, the mathematics imposes a severe restriction on the rotation speed if the field is large, which leads to a more general investigation of the flow for very large field strength, i.e. for a thin “atmosphere”. Compressibility effects can be quite sizeable in this final solution. Though a thin recirculating surface layer still exists in only slightly modified form, the fluid outside this layer is much less tenuous than for the case when the rotation is slow. The flow in the region outside the boundary layer involves solution of two quasilinear elliptic equati