  We show that photophoretic aircraft would greatly benefit from a
three-dimensional (3D) hollow geometry that pumps ambient air through sidewalls
to create a high-speed jet. To identify optimal geometries, we developed a
theoretical expression for the lift force based on both Stokes (low-Re) and
momentum (high-Re) theory and validated it using finite-element fluid-dynamics
simulations. We then systematically varied geometric parameters, including
Knudsen pump porosity, to minimize the operating altitude or maximize the
payload. Assuming that the large vehicles can be made from previously
demonstrated nanocardboard material, the minimum altitude is 55 km while the
payload can reach 1 kilogram for 3D structures with 10-meter diameter at 80 km
altitude. In all cases, the maximum areal density of the sidewalls cannot
exceed a few grams per square meter, demonstrating the need for ultralight
porous materials.
