  We address in detail the radiation forces on spherical dust envelopes around
luminous stars, and numerical solutions for these forces, as a first step
toward more general dust geometries. Two physical quantities, a normalized
force and a force-averaged radius, suffice to capture the overall effects of
radiation forces; these combine to yield the radiation term in the virial
theorem. In addition to the optically thin and thick regimes, the wavelength
dependence of dust opacity allows for an intermediate case in which starlight
is easily trapped but infrared radiation readily escapes. Scattering adds a
non-negligible force in this intermediate regime. We address all three regimes
analytically and provide approximate formulae for the force parameters, for
arbitrary optical depth and inner dust temperature. Turning to numerical codes,
we examine the convergence properties of the Monte Carlo code Hyperion run in
Cartesian geometry. We calibrate both Hyperion and our analytical estimate
using the DUSTY code, run in spherical geometry. We find that Monte Carlo codes
tend to underestimate the radiation force when the mean free path of starlight
is not well resolved, as this causes the inner dust temperature, and therefore
the inner Rosseland opacity, to be too low. We briefly discuss implications for
more complicated radiation transfer problems.
