  We generalize the Rigid-Field Hydrodynamic equations to accommodate arbitrary
magnetic field topologies, resulting in a new Arbitrary Rigid-Field
hydrodynamic (ARFHD) formalism. We undertake a critical point calculation of
the steady-state ARFHD equations with a CAK-type radiative acceleration and
determine the effects of a dipole magnetic field on the usual CAK mass-loss
rate and velocity structure. Enforcing the proper optically-thin limit for the
radiative line-acceleration is found to decrease both the mass-loss and wind
acceleration, while rotation boosts both properties. We define
optically-thin-correction and rotation parameters to quantify these effects on
the global mass-loss rate and develop scaling laws for the surface mass-flux as
a function of surface colatitude. These scaling laws are found to agree with
previous laws derived from magnetohydrodynamic simulations of magnetospheres.
The dipole magnetosphere velocity structure is found to differ from a global
beta-velocity law, which contradicts a central assumption of the
previously-developed XADM model of X-ray emission from magnetospheres.
