  A two-parameter analytical model of the magnetic vortex in a thin disk of
soft magnetic material is constructed. The model is capable of describing the
change in evolution of net vortex state magnetization and of core position when
the vortex core interacts with a magnetic pinning site. The model employs a
piecewise, physically continuous, magnetization distribution obtained by the
merger of two extensively used one-parameter analytical models of the vortex
state in a disk. Through comparison to numerical simulations of ideal disks
with and without pinning sites, the model is found to accurately predict the
magnetization, vortex position, hysteretic transitions, and 2-D displacement of
the vortex in the presence of pinning sites. The model will be applicable to
the quantitative determination of vortex pinning energies from measurements of
magnetization.
