  We have studied the Bekenstein-Sandvik-Barrow-Magueijo (BSBM) model for the
spatial and temporal variations of the fine structure constant, alpha, with the
aid of full N-body simulations which explicitly and self-consistently solve for
the scalar field driving the alpha-evolution. We focus on the scalar field (or
equivalently alpha) inside the dark matter halos and find that the profile of
the scalar field is essentially independent of the BSBM model parameter. This
means that given the density profile of an isolated halo and the background
value of the scalar field, we can accurately determine the scalar field
perturbation in that halo. We also derive an analytic expression for the
scalar-field perturbation using the Navarro-Frenk-White halo profile, and show
that it agrees well with numerical results, at least for isolated halos; for
non-isolated halos this prediction differs from numerical result by a (nearly)
constant offset which depends on the environment of the halo.
