  We consider the sensitivity of the circular-orbit adiabatic contraction
approximation to the baryon condensation rate and the orbital structure of dark
matter halos in the $\Lambda$CDM paradigm. Using one-dimensional hydrodynamic
simulations including the dark matter halo mass accretion history and gas
cooling, we demonstrate that the adiabatic approximation is approximately valid
even though halos and disks may assemble simultaneously. We further demonstrate
the validity of the simple approximation for $\Lambda$CDM halos with isotropic
velocity distributions using three-dimensional N-body simulations. This result
is easily understood: an isotropic velocity distribution in a cuspy halo
requires more circular orbits than radial orbits. Conversely, the approximation
is poor in the extreme case of a radial orbit halo. It overestimates the
response a core dark matter halo, where radial orbit fraction is larger.
Because no astronomically relevant models are dominated by low-angular momentum
orbits in the vicinity of the disk and the growth time scale is never shorter
than a dynamical time, we conclude that the adiabatic contraction approximation
is useful in modeling the response of dark matter halos to the growth of a
disk.
