  We investigate the origin of the relations between stellar mass and optical
circular velocity for early-type (ETG) and late-type (LTG) galaxies --- the
Faber-Jackson (FJ) and Tully-Fisher (TF) relations. We combine measurements of
dark halo masses (from satellite kinematics and weak lensing), and the
distribution of baryons in galaxies (from a new compilation of galaxy scaling
relations), with constraints on dark halo structure from cosmological
simulations. The principle unknowns are the halo response to galaxy formation
and the stellar initial mass function (IMF). The slopes of the TF and FJ
relations are naturally reproduced for a wide range of halo response and IMFs.
However, models with a universal IMF and universal halo response cannot
simultaneously reproduce the zero points of both the TF and FJ relations. For a
model with a universal Chabrier IMF, LTGs require halo expansion, while ETGs
require halo contraction. A Salpeter IMF is permitted for high mass (sigma >
180 km/s) ETGs, but is inconsistent for intermediate masses, unless
V_circ(R_e)/sigma_e > 1.6. If the IMF is universal and close to Chabrier, we
speculate that the presence of a major merger may be responsible for the
contraction in ETGs while clumpy accreting streams and/or feedback leads to
expansion in LTGs. Alternatively, a recently proposed variation in the IMF
disfavors halo contraction in both types of galaxies. Finally we show that our
models naturally reproduce flat and featureless circular velocity profiles
within the optical regions of galaxies without fine-tuning.
