  We fit 54,296 sparsely-sampled asteroid lightcurves in the Palomar Transient
Factory to a combined rotation plus phase-function model. Each lightcurve
consists of 20+ observations acquired in a single opposition. Using 805
asteroids in our sample that have reference periods in the literature, we find
the reliability of our fitted periods is a complicated function of the period,
amplitude, apparent magnitude and other attributes. Using the 805-asteroid
ground-truth sample, we train an automated classifier to estimate (along with
manual inspection) the validity of the remaining 53,000 fitted periods. By this
method we find 9,033 of our lightcurves (of 8,300 unique asteroids) have
reliable periods. Subsequent consideration of asteroids with multiple
lightcurve fits indicate 4% contamination in these reliable periods. For 3,902
lightcurves with sufficient phase-angle coverage and either a reliably-fit
period or low amplitude, we examine the distribution of several phase-function
parameters, none of which are bimodal though all correlate with the bond albedo
and with visible-band colors. Comparing the theoretical maximal spin rate of a
fluid body with our amplitude versus spin-rate distribution suggests that, if
held together only by self-gravity, most asteroids are in general less dense
than 2 g/cm$^3$, while C types have a lower limit of between 1 and 2 g/cm$^3$,
in agreement with previous density estimates. For 5-20km diameters, S types
rotate faster and have lower amplitudes than C types. If both populations share
the same angular momentum, this may indicate the two types' differing ability
to deform under rotational stress. Lastly, we compare our absolute magnitudes
and apparent-magnitude residuals to those of the Minor Planet Center's nominal
$G=0.15$, rotation-neglecting model; our phase-function plus Fourier-series
fitting reduces asteroid photometric RMS scatter by a factor of 3.
