We present the first phase-resolved study of the X-ray spectral properties of
the Crab Pulsar that covers all pulse phases. The superb angular resolution of
the Chandra X-ray Observatory enables distinguishing the pulsar from the
surrounding nebulosity, even at pulse minimum. Analysis of the pulse-averaged
spectrum measures interstellar X-ray extinction due primarily to photoelectric
absorption and secondarily to scattering by dust grains in the direction of the
Crab Nebula. We confirm previous findings that the line-of-sight to the Crab is
underabundant in oxygen, although more-so than recently measured. Using the
abundances and cross sections from Wilms, Allen & McCray (2000) we find [O/H] =
(3.33 +/-0.25) x 10**-4. Analysis of the spectrum as a function of pulse phase
measures the low-energy X-ray spectral index even at pulse minimum -- albeit
with large statistical uncertainty -- and we find marginal evidence for
variations of the spectral index. The data are also used to set a new (3-sigma)
upper limit to the temperature of the neutron star of log T(infinity) < 6.30.Comment: 20 Pages including 7 figures. Accepted for publication in the
Astrophysical Journa