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 \pm 0.25 ) \times 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 _ { \infty } < 6.30 .