  A model is proposed for the observed combination of power-law and thermal
emission of keV X-rays from rotationally powered pulsars. For gamma-ray pulsars
with accelerators very many stellar radii above the neutron star surface, 100
MeV curvature gamma-rays from $e^{-}$ or $e^{+}$ flowing starward out of such
accelerators are converted to electron-positron pairs on closed field lines all
around the star. These pairs strongly affect X-ray emission from near the star
in two ways. (1) The pairs are a source of synchrotron emission immediately
following their creation in regions where $B \sim 10^{10}$ G. This emission, in
the photon energy range 0.1 keV $\lesssim E_{X} \lesssim$ 5 MeV, has a
power-law spectrum with energy index 0.5 and X-ray luminosity that depends on
the backflow current, and is typically $\sim 10^{33}$ \lum. (2) The pairs
ultimately form a cyclotron resonance ``blanket'' surrounding the star except
for two holes along the open field line bundles which pass through it. In such
a blanket the gravitational pull on electron-positron pairs toward the star is
balanced by the hugely amplified push of outflowing surface emitted X-rays
wherever cyclotron resonance occurs. Because of it the neutron star is
surrounded by a leaky ``hohlraum'' of hot blackbody radiation with two small
holes, which prevents direct X-ray observation of a heated polar cap of a
gamma-ray pulsar. Weakly spin-modulated radiation from the blanket together
with more strongly spin-modulated radiation from the holes through it would
then dominate observed low energy (0.1--10 keV) emission. For non-gamma-ray
pulsars, in which no such accelerators with their accompanying extreme
relativistic backflow toward the star are ...
