We report the first X-ray detection of the radio pulsar PSR B1757-24 using
the Chandra X-Ray Observatory. We detect point-source emission at the pulsa
r position plus a faint tail extending nearly 20" east of the pulsar, in th
e same direction and with comparable morphology to the radio tail. Assuming
the point-source X-ray emission is magnetospheric, the observed X-ray tail
represents only similar to0.01% of the pulsar's spin-down luminosity. This
is significantly lower than the analogous efficiencies of most known X-ray
nebulae surrounding rotation-powered pulsars. Assuming a nonthermal spectr
um for the tail photons, we show that the tail is unlikely to be emission l
eft behind following the passage of the pulsar but rather is probably from
synchrotron-emitting pulsar wind particles having a flow velocity of 7000 k
m s(-1). We also show that there must be a significant break in the tail sy
nchrotron spectrum between the radio and X-ray bands that is intrinsic to t
he particle spectrum. No emission is detected from the shell supernova remn
ant G5.4-1.2. The upper limits on remnant emission are unconstraining.