The Large Area Telescope (LAT) onboard the Fermi satellite opened a new era for pulsar astronomy, detecting gamma-ray pulsations from more than 60 pulsars, ~40% of which are not seen at radio wavelengths. One of the most interesting sources discovered by LAT is PSR J0357+3205, a radio-quiet, middle-aged (tau_C ~0.5 Myr) pulsar standing out for its very low spin-down luminosity (Erot ~6x10^33 erg/s), indeed the lowest among non-recycled gamma-ray pulsars. A deep X-ray observation with Chandra (0.5-10 keV), coupled with sensitive optical/infrared ground-based images of the field, allowed us to identify PSR J0357+3205 as a faint source with a soft spectrum, consistent with a purely non-thermal emission (photon index Gamma=2.53+/-0.25). The absorbing column (NH=8+/-4x10^20 cm^-2) is consistent with a distance of a few hundred parsecs. Moreover, the Chandra data unveiled a huge (9 arcmin long) extended feature apparently protruding from the pulsar. Its non-thermal X-ray spectrum points to synchrotron emission from energetic particles from the pulsar wind, possibly similar to other elongated X-ray tails associated with rotation-powered pulsars and explained as bow-shock pulsar wind nebulae (PWNe). However, energetic arguments, as well as the peculiar morphology of the diffuse feature associated with PSR J0357+3205 make the bow-shock PWN interpretation rather challenging.
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