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The middle-aged PSR J0357+3205 is a nearby, radio-quiet, bright gamma-raypulsar discovered by the Fermi mission. Our previous Chandra observationrevealed a huge, very peculiar structure of diffuse X-ray emission, originatingat the pulsar position and extending for > 9' on the plane of the sky. Tobetter understand the nature of such a nebula, we have studied the propermotion of the parent pulsar. We performed relative astrometry on Chandra imagesof the field spanning a time baseline of 2.2 yr, unveiling a significantangular displacement of the pulsar counterpart, corresponding to a propermotion of 0.165"+/-0.030" yr^(-1). At a distance of ~500 pc, the space velocityof the pulsar would be of ~390 km s^(-1) assuming no inclination with respectto the plane of the sky. The direction of the pulsar proper motion is perfectlyaligned with the main axis of the X-ray nebula, pointing to a physical, yetelusive link between the nebula and the pulsar space velocity. No opticalemission in the H_alpha line is seen in a deep image collected at the Geminitelescope, which implies that the interstellar medium into which the pulsar ismoving is fully ionized.

PSR J0357+3205: a fast moving pulsar with a very unusual X-ray trail