The two tails of PSR J2055+2539 as seen by Chandra: analysis of the nebular morphology and pulsar proper motion

We analyzed two Chandra observations of PSR J2055+2539 (for a total integration time of $\sim$130 ks) in order to measure its proper motion and study its two elongated nebular features. We did not detect the proper motion, setting an upper limit of 240 mas yr$^{-1}$ (3$\sigma$ level), that translates into an upper limit on the transverse velocity of $\sim$700 km s$^{-1}$, for an assumed distance of 600 pc. A deep H$\alpha$ observation did not reveal the bow-shock associated with a classical pulsar wind nebula, thus precluding an indirect measurement of the proper motion direction. We determined the main axes of the two nebulae, which are separated by an angle of 160.8$^{\circ} \pm 0.7^{\circ}$, using a new approach based on the Rolling Hough Transformation (RHT). We analyzed the shape of the first 8' (out of the 12' seen by XMM-Newton) of the brighter, extremely collimated one. Based on a combination of our results from a standard analysis and a nebular modeling obtained from the RHT, we find that the brightest nebula is curved on an arcmin-scale, with a thickness ranging from $\sim9$" to $\sim31$" and a possible (single or multiple) helicoidal pattern. We could not constrain the shape of the fainter nebula. We discuss our results in the context of other known similar features, with particular emphasis on the Lighthouse nebula (associated with PSR J1101$-$6101). We speculate that a peculiar geometry of the powering pulsar may play an important role in the formation of such features.