  PSR J1509-5850 is a middle-aged pulsar with the period P ~ 89 ms, spin-down
power Edot = 5.1 x 10^35 erg/s, at a distance of about 3.8 kpc. We report on
deep Chandra X-ray Observatory observations of this pulsar and its pulsar wind
nebula (PWN). In addition to the previously detected tail extending up to 7'
southwest from the pulsar (the southern outflow), the deep images reveal a
similarly long, faint diffuse emission stretched toward the north (the northern
outflow) and the fine structure of the compact nebula (CN) in the pulsar
vicinity. The CN is resolved into two lateral tails and one axial tail pointing
southwest (a morphology remarkably similar to that of the Geminga PWN), which
supports the assumption that the pulsar moves towards the northeast. The
luminosities of the southern and northern outflows are about 1 x 10^33 and 4 x
10^32 erg/s, respectively. The spectra extracted from four regions of the
southern outflow do not show any softening with increasing distance from the
pulsar. The lack of synchrotron cooling suggests a high flow speed or in-situ
acceleration of particles. The spectra extracted from two regions of the
northern outflow show a hint of softening with distance from the pulsar, which
may indicate slower particle propagation. We speculate that the northern
outflow is associated with particle leakage from the bow shock apex into the
ISM, while the southern outflow represents the tail of the shocked pulsar wind
behind the moving pulsar. We estimate the physical parameters of the observed
outflows and compare the J1509-5850 PWN with PWNe of other supersonically
moving pulsars.
