We report on two Chandra observations of the 3-Myr-old pulsar B1929+10 , which reveal a faint compact ( \sim 9 ^ { \prime \prime } \times 5 ^ { \prime \prime } ) nebula elongated in the direction perpendicular to the pulsar ’ s proper motion , two patchy wings , and a possible short ( \sim 3 ^ { \prime \prime } ) jet emerging from the pulsar . In addition , we detect a tail extending up to at least 4 ^ { \prime } in the direction opposite to the pulsar ’ s proper motion , aligned with the \sim 15 ^ { \prime } -long tail detected in ROSAT and XMM-Newton observations . The overall morphology of the nebula suggests that the shocked pulsar wind is confined by the ram pressure due to the pulsar ’ s supersonic speed . The shape of the compact nebula in the immediate vicinity of the pulsar seems to be consistent with the current MHD models . However , since these models do not account yet for the change of the flow velocity at larger distances from the pulsar , they are not able to constrain the extent of the long pulsar tail . The luminosity of the whole nebula as seen by Chandra is L _ { PWN } \sim 10 ^ { 30 } ergs s ^ { -1 } in the 0.3–8 keV band , for the distance of 361 pc . Using the Chandra and XMM-Newton data , we found that the pulsar spectrum is comprised of non-thermal ( magnetospheric ) and thermal components . The non-thermal component can be described by a power-law model with photon index \Gamma \approx 1.7 and luminosity L _ { PSR } ^ { nonth } \approx 1.7 \times 10 ^ { 30 } ergs s ^ { -1 } in the 0.3–10 keV band . The blackbody fit for the thermal component , which presumably emerges from hot polar caps , gives the temperature kT \approx 0.3 keV and projected emitting area A _ { \perp } \sim 3 \times 10 ^ { 3 } m ^ { 2 } , corresponding to the bolometric luminosity L _ { bol } \sim ( 1 – 2 ) \times 10 ^ { 30 } ergs s ^ { -1 } .