  We present an updated set of constraints for the progenitor of PSR J0737-3039
and for the natal kicks imparted to pulsar B taking into account both the
evolutionary and kinematic history of the double neutron star. For this
purpose, we use recently reported scintillation velocity measurements to trace
the motion of the system in the Galaxy backwards in time as a function of the
unknown orientation Omega of the systemic velocity projected on plane of the
sky as well as the unknown radial velocity Vr. The absolute limits on the
orbital separation and the mass of pulsar B's helium star progenitor just
before its supernova explosion are 1.2Rsun < A0 < 1.7Rsun and 2.1Msun < M0 <
4.7Msun. The kick velocity is constrained to be between 60km/s and 1660km/s and
to be misaligned from the pre-SN orbital angular axis (which could be
associated with pulsar B's spin axis) by least 25 degrees. We also derive
probability distribution functions for the kick velocity imparted to pulsar B
and for the misalignment angle between pulsar A's spin and the post-supernova
orbital angular momentum for both isotropic and polar kicks. The most probable
values of both quantities depend sensitively on the unknown radial velocity. In
particular, tilt angles lower than 30-50 degrees tend to be favored for current
radial velocities of less than 500km/s in absolute value, while tilt angles
higher than 120 degrees tend to be associated with radial velocities in excess
of 1000km/s in absolute value.

Astrophysics