  The double pulsar (PSR J0737-3039A/B) provides some of the most stringent
tests of general relativity (GR) and its alternatives. The success of this
system in tests of GR is largely due to the high-precision, long-term timing of
its recycled-pulsar member, pulsar A. On the other hand, pulsar B is a young
pulsar that exhibits significant short-term and long-term timing variations due
to the electromagnetic-wind interaction with its companion and geodetic
precession. Improving pulsar B's timing precision is a key step towards
improving the precision in a number of GR tests with PSR J0737-3039A/B. In this
paper, red noise signatures in the timing of pulsar B are investigated using
roughly a four-year time span, from 2004 to 2008, beyond which time the
pulsar's radio beam precessed out of view ... The timing of pulsar B presented
in this paper depends on the size of the pulsar's orbit, which was calculated
from GR, in order to precisely account for orbital timing delays. Consequently,
our timing cannot directly be used to test theories of gravity. However, our
modelling of the beam shape and radial wind of pulsar B can indirectly aid
future efforts to time this pulsar by constraining part of the additional red
noise observed on top of the orbital delays. As such, we conclude that, in the
idealised case of zero covariance between our model's parameters and those of
the timing model, our model can bring about a factor 2.6 improvement on the
measurement precision of the mass ratio, R = mA/mB, between the two pulsars: a
theory-independent parameter, which is pivotal in tests of GR.
