We have carried out high-precision timing measurements of the binary millisecond pulsar PSR J2051$-$0827 with the Effelsberg 100-m radio telescope of the Max-Planck-Institut fur Radioastronomie and with the Lovell 76-m radio telescope at Jodrell Bank. The 6.5-yrs radio timing measurements have revealed a significant secular variation of the projected semi-major axis of the pulsar at a rate of $dot xequiv d(a_{rm 1} sin i)/dt = (-0.23pm 0.03)times 10^{-12}$, which is probably caused by the Newtonian spin-orbit coupling in this binary system leading to a precession of the orbital plane. The required misalignment of the spin and orbital angular momenta of the companion are evidence for an asymmetric supernova explosion. We have also confirmed that the orbital period is currently decreasing at a rate of $dot P_{rm b}=(-15.5 pm 0.8)times 10^{-12}$s s$^{-1}$ and have measured second and third orbital period derivatives $d^2P_{rm b}/dt^2=(+2.1 pm 0.3)times 10^{-20} {rm s^{-1}}$ and $d^3P_{rm b}/dt^3 =(3.6 pm 0.6)times 10^{-28} {rm s^{-2}}$, which indicate a quasi-cyclic orbital period variation similar to those found in another eclipsing pulsar system, PSR B1957+20. The observed variation of the orbital parameters constrains the maximal value of the companion radius to $R_{rm c max} sim 0.06 R_{odot}$ and implies that the companion is underfilling its Roche lobe by 50 %. The derived variation in the quadrupole moment of the companion is probably caused by tidal dissipation similar to the mechanism proposed for PSR B1957+20. We conclude that the companion is at least partially non-degenerate, convective and magnetically active.
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