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A millisecond pulsar in an extremely wide binary system

We report on 22 yrs of radio timing observations of the millisecond pulsar J1024$-$0719 by the telescopes participating in the European Pulsar Timing Array (EPTA). These observations reveal a significant second derivative of the pulsar spin frequency and confirm the discrepancy between the parallax and Shklovskii distances that has been reported earlier. We also present optical astrometry, photometry and spectroscopy of 2MASS J10243869$-$0719190. We find that it is a low-metallicity main-sequence star (K7V spectral type, $\mathrm{[M/H]}=-1.0$, $T_\mathrm{eff}=4050\pm50$ K) and that its position, proper motion and distance are consistent with those of PSR J1024$-$0719. We conclude that PSR J1024$-$0719 and 2MASS J10243869$-$0719190 form a common proper motion pair and are gravitationally bound. The gravitational interaction between the main-sequence star and the pulsar accounts for the spin frequency derivatives, which in turn resolves the distance discrepancy. Our observations suggest that the pulsar and main-sequence star are in an extremely wide ($P_\mathrm{b}>200$ yr) orbit. Combining the radial velocity of the companion and proper motion of the pulsar, we find that the binary system has a high spatial velocity of $384\pm45$ km s$^{-1}$ with respect to the local standard of rest and has a Galactic orbit consistent with halo objects. Since the observed main-sequence companion star cannot have recycled the pulsar to millisecond spin periods, an exotic formation scenario is required. We demonstrate that this extremely wide-orbit binary could have evolved from a triple system that underwent an asymmetric supernova explosion, though find that significant fine-tuning during the explosion is required. Finally, we discuss the implications of the long period orbit on the timing stability of PSR J1024$-$0719 in light of its inclusion in pulsar timing arrays.