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Very Long Baseline Interferometry Measured Proper Motion and Parallax of the $γ$-ray Millisecond Pulsar PSR J0218+4232

PSR J0218$+$4232 is a millisecond pulsar (MSP) with a flux density $\sim$ 0.9 mJy at 1.4 GHz. It is very bright in the high-energy X-ray and $γ$-ray domains. We conducted an astrometric program using the European VLBI Network (EVN) at 1.6 GHz to measure its proper motion and parallax. A model-independent distance would also help constrain its $γ$-ray luminosity. We achieved a detection of signal-to-noise ratio S/N > 37 for the weak pulsar in all five epochs. Using an extragalactic radio source lying 20 arcmin away from the pulsar, we estimate the pulsar's proper motion to be $μ_α\cosδ=5.35\pm0.05$ mas yr$^{-1}$ and $μ_δ=-3.74\pm 0.12$ mas yr$^{-1}$, and a parallax of $π=0.16\pm0.09$ mas. The very long baseline interferometry (VLBI) proper motion has significantly improved upon the estimates from long-term pulsar timing observations. The VLBI parallax provides the first model-independent distance constraints: $d=6.3^{+8.0}_{-2.3}$ kpc, with a corresponding $3σ$ lower-limit of $d=2.3$ kpc. This is the first pulsar trigonometric parallax measurement based solely on EVN observations. Using the derived distance, we believe that PSR J0218$+$4232 is the most energetic $γ$-ray MSP known to date. The luminosity based on even our 3$σ$ lower-limit distance is high enough to pose challenges to the conventional outer gap and slot gap models.