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The systemic recoil velocity distribution and the scale height of field millisecond pulsar systems: Implications on neutron star retention fractions in star clusters

The systemic recoil velocity ($v_\mathrm{sys}$) distribution of millisecond pulsars (MSPs) is essential for understanding the MSP formation channel(s) and for estimating the retention fractions of MSPs in star clusters. However, the determination is complicated by MSPs' long-term dynamic evolution and the scarcity of radial velocity measurements. We compiled 64 field MSP systems that are well astrometrically determined, and calculated their transverse peculiar velocities $\boldsymbol{v}_\perp$ and Galactic heights $z$. Assuming that the Galactic-longitude components $v_\mathrm{l}$ of $\boldsymbol{v}_\perp$ are statistically stable over time (the "stable-$v_\mathrm{l}$" assumption), we approached the distribution of the $v_\mathrm{l}$ components of $\boldsymbol{v}_\mathrm{sys}$ by the observed $v_\mathrm{l}$ sample. We find that the observed $v_\mathrm{l}$ can be well described by a linear combination of three normal distributions. Accordingly, the MSP $v_\mathrm{sys}$ distribution can be approximated by a linear combination of three Maxwellian components under the assumption that $\boldsymbol{v}_\mathrm{sys}$ directions are uniformly distributed. Our dynamical population synthesis analysis based on the derived $v_\mathrm{sys}$ distribution verified the "stable-$v_\mathrm{l}$" assumption in the parameter space of this work, and estimated the initial and the current Galaxy-wide scale heights of field MSP systems to be about 0.32 kpc and 0.68 kpc, respectively. According to the MSP $v_\mathrm{sys}$ distribution, $\approx14$% of all the MSPs born in a globular cluster with the nominal 50 $\mathrm{km~s^{-1}}$ central escape velocity can be retained. Therefore, the $v_\mathrm{sys}$ distribution of field MSP systems may account for the high number of MSPs discovered in globular clusters, which implies that MSPs in star clusters may follow the same formation channel(s) as field MSP systems.