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

TL;DR

This work derives the systemic recoil velocity distribution of field millisecond pulsars (MSPs) by exploiting the stable Galactic-longitude component $v_ ext{l}$ and assuming isotropic $v_ ext{sys}$ directions, then validates the result with dynamical population synthesis (DPS). The observed $v_ ext{l}$ is well described by a three-component normal distribution, which maps to a three-component Maxwell distribution for $v_ ext{sys}$; DPS confirms the stability of $v_ ext{l}$ and yields Galaxy-wide birth and evolved scale heights of $zetaz_0 \\approx 0.32$ kpc and $zetaz_1 \\approx 0.68$ kpc. The analysis implies an indicative MSP retention fraction of about 14% in globular clusters with a central escape velocity of 50 km s$^{-1}$, increasing for higher escape velocities, potentially explaining the high MSP counts in clusters without invoking distinct formation channels. The study thus supports the view that MSPs in star clusters may share the same formation channels as field MSPs and provides a framework to template $v_ ext{sys}$ distributions by formation channel for future population-synthesis and NS-retention studies.

Abstract

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.

Figures (8)

• Figure 1: The upper-left, lower-left and lower-right panels display the histograms of Galactic heights $z$, the Galactic-longitude component $v_\mathrm{l}$ and the Galactic-latitude component $v_\mathrm{b}$ of transverse peculiar velocities, respectively. Each histogram is concatenated from 10000 simulations drawn from the assumed split normal distributions for the 64 measurements (of $z$, $v_\mathrm{l}$ or $v_\mathrm{b}$; see Table \ref{['tab:MSPs_d__v_t']}). The gray dashed curves show the normalized count smoothed with kernel density estimation (using scipy.stats.Gaussian_kde). The best-fit Cauchy, Laplace and three-component normal distributions are plotted in blue and green dashed curves. Overlaid are the dynamical population synthesis results (labeled as "evolved") that best match the observed $z$, $v_\mathrm{l}$ and $v_\mathrm{b}$, based on the determined initial MSP scale height $\zeta_0=0.32$ kpc. The corresponding initial distributions are also provided except for $v_\mathrm{l}$, as the initial $v_\mathrm{l}$ distribution adopted in the dynamical population synthesis is identical to the best-fit three-component normal distribution (a linear combination of three normal distributions).
• Figure 2: The curves in the upper-left, lower-left and lower-right panels display the normalized histograms of the observed Galactic heights $z$, Galactic-longitude component $v_\mathrm{l}$ and Galactic-latitude component $v_\mathrm{b}$ of transverse peculiar velocities, respectively, for 52 binary MSPs and 12 solitary MSPs (see Table \ref{['tab:MSPs_d__v_t']}). The histograms have been smoothed with kernel density estimation using scipy.stats.Gaussian_kde. Each histogram is concatenated from 10000 simulations drawn from the assumed split normal distributions for the measurements (of $z$, $v_\mathrm{l}$ or $v_\mathrm{b}$; see Table \ref{['tab:MSPs_d__v_t']}).
• Figure 3: Upper: The three-component Maxwell distribution described by Equation \ref{['eq:Vsys_distribution']}, which shares the parameters of the best-fit three-component normal distribution (see Table \ref{['tab:PDF_candidates']}) of the observed $v_\mathrm{l}$ sample. Lower: The cumulative distribution function (CDF) calculated from the three-component Maxwell distribution shown in the upper panel. The dash-dotted vertical line marks the mode of the three-component Maxwell distribution, while the dashed lines and the dotted line correspond to the 16th, the 84th, and the 50th percentile of the CDF, respectively.
• Figure 4: The upper-left, lower-left and lower-right panels display the normalized histograms of Galactic heights $z$, the Galactic-longitude component $v_\mathrm{l}$ and the Galactic-latitude component $v_\mathrm{b}$ of transverse peculiar velocities, respectively. The normalized histograms have been smoothed with kernel density estimation (using scipy.stats.Gaussian_kde). Each histogram is concatenated from 10000 simulations drawn from the assumed split normal distributions for the measurements (of $z$, $v_\mathrm{l}$ or $v_\mathrm{b}$; see Table \ref{['tab:MSPs_d__v_t']}). The black, magenta and yellow curves correspond to the full sample (of 64 field MSPs) and two high-precision samples (see Appendix \ref{['subap:obtain_high_precision_samples']}), respectively.
• Figure 5: Gray: The histogram of the millisecond pulsar age distribution adopted in DPS (see Appendix \ref{['subap:age_distribution']}), with logarithmically spaced bins along the x-axis; Pink: The uniform distribution $\mathcal{U}\left(0.1,13.6\right)$ (Gyr) with logarithmically spaced bins along the x-axis, which is only used to examine the DPS robustness with respect to uncertainties in the MSP age distribution (see Appendix \ref{['subsubap:age_related_robustness']}).