Reconciling the Systemic Kicks of Observed Millisecond Pulsars, Spider Pulsars, and Low-mass X-ray Binaries

Abstract

Millisecond pulsars (MSPs) have been proposed as evolutionary products of low-mass X-ray binaries (LMXBs) through a stage in which they are spider pulsars (i.e., redbacks and black widows). However, recent work has found that the systemic kicks of observed MSPs are significantly lower than the kicks of LMXBs and spiders, which appears to be in tension with this evolutionary model. We argue that this tension can be relieved, at least to some degree, by considering the fact that the observed MSPs are located at relatively short distances, whereas spider pulsars are located at greater distances and LMXBs are situated even further away. We model the distance-dependent kinematic bias for dynamically old objects, which favors observing objects that have received low kicks at short distances and correct the observed systemic kicks for this bias. We find that this kinematic bias can be big enough to close the gap between the MSP and LMXB kicks, although the spider pulsars appear to come from a slightly different systemic kick distribution, but this difference is not necessarily physical. All corrected systemic kick distributions are consistent with predictions from binary population synthesis for progenitor systems with a post-supernova orbital period of $P_{\text{orb}}\leq10\,$d and a companion mass of $M_{c}\leq1\,M_{\odot}$, where the natal kicks are calibrated to the velocities of young isolated pulsars. We conclude that the difference in observed systemic kicks is not necessarily in tension with a common origin for MSPs, spider pulsars, and LMXBs.

Figures (8)

• Figure 1: Observed distributions of systemic kicks (bottom left panel) and distances (top right panel) for the MSPs (blue), spiders (purple) and LMXBs (red) in the samples of ODoherty_2023. The dark lines show the median distributions and the shaded regions correspond to the bootstrapped $90\%$ confidence intervals. The dotted line shows the fitted Beta distribution of ODoherty_2023, which is dominated by the MSPs, and the black double-headed arrow illustrates the statistical gap between the MSPs and the other objects. The bottom right panel shows the relationship between the distances and systemic kicks for the MSPs (blue circles), spider pulsars (purple triangles), and LMXBs (red squares), where the error bars show the $68\%$ confidence intervals.
• Figure 2: Distances of the kicked objects in the simulation of Disberg_2025a for $40\,$Myr$\,{<}\,t\,{<}\,1\,$Gyr (evaluated with timesteps of $1\,$Myr). The left panel shows, for each simulation containing $10^3$ objects with a kick magnitude $v_{\text{kick}}$, the distribution of distances to the Sun ($r$) in a 2D histogram with distance bins of $0.5\,$kpc and velocity bins of $10\,$km s$^{-1}$. The rows are normalized so that the distribution corresponds to $P(r\text{\textbar} v_{\text{kick}})$. The dotted line shows the median kick velocity in each distance bin. The right panel shows kick distributions integrated over certain distance ranges (colors) through Equation (\ref{['eq1']}) for a uniform kick distribution between $0$ and $1000\,$km s$^{-1}$, along with that uniform distribution itself (dotted line), which is the kick distribution observed at $t=0$ for any distance. We note that (1) the simulation considers kick velocities up to $1000\,$km s$^{-1}$, even though the figure shows kicks up to $600\,$km s$^{-1}$, and (2) $v_{\text{kick}}$ equals the natal kick for single objects but the systemic kick for binaries.
• Figure 3: Observed systemic kick distributions of ODoherty_2023, for MSPs (blue), LMXBs (red), and spider pulsars (purple). The solid lines show the median distributions and the shaded regions correspond to the $90\%$ confidence intervals. The dashed lines are determined through Equations (\ref{['eq3']}) and (\ref{['eq4']}) and a lognormal $P(v_{\text{sys}})$ distribution fitted to the lower edge of the MSP confidence interval. The dash-dotted line, in turn, uses a $P(v_{\text{sys}})$ distribution fitted to the upper edge of the spider pulsars confidence interval. The dotted line shows the kinematic bias for a uniformly distributed $P(v_{\text{sys}})$.
• Figure 4: Systemic kick distributions fitted to the observations while accounting for the kinematic bias. The left panel shows these lognormal distributions, for the MSPs and LMXBs ($\mu=5.1$ and $\sigma=0.7$, blue and red dashed line), as well as for the spider pulsars ($\mu=5.2$ and $\sigma=0.5$, purple dashed line). The dashed line shows the natal NS kick distribution of Disberg_2025b, fitted to isolated pulsar velocities, while the dotted line shows the fit of ODoherty_2023 to their observed systemic kick estimates of MSPs, spiders, and LMXBs (as shown in Figure \ref{['fig3']}). The right panel shows the same distributions, but normalized between $0$ and $300\,$km s$^{-1}$. The shaded region is the $95\%$ bootstrapped interval of COMPAS binaries with $P_{\text{orb}}\leq10\,$d and $M_{c}\leq1\,M_{\odot}$ (as discussed in the text).
• Figure A: Systemic kick distributions from ODoherty_2023 for $33$ MSPs with parallax estimates, based on DM distances (blue) and parallax distances (orange), as well as the distribution of LMXBs (red) similarly to Figure \ref{['fig3']}. The dark lines show the median distributions and the shaded regions show the bootstrapped $90\%$ confidence intervals.