Constraining the Pulsar 3D Velocity Distribution: The Impact of Spin-Velocity Alignment

Abstract

Quantifying the natal kick distribution of pulsars is essential for understanding supernova physics and binary evolution, yet measurements are historically limited by the lack of radial velocity data. Most previous studies rely on transverse velocities under the assumption of spatial isotropy. In this work, we reconstruct the intrinsic three-dimensional (3D) velocity distribution for a curated sample of 18 pulsars by explicitly incorporating the observational constraint of spin-velocity alignment. Using a hierarchical Bayesian framework that accounts for measurement uncertainties, we compare nine candidate velocity distribution models. We find that a Gamma distribution provides an adequate description of the inferred 3D velocities; however, the modest Bayes factor (1.65 relative to a single Maxwellian) indicates that the current data lack sufficient resolving power to discriminate decisively among the models considered. The Gamma model is characterized by a peak velocity of $237^{+67}_{-84}$ km s$^{-1}$. The reconstructed 3D velocities under alignment are systematically lower than those inferred under isotropy, indicating that projection effects can bias individual kick estimates high, while leaving the overall population scale largely unchanged within uncertainties. A complementary analysis of 465 pulsars with transverse velocity estimates favors a Log-Normal distribution for the full sample, while isolated young pulsars remain consistent with a Gamma-like profile. Our results underscore the importance of geometric assumptions in population inference and highlight the need for larger samples with improved distance and spin-axis measurements to place tighter constraints on natal kick physics.

Figures (9)

• Figure 1: The 3D geometry of a pulsar's rotation axis, proper motion vector, and magnetic axis. The green solid line represents the pulsar's rotation axis vector, while the red solid line denotes the pulsar's 3D velocity vector. The angle $\chi$ signifies the 3D angular separation between these two vectors. The angles $\zeta_{r}$ and $\zeta_{v}$ represent the angular separations of the pulsar's rotation axis and velocity vector from the $z$-axis, respectively. The rotation axis (PA$\rm _r$) and proper motion vector (PA$\rm _v$) projected onto the plane of the sky are represented by the green and red dashed lines, respectively, with an angular separation of $\Psi$. Note that PA$_{r}$ is equivalent to PA$_0$ used elsewhere in this paper.
• Figure 2: Velocity vector deflection angle induced by the Galactic potential as a function of characteristic age. Diamonds denote older sources ($\tau_c > 10$ Myr) for which dynamical backtracking is applied, while circles represent younger pulsars. Several young sources cluster near the coordinate origin because their small characteristic ages correspond to negligible deflection. The red star marks PSR J0538+2817.
• Figure 3: The angle $\chi$ between the pulsar spin axis and velocity vector as a function of $\zeta_v$ (the angle between the 3D velocity vector and the $z$-axis, the line-of-sight). As shown in Figure \ref{['image_1']}, the projected angle $\Psi$ between the velocity and spin axis on the 2D sky plane remains fixed, while the lack of radial velocity measurements allows the 3D velocity vector to pivot along the line of sight, resulting in a degeneracy of $\zeta_v$ spanning $0^\circ$–$180^\circ$. Each pulsar is color-coded, with three representative sources explicitly labeled in the legend. The scatter points mark the $\chi$ values calculated using Equation \ref{['eq6']} when $\zeta_{v}=\zeta_{r}$ for individual sources.
• Figure 4: Bayesian factors of 8 distribution models in comparison to the Maxwellian model for the pulsar 3D velocity distributions assuming spin-velocity alignment.
• Figure 5: Posterior parameter distributions of the Gamma model. Blue and orange contours show results for spin-velocity alignment and isotropy prior, respectively.