Measuring the crust-superfluid coupling time-scale for 105 UTMOST pulsars with a Kalman filter

TL;DR

The study addresses how crust–superfluid coupling shapes neutron-star timing noise by applying a Kalman-filter-based two-component model to 286 UTMOST pulsars. It combines per-pulsar model selection with hierarchical Bayesian inference to estimate the coupling time-scale $\tau$ and stochastic torques $Q_{\rm c}$, $Q_{\rm s}$ at both the individual-pulsar and population levels. The results show 105 pulsars favor the two-component model, with 28 exhibiting sharply peaked $\tau$ posteriors spanning $\log_{10}(\tau/{\rm s})$ from ~4.6 to ~8.6, and reveal population-level scalings $\tau \propto \Omega_{\rm c}^{0.19^{+0.50}_{-0.52}} |\dot{\Omega}_{\rm c}|^{0.18^{+0.18}_{-0.19}}$, $Q_{\rm c} \propto \Omega_{\rm c}^{1.23^{+0.80}_{-0.75}} |\dot{\Omega}_{\rm c}|^{0.49^{+0.27}_{-0.32}}$, and $Q_{\rm s} \propto \Omega_{\rm c}^{0.71^{+0.76}_{-0.78}} |\dot{\Omega}_{\rm c}|^{1.27^{+0.30}_{-0.28}}$. A comparison with glitch-based measurements suggests that the same coupling processes operate between glitches, but additional angular-momentum reservoirs participate during glitches, consistent with multi-component interior physics. The work demonstrates a rigorous path to linking microphysical crust–superfluid dynamics with macroscopic timing behavior and provides a framework for incorporating chromatic noise and larger, multi-band datasets in the future.

Abstract

Crust-superfluid coupling plays an important role in neutron star rotation, particularly with respect to timing noise and glitches. Here, we present new timing-noise-based estimates of the crust-superfluid coupling time-scale \(τ\) for 105 radio pulsars in the UTMOST dataset, by Kalman filtering the pulse times of arrival. The 105 objects are selected because they favor a two-component, crust-superfluid model over a one-component model with log Bayes factor \(\ln \mathfrak{B}_{\rm BF} \geq 5\). The median estimate of \(τ\) ranges from \(10^{4.6\pm0.4}\)\,s for PSR J2241$-$5236 to \(10^{7.7^{+0.7}_{-0.4}}\)\,s for PSR J1644$-$4559 among 28 out of 105 objects with sharply peaked \(τ\) posteriors. A hierarchical Bayesian analysis is performed on 101 out of 105 objects that are canonical (i.e.\ neither recycled nor magnetars) and reside in the populous core of the \(Ω_{\rm c}\)-\(\dotΩ_{\rm c}\) plane. It returns the population-level scaling \(τ\propto Ω_{\rm c}^{0.19^{+0.50}_{-0.52}} |\dotΩ_{\rm c}|^{0.18^{+0.18}_{-0.19}}\), where \(Ω_{\rm c}\) and \(\dotΩ_{\rm c}\) are the angular velocity and spin-down rate of the crust respectively. The variances of the stochastic crust and superfluid torques are also estimated hierarchically, with \(Q_{\rm c} \propto Ω_{\rm c}^{1.23^{+0.80}_{-0.75}} |\dotΩ_{\rm c}|^{0.49^{+0.27}_{-0.32}}\) and \(Q_{\rm s} \propto Ω_{\rm c}^{0.71^{+0.76}_{-0.78}} |\dotΩ_{\rm c}|^{1.27^{+0.30}_{-0.28}}\) respectively. Implications for the physical origin of crust-superfluid coupling, e.g.\ through mutual friction, are discussed briefly.

Figures (12)

• Figure 1: Crust-superfluid coupling time-scale per pulsar: histogram of the medians of the $\log_{10} (\tau \, \rm{s}^{-1})$ posteriors for the 105 UTMOST objects with $\ln \mathfrak{B}_{\rm BF} \geq 5$ (blue bars) and the 28 objects with $\ln \mathfrak{B}_{\rm BF} \geq 5$ and peaky $\tau$ posteriors (orange bars). The counts in each bin are recorded above each bar.
• Figure 2: Crust and superfluid noise variances per pulsar: histograms of the posterior medians of $\log_{10} Q_{\rm c}$ (top left panel; units rad$^2$s$^{-3}$), $\log_{10} Q_{\rm s}$ (top middle panel; units rad$^2$s$^{-3}$), and $\log_{10} \sigma_{\rm TN}^2$ (top right panel; units s$^2$) for the 105 UTMOST pulsars with $\ln \mathfrak{B}_{\rm BF} \geq 5$ (blue bars) and the 28 objects with $\ln \mathfrak{B}_{\rm BF} \geq 5$ and sharply peaked $\tau$ posteriors (orange bars). The counts in each bin are recorded above each bar. Note that the orange histograms count objects whose posteriors are peaky in $\tau$, not peaky in $Q_{\rm c}$ or $Q_{\rm s}$, to facilitate a like-for-like comparison with Fig. \ref{['fig:tau_histogram_pop_uninform']}. The bottom panels show two-dimensional histograms of the posterior medians of $\log_{10} Q_{\rm s}$ versus $\log_{10} Q_{\rm c}$ (bottom left panel) and $\log_{10} Q_{\rm s}$ versus $\log_{10} \sigma_{\rm TN}^2$ (bottom right panel) for the 105 pulsars with $\ln \mathfrak{B}_{\rm BF} \geq 5$. The contours enclose the $1$-, $1.5$-, and $2$-sigma confidence regions. Point estimates (blue dots) are overlaid for the 105 pulsars, to help visualize the probability mass density.
• Figure 3: Venn diagram of 105 UTMOST pulsars with $\ln \mathfrak{B}_{\rm BF} \geq 5$ that have peaky posteriors in $\tau$ (top left circle), $Q_{\rm c}$ (top right circle), and $Q_{\rm s}$ (bottom circle). Regions are labelled with their cardinalities, except for the central region, which is labelled with the names of the 14 objects it contains.
• Figure 4: Corner plot of the posterior distribution of the hyperparameters $C_{2, \theta}$, $a_{\theta}$, $b_{\theta}$, and $\sigma_{\theta}$ for the hierarchical regression model described by \ref{['eq:linear_regression_model']} for $\theta \in \{\tau, Q_{\rm c}, Q_{\rm s}\}$ after analyzing 101 canonical pulsars with $\ln \mathfrak{B}_{\rm BF} \geq 5$ in the populous core of the $\Omega_{\rm c}$-$\dot{\Omega}_{\rm c}$ plane, i.e. excluding recycled pulsars and magnetars. The contours indicate the 1-, 2-, and 3-sigma credible regions for a two-dimensional slice of the posterior after marginalizing over the other 10 parameters. The vertical dashed lines in the one-dimensional histograms, after marginalizing over the other 11 parameters, bracket the 68% confidence intervals. The heading above each rightmost panel quotes the median and 68% credible interval. Every panel confirms that the posterior distribution converges numerically and is unimodal.
• Figure 5: Histograms of the population-informed posterior medians of $\log_{10} \tau$ (top panel; units s), $\log_{10} Q_{\rm c}$ (bottom left panel; units $\rm{rad}^2 \rm{s}^{-3}$), $\log_{10} Q_{\rm s}$ (bottom middle panel; units $\rm{rad}^2 \rm{s}^{-3}$), and $\log_{10} \sigma_{\rm TN}^2$ (bottom right panel; units $\rm{s}^{2}$) for 101 canonical pulsars with $\ln \mathfrak{B}_{\rm BF} \geq 5$ in the populous core of the $\Omega_{\rm c}$-$\dot{\Omega}_{\rm c}$ plane, assuming the hierarchical regression model \ref{['eq:linear_regression_model']}. The counts in each bin are recorded above each bar. The heading above each panel quotes the median and 68% credible interval.