Hierarchical Bayesian estimation of population-level torque law parameters from $68$ young radio pulsars observed with the Murriyang telescope

TL;DR

This work uses a Brownian spin-down model in combination with a hierarchical Bayesian framework to infer the population distribution of torque-law parameters from timing data of 68 young pulsars observed with Murriyang. By separating secular evolution of the torque, encapsulated in $(n_{\rm pl}+\dot{K}_{\dim})$, from stochastic timing noise quantified by $\chi^{(m)}$, the study finds a population-level mean $\mu_{\rm pl} \approx 9.95$ and dispersion $\sigma_{\rm pl} \approx 10.89$, with most braking-index anomalies dominated by stochastic noise yet a non-negligible fraction showing secular contributions. Per-pulsar results span a wide range, with many objects exhibiting strong stochastic anomalies ($\chi^{(m)}s^{(m)}$ large) and several pulsars where the secular term produces nonzero braking indices at high confidence. The analysis suggests that the characteristic time-scale for torque evolution $\tau_{K}$ is typically comparable to the spin-down time-scale $\tau_{\rm sd}$, and it discusses three astrophysical interpretations, including single or dual secular mechanisms and possible secular evolution of $K$, while acknowledging limitations due to sample size. Overall, the method provides a principled way to decompose braking-index measurements into physical torque components and offers guidance for interpreting timing-noise-dominated indices in future pulsar surveys.

Abstract

Abridged. The measured braking index, $n=ν\ddotν/\dotν^2$, of a rotation-powered pulsar with spin frequency $ν$ and braking torque $K ν^{n_{\rm pl}}$, features secular and stochastic anomalies arising from $\dot{K} \neq 0$ and random torque noise respectively. Previous studies quantified the variance $\langle n^{2} \rangle = (n_{\rm pl}+\dot{K}_{\rm dim})^{2}+σ_{\rm dim}^{2}$, where the secular anomaly, $\dot{K}_{\rm dim}$, is inversely proportional to the characteristic time-scale $τ_{K}$ over which $K$ varies; the stochastic anomaly, $σ_{\rm dim}^{2} = σ_{\ddotν}^{2}ν^{2}γ_{\ddotν}^{-2}\dotν^{-4}T_{\rm obs}^{-1}$, is a function of the timing noise amplitude $σ_{\ddotν}$, a damping time-scale $γ_{\ddotν}^{-1}$ and the total observing time $T_{\rm obs}$; and the average is taken over an ensemble of random realizations of the noise process. Here, we use a hierarchical Bayesian scheme, based on the formula for $\langle n^{2} \rangle$, to infer the population-level distribution of $n_{\rm pl}+\dot{K}_{\rm dim}$ for a sample of $68$ young radio pulsars, observed for $\gtrsim 10~{\rm years}$ with Murriyang, the 64-m Parkes radio telescope. Upon assuming that the $n_{\rm pl}+\dot{K}_{\rm dim}$ values are drawn from a population-level Gaussian, $N(μ_{\rm pl}, σ_{\rm pl})$, the Bayesian scheme returns the mean $μ_{\rm pl} = 9.95^{+5.58}_{-5.26}$ and standard deviation $σ_{\rm pl}=10.89^{+5.14}_{-3.69}$. At a per-pulsar level it returns posterior medians satisfying $-13.86 \leq n_{\rm pl}+\dot{K}_{\rm dim} \leq 30.38$. The secular anomaly dominates the stochastic anomaly, with posterior medians satisfying $|n_{\rm pl} + \dot{K}_{\rm dim}| \geq σ_{\rm dim}$ in 10 out of 68 objects.

Figures (6)

• Figure 1: Posterior distribution $p(\mu_{\rm pl}, \sigma_{\rm pl} \vert D)$ (central panel) for the population-level hyperparameters $\mu_{\rm pl}$ and $\sigma_{\rm pl}$ for a subsample of P574 pulsars ($M=68$; see Table \ref{['AppB:TableD^m']}). The contours of $p(\mu_{\rm pl}, \sigma_{\rm pl} \vert D)$ mark the $30\%,50\%$,$70\%$, and $90\%$ credible intervals. The one-dimensional posteriors correspond to $p(\mu_{\rm pl}, \sigma_{\rm pl} \vert D)$ marginalized over $\mu_{\rm pl}$ (right panel) and $\sigma_{\rm pl}$ (upper panel). For both the upper and right panels, the marginalized posteriors are plotted as blue curves, while the priors are plotted as green curves. The surtitles for the upper and right panels display the corresponding inferred median (central value) and $90\%$ credible interval (error bars; grey dashed lines).
• Figure 2: Posterior predictive check: distribution of $n_{\rm pl}+\dot{K}_{\rm dim}$ values without distinguishing between the $68$ exchangeable objects in the analysis, given by equation (\ref{['Eq_subsecII:ppc_n_pop']}). The posterior distribution $p[(n_{\rm pl}+\dot{K}_{\rm dim})^{({\rm pop})} \vert D]$ (blue histogram) is obtained by combining the inferred $p(\mu_{\rm pl},\sigma_{\rm pl}\vert D)$ and the population-level prior $\pi[(n_{\rm pl}+\dot{K}_{\rm dim})^{({\rm pop})} \vert \mu_{\rm pl}, \sigma_{\rm pl}]$ via equation (\ref{['Eq_subsecII:ppc_n_pop']}). The population-level prior $\pi[(n_{\rm pl}+\dot{K}_{\rm dim})^{(\rm pop)} \vert \psi ]$ [equation (\ref{['eq_SecII:npl_Kdim_dist']})] is plotted as a green histogram.
• Figure 3: Location of the $68$ P574 pulsars analyzed in this paper in the plane spanned by period $P=\nu^{-1}$ (horizontal axis; logarithmic scale) versus spin-down rate $\dot{P}=-\nu^{-2}\dot{\nu}$ (vertical axis; logarithmic scale). Dashed, black lines indicate constant characteristic age $\tau_{\rm sd}$, surface magnetic field $B$, and spin-down luminosity $\dot{E}$, in units of ${\rm yr}$, ${\rm G}$, and ${\rm erg\,s}^{-1}$ respectively. The color scheme indicates the inferred stochastic braking index anomaly $\chi s$ (logarithmic scale). A blue square decorating a data point indicates a pulsar that has glitched at least once, while a red star indicates a pulsar with a previously published measurement of $n$ from ParthasarathyJohnston2020. We report the inferred posterior median $n_{\rm pl}+\dot{K}_{\rm dim}$ and $90\%$ credible intervals, alongside a few representative objects which satisfy $n_{\rm pl}+\dot{K}_{\rm dim} \geq \chi s$ i.e. for which the stochastic anomaly is smaller than the secular anomaly.
• Figure 4: Cross-correlations between the rotational parameters $\nu,\dot{\nu}, \tau_{\rm sd}$, and $\ddot{\nu}_{\rm dip}$ (top to bottom rows respectively) and the second frequency derivative measured by $\mathrm{{ TEMPO2}}$ ($\vert \ddot{\nu} \vert$; left column) and the hierarchical Bayesian scheme ($\vert \ddot{\nu}_{\rm pl}\vert$; right column). In the last row, the red, dashed line indicates $\ddot{\nu}_{\rm dip}=\vert \ddot{\nu} \vert$ and $\ddot{\nu}_{\rm dip}=\vert \ddot{\nu}_{\rm pl} \vert$. PSR J1513$-$5908, with $n_{\rm pl}+\dot{K}_{\rm dim}=2.83^{+0.01}_{-0.01}$, is the closest object to the red, dashed line.
• Figure 5: Correcting for the stochastic anomaly in braking index measurements: values of $n_{\rm pl}+\dot{K}_{\rm dim}$ inferred by the hierarchical Bayesian scheme, which exclude the stochastic anomaly (horizontal axis; see Section \ref{['subsecII:Bayesframework']}), versus traditional $\mathrm{{ TEMPONEST}}$ measurements of $n$, reported by ParthasarathyJohnston2020 (vertical axis; logarithmic scale). The blue, dashed curve indicates $n=n_{\rm pl}+\dot{K}_{\rm dim}$. The color scale indicates $\chi s$, the root-mean-square amplitude of the stochastic anomaly [see equation (\ref{['Eq_SecI:Variance_n']})]. All pulsars satisfying $\chi s \leq 1$ are assigned the color at the lower end of the scale. Objects that lie further from the dashed curve correspond to higher $\chi s$ values, i.e. their stochastic anomaly dominates their secular anomaly VargasMelatos2024.