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

Abstract

Abridged. The measured braking index, $n=\nu \ddot{\nu}/\dot{\nu}^2$, of a rotation-powered pulsar with spin frequency $\nu$ and braking torque $K \nu^{n_{\rm pl}}$, features secular and stochastic anomalies arising from $\dot{K} \neq 0$ and random torque noise respectively. Previous studies quantified the variance $\langle n^{2} \rangle = (n_{\rm pl}+\dot{K}_{\rm dim})^{2}+\sigma_{\rm dim}^{2}$, where the secular anomaly, $\dot{K}_{\rm dim}$, is inversely proportional to the characteristic time-scale $\tau_{K}$ over which $K$ varies; the stochastic anomaly, $\sigma_{\rm dim}^{2} = \sigma_{\ddot{\nu}}^{2}\nu^{2}\gamma_{\ddot{\nu}}^{-2}\dot{\nu}^{-4}T_{\rm obs}^{-1}$, is a function of the timing noise amplitude $\sigma_{\ddot{\nu}}$, a damping time-scale $\gamma_{\ddot{\nu}}^{-1}$ 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, $N(\mu_{\rm pl}, \sigma_{\rm pl})$, the Bayesian scheme returns the mean $\mu_{\rm pl} = 9.95^{+5.58}_{-5.26}$ and standard deviation $\sigma_{\rm pl}=10.89^{+5.14}_{-3.69}$. 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 $|n_{\rm pl} + \dot{K}_{\rm dim}| \geq \sigma_{\rm dim}$ in 10 out of 68 objects.