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

ABSTRACT The measured braking index, $n=\nu \ddot{\nu }/\dot{\nu }^2$, of a rotation-powered pulsar with spin frequency $\nu$ (where an overdot symbolizes a time derivative) and braking torque $K \nu ^{n_{\mathrm pl}}$, features secular and stochastic anomalies arising from $\dot{K} \ne 0$ and random torque noise respectively. Previous studies quantified the variance $\langle n^{2} \rangle = (n_{\mathrm pl}+\dot{K}_{\rm dim})^{2}+\sigma _{\rm dim}^{2}$, 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 $\gtrsim 10~{\rm yr}$ 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, ${\cal N}(\mu _{\rm pl}, \sigma _{\rm pl})$, 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 $-13.86 \le n_{\rm pl}+\dot{K}_{\rm dim} \le 30.38$. The secular anomaly dominates the stochastic anomaly, with posterior medians satisfying $|n_{\rm pl} + \dot{K}_{\rm dim}| \ge \sigma _{\rm dim}$ in 10 out of 68 objects. The inference results imply that some mechanism other than electromagnetic or gravitational radiation reaction with $\dot{K} = 0$ operates in at least 66 out of 68 analysed pulsars, in line with previous observational studies. The results are also consistent with $\tau _K \lesssim \nu /\vert \dot{\nu }\vert$, with implications for the physical mechanism causing $\dot{K} \ne 0$.