Stochastic and secular anomalies in pulsar braking indices

TL;DR

This paper investigates how stochastic and secular variations affect pulsar braking index measurements, providing a quantitative framework to interpret anomalous indices and distinguish underlying causes using Bayesian timing and simulations.

Contribution

It introduces a predictive formula for the variance of pulsar braking indices considering stochastic and secular effects, aiding interpretation of anomalies in pulsar timing data.

Findings

Derived a formula for the variance of braking index considering noise and secular changes.

Showed that large anomalies occur when secular or stochastic variations dominate.

Quantified how phase residuals are affected by different types of anomalies.

Abstract

Stochastic and secular variations in the spin frequency $\nu$ of a rotation-powered pulsar complicate the interpretation of the measured braking index, $n$, in terms of a power-law spin-down torque $\propto \nu^{n_{\rm pl}}$. Both categories of variation can lead to anomalous braking indices, with $\vert n \vert = \vert \nu \ddot{\nu} / \dot{\nu}^2 \vert \gg 1$, where the overdot symbolizes a derivative with respect to time. Here we quantify the combined effect of stochastic and secular deviations from pure power-law spin down on measurements of $n$. Through analytic calculations, Monte Carlo simulations involving synthetic data, and modern Bayesian timing techniques, it is shown that the variance of $n$ satisfies the predictive, falsifiable formula $\langle n^{2} \rangle = (n_{\rm pl}+\dot{K}_{\rm dim})^{2}+\sigma_{\rm dim}^{2}$, where $\dot{K}_{\rm dim}$ is inversely proportional to the time-scale $\tau_K$ over which the proportionality constant of the power-law spin-down torque varies, $\sigma_{\rm dim}$ is proportional to the timing noise amplitude and inversely proportional to the square root of the total observing time, and the average is over an ensemble of random realizations of the timing noise process. The anomalous regime $\langle n^2 \rangle \gg 1$ occurs for $\dot{K}_{\rm dim} \gg 1$, $\sigma_{\rm dim} \gg 1$, or both. The sign of $n$ depends in part on the sign of $\dot{K}_{\rm dim}$, so it is possible to measure unequal numbers of positive and negative $n$ values in a large sample of pulsars. The distinguishable impact of stochastic and secular anomalies on phase residuals is quantified to prepare for extending the analysis of synthetic data to real pulsars.