# Refinement of the timing-based estimator of pulsar magnetic fields

## TL;DR

This paper refines the classical pulsar magnetic field estimator by incorporating corrections based on neutron star properties, leading to more accurate and unbiased magnetic field estimates with quantified uncertainties.

## Contribution

The authors introduce a new correction to the traditional magneto-dipolar formula that accounts for the equation of state, obliquity, and mass, improving the accuracy of pulsar magnetic field estimates.

## Key findings

- The correction $
abla_B$ is nearly normally distributed with a mean between -0.5 and -0.25 dex.
- The standard deviation of $
abla_B$ is approximately 0.06 to 0.09 dex.
- A generalized estimator $B^*
oughly 3B_{md}/7$ provides an unbiased magnetic field estimate with about 30% uncertainty.

## Abstract

Numerical simulations of realistic non-vacuum magnetospheres of isolated neutron stars have shown that pulsar spin-down luminosities depend weakly on the magnetic obliquity. This result provides the opportunity to estimate the surface magnetic field for a given radiopulsar with quite a high accuracy. In the current work, we present a refinement of the classical `magneto-dipolar' formula for pulsar magnetic fields $B_{\rm md} = (3.2\times 10^{19}\mbox{ G})\sqrt{P\dot P}$, where $P$ is the neutron star spin period. The new, robust timing-based estimator is introduced as $\log B = \log B_{\rm md} + \Delta_{\rm B}(M, \alpha)$, where the correction $\Delta_{\rm B}$ depends on the equation of state (EOS) of dense matter, the individual pulsar obliquity $\alpha$ and the mass $M$. Adopting state-of-the-art statistics for $M$ and $\alpha$ we calculate the distributions of $\Delta_{\rm B}$ for a representative subset of 22 EOSs that do not contradict observations. It has been found that $\Delta_{\rm B}$ is distributed nearly normally, with the average in the range -0.5 to -0.25 dex and standard deviation $\sigma[\Delta_{\rm B}] \approx$ 0.06 to 0.09 dex, depending on the adopted EOS. The latter quantity represents a formal uncertainty of the corrected estimation of $\log B$ because $\Delta_{\rm B}$ is weakly correlated with $\log B_{\rm md}$. At the same time, if it is assumed that every considered EOS has the same chance of occurring in nature, then another, more generalized, estimator $B^* \approx 3B_{\rm md}/7$ can be introduced providing an unbiased value of the pulsar surface magnetic field with $\sim$30 per cent uncertainty with 68 per cent confidence. Finally, we discuss the possible impact of pulsar timing irregularities on the timing-based estimation of $B$ and review the astrophysical applications of the obtained results.