Testing Models of Magnetic Field Evolution of Neutron Stars with the Statistical Properties of Their Spin Evolutions

We test models for the evolution of neutron star (NS) magnetic fields (B). Our model for the evolution of the NS spin is taken from an analysis of pulsar timing noise presented by Hobbs et al. (2010). We first test the standard model of a pulsar’s magnetosphere in which B does not change with time and magnetic dipole radiation is assumed to dominate the pulsar’s spin-down. We find this model fails to predict both the magnitudes and signs of the second derivatives of the spin frequencies ($\ddot{\nu}$). We then construct a phenomenological model of the evolution of $B$, which contains a long term decay (LTD) modulated by short term oscillations (STO); a pulsar’s spin is thus modified by its B-evolution. We find that an exponential LTD is not favored by the observed statistical properties of $\ddot{\nu}$ for young pulsars and fails to explain the fact that $\ddot{\nu}$ is negative for roughly half of the old pulsars. A simple power-law LTD can explain all the observed statistical properties of $\ddot{\nu}$. Finally we discuss some physical implications of our results to models of the B-decay of NSs and suggest reliable determination of the true ages of many young NSs is needed, in order to constrain further the physical mechanisms of their B-decay. Our model can be further tested with the measured evolutions of $\dot{\nu}$ and $\ddot{\nu}$ for an individual pulsar; the decay index, oscillation amplitude and period can also be determined this way for the pulsar.

💡 Research Summary

The paper investigates how the magnetic field (B) of neutron stars evolves over time by comparing theoretical models with the statistical properties of pulsar spin derivatives, specifically the second derivative of the spin frequency ( \ddotν ). The authors begin by testing the conventional magnetospheric model in which the magnetic field is assumed constant and spin‑down is driven solely by magnetic dipole radiation. Under this assumption the spin‑down law predicts (\dotν \propto -ν^{n}) with a braking index n≈3, and consequently \ddotν should be positive and of a magnitude that scales directly with \dotν and ν. Observational data, however, show a broad distribution of \ddotν values: many young pulsars have large positive \ddotν, while roughly half of the older pulsars exhibit negative \ddotν. This discrepancy demonstrates that a static‑B model cannot simultaneously reproduce both the sign and the amplitude of the observed \ddotν.

To address the failure, the authors construct a phenomenological model in which the magnetic field itself evolves. The evolution consists of two components: a long‑term decay (LTD) and a short‑term oscillation (STO). The LTD is examined in two functional forms: (i) an exponential decay (B(t)=B_{0}\exp(-t/τ)) and (ii) a power‑law decay (B(t)=B_{0}(t/t_{0})^{-\alpha}). The STO is modeled as a sinusoidal modulation (B(t)=B_{0}