Rotational evolution of deformed magnetized neutron stars: implications for obliquity distribution and braking indices statistics

TL;DR

This work addresses why pulsars exhibit anomalous braking indices and nontrivial magnetic obliquity distributions by modeling the long-term rotational evolution of deformed neutron stars under external torques and magnetic-field decay. It develops a formalism combining free precession from crustal deformations with radiative torques, derives a secular evolution equation for a global precession parameter $\lambda$, and performs population-level simulations to connect theory with timing observations. The results show that even modest deformations ($\varepsilon_d \sim 10^{-12}-10^{-10}$) can produce large, quasi-periodic variations in the magnetic angle $\chi$, reproduce the observed braking-index spread and its age correlation, and naturally yield a broad or isotropic distribution of obliquities depending on magnetic-field decay. This framework provides a unified physical link between interior NS structure, magnetospheric torques, and pulsar timing phenomenology, with testable predictions for polarization evolution and long-term timing behavior.

Abstract

The rotational evolution of a strongly magnetized neutron star (NS), accreting or isolated, is driven by external torques of different nature. In addition to the torques, even the tiniest deformations of the NS crust can affect its rotation through asymmetries in its inertia tensor. Several factors may be responsible for the deformations, including strong magnetic fields, internal stresses, or local heating. The main effect produced by the deformations is the so-called free precession: the motion of the rotational axis with respect to the crust. We consider the evolution of a triaxially deformed isolated NS with a strong dipolar magnetic field for a broad range of parameters, taking into account the magnetic field decay. We show that the combination of pulsar torques and free precession results in a considerable broadening of the distribution of magnetic obliquity angles (the angle between the magnetic and rotational axes) and creates a population of objects where the rotational axis does not align with the magnetic axis at all but enters a limit-cycle regime. The combination of free precession and magnetic torques can also explain the observed distribution in pulsar braking indices by creating a periodic oscillation in the magnetic obliquity.

Figures (16)

• Figure 1: Principal axes of a deformed NS and associated coordinate systems. The angles $\theta$ and $\varphi$ are the polar and azimuthal angles, respectively, measured in the principal frame. The magnetic axis $\pmb m$ is represented by the red arrow. The auxiliary basis $\pmb{s}_{1-3}$ constructed on vectors $\pmb \Omega$ and $\pmb m$ is shown by green arrows.
• Figure 2: The classical period -- period derivative diagram for known isolated radiopulsars. Levels of constant effective radiative deformation $\hat{\varepsilon}_\mu$ are shown by solid black lines. Gray dashed lines represent constant magnetic fields, obtained from the relation $B_\mathrm{md} = 3.2\times 10^{19} \sqrt{P \dot P}$ G, and constant age, $\tau_\mathrm{ch} = P/2\dot P$. Assuming that all the NSs have comparable deformations $\varepsilon_\mathrm{d}$, one concludes that precession of highly magnetized stars ($B_\mathrm{md} \sim 10^{13}..10^{14}$ G) is dominated by the radiative torque, even if $\varepsilon_\mathrm{d} \sim 10^{-10}..10^{-9}$. On the other hand, even a small deformation $\varepsilon_\mathrm{d} \sim 10^{-17}..10^{-16}$ is sufficient to make free precession dominant in a millisecond pulsar rotation.
• Figure 3: Set of trajectories of the spin axis for a freely precessing star, represented by the contours of $\lambda_\mathrm{0,norm} = const$. The numbers that label the curves are values of $\lambda_{\rm 0,norm}$. Four different values of the ratio $\varepsilon_{3}/\varepsilon_1$ are shown. In each case the spin vector $\pmb \Omega$ follows one of the contour lines ( the polhodes) with precession period $\sim 2\pi/\varepsilon_\mathrm{d}\Omega$. The bold plus signs at the centers of the closed sets of contours correspond to minima of $\lambda_0$. The maxima occur at the polar angles $\theta = 0$ and $180^\circ$. The crosses depict the positions of the saddle points (X-points). Case (a) with $\varepsilon_{3} = 0$ represents a prolate biaxial deformation along the $\pmb{e}_1$ axis. In case (b) both deformations have the same amplitude: $\varepsilon_1 = -\varepsilon_3$. The other two cases, $\varepsilon_3/\varepsilon_1 = -10$ and $\varepsilon_3/\varepsilon_1 = -0.1$, correspond to deformations predominantly along $\pmb{e}_3$ and $\pmb{e}_1$, respectively.
• Figure 4: Precession trajectories (contours of $\lambda_\mathrm{norm})$ for a triaxial star with $\varepsilon_1 = -\varepsilon_3$ and nonzero radiative torque $n_\mu$ with ratios $\varepsilon_\mu/\varepsilon_\mathrm{d} = -0.1,-1,-3$ and $-10$. In all plots, the north magnetic pole has the coordinates $\varphi_\mathrm{m}=250^\circ$, $\theta_\mathrm{m} = 50^\circ$. For different orientations of the magnetic axis, the configurations are qualitatively the same. The north (N) and south (S) magnetic poles are represented by blue and red stars, respectively. The dotted red line depicts the magnetic equator, while other lines and marks have the same meaning as in Figure \ref{['fig:lambda_free_precession']}.
• Figure 5: Qualitative summary for asymptotic evolutionary states of a precessing magnetized neutron star. Here, $\pmb\Omega$ denotes the angular velocity vector and $\chi$ the magnetic angle. For a biaxial star, $\alpha$ is the angle between the magnetic and symmetry axes.