Relic of quadrupole deformation produced in a hot neutron star era

TL;DR

This paper probes whether quadrupole deformation produced in a hot neutron-star era can persist after crust solidification. By modeling a fluid hot state with a quadrupolar driving force and tracking the decay of deformation as the star cools, then computing the residual deformation sustained by crustal elasticity after solidification, the authors show that a relic ellipticity remains in the crust at a level of a few percent of the initial deformation. The relic’s spatial pattern is imprinted by the elastic crust and depends on the initial forcing and crustal rigidity, offering a path to link early NS deformation with future continuous gravitational-wave observations. While the crust-only approach captures essential physics, uncertainties in the initial forcing and core-mantle coupling necessitate further work to make robust GW predictions.

Abstract

A newly born neutron star is expected to exhibit significant deviations from spherical symmetry, which decay with time. Determining how much deformation remains at present is crucial for gravitational-wave astronomy. This study is the first investigation into the evolution of quadrupole deformation during the solid crust formation phase to obtain a plausible value at present. The equilibrium structure before solidification is modeled using a fluid description, and the deformation is introduced through an assumed driving force. As the star cools, this force weakens, leading to a gradual decay of the deformation. Eventually, the deformation vanishes in the fluid region but partially remains in the crust, sustained by elastic forces, after solidification. By comparing the equilibrium models before and after solidification, we estimate the residual ellipticity and demonstrate that the spatial profile of the elastic shear is imprinted in the crust. The relic ellipticity is only a few percent of the original value, with its absolute magnitude depending on the deformation mechanism during the hot era, which cannot be specified owing to the lack of elaborate models. This work provides a first step toward linking early neutron star deformation with future gravitational-wave observations.

Figures (5)

• Figure 1: The force density vector ${\vec{f} }$ is schematically described by arrows in the meridian plane of the crust. Positive functions ($A_2>0, B_2>0$) in eq. (\ref{['explictradfnH.eqn']}) result in prolate deformation, as shown in the left panel, whereas negative functions ($A_2<0, B_2<0$) oblate in the right panel.
• Figure 2: Density perturbation $\zeta^{-1}\delta \rho_{a2}$ (left panel) and pressure perturbation $\zeta^{-1}\delta p_{a2}$ (right panel) are shown as a function of the radius for Models I--III from top to bottom. In the right panels, $\zeta^{-1}Q_{2}$ and $\zeta^{-1}c_s ^2 \delta \rho_{a2}$ are shown using dashed curves.
• Figure 3: $\zeta^{-1}\delta \rho_{b2}$ (left panel) and $\zeta^{-1}\delta p_{b2}$ (right panel) as a function of the radius. In the right panels, $-\zeta^{-1}Q_{2}$ and $\zeta^{-1} c_s ^2 \delta \rho_{b2}$ are also shown by dashed curves. From top to bottom, the models correspond to I, II, and III.
• Figure 4: The magnitude of the normalized shear strain $\sigma/\sigma_{\rm{max}}$ is shown by contours in the meridian plane. The models correspond to I, II, and III from top to bottom.
• Figure 5: Fraction $(\epsilon _{a}+\epsilon _{b})/\epsilon _{a}$ as a function of the shear modulus $\mu_{c,30}\equiv \mu_c/(10^{30} {\text{erg~cm}}^{-3})$. Dots, triangles, and squares represent Models I, II, and III, respectively.