# Incompressible analytical models for spinning-down pulsars

## TL;DR

This paper develops analytical Newtonian models to study the deformation and strain in spinning-down neutron stars, revealing how mass, density distribution, and gravitational potential influence the star's response to decreasing rotation.

## Contribution

It introduces two analytical models for neutron star deformation during spin-down, including a refined model with different core and crust densities and self-consistent gravity effects.

## Key findings

- Strain decreases with increasing stellar mass.
- Maximum strain occurs at the equator in the first model and at the poles in the refined model.
- Neglecting gravitational potential perturbations underestimates strain by 40%.

## Abstract

We study a class of Newtonian models for the deformations of non-magnetized neutron stars during their spin-down. The models have all an analytical solution, and thus allow to understand easily the dependence of the strain on the star's main physical quantities, such as radius, mass and crust thickness. In the first ``historical'' model the star is assumed to be comprised of a fluid core and an elastic crust with the same density. We compare the response of stars with different masses and equations of state to a decreasing centrifugal force, finding smaller deformations for heavier stars: the strain angle is peaked at the equator and turns out to be a decreasing function of the mass.We introduce a second, more refined, model in which the core and the crust have different densities and the gravitational potential of the deformed body is self-consistently accounted for. Also in this case the strain angle is a decreasing function of the stellar mass, but its maximum value is at the poles and is always larger than the corresponding one in the one-density model by a factor of two. Finally, within the present analytic approach, it is possible to estimate easily the impact of the Cowling approximation: neglecting the perturbations of the gravitational potential, the strain angle is 40\% of the one obtained with the complete model.