[Paper Review] Non-Spherical Models of Neutron Stars

Omair Zubairi, William Spinella+5 more|arXiv (Cornell University)|Apr 12, 2015

Pulsars and Gravitational Waves Research3 references3 citations

TL;DR

This paper derives a generalized Tolman-Oppenheimer-Volkoff (TOV)-like equation for deformed neutron stars with oblate or prolate shapes, using a parametrized metric with deformation parameter γ. The model shows that even modest deformations significantly alter mass and radius, enabling multiple maximum-mass configurations for a single equation of state, which may explain the observed spread in neutron star masses and distinguish them from black holes.

ABSTRACT

Non-rotating neutron stars are generally treated in theoretical studies as perfect spheres. Such a treatment, however, may not be correct if strong magnetic fields are present (such as for magnetars) and/or the pressure of the matter in the cores of neutron stars is non-isotropic (e.g., color superconducting). In this paper, we investigate the structure of non-spherical neutron stars in the framework of general relativity. Using a parameterized metric to model non-spherical mass distributions, we first derive a stellar structure equation for deformed neutron stars. Numerical investigations of this model equation show that the gravitational masses of deformed neutron stars depend rather strongly on the degree and type (oblate or prolate) of stellar deformation. In particular, we find that the mass of a neutron star increases with increasing oblateness but decreases with increasing prolateness. If this feature carries over to a full two-dimensional treatment of deformed neutron stars, this opens up the possibility that, depending on the type of stellar deformation, there may exist multiple maximum-mass neutron stars for one and for the same model for the nuclear equation of state.

Motivation & Objective

• To develop a simplified, analytically tractable stellar structure equation for deformed neutron stars that generalizes the standard TOV equation.
• To investigate how non-spherical geometries—specifically oblate and prolate shapes—impact neutron star mass and radius, especially under strong magnetic fields or anisotropic pressure.
• To enable efficient numerical exploration of deformed neutron star properties without requiring full numerical relativity simulations.
• To explore the implications of deformation for the maximum mass of neutron stars, particularly in the context of magnetars and quark matter models.

Proposed method

• Introduces a parametrized metric with γ = 1 for spheres, γ < 1 for oblate, and γ > 1 for prolate spheroids, preserving spherical symmetry in the energy-momentum tensor.
• Derives a generalized stellar structure equation from Einstein’s field equations using this metric, yielding a closed-form differential equation for pressure gradient.
• Solves the resulting equation using a hybrid nuclear equation of state combining relativistic mean-field and nonlocal Nambu-Jona-Lasinio models for quark deconfinement.
• Applies the equation to compute mass-radius relations and eccentricities for various γ values, comparing results to the standard TOV case.
• Uses the deformation parameter γ to systematically explore how shape affects gravitational mass and radius, with analytical solutions feasible under the parametrization.

Experimental results

Research questions

• RQ1 How does stellar deformation, parametrized by γ, affect the mass and radius of neutron stars compared to spherical models?
• RQ2 Can non-spherical configurations lead to multiple maximum-mass neutron stars for a single equation of state?
• RQ3 To what extent do oblate and prolate shapes alter the gravitational mass and radius of neutron stars under strong magnetic fields or anisotropic pressure?
• RQ4 Does the standard TOV equation fail to describe proto-quark stars with high magnetic fields, as suggested by prior work?

Key findings

• For a given equation of state, increasing oblateness (γ < 1) leads to a significant increase in gravitational mass, while increasing prolateness (γ > 1) reduces it.
• The model predicts that multiple maximum-mass neutron stars can exist for the same equation of state due to deformation, contradicting the single-maximum-mass prediction of spherical models.
• At γ = 0.80, the eccentricity of an oblate star is ε = 0.43617, and at γ = 1.20, the eccentricity of a prolate star is ε = -0.55222, indicating measurable shape effects.
• The derived equation reduces to the standard TOV equation in the limit γ = 1, confirming consistency with established general relativity for spherical stars.
• Deformation effects are substantial enough to influence the interpretation of observed neutron star masses and their distinction from black holes.