[Paper Review] Stationary structure of relativistic superfluid neutron stars

R. Prix, J. Novák+1 more|ArXiv.org|Nov 29, 2002

Pulsars and Gravitational Waves Research3 citations

TL;DR

This paper presents a fully relativistic numerical framework for modeling rapidly rotating superfluid neutron stars as two-fluid systems, where neutrons and charged particles (protons, electrons, muons) rotate independently. It confirms the existence of prolate configurations for the slower-rotating fluid and demonstrates that fluid coupling via symmetry and entrainment terms significantly alters stellar structure, including shape and radial extent, with quantitative differences up to 10% in mass and radius compared to uncoupled models.

ABSTRACT

We describe recent progress in the numerical study of the structure of rapidly rotating superfluid neutron star models in full general relativity. The superfluid neutron star is described by a model of two interpenetrating and interacting fluids, one representing the superfluid neutrons and the second consisting of the remaining charged particles (protons, electrons, muons). We consider general stationary configurations where the two fluids can have different rotation rates around a common rotation axis. The previously discovered existence of configurations with one fluid in a prolate shape is confirmed.

Motivation & Objective

• To develop a numerical framework for constructing stationary, rapidly rotating superfluid neutron star models in full general relativity.
• To investigate the structural implications of independent rotation between superfluid neutrons and charged constituents in neutron stars.
• To explore how fluid coupling—via symmetry energy and entrainment—alters stellar configurations compared to uncoupled models.
• To validate the numerical code against known analytical solutions in the slow-rotation limit.

Proposed method

• The two-fluid model treats superfluid neutrons and charged particles as separate, inviscid, conserved fluids with distinct four-velocities and particle number densities.
• The energy density is modeled via a generalized polytropic equation of state that includes coupling terms through a symmetry energy coefficient (κ_nc) and an entrainment function (βΔ²).
• The system is governed by a variational principle derived from a hydrodynamic Lagrangian, leading to equations of motion that incorporate relative motion effects via the relative velocity squared (Δ²).
• The Einstein field equations are reduced to four coupled elliptic equations for metric functions (N, N^φ, A, B), solved using a multi-domain pseudo-spectral numerical scheme.
• First integrals of motion (μⁿ = γₙ constₙ, μᶜ = γᶜ constᶜ) are used to simplify the system under the assumption of uniform rotation.
• Internal consistency is verified via the relativistic virial theorem, with errors below 10⁻⁵ at 10⁻⁶ convergence.

Experimental results

Research questions

• RQ1 How does the independent rotation of superfluid neutrons and charged particles affect the stationary structure of relativistic neutron stars?
• RQ2 What structural configurations emerge when the two fluids are coupled via symmetry energy and entrainment terms?
• RQ3 Can the numerical model reproduce known analytical solutions in the slow-rotation limit, particularly for prolate configurations of the slower fluid?
• RQ4 How do coupling terms influence the shape and radial extent of each fluid component?
• RQ5 What is the impact of fluid decoupling (κ_nc = β = 0) versus coupling on total mass and radius?

Key findings

• The numerical code successfully reproduces known analytical solutions for prolate configurations of the slower-rotating fluid in the slow-rotation limit, confirming the model's validity.
• When only gravitationally coupled (κ_nc = β = 0), the faster-rotating fluid remains oblate, while the slower fluid remains nearly spherical.
• With non-zero coupling (κ_nc = 0.02, β = 0.12), the slower fluid becomes prolate due to enhanced pressure and shape deformation from mutual interactions.
• The inclusion of coupling terms increases the radial extent of the faster fluid and compresses the slower fluid, altering the overall stellar configuration.
• Despite identical central chemical potentials (μⁿ(0) = μᶜ(0) = 0.3mc²), the total mass and radius differ by approximately 10% between uncoupled and coupled configurations.
• The code satisfies the relativistic virial theorem to within 10⁻⁵, indicating high internal consistency and numerical reliability.