[Paper Review] The nuclear symmetry energy, the inner crust, and global neutron star modeling

William G. Newton, Michael Gearheart+2 more|arXiv (Cornell University)|Dec 9, 2011

Pulsars and Gravitational Waves Research5 references5 citations

TL;DR

This paper investigates the role of the nuclear symmetry energy in shaping neutron star inner crust structure and global properties, using a combination of mean-field models and advanced many-body methods. It demonstrates that the symmetry energy parameters J and L critically determine crust thickness, pasta phase transitions, and neutron star observables like moment of inertia and torsional oscillation frequencies.

ABSTRACT

The structure and composition of the inner crust of neutron stars, as well as global stellar properties such as radius and moment of inertia, have been shown to correlate with parameters characterizing the symmetry energy of nuclear matter such as its magnitude J and density dependence L at saturation density. It is thus mutually beneficial to nuclear physicists and astrophysicists to examine the combined effects of such correlations on potential neutron star observables in the light of recent experimental and theoretical constraints on J, L, and relationships between them. We review some basic correlations between these nuclear and astrophysical observables, and illustrate the impact of recent progress in constraining the J-L parameter space on the composition of the inner crust, crust-core transition density and pressure, and extent of the hypothesized pasta region. We use a simple compressible liquid drop model in conjunction with a simple model of nuclear matter which allows for independent, smooth, variation of the J and L. We extend the model into the core using the same nuclear matter model to explore the effects on global crust and core properties, and on potential observables such as crust oscillation frequencies and mechanically supported crust deformation. Throughout we illustrate the importance of the relationship between J and L implicit in a particular model of nuclear matter to the predictions of neutron star properties.

Motivation & Objective

• To understand how the nuclear symmetry energy governs the structure and stability of neutron star inner crust and mantle.
• To examine the sensitivity of neutron star observables—such as moment of inertia and crust oscillation frequencies—to the symmetry energy parameters J and L.
• To compare the predictive power of simplified models (like the CLDM) with more realistic microscopic approaches (e.g., 3D-Hartree-Fock) in describing nuclear pasta phases and crust-core transitions.
• To assess the limitations of mean-field approximations in capturing shell effects, long-range correlations, and self-organized structures in dense neutron-rich matter.

Proposed method

• Uses the nuclear matter equation of state (EOS) and its expansion in isospin asymmetry via the symmetry energy S(n), defined as the second derivative of energy with respect to isospin asymmetry δ.
• Applies the parabolic approximation (PA) to estimate pure neutron matter energy and pressure, with key parameters J (symmetry energy at saturation), L (slope), and K_sym (curvature) as central inputs.
• Employs the compressible liquid drop model (CLDM) for efficient exploration of crust properties across J-L parameter space, despite its limitations in capturing shell and long-range effects.
• Compares CLDM results with more accurate 3D-Hartree-Fock (3DHF) calculations using Skyrme interactions (e.g., SLy4, SII), which self-consistently include surface, shell, and shape effects.
• Utilizes the dynamical method of [85] and 3DHF to compute transition densities between nuclear phases (e.g., spherical → pasta → uniform), validating phase sequence predictions.
• Incorporates self-consistent pairing and electron screening effects in advanced models to improve realism in crust and mantle descriptions.

Experimental results

Research questions

• RQ1 How do the symmetry energy parameters J and L influence the thickness and composition of the neutron star inner crust?
• RQ2 What is the role of shell effects and nuclear shell closures in determining the geometry and stability of nuclear pasta phases?
• RQ3 To what extent do mean-field models like CLDM accurately predict crust-core transition densities compared to microscopic 3DHF calculations?
• RQ4 How do long-range effects such as electron screening and self-organization of pasta phases affect the mechanical and transport properties of the neutron star mantle?
• RQ5 What constraints do neutron star observables (e.g., moment of inertia, torsional oscillation frequencies) place on the J-L parameter space?

Key findings

• The pressure of pure neutron matter at saturation density is directly proportional to the symmetry energy slope: P_PNM(n₀) = n₀L/3.
• Crust thickness, moment of inertia, and torsional crust oscillation frequencies all show strong dependence on the symmetry energy parameters J and L.
• 3D-Hartree-Fock calculations confirm the canonical pasta phase sequence: cylindrical (spaghetti) → slab (lasagna) → cylindrical bubble → spherical bubble, with shape evolution tracked via neutron density maps.
• Transition densities from 3DHF models for SLy4, SII, SkM*, and SkMp parameterizations show good agreement with the dynamical method of [85], validating phase sequence predictions.
• The CLDM provides a reasonable guide for J-L dependence but fails to capture shell effects, long-range correlations, and breakdown of the Wigner-Seitz approximation in the mantle.
• Microscopic models like 3DHF and extended Thomas-Fermi methods are essential for self-consistently including surface, shell, and pairing effects, especially in the pasta phase region.