[Paper Review] Crustal heating in accreting neutron stars from the nuclear energy-density functional theory. I. Proton shell effects and neutron-matter constraint

A. F. Fantina, J. L. Zdunik+4 more|Lirias (KU Leuven)|Jun 11, 2018

Pulsars and Gravitational Waves Research85 references3 citations

TL;DR

This study uses nuclear energy-density functional (EDF) theory to model crustal heating in accreting neutron stars, explicitly incorporating proton shell effects and neutron-matter constraints. It finds that shell closures—particularly at Z=14—dominate heat deposition in the inner crust, yielding a total crustal heating of 1.5–1.7 MeV, significantly higher than previous models due to realistic nuclear structure effects.

ABSTRACT

Observations of soft X-ray transients in quiescence suggest the existence of heat sources in the crust of accreting neutron stars. Heat is thought to be released by electroweak and nuclear processes triggered by the burying of ashes of X-ray bursts. The heating is studied using a fully quantum approach taking consistently into account nuclear shell effects. We have followed the evolution of ashes made of $^{56}$Fe employing the nuclear energy-density functional theory. Both the outer and inner crusts are described using the same functional, thus ensuring a unified and thermodynamically consistent treatment. To assess the role of the neutron-matter constraint, we have employed the set of accurately calibrated Brussels-Montreal functionals BSk19, BSk20, and BSk21 and for comparison the SLy4 functional. Due to nuclear shell effects, the fully accreted crust is found to be much less stratified than in previous studies. In particular, large regions of the inner crust contain clusters with the magic number $Z=14$. The heat deposited in the outer crust is tightly constrained by experimental atomic mass data. The shallow heating we obtain does not exceed $0.2$~MeV and is therefore not enough to explain the cooling of some soft X-ray transients. The total heat released in the crust is very sensitive to details of the nuclear structure and is predicted to lie in the range from $1.5$~MeV to $1.7$~MeV. The evolution of an accreted matter element and therefore the location of heat sources are governed to a large extent by the existence of nuclear shell closures. Ignoring these effects in the inner crust, the total heat falls to $\sim 0.6$~MeV. The neutron-matter constraint is also found to play a key role. The large amount of heat obtained by Steiner et al. (2012) could thus be traced back to unrealistic neutron-matter equations of state.

Motivation & Objective

• To understand the origin and distribution of crustal heating in accreting neutron stars during quiescence.
• To assess the role of nuclear shell effects—especially proton shell closures—on heat deposition in the inner crust.
• To evaluate the impact of neutron-matter equation of state (EoS) constraints on total crustal heating.
• To reconcile discrepancies between earlier models (e.g., HZ, Steiner 2012) and observational constraints from soft X-ray transients.
• To provide a thermodynamically consistent, unified description of the outer and inner crust using the same EDF.

Proposed method

• Employed the nuclear energy-density functional (EDF) theory with Brussels-Montreal functionals (BSk19, BSk20, BSk21) and SLy4 for comparison.
• Tracked the evolution of 56Fe ashes through crustal layers using self-consistent EDF calculations, including pairing and medium effects.
• Calculated heat release via electron captures, neutron emissions/absorptions, and pycnonuclear reactions along the accretion path.
• Used experimental atomic mass data to constrain reactions in the outer crust, ensuring consistency with observed nuclear masses.
• Applied analytical formulas for neutron-drip density and pressure (Chamel & Fantina 2016a) to define crustal layer boundaries.
• Compared results with the compressible liquid-drop model (CLDM) of Mackie & Baym (1977) to isolate the impact of shell effects.

Experimental results

Research questions

• RQ1 How do proton shell effects, particularly at Z=14, influence the distribution and magnitude of crustal heating in the inner crust?
• RQ2 To what extent does the neutron-matter equation of state affect the total heat deposited in the crust?
• RQ3 Why do previous models (e.g., HZ, Steiner 2012) overestimate crustal heating compared to observational constraints?
• RQ4 How does the inclusion of nuclear shell effects in EDF theory alter the stratification and composition of the accreted crust compared to the cold-catalyzed crust?
• RQ5 What is the role of experimental atomic mass data in constraining heat deposition in the outer crust?

Key findings

• Nuclear shell effects, especially the magic number Z=14, lead to large, extended regions of clusters in the inner crust, reducing stratification compared to previous models.
• The total heat deposited in the crust is tightly constrained to 1.5–1.7 MeV by EDF calculations, with BSk19–BSk21 and SLy4 yielding consistent results.
• Outer crust heating is limited to ≤0.2 MeV due to experimental mass constraints, insufficient to explain shallow heating in some soft X-ray transients.
• The absence of shell effects in the CLDM (MB) model reduces total heating to ~0.6 MeV, highlighting the critical role of shell closures.
• The high heating values reported by Steiner et al. (2012) stem from unrealistic neutron-matter EoS and empirical shell parametrizations, not physical reality.
• The EDF-based crustal heating is in good agreement with Lau et al. (2018) at densities ~1.6×10¹² g cm⁻³, but diverges at higher densities due to consistent treatment of medium-modified shell effects.