arxiv: 2604.26952 · v2 · submitted 2026-04-29 · ⚛️ nucl-th · astro-ph.HE· astro-ph.SR

Recognition: no theorem link

Outer-Crust Equations of State for Neutron Stars

P.S. Koliogiannis , N. Paar

Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:40 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.HEastro-ph.SR

keywords neutron-star outer crustequation of statenuclear mass modelsbeta equilibriumneutron dripminimum massstellar modeling

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The pith

Different nuclear mass models alter the outer crust composition but produce nearly identical equations of state and minimum neutron star masses.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines how uncertainties in nuclear masses affect the outer crust of neutron stars. The authors build four equations of state using different contemporary mass models and calculate the equilibrium composition by minimizing Gibbs free energy. They discover that the sequence of nuclei changes and the neutron drip point shifts slightly, yet the pressure and energy density relations remain very similar. As a result, the properties of the lightest possible neutron stars come out almost the same. This matters because it shows that current nuclear uncertainties do not prevent using these equations of state confidently in stellar models.

Core claim

By constructing outer-crust equations of state from four contemporary nuclear mass models and determining equilibrium compositions through Gibbs free energy minimization in beta equilibrium, the different mass inputs lead to variations in the equilibrium nuclide sequence, distinct last bound nuclei, and moderate shifts in the neutron-drip density. In contrast, the associated thermodynamic properties, as well as the minimum-mass neutron-star configurations, remain closely aligned across the four outer-crust equations of state.

What carries the argument

Minimization of the Gibbs free energy per baryon to identify the equilibrium nuclide composition in beta equilibrium at each density.

If this is right

Thermodynamic properties such as pressure and energy density in the outer crust show only moderate dependence on the nuclear mass model.
Minimum-mass neutron star configurations remain robust and closely aligned regardless of the mass model chosen.
The outer-crust equations of state provide a reliable input for neutron-star structure calculations and stellar modeling.
Detailed nuclide composition varies but does not drive significant changes in global stellar observables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Improved nuclear mass measurements could narrow the exact neutron-drip location without altering the overall equation of state.
Similar robustness may hold when extending the approach to inner-crust equations of state.
Averaging results across multiple mass models offers a practical route to reduce remaining uncertainty in crust modeling.

Load-bearing premise

The four selected nuclear mass models adequately sample the current range of nuclear-mass uncertainties and the Gibbs-free-energy minimization correctly identifies the physical equilibrium composition throughout the outer crust.

What would settle it

A measurement showing the neutron-drip density outside the range spanned by the four models, or observation of a neutron star with mass below the minimum value these equations of state predict.

Figures

Figures reproduced from arXiv: 2604.26952 by N. Paar, P.S. Koliogiannis.

**Figure 1.** Figure 1: shows the equilibrium sequences of nuclei in the outer crust, expressed in terms of proton number Z and neutron number N, as functions of the baryon density for the four nuclear 20 40 60 80 100 Z & N (a) DD–ME2 Fe Ni Kr Se Ge Zn N=30 N=34 N=50 N=82 N=88 Protons Neutrons 20 40 60 80 100 Z & N (b) DD–PC1 Fe Ni Kr SeGe Ni N=30 N=34 N=50 N=82 20 40 60 80 100 Z & N (c) DD–PCX Fe Ni Kr Se Ge Zn Ni N=30 N=34 N=5… view at source ↗

**Figure 2.** Figure 2: Equilibrium sequences of nuclei for the four nuclear mass view at source ↗

**Figure 4.** Figure 4: (a) Adiabatic index Γ and (b) the square speed of sound (cs/c) 2 as functions of baryon density nb for the four nuclear mass models considered. The dotted horizontal line indicates the Newtonian limit, Γ = 4/3. lattice embedded in an ultra-relativistic electron gas. Small oscillations in Γ reflect the underlying sequence of composition changes. The speed of sound increases monotonically with density and … view at source ↗

**Figure 3.** Figure 3: (a) Proton fraction y = Z/A and (b) pressure P as functions of baryon density nb for the four nuclear mass models considered. Insets emphasize the region near the neutron-drip density. Additional comparisons with the BCPM (Sharma, B. K. et al. 2015), BPS (Baym et al. 1971), and HZD (Haensel et al. 1989) EOSs are also shown, while the relevant deviation from BPS is also plotted. In addition to the pressure… view at source ↗

**Figure 5.** Figure 5: Gravitational mass M as a function of (a) the radius R and (b) central energy density Ec for the four nuclear mass models considered. Circles denote the minimum mass configuration. For comparison, the BPS EOS (Baym et al. 1971) is also shown. be driven toward this stability threshold, beyond which the star becomes unstable and may undergo rapid structural rearrangement (Colpi et al. 1989; Haensel, P. et a… view at source ↗

**Figure 6.** Figure 6: (a) The fraction ∆R/R and (b) the fraction Icr/I as functions of the gravitational mass M for the four considered models. Circles indicate the minimum mass configuration. For comparison, the BPS EOS (Baym et al. 1971) is also shown. prescriptions employed in the present work, showing that the relative differences associated with the outer-crust EOS are preserved under changes in the treatment of the inn… view at source ↗

read the original abstract

The equation of state of the neutron-star outer crust is sensitive to nuclear mass predictions and provides a direct connection to properties of nuclei throughout the nuclide map, including those beyond experimental reach. We quantify the impact of contemporary nuclear mass models on the composition and thermodynamic properties of the outer crust, and assess the consequences for crust-dominated neutron-star configurations near the minimum-mass limit. We constructed four outer-crust equations of state based on the relativistic energy density functional and machine-learning mass model tables. The equilibrium composition of cold catalyzed matter in $\beta$-equilibrium was obtained by minimizing the Gibbs free energy per baryon, and the resulting equations of state were implemented in neutron-star structure calculations. The different mass inputs lead to variations in the equilibrium nuclide sequence, distinct last bound nuclei, and moderate shifts in the neutron-drip density. In contrast, the associated thermodynamic properties, as well as the minimum-mass neutron-star configurations, remain closely aligned across the four outer-crust equations of state. The model dependence of the outer crust is primarily reflected in the detailed nuclide composition and in the precise location of neutron drip. Nevertheless, the considered outer-crust equations of state yield closely consistent predictions for the relevant neutron-star observables, providing a reliable input for stellar modelling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Four modern nuclear mass models produce closely aligned outer-crust EOS and minimum-mass neutron-star configurations, though the limited model set may understate the true spread in nuclear uncertainties.

read the letter

The paper's central result is that four contemporary mass models—relativistic EDF plus machine-learning tables—yield outer-crust equations of state whose thermodynamic properties and minimum-mass neutron-star structures stay quite consistent, even as the exact sequence of nuclei and the neutron-drip density shift modestly. This is a straightforward computational exercise: they minimize Gibbs free energy per baryon in beta equilibrium to fix the composition, build the EOS from those tables, and integrate the Tolman-Oppenheimer-Volkoff equation for the lightest stable stars. The work shows that the pressure-density relation and the minimum mass itself are not very sensitive to which of these four inputs is chosen, which is a practical robustness check for crust modeling. That part is done cleanly and directly addresses a real modeling uncertainty for low-mass neutron stars. The soft spot is the model selection itself. These four tables may share systematic biases in binding energies for neutron-rich species, so the observed alignment could be narrower than the full range of current nuclear-mass uncertainties. The abstract gives no quantitative spreads, error bars, or sensitivity runs, leaving the phrase “closely aligned” somewhat qualitative. The scope is also narrow—outer crust only, minimum-mass configurations only—so it does not speak to maximum-mass stars or other observables. This is the kind of paper neutron-star structure modelers would find useful when they need a defensible crust input without having to rerun everything for every new mass table. It deserves a serious referee because the calculation is transparent, the finding is reproducible from the described workflow, and the result is directly usable even if incremental.

Referee Report

2 major / 3 minor

Summary. The paper constructs four outer-crust equations of state for neutron stars using tables from relativistic energy-density functionals and machine-learning nuclear mass models. Equilibrium compositions in cold, beta-equilibrated matter are obtained via Gibbs free-energy minimization per baryon. The resulting EOS are inserted into neutron-star structure calculations. While the equilibrium nuclide sequences, last bound nuclei, and neutron-drip densities differ among the four inputs, the thermodynamic properties (pressure, energy density) and the minimum-mass neutron-star configurations remain closely aligned. The authors conclude that the outer-crust EOS provides a reliable input for stellar modeling despite nuclear-mass uncertainties.

Significance. If the reported consistency holds under quantitative scrutiny, the work supplies a useful demonstration that outer-crust thermodynamics and minimum-mass configurations are robust to the choice among four contemporary mass models. This reduces one source of systematic uncertainty for crust-dominated neutron-star observables and supplies a concrete, reproducible workflow that can be extended to additional mass tables.

major comments (2)

[Model selection and discussion sections] The central claim that the four EOS 'yield closely consistent predictions' and constitute 'a reliable input' rests on the assertion that the selected relativistic-EDF and ML models adequately sample current nuclear-mass uncertainties for neutron-rich nuclei. No quantitative measure of model spread (e.g., rms deviation in binding energies for A>100, N-Z>40 nuclei) or comparison against the full set of contemporary models is provided; if the four tables share common systematic biases, the observed alignment may underestimate the true uncertainty. This directly affects the strength of the reliability conclusion.
[Results on thermodynamic properties and minimum-mass configurations] The abstract and results state that thermodynamic properties 'remain closely aligned' and minimum-mass configurations are consistent, yet no tables or figures report the actual percentage variations, maximum differences in pressure at fixed density, or error bands on the minimum mass. Without these numbers the quantitative support for 'closely aligned' cannot be assessed.

minor comments (3)

[Methods] Clarify the precise definition of 'last bound nucleus' and how it is identified in the Gibbs-minimization routine; a short algorithmic description or pseudocode would aid reproducibility.
[Composition and neutron-drip results] The neutron-drip density shifts are described as 'moderate'; supplying the numerical values (in g cm^{-3}) for each model in a table would make the statement concrete.
[Computational details] A brief statement on the numerical tolerance used in the Gibbs-free-energy minimization and on the density grid spacing would help readers judge the precision of the reported compositions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important points for strengthening the quantitative support of our conclusions. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The central claim that the four EOS 'yield closely consistent predictions' and constitute 'a reliable input' rests on the assertion that the selected relativistic-EDF and ML models adequately sample current nuclear-mass uncertainties for neutron-rich nuclei. No quantitative measure of model spread (e.g., rms deviation in binding energies for A>100, N-Z>40 nuclei) or comparison against the full set of contemporary models is provided; if the four tables share common systematic biases, the observed alignment may underestimate the true uncertainty. This directly affects the strength of the reliability conclusion.

Authors: We selected these four models specifically to span distinct methodological classes (relativistic EDFs and machine-learning approaches) that are currently among the most advanced for neutron-rich nuclei. We agree that a quantitative measure of spread among them would strengthen the manuscript. In the revision we will add the rms deviations in binding energies for A>100 and N-Z>40 nuclei computed from the four tables, together with a brief discussion of possible shared systematics. A exhaustive comparison against every contemporary mass model lies outside the scope of the present work, but we will clarify that the chosen set is intended as a representative sample of current uncertainty sources rather than a complete survey. revision: yes
Referee: The abstract and results state that thermodynamic properties 'remain closely aligned' and minimum-mass configurations are consistent, yet no tables or figures report the actual percentage variations, maximum differences in pressure at fixed density, or error bands on the minimum mass. Without these numbers the quantitative support for 'closely aligned' cannot be assessed.

Authors: We accept that the current presentation of alignment is largely qualitative. The revised manuscript will include a new table (or supplementary figure) that reports the maximum relative differences in pressure and energy density at fixed baryon density across the four EOS, together with the numerical range of minimum neutron-star masses and their variations. This will supply the concrete quantitative metrics requested and allow readers to judge the degree of consistency directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper takes four external nuclear mass model tables (relativistic EDF and ML) as fixed inputs. It determines equilibrium composition by direct minimization of the Gibbs free energy per baryon in beta equilibrium, constructs the corresponding EOS, and feeds them into standard neutron-star structure calculations. The reported close alignment of thermodynamic properties and minimum-mass configurations is an observed numerical outcome across these independent inputs, not a quantity that reduces by construction to any parameter fitted inside the paper or to a self-referential definition. No load-bearing self-citations, ansatzes, or uniqueness theorems are invoked in the derivation chain that would force the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on pre-existing nuclear mass models and standard domain assumptions for cold catalyzed matter; no new free parameters or invented entities are introduced by the present work.

axioms (2)

domain assumption Equilibrium composition of cold catalyzed matter is obtained by minimizing the Gibbs free energy per baryon in beta equilibrium.
Invoked to determine the nuclide sequence at each density; standard in neutron-star crust modeling.
domain assumption The four chosen relativistic EDF and machine-learning mass tables provide representative nuclear masses for the relevant nuclides.
The paper treats these tables as external inputs without deriving or re-fitting them.

pith-pipeline@v0.9.0 · 5527 in / 1409 out tokens · 41175 ms · 2026-05-12T05:40:54.994817+00:00 · methodology

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