The reviewed record of science sign in
Pith

arxiv: 2605.26692 · v1 · pith:C3R637R5 · submitted 2026-05-26 · astro-ph.HE · nucl-th

From supernovae to neutron stars: crust formation time

Yudai Suwa (U. Tokyo & YITP) , Ken'ichiro Nakazato (Kyushu U.) This is my paper

Reviewed by Pith2026-06-29 16:04 UTCgrok-4.3pith:C3R637R5open to challenge →

classification astro-ph.HE nucl-th
keywords protoneutron starneutron star crustcrust formation timeneutrino coolingCoulomb crystallizationanalytic estimatesupernova
0
0 comments X

The pith

A simple diffusion model gives closed expressions for when a neutron star crust first forms, typically 100 to 500 seconds after birth.

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

The paper develops a simple analytic estimate for the onset time of crust formation during the late post-convective cooling phase of a protoneutron star. It combines a diffusion-based neutrino luminosity with entropy evolution and an approximately isentropic interior structure to find the time-dependent density and temperature at the neutrinosphere. The Coulomb crystallization condition is then imposed to mark when the temperature falls below the threshold at that density. This produces closed-form expressions for the entropy at crystallization and the crust-formation time that depend explicitly on protoneutron star mass and radius, a cooling normalization, and composition parameters such as ionic charge and heavy-nuclei fraction. A reader would care because the expressions supply a quick, parameter-dependent benchmark that does not require full numerical simulations.

Core claim

Using a diffusion-based neutrino luminosity and the resulting entropy evolution together with an approximately isentropic interior structure, the time-dependent density and temperature at the neutrinosphere are obtained. Imposing the Coulomb crystallization condition for heavy nuclei then determines when the neutrinosphere temperature first falls below the crystallization threshold, yielding closed expressions for the entropy at crystallization and the crust-formation time with explicit dependence on PNS mass and radius, diffusion normalization, and composition parameters such as Z and heavy-nuclei mass fraction. For canonical microphysics, the first solid phase typically appears at t_crust

What carries the argument

The Coulomb crystallization condition expressed through the Coulomb coupling parameter, applied at the neutrinosphere density and temperature obtained from the diffusion luminosity and isentropic structure.

If this is right

  • The crust-formation time depends explicitly on the protoneutron star mass and radius.
  • The time depends on an effective diffusion or cooling normalization.
  • The time depends on composition parameters such as ionic charge Z and heavy-nuclei mass fraction.
  • For canonical microphysics the first solid phase appears at t_crust ~ 100-500 s.

Where Pith is reading between the lines

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

  • The analytic scalings could be inserted directly into population-synthesis codes to explore how crust onset varies across different supernova progenitors.
  • Early-time neutrino or gravitational-wave observations of young neutron stars might eventually be compared against the predicted 100-500 s window.
  • The closed expressions make it straightforward to test how changes in the assumed cooling normalization shift the crystallization epoch.

Load-bearing premise

The model assumes a diffusion-based neutrino luminosity together with an approximately isentropic interior structure to obtain the time-dependent density and temperature at the neutrinosphere.

What would settle it

A numerical simulation of protoneutron star cooling with the same canonical microphysics that finds the first solid phase forming at a time well outside the 100-500 s window would falsify the analytic estimate.

Figures

Figures reproduced from arXiv: 2605.26692 by Ken'ichiro Nakazato (Kyushu U.), Yudai Suwa (U. Tokyo & YITP).

Figure 1
Figure 1. Figure 1: Temperature–density trajectories during the late PNS cooling phase and the Coulomb crystallization threshold. Solid curves show the model temperature profile T(ρ) at t = 50, 100, and 200 s for the fiducial parameters MPNS = 1.4M⊙, RPNS = 12 km, Etot = 1053 erg, and gβ = 1.6. For each epoch, the interior follows the isentropic relation (Eq. 7), while the outer atmosphere is approximated as isothermal at Tν … view at source ↗
read the original abstract

A neutron star is born as a hot, lepton-rich protoneutron star (PNS) and cools via neutrino emission, eventually allowing heavy ions in the outer layers to crystallize into a solid crust. We develop a simple analytic estimate for the onset time of this crust formation during the late, post-convective PNS cooling phase. Using a diffusion-based neutrino luminosity and the resulting entropy evolution together with an approximately isentropic interior structure, we obtain the time-dependent density and temperature at the neutrinosphere. We then impose the Coulomb crystallization condition for heavy nuclei, expressed through the Coulomb coupling parameter, and determine when the neutrinosphere temperature first falls below the crystallization threshold evaluated at the neutrinosphere density. This procedure yields closed expressions for the entropy at crystallization and the corresponding crust-formation time, with explicit dependence on the PNS mass and radius, an effective diffusion/cooling normalization, and composition parameters such as the ionic charge $Z$ and heavy-nuclei mass fraction. For canonical microphysics, we find that the first solid phase typically appears at $t_{\mathrm{crust}}\sim 100$-$500\,\mathrm{s}$. These closed-form scalings provide a useful late-time analytic benchmark for the onset of crust formation and clarify its dependence on PNS and composition parameters.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a simple analytic estimate for the onset time of crust formation in a cooling protoneutron star during the late post-convective phase. It adopts a diffusion-based neutrino luminosity to evolve entropy, assumes an approximately isentropic interior to obtain time-dependent density and temperature at the neutrinosphere, and imposes the Coulomb crystallization condition (Γ_Coulomb=175) to determine when the first solid phase appears. This yields closed expressions for the entropy at crystallization and the crust-formation time t_crust, with explicit dependence on PNS mass and radius, an effective diffusion/cooling normalization, and composition parameters such as ionic charge Z and heavy-nuclei mass fraction, giving t_crust ~100-500 s for canonical microphysics.

Significance. If the modeling assumptions hold, the closed-form scalings constitute a useful late-time analytic benchmark for crust formation onset and its parametric dependence. The explicit expressions are a strength that can facilitate direct comparison with numerical PNS cooling simulations.

major comments (2)
  1. [Abstract (derivation of neutrinosphere T(ρ,t))] Abstract and main derivation: the time-dependent neutrinosphere density and temperature are obtained by mapping global entropy (from diffusion luminosity) to local conditions under the isentropic-interior assumption; this mapping is load-bearing for the t_crust expressions but is not derived from the neutrino transport equation, and no quantification of shifts arising from residual deleptonization gradients or convective remnants is provided.
  2. [Abstract (parameter dependence)] Abstract: the effective diffusion/cooling normalization is an input parameter (not derived from first principles within the paper) that directly controls the output t_crust; this introduces a circularity burden for the central claim of providing closed expressions as a benchmark, and no sensitivity tests or error estimates on this parameter are reported.
minor comments (1)
  1. [Abstract] The abstract states the result for 'canonical microphysics' but does not list the precise numerical values adopted for the free parameters when quoting the 100-500 s range.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the positive assessment of the analytic expressions as a potential benchmark. We address each major comment below, with revisions proposed where the points identify genuine gaps in the current presentation.

read point-by-point responses

  1. Referee: Abstract and main derivation: the time-dependent neutrinosphere density and temperature are obtained by mapping global entropy (from diffusion luminosity) to local conditions under the isentropic-interior assumption; this mapping is load-bearing for the t_crust expressions but is not derived from the neutrino transport equation, and no quantification of shifts arising from residual deleptonization gradients or convective remnants is provided.

    Authors: The isentropic-interior mapping is an explicit modeling assumption adopted for the late post-convective phase, where the manuscript states that convection has subsided. It is not derived from the neutrino transport equation because the work aims at closed-form scalings rather than a full numerical solution of the transport problem. We agree that residual deleptonization or convective remnants could introduce shifts, but quantifying those shifts would require direct comparison against detailed transport simulations, which lies outside the scope of an analytic estimate. In revision we will add an expanded caveats subsection that (i) reiterates the assumption and its regime of applicability, (ii) cites existing literature on the timescale when the PNS interior becomes approximately isentropic, and (iii) notes that any residual gradients would most likely produce only modest corrections to the reported 100–500 s window. revision: partial

  2. Referee: Abstract: the effective diffusion/cooling normalization is an input parameter (not derived from first principles within the paper) that directly controls the output t_crust; this introduces a circularity burden for the central claim of providing closed expressions as a benchmark, and no sensitivity tests or error estimates on this parameter are reported.

    Authors: The normalization is indeed an effective parameter that encodes the integrated diffusion physics and is not derived ab initio in the paper. Its value directly sets the absolute scale of t_crust, so the concern about circularity for a benchmark claim is valid. To address it we will insert a new sensitivity subsection that varies the normalization over a factor-of-two range around the fiducial value (consistent with the spread seen in published cooling calculations) and tabulates the resulting range in t_crust. This will supply explicit error estimates and make the benchmark utility more transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; analytic expressions derived from explicitly stated approximations

full rationale

The paper states its inputs upfront (diffusion-based neutrino luminosity, approximately isentropic interior, Coulomb crystallization threshold Γ=175) and derives closed-form expressions for entropy at crystallization and t_crust in terms of PNS mass/radius, the effective normalization, Z, and heavy-nuclei fraction. No step reduces by construction to a fitted parameter renamed as prediction, no self-citation chain is load-bearing for the central claim, and no ansatz is smuggled via prior work. The ~100-500 s range is presented as the outcome for canonical choices of those inputs, not as an independent first-principles result. The derivation chain is therefore self-contained against the paper's own stated assumptions.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The model rests on standard domain assumptions in protoneutron-star cooling and nuclear crystallization; no new entities are introduced.

free parameters (3)
  • effective diffusion/cooling normalization
    Introduced to parameterize neutrino luminosity in the diffusion approximation.
  • ionic charge Z
    Composition parameter entering the Coulomb coupling condition.
  • heavy-nuclei mass fraction
    Composition parameter entering the Coulomb coupling condition.
axioms (3)
  • domain assumption diffusion-based neutrino luminosity governs late-time cooling
    Used to obtain time-dependent temperature and density at the neutrinosphere.
  • domain assumption approximately isentropic interior structure
    Assumed to relate entropy evolution to neutrinosphere conditions.
  • domain assumption Coulomb crystallization condition for heavy nuclei
    Imposed to mark the onset of solid crust formation.

pith-pipeline@v0.9.1-grok · 5778 in / 1499 out tokens · 42617 ms · 2026-06-29T16:04:27.465879+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Bethe, H. A. 1990, Reviews of Modern Physics, 62, 801

    1990

  2. [2]

    Burrows, A., & Lattimer, J. M. 1986, The Astrophysical Journ al, 307, 178

    1986

  3. [3]

    2008, Living Reviews in Relativity, 11, 10 Dinh Thi, H., Fantina, A

    Chamel, N., & Haensel, P . 2008, Living Reviews in Relativity, 11, 10 Dinh Thi, H., Fantina, A. F., & Gulminelli, F. 2023a, Astrono my and Astrophysics, 677, A174 —. 2023b, Astronomy and Astrophysics, 672, A160

    2008

  4. [4]

    C., Mezzacappa, A., Thielemann , F.-K., & Liebendörfer, M

    Fischer, T., Whitehouse, S. C., Mezzacappa, A., Thielemann , F.-K., & Liebendörfer, M. 2010, Astronomy and Astrophysics, 517, A8 0

    2010

  5. [5]

    Y ., & Y akovlev, D

    Haensel, P ., Potekhin, A. Y ., & Y akovlev, D. G. 2007, Astroph ysics and Space Science Library, V ol. 326, Neutron Stars 1: Equation o f State and Structure (Springer) Hüdepohl, L., Müller, B., Janka, H.-T., Marek, A., & Raffelt , G. G. 2010, Physical Review Letters, 104, 251101

    2007

  6. [6]

    2012, Annual Review of Nuclear and Particle Sci ence, 62, 407 —

    Janka, H.-T. 2012, Annual Review of Nuclear and Particle Sci ence, 62, 407 —. 2025, Annual Review of Nuclear and Particle Science, 75, 4 25

    2012

  7. [7]

    2016, Rivista del Nuovo C imento, 39, 1

    Mirizzi, A., Tamborra, I., Janka, H.-T., Saviano, N., Schol berg, K., Bollig, R., Hüdepohl, L., & Chakraborty, S. 2016, Rivista del Nuovo C imento, 39, 1

    2016

  8. [8]

    2020, Monthly Notices of the Royal Astronomical Society, 492, 5764

    Nagakura, H., Burrows, A., Radice, D., & V artanyan, D. 2020, Monthly Notices of the Royal Astronomical Society, 492, 5764

    2020

  9. [9]

    2013, The Astrophysical Journal Supplement Series, 205, 2

    Nakazato, K., Sumiyoshi, K., Suzuki, H., Totani, T., Umeda, H., & Y amada, S. 2013, The Astrophysical Journal Supplement Series, 205, 2

    2013

  10. [10]

    2018, Physical Revie w C, 97, 035804

    Nakazato, K., Suzuki, H., & Togashi, H. 2018, Physical Revie w C, 97, 035804

    2018

  11. [11]

    2025, Progress of Theoretical and Experimental Physics, 2025, 123E01

    Nakazato, K., Togashi, H., Sumiyoshi, K., & Suzuki, H. 2025, Progress of Theoretical and Experimental Physics, 2025, 123E01

    2025

  12. [12]

    2022, Monthly Notices of t he Royal Astronomical Society, 511, 356

    Pascal, A., Novak, J., & Oertel, M. 2022, Monthly Notices of t he Royal Astronomical Society, 511, 356

    2022

  13. [13]

    A., Reddy, S., Prakash, M., Lattimer, J

    Pons, J. A., Reddy, S., Prakash, M., Lattimer, J. M., & Mirall es, J. A. 1999, The Astrophysical Journal, 513, 780

    1999

  14. [14]

    A., & Viganò, D

    Pons, J. A., & Viganò, D. 2019, Living Reviews in Computation al Astrophysics, 5, 3

    2019

  15. [15]

    F., Shen, G., Cirigliano, V ., Pons, J

    Roberts, L. F., Shen, G., Cirigliano, V ., Pons, J. A., Reddy, S., & Woosley, S. E. 2012, Physical Review Letters, 108, 061103

    2012

  16. [16]

    An Exploration of the Equation of State Dependence of Core-Collapse Supernova Explosion Outcomes and Signatures

    Rusakov, A., Burrows, A. S., Wang, T., & V artanyan, D. 2026, arXiv e-prints, arXiv:2602.09025

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  17. [17]

    L., & Teukolsky, S

    Shapiro, S. L., & Teukolsky, S. A. 1983, Black Holes, White Dw arfs, and Neutron Stars: The Physics of Compact Objects (New Y ork: Wil ey- Interscience)

    1983

  18. [18]

    2014, Publications of the Astronomical Society of J apan, 66, L1

    Suwa, Y . 2014, Publications of the Astronomical Society of J apan, 66, L1

    2014

  19. [19]

    2021, Pro gress of Theoretical and Experimental Physics, 2021, 013E01

    Suwa, Y ., Harada, A., Nakazato, K., & Sumiyoshi, K. 2021, Pro gress of Theoretical and Experimental Physics, 2021, 013E01

    2021

  20. [20]

    Suwa, Y ., Sumiyoshi, K., Nakazato, K., Takahira, Y ., Koshio, Y ., Mori, M., & Wendell, R. A. 2019, The Astrophysical Journal, 881, 139 Y amada, S., et al. 2024, Proceedings of the Japan Academy, Se ries B, 100, 190

    2019