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arxiv: 2604.18136 · v1 · submitted 2026-04-20 · 🌌 astro-ph.HE · astro-ph.SR· nucl-th· physics.plasm-ph

Equation of State for warm Neutron Star outer crusts

David Barba-Gonz\'alez , Conrado Albertus , M. \'Angeles P\'erez-Garc\'ia This is my paper

Pith reviewed 2026-05-10 04:18 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.SRnucl-thphysics.plasm-ph
keywords neutron star outer crustequation of statemolecular dynamicswarm plasmathermal effectsadiabatic indexion screening
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The pith

Even at moderate temperatures, thermal effects from ions shape the equation of state in the higher-density parts of a neutron star outer crust.

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

The paper computes the pressure-density-temperature relation for the warm outer crust of a neutron star by running molecular dynamics simulations of an ion plasma. It starts from a known cold composition of nuclei at each density, then adds thermal motion while treating ions as finite-size Gaussian charges and including electron screening via an Ewald summation technique. The resulting equation of state shows that ion thermal contributions become important near the transition to the inner crust even for temperatures between 1 and 5 MeV, when parametrized by an effective thermal adiabatic index. This matters for neutron star models because the crust influences cooling, oscillations, and the interface with denser layers. Tabulated values and a neural-network fit are supplied to make the results directly usable in astrophysical calculations.

Core claim

Using molecular dynamics simulations of a one-component plasma with the cold composition taken from prior work, the pressure P(n_B, T) is obtained for baryon densities between 7.48 times 10^{-10} and 2.09 times 10^{-4} fm^{-3} and temperatures 1 to 5 MeV. Electron screening and finite-size Gaussian modeling of ions are included through an efficient Ewald summation. The results demonstrate that thermal effects of ions are key in the higher-density region closer to the inner crust when described with a thermal effective parametrization based on the thermal adiabatic index Gamma_th.

What carries the argument

Molecular dynamics simulations of ions modeled as finite-size Gaussian charge distributions with electron screening and Ewald energy summation, applied to a fixed cold one-component plasma composition at each density.

Load-bearing premise

The cold one-component plasma composition remains a valid input when temperature is added and the Gaussian finite-size plus screening model captures the dominant ion interactions.

What would settle it

A comparison of the simulated pressures at the quoted densities and 1-5 MeV temperatures against independent quantum molecular dynamics or path-integral Monte Carlo results that differ by more than the numerical uncertainties reported here would falsify the classical model's adequacy.

Figures

Figures reproduced from arXiv: 2604.18136 by Conrado Albertus, David Barba-Gonz\'alez, M. \'Angeles P\'erez-Garc\'ia.

Figure 1
Figure 1. Figure 1: Energy (from Eq. (12)) as a function of baryonic density for ions. Solid lines correspond to the results of our microscopical simulations, while the ideal gas limit for each given temperature is shown as a dashed line, to showcase the system reaching it at sufficiently small densities. for moderate system sizes. As seen in the Table (1) in the appendix, we span a range of densities comprising the whole out… view at source ↗
Figure 2
Figure 2. Figure 2: Points depict MD results for pressure as a function of baryonic density within ions for different temperatures. Smooth, solid lines are predictions from our neural network prescription, with goodness of fit R 2 > 0.989718 when compared to the simu￾lation’s results. Fig. (1) there are two clear sections in density, we have sepa￾rated the Neural Network in two different sets of parameters trained over the MD… view at source ↗
Figure 4
Figure 4. Figure 4: Thermal index Γth as a function of baryonic density for several selected temperatures, as coming directly from MD. We in￾clude (blue dashed line) the same Γth for an electron gas, to show￾case ionic effects. ϵi(T = 0) = −C1Z 5/3 kF,e; Pi(T = 0) = − nB,i 3 C1 Z 5/3 A kF,e. (20) Here C1 is the Madelung constant, which for the ground state of the ionic system, i.e. a bcc lattice (Barba-González et al. 2022) i… view at source ↗
Figure 5
Figure 5. Figure 5: Pressure versus baryonic density for several equations of state in the literature, together with our data (USALWP) for T = 1, 2 MeV. We include equations of state DD2 at T = 1 MeV (Hempel & Schaffner-Bielich 2010) (green), and cold PCP-BSk24 (Goriely et al. 2013) (red), GMSR-H1 (Grams et al. 2022) (purple), BL-unified (Carreau et al. 2019) (brown); and in pink the EoS whose composition we have utilized (Mu… view at source ↗
read the original abstract

We describe the equation of state (EoS) of a warm ion plasma as obtained by performing microscopic many-body simulations using Molecular Dynamics computational techniques. Using the cold one-component plasma (OCP) composition in the Neutron Star (NS) outer crust assumed in Murarka et al. (2022) with a representative heavy nucleus for each density, we refine previous calculations. We include electron screening and modeling of ions as finite-size Gaussian distributions in the interaction potential, together with an efficient Ewald energy summation procedure. From this, the EoS relation $P(n_B,T)$ is obtained as a function of baryonic density and temperature in the NS outer crust under conditions $n_B\in[7.48\times 10^{-10},2.09\times10^{-4}]$ $ \rm fm^{-3}$ , $k_{B}T\in[1,5]$ MeV. In order to improve the usability of our results we provide tabulated data values along with a neural network parametrization available in the Zenodo repository, see https://zenodo.org/records/15348712. We find that even at moderate temperatures, thermal effects of ions are key in the higher density region closer to the inner crust, when described using a thermal effective parametrization based on the thermal adiabatic index $\Gamma_{th}$. We compare our results with other EoS in the literature performing a critical discussion.

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 / 3 minor

Summary. The manuscript reports molecular dynamics simulations of the equation of state for warm neutron star outer crusts. It adopts the cold one-component plasma compositions (specific Z,A per density) from Murarka et al. (2022), models ions as finite-size Gaussians with electron screening, employs Ewald summation, and computes P(n_B,T) over n_B in [7.48×10^{-10}, 2.09×10^{-4}] fm^{-3} and k_B T in [1,5] MeV. Tabulated data and a neural-network parametrization are provided on Zenodo; the central result is that ion thermal effects remain important at higher densities near the inner crust when the thermal pressure is described via the adiabatic index Γ_th, with comparisons to other literature EoS.

Significance. If the results hold, this supplies a microscopic, simulation-based EoS for the warm outer crust that can be used in neutron-star cooling and merger modeling. Credit is due for the direct MD approach with Ewald summation, the inclusion of finite-size and screening corrections, and especially for the public tabulated data plus neural-network fit, which directly supports reproducibility and downstream applications.

major comments (2)
  1. [Methods section (composition input)] Methods section (composition input): The cold OCP nuclei (Z,A) from Murarka et al. (2022) are fixed as input for all finite-T runs without any reported re-minimization of the Helmholtz free energy at T=1–5 MeV. This assumption is load-bearing for the abstract claim that 'even at moderate temperatures, thermal effects of ions are key in the higher density region... when described using a thermal effective parametrization based on the thermal adiabatic index Γ_th', because thermal contributions can shift the equilibrium nucleus (or favor mixtures) near the inner-crust boundary, altering both Γ and the fractional thermal pressure and thereby changing whether Γ_th still captures the dominant correction.
  2. [Results section (EoS curves and Γ_th)] Results section (EoS curves and Γ_th): No convergence tests with respect to particle number N, simulation cell size, time-step, or Ewald cutoff, and no statistical error estimates on the derived pressures or energies, are presented. This directly affects the quantitative assertion that thermal effects are 'key' at high density, since uncontrolled numerical uncertainties could be comparable to the reported thermal pressure fraction.
minor comments (3)
  1. [Data availability statement] The Zenodo repository link is welcome, but the deposited files should include the exact input parameters, random seeds, and post-processing scripts used to generate the tabulated EoS and the neural-network weights.
  2. [Figure captions] Figure captions for the EoS and Γ_th plots should explicitly state the number of particles, equilibration time, and production run length so that readers can assess the statistical quality of the data.
  3. [Methods (interaction potential)] A short paragraph comparing the Gaussian finite-size model to the point-particle limit (or to more detailed charge distributions) would clarify the systematic uncertainty introduced by the interaction potential.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive review of our manuscript. We address each major comment point by point below, indicating where revisions will be made to strengthen the work.

read point-by-point responses

  1. Referee: Methods section (composition input): The cold OCP nuclei (Z,A) from Murarka et al. (2022) are fixed as input for all finite-T runs without any reported re-minimization of the Helmholtz free energy at T=1–5 MeV. This assumption is load-bearing for the abstract claim that 'even at moderate temperatures, thermal effects of ions are key in the higher density region... when described using a thermal effective parametrization based on the thermal adiabatic index Γ_th', because thermal contributions can shift the equilibrium nucleus (or favor mixtures) near the inner-crust boundary, altering both Γ and the fractional thermal pressure and thereby changing whether Γ_th still captures the dominant correction.

    Authors: We acknowledge that fixing the cold OCP compositions without re-minimizing the Helmholtz free energy at finite temperature is an approximation. Our study focuses on computing the thermal EoS via MD for the established cold compositions to isolate ion thermal effects, rather than performing a full finite-T composition optimization which would require additional free-energy calculations for varying (Z,A) at each point. We will revise the Methods section to explicitly discuss this choice and its limitations, and we will qualify the abstract and relevant claims to note that the reported importance of thermal effects holds for the fixed compositions adopted from the cold case. This addresses the concern without altering the core simulation results. revision: partial

  2. Referee: Results section (EoS curves and Γ_th): No convergence tests with respect to particle number N, simulation cell size, time-step, or Ewald cutoff, and no statistical error estimates on the derived pressures or energies, are presented. This directly affects the quantitative assertion that thermal effects are 'key' at high density, since uncontrolled numerical uncertainties could be comparable to the reported thermal pressure fraction.

    Authors: We agree that explicit convergence tests and error estimates are necessary to support the quantitative claims. In the revised manuscript we will add a dedicated subsection (or appendix) presenting convergence studies varying N, cell size, time-step, and Ewald parameters, along with statistical uncertainties on P and energy obtained via block averaging over independent runs. These additions will directly bolster the assertion regarding the importance of thermal effects at higher densities. revision: yes

Circularity Check

0 steps flagged

No circularity: EoS derived from independent MD simulations

full rationale

The paper obtains P(n_B, T) directly from molecular dynamics simulations of the ion plasma with fixed cold OCP composition taken as an external input from a non-overlapping citation (Murarka et al. 2022). The central claim that ion thermal effects remain important at higher densities (via Γ_th parametrization) follows from those simulation outputs rather than any algebraic reduction, self-referential fit, or redefinition of the input composition. No load-bearing step equates a derived quantity to its own inputs by construction; the derivation is self-contained against the stated microscopic model.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The calculation rests on the cold OCP composition taken from an external reference and on standard plasma-physics modeling choices for the ion-ion potential; no new particles or forces are introduced.

free parameters (1)
  • Ion composition per density bin
    Representative heavy nucleus chosen for each baryon density from Murarka et al. (2022)
axioms (2)
  • domain assumption The one-component plasma with a single representative nucleus per density slice adequately represents the outer-crust composition even at finite temperature.
    Used as fixed input for all MD runs.
  • domain assumption Gaussian charge distributions plus linear electron screening capture the dominant short-range ion interactions.
    Built into the pair potential for the simulations.

pith-pipeline@v0.9.0 · 5575 in / 1355 out tokens · 58398 ms · 2026-05-10T04:18:52.217673+00:00 · methodology

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Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    P., Abbott, R., Abbott, T

    Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017a, The Astrophysical Journal Letters, 848, L12, doi: 10.3847/2041-8213/aa91c9 —. 2017b, The Astrophysical Journal Letters, 848, L12, doi: 10.3847/2041-8213/aa91c9 —. 2020, The Astrophysical Journal Letters, 892, L3, doi: 10.3847/2041-8213/ab75f5

    work page doi:10.3847/2041-8213/aa91c9 2041

  2. [2]

    H., W., et al

    Audi, G., M., W., A. H., W., et al. 2012, Chinese Physics C, 36, 002, doi: 10.1088/1674-1137/36/12/002 Barba-González, D., Albertus, C., & Pérez-García, M. A. 2022, Phys. Rev. C, 106, 065806, doi: 10.1103/PhysRevC.106.065806 Barba-González, D., Albertus, C., & Pérez-García, M. A. 2024, Monthly Notices of the Royal Astronomical Society, 528, 3498, doi: 10....

    work page doi:10.1088/1674-1137/36/12/002 2012

  3. [3]

    A., & Pethick, C

    Baym, G., Bethe, H. A., & Pethick, C. J. 1971, Nuclear Physics A, 175, 225, doi: https://doi.org/10.1016/0375-9474(71)90281-8 10

    work page doi:10.1016/0375-9474(71)90281-8 1971

  4. [4]

    Burrows, A., & Lattimer, J. M. 1984, ApJ, 285, 294, doi: 10.1086/162505

    work page doi:10.1086/162505 1984

  5. [5]

    E., Forsman, C

    Caplan, M. E., Forsman, C. R., & Schneider, A. S. 2021, PhRvC, 103, 055810, doi: 10.1103/PhysRevC.103.055810

    work page doi:10.1103/physrevc.103.055810 2021

  6. [6]

    2019, The European Physical Journal A, 55, 188, doi: 10.1140/epja/i2019-12884-1

    Carreau, T., Gulminelli, F., & Margueron, J. 2019, The European Physical Journal A, 55, 188, doi: 10.1140/epja/i2019-12884-1

    work page doi:10.1140/epja/i2019-12884-1 2019

  7. [7]

    2007, Phys

    Chamel, N., Naimi, S., Khan, E., & Margueron, J. 2007, Phys. Rev. C, 75, 055806, doi: 10.1103/PhysRevC.75.055806

    work page doi:10.1103/physrevc.75.055806 2007

  8. [8]

    2024, Annual Review of Nuclear and Particle Science, 74, 141, doi: https://doi.org/10.1146/annurev-nucl-121423-091501

    Crawford, H., Fossez, K., König, S., & Spyrou, A. 2024, Annual Review of Nuclear and Particle Science, 74, 141, doi: https://doi.org/10.1146/annurev-nucl-121423-091501

    work page doi:10.1146/annurev-nucl-121423-091501 2024

  9. [9]

    2024, A&A, 687, A236, doi: 10.1051/0004-6361/202450305 Dinh Thi, H., Fantina, A

    Dehman, C., Centelles, M., & Viñas, X. 2024, A&A, 687, A236, doi: 10.1051/0004-6361/202450305 Dinh Thi, H., Fantina, A. F., & Gulminelli, F. 2023, A&A, 677, A174, doi: 10.1051/0004-6361/202346606

    work page doi:10.1051/0004-6361/202450305 2024

  10. [10]

    2018, Phys

    Endrizzi, A., Logoteta, D., Giacomazzo, B., et al. 2018, Phys. Rev. D, 98, 043015, doi: 10.1103/PhysRevD.98.043015

    work page doi:10.1103/physrevd.98.043015 2018

  11. [11]

    Goriely, S., Chamel, N., & Pearson, J. M. 2013, Phys. Rev. C, 88, 024308, doi: 10.1103/PhysRevC.88.024308

    work page doi:10.1103/physrevc.88.024308 2013

  12. [13]

    2024, European Physical Journal A, 60, 90, doi: 10.1140/epja/s10050-024-01309-3

    Grams, G., & Margueron, J. 2024, European Physical Journal A, 60, 90, doi: 10.1140/epja/s10050-024-01309-3

    work page doi:10.1140/epja/s10050-024-01309-3 2024

  13. [14]

    2022, The European Physical Journal A, 58, 56, doi: 10.1140/epja/s10050-022-00706-w

    Grams, G., Margueron, J., Somasundaram, R., & Reddy, S. 2022, The European Physical Journal A, 58, 56, doi: 10.1140/epja/s10050-022-00706-w

    work page doi:10.1140/epja/s10050-022-00706-w 2022

  14. [15]

    Guo, L., Hempel, M., Schaffner-Bielich, J., & Maruhn, J. A. 2007, Phys. Rev. C, 76, 065801, doi: 10.1103/PhysRevC.76.065801

    work page doi:10.1103/physrevc.76.065801 2007

  15. [16]

    2016, Physical Review C, 94, doi: 10.1103/physrevc.94.025805

    Harutyunyan, A., & Sedrakian, A. 2016, Physical Review C, 94, doi: 10.1103/physrevc.94.025805

    work page doi:10.1103/physrevc.94.025805 2016

  16. [17]

    2010, NuPhA, 837, 210, doi: 10.1016/j.nuclphysa.2010.02.010

    Hempel, M., & Schaffner-Bielich, J. 2010, NuPhA, 837, 210, doi: 10.1016/j.nuclphysa.2010.02.010

    work page doi:10.1016/j.nuclphysa.2010.02.010 2010

  17. [18]

    C., Richard, R

    Holden, Z. C., Richard, R. M., & Herbert, J. M. 2013, JChPh, 139, 244108, doi: 10.1063/1.4850655

    work page doi:10.1063/1.4850655 2013

  18. [19]

    J., Pérez-García, M

    Horowitz, C. J., Pérez-García, M. A., Berry, D. K., & Piekarewicz, J. 2005, Phys. Rev. C, 72, 035801, doi: 10.1103/PhysRevC.72.035801

    work page doi:10.1103/physrevc.72.035801 2005

  19. [20]

    2004, Phys

    Piekarewicz, J. 2004, Phys. Rev. C, 70, 065806, doi: 10.1103/PhysRevC.70.065806

    work page doi:10.1103/physrevc.70.065806 2004

  20. [21]

    2003, Nuclear Physics A, 723, 517, doi: https://doi.org/10.1016/S0375-9474(03)01324-1

    Ishizuka, C., Ohnishi, A., & Sumiyoshi, K. 2003, Nuclear Physics A, 723, 517, doi: https://doi.org/10.1016/S0375-9474(03)01324-1

    work page doi:10.1016/s0375-9474(03)01324-1 2003

  21. [22]

    2024, Artificial Intelligence Review, 57, 256, doi: 10.1007/s10462-024-10874-4

    Jiao, L., Song, X., You, C., et al. 2024, Artificial Intelligence Review, 57, 256, doi: 10.1007/s10462-024-10874-4

    work page doi:10.1007/s10462-024-10874-4 2024

  22. [23]

    2013, International Journal of Mass Spectrometry, 349-350, 63, doi: https://doi.org/10.1016/j.ijms.2013.02.015

    Kreim, S., Hempel, M., Lunney, D., & Schaffner-Bielich, J. 2013, International Journal of Mass Spectrometry, 349-350, 63, doi: https://doi.org/10.1016/j.ijms.2013.02.015

    work page doi:10.1016/j.ijms.2013.02.015 2013

  23. [24]

    , keywords =

    Levan, A. J., Gompertz, B. P., Salafia, O. S., et al. 2024, Nature, 626, 737, doi: 10.1038/s41586-023-06759-1

    work page doi:10.1038/s41586-023-06759-1 2024

  24. [25]

    P., & Providência, C

    Menezes, D. P., & Providência, C. 1999, Phys. Rev. C, 60, 024313, doi: 10.1103/PhysRevC.60.024313

    work page doi:10.1103/physrevc.60.024313 1999

  25. [26]

    2022, Journal of Cosmology and Astroparticle Physics, 2022, 045, doi: 10.1088/1475-7516/2022/01/045

    Murarka, U., Banerjee, K., Malik, T., & Providência, C. 2022, Journal of Cosmology and Astroparticle Physics, 2022, 045, doi: 10.1088/1475-7516/2022/01/045

    work page doi:10.1088/1475-7516/2022/01/045 2022

  26. [27]

    V., Rosswog S., Ujevic M., 2023, @doi [ ] 10.1103/PhysRevD.107.023016 , https://ui.adsabs.harvard.edu/abs/2023PhRvD.107b3016N 107, 023016

    Neuweiler, A., Dietrich, T., Bulla, M., et al. 2023, PhRvD, 107, 023016, doi: 10.1103/PhysRevD.107.023016

    work page doi:10.1103/physrevd.107.023016 2023

  27. [28]

    G., Cantu, S., Wang, S., et al

    Newton, W. G., Cantu, S., Wang, S., et al. 2022, Phys. Rev. C, 105, 025806, doi: 10.1103/PhysRevC.105.025806

    work page doi:10.1103/physrevc.105.025806 2022

  28. [29]

    Á., & Providência, C

    Pais, H., Albertus, C., Pérez-García, M. Á., & Providência, C. 2023, A&A, 679, A113, doi: 10.1051/0004-6361/202347496

    work page doi:10.1051/0004-6361/202347496 2023

  29. [30]

    , author Chamel, N

    Pearson, J. M., Chamel, N., Potekhin, A. Y ., et al. 2018, Monthly Notices of the Royal Astronomical Society, 481, 2994, doi: 10.1093/mnras/sty2413

    work page doi:10.1093/mnras/sty2413 2018

  30. [31]

    K., & Cescutti, G

    Perego, A., Thielemann, F. K., & Cescutti, G. 2021, r-Process Nucleosynthesis from Compact Binary Mergers (Springer Singapore), 1–56, doi: 10.1007/978-981-15-4702-7_13-1 P˛ ecak, D., Zdanowicz, A., Chamel, N., Magierski, P., & Wlazłowski, G. 2024, Phys. Rev. X, 14, 041054, doi: 10.1103/PhysRevX.14.041054 Pérez-García, M. Á., Tsushima, K., & Valcarce, A. 2...

    work page doi:10.1007/978-981-15-4702-7_13-1 2021

  31. [32]

    and Bernuzzi, Sebastiano and Roberts, Luke F

    Radice, D., Perego, A., Hotokezaka, K., et al. 2018, The Astrophysical Journal, 869, 130, doi: 10.3847/1538-4357/aaf054

    work page doi:10.3847/1538-4357/aaf054 2018

  32. [33]

    R., Nacu, F., & Oertel, M

    Raduta, A. R., Nacu, F., & Oertel, M. 2021, The European Physical Journal A, 57, 329, doi: 10.1140/epja/s10050-021-00628-z

    work page doi:10.1140/epja/s10050-021-00628-z 2021

  33. [34]

    2008, PhRvC, 78, 025807, doi: 10.1103/PhysRevC.78.025807

    Roca-Maza, X., & Piekarewicz, J. 2008, PhRvC, 78, 025807, doi: 10.1103/PhysRevC.78.025807

    work page doi:10.1103/physrevc.78.025807 2008

  34. [35]

    , keywords =

    Ronchini, S., Branchesi, M., Oganesyan, G., et al. 2022, A&A, 665, A97, doi: 10.1051/0004-6361/202243705

    work page doi:10.1051/0004-6361/202243705 2022

  35. [36]

    AnP , keywords =

    Rosswog, S., & Korobkin, O. 2024, Annalen der Physik, 536, 2200306, doi: https://doi.org/10.1002/andp.202200306 Rüster, S. B., Hempel, M., & Schaffner-Bielich, J. 2006, Phys. Rev. C, 73, 035804, doi: 10.1103/PhysRevC.73.035804

    work page doi:10.1002/andp.202200306 2024

  36. [37]

    2015, Physics of Particles and Nuclei, 46, 633, doi: 10.1134/S1063779615040061

    Typel, S., Oertel, M., & Klähn, T. 2015, Physics of Particles and Nuclei, 46, 633, doi: 10.1134/S1063779615040061

    work page doi:10.1134/s1063779615040061 2015

  37. [38]

    Typel, S., Röpke, G., Klähn, T., Blaschke, D., & Wolter, H. H. 2010, PhRvC, 81, 015803, doi: 10.1103/PhysRevC.81.015803

    work page doi:10.1103/physrevc.81.015803 2010

  38. [39]

    G., Kaminker, A

    Yakovlev, D. G., Kaminker, A. D., Potekhin, A. Y ., & Haensel, P. 2020, Monthly Notices of the Royal Astronomical Society, 500, 4491–4505, doi: 10.1093/mnras/staa3547 11 APPENDIX T (MeV) ϵ MeV f m3 Pi MeV f m3 T (MeV) ϵ MeV f m3 Pi MeV f m3 Z An B,i f m−3 1 1.9005×10 −11 1.3022×10 −11 2 3.9410×10 −11 2.6348×10 −11 26 56 7.4878×10 −10 1 1.1747×10 −10 8.677...

    work page doi:10.1093/mnras/staa3547 2020