# Unified Equation of State for Neutron Stars Based on the Gogny Interaction

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Gogny Interactions Adapted for Astrophysical Calculations

## 3. Neutron Star Crust with Gogny Forces

#### 3.1. Variational Wigner-Kirkwood Method in Finite Nuclei

#### 3.2. Shell and Pairing Effects

#### 3.3. Outer Crust

#### 3.4. Inner Crust

#### 3.5. Core–Crust Transition

## 4. Global Properties of Neutron Stars Predicted by Gogny Forces

#### 4.1. The Tolman–Oppenheimer–Volkov Equations

#### 4.2. Moment of Inertia

#### 4.3. Tidal Deformability

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Differences between the computed and the experimental binding energies of 620 even–even nuclei. Theoretical calculations are performed with the Gogny D1S (

**left panel**) and D1M (

**right panel**) interactions. The experimental values are taken from [11].

**Figure 2.**Equation of state (total pressure in logarithmic scale against baryon density) for neutron star matter computed with the D1M${}^{*}$, D1M, D1N and D1S Gogny interactions, with the BCPM energy density functional and with the SLy4 and BSk22 Skyrme forces. Constraints coming from collective flow in heavy-ion collisions are also included [25].

**Figure 3.**Symmetry energy, defined as Equation (7) with $k=1$ against the baryon density predicted by the D1M${}^{*}$, D1M, D1S and D1N Gogny interactions, the BCPM energy density functional and the SLy4 and BSk22 Skyrme forces. Some constraints coming from isobaric analog states (IAS) (green), from IAS plus neutron skins (IAS + n.skin) (yellow), electric dipole polarizability ${\alpha}_{D}$ in ${}^{208}$Pb (${\alpha}_{D}$ in ${}^{208}$Pb) (dashed red) and heavy-ion collisions (dashed blue) are also included [26,27,28].

**Figure 5.**Difference between the binding energies provided by the D1M${}^{*}$ and D1M force $\Delta B$ (in MeV) plotted as a function of the shifted neutron number N-${N}_{0}$ for isotopic chains covering the periodic table. The values of the atomic number Z and neutron reference number ${N}_{0}$ are given in each panel. The vertical scale covers from +3.5 MeV to −3.5 MeV, with long ticks every MeV and short ticks every half MeV. The horizontal line in each panel at $\Delta B$ = 0 is plotted guide the eye.

**Figure 6.**Neutron numbers N and proton numbers Z for the outer crust of NSs with the experimental masses from the AME2016 [78] tabulation plus the recently measured masses of ${}^{75-79}$Cu [79] aided by theoretical HFB calculations when experimental values are not available, using the D1M and D1M${}^{*}$ Gogny forces and the BCPM energy density functional.

**Figure 7.**Binding energy per nucleon excluding the bare nucleon mass as a function of proton numbers at different average densities ${\rho}_{av}$ of the inner crust calculated with D1M* Gogny interaction.

**Figure 8.**Neutron and proton density distribution inside the Wigner–Seitz cells obtained with variational Wigner–Kirkwood method at different average densities ${\rho}_{av}$ obtained with D1M* Gogny interaction.

**Figure 9.**Transition density (left panels) and transition pressure (right panels) against the slope of the symmetry energy computed for some Skyrme and Gogny interactions. The upper panels correspond to the values obtained using the thermodynamical method whereas the lower panels display the results extracted using the dynamical method.

**Figure 10.**

**Left**: Unified EoS computed with the the D1M and D1M* Gogny force and with the BCPM energy density functional.

**Right**: Particle fractions and the proton fraction corresponding to the onset of the direct Urca (DU) process (see text for details) as functions of the nucleonic density from the D1M and D1M* interactions.

**Figure 11.**Mass-radius relation obtained using the D1M${}^{*}$ and the D1M Gogny forces and the BCPM energy density functional. Constraints from the measurements of $M\approx 2{M}_{\odot}$ (yellow and grey) [14,15], from cooling tails of type-I X-ray bursts in three low-mass X-ray binaries and a Bayesian analysis (green) [92], from five quiescent low-mass X-ray binaries and five photospheric radius expansion X-ray bursters after a Bayesian analysis (blue) [93] and from a Bayesian analysis with the data from the GW170817 detection of gravitational waves from a binary NS merger (red) [94] are shown. Finally, the very recent constraints coming from the NICER mission are also included [95,96].

**Figure 12.**Crustal mass (

**left**), crustal radius (

**center**), and crustal fraction of the moment of inertia ($\Delta {I}_{\mathrm{crust}}/I$) (

**right**) obtained with the D1M${}^{*}$, D1M and BCPM interactions.

**Figure 13.**Left: Total moment of inertia against the total mass of neutron stars computed using the D1M${}^{*}$ and D1M Gogny forces and the BCPM energy density functional. The constraint proposed in [104] is also displayed. Right: Mass weighted tidal deformability (for symmetric binaries) against the chirp mass of binary neutron star systems obtained using the same interactions as in the left panel. The constraint for $\tilde{\Lambda}$ coming from the GW170817 event is also included [109,110].

**Table 1.**Parameters of the D1M, D1M${}^{*}$ and D1M${}^{**}$ Gogny forces. The coefficients ${W}_{i}$, ${B}_{i}$, ${H}_{i}$ and ${M}_{i}$ are given in MeV, ${\mu}_{i}$ in fm and ${t}_{3}$ in MeV fm${}^{4}$. The values of the other parameters of the modified interactions are the same as in the D1M force (namely, ${x}_{3}$ = 1, $\alpha $ = 1/3 and ${W}_{LS}=115.36$ MeV fm${}^{5}$).

D1M | ${W}_{i}$ | ${B}_{i}$ | ${H}_{i}$ | ${M}_{i}$ | ${\mu}_{i}$ |

i = 1 | −12,797.57 | 14,048.85 | −15,144.43 | 11,963.81 | 0.50 |

i = 2 | 490.95 | −752.27 | 675.12 | −693.57 | 1.00 |

${t}_{3}$ | ${x}_{3}$ | $\alpha $ | ${W}_{LS}$ | ||

1562.22 | 1 | 1/3 | 115.36 | ||

D1M${}^{*}$ | ${W}_{i}$ | ${B}_{i}$ | ${H}_{i}$ | ${M}_{i}$ | ${\mu}_{i}$ |

i = 1 | −17,242.0144 | 19,604.4056 | −20,699.9856 | 16,408.3344 | 0.50 |

i = 2 | 675.3860 | −982.8150 | 905.6650 | −878.0060 | 1.00 |

${t}_{3}$ | ${x}_{3}$ | $\alpha $ | ${W}_{LS}$ | ||

1561.22 | 1 | 1/3 | 115.36 | ||

D1M${}^{**}$ | ${W}_{i}$ | ${B}_{i}$ | ${H}_{i}$ | ${M}_{i}$ | ${\mu}_{i}$ |

i = 1 | −15,019.7922 | 16,826.6278 | −17,922.2078 | 14,186.1122 | 0.50 |

i = 2 | 583.1680 | −867.5425 | 790.3925 | −785.7880 | 1.00 |

${t}_{3}$ | ${x}_{3}$ | $\alpha $ | ${W}_{LS}$ | ||

1562.22 | 1 | 1/3 | 115.36 |

**Table 2.**Nuclear matter properties predicted by the D1M${}^{*}$, D1M${}^{**}$ and D1M Gogny interactions and by the BCPM energy density functional.

${\mathit{\rho}}_{0}$ | ${\mathit{E}}_{0}$ | ${\mathit{K}}_{0}$ | ${\mathit{m}}^{*}/\mathit{m}$ | ${\mathit{E}}_{\mathbf{sym}}\left({\mathit{\rho}}_{0}\right)$ | ${\mathit{E}}_{\mathbf{sym}}\left(0.1\right)$ | L | |
---|---|---|---|---|---|---|---|

(fm${}^{-3}$) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) | ||

D1M | 0.1647 | −16.02 | 224.98 | 0.746 | 28.55 | 23.80 | 24.83 |

D1M${}^{*}$ | 0.1650 | −16.06 | 225.38 | 0.746 | 30.25 | 23.82 | 43.18 |

D1M${}^{**}$ | 0.1647 | −16.02 | 224.98 | 0.746 | 29.37 | 23.80 | 33.91 |

BCPM | 0.1600 | −16.00 | 213.75 | 1.000 | 31.92 | 24.20 | 52.96 |

**Table 3.**Core–crust transition density ${\rho}_{t}$, pressure ${P}_{t}$ and and isospin asymmetry ${\delta}_{t}$ predicted by the D1M, D1M${}^{*}$ and D1M${}^{**}$ Gogny forces and the BCPM energy density functional.

${\mathit{\rho}}_{\mathit{t}}$ | ${\mathit{P}}_{\mathit{t}}$ | ${\mathit{\delta}}_{\mathit{t}}$ | |
---|---|---|---|

(fm${}^{-3}$) | (MeVfm${}^{-3}$) | ||

D1M | |||

${V}_{ther}$ | 0.1027 | 0.3390 | 0.9241 |

${V}_{dyn}$ | 0.0949 | 0.2839 | 0.9257 |

D1M${}^{*}$ | |||

${V}_{ther}$ | 0.0909 | 0.3301 | 0.9275 |

${V}_{dyn}$ | 0.0838 | 0.2702 | 0.9300 |

D1M${}^{**}$ | |||

${V}_{ther}$ | 0.0960 | 0.3368 | 0.9257 |

${V}_{dyn}$ | 0.0886 | 0.2786 | 0.9279 |

BCPM | |||

${V}_{ther}$ | 0.0889 | 0.5137 | 0.9339 |

${V}_{dyn}$ | 0.0816 | 0.4132 | 0.9382 |

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**MDPI and ACS Style**

Viñas, X.; Gonzalez-Boquera, C.; Centelles, M.; Mondal, C.; Robledo, L.M.
Unified Equation of State for Neutron Stars Based on the Gogny Interaction. *Symmetry* **2021**, *13*, 1613.
https://doi.org/10.3390/sym13091613

**AMA Style**

Viñas X, Gonzalez-Boquera C, Centelles M, Mondal C, Robledo LM.
Unified Equation of State for Neutron Stars Based on the Gogny Interaction. *Symmetry*. 2021; 13(9):1613.
https://doi.org/10.3390/sym13091613

**Chicago/Turabian Style**

Viñas, Xavier, Claudia Gonzalez-Boquera, Mario Centelles, Chiranjib Mondal, and Luis M. Robledo.
2021. "Unified Equation of State for Neutron Stars Based on the Gogny Interaction" *Symmetry* 13, no. 9: 1613.
https://doi.org/10.3390/sym13091613