# Nuclear Symmetry Energy Effects on Neutron Star Properties within Bogoliubov Quark–Meson Coupling Model

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## Abstract

**:**

## 1. Introduction

## 2. Bogoliubov Quark Meson Coupling Model

## 3. Hadronic Matter

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The symmetry energy (panel (

**a**)) and the symmetry energy slope (panel (

**b**)) as a function of the baryonic density for several values of the mixing term ${\Lambda}_{\omega {b}_{3}}$.

**Figure 2.**The pure neutron matter pressure as a function of the baryonic density for several values of mixing term ${\Lambda}_{\omega {b}_{3}}$ compared with chEFT neutron matter constraints. The dark and the light blue bands represent, respectively, the $1\sigma $ and the $2\sigma $ constraint from chEFT calculations [45].

**Figure 3.**Baryonic and leptonic particle fractions as a function of the baryonic density for the parameter $\delta =0.0$ (panel (

**a**)) and npe$\mu $ matter (panel (

**b**)), and several values of the mixing term ${\Lambda}_{\omega {b}_{3}}$: solid line ${\Lambda}_{\omega {b}_{3}}=0.0$, dashed line 0.1, dotted line 0.2, dash-dotted line 0.3, double-dotted dashed line 0.4.

**Figure 4.**Baryonic and leptonic particle fractions as a function of the baryonic density, for several values of parameter $\delta $ shown in panels (

**a**–

**e**); and for several values of the mixing term ${\Lambda}_{\omega {b}_{3}}$, solid line ${\Lambda}_{\omega {b}_{3}}=0.0$, dashed line 0.1, dotted line 0.2, dash-dotted line 0.3 double-dot dashed line 0.4. It was shown in ref. [28] that for $\delta >0.2$, the onset of hyperons is shifted to densities above 1.2 fm${}^{-3}$. The central baryonic density lies between 0.9 and 1.1 fm${}^{-3}$, depending on hyperonic content.

**Figure 5.**Onset density of hyperon as a function of the slope L, for several values of parameter $\delta $.

**Figure 7.**EoS (

**left panel**) and mass–radius curves obtained from the integration of the TOV equations (

**right**), for different values of the $\delta $ parameter and several values of the mixing term ${\Lambda}_{\omega {b}_{3}}$. The curves stop at the maximum mass configuration. The family of stars for nucleonic stars constituted by $npe\mu $ matter is also represented. Also shown are the constraints from NICER observations of pulsars PSR J0740+6620 and PSR J0030+0451 [10,11,12,13,14].

**Figure 8.**Mass-$\Lambda $ curves for different values of $\delta $ and L. Two astronomical observation constraints are also shown, GW190425 and GW170817.

**Table 1.**Parameters and properties of the model: ${g}_{\sigma}^{q}=4.0539996$, ${g}_{\omega}^{q}=3.0824732$, ${\rho}_{0}=0.145$ ${\mathrm{fm}}^{-3}$, ${e}_{sym}=29.0$ MeV, K = 315 MeV.

${\mathbf{\Lambda}}_{\mathit{\omega}{\mathit{b}}_{3}}$ | ${\mathit{g}}_{{\mathit{b}}_{3}}^{\mathit{q}}$ | L (MeV) |
---|---|---|

$0.0$ | 3.953289 | 79.5 |

$0.1$ | 4.23752 | 67.8 |

$0.2$ | 4.593438 | 56.2 |

$0.3$ | 5.057328 | 44.6 |

$0.4$ | 5.697799 | 32.9 |

$\mathit{\delta}$ | ${\mathbf{\Lambda}}_{\mathit{\omega}{\mathit{b}}_{3}}$ | L (MeV) | ${\mathit{\rho}}_{{\mathit{\Lambda}}^{0}}$ [fm${}^{-3}$] | ${\mathit{\rho}}_{{\mathbf{\Xi}}^{0}}$ [fm${}^{-3}$] | ${\mathit{\rho}}_{{\mathbf{\Xi}}^{-}}$ [fm${}^{-3}$] |
---|---|---|---|---|---|

0.0 | 0.0 | 79.5 | 0.292 | 0.613 | 0.340 |

0.1 | 67.8 | 0.310 | 0.837 | 0.358 | |

0.2 | 56.2 | 0.323 | 0.890 | 0.370 | |

0.3 | 44.6 | 0.334 | 0.909 | 0.379 | |

0.4 | 32.9 | 0.343 | 0.920 | 0.385 | |

0.05 | 0.0 | 79.5 | 0.326 | - | 0.340 |

0.1 | 67.8 | 0.353 | - | 0.427 | |

0.2 | 56.2 | 0.374 | - | 0.445 | |

0.3 | 44.6 | 0.389 | - | 0.455 | |

0.4 | 32.9 | 0.400 | - | 0.463 | |

0.1 | 0.0 | 79.5 | 0.374 | - | 0.499 |

0.1 | 67.8 | 0.418 | - | 0.550 | |

0.2 | 56.2 | 0.4507 | - | 0.581 | |

0.3 | 44.6 | 0.472 | - | 0.598 | |

0.4 | 32.9 | 0.485 | - | 0.608 | |

0.15 | 0.0 | 79.5 | 0.445 | - | 0.771 |

0.1 | 67.8 | 0.530 | - | - | |

0.2 | 56.2 | 0.592 | - | - | |

0.3 | 44.6 | 0.626 | - | - | |

0.4 | 32.9 | 0.647 | - | - | |

0.2 | 0.0 | 79.5 | 0.566 | - | - |

**Table 3.**Properties of the stable neutron star with maximum mass, for several values of $\delta $ and ${\Lambda}_{\omega {b}_{3}}$.

$\mathit{\delta}$ | ${\mathbf{\Lambda}}_{\mathit{\omega}{\mathit{b}}_{3}}$ | ${\mathit{M}}_{\mathbf{max}}$ $\left[{\mathit{M}}_{\odot}\right]$ | ${\mathit{M}}_{\mathbf{max}}^{\mathit{b}}$ $\left[{\mathit{M}}_{\odot}\right]$ | R [km] | ${\mathit{E}}_{0}$ [fm${}^{-\mathbf{4}}$] | ${\mathit{u}}^{\mathit{c}}={\mathit{\rho}}^{\mathit{c}}/{\mathit{\rho}}_{0}$ | ${\mathit{R}}_{1.4}$ [km] | ${\mathbf{\Lambda}}_{1.4}$ | ${\mathit{R}}_{1.6}$ [km] | ${\mathbf{\Lambda}}_{1.6}$ | ${\mathit{R}}_{2.0}$ [km] | ${\mathbf{\Lambda}}_{2.0}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.0 | 0.0 | 1.97 | 2.28 | 10.91 | 7.25 | 7.674 | 13.731 | 934.72 | 13.492 | 344.43 | - | - |

0.1 | 1.98 | 2.30 | 10.70 | 7.46 | 7.865 | 13.240 | 703.90 | 12.931 | 246.80 | - | - | |

0.2 | 1.99 | 2.32 | 10.66 | 7.40 | 7.821 | 12.907 | 615.35 | 12.691 | 227.53 | - | - | |

0.3 | 2.00 | 2.33 | 10.67 | 7.32 | 7.764 | 12.745 | 576.27 | 12.590 | 221.80 | - | - | |

0.4 | 2.00 | 2.34 | 10.65 | 7.26 | 7.720 | 12.568 | 555.19 | 12.478 | 219.74 | 10.80 | - | |

0.05 | 0.0 | 2.08 | 2.43 | 11.42 | 6.43 | 6.882 | 13.752 | 947.47 | 13.680 | 394.49 | 12.62 | 40.95 |

0.1 | 2.06 | 2.41 | 11.09 | 6.81 | 7.235 | 13.283 | 726.25 | 13.155 | 290.46 | 12.03 | 27.27 | |

0.2 | 2.07 | 2.43 | 11.04 | 6.75 | 7.197 | 12.936 | 631.51 | 12.855 | 257.87 | 11.95 | 27.76 | |

0.3 | 2.08 | 2.44 | 11.02 | 6.72 | 7.181 | 12.761 | 586.53 | 12.716 | 244.82 | 11.95 | 28.82 | |

0.4 | 2.08 | 2.45 | 11.00 | 6.66 | 7.135 | 12.582 | 561.35 | 12.580 | 239.44 | 11.93 | 29.79 | |

0.1 | 0.0 | 2.16 | 2.53 | 11.73 | 5.97 | 6.429 | 13.750 | 948.13 | 13.693 | 399.85 | 13.11 | 59.61 |

0.1 | 2.12 | 2.48 | 11.30 | 6.45 | 6.891 | 13.289 | 726.08 | 13.173 | 295.53 | 12.44 | 38.27 | |

0.2 | 2.12 | 2.50 | 11.21 | 6.44 | 6.896 | 12.937 | 631.23 | 12.864 | 260.34 | 12.27 | 36.18 | |

0.3 | 2.13 | 2.52 | 11.19 | 6.42 | 6.881 | 12.763 | 586.85 | 12.727 | 246.40 | 12.21 | 36.07 | |

0.4 | 2.14 | 2.53 | 11.15 | 6.40 | 6.866 | 12.586 | 561.33 | 12.580 | 239.44 | 12.15 | 36.34 | |

0.15 | 0.0 | 2.20 | 2.58 | 11.83 | 5.84 | 6.285 | 13.746 | 948.13 | 13.698 | 399.97 | 13.19 | 63.85 |

0.1 | 2.14 | 2.51 | 11.35 | 6.35 | 6.790 | 13.284 | 726.08 | 13.177 | 295.52 | 12.50 | 40.43 | |

0.2 | 2.14 | 2.53 | 11.24 | 6.41 | 6.842 | 12.939 | 631.23 | 12.864 | 260.34 | 12.30 | 37.36 | |

0.3 | 2.15 | 2.54 | 11.20 | 6.42 | 6.858 | 12.763 | 586.85 | 12.723 | 246.40 | 12.23 | 36.86 | |

0.4 | 2.15 | 2.55 | 11.16 | 6.39 | 6.844 | 12.581 | 561.33 | 12.584 | 239.44 | 12.17 | 36.94 | |

0.2 | 0.0 | 2.21 | 2.60 | 11.84 | 5.81 | 6.256 | 13.748 | 948.13 | 13.697 | 399.97 | 13.20 | 64.33 |

0.1 | 2.14 | 2.52 | 11.36 | 6.34 | 6.776 | 13.288 | 726.08 | 13.175 | 295.52 | 12.50 | 40.54 | |

0.2 | 2.14 | 2.53 | 11.24 | 6.41 | 6.843 | 12.939 | 631.23 | 12.866 | 260.34 | 12.30 | 37.37 | |

0.3 | 2.15 | 2.54 | 11.20 | 6.42 | 6.859 | 12.762 | 586.85 | 12.727 | 246.40 | 12.23 | 36.86 | |

0.4 | 2.15 | 2.55 | 11.15 | 6.40 | 6.844 | 12.585 | 561.33 | 12.579 | 239.44 | 12.17 | 36.94 | |

npe$\mu $ | 0.0 | 2.21 | 2.60 | 11.84 | 5.84 | 6.272 | 13.746 | 948.13 | 13.696 | 399.97 | 13.20 | 64.33 |

0.1 | 2.14 | 2.52 | 11.36 | 6.36 | 6.790 | 13.283 | 726.08 | 13.177 | 295.52 | 12.50 | 40.54 | |

0.2 | 2.14 | 2.53 | 11.24 | 6.43 | 6.856 | 12.939 | 631.23 | 12.866 | 260.34 | 12.30 | 37.37 | |

0.3 | 2.15 | 2.54 | 11.20 | 6.42 | 6.859 | 12.762 | 586.85 | 12.727 | 246.40 | 12.23 | 36.86 | |

0.4 | 2.15 | 2.55 | 11.15 | 6.40 | 6.844 | 12.585 | 561.33 | 12.579 | 239.44 | 12.17 | 36.94 |

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**MDPI and ACS Style**

Boukari, O.; Rabhi, A.
Nuclear Symmetry Energy Effects on Neutron Star Properties within Bogoliubov Quark–Meson Coupling Model. *Symmetry* **2023**, *15*, 1742.
https://doi.org/10.3390/sym15091742

**AMA Style**

Boukari O, Rabhi A.
Nuclear Symmetry Energy Effects on Neutron Star Properties within Bogoliubov Quark–Meson Coupling Model. *Symmetry*. 2023; 15(9):1742.
https://doi.org/10.3390/sym15091742

**Chicago/Turabian Style**

Boukari, Olfa, and Aziz Rabhi.
2023. "Nuclear Symmetry Energy Effects on Neutron Star Properties within Bogoliubov Quark–Meson Coupling Model" *Symmetry* 15, no. 9: 1742.
https://doi.org/10.3390/sym15091742