# Thermodynamics of Hot Neutron Stars and Universal Relations

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Thermal Equilibrium in General Relativity

## 3. Binding Energy

^{56}Fe/56 = 930.412 MeV/c

^{2}.

## 4. Tidal Deformability

## 5. Rescaled Entropy

## 6. Results and Discussion

#### 6.1. Hot Equations of State

#### 6.2. Rescaled Entropy vs. Compactness

#### 6.3. Tidal Deformability vs. Compactness

#### 6.4. Binding Energy vs. Tidal Deformability

#### 6.5. Binding Energy vs. Compactness

#### 6.6. Binding Energy, Compactness and Tidal Deformability

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EOS | Equation of state |

SN | Supernova |

GW | Gravitational waves |

BNS | Binary neutron stars |

GR | General relativity |

TE | Tolman–Ehrenfest |

TOV | Tolman–Openheimer–Volkoff |

MDI | Momentum-dependent interaction |

LHS | Left-hand side |

RHS | Right-hand side |

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**Figure 3.**The rescaled entropy as a function of the star’s compactness for cold and hot (isothermal and isentropic) EOSs. The circular points correspond to cold EOS, whereas the rectangular and the triangular points stand for isentropic and isothermal neutron stars, respectively. The solid line is the prediction of the Tolman VII solution.

**Figure 4.**Universal relation connecting the tidal deformability and the star’s compactness. The circular points correspond to cold EOS, whereas the rectangular and the triangular points stand for isentropic and isothermal neutron stars, respectively. The blue solid line corresponds to the Love-C fit of Maselli et al. [33]. We employed the parameter set from the review study of Yagi and Yunes [35].

**Figure 5.**Top panel: The relation between the tidal deformability and the star’s binding energy for cold and hot neutron stars. The circular points correspond to cold EOS, whereas the rectangle points correspond to isentropic neutron stars. The solid line stands for the fit of Equation (31), which is performed for cold EOSs. Bottom panel: Relative error for the fit of Equation (31).

**Figure 6.**Top panel: Relation between the dimensionless tidal deformability and the star’s binding energy for cold neutron stars and linear fitting. The circular points correspond to cold EOS. The solid line stands for the fit of Equation (32), which was performed for cold EOSs, in the mass range $M\in \left[1{M}_{\odot},{M}_{max}\right]$. The cross points represent the 1.4 ${M}_{\odot}$ configuration for each EOS. The shaded area shows the upper and lower values of the dimensionless tidal deformability (${\Lambda}_{1.4}={190}_{-120}^{+390}$) of a 1.4 ${M}_{\odot}$ neutron star from the analysis of the GW170817 event [37]. Bottom panel: Relative error for the fit of Equation (29).

**Figure 7.**Universal relation between the neutron star compactness and the rescaled binding energy. The circular and the rectangular points stand for cold and isentropic EOSs, respectively. The corresponding fit of Lattimer and Prakash [45] of Equation (29) and the fit of Breu and Rezzolla of Equation (30) are also plotted for completeness.

**Figure 8.**Top panel: Universal relation connecting the dimensionless tidal deformability, the star’s binding energy and the compactness. The universality extends to finite isentropic EOSs as well. The circular and the rectangular points correspond to cold and isentropic EOSs, respectively. The triangular points correspond to isothermal EOSs. The solid line stands for the fit of Equation (33), which was performed for cold and hot EOSs. Bottom panel: Relative error for the fit of Equation (33).

**Figure 9.**Universal relation connecting the dimensionless tidal deformability, the star’s binding energy and the compactness. The plot contains only the data for the 1.4 ${M}_{\odot}$ configuration of each EOS. The circular and the rectangular points correspond to cold and isentropic EOSs, respectively. The solid line stands for the fit of Equation (33), which is performed for cold and hot EOSs. The shaded area shows the upper and lower values of the dimensionless tidal deformability (${\Lambda}_{1.4}={190}_{-120}^{+390}$) of a 1.4 ${M}_{\odot}$ neutron star from analysis of GW170817 event [37].

**Table 1.**Fit parameters for the ${E}_{b}$/$\left(M{c}^{2}\right)$–$\Lambda $ relation of Equation (31). The ${R}^{2}$ index is also included for completeness.

${\mathit{a}}_{0}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{R}}^{2}$ |
---|---|---|---|---|

−0.08399 | 1.52078 | −2.91006 | 1.85495 | 0.987 |

**Table 2.**Fit parameters for the ${E}_{b}$/$\left(M{c}^{2}\right)$–$\Lambda $ linear relation of Equation (32). The ${R}^{2}$ index is also included for completeness.

${\mathit{b}}_{0}$ | ${\mathit{b}}_{1}$ | ${\mathit{R}}^{2}$ |
---|---|---|

0.22350 | −0.02017 | 0.960 |

${\Lambda}_{1.4}$ | ${\mathit{E}}_{\mathit{b}}/\left({\mathit{M}\mathit{c}}^{2}\right)$ | ${\mathit{E}}_{\mathit{b}}$$\left({10}^{53}\mathit{e}\mathit{r}\mathit{g}\right)$ |
---|---|---|

70 | $0.1378\pm 3.1\times {10}^{-3}$ | $3.4489\pm 0.0793$ |

580 | $0.0952\pm 3.9\times {10}^{-3}$ | $2.3814\pm 0.0977$ |

${\mathit{C}}_{0}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{R}}^{2}$ |
---|---|---|---|---|

0.66656 | 0.05855 | −0.00402 | 0.00010 | 0.99969 |

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**MDPI and ACS Style**

Laskos-Patkos, P.; Koliogiannis, P.S.; Kanakis-Pegios, A.; Moustakidis, C.C.
Thermodynamics of Hot Neutron Stars and Universal Relations. *Universe* **2022**, *8*, 395.
https://doi.org/10.3390/universe8080395

**AMA Style**

Laskos-Patkos P, Koliogiannis PS, Kanakis-Pegios A, Moustakidis CC.
Thermodynamics of Hot Neutron Stars and Universal Relations. *Universe*. 2022; 8(8):395.
https://doi.org/10.3390/universe8080395

**Chicago/Turabian Style**

Laskos-Patkos, Pavlos, Polychronis S. Koliogiannis, Alkiviadis Kanakis-Pegios, and Charalampos C. Moustakidis.
2022. "Thermodynamics of Hot Neutron Stars and Universal Relations" *Universe* 8, no. 8: 395.
https://doi.org/10.3390/universe8080395