# Cooling Process of White Dwarf Stars in Palatini f(R) Gravity

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## Abstract

**:**

## 1. Introduction

## 2. Basic Formalism of Palatini $\mathbf{f}\left(\mathbf{R}\right)$ Gravity and Hydrostatic Balance Equations

## 3. Temperature Gradient Equation and Cooling Timescale of White Dwarfs in $\mathbf{f}\left(\mathbf{R}\right)$ Gravity

## 4. Mass–Radius Relation and Cooling Age of White Dwarfs

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | Although, one can rewrite the equations as second-order differential equations for the metric, and the additional one for the curvature scalar, which arises to a dynamical field in this theory. |

2 | This fact arises as a conclusion from the field equations and it will be evident in the next section, whereas in the metric formalism, the connection is assumed to be the Levi–Civita one of the spacetime metric. |

3 | To see the relativistic hydrostatic equilibrium equation for Palatini $f\left(R\right)$ gravity, see [88]. |

4 | Let us notice that this form differs slightly from the one obtained in [43]. This is so because of different assumptions on the matter description and its behavior under the conformal transformation. |

5 | Of course, there are other processes, such as magnetic field [91,92], noncommutative geometry [93,94], ungravity effect [95], consequence of total lepton number violation [96], generalized Heisenberg uncertainty principle [97] and many more which can also explain massive white dwarfs but they are failing to explain the sub-Chandrasekhar mass-limit. |

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**Figure 1.**Mass–radius relation of $f\left(R\right)$ gravity inspired WDs. The GR case is represented by $\alpha =0$.

**Figure 2.**Cooling timescale as a function of the mass of the WDs for ${L}_{*}={10}^{-3}{L}_{\odot}$. Notice that modified gravity allows the white dwarf stars to have higher masses than the Chandrasekhar limit.

**Figure 3.**Cooling timescale as a function of the luminosity for WDs with different masses. Note that $2.9\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ WD is possible only when $\alpha <0$. It is worth mentioning that the $\alpha $ value is not required explicitly to obtain this plot, because from Figure 2, it is evident that for a specific luminosity, cooling timescale depends only on the mass, and not the radius. Moreover, the $0.2\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ WD can be obtained regardless of $\alpha $ being positive, negative or zero, and hence it is not necessary to precisely specify a value of $\alpha $ in order to determine its cooling timescale.

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Kalita, S.; Sarmah, L.; Wojnar, A.
Cooling Process of White Dwarf Stars in Palatini *f*(*R*) Gravity. *Universe* **2022**, *8*, 647.
https://doi.org/10.3390/universe8120647

**AMA Style**

Kalita S, Sarmah L, Wojnar A.
Cooling Process of White Dwarf Stars in Palatini *f*(*R*) Gravity. *Universe*. 2022; 8(12):647.
https://doi.org/10.3390/universe8120647

**Chicago/Turabian Style**

Kalita, Surajit, Lupamudra Sarmah, and Aneta Wojnar.
2022. "Cooling Process of White Dwarf Stars in Palatini *f*(*R*) Gravity" *Universe* 8, no. 12: 647.
https://doi.org/10.3390/universe8120647