# Gravitational Instability Caused by the Weight of Heat

## Abstract

**:**

## 1. Introduction

#### A Thought Experiment

## 2. Hydrostatic Equilibrium of Relativistic Self-Gravitating Ideal Gas

#### 2.1. Tolman–Oppenheimer–Volkoff Equation

#### 2.2. Equation of State

## 3. Gravothermal Instability

## 4. Core-Collapse Supernova

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Upper panels: The caloric curves $\tilde{\beta}=\tilde{\beta}\left(E\right)$ for fixed rest compactness $\xi =2G\mathcal{M}/R{c}^{2}$, $\mathcal{M}=mN$. Both N and R may be regarded as constant in these plots. We denote $E=M{c}^{2}-\mathcal{M}{c}^{2}$ the gravothermal energy of the system. (

**a**) The low-energy gravothermal instability for the values of rest compactness, $\xi =0.1$ and $0.15$, together with the Newtonian limit $\xi \to 0$, corresponding to Antonov instability. The instability occurs at minimum energy (point A for $\xi =0.1$). (

**b**) The high-energy or relativistic gravothermal instability, for the values of rest compactness $\xi =0.1$, $0.15$. The instability occurs at maximum energy (point $\Delta $ for $\xi =0.1$). In the Newtonian limit, $\xi \to 0$ it is ${E}_{\Delta}\to \infty $. The two spirals, at low and high energies are connected with a stable series of equilibria, not shown here, but depicted in Figure 3 for $\xi =0.25$. Lower panels: The specific heat w.r.t. the density contrast for $\xi =0.1$. Both panels (

**c**,

**d**) depict the same diagram. In panel (c), the low-energy gravothermal instability is highlighted. The point $\Sigma $ denotes the threshold, beyond which thermal energy takes over and any increase of energy causes an increase and not decrease of density contrast (the system becoming less homogeneous). Both low- and high-energy instabilities, at points A, $\Delta $, respectively, occur as we move from the stable negative heat branch to positive specific heat, designating a core–halo structure.

**Figure 2.**The mass-energy density distribution of the equilibrium point A (

**a**) and $\Delta $ (

**b**) for rest compactness $\xi =0.1$. At A, the low-energy gravothermal instability sets in, whereas at $\Delta $, the high-energy gravothermal instability. In the low energy case, the core is dominated by the rest mass energy, whereas in the high energy case, the core is dominated by thermal mass-energy.

**Figure 3.**The double spiral of the caloric curve $\tilde{\beta}=\tilde{\beta}\left(E\right)$ for rest compactness $\xi =0.25$ of the relativistic classical ideal gas reflecting the gravothermal instability. The upper spiral is a manifestation of the low-energy gravothermal instability, and the lower spiral of the high-energy or relativistic gravothermal instability [5]. The two spirals are connected with a stable series of equilibria. As $\xi $ increases, the spirals approach each other and merge to a single point for $\xi =0.35$. Beyond this point, no equilibrium is attainable.

**Figure 4.**(

**a**) The critical radius at which a gravothermal instability—for low or high energy—sets in with respect to the gravothermal energy. There appears a minimum gravothermal energy ${E}_{\mathrm{min}}=-0.015Nm{c}^{2}$ at point I below which no equilibria exist. For radii bigger than I, the low-energy gravothermal instability sets in, while for radii smaller than I, the relativistic gravothermal instability sets in. The ultimate minimum radius is $2{R}_{S}$. (

**b**) The critical compactness w.r.t. the rest compactness. Above the maximum value, the high-energy gravothermal instability sets in, whereas below the minimum value, the low-energy one sets in. Point I denotes the maximum possible rest compactness under any conditions that equals $0.35$. The maximum possible compactness is $0.5$.

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**MDPI and ACS Style**

Roupas, Z.
Gravitational Instability Caused by the Weight of Heat. *Symmetry* **2019**, *11*, 1435.
https://doi.org/10.3390/sym11121435

**AMA Style**

Roupas Z.
Gravitational Instability Caused by the Weight of Heat. *Symmetry*. 2019; 11(12):1435.
https://doi.org/10.3390/sym11121435

**Chicago/Turabian Style**

Roupas, Zacharias.
2019. "Gravitational Instability Caused by the Weight of Heat" *Symmetry* 11, no. 12: 1435.
https://doi.org/10.3390/sym11121435