# Relativistic Neutron Stars: Rheological Type Extensions of the Equations of State

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## Abstract

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## 1. Introduction

## 2. Prologue: On the Rheological Type Extension of Time Dependent Constitutive Equations

#### 2.1. Classical Constitutive Equations in Rheology

#### 2.2. Generalization of the Time Dependent EoS with Parametric Representation

## 3. The Formalism of the EoS Extension in the Framework of Static Models

#### 3.1. The Standard Elements of the Model: Metric, Einstein Equations, and the Equation of Hydrostatic Equilibrium

#### 3.2. Extended Equations of State for the Static Configurations

#### 3.2.1. Convective and Directional Derivatives

#### 3.2.2. Static Analogs of the Rheologically Extended Constitutive Equations

## 4. The Model of Cold Isotropic Neutron Condensate

#### 4.1. Extended Equation of Hydrostatic Equilibrium: The General Relativistic Model

#### 4.2. Rheological Type Generalization of the Non-Relativistic Lane–Emden Equation

#### 4.2.1. Behavior of the Function $\mathsf{\Theta}\left(x\right)$ near the Center

#### 4.2.2. Behavior of the Pressure $P\left(x\right)$

#### 4.2.3. The Mass/Radius Ratio

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Plot of the function $P(x,{\mathsf{\Gamma}}_{*})/{P}_{*}$ in the domain of the first nulls; this plot depicts the profiles of the reduced pressure as the function of the dimensionless rheological parameter ${\mathsf{\Gamma}}_{*}$. For negative ${\mathsf{\Gamma}}_{*}$, the radius of the star, predetermined by the condition $P\left({x}_{1}\right)=0$, becomes smaller than the radius predicted by the Lane–Emden theory; for positive ${\mathsf{\Gamma}}_{*}$ there are no roots of the function $P(x,{\mathsf{\Gamma}}_{*})$.

**Table 1.**The value ${x}_{1}$ relates to the first zero of the function $P\left(x\right)$; it determines the radius of the object and depends on the value of the guiding parameter ${\mathsf{\Gamma}}_{*}$. The function $\mathcal{F}\left({x}_{1}\right)={x}_{1}^{5}\left[(5/4){\mathsf{\Gamma}}_{*}{\mathsf{\Theta}}^{\prime \prime}\left(x\right)-{\mathsf{\Theta}}^{\prime}\left(x\right)\right]$ enters the mass/radius ratio (51). The case ${\mathsf{\Gamma}}_{*}=0$ corresponds to the Lane–Emden model. For the fixed mass $M\left({R}_{0}\right)$, the radius of the object, ${R}_{0}$, decreases, if the modulus of the guiding parameter, $|{\mathsf{\Gamma}}_{*}|$, grows.

${\mathsf{\Gamma}}_{*}=0$ | ${x}_{1}=3.6537$ | $\mathcal{F}\left({x}_{1}\right)=132.384$ |

${\mathsf{\Gamma}}_{*}=-0.1$ | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{x}_{1}=3.1632$ | $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathcal{F}\left({x}_{1}\right)=197.746$ |

${\mathsf{\Gamma}}_{*}=-0.2$ | ${x}_{1}=2.7962$ | $\mathcal{F}\left({x}_{1}\right)=125.438$ |

${\mathsf{\Gamma}}_{*}=-0.3$ | ${x}_{1}=2.5154$ | $\mathcal{F}\left({x}_{1}\right)=84.266$ |

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**MDPI and ACS Style**

Balakin, A.; Ilin, A.; Kotanjyan, A.; Grigoryan, L.
Relativistic Neutron Stars: Rheological Type Extensions of the Equations of State. *Symmetry* **2019**, *11*, 189.
https://doi.org/10.3390/sym11020189

**AMA Style**

Balakin A, Ilin A, Kotanjyan A, Grigoryan L.
Relativistic Neutron Stars: Rheological Type Extensions of the Equations of State. *Symmetry*. 2019; 11(2):189.
https://doi.org/10.3390/sym11020189

**Chicago/Turabian Style**

Balakin, Alexander, Alexei Ilin, Anna Kotanjyan, and Levon Grigoryan.
2019. "Relativistic Neutron Stars: Rheological Type Extensions of the Equations of State" *Symmetry* 11, no. 2: 189.
https://doi.org/10.3390/sym11020189