# Quark Deconfinement in Rotating Neutron Stars

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## Abstract

**:**

## 1. Introduction

## 2. Models for the Nuclear Equations of State

#### 2.1. Hadronic Matter

#### 2.2. Deconfined Quark Phase

#### 2.3. Quark-Hadron Mixed Phase

## 3. Treatment of Rotating Neutron Stars in General Relativity Theory

## 4. Results

## 5. Discussion and Summary

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Histogram for the frequencies of all 2510 pulsars which have frequency data in version 1.53 of The Australia Telescope National Facility Pulsar Catalog [23,24]. The most rapidly rotating neutron star observed to date is the pulsar J1748-2446ad, which rotates at a frequency of 716 Hz [25] (rotational period of 1.4 milliseconds).

**Figure 2.**Comparison of the equations of state used in this work, GM1 and DD2, with models recently suggested in the literature (see text for details). The solid dots mark the beginning and the end of the quark-hadron mixed phases for GM1 and DD2. The repulsive interaction among quarks is controlled by the vector coupling constant, ${G}_{V}$.

**Figure 3.**Mass–central density (left) and mass–radius relationships of non-rotating neutron stars for the nuclear equations of state (EoS) used in this work. (${\u03f5}_{0}=140$ MeV/fm${}^{3}$ denotes the density of infinite nuclear matter).

**Figure 4.**Particle populations inside of rotating neutron stars, in equatorial (

**left**) and polar (

**right**) directions, computed for the GM1 EoS. The vector interaction among quarks is ${G}_{V}=0.09\phantom{\rule{0.166667em}{0ex}}{G}_{S}$. The stellar frequency, Ω, ranges from zero to the Kepler frequency, ${\mathsf{\Omega}}_{\mathrm{K}}=1361$ Hz. The gravitational mass of the non-rotating star is $2.10$ ${M}_{\odot}$, which increases to $2.20$ ${M}_{\odot}$ for rotation at $\mathsf{\Omega}={\mathsf{\Omega}}_{\mathrm{K}}$.

**Figure 5.**Change of the interior composition of a 2 ${M}_{\odot}$ neutron star caused by rotation, computed for the GM1 EoS. The vector interaction among quarks is ${G}_{V}=0.09\phantom{\rule{0.166667em}{0ex}}{G}_{S}$. The star on the left (right) hand-side is non-rotating (rotating at the Kepler frequency, $\mathsf{\Omega}={\mathsf{\Omega}}_{\mathrm{K}}$). The baryon number of both stars is the same (${log}_{10}A=57.51$).

**Figure 6.**Heat map showing the percent (column on the right) of total mass of a neutron star made up of deconfined quark matter, as predicted by the GM1 (${G}_{V}=0.05\phantom{\rule{0.166667em}{0ex}}{G}_{S}$) EoS. The white solid lines show the rotational evolution of neutron stars with constant baryon numbers A (the reported figures being ${log}_{10}A$). Also shown are the observed masses of pulsars J1614-2230 and J0348+0432 and the trend line (dashed white) fit, which separates confined from deconfined matter. A fit of the trend line is given by Equation (28).

**Table 1.**Properties of infinite nuclear matter at saturation density computed for parameter sets GM1 [36] and DD2 [37]. Shown are the saturation density ${\rho}_{0}$, energy per nucleon $E/N$, nuclear incompressibility K, effective nucleon mass ${m}_{N}^{*}/{m}_{N}$, asymmetry energy ${a}_{sy}$, and the density derivative of the symmetry energy, L.

Nuclear Matter Property | Units | GM1 | DD2 |
---|---|---|---|

${\rho}_{0}$ | fm${}^{-3}$ | 0.153 | 0.149 |

E/N | MeV | $-16.3$ | $-16.0$ |

K | MeV | 300 | 243 |

${m}_{N}^{*}/{m}_{N}$ | 0.70 | 0.56 | |

${a}_{sy}$ | MeV | 32.5 | 32.7 |

L | MeV | 91.96 | 55.04 |

**Table 2.**Parameters for the empirical deconfinement threshold curve for each equation of state (EoS) with the form shown in Equation (28).

EoS | a (${\mathit{M}}_{\odot}\phantom{\rule{3.33333pt}{0ex}}{\mathit{s}}^{2}$) | c (${\mathit{M}}_{\odot}$) |
---|---|---|

GM1 (${G}_{V}=0.05\phantom{\rule{0.166667em}{0ex}}{G}_{S}$) | $2.48\times {10}^{-7}$ | $1.91$ |

GM1 (${G}_{V}=0$) | $2.75\times {10}^{-7}$ | $1.71$ |

DD2 (${G}_{V}=0$) | $2.56\times {10}^{-7}$ | $1.89$ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Mellinger, R.D.; Weber, F.; Spinella, W.; Contrera, G.A.; Orsaria, M.G.
Quark Deconfinement in Rotating Neutron Stars. *Universe* **2017**, *3*, 5.
https://doi.org/10.3390/universe3010005

**AMA Style**

Mellinger RD, Weber F, Spinella W, Contrera GA, Orsaria MG.
Quark Deconfinement in Rotating Neutron Stars. *Universe*. 2017; 3(1):5.
https://doi.org/10.3390/universe3010005

**Chicago/Turabian Style**

Mellinger, Richard D., Fridolin Weber, William Spinella, Gustavo A. Contrera, and Milva G. Orsaria.
2017. "Quark Deconfinement in Rotating Neutron Stars" *Universe* 3, no. 1: 5.
https://doi.org/10.3390/universe3010005