# Equations of State for Hadronic Matter and Mass-Radius Relations of Neutron Stars with Strong Magnetic Fields

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulations

#### 2.1. Equation of State

#### 2.2. Magnetic Fields

#### 2.3. EoS of Hadronic Matter with Magnetic Fields

## 3. Results

#### 3.1. Comparison of the Case without Magnetic Fields

#### 3.2. Changing $\alpha $ and $\gamma $ Parameters of the Magnetic Field Function

- For $\gamma =2$, $\alpha =$0.05, 0.06, and 0.07;
- For $\gamma =3$, $\alpha =$0.02, 0.03, and 0.04;
- For $\gamma =4$, $\alpha =$0.01.

**Figure 4.**MR relations of GM1 EoS with magnetic fields. The strength of surface magnetic field strength ${B}_{s}$ is ${10}^{12}$ G, and the central magnetic field strength ${B}_{0}$ is $2.5\times {10}^{18}$ G. The arrow and colored hatched areas are the same as in Figure 1.

- For $\gamma =2$, $\alpha $ = 0.09;
- For $\gamma =3$, $\alpha $ = 0.03, 0.04, and 0.05;
- For $\gamma =4$, $\alpha $ = 0.01 and 0.02;
- For $\gamma =5$, $\alpha $ = 0.01.

- For $\gamma =1$, $\alpha =$0.01;
- For $\gamma =2$, $\alpha =$0.01.

**Table 5.**Maximum mass (${M}_{\mathrm{max}}$) in unit of ${M}_{\odot}$ and radius (km) at $M=1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{1.4{M}_{\odot}}$) and radius (km) at $M=2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) for GM1 EoS with magnetic fields (${B}_{s}={10}^{12}$ G, ${B}_{0}=2.5\times {10}^{18}$ G).

$\mathit{\gamma}=$ 2 | $\mathit{\gamma}=$ 3 | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | $\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ |

0.01 | 1.810 | 13.50 | − | 0.01 | 1.977 | 13.54 | − |

0.02 | 1.867 | 13.57 | − | 0.02 | 2.130 | 13.68 | 12.62 |

0.03 | 1.927 | 13.66 | − | 0.03 | 2.245 | 13.82 | 13.37 |

0.04 | 1.989 | 13.78 | − | 0.04 | 2.339 | 13.96 | 13.81 |

0.05 | 2.049 | 13.90 | − | 0.05 | 2.421 | 14.09 | 14.13 |

0.06 | 2.107 | 14.03 | 12.93 | 0.06 | 2.493 | 14.22 | 14.40 |

0.07 | 2.161 | 14.16 | 13.47 | 0.07 | 2.558 | 14.35 | 14.62 |

0.08 | 2.214 | 14.30 | 13.85 | 0.08 | 2.619 | 14.47 | 14.82 |

0.09 | 2.265 | 14.44 | 14.16 | 0.09 | 2.671 | 14.59 | 15.00 |

0.10 | 2.313 | 14.59 | 14.43 | 0.10 | 2.719 | 14.71 | 15.15 |

$\mathit{\gamma}=$ 4 | $\mathit{\gamma}=$ 5 | ||||||

$\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | $\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ |

0.01 | 2.247 | 13.64 | 13.07 | 0.01 | 2.494 | 13.76 | 13.82 |

0.02 | 2.431 | 13.83 | 13.81 | 0.02 | 2.679 | 13.98 | 14.35 |

0.03 | 2.557 | 13.99 | 14.22 | 0.03 | 2.799 | 14.15 | 14.66 |

0.04 | 2.653 | 14.14 | 14.51 | 0.04 | 2.889 | 14.29 | 14.88 |

0.05 | 2.732 | 14.27 | 14.73 | 0.05 | 2.960 | 14.42 | 15.07 |

0.06 | 2.800 | 14.40 | 14.92 | 0.05 | 3.020 | 14.53 | 15.23 |

0.07 | 2.862 | 14.51 | 15.08 | 0.07 | 3.072 | 14.63 | 15.37 |

0.08 | 2.915 | 14.62 | 15.23 | 0.08 | 3.118 | 14.72 | 15.50 |

0.09 | 2.962 | 14.71 | 15.37 | 0.09 | 3.160 | 14.80 | 15.61 |

0.10 | 3.005 | 14.81 | 15.49 | 0.10 | 3.197 | 14.88 | 15.71 |

**Figure 5.**MR relations of GM3 EoS with magnetic fields. The strength of surface magnetic field strength ${B}_{s}$ is ${10}^{12}$ G, and the central magnetic field strength ${B}_{0}$ is $2.5\times {10}^{18}$ G. The arrow and colored hatched areas are the same as in Figure 1.

**Table 6.**Maximum mass (${M}_{\mathrm{max}}$) in unit of ${M}_{\odot}$ and radius (km) at $M=1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) and radius (km) at $M=2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) for GM3 EoS with magnetic fields (${B}_{s}={10}^{12}$ G, ${B}_{0}=2.5\times {10}^{18}$ G).

$\mathit{\gamma}=$ 2 | $\mathit{\gamma}=$ 3 | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | $\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ |

0.01 | 1.357 | − | − | 0.01 | 1.660 | 12.31 | − |

0.02 | 1.458 | 11.63 | − | 0.02 | 1.863 | 13.00 | − |

0.03 | 1.557 | 12.46 | − | 0.03 | 2.011 | 13.38 | − |

0.04 | 1.647 | 12.95 | − | 0.04 | 2.130 | 13.64 | 12.92 |

0.05 | 1.730 | 13.30 | − | 0.05 | 2.233 | 13.85 | 13.54 |

0.06 | 1.808 | 13.58 | − | 0.06 | 2.322 | 14.03 | 13.95 |

0.07 | 1.881 | 13.82 | − | 0.07 | 2.402 | 14.19 | 14.27 |

0.08 | 1.951 | 14.03 | − | 0.08 | 2.473 | 14.34 | 14.54 |

0.09 | 2.018 | 14.22 | − | 0.09 | 2.536 | 14.48 | 14.77 |

0.10 | 2.081 | 14.40 | 13.19 | 0.10 | 2.594 | 14.61 | 14.96 |

$\mathit{\gamma}=$ 4 | $\mathit{\gamma}=$ 5 | ||||||

$\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | $\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ |

0.01 | 2.021 | 13.04 | 0.01 | 2.768 | 13.84 | 14.36 | |

0.02 | 2.252 | 13.49 | 13.22 | 0.02 | 2.949 | 14.13 | 14.80 |

0.03 | 2.404 | 13.76 | 13.83 | 0.03 | 3.060 | 14.31 | 15.07 |

0.04 | 2.520 | 13.96 | 14.22 | 0.04 | 3.139 | 14.45 | 15.27 |

0.05 | 2.614 | 14.13 | 14.51 | 0.05 | 3.201 | 14.57 | 15.42 |

0.06 | 2.694 | 14.28 | 14.74 | 0.06 | 3.253 | 14.66 | 15.55 |

0.07 | 2.763 | 14.42 | 14.94 | 0.07 | 3.297 | 14.75 | 15.66 |

0.08 | 2.824 | 14.54 | 15.11 | 0.08 | 3.336 | 14.82 | 15.76 |

0.09 | 2.878 | 14.65 | 15.26 | 0.09 | 3.370 | 14.89 | 15.85 |

0.10 | 2.927 | 14.75 | 15.40 | 0.10 | 3.401 | 14.95 | 15.93 |

**Figure 6.**MR relations of NL3$\omega \rho $-a EoS with magnetic fields. The strength of surface magnetic field strength ${B}_{s}$ is ${10}^{12}$ G, and the central magnetic field strength ${B}_{0}$ is $2.5\times {10}^{18}$ G. The arrow and colored hatched areas are the same as in Figure 1.

**Table 7.**Maximum mass (${M}_{\mathrm{max}}$) in unit of ${M}_{\odot}$ and radius (km) at $M=1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) and radius (km) at $M=2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) for NL3$\omega \rho $-a EoS with magnetic fields (${B}_{s}={10}^{12}$ G, ${B}_{0}=2.5\times {10}^{18}$ G).

$\mathit{\gamma}=$ 1 | $\mathit{\gamma}=$ 2 | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | $\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ |

0.01 | 2.177 | 13.63 | 14.07 | 0.01 | 2.188 | 13.63 | 14.08 |

0.02 | 2.182 | 13.66 | 14.10 | 0.02 | 2.220 | 13.65 | 14.14 |

0.03 | 2.190 | 13.71 | 14.15 | 0.03 | 2.262 | 13.69 | 14.21 |

0.04 | 2.200 | 13.78 | 14.21 | 0.04 | 2.310 | 13.74 | 14.29 |

0.05 | 2.212 | 13.87 | 14.29 | 0.05 | 2.359 | 13.81 | 14.38 |

0.06 | 2.227 | 13.98 | 14.39 | 0.06 | 2.409 | 13.88 | 14.47 |

0.07 | 2.244 | 14.10 | 14.49 | 0.07 | 2.457 | 13.96 | 14.55 |

0.08 | 2.262 | 14.25 | 14.61 | 0.08 | 2.503 | 14.05 | 14.65 |

0.09 | 2.282 | 14.42 | 14.75 | 0.09 | 2.549 | 14.16 | 14.75 |

0.10 | 2.303 | 14.61 | 14.88 | 0.10 | 2.592 | 14.26 | 14.86 |

$\mathit{\gamma}=$ 3 | $\mathit{\gamma}=$ 4 | ||||||

$\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | $\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ |

0.01 | 2.268 | 13.63 | 14.13 | 0.01 | 2.485 | 13.64 | 14.22 |

0.02 | 2.394 | 13.67 | 14.24 | 0.02 | 2.664 | 13.70 | 14.37 |

0.03 | 2.500 | 13.72 | 14.34 | 0.03 | 2.782 | 13.78 | 14.50 |

0.04 | 2.589 | 13.79 | 14.44 | 0.04 | 2.869 | 13.86 | 14.63 |

0.05 | 2.665 | 13.86 | 14.55 | 0.05 | 2.939 | 13.95 | 14.75 |

0.06 | 2.731 | 13.94 | 14.65 | 0.06 | 2.997 | 14.04 | 14.87 |

0.07 | 2.789 | 14.03 | 14.76 | 0.07 | 3.047 | 14.12 | 14.98 |

0.08 | 2.841 | 14.12 | 14.87 | 0.08 | 3.089 | 14.21 | 15.09 |

0.09 | 2.887 | 14.21 | 14.97 | 0.09 | 3.127 | 14.29 | 15.19 |

0.10 | 2.930 | 14.30 | 15.08 | 0.10 | 3.161 | 14.37 | 15.29 |

**Figure 7.**MR relations of NL3$\omega \rho $-b EoS with magnetic fields. The strength of surface magnetic field strength ${B}_{s}$ is ${10}^{12}$ G, and the central magnetic field strength ${B}_{0}$ is $2.5\times {10}^{18}$ G. The arrow and colored hatched areas are the same as in Figure 1.

**Table 8.**Maximum mass (${M}_{\mathrm{max}}$) in unit of ${M}_{\odot}$ and radius (km) at $M=1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) and radius (km) at $M=2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) for NL3$\omega \rho $-b EoS with magnetic fields (${B}_{s}={10}^{12}$ G, ${B}_{0}=2.5\times {10}^{18}$ G).

$\mathit{\gamma}=$ 1 | $\mathit{\gamma}=$ 2 | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | $\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ |

0.01 | 2.373 | 13.63 | 14.19 | 0.01 | 2.381 | 13.63 | 14.19 |

0.02 | 2.377 | 13.66 | 14.21 | 0.02 | 2.405 | 13.65 | 14.23 |

0.03 | 2.383 | 13.71 | 14.25 | 0.03 | 2.438 | 13.69 | 14.27 |

0.04 | 2.390 | 13.78 | 14.30 | 0.04 | 2.475 | 13.74 | 14.33 |

0.05 | 2.400 | 13.87 | 14.37 | 0.05 | 2.512 | 13.81 | 14.39 |

0.06 | 2.411 | 13.98 | 14.45 | 0.06 | 2.551 | 13.88 | 14.47 |

0.07 | 2.424 | 14.10 | 14.54 | 0.07 | 2.588 | 13.96 | 14.56 |

0.08 | 2.438 | 14.25 | 14.65 | 0.08 | 2.625 | 14.05 | 14.65 |

0.09 | 2.453 | 14.42 | 14.77 | 0.09 | 2.661 | 14.16 | 14.75 |

0.10 | 2.469 | 14.61 | 14.90 | 0.10 | 2.696 | 14.26 | 14.86 |

$\mathit{\gamma}=$ 3 | $\mathit{\gamma}=$ 4 | ||||||

$\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | $\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ |

0.01 | 2.440 | 13.63 | 14.21 | 0.01 | 2.606 | 13.64 | 14.25 |

0.02 | 2.536 | 13.67 | 14.27 | 0.02 | 2.751 | 13.70 | 14.37 |

0.03 | 2.620 | 13.72 | 14.35 | 0.03 | 2.848 | 13.78 | 14.50 |

0.04 | 2.691 | 13.79 | 14.45 | 0.04 | 2.924 | 13.86 | 14.63 |

0.05 | 2.754 | 13.86 | 14.55 | 0.05 | 2.984 | 13.95 | 14.75 |

0.06 | 2.808 | 13.94 | 14.65 | 0.06 | 3.033 | 14.04 | 14.87 |

0.07 | 2.855 | 14.03 | 14.76 | 0.07 | 3.076 | 14.12 | 14.98 |

0.08 | 2.900 | 14.12 | 14.87 | 0.08 | 3.114 | 14.21 | 15.09 |

0.09 | 2.940 | 14.21 | 14.97 | 0.09 | 3.149 | 14.29 | 15.19 |

0.10 | 2.977 | 14.30 | 15.08 | 0.10 | 3.180 | 14.37 | 15.29 |

- For $\gamma =2$, $\alpha =$0.05 and 0.06;
- For $\gamma =3$, $\alpha =$0.02, 0.03, and 0.04;
- For $\gamma =4$, $\alpha =$0.01.

**Figure 8.**MR relations of TM2$\omega \rho $-a EoS with magnetic fields. The strength of surface magnetic field strength ${B}_{s}$ is ${10}^{12}$ G, and the central magnetic field strength ${B}_{0}$ is $2.5\times {10}^{18}$ G. The arrow and colored hatched areas are the same as in Figure 1.

- 1.
- For $\gamma =2$, $\alpha =$0.03, 0.04, 0.05, and 0.06;
- 2.
- For $\gamma =3$, $\alpha =$0.01, 0.02, 0.03, and 0.04;

**Table 9.**Maximum mass (${M}_{\mathrm{max}}$) in unit of ${M}_{\odot}$ and radius (km) at $M=1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) and radius (km) at $M=2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) for TM2$\omega \rho $-a EoS with magnetic fields (${B}_{s}={10}^{12}$ G, ${B}_{0}=2.5\times {10}^{18}$ G).

$\mathit{\gamma}$ | $\mathit{\alpha}$ | ${\mathit{M}}_{\mathbf{max}}$ | ${\mathit{R}}_{1.4\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}_{\odot}}$ | ${\mathit{R}}_{2.072\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}_{\odot}}$ |
---|---|---|---|---|

2 | 0.05 | 1.983 | 13.59 | − |

2 | 0.06 | 2.048 | 13.71 | 12.56 |

3 | 0.02 | 2.072 | 13.38 | 12.12 |

3 | 0.03 | 2.210 | 13.51 | 13.35 |

3 | 0.04 | 2.320 | 13.65 | 13.75 |

4 | 0.01 | 2.212 | 13.35 | 13.06 |

4 | 0.02 | 2.425 | 13.53 | 13.73 |

4 | 0.03 | 2.561 | 13.71 | 14.06 |

5 | 0.01 | 2.497 | 13.46 | 13.73 |

5 | 0.02 | 2.691 | 13.71 | 14.16 |

6 | 0.01 | 2.711 | 13.61 | 14.08 |

- 3.
- For $\gamma =4$, $\alpha =$0.01.

**Figure 9.**MR relations of TM2$\omega \rho $-b EoS with magnetic fields. The strength of surface magnetic field strength ${B}_{s}$ is ${10}^{12}$ G, and the central magnetic field strength ${B}_{0}$ is $2.5\times {10}^{18}$ G. The arrow and colored hatched areas are the same as in Figure 1.

**Table 10.**Maximum mass (${M}_{\mathrm{max}}$) in unit of ${M}_{\odot}$ and radius (km) at $M=1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) and radius (km) at $M=2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) for TM2$\omega \rho $-b EoS with magnetic fields (${B}_{s}={10}^{12}$ G, ${B}_{0}=2.5\times {10}^{18}$ G).

$\mathit{\gamma}$ | $\mathit{\alpha}$ | ${\mathit{M}}_{\mathbf{max}}$ | ${\mathit{R}}_{1.4\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}_{\odot}}$ | ${\mathit{R}}_{2.072\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}_{\odot}}$ |
---|---|---|---|---|

2 | 0.03 | 2.018 | 13.38 | − |

2 | 0.04 | 2.077 | 13.47 | 12.51 |

2 | 0.05 | 2.135 | 13.59 | 13.18 |

2 | 0.06 | 2.192 | 13.71 | 13.52 |

3 | 0.01 | 2.056 | 13.28 | − |

3 | 0.02 | 2.207 | 13.38 | 13.15 |

3 | 0.03 | 2.321 | 13.51 | 13.55 |

3 | 0.04 | 2.416 | 13.65 | 13.81 |

4 | 0.01 | 2.322 | 13.35 | 13.32 |

4 | 0.02 | 2.503 | 13.53 | 13.76 |

4 | 0.03 | 2.620 | 13.71 | 14.06 |

5 | 0.01 | 2.565 | 13.46 | 13.74 |

5 | 0.02 | 2.731 | 13.71 | 14.16 |

6 | 0.01 | 2.748 | 13.61 | 14.08 |

**Figure 10.**MR relations of DDME2-a (

**left**) and DDME2-b (

**right**) EoSs with magnetic fields. The strength of surface magnetic field strength ${B}_{s}$ is ${10}^{12}$ G, and the central magnetic field strength ${B}_{0}$ is $2.5\times {10}^{18}$ G. The arrow and colored hatched areas are the same as in Figure 1.

**Table 11.**Maximum mass (${M}_{\mathrm{max}}$) in unit of ${M}_{\odot}$ and radius (km) at $M=1.4{M}_{\odot}$ (${R}_{1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) and radius (km) at $M=2.072{M}_{\odot}$ (${R}_{2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) for DDME2-a EoS with magnetic fields (${B}_{s}={10}^{12}$ G, ${B}_{0}=2.5\times {10}^{18}$ G).

$\mathit{\gamma}=1$ | $\mathit{\gamma}=2$ | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | $\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ |

0.01 | 2.163 | 14.18 | 14.43 | 0.01 | 2.174 | 14.18 | 14.45 |

0.02 | 2.168 | 14.22 | 14.47 | 0.02 | 2.205 | 14.21 | 14.51 |

0.03 | 2.176 | 14.27 | 14.52 | 0.03 | 2.247 | 14.24 | 14.59 |

0.04 | 2.187 | 14.34 | 14.59 | 0.04 | 2.295 | 14.30 | 14.68 |

0.05 | 2.200 | 14.44 | 14.68 | 0.05 | 2.345 | 14.36 | 14.76 |

**Table 12.**Maximum mass (${M}_{\mathrm{max}}$) in unit of ${M}_{\odot}$ and radius (km) at $M=1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{1.4\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) and radius (km) at $M=2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ (${R}_{2.072\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}}$) for DDME2-b EoS with magnetic fields (${B}_{s}={10}^{12}$ G, ${B}_{0}=2.5\times {10}^{18}$ G).

$\mathit{\gamma}=$ 1 | $\mathit{\gamma}=$ 2 | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | $\mathbf{\alpha}$ | ${\mathbf{M}}_{\mathrm{max}}$ | ${\mathbf{R}}_{\mathbf{1}.\mathbf{4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ | ${\mathbf{R}}_{\mathbf{2}.\mathbf{072}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{\odot}}$ |

0.01 | 2.362 | 14.18 | 14.59 | 0.01 | 2.370 | 14.18 | 14.60 |

0.02 | 2.366 | 14.22 | 14.62 | 0.02 | 2.394 | 14.21 | 14.62 |

0.03 | 2.372 | 14.27 | 14.66 | 0.03 | 2.427 | 14.24 | 14.67 |

0.04 | 2.381 | 14.34 | 14.71 | 0.04 | 2.465 | 14.30 | 14.73 |

0.05 | 2.391 | 14.44 | 14.78 | 0.05 | 2.505 | 14.36 | 14.79 |

- For $\gamma =1$, $\alpha =$0.01, 0.02, and 0.03;
- For $\gamma =2$, $\alpha =$0.01.

## 4. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**MR relations of 12 EoSs without magnetic fields. The unstable region in each EoS is not shown. The orange and light green dashed areas indicate pulsars PSR J0740+6620 and PSR J0030+0451 (68% (thick) and 90% (thin) credibility), respectively. The arrow indicates the upper limit of radius for 1.4 ${M}_{\odot}$ from the gravitational wave event GW170817. The colored lines represent the following: black (solid line), GM1 EoS; green (solid line), TM1-a EoS; green (dashed line), TM1-b EoS; light blue (solid line), TM2$\omega \rho $-a EoS; light blue (dashed line), TM2$\omega \rho $-b EoS; purple (solid line), NL3-a EoS; purple (dashed line), NL3-b EoS; dark blue (solid line), NL3$\omega \rho $-a EoS; dark blue (dashed line), NL3$\omega \rho $-b EoS; red (solid line), DDME2-a EoS; red (dashed line), DDME2-b EoS.

**Figure 2.**B($\rho $) as a function of baryon number density $\rho $ for various $\alpha $ parameters with ${B}_{0}=2.5\times {10}^{18}$ G and $\gamma =2$. The colored lines represent the following: black (solid line), magnetic fields $B\left(\rho \right)$ with $\alpha =0.01$; purple (solid line), magnetic fields $B\left(\rho \right)$ with $\alpha =0.02$; green (solid line), magnetic fields $B\left(\rho \right)$ with $\alpha =0.03$; light blue (solid line), magnetic fields $B\left(\rho \right)$ with $\alpha =0.04$; orange (solid line), magnetic fields $B\left(\rho \right)$ with $\alpha =0.05$; dark blue (solid line), magnetic fields $B\left(\rho \right)$ with $\alpha =0.06$; black (dashed line), magnetic fields $B\left(\rho \right)$ with $\alpha =0.07$; purple (dashed line), magnetic fields $B\left(\rho \right)$ with $\alpha =0.08$; green (dashed line), magnetic fields $B\left(\rho \right)$ with $\alpha =0.09$; brown (solid line), magnetic fields $B\left(\rho \right)$ with $\alpha =0.10$.

**Figure 3.**B($\rho $) as a function of baryon number density $\rho $ for various $\gamma $ parameters with ${B}_{0}=2.5\times {10}^{18}$ G and $\alpha =0.05$. The colored lines represent the following: black (solid line), magnetic fields $B\left(\rho \right)$ with $\gamma =1$; purple (solid line), magnetic fields $B\left(\rho \right)$ with $\gamma =2$; green (solid line), magnetic fields $B\left(\rho \right)$ with $\gamma =3$; light blue (solid line), magnetic fields $B\left(\rho \right)$ with $\gamma =4$; orange (solid line), magnetic fields $B\left(\rho \right)$ with $\gamma =5$; yellow (solid line), magnetic fields $B\left(\rho \right)$ with $\gamma =6$; dark blue (solid line), magnetic fields $B\left(\rho \right)$ with $\gamma =7$; red (solid line), magnetic fields $B\left(\rho \right)$ with $\gamma =8$; purple (dashed line), magnetic fields $B\left(\rho \right)$ with $\gamma =9$; brown (solid line), magnetic fields $B\left(\rho \right)$ with $\gamma =10$.

**Table 1.**Nuclear properties at saturation number density (${\rho}_{0}$ (fm${}^{-3}$)) predicted by seven kinds of EoSs used in this study [20,28]: energy per nucleon ($B/A$ (MeV)), incompressibility (K (MeV)), symmetry energy (J (MeV)), its slope parameter (L (MeV)), and curvature (${K}_{sym}$ (MeV)) at the saturation point of uniform symmetric nuclear matter [29].

GM1 | GM3 | TM1 | TM2$\mathit{\omega}\mathit{\rho}$ | NL3 | NL3$\mathit{\omega}\mathit{\rho}$ | DDME2 | |
---|---|---|---|---|---|---|---|

${\rho}_{0}$ | 0.153 | 0.153 | 0.146 | 0.146 | 0.148 | 0.148 | 0.152 |

$B/A$ | $-16.3$ | $-16.3$ | $-16.3$ | $-16.4$ | $-16.30$ | $-16.30$ | $-16.14$ |

K | 300 | 240 | 281.2 | 281.7 | 271.8 | 271.8 | 250.9 |

J | 32.5 | 32.5 | 36.9 | 32.1 | 37.4 | 31.7 | 32.3 |

L | 93.9 | 89.7 | 111.2 | 54.8 | 118.9 | 55.5 | 51.2 |

${K}_{sym}$ | 17.9 | $-6.5$ | 33.8 | $-70.5$ | $101.6$ | $-7.6$ | $-87.1$ |

**Table 2.**Adopted ratios of $\sigma $-$\Lambda $ coupling, $\sigma $-$\Sigma $ coupling, and $\sigma $-$\Xi $ coupling in EoS parameter sets.

EoS | ${\mathit{R}}_{\mathit{\sigma}\mathit{\Lambda}}$ | ${\mathit{R}}_{\mathit{\sigma}\mathit{\Sigma}}$ | ${\mathit{R}}_{\mathit{\sigma}\mathit{\Xi}}$ |
---|---|---|---|

TM1-a | 0.64 | 0.56 | 0.32 |

TM1-b | 0.91 | 0.56 | 0.32 |

TM2$\omega \rho $-a | 0.64 | 0.55 | 0.32 |

TM2$\omega \rho $-b | 0.93 | 0.55 | 0.32 |

NL3-a | 0.67 | 0.59 | 0.33 |

NL3-b | 0.97 | 0.59 | 0.33 |

NL3$\omega \rho $-a | 0.67 | 0.59 | 0.33 |

NL3$\omega \rho $-b | 0.97 | 0.59 | 0.33 |

DDME2-a | 0.69 | 0.60 | 0.34 |

DDME2-b | 0.98 | 0.60 | 0.34 |

**Table 3.**Maximum masses, radius at M = 1.4 ${M}_{\odot}$, and radius at M = 2.072 ${M}_{\odot}$ for 12 EoSs without magnetic fields.

EoS | M_{max} (${\mathit{M}}_{\odot}$) | ${\mathit{R}}_{1.4{\mathit{M}}_{\odot}}$ (km) | ${\mathit{R}}_{2.072{\mathit{M}}_{\odot}}$ (km) |
---|---|---|---|

GM1 | 1.784 | 13.5 | − |

GM3 | 1.299 | − | − |

TM1-a | 1.540 | 14.1 | − |

TM1-b | 1.787 | 14.1 | − |

TM2$\omega \rho $-a | 1.663 | 13.2 | − |

TM2$\omega \rho $-b | 1.891 | 13.2 | − |

NL3-a | 2.122 | 14.4 | 14.4 |

NL3-b | 2.336 | 14.4 | 14.8 |

NL3$\omega \rho $-a | 2.175 | 13.6 | 14.1 |

NL3$\omega \rho $-b | 2.372 | 13.6 | 14.2 |

DDME-a | 2.161 | 14.2 | 14.4 |

DDME-b | 2.360 | 14.2 | 14.6 |

**Table 4.**Strengths of the magnetic field at saturation density ${\rho}_{0}=0.153$ fm${}^{-3}$ ( $B\left({\rho}_{0}\right)$ in unit of Gauss) for 12 EoSs with ${B}_{0}=2.5\times {10}^{18}$ G.

$\mathit{\alpha}$ | $\mathit{B}\left({\mathit{\rho}}_{0}\right)$ |
---|---|

0.01 | $2.59\times {10}^{16}$ |

0.02 | $5.05\times {10}^{16}$ |

0.03 | $7.79\times {10}^{16}$ |

0.04 | $9.90\times {10}^{16}$ |

0.05 | $1.23\times {10}^{17}$ |

0.06 | $1.47\times {10}^{17}$ |

0.07 | $1.70\times {10}^{17}$ |

0.08 | $1.93\times {10}^{17}$ |

0.09 | $2.16\times {10}^{17}$ |

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Watanabe, C.; Yoshinaga, N.; Ebata, S.
Equations of State for Hadronic Matter and Mass-Radius Relations of Neutron Stars with Strong Magnetic Fields. *Universe* **2022**, *8*, 48.
https://doi.org/10.3390/universe8010048

**AMA Style**

Watanabe C, Yoshinaga N, Ebata S.
Equations of State for Hadronic Matter and Mass-Radius Relations of Neutron Stars with Strong Magnetic Fields. *Universe*. 2022; 8(1):48.
https://doi.org/10.3390/universe8010048

**Chicago/Turabian Style**

Watanabe, Chinatsu, Naotaka Yoshinaga, and Shuichiro Ebata.
2022. "Equations of State for Hadronic Matter and Mass-Radius Relations of Neutron Stars with Strong Magnetic Fields" *Universe* 8, no. 1: 48.
https://doi.org/10.3390/universe8010048