# Beta Equilibrium under Neutron Star Merger Conditions

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## Abstract

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## 1. Introduction

## 2. Beta Equilibration

## 3. Nuclear Matter Models

- ${M}_{\mathrm{max}}>{2.072}_{-0.066}^{+0.067}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ from NICER and XMM analysis of PSR J0740+6620 [43];
- ${M}_{\mathrm{max}}=1.{928}_{-0.017}^{+0.017}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ from NANOGrav analysis of PSR J1614-2230 [44];
- ${M}_{\mathrm{max}}=2.{01}_{-0.14}^{+0.14}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}$ from pulsar timing analysis of PSR J0348+0432 [45].

## 4. Beta Equilibration via Direct Urca

#### 4.1. Neutron Decay

#### 4.2. Electron Capture

## 5. Beta Equilibration via Modified Urca

#### 5.1. Neutron Decay

#### 5.2. Electron Capture

## 6. Results

#### 6.1. Beta Equilibrium at Nonzero Temperature

- At low temperatures $T\lesssim 1\phantom{\rule{0.166667em}{0ex}}\mathrm{MeV}$, the Fermi surface approximation is valid and beta equilibrium is achieved with a negligible correction $\Delta \mu $ (see Section 2).
- At the temperature rises through the neutrino-transparent regime, the value of $\Delta \mu $ rises.
- We only provide results for temperatures up to $5\phantom{\rule{0.166667em}{0ex}}\mathrm{MeV}$ because at temperatures of around 5 to 10 MeV the neutrino mean free path will become smaller than the star, invalidating our assumption of neutrino transparency.
- The figures indicate that the nonzero-temperature correction reaches values of 10 to 20 MeV before neutrino trapping sets in.
- The density dependence of $\Delta \mu $ appears very different for different EoSs. For IUF the largest values are reached at moderate densities, near the direct Urca threshold. For SFHo, $\Delta \mu $ has a minimum at those densities.

#### 6.2. Urca Rates

#### 6.3. Direct Urca Suppression Factors

#### 6.3.1. Neutron Decay

#### 6.3.2. Electron Capture

#### 6.4. Nonrelativistic Rate vs. Relativistic Rate

#### 6.4.1. Direct Urca Neutron Decay

#### 6.4.2. Modified Urca Neutron Decay

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The SFHo Relativistic Mean Field Theory

**Table A1.**SFHo parameter values taken from CompOSE (https://compose.obspm.fr/eos/34, accessed on 27 April 2021). The last three masses are taken from [42].

Quantity | Unit | Value |
---|---|---|

${c}_{\sigma}$ | fm | 3.1791606374 |

${c}_{\omega}$ | fm | 2.2752188529 |

${c}_{\rho}$ | fm | 2.4062374629 |

b | $7.3536466626\times {10}^{-3}$ | |

c | $-3.8202821956\times {10}^{-3}$ | |

$\zeta $ | $-1.6155896062\times {10}^{-3}$ | |

$\xi $ | $4.1286242877\times {10}^{-3}$ | |

${a}_{1}$ | ${\mathrm{fm}}^{-1}$ | $-1.9308602647\times {10}^{-1}$ |

${a}_{2}$ | $5.6150318121\times {10}^{-1}$ | |

${a}_{3}$ | fm | $2.8617603774\times {10}^{-1}$ |

${a}_{4}$ | ${\mathrm{fm}}^{2}$ | 2.7717729776 |

${a}_{5}$ | ${\mathrm{fm}}^{3}$ | 1.2307286924 |

${a}_{6}$ | ${\mathrm{fm}}^{4}$ | $6.1480060734\times {10}^{-1}$ |

${b}_{1}$ | 5.5118461115 | |

${b}_{2}$ | ${\mathrm{fm}}^{2}$ | −1.8007283681 |

${b}_{3}$ | ${\mathrm{fm}}^{4}$ | $4.2610479708\times {10}^{2}$ |

${m}_{\sigma}$ | ${\mathrm{fm}}^{-1}$ | 2.3689528914 |

${m}_{\omega}$ | ${\mathrm{fm}}^{-1}$ | 3.9655047020 |

${m}_{\rho}$ | ${\mathrm{fm}}^{-1}$ | 3.8666788766 |

${m}_{n}$ | MeV | $939.565346$ |

${m}_{p}$ | MeV | $938.272013$ |

M | MeV | 939 |

## Appendix B. Direct Urca Neutron Decay Rate

#### Limits of Angular Integration

## Appendix C. Modified Urca Neutron Decay Rate

## References

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**Figure 1.**Direct Urca momentum surplus ${k}_{Fp}+{k}_{Fe}-{k}_{Fn}$ for IUF and SFHo equations of state at $T=0\phantom{\rule{0.166667em}{0ex}}$. When the surplus is negative, direct Urca is forbidden. IUF has an upper density threshold above which direct Urca is allowed; SFHo does not.

**Figure 2.**Density dependence of the neutron’s (Dirac) effective mass and Fermi momentum for the IUF and SFHo EoSs, showing that neutrons at the Fermi surface become relativistic at densities above 2 to 3 ${n}_{0}$.

**Figure 3.**Density dependence of the proton’s (Dirac) effective mass and Fermi momentum for the IUF and SFHo EoSs, showing that protons at the Fermi surface become relativistic starting at densities between $3-6{n}_{0}$.

**Figure 4.**Direct Urca neutron decay rate calculated using relativistic, nonrelativistic and the vacuum dispersion relations at $T=3\phantom{\rule{0.166667em}{0ex}}$MeV for IUF.

**Figure 5.**Modified Urca rate calculated using relativistic and nonrelativistic dispersion relations at $T=3\phantom{\rule{0.166667em}{0ex}}$MeV for IUF. (n) stands for neutron-spectator modified Urca and (p) stands for proton-spectator modified Urca.

**Figure 6.**Nonzero-temperature correction $\Delta \mu $ required for beta equilibrium Equation (7) with the IUF EoS.

**Figure 7.**Nonzero-temperature correction $\Delta \mu $ required for beta equilibrium Equation (7) with the SFHo EoS.

**Figure 8.**Urca (direct plus modified) rates for IUF and SFHo EoSs at $T=3\phantom{\rule{0.166667em}{0ex}}\mathrm{MeV}$. When $\Delta \mu =0$ (dashed lines) the rates for neutron decay (nd) and electron capture (ec) do not balance. With the correct choice of $\Delta \mu $ (Figure 6 and Figure 7) the neutron decay and electron capture rates (solid lines) become equal, and the system is in beta equilibrium.

**Figure 9.**Urca rates calculated using the IUF EoS at $T=3\phantom{\rule{0.166667em}{0ex}}$MeV. Because $\Delta \mu =0$ there is a large mismatch between the direct Urca rates for neutron decay and electron capture. Modified Urca (with neutron spectator (n) and proton spectator (p)) rates are calculated in the Fermi surface approximation and therefore match automatically.

**Figure 10.**Urca rates calculated using the SFHo EoS at $T=3\phantom{\rule{0.166667em}{0ex}}$MeV. Because $\Delta \mu =0$ there is a large mismatch between the direct Urca rates for neutron decay and electron capture. Modified Urca (with neutron spectator (n) and proton spectator (p)) rates are calculated in the Fermi surface approximation and therefore match automatically.

**Figure 11.**The optimal kinematics for neutron decay for the IUF EoS.

**Left panel**: the least suppressed kinematic arrangement, showing the energy distance $\gamma $ of each particle from its Fermi surface.

**Right panel**: the Fermi-Dirac suppression factor, ${e}^{-|{\gamma}_{e}|/T}{e}^{-|{\gamma}_{n}|\Theta \left({\gamma}_{n}\right)/T}$ which is dominated by the difficulty of finding an electron hole at energy ${\gamma}_{e}$ below its Fermi surface.

**Figure 12.**The optimal kinematics for electron capture for the IUF EoS.

**Left panel**: the least suppressed kinematic arrangement, showing the energy distance $\gamma $ of each particle from its Fermi surface.

**Right panel**: the overall Fermi-Dirac suppression factor, ${e}^{-|{\gamma}_{p}|/T}{e}^{-|{\gamma}_{e}|\Theta \left({\gamma}_{e}\right)/T}{e}^{-|{\gamma}_{n}|\Theta (-{\gamma}_{n})/T}$, which is dominated by the difficulty of finding a proton at energy ${\gamma}_{p}$ above its Fermi surface.

**Figure 13.**The optimal kinematics for neutron decay at $T=3\phantom{\rule{0.166667em}{0ex}}$MeV for SFHo, obtained by maximizing the Fermi-Dirac products. The suppression factor, ${e}^{-|{\gamma}_{e}|/T}{e}^{-|{\gamma}_{n}|\Theta \left({\gamma}_{n}\right)/T}$ is dominated by the difficulty of finding an electron hole below its Fermi surface.

**Figure 14.**The optimal kinematics for electron capture at $T=3\phantom{\rule{0.166667em}{0ex}}$MeV for SFHo, obtained by maximizing the Fermi-Dirac products. The suppression factor, ${e}^{-|{\gamma}_{p}|/T}{e}^{-|{\gamma}_{e}|\Theta \left({\gamma}_{e}\right)/T}{e}^{-|{\gamma}_{n}|\Theta (-{\gamma}_{n})/T}$, is dominated by the difficulty of finding a proton above its Fermi surface.

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**MDPI and ACS Style**

Alford, M.G.; Haber, A.; Harris, S.P.; Zhang, Z.
Beta Equilibrium under Neutron Star Merger Conditions. *Universe* **2021**, *7*, 399.
https://doi.org/10.3390/universe7110399

**AMA Style**

Alford MG, Haber A, Harris SP, Zhang Z.
Beta Equilibrium under Neutron Star Merger Conditions. *Universe*. 2021; 7(11):399.
https://doi.org/10.3390/universe7110399

**Chicago/Turabian Style**

Alford, Mark G., Alexander Haber, Steven P. Harris, and Ziyuan Zhang.
2021. "Beta Equilibrium under Neutron Star Merger Conditions" *Universe* 7, no. 11: 399.
https://doi.org/10.3390/universe7110399