# General Relativistic Mean-Field Dynamo Model for Proto-Neutron Stars

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Mean-Field Dynamo

#### 2.1. Classical Theory

**U**is the fluid velocity and ${\mathbf{E}}^{\prime}$ is the electric field in the frame comoving with the fluid—into the Faraday’s law of induction, one can write a single evolution equation for

**B**, which is called the induction equation:

#### 2.2. Mean-Field Dynamo in 3 + 1 Resistive GRMHD

**J**appears as a source term. Writing Equation (5) in the 3 + 1 form we find

## 3. Model and Numerical Set-Up

`XNS`code [7,32,33] with an uniform grid in both radial (256 points in the range $[0,30]$ (geometrized units), with the star resolved in about half of the points) and $\theta $ (64 points in the range $[0,\pi ]$) directions. A polytropic relationship between pressure and density is, i.e.,

`ECHO`code [34]. Therefore, at the initial time (i.e., at $t=0$), in addition to the configuration of the stellar fluid just described, a small magnetic field (added “by hand”) and the profiles of the $\xi $ and $\eta $ parameters must be inserted (see below).

## 4. Numerical Results

`ECHO`code using a third order scheme in time and a fifth order scheme in space. Following the prescription of Tomei et al. in [29], we define the average on the star of any quantity $f=f(r,\theta )$ as

#### 4.1. Dependence on the $\alpha $-Dynamo Number

#### 4.2. Dependence on the Initial Configuration of the Magnetic Field

## 5. Summary and Conclusions

`ECHO`code [34] in the upgraded version to include non-ideal resistive and dynamo effects in the Ohm’s law [26,30]. We find that the exponential growth rates follow a quasi-linear dependence on the $\alpha $-dynamo number ${C}_{\xi}$ for both poloidal and toroidal components of the magnetic field, with a slope of about $(7\pm 1)\times {10}^{-2}$ ms${}^{-1}$. Moreover, the dynamo action occurs in a time interval that is much smaller than that on which the evolution of the neutron-finger instability occurs, in support of the hypothesis that the tachocline does not evolve during the action of the dynamo.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Özel, F.; Psaltis, D.; Narayan, R.; Santos Villarreal, A. On the Mass Distribution and Birth Masses of Neutron Stars. Astrophys. J.
**2012**, 757, 55. [Google Scholar] [CrossRef] [Green Version] - Rezzolla, L.; Most, E.R.; Weih, L.R. Using Gravitational-wave Observations and Quasi-universal Relations to Constrain the Maximum Mass of Neutron Stars. Astrophys. J.
**2018**, 852, L25. [Google Scholar] [CrossRef] - Cromartie, H.T.; Fonseca, E.; Ransom, S.M.; Demorest, P.B.; Arzoumanian, Z.; Blumer, H.; Brook, P.R.; DeCesar, M.E.; Dolch, T.; Ellis, J.A.; et al. Relativistic Shapiro delay measurements of an extremely massive millisecond pulsar. Nat. Astron.
**2019**, 4, 72–76. [Google Scholar] [CrossRef] [Green Version] - Kiuchi, L.; Yoshida, S. Relativistic stars with purely toroidal magnetic fields. Phys. Rev. D
**2008**, 78, 044045. [Google Scholar] [CrossRef] [Green Version] - Ciolfi, R.; Ferrari, V.; Gualtieri, L.; Pons, J.A. Relativistic models of magnetars: The twisted torus magnetic field configuration. Mon. Not. R. Astron. Soc.
**2009**, 397, 913–924. [Google Scholar] [CrossRef] [Green Version] - Frieben, J.; Rezzolla, L. Equilibrium models of relativistic stars with a toroidal magnetic field. Mon. Not. R. Astron. Soc.
**2012**, 427, 3406–3426. [Google Scholar] [CrossRef] [Green Version] - Pili, A.G.; Bucciantini, N.; Del Zanna, L. Axisymmetric equilibrium models for magnetized neutron stars in General Relativity under the Conformally Flat Condition. Mon. Not. R. Astron. Soc.
**2014**, 439, 3541–3563. [Google Scholar] [CrossRef] [Green Version] - Bucciantini, N.; Pili, A.G.; Del Zanna, L. The role of currents distribution in general relativistic equilibria of magnetized neutron stars. Mon. Not. R. Astron. Soc.
**2015**, 447, 3278–3290. [Google Scholar] [CrossRef] [Green Version] - Dall’Osso, S.; Shore, S.N.; Stella, L. Early evolution of newly born magnetars with a strong toroidal field. Mon. Not. R. Astron. Soc.
**2009**, 398, 1869–1885. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. Phys. Rev. Lett.
**2017**, 119, 1869–1885. [Google Scholar] [CrossRef] [Green Version] - Usov, V. Millisecond pulsars with extremely strong magnetic fields as a cosmological source of gamma-ray bursts. Nature
**1992**, 357, 472–474. [Google Scholar] - Bucciantini, N.; Quataert, E.; Metzger, B.D.; Thompson, T.A.; Arons, J.; Del Zanna, L. Magnetized relativistic jets and long-duration GRBs from magnetar spin-down during core-collapse supernovae. Mon. Not. R. Astron. Soc.
**2009**, 396, 2038–2050. [Google Scholar] [CrossRef] [Green Version] - Bucciantini, N.; Metzger, B.D.; Thompson, T.A.; Quataert, E. Short gamma-ray bursts with extended emission from magnetar birth: Jet formation and collimation. Mon. Not. R. Astron. Soc.
**2012**, 419, 1537–1545. [Google Scholar] [CrossRef] [Green Version] - Mösta, P.; Ott, C.D.; Radice, D.; Roberts, L.F.; Schnetter, E.; Haas, R. A large-scale dynamo and magnetoturbulence in rapidly rotating core-collapse supernovae. Nature
**2015**, 518, 376–379. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ciolfi, R.; Kastaun, W.; Kalinani, J.V.; Giacomazzo, B. First 100 ms of a long-lived magnetized neutron star formed in a binary neutron star merger. Phys. Rev. D
**2019**, 100, 023005. [Google Scholar] [CrossRef] [Green Version] - Duncan, R.C.; Thompson, C. Formation of very strongly magnetized neutron stars—Implications for gamma-ray bursts. Astrophys. J.
**1992**, 392, L9–L13. [Google Scholar] [CrossRef] - Thompson, C.; Duncan, R.C. Neutron Star Dynamos and the Origins of Pulsar Magnetism. Astrophys. J.
**1993**, 408, 194–217. [Google Scholar] [CrossRef] - Raynaud, R.; Guilet, J.; Janka, H.-T.; Gastine, T. Magnetar formation through a convective dynamo in protoneutron stars. Sci. Adv.
**2020**, 6, eaay2732. [Google Scholar] [CrossRef] [Green Version] - Akiyama, S.; Wheeler, J.C.; Meier, D.L.; Lichtenstadt, I. The Magnetorotational Instability In Core-Collapse Supernova Explosions. Astrophys. J.
**2003**, 584, 954–970. [Google Scholar] [CrossRef] [Green Version] - Ardeljan, N.V.; Bisnovatyi-Kogan, G.S.; Moiseenko, S.G. Magnetorotational supernovae. Mon. Not. R. Astron. Soc.
**2005**, 359, 333–344. [Google Scholar] [CrossRef] [Green Version] - Obergaulinger, M.; Cerdá-Durán, P.; Müller, E.; Aloy, M.A. Semi-global simulations of the magneto-rotational instability in core collapse supernovae. Astron. Astrophys.
**2009**, 498, 241–271. [Google Scholar] [CrossRef] [Green Version] - Reboul-Salze, A.; Guilet, J.; Raynaud, R.; Bugli, M. A global model of the magnetorotational instability in protoneutron stars. arXiv
**2020**, arXiv:2005.03567v1. [Google Scholar] - Bonanno, A.; Rezzolla, L.; Urpin, V. Mean-field dynamo action in protoneutron stars. Astron. Astrophys.
**2003**, 410, L33–L36. [Google Scholar] [CrossRef] [Green Version] - Brandenburg, A.; Subramanian, K. Astrophysical magnetic fields and nonlinear dynamo. Phys. Rep.
**2005**, 417, 1–209. [Google Scholar] [CrossRef] [Green Version] - Naso, L.; Rezzolla, L.; Bonanno, A.; Paternò, L. Magnetic field amplification in proto-neutron stars—The role of the neutron-finger instability for dynamo excitation. Astron. Astrophys.
**2008**, 479, 167–176. [Google Scholar] [CrossRef] [Green Version] - Bucciantini, N.; Del Zanna, L. A fully covariant mean-field dynamo closure for numerical 3+1 resistive GRMHD. Mon. Not. R. Astron. Soc.
**2013**, 428, 71–85. [Google Scholar] [CrossRef] - Parker, E.N. Hydromagnetic Dynamo Models. Astrophys. J.
**1955**, 122, 293. [Google Scholar] [CrossRef] - Bugli, M.; Del Zanna, L.; Bucciantini, N. Dynamo action in thick discs around Kerr black holes: High-order resistive GRMHD simulations. Mon. Not. R. Astron. Soc.
**2014**, 440, L41–L45. [Google Scholar] [CrossRef] [Green Version] - Tomei, N.; Del Zanna, L.; Bugli, M.; Bucciantini, N. General relativistic magnetohydrodynamic dynamo in thick accretion discs: Fully non-linear simulations. Mon. Not. R. Astron. Soc.
**2020**, 491, 2346–2359. [Google Scholar] - Del Zanna, L.; Bucciantini, N. Covariant and 3+1 equations for dynamo-chiral general relativistic magnetohydrodynamics. Mon. Not. R. Astron. Soc.
**2018**, 479, 657–666. [Google Scholar] [CrossRef] - Del Zanna, L.; Papini, E.; Landi, S.; Bugli, M.; Bucciantini, N. Fast reconnection in relativistic plasmas: The magnetohydrodynamics tearing instability revisited. Mon. Not. R. Astron. Soc.
**2016**, 460, 3753–3765. [Google Scholar] [CrossRef] - Bucciantini, N.; Del Zanna, L. GRMHD in axisymmetric dynamical spacetimes—The X-ECHO code. Astron. Astrophys.
**2011**, 528, A101. [Google Scholar] [CrossRef] [Green Version] - Pili, A.G.; Bucciantini, N.; Del Zanna, L. General relativistic models for rotating magnetized neutron stars in conformally flat space-time. Mon. Not. R. Astron. Soc.
**2017**, 470, 2469–2493. [Google Scholar] [CrossRef] - Del Zanna, L.; Zanotti, O.; Bucciantini, N.; Londrillo, P. ECHO: An Eulerian Conservative High Order scheme for general relativistic magnetohydrodynamics and magnetodynamics. Astron. Astrophys.
**2007**, 473, 11–30. [Google Scholar] [CrossRef] [Green Version] - Miralles, J.A.; Pons, J.A.; Urpin, V.A. Convective Instability in Proto–Neutron Stars. Astrophys. J.
**2000**, 543, 1001–1006. [Google Scholar] [CrossRef] [Green Version] - Mastrano, A.; Lasky, P.D.; Melatos, A. Neutron star deformation due to multipolar magnetic fields. Mon. Not. R. Astron. Soc.
**2013**, 447, 3475–3485. [Google Scholar] [CrossRef] [Green Version] - Mastrano, A.; Suvorov, A.G.; Melatos, A. Neutron star deformation due to poloidal-toroidal magnetic fields of arbitrary multipole order: A new analytic approach. Mon. Not. R. Astron. Soc.
**2015**, 447, 3475–3485. [Google Scholar] [CrossRef] [Green Version] - Drago, A.; Lavagno, A.; Pagliara, G. Can very compact and very massive neutron stars both exist? Phys. Rev. D
**2014**, 89, 043014. [Google Scholar] [CrossRef] [Green Version] - Drago, A.; Pagliara, G. Combustion of a hadronic star into a quark star: The turbulent and the diffusive regimes. Phys. Rev. C
**2015**, 92, 045801. [Google Scholar] [CrossRef] [Green Version] - Pili, A.G.; Bucciantini, N.; Drago, A.; Pagliara, G.; Del Zanna, L. Quark deconfinement in the proto-magnetar model of long gamma-ray bursts. Mon. Not. R. Astron. Soc.
**2016**, 462, L26–L30. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Time evolution in the Run1 for the average value of poloidal (${B}_{pol}$) and toroidal (${B}_{tor}$) components of the magnetic field in the first 5 ms. Values in Gauss (G) along the vertical axis.

**Figure 2.**Color maps of the poloidal (${B}_{pol}$, left side of each figure) and toroidal (${B}_{tor}$, right side of each figure) components of the magnetic field at four different times for Run1. The blue solid curve is the surface of the star. In the colorbar, values are in order of (

**a**) ${10}^{12}$ G; (

**b**) 1013 G; (

**c**) ${10}^{14}$ G; (

**d**) ${10}^{15}$ G.

**Figure 3.**Dependence of growth rates ${\omega}_{P}$ (blue squares) and ${\omega}_{T}$ (red stars) on the dynamo number ${C}_{\xi}$. Values in ms${}^{-1}$ along the vertical axis.

**Figure 4.**Time evolution in the Run1 for the average value of poloidal (${B}_{pol}$) and toroidal (${B}_{tor}$) components of the magnetic field in the first 5 ms in the case of a purely dipolar initial magnetic field. Values in Gauss (G) along the vertical axis.

**Table 1.**Parameters of the fluid model in geometrized units ($c=G={M}_{\odot}=1$). The corresponding values in cgs units are reported in the text.

K | $\mathit{\gamma}$ | ${\mathit{\rho}}_{\mathit{c}}$ | ${\Omega}_{\mathit{c}}$ | ${\mathit{A}}^{2}$ | ${\Omega}_{\mathit{eq}}$ | ${\mathit{R}}_{\mathit{eq}}$ |
---|---|---|---|---|---|---|

100 | 2 | $1.28\times {10}^{-3}$ | $2.633\times {10}^{-2}$ | 70 | $9.660\times {10}^{-3}$ | 8.38 |

${\mathit{\xi}}_{0}$ | $\mathit{\eta}$ | ${\mathit{C}}_{\mathit{\xi}}$ | ${\mathit{C}}_{\Omega}$ | ${\mathit{R}}_{\mathit{c}}/\mathit{R}$ | ${\mathit{\alpha}}_{\mathit{q}}$ | |
---|---|---|---|---|---|---|

Run1 | $1.0\times {10}^{-2}$ | $1.0\times {10}^{-3}$ | $0.084\times {10}^{3}$ | $1.170\times {10}^{3}$ | 0.4 | $1.0\times {10}^{-5}$ |

Run2 | $5.0\times {10}^{-2}$ | $1.0\times {10}^{-3}$ | $0.419\times {10}^{3}$ | $1.170\times {10}^{3}$ | 0.4 | $1.0\times {10}^{-5}$ |

Run3 | $1.0\times {10}^{-1}$ | $1.0\times {10}^{-3}$ | $0.838\times {10}^{3}$ | $1.170\times {10}^{3}$ | 0.4 | $1.0\times {10}^{-5}$ |

Run4 | $2.0\times {10}^{-1}$ | $1.0\times {10}^{-3}$ | $1.676\times {10}^{3}$ | $1.170\times {10}^{3}$ | 0.4 | $1.0\times {10}^{-5}$ |

Run5 | $3.0\times {10}^{-1}$ | $1.0\times {10}^{-3}$ | $2.514\times {10}^{3}$ | $1.170\times {10}^{3}$ | 0.4 | $1.0\times {10}^{-5}$ |

**Table 3.**Exponential growth rates (in ms${}^{-1}$) for both poloidal (${\omega}_{P}$) and toroidal (${\omega}_{T}$) components of the magnetic filed for all runs. The corresponding value of the $\alpha $-dynamo number ${C}_{\xi}$ is also reported.

${\mathit{C}}_{\mathit{\xi}}$ | ${\mathit{\omega}}_{\mathit{P}}$ [ms${}^{-1}$] | ${\mathit{\omega}}_{\mathit{T}}$ [ms${}^{-1}$] | |
---|---|---|---|

Run1 | $0.084\times {10}^{3}$ | $2.75\pm 0.02$ | $2.67\pm 0.03$ |

Run2 | $0.419\times {10}^{3}$ | $34.6\pm 0.2$ | $33.5\pm 0.1$ |

Run3 | $0.838\times {10}^{3}$ | $45.8\pm 0.3$ | $45.9\pm 0.2$ |

Run4 | $1.676\times {10}^{3}$ | $152\pm 2$ | $146\pm 1$ |

Run5 | $2.514\times {10}^{3}$ | $169\pm 4$ | $163\pm 3$ |

**Table 4.**Exponential growth rates (in ms${}^{-1}$) for both poloidal (${\omega}_{P}$) and toroidal (${\omega}_{T}$) components of the magnetic filed for Run1. The subscript ${}_{t}$ (${}_{d}$) is for the purely toroidal (dipolar) initial configuration of the magnetic field.

${\mathit{C}}_{\mathit{\xi}}$ | ${\mathit{\omega}}_{\mathit{P}}$ [ms${}^{-1}$] | ${\mathit{\omega}}_{\mathit{T}}$ [ms${}^{-1}$] | |
---|---|---|---|

Run1${}_{t}$ | $0.084\times {10}^{3}$ | $2.75\pm 0.02$ | $2.67\pm 0.03$ |

Run1${}_{d}$ | $0.084\times {10}^{3}$ | $2.69\pm 0.02$ | $2.64\pm 0.01$ |

© 2020 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/).

## Share and Cite

**MDPI and ACS Style**

Franceschetti, K.; Del Zanna, L.
General Relativistic Mean-Field Dynamo Model for Proto-Neutron Stars. *Universe* **2020**, *6*, 83.
https://doi.org/10.3390/universe6060083

**AMA Style**

Franceschetti K, Del Zanna L.
General Relativistic Mean-Field Dynamo Model for Proto-Neutron Stars. *Universe*. 2020; 6(6):83.
https://doi.org/10.3390/universe6060083

**Chicago/Turabian Style**

Franceschetti, Kevin, and Luca Del Zanna.
2020. "General Relativistic Mean-Field Dynamo Model for Proto-Neutron Stars" *Universe* 6, no. 6: 83.
https://doi.org/10.3390/universe6060083