## Seed_Magnetic_Fields_BNS: An Einstein Toolkit thorn for seeding magnetic fields to binary neutron star systems

Feb 2, 2022

### Abstract

Seed_Magnetic_Fields_BNS is designed to seed a poloidal magnetic field to binary neutron star (BNS) initial data.

### 1 Overview

We provide a basic overview of the mathematical expressions implemented by this thorn, as well as key references.

#### 1.1 Poloidal $$\vec {A}$$: A_field_type = poloidal_A_interior.

We seed a poloidal magnetic field in the interior of the two neutron stars (NSs) following the prescription described in Appendix C of [1], namely \begin {align} A_{x} &= -yA_{\rm b}\max \left (P-P_{\rm cut},0\right )\;,\\ A_{y} &= +xA_{\rm b}\max \left (P-P_{\rm cut},0\right )\;,\\ A_{z} &= 0\;. \end {align}

It is recommended to set $$P_{\rm cut}$$ to about 4% of the initial maximum pressure. Furthermore, $$A_{\rm b}$$ should be adjusted in order to obtain the desired initial magnetic-to-gas pressure ratio.

#### 1.2 Dipolar $$\vec {A}$$: A_field_type = dipolar_A_everywhere

We seed a dipolar magnetic field onto the BNS system by extending the prescription of [2] from one to two NSs. Namely, we consider that each neutron star contributes the following term to the vector potential $$\vec {A}$$ in spherical basis: \begin {equation} A^{\phi ,n} = \frac {\pi r_{0,n}^{2}I_{0,n}\varpi ^{2}}{\left (r_{0,n}^{2} + r_{n}^{2}\right )^{3/2}} \left [1 + \frac {15}{8}\frac {r_{0,n}^{2}\left (r_{0,n}^{2} + \varpi _{n}^{2}\right )}{\left (r_{0,n}^{2}+r_{n}^{2}\right )}\right ]\;, \end {equation} where $$n=1,2$$ identifies the neutron star, $$r_{0,n}^{2} \equiv r^{\mathrm {NS}}_{n}r^{\mathrm {NS}}_{0,n}$$, where $$r^{\mathrm {NS}}_{n}$$ are the NS radii and $$r^{\mathrm {NS}}_{0,n}$$ the radii of the current loops, $$r_{n}^{2} = \left (x-x^{\mathrm {NS}}_{n}\right )^{2}+y^{2}+z^{2}$$, $$x^{\mathrm {NS}}_{n}$$ is the position of each neutron star (assumed to be in the $$x$$-axis), and $$\varpi _{n}^{2} = \left (x-x_{n}\right )^{2}+y^{2}$$.

We then set the vector potential in the Cartesian basis using the standard formulae \begin {align} A^{x} &= -\left (y+\frac {\Delta y}{2}\right )\left (A^{\phi ,1} + A^{\phi ,2}\right )\;,\\ A^{y} &= \left (x_{1}+\frac {\Delta x}{2}\right )A^{\phi ,1} + \left (x_{2}+\frac {\Delta x}{2}\right )A^{\phi ,2}\;. \end {align}

Note that the behavior described above is valid when using staggered vector potentials (see Table 1 for details).

 Variable Storage location Non-staggered $$\left (x_{i},y_{j},z_{k}\right )$$ $$A^{x}$$ $$\left (x_{i},y_{j}+\frac {\Delta y}{2},z_{k}+\frac {\Delta z}{2}\right )$$ $$A^{y}$$ $$\left (x_{i}+\frac {\Delta x}{2},y_{j},z_{k}+\frac {\Delta z}{2}\right )$$ $$A^{z}$$ $$\left (x_{i}+\frac {\Delta x}{2},y_{j}+\frac {\Delta y}{2},z_{k}\right )$$

Table 1: Storage location on grid of vector potential $$A^{i}$$.

### 2 Basic usage

The following lines provide an example of how to use this thorn to use this thorn to seed a poloidal magnetic field to the interior of two stars located at $$x=\pm 15$$ (in code units). The star radii are not specified explicitly, and we use the default radius $$13.5$$ as a good enough estimate.

ActiveThorns = "Seed_Magnetic_Fields_BNS"
Seed_Magnetic_Fields_BNS::enable_IllinoisGRMHD_staggered_A_fields = "yes"
Seed_Magnetic_Fields_BNS::A_field_type = "poloidal_A_interior"
Seed_Magnetic_Fields_BNS::have_two_NSs_along_x_axis = "yes"
Seed_Magnetic_Fields_BNS::x_c1 = +15
Seed_Magnetic_Fields_BNS::x_c2 = -15
Seed_Magnetic_Fields_BNS::A_b = 49529.954
Seed_Magnetic_Fields_BNS::n_s = 2.0
# Set to 4% initial max pressure, which is 0.000113824867150113*0.04 = .00000455299468600452
Seed_Magnetic_Fields_BNS::P_cut = 0.00000455299468600452


### References

[1]   Z. B. Etienne, V. Paschalidis, R. Haas, P. Mösta, & S. L. Shapiro, IllinoisGRMHD: an open-source, user-friendly GRMHD code for dynamical spacetimes, Classical and Quantum Gravity, 32(17), 175009 (2015).

[2]   V. Paschalidis, Z. B. Etienne, & S. L. Shapiro, General-relativistic simulations of binary black hole-neutron stars: precursor electromagnetic signals, Physical Review D, 88(2), 021504 (2013).

### 3 Parameters

 a_b Scope: restricted REAL Description: Magnetic field strength parameter. Range Default: 1e-3 *:* Any real

 a_field_type Scope: restricted KEYWORD Description: Which field structure to use. Range Default: poloidal_A_interior see [1] below Dipole magnetic field, interior to the star see [1] below Dipole magnetic field everywhere

[1]

poloidal\_A\_interior


[1]

dipolar\_A\_everywhere


 enable_illinoisgrmhd_staggered_a_fields Scope: restricted BOOLEAN Description: Define A fields on an IllinoisGRMHD staggered grid Default: no

 have_two_nss_along_x_axis Scope: restricted BOOLEAN Description: Do we have two NSs centered along x-axis? Default: no

 i_zero_ns1 Scope: restricted REAL Description: Magnetic field loop current of NS1. Range Default: 0.0 0.0:*)

 i_zero_ns2 Scope: restricted REAL Description: Magnetic field loop current of NS2. Range Default: 0.0 0.0:*)

 n_s Scope: restricted REAL Description: Magnetic field strength pressure exponent. Range Default: 1.0 *:* Any real

 p_cut Scope: restricted REAL Description: Cutoff pressure, below which vector potential is set to zero. Typically set to 4% of the maximum initial pressure. Range Default: 1e-5 0:* Positive

 r_ns1 Scope: restricted REAL Description: Radius of NS1. Does not have to be perfect, but must not overlap other star. Range Default: 13.5 0:* Any positive

 r_ns2 Scope: restricted REAL Description: Radius of NS2. Does not have to be perfect, but must not overlap other star. Range Default: 13.5 0:* Any positive

 r_zero_ns1 Scope: restricted REAL Description: Current loop radius of NS1. Range Default: 1.0 0.0:*)

 r_zero_ns2 Scope: restricted REAL Description: Current loop radius of NS2. Range Default: 1.0 0.0:*)

 x_c1 Scope: restricted REAL Description: x coordinate of NS1 center Range Default: -15.2 *:* Any real

 x_c2 Scope: restricted REAL Description: x coordinate of NS2 center Range Default: 15.2 *:* Any real

### 4 Interfaces

#### General

Implements:

seed_magnetic_fields_privt

Inherits:

grid

hydrobase

### 5 Schedule

This section lists all the variables which are assigned storage by thorn WVUThorns_Diagnostics/Seed_Magnetic_Fields_BNS. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

#### Storage

 Always: HydroBase::rho[1] HydroBase::press[1] HydroBase::eps[1] HydroBase::vel[1] HydroBase::Bvec[1] HydroBase::Avec[1] HydroBase::Aphi[1]

#### Scheduled Functions

HydroBase_Initial

seed_magnetic_fields_privt

set up binary neutron star seed magnetic fields.

 After: meudon_bin_ns_initialise Before: illinoisgrmhd_id_converter Language: c Type: function