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).

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

3 Parameters

[1]

poloidal\_A\_interior

[1]

dipolar\_A\_everywhere

4 Interfaces

General

Implements:

seed_magnetic_fields_privt

Inherits:

grid

admbase

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

Scheduled Functions

HydroBase_Initial

  seed_magnetic_fields_privt

  set up binary neutron star seed magnetic fields.