## Hydro_RNSID

August 1 2017

### Abstract

Hydro_RNSID - rotating relativistic neutron stars.

### 1 Introduction

This thorn generates neutron star initial data for the GRHydro code. As with the Einstein Toolkit code itself, please feel free to add, alter or extend any part of this code. However please keep the documentation up to date (even, or especially, if it’s just to say what doesn’t work).

This thorn effectively takes the public domain code RNSID written by Nik Stergioulas and interpolates the output onto a Cartesian grid. This porting is based on an initila porting to Whisky by Luca Baiotti and Ian Hawke and has been adapted to GRHydro and Einstein Toolkit.

### 2 RNSID

RNSID, or rotating neutron star initial data, is a code based on the Komatsu-Eriguch-Hachisu (KEH) method for constructing models of rotating neutron stars. It allows for polytropic or tabulated equations of state. For more details of the how the code works see [3], [4] (appendix A is particularly helpful) or especially [5] which is the most up to date and lists other possible methods of constructing rotating neutron star initial data.

In short Hydro_RNSID is a thorn that generate initial model for rotating isolated stars described by a zero-temperature tabulated Equation of State or an iso-entrophic politropic EOS. The activation of the thorn for genereting ID (The thorns “Hydro_Base” and “GRHydro” are the two prerequisites)

The model are generated specifing the central baryonic density (rho_central), the oblatness of the Star (axes_ratio) and the rotational profile (rotation_type). Currently two kinds of rotational profiles are implemented: “uniform” for uniformly rotating stars and “diff” for differentially rotating stars, described by the j-law profile (parametrized by the parameter A_diff=$Â$):

 ${\mathrm{\Omega }}_{c}-\mathrm{\Omega }=\frac{1}{{Â}^{2}{r}_{e}^{2}}\left[\frac{\left(\mathrm{\Omega }-\omega \right){r}^{2}{\mathrm{sin}}^{2}𝜃{e}^{-2\nu }}{1-{\left(\mathrm{\Omega }-\omega \right)}^{2}{r}^{2}{\mathrm{sin}}^{2}𝜃{e}^{-2\nu }}\right]$ (1)

where ${r}_{e}$ is the equatorial radius of the star and $\mathrm{\Omega }$ is the rotational angular velocity $\mathrm{\Omega }={u}^{\varphi }∕{u}^{0}$ and ${\mathrm{\Omega }}_{c}$ is $\mathrm{\Omega }$ at the center of the star.

### 3 Parameters of Thorn

Here one can find definition of the main parameter the determine the behaviour of the Thorn. The activation of the RNSID initial data is achieved by the following line:

ActiveThorns="Hydro_Base GRHydro Hydro_RNSID"
#####
##### Setting for activating the ID
#####


The correspongig section of the parameter file is:

#####
##### Basic Setting
#####
Hydro_rnsid::rho_central   = 1.28e-3  # central baryon density (G=c=1)
Hydro_rnsid::axes_ratio    = 1        # radial/equatorial axes ratio
Hydro_rnsid::rotation_type = diff     # uniform = uniform rotation
Hydro_rnsid::A_diff        = 1        # Parameter of the diff rot-law.
Hydro_rnsid::accuracy      = 1e-10    # accuracy goal for convergence


Than a section for setting the Equation of State (EOS) should be added. If this section is missing a “poly” EOS will be used with default parameters. The two possibilities are:

Isentropic Polytrope:

In this case the base setting for the initial data are specified giving the following parameters:

     #####
##### Setting for polytrope
#####
Hydro_rnsid::eos_type  = "poly"
Hydro_rnsid::RNS_Gamma = 2.0
Hydro_rnsid::RNS_K     = 165


They correspond at the following implementation of the EOS that it is consistent with the 1st Law of thermodinamics.

$\begin{array}{rcll}p& =& K\cdot {\rho }^{\Gamma }& \text{(2)}\text{}\text{}\\ 𝜖& =& \frac{K}{\Gamma -1}\cdot {\rho }^{\Gamma -1}& \text{(3)}\text{}\text{}\end{array}$

and for the above choice of parameters corresponding to the choice $K=165$ (in units where $G=c={M}_{\odot }=1$) and $\Gamma =2$.

Tabulated EOS:

In this case the (cold) EOS used to generate the initial data is read from a file

     #####
##### Setting for tabulated EOS
#####
Hydro_rnsid::eos_type  = "tab"
Hydro_rnsid::eos_file  = "full_path_name_of_the tabulated_EOS_file"


The syntax of the tabulated file is the same as for the original RNSID program and assumes that all quantities are expressed in the cgs system of units. The first line contains the number of tabulated values ($N$) while the next $N$ lines contain the values: $e=\rho \left(1+𝜀\right)$, $p$, $\mathrm{log}h={c}^{2}{\mathrm{log}}_{e}\left(\left(e+p\right)∕\rho \right)$, and $\rho$, respectively.

An additional section allows one to start initial data from a previously generated binary file or to save the data generated at this time. Usually the best way to proceed is to specify where the initial data file should be located.

#####
##### Setting for recover and saving of 2d models
#####
Hydro_rnsid::save_2Dmodel    = "yes"   # other possibility is no (default)
Hydro_rnsid::recover_2Dmodel = "yes"   # other possibility is no (default)
Hydro_rnsid::model2D_file    = "full_file_name"


For examples of initial data generated using RNSID and their evolutions, see [12]. In the par directory, examples are provided as a perl file that produces the corresponding Cactus par files. These examples correspond to the evolutions described in [12].

### 4 Utility program

Together with the Thorn, we distribute a self-executable version of the initial data routine RNSID that accepts the same parameters as the thorn and is able to create a binary file of the 2d initial data that can be directly imported into the evolution code. Moreover, the program RNS_readID is provided that reads a 2d initial data file and produces an hdf5 version of the data interpolated onto a 3d grid.

### References

[1]   J. A. Font, N. Stergioulas and K. D. Kokkotas. Nonlinear hydrodynamical evolution of rotating relativistic stars: Numerical methods and code tests. Mon. Not. Roy. Astron. Soc., 313, 678, 2000.

[2]   F. Löffler, R. De Pietri, A. Feo, F. Maione and L. Franci, Stiffness effects on the dynamics of the bar-mode instability of neutron stars in full general relativity. Phys. Rev., D 91, 064057, 2015 (arXiv:1411.1963).

[3]   N. Stergioulas and J. L. Friedmann. Comparing models of rapidly rotating relativistic stars constructed by two numerical methods. ApJ., 444, 306, 1995.

[4]   N. Stergioulas. The structure and stability of rotating relativistic stars. PhD thesis, University of Wisconsin-Milwaukee, 1996.

[5]   N. Stergioulas. Rotating Stars in Relativity Living Rev. Relativity, 1, 1998. [Article in online journal], cited on 18/3/02, http://www.livingreviews.org/Articles/Volume1/1998-8stergio/index.html.

### 5 Parameters

 a_diff Scope: private REAL Description: constant A in differential rotation law Range Default: 1.0 0.0: Any positive number

 accuracy Scope: private REAL Description: rnsid accuracy in convergence Range Default: 1.0e-7 0: Any positive number

 axes_ratio Scope: private REAL Description: rnsid axes ratio Range Default: 1 0: Any positive number

 cf Scope: private REAL Description: Convergence factor Range Default: 1.0 0: Any positive number

 eos_file Scope: private STRING Description: Equation of state table Range Default: (none) .* EOS table file

 eos_type Scope: private KEYWORD Description: Specify type of equation of state Range Default: poly poly Polytropic EOS tab Tabulated EOS

 model2d_file Scope: private STRING Description: Name of 2D model file Range Default: model2D.dat .* Default 2D model file

 recover_2dmodel Scope: private KEYWORD Description: Recover 2D model? Range Default: no yes recover 2D model no don’t recover 2D model

 rho_central Scope: private REAL Description: Central Density for Star Range Default: 1.24e-3 :

 rns_atmo_tolerance Scope: private REAL Description: A point is set to atmosphere if rho < (1+RNS_atmo_tolerance)*RNS_rho_min Range Default: 0.00001 0.0: Zero or larger. A useful value could be 0.0001

 rns_gamma Scope: private REAL Description: If we’re using a different EoS at run time, this is the RNS Gamma Range Default: 2 *:* Will be ignored if negative

 rns_k Scope: private REAL Description: If we’re using a different EoS at run time, this is the RNS K Range Default: 100 *:* Will be ignored if negative

 rns_lmax Scope: private INT Description: max. term in Legendre poly. Range Default: 10 1: Any positive, non zero number

 rns_rho_min Scope: private REAL Description: A minimum rho below which evolution is turned off (atmosphere). Range Default: 1.0e-14 0.0: Atmosphere detection for RNSID

 rotation_type Scope: private KEYWORD Description: Specify type of rotation law Range Default: uniform uniform uniform rotation diff KEH differential rotation law

 save_2dmodel Scope: private KEYWORD Description: Save 2D model? Range Default: no yes save 2D model no don’t save 2D model

 zero_shift Scope: private KEYWORD Description: Set shift to zero? Range Default: no yes set shift to zero no don’t set shift to zero

 initial_hydro Scope: shared from HYDROBASE KEYWORD Extends ranges: hydro_rnsid Construnct stationary initial data with rnsid

 timelevels Scope: shared from HYDROBASE INT

Implements:

hydro_rnsid

Inherits:

hydrobase

FishEye.h

Boundary.h

Symmetry.h

### 7 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinInitialData/Hydro_RNSID. 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.

#### Scheduled Functions

CCTK_PARAMCHECK (conditional)

hydro_rnsid_checkparameters

check parameters

 Language: c Type: function

HydroBase_Initial (conditional)

hydro_rnsid_init

create rotating neutron star initial data