Hydro_RNSID

March 31, 2019

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=$Â$):

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

where ${r}_{e}$ is the equatorial radius of the star and $\Omega$ is the rotational angular velocity $\Omega ={u}^{\varphi }∕{u}^{0}$ and ${\Omega }_{c}$ is $\Omega$ 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: $e=\rho \left(1+𝜀\right)$, $p$, $logh={c}^{2}{log}_{e}\left(\left(e+p\right)∕\rho \right)$, 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 [1, 2]. 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 [1, 2].

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.