xx HydroRNSID

HydroxRNSID

Nik Stergioulas, Roberto De Pietri, Frank Lxfller

August 1 2017

Abstract

HydroxRNSID - 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 seex[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 HydroxRNSID 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 “HydroxBase” and “GRHydro” are the two prerequisites)

The model are generated specifing the central baryonic density (rhoxcentral), the oblatness of the Star (axesxratio) and the rotational profile (rotationxtype). 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 Axdiff=): where is the equatorial radius of the star and is the rotational angular velocity and is 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_BasexGRHydroxHydro_RNSID"
#####
#####xSettingxforxactivatingxthexID
#####
ADMBase::initial_dataxx=x"hydro_rnsid"
ADMBase::initial_lapsex=x"hydro_rnsid"
ADMBase::initial_shiftx=x"hydro_rnsid"

The correspongig section of the parameter file is:

#####
#####xBasicxSetting
#####
Hydro_rnsid::rho_centralxxx=x1.28e-3xx#xcentralxbaryonxdensityx(G=c=1)
Hydro_rnsid::axes_ratioxxxx=x1xxxxxxxx#xradial/equatorialxaxesxratio
Hydro_rnsid::rotation_typex=xdiffxxxxx#xuniformx=xuniformxrotation
Hydro_rnsid::A_diffxxxxxxxx=x1xxxxxxxx#xParameterxofxthexdiffxrot-law.
Hydro_rnsid::accuracyxxxxxx=x1e-10xxxx#xaccuracyxgoalxforxconvergence

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:

#####
#####xSettingxforxpolytrope
#####
Hydro_rnsid::eos_typexx=x"poly"
Hydro_rnsid::RNS_Gammax=x2.0
Hydro_rnsid::RNS_Kxxxxx=x165

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

and for the above choice of parameters corresponding to the choice (in units where ) and .

Tabulated EOS:

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

#####
#####xSettingxforxtabulatedxEOS
#####
Hydro_rnsid::eos_typexx=x"tab"
Hydro_rnsid::eos_filexx=x"full_path_name_of_thextabulated_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 () while the next lines contain the values: , , , and , 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.

#####
#####xSettingxforxrecoverxandxsavingxofx2dxmodels
#####
Hydro_rnsid::save_2Dmodelxxxx=x"yes"xxx#xotherxpossibilityxisxnox(default)
Hydro_rnsid::recover_2Dmodelx=x"yes"xxx#xotherxpossibilityxisxnox(default)
Hydro_rnsid::model2D_filexxxx=x"full_file_name"

For examples of initial data generated using RNSID and their evolutions, seex[1,x2]. 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 inx[1,x2].

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 RNSxreadID is provided that reads a 2d initial data file and produces an hdf5 version of the data interpolated onto a 3d grid.

References

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

[2]xxxF.xLxffler, R.xDe Pietri, A.xFeo, F.xMaione and L.xFranci, 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]xxxN. Stergioulas and J.xL. Friedmann. Comparing models of rapidly rotating relativistic stars constructed by two numerical methods. ApJ., 444, 306, 1995.

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

[5]xxxN.

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

11axdiff 10Scope: privatex01 REAL

11Description: constant A in differential rotation law

11 Range 10 x01 Default: 1.0
110.0:  10Any positive number

11

11accuracy  10Scope: privatex01  REAL

11Description: rnsid accuracy in convergence

11 Range 10 x01 Default: 1.0e-7
110:  10Any positive number

11

11axesxratio  10Scope: privatex01  REAL

11Description: rnsid axes ratio

11 Range 10 x01 Default: 1
110:  10Any positive number

11

11cf  10Scope: privatex01  REAL

11Description: Convergence factor

11 Range 10 x01 Default: 1.0
110:  10Any positive number

11eosxfile  10Scope: privatex01  STRING

11Description: Equation of state table

11 Range 10 x01 Default: (none)
11.*  10EOS table file

11

11eosxtype  10Scope: privatex01  KEYWORD

11Description: Specify type of equation of state

11 Range 10 x01 Default: poly
11poly  10Polytropic EOS
11tab  10Tabulated EOS

11

11model2dxfile  10Scope: privatex01  STRING

11Description: Name of 2D model file

11 Range 10 x01 Default: model2D.dat
11.*  10Default 2D model file

11

11recoverx2dmodel 10Scope: privatex01  KEYWORD

11Description: Recover 2D model?

11 Range 10 x01 Default: no
11yes  10recover 2D model
11no  10don’t recover 2D model

11

11rhoxcentral  10Scope: privatex01  REAL

11Description: Central Density for Star

11  Range  10  x01  Default: 1.24e-3

11:  10

11

11rnsxatmoxtolerance 10Scope: privatex01  REAL

11Description: A point is set to atmosphere if rho < (1+RNSxatmoxtolerance)*RNSxrhoxmin

11 Range 10 x01 Default: 0.00001
110.0:  10Zero or larger. A useful value could be 0.0001

11

11rnsxgamma  10Scope: privatex01  REAL

11Description: If we’re using a different EoS at run time, this is the RNS Gamma

11 Range 10 x01 Default: 2
11*:*  10Will be ignored if negative

11

11rnsxk  10Scope: privatex01  REAL

11Description: If we’re using a different EoS at run time, this is the RNS K

11 Range 10 x01 Default: 100
11*:*  10Will be ignored if negative

11

11rnsxlmax  10Scope: privatex01  INT

11Description: max. term in Legendre poly.

11 Range 10 x01 Default: 10
111:  10Any positive, non zero number

11

11rnsxrhoxmin  10Scope: privatex01  REAL

11Description: A minimum rho below which evolution is turned off (atmosphere).

11 Range 10 x01 Default: 1.0e-14
110.0:  10Atmosphere detection for RNSID

11rotationxtype 10Scope: privatex01  KEYWORD

11Description: Specify type of rotation law

11 Range 10 x01 Default: uniform
11uniform  10uniform rotation
11diff  10KEH differential rotation law

11

11savex2dmodel 10Scope: privatex01  KEYWORD

11Description: Save 2D model?

11 Range 10 x01 Default: no
11yes  10save 2D model
11no  10don’t save 2D model

11

11zeroxshift  10Scope: privatex01  KEYWORD

11Description: Set shift to zero?

11 Range 10 x01 Default: no
11yes  10set shift to zero
11no  10don’t set shift to zero

11

11initialxhydro 10Scope: shared from HYDROBASEx01  KEYWORD

11Extends ranges:

11hydroxrnsid  10Construnct stationary initial data with rnsid

11

11timelevels  10Scope: shared from HYDROBASEx01INT

11

6 Interfaces

General Implements: hydroxrnsid

Inherits: admbase hydrobase

Uses header: FishEye.h Boundary.h Symmetry.h 7 Schedule This section lists all the variables which are assigned storage by thorn EinsteinInitialData/HydroxRNSID. 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 x

11x11Conditional:
11x 11ADMBase::metric[2] ADMBase::curv[2] ADMBase::lapse[2] ADMBase::shift[2]

11x 11x

11

Scheduled Functions

CCTKxPARAMCHECK (conditional) x hydroxrnsidxcheckparameters x check parameters x

11x11Language:11c
11 x 11Type: 11function

11

HydroBasexInitial (conditional) x hydroxrnsidxinit x create rotating neutron star initial data x

11x11Language:11c
11 x 11Sync:  11admbase::metric
11 x 11x  11admbase::curv
11 x 11x  11admbase::lapse
11 x 11x  11admbase::shift
11 x 11x  11hydrobase::rho
11 x 11x  11hydrobase::press
11 x 11x  11hydrobase::eps
11 x 11x  11hydrobase::vel
11 x 11Type: 11function