Equations of State (EOS)

The FUKA solvers support both piecewise polytropic and tabulated EOS in generating ID for Neutron Stars. For both cases, a file is required which includes either the tabulated EOS or describes the piecewise polytrope, as described below.

Piecewise Polytropic Equations of State

The file format for storing a piecewise polytrope is quite basic. All lines starting with a # or are blank are ignored. Most importantly, the order of parameters is strictly defined

#h4 pwpolytrope
num_pieces: 4
rhomin: 1.e-19
rhomax: 1.0
ktab0: 3.99873692e-8
Ptab0: 0
 
gamma_tab
1.35692395 2.909 2.246 2.144
 
rho_tab
0.0 8.87824e+14 5.01187234e+14 1.0e+15
 
#units can be:
#geometrised
#cgs
#cgs_cgs_over_c2
units: cgs_cgs_over_c2

In the first block is an ordered list (the order matters) of:
- the number of pieces used to construct the piecewise polytrope
- the extrema of the allowed densities in geometrized units. Note, values above or below the min/max will be set to the min/max value.
- K_0 constant factor at the initial boundary and P0 (pressure at the initial boundary)
secondly the values of the polytropic exponent for each piece is defined in a single row
thirdly the values of the density are defined for each piece within a single row
finally the units must be specified for the input values of K_0 and rho_tab

Tabulated Equations of State

The format for tabulated EOS' that are currently supported is the well known format used by the LORENE spectral solver. Details on the format of the table can be found here

EOS Framework

The backend that handles the EOS is constructed in two parts

1. EOS Management

As for FUKAv2.3, an EOS wrapper was written by Samuel Tootle to interface between multiple potential EOS backends which currently includes Margherita and now the GRHayL.

MargheritaEOS

The construction and evaluation of the EOS is handled by default by a standalone version of the Margherita EOS framework developed by Elias R. Most For code details see $HOME_KADATH/include/EOS/standalone/

GRHayLEOS

In FUKAv2.3, support has been added to link against the GRHayL library to leverage the GRHayLEOS module. While this includes support for hybid EOS', it provides additional support to ingest 3D temperature dependent EOSs in stellar collapse format. More information can be found here.

2. EOS Interface

To interface between KADATH and the EOS backend, a user-defined module was written by Samuel Tootle and L. Jens Papenfort allowing the use of an EOS manager within KADATH's System_of_equations framework. For details see $HOME_KADATH/include/EOS/EOS.hh.

To initialize an EOS manager, the following parameters need to be set within the config file

{"eostype",EOSTYPE}, // Options: Cold_PWPoly, Cold_Table, (for GRHayL: ghl_eos_hybrid, ghl_eos_tabulated)
{"eosfile",EOSFILE}, // Polytrope or Tabulated EOS file
{"h_cut",HCUT}, // value to cut the specific enthalpy (0 is default)
{"interpolation_pts", INTERP_PTS} // number of points to use for interpolating the table

If no path is specified for the eosfile, e.g. eosfile togashi.lorene, the EOS framework will only search the $HOME_KADATH/eos/ directory for the EOS file specified. In the event you want to specify an EOS not located in the default directory, a path must be included. For example, to read an EOS from the current directory, one would need to write eos_file ./togashi.lorene.

3. EOS File Formats

Polytropes

Both Margherita and GRHayL (eostype: Cold_PWPoly, ghl_eos_simple, ghl_eos_hybrid) support poltryopic equations of state which can specified in a <eos>.polytrope file. See, e.g. eos/apr4.polytrope and eos/gam2.polytrope.

Cold Tables

Margherita supports ingesting 1D cold slices in the common LORENE format. Examples of these tables include eos/dd2.lorene and eos/togashi.lorene, for instance. These tables can be used for computing initial data with eostype Cold_Table.

Stellar Collapse

Support for stellar collapse tables is now possible when compiling with GRHayL. Use of stellar collapse tables requires specifying a configuration file, e.g. dd2.stellarcollapse, which stores the location of the 3D table along with initialization parameters such as table bounds and the temperature slice to use (T_beta), in units of MeV.

For cold beta equilibrium slices, excellent agreement has been found between dd2.lorene and dd2.stellarcollapse computed initial data. Use of stellar collapse tables to compute initial data using a hot, beta equilibrium slice has not yet been fully explored.