Freddi — compute FRED-like light curves of LMXB

Table of contents

Overview

The code solves 1-D evolution equation of Shakura-Sunyaev accretion disk. The code is developed to simulate fast-rise exponential-decay (FRED) light curves of low mass X-ray binaries (LMXBs) for the paper “Determination of the turbulent parameter in the accretion disks: effects of self-irradiation in 4U 1543-47 during the 2002 outburst” by Lipunova & Malanchev (2017) 2017MNRAS.468.4735L.

Freddi is written on C++ and available as a couple of binary executables and a Python module.

Note that the original Freddi version 1 introduced in Lipunova & Malanchev (2017) 2017MNRAS.468.4735L is still available in the v1 git branch.

Installation

Executables

Freddi is represented by two binary executables: the black hole version freddi and the neutron star version freddi-ns.

Docker

If you are familiar with Docker then you can use pre-compiled binaries inside Docker container:

docker run -v "`pwd`":/data --rm -ti ghcr.io/hombit/freddi freddi -d/data
docker run -v "`pwd`":/data --rm -ti ghcr.io/hombit/freddi freddi-ns --Bx=1e8 -d/data

Build from source

Freddi has following build dependencies:

Get requirements on Debian based systems (e.g. Ubuntu):

apt-get install g++ cmake libboost-all-dev

On Red-Hat based systems (e.g. Fedora):

dnf install gcc-c++ cmake boost-devel

On macOS via Homebrew:

brew install cmake boost

Get Freddi source code and compile it:

git clone https://github.com/hombit/freddi
cd freddi
mkdir cmake-build
cd cmake-build
cmake .. # -DSTATIC_LINKING=TRUE
cmake --build .

Uncomment -DSTATIC_LINKING=TRUE to link against static Boost libraries

Now you should have both freddi and freddi-ns executables in the build directory. You can install these binaries and the default configuration file freddi.ini by running

cmake --install . --prefix=PREFIX  # replace with preferable location

Freddi is known to be built on Linux and macOS.

Python

PyPI version

Python 2 isn’t supported, use Python 3 instead.

Freddi pre-compiled x86-64 Linux packages for several Python versions are available on https://pypi.org/project/freddi/ and can be used as is, while for other configurations you should have C++ compiler and Boost libraries in your system before running this command:

# Please upgrade your pip
python3 -m pip install -U pip
# Depending on your Python setup, you need or need not --user flag
python3 -m pip install --user freddi

astropy is an optional requirement which must be installed to use dimensional input via Freddi.from_astropy

Usage

Executables

Freddi runs from the command line with inline options and/or with a configuration file. Freddi outputs file freddi.dat with distribution of various physical values over time. If --fulldata is specified then files freddi_%d.dat for each time step are created in the same directory with snapshot radial distributions. These data-files contain whitespace-separated data columns with header lines starting with # symbol. You can set another prefix instead of freddi with --prefix option and change the output directory with --dir option. If you choose the Docker way and would like to specify the directory, then avoid using --dir option and just replace "`pwd`" with some local path (for more details see Docker documentation).

Options

The full list of command line options is viewed with --help option. Default values are given in brackets.

./freddi --help
expand ``` Freddi: numerical calculation of accretion disk evolution: General options: -h [ --help ] Produce help message --config arg Set additional configuration filepath --prefix arg (=freddi) Set prefix for output filenames. Output file with distribution of parameters over time is PREFIX.dat --stdout Output temporal distribution to stdout instead of PREFIX.dat file -d [ --dir ] arg (=.) Choose the directory to write output files. It should exist --precision arg (=12) Number of digits to print into output files --tempsparsity arg (=1) Output every k-th time moment --fulldata Output files PREFIX_%d.dat with radial structure for every time step. Default is to output only PREFIX.dat with global disk parameters for every time step Basic binary and disk parameter: -a [ --alpha ] arg Alpha parameter of Shakura-Sunyaev model --alphacold arg Alpha parameter of cold disk, currently it is used only for Sigma_minus, see --Qirr2Qvishot. Default is --alpha values divided by ten -M [ --Mx ] arg Mass of the central object, in the units of solar masses --kerr arg (=0) Dimensionless Kerr parameter of the black hole --Mopt arg Mass of the optical star, in units of solar masses --rochelobefill arg (=1) Dimensionless factor describing a size of the optical star. Polar radius of the star is rochelobefill * (polar radius of critical Roche lobe) --Topt arg (=0) Thermal temperature of the optical star, in units of kelvins -P [ --period ] arg Orbital period of the binary system, in units of days --rin arg Inner radius of the disk, in the units of the gravitational radius of the central object GM/c^2. If it isn't set then the radius of ISCO orbit is used defined by --Mx and --kerr values -R [ --rout ] arg Outer radius of the disk, in units of solar radius. If it isn't set then the tidal radius is used defined by --Mx, --Mopt and --period values --risco arg Innermost stable circular orbit, in units of gravitational radius of the central object GM/c^2. If it isn't set then the radius of ISCO orbit is used defined by --Mx and --kerr values Parameters of the disk mode: -O [ --opacity ] arg (=Kramers) Opacity law: Kramers (varkappa ~ rho / T^7/2) or OPAL (varkappa ~ rho / T^5/2) --Mdotout arg (=0) Accretion rate onto the disk through its outer radius --boundcond arg (=Teff) Outer boundary movement condition Values: Teff: outer radius of the disk moves inwards to keep photosphere temperature of the disk larger than some value. This value is specified by --Thot option Tirr: outer radius of the disk moves inwards to keep irradiation flux of the disk larger than some value. The value of this minimal irradiation flux is [Stefan-Boltzmann constant] * Tirr^4, where Tirr is specified by --Thot option --Thot arg (=0) Minimum photosphere or irradiation temperature at the outer edge of the hot disk, Kelvin. For details see --boundcond description --Qirr2Qvishot arg (=0) Minimum Qirr / Qvis ratio at the outer edge of the hot disk to switch evolution from temperature-based regime to Sigma_minus-based regime (see Eq. A.1 in Lasota et al. 2008, --alphacold value is used as alpha parameter) --initialcond arg (=powerF) Type of the initial condition for viscous torque F or surface density Sigma Values: [xi = (h - h_in) / (h_out - h_in)] powerF: F ~ xi^powerorder, powerorder is specified by --powerorder option linearF: F ~ xi, specific case of powerF but can be normalised by --Mdot0, see its description for details powerSigma: Sigma ~ xi^powerorder, powerorder is specified by --powerorder option sineF: F ~ sin( xi * pi/2 ) gaussF: F ~ exp(-(xi-mu)**2 / 2 sigma**2), mu and sigma are specified by --gaussmu and --gausssigma options quasistat: F ~ f(h/h_out) * xi * h_out/h, where f is quasi-stationary solution found in Lipunova & Shakura 2000. f(xi=0) = 0, df/dxi(xi=1) = 0 --F0 arg Initial maximum viscous torque in the disk, dyn*cm. Can be overwritten via --Mdisk0 and --Mdot0 --Mdisk0 arg Initial disk mass, g. If both --F0 and --Mdisk0 are specified then --Mdisk0 is used. If both --Mdot0 and --Mdisk0 are specified then --Mdot0 is used --Mdot0 arg Initial mass accretion rate through the inner radius, g/s. If --F0, --Mdisk0 and --Mdot0 are specified then --Mdot0 is used. Works only when --initialcond is set to linearF, sinusF or quasistat --powerorder arg Parameter for the powerlaw initial condition distribution. This option works only with --initialcond=powerF or powerSigma --gaussmu arg Position of the maximum for Gauss distribution, positive number not greater than unity. This option works only with --initialcond=gaussF --gausssigma arg Width of for Gauss distribution. This option works only with --initialcond=gaussF --windtype arg (=no) Type of the wind no: no wind SS73C: super-Eddington spherical wind from Shakura-Sunyaev 1973 ShieldsOscil1986: toy wind model from Shields et al. 1986 which was used to obtain oscillations in the disk luminosity. Requires --windC_w and --windR_w to be specified Janiuk2015: super-Eddington wind from Janiuk et al 2015. Requires --windA_0 and --windB_1 to be specified Shields1986: thermal wind from Begelman et al. 1983 and Shields et al. 1986. Requires --windXi_max, --windT_ic and --windPow to be specified Woods1996AGN: thermal AGN wind from Woods et al. 1996. Requires --windC_0 and --windT_ic to be specified Woods1996: thermal wind from Woods et al. 1996. Requires --windXi_max, --windT_ic and --windPow to be specified toy: a toy wind model used in arXiv:2105.11974, the mass loss rate is proportional to the central accretion rate. Requires --windC_w to be specified --windC_w arg The ratio of the mass loss rate due to wind to the central accretion rate, |Mwind|/Macc --windR_w arg The ratio of the wind launch radius to the outer disk radius, Rwind/Rout --windA_0 arg Dimensionless parameter characterizing the strength of the super-Eddington wind in the framework of the model Janiuk et al. 2015. Effective value range from 10 to 25 --windB_1 arg The quantity is of the order of unity. Characterizes the relationship between the change in energy per particle and virial energy. E = B_1 * k * T --windXi_max arg Ionization parameter, the ratio of the radiation and gas pressures --windT_ic arg Inverse Compton temperature, K. Characterizes the hardness of the irradiating spectrum --windPow arg Multiplicative coefficient to control wind power --windC_0 arg Characteristic column density of the wind mass loss rate from Woods et al. 1996 model, g/(s*cm^2). For AGN approx value is 3e-13 g/(s*cm^2) Parameters of self-irradiation. Qirr = Cirr * (H/r / 0.05)^irrindex * L * psi / (4 pi R^2), where psi is angular distrbution of X-ray radiation: --Cirr arg (=0) Irradiation factor for the hot disk --irrindex arg (=0) Irradiation index for the hot disk --Cirrcold arg (=0) Irradiation factor for the cold disk --irrindexcold arg (=0) Irradiation index for the cold disk --h2rcold arg (=0) Seme-height to radius ratio for the cold disk, it affects disk shadow in star --angulardistdisk arg (=plane) Angular distribution of the disk X-ray radiation. Values: isotropic, plane Parameters of flux calculation: --colourfactor arg (=1.7) Colour factor to calculate X-ray flux --emin arg (=1) Minimum energy of X-ray band, keV --emax arg (=12) Maximum energy of X-ray band, keV --staralbedo arg (=0) Part of X-ray radiation reflected by optical star, (1 - albedo) heats star's photosphere. Used only when --starflux is specified -i [ --inclination ] arg (=0) Inclination of the system, degrees --ephemerist0 arg (=0) Ephemeris for the time of the minimum of the orbital light curve T0, phase zero corresponds to inferior conjunction of the optical star, days --distance arg Distance to the system, kpc --colddiskflux Add Fnu for cold disk into output file. Default output is for hot disk only --starflux Add Fnu for irradiated optical star into output file. See --Topt, --starlod and --h2rcold options. Default is output for the hot disk only --lambda arg Wavelength to calculate Fnu, Angstrom. You can use this option multiple times. For each lambda one additional column with values of spectral flux density Fnu [erg/s/cm^2/Hz] is produced --passband arg Path of a file containing tabulated passband, the first column for wavelength in Angstrom, the second column for transmission factor, columns should be separated by spaces Parameters of disk evolution calculation: --inittime arg (=0) Initial time moment, days -T [ --time ] arg Time interval to calculate evolution, days --tau arg Time step, days --Nx arg (=1000) Size of calculation grid --gridscale arg (=log) Type of grid for angular momentum h: log or linear --starlod arg (=3) Level of detail of the optical star 3-D model. The optical star is represented by a triangular tile, the number of tiles is 20 * 4^starlod ``` 
./freddi-ns --help
expand ``` Freddi NS: numerical calculation of accretion disk evolution: General options: -h [ --help ] Produce help message --config arg Set additional configuration filepath --prefix arg (=freddi) Set prefix for output filenames. Output file with distribution of parameters over time is PREFIX.dat --stdout Output temporal distribution to stdout instead of PREFIX.dat file -d [ --dir ] arg (=.) Choose the directory to write output files. It should exist --precision arg (=12) Number of digits to print into output files --tempsparsity arg (=1) Output every k-th time moment --fulldata Output files PREFIX_%d.dat with radial structure for every time step. Default is to output only PREFIX.dat with global disk parameters for every time step Basic binary and disk parameter: -a [ --alpha ] arg Alpha parameter of Shakura-Sunyaev model --alphacold arg Alpha parameter of cold disk, currently it is used only for Sigma_minus, see --Qirr2Qvishot. Default is --alpha values divided by ten -M [ --Mx ] arg Mass of the central object, in the units of solar masses --kerr arg (=0) Dimensionless Kerr parameter of the black hole --Mopt arg Mass of the optical star, in units of solar masses --rochelobefill arg (=1) Dimensionless factor describing a size of the optical star. Polar radius of the star is rochelobefill * (polar radius of critical Roche lobe) --Topt arg (=0) Thermal temperature of the optical star, in units of kelvins -P [ --period ] arg Orbital period of the binary system, in units of days --rin arg Inner radius of the disk, in the units of the gravitational radius of the central object GM/c^2. If it isn't set then the radius of ISCO orbit is used defined by --Mx and --kerr values -R [ --rout ] arg Outer radius of the disk, in units of solar radius. If it isn't set then the tidal radius is used defined by --Mx, --Mopt and --period values --risco arg Innermost stable circular orbit, in units of gravitational radius of the central object GM/c^2. If it isn't set then the radius of ISCO orbit is used defined by --Mx and --kerr values Parameters of the disk mode: -O [ --opacity ] arg (=Kramers) Opacity law: Kramers (varkappa ~ rho / T^7/2) or OPAL (varkappa ~ rho / T^5/2) --Mdotout arg (=0) Accretion rate onto the disk through its outer radius --boundcond arg (=Teff) Outer boundary movement condition Values: Teff: outer radius of the disk moves inwards to keep photosphere temperature of the disk larger than some value. This value is specified by --Thot option Tirr: outer radius of the disk moves inwards to keep irradiation flux of the disk larger than some value. The value of this minimal irradiation flux is [Stefan-Boltzmann constant] * Tirr^4, where Tirr is specified by --Thot option --Thot arg (=0) Minimum photosphere or irradiation temperature at the outer edge of the hot disk, Kelvin. For details see --boundcond description --Qirr2Qvishot arg (=0) Minimum Qirr / Qvis ratio at the outer edge of the hot disk to switch evolution from temperature-based regime to Sigma_minus-based regime (see Eq. A.1 in Lasota et al. 2008, --alphacold value is used as alpha parameter) --initialcond arg (=powerF) Type of the initial condition for viscous torque F or surface density Sigma Values: [xi = (h - h_in) / (h_out - h_in)] powerF: F ~ xi^powerorder, powerorder is specified by --powerorder option linearF: F ~ xi, specific case of powerF but can be normalised by --Mdot0, see its description for details powerSigma: Sigma ~ xi^powerorder, powerorder is specified by --powerorder option sineF: F ~ sin( xi * pi/2 ) gaussF: F ~ exp(-(xi-mu)**2 / 2 sigma**2), mu and sigma are specified by --gaussmu and --gausssigma options quasistat: F ~ f(h/h_out) * xi * h_out/h, where f is quasi-stationary solution found in Lipunova & Shakura 2000. f(xi=0) = 0, df/dxi(xi=1) = 0 quasistat-ns: ??? --F0 arg Initial maximum viscous torque in the disk, dyn*cm. Can be overwritten via --Mdisk0 and --Mdot0 --Mdisk0 arg Initial disk mass, g. If both --F0 and --Mdisk0 are specified then --Mdisk0 is used. If both --Mdot0 and --Mdisk0 are specified then --Mdot0 is used --Mdot0 arg Initial mass accretion rate through the inner radius, g/s. If --F0, --Mdisk0 and --Mdot0 are specified then --Mdot0 is used. Works only when --initialcond is set to linearF, sinusF or quasistat --powerorder arg Parameter for the powerlaw initial condition distribution. This option works only with --initialcond=powerF or powerSigma --gaussmu arg Position of the maximum for Gauss distribution, positive number not greater than unity. This option works only with --initialcond=gaussF --gausssigma arg Width of for Gauss distribution. This option works only with --initialcond=gaussF --windtype arg (=no) Type of the wind no: no wind SS73C: super-Eddington spherical wind from Shakura-Sunyaev 1973 ShieldsOscil1986: toy wind model from Shields et al. 1986 which was used to obtain oscillations in the disk luminosity. Requires --windC_w and --windR_w to be specified Janiuk2015: super-Eddington wind from Janiuk et al 2015. Requires --windA_0 and --windB_1 to be specified Shields1986: thermal wind from Begelman et al. 1983 and Shields et al. 1986. Requires --windXi_max, --windT_ic and --windPow to be specified Woods1996AGN: thermal AGN wind from Woods et al. 1996. Requires --windC_0 and --windT_ic to be specified Woods1996: thermal wind from Woods et al. 1996. Requires --windXi_max, --windT_ic and --windPow to be specified toy: a toy wind model used in arXiv:2105.11974, the mass loss rate is proportional to the central accretion rate. Requires --windC_w to be specified --windC_w arg The ratio of the mass loss rate due to wind to the central accretion rate, |Mwind|/Macc --windR_w arg The ratio of the wind launch radius to the outer disk radius, Rwind/Rout --windA_0 arg Dimensionless parameter characterizing the strength of the super-Eddington wind in the framework of the model Janiuk et al. 2015. Effective value range from 10 to 25 --windB_1 arg The quantity is of the order of unity. Characterizes the relationship between the change in energy per particle and virial energy. E = B_1 * k * T --windXi_max arg Ionization parameter, the ratio of the radiation and gas pressures --windT_ic arg Inverse Compton temperature, K. Characterizes the hardness of the irradiating spectrum --windPow arg Multiplicative coefficient to control wind power --windC_0 arg Characteristic column density of the wind mass loss rate from Woods et al. 1996 model, g/(s*cm^2). For AGN approx value is 3e-13 g/(s*cm^2) Parameters of accreting neutron star: --nsprop arg (=dummy) Neutron star geometry and radiation properties name and specifies default values of --Rx, --Risco and --freqx Values: dummy: NS radiation efficiency is R_g * (1 / R_x - 1 / 2R_in), default --freqx is 0, default Rx is 1e6, default Risco is Kerr value newt: NS radiation efficiency is a functions of NS frequency, that's why --freqx must be specified explicitly sibgatullin-sunyaev2000: NS radiation efficiency, and default values of Rx and Risco are functions of NS frequency, that's why --freqx must be specified explicitly --freqx arg Accretor rotation frequency, Hz. This parameter is not linked to --kerr, agree them yourself --Rx arg Accretor radius, cm --Bx arg Accretor polar magnetic induction, G --hotspotarea arg (=1) ??? Relative area of hot spot on the accretor --epsilonAlfven arg (=1) Factor in Alfven radius formula --inversebeta arg (=0) ??? --Rdead arg (=0) Maximum inner radius of the disk that can be obtained, it characterises minimum torque in the dead disk, cm --fptype arg (=no-outflow) ??? Accretor Mdot fraction mode fp. Values: no-outflow: all the matter passing inner disk radius falling onto neutron star, fp = 1 propeller: all the matter flows away from both disk and neutron star, fp = 0 corotation-block: like 'no-otflow' when Alfven radius is smaller than corotation radius, like 'propeller' otherwise geometrical: generalisation of 'corotation-block' for the case of not co-directional of disk rotation axis and neutron star magnetic field axis. Requires --fp-geometrical-chi to be specified eksi-kutlu2010: ??? romanova2018: ???, requires --romanova2018-par1 and --romanova2018-par2 to be specified --fp-geometrical-chi arg angle between disk rotation axis and neutron star magnetic axis for --fptype=geometrical, degrees --romanova2018-par1 arg ??? par1 value for --fptype=romanova201 8 and --kappattype=romanova2018 --romanova2018-par2 arg ??? par2 value for --fptype=romanova201 8 and --kappattype=romanova2018 --kappattype arg (=const) kappa_t describes how strong is interaction between neutron star magnitosphere and disk, magnetic torque is kappa_t(R) * mu^2 / R^3. This parameter describes type of radial destribution of this parameter Values: const: doesn't depend on radius, kappa_t = value. Requires --kappat-const-value to be specified corstep: kappa_t is 'in' inside corotation radius, and 'out' outside. Requires --kappat-corstep-in and --kappat-corstep-out to be specified romanova2018: similar to corstep option, but the outside value is reduced by the portion taken away by wind (see Table 2 of Romanova+2018,NewA,62,94). Requires --kappat-romanova2018-in, --kappat-romanova2018-out --romanova2018-par1 and --romanova-par2 to be specified --kappat-const-value arg (=0.33333333333333331) kappa_t value for --kappattype=const --kappat-corstep-in arg (=0.33333333333333331) kappa_t value inside corotation radius for --kappattype=corstep --kappat-corstep-out arg (=0.33333333333333331) kappa_t value outside corotation radius for --kappattype=corstep --kappat-romanova2018-in arg (=0.33333333333333331) kappa_t value inside corotation radius for --kappattype=romanova2018 --kappat-romanova2018-out arg (=0.33333333333333331) kappa_t value outside corotation radius for --kappattype=romanova2018 --nsgravredshift arg (=off) Neutron star gravitational redshift type. Values: off: gravitational redshift doesn't taken into account on: redshift is (1 - R_sch / Rx), where R_sch = 2GM/c^2 Parameters of self-irradiation. Qirr = Cirr * (H/r / 0.05)^irrindex * L * psi / (4 pi R^2), where psi is angular distrbution of X-ray radiation: --Cirr arg (=0) Irradiation factor for the hot disk --irrindex arg (=0) Irradiation index for the hot disk --Cirrcold arg (=0) Irradiation factor for the cold disk --irrindexcold arg (=0) Irradiation index for the cold disk --h2rcold arg (=0) Seme-height to radius ratio for the cold disk, it affects disk shadow in star --angulardistdisk arg (=plane) Angular distribution of the disk X-ray radiation. Values: isotropic, plane --angulardistns arg (=isotropic) Angular distribution type of the neutron star X-ray radiation. Values: isotropic, plane Parameters of flux calculation: --colourfactor arg (=1.7) Colour factor to calculate X-ray flux --emin arg (=1) Minimum energy of X-ray band, keV --emax arg (=12) Maximum energy of X-ray band, keV --staralbedo arg (=0) Part of X-ray radiation reflected by optical star, (1 - albedo) heats star's photosphere. Used only when --starflux is specified -i [ --inclination ] arg (=0) Inclination of the system, degrees --ephemerist0 arg (=0) Ephemeris for the time of the minimum of the orbital light curve T0, phase zero corresponds to inferior conjunction of the optical star, days --distance arg Distance to the system, kpc --colddiskflux Add Fnu for cold disk into output file. Default output is for hot disk only --starflux Add Fnu for irradiated optical star into output file. See --Topt, --starlod and --h2rcold options. Default is output for the hot disk only --lambda arg Wavelength to calculate Fnu, Angstrom. You can use this option multiple times. For each lambda one additional column with values of spectral flux density Fnu [erg/s/cm^2/Hz] is produced --passband arg Path of a file containing tabulated passband, the first column for wavelength in Angstrom, the second column for transmission factor, columns should be separated by spaces Parameters of disk evolution calculation: --inittime arg (=0) Initial time moment, days -T [ --time ] arg Time interval to calculate evolution, days --tau arg Time step, days --Nx arg (=1000) Size of calculation grid --gridscale arg (=log) Type of grid for angular momentum h: log or linear --starlod arg (=3) Level of detail of the optical star 3-D model. The optical star is represented by a triangular tile, the number of tiles is 20 * 4^starlod ```

Write which options are mandatory

Also you can use freddi.ini configuration file to store options. This INI file contains lines option=value, option names are the as provided by the help message above. Command line option overwrites configuration file option. For example, see default freddi.ini.

Paths where this file is searched are ./freddi.ini (execution path), $HOME/.config/freddi/freddi.ini, /usr/local/etc/freddi.ini and /etc/freddi.ini. You can provide configuration file to Docker container as a volume: -v "`pwd`/freddi.ini":/etc/freddi.ini.

Output values

Freddi outputs time; the accretion rate; the mass of the hot part of the disk; the outer radius of the hot zone; the irradiation factor; the relative half-height, effective and irradiation temperature, ratio of the irradiation to viscous flux at the outer radius of the hot zone; X-ray luminosity (erg/s) in the band from E_min to E_max (--emin and --emax options); the optical magnitudes in U, B, V, R, I, and J band (Allen’s Astrophysical Quantities, Cox 2015); the spectral density flux (erg/s/cm^2/Hz) at some wavelengths set by one or more --lambda options.

Snapshot distributions at each time step, if produced, contain the following data: radial coordinate in terms of the specific angular momentum, radius, viscous torque, surface density, effective temperature Teff, viscous temperature Tvis, irradiation temperature Tirr, and the absolute half-height of the disk.

Example

The following arguments instruct Freddi to calculate the decay of the outburst in the disk with the constant outer radius equal to 1 solar radius. The Kerr black hole at the distance of 5 kpc has the mass of 9 solar masses, and the Kerr’s parameter is 0.4. The outer disk is irradiated with Cirr=1e-3. Discuss all options used in the example*

./freddi --alpha=0.5 --Mx=9 --rout=1 --period=0.5 --Mopt=0.5 --time=50 \
  --tau=0.25 --dir=data/ --F0=2e+37 --colourfactor=1.7 --Nx=1000 \
  --distance=5 --gridscale=log --kerr=0.4 --Cirr=0.001 --opacity=OPAL \
  --initialcond=quasistat --windtype=Woods1996 --windXi_max=10 --windT_ic=1e8 \
  --windPow=1

Python

Python bindings can be used as a convenient way to run and analyse Freddi simulations.

Initializing

You can prepare simulation set-up initializing Freddi (for black hole accretion disk) or FreddiNeutronStar (for NS) class instance. These classes accept keyword-only arguments which have the same names and meanings as command line options, but with three major exceptions:

  1. Python package doesn’t provide any file output functionality, that’s why output arguments like config, dir, fulldata, starflux, lambda or passband are missed;
  2. all values are assumed to be in CGS units, but you can use Freddi.from_asrtopy for dimensional values (see details bellow);
  3. parameters of wind, NS fp and NS kappa models are passed as dictionaries (see specification bellow).

The following code snippet would set-up roughly the same simulation as the command-line example

from freddi import Freddi

freddi = Freddi(
    alpha=0.5, Mx=9*2e33, rout=1*7e10, period=0.5*86400, Mopt=0.5*2e33,
    time=50*86400, tau=0.25*86400, F0=2e+37, colourfactor=1.7, Nx=1000,
    distance=5*3e21, gridscale='log', kerr=0.4, Cirr=0.001, opacity='OPAL',
    initialcond='quasistat', wind='Woods1996',
    windparams=dict(Xi_max=10, T_iC=1e8, W_pow=1),
)

Alternatively we can do the same using from_astropy class-method which casts all astropy.units.Quantity objects to CGS values. Note that dimensionality isn’t checked, and technically it just does arg.cgs.value for every Quantity argument.

import astropy.units as u
from freddi import Freddi

freddi = Freddi.from_astropy(
    alpha=0.5, Mx=9*u.Msun, rout=1*u.Rsun, period=0.5*u.day, Mopt=0.5*u.Msun,
    time=50*u.day, tau=0.25*u.day, F0=2e+37, colourfactor=1.7, Nx=1000,
    distance=5*u.kpc, gridscale='log', kerr=0.4, Cirr=0.001, opacity='OPAL',
    initialcond='quasistat', wind='Woods1996',
    windparams=dict(Xi_max=10, T_iC=1e8*u.K, W_pow=1),
)

Wind model parameters are specified by windparams argument which should be a dict instance with string keys and numeric values. Command option to windparams keys relation is: --windC_w -> C_w, --windR_w -> R_w, --windA_0 -> A_0, --windB_1 -> B_1, --windXi_max -> Xi_max, windT_ic -> T_ic, --windPow -> Pow, windC_0 -> C_0.

Neutron star f_p model parameters are specified by fpparams mapping with the same structure as windparams. Command options to fpparams keys relation is: --fp-geometrical-chi -> chi, romanova2018-par1 -> par1, romanova2018-par2 -> par2.

Neutron star kappa_t model parameters are specified by kappatparams mapping with the same structure as windparams. Command options to the mapping keys relation is: --kappat-const-value -> value, --kappat-corstep-in -> in, kappat-corstep-out -> out, --kappat-romanova2018-in -> in, --kappat-romanova2018-out -> out, romanova2018-par1 -> par1, --romanova2018-par2 -> par2

Running

There are two ways to run a simulation: iterating over time steps, and run the whole simulation in one shot. Note that in both cases your Freddi object is mutating and represents the current state of the accretion disk.

Here we use iterator interface which yields another Freddi object for each time moment.

import astropy.units as u
from freddi import Freddi

freddi = Freddi.from_astropy(
    alpha=0.5, Mx=9*u.Msun, rout=1*u.Rsun, period=0.5*u.day, Mopt=0.5*u.Msun,
    time=20*u.day, tau=1.0*u.day, Mdot0=5e18, distance=10*u.kpc,
    initialcond='quasistat',
)

for state in freddi:
    print(f't = {state.t} s, Mdot = {state.Mdot:g} g/s')
assert state.t == freddi.t

In this example we run a simulation via .evolve() method which returns EvolutionResult object keeping all evolution states internally and providing temporal distribution of disk’s properties.

import astropy.units as u
import matplotlib.pyplot as plt
from freddi import Freddi

freddi = Freddi.from_astropy(
    alpha=0.5, Mx=9*u.Msun, rout=1*u.Rsun, period=0.5*u.day, Mopt=0.5*u.Msun,
    time=20*u.day, tau=1.0*u.day, Mdot0=5e18, distance=10*u.kpc,
    initialcond='quasistat',
)

result = freddi.evolve()
assert result.t[-1] == freddi.t

# Plot Mdot(t)
plt.figure()
plt.title('Freddi disk evolution: accretion rate')
plt.xlabel('t, day')
plt.ylabel(r'$\dot{M}$, g/cm')
plt.plot(result.t / 86400, result.Mdot)
plt.show()

# Plot all F(h) profiles
plt.figure()
plt.title('Freddi disk evolution: viscous torque')
plt.xlabel('r, cm')
plt.ylabel('F, dyn cm')
plt.xscale('log')
plt.yscale('log')
plt.plot(result.R.T, result.F.T)
plt.show()

# Plot evolution of effective temperature of the outer hot disk ring
plt.figure()
plt.title('Freddi disk evolution: outer effective temperature')
plt.xlabel('t, day')
plt.ylabel('T, K')
plt.plot(result.t / 86400, result.last_Tph)
plt.show()

Properties and methods

Freddi, FreddiNeutronStar and EvolutionResult objects contain dozens of properties returning various physical values like t for time moment, Mdot for accretion rate onto central object, R for radius, F for torque, Tph for effective temperature and so on. first_* and last_* properties are used to access innermost and outermost values of radial-distributed quantities. The complete list of properties can be obtained by dir(Freddi) or dir(FreddiNeutronStar). Note that the most properties are lazy-evaluated and require some time to access first time. EvolutionRadius provides all the same properties as underlying Freddi or FreddiNeutronStar objects but with additional array dimension for temporal distribution, so if Freddi.Lx is a scalar then EvolutionResult.Lx is 1-D numpy array of (Nt,) shape, if Freddi.Sigma is 1-D array of (Nx,) shape, then EvolutionResult.Sigma is 2-D array of (Nt, Nx) shape. Also, note that if disk shrinks during a simulation, the missing values of EvolutionResult properties are filled by NaN.

All three classes have flux(lmbd, region='hot', phase=None) method which can be used to find spectral flux density per unit frequency for optical emission. lmbd argument can be a scalar or a multidimensional numpy array of required wavelengths in cm; region could be one of “hot” (hot disk), “cold” (cold disk), “disk” (“hot” + “cold”), “star” (companion star), and “all” (“hot” + “cold” + “star”); phase is a binary system orbital phase in radians, it is required for region="star" and region="all" only, it can be calculated as 2π t / period + constant.

All properties and methods return values in CGS units.

Physical Background

Freddi — Fast Rise Exponential Decay: accretion Disk model Implementation — is designed to solve the differential equation of the viscous evolution of the Shakura-Sunyaev accretion disk in a stellar binary system. Shakura-Sunyaev disk is the standard model of accretion of plasma onto the cosmic bodies, like neutron stars or black holes. Viscous evolution of the accretion disks exibits itself, for example, in X-ray outbursts of binary stars. Usually, the outbursts last for several tens of days and many of them are observed by orbital observatories.

The basic equation of the viscous evolution relates the surface density and viscous stresses and is of diffusion type. Evolution of the accretion rate can be found on solving the equation. The distribution of viscous stresss defines the emission from the source.

The standard model for the accretion disk is implied, which is developed by Shakura & Sunyaev (1973). The inner boundary of the disk is at the ISCO or can be explicitely set. The boundary conditions in the disk are the zero stress at the inner boundary and the zero accretion rate at the outer boundary. The conditions are suitable during the outbursts in X-ray binary transients with black holes.

In a binary system, the accretion disk is radially confined. In Freddi, the outer radius of the disk can be set explicitely or calculated as the position of the tidal truncation radius following Paczynski (1997) for small mass ratios of the black using the approximation by Suleimanov et al. (2008).

The parameters at the disk central plane are defined by the analytic approximations (Suleimanov et al. 2007), valid for the effective surface temperatures from 10 000 to 100 000 K, approximately. It is assumed that the gas pressure dominates, the gas is completely ionized, and the photon opacity is defined by the free-free and free-bound transitions. Opacity law is for the solar element abundancies and can be chosen from two types: (1) Kramers’ opacity: kappa = 5e24 rho/T\^(7/2) cm2/g (2) approximation to OPAL tables: kappa = 1.5e20 rho/T\^(5/2) cm2/g (Bell & Lin 1994)

The disk at each radius is in the “hot” state if the gas is completely ionized. Otherwise, the disk is considered to be “cold” locally. Alpha-parameter in the cold parts of the disk is appreciably lower than in the hot parts. Thus the viscous evolution of the disk should proceed more effectively in the hot parts of the disk. To simulate this, Freddi has an option to control the outer radius of the hot evolving disk. We assume that the evolution goes through the quasi-stationary states in the hot zone of variable size. By default, the hot zone has the constant size, equal to the tidal radius.

The initial distribution of the matter in the disk should be specified with --initialcond option. Freddi can start from several types of initial distributions: power-law distribution of the surface density --initialcond=powerSigma, power-law --initialcond=powerF or sinus-law --initialcond=sinusF distribution of the viscous torque, quasi-stationary distribution --initialcond=quasistat. The choice of the initial distribution defines what type of evolution is to be calculated.

Starting from the quasi-stationary or sinusF distribution, the solution describes the decaying part of the outburst. Zero-time accretion rate through the inner edge can be set. In other cases, the rise to the peak is also computed. Then, initial value of viscous torque at the outer radius (can be set by --F0) defines uniquely the initial mass of the disk.

Self-irradiation by the central X-rays heats the outer parts of the disk. A fraction of the bolometric flux is supposed to illuminate the disk surface. This results in the larger size of the hot disk if such model is assumed. Also, the optical flux is increased because the flux outgoing from the disk surface is proportional to Teff\^4 = Tvis\^4+Tirr\^4. Self-irradiation of the disk is included in the computation if irradiation parameter is not zero. The simplest way is to set a constant irradiation factor --Cirr (the studies of X-ray novae suggest the range for Cirr 1e-5—5e-3).

Observed flux depends on the distance to the source and the inclination of the disk plane. The inclination angle is the angle between the line of sight and the normal to the disk. The flux, emitted from the disk surface, is defined by the sum of the visous and irradiating flux, where the viscous flux is calculated taking into account general relativity effects near the black hole, following Page & Thorne (1974) and Riffert & Herold (1995).

Accretion disk wind

Presumably, during an outburst there is an outflow in the form of a wind from the accretion disk around the compact object. The presence of such a wind in the LMXBs is supported by modern observations indicating the expansion of ionized matter. Such an outflow of matter, being an additional source of angular momentum transfer in the disk, can strongly influence its viscous evolution.

However, the nature of such winds and their physical characteristics are an open question. Namely, there are three mechanisms which are considered: heating of matter by central radiation in optically thin regions of the disk (Begelman et al. 1983, Shields et al. 1986, Woods et al. 1996), the pressure of the magnetic field of the disk (Blandford & Payne 1982, Habibi & Abbassi 2019, Nixon & Pringle 2019) and the pressure of local radiation at super-Eddington accretion rates (Shakura & Sunyaev 1973, Proga & Kallman 2002).

Freddi is modernized in such a way that it is able to solve the viscous evolution equation with an inhomogeneous term that is responsible for the presence of the disk wind. This term is the dependence of the surface density of the wind mass-loss rate on the distance along the disk’s surface. Different forms of such dependence correspond to different wind models, and to different classes within Freddi.

One can choose a wind model by setting the --windtype option. The thermal wind model (Woods et al. 1996), which implies that the outflow of matter occurs due to the heating of the outer parts of the disk by a central radiation source, can be chosen by setting --windtype=Woods1996. The option --windtype=Janiuk15 corresponds to the model from work Janiuk et al. (2015) where the wind is started in the super-Eddington regime. When choosing option --windtype=Janiuk15, the you must also specify the values of the super-Eddington wind parameters with --windA0 and --windB1 options. You can also select the --windtype=toy option, which corresponds to a toy wind model when the user sets the wind strength relatively to the accretion rate using the option --windPow.

Compton-heated wind

At the moment, Freddi is more focused on simulating outbursts taking into account the thermal wind (--windtype=Woods1996 option). For a better understanding, let’s discuss a little the physics of the process of launching such a wind and its parameters in the code.

In the standard accretion disk model by Shakura & Sunyaev (1973) the disk is concave, and, as a result, the disk surface is exposed to the central radiation, which heats the disk material. As a result, the heated matter, starting from a certain radius, begins to leave the accretion disk. This process of heating the matter of the accretion disk by means of Compoton processes was developed in Begelman et al. (1983) and Shields et al. (1986). In a later work Woods et al. (1996), two-dimensional magnetohydrodynamic calculations were performed and the results of Shields et al. (1986) were generalized. Woods et al. (1996) give an expression for the surface density of the mass loss rate as a function of distance along the disk’s surface. This function is used in Freddi to taking thermal wind into account.

Choosing option --windtype=Woods1996, it is necessary to set the value of the ionization parameter Xi (which is proportional to the ratio of the radiation and gas pressures) by the option --windXi_max and the Compoton temperature T_ic (which determines the hardness of the irradiating spectrum and the size of the region where the wind operates) by the option --windT_ic.

Companion star irradiation

We use a simple model of irradiated star to simulate periodic variability and X-ray thermalization by a companion’s photosphere. Our model assumes that the companion star’s shape corresponds to equipotential surface which size is set by --rochelobefill option, unity means that star fills its Roche lobe, any smaller value decreases star’s polar radius correspondingly. Technically, star’s surface is built from 20 * 4^starlod triangles, use --starlod to set level of detail, --starlod=3 should give few percent precision. Every triangle has black-body spectrum with bolometric luminosity given by a sum of star’s own luminosity (set by --Topt) and irradiation flux multiplied by unity minus albedo (set by --staralbedo).

Please note that the model is limited and doesn’t implement limb darking or eclipsing.

Development guide

Source code and tests

Freddi uses Cmake as a build system.

The C++ source code is located in cpp folder which has following structure:

Note, that we require C++17 standard (while not having idiomatic C++17 code), and require code to be compiled by modern GCC and CLang on Linux. Please write unit tests where you can and use ctest to check they pass.

The Python project is specified by pyproject.toml (which just lists build requirements), setup.py and MANIFEST.in files, we use scikit-build as a build system. scikit-build uses Python-related section of CMakeLists.txt to build C++ source code into Python extension, and accomplish it with Python files located in python/freddi directory. Use python setup.py build_ext to build the extension, optionally with -DSTATIC_LINKING=TRUE to link Boost::Filesystem, Boost::Python and Boost::NumPy statically. Please, pay attention to two last libraries, because they should be built against the same Python version as you use.

python/test contains some tests, you can run them by python3 setup.py test.

The regression test data are located in python/test/data. Sometimes you need to update these regression data, for example when you introduce new command-line option with a default value, add new output column or fix some bug in physical model. For these purposes you can use generate_test_data.sh script located in this folder.

Dockerfile is used to build a Docker image with statically-linked binaries, and Dockerfile.python is used to build a Docker image with manylinux-compatible Python wheels.

Continuous integration

We use Github Actions as a continuous integration (CI) system. The workflow file is located in .github/workflows/main.yml and a couple of auxiliary files are located in .ci folder. CI allows us to test new commits to prevent different bugs:

This Readme

Please keep Readme updated. You can update the help messages in the Usage section using .ci/update-help-readme.py script.

Release new version

Check-list:

Questions and comments

If you have any problems, questions, or comments, please address them to Issues or to hombit\@gmail.com

License

Copyright (c) 2016–2021, Konstantin L. Malanchev, Galina V. Lipunova & Artur L. Avakyan.

Freddi is distributed under the terms of the GPLv3.

Please, accompany any results obtained using this code with reference to Lipunova & Malanchev (2017) 2017MNRAS.468.4735L, and for the case of windy calculations please also refer Avakyan et al. (2021) 2021arXiv210511974A.

BibTex

@ARTICLE{2017MNRAS.468.4735L,
   author = { {Lipunova}, G.~V. and {Malanchev}, K.~L.},
    title = "{Determination of the turbulent parameter in accretion discs: effects of self-irradiation in 4U 1543{\minus}47 during the 2002 outburst}",
  journal = {\mnras},
archivePrefix = "arXiv",
   eprint = {1610.01399},
 primaryClass = "astro-ph.HE",
 keywords = {accretion, accretion discs, methods: numerical, binaries: close, stars: black holes, X-rays: individual: 4U 1543-47},
     year = 2017,
    month = jul,
   volume = 468,
    pages = {4735-4747},
      doi = {10.1093/mnras/stx768},
   adsurl = {http://adsabs.harvard.edu/abs/2017MNRAS.468.4735L},
  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2021arXiv210511974A,
       author = { {Avakyan}, A.~L. and {Lipunova}, G.~V. and {Malanchev}, K.~L. and {Shakura}, N.~I.},
        title = "{Change in the orbital period of a binary system due to an outburst in a windy accretion disc}",
      journal = {arXiv e-prints},
     keywords = {Astrophysics - High Energy Astrophysical Phenomena},
         year = 2021,
        month = may,
          eid = {arXiv:2105.11974},
        pages = {arXiv:2105.11974},
archivePrefix = {arXiv},
       eprint = {2105.11974},
 primaryClass = {astro-ph.HE},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2021arXiv210511974A},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}