Name Last Update

# Model description

The KYNrefrev model computes the time dependent reflection spectra of the disc as a response to a flash of primary power-law radiation from a point source located on the axis of the black-hole accretion disc.

Assumptions of the model:

• central Kerr black hole,

• Keplerian, geometrically thin, optically thick, ionised disc with different radial density profiles,

• stationary hot point-like patch of plasma located on the system rotation axis and emitting isotropic power-law radiation,

• full relativistic ray-tracing code in vacuum is used for photon paths from the corona to the disc and to the observer and from the disc to the observer,

• re-processing in the ionised accretion disc is computed for each radius from REFLIONX tables for constant density slab illuminated by power-law radiation,

• the ionisation of the disc is set for each radius according to the amount of the incident primary flux and the density of the accretion disc,

• several limb brightening/darkening prescriptions for directionality of the re-processed emission are used.

Output of the code:

• time dependent spectra (only when used outside of XSPEC) of the disc response and observed primary flash,

• integrated spectrum,

• light curve for a given energy band,

• lag as a function of frequency between given energy bands,

• lag as a function of energy for different frequencies.

# Installation

## Usage in XSPEC

The code is compiled inside XSPEC with the following command (assuming all the source files and FITS tables are in the directory /path/to/KYNrefrev):

• initpackage kynrefrev lmodel-kynrefrev.dat /path/to/KYNrefrev.

To use the KYNrefrev model inside XSPEC, first the package needs to be loaded and directory with KYNrefrev set:

• lmod kynrefrev /path/to/KYNrefrev,

• xset KYDIR /path/to/KYNrefrev.

Then the model may be used:

• mo kynrefrev.

Note: In case of segmentation fault, one may need to increase the stack size, e.g. with the command ulimit -s unlimited or ulimit -s 65532.

## Usage outside of XSPEC

• One also needs the Makefile and libxspec library included in the directory 'other'.

• The library to work with FITS files (libcfitsio.so) is needed, thus one needs to define the name of the library and path to it in the provided Makefile.

• The model parameters have to be changed inside the source file.

• Compile with the make command:

• make kynrefrev
• Run the code:

• ./kynrefrev.
• The model creates various files described below.

Note: In case of segmentation fault, one may need to increase the stack size, e.g. with the command ulimit -s unlimited or ulimit -s 65532.

# Parameters of the model

## Definition in XSPEC

• The meaning of the input parameters are also explained at the beginning of the kynrefrev.c file. The parameters when the code runs under XSPEC are defined in the usual way as for other XSPEC models. The parameter definitions when run outside of XSPEC must be changed directly inside the source code. Summary of the parameters:

• par1 ... a/M
• black hole angular momentum (-1 ≤ a/M ≤ 1)
• par2 ... theta_o
• observer inclination in degrees (0°-pole, 90°-disc)
• par3 ... rin
• inner edge of non-zero disc emissivity (in GM/c2 or in rmso)
• par4 ... ms
• switch for inner edge
• 0: we integrate from inner edge = par3
• 1: if the inner edge of the disc is below marginally stable orbit (MSO) then we integrate emission above MSO only
• 2: we integrate from inner edge given in units of MSO, i.e. inner edge = par3 × rmso (the same applies for outer edge)
• par5 ... rout
• outer edge of non-zero disc emissivity (in GM/c2 or in rmso)
• par6 ... phi
• lower azimuth of non-zero disc emissivity (degrees)
• par7 ... dphi
• (phi + dphi) is upper azimuth of non-zero disc emissivity 0° ≤ dphi ≤ 360°
• par8 ... M/M8
• black hole mass in units of 108 solar masses
• par9 ... height
• height on the axis (measured from the center) at which the primary source is located (GM/c2)
• par10 ... PhoIndex
• power-law energy index of the primary flux
• par11 ... L/LEdd
• dE/dt;, the intrinsic local (if negative) or the observed (if positive) primary isotropic flux in the X-ray energy range 2-10keV in units of LEdd
• par12 ... Np:Nr
• ratio of the primary to the reflected normalization
• 1: self-consistent model for isotropic primary source
• 0: only reflection, primary source is hidden
• if positive then L/LEdd (par11) means the luminosity towards the observer
• if negative then L/LEdd (par11) means the luminosity towards the disc
• par13 ... density
• density profile normalization in 1015 cm-3
• par14 ... den_prof
• par15 ... abun
• Fe abundance (in solar abundance)
• par16 ... alpha
• position of the cloud centre in GM/c2 in alpha coordinate (alpha being the impact parameter in φ-direction, positive for approaching side of the disc)
• par17 ... beta
• position of the cloud centre in GM/c2 in beta coordinate (beta being the impact parameter in θ-direction, positive in up direction, i.e. above the disc)
• par18 ... rcloud
• radius of the obscuring cloud
• the meaning of cloud is inverted for negative values of rcloud, i.e. only the radiation transmitted through the cloud is computed
• par19 ... zshift
• overall Doppler shift
• par20 ... limb
• 0: for isotropic emission (flux ~ 1)
• 1: for Laor's limb darkening (flux ~ 1+2.06μ)
• 2: for Haardt's limb brightening (flux ~ ln (1+1/μ))
• par21 ... tab
• which reflion table to use
• 1: reflion (the old one, lower cut-off energy at 1eV, not good for PhoIndex > 2)
• 2: reflionx (the newer one, lower cut-off energy at 100eV)
• par22 ... sw
• switch for the way how to compute the refl. spectra
• 1: use the computed ionisation parameter, ξ, for the interpolation in reflion, i.e. use proper total incident intensity with the shifted cut-offs
• 2: use the ionisation parameter, ξ, correspondent to the computed normalization of the incident flux, i.e. do not shift the cut-offs when computing the total incident intensity
• par23 ... ntable
• defines fits file with tables (0 ≤ ntable ≤ 99), currently the tables with ntable=80 are correct for this model
• number of grid points in radius
• if negative than the number of radial grid points is dependent on height as -nrad / height 0.66
• par25 ... division
• type of division in radial integration
• 0: equidistant radial grid (constant linear step)
• 1: exponential radial grid (constant logarithmic step)
• >1: mixed radial grid with a constant logarithmic step in the inner region and with a constant linear step in the outer region; the total nradius (par24) number of points is divided in the 3:2 ratio in these regions; the value of par25 gives the transition radius between these regions (in GM/c2)
• par26 ... nphi
• number of grid points in azimuth
• par27 ... deltaT
• length of the time bin (GM/c3)
• par28 ... nt
• number of time subbins per one time bin
• par29 ... t1/f1/E1
• the time to be used in XSPEC for the spectrum (0 means average spectrum, i.e. divided by the flare duration)
• the frequency to be used in XSPEC for the energy dependent Fourier transform (0 means average values in the range of 0 to the first wrapping frequency)
• positive values are in sec or Hz
• negative values are in GM/c3 or (GM/c3)-1
• if different than par30, the value gives the lower end of the time/frequency interval of interest
• if same as par30, then the functions are computed for this value of the time/frequency of interest
• in case of frequency dependent lags it defines the lower value of the energy band of interest in keV
• par30 ... t2/f2/E2
• used only if different than par29 and if par29 is nonzero
• its value gives the upper end of the time/frequency interval of interest
• positive values are in sec or Hz
• negative values are in GM/c3 or (GM/c3)-1
• in case of frequency dependent lags it defines the upper value of the energy band of interest in keV
• par31 ... Eref1
• it defines the lower value of the reference energy band for lag or amplitude energy dependence as well as in case of frequency dependent lags and amplitudes
• if zero no reference band is used
• if negative:
• for lag-energy spectra, the whole energy band is used as a reference band, always excluding the current energy bin
• for lag-frequency dependence, the energy reference band is abs(par31) to abs(par32) excluding overlaping part with energy band of interest abs(par29) to abs(par30)
• par32 ... Eref2
• it defines the upper value of the reference energy band for lag-energy dependence as well as in case of frequency dependent lags
• par33 ... dt/Af
• lag shift for lag-energy dependence in case of par35=+6
• multiplicative factor in case of adding empirical hard lags Af×fqf, used for par35=+16; if par33=-1 then the following hard lags prescription is used (see Epitropakis & Papadakis, 2017): 100 * log10(Eref/E) * (f/1e-4)-1 s with Eref being middle of the reference energy band and E middle of the energy band of interest
• par34 ... Amp/qf
• multiplicative factor for the amplitude-energy dependence in case of par35=+5
• powerlaw index in case of adding empirical hard lags Af×fqf, used for par35=+16
• par35 ... xsw
• defines output in the XSPEC (photar array)
• 0: spectrum for time interval defined by par29 and par30
• the following values correspond to energy dependent Fourier transform at the frequency band defined by par29 and par30:
• -1: real part of FT of the relative reflection
• -2: imaginary part of FT of the relative reflection
• -3: amplitude of FT of the relative reflection
• -4: phase of FT of the relative reflection
• -5: amplitude for the relative reflection divided by amplitude in the reference energy band defined by par31 and par32 (integration in frequencies is done in real and imaginary parts first and then the amplitudes are computed)
• -6: lag for the relative reflection with respect to reference energy band defined by par31 and par32 (integration in frequencies is done in real and imaginary parts first and then the lags are computed with frequency at half of the wrapping frequency or middle of the frequency band)
• -7: amplitude for the relative reflection divided by amplitude in the reference energy band defined by par31 and par32 (integration in frequencies here is done in amplitudes directly)
• -8: lag for the relative reflection with respect to reference energy band defined by par31 and par32 (integration in frequencies here is done in lags directly)
• 1: real part of FT including primary radiation
• 2: imaginary part of FT including primary radiation
• 3: amplitude of FT including primary radiation
• 4: phase of FT including primary radiation
• 5: amplitude including the primary radiation divided by amplitude in the reference energy band defined by par31 and par32 (integration in frequencies is done in real and imaginary parts first and then the amplitudes are computed)
• 6: lag diluted by primary radiation with respect to reference energy band defined by par31 and par32 (integration in frequencies is done in real and imaginary parts first and then the lags are computed with frequency at half of the wrapping frequency or middle of the frequency band)
• 7: amplitude including the primary radiation divided by amplitude in the reference energy band defined by par31 and par32 (integration in frequencies here is done in amplitudes directly)
• 8: lag diluted by primary radiation with respect to reference energy band defined by par31 and par32 (integration in frequencies here is done in lags directly)
• the following values correspond to frequency dependent Fourier transform for the energy band of interest defined by par29 and par30:
• -11: real part of FT of the relative reflection
• -12: imaginary part of FT of the relative reflection
• -13: amplitude of FT of the relative reflection
• -14: phase of FT of the relative reflection
• -15: amplitude for the relative reflection divided by amplitude in the reference energy band defined by par31 and par32 (rebinning here is done in real and imaginary parts first and then the amplitudes are computed)
• -16: lag for the relative reflection with respect to reference energy band defined by par31 and par32 (rebinning here is done in real and imaginary parts first and then the lags are computed)
• -17: amplitude for the relative reflection divided by amplitude in the reference energy band defined by par31 and par32 (rebinning here is done in amplitudes directly)
• -18: lag for the relative reflection with respect to reference energy band defined by par31 and par32 (rebinning here is done in lags directly)
• 11: real part of FT including primary radiation
• 12: imaginary part of FT including primary radiation
• 13: amplitude of FT including primary radiation
• 14: phase of FT including primary radiation
• 15: amplitude including the primary radiation divided by amplitude in the reference energy band defined by par31 and par32 (rebinning here is done in real and imaginary parts first and then the amplitudes are computed)
• 16: lag diluted by primary radiation with respect to reference energy band defined by par31 and par32 (rebinning here is done in real and imaginary parts first and then the lags are computed)
• 17: amplitude including the primary radiation divided by amplitude in the reference energy band defined by par31 and par32 (rebinning here is done in amplitudes directly)
• 18: lag diluted by primary radiation with respect to reference energy band defined by par31 and par32 (rebinning here is done in lags directly)
• how many threads should be used for computations
• par37 ... norm
• has to be set to unity!

## Definition outside XSPEC

• The model parameters need to be defined inside the kynrefrev.c code when run outside of XSPEC (the code needs to be recompiled after changing them):

• energy in the following lines:

      #define NE     30
#define E_MIN  0.3
#define E_MAX  80.

• choose the energy bands of interest in the following lines:

      #define NBANDS 5
.
.
.
//definition of energy band of interest, reference band is defined as the last
//one, usually the whole energy range
ener_low[0] = 0.3;
ener_high[0] = 0.8;
ener_low[1] = 1.;
ener_high[1] = 3.;
ener_low[2] = 3.;
ener_high[2] = 9.;
ener_low[3] = 12.;
ener_high[3] = 40.;
ener_low[4] = E_MIN;
ener_high[4] = E_MAX;

• all basic parameters of the model (physical ones as well as those defining resolution grid for computations) are defined in the following lines:

      param[ 0] = 1.;       // a/M
param[ 1] = 30.;      // thetaO
param[ 2] = 1.;       // rin
param[ 3] = 1.;       // ms
param[ 4] = 1000.;    // rout
param[ 5] = 0.;       // phi0
param[ 6] = 360.;     // dphi
param[ 7] = 0.1;      // M/M8
param[ 8] = 3.;       // height
param[ 9] = 2.;       // PhoIndex
param[10] = 0.001;    // L/Ledd
param[11] = 1.;       // Np:Nr
param[12] = 1.;       // density
param[13] = 0.;       // den_prof
param[14] = 1.;       // abun
param[15] = -6.;      // alpha
param[16] = 0.;       // beta
param[17] = 0.;       // rcloud
param[18] = 0.;       // zshift
param[19] = 0.;       // limb
param[20] = 2.;       // tab
param[21] = 2.;       // sw
param[22] = 80.;      // ntable
param[24] = -1.;      // division
param[25] = 180.;     // nphi
param[26] = 1.;       // deltaT
param[27] = 1.;       // nt
param[28] = 2.e-4;    // t1/f1/E1
param[29] = 8.e-4;    // t2/f2/E2
param[30] = -1.;      // Eref1
param[31] = 3.;       // Eref2
param[32] = 0.;       // dt/Af
param[33] = 1.;       // Amp/qf
param[34] = 6.;       // xsw
param[36] = 1.;       // norm

• some parameters are later changed in the loops for convenience (to create files for grid of parameters), see lines as:

      for (ia=0;ia<=1;ia++){
param[0] = (double) ia;
for (iinc=20;iinc<=80;iinc+=20){
param[1] = (double) iinc;
for (ih=1;ih<=20;ih++){
param[8] = 1.5 * (100./1.5)**((ih-1.)/19.);


# Output files created

• The output files are created only when the code is run outside XSPEC.
• The following naming scheme is used for the output files:
• AAA is 100× the horizon value (thus 100 means a=1 and 200 means a=0),
• BB is the inclination in degrees,
• CCCC is 10× the height (e.g. 0030 means h=3),
• u1 is used for phase unwrapped in frequency dependence,
• u2 is used for phase unwrapped in energy dependence.
• All spectra and light curves are always computed in photon numbers and per keV and per second, time is in seconds and frequency in Hz.
• The output files created by kynrefrev.c code:
• below by relative reflection it is meant the disc response divided by the total primary flux in the flash.
• kynrefrev_photar.dat → the values as would be given inside XSPEC,
• kynrefrev_AAA_BB_CCCC.txt → the summary of parameter values,
• kynrefrev_AAA_BB_CCCC_far.dat → the time evolving observed reflection spectrum where the energy changes with rows and the time changes with columns,
• kynrefrev_AAA_BB_CCCC_flux_prim.dat → the total observed primary flux (i.e. integrated in energy) per second, it is constant during the duration of the flare,
• kynrefrev_AAA_BB_CCCC_lc.dat → the light curve of the observed reflection (2nd column) where we integrated over the whole energy range, the 1st column contains the time,
• kynrefrev_AAA_BB_CCCC_spectrum.dat → the time integrated spectrum of the observed reflection (2nd column) and the observed primary (3rd column), both are divided by the flare duration, the 1st column contains the central value of the energy bins in keV.
• kynrefrev_AAA_BB_CCCC_bands_lc.dat → the light curves for the observed reflection (2nd and higher columns) where in each column the light curve is integrated over different energy band (defined in the code), the 1st column contains the time,
• kynrefrev_AAA_BB_CCCC_bands_prim.dat → the observed primary flux per second, it is constant during the duration of the flare where in each column the flux is integrated over different energy band (defined in the code).
• kynrefrev_AAA_BB_CCCC_real.dat → the real part of the FFT of the relative reflection with frequency changing with rows and energy with columns,
• kynrefrev_AAA_BB_CCCC_imag.dat → the imaginary part of the FFT of the relative reflection with frequency changing with rows and energy with columns,
• kynrefrev_AAA_BB_CCCC_ampl.dat → the amplitude of the FFT of the relative reflection with frequency changing with rows and energy with columns,
• kynrefrev_AAA_BB_CCCC_phase.dat, kynrefrev_AAA_BB_CCCC_phase_u1.dat, kynrefrev_AAA_BB_CCCC_phase_u2.dat → the phase of the FFT of the relative reflection with frequency changing with rows and energy with columns,
• kynrefrev_AAA_BB_CCCC_real_tot.dat → the real part of the FFT of the total signal (reflection response plus primary flash) with frequency changing with rows and energy with columns,
• kynrefrev_AAA_BB_CCCC_imag_tot.dat → the imaginary part of the FFT of the total signal (reflection response plus primary flash) with frequency changing with rows and energy with columns,
• kynrefrev_AAA_BB_CCCC_ampl_tot.dat → the amplitude of the FFT of the total signal (reflection response plus primary flash) with frequency changing with rows and energy with columns,
• kynrefrev_AAA_BB_CCCC_phase_tot.dat, kynrefrev_AAA_BB_CCCC_phase_tot_u1.dat, kynrefrev_AAA_BB_CCCC_phase_tot_u2.dat → the phase of the FFT of the total signal (reflection response plus primary flash) with frequency changing with rows and energy with columns,
• kynrefrev_AAA_BB_CCCC_bands_real.dat → the real part, as a function of frequency, of the FFT of the relative reflection integrated in different energy bands as defined in the code (2nd and higher columns), the 1st column contains the frequency,
• kynrefrev_AAA_BB_CCCC_bands_imag.dat → the imaginary part, as a function of frequency, of the FFT of the relative reflection integrated in different energy bands as defined in the code (2nd and higher columns), the 1st column contains the frequency,
• kynrefrev_AAA_BB_CCCC_bands_ampl.dat → the amplitude, as a function of frequency, of the FFT of the relative reflection integrated in different energy bands as defined in the code (2nd and higher columns), the 1st column contains the frequency,
• kynrefrev_AAA_BB_CCCC_bands_phase.dat, kynrefrev_AAA_BB_CCCC_bands_phase_u1.dat → the phase, as a function of frequency, of the FFT of the relative reflection integrated in different energy bands as defined in the code (2nd and higher columns), the 1st column contains the frequency,
• kynrefrev_AAA_BB_CCCC_bands_real_tot.dat → the real part, as a function of frequency, of the FFT of the total signal (reflection response plus primary flash) integrated in different energy bands as defined in the code (2nd and higher columns), the 1st column contains the frequency,
• kynrefrev_AAA_BB_CCCC_bands_imag_tot.dat → the imaginary part, as a function of frequency, of the FFT of the total signal (reflection response plus primary flash) integrated in different energy bands as defined in the code (2nd and higher columns), the 1st column contains the frequency,
• kynrefrev_AAA_BB_CCCC_bands_ampl_tot.dat → the amplitude, as a function of frequency, of the FFT of the total signal (reflection response plus primary flash) integrated in different energy bands as defined in the code (2nd and higher columns), the 1st column contains the frequency,
• kynrefrev_AAA_BB_CCCC_bands_phase_tot.dat, kynrefrev_AAA_BB_CCCC_bands_phase_tot_u1.dat → the phase, as a function of frequency, of the FFT of the total signal (reflection response plus primary flash) integrated in different energy bands as defined in the code (2nd and higher columns), the 1st column contains the frequency,
• kynrefrev_AAA_BB_CCCC_freq_wrap.dat → the first wrapping frequency for the phase computed for the relative reflection (the lowest one from all energy bins)
• kynrefrev_AAA_BB_CCCC_fft_tot_int.dat → energy dependent average values of the FFT of the total signal (real part, imaginary part, amplitude, phase, unwrapped phase, delay, ratio of the amplitudes for the energy band of interest and reference energy band, delay between the energy band of interest and reference energy band computed from wrapped and unwrapped phases and directly averaged delay between the two energy bands as well as the ratio of the amplitudes and delay difference between the energy band of interest and reference energy band where reference energy band always excludes the current energy bin), the average is computed in the range of 0 to the first wrapping frequency, the 1st column contains the central value of energy bins in keV; the FFT here is averaged over frequencies for real and imaginary parts first and then, from the result, all the rest quantities are computed except for the directly averaged delay, where the delay is computed first from real and imaginary parts of the FFT for each frequency and only then it is averaged (just for comparison).
• kynrefrev_AAA_BB_CCCC_fft_tot_fband.dat → energy dependent average values of the FFT of the total signal (real part, imaginary part, amplitude, phase, unwrapped phase, delay, ratio of the amplitudes for the energy band of interest and reference energy band, delay between the energy band of interest and reference energy band computed from wrapped and unwrapped phases and directly averaged delay between the two energy bands as well as the ratio of the amplitudes and delay difference between the energy band of interest and reference energy band where reference energy band always excludes the current energy bin), the average is computed in the frequency range given by param[28] and param[29], the 1st column contains the central value of energy bins in keV; the FFT here is averaged over frequencies for real and imaginary parts first and then, from the result, all the rest quantities are computed except for the directly averaged delay, where the delay is computed first from real and imaginary parts of the FFT for each frequency and only then it is averaged (just for comparison).