X-ray Reflection spectroscopy#
The geometrical configuration, models avaliable and an example of analyzing data.
Contact: honghui.liu<at>uni-tuebingen.de
Material#
First broad iron line calculated for accreting black holes: Fabian et al, 1989, MNRAS, 238, 729 link
First broad iron line detected: Tanaka et al, 1995, Nature, 375, 659 link
X-ray reflection spectroscopy for black hole spins: Reynolds 2014, SSRv, 183, 277 link; Bambi et al. 2021, SSRv, 217, 65B link;
The disk-corona system#
Figure 1. The disk-corona system and the parameters required to calculate the full reflection spectrum.
It is necessary to know how the incident radiation is distributed on the disk. Or equivalently, the intensity of the incident radiation as a function of disk radius (the emissivity profile). In principle, this distribution can be self-consistently calculated if the disk-corona geometry is known (e.g. if the corona is a point source above a black hole with spin \(a_{*}\) at height \(h\)). Here the corona is the X-ray source illuminating the accretion disk (the source that emits the power-law radiation to the accretion disk). In practice, a phenomenological emissivity profile in the form of broken power-law is commenly used to fit the reflection spectra. Please distinguish the power-law emission from the corona and the power-law-like emissivity profile! Power-law is just a kind of function that has format: \(f(x)=ax^{-k}\).
Then, the disk geometry and its physical properties (inclination, inner and outer radius, density, temperature, ionization, element abundances) are also key ingredients. The inclination of the inner disk, together with the inner radius, will affect the strength of Doppler shifting on the photons emitted from the disk. The inner radius of the disk is also important, because there the gravitational redshift of the photons is the strongest.
The last component is the spacetime. The photons emitted in the vicinity of black holes will inevitably affected by the strong gravity field. This must be taken into account when modeling the reflected spectra from the inner accretion disk. It is also for this reason that the relativistic reflection spectrum can be used to probe the strong gravity field (e.g., measure the black hole spin) and to test gravity theory. For a Kerr black hole, the spacetime metric can be fully described by three parameters: black hole mass (\(M\)), black hole spin (\(a_{*}\)) and electric charge (\(Q\)). The influence of electric charge on the reflection spectrum is negligible. The black hole mass is not needed in the calculation of reflection spectrum when measuring length scale in gravitational radius (\(r_g=GM/c^2\)). In the end, there is only one parameter, the black hole spin (\(a_*\)), left in the reflection model.
import xspec
xspec.AllModels.lmod("relxill", '/Users/honghui/projects/models/relxill')
model = xspec.Model("relxill")
---------------------------------------------------------------------------
ModuleNotFoundError Traceback (most recent call last)
Cell In[1], line 1
----> 1 import xspec
2 xspec.AllModels.lmod("relxill", '/Users/honghui/projects/models/relxill')
4 model = xspec.Model("relxill")
ModuleNotFoundError: No module named 'xspec'
Local reflection and relativistic effects#
Figure 2. (Left) The broadening of a 6.4 keV line by the accretion disk. (Right) The reflection spectrum in the rest-frame of the accretion disk (black) and the relativisticlly broadened observed spectrum (red).
Calculate the observed reflection spectrum with (1) the local reflection; (2) the emssivity profile and (3) the transfer function.
The X-ray reflection spectroscopy can be applied to both X-ray binary and active galactic nuclei systems.
Models available#
Broad line models#
relline: https://heasarc.gsfc.nasa.gov/docs/software/xspec/manual/node227.htmllaor: https://heasarc.gsfc.nasa.gov/docs/software/xspec/manual/node195.htmlkynrline: https://projects.asu.cas.cz/stronggravity/kyn#kynrline
Local reflection models#
xillverand its variations (e.g.,xillverCp,xillverNS): https://www.sternwarte.uni-erlangen.de/~dauser/research/relxill/.The difference between these variations is the incident spectrum.
reflionxand its variations: https://github.com/honghui-liu/reflionx_tables.DAO: https://github.com/ABHModels/DAO
Convolution models for relativistic effects#
relconv. To use the model in XSPEC:relconv*xillver.relconv*atable{reflionx_HD_nthcomp_v2_log.fits}In this configuration, the whole disk has the same density, ionization parameter and emission angle.
KYNconv: https://projects.asu.cas.cz/stronggravity/kyn#kynconv
Full reflection model#
relxill: https://www.sternwarte.uni-erlangen.de/~dauser/research/relxill/Kerr spacetime with BH spins.
Other flavors:
relxillcp,relxilllp,relxilllpcp,relxillNS, etc.
relxill_nk: https://github.com/ABHModels/relxill_nkIn this model, the spacetime geometry near the BH can be non-Kerr.
KYNSTOKES: https://projects.asu.cas.cz/dovciak/kynstokesX-ray Relativistic reflection models with both spectral and polarimetric properties.
RELTRANS: https://reltransdocs.readthedocs.io/en/latest/X-ray Relativistic reflection models with both spectral and reverberation lag properties.
relfkerr: https://users.camk.edu.pl/mitsza/reflkerr/
Hands-on#
import os
#import bxa.xspec as bxa
from IPython.display import display, HTML, Image
import xspec
%load_ext wurlitzer
import IPython.display
os.environ['HEADASNOQUERY'] = ''
os.environ['HEADASPROMPT'] = '/dev/null'
## Define functions for easy control and plot
import seaborn as sns
colors = sns.color_palette()
import matplotlib.pyplot as plt
import matplotlib
import numpy as np
import matplotlib_inline
matplotlib_inline.backend_inline.set_matplotlib_formats("svg")
def delete_head(infile, skip_head=3):
import subprocess as sp
comm = f"sed -i -e \"1,{skip_head}d\" {infile}"
sp.check_call(comm, shell=True)
return True
def rewrite_ipl(infile:str):
"""
Rewrite the output file of iplot in XSPEC. Each data group will be stored in
the file 'temp-N.txt' (N starts from 0).
Parameters:
-----------
infile: str
The filename of the input file
Output:
-------
Total number of files
"""
j = 0
fw = open('temp-0.txt','w+')
with open(infile,'r') as f:
for i in f:
if (i.split()[0]=='NO'):
fw.close()
j = j + 1
outfile=f"temp-{j}.txt"
fw = open(outfile,'w+')
else:
fw.write(i)
fw.close()
delete_head('temp-0.txt', skip_head=3)
return j+1
def plot_format(ax, chi=True):
ax.set_xscale('log')
ax.set_xlim(1.0,100.0)
ax.xaxis.set_ticks_position('both')
#ax.xaxis.set_ticks_position('both')
#ax.xaxis.set_major_formatter(matplotlib.ticker.ScalarFormatter())
ax.xaxis.set_minor_formatter(matplotlib.ticker.NullFormatter())
size=18
ax.tick_params(axis='x',which='major', labelsize=size,
direction='in', length=7, width=1.2)
ax.tick_params(axis='x',which='minor', labelsize=size,
direction='in', length=5, width=1.2)
ax.tick_params(axis='y',which='major', labelsize=size,
direction='in', length=7, width=1.2)
ax.tick_params(axis='y',which='minor', labelsize=size,
direction='in', length=5, width=1.2)
ax.set_xlabel("Energy", fontsize=size+2)
def plot_eeuf_ra(nspec=1, naddmodel=2, delchi=False):
save_data = 'specfit.qdp'
if (os.path.isfile(save_data)):
os.remove(save_data)
xspec.Plot.commands = ()
xspec.Plot.device = '/null'
xspec.Plot.xAxis = "keV"
xspec.Plot.add = True
xspec.Plot.addCommand(f'wd {save_data}')
if delchi:
xspec.Plot("eeuf de")
hline = 0.0
yl = "DelChi"
else:
xspec.Plot("eeuf ra")
hline= 1.0
yl = "Data/Model"
plt.close("all")
fig = plt.figure(figsize=(8, 8))
gs = matplotlib.gridspec.GridSpec(2, 1, height_ratios=[5, 2],hspace=0.0)
ax = [None]*2
i = 0
for i in range(2):
ax[i] = plt.subplot(gs[i])
#ax[1] = plt.subplot(gs[1])
rewrite_ipl(save_data)
i = 0
for i in range(nspec):
infile = f"temp-{i}.txt"
data = np.genfromtxt(infile, unpack=True, skip_header=0)
ax[0].errorbar(x=data[0], y=data[2], xerr=data[1], yerr=data[3],
lw = 1.0, ls='None', color=colors[i], fmt='o', ms=5.0)
ax[0].plot(data[0], data[4], ls='-',lw=1.0, color=colors[i], label="Total")
for j in range(naddmodel):
ax[0].plot(data[0], data[5+j], ls='--',lw=1.0, color=colors[i])
ax[0].set_yscale('log')
infile = f"temp-{i+nspec}.txt"
data = np.genfromtxt(infile, unpack=True, skip_header=0)
ax[1].errorbar(x=data[0], y=data[2], xerr=data[1], yerr=data[3],
lw = 1.0, ls='None', color=colors[i], fmt='o', ms=5.0)
plot_format(ax[0])
ax[0].tick_params(axis='x',which='major', labelsize=0)
ax[0].tick_params(axis='x',which='minor', labelsize=0)
ax[0].set_ylabel(r'keV$^2$ (phs cm$^{-2}$ s$^{-1}$ keV$^{-1}$)', fontsize=18)
ax[1].set_ylabel(yl, fontsize=18)
ax[1].axhline(y=hline, color='grey', ls='--', lw=1.5)
plot_format(ax[1])
return ax
def plot_ldat(nspec=1):
save_data = 'specfit.qdp'
if (os.path.isfile(save_data)):
os.remove(save_data)
xspec.Plot.commands = ()
xspec.Plot.device = '/null'
xspec.Plot.xAxis = "keV"
xspec.Plot.add = True
xspec.Plot.addCommand(f'wd {save_data}')
xspec.Plot.area = True ## devided by effetive area
xspec.Plot.setRebin(minSig=50, maxBins=10)
xspec.Plot("ld")
plt.close("all")
fig, ax = plt.subplots(figsize=(10,6))
rewrite_ipl(save_data)
for i in range(nspec):
infile = f"temp-{i}.txt"
data = np.genfromtxt(infile, unpack=True, skip_header=0)
x = data[0]
y = data[2]
xerr = data[1].T
yerr = data[3].T
ax.errorbar(x=x, y=y, xerr=xerr, yerr=yerr, lw = 1.0, ls='None',
color=colors[i], fmt='o', ms=5.0)
ax.set_yscale('log')
ax.set_ylabel("Counts/s/keV/cm^2", fontsize=18)
plot_format(ax)
# get_free: Get list of free parameters (corrected version)
def get_free():
"""
Return and print a list of free parameter indices.
"""
lis = []
try:
# Try to access the first model to check if any model is defined
model = xspec.AllModels(1)
nopar = model.nParameters # Number of parameters in the first model
except IndexError:
raise ValueError("No model defined. Please define a model first (e.g., AllModels += 'pow').")
# Iterate over parameters in the first model
for j in range(1, nopar + 1):
param = model(j) # Access parameter by index
if not param.frozen: # Check if thawed
if param.link=='': #Check if it is linked
lis.append(j)
print(lis)
return lis
def error_free():
# Save initial chi-square
chi1 = xspec.Fit.statistic
#xspec.Fit.query = "yes"
xspec.Xset.parallel.error = 5
lis = get_free()
delchi = 2.706
# Save the current best-fit parameter values
#best1 = get_best()
for j in lis:
xspec.Fit.error(f"stopat 20 0.01 maximum 100.0 {delchi},,{j}")
chi2 = xspec.Fit.statistic
#best2 = get_best()
# Compare old and new best-fit values
#print_old_new(best1, best2)
print(f">>>Old chi^2: {chi1}")
print(f">>>New chi^2: {chi2}")
return chi1, chi2
Load data#
From FPMA and FPMB
os.chdir("../data/")
xspec.AllData("1:1 fpma_optmin20.pha 2:2 fpmb_optmin20.pha")
os.chdir("../notebooks/")
Warning: RMF DETNAM keyword (DET0) is not consistent with that in spectrum (NONE)
Warning: RMF DETNAM keyword (DET0) is not consistent with that in spectrum (NONE)
2 spectra in use
Spectral Data File: fpma_optmin20.pha Spectrum 1
Net count rate (cts/s) for Spectrum:1 2.769e+02 +/- 1.183e-01 (99.8 % total)
Assigned to Data Group 1 and Plot Group 1
Noticed Channels: 1-378
Telescope: NuSTAR Instrument: FPMA Channel Type: PI
Exposure Time: 1.986e+04 sec
Using fit statistic: chi
Using Background File nu30101022002A01_bk.pha
Background Exposure Time: 1.986e+04 sec
Using Response (RMF) File nu30101022002A01_sr.rmf for Source 1
Using Auxiliary Response (ARF) File nu30101022002A01_sr.arf
Spectral Data File: fpmb_optmin20.pha Spectrum 2
Net count rate (cts/s) for Spectrum:2 2.538e+02 +/- 1.115e-01 (99.7 % total)
Assigned to Data Group 2 and Plot Group 2
Noticed Channels: 1-370
Telescope: NuSTAR Instrument: FPMB Channel Type: PI
Exposure Time: 2.05e+04 sec
Using fit statistic: chi
Using Background File nu30101022002B01_bk.pha
Background Exposure Time: 2.05e+04 sec
Using Response (RMF) File nu30101022002B01_sr.rmf for Source 1
Using Auxiliary Response (ARF) File nu30101022002B01_sr.arf
Fit statistic : Chi-Squared 1.066819e+10 using 378 bins, spectrum 1, group 1.
***Warning: Chi-square may not be valid due to bins with zero variance
in spectrum number: 1
Chi-Squared 1.072276e+10 using 370 bins, spectrum 2, group 2.
***Warning: Chi-square may not be valid due to bins with zero variance
in spectrum number: 2
Total fit statistic 2.139095e+10 with 741 d.o.f.
Test statistic : Chi-Squared 2.139095e+10 using 748 bins.
***Warning: Chi-square may not be valid due to bins with zero variance
in spectrum number(s): 1 2
Null hypothesis probability of 0.000000e+00 with 741 degrees of freedom
Current data and model not fit yet.
Ingore uneffective energy ranges#
xspec.AllData.ignore("*: **-3.0 79.0-**")
10 channels (1-10) ignored in spectrum # 1
10 channels (1-10) ignored in spectrum # 2
97 channels (282-378) ignored in spectrum # 1
87 channels (284-370) ignored in spectrum # 2
Fit statistic : Chi-Squared 1.008206e+10 using 271 bins, spectrum 1, group 1.
Chi-Squared 1.002355e+10 using 273 bins, spectrum 2, group 2.
Total fit statistic 2.010561e+10 with 537 d.o.f.
Test statistic : Chi-Squared 2.010561e+10 using 544 bins.
Null hypothesis probability of 0.000000e+00 with 537 degrees of freedom
Current data and model not fit yet.
View the data#
## Plot to see the raw data
plot_ldat(nspec=2)
Plot command list is now empty
***Warning: Fit is not current.
Load a simple continuum model#
model = xspec.Model("const*tbabs*(diskbb+cutoffpl)")
========================================================================
Model constant<1>*TBabs<2>(diskbb<3> + cutoffpl<4>) Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
Data group: 1
1 1 constant factor 1.00000 +/- 0.0
2 2 TBabs nH 10^22 1.00000 +/- 0.0
3 3 diskbb Tin keV 1.00000 +/- 0.0
4 3 diskbb norm 1.00000 +/- 0.0
5 4 cutoffpl PhoIndex 1.00000 +/- 0.0
6 4 cutoffpl HighECut keV 15.0000 +/- 0.0
7 4 cutoffpl norm 1.00000 +/- 0.0
Data group: 2
8 1 constant factor 1.00000 = p1
9 2 TBabs nH 10^22 1.00000 = p2
10 3 diskbb Tin keV 1.00000 = p3
11 3 diskbb norm 1.00000 = p4
12 4 cutoffpl PhoIndex 1.00000 = p5
13 4 cutoffpl HighECut keV 15.0000 = p6
14 4 cutoffpl norm 1.00000 = p7
________________________________________________________________________
tbvabs Version 2.3
Cosmic absorption with grains and H2, modified from
Wilms, Allen, & McCray, 2000, ApJ 542, 914-924
Questions: Joern Wilms
joern.wilms@sternwarte.uni-erlangen.de
joern.wilms@fau.de
http://pulsar.sternwarte.uni-erlangen.de/wilms/research/tbabs/
PLEASE NOTICE:
To get the model described by the above paper
you will also have to set the abundances:
abund wilm
Note that this routine ignores the current cross section setting
as it always HAS to use the Verner cross sections as a baseline.
Fit statistic : Chi-Squared 389010.7 using 271 bins, spectrum 1, group 1.
Chi-Squared 381493.4 using 273 bins, spectrum 2, group 2.
Total fit statistic 770504.1 with 537 d.o.f.
Test statistic : Chi-Squared 770504.1 using 544 bins.
Null hypothesis probability of 0.0e+00 with 537 degrees of freedom
Current data and model not fit yet.
Perform a fit#
xspec.AllModels(1).setPars({1: "1.0 -0.1", 2:"0.6 -0.1"})
xspec.AllModels(2).setPars({1: "1.0 0.01"})
# perform a fit
xspec.Xset.abund = "wilm"
xspec.Fit.statMethod = 'chi'
xspec.Fit.query = 'yes'
xspec.Fit.statTest = 'chi'
xspec.Fit.renorm()
xspec.Fit.perform()
Fit statistic : Chi-Squared 338.28 using 271 bins, spectrum 1, group 1.
Chi-Squared 343.68 using 273 bins, spectrum 2, group 2.
Total fit statistic 681.96 with 532 d.o.f.
Test statistic : Chi-Squared 681.96 using 544 bins.
Null hypothesis probability of 1.09e-05 with 532 degrees of freedom
Current data and model not fit yet.
Fit statistic : Chi-Squared 338.28 using 271 bins, spectrum 1, group 1.
Chi-Squared 4230.25 using 273 bins, spectrum 2, group 2.
Total fit statistic 4568.53 with 532 d.o.f.
Test statistic : Chi-Squared 4568.53 using 544 bins.
Null hypothesis probability of 0.00e+00 with 532 degrees of freedom
Current data and model not fit yet.
Solar Abundance Vector set to wilm: Wilms, J., Allen, A. & McCray, R. ApJ 542 914 (2000) (abundances are set to zero for those elements not included in the paper).
Fit statistic : Chi-Squared 338.28 using 271 bins, spectrum 1, group 1.
Chi-Squared 4230.25 using 273 bins, spectrum 2, group 2.
Total fit statistic 4568.53 with 532 d.o.f.
Test statistic : Chi-Squared 4568.53 using 544 bins.
Null hypothesis probability of 0.00e+00 with 532 degrees of freedom
Current data and model not fit yet.
Default fit statistic is set to: Chi-Squared
This will apply to all current and newly loaded spectra.
Fit statistic : Chi-Squared 338.28 using 271 bins, spectrum 1, group 1.
Chi-Squared 4230.25 using 273 bins, spectrum 2, group 2.
Total fit statistic 4568.53 with 532 d.o.f.
Test statistic : Chi-Squared 4568.53 using 544 bins.
Null hypothesis probability of 0.00e+00 with 532 degrees of freedom
Current data and model not fit yet.
Test statistic is set to: Chi-Squared
Fit statistic : Chi-Squared 338.28 using 271 bins, spectrum 1, group 1.
Chi-Squared 4230.25 using 273 bins, spectrum 2, group 2.
Total fit statistic 4568.53 with 532 d.o.f.
Test statistic : Chi-Squared 4568.53 using 544 bins.
Null hypothesis probability of 0.00e+00 with 532 degrees of freedom
Current data and model not fit yet.
Fit statistic : Chi-Squared 1309.92 using 271 bins, spectrum 1, group 1.
Chi-Squared 1323.02 using 273 bins, spectrum 2, group 2.
Total fit statistic 2632.94 with 532 d.o.f.
Test statistic : Chi-Squared 2632.94 using 544 bins.
Null hypothesis probability of 2.11e-274 with 532 degrees of freedom
Current data and model not fit yet.
Parameters
Chi-Squared |beta|/N Lvl 3:Tin 4:norm 7:norm 8:Index1 11:a 12:Incl 16:gamma 17:logxi 18:Afe 19:Ecut 21:norm 22:factor
686.763 5750.56 -3 0.560168 2607.66 2.49630 7.64878 0.800207 50.0655 1.91601 3.20841 4.49867 1000.00 0.0375962 0.971796
681.747 2739.12 -4 0.557243 2715.61 2.50627 7.21399 0.786422 49.2022 1.91562 3.20761 4.55888 1000.00 0.0365919 0.972122
681.678 207.542 -5 0.552980 2849.97 2.50618 7.11175 0.775938 48.7153 1.91527 3.21011 4.57569 1000.00 0.0361212 0.972123
681.67 226.412 -4 0.549968 2953.52 2.50788 6.98325 0.767686 48.3078 1.91527 3.21109 4.58744 1000.00 0.0357772 0.972125
681.663 330.995 -1 0.549958 2953.89 2.50798 6.98686 0.767155 48.3101 1.91528 3.21123 4.58851 1000.00 0.0357861 0.972130
==========================================================================================
Variances and Principal Axes
3 4 7 8 11 12 16 17 18 19 21 22
2.3102E-09| 0.0472 0.0000 0.0441 -0.0004 -0.0128 -0.0002 -0.2270 0.0659 -0.0027 -0.0000 0.9666 0.0743
1.7782E-07| -0.4600 -0.0000 0.0426 -0.0001 0.0055 0.0001 -0.8497 -0.1749 -0.0118 0.0000 -0.1720 0.0647
3.7718E-07| -0.1839 -0.0000 -0.0189 0.0001 -0.0058 0.0001 0.0156 -0.0486 0.0031 -0.0000 0.0918 -0.9771
6.2109E-07| 0.8453 0.0000 0.0788 0.0004 0.0488 -0.0017 -0.4589 0.1012 -0.0152 0.0000 -0.1446 -0.1869
6.4119E-06| -0.1262 -0.0000 0.2591 -0.0194 -0.6116 0.0216 -0.0588 0.7293 -0.0235 -0.0000 -0.0757 -0.0220
1.0249E-05| -0.1471 -0.0000 0.2339 0.0141 0.7878 -0.0216 -0.0101 0.5473 -0.0398 -0.0000 -0.0319 -0.0120
3.0663E-04| 0.0155 0.0000 0.9199 0.0474 -0.0354 -0.0102 0.1094 -0.3470 -0.1325 -0.0002 0.0059 -0.0014
9.0077E-03| -0.0015 0.0000 -0.1425 0.0355 -0.0152 -0.0354 0.0047 0.0086 -0.9884 -0.0003 0.0041 0.0000
9.2616E-02| 0.0017 0.0003 0.0357 -0.9231 0.0305 0.3772 0.0065 -0.0236 -0.0524 -0.0006 0.0014 0.0000
1.9443E+00| 0.0005 0.0028 -0.0103 0.3792 0.0194 0.9249 -0.0009 0.0022 -0.0183 -0.0041 0.0007 0.0000
3.9566E+05| 0.0000 -0.7458 0.0001 0.0013 0.0001 0.0047 0.0000 -0.0000 -0.0004 0.6662 0.0000 -0.0000
6.8964E+05| -0.0000 0.6662 0.0001 0.0003 0.0000 0.0012 0.0000 -0.0000 -0.0003 0.7458 0.0000 0.0000
------------------------------------------------------------------------------------------
================================================================================================================================================
Covariance Matrix
1 2 3 4 5 6 7 8 9 10 11 12
4.946e-04 -1.578e+01 -8.494e-04 4.232e-03 3.441e-04 1.335e-02 -2.173e-04 1.927e-04 1.879e-03 -7.561e+00 -3.079e-05 -2.956e-07
-1.578e+01 5.261e+05 1.728e+01 -2.443e+02 -1.929e+01 -8.373e+02 3.964e+00 -1.196e+00 -1.721e+01 1.461e+05 7.382e-02 9.703e-03
-8.494e-04 1.728e+01 6.990e-03 3.817e-02 3.495e-03 1.729e-01 1.878e-03 -3.017e-03 -2.260e-02 5.896e+01 4.926e-04 -1.760e-07
4.232e-03 -2.443e+02 3.817e-02 1.043e+00 6.486e-02 3.224e+00 1.387e-02 -2.348e-02 -2.370e-01 4.763e+02 5.386e-03 -1.989e-06
3.441e-04 -1.929e+01 3.495e-03 6.486e-02 4.961e-03 2.359e-01 1.149e-03 -2.079e-03 -1.833e-02 3.681e+01 4.170e-04 -1.629e-07
1.335e-02 -8.373e+02 1.729e-01 3.224e+00 2.359e-01 1.137e+01 5.706e-02 -1.008e-01 -9.088e-01 1.850e+03 2.045e-02 -6.765e-06
-2.173e-04 3.964e+00 1.878e-03 1.387e-02 1.149e-03 5.706e-02 5.406e-04 -8.572e-04 -7.156e-03 1.722e+01 1.503e-04 8.816e-08
1.927e-04 -1.196e+00 -3.017e-03 -2.348e-02 -2.079e-03 -1.008e-01 -8.572e-04 1.445e-03 1.142e-02 -2.667e+01 -2.457e-04 -6.412e-08
1.879e-03 -1.721e+01 -2.260e-02 -2.370e-01 -1.833e-02 -9.088e-01 -7.156e-03 1.142e-02 1.075e-01 -2.293e+02 -2.153e-03 -8.997e-07
-7.561e+00 1.461e+05 5.896e+01 4.763e+02 3.681e+01 1.850e+03 1.722e+01 -2.667e+01 -2.293e+02 5.592e+05 4.832e+00 3.152e-03
-3.079e-05 7.382e-02 4.926e-04 5.386e-03 4.170e-04 2.045e-02 1.503e-04 -2.457e-04 -2.153e-03 4.832e+00 4.594e-05 5.743e-09
-2.956e-07 9.703e-03 -1.760e-07 -1.989e-06 -1.629e-07 -6.765e-06 8.816e-08 -6.412e-08 -8.997e-07 3.152e-03 5.743e-09 3.879e-07
------------------------------------------------------------------------------------------------------------------------------------------------
========================================================================
Model constant<1>*TBabs<2>(diskbb<3> + cutoffpl<4> + relxill<5>) Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
Data group: 1
1 1 constant factor 1.00000 frozen
2 2 TBabs nH 10^22 0.600000 frozen
3 3 diskbb Tin keV 0.549958 +/- 2.22387E-02
4 3 diskbb norm 2953.89 +/- 725.346
5 4 cutoffpl PhoIndex 1.91528 = p16
6 4 cutoffpl HighECut keV 1000.00 = p19
7 4 cutoffpl norm 2.50798 +/- 8.36083E-02
8 5 relxill Index1 6.98686 +/- 1.02125
9 5 relxill Index2 6.98686 = p8
10 5 relxill Rbr 15.0000 frozen
11 5 relxill a 0.767155 +/- 7.04344E-02
12 5 relxill Incl deg 48.3101 +/- 3.37235
13 5 relxill Rin -1.00000 frozen
14 5 relxill Rout 400.000 frozen
15 5 relxill z 0.0 frozen
16 5 relxill gamma 1.91528 +/- 2.32507E-02
17 5 relxill logxi 3.21123 +/- 3.80088E-02
18 5 relxill Afe 4.58851 +/- 0.327874
19 5 relxill Ecut keV 1000.00 +/- 747.769
20 5 relxill refl_frac -1.00000 frozen
21 5 relxill norm 3.57861E-02 +/- 6.77775E-03
Data group: 2
22 1 constant factor 0.972130 +/- 6.22847E-04
23 2 TBabs nH 10^22 0.600000 = p2
24 3 diskbb Tin keV 0.549958 = p3
25 3 diskbb norm 2953.89 = p4
26 4 cutoffpl PhoIndex 1.91528 = p5
27 4 cutoffpl HighECut keV 1000.00 = p6
28 4 cutoffpl norm 2.50798 = p7
29 5 relxill Index1 6.98686 = p8
30 5 relxill Index2 6.98686 = p9
31 5 relxill Rbr 15.0000 = p10
32 5 relxill a 0.767155 = p11
33 5 relxill Incl deg 48.3101 = p12
34 5 relxill Rin -1.00000 = p13
35 5 relxill Rout 400.000 = p14
36 5 relxill z 0.0 = p15
37 5 relxill gamma 1.91528 = p16
38 5 relxill logxi 3.21123 = p17
39 5 relxill Afe 4.58851 = p18
40 5 relxill Ecut keV 1000.00 = p19
41 5 relxill refl_frac -1.00000 = p20
42 5 relxill norm 3.57861E-02 = p21
________________________________________________________________________
Fit statistic : Chi-Squared 337.83 using 271 bins, spectrum 1, group 1.
Chi-Squared 343.83 using 273 bins, spectrum 2, group 2.
Total fit statistic 681.66 with 532 d.o.f.
Test statistic : Chi-Squared 681.66 using 544 bins.
Null hypothesis probability of 1.13e-05 with 532 degrees of freedom
View the residuals#
ax = plot_eeuf_ra(nspec=2, naddmodel=2, delchi=False)
ax[0].set_ylim(1e-1)
Plot command list is now empty
(0.1, 43.792224447732075)
In the plot above, we can see that when fitting the data without a reflection model, there are large residuals around the iron line region (6-7 keV) and the Compton hump region (20-30 keV). To resolve these residuals, we should include a full reflection model (see below).
Load the RELXILL package#
# Load RELXILL model package
xspec.AllModels.lmod("relxill", '/Users/honghui/projects/models/relxill')
Model package relxill successfully loaded.
model2 = xspec.Model("const*tbabs*(diskbb+cutoffpl+relxill)")
========================================================================
Model constant<1>*TBabs<2>(diskbb<3> + cutoffpl<4> + relxill<5>) Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
Data group: 1
1 1 constant factor 1.00000 +/- 0.0
2 2 TBabs nH 10^22 1.00000 +/- 0.0
3 3 diskbb Tin keV 1.00000 +/- 0.0
4 3 diskbb norm 1.00000 +/- 0.0
5 4 cutoffpl PhoIndex 1.00000 +/- 0.0
6 4 cutoffpl HighECut keV 15.0000 +/- 0.0
7 4 cutoffpl norm 1.00000 +/- 0.0
8 5 relxill Index1 3.00000 frozen
9 5 relxill Index2 3.00000 frozen
10 5 relxill Rbr 15.0000 frozen
11 5 relxill a 0.998000 +/- 0.0
12 5 relxill Incl deg 30.0000 +/- 0.0
13 5 relxill Rin -1.00000 frozen
14 5 relxill Rout 400.000 frozen
15 5 relxill z 0.0 frozen
16 5 relxill gamma 2.00000 +/- 0.0
17 5 relxill logxi 3.10000 +/- 0.0
18 5 relxill Afe 1.00000 +/- 0.0
19 5 relxill Ecut keV 300.000 frozen
20 5 relxill refl_frac 3.00000 +/- 0.0
21 5 relxill norm 1.00000 +/- 0.0
Data group: 2
22 1 constant factor 1.00000 = p1
23 2 TBabs nH 10^22 1.00000 = p2
24 3 diskbb Tin keV 1.00000 = p3
25 3 diskbb norm 1.00000 = p4
26 4 cutoffpl PhoIndex 1.00000 = p5
27 4 cutoffpl HighECut keV 15.0000 = p6
28 4 cutoffpl norm 1.00000 = p7
29 5 relxill Index1 3.00000 = p8
30 5 relxill Index2 3.00000 = p9
31 5 relxill Rbr 15.0000 = p10
32 5 relxill a 0.998000 = p11
33 5 relxill Incl deg 30.0000 = p12
34 5 relxill Rin -1.00000 = p13
35 5 relxill Rout 400.000 = p14
36 5 relxill z 0.0 = p15
37 5 relxill gamma 2.00000 = p16
38 5 relxill logxi 3.10000 = p17
39 5 relxill Afe 1.00000 = p18
40 5 relxill Ecut keV 300.000 = p19
41 5 relxill refl_frac 3.00000 = p20
42 5 relxill norm 1.00000 = p21
________________________________________________________________________
Fit statistic : Chi-Squared 1.016496e+10 using 271 bins, spectrum 1, group 1.
Chi-Squared 1.011168e+10 using 273 bins, spectrum 2, group 2.
Total fit statistic 2.027664e+10 with 530 d.o.f.
Test statistic : Chi-Squared 2.027664e+10 using 544 bins.
Null hypothesis probability of 0.000000e+00 with 530 degrees of freedom
Current data and model not fit yet.
xspec.AllModels(1).setPars({1: "1.0 -0.1", 2:"0.6 -0.1"})
xspec.AllModels(2).setPars({1: "1.0 0.01"})
xspec.AllModels(1).setPars({5: "=p16", 6:"=19", 9:"=p8"})
xspec.AllModels(1).setPars({8: "3.0 0.01", 19:"300.0 0.1", 20:"-1.0 -0.1"})
Fit statistic : Chi-Squared 1.032585e+10 using 271 bins, spectrum 1, group 1.
Chi-Squared 1.026953e+10 using 273 bins, spectrum 2, group 2.
Total fit statistic 2.059537e+10 with 532 d.o.f.
Test statistic : Chi-Squared 2.059537e+10 using 544 bins.
Null hypothesis probability of 0.000000e+00 with 532 degrees of freedom
Current data and model not fit yet.
Fit statistic : Chi-Squared 1.032585e+10 using 271 bins, spectrum 1, group 1.
Chi-Squared 1.026953e+10 using 273 bins, spectrum 2, group 2.
Total fit statistic 2.059537e+10 with 531 d.o.f.
Test statistic : Chi-Squared 2.059537e+10 using 544 bins.
Null hypothesis probability of 0.000000e+00 with 531 degrees of freedom
Current data and model not fit yet.
Fit statistic : Chi-Squared 9.951712e+09 using 271 bins, spectrum 1, group 1.
Chi-Squared 9.897553e+09 using 273 bins, spectrum 2, group 2.
Total fit statistic 1.984926e+10 with 533 d.o.f.
Test statistic : Chi-Squared 1.984926e+10 using 544 bins.
Null hypothesis probability of 0.000000e+00 with 533 degrees of freedom
Current data and model not fit yet.
Fit statistic : Chi-Squared 3.903736e+08 using 271 bins, spectrum 1, group 1.
Chi-Squared 3.901063e+08 using 273 bins, spectrum 2, group 2.
Total fit statistic 7.804799e+08 with 532 d.o.f.
Test statistic : Chi-Squared 7.804799e+08 using 544 bins.
Null hypothesis probability of 0.000000e+00 with 532 degrees of freedom
Current data and model not fit yet.
xspec.AllModels.show()
Parameters defined:
========================================================================
Model constant<1>*TBabs<2>(diskbb<3> + cutoffpl<4> + relxill<5>) Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
Data group: 1
1 1 constant factor 1.00000 frozen
2 2 TBabs nH 10^22 0.600000 frozen
3 3 diskbb Tin keV 1.00000 +/- 0.0
4 3 diskbb norm 1.00000 +/- 0.0
5 4 cutoffpl PhoIndex 2.00000 = p16
6 4 cutoffpl HighECut keV 300.000 = p19
7 4 cutoffpl norm 1.00000 +/- 0.0
8 5 relxill Index1 3.00000 +/- 0.0
9 5 relxill Index2 3.00000 = p8
10 5 relxill Rbr 15.0000 frozen
11 5 relxill a 0.998000 +/- 0.0
12 5 relxill Incl deg 30.0000 +/- 0.0
13 5 relxill Rin -1.00000 frozen
14 5 relxill Rout 400.000 frozen
15 5 relxill z 0.0 frozen
16 5 relxill gamma 2.00000 +/- 0.0
17 5 relxill logxi 3.10000 +/- 0.0
18 5 relxill Afe 1.00000 +/- 0.0
19 5 relxill Ecut keV 300.000 +/- 0.0
20 5 relxill refl_frac -1.00000 frozen
21 5 relxill norm 1.00000 +/- 0.0
Data group: 2
22 1 constant factor 1.00000 +/- 0.0
23 2 TBabs nH 10^22 0.600000 = p2
24 3 diskbb Tin keV 1.00000 = p3
25 3 diskbb norm 1.00000 = p4
26 4 cutoffpl PhoIndex 2.00000 = p5
27 4 cutoffpl HighECut keV 300.000 = p6
28 4 cutoffpl norm 1.00000 = p7
29 5 relxill Index1 3.00000 = p8
30 5 relxill Index2 3.00000 = p9
31 5 relxill Rbr 15.0000 = p10
32 5 relxill a 0.998000 = p11
33 5 relxill Incl deg 30.0000 = p12
34 5 relxill Rin -1.00000 = p13
35 5 relxill Rout 400.000 = p14
36 5 relxill z 0.0 = p15
37 5 relxill gamma 2.00000 = p16
38 5 relxill logxi 3.10000 = p17
39 5 relxill Afe 1.00000 = p18
40 5 relxill Ecut keV 300.000 = p19
41 5 relxill refl_frac -1.00000 = p20
42 5 relxill norm 1.00000 = p21
________________________________________________________________________
Perform a reflection fitting#
xspec.Fit.renorm()
xspec.Fit.perform()
Fit statistic : Chi-Squared 55529.85 using 271 bins, spectrum 1, group 1.
Chi-Squared 47949.58 using 273 bins, spectrum 2, group 2.
Total fit statistic 103479.4 with 532 d.o.f.
Test statistic : Chi-Squared 103479.4 using 544 bins.
Null hypothesis probability of 0.0e+00 with 532 degrees of freedom
Current data and model not fit yet.
Parameters
Chi-Squared |beta|/N Lvl 3:Tin 4:norm 7:norm 8:Index1 11:a 12:Incl 16:gamma 17:logxi 18:Afe 19:Ecut 21:norm 22:factor
100199 27495.5 1 1.74044 0.592688 0.100600 3.00116 0.998000 30.0180 2.00008 3.10021 0.998716 300.079 0.100125 0.999855
95087.1 59928.9 0 1.82179 0.743443 0.103002 3.01500 0.998000 30.2700 2.00118 3.10173 0.988915 303.487 0.100063 0.997987
66673.2 60801.7 0 2.19333 1.58105 0.122718 3.10578 0.998000 32.3575 2.00745 3.11272 0.929116 357.204 0.0996281 0.986695
34879.9 153370 -1 2.30463 2.47087 0.217027 3.25213 0.998000 39.1647 2.01640 3.15826 0.791231 761.959 0.0986573 0.973348
26749.7 177021 0 2.28580 2.42519 0.220750 3.27480 0.998000 39.7586 2.01616 3.15997 0.783623 952.765 0.0985112 0.971965
25975.8 10339.7 0 2.27753 2.43427 0.231466 3.27689 0.998000 40.0920 2.01461 3.16383 0.774641 989.051 0.0985884 0.972315
25569.6 44506.6 0 2.26673 2.43258 0.239995 3.28073 0.998000 40.4616 2.01339 3.16717 0.769645 998.158 0.0986183 0.972281
25156.9 60833 0 2.25451 2.42452 0.247408 3.28545 0.998000 40.8411 2.01228 3.17016 0.766382 999.120 0.0986258 0.972132
24657.4 67010.7 0 2.24158 2.41332 0.254295 3.29036 0.998000 41.2344 2.01117 3.17294 0.763835 999.641 0.0986254 0.971989
24145.1 67944 0 2.22830 2.40096 0.260872 3.29515 0.998000 41.5968 2.01003 3.17558 0.761759 999.911 0.0986221 0.971879
23718.2 66415.3 0 2.21493 2.38884 0.267236 3.29965 0.998000 41.9095 2.00886 3.17814 0.760083 999.980 0.0986175 0.971802
23293.8 66896.8 0 2.20145 2.37695 0.273367 3.30410 0.998000 42.2143 2.00768 3.18062 0.758781 999.997 0.0986104 0.971742
22889.5 66716 0 2.18795 2.36568 0.279315 3.30846 0.998000 42.5153 2.00647 3.18303 0.757789 999.999 0.0986019 0.971699
22495.7 66655.8 0 2.17442 2.35509 0.285076 3.31282 0.998000 42.8112 2.00524 3.18537 0.757081 1000.00 0.0985917 0.971665
22115.8 66347 0 2.16088 2.34533 0.290652 3.31720 0.998000 43.0955 2.00399 3.18765 0.756653 1000.00 0.0985796 0.971636
21770.2 65791.5 0 2.14738 2.33662 0.296049 3.32160 0.998000 43.3655 2.00272 3.18988 0.756495 1000.00 0.0985654 0.971610
21436.7 65680.9 0 2.13393 2.32901 0.301277 3.32606 0.998000 43.6319 2.00143 3.19206 0.756571 1000.00 0.0985492 0.971585
21122.8 65543 0 2.12053 2.32255 0.306333 3.33061 0.998000 43.8906 2.00012 3.19419 0.756877 1000.00 0.0985309 0.971561
20600.9 65803.8 0 2.10716 2.31714 0.311207 3.33530 0.998000 44.1428 1.99879 3.19627 0.757420 1000.00 0.0985099 0.971534
20230.2 55575.2 0 2.09500 2.31931 0.316400 3.33898 0.998000 44.3781 1.99742 3.19847 0.757930 1000.00 0.0985002 0.971586
19912.6 52227.3 0 2.08344 2.32585 0.321585 3.34241 0.998000 44.6047 1.99607 3.20071 0.758722 1000.00 0.0984925 0.971638
19614.5 51188.8 0 2.07224 2.33522 0.326667 3.34583 0.998000 44.8291 1.99475 3.20294 0.759808 1000.00 0.0984843 0.971673
19329.3 50503.8 0 2.06132 2.34689 0.331648 3.34931 0.998000 45.0540 1.99344 3.20514 0.761106 1000.00 0.0984756 0.971697
19057.7 49986.1 0 2.05062 2.36046 0.336510 3.35291 0.998000 45.2734 1.99215 3.20731 0.762589 1000.00 0.0984659 0.971710
18788.9 49801.2 0 2.04009 2.37556 0.341231 3.35667 0.998000 45.4837 1.99087 3.20944 0.764262 1000.00 0.0984544 0.971711
18541.7 48908.8 0 2.02981 2.39254 0.345850 3.36053 0.998000 45.6843 1.98959 3.21153 0.766059 1000.00 0.0984421 0.971709
18308.7 48451.9 0 2.01973 2.41120 0.350359 3.36449 0.998000 45.8778 1.98832 3.21358 0.767972 1000.00 0.0984288 0.971704
16108.6 53006.8 -1 1.91905 2.54003 0.387911 3.42522 0.998000 47.6450 1.97672 3.23062 0.783098 1000.00 0.0981179 0.971467
11882.7 13182.8 -2 1.45666 4.29877 0.622409 3.69472 0.998000 53.5643 1.93298 3.29899 0.878110 1000.00 0.0958211 0.971582
10252.1 5540.5 -1 1.45605 7.22509 0.636152 3.74083 0.998000 54.2820 1.92984 3.30174 0.922919 1000.00 0.0954821 0.971523
9746.6 2973.56 -1 1.43307 9.21337 0.649973 3.78797 0.998000 54.9556 1.92229 3.30312 0.935069 1000.00 0.0951597 0.971507
9307.66 909.111 -1 1.41593 11.2211 0.662910 3.82751 0.998000 55.6208 1.91540 3.30408 0.941392 1000.00 0.0948461 0.971527
9264.45 855.925 -2 1.29343 21.7164 0.766017 3.94203 0.998000 58.6066 1.89180 3.29845 0.932441 1000.00 0.0927556 0.971644
7029.07 77769.6 -2 1.25753 33.1717 0.851843 3.90892 0.998000 60.3175 1.88436 3.28657 0.903046 1000.00 0.0911384 0.971795
6613.3 17299.2 -3 1.09759 48.2793 1.50497 4.19803 0.998000 61.9731 1.92594 3.20036 1.01654 1000.00 0.0832335 0.971879
4869.98 28650.1 -2 1.04175 66.5281 1.57949 4.62935 0.998000 61.1392 1.91274 3.21330 2.00145 1000.00 0.0827776 0.972002
4468.5 6826.69 -2 0.960872 84.6690 1.65601 4.78658 0.998000 60.6784 1.91310 3.22786 2.50165 1000.00 0.0817143 0.971975
4306.16 5977.01 -2 0.867717 122.449 1.72397 4.84686 0.998000 60.2654 1.91521 3.23337 2.78530 1000.00 0.0802421 0.971969
4015.66 16408.2 -1 0.864452 140.061 1.73016 4.85112 0.998000 60.2179 1.91538 3.23313 2.80880 1000.00 0.0800588 0.971974
4009.14 137.059 -2 0.773333 213.879 1.79326 4.85312 0.998000 59.6832 1.91811 3.23340 3.01894 1000.00 0.0784361 0.971989
3604.25 19572.4 -1 0.772175 253.188 1.79912 4.85165 0.998000 59.6094 1.91829 3.23269 3.03579 1000.00 0.0782397 0.972012
3557.27 483.161 -1 0.763909 274.408 1.80482 4.84146 0.998000 59.5661 1.91833 3.23132 3.04835 1000.00 0.0780259 0.971977
3514.57 175.109 -1 0.755528 294.998 1.81074 4.83059 0.998000 59.5024 1.91852 3.23029 3.06223 1000.00 0.0778231 0.971986
3473.19 202.914 -1 0.747231 316.036 1.81676 4.82087 0.998000 59.4252 1.91881 3.22947 3.07704 1000.00 0.0776235 0.971997
3432.66 204.276 -1 0.739049 337.954 1.82284 4.81212 0.998000 59.3396 1.91916 3.22878 3.09228 1000.00 0.0774249 0.972006
3392.53 335.564 -1 0.730985 361.031 1.82894 4.80408 0.998000 59.2494 1.91955 3.22815 3.10772 1000.00 0.0772251 0.972013
3352.81 263.356 -1 0.723054 385.497 1.83507 4.79619 0.998000 59.1564 1.91996 3.22758 3.12308 1000.00 0.0770241 0.972020
3313.43 320.701 -1 0.715257 411.523 1.84122 4.78857 0.998000 59.0622 1.92037 3.22702 3.13834 1000.00 0.0768206 0.972025
3274.24 406.82 -1 0.707597 439.244 1.84736 4.78107 0.998000 58.9661 1.92079 3.22647 3.15361 1000.00 0.0766140 0.972030
3235.64 192.558 -1 0.700091 468.818 1.85354 4.77327 0.998000 58.8694 1.92120 3.22594 3.16864 1000.00 0.0764065 0.972036
3197.51 268.575 -1 0.692726 500.321 1.85970 4.76553 0.998000 58.7714 1.92162 3.22541 3.18363 1000.00 0.0761958 0.972041
3159.87 368.614 -1 0.685509 533.880 1.86586 4.75793 0.998000 58.6738 1.92202 3.22487 3.19854 1000.00 0.0759822 0.972045
3122.76 299.756 -1 0.678449 569.620 1.87204 4.75035 0.998000 58.5771 1.92242 3.22435 3.21328 1000.00 0.0757673 0.972051
3086.06 391.476 -1 0.671541 607.622 1.87822 4.74299 0.998000 58.4817 1.92281 3.22382 3.22797 1000.00 0.0755504 0.972056
3049.77 381.329 -1 0.664787 647.985 1.88440 4.73565 0.998000 58.3865 1.92319 3.22330 3.24265 1000.00 0.0753321 0.972062
3048.48 375.434 -2 0.604135 1006.68 1.94497 4.72241 0.998000 57.6799 1.92586 3.22247 3.41477 1000.00 0.0733091 0.972103
2969.66 15382.6 -2 0.554349 1763.08 2.00191 4.67424 0.998000 56.9961 1.92788 3.21858 3.55429 1000.00 0.0711651 0.972170
2593.84 20252.5 -2 0.518372 3051.22 2.05638 4.61127 0.998000 56.2241 1.92970 3.21494 3.68594 1000.00 0.0689047 0.972220
2238.15 16263.8 -2 0.493240 4828.76 2.10835 4.53690 0.998000 55.4186 1.93124 3.21147 3.80661 1000.00 0.0665563 0.972246
2000.94 11038.4 -2 0.475066 6905.86 2.15791 4.45484 0.998000 54.5790 1.93248 3.20837 3.91960 1000.00 0.0641864 0.972262
1835.28 7009.33 -2 0.461546 9102.73 2.20445 4.35396 0.998000 53.6498 1.93348 3.20547 4.01846 1000.00 0.0617042 0.972268
1716.22 3011.74 -2 0.451423 11266.0 2.24812 4.25346 0.998000 52.7137 1.93419 3.20270 4.11172 1000.00 0.0592149 0.972276
1611.77 2490.54 -2 0.443893 13266.6 2.28889 4.14616 0.998000 51.7161 1.93465 3.20034 4.19733 1000.00 0.0567188 0.972281
1558.49 1632.85 -3 0.421165 19599.3 2.47886 3.72593 0.998000 47.6834 1.93104 3.20109 4.72003 1000.00 0.0441024 0.972276
1127.19 27361 -3 0.438826 15579.8 2.53376 3.26496 0.998000 42.8941 1.92482 3.20851 4.89504 1000.00 0.0360668 0.972261
934.004 21129.6 -2 0.443570 14992.6 2.53665 3.16792 0.998000 40.8015 1.92400 3.20820 4.82002 1000.00 0.0347860 0.972224
896.975 1802.74 -2 0.448107 13889.6 2.53853 3.08513 0.998000 39.4713 1.92304 3.20976 4.77382 1000.00 0.0335226 0.972211
872.714 253.075 -2 0.453006 12775.5 2.53910 3.02726 0.998000 38.5586 1.92170 3.21231 4.75504 1000.00 0.0324403 0.972206
853.818 726.439 -2 0.457911 11747.8 2.53839 2.98295 0.998000 37.7786 1.92025 3.21538 4.75034 1000.00 0.0315331 0.972200
838.906 539.105 -2 0.462674 10838.7 2.53651 2.94700 0.998000 37.1219 1.91880 3.21851 4.74965 1000.00 0.0307674 0.972193
826.646 742.664 -2 0.467234 10045.3 2.53368 2.91505 0.998000 36.5444 1.91740 3.22168 4.74729 1000.00 0.0301009 0.972186
816.495 513.219 -2 0.471543 9357.14 2.53018 2.88840 0.998000 36.0384 1.91609 3.22478 4.74449 1000.00 0.0295374 0.972181
808.14 613.712 -2 0.475590 8761.97 2.52611 2.86340 0.998000 35.5276 1.91488 3.22776 4.73795 1000.00 0.0290320 0.972175
801.381 289.339 -2 0.479367 8246.21 2.52185 2.84332 0.998000 35.1303 1.91375 3.23053 4.73204 1000.00 0.0286146 0.972172
795.608 309.384 -2 0.482872 7800.42 2.51736 2.82409 0.998000 34.7164 1.91273 3.23315 4.72299 1000.00 0.0282354 0.972168
791.235 295.782 -2 0.486126 7412.15 2.51297 2.80816 0.998000 34.4236 1.91178 3.23556 4.71487 1000.00 0.0279262 0.972166
787.109 206.174 -2 0.489125 7074.96 2.50855 2.79276 0.998000 34.1325 1.91090 3.23789 4.70473 1000.00 0.0276423 0.972163
783.637 349.274 -2 0.491901 6779.58 2.50420 2.77809 0.998000 33.8306 1.91009 3.24013 4.69357 1000.00 0.0273820 0.972161
781.002 117.009 -2 0.494463 6519.73 2.50009 2.76618 0.998000 33.6014 1.90934 3.24213 4.68417 1000.00 0.0271660 0.972160
778.927 89.1477 -2 0.496811 6291.93 2.49618 2.75559 0.998000 33.3912 1.90868 3.24395 4.67482 1000.00 0.0269769 0.972159
776.755 66.4617 -2 0.498959 6092.10 2.49242 2.74517 0.998000 33.1649 1.90806 3.24570 4.66456 1000.00 0.0267953 0.972157
776.61 61.1541 -3 0.509676 5107.62 2.47033 2.69734 0.998000 32.1957 1.90480 3.25515 4.60828 1000.00 0.0259797 0.972154
770.314 4798.09 -4 0.520185 4421.73 2.45019 2.65082 0.998000 31.0568 1.90178 3.26431 4.55292 1000.00 0.0252060 0.972159
766.295 3530.42 -5 0.521616 4392.81 2.44918 2.64014 0.998000 30.5865 1.90128 3.26594 4.55168 1000.00 0.0249752 0.972158
766.279 2.50699 -6 0.521913 4375.81 2.44990 2.63873 0.998000 30.5174 1.90131 3.26575 4.55245 1000.00 0.0249473 0.972154
766.277 16.8319 -7 0.521946 4374.02 2.45024 2.63912 0.998000 30.5352 1.90136 3.26553 4.55265 1000.00 0.0249583 0.972154
766.266 6.75624 0 0.521946 4374.03 2.45023 2.63919 0.997671 30.5364 1.90136 3.26553 4.55261 999.282 0.0249580 0.972153
766.258 87.8721 0 0.521946 4374.08 2.45023 2.63931 0.997455 30.5393 1.90136 3.26552 4.55254 998.945 0.0249576 0.972152
==========================================================================================
Variances and Principal Axes
3 4 7 8 11 12 16 17 18 19 21 22
1.1835E-09| 0.0360 0.0000 0.0323 -0.0041 -0.0020 -0.0001 -0.1626 0.0438 -0.0016 -0.0000 0.9831 0.0529
1.5573E-07| -0.4387 -0.0000 0.0573 0.0096 0.0047 0.0001 -0.8731 -0.1283 -0.0122 0.0000 -0.1295 0.0922
3.7423E-07| 0.2832 0.0000 0.0231 0.0107 0.0026 0.0000 -0.0365 0.0489 -0.0035 0.0000 -0.0706 0.9542
5.2563E-07| 0.8404 0.0000 0.0708 0.0384 0.0069 -0.0001 -0.4424 0.0672 -0.0128 0.0000 -0.0941 -0.2790
9.1691E-06| -0.1367 -0.0000 0.3649 0.0051 0.0396 0.0036 -0.0357 0.9168 -0.0397 -0.0000 -0.0526 -0.0208
1.7713E-04| -0.0296 -0.0000 0.0480 0.9923 0.0135 -0.0226 0.0335 -0.0318 -0.0956 0.0000 0.0104 -0.0002
2.3113E-04| -0.0103 -0.0000 -0.9131 0.0712 -0.0588 0.0097 -0.1087 0.3646 0.1131 0.0002 -0.0036 0.0018
1.0301E-02| 0.0008 0.0001 -0.1405 -0.0879 0.2735 0.0122 0.0030 0.0039 -0.9474 -0.0005 0.0036 0.0000
1.9415E-01| 0.0006 -0.0001 -0.0261 0.0225 0.8866 0.3788 0.0014 -0.0178 0.2625 -0.0001 0.0042 -0.0000
1.5353E+01| -0.0002 0.0032 0.0219 0.0154 -0.3659 0.9250 0.0015 -0.0009 -0.0984 -0.0086 -0.0011 0.0000
2.2074E+05| -0.0000 -0.3828 0.0003 0.0000 -0.0032 0.0085 0.0000 -0.0001 -0.0014 0.9238 -0.0000 -0.0000
1.1321E+06| -0.0000 0.9238 0.0000 -0.0000 0.0000 0.0004 0.0000 -0.0000 -0.0001 0.3828 0.0000 0.0000
------------------------------------------------------------------------------------------
================================================================================================================================================
Covariance Matrix
1 2 3 4 5 6 7 8 9 10 11 12
3.725e-04 -1.865e+01 -1.100e-03 1.683e-04 2.554e-03 -1.683e-02 -2.754e-04 4.213e-04 4.193e-03 -8.398e+00 -3.207e-05 -2.311e-07
-1.865e+01 9.985e+05 1.946e+01 -1.538e+01 3.075e+02 -2.874e+02 9.097e+00 -1.487e+01 -3.537e+01 3.223e+05 2.551e+00 1.327e-02
-1.100e-03 1.946e+01 2.747e-02 6.696e-03 -3.241e-01 8.586e-01 3.615e-03 -4.787e-03 -1.262e-01 7.561e+01 -6.839e-04 -7.125e-07
1.683e-04 -1.538e+01 6.696e-03 4.401e-03 -1.067e-01 2.773e-01 4.923e-04 -4.623e-04 -2.948e-02 1.599e+00 -3.215e-04 -1.455e-07
2.554e-03 3.075e+02 -3.241e-01 -1.067e-01 4.434e+00 -1.107e+01 -3.609e-02 4.171e-02 1.559e+00 -6.309e+02 1.134e-02 7.133e-06
-1.683e-02 -2.874e+02 8.586e-01 2.773e-01 -1.107e+01 2.931e+01 1.032e-01 -1.314e-01 -4.043e+00 1.912e+03 -2.599e-02 -1.436e-05
-2.754e-04 9.097e+00 3.615e-03 4.923e-04 -3.609e-02 1.032e-01 5.674e-04 -8.121e-04 -1.663e-02 1.352e+01 -5.077e-05 2.630e-08
4.213e-04 -1.487e+01 -4.787e-03 -4.623e-04 4.171e-02 -1.314e-01 -8.121e-04 1.255e-03 2.120e-02 -1.998e+01 3.398e-05 -5.285e-08
4.193e-03 -3.537e+01 -1.262e-01 -2.948e-02 1.559e+00 -4.043e+00 -1.663e-02 2.120e-02 6.180e-01 -3.453e+02 3.377e-03 1.466e-06
-8.398e+00 3.223e+05 7.561e+01 1.599e+00 -6.309e+02 1.912e+03 1.352e+01 -1.998e+01 -3.453e+02 3.543e+05 -3.554e-01 1.647e-03
-3.207e-05 2.551e+00 -6.839e-04 -3.215e-04 1.134e-02 -2.599e-02 -5.077e-05 3.398e-05 3.377e-03 -3.554e-01 3.549e-05 2.725e-08
-2.311e-07 1.327e-02 -7.125e-07 -1.455e-07 7.133e-06 -1.436e-05 2.630e-08 -5.285e-08 1.466e-06 1.647e-03 2.725e-08 3.879e-07
------------------------------------------------------------------------------------------------------------------------------------------------
========================================================================
Model constant<1>*TBabs<2>(diskbb<3> + cutoffpl<4> + relxill<5>) Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
Data group: 1
1 1 constant factor 1.00000 frozen
2 2 TBabs nH 10^22 0.600000 frozen
3 3 diskbb Tin keV 0.521946 +/- 1.93011E-02
4 3 diskbb norm 4374.08 +/- 999.262
5 4 cutoffpl PhoIndex 1.90136 = p16
6 4 cutoffpl HighECut keV 998.945 = p19
7 4 cutoffpl norm 2.45023 +/- 0.165726
8 5 relxill Index1 2.63931 +/- 6.63426E-02
9 5 relxill Index2 2.63931 = p8
10 5 relxill Rbr 15.0000 frozen
11 5 relxill a 0.997455 +/- 2.10577
12 5 relxill Incl deg 30.5393 +/- 5.41400
13 5 relxill Rin -1.00000 frozen
14 5 relxill Rout 400.000 frozen
15 5 relxill z 0.0 frozen
16 5 relxill gamma 1.90136 +/- 2.38197E-02
17 5 relxill logxi 3.26552 +/- 3.54213E-02
18 5 relxill Afe 4.55254 +/- 0.786124
19 5 relxill Ecut keV 998.945 +/- 595.216
20 5 relxill refl_frac -1.00000 frozen
21 5 relxill norm 2.49576E-02 +/- 5.95776E-03
Data group: 2
22 1 constant factor 0.972152 +/- 6.22835E-04
23 2 TBabs nH 10^22 0.600000 = p2
24 3 diskbb Tin keV 0.521946 = p3
25 3 diskbb norm 4374.08 = p4
26 4 cutoffpl PhoIndex 1.90136 = p5
27 4 cutoffpl HighECut keV 998.945 = p6
28 4 cutoffpl norm 2.45023 = p7
29 5 relxill Index1 2.63931 = p8
30 5 relxill Index2 2.63931 = p9
31 5 relxill Rbr 15.0000 = p10
32 5 relxill a 0.997455 = p11
33 5 relxill Incl deg 30.5393 = p12
34 5 relxill Rin -1.00000 = p13
35 5 relxill Rout 400.000 = p14
36 5 relxill z 0.0 = p15
37 5 relxill gamma 1.90136 = p16
38 5 relxill logxi 3.26552 = p17
39 5 relxill Afe 4.55254 = p18
40 5 relxill Ecut keV 998.945 = p19
41 5 relxill refl_frac -1.00000 = p20
42 5 relxill norm 2.49576E-02 = p21
________________________________________________________________________
Fit statistic : Chi-Squared 381.34 using 271 bins, spectrum 1, group 1.
Chi-Squared 384.91 using 273 bins, spectrum 2, group 2.
Total fit statistic 766.26 with 532 d.o.f.
Test statistic : Chi-Squared 766.26 using 544 bins.
Null hypothesis probability of 1.04e-10 with 532 degrees of freedom
View the model and residuals#
ax = plot_eeuf_ra(nspec=2, naddmodel=3, delchi=False)
ax[0].set_ylim(1e-1, 30.0)
Plot command list is now empty
(0.1, 30.0)
ax = plot_eeuf_ra(nspec=2, naddmodel=3, delchi=True)
ax[0].set_ylim(1e-1, 30.0)
Plot command list is now empty
(0.1, 30.0)
get_free()
[3, 4, 7, 8, 11, 12, 16, 17, 18, 19, 21]
[3, 4, 7, 8, 11, 12, 16, 17, 18, 19, 21]
Run error command#
## Run `error` for each parameter to obtain the uncertainties.
# error_free()
## The upper command takes too much time,
## Uncomment and run the error command above and plotting command below.
# xspec.Fit.error(f"stopat 20 0.01 maximum 100.0 2.706,,11")
Parameter Confidence Range (2.706)
Parameters
Chi-Squared |beta|/N Lvl 3:Tin 4:norm 7:norm 8:Index1 11:a 12:Incl 16:gamma 17:logxi 18:Afe 19:Ecut 21:norm 22:factor
757.043 24.8394 -3 0.548707 3377.71 2.36098 2.71777 0.929657 30.4739 1.87166 3.30541 5.04087 517.071 0.0210667 0.972144
755.976 488.237 -1 0.548895 3387.88 2.36036 2.71573 0.928876 30.4032 1.87169 3.30481 5.00555 520.160 0.0210577 0.972121
755.937 56.4814 -1 0.548922 3389.17 2.36024 2.71449 0.929581 30.3675 1.87164 3.30463 5.00798 520.955 0.0210544 0.972137
755.916 147.591 -2 0.549191 3384.80 2.35923 2.71438 0.929749 30.3450 1.87125 3.30455 5.02006 515.817 0.0210281 0.972142
755.728 359.863 -2 0.549570 3377.16 2.35782 2.71465 0.923185 30.3666 1.87105 3.30435 4.99280 512.328 0.0209924 0.972141
755.639 99.436 -1 0.549591 3375.83 2.35785 2.71483 0.923932 30.3609 1.87099 3.30444 5.00059 512.284 0.0209905 0.972148
755.637 185.763 -1 0.549610 3374.61 2.35787 2.71495 0.924735 30.3613 1.87094 3.30452 5.00414 511.648 0.0209880 0.972148
==========================================================================================
Variances and Principal Axes
3 4 7 8 11 12 16 17 18 19 21 22
9.5442E-10| 0.0360 0.0000 0.0303 -0.0031 -0.0041 -0.0001 -0.1496 0.0329 -0.0008 0.0000 0.9859 0.0472
1.3527E-07| -0.3962 -0.0000 0.0737 0.0064 0.0159 0.0001 -0.8985 -0.0374 -0.0072 0.0000 -0.1281 0.1104
3.6514E-07| 0.4541 0.0000 0.0349 0.0154 0.0185 -0.0001 -0.0808 0.0639 -0.0033 0.0000 -0.0741 0.8808
4.7011E-07| 0.7869 0.0000 0.0688 0.0314 0.0368 -0.0002 -0.3901 0.0839 -0.0072 0.0000 -0.0707 -0.4576
1.2403E-05| 0.1207 0.0000 -0.4472 0.0005 -0.0477 -0.0035 -0.0562 -0.8822 0.0241 0.0001 0.0293 0.0179
1.6147E-04| -0.0360 -0.0000 -0.1739 0.9536 0.2227 -0.0225 0.0086 0.0699 -0.0622 0.0001 0.0094 0.0004
4.1349E-04| -0.0035 -0.0000 -0.8552 -0.1408 -0.1893 0.0235 -0.0914 0.4499 0.0407 0.0004 -0.0037 0.0009
3.8865E-03| 0.0176 -0.0000 0.1604 0.2619 -0.9006 -0.0014 -0.0090 -0.0214 0.3060 0.0011 -0.0089 0.0000
1.4880E-01| -0.0063 -0.0007 -0.0112 -0.0112 0.3120 0.1627 -0.0011 0.0130 0.9358 0.0027 0.0020 -0.0000
1.5878E+00| 0.0010 0.0009 0.0170 0.0273 -0.0433 0.9861 0.0023 -0.0144 -0.1563 -0.0057 0.0002 0.0000
1.2113E+04| -0.0000 -0.0990 0.0004 -0.0001 -0.0001 0.0052 0.0001 -0.0002 -0.0038 0.9951 0.0000 -0.0000
1.1778E+06| -0.0000 0.9951 0.0000 -0.0000 0.0002 -0.0002 0.0000 0.0000 0.0004 0.0990 0.0000 -0.0000
------------------------------------------------------------------------------------------
================================================================================================================================================
Covariance Matrix
1 2 3 4 5 6 7 8 9 10 11 12
1.070e-03 -3.511e+01 -3.269e-04 7.128e-04 -8.914e-03 8.354e-03 -2.826e-04 -3.677e-04 -1.538e-02 -3.671e+00 -8.062e-05 8.598e-08
-3.511e+01 1.166e+06 9.677e+00 -2.050e+01 2.826e+02 -2.691e+02 8.780e+00 1.238e+01 5.013e+02 1.148e+05 2.549e+00 -1.444e-03
-3.269e-04 9.677e+00 2.864e-03 8.809e-05 -1.624e-04 4.897e-02 6.595e-04 -1.443e-03 -1.953e-02 5.771e+00 5.429e-05 -3.427e-07
7.128e-04 -2.050e+01 8.809e-05 2.231e-03 -8.122e-03 3.827e-02 -2.333e-04 -5.539e-04 -1.034e-02 -3.751e+00 -6.187e-05 3.875e-07
-8.914e-03 2.826e+02 -1.624e-04 -8.122e-03 8.918e-02 -1.296e-01 1.874e-03 4.799e-03 1.776e-01 2.701e+01 7.246e-04 -2.540e-06
8.354e-03 -2.691e+02 4.897e-02 3.827e-02 -1.296e-01 1.937e+00 8.070e-03 -3.778e-02 -5.753e-01 3.677e+01 1.924e-04 5.993e-06
-2.826e-04 8.780e+00 6.595e-04 -2.333e-04 1.874e-03 8.070e-03 2.083e-04 -2.296e-04 -1.689e-03 2.123e+00 2.955e-05 -1.830e-08
-3.677e-04 1.238e+01 -1.443e-03 -5.539e-04 4.799e-03 -3.778e-02 -2.296e-04 1.074e-03 1.990e-02 -1.233e+00 8.351e-06 -1.664e-07
-1.538e-02 5.013e+02 -1.953e-02 -1.034e-02 1.776e-01 -5.753e-01 -1.689e-03 1.990e-02 5.575e-01 3.406e+00 9.711e-04 -5.823e-06
-3.671e+00 1.148e+05 5.771e+00 -3.751e+00 2.701e+01 3.677e+01 2.123e+00 -1.233e+00 3.406e+00 2.353e+04 3.444e-01 -4.222e-04
-8.062e-05 2.549e+00 5.429e-05 -6.187e-05 7.246e-04 1.924e-04 2.955e-05 8.351e-06 9.711e-04 3.444e-01 7.259e-06 -2.059e-08
8.598e-08 -1.444e-03 -3.427e-07 3.875e-07 -2.540e-06 5.993e-06 -1.830e-08 -1.664e-07 -5.823e-06 -4.222e-04 -2.059e-08 3.880e-07
------------------------------------------------------------------------------------------------------------------------------------------------
========================================================================
Model constant<1>*TBabs<2>(diskbb<3> + cutoffpl<4> + relxill<5>) Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
Data group: 1
1 1 constant factor 1.00000 frozen
2 2 TBabs nH 10^22 0.600000 frozen
3 3 diskbb Tin keV 0.549610 +/- 3.27102E-02
4 3 diskbb norm 3374.61 +/- 1080.00
5 4 cutoffpl PhoIndex 1.87094 = p16
6 4 cutoffpl HighECut keV 511.648 = p19
7 4 cutoffpl norm 2.35787 +/- 5.35186E-02
8 5 relxill Index1 2.71495 +/- 4.72351E-02
9 5 relxill Index2 2.71495 = p8
10 5 relxill Rbr 15.0000 frozen
11 5 relxill a 0.924735 +/- 0.298624
12 5 relxill Incl deg 30.3613 +/- 1.39189
13 5 relxill Rin -1.00000 frozen
14 5 relxill Rout 400.000 frozen
15 5 relxill z 0.0 frozen
16 5 relxill gamma 1.87094 +/- 1.44321E-02
17 5 relxill logxi 3.30452 +/- 3.27661E-02
18 5 relxill Afe 5.00414 +/- 0.746629
19 5 relxill Ecut keV 511.648 +/- 153.393
20 5 relxill refl_frac -1.00000 frozen
21 5 relxill norm 2.09880E-02 +/- 2.69431E-03
Data group: 2
22 1 constant factor 0.972148 +/- 6.22876E-04
23 2 TBabs nH 10^22 0.600000 = p2
24 3 diskbb Tin keV 0.549610 = p3
25 3 diskbb norm 3374.61 = p4
26 4 cutoffpl PhoIndex 1.87094 = p5
27 4 cutoffpl HighECut keV 511.648 = p6
28 4 cutoffpl norm 2.35787 = p7
29 5 relxill Index1 2.71495 = p8
30 5 relxill Index2 2.71495 = p9
31 5 relxill Rbr 15.0000 = p10
32 5 relxill a 0.924735 = p11
33 5 relxill Incl deg 30.3613 = p12
34 5 relxill Rin -1.00000 = p13
35 5 relxill Rout 400.000 = p14
36 5 relxill z 0.0 = p15
37 5 relxill gamma 1.87094 = p16
38 5 relxill logxi 3.30452 = p17
39 5 relxill Afe 5.00414 = p18
40 5 relxill Ecut keV 511.648 = p19
41 5 relxill refl_frac -1.00000 = p20
42 5 relxill norm 2.09880E-02 = p21
________________________________________________________________________
Fit statistic : Chi-Squared 375.85 using 271 bins, spectrum 1, group 1.
Chi-Squared 379.79 using 273 bins, spectrum 2, group 2.
Total fit statistic 755.64 with 532 d.o.f.
Test statistic : Chi-Squared 755.64 using 544 bins.
Null hypothesis probability of 5.36e-10 with 532 degrees of freedom
***Warning: New best fit found, fit parameters will be set to new values.
Parameters
Chi-Squared |beta|/N Lvl 3:Tin 4:norm 7:norm 8:Index1 11:a 12:Incl 16:gamma 17:logxi 18:Afe 19:Ecut 21:norm 22:factor
691.377 1.98791 -2 0.552462 3557.46 2.16752 5.15418 0.459109 38.5488 1.83855 3.37303 6.75622 359.988 0.0219034 0.972155
689.866 900.488 -3 0.553800 3454.78 2.16609 5.33984 0.494991 39.4342 1.83928 3.37233 6.77174 363.690 0.0224482 0.972147
689.378 392.882 -4 0.554862 3334.95 2.19671 5.42159 0.537457 40.3219 1.84565 3.35432 6.41563 382.773 0.0232685 0.972143
688.238 2220.47 -4 0.554116 3324.64 2.22538 5.45981 0.552069 40.6846 1.85051 3.33972 6.15944 397.753 0.0236907 0.972141
687.9 800.413 -4 0.552778 3349.26 2.24769 5.47245 0.559948 40.8873 1.85407 3.32900 5.98547 409.003 0.0239460 0.972140
687.78 367.081 -4 0.551508 3377.05 2.26608 5.48296 0.565836 41.0435 1.85701 3.32039 5.85453 418.595 0.0241624 0.972140
687.706 52.938 -3 0.551469 3375.92 2.26845 5.48904 0.567489 41.0798 1.85729 3.31917 5.83526 419.465 0.0241838 0.972140
687.688 94.7232 -3 0.551418 3375.61 2.27068 5.48915 0.568141 41.0963 1.85761 3.31804 5.81850 420.383 0.0242034 0.972140
687.671 96.5557 -3 0.551345 3376.10 2.27288 5.48970 0.568838 41.1136 1.85794 3.31695 5.80228 421.345 0.0242243 0.972140
687.656 92.3894 -3 0.551270 3376.50 2.27506 5.49102 0.569708 41.1343 1.85826 3.31587 5.78638 422.337 0.0242477 0.972140
687.647 111.946 -3 0.551190 3377.12 2.27724 5.49240 0.570547 41.1547 1.85859 3.31480 5.77054 423.338 0.0242709 0.972140
==========================================================================================
Variances and Principal Axes
3 4 7 8 11 12 16 17 18 19 21 22
1.1564E-09| 0.0407 0.0000 0.0342 -0.0004 -0.0045 -0.0001 -0.1655 0.0358 -0.0004 -0.0000 0.9827 0.0521
1.3400E-07| -0.4183 -0.0000 0.0745 -0.0001 0.0024 0.0004 -0.8879 -0.0246 -0.0029 0.0000 -0.1394 0.1048
3.6732E-07| -0.4332 -0.0000 -0.0329 -0.0010 -0.0096 0.0003 0.0848 -0.0594 0.0018 -0.0000 0.0828 -0.8908
4.7202E-07| 0.7888 0.0000 0.0699 0.0023 0.0223 -0.0009 -0.4068 0.0865 -0.0044 0.0001 -0.0834 -0.4387
1.3059E-05| -0.1109 -0.0000 0.4552 -0.0140 -0.0860 0.0091 0.0689 0.8757 -0.0033 -0.0002 -0.0311 -0.0166
5.9886E-05| -0.0306 -0.0000 0.0378 0.0772 0.9904 -0.0640 0.0175 0.0744 -0.0287 0.0000 0.0047 -0.0002
4.9059E-04| 0.0136 0.0000 0.8796 0.0521 -0.0022 -0.0013 0.0799 -0.4609 0.0669 -0.0013 -0.0008 -0.0007
3.8312E-02| -0.0023 -0.0002 -0.0055 0.8102 -0.0947 -0.2483 0.0017 0.0067 -0.5224 -0.0018 -0.0000 -0.0000
1.6051E-01| 0.0002 0.0005 0.0672 -0.4639 0.0286 0.2507 0.0097 -0.0461 -0.8451 -0.0128 0.0005 0.0000
9.7140E-01| -0.0006 0.0014 -0.0201 0.3457 0.0358 0.9332 -0.0016 0.0102 0.0864 -0.0186 0.0005 0.0000
4.3251E+03| -0.0001 -0.0598 0.0017 0.0020 0.0008 0.0202 0.0003 -0.0008 -0.0101 0.9980 0.0000 -0.0000
3.8363E+05| -0.0000 0.9982 0.0001 -0.0000 -0.0000 -0.0003 0.0000 -0.0000 -0.0004 0.0598 0.0000 0.0000
------------------------------------------------------------------------------------------
================================================================================================================================================
Covariance Matrix
1 2 3 4 5 6 7 8 9 10 11 12
2.883e-04 -1.015e+01 -1.187e-03 -6.873e-04 2.383e-05 -2.879e-03 -2.165e-04 4.135e-04 6.895e-03 -8.766e-01 -1.516e-05 -2.023e-07
-1.015e+01 3.823e+05 2.804e+01 -5.964e+00 -9.711e+00 -1.195e+02 5.251e+00 -8.030e+00 -1.563e+02 2.263e+04 2.799e-01 8.138e-03
-1.187e-03 2.804e+01 1.572e-02 2.523e-03 4.752e-03 1.222e-01 2.623e-03 -7.085e-03 -9.537e-02 8.925e+00 2.112e-04 -2.880e-07
-6.873e-04 -5.964e+00 2.523e-03 1.940e-01 1.425e-02 4.676e-01 1.243e-03 3.416e-04 -1.124e-02 8.511e+00 3.761e-04 -1.610e-06
2.383e-05 -9.711e+00 4.752e-03 1.425e-02 4.826e-03 1.078e-01 8.424e-04 -2.360e-03 -3.013e-02 2.908e+00 1.085e-04 -3.774e-07
-2.879e-03 -1.195e+02 1.222e-01 4.676e-01 1.078e-01 2.658e+00 2.218e-02 -5.759e-02 -7.834e-01 8.034e+01 2.783e-03 -8.338e-06
-2.165e-04 5.251e+00 2.623e-03 1.243e-03 8.424e-04 2.218e-02 4.447e-04 -1.167e-03 -1.608e-02 1.546e+00 3.714e-05 3.034e-08
4.135e-04 -8.030e+00 -7.085e-03 3.416e-04 -2.360e-03 -5.759e-02 -1.167e-03 3.309e-03 4.398e-02 -3.822e+00 -9.494e-05 -2.751e-09
6.895e-03 -1.563e+02 -9.537e-02 -1.124e-02 -3.013e-02 -7.834e-01 -1.608e-02 4.398e-02 6.372e-01 -5.299e+01 -1.322e-03 -1.067e-06
-8.766e-01 2.263e+04 8.925e+00 8.511e+00 2.908e+00 8.034e+01 1.546e+00 -3.822e+00 -5.299e+01 5.678e+03 1.336e-01 1.506e-04
-1.516e-05 2.799e-01 2.112e-04 3.761e-04 1.085e-04 2.783e-03 3.714e-05 -9.494e-05 -1.322e-03 1.336e-01 3.717e-06 -6.641e-09
-2.023e-07 8.138e-03 -2.880e-07 -1.610e-06 -3.774e-07 -8.338e-06 3.034e-08 -2.751e-09 -1.067e-06 1.506e-04 -6.641e-09 3.879e-07
------------------------------------------------------------------------------------------------------------------------------------------------
========================================================================
Model constant<1>*TBabs<2>(diskbb<3> + cutoffpl<4> + relxill<5>) Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
Data group: 1
1 1 constant factor 1.00000 frozen
2 2 TBabs nH 10^22 0.600000 frozen
3 3 diskbb Tin keV 0.551190 +/- 1.69780E-02
4 3 diskbb norm 3377.12 +/- 618.281
5 4 cutoffpl PhoIndex 1.85859 = p16
6 4 cutoffpl HighECut keV 423.338 = p19
7 4 cutoffpl norm 2.27724 +/- 0.125388
8 5 relxill Index1 5.49240 +/- 0.440463
9 5 relxill Index2 5.49240 = p8
10 5 relxill Rbr 15.0000 frozen
11 5 relxill a 0.570547 +/- 6.94700E-02
12 5 relxill Incl deg 41.1547 +/- 1.63033
13 5 relxill Rin -1.00000 frozen
14 5 relxill Rout 400.000 frozen
15 5 relxill z 0.0 frozen
16 5 relxill gamma 1.85859 +/- 2.10881E-02
17 5 relxill logxi 3.31480 +/- 5.75237E-02
18 5 relxill Afe 5.77054 +/- 0.798241
19 5 relxill Ecut keV 423.338 +/- 75.3535
20 5 relxill refl_frac -1.00000 frozen
21 5 relxill norm 2.42709E-02 +/- 1.92796E-03
Data group: 2
22 1 constant factor 0.972140 +/- 6.22814E-04
23 2 TBabs nH 10^22 0.600000 = p2
24 3 diskbb Tin keV 0.551190 = p3
25 3 diskbb norm 3377.12 = p4
26 4 cutoffpl PhoIndex 1.85859 = p5
27 4 cutoffpl HighECut keV 423.338 = p6
28 4 cutoffpl norm 2.27724 = p7
29 5 relxill Index1 5.49240 = p8
30 5 relxill Index2 5.49240 = p9
31 5 relxill Rbr 15.0000 = p10
32 5 relxill a 0.570547 = p11
33 5 relxill Incl deg 41.1547 = p12
34 5 relxill Rin -1.00000 = p13
35 5 relxill Rout 400.000 = p14
36 5 relxill z 0.0 = p15
37 5 relxill gamma 1.85859 = p16
38 5 relxill logxi 3.31480 = p17
39 5 relxill Afe 5.77054 = p18
40 5 relxill Ecut keV 423.338 = p19
41 5 relxill refl_frac -1.00000 = p20
42 5 relxill norm 2.42709E-02 = p21
________________________________________________________________________
Fit statistic : Chi-Squared 342.11 using 271 bins, spectrum 1, group 1.
Chi-Squared 345.54 using 273 bins, spectrum 2, group 2.
Total fit statistic 687.65 with 532 d.o.f.
Test statistic : Chi-Squared 687.65 using 544 bins.
Null hypothesis probability of 5.60e-06 with 532 degrees of freedom
***Warning: New best fit found, fit parameters will be set to new values.
Parameters
Chi-Squared |beta|/N Lvl 3:Tin 4:norm 7:norm 8:Index1 11:a 12:Incl 16:gamma 17:logxi 18:Afe 19:Ecut 21:norm 22:factor
682.448 6.02865 -2 0.520665 4281.56 2.51517 6.19945 0.689467 45.2312 1.91398 3.22292 4.67778 1000.00 0.0330553 0.972145
682.43 25.2124 -2 0.520711 4278.19 2.51549 6.22002 0.691278 45.2954 1.91406 3.22279 4.67836 1000.00 0.0331134 0.972145
682.418 301.16 -1 0.520722 4278.14 2.51551 6.22148 0.691521 45.3020 1.91406 3.22275 4.67819 1000.00 0.0331179 0.972144
682.41 264.556 -1 0.520733 4277.88 2.51552 6.22257 0.691731 45.3082 1.91407 3.22272 4.67813 1000.00 0.0331226 0.972144
==========================================================================================
Variances and Principal Axes
3 4 7 8 11 12 16 17 18 19 21 22
2.0304E-09| 0.0446 0.0000 0.0414 -0.0004 -0.0084 -0.0002 -0.2134 0.0612 -0.0024 -0.0000 0.9706 0.0695
1.6914E-07| -0.4597 -0.0000 0.0468 0.0001 0.0062 0.0002 -0.8531 -0.1619 -0.0120 0.0000 -0.1636 0.0752
3.7604E-07| 0.2223 0.0000 0.0207 0.0001 0.0045 -0.0001 -0.0255 0.0500 -0.0034 0.0000 -0.0892 0.9690
5.6690E-07| 0.8420 0.0000 0.0735 0.0010 0.0301 -0.0013 -0.4591 0.0813 -0.0142 0.0000 -0.1316 -0.2233
7.3116E-06| 0.1583 0.0000 -0.3363 0.0148 0.2047 -0.0120 0.0555 -0.8993 0.0367 0.0000 0.0760 0.0249
2.0561E-05| -0.0563 -0.0000 0.1151 0.0354 0.9751 -0.0399 0.0122 0.1692 -0.0353 0.0000 -0.0018 -0.0028
2.9576E-04| 0.0138 0.0000 0.9185 0.0476 -0.0546 -0.0038 0.1096 -0.3506 -0.1263 -0.0002 0.0058 -0.0014
9.4183E-03| 0.0014 -0.0000 0.1380 -0.0559 0.0242 0.0312 -0.0047 -0.0075 0.9880 0.0003 -0.0038 -0.0000
8.0871E-02| -0.0019 -0.0002 -0.0325 0.9173 -0.0469 -0.3871 -0.0063 0.0237 0.0700 0.0006 -0.0013 -0.0000
1.0752E+00| -0.0001 -0.0013 0.0138 -0.3896 -0.0241 -0.9205 0.0013 -0.0045 0.0056 0.0036 -0.0006 -0.0000
4.3916E+05| 0.0000 0.3793 -0.0001 -0.0009 -0.0001 -0.0037 -0.0000 0.0000 0.0004 -0.9253 -0.0000 0.0000
1.2862E+06| -0.0000 0.9253 0.0000 0.0000 0.0000 0.0002 0.0000 -0.0000 -0.0001 0.3793 0.0000 0.0000
------------------------------------------------------------------------------------------
================================================================================================================================================
Covariance Matrix
1 2 3 4 5 6 7 8 9 10 11 12
4.334e-04 -2.181e+01 -9.370e-04 -1.024e-03 4.717e-06 -3.922e-03 -2.591e-04 3.061e-04 2.927e-03 -8.971e+00 -5.008e-05 -2.801e-07
-2.181e+01 1.164e+06 3.227e+01 -1.025e+02 -1.601e+01 -4.179e+02 8.364e+00 -7.892e+00 -8.378e+01 2.973e+05 1.197e+00 1.548e-02
-9.370e-04 3.227e+01 6.482e-03 2.961e-02 3.688e-03 1.461e-01 1.769e-03 -2.792e-03 -2.055e-02 5.683e+01 4.235e-04 -2.794e-07
-1.024e-03 -1.025e+02 2.961e-02 5.807e-01 4.439e-02 1.817e+00 1.062e-02 -1.524e-02 -1.524e-01 3.802e+02 3.376e-03 -2.270e-06
4.717e-06 -1.601e+01 3.688e-03 4.439e-02 4.936e-03 1.832e-01 1.194e-03 -1.950e-03 -1.635e-02 3.932e+01 3.590e-04 -2.523e-07
-3.922e-03 -4.179e+02 1.461e-01 1.817e+00 1.832e-01 7.029e+00 4.762e-02 -7.536e-02 -6.553e-01 1.593e+03 1.413e-02 -8.307e-06
-2.591e-04 8.364e+00 1.769e-03 1.062e-02 1.194e-03 4.762e-02 5.181e-04 -7.977e-04 -6.564e-03 1.689e+01 1.301e-04 5.673e-08
3.061e-04 -7.892e+00 -2.792e-03 -1.524e-02 -1.950e-03 -7.536e-02 -7.977e-04 1.291e-03 9.943e-03 -2.539e+01 -2.006e-04 -3.512e-08
2.927e-03 -8.378e+01 -2.055e-02 -1.524e-01 -1.635e-02 -6.553e-01 -6.564e-03 9.943e-03 9.283e-02 -2.149e+02 -1.721e-03 -6.493e-07
-8.971e+00 2.973e+05 5.683e+01 3.802e+02 3.932e+01 1.593e+03 1.689e+01 -2.539e+01 -2.149e+02 5.610e+05 4.289e+00 2.067e-03
-5.008e-05 1.197e+00 4.235e-04 3.376e-03 3.590e-04 1.413e-02 1.301e-04 -2.006e-04 -1.721e-03 4.289e+00 3.464e-05 -1.450e-09
-2.801e-07 1.548e-02 -2.794e-07 -2.270e-06 -2.523e-07 -8.307e-06 5.673e-08 -3.512e-08 -6.493e-07 2.067e-03 -1.450e-09 3.879e-07
------------------------------------------------------------------------------------------------------------------------------------------------
========================================================================
Model constant<1>*TBabs<2>(diskbb<3> + cutoffpl<4> + relxill<5>) Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
Data group: 1
1 1 constant factor 1.00000 frozen
2 2 TBabs nH 10^22 0.600000 frozen
3 3 diskbb Tin keV 0.520733 +/- 2.08171E-02
4 3 diskbb norm 4277.88 +/- 1079.06
5 4 cutoffpl PhoIndex 1.91407 = p16
6 4 cutoffpl HighECut keV 1000.000 = p19
7 4 cutoffpl norm 2.51552 +/- 8.05132E-02
8 5 relxill Index1 6.22257 +/- 0.762013
9 5 relxill Index2 6.22257 = p8
10 5 relxill Rbr 15.0000 frozen
11 5 relxill a 0.691731 +/- 7.02536E-02
12 5 relxill Incl deg 45.3082 +/- 2.65113
13 5 relxill Rin -1.00000 frozen
14 5 relxill Rout 400.000 frozen
15 5 relxill z 0.0 frozen
16 5 relxill gamma 1.91407 +/- 2.27609E-02
17 5 relxill logxi 3.22272 +/- 3.59317E-02
18 5 relxill Afe 4.67813 +/- 0.304676
19 5 relxill Ecut keV 1000.000 +/- 748.996
20 5 relxill refl_frac -1.00000 frozen
21 5 relxill norm 3.31226E-02 +/- 5.88529E-03
Data group: 2
22 1 constant factor 0.972144 +/- 6.22820E-04
23 2 TBabs nH 10^22 0.600000 = p2
24 3 diskbb Tin keV 0.520733 = p3
25 3 diskbb norm 4277.88 = p4
26 4 cutoffpl PhoIndex 1.91407 = p5
27 4 cutoffpl HighECut keV 1000.000 = p6
28 4 cutoffpl norm 2.51552 = p7
29 5 relxill Index1 6.22257 = p8
30 5 relxill Index2 6.22257 = p9
31 5 relxill Rbr 15.0000 = p10
32 5 relxill a 0.691731 = p11
33 5 relxill Incl deg 45.3082 = p12
34 5 relxill Rin -1.00000 = p13
35 5 relxill Rout 400.000 = p14
36 5 relxill z 0.0 = p15
37 5 relxill gamma 1.91407 = p16
38 5 relxill logxi 3.22272 = p17
39 5 relxill Afe 4.67813 = p18
40 5 relxill Ecut keV 1000.000 = p19
41 5 relxill refl_frac -1.00000 = p20
42 5 relxill norm 3.31226E-02 = p21
________________________________________________________________________
Fit statistic : Chi-Squared 337.56 using 271 bins, spectrum 1, group 1.
Chi-Squared 344.85 using 273 bins, spectrum 2, group 2.
Total fit statistic 682.41 with 532 d.o.f.
Test statistic : Chi-Squared 682.41 using 544 bins.
Null hypothesis probability of 1.04e-05 with 532 degrees of freedom
***Warning: New best fit found, fit parameters will be set to new values.
Apparent non-monotonicity in statistic space detected.
Current bracket values 0.691731, 0.596995
and delta stat 0, 3.45192
but latest trial 0.621033 gives 3.49642
Suggest that you check this result using the steppar command.
Parameters
Chi-Squared |beta|/N Lvl 3:Tin 4:norm 7:norm 8:Index1 11:a 12:Incl 16:gamma 17:logxi 18:Afe 19:Ecut 21:norm 22:factor
681.964 0.981312 4 0.564018 2525.45 2.50233 7.61735 0.807340 50.3371 1.91563 3.20462 4.52607 1000.00 0.0375577 0.972119
681.964 0.981311 9 0.564018 2525.45 2.50233 7.61735 0.807340 50.3371 1.91563 3.20462 4.52607 1000.00 0.0375577 0.972119
==========================================================================================
Variances and Principal Axes
3 4 7 8 11 12 16 17 18 19 21 22
2.5073E-09| 0.0494 0.0000 0.0459 -0.0003 -0.0168 -0.0002 -0.2360 0.0690 -0.0029 -0.0000 0.9637 0.0775
1.8183E-07| -0.4678 -0.0000 0.0398 -0.0002 0.0033 0.0001 -0.8439 -0.1818 -0.0116 0.0000 -0.1761 0.0568
3.7723E-07| -0.1739 -0.0000 -0.0188 0.0001 -0.0088 0.0001 0.0196 -0.0473 0.0032 -0.0000 0.0965 -0.9785
6.3627E-07| 0.8393 0.0000 0.0807 0.0003 0.0764 -0.0020 -0.4643 0.1067 -0.0158 0.0000 -0.1524 -0.1809
4.1949E-06| -0.0129 0.0000 0.1094 -0.0169 -0.9292 0.0237 -0.0540 0.3422 -0.0043 -0.0000 -0.0573 -0.0147
8.4314E-06| -0.2087 -0.0000 0.3288 -0.0002 0.3589 -0.0050 -0.0360 0.8435 -0.0443 -0.0000 -0.0664 -0.0207
3.1207E-04| 0.0164 0.0000 0.9207 0.0439 -0.0263 -0.0132 0.1091 -0.3456 -0.1337 -0.0002 0.0060 -0.0013
8.8025E-03| -0.0015 0.0000 -0.1431 0.0279 -0.0111 -0.0358 0.0049 0.0086 -0.9885 -0.0003 0.0043 0.0000
1.0541E-01| 0.0016 0.0005 0.0343 -0.9202 0.0221 0.3859 0.0062 -0.0224 -0.0453 -0.0006 0.0014 0.0000
3.3941E+00| 0.0006 0.0043 -0.0076 0.3875 0.0159 0.9214 -0.0006 0.0008 -0.0215 -0.0049 0.0007 0.0000
2.7379E+05| 0.0000 -0.9028 0.0000 0.0017 0.0001 0.0057 0.0000 -0.0000 -0.0003 0.4301 0.0000 -0.0000
6.4048E+05| -0.0000 0.4301 0.0001 0.0007 0.0000 0.0025 0.0000 -0.0000 -0.0003 0.9028 0.0000 0.0000
------------------------------------------------------------------------------------------
================================================================================================================================================
Covariance Matrix
1 2 3 4 5 6 7 8 9 10 11 12
5.068e-04 -1.291e+01 -9.196e-04 5.522e-03 3.271e-04 1.618e-02 -2.368e-04 1.943e-04 1.917e-03 -8.268e+00 -3.390e-05 -2.896e-07
-1.291e+01 3.416e+05 1.587e+01 -2.317e+02 -1.376e+01 -7.352e+02 3.843e+00 -1.117e+00 -1.709e+01 1.424e+05 1.562e-01 7.546e-03
-9.196e-04 1.587e+01 7.659e-03 5.118e-02 3.485e-03 2.073e-01 2.030e-03 -3.329e-03 -2.470e-02 6.283e+01 5.686e-04 -1.154e-07
5.522e-03 -2.317e+02 5.118e-02 1.676e+00 8.245e-02 4.897e+00 1.784e-02 -3.360e-02 -3.243e-01 5.907e+02 7.762e-03 -1.218e-06
3.271e-04 -1.376e+01 3.485e-03 8.245e-02 4.680e-03 2.707e-01 1.116e-03 -2.179e-03 -1.890e-02 3.486e+01 4.526e-04 -8.734e-08
1.618e-02 -7.352e+02 2.073e-01 4.897e+00 2.707e-01 1.577e+01 6.664e-02 -1.276e-01 -1.124e+00 2.101e+03 2.671e-02 -4.084e-06
-2.368e-04 3.843e+00 2.030e-03 1.784e-02 1.116e-03 6.664e-02 5.720e-04 -9.318e-04 -7.631e-03 1.793e+01 1.697e-04 1.079e-07
1.943e-04 -1.117e+00 -3.329e-03 -3.360e-02 -2.179e-03 -1.276e-01 -9.318e-04 1.633e-03 1.276e-02 -2.849e+01 -2.912e-04 -8.825e-08
1.917e-03 -1.709e+01 -2.470e-02 -3.243e-01 -1.890e-02 -1.124e+00 -7.631e-03 1.276e-02 1.169e-01 -2.400e+02 -2.501e-03 -1.143e-06
-8.268e+00 1.424e+05 6.283e+01 5.907e+02 3.486e+01 2.101e+03 1.793e+01 -2.849e+01 -2.400e+02 5.726e+05 5.346e+00 3.826e-03
-3.390e-05 1.562e-01 5.686e-04 7.762e-03 4.526e-04 2.671e-02 1.697e-04 -2.912e-04 -2.501e-03 5.346e+00 5.659e-05 1.233e-08
-2.896e-07 7.546e-03 -1.154e-07 -1.218e-06 -8.734e-08 -4.084e-06 1.079e-07 -8.825e-08 -1.143e-06 3.826e-03 1.233e-08 3.879e-07
------------------------------------------------------------------------------------------------------------------------------------------------
========================================================================
Model constant<1>*TBabs<2>(diskbb<3> + cutoffpl<4> + relxill<5>) Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
Data group: 1
1 1 constant factor 1.00000 frozen
2 2 TBabs nH 10^22 0.600000 frozen
3 3 diskbb Tin keV 0.564018 +/- 2.25113E-02
4 3 diskbb norm 2525.45 +/- 584.491
5 4 cutoffpl PhoIndex 1.91563 = p16
6 4 cutoffpl HighECut keV 1000.00 = p19
7 4 cutoffpl norm 2.50233 +/- 8.75159E-02
8 5 relxill Index1 7.61735 +/- 1.29471
9 5 relxill Index2 7.61735 = p8
10 5 relxill Rbr 15.0000 frozen
11 5 relxill a 0.807340 +/- 6.84121E-02
12 5 relxill Incl deg 50.3371 +/- 3.97152
13 5 relxill Rin -1.00000 frozen
14 5 relxill Rout 400.000 frozen
15 5 relxill z 0.0 frozen
16 5 relxill gamma 1.91563 +/- 2.39161E-02
17 5 relxill logxi 3.20462 +/- 4.04046E-02
18 5 relxill Afe 4.52607 +/- 0.341924
19 5 relxill Ecut keV 1000.00 +/- 756.720
20 5 relxill refl_frac -1.00000 frozen
21 5 relxill norm 3.75577E-02 +/- 7.52257E-03
Data group: 2
22 1 constant factor 0.972119 +/- 6.22793E-04
23 2 TBabs nH 10^22 0.600000 = p2
24 3 diskbb Tin keV 0.564018 = p3
25 3 diskbb norm 2525.45 = p4
26 4 cutoffpl PhoIndex 1.91563 = p5
27 4 cutoffpl HighECut keV 1000.00 = p6
28 4 cutoffpl norm 2.50233 = p7
29 5 relxill Index1 7.61735 = p8
30 5 relxill Index2 7.61735 = p9
31 5 relxill Rbr 15.0000 = p10
32 5 relxill a 0.807340 = p11
33 5 relxill Incl deg 50.3371 = p12
34 5 relxill Rin -1.00000 = p13
35 5 relxill Rout 400.000 = p14
36 5 relxill z 0.0 = p15
37 5 relxill gamma 1.91563 = p16
38 5 relxill logxi 3.20462 = p17
39 5 relxill Afe 4.52607 = p18
40 5 relxill Ecut keV 1000.00 = p19
41 5 relxill refl_frac -1.00000 = p20
42 5 relxill norm 3.75577E-02 = p21
________________________________________________________________________
Fit statistic : Chi-Squared 338.28 using 271 bins, spectrum 1, group 1.
Chi-Squared 343.68 using 273 bins, spectrum 2, group 2.
Total fit statistic 681.96 with 532 d.o.f.
Test statistic : Chi-Squared 681.96 using 544 bins.
Null hypothesis probability of 1.09e-05 with 532 degrees of freedom
***Warning: New best fit found, fit parameters will be set to new values.
11 0.638479 0.887005 (-0.168861,0.0796647)
ax = plot_eeuf_ra(nspec=2, naddmodel=3, delchi=False)
ax[0].set_ylim(1e-1, 30.0)
Plot command list is now empty
(0.1, 30.0)
ax = plot_eeuf_ra(nspec=2, naddmodel=3, delchi=True)
ax[0].set_ylim(1e-1, 30.0)
Plot command list is now empty
(0.1, 30.0)
# An easier way to make plot
xspec.Plot.commands = ()
xspec.Plot.device="/svg"
#xspec.Plot.addCommand("log y")
xspec.Plot("eeuf ra")
Plot command list is now empty
Some advice#
Note that this tutorial only gives an minimal example of reflection fitting. It has no guarantee that the results shown here is the final best-fit. Due to the complexity of reflection models, the fitting can easily be traped in local minimums.
It is recommended to run steppar on each parameter or even pairs of parameters to check for lower minimums. Running error command sometimes also helps to find better fit.
Some guidelines to assess a fitting:
Are there any residuals in the ratio plot?
Are all parameters constrained? How are the parameters degenerated, e.g., MCMC simulation?
Are the parameters physically acceptable?
Are the parameters consistent with literature? If not, why?
Are the results model-dependent? For example, if changing flavors of the reflection model, how does the parameters change?