11institutetext:  SRON Netherlands Institute for Space Research, Niels Bohrweg
4, 2333 CA Leiden, the Netherlands 22institutetext: RIKEN High Energy
Astrophysics Laboratory, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
33institutetext: NASA/Goddard Space Flight Center, 8800 Greenbelt Rd,
Greenbelt, MD 20771, USA 44institutetext: Max-Planck-Institut f$\rm\ddot{u}$r
Kernphysik, Heidelberg, D-69117 Heidelberg, Germany 55institutetext:
Department of Physics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-
Hiroshima, Hiroshima 739-8526, Japan 66institutetext: Department of Physics,
University of Strathclyde, Glasgow, G4 0NG, UK 77institutetext: Astronomical
Institute “Anton Pannekoek”, Science Park 904, 1098 XH Amsterdam, University
of Amsterdam, The Netherlands 88institutetext: INAF - IASF Palermo, Via U. La
Malfa 153, 90146 Palermo PA, Italy 99institutetext: Department of Theoretical
Physics and Astrophysics, Faculty of Science, Masaryk University,
Kotl$\rm\acute{a}$$\rm\breve{r}$sk$\rm\acute{a}$ 2, Brno, 611 37, Czech
Republic 1010institutetext: Leiden Observatory, Leiden University, PO Box
9513, 2300 RA Leiden, the Netherlands 1111institutetext: Kavli Institute for
the Physics and Mathematics of the Universe (WPI), University of Tokyo,
Kashiwa 277-8583, Japan 1212institutetext: ESTEC/ESA, Keplerlaan 1, 2201AZ
Noordwijk, The Netherlands 1313institutetext: Laboratory of Instrumentation,
Biomedical Engineering and Radiation Physics (LIBPhys-UNL), Department of
Physics, NOVA School of Science and Technology, NOVA University Lisbon,
2829-516 Caparica, Portugal 1414institutetext: Space Science Laboratory,
University of California, Berkeley, CA 94720, USA

# X-ray spectra of the Fe-L complex III: systematic uncertainties in the
atomic data

Liyi Gu 11221010 Chintan Shah 3344 Junjie Mao 556611 A.J.J. Raassen 1177 Jelle
de Plaa 11 Ciro Pinto 88 Hiroki Akamatsu 11 Norbert Werner 9955 Aurora
Simionescu 1110101111 François Mernier 121211 Makoto Sawada 22 Pranav Mohanty
1010 Pedro Amaro 1313 Ming Feng Gu 1414 F. Scott Porter 33 José R. Crespo
López-Urrutia 44 Jelle S. Kaastra 111010

There has been a growing request from the X-ray astronomy community for a
quantitative estimate of systematic uncertainties originating from the atomic
data used in plasma codes. Though there have been several studies looking into
atomic data uncertainties using theoretical calculations, in general, there is
no commonly accepted solution for this task. We present a new approach for
estimating uncertainties in the line emissivities for the current models of
collisional plasma, mainly based upon dedicated analysis of observed high
resolution spectra of stellar coronae and galaxy clusters. We find that the
systematic uncertainties of the observed lines consistently show anti-
correlation with the model line fluxes, after properly accounting for the
additional uncertainties from the ion concentration calculation. The strong
lines in the spectra are in general better reproduced, indicating that the
atomic data and modeling of the main transitions are more accurate than those
for the minor ones. This underlying anti-correlation is found to be roughly
independent on source properties, line positions, ion species, and the line
formation processes. We further apply our method to the simulated XRISM and
Athena observations of collisional plasma sources and discuss the impact of
uncertainties on the interpretation of these spectra. The typical
uncertainties are $1-2$% on temperature and $3-20$% on abundances of O, Ne,
Fe, Mg, and Ni.

###### Key Words.:

Atomic data – Techniques: spectroscopic – Stars: coronae – Galaxies: clusters:
intracluster medium

##  1 Introduction

Studies of astrophysical sources involve analyses of spectra for diagnostics
of plasma parameters: density, temperature, ionization states, chemical
composition, dynamics, and the underlying energy source. Interpretation of
astrophysical spectra requires a huge atomic database including data such as
energy levels, transition probabilities, excitation rates, and the ionization
balance of plasma. Most of these data are obtained in theoretical calculations
with only a few benchmarks performed with laboratory measurements (Kaastra et
al. 1996; Dere et al. 1997; Smith et al. 2001; Foster et al. 2012; de Plaa et
al. 2019; Del Zanna & Young 2020). There is an increasing demand by the
astronomical community that the plasma modeling should include an uncertainty
estimate alongside the numerical values provided, which are essential for
astronomers to know how the atomic data could affect the accuracy of their
final results obtained from the spectra.

Despite the obvious need, so far there is no straightforward way to assess the
uncertainties of atomic data and how these propagate into the spectral
parameters. Recently, a growing effort has been made: there are several
studies attempting to define uncertainties on existing atomic data based on
the spread of different fundamental theoretical calculations (Bautista et al.
2013; Loch et al. 2013; Mehdipour et al. 2016; Yu et al. 2018; Hitomi
Collaboration et al. 2018), or from perturbation terms in solving the many-
body Schrödinger equation (Chung et al. 2016). These uncertainties can be
propagated through plasma modeling using a quasi-analytic algorithm (Bautista
et al. 2013), a Monte-Carlo method (Loch et al. 2013; Hitomi Collaboration et
al. 2018; Del Zanna et al. 2019), or affect the interpretation of
observational data with a Bayesian approach (Yu et al. 2018).

These theoretical approaches often require sampling of a large amount of
relevant level and transition calculations, taking into account complexities
such as correlated uncertainties in different transitions. This makes most of
the theoretical approaches too computational demanding for practical analysis
of the observed spectra. Here we propose an alternative solution. The model
uncertainties could in principle be inferred by comparing with the real data,
for instance, through a statistical sampling of discrepancies between
theoretical models and well-calibrated, high-quality spectra taken from
laboratory measurements and/or observations, which can be regarded as the
absolute true values within their quoted uncertainties. The observational
constraint on model uncertainties will be useful if (1) the sampling size of
spectral features (e.g., emission lines) is statistically significant, and (2)
the observed discrepancies are not driven by other types of uncertainties,
e.g., statistical uncertainty and systematic uncertainty from instrumental
calibration. Although this approach may not explore the scope of detailed
physics (e.g., underlying correlations between the line flux uncertainties,
Loch et al. 2013) that are accessible only by the theoretical method, it might
provide a relevant benchmark for the latter. In this paper, we explore the
uncertainty assessment based on observed spectra.

This is the third part of a series of papers centered on the Fe-L shell
modeling for X-ray astrophysics. In the first (Gu et al. 2019, paper I) and
the second (Gu et al. 2020, paper II) parts, we presented respectively a
theoretical model of the Fe-L complex spectrum for collisional plasma and an
experimental benchmark of several key transitions. In paper II, we further
carried out a comprehensive analysis of the 600 ks Chandra High Energy
Transmission Grating (HETG) of the Capella corona with a peak temperature of
$0.5-0.6$ keV using the advanced model. The Capella data features a photon-
rich ($\sim 1.1\times 10^{6}$ counts), line-rich ($>750$ lines), and well-
resolved ($\sim 1.2$ eV resolution at 800 eV and $\sim 34.7$ eV at 6000 eV)
spectrum. The instrument calibration and astrophysical modeling (see § 2) are
reasonably well understood. These make the Capella data one of the best
candidates for the study of model uncertainties. In order to cover the high
temperature range, we also include in the test the 110 ks HETG data of HR 1099
corona and the 289 ks Hitomi data of the Perseus cluster. HR 1099 is a RS CVn
binary with a broad coronal temperature distribution within $1-3$ keV
(Huenemoerder et al. 2013). The Perseus cluster of galaxies is the brightest
cluster in the X-ray sky, the main source is the diffuse intracluster medium
with a peak temperature of 4 keV (Hitomi Collaboration et al. 2016).

This paper is organized as follows. Section 2 presents the new approach to
assess the uncertainties in line fluxes of a collisional spectral model based
on observed data. We attempt to decouple it from the uncertainties in the
ionization concentration which constitutes another major error component in
the model. Section 3 applies the obtained uncertainties to the science
interpretation of spectra to be obtained with future XRISM and Athena missions
(Guainazzi & Tashiro 2020). Throughout the paper, the errors are given at a
68% confidence level.

Figure 1:  The charge state distributions of Fe ions calculated from the
multi-temperature models of Capella (a) and HR 1099 (b). The black and red
dashed lines are obtained from the fits with the best physical model (i.e.,
model 1 in § 2.2) and with the file model which best describes the
observation. The lower panels show the ratios between the two models.

##  2 Method and results

In Paper II, we carried out a global, self-consistent modeling and detailed
fit to the Chandra HETG data of Capella, based on the SPEX code (Kaastra et
al. 1996) with improvements on atomic database made in Papers I and II. The
collisional excitation and dielectronic recombination cross sections of the Fe
L-shell species are updated to the modern R-matrix calculations (Fe XVII from
Liang & Badnell 2010, Fe XVIII from Witthoeft et al. 2006, Fe XIX from Butler
& Badnell 2008, Fe XX from Witthoeft et al. 2007, Fe XXI from Badnell &
Griffin 2001, Fe XXII from Liang et al. 2012, Fe XXIII from Fernández-Menchero
et al. 2014, and Fe XXIV from Liang & Badnell 2011), augmented by a large
scale distorted wave calculation when R-matrix results are not available. The
measured cross sections of Fe XVI and Fe XVII excitation and dielectronic
recombination from an Electron Beam Ion Trap experiment (Shah et al. 2019)
have also been incorporated. These improvements are included in SPEX version
3.06. Results on the Capella spectrum were shown for the fits with three kinds
of models ordered by the level of sophistication: the ‘baseline’, the
‘advanced’, and the ‘ultimate’ models. The ‘baseline’ model utilizes a
combination of 18 collisional ionization equilibrium (CIE) components,
characterized by the common elemental abundances, to approximate the multi-
temperature structure of the coronal plasma. The model further takes into
account the effect of interstellar absorption, resonant scattering, and
astrophysical turbulence, as well as various systematic uncertainties from the
instrumental calibration on, e.g., the effective area, energy scale, and line
spread function. The ‘advanced’ model improves further from the ‘baseline’, by
allowing the elemental abundances of different CIE components to vary freely,
decoupling the temperatures used for ion concentration calculation from the
ones for spectral evaluation, and setting the plasma density as a free
parameter. The ‘ultimate’ model is basically the same as the ‘advanced’ model,
except that the former further applies a fix to possible wavelength errors in
the code by comparing the data in SPEX with those in APEC and Chianti. The fit
with the ultimate model becomes the best of the three, however, the final
C-statistic value (40281 for expected value of 7137) remains formally
unacceptable, revealing remaining uncertainties in the best available atomic
data and plasma codes.

The differential emission measure distribution obtained with the ultimate
model shows a peak at $\sim 0.6$ keV (see Figure 11 in paper II), which in
general agrees with the previous measurements using the Chandra HETG and XMM-
Newton Reflection Grating Spectrometer (RGS) data (e.g., Gu et al. 2006). The
chemical abundances of C, O, Ne, and Ni are found to be sub-solar, while the
abundances of N, Na, Mg, Al, and Cr are solar or above (see Table 4 in paper
II). These results agree within the uncertainties with the values reported in
Gu et al. (2006), except for the Ni abundance which is 40% higher in their
work. The Fe, Si, S, Ar, and Ca abundances are set free to vary among
different CIE components (Figure 12 in paper II), so that they cannot be
compared directly with the previous reports in which these abundances were
tied among all the components. The electron density of the stellar corona is
determined to be $<1.4\times 10^{9}$ cm-3, which is in good agreement with the
previous results, e.g., $<2.4\times 10^{9}$ cm-3 by Ness et al. (2001) and
$<7\times 10^{9}$ cm-3 by Mewe et al. (2001) obtained with the Chandra Low
Energy Transmission Grating Spectrometer (LETGS) data.

Here we introduce a new method to assess the atomic uncertainty on line
emissivities using the data from real observations. First, we revise the
‘ultimate’ spectral model obtained in Paper II, incorporating additional
degrees of freedom that could set the ion concentration and the emissivities
of strong lines as free parameters in the fits (see § 2.2 for details). By
fitting the revised model to the observed high-quality spectra, we can anchor
the obtained deviations between theoretical values and the actual data to the
underlying uncertainties in the atomic modeling.

This work focuses on a set of ions of interest (IOIs), which are Fe XVI $-$ Fe
XXIV, Ni XIX, N VII, and He- and H- like O, Ne, Na, Mg, Al, Si, and S. Though
the Ar and Ca lines are also visible in the Capella spectrum, they are not
included in the IOIs as the quality of Chandra grating spectrum at these lines
are not sufficient for a robust study.

Figure 2: Line flux uncertainties in the modeling of the Capella HETG
spectrum. (a) Absolute fluxes of the additional Gaussian components (or the
differential fluxes between model 2 and model 1, see § 2.2 for details)
plotted against the theoretical fluxes of the corresponding emission lines.
Each point represents one line. The black and red data points are the Gaussian
components with positive and negative normalizations, respectively. Arrows
show the upper limits. (b) Fractional contributions of the Gaussian components
to the total fluxes shown as a function of line emissivities. (c) Same as (b),
but shows a comparison between the results with line widths fixed (black) and
line widths free to fit (green). The black data points include both the black
and red data points in (b). (d) Histogram of the Gaussian contribution. The
black and red data are the results with positive and negative Gaussian
normalizations from the fit with fixed line widths, and the green data are the
ones with the free line widths.

Figure 3: Fractional uncertainties of Fe XVIII and Fe XIX lines obtained in
Desai et al. (2005, UD) with the Capella spectrum compared with those derived
in the current work (UC). Black and red data points represent Fe XVIII and Fe
XIX lines, respectively.

Figure 4: (a) Systematic uncertainties of the line flux modeling of the
Capella spectrum, derived by subtracting the statistical uncertainties from
the total uncertainties obtained above (Figure 2). (b) Variances of the
systematic uncertainties as a function of the theoretical line emissivities.
The solid line shows the analytic fit with Eq. 1.

Figure 5: (a & b) Same as Fig. 2 panels a and b but for HR 1099. (c & d) Same
as Fig. 4 panels a and b but for HR 1099.

Figure 6: (a & b) Same as Fig. 2 panels a and b but for the Perseus cluster
with the Hitomi data. (c & d) Same as Fig. 4 panels a and b but for the
Perseus cluster.

Figure 7: (left) Variances of the systematic uncertainties in the line
intensities for Capella (black), HR 1099 (red), and the Perseus cluster
(blue). The solid line shows the analytic fit to the combined data (see Table
1). (right) Systematic uncertainties obtained with the Capella spectrum
plotted against those obtained with the HR 1099 data for the same set of
transitions.

Figure 8: The fractional line flux uncertainties of Capella with the original
model (Fig. 2, black) compared with the uncertainties obtained with the model
excluding the weak transitions (red). See text for details.

Figure 9: Fractional Gaussian contributions to the total line fluxes plotted
as a function of line wavelengths for the Capella (a) and HR 1099 (b) data.
The black and red data points are the Gaussian components with positive and
negative normalizations, respectively.

Figure 10: Variances of systematic uncertainties plotted as a function of line
emissivities for individual ions based on the Capella (left) and HR 1099
(right) spectra. The solid lines are the same as in Figs. 4 (b) and 5 (d).

Figure 11: Fractional Gaussian contributions to the total line fluxes plotted
as a function of contributions to the upper level formation by direct
collisional excitation for Capella (a) and HR 1099 (b). The black and red data
points are the Gaussian components with positive and negative normalizations,
respectively.

Figure 12: Same as Fig. 11 but for radiative recombination.

Figure 13: Same as Fig. 11 but for dielectronic recombination.

Figure 14: Same as Fig. 11 but for radiative cascade from higher levels.

###  2.1 Data

This work is based on high quality spectroscopic observations of Capella, HR
1099, and the Perseus cluster, observed with the Chandra HETG and the Hitomi
Soft X-ray Spectrometer (SXS). Capella and HR 1099 are both representative
bright stellar corona objects: the former has a peak temperature of $0.5-0.6$
keV (Phillips et al. 2001; Behar et al. 2001; Desai et al. 2005; Gu et al.
2006) and the latter is substantially hotter ($\sim 1-3$ keV, Ayres et al.
2001; Huenemoerder et al. 2013). Capella is the best target for testing models
of Fe XVI $-$ Fe XX, while HR 1099 is appropriate for higher ionization
states, e.g., Fe XX $-$ Fe XXIV. The Medium Energy Grating (MEG) spectra in
the wavelength range of $3-32$ Å are used for the two stellar objects. MEG
data have an energy resolution of 1.2 eV at 800 eV, where the Capella spectrum
peaks. The High Energy Grating (HEG) is more suited for sources of higher
energies, since it cannot cover energies below 800 eV. Due to the instrumental
efficiency, the HEG count rates of the two stellar objects in $5-20$ Å are
lower, by a factor of 3.8 (Capella) and 2.8 (HR 1099), than those of the MEG
data. To prevent systematic uncertainties by cross calibration between MEG and
HEG, we only use the MEG data for the two coronal objects. Similarly, we do
not include the data from Low Energy Transmission Grating (LETG) in this work.
Although LETG provides the best spectral resolution for soft X-ray at $>50$ Å,
its resolution gradually gets worse at shorter wavelengths. At 800 eV the LETG
resolution becomes 2.8 eV. Therefore the MEG has the best combination of
spectral resolution and data statistics for the source spectra. The raw data
were reduced using the CIAO v4.10 and calibration database (CALDB) v4.8. The
$chandra\\_repro$ script is used for the data screening and production of
spectral files. The multiple Capella spectra and the associated response files
are combined using the CIAO $combine\\_grating\\_spectra$ tool. The spectral
analysis is carried out with SPEX version 3.06 (Kaastra et al. 1996, 2020).

Our work relies on the critical assumption that most of the uncertainties in
instrumental calibration can be properly dealt with in the analysis. Previous
reports show that the systematic uncertainties are about 8% in the HETG
effective area calibration and $10^{-5}$ in the wavelength scale
111https://cxc.harvard.edu/cal/summary/Calibration_Status_Report.html. To
overcome the possible deviations in the calibration, we incorporate specific
functions as follows in the modeling of the grating spectra. The possible
residual calibration errors in the MEG effective area for continuum are
corrected by multiplying the main spectral components by the SPEX knak model,
which defines a set of piecewise power-law correction functions with grid
points of 1, 3, 6, 10, 14, 18, 26, 32, and 38 Å. We also incorporate a neutral
oxygen absorption model to model the instrumental uncertainty at the O I edge
($22.6-22.9$ Å). By making several iterations between a fit with 100 eV-wide
bins and a fit with the optimal binning, the best-fit knak and O I edge models
are determined. The systematic uncertainties in the wavelength scale have been
corrected by applying a redshift component on the astrophysical model. The MEG
line spread function is modeled with the arbitrary line broadening model vpro,
with a profile shape calculated from the observed O VIII Ly$\alpha$ line at
$\sim 19$ Å. A Wiener filter has been applied to the background dataset
obtained with the standard pipeline. This filter minimizes the background
noise by using a Fourier transform.

Above instrumental modeling and fine-tuning are introduced in order to provide
a platform that allows a fair comparison between two different sets of atomic
line modeling. The priority is therefore to correct the possible energy-
dependent calibration residuals and biases throughout the wavelength range,
rather than to achieve the absolute calibration precision that is otherwise
needed for the measurements of astrophysical parameters. For instance,
although the knak component might not be able to provide the absolute correct
value for the instrumental effective area, it is sufficient to remove the
wavelength-dependent biases in the continuum modeling.

A similar method has been applied to the Hitomi observations of the Perseus
cluster, which has a peak temperature of $\sim 4$ keV. The micro-calorimeter
data has a resolution of $\sim 5$ eV in the energy range of $2-10$ keV
($1.2-6.2$ Å). The data screening and calibration corrections are identical to
those reported in Hitomi Collaboration et al. (2018). The Hitomi data is used
as an essential crosscheck at shorter wavelengths for our analysis with the
stellar sources.

###  2.2 Spectral modeling

We utilize the ‘ultimate’ model described at the beginning of § 2 as a
template (hereafter model 1) for the Capella and HR 1099 spectra. The
differential emission measure of the quiescent coronal plasma can be well
approximated by the sum of multiple CIE components (e.g., Brickhouse et al.
2000; Desai et al. 2005). Model 1 contains 18 independent CIE components
forming a quasi-continuous emission measure distribution. The temperatures of
the 18 components are set by Eq. 5 of Paper II to the fixed values of 0.12
keV, 0.15 keV, 0.18 keV, 0.21 keV, 0.25 keV, 0.30 keV, 0.36 keV, 0.43 keV,
0.51 keV, 0.61 keV, 0.75 keV, 0.92 keV, 1.15 keV, 1.47 keV, 1.94 keV, 2.70
keV, 4.07 keV, and 7.00 keV. The emission measure of each component is set
free to vary in the fits. The Fe abundance of each component can also vary
independently, and the Si, S, Ar, and Ca are fit quasi-independently by
properly grouping of the temperature components. As shown in Figures 11 and 12
of Paper II, both the emission measure and abundances of each temperature
component can be well constrained from the fit, because the former can be
derived from the total line plus continuum emissivity, while the latter is
determined mostly from several relevant lines. Our model further takes into
account a set of astrophysical effects, including the density-sensitive lines,
turbulent broadening at each temperature, systematic line shift, systematic
bias from ionization equilibrium, the neutral and ionized interstellar
absorption, and resonant scattering.

The model applied to the Hitomi spectrum of the Perseus cluster is composed of
three independent temperature components (Hitomi Collaboration et al. 2018),
each with temperature, emission measure, turbulence velocity, and Fe abundance
free to fit. Our model takes into account effects of the AGN emission,
resonance scattering, charge exchange, and the Galactic absorption.

An accurate continuum modeling is essential for assessing line uncertainties.
The continuum of model 1 is composed by the Bremsstrahlung, two photon
emission, and free-bound radiation components with a quasi-continuous electron
temperature distribution described above. As shown in Figure B.1$-$3 of Paper
II, the model continuum agrees within $\sim 10$% with the Capella spectrum in
$4-18$ Å. The continuum at $>18$ Å seems to be overestimated by $20$% on
average, but this affects only $<10$% of the total lines since most L-shell
transitions are in the range of $8-18$ Å. As shown in Figure 23 of Hitomi
Collaboration et al. (2018), the continuum model of the Perseus cluster is in
agreement within $5-10$% with the data in the entire band of the observed
data. The non-thermal component is found negligible in Capella and the Perseus
cluster. The possible small residuals due to instrumental effective area
calibration issues have been corrected with the knak component described in §
2.1.

####  2.2.1 Uncertainties on ion concentration

Before examining the uncertainties in the model line fluxes, we would like to
address the possible deviations in the ion concentration (or charge state
distribution) calculation. The uncertainties in ion concentration (with
respect to the present equilibrium values, e.g., Urdampilleta et al. 2017) are
in general expected to be another major component in the total error budget
(e.g., Foster & Heuer 2020). To estimate uncertainties in ion concentration,
first we need to verify the assumption of collisional ionization equilibrium
for the sources. This could be done by setting rt of the SPEX model ($\it rt$
= 1 for an equilibrium case) a free parameter in the fit, in such a way the
temperature used for calculating the ionization balance is decoupled from the
temperature for the evaluation of rate coefficients (e.g., the excitation
rates). For Capella and HR 1099, rt is determined to be $0.99\pm 0.01$ and
$1.0\pm 0.01$, and Hitomi Collaboration et al. (2018) reported a near unity
value of $0.98\pm 0.01$ for the Perseus cluster. This means that the non-
equilibrium effects are negligible for the three test sources.

As shown in Figure 1, the total ion concentration with model 1 can be
determined by combining the ion concentration of each component weighed by the
emission measure derived from the fit. To compare the model concentration with
the observed one, we re-fit the spectra by allowing the model component of
each IOI free to vary. This can be done by the SPEX file model, which reads a
spectral model from an ASCII file. The emission of IOIs are now modeled by
multiple file models, each contains the line plus continuum of one IOI from
all the spectral components. The remaining non-IOI spectra are converted to
another file model. The astrophysical and instrumental corrections are applied
to all the file models. This would allow us to determine the absolute ion
concentration directly from the observation which does not depend on the
temperatures of the CIE components. Through a fit of the Capella and HR 1099
spectra, we determine the normalizations of the file components, and thus the
deviations on ion concentrations from the values with model 1. For the Fe IOIs
shown in Figure 1, the measured deviation is less than 20% at worst, with an
average value of 8% for Capella and 10% for HR 1099. Similar values can be
obtained with the other IOIs. The current estimate is also in good agreement
with the uncertainties obtained in Hitomi Collaboration et al. (2018) for the
Perseus cluster.

It should be noted that the measured deviation might not come fully from the
ion concentration uncertainty alone. For several IOIs, the thermal emission is
dominated by the strong emission lines. A possible systematic bias in the
emissivities of the strongest transitions could therefore cause potential
deviation in the measured concentration of the relevant IOI. In addition, the
emissivities of radiative and dielectronic recombination and innershell
ionization transitions are determined by the concentration of the neighbour
ions. The changes of their emissivities due to the changes in relative ion
concentration are not accounted in the current approach. As both effects would
introduce extra deviations to the IOI concentration measurement, we consider
the deviation shown in Figure 1 as a crude approximation to the actual atomic
uncertainty in ion concentration.

####  2.2.2 Uncertainties on line intensities

Next we address the uncertainties on line emissivities. First, we define the
lines of interest (LOIs), which are set as a collection of the strongest X-ray
lines of the IOIs. For each Fe ion, we pick up the top 200 lines, and exclude
those in the UV range. For the other elements, we do the same but for top 50
lines. These numbers of lines are determined empirically to make sure that all
the lines visible in the three test spectra are included. The LOIs are
selected at the average temperatures (weighted by the emission measure of each
component) of the objects. A full list of the LOIs selected based above
criteria is provided in Table LABEL:tab:longtable.

To determine the emissivity uncertainty of each LOI, we add for each LOI an
emission line with a Gaussian shape to our ultimate spectral model. The line
energy is fixed to the energy of the target LOI, the width is determined as
described below, and the normalization is directly determined from a spectral
fit and represents the difference in flux between the prediction by the
ultimate model and the measured spectrum. A part of the LOIs cannot be
resolved as they are blended within the instrumental resolution. In such
cases, we use one Gaussian component for one line blend, and set the Gaussian
central wavelength to the average of the unresolved LOIs weighted by the
emissivities. The line widths of the resolved Gaussian components are fixed to
the thermal plus turbulent broadening of the target LOIs, while for the
blended lines, the widths are set free. We have tested to let all the line
widths free, and found that the best-fit line fluxes vary typically by $\leq
20$%. As shown later in Figure 4, the variation of the line width has a
negligible effect on our conclusion.

The Gaussian components are further corrected for the known astrophysical and
instrumental effects, including the ionized/neutral absorption, the systematic
line shift, and the residual calibration errors on the effective area as well
as on the line spread function as described in § 2.1. Parameters of these
models are fixed to the values determined from the original fits with model 1.
We define model 2 as a co-addition of model 1, which is now converted to a
file model as to take into account the uncertainties on ion concentration, and
the set of Gaussian components. The normalizations of the Gaussian components
are the parameters of main interest, the positive normalizations account for
lines where the model underestimates the data, while the negative
normalizations (or absorption-like components) represent the opposite.

Although the additional Gaussian components with free widths for the blends
result in a large number of free parameters with model 2 (656 for Capella and
HR 1099, and 186 for the Perseus cluster), most of the Gaussian components are
well constrained, or have well defined upper bounds on fluxes (Figs. 2, 5, and
6). This is because that, at first, all the Gaussian components, including
those for the blends, are well resolved in the test spectra and fit
independently; and secondly, model 2 contains nearly zero degrees of freedom
on the astrophysical model (e.g., emission measure, temperature, and
abundances) by converting it to a set of file models.

Figures 17 and 18 plot a comparison of model 1 and model 2 for the Capella
HETG spectrum. It reveals significant discrepancies between model 1 and the
observed spectrum in a range of positions, most of those likely originate from
issues in the atomic data that calculate the line fluxes. Most of the
discrepancies appear to be solved with model 2, which compensates for the
mismatches by fitting the extra Gaussian components. As seen in the lower
panels of the figures, these Gaussian components contribute up to $\sim 20$%
of the corresponding line fluxes, and vary strongly, in both positive and
negative ranges, among different LOIs or line blends.

For Capella, the C-stat has been improved from 40281.2 with model 1 to 13520.3
with model 2, for an ideal expectation of 6555. As shown in Figures 17 and 18,
model 2 has fixed the issues in the previous fits of Fe XVIII lines in $14-16$
Å, Fe XIX-XXI lines in $12-14$ Å, Fe XXII-XXIII lines around 11.8 Å, and Fe
XVII-XIX lines in $10-11$ Å. Although model 2 has accurately reproduced most
of the lines in the spectrum, the overall fit is still formally unacceptable.
The remaining discrepancies should be understood as residual errors in
instrument calibration, combined with the minor inadequacies of the atomic
data (in particular on wavelengths, see Paper II). Yet there are a few
observed weak lines, such as those at 9.64 Å, 9.89 Å, 10.68 Å, 10.86 Å, 16.62
Å, and 17.80 Å, still missing in the present modeling. Ignoring the spectral
bins containing the missing lines could further reduce the C-stat to 10343.

As stated earlier, the present work is based on a systematic comparison
between model 1 and model 2 on the known atomic transitions. We will
demonstrate later (§ 2.2.3) that the remaining issues in the fit with model 2,
such as the missing weak transitions at a few places, have a negligible
influence on the systematic comparison. It would therefore be valid to
consider the Gaussian components, which quantify the difference between model
1 and model 2, as an approximation to the total uncertainties in the SPEX line
modeling with respect to the actual data. In panel (a) of Figure 2, the model
2 minus model 1 differential flux (i.e., flux of each Gaussian component added
to model 2) is plotted against the model 1 line flux for each LOI. It shows
that the differential fluxes are positively correlated with model line fluxes;
a flat distribution assuming all Gaussian lines have a common flux can be
excluded at $>5\sigma$. This implies that the observed differential fluxes
cannot be fully explained by the systematic uncertainties due to the
instrumental calibration, which do not depend on the fluxes of the observed
lines.

By dividing the Gaussian fluxes by the total fluxes of the corresponding
lines, we further obtain the fractional uncertainties. As shown in panel (b)
of Figure 2, the uncertainties show anti-correlation with the line
emissivities; for the strong transitions with emissivities $\geq 5\times
10^{41}$ photons s-1, the fractional uncertainties are found to be around 10%,
while for the weak lines $\sim 10^{39}$ photons s-1, the uncertainties
increase to unity or even larger. In addition, panel (c) of Figure 2 shows
that the variation of Gaussian line widths has a nearly negligible effect on
the observed uncertainty-emissivity relation.

In Figure 3 we compare the fractional uncertainties for several Fe XVIII and
Fe XIX transitions reported in Desai et al. (2005) and those obtained in the
our work. It can be found that the present uncertainties are systematically
smaller than those from Desai et al. (2005), for instance, the discrepancies
between the model and data on Fe XVIII resonance line at 14.208 Å and Fe XIX
line at 13.518 Å have been reduced from 30% and 98% (Desai et al. 2005) to the
present values of 13% and 5%. This means that the current modeling of the
Capella spectrum, though still far from ideal, has already been improved
significantly from the those used in Desai et al. (2005) on in particular the
atomic database.

The Gaussian fluxes shown in Figure 2 can be treated as a combination of
statistical and systematic uncertainties. By subtracting the statistical
uncertainties in quadrature from the total values, we estimate the systematic
uncertainties from the line modeling (Fig. 4 left panel). It can be seen that
the contributions of statistical uncertainties are minor for most of LOIs,
thanks to the high quality of the Capella spectrum. To describe the
uncertainty-emissivity relation, we divide the emissivity range into a number
of emissivity bins, and assume for each bin that the distribution of
systematic uncertainty follows a Gaussian function with zero mean value. As
seen in panel (d) of Figure 2, the total uncertainty does show a distribution
that can be described by a combination of multiple Gaussian components peaked
at zero with different variances. The derived variances of the systematic
uncertainties are plotted in the right panel of Figure 4 as a function of
emissivity. It turns out that the systematic uncertainty-emissivity relation
could be approximated by a simple power-law function,

$\sigma=a\times\left(\frac{I}{10^{41}}\right)^{b},$ (1)

where $\sigma$ is the variance of systematic uncertainties, $I$ is line
emissivity in unit of photons s-1, $a$ and $b$ are the free parameters. The
line emissivities are calculated for a standard CIE model with proto-solar
abundances (Lodders et al. 2009) and a fixed emission measure of $10^{64}$
m-3. As shown in Table 1, the parameters $a$ and $b$ are found to be 1.020 and
-0.563 for Capella. The subsets with positive and negative Gaussian
normalizations (black and red data points in Fig. 2) can be described by the
same power-law function as the combined set.

Table 1: Fit parameters and errors of the observed uncertainty-emissivity relations with Eq. 1. |  $T_{\rm peak}$ (keV) | a | b
---|---|---|---
Capella | 0.5 | 1.020 (0.150) | -0.563 (0.067)
HR 1099 | 1.5 | 0.212 (0.084) | -0.762 (0.217)
Perseus | 4.0 | 0.103 (0.183) | -0.946 (0.511)
All | $-$ | 0.332 (0.058) | -0.623 (0.104)

In addition to Capella, we have applied the same exercise to the HR 1099
(Chandra HETG) and the Perseus cluster (Hitomi) spectra. The differential
fluxes obtained in model 2, the systematic uncertainties, and the variances
are plotted against the line emissivity in Figs. 5 and 6. These two objects
have higher peak temperatures (1.5 keV for HR 1099 and 4 keV for Perseus) than
Capella (0.5 keV). One should also note that the Hitomi spectrum of the
Perseus cluster contains only the K-shell lines above 2 keV, whereas the
Capella spectrum is dominated by the Fe-L shell lines. Despite of the
differences, the two objects exhibit similar trends in the systematic
uncertainty-emissivity diagram as Capella (Figure. 7). This reinforces the
general picture that the strong emission lines are consistently much better
modeled than the weak ones. This picture seems to hold for both the L-shell
and the K-shell lines, though the latter is vastly overnumbered by the former
in the current test. As shown in Table 1, the power-law fits to the HR 1099
and Perseus variances reveal marginal difference from Capella: the best-fit
relations of HR 1099 and Perseus cluster appear to be slightly steeper than
Capella, implying for smaller systematic uncertainties in the modeling of
strong lines for the objects with higher average temperatures. A similar hint
can be inferred from the right panel of Figure 7, where we compare the
systematic uncertainties for the common lines that appear in both Capella and
HR 1099 spectra. For the same transition, the fractional error obtained with
Capella (low temperature) is systematically higher than that with HR 1099
(high temperature), indicating that the uncertainty on line flux is likely a
temperature-dependent variable rather than a constant. Verifying this possible
dependence would require a follow-up study with a systematic spectroscopic
sample to cover both the L-shell and K-shell emissions.

####  2.2.3 Influence of the weak transitions

As described earlier, a few weak emission features in the Capella spectrum are
not fully accounted for by the present model, indicating that the atomic data
for the minor transitions are not yet complete. As the missing line issue
could also occur to the blends with other lines, it thus becomes vital to
evaluate the effect of weak lines on the results obtained so far.

In Figure 8 we compare the original fractional uncertainties of Capella with
the uncertainties obtained with a modified model, in which the Gaussian
components of the known transitions of emissivity below $1.5\times 10^{39}$
photons per second are removed in the fit. Ignoring these components means
many weak lines are wrongly modeled, even so, it seems that they do not alter
much the uncertainties of the strong transitions. The uncertainty-emissivity
relation of the strong transitions remains largely intact in the comparison;
parameters $a$ and $b$ from Eq. 1 change by 6% and 1% from the original values
in Table 1. Therefore, our results obtained so far should be robust against
the present limitation in atomic data on weak transitions.

###  2.3 Dependence on various factors

In order to understand the origin of the observed uncertainty-emissivity
relation, here we examine its possible dependence on several variables
including the line wavelength, ion species, and dominant line formation
processes. As shown in Figure 9, the total uncertainty on the line flux is
plotted as a function of wavelength for each individual line in the Capella
and HR 1099 spectra. We find no clear evidence for dependence on line
positions; the relative errors seem to be equally distributed across the
energy band, except for a small group of lines at long wavelengths ($\sim 19$
Å) where the uncertainties are larger than the average. Figure 18 shows that
the present spectral model of Capella does not fully reproduce the observed
continuum between $17-19$ Å, which might partially explain the large line
uncertainties. However, this potential bias only affects a small subset of the
lines, which is minor to the uncertainty-emissivity relation obtained with the
entire set. Therefore, we conclude that the systematic uncertainties on line
fluxes are independent on the line positions.

Next we divide the entire line sample into groups by the ion species. As shown
in Figure 10, the Fe ion groups can be found at different positions in the
line emissivity range, which is primarily determined by the ion concentration
of the source. In general, the distribution of each individual group appears
to follow the combined distribution described by the analytic form (Eq. 1).
There might be a small number of minor biases in individual groups, such as Fe
XIX, which shows a flatter uncertainty-emissivity relation than the average
one for HR 1099, though for Capella it agrees well with the average relation.
We also find good agreement between the Fe group and the non-Fe group.
Therefore, it is likely that the same power-law dependence of spectral
uncertainties on line emissivity can be applied to most of the ions in the
present collisional plasma model.

Finally let us consider the effect of line formation processes. We calculate
the contributions to the upper levels of the LOIs by various processes:
collisional excitation, radiative recombination, dielectronic recombination,
and the cascades from these processes to lower levels. The fractional
contributions calculated for Capella are shown in Table LABEL:tab:longtable.
As seen in Figure 11, there is no obvious correlation between the line
uncertainties and the collisional excitation contribution for Capella and HR
1099. The same is found for the dielectronic recombination (though with much
less data; Fig. 13) and the cascade contribution (Fig. 14). As for the
radiative recombination (Fig. 12), we can see a hint for a weak positive
correlation against the line uncertainty, however, the current significance is
rather low, as the data points with large radiative recombination contribution
are very sparse. Overall, it suggests that the observed uncertainty-emissivity
relation is unlikely to be fully ascribed to one specific line formation
process, but rather caused by the atomic uncertainties in the theoretical
calculations of multiple relevant processes.

It should be noted that some line formation processes might be further
affected by the physical properties of stellar coronae, for example, the
finite density and electric or magnetic fields. As discussed in, e.g., Mewe
(1999), the dielectronic recombination rate might be suppressed in high
density plasma due to the ionization of doubly excited states, while it can be
enhanced by a factor of $5-10$ by the influence of external electric fields.
These two effects might explain some individual scatters observed in the
Capella and HR 1099 spectra, however, they alone cannot explain the
uncertainty-emissivity relation due to the scarcity of strong dielectronic
recombination lines in the observed spectra (Fig. 13).

Figure 15: (a) G ratio of O VII as a function of temperature with associated
uncertainties calculated for low density plasma. (b) R ratio of O VII as a
function of electron density with an electron temperature of $10^{6}$ K. The
black data with errors show our results, the red curves and shaded areas
represent the peaks and the associated errors reported in Loch et al. (2013).
The blue and green curves show the results with AtomDB (Smith et al. 2001;
Foster et al. 2012).

Figure 16: A part of the simulated XRISM (left) and Athena (right) spectra for
a 1 keV CIE model, with the associated uncertainties shown in thin grey
curves. The original model is plotted in red.

As a summary, we discover a relation between the systematic uncertainties and
the line emissivities based on the high-quality spectra of Capella, HR 1099,
and the Perseus cluster. This power-law like relation holds for spectra with
different instruments, objects with different temperature, and lines at
different positions and from different ions. It cannot be explained by errors
in individual line formation process. The observed relation might describe a
universal feature of the state-of-the-art atomic database.

##  3 Applications

In this paper, we present a study of systematic uncertainties in modeling of
collisional sources using observational data. A key application of the results
(e.g., Eq. 1) would be to provide quick and reliable estimate of atomic
uncertainties for a range of relevant observables, including in particular the
line ratios (e.g., He-like triplet), emission measure, temperature, and
abundances. These parameters are the key science outcomes from the present and
future X-ray spectroscopic instruments, their intrinsic uncertainties are
vital, albeit so far often missing, to our understanding of the astrophysical
sources.

###  3.1 He$\alpha$ triplet line ratios

At first we apply our phenomenological relation (Eq. 1) to evaluate the
systematics on the He-like line ratios. In Figure 15, we show the
uncertainties associated with the O VII He$\alpha$ triplet line ratios
compared with the previous results. For a low-density plasma, the error on the
G ratio is found to be $\sim 35$% at 0.05 keV and $\sim 5$% at 0.5 keV. The
uncertainty on the R ratio with a temperature of $10^{6}$ K is $\sim 15$% at
an electron density of $1\times 10^{9}$ cm-3 and diminishes with increasing
density. These values in general agree with the uncertainties for the same
transitions reported in Loch et al. (2013), though the two are derived with
very different approaches. The uncertainties of Loch et al. (2013) were
obtained from a Monte-Carlo calculation with fundamental atomic constants,
while our values come directly from observations.

We attempted to do the same comparison with the published results on the lines
in ultraviolet (Yu et al. 2018; Del Zanna et al. 2019), however, the present
quality of atomic database in SPEX for these ultraviolet lines is still
insufficient to provide useful constraint on their emissivities.

###  3.2 Atomic uncertainties for XRISM and Athena

Next we apply the obtained line uncertainties to realistic simulations with
complicated spectral models, and estimate the induced errors on the primary
model parameters, e.g., plasma temperature, emission measure, and elemental
abundances. We simulate a set of spectral model by varying all the model line
intensities using the observed uncertainty-emissivity relation (Eq. 1 and
Table 1). For simplicity we use the average relation with $a=0.332$ and
$b=-0.623$. To avoid negative or absorption features, the lower boundary of
new line emissivities is set to zero. The upper limit on the fractional
uncertainty is set to 100 for the weakest transitions, which is approximately
the maximal uncertainty observed in the Capella spectrum (Fig. 2). We test our
method on the collisional ionization equilibrium spectrum for a set of
temperature grids from $0.5-2.0$ keV (Table 2), which is about the temperature
range of Capella and HR 1099 where the Fe-L lines dominate the spectrum. We
intend to limit our exercise to relatively low temperatures as the obtained
uncertainty-emissivity relation is primarily determined by the Fe L-shell
lines from the observed spectra.

The simulated models are folded with the instrumental responses of XRISM
Resolve and Athena X-ray Integral Field Unit (X-IFU). XRISM (due to launch in
the early 2020s) and Athena (early 2030s) are two future X-ray observatories
exploring the hot and energetic baryons in the Universe. These two missions
will enable very well-resolved X-ray spectroscopy of various X-ray sources and
will push the atomic modeling to its limit. In Figure 16 we illustrate a part
of the spectrum, as well as the associated systematic uncertainty, of a
turbulence-free 1 keV CIE model. We run the simulation 1000 times for each
temperature grid and instrument, fit the randomized data with the original
spectral model, and summarize in Table 2 the obtained standard deviations of
the primary model parameters: emission measure, temperature, and abundances.
For the abundances, we show the atomic uncertainties on Fe as well as several
representative elements in the Fe-L region (O, Ne, Mg, and Ni). For the
latter, the uncertainties originate from the errors on their own emission as
well as on the blended neighbor Fe-L lines.

Table 2: Fractional systematic uncertainties on model parameters based on the simulation described in § 3. kT | $\sigma_{\rm EM}$ | $\sigma_{\rm kT}$ | $\sigma_{\rm O}$ | $\sigma_{\rm Ne}$ | $\sigma_{\rm Fe}$ | $\sigma_{\rm Mg}$ | $\sigma_{\rm Ni}$
---|---|---|---|---|---|---|---
(keV) | XRISM
0.5 | 0.038 | 0.011 | 0.047 | 0.058 | 0.044 | 0.078 | 0.076
0.75 | 0.063 | 0.007 | 0.067 | 0.081 | 0.072 | 0.066 | 0.092
1.0 | 0.066 | 0.007 | 0.059 | 0.097 | 0.065 | 0.065 | 0.130
1.25 | 0.071 | 0.012 | 0.072 | 0.100 | 0.067 | 0.081 | 0.167
1.5 | 0.060 | 0.021 | 0.060 | 0.098 | 0.067 | 0.093 | 0.133
1.75 | 0.045 | 0.023 | 0.054 | 0.097 | 0.057 | 0.104 | 0.171
2.0 | 0.032 | 0.022 | 0.061 | 0.097 | 0.050 | 0.113 | 0.200
| Athena
0.5 | 0.033 | 0.011 | 0.029 | 0.047 | 0.043 | 0.067 | 0.050
0.75 | 0.039 | 0.007 | 0.040 | 0.059 | 0.039 | 0.048 | 0.056
1.0 | 0.044 | 0.005 | 0.048 | 0.072 | 0.035 | 0.053 | 0.091
1.25 | 0.051 | 0.008 | 0.065 | 0.077 | 0.038 | 0.071 | 0.112
1.5 | 0.052 | 0.019 | 0.080 | 0.087 | 0.053 | 0.092 | 0.128
1.75 | 0.041 | 0.023 | 0.078 | 0.091 | 0.053 | 0.103 | 0.153
2.0 | 0.031 | 0.022 | 0.070 | 0.092 | 0.049 | 0.111 | 0.175

As shown in Table 2, the uncertainties on line flux have propagated into $\sim
3-7$% errors on the emission measure (hence $2-4$% on gas density), $\sim
1-2$% on temperature, $\sim 4-7$% on O and Fe abundances, $\sim 6-10$% on Ne
and Mg abundances, and $\sim 8-20$% on Ni abundance, for a XRISM-like
spectrum. Some of these parameters are better constrained with Athena X-IFU,
as X-IFU has a significantly better spectral resolution that helps to de-blend
the lines in a crowded complex. The atomic uncertainties are further found to
be temperature-dependent: the fractional temperature error increases by a
factor of three from 1 keV to 2 keV, while the abundance errors show peaks
around $1.25-1.5$ keV. The different behaviors of temperature and abundance
errors show that they might originate from different sets of lines. Note that
the obtained systematic uncertainties likely represent a lower limit, as (1)
the continuum uncertainties, and the errors on the ionization balance
calculation, are not yet included; and (2) in the present simulation the
discrepancies on the line intensities are assumed to be fully random, which
might not hold in reality.

##  4 Conclusion

We present an observational constraint that could be used to calculate the
systematic uncertainties in spectral model of sources in collisional
ionization equilibrium. Our method is based on statistical properties of the
discrepancies between model line fluxes and observed values. The uncertainties
are found to be about 10% for the strong emission lines, and significantly
increase towards low fluxes. The observed uncertainty-emissivity relation can
be approximated by an analytic form, which holds for lines with different
wavelengths, ion species, and formation processes. Applying the observed
uncertainties to the simulated XRISM and Athena spectra yields $4-20$%
systematic errors on the elemental abundances measured from these spectra. In
the future this work will be extended to the other spectral components
(continuum and absorption features), and to astrophysical sources in photo-
ionization and non-equilibrium ionization status. It should be emphasized that
our approach based on observational data can provide absolute uncertainties of
the target atomic constants, however, it cannot be used to illustrate the
underlying correlations between the uncertainties of different transitions.
Ideally, it will require fundamental theoretical calculations, benchmarked by
the observational results, to derive a full picture of the atomic
uncertainties.

The atomic uncertainties estimated using the new approach have been
implemented in the aerror command of the SPEX code.

###### Acknowledgements.

L.G. is supported by the RIKEN Special Postdoctoral Researcher Program. SRON
is supported financially by NWO, the Netherlands Organization for Scientific
Research. Work by C. Shah was supported by the Max-Planck-Gesellschaft (MPG),
the Deutsche Forschungsgemeinschaft (DFG) Project No. 266229290, and by an
appointment to the NASA Postdoctoral Program at the NASA Goddard Space Flight
Center, administered by Universities Space Research Association under contract
with NASA. J.M. acknowledges the support from STFC (UK) through the University
of Strathclyde UK APAP network grant ST/R000743/1. P. Amaro acknowledges the
support from Fundação para a Ciência e a Tecnologia (FCT), Portugal, under
Grant No. UID/FIS/04559/2020(LIBPhys). The research leading to these results
has received funding from the European Union’s Horizon 2020 Programme under
the AHEAD2020 project (grant agreement n. 871158).

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##  Appendix A Modeling details of the Capella and HR 1099 spectra

In Figures 17 and 18, we plot the Chandra HETG data fit with a plasma model
and a file plus Gaussian line model (see details in § 2). We also show the
ratios between the two models. For each line of interest, we list its
position, flux, uncertainty, and line formation properties in Table
LABEL:tab:longtable.

Figure 17: Stacked Chandra grating spectrum of Capella in $5.0-14.0$ Å
(wavelength region divided by panels) fit with model 1 (ultimate model from
paper II, blue) and model 2 (file model plus multiple Gaussian components,
red). The relative discrepancies between model 1 and model 2 are shown in the
lower panel. The selected lines of interest are marked with thin vertical
lines. The numbers on the top are the associated line IDs (see Table
LABEL:tab:longtable for details), the colors of the numbers indicate the
dominant line formation process: black $-$ direct excitation; red $-$
radiative cascade; blue $-$ dielectronic recombination.

Figure 18: Same as Fig 17 but for $14-20$ Å.  Table 3:  List of lines of interest Ion | ID | Wavelength | EX | CEX | RR | CRR | DR | CDR | $I_{\rm Capella}$ | $U_{\rm Capella}$ | $I_{\rm HR1099}$ | $U_{\rm HR1099}$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | ($\AA$) |  |  |  |  |  |  | (ph m-2 s-1) |  | (ph m-2 s-1) |
Fe XVI | 1 | 40.1517 | 0.543 | 0.008 | 0.041 | 0.016 | 0.000 | 0.385 | 0.8458 | 0.0448 | 0.2479 | 0.1462
Fe XVI | 2 | 39.8258 | 0.608 | 0.002 | 0.045 | 0.018 | 0.000 | 0.319 | 0.4057 | 0.1374 | 0.1163 | 0.4795
Fe XVI | 3 | 15.2008 | 0.000 | 0.000 | 0.000 | 0.000 | 0.934 | 0.066 | 0.4529 | 0.9431 | 0.0616 | 0.5325
Fe XVI | 4 | 15.2442 | 0.365 | 0.002 | 0.000 | 0.000 | 0.607 | 0.025 | 0.7161 | 0.0719 | 0.0962 | 0.1186
Fe XVI | 3 | 15.2109 | 0.965 | 0.002 | 0.000 | 0.000 | 0.021 | 0.012 | 1.2664 | 0.9431 | 0.1756 | 0.5325
Fe XVI | 3 | 15.2140 | 0.000 | 0.000 | 0.000 | 0.000 | 0.973 | 0.027 | 0.3508 | 0.9431 | 0.0456 | 0.5325
Fe XVI | 5 | 36.7478 | 0.633 | 0.007 | 0.073 | 0.030 | 0.000 | 0.248 | 0.3032 | 21.4497 | 0.0741 | 12.7918
Fe XVI | 4 | 15.2597 | 0.000 | 0.000 | 0.000 | 0.000 | 0.935 | 0.064 | 0.2516 | 0.0719 | 0.0329 | 0.1186
Fe XVI | 6 | 15.0960 | 0.182 | 0.001 | 0.000 | 0.000 | 0.760 | 0.056 | 0.3394 | 0.8149 | 0.0452 | 0.2251
Fe XVI | 7 | 15.3150 | 0.000 | 0.000 | 0.000 | 0.000 | 0.957 | 0.043 | 0.2498 | 1.4674 | 0.0335 | 0.2118
Fe XVI | 6 | 15.0794 | 0.000 | 0.000 | 0.000 | 0.000 | 0.990 | 0.010 | 0.3070 | 0.8149 | 0.0457 | 0.2251
Fe XVI | 6 | 15.0752 | 0.000 | 0.000 | 0.000 | 0.000 | 0.997 | 0.003 | 0.3107 | 0.8149 | 0.0463 | 0.2251
Fe XVI | 8 | 35.1134 | 0.000 | 0.007 | 0.100 | 0.021 | 0.000 | 0.860 | 0.3103 | 2.1108 | 0.0709 | 11.3896
Fe XVI | 9 | 15.0260 | 0.000 | 0.000 | 0.000 | 0.000 | 0.995 | 0.005 | 0.3035 | 0.1278 | 0.0454 | 0.0230
Fe XVI | 6 | 15.0916 | 0.000 | 0.000 | 0.000 | 0.000 | 0.991 | 0.009 | 0.2497 | 0.8149 | 0.0371 | 0.2251
Fe XVI | 9 | 15.0472 | 0.000 | 0.000 | 0.000 | 0.000 | 0.997 | 0.003 | 0.2931 | 0.1278 | 0.0446 | 0.0230
Fe XVI | 9 | 15.0309 | 0.000 | 0.000 | 0.000 | 0.000 | 0.999 | 0.001 | 0.2940 | 0.1278 | 0.0448 | 0.0230
Fe XVI | 5 | 36.8018 | 0.662 | 0.022 | 0.071 | 0.021 | 0.000 | 0.216 | 0.1502 | 21.4497 | 0.0365 | 12.7918
Fe XVI | 1 | 40.2348 | 0.000 | 0.026 | 0.082 | 0.025 | 0.000 | 0.861 | 0.1891 | 0.0448 | 0.0622 | 0.1462
Fe XVI | 9 | 15.0471 | 0.000 | 0.000 | 0.000 | 0.000 | 0.999 | 0.001 | 0.2363 | 0.1278 | 0.0360 | 0.0230
Table 3: continued.

  1. 1.

Full data table can be found via the link to the machine-readable version.

  2. 2.

ID: line ID (see Figs. 17 and 18)

  3. 3.

EX/CEX/RR/CRR/DR/CDR: contributions of upper level population from direct
excitation, excitation followed by cascades, radiative recombination,
radiative recombination followed by cascades, dielectronic recombination,
dielectronic recombination followed by cascades

  4. 4.

$I_{\rm Capella}$, $I_{\rm HR\leavevmode\nobreak\ 1099}$: line fluxes from the
best-fit models (model 1)

  5. 5.

$U_{\rm Capella}$, $U_{\rm HR\leavevmode\nobreak\ 1099}$: fractional
systematic uncertainties of the lines from the Gaussian fits (model 2)