# Spectral Implications of Atomic Uncertainties in Optically-thin Hot Plasmas

Keri Heuer Department of Physics, Drexel University,
3141 Chestnut St, Philadelphia, PA 19104, USA  Center for Astrophysics —
Harvard & Smithsonian,
60 Garden Street, Cambridge, MA 02138, USA  Adam R. Foster Center for
Astrophysics — Harvard & Smithsonian,
60 Garden Street, Cambridge, MA 02138, USA  Randall Smith Center for
Astrophysics — Harvard & Smithsonian,
60 Garden Street, Cambridge, MA 02138, USA Keri Heuer<EMAIL_ADDRESS>
(Received September 28, 2020; Accepted November 15, 2020; )

###### Abstract

Two new high-resolution X-ray spectroscopy missions, XRISM and Athena, will
observe deeper and with higher X-ray resolution than ever before possible.
Interpreting these new X-ray spectra will require understanding the impact
that uncertainties on fundamental atomic quantities such as collisional cross
sections, transition rates, and wavelengths have on spectral models. As
millions of values are required to generate even a simple model of an
optically-thin hot plasma, most such rates exist only as theoretical
calculations. We have developed methods to estimate the uncertainty in the
final spectral calculations based on published experimental data and plausible
approximations to the uncertainties in the underlying atomic data. We present
an extension to the pyatomdb code that implements these methods and
investigate the sensitivity of selected strong diagnostic lines in the X-ray
bandpass (0.3-12 keV).

X-ray astronomy, atomic spectroscopy, atomic data benchmarking

††journal: APJ

##  1 Introduction

X-ray-emitting optically-thin collisional plasmas exist throughout
astrophysics on a range of scales, from stellar coronae to galaxy clusters.
The X-ray emission arises from a range of processes, most prominently
including bremsstrahlung, ion line emission, and radiative recombination.
Diagnosing basic underlying plasma parameters does not always require high-
resolution data. At electron temperatures above 2 keV, bremsstrahlung
radiation dominates the emission from astrophysical plasmas with a broad and
well-understood continuum. Even at lower temperatures, early X-ray grating
spectrometers were, despite limited resolution, able to make accurate
inferences about the underlying system. Holt et al. (1979) used Einstein
Solid-State Spectrometer ($R\sim 140$) observations of Capella to correctly
conclude the corona has at least two components between $6-24\times 10^{6}$K
and to detect Mg, Si, S, and Fe. These results were primarily based on
detecting a continuum that could not be fit with a single bremsstrahlung
component along with models for the strong lines from He-like ions of Mg, Si,
S, and Fe.

More detailed analyses with higher-resolution spectrometers, however, require
both more and better atomic data. In some cases, these data can be directly
tied to laboratory measurements. Charge state distributions (CSDs111We use
CSD, rather than ionization balance because the latter implies the system is
in some type of equilibrium, which may or may not be true.) in hot plasmas,
for example, depend upon a modest number of ionization and recombination rate
coefficients. Many different collections have been used throughout the history
of X-ray astronomy. Early compilations such as Jordan (1969) and Raymond &
Smith (1977) were updated (Sutherland & Dopita, 1993; Mazzotta et al., 1998)
as newer data became available (e.g. dielectronic recombination rates) or it
was realized that processes such as excitation-ionization were dramatically
underestimated. The process has continued (Bryans et al., 2006, 2009), and
these updates have at times significantly changed the conclusions observers
draw. The update of the AtomDB atomic database Smith et al. (2001); Foster et
al. (2012) from the Mazzotta et al. (1998) CSD to the calculations of Bryans
et al. (2006, 2009) showed that the ISM of the elliptical galaxy NGC 4649
could be explained with a single temperature, rather than requiring a
physically-unmotivated two component model (Loewenstein & Davis, 2012).
Nonetheless, for CSD calculations at least, the process seems to be
converging. The most recent compilation of ionization rates by Urdampilleta et
al. (2017) showed “differences less than 10-20% for more than 85% of the
elements,” a substantial improvement from previous updates, although the
prospect of including a $\sim 15\%$ error into collisional spectral models
suggests work remains to be done.

The situation with line emission remains less well understood. High-resolution
X-ray grating observations of stars with Chandra and XMM-Newton have revealed
some problems with the existing collisional plasma models (e.g. Huenemoerder
et al., 2003). However, stellar coronae have complex temperature distributions
and densities high enough to impact line emission, making it challenging to
determine if a poor fit is due to the atomic model or the physical parameters
used. Recently, the Hitomi spectra of the Perseus cluster (Hitomi
Collaboration et al., 2016) highlighted the need to explore the sensitivity of
measurements to atomic line emission data. Unlike a highly-dynamic magnetized
corona, the diffuse emission from a galaxy cluster like Perseus is relatively
simple to model, and showed a number of flaws in both the AtomDB and SPEX
(Kaastra et al., 1996) models used to fit it. The Hitomi Collaboration made an
initial effort in their atomic data paper (Hitomi Collaboration et al., 2018,
hereafter “HADP2018”) to estimate uncertainties in fluxes from lines of H- and
He-like ions using disagreements between SPEX 3.03, ATOMDB 3.0.8 and other
databases. HADP2018 thus demonstrated uncertainty estimates were both feasible
and useful but focused only on a few lines.

Here we expand upon this approach to consider uncertainties in the CSD as well
as emission lines, based on the wavelengths, rates, and cross sections in
AtomDB database. Using variableapec
1.0.2222http://github.com/AtomDB/variableapec, we perturb values such as
collisional cross sections and recalculate the CSD and level populations. This
allows an encapsulation of an overall uncertainty into a single quantity, such
as the line emissivity for a collisional plasma ($\epsilon$(n,T) in units of
photons cm3 s-1). While the size of our perturbations are motivated by
existing measurements, we emphasize that these calculations are at best
estimates. In particular, we have no method to include correlated errors, such
as would occur from changes to a structure calculation of an ion’s electronic
energy levels (e.g. Loch et al., 2013). Nonetheless, we hope that the results
can be used by observers to estimate potential uncertainties in their model
parameters from the underlying atomic data.

##  2 CSD Uncertainty Calculation

###  2.1 Method

We use related but distinct methods to calculate uncertainties in the CSD and
the line emissivities. The rationale for this separation is a combination of
issues. First, for a given element $Z$ and electron temperature $T_{e}$, the
CSD calculation involves $\sim Z$ ionization and recombination rates.
Depending upon the ion and the level of detail in the calculation, metastable
levels and multiple-ionization rates may be included, but in any event there
is limited amount of atomic data required. This cannot be said of line
emission calculations, which rely upon theoretical calculations of atomic
structure with hundreds to thousands of energy levels per ion with collisional
and radiative transitions potentially allowed between any of these. Secondly,
the CSD rates have some of the best experimental measurements of any values
required for collisional plasma models, while electronic excitation cross
sections and radiative transitions are dominated by theoretical calculations
and have at best spot measurements.

The underlying atomic processes driving the CSD are measured experimentally
and calculated theoretically as cross sections as a function of electron
energy. These process include direct ionization, excitation-autoionization,
radiative recombination, and dielectronic recombination (see e.g. Smith &
Brickhouse, 2014). For a thermal plasma model, theoretical calculations are
typically compared to experimental measurements (and perhaps adjusted) and
then integrated over a Boltzmann distribution to derive the ionization or
recombination rate as a function of temperature.

Bryans et al. (2006) updated the largely theoretical calculation of ionization
rates by Dere (2007) and recombination rates by Badnell (2006). Urdampilleta
et al. (2017) has since provided another update of ionization rates with an
estimate of uncertainty in experimental cross sections. The various rate
errors from different experimental data sets can result in vast differences of
CSD for an otherwise identical plasma. These updates prompt a robust study of
the effect of atomic uncertainties on the CSD, which up until now has yet to
be investigated in depth.

To estimate a reasonable systematic uncertainty, we conducted a thorough
literature search of experiments measuring energy-dependent cross sections
$\sigma(E)$ or Maxwellian-averaged rate coefficients $\Upsilon(T_{e})$333As
both types of data are regularly provided, we refer to ionization or
recombination ‘values’ to indicate when either is possible.. Tables 1 and 2
list systematic uncertainties for ionization and recombination values
respectively from the most recent experiments for highly charged Fe and O
ions. These also list the type of experiment and for Table 1 notes if the
result is an energy-dependent electron-impact ionization (EII) cross section
or a Maxwellian-averaged ionization (I) rate coefficient. In most cases papers
listed only a single systematic error, but in cases where multiple errors were
listed, e.g. at different energies, the value closest to the threshold for
ionization was selected. Based on this catalog of multiple systematic
uncertainties for each ion, we use the minimum error found for each ion’s
ionization and recombination rate coefficient for our thermal CSD calculations
using the collisional equilibrium model. The minimum was chosen in order to
not overestimate the effects of systematic errors. For our other calculations
looking at the sensitivity of ionic concentration to error magnitude
independent of this tabulation, we considered a range of perturbations,
typically from 5-40%, and use the same change for both ionization and
recombination rate coefficients. Our calculated changes in ionic concentration
are thus the minimum due to systematic uncertainties. In cases where multiple
systematic errors are listed at different energies, the resulting change in
concentration can therefore be larger.

We also note any published comparisons to rates (“R”, either cross sections or
rate coefficients) in theoretical data sets since it is informative when
updated experimental results are compared to commonly-used theoretical values.
$\Delta$R values are defined as $(R_{exp}-R_{th})/R_{exp}$ where these values
are available; listed values from the literature (e.g., from text such as “Our
rates agree to theory within 10%”) are assumed to mean the same. These are
listed only for the most recent theoretical data set if a reference has
compared their experimental data to multiple sources; see respective
references for comparisons with older theories. Where references have
calculated their own theoretical values for comparison with experimentally-
derived rate coefficients, we list the source for $\Delta$R as self.

To understand the effect of atomic uncertainties on the CSD, we first perturb
a single rate coefficient by the same amount at all temperatures and calculate
new ion populations for temperatures where the fractional ionic concentration
is greater than $10^{-4}$. CSD curves generally appear to be Gaussian,
although the wings and details depend upon the specific ion. To quantify the
impact of the uncertainty on the inferred temperature of the plasma, we
considered the fractional change in ionic concentration at five temperatures:
where the ion fraction is rising as a function of temperature to 1%, 10%, and
then peak, and as it drops back down again to 10% and 1% [see Figure 1].
Comparing the rate of change in ionic concentration as a function of
fractional change, defined as half the change in emissivity calculated from
the minimum and maximum CSD, on a given rate coefficient at these five
temperatures indicates the temperatures where an ion is most affected by
uncertainties [see Figure 3]. We repeat this process for various levels of
uncertainties to compare the resulting CSD and sensitivities to perturbation
magnitude.

Figure 1: Five temperatures of interest for Fe xvii indicated by horizontal
dashed lines on the low-temperature (‘rising’) side at 1% and 10% ionic
concentration, the peak temperature, and the high-temperature (‘falling’) side
at 1% and 10%.

Using random errors selected from a flat distribution (chosen as these are
systematic, not statistical, errors), we then jointly perturb all rate
coefficients for an element and use Monte Carlo calculations to derive the
range of potential population values for each ion over a range of plasma
temperatures. For ions where no errors have been published, we employ an
average of the errors across all ions in Tables 1 and 2. Errors ($\sigma$) are
selected from the minimum error on the rate coefficient we employ from Tables
1 and 2. We then compare the non-perturbed ion population over Monte Carlo
approximations with $\pm\sigma$ error bars to derive a range of potential
population values for an ion over a range of plasma temperatures.

Table 1: Systematic uncertainties for experimentally measured ionization rate coefficients. $\Delta$R is defined as the published comparison to the theoretical data set listed as the “comparison”. If the reference calculated their own theoretical values, the comparison source is listed as “self”. Ion | Error | TypeaaElectron-impact ionization (EII) or effective ionization rate (I). | MethodbbCrossed-beam apparatus (cb), plasma spectroscopy/$\theta$-pinch (ps), electron beam ion trap (e), tokamak (t). | Reference | $\Delta$R | Comparison
---|---|---|---|---|---|---
O 7+ | 10% | EII | cb | Aichele et al. (1998) |  |
O 6+ | 29% | EII | ps | Rowan & Roberts (1979) | 35% | Summers (1974)
| 20% | EII | ps | Kallne & Jones (1977) |  |
O 5+ | 11% | EII | cb | Crandall et al. (1986) | 50% | Jakubowicz & Moores (1981)
| 8% | EII | cb | Rinn et al. (1987) | 9% | Pindzola et al. (1986)
| 8% | EII | cb | Müller et al. (2000) |  |
O 4+ | 8% | EII | cb | Fogle et al. (2008) | 17% | Dere (2007)
| 7% | EII | cb | Loch et al. (2003) |  |
| 6% | EII | cb | Falk et al. (1983) | 5% | Younger (1981)
Fe 25+ | 15% | EII | e | O’Rourke et al. (2001) |  |
Fe 23+ | 13% | EII | e | Wong et al. (1993) | 4% | Zhang & Sampson (1990)
Fe 21+ | 40% | I | t | Wang et al. (1988) | 33% | Golden & Sampson (1977)
Fe 20+ | 40% | I | t | Wang et al. (1988) | 0% | Golden & Sampson (1977)
Fe 19+ | 40% | I | t | Wang et al. (1988) | 64% | Golden & Sampson (1977)
Fe 18+ | 40% | I | t | Wang et al. (1988) | 20% | Golden & Sampson (1977)
Fe 17+ | 40% | I | t | Wang et al. (1988) |  |
| 16% | EII | cb | Hahn et al. (2013) | 4-17% | Dere (2007)
| 15% | EII | cb | Hahn (2014) |  |
Fe 16+ | 15% | EII | cb | Hahn (2014) | 10-15% | Dere (2007)
| 40% | I | t | Wang et al. (1988) |  |
| 16% | EII | cb | Hahn et al. (2013) | 1-19% | Dere (2007)
Fe 15+ | 20% | EII | cb | Hahn (2014) |  |
| 40% | I | t | Wang et al. (1988) | 15% | Pindzola et al. (1986)
| 20% | EII | cb | Müller (1999) | 20% | Arnaud & Raymond (1992)
Fe 14+ | 26% | EII | cb | Bernhardt et al. (2014) | 35-70% | Arnaud & Raymond (1992)
Fe 13+ | 16% | EII | cb | Hahn (2014) | 35% | Gregory et al. (1987)
| 6% | EII | cb | Hahn et al. (2011a) | 15% | Dere (2007)
| 16% | EII | cb | Hahn et al. (2013) | 35% | Arnaud & Raymond (1992)
Fe 12+ | 6% | EII | cb | Hahn et al. (2011b) | 15% | Dere (2007)
Fe 11+ | 16% | EII | cb | Hahn et al. (2011a) | 15%, 30% | self, Arnaud & Raymond (1992)
| 15% | EII | cb | Hahn (2014) |  |
| 14% | EII | cb | Gregory et al. (1987) | 30% | Pindzola et al. (1986)
Fe 10+ | 9% | EII | cb | Hahn et al. (2012) | 19% | Dere (2007)
Fe 9+ | 15% | EII | cb | Hahn (2014) |  |
| 9% | EII | cb | Hahn et al. (2012) | 16% | Dere (2007)
Fe 8+ | 16% | EII | cb | Hahn et al. (2016) | 15-40% | Dere (2007)
Fe 7+ | 12% | EII | cb | Hahn et al. (2015) | 10% | Dere (2007)

Table 2: Systematic uncertainties for experimental recombination rate coefficients. $\Delta$R is defined as in Table 1. Ion | Error | TypeaaDielectronic recombination (DR) or radiative recombination (RR). | MethodbbCrossed-beam apparatus (cb), plasma spectroscopy/$\theta$-pinch (ps). | Reference |  $\Delta$R | Comparison
---|---|---|---|---|---|---
O 7+ | 25% | DR | cb | Kilgus et al. (1990) | 20% | Bell & Bell (1982)
| 18% | RR | cb | Andersen & Bolko (1990) |  |
O 6+ | 20% | DR | cb | Andersen et al. (1990) |  |
O 5+ | 11% | DR | cb | Böhm et al. (2002) | 15-25% | self
| 20% | DR | cb | Andersen et al. (1990) |  |
| 35% | DR | cb | Dittner et al. (1987a) | 30-60% | Pindzola et al. (1986)
| 18% | RR | cb | Andersen & Bolko (1990) |  |
O 4+ | 30% | DR | cb | Dittner et al. (1987b) |  |
Fe 24+ | 20% | DR | cb | Gwinner et al. (2001) |  |
Fe 22+ | 20% | DR | cb | Savin et al. (2006) | 20% | self
| 20% | DR | cb | Schippers et al. (2010) |  |
| 20% | DR | cb | Gwinner et al. (2001) |  |
Fe 21+ | 20% | DR | cb | Schippers et al. (2010) |  |
| 20% | DR | cb | Savin et al. (2013) |  |
| 20% | DR | cb | Gwinner et al. (2001) |  |
Fe 20+ | 20% | DR | cb | Schippers et al. (2010) |  |
| 20% | DR | cb | Savin et al. (2013) |  |
| 20% | DR | cb | Gwinner et al. (2001) |  |
Fe 19+ | 20% | DR | cb | Savin et al. (2002) | 15%, 30% | self, Zatsarinny et al. (2004b) cc30% scatter quoted from Zatsarinny et al. (2004a)
| 20% | DR | cb | Schippers et al. (2010) |  |
| 20% | DR | cb | Gwinner et al. (2001) |  |
Fe 18+ | 20% | DR | cb | Savin et al. (2002) | 20%, 40% | self, Wang et al. (1988)
| 20% | DR | cb | Schippers et al. (2010) |  |
| 20% | DR | cb | Gwinner et al. (2001) | 160% | Savin et al. (1999)
Fe 17+ | 20% | DR | cb | Savin et al. (1997) | 200%, 30% |  Chen (1988), self
| 20% | DR | cb | Schippers et al. (2010) | 30% | self, Savin et al. (1999)
| 20% | DR | cb | Gwinner et al. (2001) | 200% | Savin et al. (1999)
Fe 16+ | 22% | DR | cb | Schippers et al. (2010) |  |
| 20% | DR | cb | Schmidt et al. (2009) | 14% | Zatsarinny et al. (2004a)
Fe 15+ | 20% | DR | cb | Linkemann et al. (1995) | 200% | Chen et al. (1990)
| 20% | DR | cb | Schippers et al. (2010) |  |
Fe 14+ | 26% | DR | cb | Bernhardt et al. (2014) | 26% | Altun et al. (2007)
| 29% | DR | cb | Lukić et al. (2007) | 19-37%, 300% | self, Arnaud & Raymond (1992)
| 29% | DR | cb | Schippers et al. (2010) |  |
Fe 13+ | 18% | DR | cb | Schmidt et al. (2006) | 21% | Arnaud & Raymond (1992)
| 29% | DR | cb | Schippers et al. (2010) |  |
Fe 12+ | 18% | DR | cb | Hahn (2014) | 30% | Badnell (2006)
Fe 11+ | 13% | DR | cb | Novotný et al. (2012) | 30% | Badnell (2006)
Fe 10+ | 30% | DR | ps | Brooks et al. (1980) | 70% | Jacobs et al. (1977)
| 25% | DR | cb | Schippers et al. (2010) |  |
| 25% | DR | cb | Lestinsky et al. (2009) | 25% | Badnell (2006)
Fe 9+ | 30% | DR | ps | Brooks et al. (1980) | 150% | Jacobs et al. (1977)
| 25% | DR | cb | Schippers et al. (2010) |  |
| 25% | DR | cb | Lestinsky et al. (2009) | 25% | Badnell (2006)
Fe 8+ | 35% | DR | cb | Schmidt et al. (2008) |  |
| 30% | DR | ps | Brooks et al. (1980) | 13% | Jacobs et al. (1977)
| 29% | DR | cb | Schippers et al. (2010) |  |
Fe 7+ | 26% | DR | cb | Schmidt et al. (2008) | 120% | Arnaud & Raymond (1992)
| 25% | DR | cb | Schippers et al. (2010) |  |

###  2.2 Results

For simplicity, we focus on results for oxygen and iron ions, but the
variableapec tools developed for this project can be used for any atom or line
in the AtomDB database with user-defined errors (see Appendix). We note that a
complete review of all systematic errors in ionization and recombination rates
for astrophysically abundant ions, along the lines of Tables 1 and 2 would be
welcomed, but is beyond the scope of this initial effort. The primary
challenge is to determine a reasonable set of estimated uncertainties for the
underlying atomic data. The availability of experimental data depends upon a
number of factors, including the ease of use (noble gases are easier to work
with than metals) and their astrophysical abundance (a proxy for their
importance).

Figure 2: [Top] Oxygen CSD as a function of temperature. [Bottom] Maximum
relative changes in the CSD after perturbing the O6+ ionization rate by
$\pm$15%.

Changing even a single rate, such as the total ionization rate
I(O${}^{6+}\rightarrow$ O7+), will potentially affect the population of a
number of ions. Figure 2 shows the impact of a $\pm 15$% change in this rate
on the O${}^{5+}-$O8+ ion populations. To avoid numerical issues, the
calculations cut off at populations less than $10^{-5}$, resulting in the
sharp bounds seen in the lower half of Figure 2. The relative impact of this
one uncertainty is largest where the ion has a low ionic concentration,
showing that inferences made about temperature using an ion whose population
is $<5\%$ will potentially have large uncertainties, even though the apparent
sensitivity is high due to a ion population that rapidly changes with
temperature.

This effect is explicitly shown in Figure 3, which plots the changes in the
ionic concentration of O6+ ([Left]) and O7+ ([Right]) at the 5 temperatures
shown in Figure 1 after modifying the O6+ ionization rate. The lack of change
at many temperatures shows that, in equilibrium, the exact value of each rate
matters only over a small range of temperatures. The ionization rate out of
O6+, for example, only matters if the plasma contains O6+ to ionize. In
practice, the most sensitive range for ionization is usually close to the
threshold, where calculations and measurements are most difficult.

Figure 3: [Left] Relative changes in O6+ ion population at selected
temperatures from a change in the O6+ ionization rate ranging from a -20%
decrease to a +20% increase. No change is seen at temperatures at or below the
peak of the ionic concentration (and so are not plotted), as there the ionic
concentration is dominated the balance between recombination with ionization
from the lower ion O5+. At temperatures above peak where the ionic
concentration has dropped to 10% or 1%, changes to the ionization rate
directly affects the ionic concentration. [Right] Same for O7+ due to the
perturbation in O6+ ionization rate.

Figure 4 plots the perturbed CSD for all O and Fe atoms from Monte Carlo
calculations and a flat distribution of errors using the systematic
uncertainties for each ion in Tables 1 and 2. For ions with multiple errors
reported, we employ the smallest experimental error measured. For ions with no
error listed, we average the errors from adjacent ions and for ions below O4+
and Fe7+, we use the errors of O4+ and Fe7+. The top panel of Figure 4 shows
the original CSD with the range of perturbed CSD, while the bottom panel shows
the error in ionic concentration for temperatures where the original ionic
concentration is greater than $10^{-3}$. Systematic uncertainties can result
in an error on the ionic concentration up to a factor of two for certain ions
at temperatures where the ion is of low concentration. ‘Transition’ ions such
as Li-like O vii or Na-like Fe xvi demonstrate strong sensitivity to
temperature variations, making them often used for astrophysical analysis of
spectra. Similarly the Fe L-shell ions (Fe xvii \- Fe xxiv) provide a powerful
diagnostic due to their dense and close-spacing in temperature. Unfortunately,
this shows that CSD calculations for these ions also suffer disproportionately
from systematic errors.

Figure 4: [Left] CSD from perturbing all O rates with the minimum systematic
errors listed in Tables 1 and 2 from 100 Monte Carlo runs. [Right] Same, for
Fe rates.

HADP2018 considered a range of different atomic data compilations to estimate
the underlying errors in the data. Figure 5 shows the comparison of CSD for
select O and Fe ions from commonly used compilations with the perturbed CSD
range using the same errors employed for Figure 4, itself based on the Bryans
et al. (2009) rates. The difference between the various data sets generally
lies well within error range. O4+ has the greatest difference in ionic
concentration, with older data sets of Mazzotta et al. (1998) and Arnaud &
Rothenflug (1985) lying outside the CSD calculated using the minimum
experimental errors for these ions. Additionally, the temperature of peak
ionic concentration is slightly shifted for these two data sets. A visual
comparison shows that recent theoretical data sets are in excellent agreement,
in fact better than the systematic uncertainties in the underlying laboratory
measurements. This suggests that future work needs to include more precise
experimental data rather than just theoretical calculations.

Figure 5: Comparison of CSD for selected O [left] and Fe ions [right] from
different data sets with the perturbed CSD range from minimum systematic
uncertainties shown in Figure 4.

####  2.2.1 Ionic Concentration Calculations

A common use of CSDs is to determine the relative abundance of an element
based on emission or absorption from a single ion. More complex calculations
will involve multiple ions, potentially including a combination of plasma
temperatures. In the context of analyzing spectral data from such complex
plasmas, the exact use of our results is beyond the scope of this paper and
will require future work. For now, we can use our Monte Carlo technique to
determine which individual ions are most sensitive to uncertainties even
though we may not know the total errors on the ionization and recombination
rates.

Figure 6: [Left] Sensitivity of peak ionic concentration for all O ions from
perturbing all rate coefficients as a function of the fractional change from
1000 Monte Carlo calculations. The dashed black line shows a sample curve with
unit slope, showing that the overall change in peak ionic concentration for
ions is lower than the errors on the total rates. [Right] Sensitivity of O5+
ionic concentration over all temperatures where the ionic concentration is
greater than 10e-3 to perturbing all rate coefficients between 5 and 40% from
the same 1000 Monte Carlo runs. The vertical line is drawn at the temperature
of the ion’s peak concentration.

As O5+ is both astrophysically relevant and the ion most sensitive to
perturbations in rate coefficients, we investigate its peak ionic
concentration sensitivity by perturbing only its ionization or recombination
rate coefficient and then perturbing all oxygen rate coefficients via Monte
Carlo calculations. We find that the change in peak ionic concentration is
generally independent of which rate coefficient is varied, i.e. changing
either the ionization or recombination rate coefficient yields the same
fractional change in peak ionic concentration. Experimental recombination rate
coefficients, however, have a larger systematic error on average than
ionization rate coefficients, so once both rate coefficients get perturbed,
the change in peak ionic concentration will be greater than the change from
varying an individual rate coefficient.

Figure 6 ([Left]) plots the fractional change in peak ionic concentration for
each oxygen ion as a function of the maximum assumed range used in the Monte
Carlo calculations, taken to be the same for both ionization and
recombination. Ions with large peak ionic concentrations (e.g. neutral O or
O6+) naturally show small changes in this plot, since the peak is near unity
in all cases. Less abundant transition ions such as Li-like ion O5+, however,
have changes that amplify any systematic uncertainties in the atomic rates.
For O5+ specifically, a change of 20% on rate coefficients can lead to less
than a 10% change in ionic concentration at the temperature O5+ peaks at,
while it results in changes $\sim$ 17% at high temperatures and up to the full
20% at low temperatures (Figure 6 [Right]).

####  2.2.2 Temperature Measurements via CSD

Another common use of CSDs is to measure a plasma temperature by considering
the ratio of lines from adjacent ions, such as O vii and O viii. Tables 3 and
4 list the implied temperature shifts in keV for O and Fe respectively from
perturbing all rate coefficients with typical systematic errors. The
temperature shifts listed are measured from the temperatures where the ionic
concentration in equilibrium is half of the peak value, both on the rising and
falling sides [see Figure 7]. These shifts can be treated as a type of
systematic error on any temperature measurement made from these ions. We
determined the temperature shifts from estimating the plasma temperature at
half peak ionic concentration after performing Monte Carlo CSD calculations.
As they were not determined through spectral fitting, these temperature shifts
should be treated as a calculational exercise of estimating possible
uncertainties in plasma temperature diagnostics due to typical systematics.

Studying the fractional changes in peak ionic concentration of ions over
various magnitudes of perturbations to the rate coefficients is important for
a quantitative suggestion for optimal error estimates. Optimizing temperature-
dependent CSD errors is especially relevant when looking at specific
temperature regions, such as the temperature ranges over which plasmas are
formed through collisional ionization or photoionization.

Figure 7: Temperature shifts (black error bars) for Fe $25^{+}$ from the unperturbed to perturbed CSD from Figure 4 [right] at half peak ionic concentration (the horizontal dashed line). The temperature range plotted is where the ionic concentration is greater or equal to 10%. Table 3: Implied absolute temperature shifts (in keV) and relative temperature shifts (%) from perturbing all O ionization and recombination rates with the minimum systematic errors of Tables 1 and 2 at half peak ionic concentration on the unperturbed rising ($T_{rise}$) and falling ($T_{fall}$) CSD curve. Ion | $T_{rise}$ | Rising Shift | $T_{fall}$ | Falling Shift
---|---|---|---|---
| keV | keV | % | keV | keV | %
O4+ | 0.0148 | 0.0004 | 2.7 | 0.0260 | 0.0006 | 2.3
O5+ | 0.0209 | 0.0006 | 2.9 | 0.0331 | 0.0008 | 2.4
O6+ | 0.0271 | 0.0004 | 1.5 | 0.1704 | 0.0052 | 3.1
O7+ | 0.1398 | 0.0052 | 3.7 | 0.2979 | 0.0078 | 2.6
O8+ | 0.2471 | 0.0063 | 2.5 | – | – | –
Table 4: Implied absolute temperature shifts (in keV) and relative temperature shifts (%) from perturbing all Fe ionization and recombination rates with the minimum systematic errors of Tables 1 and 2 at half peak ionic concentration on the unperturbed rising ($T_{rise}$) and falling ($T_{fall}$) CSD curve. Ion | $T_{rise}$ | Rising Shift | $T_{fall}$ | Falling Shift
---|---|---|---|---
| keV | keV | % | keV | keV | %
Fe7+ | 0.0230 | 0.0009 | 3.9 | 0.0572 | 0.0031 | 5.4
Fe8+ | 0.0402 | 0.0016 | 4.0 | 0.0944 | 0.0036 | 3.8
Fe9+ | 0.0678 | 0.0030 | 4.4 | 0.1238 | 0.0029 | 2.3
Fe10+ | 0.0881 | 0.0030 | 3.4 | 0.1456 | 0.0033 | 2.3
Fe11+ | 0.1065 | 0.0029 | 2.7 | 0.1652 | 0.0026 | 1.6
Fe12+ | 0.1236 | 0.0026 | 2.1 | 0.1850 | 0.0039 | 2.1
Fe13+ | 0.1399 | 0.0026 | 1.9 | 0.2094 | 0.0065 | 3.1
Fe14+ | 0.1563 | 0.0028 | 1.8 | 0.2526 | 0.0180 | 7.1
Fe15+ | 0.1777 | 0.0048 | 2.7 | 0.3935 | 0.0266 | 6.8
Fe16+ | 0.2090 | 0.0065 | 3.1 | 0.6231 | 0.0231 | 3.7
Fe17+ | 0.4092 | 0.0178 | 4.3 | 0.8572 | 0.0344 | 4.0
Fe18+ | 0.5605 | 0.0205 | 3.7 | 1.0023 | 0.0472 | 4.7
Fe19+ | 0.6786 | 0.0340 | 5.0 | 1.1325 | 0.0664 | 5.9
Fe20+ | 0.7820 | 0.0421 | 5.4 | 1.2741 | 0.0749 | 5.9
Fe21+ | 0.8814 | 0.0534 | 6.1 | 1.4660 | 0.0866 | 5.9
Fe22+ | 0.9894 | 0.053 | 5.4 | 1.8313 | 0.1393 | 7.6
Fe23+ | 1.1337 | 0.0478 | 4.2 | 3.2656 | 0.2950 | 9.0
Fe24+ | 1.4706 | 0.0586 | 4.0 | 7.6136 | 0.3788 | 5.0
Fe25+ | 4.7617 | 0.2378 | 5.0 | 23.8883 | 2.7231 | 11.4
Fe26+ | 12.7911 | 0.9198 | 7.2 | – | – | –

##  3 Line Uncertainty Calculation

###  3.1 Method

Unlike an element’s CSD with only $O(Z)$ relevant values, determining ionic
line emission requires calculating the ion’s energy level population,
potentially including thousands or millions of individual calculations. In
most cases, no laboratory measurement has ever been made so error estimates
rely on comparing different theoretical methods against each other - as done
in HADP2018 - or extrapolating from the few lab results that do exist. Even if
this experimental data were available, theory suggests that some rates will be
correlated or anti-correlated (Loch et al., 2013). Thus creating a table of
line emissivities and errors is both impractical and unusable for quantitative
analysis. Nonetheless, observers require some estimate of how uncertainties in
atomic data affect models of plasma emission.

For a given ion, the emissivity $\Lambda_{ij}(T_{e})$ of a line with
wavelength $\lambda_{ij}$ due to a transition from level $i\rightarrow j$ is
simply $\Lambda(T_{e})=A_{ij}p_{i}(T_{e})$ where $A_{ij}$ is the radiative
transition rate and $p_{i}(T_{e})$ the population of level $i$ at electron
temperature $T_{e}$. Calculating $p_{i}(T_{e})$, however, requires considering
all of the processes that populate and depopulate that level; see Foster et
al. (2012) and http://www.atomdb.org/physics.php for details. To probe the
non-linear dependence of line emissivities to atomic rate coefficients, we
employ variableapec to perturb a single rate coefficient, construct a new
collisional radiative matrix, and recalculate line emissivities. We then study
how line emissivities vary over a range of perturbation magnitudes and
temperatures to understand where atomic uncertainties matter most. HADP2018
estimated the perturbation amount from differences between atomic codes in
collisional excitation rates and transition probabilities. We instead estimate
a perturbation amount associated with the uncertainty from atomic calculations
of rate coefficients or experimental errors. Note that we quantify the effect
of atomic uncertainties on line emissivities by calculating changes to the
line emissivity. We use the word “change” rather than “error” to reference
line emissivities (or other values) we have calculated theoretically, as we
employ the word “error” in reference to measured errors (e.g. errors on CSD
rates or experimental errors on transition rates). Uncertainties are used
throughout to mean general unknowns in the fundamental atomic data.

The variableapec code will perturb rate coefficients for collisional
excitation to an individual level ($i$) or radiative transition rates $A_{ij}$
at the temperature where the ion’s concentration peaks. It will then determine
the fractional changes in emissivity for all transitions from level $i$ as
well as any others affected. The code outputs the fractional change in
emissivity d$\epsilon$/$\epsilon$, i.e. the averaged fractional change in
emissivity calculated from the positive and negative error on the rate. It
also calculates the partial derivatives of the resulting line emissivities
($\epsilon$) with respect to the change in the rate $d\epsilon/dr$. This value
shows how sensitive each line is to uncertainties in the underlying rates.

For this phase of the study, we considered the low density case only. At
densities where the electronic collision cross section competes with the
radiative transition rate, it is necessary to also consider perturbations to
the $A$ value. We found that at these larger densities such perturbations can
affect thousands of lines by an observable amount and thus we leave this
project for a future paper.

###  3.2 Results

The challenge of calculating line emissivities in a collisional plasma varies
substantially depending upon the line. An electric dipole transition involving
low-lying excited levels, such as the He-like resonance line
$1s2p{}^{1}$P${}_{1}\rightarrow 1s^{2}{}^{1}$S0, may have an emission entirely
dominated by direct collisional excitation from the ground state followed by a
radiative transition to ground. As typical $A$ values for such transitions are
large, at densities below $10^{8}$ cm-3, the line emissivity uncertainty is
governed by this one excitation cross section. More complex situations,
however, are common as an excited level may be populated by excitation
ionization, recombination, and cascades as well as direct excitation.
Determining experimental uncertainties for each of these rates is impractical,
and in most cases even theoretical uncertainties are estimated simply by
comparing the values obtained from using different methods.

We therefore focus on understanding the variation in the final emissivities as
the underlying excitation or radiative rates are changed, expressed as partial
derivatives. The variableapec code can calculate these for any transition in
the AtomDB; we provide examples here for select transitions in the Fe xvii and
Fe xix line complexes. Despite the fact these key transitions in particular
have been poorly measured historically, Fe xvii and Fe xix have strong lines
that are commonly used temperature diagnostics (Gu et al., 2020). We perturb
the atomic data driving strong lines in Fe xvii and Fe xix by 5% and 20% and
compare fractional emissivity changes. At low density, most line emissivities
are not sensitive to perturbations in $A$ value. For example, while
perturbations to the $A$ value of strong Fe xvii lines do not affect any other
lines, varying the $A$ value of two Fe xix lines by 20% affects those lines by
only 2% but two other X-ray lines by 20%. However, a 20% perturbation to the
direct excitation rate of a strong line at the temperature of peak ionic
concentration can increase both that line’s emissivity and others up to 20%.

We varied the direct excitation rate (from ground) for a number of Fe xvii
lines (3C, 3D, 3E, 3F, M2, 3G) and found that, out of this group of lines,
only the line whose rate was perturbed changed. Varying the rates by 5%
changed the M2, 3G, and 3F lines by 1% or less and the 3E, 3D, and 3C lines by
4-5%, and proportionally larger for 20% changes. For the 3E, 3D, and 3C lines,
a 20% perturbation to the direct excitation rates also led to 16-20%
variations in some strong EUV (100-500Å), although no X-ray lines were
significantly affected.

We then perturbed the direct excitation rates (again from ground) by 20% for
four strong lines of Fe xix: O19 (53$\rightarrow$1), O24 (68$\rightarrow$1),
O25 (71$\rightarrow$1), O26 (74$\rightarrow$1), using the nomenclature of
Brown et al. (2002) with the AtomDB v3.0.9 levels in parentheses. As with Fe
xvii, the emissivity of these strong lines only changes when their own direct
excitation rate is varied and by 17-20%.

The partial derivative analysis reveals additional lines affected by a
perturbation to the direct excitation rate of a single line. To investigate
the correlation of line emissivity errors, we study the effect of various
magnitudes of errors on emissivity changes over a range of temperatures. For
Fe xvii, we found that the EUV line at 263.63 Å is the strongest line impacted
by perturbations to the direct excitation rate of the 3E line
(17$\rightarrow$1) (Figure 8 [Left]). At peak ion temperature, varying the
direct excitation rate of the 3E line by 20% results in a 17% change in the
emissivity (1.79$\times 10^{-13}$ ph cm3 s-1) of the 17$\rightarrow$6
transition (263.63 Å). We perturb the 3E line by various magnitudes at the
designated five temperatures of interest (see Figure 1) to study the
correlation between these two lines. The fractional change in emissivity is
greatest at low temperatures where the ionic concentration is between 1-10%.
The maximum difference in fractional change in emissivity across the
temperature range increases with perturbation magnitude. We see that the
17$\rightarrow$6 line emissivity is then sensitive to both temperature and any
uncertainties in the 3E direct excitation rate.

We repeat the partial derivative analysis for Fe xix to look for any X-ray
lines that may be affected by perturbations to the direct excitation rates of
key Fe xix lines. We find that the O20 line from Brown et al. (2002)
(74$\rightarrow$4) is affected by an observable amount with perturbations to
the direct excitation rate of O25 (74$\rightarrow$1). Unlike the Fe xvii line
at 263.63 Å, this line is not temperature sensitive: the fractional change in
the emissivity of O20 is approximately equal to the error on the direct
excitation rate of O25 for all temperatures. This is a subtle but important
detail in that it shows that the O20 line is primarily populated by direct
excitation from the ground level. If recombination or ionization contributed
more to the line emissivity, the temperature of the plasma would be more
significant. Conversely, ionization and recombination of Fe xvi and Fe xviii
into Fe xvii contributes to the 17$\rightarrow$6 line emissivity, so the
plasma temperature factors significantly into the fractional change in
emissivity.

This is evident in Figure 8 [Right], where we show the Fe xvii
17$\rightarrow$6 line sensitivity to the 17$\rightarrow$1 direct excitation
rate as well as the sensitivity from employing the maximum possible CSD values
due to the typical systematics listed in Tables 1 and 2 in emissivity
calculations. We find that the CSD error, i.e. the contribution of the
perturbed CSD to the increase in fractional change in emissivity from the non-
perturbed case, is greatest at temperatures where the ionic concentration is
between 1-10% and smallest at the temperature of peak ionic concentration.
This matches expectations given the sensitivity of ionic concentrations to
perturbations to the CSD shown in Figure 4.

Figure 8: [Left] Fe xvii 17$\rightarrow$6 sensitivity over various
temperatures at low density due only to changes to the 17$\rightarrow$1 direct
excitation rate. [Right] Same (dashed) compared to the sensitivity from also
including the maximum perturbed CSD due to systematics (solid).

We also investigate the effect of atomic uncertainties on density diagnostics
for line ratios, such as the density-sensitive Fe xxii I(11.92 Å)/I(11.77 Å)
line ratio as studied in Mauche et al. (2003) and Miller et al. (2006). We
changed the $A$ values of the two lines individually by 10%, 20% and 30% and
recalculated the final line ratio, finding a resulting change to the final
line ratio that is about a sixth of the perturbation magnitude in all cases
[see Figure 9]. Unlike other temperature-sensitive line ratios, the final
fractional change is not density-sensitive. Perturbing these two lines,
however, can result in a relative $\pm$ error on density diagnostics that is
asymmetric. Looking at the density where the line ratio equals 4, a 10%
perturbation gives a (-4.6%, +3.8%) error on the unperturbed density of
$4.86\times 10^{13}$ cm-3, while 20% and 30% perturbations give (-10.3%,
+6.9%) and (-17.7%, +9.6%) errors respectively.

Figure 9: The Fe xxii I(11.92 Å)/I(11.77 Å) line ratio at $T_{e}=12.8\times
10^{6}$ K for three different changes to the $A$ value of the transitions. The
error bars on the bottom panel show the uncertainty of the density diagnostic
for when the line ratio equals 4.

##  4 Discussion and Conclusions

The problem of providing EUV/X-ray spectroscopists with practical error
estimates has been long-standing and will only become more acute as newer
facilities come online with higher resolution and more sensitivity.
Unfortunately, no theoretical method yet exists that can generate reliable
error estimates for atomic calculations, including the effect of correlations.
The goal of this paper is not to solve the problem but rather to provide
observers with practical methods to estimate the uncertainty involved in a
specific calculation. We also hope to motivate further and more accurate lab
measurements of atomic rates, a problem that requires significant and long-
term funding. As our results show, it is important to consider all the
parameters that contribute to an analysis, not merely statistical errors on
the spectrum.

Our primary focus was uncertainties in CSD calculations as this involves a
limited number of rates, most of which have at least some experimental
measurements. In conducting a literature search for rate coefficients and
systematic uncertainties, we found that while a number of papers explicitly
detail the components of their systematic errors and the justification for
each value, more than a few provide little to no such information. We admit to
being surprised, however, that even using only the published systematic errors
showed that most modern CSD calculations have converged to lie within their
bounds. It will be difficult to make significant improvements, then, without
more accurate measurements for theory to attempt to match. Of course, this
does not test the effect of processes that have been so far excluded from the
CSD calculation entirely, such as electron-impact multiple-ionization effects
(Hahn et al., 2017). Direct measurements of CSDs as a function of temperature
would be most useful to ensure the CSD calculations contain all the key
processes.

We have also investigated the uncertainties involved in the calculation of
emission line fluxes. The variableapec code allows users to recreate
calculations with a range of perturbations in order to find the lines most
sensitive to atomic uncertainties and which uncertainties are correlated. The
code will also show the different components that enter into the line flux
beyond simple direct excitation, such as ionization, recombination, and
cascade effects. While we only focused here on lines sensitive to
perturbations in excitation rates, emission and absorption lines will also be
affected by changes to $A$ values, especially when using lines as density
diagnostics.

Our methods aim to address observers’ needs by explicitly showing the effect
individual uncertainties have on a final error. In some cases, it can also
show how specific line ratios are less affected by correlations or errors than
others, aiding in analysis even if direct laboratory measurements are not
available. This will also motivate identification of specific atomic data that
requires more precise lab measurements or theoretical atomic calculations for
better accuracy. Ultimately, optimizing spectral modeling to extract the
maximum from an observed spectrum will involve determining which atomic data
are ill-measured, the ions most sensitive to atomic uncertainties, and the
temperature regions where uncertainties most greatly impact observable plasma
features.

We hope the variableapec tools developed for this paper will be useful for
people to calculate their own estimate uncertainties given their own
interpretations of underlying experimental uncertainties [see Appendix for
examples of routine commands]. We have completed the CSD error estimates for
Fe and O and made these available as part of the Github distribution. We
invite contributions to the open-source project to add other ions and elements
to the database.

The authors gratefully acknowledge support from NASA Astrophysics Grant
80NSSC18K0409.

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The tools and CSD error estimates used for this paper are available on the
GitHub distribution. variableapec requires the pyatomdb package and Python 3.
Below are examples of how to access the CSD error estimates we have completed
and reproduce some of the figures in this paper using routines that can be
used for other elements and lines.

The systematic uncertainties we have gathered are listed in a dictionary with
the element number Z and a dictionary of ionization and recombination errors
for each ion as key-value pairs. Currently we have only completed CSD error
estimates for O and Fe. We welcome any contributions to add errors for other
ions and elements.

>>> oxygen = variableapec.systematics[8]

>>> iron = variableapec.systematics[26]

To perturb all rate coefficients via Monte Carlo calculations like Figure 4,
supply the element number, ionization and recombination errors, and the number
of runs (default is 100):

>>> Z, max_ionize, max_recomb = 26, iron[‘ionize’], iron[‘recomb’]

>>> variableapec.monte_carlo_csd(Z, max_ionize, max_recomb, runs=500)

To compare the rate of change in ionic concentration from varying a single
rate coefficient with multiple perturbation magnitudes like Figure 3, supply
the element, ion, type of rate coefficient to vary, and a list of positive
rate errors.

>>> Z, z1, rate_type, errors = 8, 7, ‘i’, [0.05, 0.1, 0.15, 0.2]

>>> variableapec.csd_slope(Z, z1, rate_type, errors)

To perturb all rate coefficients via Monte Carlo calculations and calculate
the change in peak ionic concentration as a function of error on rate
coefficients like Figure 6 [left], supply the element and the fractional
errors to perturb all rate coefficients by. If a z1_interesting is given, the
routine will create a second plot from the same Monte Carlo calculations of
the change in ionic concentration of that ion over temperatures where the
unperturbed concentration is greater than $10^{-3}$ (see Figure 6 [right]). If
no z1_interesting given, only the change in peak ionic concentration will be
plotted for z1_plot, which is a list of all ions by default:

>>> Z, errors, z1_interesting = 8, [0.05, 0.1, 0.15, 0.2, 0.25, 0.3], 6

>>> variableapec.ion_sensitivity(Z, errors, z1_interesting=z1_interesting)

To check the sensitivity of a line to a perturbation in the direct excitation
rate or $A$ value of another transition, supply the element, ion, sensitive
transition, rate type and rate errors, the transition to be varied, and the
temperatures or densities to look at. The default density is 1 cm-3 and the
default temperatures are the five temperatures of interest [see Figure 1].

To check the sensitivity due to only changes in direct excitation rate like
Figure 8 [left]:

>>> Z, z1, up, lo, rate_type, rate_errors, transition_list = 26, 17, 17, 6,
‘exc’, [0.05, 0.1, 0.15, 0.2, 0.25, 0.3], [(17,1)]

>>> variableapec.line_sensitivity(Z, z1, up, lo, rate_type, rate_error,
transition_list)

To check the sensitivity to changes in direct excitation rate in addition to
CSD errors like Figure 8 [right]:

>>> Z, z1, up, lo, rate_type, rate_errors, transition_list, max_ionize,
max_recomb = 26, 17, 17, 6, ‘exc’, [0.05, 0.1, 0.15, 0.2, 0.25, 0.3],
[(17,1)], iron[‘max_ionize’], iron[‘max_recomb’]

>>> variableapec.line_sensitivity_csd(Z, z1, up, lo, rate_type, errors,
transition_list, max_ionize, max_recomb)