11institutetext: INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica di
Milano, via A. Corti 12, 20133 Milano, Italy
11email<EMAIL_ADDRESS>22institutetext: Dipartimento di Fisica,
Università degli Studi di Milano, via G. Celoria 16, 20133 Milano, Italy
33institutetext: INAF – Osservatorio Astronomico di Brera, via E. Bianchi 46,
23807 Merate, Italy

# Does the Fe L-shell blend bias abundance measurements in intermediate-
temperature clusters?

Giacomo Riva 1122 Simona Ghizzardi 11 Silvano Molendi 11 Iacopo Bartalucci 11
Sabrina De Grandi 33 Fabio Gastaldello 11 Claudio Grillo 1122 Mariachiara
Rossetti 11

(Received XXX; accepted XXX)

In intermediate-mass ($M=2-4\times 10^{14}$ M⊙, or equivalently $T=2.5-4.5$
keV) galaxy clusters, abundance measurements are almost-equally driven by Fe K
and L transitions, at $\sim 6.7$ keV and $0.9-1.3$ keV, respectively. While
K-shell-derived measurements are considered reliable, the resolution of the
currently available instrumentation, as well as our current knowledge of the
atomic processes, makes the modelling of the L-line complex challenging,
resulting in potential biases for abundance measurements. In this work, we
study with unprecedented accuracy systematics related to the modelling of the
Fe L-line complex that may influence iron-abundance measurements in the
intermediate-mass range. To this aim, we select a sample of three bright and
nearby galaxy clusters, with long XMM-Newton observations available and
temperature in the $2.5-4.5$ keV range. We fit spectra extracted from
concentric rings with APEC and APEC+APEC models, by alternatively excluding
the L and K bands, and derive the fractional difference of the metal
abundances, $\Delta Z/Z$, as indication of the consistency between K- and
L-shell-derived measurements. The $\Delta Z/Z$ distribution is then studied as
a function of the cluster radius, ring temperature and X-ray flux. The
L-blend-induced systematics, measured through an individual fit of each MOS
and pn spectrum, remain constant at a $5-6\%$ value in the whole $2.5-4.5$ keV
temperature range. Conversely, a joint fit of MOS and pn spectra leads to a
slight excess of $1-2\%$ in the above estimate. No significant dependence on
the ring X-ray flux is highlighted. The measured $5-8\%$ value indicates a
modest contribution of the systematics to the derived iron abundances, giving
confidence for future measurements. To date, these findings represent the
best-achievable estimate of the systematics in analysis, while future
microcalorimeters will significantly improve our understanding of the atomic
processes underlying the Fe L emissions.

###### Key Words.:

X-rays: galaxies: clusters - galaxies: clusters: intracluster medium -
galaxies: abundances

##  1 Introduction

Table 1: Cluster properties and information of the XMM-Newton observations.

Cluster | Obs. ID | MOS Exposure | pn Exposure | $R_{500}$ | $T$ | Flux ($0.1-2.4$ keV) | $z$
---|---|---|---|---|---|---|---
|  | (ks) | (ks) | (Mpc) | (keV) | (erg s-1 cm-2) |
Perseus | $0305780101$ | 118 | 94 | $1.3$ | $3.5-4.5$ | $9.61\times 10^{-10}$ | $0.018$
2A 0335+096 | $0147800201$ | 79 | 76 | $0.9$ | $2.4-3.5$ | $9.16\times 10^{-11}$ | $0.036$
Sérsic 159-03 | $0147800101$ | 86 | 77 | $0.8$ | $2.5-2.7$ | $2.49\times 10^{-11}$ | $0.056$
111”MOS Exposure” and ”pn Exposure” are the effective exposure times of MOS
and pn cameras, after the soft-proton cleaning. Estimates for $R_{500}$ are
taken from Giacintucci et al. (2019) for Perseus and 2A 0335+096 and from Sun
et al. (2009) for Sérsic 159-03. In the Table we list, for each cluster, the
temperature ranges covered by the various regions for which we extracted and
fitted the spectra. These temperatures lie in the range we are interested in.
Regions with temperature higher than $4.5$ keV are not considered for Perseus,
as explained in Sect. 3.2.1. Perseus flux and redshift are taken from Ebeling
et al. 2002; for both 2A 0335+096 and Sérsic 159-03, we refer to de Plaa et
al. 2007 and Sanders et al. 2011 for the flux and the redshift, respectively.

During the last decades, several studies have focussed on the measurement of
the metallicity in galaxy clusters and groups (see Mernier et al. 2018 and
Gastaldello et al. 2021 for a review). Since they are the largest structures
to have clearly decoupled from the Hubble flow, studies of metal abundance in
the Intra-Cluster Medium (ICM) provide important clues on the cluster
formation and evolution processes, as well as on the enrichment of the
Universe. Since the $1950$’s, when the first complete theories of stellar
nucleosynthesis (Cameron 1957; Burbidge et al. 1957) were developed, it has
been generally accepted that most of the chemical elements heavier than helium
are produced in stars at different stages (particularly at the end) of their
lifetimes (see Nomoto et al. 2013 for a review). Although the synthesis
processes of the heavy elements in the Universe are known fairly well, the
general picture of the enrichment history in galaxy clusters and groups is
still far from being completely clear.

Galaxy-cluster outskirts have recently enjoyed particular attention, since
they provide direct information on the ICM (its formation, accretion and
contribution to the growth of large-scale structures; Walker et al. 2019).
However, iron-abundance measurements are not easy to perform in clusters,
especially in their outermost, low-surface-brightness regions. Reliable
measurements have typically been carried out up to $R\sim
0.6\,R_{500}$222$R_{\Delta}$ is the radius inside which the cluster mass
density is $\Delta$ times the critical density of the Universe
$\rho_{\text{c}}(z)=3H_{0}^{2}E(z)^{2}/8\pi G$, where
$E(z)=\sqrt{\Omega_{\text{m}}(1+z)^{3}+\Omega_{\Lambda}}$ (in a flat Universe)
and $z$ is the redshift. $M_{\Delta}$ is then defined as the total mass within
$R_{\Delta}$., sampling about one third of the total gas mass in clusters (as
discussed in Molendi et al. 2016) and only on archival samples without
guarantee of representativity (Mernier et al. 2017; Leccardi & Molendi 2008a;
Lovisari & Reiprich 2019). In this context, a very-large programme on XMM-
Newton (the XMM-Newton Cluster Outskirts Project, X-COP; Eckert et al. 2017)
has been carried out to advance our knowledge of the physical conditions in
the cluster outskirts, pushing the analyses up to $R_{500}$, for a sample of
12 massive ($M_{500}>4\times 10^{14}$ M⊙) galaxy clusters selected in the
Planck all-sky survey of Sunyaev-Zeldovich (SZ, Sunyaev & Zeldovich 1972)
sources (Planck Collaboration et al. 2014; Planck Collaboration et al. 2016).

Within this programme, Ghizzardi et al. (2021a) derived for the first time
robust metallicity profiles up to $R_{500}$ for a sample of galaxy clusters
and identified a main systematic error that affects iron-abundance
measurements in the cluster outskirts. In the central regions of massive
clusters (as for the X-COP sample, where temperatures are higher than $4$
keV), iron-abundance measurements are mostly driven by the Fe K transitions
(mainly by Fe XXV and Fe XXVI K$\alpha$ lines) at $\sim 6.7$ keV in the ICM
spectrum, while the Equivalent Width (EW) of the Fe L-line
complex333Hereafter, we will refer to the just-mentioned transitions simply as
K$\alpha$ and L-shell, respectively. ($0.9-1.3$ keV) is smaller (see Fig. 3 of
their work). However, going towards the cluster outskirts, the EW of the
K$\alpha$ reduces with respect to the total continuum, which is dominated by
the background emission, and the L-shell assumes a primary role in the fit
(see Fig. 4 of their work). The L-line-derived abundance measurements in such
high-background regime were found to be really puzzling in some cases, making
the reliability of the L-shell in abundance measurements questionable. The
exclusion of the L-shell ($0.9-1.3$ keV) energy band was found to lead to
more-reliable abundance profiles, renewing some old but still relevant
questions: are the L-shell-derived abundance measurements unreliable in
general, or do they become so only in specific and complex situations?

At the spectral resolution of currently available CCD cameras, the single
lines that build up the L-shell (see Hwang et al. 1997; Mazzotta et al. 1998;
Gu et al. 2019; Gu et al. 2020; Heuer et al. 2021) are not resolved and are
actually seen as a blend (see Fig. 1 in Gastaldello & Molendi 2002). The large
number of Fe emission lines peaking in the $0.9-1.3$ keV energy range, as well
as our current knowledge of the atomic processes, makes the modelling of the
L-shell challenging, resulting in potential biases. Conversely, the Fe
K$\alpha$ at $\sim 6.7$ keV is relatively well understood, especially after
the data release of the SXS spectrum of Perseus (Hitomi Collaboration et al.
2018). In the late 1990’s, Hwang et al. (1997) quantified possible L-shell-
induced systematics, by comparing K$\alpha$\- and L-blend-derived iron
abundances for a few galaxy clusters (with ASCA observations) in the $2-4$ keV
temperature range. Their research did not highlight any significant
discrepancies between the K- and L-derived measurements, within the $90\%$
confidence level. However, the large error bars of their results do not allow
a precise assessment of the reliability of the L-shell-derived abundance
measurements.

Currently, we can rely on satellites with advanced characteristics (e.g. the
large effective area of XMM-Newton) and also on atomic codes (e.g.
AtomDB444http://www.atomdb.org., implemented in the APEC model as part of the
XSPEC555https://heasarc.gsfc.nasa.gov/xanadu/xspec. fitting package; Smith et
al. 2001; Foster et al. 2012; SPEXACT, available in the
SPEX666https://www.sron.nl/astrophysics-spex. fitting package; Kaastra et al.
1996; Kaastra et al. 2018) which describe the atomic processes that produce
the emission lines observed in the ICM spectra with greater precision than in
the past. These instruments allow us to study with high accuracy systematics
related to the L-shell that affect iron-abundance measurements. In this work,
we compare K$\alpha$\- and L-shell-derived abundance measurements for a sample
of galaxy clusters observed with XMM-Newton and with temperatures in the
$2.5-4.5$ keV range. In this range, the EWs of the K$\alpha$ and L-shell (at
these temperatures primarily due to Fe XXIII and XXIV) emissions are
comparable, giving us the possibility to test the atomic codes underlying
abundance measurements.

The paper is organised as follows: in Sect. 2, we present our sample and the
data reduction; Section 3 is dedicated to the X-ray data analysis: here we
define the strategies that are used to estimate the L-shell-induced
systematics; in Sect. 4 we analyse a possible dependence of the systematics on
the temperature and the flux, while Section 5 is dedicated to the discussion
of the results; finally, in Sect. 6 we summarise our main findings.

Throughout the paper, we assume a $\Lambda$ cold-dark-matter cosmology with
$H_{0}=70$ km s-1 Mpc-1, $\Omega_{\text{m}}=0.3$, $\Omega_{\Lambda}=0.7$ and
$E(z)=\sqrt{\Omega_{\text{m}}(1+z)^{3}+\Omega_{\Lambda}}$ for the evolution of
the Hubble parameter. The solar abundance table is set to Asplund et al.
(2009), while all the quoted errors hereafter are at the $1\sigma$ confidence
level.

##  2 Selection of the sample and data processing

In this work, we restrict our analysis to a sample of nearby and bright
clusters, with i) long XMM-Newton observations ($\gtrsim 120$ ks), ii) good
statistics, iii) regions with temperature in the $2.5-4.5$ keV range and iv)
high source-to-background (S/B) ratio, to focus on the contribution to the
systematics related to the underlying atomic model in the above temperature
range and to neglect contributions from the background contamination. To this
aim, we select the following galaxy clusters: Perseus, 2A 0335+096 and Sérsic
159-03 (Table 1).

For each cluster, we only consider the longest available observation in the
XMM-Newton Science Archive777https://www.cosmos.esa.int/web/xmm-newton/xsa.
(XSA), to avoid additional systematics deriving from a combination of
different observations of the same cluster. The identification numbers of the
selected observations are listed in Table 1, together with the cluster
redshifts, temperatures and X-ray fluxes in the $0.1-2.4$ keV band. Given the
redshifts of the three clusters (see Table 1), $1^{\prime}$ corresponds to 22
kpc, 43 kpc and 65 kpc for Perseus, 2A 0335+096 and Sérsic 159-03,
respectively.

X-ray data are reduced using the XMM-Newton Science Analysis
Software888https://www.cosmos.esa.int/web/xmm-newton/sas. (SAS). The main
steps of the data processing are the following:

  * •

we start by producing calibrated event files for each observation (with
emchain and epchain tasks);

  * •

we extract light curves in the hard ($10-12$ keV for MOS detectors and $10-13$
keV for pn) band by using a $100$-s binning. We filter out time intervals
affected by soft-proton flares by applying the appropriate threshold for each
instrument. For MOS detectors we adopt a threshold of $0.25$, $0.17$ and
$0.16$ cts s-1 for Perseus, 2A 0335+096 and Sérsic 159-03, respectively, while
the threshold is set to $0.70$, $0.43$, $0.42$ cts s-1 for the three pn light
curves;

  * •

following Leccardi & Molendi (2008b), we then reject time periods in which
count rates exceed $3\sigma$ from the median of the light curve in the $2-5$
keV band, in order to eliminate contributions of softer flares. The effective
exposure times of MOS and pn cameras after the soft-proton cleaning are listed
in Table 1;

  * •

finally, we select by eye and excise the bright point sources inside the
regions from which spectra are extracted (see next section).

A characterisation of the local sky background cannot be considered for
Perseus, 2A 0335+096 and Sérsic 159-03, since their thermal emissions fill the
whole XMM-Newton FoV. For this reason and since we are interested in studying
the L-shell systematics in condition of relatively high surface brightness,
where the background subtraction is less critical than for external low-
surface-brightness regions, we use blank-sky fields instead of proceeding with
a more-detailed modelling of the different background components. In Sect. 4.1
we investigate the impact of this choice on our results. The blank-sky fields
for EPIC MOS and pn were produced by Leccardi & Molendi (2008b). We perform a
background rescaling for each observation to account for temporal variations
of the background (Leccardi & Molendi 2008b; Ghizzardi et al. 2014).

##  3 Data analysis

###  3.1 Spectral fitting

We extract spectra from concentric rings (with centre located at the X-ray
emission peak), to derive radial abundance profiles for each cluster. Spectra
are extracted up to $7^{\prime}$, $9^{\prime}$ and $5^{\prime}$ for Perseus,
2A 0335+096 and Sérsic 159-03, respectively (Fig. 1), according to the
following pattern: rings are i) $0.5^{\prime}$-wide within $1^{\prime}$, ii)
$1^{\prime}$-wide up to $5^{\prime}$, and iii) $2^{\prime}$-wide at larger
radii. Given the statistics of the three selected clusters, the above ring
angular dimensions allow to derive temperature and abundance radial profiles
with good precision. As detailed below, the external limit for Perseus is set
by the high temperature at large radii, while in the case of 2A 0335+096 and
Sérsic 159-03 we are limited by the presence of an increasingly dominating
background.

(a)

(b)

(c)

Figure 1: Surface-brightness map (MOS 2 camera) of Perseus (top left), 2A
0335+096 (top right) and Sérsic 159-03 (bottom), extracted in the $0.15-12$
keV band. X-ray point sources selected during the data reduction are excised.
Rings from which the X-ray spectra are extracted are shown in magenta. Given
the redshifts of the three clusters (see Table 1), $1^{\prime}$ corresponds to
22 kpc, 43 kpc and 65 kpc for Perseus, 2A 0335+096 and Sérsic 159-03,
respectively.

Spectra are fitted using XSPEC v.12.9.1p (Arnaud 1996). At first, MOS and pn
spectra of each region are fitted separately. We then perform a joint fit of
the three cameras. The spectra are grouped to ensure a minimum of 25 counts
per spectral channel. Unless otherwise stated, the energy range considered in
the fitting is $0.7-10$ keV. We adopted the following models:

  * •

an absorbed (phabs) single-temperature APEC model (hereafter called ”1T
model”), with temperature, Galactic hydrogen column density $N_{\text{H}}$,
abundance, redshift and normalisation free to vary;

  * •

an absorbed (phabs) double-temperature APEC+APEC model, with two abundances
linked together (hereafter ”2T 1Z model”). Also in this case, temperatures,
Galactic hydrogen column density, abundance, redshift and normalisations are
free to vary.

As further addressed in Sect. 3.2.2, the single-temperature (”1T”) model is
adequate for cluster rings that can be considered nearly isothermal, while the
double-temperature (”2T”) model better describes the data in case of
multiphase plasma. The 2T model is also useful to mimic and describe cases
where multiphaseness is artificial, due to projection or to contamination from
adjacent rings. For this reason we do not correct data for PSF nor deproject
profiles and use the 2T model to take into account these effects.

###  3.2 Estimate of the systematic error

The estimate of the systematic error linked to the Fe L-shell follows three
major steps, that will be addressed in the following subsections:

  1. 1.

we exclude cluster rings whose properties are not adequate for our purpose;

  2. 2.

for each spectrum, we then select the model that better describes the data;

  3. 3.

finally we define the strategies for the estimate of the systematics.

####  3.2.1 Selection of the rings

As already discussed, the aim of this work is the study of systematics related
to the L-shell at intermediate temperatures ($T=2.5-4.5$ keV) and in condition
of negligible background contamination. For this reason, starting from the
general ring selection defined at the beginning of Sect. 3.1, we take the
following measures:

  * •

the innermost central bin ($0^{\prime}-0^{\prime}.5$) is not considered for
any cluster. Projection effects play a significant role in the central
regions, particularly so in the innermost bin, where the temperature gradient
becomes very steep: here neither the 1T nor the 2T 1Z models adequately
describe the spectra in this circumstance. Since our focus is the study of
L-shell-induced systematics rather than modelling the temperature distribution
in such a central region, for each cluster we reject the central bin from the
analysis. Moreover, Perseus and Sérsic 159-03 are experiencing in their cores
a phase of AGN feedback (Gendron-Marsolais et al. 2016; Werner et al. 2011;
Farage et al. 2012), which dominates the X-ray emission in the innermost
region.

  * •

radial bins whose temperature exceeds the range of interest are excluded from
the analysis. This requirement places limits only on Perseus, whose
temperature exceeds $5$ keV beyond $3^{\prime}$;

  * •

radial bins where the background emission is comparable to the source emission
in the K$\alpha$ region ($\sim 6.7$ keV) are excluded from the analysis. This
choice places limits on 2A 0335+096 and Sérsic 159-03, where radii larger than
$5^{\prime}$ and $3^{\prime}$, respectively, are not considered.

After the above considerations, we focus on the spectra extracted from the
following rings:

  * •

$0^{\prime}.5-1^{\prime}$, $1^{\prime}-2^{\prime}$, $2^{\prime}-3^{\prime}$
for Perseus;

  * •

$0^{\prime}.5-1^{\prime}$, $1^{\prime}-2^{\prime}$, $2^{\prime}-3^{\prime}$,
$3^{\prime}-4^{\prime}$, $4^{\prime}-5^{\prime}$ for 2A 0335+096;

  * •

$0^{\prime}.5-1^{\prime}$, $1^{\prime}-2^{\prime}$, $2^{\prime}-3^{\prime}$
for Sérsic 159-031.

####  3.2.2 Selection of the model

In order to select which model provides the best description of the data for
each spectrum, we fit all the spectra with both the 1T and the 2T 1Z models.
As a first step, we qualitatively analyse how the residuals (i.e. the ratio
between the data and the model) vary, using the two models, at high energy
(beyond 5 keV) and around the L-shell and the K$\alpha$ energy bands. The
presence of an excess in the residuals at high energy, as well as a tension
between the residuals around the L-shell and the K$\alpha$, is a clear
indication that a double-temperature model is needed in the fitting. This
qualitative inspection allows us to assess that the 2T 1Z model is required at
small radii ($r<3^{\prime}$, $<2^{\prime}$ and $<2^{\prime}$ for Perseus, 2A
0335+096 and Sérsic 159-03, respectively), while the 1T model correctly
describes 2A 0335+096 and Sérsic 159-03 data at external radii.

In order to quantitatively support our preliminary inferences, we compare the
$\chi^{2}/$d.o.f. values obtained for each fit. Such comparison confirms that,
for inner radii, the double-temperature model is needed, while for external
radii the clusters in our sample can be considered nearly isothermal.

####  3.2.3 Estimate of the systematics

To estimate the systematic error related to the L-shell, we follow two
slightly different strategies. We fit each spectrum with the selected (1T or
2T 1Z, see previous section) model, excluding the L-shell ($0.9-1.3$ keV) and
the K$\alpha$ ($6.3-6.9$ keV, $6.2-6.8$ keV and $6.1-6.7$ keV999The K$\alpha$
energy band slightly changes with the cluster redshift. for Perseus, 2A
0335+096 and Sérsic 159-03, respectively) bands alternatively, with
temperature(s):

  1. 1.

free to vary (hereafter, this will be referred to as the ”$1^{\text{st}}$
strategy”);

  2. 2.

frozen to the best-fit values derived by fitting over the whole $0.7-10$ keV
energy range (hereafter, this will be referred to as the ”$2^{\text{nd}}$
strategy”).

In this way, the iron abundance can be measured through the L-shell and the
K$\alpha$ emissions separately, providing an estimate of the difference
between the two. Since the physics underlying the K$\alpha$ emission is
simpler and better understood than that for the L-shell blend (as discussed in
the introduction), we shall attribute the above difference to a systematic
error in the L-shell measurement. This error is quantified as:

$\frac{\Delta
Z}{Z}=\frac{Z_{\text{L-shell}}-Z_{\text{K$\alpha$}}}{Z_{\text{K$\alpha$}}},$
(1)

where $Z_{\text{L-shell}}$ and $Z_{\text{K}\alpha}$ are the iron abundances
measured after excluding the K$\alpha$ and the L-shell energy bands,
respectively. Here, $Z_{\text{K}\alpha}$ is considered as a reference in the
iron-abundance measurement. The values obtained for $Z_{\text{K}\alpha}$ and
$Z_{\text{L-shell}}$ for each cluster and for both the strategies are listed
in Appendix A.

(a)

(b)

(c)

Figure 2: Radial profile of the systematic error $\Delta Z/Z$ for 2A 0335+096.
In the top and the central panels, we show the results obtained through an
individual fit of the MOS and pn spectra of each region, following the
$1^{\text{st}}$ and the $2^{\text{nd}}$ strategies, respectively. In the
bottom panel, we show the same profiles, with the MOS and pn spectra of each
region considered jointly in the fitting. The weighted averages of the results
are shown, together with their $1\sigma$ errors, as the pink and the yellow
regions, for the $1^{\text{st}}$ and the $2^{\text{nd}}$ strategies,
respectively.

The above strategies are applied to each cluster of the sample and systematics
are studied as a function of the cluster radii. As an example, in Fig. 2 (top
and centre) we show the results obtained for 2A 0335+096, following the
$1^{\text{st}}$ and the $2^{\text{nd}}$ strategies and by separately fitting
MOS and pn spectra; values obtained through a joint fit of the three detector
spectra are displayed in Fig. 2 (bottom). Figure 2 shows that the systematic
error $\Delta Z/Z$ related to the L-shell does not significantly vary with the
cluster radius and can be considered constant, both looking at the individual
and the joint fits of the MOS and pn spectra. In bright galaxy clusters, where
statistical errors become tiny, the abundances measured by individually
fitting MOS and pn data are usually found to be significantly different, due
to the well-known cross-calibration issues of the three XMM-Newton detectors
(Read et al. 2014). However, and interestingly, MOS and pn cameras lead to
$\Delta Z/Z$ values consistent with each other, within the statistical errors,
as shown in Fig. 2. The results obtained for Perseus and Sérsic 159-03 are
quite similar and displayed in Appendix B.

The weighted averages of the results obtained with both the individual and the
joint fits for each cluster are shown in Table 10. The average values derived
considering the three detectors together are indicated as ”EPIC”, while those
derived separately are indicated as ”MOS 1”, ”MOS 2” and ”pn”. The total
(”EPIC”) averages for 2A 0335+096, obtained through the $1^{\text{st}}$ and
the $2^{\text{nd}}$ strategies with both the individual and the joint fits,
are also shown, with the corresponding $1\sigma$ errors, as the pink and the
yellow regions (respectively) in Fig. 2.

Table 2: Weighted-averaged $\Delta Z/Z$ values, estimated for Perseus, 2A
0335+096 and Sérsic 159-03.

Cluster | Detector | Individual fit | Joint fit
---|---|---|---
|  |  $1^{\text{st}}$ strategy |  $2^{\text{nd}}$ strategy |  $1^{\text{st}}$ strategy |  $2^{\text{nd}}$ strategy
| MOS 1 | $0.056\pm 0.011$ | $0.027\pm 0.011$ | $-$ | $-$
Perseus | MOS 2 | $0.044\pm 0.016$ | $0.048\pm 0.012$ | $-$ | $-$
| pn | $0.076\pm 0.009$ | $0.073\pm 0.010$ | $-$ | $-$
| EPIC | $0.064\pm 0.007$ | $0.051\pm 0.006$ | $0.081\pm 0.006$ | $0.062\pm 0.006$
| MOS 1 | $0.020\pm 0.031$ | $0.033\pm 0.029$ | $-$ | $-$
2A 0335+096 | MOS 2 | $0.030\pm 0.030$ | $0.032\pm 0.027$ | $-$ | $-$
| pn | $0.067\pm 0.025$ | $0.048\pm 0.018$ | $-$ | $-$
| EPIC | $0.043\pm 0.016$ | $0.041\pm 0.013$ | $0.068\pm 0.014$ | $0.052\pm 0.013$
| MOS 1 | $-0.033\pm 0.070$ | $-0.031\pm 0.063$ | $-$ | $-$
Sérsic 159-03 | MOS 2 | $0.112\pm 0.066$ | $0.128\pm 0.065$ | $-$ | $-$
| pn | $-0.020\pm 0.046$ | $0.039\pm 0.043$ | $-$ | $-$
| EPIC | $0.010\pm 0.033$ | $0.042\pm 0.031$ | $0.042\pm 0.032$ | $0.048\pm 0.029$
101010$\Delta Z/Z$ values are obtained both considering the three XMM-Newton
camera independently (indicated with ”MOS 1”, ”MOS 2” and ”pn”) and together
(”EPIC”) in the average (see text).

Looking at the values listed in Table 10, we notice that the weighted averages
of $\Delta Z/Z$ suggest a slightly increasing trend from Sérsic 159-03 to
Perseus. However, large error bars do not allow to make further considerations
and the systematic error estimated for each cluster is consistent with being
constant at the same ($\sim 5-6\%$ and $\sim 6-8\%$ for the individual and the
joint fits, respectively) value. The joint fits are found to lead to a slight
excess of $\sim 1-2\%$ in the estimate of the systematics linked to the
L-shell, with respect to the results obtained with the detectors considered
separately. This excess is likely to be related to the above mentioned cross-
calibration issues of the three XMM-Newton detectors (Read et al. 2014),
resulting in an additional systematic error.

##  4 Dependence on temperature and flux

###  4.1 Systematics vs Temperature

The analysis, independently run on the three clusters of our sample, has shown
that the systematics related to the L-shell remain constant, both considering
different radii of the same cluster (see Fig. 2) and the three clusters
themselves (see Table 10). Since each cluster ring has its own temperature,
the above result also gives us an indication that systematics seem not to vary
with the temperature in the considered range and to be constant with a $\Delta
Z/Z\sim 5-6\%$ (or at least not higher than $8\%$) value. In order to study
the dependence, if any, of the systematic error $\Delta Z/Z$ on the
temperature, we use the weighted average of $\Delta Z/Z$ (averaged over the
three detectors) for each cluster ring. The obtained values are plotted in
Fig. 3 (top left and right, for the $1^{\text{st}}$ and the $2^{\text{nd}}$
strategies respectively), as a function of the ring temperatures. The
temperatures considered in this part of the analysis are obtained for each
ring through the 1T model111111An alternative way of calculating the
temperatures for the rings needing the 2T 1Z model is to consider a weighted-
averaged temperature on the normalisations of the APEC+APEC model. Such
alternative way does not significantly modify the measured temperatures., with
independent fits for each XMM-Newton camera spectrum, and then averaged over
the three detectors121212The gap between the temperatures measured for the
$4^{\text{th}}$ and $5^{\text{th}}$ bins of 2A 0335+096 has been slightly
widened, to avoid an overlap of the results..

(a)

(b)

(c)

(d)

Figure 3: Systematic error $\Delta Z/Z$ linked to the L-shell as a function of
the temperature. The top and the bottom rows refer to the different fitting
option (individual and joint fit of the MOS and pn spectra of each region,
respectively), while the left and the right columns refer to the
$1^{\text{st}}$ and the $2^{\text{nd}}$ strategies, respectively. The pink and
the yellow regions represent the weighted averages of the results, together
with their $1\sigma$ errors.

The three clusters of our sample allow to homogeneously study the $2.5-4.5$
keV range. Perseus gives information about the highest temperatures, i.e. the
$3.5-4.5$ keV range. 2A 0335+096 allows to study the intermediate $3.0-3.5$
keV range, while Sérsic 159-03 (together with the innermost ring for 2A
0335+096) gives an estimate of the systematic error $\Delta Z/Z$ at lower
temperatures. An inspection of Fig. 3 (top panels) shows that the systematic
error $\Delta Z/Z$ linked to the L-shell remains constant over the studied
temperature range. The average values of the results (also reported, together
with their $1\sigma$ error bars, as the pink and the yellow regions in Fig. 3,
top panels), for the $1^{\text{st}}$ and the $2^{\text{nd}}$ strategies,
respectively, are the following:

  1. 1.

$\Delta Z/Z=0.060\pm 0.006$;

  2. 2.

$\Delta Z/Z=0.049\pm 0.005$.

Considering MOS and pn spectra separately in the fitting leads to a $\sim
5-6\%$ systematic error associated to the Fe L-shell, in the whole $2.5-4.5$
keV temperature range. The result obtained through the $2^{\text{nd}}$
strategy is slightly lower than that derived through the $1^{\text{st}}$ one
and they are consistent with each other within their error bars. This finding
suggests that temperatures free to vary do not significantly affect our
results and seem not to introduce additional systematics in the iron-abundance
measurements.

We then repeat the same estimate, this time considering a joint fit of the
three camera spectra. The results are shown in Fig. 3 (bottom left and right,
for the $1^{\text{st}}$ and the $2^{\text{nd}}$ strategies, respectively).
Also the $\Delta Z/Z$ values estimated through the joint fit remain constant
over the studied temperature range, with values:

  1. 1.

$\Delta Z/Z=0.078\pm 0.006$;

  2. 2.

$\Delta Z/Z=0.060\pm 0.005$,

for the $1^{\text{st}}$ and the $2^{\text{nd}}$ strategies, respectively. As
already pointed out, a joint fit of the MOS and pn spectra leads to a $\sim
1-2\%$ excess in the estimate of the systematic error linked to the L-shell,
with respect to an individual fit of each spectrum. We already suggested that
this behaviour is likely to be related to a cross-calibration issue of the
three detectors (Read et al. 2014), leading to an overestimate of the
systematics we are interested in.

Although our study is mainly focussed on bright cluster regions, where the
background contribution to the total emission is negligible, two of the
considered rings (one for 2A 0335+096 and one for Sérsic 159-03, marked with
red circles in Fig. 3) also allow us to explore regions where the background
emission starts to play a role in the K$\alpha$ region (S/B $\sim 5$ and $\sim
3$ for the most external ring of 2A 0335+096 and Sérsic 159-03, respectively,
in the $6-7$ keV range). Interestingly, as we see in Fig. 3, the systematics
measured for such rings are in agreement, within their $1\sigma$ errors, with
the average values. This finding suggests that a moderate contamination by the
background emission seems not to lead to additional contributions to the
systematics we are interested in.

At this point, one may wonder if using blank sky fields instead of a detailed
description of each background component leads to significantly biased
results, especially for the most external rings of 2A 0335+096 and Sérsic
159-03, where the S/B ratio is smaller. To investigate further this issue, we
refit spectra assuming a possible over- and under-estimation of the background
of $10\%$, in order to evaluate potential systematics introduced by an
incorrect estimate of the background contribution131313A $10\%$ variation of
the background contribution is above the current uncertainties related to the
blank sky fields, which are considered to be about $5\%$. However, this is
useful to place strong limits on background-related systematics.. The $\Delta
Z/Z$ values measured after modifying the background emission (and through the
$2^{\text{nd}}$ strategy; top right panel in Fig. 3) show little scatter ($\pm
1\%$ and $\pm 3\%$ for the most external rings of Sérsic 159-03 and 2A
0335+096, respectively, well below the $\pm 9\%$ and $\pm 7\%$ statistical
uncertainties), with respect to the standard (non-modified) value. It should
be noticed that these tiny variations represent an upper limit to background-
related effects on our results, since central rings are less sensitive to
moderate variations of the background contamination, given their significantly
higher S/B ratio. The above considerations show that the way we treated the
background emission does not significantly influence our results and is
sufficient for our purpose. However, in order to reliably estimate the
systematics linked to the L-shell in conditions with stronger background
emission, a precise description of the different background components should
be considered.

###  4.2 Systematics vs Flux

(a)

(b)

(c)

(d)

Figure 4: Systematic error $\Delta Z/Z$ linked to the L-shell as a function of
the X-ray flux. The division in the four panels is the same as in Fig. 3. The
pink and the yellow regions represent the weighted averages of the results,
together with their $1\sigma$ errors.

We also study a possible dependence of the $\Delta Z/Z$ values on the X-ray
flux. In fact, in condition of high source count rates, additional systematic
errors may be included in the analysis (systematics linked to the modelling,
calibration issues of the instruments etc; see Mernier et al. 2018 for a
review of the systematic errors that may affect the ICM abundances) and lead
to an overestimate of the systematics related to the L-shell. The X-ray flux
of each ring is calculated through the XSPEC flux
command141414https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node96.html. in
the $0.7-10$ keV energy range (the same used in the spectral fitting) for the
pn-camera spectra. The dependence of the systematics on the X-ray flux is
shown in Fig. 4. As already discussed for Fig. 3, left and right panels of
Fig. 4 are obtained through the $1^{\text{st}}$ and the $2^{\text{nd}}$
strategies, respectively, while the top and bottom division refers to the
different fitting option of the spectra (individual and joint fit of the MOS
and pn spectra of each region, respectively). In Fig, 4, we also show the
average $\Delta Z/Z$ values, already discussed in Sect. 4.1, as the pink and
the yellow regions.

An inspection of Fig. 4 shows no evident variations with the flux and the
systematic error is consistent with being constant, within the statistical
errors, over the considered flux range. In the low-flux range, large
statistical errors do not allow to derive precise conclusions, while at higher
flux statistical errors are tiny and systematics becomes dominant.

##  5 Discussion

In this work, we investigate systematic errors related to the Fe L-shell blend
($0.9-1.3$ keV energy range), which may affect iron-abundance measurements in
intermediate-mass ($M_{500}=2-4\times 10^{14}$ M⊙, with temperature
$T=2.5-4.5$ keV) galaxy clusters. The resolution of the currently available
instrumentation, as well as our knowledge of the atomic processes, makes the
modelling of the Fe L-shell particularly challenging, resulting in potential
biases for our iron-abundance measurements. As already mentioned in the
introduction, at temperatures higher than $\sim 4$ keV, iron-abundance
measurements are mostly driven by the K$\alpha$ emission at $\sim 6.7$ keV
while, going down to the intermediate $2.5-4.5$ keV temperature range, the EW
of the K$\alpha$ line reduces and the L-shell starts to assume a leading role
in the fitting.

The increasing importance of the L-shell blend in the fitting is highlighted
in Fig. 5, where we show the ratio between the statistical errors associated
to $Z_{\text{K}\alpha}$ and $Z_{\text{L-shell}}$ (Sect. 3.2.3) for the
clusters in our sample, obtained through the $2^{\text{nd}}$ strategy (Sect.
3.2.3) and with a joint fit of the MOS and pn spectra of each region.

Figure 5: Ratio between statistical errors associated to the K$\alpha$\- and
L-shell-derived measurements, as a function of the ring temperatures, for the
three clusters of our sample. Violet area represents $\sigma_{K}/\sigma_{L}$
values (with their $1\sigma$ scatter) derived through high-S/B simulated
spectra, as explained in the text. Errors are estimated following the
$2^{\text{nd}}$ strategy and through the joint fit of MOS and pn camera
spectra of each region. For temperatures lower than $\sim 3.5$ keV, the above
ratio is larger than $1$, meaning that the L-shell drives the abundance
measurements.

To further highlight the interplay between the K$\alpha$ and the L-shell
emissions at different temperatures, we also produced high-S/B simulated
spectra, characterised by single temperature, fixed metallicity ($Z=0.4$ Z⊙)
and normalisation equal to $0.1$ (the same for Perseus, i.e. one of the
brightest clusters of the X-ray sky). We sampled the $2.3-4.7$ keV range every
0.1 keV, and for each step we produced 100 mock spectra with that temperature.
By doing this, we were able to measure the median value of the simulated
$\sigma_{K}/\sigma_{L}$ ratio (together with its $1\sigma$ uncertainty; violet
region in Fig. 5) for each temperature, through the $2^{\text{nd}}$ strategy
and with a joint fit of MOS and pn spectra of each region. An inspection of
Fig. 5 shows that our simulation manages to reproduce the measured
$\sigma_{K}/\sigma_{L}$ trend. We found that, at temperatures lower than $\sim
3.5$ keV, the $\sigma_{K}/\sigma_{L}$ ratio becomes larger than $1$, meaning
that the L-shell blend has larger statistics than the K$\alpha$ and represents
the leading line emission for the iron-abundance measurement.

The three clusters in our sample (Table 1) allow us to homogeneously study the
intermediate $2.5-4.5$ keV range (Fig. 3), particularly in condition of
negligible background emission, and so to explore a temperature range where
the transition between K$\alpha$\- and L-shell-driven fits is observed. In
such a favourable situation, where the K$\alpha$ and the L-shell emissions
have similar EWs, to estimate the systematics linked to the modelling of the
L-shell, we can simply compare the L- to the K-derived iron-abundance
measurements, since the latter are currently considered reliable, as
previously discussed. In the case of an individual fit of each detector
spectrum, the L-shell is found to lead to $\sim 5-6\%$ systematics (Fig. 3,
top), in the whole $2.5-4.5$ keV temperature range. We point out that this is
actually an upper limit to the systematic error linked to the modelling of the
L-shell, since this value may include the contribution of other systematics.
Conversely, the joint fits of the three detectors lead to $\sim 6-8\%$
systematics (Fig. 3, bottom), a $\sim 1-2\%$ excess which is likely due to the
cross-calibration issues of the EPIC detectors, as already discussed in Sects.
3.2.3 and 4.1. The strategies we have adopted in Sect. 3.2.3 represent the
most-straightforward way to quantify the L-blend-induced systematics and,
thanks to a carefully selected sample of galaxy clusters, they lead us to the
most accurate (and precise) results that can be obtained with the currently
available instrumentation. Indeed, in our work we carefully focus on a sample
of clusters with long XMM-Newton observations and high flux in the $0.1-2.4$
keV band (Table 1), at low redshifts ($z\sim 0.018-0.056$) and in the
$2.5-4.5$ keV temperature range. As already discussed, such conditions allowed
us to easily compare K$\alpha$\- and L-shell-derived measurements.

When performing abundance measurements on galaxy clusters, both the K$\alpha$
and L-shell emission bands are generally used in the fitting, to maximise the
available statistic. Thus the measured $5-8\%$ value must be considered as an
upper limit to the contribution of the L-shell-induced systematics to the
derived iron abundances. This consideration gives us confidence for future
measurements in intermediate-mass ($M_{500}=2-4\times 10^{14}$ M⊙) galaxy
clusters, at least in low-background-emission regions. In addition, when
studying individual galaxy clusters, the precision of the results is usually
limited by the poor data quality, with the statistical errors associated to
the measured abundances being larger than the derived $5-8\%$ systematics.
This means that abundance measurements are usually mostly influenced by the
low statistics, while systematics do not significantly affect the results,
making the abundance measurements reliable. However, systematic errors will be
dominant when averaging abundance measurements over a sufficiently large
sample of clusters. For this reason, when performing iron-abundance
measurements over a large sample of intermediate-mass clusters with
temperatures lower than $\sim 3.5$ keV, where the L-shell becomes the leading
emission, the L-shell-induced systematics should no longer be neglected.

A detailed analysis of the L-shell systematics is of primary importance for
future iron-abundance measurements in galaxy-cluster outskirts. As mentioned
above, cluster external regions provide important clues on the physical
properties of the ICM, and so they have recently enjoyed particular attention.
Ghizzardi et al. (2021a, b) focussed on the outskirts of massive clusters, but
no reliable measurements have been carried out in the intermediate-mass (and
temperature) range so far. These temperatures, together with the increasing
background contribution which rises in the cluster outskirts, make the L-shell
the leading emission for iron-abundance measurements. For this reason, an
accurate characterisation of the systematics related to the L-shell is
required before proceeding with future iron-abundance measurements. In the
cluster outskirts, additional systematics (particularly related to the
background emission) may influence the measured abundances. However, our work
places a limit on the L-shell-induced systematics and represents a good
starting point for future iron-abundance measurements.

Future X-ray missions, such as XRISM (Tashiro et al. 2018; Guainazzi & Tashiro
2018) and ATHENA (Barret et al. 2013; Cucchetti et al. 2019), will carry
microcalorimeters and this will allow to perform high-resolution spectroscopy,
giving the opportunity to test, with unprecedented detail, the atomic codes
underlying iron-abundance measurements. Systematic errors related to the
spectral resolution of the instrumentation will be less important, while those
related to the theoretical models which describe the emission lines will
become dominant.

##  6 Summary

We estimated a systematic error related to the modelling of the Fe L-shell
that affects iron-abundance measurements in the $2.5-4.5$ keV temperature
range. The main results of our work are the following:

  * •

the analysis carried out on three clusters shows that L-induced systematics
range between $5-8\%$, depending on the strategy and on the spectral fitting
option (Fig. 2). This result indicates that systematics have a modest
contribution to the measured abundances;

  * •

considering MOS and pn spectra of each region individually in the fitting
leads to $5-6\%$ systematics, in the whole $2.5-4.5$ keV temperature range
(Fig. 3, top);

  * •

a joint fit of the MOS and pn spectra of each region leads on average to a
$1-2\%$ excess in the estimate of the systematics, with respect to the
individual fits (Fig. 3, bottom);

  * •

the systematics show no evident dependence on the X-ray flux (Fig. 4).

Finally and especially when averaging iron abundances over a sufficiently
large sample of clusters with temperatures lower than $\sim 3.5$ keV, we
suggest to take into account the measured $5-8\%$ systematic contribution
related to the L-shell to the derived abundances, in order to correct L-shell-
induced systematics.

###### Acknowledgements.

We thank the anonymous referee for useful suggestions. GR acknowledges funding
from Istituto Nazionale di Astrofisica (INAF) and from the AHEAD project
(grant agreement n. 654215), which is part of the EU-H2020 programme. We
acknowledge financial support contributions from contract n.
4000116655/16/NL/BW by ESA and from contract ASI-INAF n. 2019-27-HH.0. The
results reported in this article are based on data obtained from the XMM-
Newton observatory, an ESA science mission with instruments and contributions
directly funded by ESA Member States and USA (NASA).

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##  Appendix A Iron-abundance measurements

In Table 3 and Table A, we report the derived iron abundances for Perseus, 2A
0335+096 and Sérsic 159-03, measured after the exclusion of the L-shell
($Z_{\text{K}\alpha}$) and the K$\alpha$ ($Z_{\text{L-shell}}$) and obtained
either through the individual (Table 3) and the joint (Table A) fit of MOS and
pn spectra.

Table 3:  Iron-abundance values measured for Perseus, 2A 0335+096 and Sérsic
159-03 after the exclusion of the L-shell ($Z_{\text{K}\alpha}$) and the
K$\alpha$ ($Z_{\text{L-shell}}$) and by individually fitting MOS and pn
spectra of each region.

Cluster | Strategy |  Rings ( ′) |  Z${}_{\text{K}\alpha}$ (Z⊙) |  Z${}_{\text{L-shell}}$ (Z⊙)
---|---|---|---|---
|  |  | MOS 1 | MOS 2 | pn | MOS 1 | MOS 2 | pn
|  | $0.5-1$ | $0.695\pm 0.012$ | $0.636\pm 0.011$ | $0.582\pm 0.009$ | $0.714\pm 0.010$ | $0.683\pm 0.021$ | $0.683\pm 0.008$
| $1$ | $1-2$ | $0.723\pm 0.009$ | $0.659\pm 0.008$ | $0.650\pm 0.006$ | $0.758\pm 0.008$ | $0.710\pm 0.014$ | $0.691\pm 0.006$
Perseus |  | $2-3$ | $0.697\pm 0.010$ | $0.662\pm 0.009$ | $0.630\pm 0.007$ | $0.763\pm 0.011$ | $0.652\pm 0.016$ | $0.654\pm 0.007$
|  | $0.5-1$ | $0.698\pm 0.012$ | $0.655\pm 0.012$ | $0.590\pm 0.010$ | $0.713\pm 0.010$ | $0.695\pm 0.010$ | $0.654\pm 0.011$
| $2$ | $1-2$ | $0.744\pm 0.010$ | $0.680\pm 0.008$ | $0.661\pm 0.007$ | $0.750\pm 0.007$ | $0.718\pm 0.008$ | $0.704\pm 0.006$
|  | $2-3$ | $0.719\pm 0.010$ | $0.664\pm 0.009$ | $0.629\pm 0.007$ | $0.757\pm 0.008$ | $0.679\pm 0.012$ | $0.672\pm 0.007$
|  | $0.5-1$ | $0.706\pm 0.027$ | $0.701\pm 0.027$ | $0.726\pm 0.019$ | $0.676\pm 0.016$ | $0.680\pm 0.018$ | $0.763\pm 0.030$
|  | $1-2$ | $0.628\pm 0.023$ | $0.649\pm 0.022$ | $0.632\pm 0.015$ | $0.668\pm 0.027$ | $0.699\pm 0.029$ | $0.638\pm 0.027$
| $1$ | $2-3$ | $0.573\pm 0.058$ | $0.536\pm 0.026$ | $0.566\pm 0.019$ | $0.675\pm 0.038$ | $0.612\pm 0.036$ | $0.623\pm 0.018$
|  | $3-4$ | $0.453\pm 0.038$ | $0.572\pm 0.035$ | $0.520\pm 0.025$ | $0.520\pm 0.029$ | $0.575\pm 0.045$ | $0.549\pm 0.025$
2A 0335+096 |  | $4-5$ | $0.573\pm 0.049$ | $0.498\pm 0.044$ | $0.443\pm 0.029$ | $0.547\pm 0.049$ | $0.509\pm 0.055$ | $0.546\pm 0.032$
|  | $0.5-1$ | $0.706\pm 0.027$ | $0.696\pm 0.026$ | $0.728\pm 0.019$ | $0.677\pm 0.022$ | $0.680\pm 0.017$ | $0.745\pm 0.011$
|  | $1-2$ | $0.628\pm 0.023$ | $0.645\pm 0.022$ | $0.632\pm 0.015$ | $0.660\pm 0.021$ | $0.689\pm 0.021$ | $0.668\pm 0.011$
| $2$ | $2-3$ | $0.544\pm 0.028$ | $0.536\pm 0.026$ | $0.568\pm 0.021$ | $0.618\pm 0.026$ | $0.593\pm 0.023$ | $0.604\pm 0.017$
|  | $3-4$ | $0.453\pm 0.034$ | $0.572\pm 0.035$ | $0.525\pm 0.025$ | $0.507\pm 0.032$ | $0.569\pm 0.034$ | $0.539\pm 0.025$
|  | $4-5$ | $0.569\pm 0.050$ | $0.498\pm 0.045$ | $0.448\pm 0.032$ | $0.543\pm 0.042$ | $0.507\pm 0.038$ | $0.527\pm 0.029$
|  | $0.5-1$ | $0.452\pm 0.038$ | $0.433\pm 0.031$ | $0.403\pm 0.025$ | $0.433\pm 0.024$ | $0.480\pm 0.023$ | $0.415\pm 0.012$
| $1$ | $1-2$ | $0.331\pm 0.030$ | $0.337\pm 0.032$ | $0.365\pm 0.021$ | $0.332\pm 0.025$ | $0.365\pm 0.019$ | $0.343\pm 0.014$
Sérsic 159-03 |  | $2-3$ | $0.273\pm 0.043$ | $0.266\pm 0.040$ | $0.308\pm 0.035$ | $0.249\pm 0.023$ | $0.333\pm 0.043$ | $0.291\pm 0.022$
|  | $0.5-1$ | $0.455\pm 0.033$ | $0.429\pm 0.031$ | $0.398\pm 0.021$ | $0.433\pm 0.021$ | $0.485\pm 0.024$ | $0.429\pm 0.011$
| $2$ | $1-2$ | $0.332\pm 0.029$ | $0.332\pm 0.028$ | $0.352\pm 0.022$ | $0.334\pm 0.021$ | $0.367\pm 0.020$ | $0.357\pm 0.011$
|  | $2-3$ | $0.271\pm 0.042$ | $0.267\pm 0.042$ | $0.315\pm 0.036$ | $0.255\pm 0.023$ | $0.317\pm 0.025$ | $0.297\pm 0.018$

Table 4: Same as Table 3, with the difference that MOS and pn spectra are
jointly fitted.

Cluster | Rings ( ′) |  $1^{\text{st}}$ strategy |  $2^{\text{nd}}$ strategy
---|---|---|---
|  |  Z${}_{\text{K}\alpha}$ (Z⊙) |  Z${}_{\text{L-shell}}$ (Z⊙) |  Z${}_{\text{K}\alpha}$ (Z⊙) |  Z${}_{\text{L-shell}}$ (Z⊙)
| $0.5-1$ | $0.618\pm 0.006$ | $0.677\pm 0.006$ | $0.620\pm 0.006$ | $0.672\pm 0.005$
Perseus | $1-2$ | $0.669\pm 0.004$ | $0.723\pm 0.004$ | $0.677\pm 0.004$ | $0.719\pm 0.004$
| $2-3$ | $0.643\pm 0.005$ | $0.686\pm 0.008$ | $0.663\pm 0.005$ | $0.692\pm 0.006$
| $0.5-1$ | $0.713\pm 0.013$ | $0.731\pm 0.008$ | $0.711\pm 0.014$ | $0.738\pm 0.006$
| $1-2$ | $0.628\pm 0.012$ | $0.675\pm 0.008$ | $0.628\pm 0.011$ | $0.663\pm 0.008$
2A 0335+096 | $2-3$ | $0.548\pm 0.015$ | $0.633\pm 0.013$ | $0.552\pm 0.013$ | $0.591\pm 0.012$
| $3-4$ | $0.511\pm 0.018$ | $0.545\pm 0.017$ | $0.512\pm 0.017$ | $0.536\pm 0.016$
| $4-5$ | $0.485\pm 0.024$ | $0.554\pm 0.026$ | $0.488\pm 0.023$ | $0.523\pm 0.021$
| $0.5-1$ | $0.423\pm 0.016$ | $0.440\pm 0.010$ | $0.421\pm 0.015$ | $0.445\pm 0.007$
Sérsic 159-03 | $1-2$ | $0.338\pm 0.016$ | $0.357\pm 0.008$ | $0.341\pm 0.014$ | $0.357\pm 0.008$
| $2-3$ | $0.285\pm 0.022$ | $0.286\pm 0.015$ | $0.288\pm 0.024$ | $0.287\pm 0.013$

##  Appendix B Perseus and Sérsic 159-03

In Fig. 6 and Fig. 7, we show the derived $\Delta Z/Z$, as a function of the
cluster radius, for Perseus and Sérsic 159-03, respectively. $\Delta Z/Z$
values are obtained either by individually fitting each detector spectrum and
from a joint fit.

(a)

(b)

(c)

Figure 6: Radial profile of the systematic error $\Delta Z/Z$ for Perseus. In
the top and the centre panels, we show the results obtained through an
individual fit of the MOS and pn spectra of each region, following the
$1^{\text{st}}$ and the $2^{\text{nd}}$ strategies, respectively. In the
bottom panel, we show the same profiles, with the difference that MOS and pn
spectra are considered jointly in the fitting. The weighted averages of the
results are shown, together with their $1\sigma$ errors, as the pink and the
yellow regions, for the $1^{\text{st}}$ and the $2^{\text{nd}}$ strategies,
respectively.

(a)

(b)

(c)

Figure 7: Same as Fig. 6, for Sérsic 159-03.

##  Appendix C Temperature values

In Table 5, we report the measured temperatures for Perseus, 2A 0335+096 and
Sérsic 159-03, by individually fitting MOS and pn spectra of each region,
which is a useful sanity check. For spectra that were better described by the
double-temperature model (see Sect. 3.2.2), we report a weighted-averaged
temperature on the normalisations of the APEC+APEC model, calculated as
follows:

$T=\frac{T_{\text{cold}}\times\text{norm}_{\text{cold}}+T_{\text{hot}}\times\text{norm}_{\text{hot}}}{\text{norm}_{\text{cold}}+\text{norm}_{\text{hot}}},$
(2)

where $T_{\text{cold}}$ and $T_{\text{hot}}$ are the lowest and the highest
derived temperatures through the 2T 1Z model, while
$\text{norm}_{\text{cold}}$ and $\text{norm}_{\text{hot}}$ are their
normalisations, respectively.

Table 5:  Temperature values measured for Perseus, 2A 0335+096 and Sérsic
159-03 by individually fitting MOS and pn spectra of each region.

Cluster |  Rings ( ′) | Temperature (keV)
---|---|---
|  | MOS 1 | MOS 2 | pn
| $0.5-1$ | $3.924\pm 0.087$ | $3.933\pm 0.119$ | $3.696\pm 0.055$
Perseus | $1-2$ | $4.369\pm 0.088$ | $4.161\pm 0.073$ | $4.042\pm 0.036$
| $2-3$ | $5.212\pm 0.165$ | $4.900\pm 0.226$ | $4.671\pm 0.072$
| $0.5-1$ | $2.565\pm 0.124$ | $2.539\pm 0.156$ | $2.431\pm 0.047$
| $1-2$ | $3.248\pm 0.288$ | $3.266\pm 0.428$ | $3.039\pm 0.138$
2A 0335+096 | $2-3$ | $3.326\pm 0.024$ | $3.230\pm 0.033$ | $3.198\pm 0.020$
| $3-4$ | $3.430\pm 0.055$ | $3.439\pm 0.042$ | $3.347\pm 0.023$
| $4-5$ | $3.493\pm 0.078$ | $3.390\pm 0.068$ | $3.357\pm 0.052$
| $0.5-1$ | $2.750\pm 0.353$ | $2.532\pm 0.103$ | $2.688\pm 0.187$
Sérsic 159-03 | $1-2$ | $2.768\pm 0.414$ | $2.666\pm 0.213$ | $2.729\pm 0.345$
| $2-3$ | $2.564\pm 0.034$ | $2.536\pm 0.039$ | $2.585\pm 0.028$
151515For cluster rings that were better described by the 2T 1Z model, we
report a weighted-averaged temperature on the normalisations of the APEC+APEC
model, as described in the text.