# A Systematic Comparison of Galaxy Cluster Temperatures Measured with NuSTAR
and Chandra

A. N. Wallbank1, B. J. Maughan1, F. Gastaldello2, C. Potter3, and D. R. Wik3
1H. H. Wills Physics Laboratory, University of Bristol, Tyndall Ave, Bristol
BS8 1TL, UK. 2INAF – IASF Milano, Via E. Bassini 15, 20133 Milano, Italy
3Department of Physics & Astronomy, University of Utah, 115 South 1400 East,
Salt Lake City, UT 84112, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Temperature measurements of galaxy clusters are used to determine their
masses, which in turn are used to determine cosmological parameters. However,
systematic differences between the temperatures measured by different
telescopes imply a significant source of systematic uncertainty on such mass
estimates. We perform the first systematic comparison between cluster
temperatures measured with Chandra and NuSTAR. This provides a useful
contribution to the effort of cross-calibrating cluster temperatures due to
the harder response of NuSTAR compared with most other observatories. We
measure average temperatures for 8 clusters observed with NuSTAR and Chandra.
We fit the NuSTAR spectra in a hard ($3-10$ keV) energy band, and the Chandra
spectra in both the hard and a broad ($0.6-9$ keV) band. We fit a power-law
cross-calibration model to the resulting temperatures. At a Chandra
temperature of 10 keV, the average NuSTAR temperature was $(10.5\pm 3.7)\%$
and $(15.7\pm 4.6)\%$ lower than Chandra for the broad and hard band fits
respectively. We explored the impact of systematics from background modelling
and multiphase temperature structure of the clusters, and found that these did
not affect our results. Our sample are primarily merging clusters with complex
thermal structures so are not ideal calibration targets. However, given the
harder response of NuSTAR it would be expected to measure a higher average
temperature than Chandra for a non-isothermal cluster, so we interpret our
measurement as a lower limit on the difference in temperatures between NuSTAR
and Chandra.

###### keywords:

cosmology: observations – galaxies: clusters: general – X-rays: galaxies:
clusters

††pubyear: ????††pagerange: A Systematic Comparison of Galaxy Cluster
Temperatures Measured with NuSTAR and Chandra–B

##  1 Introduction

Galaxy clusters are the largest gravitationally bound objects in the Universe.
Massive galaxy clusters are ideal targets for studying cosmology as their
number density is sensitive to cosmological parameters (see e.g. Allen et al.,
2011, for a review). Cluster masses are essential for such cosmological
studies. However, since clusters are dominated by dark matter, their masses
must be inferred from observable properties (the main methods used are
reviewed in Pratt et al., 2019). The X-ray emission from the intra-cluster
medium (ICM) provides several observable quantities which may be used for this
purpose. These include temperatures and densities of the ICM, radial profiles
of which can be used to infer the total mass of a cluster assuming hydrostatic
equilibrium. Where the X-ray data quality does not allow detailed
measurements, scaling relations between ICM quantities (such as luminosity,
temperature or gas mass) may be used to estimate masses if they are accurately
calibrated.

There has been significant attention paid recently to the accuracy of cluster
masses obtained from hydrostatic equilibrium analyses. If hydrostatic masses
were systemtatically underestimated, this could help to resolve the
discrepancy that has been observed in some of the cosmological parameters
(principally $\sigma_{8}$) obtained from Planck measurements of the cosmic
microwave background (CMB) anisotropies versus those obtained from number
densities of massive clusters detected by Planck (Planck Collaboration et al.,
2020a).

Hydrostatic masses may be underestimated if there is significant non-thermal
pressure support in the ICM, since the X-ray observations only measure the
thermal pressure (Laganá et al., 2010; Barnes et al., 2021; Ettori & Eckert,
2022). This is known as the hydrostatic bias, and conventionally refers to the
amount by which hydrostatic masses are biased low compared to the true mass.
Comparisons between hydrostatic masses and masses obtained via weak
gravitational lensing measurements suggest that that a hydrostatic bias may be
present but the size of the bias remains uncertain. Most analyses point
towards hydrostatic masses being underestimated by $\sim(0-20)\%$ (von der
Linden et al., 2014; Okabe & Smith, 2016; Herbonnet et al., 2020; Salvati et
al., 2021; Logan et al., 2022), but underestimates as large as $(30-40)\%$ may
be required to fully align the Planck CMB and cluster number counts
constraints on $\sigma_{8}$ (Planck Collaboration et al., 2020b).

The calibration of X-ray telescopes is another possible source of systematic
uncertainty on hydrostatic masses. In particular if ICM temperatures were
systematically underestimated, this would lead to underestimates of
hydrostatic masses. Absolute calibration of X-ray telescopes in orbit is
challenging due to the lack of standard, unvarying astrophysical calibration
sources, and the cross-calibration of different X-ray telescopes reveals some
inconsistencies. One of the key instrumental properties that needs to be
calibrated is the effective area of the X-ray telescopes. This is energy-
dependent, and miscalibration at different energies would distort the slope of
observed X-ray spectra, potentially biasing quantities such as ICM
temperatures that are derived from spectral fits. Restricting spectral fitting
to different X-ray energy bands can then help to reveal any energy-dependent
problems with the effective area calibration. As discussed below, it is
difficult to ascribe temperature differences directly to the effective area
calibration due to the presence of other instrumental effects, such as the
point spread function (PSF) in the case of extended sources, and detector
gain. The contributions of multiple temperature components in the ICM to the
observed spectrum also complicate the interpretation of temperature
differences.

Cross-calibration work has been supervised by the International Astrophysical
Consortium for High Energy Calibration (IACHEC), including the use of galaxy
clusters as calibration sources. Nevalainen et al. (2010) and Schellenberger
et al. (2015) both performed a cross-calibration using galaxy clusters
observed with both Chandra and XMM-Newton, focussing on the ICM temperature
measurements. Nevalainen et al. (2010) found that temperatures measured with
XMM-Newton were systematically lower than those measured with Chandra by
10-15% on average, but the difference was not significant if the spectral
fitting was restricted to higher X-ray energies ($\gtrsim 2$ keV).

Schellenberger et al. (2015) found qualitatively similar results using a much
larger sample, with the disagreement between Chandra and XMM-Newton increasing
with cluster temperature. Chandra temperatures were $\sim 20\%$ higher than
those measured using XMM-Newton at a cluster temperature of $10$ keV. Again
the discrepancy was found to be dominated by the inclusion of softer X-rays
($\lesssim 2$ keV) in the spectral fitting. This work also tested the internal
cross-calibration of the observatories’ instruments. There was good agreement
between temperatures measured by Chandra Advanced CCD Imaging Spectrometer
(ACIS) I and S arrays (which use the same optics) but inconsistencies between
temperatures from the three XMM-Newton European Photon Imaging Camera (EPIC)
detectors (which each use a different telescope).

A cross-calibration of Suzaku and XMM-Newton using galaxy clusters has also
been performed (Kettula et al., 2013). The results showed that Suzaku
systematically measured the temperature of galaxy clusters to be 2-6% lower
than those from XMM-Newton when the analysis was performed in the hard energy
band of 2.0-7.0 keV and up to a 12% difference for temperatures measured in
the soft energy band of 0.5-2.0 keV. However, it was determined that up to
half of this difference could be caused by Suzaku’s PSF scattering (typically
softer) photons from the core regions.

The results from these cross-calibrations highlight the uncertainty in
measurements between the different X-ray observatories and the importance of
understanding how different detectors are calibrated.

NuSTAR presents a relatively untapped opportunity for the cross-calibration of
cluster temperatures. The hard response of NuSTAR makes it more sensitive to
the exponential cut-off of the bremsstrahlung continuum that dominates the ICM
emission for hot clusters. This enables it to more accurately measure
temperatures of hot clusters compared to e.g. Chandra, XMM-Newton or Suzaku.
NuSTAR has been used for spectral analysis of a relatively small number of
galaxy clusters, often with the aim of constraining any non-thermal component
of the emission (e.g. Wik et al., 2014; Cova et al., 2019; Rojas Bolivar et
al., 2021).

In this paper, we present the first systematic comparison of galaxy cluster
temperatures measured with NuSTAR and Chandra. We compare the temperatures of
a sample of eight galaxy clusters which have been observed with both NuSTAR
and Chandra. The paper is organised as follows. We give an overview of the
cluster sample in section 2. Section 3 discusses NuSTAR and Chandra data
processing as well as the background and spectral analysis. The temperature
measurements and cross-calibration results are presented in section 4. The
results are then discussed in more detail in section 5, and our conclusions
are presented in section 6. Throughout this paper, when referring to Chandra
and XMM-Newton temperatures, we mean those measured with the Chandra ACIS and
XMM-Newton EPIC detectors.

##  2 Cluster sample

In principle, galaxy clusters make good X-ray calibration sources since they
are bright and non-varying, and generally unaffected by pile-up. However, the
ICM can be thermally complex with multiphase gas projected along the line of
sight. If a model with a single temperature component is fitted to an X-ray
spectrum from a region of ICM, then the measured temperature will be an
average of the different temperature components of the ICM along the line of
sight, weighted by the emissivity of each component combined with the energy-
dependent effective area of the telescope (Mazzotta et al., 2004; Vikhlinin,
2006). If a galaxy cluster were perfectly isothermal, then all correctly-
calibrated X-ray telescopes would recover the same temperature. However, when
the observed spectrum comprises a combination of temperature components, then
even perfectly calibrated telescopes would recover different average
temperatures if their sensitivities were different at different X-ray
energies. For example, a telescope with a hard response (like NuSTAR) would
recover a higher average temperature than a telescope with a softer response
(like Chandra).

For cross-calibration purposes it would be ideal, therefore, to select a
sample of relaxed galaxy clusters which contain large volumes of approximately
isothermal gas. However, the relatively small number of clusters that have
been observed by NuSTAR are almost exclusively merging clusters. This is by
design, since such systems are expected to contain very hot ICM regions
related to merger shocks and radio halos (with possible associated non-thermal
X-ray emission) so make good targets for NuSTAR observations.

As of January 2020, the NuSTAR public archive contained eight galaxy clusters
that we deemed were suitable for our purposes. In particular, we required the
clusters to be bright but sufficiently distant that the majority of the ICM
emission was contained within the field of the Focal Plane Modules, giving a
sample of eight clusters. All of these clusters had previously been observed
by Chandra, many of them with a large number of observations. A subset of the
available observations were chosen to provide a similar data quality to the
NuSTAR data (as will be seen later, the statistical uncertainty on the Chandra
temperature measurements is sufficiently small that there is no benefit to
utilising all of the available data). The resulting sample is presented in
Table 1.

Cluster Name | NuSTAR | $t_{N}$ | Chandra | $t_{C}$ | Redshift | NH | RA | Dec
---|---|---|---|---|---|---|---|---
| ObsID | ks | ObsID | ks | z | (1022 cm-2) | degrees | degrees
Abell 2146 | 70401001002 | 226 | 12247, 12245 | 114 | 0.232 | 0.0337 | 239.042 | 66.355
Abell 2163 | 70101002002 | 99 | 1653 | 71 | 0.203 | 0.2060 | 243.942 | -6.143
Abell 2256 | 70001053002 | 86 | 16514, 16129 | 89 | 0.058 | 0.0494 | 255.975 | 78.636
Abell 523 | 70301001002, 70102001004 | 184 | 15321 | 30 | 0.104 | 0.1590 | 74.779 | 8.753
Abell 665 | 70201002002, 70201003002 | 164 | 13201, 12286 | 89 | 0.183 | 0.0507 | 127.729 | 65.853
Abell 754 | 70201001002 | 104 | 10743 | 94 | 0.054 | 0.0579 | 137.346 | -9.684
1E 0657-56 | 70001055002, 70001055002 | 251 | 3184, 4986 | 121 | 0.296 | 0.0643 | 104.621 | -55.944
RX J1347.5-1145 | 70301001002, 70301001004 | 142 | 3592 | 57 | 0.451 | 0.0581 | 206.879 | -11.753
Table 1: Summary of the clusters analysed in this work. The columns give the
cluster name, NuSTAR ObsID(s) and total cleaned exposure time ($t_{N}$),
Chandra ObsID(s)and total cleaned exposure time ($t_{C}$), redshift, hydrogen
column density (taken from Willingale et al., 2013), right ascension and
declination. The coordinates given correspond to the centre of the source
region used to extract the spectra for our analysis.

##  3 Data Processing and Analysis

###  3.1 NuSTAR data processing

The NuSTAR observations of each cluster were processed with the latest
calibration as of May 2020 (CALDB version 20200429111Subsequent to the
completion of our analysis, NuSTAR CALDB 20211020 was released, which included
an update to the effective area (Madsen et al., 2021). The change to the
effective area is almost constant with energy below $10$ keV, so is not
expected to significantly impact cluster temperature measurments. We confirmed
this by repeating our analysis for A2146 with the new calibration and found
that temperature changed by $2\%$.). For four clusters (Abell 523, Abell 665,
1E 0657-56 and RX J1347.5-1145), there were two NuSTAR observations, and in
these cases, both observations were processed and used in the analysis. The
data reduction was performed for both focal plane modules A and B (FPMA and
FPMB) for each cluster. The standard
nupipline222https://heasarc.gsfc.nasa.gov/docs/nustar/analysis/nustar_swguide.pdf
processes were run on the observations, with the additional arguments
saamode=STRICT and TENTACLE=yes, closely following the analysis done in Cova
et al. (2019).

NuSTAR has a complex background, which must be accounted for when performing
spectral analyses. The approach we use closely follows that described in Wik
et al. (2014) and consists of extracting background spectra from the target
observation, fitting a detailed model to those, and then simulating background
spectra for the source region based on this model.

We briefly summarise the main components of the NuSTAR background below. A
more detailed description can be found in Wik et al. (2014); Harrison et al.
(2013); Madsen et al. (2017).

The focused cosmic X-ray background (“fCXB”) is due to unresolved sources in
the field of view (FOV) and can be important below $\sim$15 keV. However,
$\sim 90\%$ of the cosmic X-ray background (CXB) photons detected in the FOV
are due to stray light that seeps through the aperture stops, creating a
spatial gradient across the FOV. This is the dominant background component at
energies below $\sim 15$ keV, and is known as the “aperture” background. Stray
light (from the CXB, the Sun and Earth’s albedo) is also present in the FOV
due to reflection from surfaces of the spacecraft. This component can undergo
significant fluctuations due to solar activity at the observation time. Above
15-20 keV, the instrument or internal background dominates. This component
differs between the FPMA and FPMB detectors but is spatially uniform and
consists of a continuum plus strong fluorescent and activation lines.

To model the background nuskybgd was used (Wik et al., 2014). This tool fits a
model to background spectra extracted from the target observation and then
produces a simulated background at the detector position of the source. This
includes the spatial variation of the different background components where
necessary, and geometric corrections for the different region sizes. This
simulated background is then used in the analysis of the source spectrum.

Background regions were defined in each corner of the FOV, avoiding areas of
high cluster emission when possible, similar to the approach used in Cova et
al. (2019). For observations where the cluster was located in one of the
corners of the FOV, background regions were defined in the remaining three
corners. The same regions were used for both FPMA and FPMB. Figure 1 presents
an example of an NuSTAR image of Abell 523, in the energy band of $3-20$ keV,
indicating the regions used to extract background spectra.

Because the target clusters are bright and relatively nearby, their emission
fills a large fraction of the NuSTAR FOV, and so it is possible that emission
from the cluster is present in the background regions at a non-negligible
level. We accounted for this by including an additional APEC (Smith et al.,
2001) component in the background model to describe the ICM emission. The
temperature, abundance and normalisation of this component were free to fit
(with the values tied for matching regions on the FPMA and FPMB detectors). In
general, the parameters of this model were not well constrained, and the
normalisation was degenerate with other background model components. (The
process was also performed with the temperature and abundance of the APEC
component in the background region fixed at the values measured for the
cluster, with consistent results.)

In all cases, the impact of this additional APEC component in the background
model on the final temperatures measured for the clusters was negligible (the
temperatures changed by $<3\%$ compared to the case where no additional APEC
component was included, and the change was random in direction for the eight
clusters rather than a systematic increase or decrease). Our final temperature
measurements used the background models where the additional APEC component
was included, but this has no significant effect on our results.

Figure 1: NuSTAR image of Abell 523, ObsID 70102001002, showing the four
regions used to model the background with nuskybgd. The image is from detector
FPMA in the energy band 3-20 keV and has been smoothed with a Gaussian with
$\sigma=6.25"$. The background has not been subtracted in this image, and the
strong gradient in the aperture background is visible across the field.

Once the simulated background spectra were produced, the source spectra were
then extracted and associated response matrices (RMF) and auxiliary response
files (ARF) were produced using nuproducts. The parameter extended=yes was
used, which weights the ARF based on the distribution of events within the
chosen extraction region333This weighting was calculated using events detected
over the whole 3-79 keV energy band (i.e. the keywords pilowarf and pihigharf
were not used), which results in the distribution being flatter than the
cluster emission. This could have the effect of weighting the ARF too much
towards the outer parts of the field of view and hence underestimating the
effective area at higher energies. However the effect is negligible for our
analysis; the effective area at 10 keV changes by $\sim 0.1\%$ when the ARF is
weighted by the events in the $3-10$ keV band.. The cluster source spectra
were extracted from a circular region of radius $250"$ centred on the emission
centroid determined from the NuSTAR image. This radius was chosen to produce a
high signal to noise spectrum for the clusters studied, and was used for
consistency for all clusters (although, as seen in Appendix B, the results are
not sensitive to the choice of region). The same regions were used for both
FPMA and FPMB detectors.

###  3.2 Chandra data processing

The Chandra data were processed following the reduction and analysis
procedures presented in Maughan et al. (2008, 2012), and we summarise the main
steps below. For all clusters, Chandra ACIS-I observations were used. In this
work CIAO version 4.12 and CALDB version 4.9.1 was used.

The data were reprocessed from level 1 events and lightcurves were produced
(in the $0.3-12$ keV band) and filtered to remove periods of high background.
Blank-sky backgrounds events lists were used in the spectral analysis and were
processed in a consistent way to the source observations. Exposure corrected
images in the $0.3-7$ keV energy band were used for the detection of point-
like sources. All the point sources were excluded from the subsequent
analysis.

We note that in all cases, any point sources detected in the Chandra data that
fell within the $250"$ source region used for the spectral analysis were not
resolved in the NuSTAR observations, and so were not excluded in the NuSTAR
spectral analysis. However, in all cases these point sources were faint.
Within the $250"$ source spectrum region, the count rate from resolved point
sources measured with Chandra was at most $3\%$ of the total emission, in any
of the energy bands used for the Chandra and NuSTAR spectral analysis. We
verified that excluding the point sources in the Chandra analysis did not
impact the temperatures by measuring the temperature of A2146 (which had the
maximum $3\%$ point source contamination) without excluding the sources.
Retaining the point sources increased the Chandra temperature by $1\%$.

Radial surface-brightness profiles of the cluster emission were used to define
the radii beyond which the cluster emission was negligible compared to the
background in the Chandra observations. This enabled the definition of local
background regions within which the local and blank-sky background spectra
could be compared.

The background spectra were normalised to match the count rate in the hard
X-ray band ($9.5-12$ keV). The residuals between the local and blank-sky
background spectra were then primarily due to differences in soft Galactic
foreground emission at $<2$ keV. To account for this, the residuals were fit
with an unabsorbed APEC model with a temperature of $0.18$ keV and solar
abundance, which was then included in the subsequent analysis of the source
spectrum.

Since the target clusters are bright and relatively nearby, in some cases the
cluster emission fills a large fraction of the Chandra FOV, and may be present
in the regions defined for the background analysis. As discussed in §5, due to
the high signal-to-noise of the source spectra, any cluster emission in the
background region is not expected to significantly influence the temperature
measurement.

The cluster spectra were then extracted for each observation using the $250"$
circular source regions defined for the NuSTAR analysis. For illustration, a
comparison of the NuSTAR and Chandra images of 1E 0657-56 is shown in Figure
2, with the spectral extraction region shown in magenta, and the NuSTAR FOV
overlaid on the Chandra image in white. The images of all clusters are
presented in Appendix A.

(a)

(b)

Figure 2: NuSTAR (top) and Chandra (bottom) image of 1E 0657-56, ObsID
70001055002 and 3184, respectively. The NuSTAR observation is a nuskybgd
background-subtracted image in the $3-20$ keV energy band, while the Chandra
image is in the $0.7-2.0$ keV energy band. Both images are smoothed with a
Gaussian with $\sigma=6.25"$. The magenta circle is the region in which the
source spectra were extracted. The white square shows the region covered by
the NuSTAR detectors overlaid on the Chandra image.

###  3.3 Spectral fitting

The cluster spectra were modelled with a single temperature, absorbed APEC
model convolved with the appropriate ARF and RMF for the regions used. The
spectra were fit in XSPEC (Arnaud, 1996), using the cstat statistic, with the
normalisation, temperature and metallicity free to vary when fitting, while
the NH and redshift were frozen. The absorption was modelled with phabs, and
the values for NH were derived from Willingale et al. (2013), with XSPEC’s
default abundance table from Anders & Grevesse (1989).

For the NuSTAR analysis, the spectra were fit between $3-10$ keV (hereafter
referred to as the hard band), which is close to optimal given the instrument
response and typical cluster spectrum. For Chandra, the spectra were fit in
the hard band for consistency with NuSTAR, and in the $0.6-9$ keV band
(hereafter referred to as the broad band), which is more suitable for Chandra
analyses given the softer response.

For the NuSTAR observations, the temperature was measured for the FPMA and
FPMB detectors individually, as well as by fitting a single source model to
the spectra from the two detectors simultaneously. Clusters with two NuSTAR
observations (Abell 523, Abell 665, 1E 0657-56 and RX J1347.5-1145) were fit
individually and combined, with the same being done for Chandra data when two
observations were used (Abell 665, Abell 2146, Abell 2256, and 1E 0657-56).

###  3.4 Notes on individual clusters

In this section we note any departures that were required from the standard
analysis described in the preceding sections.

####  3.4.1 Abell 754

For Abell 754, the location of the cluster within the NuSTAR FOV meant that
the $250"$ circular source region extended off the edge of the detectors in
both observations. The part of the region that fell off the detector in any of
the data was excluded from all of the NuSTAR and Chandra spectral extractions.

####  3.4.2 Abell 2163

For the Chandra analysis of A2163, the lightcurve showed a gradient over the
duration of the observation, indicating possible a low-level background flare.
The lightcurve was manually cleaned to select a 20 ks period where the
background appeared stable. After this filtering, the comparison between the
local and blank-sky background showed a strong excess in the local background
below 2 keV, with some hints of excess emission up to $\sim 7$ keV, which
could indicate residual flare contamination. Abell 2163 is also located near
the North Polar Spur (NPS), giving it a particularly complex soft background
component and strong NH gradient across the cluster (Pratt et al., 2001;
Bourdin et al., 2011; Rojas Bolivar et al., 2021).

As a result we consider the Chandra temperature measurement for this cluster
to be less reliable than for the other clusters in our sample, particularly
for the spectral fit in the broad band, and hence we exclude the broad-band
temperature from our main analysis.

####  3.4.3 Abell 2256

For this cluster, the local background spectrum in the Chandra data had a
significant excess above the blank-sky background below $\sim$2 keV. Unlike
Abell 2163 there was no indication of residual flaring in the lightcurve. This
is a bright, nearby cluster with significant emission from the ICM across most
of the Chandra FOV (see Figure 9). We thus interpret the excess in the local
background as being due to emission from the ICM and/or a particularly strong
soft Galactic foreground, which was not modelled completely by the APEC
component in the standard data processing. To account for this, the
temperature of the APEC component used to model the soft Galactic foreground
was unfrozen, and the best fitting values (0.86 keV and 0.94 keV for
observations 16514 and 16129, respectively) were used in the subsequent
analysis.

We note that if the excess in the local background were due to contamination
of the background spectrum by emission from the ICM, then this could lead to
the flux of the soft Galactic foreground model component being overestimated
and hence the cluster temperature being overestimated. However, given the very
high signal to noise of the cluster spectrum within $250"$, this is not
expected to have a significant effect.

##  4 Results

###  4.1 Temperatures

Cluster Name | $kT_{C,(0.6-9)}$ | $kT_{C}$ | $kT_{N}$
---|---|---|---
| keV | keV | keV
Abell 2146 | $7.08\pm 0.14$ | $8.90\pm 0.66$ | $6.83\pm 0.11$
Abell 2163 | $16.36\pm 0.70^{\dagger}$ | $12.23\pm 1.15$ | $9.72\pm 0.13$
Abell 2256 | $7.81\pm 0.15$ | $7.38\pm 0.41$ | $6.42\pm 0.08$
Abell 523 | $5.30\pm 0.36$ | $7.15\pm 2.55$ | $4.94\pm 0.11$
Abell 665 | $8.66\pm 0.23$ | $8.29\pm 0.62$ | $7.45\pm 0.12$
Abell 754 | $9.09\pm 0.17$ | $9.25\pm 0.47$ | $8.58\pm 0.14$
1E 0657-56 | $13.57\pm 0.36$ | $14.57\pm 0.96$ | $12.58\pm 0.18$
RX J1347.5-1145 | $14.71\pm 0.46$ | $15.80\pm 1.09$ | $12.64\pm 0.28$

Table 2: The temperature measurements for the cluster sample. $kT_{C,(0.6-9)}$
denotes the temperatures from the Chandra data when fit in the 0.6-9 keV
energy band, while $kT_{C}$ denotes the Chandra temperatures measured in the
3-10 keV band. $kT_{N}$ are the temperatures measured from the NuSTAR data
when fit in the 3-10 keV energy band. †We consider the $kT_{C,(0.6-9)}$
measurement to be unreliable for this cluster (see §3.4.2)

.

The global temperatures measured for each cluster are presented in Table 2.
$kT_{C,(0.6-9)}$ denotes the temperatures from the Chandra data when fit in
the broad band, while $kT_{C}$ denotes the Chandra temperatures measured in
the hard band. $kT_{N}$ denotes the temperatures measured from the NuSTAR data
when fit in the hard band. The temperatures reported here are from joint
spectral fits to all available detectors and observations for each cluster.
When fit individually, the differences in temperatures between detectors and
observations for a given cluster were consistent with the statistical
uncertainties.

Figure 3: Cluster temperatures measured by NuSTAR in the hard ($3-10$ keV)
band are plotted against those measured by Chandra in the broad ($0.6-9$ keV)
band. Note that Abell 2163 is omitted from this plot as the Chandra broad-band
temperature was considered to be unreliable (see §3.4.2).

Figure 3 presents the comparison between the hard-band NuSTAR temperatures and
those measured with Chandra in the broad band. This is intended to reflect the
optimal energy bands used for temperature measurements with each observatory.
This comparison clearly shows that the temperatures measured by Chandra are
systematically higher than those measured by NuSTAR for the clusters in our
sample.

While the $0.6-9$ keV band is most appropriate for spectral analysis with
Chandra due to the signal-to-noise of the cluster and effective area of the
telescope, a more direct comparison with NuSTAR temperatures can be made using
the Chandra temperatures made in the same hard band used for the NuSTAR
analysis. In the presence of multi-phase gas in the ICM, one would expect the
temperature measured with Chandra in the hard band to be higher than that
measured in the broad band. This comparison is shown in Figure 4, which shows
a tendency for the Chandra hard-band temperatures to be higher than the broad-
band temperatures, but the effect is not statistically significant. As noted
in §3.4.2, Abell 2163 is a significant outlier due to the unreliable Chandra
temperature measurement in the broad band. It is also apparent from Figure 4
that the statistical error on the temperature is larger when fit in the hard
band, particularly for the coolest cluster (Abell 523). This is unsurprising
given the lower signal to noise in the Chandra data above 3 keV.

The clusters in our sample are well-known and have multiple temperature
measurements in the literature from Chandra, and in many cases from NuSTAR.
However, the complex thermal structures of these clusters (again, they were
mainly selected for NuSTAR observation due to their disturbed dynamical state)
means that temperature measurements will be sensitive to the region used for
the spectral extraction and the energy range used for spectral fitting. For
this reason, there is significant variation in temperature measurements from
different analyses, and we forego making any detailed comparison between our
temperatures and those in the literature.

Figure 4: The Chandra temperatures measured for spectra fit between 3-10 keV
are plotted against those measured with Chandra in the $0.6-9$ keV band. Note
that Abell 2163 is a significant outlier due to the unreliable temperature
measurement in the $0.6-9$ keV band (see §3.4.2)

###  4.2 Cross-calibration

Chandra Energy Band | $\alpha$ | $\beta$ | $\epsilon$ |  $\Delta_{10keV}$ $(\%)$
---|---|---|---|---
0.6-9 keV | $0.952\pm 0.018$ | $0.98\pm 0.13$ | $0.002^{+0.003}_{-0.001}$ | $10.5\pm 3.7$
3-10 keV | $0.926\pm 0.024$ | $0.99\pm 0.19$ | $0.002^{+0.006}_{-0.001}$ | $15.7\pm 4.6$
Table 3:  The cross-calibration between Chandra and NuSTAR. $\alpha$ and
$\beta$ are the intercept and gradient from equation 1. $\epsilon$ is the
intrinsic scatter from Linmix, and $\Delta_{10keV}$ is the percentage
difference between the two detectors when the instrument X is at 10 keV.
Energy BandC is the energy band in which a model was fit to the Chandra
observation spectra.

To quantify the cross-calibration between NuSTAR and Chandra temperatures, we
follow Schellenberger et al. (2015) and fit a power law model to the data. The
fitting was performed using a linear model in log space using Linmix, which
performs a Markov chain Monte-Carlo sampling of the model likelihood, and
accounts for measurement errors in both axes and intrinsic scatter (Kelly,
2007). The form of the scaling relationship used for this analysis is

${\log_{10}\left(kT_{N}\right)=\beta\times\log_{10}\left(\frac{kT_{C}}{10~{}{\rm
keV}}\right)+\alpha}$ (1)

The fit is also performed with $kT_{C,(0.6-9)}$ in place of $kT_{C}$ when
comparing NuSTAR temperatures with broad-band temperatures from Chandra. When
using broad-band Chandra temperatures, Abell 2163 was excluded as discussed in
§3.4.2.

The results of the fits are presented in Table 3, and the best fitting model
for the broad-band Chandra temperatures is plotted along with the data in
Figure 5. The uncertainties on the fit parameters are derived from the median
and 16th and 84th percentiles of parameter chains sampled by Linmix,
corresponding to 1$\sigma$ uncertainty. For both Chandra energy bands, the
power law slope is consistent with unity and the intrinsic scatter between
NuSTAR and Chandra is negligible. However there is evidence for the Chandra
temperatures being systematically higher. We quantify this by using the best
fitting model to calculate the percentage difference between the NuSTAR and
Chandra temperatures at a Chandra temperature of 10 keV.

We find that the broad-band Chandra temperatures are on average $(10.5\pm
3.7)\%$ higher at a Chandra temperature of 10 keV than those measured with
NuSTAR (in the $3-10$ keV band). Similarly, the hard-band Chandra temperatures
are found to be $(15.7\pm 4.6)\%$ higher than NuSTAR temperatures at a Chandra
temperature of 10 keV.

Figure 5 also shows the cross-calibration model for Chandra and XMM-Newton
temperatures derived by Schellenberger et al. (2015) (for a different set of
clusters). This gives an indication of the relative temperature calibrations
of the three telescopes, and is discussed further in §5. Figure 6 shows the
comparison between the cross-calibration fits for both the broad and hard-band
Chandra temperatures.

Figure 5: The cross-calibration between NuSTAR temperatures measured in the
$3-10$ keV band and broad-band Chandra temperatures (measured in the $0.6-9$
keV band). The best fitting power-law model to the data is shown in red, and
the black dashed line indicates a perfect agreement between the temperatures.
Abell 2163 is omitted from this plot as discussed in §3.4.2. The green dashed
line shows, for illustrative purposes, the best fitting power-law to the
cross-calibration of temperatures measured with XMM-Newton versus Chandra
derived in Schellenberger et al. (2015) (we show their “ACIS-Combined XMM
Full” model, measured in the $0.7-7$ keV energy band, with XMM-Newton
temperatures in place of NuSTAR temperatures on the vertical axis). Note that
the comparison with the result from Schellenberger et al. (2015) is not
precise due to the different energy bands used for their temperature
measurements, but illustrates the relative cross-calibration of the
temperatures between the three telescopes. Figure 6: The cross-calibration
between NuSTAR and Chandra for both broad-band (red points) and hard-band
(blue points) Chandra temperatures. The NuSTAR spectra were fit between $3-10$
keV in all cases. The broad-band Chandra temperature for Abell 2163 was
excluded, as discussed in §3.4.2.

##  5 Discussion

Our comparison of cluster temperature measurements with NuSTAR and Chandra
shows that the Chandra temperatures are systematically higher. In the
following we discuss possible sources of systematic uncertainty on our
temperature measurements, and the limitations and implications of our cross-
calibration measurement.

###  5.1 Systematics on temperature measurements

We explored the possible impact on our results of uncertainties on the
background modelling in the spectral analysis. We found that due to the high
signal-to-noise of the cluster spectra, our results were insensitive to these
systematics, but we briefly summarise here the main tests that were performed.

For the NuSTAR analysis, the background spectrum is determined by fitting a
model to spectra extracted from local background regions. In our standard
analysis the background model is fit jointly to the spectra from all of the
background regions, and the uncertainty on the resulting background model
parameters is not propagated to the final cluster temperature. We tested the
robustness of our temperatures by allowing the normalisations of the
“aperture” background and “fCXB” components to fit independently to the
spectra from each background region and then used the variation between
regions as an estimate of the uncertainty on those components. The cluster
temperatures were remeasured with the normalisation of those background
components varied within these ranges.

For both the aperture and fCXB components, the resulting variation in cluster
temperatures was less than or equal to the statistical uncertainty on the
temperature measurement (with the exceptions of A523 where the aperture
background gave a systematic of $\pm 6\%$ on the temperature compared to a
$2\%$ statistical error, and A2256 where the fCXB gave a systematic of $\pm
4\%$ compared to a $1\%$ statistical error on the temperature. The change in
normalisation of both background components is anti-correlated with the change
in temperature, so if one of the background components were reduced by
$1\sigma$ for all clusters, the temperatures would increase by an average
amount that was less than the statistical errors. These systematics are thus
subdominant to the systematic difference between NuSTAR and Chandra
temperatures.

As discussed in §3.1 we also considered the uncertainty on the NuSTAR model
background due to the possible presence of cluster emission in the background
regions, and found this had negligible impact on the NuSTAR temperatures. The
inclusion (or not) of this extra background component produced a change in
temperature that was always smaller than the statistical errors, and did not
systematically increase or decrease the temperatures.

Furthermore, for all clusters, when the temperatures were measured
independently for the FPMA and FPMB detectors, or independently for multiple
observations, the agreement was good (e.g. the mean ratio of the FPMA to FPMB
temperatures, computed in log space, was $0.99$ with a standard deviation of
$0.06$).

For each cluster, the largest difference in temperature from any of the
individual systematics investigated above was used as an estimate of the
systematic temperature error for NuSTAR. The cross-calibration fit was
repeated using these systematic errors in place of the statistical errors on
the temperatures. This made a negligible difference to the values of the best-
fitting parameters or their uncertainties (as presented in table 3). We thus
used only the statistical uncertainties on the temperatures in our analysis.

For Chandra, the availability of blank-sky background spectra, and the ability
to compare them with the in-field background meant the systematic
uncertainties due to the background modelling were minimal.

###  5.2 Chandra/NuSTAR cross-calibration

Our cross-calibration analysis shows that for all clusters in the sample,
NuSTAR measures their temperatures to be lower than Chandra. This is
unexpected, since the harder response of NuSTAR should make it more sensitive
to any hotter components present in the cluster spectrum. If both telescopes
were perfectly calibrated then for any non-isothermal cluster, it would be
expected that NuSTAR would measure a higher temperature than Chandra.

The use of different energy bands for the spectral fitting between the two
telescopes would also contribute to systematic temperature differences. The
broader band typically used for Chandra spectral analysis ($0.6-9$ keV here),
compared with the harder band ($3-10$ keV here) used for NuSTAR should further
exacerbate any temperature discrepancy, increasing the contribution of lower
temperature components in a multiphase ICM to the global Chandra temperature
measurement.

Our results show that regardless of the energy band used, the Chandra
temperatures were systematically higher than those measured with NuSTAR. When
the broad band was used, the Chandra temperatures were $(10.5\pm 3.7)\%$
higher at a Chandra temperature of 10 keV. When the hard band was used for the
Chandra analysis, the Chandra temperatures increased (as would be expected)
and were higher than the NuSTAR temperatures by $(15.7\pm 4.6)\%$ at a Chandra
temperature of 10 keV.

The fact that the temperature difference is significant when the data from
both telescopes were fit in the hard ($3-10$ keV) band is interesting. This
contrasts to some extent with previous cross-calibration work that has
identified the calibration of X-ray telescopes at lower energies as largely
responsible for temperature differences.

For example, Schellenberger et al. (2015) found strong evidence that cluster
temperatures from Chandra and XMM-Newton were inconsistent (at $>5\sigma$)
when the spectra were fit in either soft ($0.7-2$ keV) or broad ($0.7-7$ keV)
energy bands. When temperatures were measured in a hard ($2-7$ keV) band, the
inconsistency was less significant (at the $\approx 1-4\sigma$ level depending
on the combination of XMM-Newton detectors used). For these hard-band
temperatures, the average XMM-Newton temperature was between $0\%$ and $10\%$
lower than the average Chandra temperature at a Chandra temperature of $10$
keV.

The disagreement we find between Chandra and NuSTAR when Chandra temperatures
are measured in the hard band suggests that temperature difference is not
driven by factors affecting the Chandra measurements at low energies. This
disfavours some possible origins for the temperature difference. For example,
the uncertainty associated with the soft Galactic foreground modelling, or
variations in the absorbing column (e.g. §3.4.2 and 3.4.3) will not impact the
temperatures measured in the hard band. Similarly, uncertainties on the
modelling of the contamination build-up on Chandra’s optical blocking filter
(Weisskopf et al., 2000) should not impact the hard-band temperatures.

###  5.3 Implications for Chandra/XMM-Newton cross-calibration

Figure 5 presents the scaling relationship between Chandra and XMM-Newton
derived in Schellenberger et al. (2015) alongside the NuSTAR-Chandra cross-
calibration derived in our analysis. Here we show the “ACIS-Combined XMM Full”
model from Schellenberger et al. (2015), measured in the $0.7-7$ keV energy
band, while our Chandra and NuSTAR temperatures are measured in the $0.6-9$
keV and $3-10$ keV bands, respectively. The scaling relationship from
Schellenberger et al. (2015) is not directly comparable with our results due
to a number of differences in the analyses (e.g. the comparison was made for
different clusters, using a different definition for regions for the spectral
extraction, and different energy bands for the spectral fitting), but it
provides a useful indication of the relative direction and size of the the
differences between the temperatures measured with different instruments.

Taken at face value, this comparison suggests that both NuSTAR and XMM-Newton
temperatures are systematically lower than those measured with Chandra (in the
conventionally used broad band). Furthermore, while we lack the data to
directly compare NuSTAR and XMM-Newton temperatures for the same clusters, the
comparisons of each with Chandra in Figure 5 imply that the NuSTAR
temperatures would be systematically higher than XMM-Newton. Hotter NuSTAR
temperatures are expected in the presence of multi-phase ICM, given the harder
NuSTAR response relative to XMM-Newton. The implication of this indirect
comparison (that NuSTAR temperatures are hotter than XMM-Newton but cooler
than Chandra) thus qualitatively favours consistency between NuSTAR and XMM-
Newton, with Chandra being the outlier.

The picture is more complicated when hard-band temperatures are considered.
This is illustrated in Figure 7, which shows the cross-calibration between
Chandra hard-band temperatures and NuSTAR, along with the best-fitting
relation between XMM-Newton and Chandra temperatures measured in the hard band
from Schellenberger et al. (2015, we show their “ACIS-Combined XMM Hard”
model, measured in the $2-7$ keV band). Again, this relation was derived for
different clusters with a different hard-band definition, so is intended only
as an illustration of the relative calibrations. This comparison implies that
the hard-band temperatures measured with XMM-Newton may be hotter than those
measured with NuSTAR, however the systematic differences between these studies
make it difficult at present to explore this in more detail.

The relatively good agreement between hard-band Chandra and XMM-Newton
temperatures found by Schellenberger et al. (2015) and illustrated in Figure 7
imply that NuSTAR may be the outlier in this band. It is thus reasonable to
consider whether the differences we find between NuSTAR and Chandra
temperatures could be driven by uncertainties in the NuSTAR calibration.
However, in their recent NuSTAR calibration update (CALDB 20211020), Madsen et
al. (2021) found good agreement between measurements of the Crab made with
light that passed through the optics compared with stray light (which has a
trivial effective area). Although we did not use CALDB 20211020 for our main
analysis, we have verified that this CALDB update has negligible effect on our
cluster temperatures, and so this increases confidence that the discrepancy we
have found is not due to NuSTAR calibration uncertainties.

Furthermore, two new (optional) updates to the XMM-Newton effective area have
recently been released which improve the internal agreement between the MOS
and pn detectors, and also produce agreement with NuSTAR in the hard
band444https://xmmweb.esac.esa.int/docs/documents/CAL-TN-0018.pdf. This is
expected to move XMM-Newton hard-band temperatures into better agreement with
NuSTAR. We thus conclude that the apparent agreement between hard-band Chandra
and XMM-Newton temperatures, and discrepancy with NuSTAR implied by the
comparison between our work and Schellenberger et al. (2015) does not
constitute strong evidence that the difference is driven by uncertainties in
the NuSTAR calibration.

Figure 7: The cross-calibration between NuSTAR and Chandra temperatures
measured in the $3-10$ keV band. The best fitting power-law model to the data
is shown in blue, and the black dashed line indicates a perfect agreement
between the temperatures. The dashed pink line shows, for illustrative
purposes, the best fitting power-law to the cross-calibration of temperatures
measured with XMM-Newton versus Chandra derived in Schellenberger et al.
(2015) (we show their “ACIS-Combined XMM hard” model, with XMM-Newton
temperatures in place of NuSTAR temperatures on the vertical axis). Note that
the comparison with the result from Schellenberger et al. (2015) is not
precise due to the different energy bands used for their temperature
measurements, but illustrates the relative cross-calibration of the
temperatures between the three telescopes.

###  5.4 Impact of multi-phase ICM

As discussed previously, systematic differences in global temperatures are
expected even for perfectly calibrated instruments if their energy responses
differ and the source has multiple temperature components. It is therefore
instructive to measure the temperature structure of the ICM in our target
clusters. This was done by measuring radial temperature profiles from the
Chandra data.

The profiles were constructed by extracting and fitting spectra following the
methods described in §3.2 and 3.3. Spectra were extracted from annular regions
centred on the same location used for the global temperature measurement, with
the widths of each annulus set such that the signal to noise of the resulting
spectrum was at least 30.

The measured temperature profile of Abell 2256 is shown as an example in
Figure 8, with the profiles of the other clusters shown in Appendix B. On the
temperature profile plots we also show the global temperatures measured by
Chandra and NuSTAR. For these comparisons the Chandra temperatures were
measured in the $0.6-9$ keV band, while the $3-10$ keV band was used for
NuSTAR.

Figure 8: The Chandra temperature profile of Abell 2256 is plotted in green.
The global temperature across a circular region of radius 250 arcseconds is
also shown for Chandra (red) and NuSTAR (blue). Chandra and NuSTAR
temperatures are measured in the $0.6-9$ keV and $3-10$ keV bands
respectively.

The temperature profiles show that for each cluster, the global NuSTAR
temperature is lower than the Chandra temperature found in the majority of the
radial bins that overlap with the global temperature region. This underlines
the systematic nature of the temperature difference. While each temperature
bin will contain multiphase gas across its radial extent and projected along
the line of sight, the profiles do not suggest that the global Chandra
temperature is higher than NuSTAR due to their different weighting of
different temperature components across the 250 arcsecond global temperature
region.

In principle, the global NuSTAR and Chandra temperature could differ due to
X-ray photons scattered into the global temperature extraction region by the
large NuSTAR PSF. In order for this to produce systematically lower NuSTAR
temperatures, there would need to be bright, cool regions of emission outside
the 250 arcsecond global temperature region (possibly outside the NuSTAR FOV).
No such emission was apparent in the Chandra images of these clusters, and the
temperature profiles show no evidence for the systematic presence of large
amounts of cool gas outside the global temperature extraction region.

It is clear from the temperature profiles of the clusters in our sample that
they have complex thermal structures, which will contribute differently to the
NuSTAR and Chandra global temperatures. The optimal calibration sources would
be isothermal to avoid this effect. For example, Schellenberger et al. (2015)
removed the central regions of cool core clusters to create spectra that were
closer to isothermality. For our sample, the clusters are mainly mergers,
without strong cool cores, as can be seen in Appendix B, and so no core
regions were excluded, and no attempt was made to select isothermal regions of
the clusters.

We emphasise again that the multi-phase gas in these merging clusters should
lead to NuSTAR temperatures being higher than those from Chandra due to the
harder response of NuSTAR. Thus, while the thermal structure of the ICM in our
clusters means they are not ideal calibration sources, the systematic effect
of this would be to lessen the disagreement we find between Chandra and
NuSTAR. We can thus interpret the discrepancy we find as a lower limit on the
possible disagreement between the temperature scale of the two instruments. A
more definitive cross-calibration would require a sample of relaxed clusters
for which isothermal regions can be defined. The required NuSTAR observations
are not currently available for a significant sample of relaxed clusters, and
observing such targets would be of great value for the cross-calibration of
current and future X-ray observatories.

###  5.5 Implications of the NuSTAR temperature calibration

Our results imply a systematic uncertainty on X-ray temperature measurements
of clusters that is of order $10-15\%$. This is much larger than the typical
statistical uncertainty on temperatures for clusters with good-quality data
such as those studied here, and is one of the dominant systematics for many
applications of cluster temperatures. For example, hydrostatic masses depend
directly on temperatures and their radial gradients so are directly impacted
by temperature systematics. Schellenberger et al. (2015) showed that the
difference between their broad-band Chandra and XMM-Newton temperatures gave
rise hydrostatic masses that were $\approx 15\%$ higher when assuming the
Chandra calibration was correct compared to assuming the XMM-Newton
calibration was correct.

Measuring hydrostatic masses for our sample is beyond the scope of the current
paper (and the dynamically unrelaxed of the clusters make them unsuitable for
such analyses), but as discussed in §5.2, our NuSTAR temperatures appear more
consistent with broad-band temperatures from XMM-Newton rather than Chandra.
We thus expect that assuming the NuSTAR temperature calibration would lead to
results similar to those derived from XMM-Newton temperatures by
Schellenberger et al. (2015), with hydrostatic masses that were $\sim 15\%$
lower than those determined from Chandra. This would then mean that
determinations of the hydrostatic bias derived from Chandra data would
underestimate the level of hydrostatic bias by $\sim 15\%$. In other words,
assuming the NuSTAR temperatures were correct would not reduce the amount of
hydrostatic bias needed to fully align the Planck constraints on $\sigma_{8}$.

The impact of temperature calibration uncertainties was explored by Wan et al.
(2021) in the context of combining X-ray and Sunyaev-Zel’dovich effect (SZE)
measurements of the pressure in the ICM to constrain the Hubble constant. Wan
et al. (2021) found that the uncertainty on the temperature calibration was
the dominant systematic, with a $\sim 10\%$ decrease in X-ray temperatures
leading to an increase in $H_{0}$ of $\sim 10$ km s-1 Mpc-1. When using
external priors on the value of $H_{0}$, they found evidence that the Chandra
temperatures used in their analysis were overestimated by $\sim 10\%$ due to
calibration uncertainties. This is consistent with the results of our
comparison of Chandra and NuSTAR temperatures.

##  6 Summary and Conclusions

We performed an X-ray spectral analysis on a sample of eight bright galaxy
clusters to produce the first cross-calibration between NuSTAR and Chandra
temperature measurements of the ICM. We found that Chandra systematically
finds ICM temperatures that are higher than those measured with NuSTAR. We fit
a power-law model to the temperatures and found that at a Chandra temperature
of 10 keV, the average NuSTAR temperature was $(10.5\pm 3.7)\%$ and $(15.7\pm
4.6)\%$ lower than that of Chandra, when the Chandra spectra were fit in the
broad ($0.6-9$ keV) and hard ($3-10$ keV) energy bands, respectively (the
NuSTAR spectra were always fit in the hard band).

We examined the impact of uncertainties on the background modelling and found
that due to the high signal-to-noise of the cluster spectra, our results were
insensitive to the details of the background modelling. We also examined the
thermal structure of the clusters using temperature profiles from the Chandra
data, and found no evidence that NuSTAR temperatures were being biased by
emission from cool gas scattered from outside the spectral extraction region
by the larger NuSTAR PSF.

The fact that, when limited to the hard band for spectral fitting, the Chandra
temperature remains systematically higher than that from NuSTAR, implies that
the discrepancy is not driven by factors influencing the modelling at soft
energies. These include systematics in the modelling of the absorbing column
in the spectral analysis, and the calibration of the Chandra effective area at
soft energies to account for the ACIS contamination build-up.

We conclude that the difference is most likely due to systematic uncertainties
in the calibration of one or both of the instruments. However, given that the
presence of multiphase gas in the ICM is expected to lead NuSTAR to measure a
higher average temperature than Chandra or XMM-Newton, combined with previous
findings that Chandra temperatures are systematically higher than those of
XMM-Newton (Schellenberger et al., 2015), we cautiously conclude that the
evidence from NuSTAR favours the XMM-Newton temperature calibration when
temperatures are fit in the conventional broad bands. However, the picture is
less clear for temperatures measured in the hard band, and a more robust
conclusion will require a direct comparison of NuSTAR, Chandra, and XMM-Newton
temperatures for the same clusters. Given that the sample used for the current
analysis are thermally complex, merging systems, the definitive cross-
calibration study should be deferred to a sample of relaxed clusters for which
isothermal regions can be defined.

Our overall conclusion is that the average NuSTAR temperature measurement is
$10-15\%$ lower than that of Chandra (at $10$ keV), most likely due to
calibration differences between the observatories. Given the non-isothermal
nature of the clusters studied, this is likely to be a lower limit on the
difference in the relative temperature scales.

## Acknowledgements

BJM acknowledges support from STFC grant ST/R000700/1 and ST/V000454/1. This
research has made use of NASA’s Astrophysics Data System Bibliographic
Services and TOPCAT (Taylor, 2005).

## Data Availability

All observations for the X-ray data analysis are publicly available at the
Chandra and NuSTAR Data Archives.

## References

  * Allen et al. (2011) Allen S. W., Evrard A. E., Mantz A. B., 2011, ARA&A, 49, 409
  * Anders & Grevesse (1989) Anders E., Grevesse N., 1989, Geochim. Cosmochim. Acta, 53, 197
  * Arnaud (1996) Arnaud K. A., 1996, in Jacoby G. H., Barnes J., eds, Astronomical Society of the Pacific Conference Series Vol. 101, Astronomical Data Analysis Software and Systems V. p. 17
  * Barnes et al. (2021) Barnes D. J., Vogelsberger M., Pearce F. A., Pop A.-R., Kannan R., Cao K., Kay S. T., Hernquist L., 2021, MNRAS, 506, 2533
  * Bourdin et al. (2011) Bourdin H., et al., 2011, A&A, 527, A21
  * Cova et al. (2019) Cova F., et al., 2019, A&A, 628, A83
  * Ettori & Eckert (2022) Ettori S., Eckert D., 2022, A&A, 657, L1
  * Harrison et al. (2013) Harrison F. A., et al., 2013, ApJ, 770, 103
  * Herbonnet et al. (2020) Herbonnet R., et al., 2020, MNRAS, 497, 4684
  * Kelly (2007) Kelly B. C., 2007, The Astrophysical Journal, 665, 1489–1506
  * Kettula et al. (2013) Kettula K., Nevalainen J., Miller E. D., 2013, A&A, 552, A47
  * Laganá et al. (2010) Laganá T. F., de Souza R. S., Keller G. R., 2010, A&A, 510, A76
  * Logan et al. (2022) Logan C. H. A., Maughan B. J., Diaferio A., Duffy R. T., Geller M. J., Rines K., Sohn J., 2022, arXiv e-prints, p. arXiv:2202.08569
  * Madsen et al. (2017) Madsen K. K., Christensen F. E., Craig W. W., Forster K. W., Grefenstette B. W., Harrison F. A., Miyasaka H., Rana V., 2017, Journal of Astronomical Telescopes, Instruments, and Systems, 3, 044003
  * Madsen et al. (2021) Madsen K. K., Forster K., Grefenstette B. W., Harrison F. A., Miyasaka H., 2021, arXiv e-prints, p. arXiv:2110.11522
  * Maughan et al. (2008) Maughan B. J., Jones C., Forman W., Van Speybroeck L., 2008, ApJS, 174, 117
  * Maughan et al. (2012) Maughan B. J., Giles P. A., Randall S. W., Jones C., Forman W. R., 2012, MNRAS, 421, 1583
  * Mazzotta et al. (2004) Mazzotta P., Rasia E., Moscardini L., Tormen G., 2004, MNRAS, 354, 10
  * Nevalainen et al. (2010) Nevalainen J., David L., Guainazzi M., 2010, A&A, 523, A22
  * Okabe & Smith (2016) Okabe N., Smith G. P., 2016, MNRAS, 461, 3794
  * Planck Collaboration et al. (2020a) Planck Collaboration et al., 2020a, A&A, 641, A6
  * Planck Collaboration et al. (2020b) Planck Collaboration et al., 2020b, A&A, 641, A6
  * Pratt et al. (2001) Pratt G. W., Arnaud M., Aghanim N., 2001, in Neumann D. M., Tran J. T. V., eds, Clusters of Galaxies and the High Redshift Universe Observed in X-rays. p. 38 (arXiv:astro-ph/0105431)
  * Pratt et al. (2019) Pratt G. W., Arnaud M., Biviano A., Eckert D., Ettori S., Nagai D., Okabe N., Reiprich T. H., 2019, Space Sci. Rev., 215, 25
  * Rojas Bolivar et al. (2021) Rojas Bolivar R. A., Wik D. R., Giacintucci S., Gastaldello F., Hornstrup A., Westergaard N.-J., Madejski G., 2021, ApJ, 906, 87
  * Salvati et al. (2021) Salvati L., et al., 2021, arXiv e-prints, p. arXiv:2112.03606
  * Schellenberger et al. (2015) Schellenberger G., Reiprich T. H., Lovisari L., Nevalainen J., David L., 2015, A&A, 575, A30
  * Smith et al. (2001) Smith R. K., Brickhouse N. S., Liedahl D. A., Raymond J. C., 2001, ApJ, 556, L91
  * Taylor (2005) Taylor M. B., 2005, in Shopbell P., Britton M., Ebert R., eds, Astronomical Society of the Pacific Conference Series Vol. 347, Astronomical Data Analysis Software and Systems XIV. p. 29
  * Vikhlinin (2006) Vikhlinin A., 2006, ApJ, 640, 710
  * Wan et al. (2021) Wan J. T., Mantz A. B., Sayers J., Allen S. W., Morris R. G., Golwala S. R., 2021, MNRAS, 504, 1062
  * Weisskopf et al. (2000) Weisskopf M. C., Tananbaum H. D., Van Speybroeck L. P., O’Dell S. L., 2000, in Truemper J. E., Aschenbach B., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4012, X-Ray Optics, Instruments, and Missions III. pp 2–16 (arXiv:astro-ph/0004127), doi:10.1117/12.391545
  * Wik et al. (2014) Wik D. R., et al., 2014, ApJ, 792, 48
  * Willingale et al. (2013) Willingale R., Starling R. L. C., Beardmore A. P., Tanvir N. R., O’Brien P. T., 2013, MNRAS, 431, 394
  * von der Linden et al. (2014) von der Linden A., et al., 2014, MNRAS, 443, 1973

##  Appendix A Cluster images

The Chandra and NuSTAR images of the eight clusters in the sample are shown in
Figures 9 \- 11b. In each image, the magenta circle shows the region in which
the source spectra were extracted. The white square shows the region covered
by the NuSTAR detectors overlaid on the Chandra image.

(a)

(b)  (d)  (f)

A2146

A2163

A2256

(c)

(e)

Figure 9: Chandra (left) and NuSTAR (right) images of A2146, A2163 and A2256.
The NuSTAR images are background subtracted and in the $3-20$ keV energy band,
while the Chandra image is in the $0.7-2$ keV energy band. All images are
smoothed with a Gaussian with $\sigma=6.25"$. The magenta circle shows the
region in which the source spectra were extracted. The white square shows the
region covered by the NuSTAR detectors overlaid on the Chandra image.

(a)

(b)  (d)  (f)

A523

A665

A754

(c)

(e)

Figure 10: Chandra (left) and NuSTAR (right) images of A523, A665 and A754.
The NuSTAR images are background subtracted and in the $3-20$ keV energy band,
while the Chandra image is in the $0.7-2$ keV energy band. All images are
smoothed with a Gaussian with $\sigma=6.25"$. The magenta circle shows the
region in which the source spectra were extracted. The white square shows the
region covered by the NuSTAR detectors overlaid on the Chandra image.

(a)

(b)  (d)  Figure 11: Chandra (left) and NuSTAR (right) images of 1E 0657-56
and RX J1347.5-1145. The NuSTAR images are background subtracted and in the
$3-20$ keV energy band, while the Chandra image is in the $0.7-2$ keV energy
band. All images are smoothed with a Gaussian with $\sigma=6.25"$. The magenta
circle shows the region in which the source spectra were extracted. The white
square shows the region covered by the NuSTAR detectors overlaid on the
Chandra image.

1E 0657-56

RX J1347.5-1145

(c)

##  Appendix B Temperature profiles

Figures 12 and 13 show the temperature profiles created using Chandra for the
eight galaxy cluster in this sample, following the methods described in §3.2,
3.3 and 5.4. Overlayed are the global temperatures of the clusters measured by
Chandra and NuSTAR across the chosen source region.

(a)

(b)

(c)

(d)

Figure 12: Temperature profiles of clusters A2256, A665, A754 and A2163
plotted in green. The global temperature across a circular region of 250
arcseconds is shown for Chandra (red) and NuSTAR (blue). Chandra and NuSTAR
temperatures are measured in the $0.6-9$ keV and $3-10$ keV bands
respectively. Notes that due to problems in the background of the Abell 2163
Chandra observation at low energies (§3.4.2), the global temperature from the
fit between 0.6-9 keV was not used in our main analysis. The temperature fit
between 3-10 keV (pink) was alternatively plot.

(a)

(b)

(c)

(d)

Figure 13: Temperature profiles of clusters A523, A2146, 1E 0657-56 and RX
J1347.5-1145 plotted in green. The global temperature across a circular region
of 250 arcseconds is shown for Chandra (red) and NuSTAR (blue). Chandra and
NuSTAR temperatures are measured in the $0.6-9$ keV and $3-10$ keV bands
respectively.