© The Authors 2024

1. Introduction

Galaxy clusters, the largest virialized systems in the Universe, play a critical role in our understanding of various astrophysical and cosmological phenomena. X-ray observations of their most massive baryonic component, the hot intracluster medium, offer a wealth of information on a wide range of cluster properties. Among these, the intracluster medium temperature (T) is particularly valuable, as it is a key factor in many cosmological applications of galaxy clusters.

Widespread cosmological use of galaxy clusters comes through the halo mass function, which obtains tight cosmological constraints by modeling the number of cluster halos per mass and redshift. For such a test, an unbiased modeling of cluster masses is of immense significance. Accurate measurements of T profiles are used to determine the total hydrostatic mass of clusters, which can then be utilized to constrain cosmology (e.g., Schellenberger & Reiprich 2017b)¹. Moreover, since T is a low-scattered proxy of the total cluster mass, the latter can also be directly estimated through the mass-temperature scaling relation M − T (e.g., Lovisari et al. 2015, 2020; Bulbul et al. 2019). Another vital need for high-quality T measurements arises from the studies of cosmic isotropy with cluster scaling relations (Migkas et al. 2021). Measuring T does not require any strong cosmological assumptions. As such, the directionality of its correlation with other cosmology-dependent cluster properties constitutes a powerful test of the isotropy of cosmic expansion and the existence of bulk flows in the local Universe. Furthermore, X-ray gas T measurements are an essential component in a substantial number of astrophysical studies, including assessments of the impact of baryonic feedback on cluster scaling relations (e.g., Mittal et al. 2011) as well as the detection of the warm-hot intergalactic medium (e.g., Werner et al. 2008; Eckert et al. 2015), and many others. It is therefore evident that an unbiased determination of T is of utter importance for cluster science.

However, it is well established that discrepancies exist between cluster T measurements obtained by various X-ray telescopes. Schellenberger et al. (2015, S15 hereafter) showed that XMM-Newton/EPIC (hereafter, XMM-Newton) returned systematically lower T values than Chandra/ACIS (hereafter, Chandra) using a sample of 64 galaxy clusters observed by both telescopes. This discrepancy was found to increase as a function of T from consistency for T ≈ 1 keV clusters² to a ≈29% higher ChandraT (T_Chandra hereafter) for T ≈ 10 keV clusters. Furthermore, the temperature difference was stronger in the soft X-ray than in the hard X-ray band. Other studies that used only eight to 11 clusters found similar, often milder, cluster temperature discrepancies between XMM-Newton, Suzaku, Chandra, and NuSTAR (Nevalainen et al. 2010; Kettula et al. 2013; Wallbank et al. 2022).

If T disagreements between different X-ray telescopes are not properly taken into account, they can lead to several systematic biases. For instance, S15 showed that using T values only from XMM-Newton or Chandra without accounting for their cross-calibration issue results in an 8% difference in the estimated Ω_m when using hydrostatic masses and the cluster mass function. Moreover, using the Lovisari et al. (2020)M − T relation, a 10% bias in T shifts the total cluster masses by 16%. Finally, when testing cosmic isotropy with cluster scaling relations, a 10% systematic difference in T causes a 12% shift in H₀, inducing artificial anisotropies (Migkas et al. 2021).

Cluster science will significantly progress in the coming years thanks to the X-ray cluster catalogs provided by the extended ROentgen Survey with an Imaging Telescope Array (eROSITA; Merloni et al. 2012; Predehl et al. 2021). On board the Spectrum-Roentgen-Gamma (SRG) space observatory (Sunyaev et al. 2021) launched in July 2019, eROSITA consists of seven telescope modules (TMs), each with 54 nested mirror shells. SRG/eROSITA (hereafter, eROSITA) will conduct a four-year survey, mapping the entire X-ray sky for the first time in the 21st century. Thanks to its large effective area and broad X-ray energy range coverage, eROSITA is expected to detect all massive galaxy clusters (M ≥ 3 × 10¹⁴ M_⊙) in the observable Universe away from the Galactic plane (Merloni et al. 2012; Pillepich et al. 2012). For many thousands of these clusters, precise spectroscopic T measurements will become possible (Borm et al. 2014), significantly increasing the cluster sample size with available T values. Thus, a cross-calibration of eROSITA-measured cluster T with those from XMM-Newton and Chandra is of utter importance for studies that wish to jointly analyze data coming from these X-ray telescopes.

Recently, Turner et al. (2022) used eight clusters from the eROSITA Final Equatorial Depth Survey (eFEDS) cluster catalog (Liu et al. 2022) and found that eROSITA returns 25 ± 9% lower T than XMM-Newton in the broad X-ray band, although different energy bands were used for the two instruments. Due to the limited data, the dependence of the discrepancy on cluster temperature could not be constrained. That work provided a first indication of the cross-calibration between these two telescopes but lacked the statistical power for a more precise comparison and a deeper investigation of the discrepancy. Other studies that focused on the eROSITA analysis of single clusters also found temperature discrepancies between eROSITA, XMM-Newton, and Chandra. Sanders et al. (2022) and Whelan et al. (2022) found that eROSITA showed higher T than XMM-Newton and only slightly lower T than Chandra for their respective clusters. On the other hand, Veronica et al. (2022) found mildly lower T with eROSITA than XMM-Newton both in the broad and soft bands for another single cluster. Liu et al. (2023) found that, in the full band, eROSITA measures ∼60% and ∼45% lower T than Chandra and XMM-Newton, respectively. The discrepancy was alleviated when they measured the eROSITA T (T_eROSITA hereafter) in the 2–8 keV band, which agreed with the T_Chandra from the broad X-ray band. These results, which are based on single cluster comparisons, often contradict each other, and this indicates that such individual comparisons are probably not sufficient to characterize the cross-calibration of eROSITA with other X-ray instruments. Due to the non-negligible scatter of these relations, single cluster comparisons are subject to noise fluctuations; thus, one needs large cluster samples to robustly quantify any systematic T discrepancy. Moreover, the above studies often used different metal abundances and Galactic absorption values for different instruments when comparing their respective T. To obtain a clearer picture of the eROSITA-Chandra-XMM-Newton cross-calibration, one needs to perform spectral fits with self-consistent “nuisance” parameters across different telescopes so an artificial systematic bias in the T estimation can be avoided.

In this work, we use the data products of the first eROSITA All-Sky survey (eRASS1) to accurately assess the cross-calibration between eROSITA, XMM-Newton, and Chandra. To do so, we measured the cluster gas T using eRASS1 data for (nearly) all systems included in the preliminary extremely expanded HIgh FLux Galaxy Cluster Sample (eeHIFLUGCS) catalog, as presented in Migkas et al. (2020, M20 hereafter), and lying in the western Galactic hemisphere. We measured the core and core-excised T and metallicity Z for 120 and 51 clusters with Chandra and XMM-NewtonT measurements, respectively, from the same cluster areas and for the 0.7–7 keV energy band. For the majority of these clusters, we also compared the spectral fits for the soft (0.5–4 keV) and hard (1.5–7 keV) X-ray bands. Our results provide clear conversion functions between cluster T measurements coming from different X-ray telescopes, which are of great value for any joint cluster analysis of eROSITA, XMM-Newton, and Chandra.

The paper is organized as follows: In Sect. 2, we describe the cluster sample used in this work. In Sect. 3, we describe the data reduction and spectral analysis of the eROSITA, XMM-Newton, and Chandra data. In Sect. 4, we describe the statistical methods followed in order to compare the results across different telescopes and quantify the significance of any differences. In Sect. 5, we present the cross-calibration results between all telescopes and energy bands. In Sect. 6, we present the broken power law fits for the eROSITA-Chandra scaling relations. In Sect. 7, we discuss the possible reasons behind the observed T discrepancies and the impact of this work. Finally, in Sect. 8, the conclusions of this work are given.

2. Sample

The galaxy cluster sample used for this work was presented and described in M20. In a nutshell, it is a subsample of the eeHIFLUGCS catalog (Pacaud, et al., in prep.), which is a nearly complete, X-ray flux-limited cluster sample. eeHIFLUGCS was selected based on a re-analysis of ROSAT data of the clusters included in the Meta-Catalogue of X-ray detected Clusters of galaxies (MCXC; Piffaretti et al. 2011). After masking the Galactic plane (|b|≤20°) and the Magellanic clouds, the only selection criterion is an unabsorbed X-ray flux of f_X ≥ 5 × 10⁻¹² ergs/s/cm² in the 0.1–2.4 keV band. Only clusters with sufficient XMM-Newton or Chandra data were kept in the M20 sample. That allowed for a precise T measurement within the 0.2 − 0.5 R₅₀₀³ cluster annulus in the 0.7–7 keV band. Moreover, multiple cluster systems and clusters with strong AGN contamination were discarded. This resulted in 313 clusters in M20. In Migkas et al. (2021) we measured more properties for this sample, including the core (< 0.2R₅₀₀) T, which is also utilized here. The M20 R₅₀₀ values used in this work were taken from MCXC and they were calculated through the X-ray luminosity-total mass scaling relation of Arnaud et al. (2010) for a flat ΛCDM cosmology with H₀ = 70 km/s/Mpc and Ω_m = 0.3. For very few clusters, the MCXC R₅₀₀ were further corrected in M20 due to the additional cleaning of the MCXC X-ray luminosity (see M20 for details).

In this work, we use the western Galactic hemisphere eRASS1 data to which the German eROSITA Consortium holds proprietary rights. This results in 155 different clusters from M20, which is the sample used in this work. Out of these 155, 111 are analyzed with Chandra and 53 with XMM-Newton data (with nine clusters analyzed with both instruments). All cluster data used in this work are publicly available⁴.

3. Data analysis and cluster measurements

3.1. eROSITA

3.1.1. Data reduction

We used the first eROSITA-All Sky Survey data (eRASS:1) with processing version 010. The products of the eRASS are divided into 4, 700 sky tiles, each of the size of slightly overlapping 3.6 × 3.6° area (Predehl et al. 2021). We downloaded all sky tiles where the clusters of our sample are located, as well as some surrounding sky tiles for sky background estimation. The calibration and data reduction steps of the eRASS:1 data were performed with the extended Science Analysis Software (eSASS) version 211214 (Brunner et al. 2022; Merloni et al. 2024). The first step of the data reduction was to generate clean event lists and photon images. We ran the evtool tasks routine in the energy band 0.2 − 10.0 keV and set the parameters flag=0xc00fff30 to remove bad pixels and the strongly vignetted corners of the square CCDs, and pattern=15 to include all patterns (single, double, triple, and quadruple). The exposure maps of the corresponding event files were generated using the expmap task. If multiple eRASS sky tiles were to be merged, for instance, in the case of very extended clusters, the radec2xy task was used to first align the sky tiles before merging them with evtool task. Count-rate images were generated by dividing the photon images and their corresponding exposure maps.

3.1.2. eSASS source detection chain

We used the eSASS source detection chain to identify and exclude point sources from the eROSITA data⁵. The eSASS source detection chain consists of four consecutive tasks, relying on the results already obtained during image creation. The first step is to run the erbox task in local mode, which is based on a sliding box algorithm and detects peaks in the image by estimating a background. It supplies a first list of possible sources in the image as well as an updated detection mask excluding the source positions (cheese map). Next, the task erbackmap creates a background map from an image by masking the sources found by erbox and smoothing. Afterward, another iteration of the task erbox is performed in map mode. It uses the background map to create a more accurate list of sources. Finally, the ermldet task is used to characterize the sources as point or extended sources. It supplies a final source list and a source image. Additionally, the task catprep can be used to convert the source list into a source catalog fits file. All point and extended sources were masked with a radius large enough so the immediate surroundings of the mask converged to the local background level. There are very few cases were extended sources needed masking, due to the way the M20 sample was constructed (excluding clusters with nearby extended sources, e.g., double clusters). After the completion of the eSASS source detection chain, a visual inspection was performed to manually mask apparent sources that were missed by the algorithm.

3.1.3. Spectral analysis

To extract the source and background spectra, and their response matrix files (RMFs) and ancillary response files (ARFs), the task srctool was used. We extracted spectra of all seven telescope modules combined (referred to as TM0), as well as the combinations of TM1,2,3,4,6 (TM8), and TM5,7 (TM9). The latter are the cameras affected by the optical light leak at the very soft X-ray bands (Predehl et al. 2021). To compare the results between eROSITA, XMM-Newton, and Chandra, one wants to keep the temperature measurement method as similar as possible. For this purpose, we use the exact same spectra extraction regions for the clusters as used in M20. These are the 0.2 − 0.5R₅₀₀ annuli centered at the X-ray emission peak as seen by XMM-Newton or Chandra, as well as the cluster core region, < 0.2R₅₀₀.

For the masking, two different approaches were used. In the first case, the masking process described in Sect. 3.1.2 was followed, based solely on the eROSITA point source detection and manual masking after visual inspection. As a second approach, the same masks as in the XMM-Newton and Chandra data were used. For the eROSITA-XMM-Newton clusters, we used the exact same masks as in M20, obtained from the XMM-Newton observations. For the eROSITA-Chandra clusters, we masked the same point sources as in the Chandra data, but with a 30″ radius mask to account for the larger eROSITA point spread function (PSF). The two approaches returned nearly indistinguishable temperature results for nearly all clusters. This is due to the fact that we focus on central cluster regions (< 0.5R₅₀₀) with high surface brightness, where minor changes in the background treatment usually have a negligible impact. Therefore, we adopted the first approach as the default one, to better “simulate” the measurements of eROSITA temperatures that future studies will wish to convert to XMM-Newton or Chandra temperatures based on our findings.

The background region of each cluster was defined based on two criteria. Firstly, the inner radius should correspond to the 1.6R₅₀₀ of the respective cluster (as in M20) so that the background spectra do not include bright residual emission from the cluster. The outer radius is calculated such that the background area is at least four times as large as the source region. In some cases, the background spectra had very few counts. In those cases, the outer radii were scaled up so that each background spectrum is expected to contain at least 300 counts, and the source detection chain was reiterated.

The spectral fitting was performed with XSPEC (Arnaud 1996) version 12.12.0. The model can be described as

$$\begin{array}{cc}& \mathtt{Model}=\mathtt{constant}\times [\mathtt{ape}{c}_1+\mathtt{phabs}\times (\mathtt{ape}{\mathtt{c}}_2+\hfill \\ \hfill & \mathtt{powerlaw})]+\mathtt{phabs}\times \mathtt{ape}{\mathtt{c}}_3+\mathrm{PIB}.\hfill \end{array}$$(1)

The constant (arcmin²) term represents the cosmic X-ray background (CXB) components scaled to the areas of the source regions. The components consist of the unabsorbed thermal emission from the Local Hot Bubble (LHB; apec₁), the absorbed Milky Way Halo (MWH; apec₂), and the cosmic X-ray background from the unresolved sources (powerlaw). The absorption along the line of sight by the Milky Way is modeled by a phabs model and its parameter is set to the N_H,tot values from Willingale et al. (2013; same as in M20). The temperatures for LHB and MWH were fixed at 0.099 and 0.225 keV respectively (McCammon et al. 2002), while the photon index of the powerlaw was fixed at 1.46 (e.g.; Luo et al. 2017). The second term is for the source spectra, which is an absorbed thermal emission component (phabs×apec₃). The last term (PIB) describes the particle-induced (instrumental) background, which is modeled as in Veronica et al. (2024, see that paper for details). For this, we adapted the results of the eROSITA EDR filter wheel closed (FWC)⁶ data analysis. The FWC spectra were rescaled to the observed spectra using the 7 − 9 keV band, where the PIB dominates. The normalizations of all PIB components were left to vary. The best-fit results were then used as starting points in the subsequent spectral fittings, where the PIB normalizations were left free to vary within the 3σ range of the best-fits.

The spectral fitting was performed in three different bands: the 0.7 − 7.0 keV (full, same band used in M20), 0.5 − 4.0 keV (soft), and 1.5 − 7.0 keV (hard) bands⁷. All spectra (CXB+source) were fitted simultaneously for individual TMs for each band. Moreover, the TM8 and TM9 modules were first fitted separately. As discussed in Sect. 7.4, the T results of the two modules agree within ≲2σ⁸. Therefore, we fit simultaneously both modules, that is, we use TM0 as the default for the main analysis. Additionally, we performed all fits with the metal abundance (Z) both fixed to the M20 value and free to vary. To determine the best-fit spectral parameters, we used C-statistics (Cash 1979), which is a more suitable estimator for Poissonian counts. The Solar metal abundance table from Asplund et al. (2009) was also used.

Finally, due to its low sensitivity at high energies (≳2.5 keV), eROSITA was unable to constrain T⁹ in the full band for 11% (23) and 33% (21) of the cluster regions for which we have Chandra and XMM-NewtonT respectively. These fractions are similar for the soft band T as well. For the hard band, this fraction increases to 47% (96) and 60% (56) respectively. All T/σ_T < 1 clusters were excluded from the analysis (the negligible effects of this selection are explored in Sect. 7.6). For the eROSITA-Chandra cross-calibration, this resulted in 185 (full band), 179 (soft band), and 108 (hard band) available cluster temperatures. The median T/σ_T is 4.4, 4.2, and 2.3 respectively. For the eROSITA-XMM-Newton cross-calibration, this resulted in 71 (full band), 68 (soft band), and 38 (hard band) cluster temperatures. The median T/σ_T is 3.2, 3.0, and 1.9 respectively.

3.2. Chandra and XMM-Newton

3.2.1. Data reduction

The data reduction and analysis of the Chandra and XMM-Newton observations are described in detail in Schellenberger & Reiprich (2017a) and Ramos-Ceja et al. (2019) respectively. For Chandra the ACIS I/S instruments were used, with no grating (no HETG/LETG). For XMM-Newton, the EPIC (MOS1/MOS2/PN) instrument was used. In short, pointed observations of both Chandra and XMM-Newton were treated for solar flare contamination, bad pixels, anomalous CCD state (for XMM-Newton only; Kuntz & Snowden 2008) and out-of-time events, masking of point sources and extended emission sources not related to the cluster, vignetting, exposure time correction, and instrumental background. The HEASOFT 6.20, XMMSAS v16.0.0, and CIAO v4.8 with CALDB 4.7.6 software packages were used in the M20 analysis and the full band results. Any additional spectral fitting in this work was performed using the already available spectra from M20, produced by the same software packages. For the additional soft and hard band XMM-Newton spectral analysis, we use the same software packages. For the additional Chandra spectral fitting, the CIAO v4.13 software package was used with CALDB 4.9.4¹⁰.

3.2.2. Spectral analysis

The spectral analysis methodology for both Chandra and XMM-Newton is described in M20. In brief, we extracted two independent spectra per cluster, from within the ≤0.2R₅₀₀ and 0.2 − 0.5R₅₀₀ regions. For Chandra, the PIB was obtained from the stow event files (Schellenberger & Reiprich 2017a). It was renormalized to the 9.5–12 keV count-rate of the cluster observation before being subtracted from the source spectra. The CXB was extracted from within 1° −2° around the cluster using the seven ROSAT All-Sky Survey (RASS) bands (Snowden et al. 1997), as done during the M20 measurements. For XMM-Newton, the PIB was measured utilizing filter wheel closed observations. It was then rescaled to the cluster observation count-rates of the unexposed EPIC corners (> 925″ from the pointing’s center) using the 2.5 − 5 keV (MOS and PN) and 8 − 9 keV (MOS) energy bands. The CXB was extracted from the XMM-Newton field of view (FOV), from regions at ≳1.6 × R₅₀₀ from the cluster’s center. When this was not possible, the CXB was extracted from the outer 1′ width annulus of the FOV. In this case, an extra, free-to-vary apec component was added to the CXB model to account for residual cluster emission.

The fitted model is the same as for eROSITA (Sect. 3.1.3), without the PIB component. For both Chandra and XMM-Newton, the rescaled PIB was subtracted from the source spectra. To account for an imperfect PIB subtraction, additional Gaussian lines are added to the model, representing fluorescence line residuals (for details see Schellenberger & Reiprich 2017a; Ramos-Ceja et al. 2019; M20). The CXB-only spectra were first fitted alone. The best-fit CXB values are then used as starting points and the full model (cluster emission+CXB) is left free to vary when fitting the source spectra. All properties of each model component (e.g., normalizations, temperatures, and metallicities) were linked across the three EPIC detectors during the XMM-Newton fits. For each spectrum, different fits are performed for three energy bands, as in eROSITA. The 0.7 − 7 keV results were taken from M20. For this work, we additionally performed the 0.5 − 4 keV and 1.5 − 7 keV spectral fits for all spectra, leaving Z free to vary and fixing N_H to the M20 value. For consistency with M20, we used XSPEC v12.9.1 for both telescopes. A χ²-statistic was used for both Chandra and XMM-Newton to determine the best-fit spectral parameters, as in M20¹¹.

4. Statistical methodology

For the comparison of different temperature measurements, we adopt a linear relation in the logarithmic space of the form:

$$\begin{array}{c}\hfill {log}_{10}\frac{T_{\text{Y}}}{T_{\text{piv}}}=A+B\times {log}_{10}\frac{T_{\text{X}}}{T_{\text{piv}}},\end{array}$$(2)

where T_Y and T_X are the compared temperature distributions and A and B are the intercept and slope of the relation respectively. Moreover, T_piv = 4.5 keV for eROSITA-Chandra comparisons and T_piv = 3 keV for eROSITA-XMM-Newton comparisons (i.e., rounded median values of Chandra and XMM-NewtonT distributions). Equation (2) corresponds to a single power law in linear space. A broken (double) power law fit was also tested. However, for all temperature comparisons, it was “strongly disfavored” or “disfavored” by the Bayesian and Akaike Information Criteria (BIC and AIC respectively) since it did not improve the fit significantly to justify the introduction of extra free parameters. Nevertheless, we present the results for a broken power law fit in Sect. 6 for the eROSITA-Chandra scaling relations for which a broken power law fit showed the greatest improvement compared to a single power law fit.

For the linear regression of Eq. (2) we use a likelihood maximization method (LMM hereafter) as the default. Specifically, we maximize the log-likelihood function

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ln\mathcal{L}& =\hfill \\ \hfill =& -\frac{1}{2}{\displaystyle \sum _{i=1}^N}[\frac{{(logT{{}^{\prime }}_{\text{Y},i}-A-B\times logT{{}^{\prime }}_{\text{X},i})}^2}{{\sigma }_i^2+{\sigma }_{\text{intr}}^2}+ln({\sigma }_i^2+{\sigma }_{\text{intr}}^2)],\hfill \end{array}\end{array}$$(3)

where

$$\begin{array}{c}\hfill T{{}^{\prime }}_{\text{Y},i}=\frac{T_{\text{Y},i}}{T_{\text{piv}}},T{{}^{\prime }}_{\text{X},i}=\frac{T_{\text{X},i}}{T_{\text{piv}}},\text{and}{\sigma }_i^2={\sigma }_{log{T}_{\text{Y},i}i}^2+B^2\times {\sigma }_{log{T}_{\text{X},i}}^2,\end{array}$$(4)

and σ_intr is the intrinsic scatter of the relation. Moreover, ${\sigma }_{logT}=loge\times \frac{T_{\text{max}}-T_{\text{min}}}{2T}$ are the T measurement uncertainties in log-space, symmetrized from the linear-space 68.3% T uncertainties. The fitted parameters here are A, B, and σ_intr. The total scatter σ_tot is given by the mean quadratic sum of σ_i and σ_intr.

Additionally to LMM we also utilize the Bivariate Correlated Errors and intrinsic Scatter (BCES) package by Akritas & Bershady (1996). For every scaling relation, the best-fit values of A and B were obtained using the BCES(orth) method. The latter minimizes the orthogonal distance of the data points compared to the best-fit line, as opposed to LMM, which mostly considers the residuals on the y-axis. Due to the much larger measurement uncertainties of T_eROSITA, a minimization in the y-axis (as in LMM) is more appropriate than an orthogonal distance regression. Nevertheless, BCES(orth) serves as a measure of how sensitive our results are on the exact fitting method.

Similarly to Pratt et al. (2009), the total scatter in the y-axis σ_tot is measured using the error-weighted vertical distance to the best-fit regression line:

$$\begin{array}{c}\hfill {\sigma }_{\text{tot}}^2=\frac{1}{N-2}{\displaystyle \sum _{i=1}^N}{w}_i{(Y_i-A_{\text{orth}}-B_{\text{orth}}\times X_i)}^2,\end{array}$$(5)

where $Y={log}_{10}\frac{T_{\text{Y},i}}{T_{\text{piv}}}$ and $X={log}_{10}\frac{T_{\text{X},i}}{T_{\text{piv}}}$. Furthermore,

$$\begin{array}{c}\hfill \begin{array}{cc}& w_i=\frac{1/{\sigma }_i^2}{(1/N){\sum }_{i=1}^{N}1/{\sigma }_i^2}\text{and}{\sigma }_i^2={\sigma }_{Y_i}^2+B^2\times {\sigma }_{X_i}^2.\hfill \end{array}\end{array}$$(6)

Here, σ_intr is then given by the quadratic difference between σ_tot and σ_i. Although we use BCES(orth), we measure the scatter in the y-axis direction (instead of the orthogonal scatter) to make a direct comparison with the results of LMM.

For both LMM and BCES, the 68.3% (1σ) credible intervals of fitted model parameters are estimated by performing 10 000 bootstrap resamplings with replacement. From the posterior distribution of the best-fit parameters, we determine the 16th and 84th percentiles. In the 2-dimensional planes, the 1, 2, and 3σ ellipses are drawn by finding the ellipse with the smallest area that encompasses the 68.3%, 95%, and 99.7% of the 10 000 bootstrap data points.

Finally, following S15, we define a measure ξ of the statistical deviation between the two compared T distributions in linear space,

$$\begin{array}{c}\hfill \xi =\text{med}\frac{T_{\text{Y}}-T_{\text{X}}}{\sqrt{{\sigma }_{T_{\text{Y}}}^2+{\sigma }_{T_{\text{X}}}^2}},\end{array}$$(7)

where the median of ξ for all clusters is considered. The term ξ can take both positive and negative values, depending on which T distribution is systematically higher or lower. If there is no systematic difference between the two compared instruments, one expects ξ ≈ 0.

5. Single power law fits

In this section we present the results from the cluster T comparison between eROSITA, Chandra, and XMM-Newton. All scaling relation results are shown in Table 1.

5.1. eROSITA versus Chandra temperatures

eROSITA returns systematically lower cluster T than Chandra for all three energy bands. The discrepancy is a function of cluster T, with hot, massive clusters deviating more than low T groups. For the latter, the two instruments seem to return similar T across all energy bands. Overall, the discrepancy between eROSITA and Chandra cluster T is the strongest for the hard energy band. This is unlike what was found in previous studies (e.g, S15; Kettula et al. 2013) where X-ray instruments tend to return consistent T in the hard band (> 2 keV) while showing stronger disagreement in the soft band (< 2 keV). However, the band definition in this work is different than past studies and the soft and hard bands overlap at 1.5 − 4 keV; therefore, a direct comparison is challenging.

5.1.1. Full band

The T comparison for the full band between the two instruments is displayed in the left panel of Fig. 1. The relation shows a slope of $B=0.{781}_{-0.023}^{+0.020}$, deviating from the equality line by > 10σ (Fig. 2). The total scatter of the relation (≈30%) is dominated by the T_eROSITA uncertainties of high-T clusters, rather than the intrinsic scatter (≈10%). The latter remains rather constant with increasing T_Chandra (see Appendix A.3 for the details of scatter dependency on T). Due to the energy dependence of eROSITA’s effective area, T_eROSITA uncertainties generally increase with increasing T. Similarly, for a fixed T_Chandra, clusters seen as high-T systems by eROSITA (i.e., upscattered) show larger uncertainties than downscattered clusters that seem to have lower T.

Fig. 1. Comparison between eROSITA and Chandra temperatures for the full (left), soft (middle), and hard (right) bands. The best-fit scaling relation line by LMM (blue) and BCES(orth) (green dashed) are displayed. The equality 1:1 line is shown in red (dash dot). The blue shaded area represents the LMM statistical error plus the intrinsic scatter.

eROSITA returns ≈25% lower T than Chandra for clusters with T_Chandra = 4.5 keV. For more massive clusters with T_Chandra = 10 keV, the deviation rises to ≈38%. On the contrary, galaxy groups with T_Chandra ≲ 2 keV do not show any systematic T difference between the two telescopes. The value of ξ = −1.14 shows that individual clusters do not deviate significantly from the equality line due to their large T_eROSITA uncertainties. Therefore, the large cluster sample used here is crucial to compensate for these individual uncertainties and robustly determine the cross-calibration between eROSITA and Chandra cluster T.

The single power law fit seems to slightly overestimate the expected T_eROSITA for T_Chandra ≈ 1 keV clusters by ≈5% for LMM. Most galaxy groups at this T range seem to lie closer to the equality line. This overestimation is partially caused by a single, upscattered cluster region (the 0.2 − 0.5R₅₀₀ annulus of S0805). For BCES(orth), the expected T_eROSITA is overestimated by 18%. This strongly demonstrates that LMM provides a better fit than BCES in this case. The systematic non-Gaussian scatter at low T may suggest the need for a broken power law. This is presented in Sect. 6.

5.1.2. Soft band

The soft band T comparison is displayed in the middle panel of Fig. 1 and it is very similar to the full band T comparison. All scaling relation parameters are consistent within 1σ with the full band results, as shown in Fig. 2. Once again, LMM shows a steeper slope than BCES(orth), representing the low T systems better. The systematic residuals at low T persist in the soft band as well, while ξ shows that the median T discrepancy between the two instruments is only slightly larger than in the full band. Overall, no significant change is observed in the eROSITA-Chandra cross-calibration by limiting the spectroscopic analysis to softer energies.

Fig. 2. 1σ (68.3%) and 3σ (99.7%) confidence levels for the eROSITA-Chandra scaling relations for the full (blue), soft (pink), and hard (green) bands. The 1:1 line is represented by the black cross. The band best-fit lines for all bands deviate by ≳10σ from the 1:1 line.

5.1.3. Hard band

The T comparison for the hard band between eROSITA and Chandra is displayed in the right panel of Fig. 1. Overall, the offset in the measured T between the two instruments is stronger in the hard band than in the soft and full bands, as shown in Fig. 2. The dependency of the offset on T is also stronger with a slope of B = 0.638 ± 0.076, 1.8σ lower than in the full band. Low T systems seem consistent between the two telescopes for the hard band as well, while high-T clusters deviate more than in the full and soft bands. Specifically, T_Chandra = 1 keV clusters return the same T for both telescopes, while eROSITA measures ≈44% lower T_eROSITA for T_Chandra = 4.5 keV systems and 58% lower T_eROSITA for T_Chandra = 10 keV. ξ also has a higher value than for the other bands. The average intrinsic scatter is ≈3 times larger than in the other bands. Its high value is mostly driven by T_Chandra ≲ 5 keV, and it is rather stable within this T_Chandra range. For hotter clusters the scatter reduces by ≈50%, although the statistical uncertainties are too large to robustly support a T-dependent scatter behavior (Fig. A.3.) Finally, there is no obvious need for a broken power law in this case.

5.2. eROSITA versus XMM-Newton temperatures

eROSITA shows systematically lower cluster T than XMM-Newton for all three energy bands, with the discrepancy being a function of cluster T, similar to the eROSITA-Chandra comparison. However, the discrepancy is milder this time for the full and soft energy bands. For the hard band, the relation is very loosely constrained due to the large scatter. There is no obvious need for a broken power law fit for any band.

5.2.1. Full band

The T comparison for the full band between eROSITA and XMM-Newton is displayed in the left panel of Fig. 3. The relation shows a slope of $B=0.{825}_{-0.066}^{+0.074}$, supporting a T-dependence of the discrepancy between the two instruments at 2.4σ. Considering also the intercept $A=-0.{078}_{-0.017}^{+0.016}$, which deviates more strongly from 1:1 than the slope, the overall relation deviates from the equality line by ≈5σ (Fig. 4)¹², where the bulk of the tension comes from the intercept $A=-0.{078}_{-0.017}^{+0.016}$. The intrinsic scatter is once again low (≈9%) and relatively constant for T_XMM ≲ 4 keV clusters. For hotter systems, it drops to < 1%, although it remains within < 2σ from the average scatter. Nevertheless, this might indicate a T-dependent intrinsic scatter behavior that will be explored when the statistical uncertainties of T_eROSITA reduce with future data. The total scatter is dominated again by the T_eROSITA uncertainties.

Fig. 3. Same as in Fig. 1 but for the comparison between eROSITA and XMM-Newton temperatures.

For galaxy groups with T ≲ 1.5 keV, the two instruments agree within 5.5%. For average-sized clusters with T_XMM = 3 keV, eROSITA measures ≈16% lower T than XMM-Newton. For more massive clusters with T_XMM = 7 keV, the deviation rises to ≈28%. The value of ξ = −0.41 is mostly affected by the clusters with T_eROSITA ≥ T_XMM, which are the ones with large T_eROSITA uncertainties and not much statistical weight.

5.2.2. Soft band

The eROSITA-XMM-Newton cross-calibration in the soft band is shown in the middle panel of Fig. 3. In general, the soft band is slightly closer to the equality line than the full band (Fig. 4), although the two bands give consistent results within 1σ. The slightly better agreement is also shown from the lower ξ = −0.25 value. Overall, the soft band scaling relation deviates by 4.2σ by the equality line, with the main source of tension being the intercept value $A=-0.{065}_{-0.019}^{+0.018}$, which implies that T_XMM = 3 keV clusters show a 14% lower T_eROSITA. Furthermore, eROSITA measures a 23% lower T_eROSITA for T_XMM = 7 keV clusters, while at low-T there is no meaningful difference with the full band, with the two instruments agreeing with each other. As in the eROSITA-Chandra cross-calibration, LMM fitting returns a steeper slope than BCES(orth), representing the low T systems better. Overall, a marginally better agreement between eROSITA and XMM-Newton is observed in softer X-ray energies than in the full band.

Fig. 4. 1σ (68.3%) and 3σ (99.7%) confidence levels for the eROSITA-XMM-Newton scaling relations for the full (blue), soft (pink), and hard (green) bands. The 1:1 line is represented by the black cross. The full and soft band best-fit lines deviate by 5σ and 4.2σ respectively from the 1:1 line.

5.2.3. Hard band

The T comparison for the hard band between eROSITA and XMM-Newton is displayed in the right panel of Fig. 3. Due to the limited number of available T_eROSITA in the hard band, and their large uncertainties, the relation is not as robustly constrained as for the other bands. This is also clear from the large slope difference that LMM and BCES(orth) return. Nevertheless, one can safely conclude that eROSITA measures much lower T than XMM-Newton in the hard band. The average T_eROSITA underestimation for a T_XMM = 3 keV cluster is 55%. The slope is flatter (but highly uncertain) while the average intrinsic scatter is ≈1.5 − 2 times larger than the respective full and soft band values. The intrinsic scatter also remains stable for T_XMM ≲ 4 keV, while it decreases to almost zero for hotter clusters, as in the full band (but still within ≲1.5σ from the average). We need to note that before we constrain the scaling relation, two strong outliers were excluded despite fulfilling the criteria to be in the sample. More details can be found in Sect. A.4.

6. Broken power law fits

From Fig. 1 and the systematic residuals at T ≲ 2 keV, it seems that the full and soft band eROSITA-ChandraT scaling relations might be better described by a broken power law at low T instead of a single power law. To explore this, we fit a broken power law

$$\begin{array}{c}\hfill {log}_{10}\left(\frac{T_{\text{Y}}}{T_{\text{break}}}\right)=A_{\text{bpl}}+\{\begin{array}{c}{B}_1\times {log}_{10}\left(\frac{T_{\text{X}}}{T_{\text{break}}}\right),\text{for}{T}_{\text{X}}\le T_{\text{break}}\hfill \\ \hfill \\ B_2\times {log}_{10}\left(\frac{T_{\text{X}}}{T_{\text{break}}}\right),\text{for}{T}_{\text{X}}>T_{\text{break}},\hfill \end{array}\end{array}$$(8)

where T_break is the T where the power law changes (the so-called power law knee) and B₁ and B₂ are the slopes of the power laws before and after T_break. We note that the intercept A_bpl here corresponds to T_break and not T_piv, as in the single power law case. Therefore, A and A_bpl should not be compared. Moreover, the two power law parts have their own σ_intr fitted. The fitting is performed only with LMM since BCES(orth) does not allow for a broken power law fit. Finally, the T_break uncertainty is rather large and degenerates the fit of all other parameters as well, making the fit non-informative. Therefore, we fix T_break to its best-fit value.

We stress again that both the BIC and AIC criteria disfavor the broken power law compared to the single power law fit. This is because the improvement of the fit (i.e., an increase of model likelihood lnℒ) is not enough to justify the extra, fitted model parameters¹³. Nevertheless, this is probably due to the limited number of low-T clusters to which the first part of the broken power law is fit and not due to the lack of a real break in the scaling relation. Therefore, the broken power law results might be more accurate for converting eROSITA and ChandraT in the full and soft bands (i.e., reduced scatter compared to single power law), even if this is not strictly justified from BIC and AIC. Finally, all broken power law scaling relation results are shown in Table 2.

6.1. Full band

The fit results for eROSITA-Chandra full band comparison are shown in the top panel of Fig. 5. The break in the power law is detected at 2.7 keV, where eROSITA measures 7% lower T than Chandra. For lower T, the slope is B₁ = 0.946 ± 0.022, very close to unity. Combined with the intercept A_bpl = −0.031 ± 0.008, one sees that the low-T part is similar to the 1:1 line as T decreases, although the statistical deviation from the equality line is still ≈4σ. More specifically, eROSITA shows only ≈2 − 5% lower T for T_Chandra ≈ 1 − 2 keV clusters, lower than typical measurement uncertainties. The intrinsic scatter is also very small; 3.7% for T ≤ 2.7 keV, consistent with zero at 2σ.

Fig. 5. Same as in Fig. 1 but with a broken power law fit with LMM for the full (top) and soft (bottom) bands, as shown in Eq. (8). The break point of the power law is found at Tbreak = 2.7 keV and 1.7 keV, respectively. The low-T clusters are much better fit than the single power law fit, showing consistency between instruments. At high-T, the best-fit line deviates more from the 1:1 line than the single power law fit.

For T > 2.7 keV the slope B₂ = 0.614 ± 0.024, strongly demonstrating the more highly evolving discrepancy between the two instruments as T increases. B₂ is 24% flatter than the single power law slope and 43% lower than B₁. The intrinsic scatter is ≈2.8 times larger than the respective low-T value, at 10%. Overall, the broken power law fit brings closer the two instruments in terms of measured T compared to the single power law fit for T ≤ 5.3 keV clusters, while it increases their discrepancy for T > 5.3 keV systems.

6.2. Soft band

The soft band fit results for the eROSITA-Chandra comparison are displayed in the bottom panel of Fig. 5. The break in the power law is detected at 1.7 keV. For T ≤ 1.7 keV, T_eROSITA and T_Chandra are consistent within ≤2.5%. The low-T scaling relation with B₁ = 0.979 ± 0.042 and A_bpl = −0.011 ± 0.009 is consistent with the 1:1 line at only ≈1σ, closer than the full band fit. The intrinsic scatter is again small and relatively consistent with zero. Thus, no significant deviation from the 1:1 line is observed at the soft band for low-T clusters.

For T > 1.7 keV, the relation deviates from 1:1 as T rises, with B₂ = 0.700 ± 0.019. However, it returns a less discrepant result than the full band broken power law at high T. The intrinsic scatter is > 3 times larger than the respective low-T value, at 13%. Compared to the single power law fit, the broken power law brings closer the measured T of the two instruments at T ≤ 4.3 keV, while it increases their tension for T > 4.3 keV clusters.

7. Discussion

The accurate characterization of the cross-calibration of different X-ray telescopes is crucial to understand the systematic biases and uncertainties of the effective area calibration of X-ray instruments. Furthermore, if the cross-calibration factor between two instruments is not well-known, it hinders the joint analysis of data sets coming from different telescopes.

In the previous sections, we used a large galaxy cluster T sample for the first time to establish that eROSITA shows systematically lower cluster T compared to Chandra and XMM-Newton. This discrepancy was found to be stronger the hotter a cluster is, with T ≲ 2 keV clusters not showing significant differences between eROSITA and the other instruments. In general, the soft energy band 0.5 − 4.0 keV showed slightly better agreement between different instruments than the full 0.7 − 7.0 keV band. At the same time, restricting the spectral fitting to hard X-ray energies (1.5 − 7.0 keV) significantly increased the difference between T_eROSITA and the other T measurements. All the above indicates that the cross-calibration between eROSITA, Chandra, and XMM-Newton, is a function of spectral shape and energy range used, with the softer spectra showing better agreement.

Opposite to the eROSITA cross-calibration results, most past studies found that the cluster T differences between other X-ray instruments seem to be stronger at soft X-ray energies and more consistent at harder X-ray bands. In general, these studies failed to pinpoint a specific systematic causing these cross-calibration differences, although a wide range of possible causes was investigated (e.g., S15). The observed cluster T differences were attributed to systematic effective area calibration uncertainties. In this section, we discuss and examine the possible systematic biases behind the established discrepancy between eROSITA T with Chandra and XMM-NewtonT.

7.1. Metal abundance degeneracy with temperature

The determination of T through spectral fitting is partially influenced by the (mildly) correlated metal abundance Z. When converting T values between different X-ray instruments, accounting for the unknown Z value of the second instrument is not feasible; thus one assumes the Z from the first instrument. For this reason, we adopted the Z values from Chandra and XMM-Newton measurements to use during the eROSITA spectral fitting. However, since the best-fit T and Z values are correlated, fixing Z could potentially bias the T_eROSITA values. In other words, if eROSITA measured systematically different Z and consistent T values, fixing Z might erroneously interpret this as a T discrepancy. Furthermore, due to the so-called “Fe bias” (e.g.; Buote 2002; Mernier et al. 2018; Riva et al. 2022), forcing Z to be the same between instruments with different soft- and hard-band sensitivity ratios (e.g., eROSITA and Chandra) might lead to systematic T discrepancies if multiphase gas is present. To test if fixing eROSITA Z to the Chandra and XMM-Newton values causes a systematic bias, we repeated the entire analysis, this time leaving Z free to vary during the eROSITA spectral fits.

The best-fit results can be found in Table A.1. No scaling relation changes significantly when the new T_eROSITA are used. In fact, the full and soft band scaling relations for both Chandra and XMM-Newton shift further from the 1:1 line for T ≳ 3 keV clusters, while the low-T end maintains the relative consistency that showed for the fixed Z case. In general, the best-fit A and B slightly decrease by a few percent, while the scatter increases. The hard band is the most insensitive one to the varying Z change. To conclude, the discrepancy between eROSITA and other telescopes remains the same or increases for all comparisons when Z is left free to vary, disproving the treatment of Z as the cause of the discrepancy. This is consistent with S15, who also did not find a significant improvement in the Chandra-XMM-Newton cross-calibration when using the same or different Z for the two telescopes.

7.2. Different effective area sensitivity on soft X-ray and hard X-ray bands

7.2.1. Multi-temperature gas structure

The intracluster medium has a multi-temperature structure, which is typically a function of the distance from the cluster center. The observed spectra we use come from 2-dimensional cluster circles and annuli, which include the projection of outer cluster regions in the same line of sight. Hence, the observed spectra include several T gas components that are eventually fit by a single-T model. In addition, multiphase gas might be present at the same cluster radius (e.g, cold and hot gas clouds). In both of these cases, the different sensitivity of the eROSITA effective area at different energies compared to Chandra and XMM-Newton could result in a systematic bias in the measured single-T. As shown in Fig. 6, eROSITA’s effective area drops more rapidly from soft to hard X-ray energies compared to Chandra and XMM-Newton. Consequently, it will potentially assign more weight to the low-T components of multiphase cluster gas, leading to systematically lower measured single-T than the other two telescopes. Such a discrepancy could be also enhanced due to the Fe bias. The latter tends to underestimate single-T measurements when multiphase gas is present, with the effect being stronger for instruments more sensitive to the soft band. The bias introduced to single-T fits due to multiphase gas was discussed in detail in several past studies (e.g.; Reiprich et al. 2013). In past cross-calibration studies, this was disfavored as the reason behind the observed calibration discrepancy between XMM-Newton and Chandra (e.g., S15) and other instruments. Nevertheless, it is important to also test this for the eROSITA calibration to fully understand the importance of this possible systematic.

Fig. 6. Normalized average effective area for eROSITA TM0 (red, solid), XMM-Newton-EPIC (blue, dashed), and Chandra-ACIS (green, dot dashed), as derived from the spectra used in this work.

Observational test. To do so, we first divided our sample into two subsamples: cluster core T and cluster annulus T. The former are expected to show stronger discrepancies between eROSITA and the other instruments. This is due to a higher level of multi-temperature structure in the core spectra caused by projection effects and the presence of the (often complicated) cluster core. The comparisons are performed in the full band since the broader band is expected to be more affected by the soft-hard ratio differences of the instruments’ effective areas. The T distributions of the two subsamples are very similar (see Fig. A.2).

Comparing the T cross-calibration scaling relations for core and annulus spectra, one sees there are no statistically significant differences. As shown in the top panel of Fig. 7, the core versus annulus scaling relations deviate by ≲1σ for both the eROSITA-Chandra and eROSITA-XMM-Newton comparisons. Interestingly, the core T scaling relations for both instruments are slightly shifted toward the 1:1 line (though still significantly far away) despite their more intense bias caused by the multiphase cluster gas. One would need more data to better understand if this shift is a statistical fluctuation or a true effect.

Fig. 7. Confidence levels for the eROSITA-Chandra and eROSITA-XMM-Newton scaling relations for different subsamples, defined based on their cluster properties. Top: 1σ (68.3%) contours for the core-only and annulus-only eROSITA-Chandra and eROSITA-XMM-Newton scaling relations for the full band. The colors are explained in the figure legend. The best-fit results do not differ significantly for core and annuli cluster T for any instrument comparison. Bottom: 1σ (68.3%) and 3σ (99.7%) confidence levels for the eROSITA-Chandra scaling relations for high NH > 5.59 cm−2 (pink) and low NH < 2.45 cm−2 (blue) clusters. The median values of the subsamples are NH = 1.72 cm−2 and NH = 7.96 cm−2, respectively. The contours correspond to the full band. The 1:1 line is represented by the black cross. The best-fit results are very similar between clusters with low and high Galactic absorption.

Simulations from past studies. Moreover, ZuHone et al. (2023) used hydrodynamical simulations to assess the bias that a multi-temperature structure that resembles realistic cluster temperature profiles introduces to single-T fits of eROSITA. They found that spectroscopic single-thermal models underestimate T by ∼10 − 15% compared to the mass-weighted simulated T, with no strong T dependence of this discrepancy. This underestimation is not enough to explain the observed T differences between eROSITA and the other telescopes. On the other hand, compared to the X-ray emission-weighted simulated T, ZuHone et al. (2023) found that a spectroscopic single-thermal model underestimates T by ∼25 − 30%. If there were no such bias for Chandra and XMM-NewtonT, this discrepancy would be enough to alleviate most of the tension between T_eROSITA and the rest. However, both Chandra and XMM-Newton are expected to also return underestimated T from single-thermal fits, but likely at a lesser degree. Until a similar analysis is performed for these two X-ray instruments, it is unclear exactly how the T comparison between them and eROSITA would be affected.

A relevant test was performed by Reiprich et al. (2013) where they fit a single-T model to a two-temperature simulated gas with a hot (T_hot = 8 keV) and a cold (T_cold = 0.5 keV) component. They found that Chandra, XMM-Newton, and eROSITA are all expected to return a best-fit T lying between of the two T components, but eROSITA returns the lowest T. The reported discrepancy is similar to the T differences we observe in this work. However, to illustrate the effect, Reiprich et al. (2013) used extreme T differences that are not present in the data we used, where the expected differences, given the measured core and core-excised T and typical cluster T profiles, are much smaller. Also, as noted in S15, such extreme T differences of a two-phase plasma would result in high reduced chi-square values from the single-T fits; this is not observed in our spectral fits. Finally, in that case one would also expect the eROSITA-Chandra core-T comparison to deviate more than the annulus-T comparison, which is not the case (Fig. 7).

Simulations in this work. To test the effects of multi-temperature structure on eROSITA T more extensively, we used XSPEC to simulate multi-temperature plasma spectra, following the same approach as in Reiprich et al. (2013) and S15. We simulated spectra with two temperature components and fit them with a single temperature model, as for the real data. We used T_cold = 0.5, 0.75, 1 keV and T_hot = 1, 2, 4, 6 keV, covering a wide and realistic range of T combinations. The redshift and metallicity were kept fixed to z = 0.05 and Z = 0.3 Z_⊙ respectively, which are typical values for the cluster sample we use. Moreover, we used four emission measure ratios (EM) between the cold and hot T components, EM = 0.01, 0.05, 0.10, 0.15 (as explained later, higher EM returned unrealistic results). For every combination, we simulated 125 random spectra, with a typical number of counts for every instrument, based on the real spectra. In total, we simulated and fit 6000 spectra with a single T model for eROSITA (TM0), XMM-Newton (combined EPIC and PN-only), and Chandra (ACIS). We assumed that the calibration of each instrument is perfect and, consequently, any differences between instruments should originate due to their different energy dependence of their effective areas.

The average results for the full band eROSITA-Chandra and eROSITA-XMM-Newton comparisons are displayed in Fig. 8. It is evident that the observed cross-calibration differences are unlikely to be explained due to possible 2T thermal structure in the fitted spectra. although eROSITA returns slightly lower single T values than Chandra and XMM-Newton in all cases, the T difference is always ≤20%. The only combination for which the 2T structure could mostly explain the observed T discrepancies between eROSITA and the other instruments is the most extreme case with T_cold = 0.5 keV and EM = 0.15. However, these models on average return χ² ∼ 1.3 − 1.6, which suggests a bad fit. Such increased χ² values are not generally seen in the real data. Subsequently, such cold, bright T components in the observed spectra are disfavored as the reason for the observed T discrepancy. Higher EM values return even higher χ² values, and as such, they are not explored further. On the other hand, T_cold = 0.5 keV and EM = 0.10 models return χ² ≤ 1.15, which is rather consistent with the observed χ². Such models can alleviate most (but not all) of the T tension between eROSITA-Chandra and eROSITA-XMM-Newton. All other combinations with T_cold ≥ 0.75 keV do not show significant T differences between instruments, regardless of the EM and T_hot values. We reach the same conclusion when we only use the PN detector of XMM-Newton to fit the spectra. Finally, we repeat the analysis using the hard band fits only. In this case, the observed cross-calibration differences cannot be explained by the 2T structure in the simulated data since eROSITA returns very similar single T results to Chandra and XMM-Newton, regardless of the exact model values. The results are presented in Appendix A.5 and Fig. A.4.

Fig. 8. Ratio of the eROSITA and Chandra (left) and eROSITA and XMM-Newton (right) single-T fits to simulated spectra with two temperature components as a function of the Chandra and XMM-Newton best-fit single-T, respectively. The blue, green, and red colors correspond to Tcold = 0.5, 0.75, 1 keV, respectively. The hot component has values of Thot = 1, 2, 4, 6 keV. The circles, triangles, and squares correspond to emission measures of EM = 0.05, 0.10, 0.15 for the cold and hot components. The black curve shows the observed T difference between eROSITA-Chandra (left) and eROSITA-XMM-Newton (right), as presented in Table 1.

Conclusions on multi-temperature structure bias. Given all the above, the bias introduced by multiphase gas and different effective areas is currently disfavored as the main cause for the observed T discrepancy between eROSITA and Chandra/XMM-Newton. This is consistent with what past studies found for other instrument comparisons (e.g., S15). However, to accurately quantify the exact level of bias introduced due to more complicated multiphase gas structure, more work is needed.

7.2.2. Bias from Galactic absorption

The X-ray absorption, proxied by the N_H parameter, has a much stronger influence on the soft part of the spectra (≲1.5 keV) than the hard part, for which it is almost irrelevant. Given the different energy-dependent shapes of the effective areas of the telescopes, different N_H values can affect each telescope’s T measurements differently, introducing an N_H-dependent systematic bias in the T comparison for the soft and full bands (while the hard band would remain unaffected from such bias). To address this possibility, we compare the full band eROSITA-Chandra scaling relations between the $\frac{1}{3}$ of the sample with the lowest and highest N_H¹⁴ (N_H < 2.45 cm⁻² and N_H > 5.59 cm⁻² respectively). The two subsamples do not show significant differences in their T distributions and any difference is expected to come from the different N_H.

As displayed in the bottom panel of Fig. 7, the effect different N_H values have on the eROSITA-Chandra cross-calibration is not statistically significant, with the low and high N_H subsamples agreeing within ≲1σ. Therefore, this further supports the notion that the different effective area dependence on photon energy per telescope is not the cause of the T discrepancy.

7.3. Soft band versus hard band T per instrument

To better understand the cause of the T discrepancy dependence on the used energy band, we explore which instrument is the most affected by the energy band change. For a self-consistently calibrated instrument, and assuming multiphase gas effects are not dominant, the different energy bands should result in statistically similar T. To test this, we compare the soft and hard band temperatures (T_{0.5 − 4 keV} and T_{1.5 − 7 keV} respectively) for all three instruments. The detailed results are given in Appendix A.6 (see Fig. A.5 and Table A.2).

eROSITA shows the largest deviation between all instruments, with T_{0.5 − 4 keV} being on average 29 ± 4% higher than T_{1.5 − 7 keV}. This difference only weakly depends on T at a < 1σ level. Specifically, for eROSITA:

$$\begin{array}{c}\hfill {log}_{10}\frac{T_{1.5-7\text{keV}}}{3\text{keV}}\approx -0.148+0.928\times {log}_{10}\frac{T_{0.5-4\text{keV}}}{3\text{keV}}.\end{array}$$(9)

On the other hand, Chandra shows the highest level of agreement between T_{0.5 − 4 keV} and T_{1.5 − 7 keV}, with the former being 8 ± 1% higher than the latter on average. There is no noticeable dependence with T. Specifically, for Chandra:

$$\begin{array}{c}\hfill {log}_{10}\frac{T_{1.5-7\text{keV}}}{4.5\text{keV}}\approx -0.034+0.988\times {log}_{10}\frac{T_{0.5-4\text{keV}}}{4.5\text{keV}}.\end{array}$$(10)

Finally, XMM-Newton shows an opposite behavior compared to eROSITA and Chandra, with T_{0.5 − 4 keV} being on average 17 ± 2% lower than T_{1.5 − 7 keV}. Again, this discrepancy is not a strong function of T. Specifically, for XMM-Newton:

$$\begin{array}{c}\hfill {log}_{10}\frac{T_{1.5-7\text{keV}}}{3\text{keV}}\approx 0.068+0.936\times {log}_{10}\frac{T_{0.5-4\text{keV}}}{3\text{keV}}.\end{array}$$(11)

From all the above, it is evident that eROSITA temperatures are the most affected when one changes the used energy band. The nature of the change suggests that, at least for eROSITA and Chandra, multi-temperature gas does not play a dominant role on T, otherwise one would generally expect T_{1.5 − 7 keV} > T_{0.5 − 4 keV}. Overall, these results indicate that the increased discrepancy in the cross-calibration between eROSITA and Chandra/XMM-Newton in the hard band might be mostly attributed to possible systematic calibration uncertainties at higher X-ray energies. However, this remains to be confirmed or challenged by future analysis.

7.4. Bias introduced from eROSITA TM8 versus TM9

Next, we explore the effect that the inclusion of eROSITA TM9 data has on the T measurements. As discussed in Sect. 3.1.3, TM9 cameras are affected by the light leak. In theory, this could potentially alter the signal at low X-ray energies while the harder energies remain unaffected, biasing the fitted T. This would not explain the observed T discrepancies in the hard band, but it is worth checking if it has any effect on the other bands. To test the consistency of the two telescope modules, we fit the TM9-TM8 T scaling relation. We find there is a general agreement between the TM8 and TM9 T measurements at a ≲2.5σ level, with ξ = −0.05. Their scaling relation parameters for the full band fit are A = −0.020 ± 0.015, B = 0.921 ± 0.038, and σ_tot = 0.193 ± 0.010, while the intrinsic scatter is negligible (< 1%). The agreement improves to ≲2σ for the soft band, mostly due to the increase of measurement uncertainties.

Even though it is not statistically significant, TM9 measures slightly lower T than TM8 for T ≳ 3 keV clusters (5% and 10% difference for T_TM8 = 3 keV and T_TM8 = 6 keV respectively). To ensure our findings are insensitive to this small TM9 effect, we refit the eROSITA-Chandra and eROSITA-XMM-Newton scaling relations excluding TM9 for the full and soft bands. We find very small differences with the default analysis. For the eROSITA-Chandra and eROSITA-XMM-Newton scaling relations both A and B change by ≲2% and ≲2.5% respectively. Consequently, TM9 does not introduce any bias to the T comparisons.

7.5. Bias from non-Gaussian scatter and asymmetric T_eROSITA uncertainties

7.5.1. Non-Gaussian scatter

Due to the decrease of eROSITA’s effective area at ≳2 keV, clusters with higher T_eROSITA show larger uncertainties σ_TeROSITA. For similar Chandra or XMM-NewtonT, upscattered T_eROSITA show higher uncertainty on average than downscattered T values, making the total scatter slightly non-Gaussian. If clusters with truly high, upscattered T_eROSITA carry larger statistical uncertainties, the fit will give more weight to lower, downscattered T data points, shifting the best-fit relation away from the 1:1 relation. To test if this non-Gaussianity of σ_TeROSITA has a significant effect on the scaling relation constraints, we refit the eROSITA-Chandra/XMM-Newton scaling relations for all bands, without taking into account σ_TeROSITA, hence, we assign the same weight to all data points.

For the eROSITA-Chandra cross-calibration, A increases by ≈0.007 − 0.013 dex (1.5 − 3% in linear space) for all bands. B remains almost unchanged for the full and soft bands while it decreases by 3% for the hard band.

For the eROSITA-XMM-Newton cross-calibration the effect of non-Gaussianity is stronger. A increases by 0.044 dex (10%), 0.059 dex (13.5%), and 0.032 dex (7.6%) for the full, soft, and hard bands respectively. B increases by 1.4% and 0.3% for the full and soft bands and by 24% for the hard band. Non-Gaussianity has no significant effect on B for the full and soft bands. However, it causes an underestimation of A, resulting in an overestimation of the cross-calibration differences between eROSITA and XMM-Newton. Nevertheless, given the parameter uncertainties, this shift of A is within ≲2σ from the default analysis. Of course the performed test here is quite conservative and demonstrates the maximum impact of non-Gaussianity possible.

To conclude, the non-Gaussianity of σ_TeROSITA does not introduce any significant bias in the scaling relations between eROSITA and Chandra. On the other hand, it does partially overestimate the discrepancy between eROSITA and XMM-Newton, however, not at a statistically significant level. Moreover, this overestimation bias cannot fully alleviate the tension between T_eROSITA and T_XMM values. Non-Gaussianity effects become less relevant as more eROSITA counts are available for determining T_eROSITA and the T_eROSITA measurement uncertainties become smaller than the intrinsic scatter.

7.5.2. Symmetrization of asymmetric T_eROSITA uncertainties

In this work we adopt a widely used approach by symmetrizing T uncertainties in linear-space in order to convert them to log-space (through error propagation) before we fit scaling relations. Slightly asymmetric T uncertainties normally converge to symmetry in log-space; thus, this approach is not expected to introduce significant biases most of the time. For instance, the T_XMM and T_Chandra uncertainties used in this work are nearly symmetric, with the average log-uncertainty difference being $\mathrm{\Delta }{\left[{\sigma }_{\log T}\right]}_{-}^{+}\approx 0.002$ dex and $\mathrm{\Delta }{\left[{\sigma }_{\log T}\right]}_{-}^{+}\approx 0.006$ dex for the full (or soft) and hard bands respectively, for both telescopes. Hence, their symmetrization has a nearly zero effect on log-space uncertainties. On the other hand, the T_eROSITA uncertainties are larger and tend to be more asymmetric, that is, $\mathrm{\Delta }{\left[{\sigma }_{\log T}\right]}_{-}^{+}\approx 0.096$ dex and $\mathrm{\Delta }{\left[{\sigma }_{\log T}\right]}_{-}^{+}\approx 0.161$ dex for the full (or soft) and hard bands, respectively. Consequently, symmetrizing the T_eROSITA uncertainties might bias our best-fit estimates for the cross-calibration scaling relations.

To test this, we refit all scaling relations without symmetrizing σ _{log TeROSITA}. Instead, we consider two T_eROSITA log-errorbars: ${\sigma }_{\log T}^{+}=\log T^{+}-\log T_{\text{best}}$ and ${\sigma }_{\log T}^{-}=\log T_{\text{best}}-\log T^{-}-$, where log T_best is the best-fit T_eROSITA and T⁺ and T⁻ are its upper and lower 68.3% uncertainties respectively. For every tested A and B combinations in Eq. (3) and 4, we consider ${\sigma }_{\log T_{\text{eROSITA}}}={\sigma }_{\log T}^{+}$ (or ${\sigma }_{\log T_{\text{eROSITA}}}={\sigma }_{\log T}^{-}$) when T_eROSITA is downscattered (or upscattered) compared to the fitted line (i.e., when $log{T}_{\text{Y},i}^{\prime }-A-B\times log{T}_{\text{X},i}^{\prime }<0$ or > 0 respectively in Eq. (3)). In other words, we consider only the errorbar toward the fitted line as the standard errorbar, ignoring the errorbar on the opposite side of the data point.

For the full and soft band eROSITA-Chandra scaling relations, A and σ_intr change by ≤0.004 dex (≤1%) and ≤0.008 dex (≤1.8%)¹⁵ respectively, while B decreases by 1.6% and 3.5%. It is evident that these changes are nearly negligible compared to the overall tension of the two instruments, with only B showing a ≈0.5 − 1σ shift. For the hard band, A and B both increase by ≈5.5%, while the scatter remains similar to the default case. These are again ≲1σ shifts for both A and B. Therefore, it is safe to conclude that the symmetrization of temperature uncertainties has an insignificant effect on the eROSITA-Chandra scaling relations.

For the eROSITA-XMM-Newton scaling relations, the effect of asymmetric σ_{log TeROSITA} has a somewhat larger effect, but still statistically insignificant. Both A and B increase by ≈6%, ≈3%, and ≈8.5% for the full, soft, and hard band respectively. All these changes are within < 1σ. The similarity in the A and B shifts is noteworthy and it is attributed to some correlation between the two parameters. Additionally, σ_intr increases by < 0.01 dex (< 2.3%) for the full and soft bands, while it doubles for the hard band. The latter is the only parameter that changes at a > 1σ level. Overall, it is shown that the use of asymmetric temperature uncertainties only shifts the best-fit scaling relations within the statistical noise.

7.6. Selection bias due to low T/σ_T

Due to its low sensitivity at ≳2.5 keV, eROSITA returned a T/σ_T < 1 measurement for 4% of the Chandra and 12% of the XMM-Newton cluster T for the full and soft bands. For the hard band, this fraction increased to 37% and 58% respectively. These clusters were excluded from the default analysis. With eROSITA, high T clusters are more likely to show lower T/σ_T on average than low T clusters for the same counts. Consequently, excluding low T/σ_T clusters might lead to preferentially select clusters for which T_eROSITA was measured low. This selection effect could potentially shift the scaling relations further away from the equality line. To test this, we refit the scaling relations while considering all T measurements independently of their T/σ_T value.

The newly added data points have nearly no effect on the eROSITA-Chandra scaling relations for the full and soft bands. Both the new A and B best-fit values increase by < 1% compared to the default best-fit values. This occurs since the added data with T/σ_T < 1 values only marginally increase the sample size compared to the default case. Additionally, they carry very little statistical weight. For the hard band, the overall effect on A and B is similar, increasing only by 3% and 1% respectively. In all bands, the total scatter almost doubles when the low T/σ_T values are added to the sample.

For the eROSITA-XMM-Newton scaling relations in the full and soft bands, both A and B once again change by < 1% compared to the default best-fit values. For the hard band, the sample size increases significantly. Nevertheless, A decreases by 4% while B increases by 9.5%. These changes are within 0.5σ and do not practically improve the agreement between eROSITA and XMM-Newton.

It is evident that the data selection of T/σ_T > 1 introduces no significant bias to the final results, while it reduces the total scaling relation scatter. However, it is possible that when future eRASS data are used, more high-T_eROSITA measurements will have higher T/σ_T, and hence, a larger effect on the best-fit scaling relations. This might cause the latter to shift somewhat closer to the equality line. Nonetheless, as shown in Sect. 7.5, this is unlikely to alleviate the cross-calibration differences between eROSITA and Chandra/XMM-Newton.

7.7. eROSITA’s half-energy width and PSF effects

The on-axis half-energy width (HEW) of eROSITA is ≈15.5 − 16.5″ between 1.49 keV and 8.04 keV (Predehl et al. 2021). For the eROSITA survey data, such as eRASS1, the average HEW increases to ≈26 − 29″ (Brunner et al. 2022). This is only slightly larger than the XMM-Newton on-axis HEW (≈15 − 16″), but significantly larger than the Chandra on-axis HEW (≈0.5 − 1″). If the spectra extraction regions are not significantly larger than eROSITA’s HEW then the emission from the cluster core might scatter to the core-excised annulus spectra. At the same time, Chandra would not suffer from this due to its much lower HEW. For cool core clusters, this would bias the core-excised T toward lower values. This is a potential problem for distant clusters with small apparent R₅₀₀. However, our sample consists of nearby clusters with large apparent R₅₀₀. Characteristically, the median apparent radius of the cluster cores (< 0.2R₅₀₀) is 2.3′, which is five times the eROSITA survey HEW. At the same time, only four clusters of the eROSITA-Chandra comparison subsample (3.6% of the subsample) have a cluster core of < 65″, that is, ≲2.5× the eROSITA survey HEW. These clusters do not show any special behavior compared to the rest of the sample. In conclusion, it is clear that the eROSITA HEW has nearly no effect on the eROSITA-ChandraT comparison. The same conclusion was reached by S15 for the XMM-Newton-ChandraT comparison and a similar cluster population.

7.8. Potential systematic biases from background treatment

When one constrains cluster T, the treatment of both CXB and PIB is important. Potential biases in the background treatment can propagate to the final T and the resulting scaling relations. This is particularly true for outter cluster regions with low X-ray surface brightness. In this work, we deal with central cluster regions with generally high surface brightness. The constrained T of these regions are not expected to be significantly affected by mild changes in the background treatment, or even mild background biases. Nevertheless, here we discuss the possible existence of such potential biases and their impact on the final cross-calibration scaling relations.

7.8.1. eROSITA

For eROSITA, both the CXB and PIB were modeled together with the source spectra, after they were robustly constrained as described in Sect. 3.1.3. Therefore, there is no suspected bias in the background treatment of eROSITA data. Additionally, due to the large T_eROSITA uncertainties, even (currently unknown) mild background biases would have a negligible impact on the final scaling relations.

7.8.2. Chandra

For Chandra, the CXB is modeled simultaneously with the cluster spectra, using both Chandra and RASS data (see Sect. 3.2.2). Even in the case of mild calibration differences between ROSAT and Chandra fluxes, the inclusion of RASS data is not expected to bias the fitted T_Chandra due to the high emissivity of the used cluster regions and the large number of resulting Chandra counts, which eventually drive the simultaneous cluster+CXB fit. Moreover, the exact CXB values only weakly affect T_Chandra in central cluster regions. Recently, Rossetti et al. (2024) showed how the normalizations of the CXB components change if one uses RASS data along with XMM-Newton data. They showed that when the CXB is constrained from cluster-free sky regions (as done for Chandra in our work, where the CXB is constrained > 1° from the cluster), the use of RASS data does not significantly change the CXB constraints. Even more importantly, they showed that the inclusion of RASS data has no effect on the final T from regions where the source emission is higher than the background (which is clearly the case in our work).

Furthermore, the Chandra PIB is relatively stable with time and generally well-known. For the vast majority of Chandra source spectra we analyzed, the PIB level was lower than the source emission even at the highest end of the full band range (i.e., 5 − 7 keV). As such, subtracting PIB from the source spectra is followed by most studies using Chandra data and there are no indications this causes significant biases. From all the above, it is clear that there is no reason to believe that the background treatment noticeably biases T_Chandra.

7.8.3. XMM-Newton

For XMM-Newton, the CXB was constrained using only XMM-Newton data as described in Sect. 3.2.2 and then fitted simultaneously with the source spectra. This is a commonly used approach and it is not expected to bias the CXB estimates. As discussed before, even if we somewhat shift the normalizations of the CXB components, T_XMM would mostly be insensitive to these changes due to the bright cluster regions we analyze. The insensitivity of fitted T to mild changes of the CXB for high surface brightness regions was also clearly shown in Rossetti et al. (2024).

On the other hand, the PIB was subtracted from the source spectra, with Gaussian lines added to the model to account for any fluorescence line residuals after the PIB subtraction. However, the possible noise of the PIB normalization is not accounted for. This might introduce some bias to the fitted T_XMM, or underestimate the T_XMM uncertainties, if the PIB estimation is biased or uncertain. To test the effect of an uncertain PIB subtraction on T_XMM and eventually on the eROSITA-XMM-Newton scaling relations, we perform the following. We randomly select 15 clusters (with 30 measured T_XMM) and fluctuate the PIB by ±5% in all EPIC detectors. We then refit T_XMM. We find that, for the cluster core regions, T_XMM fluctuates by < 1% on average for 80% of the cores and < 2.5% for the rest. Therefore, the effect of an uncertain (or slightly biased) PIB subtraction would be nearly negligible for the core regions, which comprise half of the sample used to constrain the eROSITA-XMM-Newton scaling relations. For the annulus T_XMM, the average shift was ΔT_XMM ∼ 3.5%, which is typically < 1σ. Moreover, the T_XMM shift was not systematic across all clusters when the PIB level was increased or decreased, which further decreases the impact of the PIB uncertainty on the final scaling relations. The ΔT_XMM results were very similar across all three bands. This test suggests that subtracting the PIB from high surface brightness cluster regions does not significantly impact the constrained T_XMM and final eROSITA-XMM-Newton scaling relations.

Another potential bias of the XMM-Newton PIB treatment is that the unexposed corners of the PN detector are not fully shielded from FOV photons and soft protons, as shown in Marelli et al. (2021), where they mask the inner 905″ of the FOV. They also showed that the contamination is lower as one moves to the outer edges of the unexposed corners. Recently, Rossetti et al. (2024) showed that the > 905″ count-rate in the 10 − 14 keV band correlates very well between the PN and MOS2 detectors of XMM-Newton for data with low solar flare contamination, that is, the PIB estimate of XMM-Newton/PN is unbiased in this band when soft proton contamination is low. In our analysis, we mask the inner > 925″ of the XMM-Newton/PN detector, which ensures a slightly lower level of contamination, and use the count-rate in the 2.5 − 5 keV band to determine the rescaling factor from the FWC data. To check if our PIB count-rate is overestimated due to residual contamination we perform the following. Firstly, we estimate the PIB rescaling factor based on the 10 − 14 keV band from the same > 925″ region, as done for the default analysis. We did so for 25 XMM-Newton pointings with different levels of solar flare contamination, quantified by the IN/OUT ratio (see Appendix A.7 for details). In Fig. A.6 we show that our default PIB estimates are unbiased compared to the 10 − 14 keV band for IN/OUT ≲ 1.1, which applies to 81% of our XMM-Newton pointings (43 out of 53). At the highest-end of acceptable soft proton contamination, that is, 1.1 < IN/OUT < 1.15, our default PIB level estimates for the XMM-Newton/PN camera are ∼8%−15% overestimated compared to the 10 − 14 keV case. Our results are consistent with the ones from Rossetti et al. (2024), although our method for probing the contamination level slightly differs.

To quantify the effect on the final eROSITA-XMM-Newton scaling relations that the mild XMM-Newton/PN PIB bias would have, we perform the following. We consider the ten observations with 1.1 < IN/OUT < 1.15 and reduce the PIB level of the XMM-Newton/PN spectra by 8%−15%, according to the individual IN/OUT ratios. We then refit the full XMM-NewtonT_XMM and the eROSITA-XMM-Newton scaling relations. We found that the latter show completely negligible changes, < 0.3σ for all parameters (see Appendix A.7 for details). Consequently, we confirm that the residual contamination of the unexposed XMM-Newton/PN corners has no effect on the final results of this work.

7.9. Consistency of spectroscopic T_eROSITA with the eRASS1 cluster catalog

The eRASS1 galaxy cluster catalog (Bulbul et al. 2024) includes cluster T measurement determined by the MultiBand Projector 2D (MBProj2D) software¹⁶ (Sanders et al. 2018). MBProj2D is a tool that uses X-ray images of galaxy clusters in different, independent energy bands, and through forward-modeling, constrains X-ray cluster properties such as the surface brightness and temperature profiles. The cluster T provided in the eRASS1 galaxy cluster catalog is constrained using seven independent energy bands between 0.3–7.0 keV and within the entire R₅₀₀ of the cluster (for more details see Bulbul et al. 2024).

To check the consistency between our spectroscopic T_eROSITA and the official eRASS1 cluster catalog T_eRASS1, and to provide conversion functions between the latter and T_Chandra and T_XMM, we study the scaling relations between T_eRASS1 and the other T measurements. It is crucial to stress here that this comparison does not serve as a cross-calibration test between eROSITA and other instruments. In these comparisons, the cluster area used, the methodology (MBproj2D T versus spectroscopic T), and the energy bands used for T_eRASS1 and T_Chandra/T_XMM are all different. The sole purpose of this comparison is to test the applicability of our main results for the eRASS1 cluster catalog. Since T_eRASS1 refers to the entire R₅₀₀, we can only compare with a single T per cluster for Chandra, XMM-Newton, and spectroscopic eROSITA; that is, with core-only or annulus-only T. The comparison is performed only for the full band.

The detailed results are presented in Appendix A.8. In a nutshell, we find excellent agreement between our spectroscopic T_eROSITA and T_eRASS1, especially when we use the core-only T_eROSITA, with the comparison matching the 1:1 line. When we compare T_eRASS1 to T_Chandra and T_XMM, the results are consistent (within the uncertainties) with the main results presented in Sects. 5.1 and 5.2. Thus, we conclude that our results can be safely used for converting (to) T_eRASS1 as well.

7.10. Consistency with past studies and indirect eROSITA comparison with other X-ray telescopes

7.10.1. Chandra-XMM-Newton and eROSITA-XMM-Newton(MOS/PN only)

The only previous eROSITA cross-calibration test using a cluster sample was performed against XMM-Newton by Turner et al. (2022), who used T measurements from eight common eFEDS and XMM-Newton Cluster Survey (XCS) clusters. Despite the very small number of available data points and the different energy ranges used for eROSITA and XMM-Newton spectral analysis, they found $T_{\text{eROSITA}}=0.{75}_{-0.08}^{+0.10}\times T_{\text{XMM}}$, for a fixed slope of unity. This is completely consistent with our result. As discussed in Sect. 1, there are also a few studies that used individual cluster T measurements to compare eROSITA with other instruments (e.g.; Liu et al. 2023). However, a direct comparison of our results with these studies is not helpful since single T comparisons can scatter significantly around the average behavior we present in this work.

Furthermore, based on our findings, we can 1) perform an indirect consistency test with past Chandra-XMM-Newton cross-calibration studies and 2) evaluate the cross-calibration of eROSITA with the individual XMM-Newton/MOS and XMM-Newton/PN detectors.

Using the full band eROSITA-Chandra and eROSITA-XMM-Newton scaling relations, we can indirectly constrain the Chandra-XMM-NewtonT scaling relation and compare it with the findings of S15 (after adjusting for the pivot points S15 used). We find

$$\begin{array}{c}\hfill {log}_{10}{T}_{\text{XMM}}\approx 0.010+0.946\times {log}_{10}{T}_{\mathit{Chandra}},\end{array}$$(12)

which is entirely consistent with S15 within the 1σ uncertainties.

We now repeat this exercise to predict if using only the XMM-Newton/MOS or XMM-Newton/PN detectors would cause a significant shift to the eROSITA-XMM-Newton cross-calibration scaling relations. Past studies (e.g., S15) have shown that XMM-Newton/MOS and XMM-Newton/PN return similar T for T ≲ 3 keV clusters, but XMM-Newton/PN returns slightly lower T than XMM-Newton/MOS for hotter clusters, with the discrepancy reaching ≈12% for T ≈ 7 keV clusters (which is close to the maximum T_XMM we used in this work). Nevalainen & Molendi (2023) also showed that at higher energies (≳3 keV), XMM-Newton/PN and XMM-Newton/MOS show cross-calibration biases. For typical T, XMM-Newton/PN provides ∼60% of the total XMM-Newton counts. Since in this work we jointly use all the XMM-Newton detectors to constraint T_XMM, the latter is generally expected to land somewhere between the XMM-Newton/PN and XMM-Newton/MOS T values. All three T values are expected to agree within the uncertainties for the vast majority of our clusters based on their T_XMM distribution. Therefore, no major changes in the results should be observed if one uses individual XMM-Newton detectors. Nevertheless, it is interesting to explore the cross-calibration of eROSITA versus XMM-Newton/MOS or XMM-Newton/PN alone.

To do so, we use this work’s scaling relations combined with the results provided in Table 2 of S15. For the full band eROSITA-XMM-Newton/MOS and eROSITA-XMM-Newton/PN scaling relations, we find

$$\begin{array}{c}\hfill \begin{array}{c}{log}_{10}\frac{T_{\text{eROSITA}}}{3\text{keV}}\approx -0.106+0.802\times {log}_{10}\frac{T_{\text{XMM/MOS}}}{3\text{keV}},\hfill \\ {log}_{10}\frac{T_{\text{eROSITA}}}{3\text{keV}}\approx -0.090+0.878\times {log}_{10}\frac{T_{\text{XMM/PN}}}{3\text{keV}}.\hfill \end{array}\end{array}$$(13)

The XMM-Newton/PN T seem to be slightly closer to T_eROSITA than XMM-Newton/MOS T when the two EPIC detectors measure T ≳ 2.5 keV each, although the differences are minimal for most clusters. However, when XMM-Newton/MOS and XMM-Newton/PN measure T ≳ 7 keV each, the discrepancy with T_eROSITA becomes 33% and 26% respectively. This (only slightly) better agreement of XMM-Newton/PN with eROSITA is particularly interesting since the eROSITA telescope carries similar PN detectors. Nevertheless, such differences are still within the expected uncertainties and scatter; thus, better statistics (and a wider T range) are required to draw more robust conclusions from these comparisons.

7.10.2. Other instruments

Using our findings, we indirectly estimate the cross-calibration of eROSITA and other X-ray telescopes, such as Suzaku and NuSTAR. Kettula et al. (2013) found that Suzaku returned ∼12% lower T than the XMM-Newton/PN detector. Based on that, one can predict that eROSITA and SuzakuT should be closer compared to the other instruments, especially at lower T where they should be comparable. However, eROSITA is still expected to return lower T than Suzaku by ∼10 − 15% for T = 7 keV clusters. Nevertheless, it is not trivial to draw robust conclusions due to the very limited sample of Kettula et al. (2013), their use of XMM-Newton/PN-only data, and the different energy bands in which they performed the Suzaku-XMM-Newton cross-calibration. It is interesting to note here that, if multi-temperature structure was the main cause of the observed T discrepancies across different instruments, Suzaku would be expected to return higher T than XMM-Newton (and comparable to Chandra), which is not the case.

Wallbank et al. (2022) reported ≈10% lower T from NuSTAR compared to Chandra for the 0.6–9.0 keV band and T ≈ 10 keV clusters. As a result, one would expect eROSITA to measure much lower T than NuSTAR for massive clusters. On the T ≲ 2 keV end, eROSITA is expected to return rather consistent T values with NuSTAR within ≈3 − 5%. As we stressed already, these predictions simply help provide a better understanding of what to expect from future studies rather than a comparison between eROSITA, NuSTAR, and Suzaku.

7.11. Future improvements

The current analysis already put tight constraints in the eROSITA cross-calibration with Chandra and XMM-Newton and provides a good understanding of the photon energy and cluster mass dependence of the T discrepancy. These results will be highly valuable for future studies that wish to use eROSITA T combined with T values from other telescopes. Nonetheless, there is still room for several improvements to be made in the future.

First and foremost, the T_eROSITA measurement uncertainties are still large. Their decrease would result in a lower total scatter, improving the precision of the constraints. This will be achieved with the eRASS:4 data, where σ_TeROSITA will be reduced by a factor of ∼2. Additionally, the effects of the non-Gaussian scatter will also be better studied and accounted for since higher T_eROSITA will carry lower uncertainties and higher statistical weight compared to the present work. T_break will also be better constrained when using a broken power law to fit the scaling relations.

The fourfold more counts in the eRASS:4 data will also allow us to use narrower energy bands for the spectral analysis, for instance, 0.5 − 2 keV and 2 − 8 keV. Consequently, a better characterization of the effective area cross-calibration as a function of energy will be achieved, for example, by utilizing the stacked residual ratios methodology (Kettula et al. 2013). In this work, it was not possible to robustly constrain T_eROSITA using narrower energy bands due to the limited available counts. As a result, the soft and hard bands inevitably overlapped.

Given the higher number of counts in eRASS:4, a better characterization of the multiphase gas bias will also be achieved. Multi-temperature emission models will be fit to the spectra, which will reduce any dependence of the observed T discrepancy on the different effective areas of the compared telescopes. Such fits were not possible in this study due to the limited number of counts in eRASS1. Moreover, eROSITA and XMM-Newton have similar effective area shapes in the 0.5 − 2 keV band. Therefore, when more available counts enable T in this band to be constrained, eROSITA and XMM-Newton should not show any systematic discrepancies due to the presence of multiphase gas, and any remaining discrepancies would be attributed to systematic calibration uncertainties.

Another interesting future test is the dependence of the comparison to the cluster redshift. The M20 sample used in this work is a low−z sample with most clusters lying at z < 0.2. As the T comparison between eROSITA and Chandra/XMM-Newton depends on both the energy band and T (i.e., spectral shape), the T comparison might slightly change for different redshift ranges. In this work, it was not possible to perform such a test since the redshift range is small and clusters with higher T are found at larger z. As a result, larger z would deviate more than low z due to the different cluster populations. The upcoming eRASS:4 data will allow the use of similar cluster samples at different z in order to detect any possible redshift dependency of the eROSITA-Chandra/XMM-Newton cluster T comparison.

Finally, the eROSITA-XMM-Newton sample size will be increased. Due to the nature of the M20 sample, XMM-Newton data were available mostly for low flux (i.e., low eROSITA counts on average) clusters, and only for 51 of them. This resulted in 71 independent T_eROSITA and T_XMM (core and core-excised annulus). Given the vast availability of XMM-Newton cluster data and the deeper eRASS:4 data, we expect the eROSITA-XMM-NewtonT scaling relations to significantly improve within the next years. A broken power law analysis will then become feasible for these scaling relations as well.

8. Summary and conclusions

In this work, we provide the first-ever eROSITA-Chandra cross-calibration in the broad, soft, and hard X-ray energy bands using the same spectral energy ranges and 186 independent galaxy cluster T, the largest sample used in such studies to date. In the same manner, we also provide the first eROSITA-XMM-Newton calibration in the soft and hard bands and the first in the broad band using the same spectral energy for both instruments and a much larger T sample than previously used. Our work offers robust conversion factors for the measured T between eROSITA, Chandra, and XMM-Newton.

Using a single power law fit, we found that eROSITA shows a strong discrepancy with Chandra, measuring 25% and 38% lower T_eROSITA for T_Chandra = 4.5 keV and T_Chandra = 10 keV, respectively. Furthermore, we performed the first-ever (to our knowledge) broken power law fit in such a T cross-calibration scaling relation. We found the T where the power law breaks to be 2.7 keV and 1.7 keV for the full and soft band, respectively. At lower T, the values of the two telescopes were consistent within ≲5%, with a slope close to unity. For T ≳ 6 keV clusters, the broken power law further increased the tension between eROSITA and Chandra by a few percent. The hard band was the one that showed the largest discrepancy between the two instruments, with eROSITA returning ≈25 − 60% lower T_eROSITA for T_Chandra ≈ 2 − 10 keV.

Moreover, eROSITA shows lower T than XMM-Newton as well, with the discrepancy being milder than the one with Chandra. For the full band, eROSITA measures 10 − 28% lower T_eROSITA for T_XMM ≈ 2 − 7 keV clusters, while there is a slightly better agreement for cooler systems. The (dis)agreement between the two instruments improves by a few percent for the soft band. The hard band was only loosely constrained due to the large measurement uncertainties, although it is obvious that eROSITA largely underestimates T compared to XMM-Newton in the hard band.

A wide range of possible systematics was explored. Namely, we looked for possible biases introduced by a multi-temperature gas structure, the degeneracy between the temperature and the metallicity parameters, the effect different levels of Galactic absorption have on the T constraints of different telescopes, the non-Gaussianity of the scatter and the correlation between T_eROSITA and its uncertainty, sample selection effects, and the use of eROSITA TM8 and TM9 data. We could not identify a specific systematic that had a strong effect on the eROSITA cross-calibration with Chandra and XMM-Newton, and more work is needed to obtain conclusive results about some of the potential biases.

Overall, the soft band demonstrated a marginally improved agreement between instruments than the full band, whereas the hard band cross-calibration revealed significant discrepancies. Similarly, clusters with softer spectra and lower T showed much better agreement than clusters with harder spectra and high T. Subsequently, we conclude that eROSITA’s effective area calibration is more comparable to Chandra and XMM-Newton for soft X-ray energies, though there is still a systematic bias present. Our findings further point to a larger systematic bias in eROSITA’s cross-calibration with Chandra and XMM-Newton at harder energies. It is possible that this bias is overestimated due to the lower statistical weight that high T_eROSITA carry given the shallow eRASS1 data. For now, in line with earlier studies, we conclude that these discrepancies are attributable to the systematic effective area calibration uncertainties.

Finally, our work offers the first robust conversion factors of spectroscopic eROSITA T to XMM-Newton and ChandraT. Additionally, we offer conversion factors between the official eRASS1 cluster catalog T and the Chandra and XMM-NewtonT from the core and core-excised cluster regions. All of these findings will enable the simultaneous use of eROSITA cluster T with values coming from the other two telescopes. Given the unprecedented volume of cluster data eRASS1 (and eventually eRASS:4 and eRASS:8) will provide, these conversion factors are expected to be of utter importance for future cluster studies.

Acknowledgments

We thank the anonymous referee for their constructive and insightful comments. This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint Russian-German science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors, and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument was led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The eROSITA data shown here were processed using the eSASS software system developed by the German eROSITA consortium. K.M. acknowledges support in the form of the X-ray Oort Fellowship at Leiden Observatory. G.S. acknowledges support through the Chandra grant GO5-16126X. A.V. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 450861021.

References

Appendix A: Additional material and tests

A.1. Free-to-vary metal abundance and temperature distributions

In Table A.1 we present the best-fit scaling relation parameters when Z is left free to vary, as explained in Sect. 7.1. As discussed earlier, the general behavior of the scaling relations remains unaffected, with only a slight increase in the scatter being observed. Thus, the metallicity value treatment is not important for the T comparison between different instruments.

In Fig. A.2 we show the Chandra and XMM-NewtonT distributions of cluster core and core-excised regions, as discussed in Sect. 7.2.1. The T distributions are very similar. Hence, the core and core-excised cross-calibration results of these two subsamples are directly comparable and any observed differences should not be attributed to a different T range.

A.2. χ²-statistic bias compared to C-statistic

The χ²-statistic assumes Gaussian distribution of spectral counts and can return biased T values during spectral fitting, while the C-statistic is based on Poissonian count distributions and it is known to return nearly unbiased T results. However, the χ² bias reduces in amplitude as the number of available counts increases per bin (and in total) since the counts distribution can be well approximated by a Gaussian distribution. Due to that, using the χ²-statistic for fitting X-ray spectra is commonly used in X-ray astronomy. In this work, the XMM-Newton and Chandra spectra we use typically contain ≳10, 000, ≳8, 000, and ≳5, 000 counts for the full, soft, and hard bands respectively. As a result, adopting the χ²-statistic (to be consistent with the full band results from M20) should not introduce strong biases in the best-fit T compared to the C-statistic. We confirm this by fitting 20 randomly selected full band XMM-Newton spectra by using the cstat option in XSPEC. The average shift in the best-fit T is ≈0.7σ, or ≈2.9%. This is consistent with what was shown in Veronica et al. (2022), where they observed similar T shifts by changing between the above-mentioned statistics.

Humphrey et al. (2009) explored this issue in detail. They used simulated thermal plasma emission spectra and folded them with the Chandra response to determine the bias of the χ²− and C-statistics as a function of the available counts and thermal plasma properties. They found that, while the amplitude of the bias decreases with increasing counts, the statistical significance of the T shift increases until ∼10⁵ counts (Fig. 2 in that paper). This occurs because, as counts increase, σ_T drops faster than the absolute T bias. For our number of counts in the XMM-Newton and Chandra spectra, Humphrey et al. (2009) predicts a T bias of ∼0.5 − 1σ. To test the effect such a bias would have on the constrained scaling relations, we refit the latter in all bands for both the eROSITA-Chandra and eROSITA-XMM-Newton comparisons, but we increased T_Chandra and T_XMM by ∼0.5 − 1σ_T.¹⁷ The shift in the 1σ contours of all scaling relations is displayed in Fig. A.1. The best-fit parameters change by ≤1.1σ for all scaling relations. The largest change is found for the two soft band comparisons, where A changes by 0.01 dex (2.3%) and 0.013 dex (3%) for the eROSITA-Chandra and eROSITA-XMM-Newton scaling relations respectively. The slope B changes by 1.7% for both relations. For the full band comparisons, the best-fit parameters change by ≲1σ.

Fig. A.1. 1σ confidence levels for the eROSITA-Chandra (top) and eROSITA-XMM-Newton (bottom) scaling relations for the full (blue), soft (red), and hard (green) bands. The more opaque ellipses correspond to the default results, while the more transparent ellipses correspond to the best-fit results when TChandra and TXMM are increased by ∼0.5 − 1σ, as described in Sect. A.2.

Fig. A.2. Histogram of core (blue) and core-excise (green) temperature values for Chandra (right) and XMM-Newton (right). The two distributions are very similar for both instruments.

The minimal effect of this temperature shift on the scaling relations occurs because small changes in T_XMM and T_Chandra are negligible compared to the uncertainties of T_eROSITA for most clusters. Additionally, the slope of the relations (B ∼ 0.65 − 0.85 < 1) further minimizes the effect that small T_XMM and T_Chandra changes have on the results. Consequently, the exact statistic used to fit the XMM-Newton and Chandra spectra does not affect the conclusions of this work. However, we note that for future similar studies with decreased T_eROSITA uncertainties and total scatter, a consistent analysis using the C-statistic would be preferable since the mild χ²-statistic bias would have a larger impact on the final scaling relations.

A.3. Intrinsic scatter dependency on temperature

The intrinsic scatter σ_intr plays an essential role on the propagated statistical uncertainty when one converts a measured T value from one telescope to the equivalent T of another telescope. If σ_intr is not relatively constant with T, then one might need to adopt different values for such conversions, depending on the used T. To test this, we fix A and B to their full sample best-fit values and fit only σ_intr in four (three) independent, increasing-T bins for the eROSITA-Chandra (eROSITA-XMM-Newton) scaling relations. All bins have the same number of clusters per scaling relation. We display the results for the full and hard bands in Fig. A.3.

Fig. A.3. Dependence of the intrinsic scatter on TChandra and TXMM for the eROSITA-Chandra (top) and eROSITA-XMM-Newton (bottom) scaling relations, respectively. Full band results are displayed in blue, while the hard band results are shown in black and gray. The shaded bands represent the best-fit, full sample σintr with its 1σ uncertainties.

For the eROSITA-Chandra full and soft band comparisons, σ_intr does not depend on T_Chandra. For the hard band, T_Chandra < 5 keV clusters show σ_intr ≈ 0.18 dex while hotter clusters return σ_intr ≈ 0.09. This suggests that soft-spectrum clusters show a more noisy behavior when the hard band is used since eROSITA struggles more to constrain the low T of these clusters. However, all bins are within < 2σ from the full sample average σ_intr. Consequently, until lower statistical uncertainties are achieved with future data, we cannot argue in favor of a T-dependent scatter in the hard band.

For the eROSITA-XMM-Newton comparison, all three bands show similar behavior. Clusters with T_XMM < 5 keV show a constant σ_intr, consistent with the full sample average. Hotter clusters (which represent 33% of the full sample) show σ_intr ≈ 0 dex, deviating by ≈1.5σ from the average. Nevertheless, this can again suggest that hotter clusters have a lower σ_intr than cooler clusters. For now, the currently poor statistics do not allow us to draw robust conclusions on the scatter’s dependency on T.

A.4. Outliers in the eROSITA-XMM-Newton hard band comparison

There are two strong outliers in the eROSITA-XMM-Newton hard band comparison: the core of A0602 ($T_{\text{XMM}}=2.{850}_{-0.122}^{+0.158}$) and the annulus of A2721 ($T_{\text{XMM}}=6.{781}_{-0.553}^{+0.703}$). Both of these regions return T_eROSITA < 0.3 keV with T/σ_T > 2. Both hard band spectra have very low eROSITA counts and higher T models return rather consistent fits as well; thus, their obtained T_eROSITA uncertainties seem to be underestimated. Including these two data points, the LMM slope decreases to $B=0.{324}_{-0.314}^{+0.361}$ while the intrinsic scatter increases by a factor of five. Thus, these outliers have a strong effect on the best-fit results. The core of A0602 has T_eROSITA ≈ 1.9 keV in the full and soft bands while the annulus T_eROSITA of A2721 could not be constrained in these bands. Consequently, based on all the above, their hard band T seem to be problematic and they were excluded.

A.5. Multi-temperature fits with single T models for the hard band

We repeat the analysis of Sect. 7.2.1, focusing on the hard band this time. As shown in Fig. A.4, the three instruments return very similar single T constraints for all 2T combinations. Although the T_eROSITA uncertainties are very large and only < 2σ away from the observed cross-calibration differences, the general trend does not seem to explain the observed T tension, especially between eROSITA and Chandra.

Fig. A.4. Same as in Fig. 8 but for the hard band fits.

A.6. Soft versus hard band T comparison

In Sect. 7.3 we explored the self-consistency of T measurements when one uses only the soft or hard band of the same telescope. To do that, we compared T_{0.5 − 4 keV} and T_{1.5 − 7 keV} for all available cluster regions per instrument. We used 132 cluster regions for eROSITA, 189 for Chandra, and 78 for XMM-Newton. The pivot point values T_piv used in Eq. 4 are 3 keV, 4.5 keV, and 3 keV for eROSITA, Chandra, and XMM-Newton respectively. T_{0.5 − 4 keV} and T_{1.5 − 7 keV} are not fully independent since they overlap within the 1.5 − 4 keV energy range. Here, the partial covariance of their uncertainties is ignored since this test aims to provide indicative results rather than numerically precise ones.

As shown in Table A.2 and Fig. A.5, any discrepancy between T_{0.5 − 4 keV} and T_{1.5 − 7 keV} is almost constant with increasing T, for all instruments. The intrinsic scatter is ≈10% for the Chandra and XMM-NewtonT_{0.5 − 4 keV} − T_{1.5 − 7 keV} comparisons, while it almost doubles for the eROSITA comparison. In the left and middle panels of Fig. A.5, one sees that, for low T_{0.5 − 4 keV}, there is some non-Gaussianity of the scatter. However, excluding these measurements does not significantly alter the results since their statistical weight is generally low due to the increased T_{1.5 − 7 keV} uncertainties.

Fig. A.5. Comparison between T measured in the soft (0.5-4 keV) and hard (1.5-7 keV) bands for Chandra (left), XMM-Newton (middle), and eROSITA (right). The best-fit scaling relation line by LMM (blue) is displayed. The equality 1:1 line is shown in red (dash dot). The blue shaded area represents the LMM statistical error plus the intrinsic scatter.

A.7. Effects of residual contamination in the XMM-Newton/PN camera

In Sect. 7.8.3, we discuss the effect that the residual contamination in the XMM-Newton/PN unexposed corners has on the eROSITA-XMM-Newton scaling relations. In this section, we provide more details for this test.

To characterize the soft proton contamination we use the IN/OUT diagnostic as defined in Ramos-Ceja et al. (2019; see also De Luca & Molendi 2004). Briefly, this refers to the ratio of the surface brightness within the FOV after masking the central 10′ and the surface brightness of the unexposed corners. Both values are measured in the 5 − 7 keV plus 10 − 14 keV bands for XMM-Newton/PN and the 6 − 12 keV band for XMM-Newton/MOS. We estimate the IN/OUT ratio for the actual observation (IN/OUT_obs) and for the FWC data (IN/OUT_FWC). The final IN/OUT value is given by the ratio of IN/OUT_obs and IN/OUT_FWC. XMM-Newton/EPIC detectors with IN/OUT< 1.15 are considered sufficiently filtered from solar flares and they are used in this work.

In Sect. 7.8.3 we discussed our methodology for checking the bias level in our default PIB estimation of XMM-Newton/PN as a function of the latter’s IN/OUT ratio. The results are shown in Fig. A.6. This test clearly shows that there is no obvious bias for the majority of our XMM-Newton data. For XMM-Newton/PN data with 1.1< IN/OUT< 1.15 our method indeed overestimates the rescaling factor by < 15%, that is, we subtracted more photons to correct for the PIB than we should have. We then proceed to correct the PIB subtraction in the XMM-Newton/PN camera for the ten clusters in our sample with 1.1< IN/OUT< 1.15 and refit their T_XMM. We found that the core T_XMM are negligibly affected by this mild PIB bias (ΔT < 2% typically), due to the very high surface brightness of the clusters at these regions. For the core-excised T_XMM, we found that the shift was more significant (ΔT ∼ 3 − 15%) and depended on the IN/OUT ratio of XMM-Newton/PN. Moreover, the T_XMM shifts were not systematic for all clusters (i.e., not all T_XMM increased). Overall, only 11% (8 out of 71) of the T_XMM used in the eROSITA-XMM-Newton scaling relations show a ΔT_XMM > 4% shift, which is the average 1σ uncertainty for these values.

Fig. A.6. Ratio of PIB rescaling factors as they occur from the 2.5 − 5 keV (default) and the 10 − 14 keV count rates of the unexposed XMM-Newton/PN corners (top) and the entire FOV (bottom). The blue shaded region corresponds to the standard deviation of the distribution.

To quantify the effect of these T_XMM changes to the eROSITA-XMM-Newton scaling relations, we replace the default T_XMM with the newly fitted ones that account for the contamination in the XMM-Newton/PN unexposed corners. As expected, the effect is minimal. For the full and soft band eROSITA-XMM-Newton scaling relations, A increased by 0.005 dex (1%) while B increased by ≈2 − 2.5%. These correspond to < 0.3σ changes for both parameters. For the hard band, the effects were even less statistically significant. Subsequently, we show that the default PIB treatment does not introduce any noticeable bias to the final scaling relations.

Finally, we repeat the estimation of the rescaling factor using the whole FOV in the 10 − 14 keV band instead of the unexposed corners. The comparison with the default rescaling factor is shown in the bottom panel of Fig. A.6. Interestingly, the full FOV method returns a similar rescaling factor to our default methodology for all IN/OUT values.

A.8. Comparison of our measurements with the official eRASS1 cluster catalog T

To compare T_eRASS1 to the eROSITA, Chandra, and XMM-NewtonT presented in this work, we matched all the clusters from M20 with all the eRASS1 cluster catalog systems that have available T measurements. Since z correlates with T during spectral fittings, we excluded six clusters that showed a redshift difference of $\left|\frac{\mathrm{\Delta }z}{1+z_{\text{M20}}}\right|>0.01$¹⁸ between M20 and the eRASS1 cluster catalog. This resulted in 78 eRASS1-Chandra, 33 eRASS1-XMM-Newton, and 96 eRASS1-spectroscopic eROSITA clusters.

The best-fit scaling relation parameters for the T comparison between the eRASS1 cluster catalog and the other spectroscopic T measurements are presented in Table A.3.

A.8.1. eRASS1 versus Chandra temperatures

The eRASS1 versus ChandraT comparison is displayed in the left panel of Fig. A.7. Both the core and core-excised T_Chandra show similar deviations from T_eRASS1. The main difference is found in the negligible intrinsic scatter of the relation when the core T_Chandra is used. T_eRASS1, which come from the full R₅₀₀ area of the clusters, are expected to be mostly influenced by the cluster core since this is where most of the emission originates. The eRASS1-Chandra scaling relations agree slightly better than the results presented in Sect. 5.1, but within the statistical uncertainties. The eRASS1-Chandra relations still deviate from the equality line by > 10σ. T_eRASS1 are 22% (21%) lower than the core-excised (core) T_Chandra for T_Chandra = 4.5 keV. For hotter plasma with T_Chandra = 10 keV, the deviation rises to 32% (28%). Cooler gas values of T_Chandra = 2 keV disagree by 10% (21%) between the eRASS1 catalog and Chandra. However, these differences come with non-negligible uncertainties and individual clusters can deviate significantly from these averages. Overall, the conclusions from the eRASS1-ChandraT comparisons do not significantly change compared to the results presented in Sect. 5.1.

Fig. A.7. Comparison between eRASS1 and Chandra (left), XMM-Newton (middle), and spectroscopic eROSITA (right) temperatures. The comparison between the core-excised (core) TChandra/TXMM/TeRO,spec and the single TeRASS1 from the entire R500 is displayed in blue (green). The best-fit scaling relation lines are displayed with the same colors. The equality 1:1 line is shown in red (dash dot), except for the right panel, where it coincides with the green best-fit line. The blue and green shaded areas represent the statistical error plus the intrinsic scatter for the core-excised and core scaling relations respectively.

A.8.2. eRASS1 versus XMM-Newton temperatures

The eRASS1 versus XMM-NewtonT comparison is displayed in the middle panel of Fig. A.7. Due to the limited sample and the significant scatter, no statistical significant differences can be found between the core and core-excised T_XMM compared to T_eRASS1, although the core-excised T_XMM offers a best-fit relation slightly closer to the equality line. The eRASS1-XMM-Newton scaling relations are consistent with the results presented in Sect. 5.2 within the uncertainties. However, the best-fit functions deviate more than the equality line compared to the comparison with the spectroscopic T_eROSITA in 5.2, while the scatter here is also larger. For cool gas with T_XMM = 1.5 keV, T_eRASS1 is found within 2% (13%) when the core-excised (core) scaling relation is considered. Respectively, for T_XMM = 3 keV the difference is 18% (23%), while it rises to 34% (33%) for T_XMM = 7 keV.

A.8.3. eRASS1 versus spectroscopic eROSITA temperatures

The comparison between the eRASS1 and the spectroscopic eROSITA T measured in this work is displayed in the right panel of Fig. A.7. Both the core-excised and core scaling relations show excellent agreement with the equality line (the core relation practically coincides with the 1:1 line). The scatter of the relations is almost exclusively driven by the T measurement uncertainties with no significant intrinsic scatter being present.

All Tables

All Figures

	Fig. 1. Comparison between eROSITA and Chandra temperatures for the full (left), soft (middle), and hard (right) bands. The best-fit scaling relation line by LMM (blue) and BCES(orth) (green dashed) are displayed. The equality 1:1 line is shown in red (dash dot). The blue shaded area represents the LMM statistical error plus the intrinsic scatter.
In the text

	Fig. 2. 1σ (68.3%) and 3σ (99.7%) confidence levels for the eROSITA-Chandra scaling relations for the full (blue), soft (pink), and hard (green) bands. The 1:1 line is represented by the black cross. The band best-fit lines for all bands deviate by ≳10σ from the 1:1 line.
In the text

	Fig. 3. Same as in Fig. 1 but for the comparison between eROSITA and XMM-Newton temperatures.
In the text

	Fig. 4. 1σ (68.3%) and 3σ (99.7%) confidence levels for the eROSITA-XMM-Newton scaling relations for the full (blue), soft (pink), and hard (green) bands. The 1:1 line is represented by the black cross. The full and soft band best-fit lines deviate by 5σ and 4.2σ respectively from the 1:1 line.
In the text

Fig. 5. Same as in Fig. 1 but with a broken power law fit with LMM for the full (top) and soft (bottom) bands, as shown in Eq. (8). The break point of the power law is found at Tbreak = 2.7 keV and 1.7 keV, respectively. The low-T clusters are much better fit than the single power law fit, showing consistency between instruments. At high-T, the best-fit line deviates more from the 1:1 line than the single power law fit.

In the text

	Fig. 6. Normalized average effective area for eROSITA TM0 (red, solid), XMM-Newton-EPIC (blue, dashed), and Chandra-ACIS (green, dot dashed), as derived from the spectra used in this work.
In the text

Fig. 7. Confidence levels for the eROSITA-Chandra and eROSITA-XMM-Newton scaling relations for different subsamples, defined based on their cluster properties. Top: 1σ (68.3%) contours for the core-only and annulus-only eROSITA-Chandra and eROSITA-XMM-Newton scaling relations for the full band. The colors are explained in the figure legend. The best-fit results do not differ significantly for core and annuli cluster T for any instrument comparison. Bottom: 1σ (68.3%) and 3σ (99.7%) confidence levels for the eROSITA-Chandra scaling relations for high NH > 5.59 cm−2 (pink) and low NH < 2.45 cm−2 (blue) clusters. The median values of the subsamples are NH = 1.72 cm−2 and NH = 7.96 cm−2, respectively. The contours correspond to the full band. The 1:1 line is represented by the black cross. The best-fit results are very similar between clusters with low and high Galactic absorption.

In the text

Fig. 8. Ratio of the eROSITA and Chandra (left) and eROSITA and XMM-Newton (right) single-T fits to simulated spectra with two temperature components as a function of the Chandra and XMM-Newton best-fit single-T, respectively. The blue, green, and red colors correspond to Tcold = 0.5, 0.75, 1 keV, respectively. The hot component has values of Thot = 1, 2, 4, 6 keV. The circles, triangles, and squares correspond to emission measures of EM = 0.05, 0.10, 0.15 for the cold and hot components. The black curve shows the observed T difference between eROSITA-Chandra (left) and eROSITA-XMM-Newton (right), as presented in Table 1.

In the text

	Fig. A.1. 1σ confidence levels for the eROSITA-Chandra (top) and eROSITA-XMM-Newton (bottom) scaling relations for the full (blue), soft (red), and hard (green) bands. The more opaque ellipses correspond to the default results, while the more transparent ellipses correspond to the best-fit results when TChandra and TXMM are increased by ∼0.5 − 1σ, as described in Sect. A.2.
In the text

	Fig. A.2. Histogram of core (blue) and core-excise (green) temperature values for Chandra (right) and XMM-Newton (right). The two distributions are very similar for both instruments.
In the text

	Fig. A.3. Dependence of the intrinsic scatter on TChandra and TXMM for the eROSITA-Chandra (top) and eROSITA-XMM-Newton (bottom) scaling relations, respectively. Full band results are displayed in blue, while the hard band results are shown in black and gray. The shaded bands represent the best-fit, full sample σintr with its 1σ uncertainties.
In the text

	Fig. A.4. Same as in Fig. 8 but for the hard band fits.
In the text

	Fig. A.5. Comparison between T measured in the soft (0.5-4 keV) and hard (1.5-7 keV) bands for Chandra (left), XMM-Newton (middle), and eROSITA (right). The best-fit scaling relation line by LMM (blue) is displayed. The equality 1:1 line is shown in red (dash dot). The blue shaded area represents the LMM statistical error plus the intrinsic scatter.
In the text

	Fig. A.6. Ratio of PIB rescaling factors as they occur from the 2.5 − 5 keV (default) and the 10 − 14 keV count rates of the unexposed XMM-Newton/PN corners (top) and the entire FOV (bottom). The blue shaded region corresponds to the standard deviation of the distribution.
In the text

Fig. A.7. Comparison between eRASS1 and Chandra (left), XMM-Newton (middle), and spectroscopic eROSITA (right) temperatures. The comparison between the core-excised (core) TChandra/TXMM/TeRO,spec and the single TeRASS1 from the entire R500 is displayed in blue (green). The best-fit scaling relation lines are displayed with the same colors. The equality 1:1 line is shown in red (dash dot), except for the right panel, where it coincides with the green best-fit line. The blue and green shaded areas represent the statistical error plus the intrinsic scatter for the core-excised and core scaling relations respectively.

In the text