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Appendix A: Detailled data reduction

Appendix A.1: GTI filtering

In order to reduce the soft-proton (SP) background, we build good time intervals (GTI) using the light curves in the 10–12 keV band for MOS and 12–14 keV band for pn. We fit the count-rate histograms from the light curves of each instrument, binned in 100 s intervals, with a Poissonian function and we reject all time bins for which the number of counts lies outside the interval μ ± 2σ (i.e. ), where μ is the fitted average of the distribution. We repeat the same screening procedure and threshold (so-called 2σ-clipping) for 10 s binned histograms in the 0.3–10 keV band because De Luca & Molendi (2004) reported episodes of particularly soft background flares. In order to get a qualitative estimation of the residual SP flare contamination, we use the Fin_over_Fout algorithm which compares the count rates in and out of the FoV of each detector (De Luca & Molendi 2004). We found that in both observations MOS 1 displays a F_in/F_out ratio higher than 1.3, meaning that the observations have been significantly contaminated by SP events. This value is still reasonable though, and a look at the filtered light curve lead us to keep the MOS 1 datasets. Furthermore, a careful modelling of convenient SP spectral components are used in our spectral fittings as well (see Appendix B).

Appendix A.2: Resolved point sources excision

The point sources in our FoV contribute to the total flux and may bias the astrophysical results that we aim to derive from the cluster emission. Therefore, they should be discarded. We detect all the resolved point sources (RPS) with the SAS task edetect_chain and we proceed with a second check by eye in order to discard erroneous detections and possibly include a few missing candidates. It is common practice in extended source analysis to excise bright point sources from the EPIC data. We note, however, that many sources have fluxes below the detection limit S_cut and an unresolved component might remain (Appendix B.2).

A remaining problem is how to choose the excision radius in the best way. A very small excision radius may leave residual flux from the excised point sources while a very large radius may cut out a significant fraction of the cluster emission leading to decreased S/N. We define A_eff as the extraction region area for the cluster emission when the point sources are excised with a radius r_s, (A.1)where N is the number of sources, S is the flux, and A is the full detection area.

Since we are dealing with a Poissonian process, S/N can be estimated as , where C and B are the number of counts of the cluster emission and the total background, respectively. The value of C depends on the extraction area and can thus be written C = C^∗A_eff, where C^∗ is the local surface brightness of the cluster (counts/′′), and B can be divided into the instrumental or hard particle (HP) background I, an unresolved point sources (UPS) component, and the remaining excised point source flux outside the excision region. The total background can be thus written as (A.2)where EEF(r_s) is the encircled energy fraction of the PSF as a function of radius. We can finally write the total S/N as (A.3)The optimum S/N can be then computed as a function of r_s and S_cut (Eq. (A.3)). In Appendix B.2 we discuss the origin of dN/ dS. We find and adopt an optimised radius for RPS excision in our dataset of ~10′′.

Appendix A.3: RGS spectral broadening correction from MOS 1 image

Because the RGS spectrometers are slitless and the source is spatially extended in the dispersion direction, the RGS spectra are broadened. The effect of the broadening of a spectrum by the spatial extent of the source is given by (A.4)where m is the spectral order and θ is the offset angle in arcmin (see the XMM-Newton Users Handbook).

The MOS 1 DET Y direction is parallel to the RGS dispersion direction. Therefore, we extract the brightness profile of the source in the dispersion direction from the MOS 1 image and use this to account for the broadening following the method described by Tamura et al. (2004). This method is implemented through the Rgsvprof task in SPEX. As an input of this task, we choose a width of 10′ around the core and along the dispersion axis, in which the cumulative brightness profile is estimated. In order to correct for continuum and background, we use a MOS 1 image extracted within 0.5–1.8 keV (i.e. the RGS energy band). This procedure is applied to both observations and we average the two spatial profiles obtaining a single profile that will be used for the stacked RGS spectrum.

Appendix B: EPIC background modelling

We split the total EPIC background into two categories, divided further into several components:

• 1.

  Astrophysical X-ray background (AXB), from the emission ofastrophysical sources and thus folded by the response files. TheAXB includes the Local Hot Bubble (LHB), the galactic thermalemission (GTE), and the UPS.

• 2.

  Non-X-ray background (NXB), consisting of soft or hard particles hitting the CCD chips and considered as photon events. For this reason, they are not folded by the response files. The NXB contains the SP and the HP backgrounds.

In total, five components are thus carefully modelled.

Appendix B.1: Hard particle background

High energy particles are able to reach the EPIC detectors from every direction, even when the filter wheel is closed. Besides continuum emission, they also produce instrumental fluorescence lines which should be carefully modelled. Moreover, for low S/N areas, we observe a soft tail in the spectra due to the intrinsic noise of the detector chips. A good estimate of the HP background can be obtained by using Filter Wheel Closed (FWC) data which are publicly available on the XMM-Newton SOC webpage⁵. We select FWC data that were taken on 1 October 2011 and 28 April 2011 with an exposure time of 53.7 ks and 35.5 ks for MOS and pn, respectively. We removed the MOS 1 events from CCD3, CCD4, and CCD6 to be consistent with our current dataset.

Instead of subtracting directly the FWC events from our observed spectra, modelling the HP background directly allows a much more precise estimate of the instrumental lines fluxes, which are known to vary across the detector (Snowden & Kuntz 2013).

We fit the individual FWC MOS and pn continuum spectra with a broken power law F(E) = YE^{− Γ}e^η(E) where η(E) is given by (B.1)with ξ = ln(E/E₀) and (see SPEX manual). In this model, the independent parameters are A, Γ (spectra index), ΔΓ (spectral index break), E₀ (break energy), and b (break strength). Unlike the instrumental lines, this continuum does not vary strongly across the detector. Tables B.1 and B.2 show the best-fit parameters that we found for the entire FoV extraction area and the modelled instrumental lines, respectively. In addition to the broken power-law, each instrumental line is modelled with a narrow (FWHM ≤ 0.3) Gaussian function. Although a delta function is more realistic, in this case allowing a slight broadening optimises the correction for the energy redistribution on the instrumental lines.

Appendix B.2: Unresolved point sources

An important component of the EPIC background is the contribution of UPS to the total X-ray background. Its flux can be estimated using the so called log N–log S curve derived from blank field data. This curve describes how many sources are expected at a certain flux level. The source function has the form of a derivative (dN/ dS) and can be integrated to estimate the number of sources in a certain flux range, (B.2)where N is the number of sources and S is the low-flux limit.

The most common bright UPS are AGNs, but galaxies and hot stars contribute as well. Based on the Chandra deep field, Lehmer et al. (2012) find that AGNs are the most dominant in terms of number counts, but in the 0.5–2 keV band the galaxy counts become higher than the AGN counts below a few times 10^-28 W m^-2 deg^-2. The assumed spectral model of this component is a power-law with a photon index of Γ = 1.41 (see e.g. Moretti et al. 2003; De Luca & Molendi 2004). In reality, the power-law index may vary slightly between 1.4–1.5, given the uncertainties in the different surveys and estimations (Moretti et al. 2009). Based on the Chandra Deep Field South (CDF-S) data, Lehmer et al. (2012) define the (dN/ dS) relations for each source category as follows: Each relation describes a power law with a normalisation constant K and a slope β. Since the (dN/ dS) relation of AGNs shows a break, there is an additional β₂ parameter and a break flux f_b. The reference flux is defined as S_ref ≡ 10^-14 erg cm^-2 s^-1. The best-fit parameters for the studied energy bands are listed in Table 1 of Lehmer et al. (2012).

The relations above can be used to estimate the flux from sources that are not detected in our EPIC observations. The UPS component also holds for the deepest Chandra observations. Hickox & Markevitch (2006) found a detection limit of 1.4 × 10^-16 in a 1 Ms CDF-S observation and estimated the unresolved flux to be (3.4 ± 1.7) × 10^-12 erg cm^-2 s^-1 deg^-2 in the 2–8 keV band. Since Chandra has a much lower confusion limit and a narrower PSF, we do not expect EPIC to reach this detection limit even in a deep cluster observation. It is therefore not necessary to know the log N–log S curve below this flux limit to obtain a reasonable estimate for the unresolved flux.

In the flux range from 1.4 × 10^-16 up to the EPIC flux limit, we can calculate the flux using the log N–log S relation. The total unresolved flux Ω_UPS for the 2–8 keV band is then calculated using (B.6)Using the Eqs. (B.3)–(B.5)for in the integral above, the unresolved flux calculation is straightforward. Given the detection limit of our observations S_cut = 3.83 × 10^-15 W m^-2, we find a total UPS flux of 8.07 × 10^-15 W m^-2 deg^-2. This value can be used to constrain the normalisation of the power-law component describing the AXB background in cluster spectral fits. We note that this method does not take the cosmic variance into account (see e.g. Miyaji et al. 2003), which means that the normalisation may still be slightly biased.

Appendix B.3: Local Hot Bubble and Galactic thermal emission

The LHB component is thought to originate from a shock region between the solar wind and our local interstellar medium (Kuntz & Snowden 2008), while the GTE is the X-ray thermal emission from the Milky Way halo. At soft energies (below ~1 keV), the flux of these two foreground components is significant. They are both known to vary spatially across the sky, but we assume that they do not change significantly within the EPIC FoV. Both components are modelled with a CIE component where we assume the abundances to be proto-solar. Both temperatures are left free, but are expected to be within 0.1–0.7 keV. The GTE component is absorbed by a gas with hydrogen column density (N_H = 1.26 × 10²⁰ cm^-2), while the LHB component is not.

Appendix B.4: Residual soft-proton component

Even after filtering soft flare events from our raw datasets, a quiescent level of SP remains that might affect the spectra, especially at low S/N and above ~1 keV. It is extremely hard to precisely estimate the normalisation and the shape of its spectrum since SP quiescent events strongly vary with detector position and time (Snowden & Kuntz 2013). They may also depend on the attitude of the satellite. For these reasons, blank sky XMM-Newton observations are not good enough for our deep exposures. The safest way to deal with this issue is to model the spectrum by a single power law (Snowden & Kuntz 2013). Using a broken power law might be slightly more realistic, but the number of free parameters is then too high to make the fits stable. The spectral index Γ of the power law is unfortunately unpredictable and may be different for MOS and pn instruments and between different observations. Since Snowden & Kuntz (2013) reported spectral indices between ~0.1–1.4, we allow the Γ parameter in our fits to vary within this range.

Appendix B.5: Application to our datasets

We apply the procedure described above for each component on our two observations of A 4059. We extract an annular region with inner and outer radii of 6′ and 12′, respectively, and centred on the cluster core (Fig. 1, the outer two annuli), assuming that all the background components described above contribute to the detected events covered by this area. In order to get a better estimation of the foreground thermal emission (GTE and LHB), we fit a ROSAT PSPC spectrum from Zhang et al. (2011) simultaneously with our EPIC spectra. This additional observation covers an annulus centred to the core and with inner and outer radii of 28′ (~r₂₀₀) and 40′ (~r₂₀₀ + 12′), respectively, avoiding instrumental features and visible sources. We note that in this fit we also take the UPS contribution into account. Depending on the extraction area, all the normalisations (except for the UPS component, Appendix B.2) are left free, but are properly coupled between each observation and instrument.

Table B.3 shows the different background values that we found for the extracted annulus. Figure B.1 shows the result for the MOS 2 spectrum at the first observation, its best fit model, and the contribution of every modelled component. As expected, the NXB contribution is more important at high energies. Above ~5 keV, the cluster emission is much smaller than the HP background. Consequently and as already reported, the temperature and abundances measured by EPIC are harder to estimate in the outer parts of the FoV.

	Fig. B.1 EPIC MOS2 spectrum of the 6′–12′ annular region around the core (see text). The solid black line represents the total best-fit model. Its individual modelled components (background and cluster emission, solid coloured lines) are also shown.
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	Fig. B.2 Expected relative errors on the temperatures and abundances. Different cell sizes (symbols) are simulated within the inner five annuli (colours).
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Finally, we apply and adapt our best background model to the core region (Sect. 4) and the eight concentric annuli (Sect. 5). The normalisation of every background component has been scaled and corrected for vignetting if necessary. From the background parameters, only the normalisations of the HP component (initially evaluated from the 10–12 keV band, where negligible cluster emission is expected), as well as those of the instrumental fluorescent lines, are kept free for all the spectra. In the outermost annulus (9′–12′) we ignore the channels below 0.4 keV (MOS) and 0.6 keV (pn) to avoid low energy instrumental noise. For the same reason we ignore the channels below 0.4 keV (MOS) and 0.5 keV (pn) in the second outermost annulus (6′–9′). The background is also applied to and adapted for the analysis of the spectra of each map cell (Sect. 6).

Appendix C: S/N requirement for the maps

Despite their good statistics, we want to optimise the use of our data and find the best compromise between the required spatial resolution of our maps (Sect. 6) and S/N. The former is necessary when searching for inhomogeneities and kT/metal clumps (i.e. the smaller the better), the latter to ensure that the associate error bars are small enough to make our measurement significant. Clearly, these variables depend on the properties of the cluster and on the exposure time of our observations.

We perform a set of simulations to determine what the best combination of S/N and spatial resolution is for the case of A 4059. For every annulus (i.e. the ones determined in Sect. 5) we simulate a spectrum with input parameters (i.e. kT, O, Ne, Mg, Si, S, Ca, Fe, Ni, and the normalisation) corresponding to the ones determined in the radial profiles analysis.

The AXB and the HP background are added to the total spectrum by using the properties derived in Appendix B. We allow their respective normalisations to vary within ±3% in order to take into account spatial variations on the FoV. Starting from the value we derived for the radial profile, we rescale the normalisation of the simulated spectrum to the particular spatial resolutions we are interested in (here we test 15′′, 20′′, 25′′, 30′′, 40′′, 50′′, and 60′′). We then fit the spectrum as done for the real data and for all the annuli and spatial resolutions we calculate the relative errors on the temperature and Fe abundance as a function of S/N. The median values of 300 realisations are shown in Fig. B.2 with their 1σ errors.

A S/N of 100 is required to measure the abundance with a relative error lower than ~20%. With this choice the temperature will be also determined with a very good accuracy, i.e. relative errors always lower than ~5%.

© ESO, 2015