5 Spatially resolved spectroscopy

   
5.1 Global spectrum

For each instrument, a global spectrum was extracted from all events lying within $5\hbox{$.\mkern-4mu^\prime$ }1$ from the cluster emission peak. This radial range was chosen to maximise the S/N ratio, allowing us to check in detail the consistency between the three cameras. Each global spectrum was fitted with an absorbed MEKAL model with the redshift fixed at z = 0.143. The normalisation for each instrument was left as an additional free parameter. We excluded the energy bins around the strong fluorescence lines of Ni, Cu & Zn from the pn spectrum fit. These lines, present in the background, are not well subtracted by the procedure described in Sect. 2 because they do not scale perfectly with the continuum of the particle-induced background. In all fits we used the following response matrices: m1_thin1v9q20t5r6_all_15.rsp (MOS1), m2_thin1v9q20t5r6_all_15.rsp (MOS2) and epn_ef20_sY9_medium.rsp (pn).

Figure 4: XMM-Newton spectra of the cluster from annuli 5 ( $1.35\hbox {$^\prime $ }< \theta < 1.89\hbox {$^\prime $ }$, top panel) and 9 ( $5.13\hbox {$^\prime $ }<\theta < 7.16\hbox {$^\prime $ }$, bottom panel). Black (red) (green) points: EPIC/MOS1(2)[pn] data. The EPIC spectra are background subtracted and corrected for vignetting as described in Sect. 2. Solid lines: best fit isothermal model with parameters given in Table B.1.

Figure 5: The projected temperature profile of A1413. The bold black profile is the total total MOS+pn fit. For comparison we show the MOS 1 (red) MOS 2 (blue) and pn (green) separate fits. The error bars are $1\sigma$.

Fitting the data from all instruments above $0.3~\rm keV$, with the absorption fixed at the galactic value of $N_{\rm H} = 2.2 \times 10^{20}$ cm^-2, we found inconsistent values for the temperature derived with the MOS and pn cameras: kT = 6.91_-0.23^+0.23 keV (MOS1), $6.33_{-0.23}^{+0.23}~\rm keV$ (MOS2) and  $5.76_{-0.19}^{+0.19}~\rm keV$ keV (pn). A better agreement between the cameras, together with a lower $\chi^2$ value, are obtained if the $N_{\rm H}$ value is let free, but then the best fit $N_{\rm H}$values are significantly lower than the 21 cm value (see Table 3). In other words the data presents an excess at low energy as compared to an isothermal model absorbed with the galactic hydrogen column density. This effect could be due to a true soft excess component (e.g. Durret et al. 2002) and/or an artifact due to remaining calibration uncertainties. In particular, it is known that the EPIC-pn and MOS cameras show a relative flux difference which increases with energy above $4.5~\rm keV$, resulting in a MOS spectral slope flatter than the pn (Saxton 2002; Griffiths et al. 2002).

We then performed a systematic study of the effect of imposing various high and low-energy cutoffs for each instrument. The $N_{\rm H}$ is fixed to the 21 cm value. Having first found that progressive cutting of the high energy channels had a negligible effect on the derived temperatures, we then varied the low energy cutoff, for which the results are shown in Table 3.

This table shows that there is an optimum low-energy cutoff for each instrument, above which no amount of further cutting of the low-energy response will significantly affect the temperature. The temperature stabilises above a certain cutoff point for each instrument, this being $\sim$0.6 keV for the MOS cameras and $\sim$1.0 keV for the pn camera. The adoption of these low-energy cutoffs has the pleasing effect of bringing the temperatures for each instrument into agreement both with each other and with previous ASCA analysis. The combined MOS+pn global temperature is $kT = 6.85^{+ 0.15}_{- 0.16}~\rm keV$ ($90\%$ confidence for one interesting parameter, $\chi^{2}=1459.6$ for 1436 d.o.f.) in agreement with the results of Ikebe et al. (2002), who find $kT = 6.56^{+0.65}_{-0.44}~\rm keV$ and Matsumoto et al. (2001), who find $kT = 6.72 \pm 0.26~\rm keV$, and marginally consistent with the result of White (2000), who finds $kT = 7.32_{-0.24}^{+0.26}~\rm keV$.

It thus appears that the discrepancies observed by fitting the whole energy range are mostly due to some residual calibration uncertainties in the low-energy response of all instruments and/or a true soft excess. The scientific analysis of such a possible soft excess is beyond the scope of this paper. To minimise these effects, we adopted the low-energy cutoffs derived above for the spatially-resolved analysis discussed below.

   
5.2 Radial temperature profile

We produced a radial temperature profile by excluding sources and extracting spectra in annuli centred on the peak of the X-ray emission. All spectra were binned to $3 \sigma$ above background level (except the final annulus, which was binned to $2\sigma$) to allow the use of Gaussian statistics. We show the fitted spectra for annuli 5 and 9 in Fig. 4.

These spectra were fitted using the absorbed MEKAL model described above; we fitted separately the spectra from each instrument as well as making a simultaneous MOS+pn fit, as detailed in Table B.1. All the temperatures are consistent within the respective errors. It is also evident from Fig. 5 that the form of each profile is similar. As a further test, we fitted the annular spectra with an absorbed MEKAL model with the absorption left as a free parameter. This produced profiles with, again, exactly the same shape, giving us high confidence in the form of the profile which we have derived.

   
5.3 Projection and PSF effects

5.3.1 PSF correction

The PSF of XMM-Newton is a potential cause of concern, especially in the inner regions, where the bin sizes are small. To assess the effect of the PSF, we first calculate a redistribution matrix, F(i,j), where F(i,j) is the fractional flux in annulus i coming from annulus j. These redistribution factors were derived from our best model of the gas density profile, converted to emission measure profile and convolved with the XMM-Newton PSF. The fractional contribution in each bin of the emission coming from the bin, as well as all inner and outer bins are plotted in Fig. 6. The PSF mostly affects the central regions and, above  $2\hbox{$^\prime$ }$, the contamination from adjacent bins is less than $25\%$.

We have a total of 30 spectra (10 annuli $\times$ 3 cameras) to be fitted with a model consisting of 10 MEKAL models (corresponding to the 10 "true'' temperatures) absorbed by a common (frozen) absorption. The normalisations of the MEKAL models, for annulus i, are linked by the factors F(i,j), such as to reflect the contribution of each annulus j due to the PSF. In practice we ignore any contributions at the less than $1\%$ level. Each MEKAL model has 6 parameters, which, together with the absorption, makes 7. If we fit the 30 spectra simultaneously, the model has $(6 \times 10 +1) \times 30 = 1830$ parameters. XSPEC can only handle 1000 parameters, (even if most of them are frozen), so we have to find a way to reduce their total number. One way is to group the spectra, but for this to work the spectra in each group need a common normalisation.

Figure 6: Redistribution of the flux due to the XMM-Newton PSF: fraction of flux in each radial bin due to the emission of the bin (filled circles), as well as all inner (open circles) and outer bins (squares).

We typically find differences of $\sim$$20\%$ between the normalisations of MOS and pn annular spectra (MOS1 and MOS2 normalisations are the same to within $\sim$$5\%$). We fit the global spectrum discussed in Sect. 5.1 to find the overall difference in normalisation between MOS and pn. We then checked that the annular fit results were the same when the global difference in normalisation was applied. This being so, we multiplied the pn annular spectra by this factor to bring their normalisations into line with those for the MOS cameras.

We now grouped the MOS and pn spectra of each annulus, giving 10 groups of 3 spectra, leaving us with $(6 \times 10 +1) \times 10 = 610$ parameters, enabling a simultaneous fit. We froze the metallicity of each MEKAL model at the best-fit value found for each projected annulus. The free parameters in the fit are then the temperature and normalisation of each MEKAL model. The resulting PSF-corrected profile is shown in Fig. 7 ($1\sigma$ errors). The PSF corrected results are entirely consistent with the projected temperature profile, with systematic differences of about half the $1\sigma$ errors in the first 3 bins and smaller beyond. This result is not surprising, since the profile is relatively flat. Consideration of the PSF has a much smaller effect on the temperature profile of A1413 than for the bright cooling flow cluster A1835 at z=0.25 (Markevitch 2002; Majerowicz et al. 2002). In contrast to A1835, A1413 displays neither an extremely steep rise in the central gas density, nor a sharp drop in the temperature towards the center. As a result the contamination of central bins is first smaller: for A1835, more than 1/3 of the observed brightness at any radius is due to PSF scattering at smaller radii (Markevitch 2002; Majerowicz et al. 2002) while for A1413 this contamination is already less than $25\%$ at  $2\hbox{$^\prime$ }$, and decreases beyond (Fig. 6). Secondly, the smaller temperature gradient towards the center means that the redistribution biases less the temperature determinations.

5.3.2 Spectral deprojection

Figure 7: The effect of deprojection and the PSF. The projected temperature profile of A1413 (squares) compared with the profile obtained after correction for the PSF (diamonds) and the deprojected profile (triangles). Errors are $1\sigma$.

Another possible source of error in the derived profile comes from projection effects.

A deprojected temperature profile was produced by first simultaneously fitting the MOS and pn spectra of the outer annulus with a MEKAL model absorbed by a fixed galactic column density. The spectrum of the next annulus inward was then fitted with a two-temperature model with the parameters of one of the models fixed to the best-fitting values derived for the outer spectrum. The normalisation of each fixed model must account for the volume within the outer shell projected along the line of sight toward the next shell inward. Furthermore, as the gas density profile is not flat, the normalisation must also account for this effect. We model the gas density profile using the parameters from the best-fit KBB model described in Sect. 4. The normalisations are then adjusted by the emission weighted volume factors. This process was continued inward, adding a MEKAL model for each annulus, with the parameters of the outer annulus models frozen to their previously determined best-fit values. The abundances of the two outer annuli were frozen to the global value, so for these fits the free parameter is the temperature. For all other annuli both the temperature and abundance were free parameters. The deprojected temperature profile is shown compared to the projected profile in Fig. 7. In practice we find very little difference between the projected and deprojected results. The jump in the temperature of the ninth annulus is somewhat an artifact of the fitting process. In this case the software tries to compensate for the contribution of the low temperature found in the tenth (and first-fitted) annulus by putting a higher temperature in the subsequent annular bin. The error on the tenth temperature is large, the contribution of the outer emission in the ninth bin depends on the actual cluster extent and thus the deprojected ninth temperature is probably more uncertain than found in this simple procedure. Note, however, that the projected and deprojected temperatures agree well within the $1\sigma$ errors.

In summary, we find that neither a consideration of the PSF or projection effects substantially changes the form of the temperature profile. The profiles obtained by taking into account these effects are consistent with the projected profile, within the $1\sigma$errors. For all subsequent analysis, we thus used the projected profile.

   
5.4 Modelling the temperature profile

We now consider the scaled temperature profile, $\tau(x)= T(r)/T_{\rm X}$, where $T_{\rm X}$ is the average temperature and x is the scaled radius, normalised to r₂₀₀. r₂₀₀ is estimated from the average temperature $T_{\rm X}$ and the r₂₀₀-T relation of EMN96 at the cluster redshift. $T_{\rm X}$ is estimated by fitting the global spectrum, extracted from the $[0.5\hbox{$^\prime$ }{-}9\hbox{$^\prime$ }]$region, i.e. outside the possible cooling flow region (see below). We found $kT_{\rm X} = 6.49\pm0.15~\rm keV$ ($1\sigma$ error). Note that the temperature profile is determined up to $\sim$ 0.7 r₂₀₀ or $\sim$r₅₀₀.

We then modelled this projected temperature profile with a polytropic model:

$$\tau(x) = \tau_0 \left(\frac{n_{\rm H}(x r_{200})}{n_{\rm H,0}}\right)^{(\gamma - 1)}$$ (10)

where $n_{\rm H}$ is the gas density profile given by the KBB model (Eq. (5)), $\gamma$ is the polytropic index and $\tau_0$ is the normalised temperature at x=0.

Figure 8: The projected temperature profile of A1413 with the best-fit polytropic model obtained with the central bin excluded (full line). The best fit for the entire radial range is shown as a dashed line.

We fitted the profile with all ${\beta }$, $r_{\rm c}$ and $\xi$ parameters fixed at the values best fitting the surface brightness profile (Table 2), so that the free parameters for the fit are $\tau_0$ and $\gamma$. When the whole radial range is fitted, we find $\tau_0 = 1.11 \pm 0.03$ and $\gamma = 1.03 \pm 0.01$, with $\chi^2 = 7.62/8$. If we then exclude the inner point, we find a better fit with $\tau_0 = 1.22 \pm 0.03$ and $\gamma = 1.07\pm 0.01$, and $\chi^2 = 4.45/7$. The polytropic fits to the temperature profile are shown in Fig. 8.

We note that the fit is considerably better if the central point is excluded from the fit. The resulting polytropic profile rises to a peak in the centre which is not seen in the annular temperature determinations, which may lead us to believe that there is a small cooling flow (CF) at work in the very central regions. This possibility is discussed further in Sect. 6.

We further note that the derived value for the $\gamma$-parameter is very close to isothermal and moreover, is not very sensitive to whether the central bin is included in the fit. There is a tantalising hint that the temperature profile may drop further in the very outer regions, but the errors on this last data point are large enough that it is easily compatible with the derived polytropic model.