4 Gas density profile

4.1 Surface brightness profile

For each camera, we generated an azimuthally averaged surface brightness profile for both source and background observations. Weighted events from the corresponding event files were binned into circular annuli centred on the position of the cluster emission peak. We cut out serendipitous sources in the field of view and the southern sub-structure. The background subtraction was performed as described in Sect. 2. We consider the profiles in several energy bands. Due to the contribution of the instrumental Al K line around 1.5 keV, we ignored the (1.4-2.0) keV band in both cameras. To maximise the signal to noise (S/N) ratio, particularly in the outer cluster region, we choose to base the following on analysis of the (0.3-1.4) keV band.

We checked that the vignetting corrected and background subtracted profiles of the three cameras are consistent: they differ only by a normalisation factor within the error bars. We thus coadd the profiles and bin the resulting profile is the following way. Starting from the central annulus, we re-binned the data in adjacent annuli so that i) at least a S/N ratio of $3 \sigma$ is reached and ii) the width of the bin increases with radius, with $\Delta(\theta) > 0.1\theta$. Such a logarithmic radial binning insures a S/N ratio in each bin roughly constant in the outer part of the profile, when the background can still be neglected.

The resulting surface brightness profile, $S(\theta)$, is shown in Fig. 3. The cluster emission is significantly detected up to $R_{\rm det}=8.6^\prime$ or 1.7 Mpc.

Figure 3: Combined MOS1, MOS2 and pn surface brightness profile of A1413 in the $[0.3{-}1.4]~\rm keV$ energy band. The profile is background subtracted and corrected for vignetting effects. Black (red) (green) lines: best fit KBB (Eq. (5)), BB (Eq. (3)) and AB (Eq. (2)) models convolved with the XMM-Newton PSF binned as the observed profile. Dotted line: best fit ${\beta }$-model fitted to the outer region of the cluster ( $\theta >1.3\hbox {$^\prime $ }$). See Sect. 4.2 for model details and Table 2 for best fit parameter values.

   
4.2 Density profile modelling

We fitted $S(\theta)$ with various parametric models convolved with the XMM-Newton PSF (Ghizzardi 2001; Griffiths & Saxton 2002), binned into the same bins as the observed profile.

A single ${\beta }$-model cannot account for the data. When the entire radial range is fitted, the reduced $\chi^2$ is $\sim$13; for the best fit slope, $\beta=0.60$, and core radius, $\theta_{\rm c}=0.66\hbox{$^\prime$ }$. An excess of emission is readily apparent in the centre and a lower reduced $\chi^2$ is obtained when excluding the central region from the fit. The reduced $\chi^2$ decreases with increasing cut-out radius until it stabilises for  $R_{\rm cut}\sim 1.3'$. In that case we obtained $\chi^2=47$ for 31 d.o.f., with $\beta=0.71~\pm~0.02$ and $\theta_{\rm c}=1.30\hbox{$^\prime$ }~\pm~ 0.09\hbox{$^\prime$ }$. The best fit model is plotted as a dotted line in Fig. 3. The ${\beta }$ value is not surprisingly larger than the value ( $\beta=0.62$) derived by Cirimele et al. (1997) from their global fit to the ROSAT profile, but is in excellent agreement with the value $\beta=0.70\pm0.02$ obtained by Vikhlinin et al. (1999) by fitting the outer cluster region ( $\theta > 4.9'$). There is also an excellent agreement between the 1D and 2D ${\beta }$-values.

We note that the last two points ( $7.75\hbox{$^\prime$ }< \theta < 8.6\hbox{$^\prime$ }$) lie significantly below the best fit model (a $3 \sigma$ effect for the last bin). The cluster flux in the last bin is about 16% of the total background and we cannot totally exclude that this discrepancy is an artifact due to remaining systematic uncertainties in the background subtraction. This is further discussed in Sect. 9.4. These last two points are discarded in the present analysis.

For the mass analysis which follows (Sect. 8) it is convenient to have an analytical description of the gas density radial profile ( $n_{\rm H}(r)$) at all radii. We thus tried several alternative parameterisations, with behaviour at large radii similar to a ${\beta }$-model:

• AB model: A cusped profile similar to the NFW universal density profile:
  

  $$n_{\rm H}(r) = A \left ( \frac{r }{r_{\rm c}} \right)^{- \alp... ...c}}\right)^{2} \right]^{ \frac{3\beta}{2} + \frac{\alpha}{2} }$$ (2)

  
  

  where $\alpha$ is the slope at small radii.
  

• BB model: We also introduce a double isothermal ${\beta }$ model (BB), assuming that both the inner and outer gas density profile can be described by a ${\beta }$-model, but with different parameters.
  

  $$\begin{array}{lllll} r < R_{\rm cut}&n_{\rm H}(r)& = &n_{\rm ... ...m c}}\right)^2\right]^{-\frac{3 \beta}{2}}\cdot\\ \end{array}$$ (3)

  
  
  The boundary between the two regions, $R_{\rm cut}$, is a free parameter of the model and we took care that the density distribution is continious across  $R_{\rm cut}$, as well as its gradient (for continuity of the total mass profile, see Eq. (12)):
  

  $$N = n_{\rm H,0} \frac{ \left[ 1+ \left( \frac{R_{\rm cut}}{r_... ...\rm cut}}{r_{\rm c}}\right)^{2} \right]^{-\frac{3\beta}{2} } }$$ (4)

  
  
  and
  

  $$\beta_{\rm in} = \beta~\frac{1 +\left(\frac{r_{\rm c,in}}{R_{... ...^{2}} {1 +\left(\frac{r_{\rm c}}{R_{\rm cut}}\right)^{2}}\cdot$$ (5)

  
  

• KBB model: We finally consider a generalisation of the ${\beta }$-model for the inner region, allowing a more centrally peaked gas density profile in the core:
  

  $$\begin{array}{lllll} r < R_{\rm cut}&n_{\rm H}(r)& = &n_{\rm ... ..._{\rm c}}\right)^2 \right]^{-\frac{3 \beta}{2}}\\ \end{array}$$ (6)

  
  
  where $\xi<1$ and
  

  $$N = n_{\rm H,0} \frac{ \left[ 1+ \left( \frac{R_{\rm cut}}{r_... ...\rm cut}}{r_{\rm c}}\right)^{2} \right]^{-\frac{3\beta}{2} } }$$ (7)

  
  
  
  

  $$\beta_{\rm in} = \beta~\frac{1 +\left(\frac{r_{\rm c,in}}{R_{... ...\xi}} {1 +\left(\frac{r_{\rm c}}{R_{\rm cut}}\right)^{2}}\cdot$$ (8)

  
  
  The parameters $\xi$ and $r_{\rm c,in}$ are strongly correlated. An arbitrarily low value of $\xi$ can fit the data if no upper limit is put on $r_{\rm c,in}$. The lower limit on this parameter given in Table 2 is obtained by imposing $r_{\rm c,in}<1\hbox{$^\prime$ }$.

The corresponding surface brightness profile is computed numerically by integration of the emission measure along the line of sight:

$$S_{\rm X}(\theta) \propto \int_{r}^{\infty} \frac{n_{\rm H}^{2}(b)}{ \sqrt{b^{2}-r^{2}}} {\rm d}r^{2}$$ (9)

where $r=d_{\rm A}\theta$ and $d_{\rm A}$ is the angular distance. The emissivity in the considered energy band was estimated using an absorbed isothermal model at the cluster mean temperature (given Sect. 5.1), taking into account the instrument response. In the soft energy band considered, this emissivity is insensitive to the observed temperature gradient (shown Sect. 5.2). Note that the profile beyond  $R_{\rm cut}$obtained for the BB and KBB models is a classical ${\beta }$-model. The inner surface brightness profile for the BB model can be analytically computed using incomplete Beta functions.

 

 

Table 2: Results of the surface brightness profile fits.

Parameter	AB model	BB model	KBB model
$n_{\rm H,0}~(10^{-2}~{\rm cm}^{-3})$	-	2.15	3.07
$r_{\rm c}$	$1.54\hbox{$^\prime$ }$	$1.29\hbox{$^\prime$ }\pm0.10\hbox{$^\prime$ }$	$1.34\hbox{$^\prime$ }\pm0.12\hbox{$^\prime$ }$
${\beta }$	0.69	$0.71\pm 0.02$	$0.71\pm 0.02$
$R_{\rm cut}$	-	$1.47\hbox{$^\prime$ }\pm0.13\hbox{$^\prime$ }$	$1.69\hbox{$^\prime$ }^{+0.32}_{-0.22}$
$r_{\rm c,in}$	-	-	$0.41\hbox{$^\prime$ }^{a}_{-0.13}$
$\xi$	-	-	0.49+0.32-0.16a
$\alpha$	0.68	-	-
$\chi^2$/d.o.f.	112/51	70.4/48	64.8/47
$\chi^{2}_{\rm red}$	2.20	1.47	1.38

Notes: All errors are at the $90\%$ confidence level.
^a The maximum value of $r_{\rm c,in}$ is fixed to $1\hbox{$^\prime$ }$.

 

 

Table 3: Influence of the low-energy cutoff. Absorption values in bold are frozen at the galactic value.

Instrument	Band	$N_{\rm H}$	kT	$\chi^{2}/{\rm d.o.f.}$
	(keV)	( $\times 10^{20}$ cm-2)	(keV)
MOS1	> 0.3	1.04+0.34-0.31	7.51+0.40-0.30	394.3/395
	>0.3	2.19	6.91+0.23-0.23	424.2/396
	>0.6	2.19	7.15+0.25-0.25	386.7/376
	>0.8	2.19	7.27+0.26-0.26	358.5/363
	>1.0	2.19	7.20+0.30-0.30	349.5/350
MOS2	>0.3	1.00+0.33-0.32	6.94+0.29-0.29	381.9/401
	>0.3	2.19	6.33+0.23-0.23	407.7/402
	>0.6	2.19	6.54+0.24-0.24	374.7/382
	>0.8	2.19	6.67+0.25-0.25	354.6/369
	>1.0	2.19	6.67+0.29-0.29	344.1/356
pn	>0.3	0.64+0.28-0.28	6.77+0.33-0.33	836.6/811
	>0.3	2.19	5.76+0.19-0.19	906.2/812
	>0.6	2.19	6.14+0.28-0.30	818.8/754
	>0.8	2.19	6.49+0.31-0.30	729.0/714
	>1.0	2.19	6.85+0.36-0.35	675.2/673
	>1.2	2.19	7.01+0.41-0.40	636.6/633
	>1.5	2.19	7.28+0.65-0.47	582.9/572

The best fit models are plotted in Fig. 3, together with the residuals. The corresponding best fit parameters with errors and $\chi^2$ values are given in Table 2. In all cases, the outer slope, ${\beta }$, is consistently found to be similar to the slope obtained by fitting only the outer part of the profile. We found that the AB model does not provide a particularly good representation of the data: the reduced $\chi^2$ is  $\chi^{2}_{\rm red}\sim 2$ and the residual profile below $R_{\rm cut}$ clearly indicates that the gas distribution is less peaked than a cusped profile. In other words, the gas distribution possesses a core. The best fit is obtained with the KBB model, but the reduced $\chi^2$, $\chi^{2}_{\rm red}\sim 1.38$ is still larger than 1. However, the residuals are small (at the $3\%$ level on average) and might be due in part to the observed departure from spherical symmetry. As the BB model is a special case of the KBB model ($\xi$ fixed to $\xi=1$), we can compare both models using a F-test. The KBB model provides a better fit than the BB model at the $95\%$ confidence level, suggesting that the density distribution in the core is indeed more centrally peaked than for a conventional ${\beta }$-model. This KBB model is thus adopted for the remainder of the analysis.