9 Discussion

   
9.1 The shape of the temperature profile

It is instructive to compare the projected XMM-Newton temperature profile of A1413 with the composite profiles found for larger cluster samples. The most extensive samples come from ASCA and BeppoSAX data; these are, in order of publication: Markevitch et al. (1998; MFSV98), White (2000; W00) Irwin & Bregman (2000; IB00) and De Grandi & Molendi (2002; DM02). The MFSV98 ASCA-derived profile is sharply decreasing, such that for a typical $7~\rm keV$ cluster the temperature drop is characterised by a polytropic index of 1.2-1.3. W00 finds that $90\%$ of the cluster profiles in his ASCA sample are consistent with isothermality at the $3 \sigma$-level. The IB00 BeppoSAX-derived profile extends only out to $\sim$ 0.3 r₂₀₀ and is flat or even slightly increasing. In contrast, the overall DM02 profile, from a larger sample of BeppoSAX observations, is characterised by an isothermal core extending to $\sim$ 0.2 r₂₀₀. Their CF subsample exhibits a temperature drop of a factor of 1.7 between $\sim$ 0.2 r₂₀₀ and $\sim$ 0.5 r₂₀₀. The non-CF clusters exhibit a sharper temperature drop in the outer regions. DM02 suggest that an incorrect treatment of the BeppoSAX strongback may explain the discrepancy between their result and that of IB00.

Figure 12: The projected scaled temperature profile of A1413 compared to the composite CF cluster profile as obtained by De Grandi & Molendi 2002 (dot-dash lines are joining their data points plus or minus the $1\sigma$ errors). The composite profile of Markevitch et al. (1998) is shown as the shaded region (enclosing the scatter in their best fit profiles). The solid line is our best fit polytropic model (excluding the CF region).

We have derived the projected temperature profile of A1413 out to $\sim$ 0.7 r₂₀₀ or $\sim$r₅₀₀, in much finer detail than is possible with either ASCA or BeppoSAX. All indications are that A1413 is a relaxed cluster.

Our data are compared to the DM02 and MFSV98 composite profiles in Fig. 12. Although each individual data point is (marginally) consistent with the typical region defined by these composite profiles, there is an obvious systematic difference in shape. The A1413 profile does not decline sharply like the composite profile of MFSV98, or the profile of DM02 beyond $\sim$ 0.2 r₂₀₀. In Sect. 5.4, we show that the polytropic model gives an acceptable fit to the data. Excluding the central bin, the $\gamma$ value (1.07) implies an almost isothermal temperature profile, and is not compatible with that found by MFSV98. It is very similar to that found by DM02 for CF clusters, but DM02 reject the polytropic model on the grounds of poorness of fit, which is not surprising given the decline of a factor of 1.7 in the temperature of their composite profile between 0.2 and  0.5 r₂₀₀. We do not see a similar decline, and so a polytropic model is a good fit to these data. On the other hand, their best fit broken line model is a poor fit to our data: we find a $\chi^2 = 22.1/6$ for their CF best-fit, and the fit is worse for their non-CF relation ( $\chi^2 = 27.7/7$).

It is obvious that, given the extra radial range afforded by these XMM-Newton data, a flat or increasing profile, such as that of W00 or IB00 extrapolated to high radii, does not describe the A1413 data either. We emphasize again that the temperature gradient is modest: the temperature decreases by $\sim$$15(20)\%$ between 0.1 r₂₀₀ and  0.3(0.5) r₂₀₀. Fitting the temperature profile up to  0.3 r₂₀₀ (i.e., excluding the last three temperature bins, and the inner bin) with a polytrope allows us to compare our profile directly with that of IB00. We find $\tau_0 = 1.19 \pm 0.03$, $\gamma = 1.06 \pm 0.01$, consistent with the value derived for the full radial range. This gradient is in agreement with the level of isothermality found by W00 and IB00 in that radial range taking into account their errors, as well as that found by Allen et al. (2001a) from Chandra data below $r_{2500}\sim0.3~r_{200}$. Our observation is also consistent with other XMM-Newton observations of nearby clusters, e.g. the slightly decreasing XMM-Newton temperature profile of Coma ($10\%$ at  0.2 r₂₀₀, Arnaud et al. 2001a) and the temperature profile of A1795, measured up to  0.4 r₂₀₀ and found to be flat within $\pm$$10\%$ beyond the CF region ( 0.1 r₂₀₀, Arnaud et al. 2001b).

9.2 Shape of the total mass profile

In Sect. 8.3 we showed that the NFW form can describe the total mass profile of A1413. However a slightly better agreement in the center is obtained with a MQGSL profile, derived from higher resolution simulations.

With Chandra, it is possible to examine the central regions in great detail, at the expense of information at large radii. At present, it is unclear whether the NFW or MQGSL profiles provide the better description of the mass profiles derived from Chandra observations. Allen et al. (2001b) investigate several forms for the mass profile of RXJ1347.5-1145, finding that both the NFW and MQGSL provide an acceptable fit, although the NFW profile is favoured in terms of $\chi^2$. Perhaps the highest resolution examination of a cluster mass profile is that of Hydra A by David et al. (2001), who find $\rho \propto r^{-1.3}$ between 30 and 200 kpc, which is intermediate between the NFW and MQGSL forms. The addition of a mass point from H$_{\alpha}$ observations leads them to favour the NFW profile, although the result is still consistent with the MQGSL result.

One sticking point is the value of the concentration parameter from the NFW fit by David et al. (2001). They find c = 12, which is 3 times that expected for a cluster of the mass of Hydra A. Interestingly, a similar value of c was found by Arabadjis et al. (2002) from a Chandra study of EMSS 1358+6245. On the other hand, the c parameters of Allen et al. (2001a) are better in agreement with the theoretical predictions. At large radii the NFW and MQGSL profiles coincide and we can compare the c value derived from our NFW fit, $c=5.4\pm0.2$ for a $M_{200}\sim 10^{15} h_{50}^{-1}~M_{\odot}$ cluster, to numerical simulations. Teyssier (2002) derived c parameters in the range 4.9-9.5 for 5 clusters in this very mass range. The average c parameter derived by Eke et al. (1998) for hot massive clusters is $c\sim 6$. It must be noted that a relatively large dispersion on this parameter is expected from numerical simulations, with a $1\sigma \Delta(\log{c})=0.18 (50\%)$ at a given mass (Bullock et al. 2001). In conclusion, we emphasize the excellent agreement in shape between the mass profile derived for A1413 and the theoretical expectations, all the more remarkable in view of the very small statistical errors on the profile.

   
9.3 Normalisation of the M-T relation

We now examine the normalisation of the mass profile. We will classically define  $M_{\delta}$, the mass within a given radius  $r_{\delta}$, inside which the mean mass density is $\delta$ times the critical density, $\rho_{\rm c} (z)$ at the cluster redshift. For clusters obeying HE and self-similarity, the mass  $M_{\delta}(T,z)$, scales with the cluster temperature and redshift as:

$$h(z) M_{\delta}(T,z) = M_{10}(\delta) T_{10}^{3/2}$$ (23)

where h(z) is the Hubble constant normalised to its local value and  $M_{10}(\delta)$ is the normalisation at density contrast $\delta$ (here T₁₀ is arbitrarily expressed in unit of  $10~\rm keV$). The above relation is remarkably well verified by adiabatic numerical simulations, down to  $\delta\sim 200$, for both SCDM and  $\Lambda{\rm CDM}$ cosmology, with  $M_{10}(\delta)$independent of cosmology (e.g. Mathiesen 2001). The variation of the normalisation  $M_{10}(\delta)$ with $\delta$ reflects the (universal) cluster internal structure and is the same for all clusters for a given density contrast (although some scatter is present in practice).

Figure 13: The scaled mass profile of A1413 (data points), expressed as a function of density contrast, $\delta$, compared with the simulations of Evrard et al. (1996) (thick line). Filled circles: SCDM cosmology. Open circles: $\Lambda$CDM cosmology. The solid and dotted lines represent the scaled best-fit Moore et al. (1999) profile and the scaled profile derived from the best fit polytropic model, respectively (SCDM cosmology).

In Fig. 13 we show the scaled mass profile of A1413 defined as:

$$\widetilde{M}(\delta) = M(r)~h(z) \left(\frac{ kT_{\rm X}}{10~\rm keV}\right)^{-3/2}$$

where

 

$$\delta = \frac{3 M(r)}{4\pi r^{3} \rho_{\rm c}(z)}$$ (24)

with

 

$$\rho_{\rm c}(z) = h^{2}(z) \frac{3 H_0^2}{ 8 \pi G}$$ (25)

and h(z)=(1+z)^3/2 for the SCDM cosmology considered here. This profile can be compared with the normalisations  $M_{10}(\delta)$derived from numerical simulations, allowing us to check the normalisation of the $M_{\delta}{-}T$ relation at different density contrasts. The results of EMN96 are indicated with a thick line. We also plot the scaled profiles corresponding to our best-fit MQGSL model (thin line), and to our best fit polytropic model (dashed line). The density contrast is computed self-consistently for each profile using Eq. (24).

Both the data points and the MQGSL and polytropic model curves run parallel to the EMN96 profile down to  $\delta \sim 600$. This simply reflects the excellent agreement in shape of the A1413 profile with numerical simulations, as outlined above. However there is a very significant offset in normalisation: $\widetilde{M}(2500)=7.3\pm0.2\times 10^{14}~~M_{\odot}$, compared with $M_{10}(2500)=9.95 \times 10^{14}~~M_{\odot}$ from EMN96. In other words, the predicted normalisation of the M-T relation lies $\sim$$36\%$higher than the observed value, in excellent agreement with the Chandra finding of Allen et al. (2001a, $\sim$$40\%$ at  $\delta=2500$).

Below $\delta=600$ the observed profile levels off, so that the data points seem to converge towards the EMN96 predictions at small $\delta$. At  $\delta=500$ $\widetilde{M}(500)=1.8^{+0.3}_{-0.2} \times 10^{15}~M_{\odot}$, only $20\%$ lower than the EMN96 value of $M_{10}(500)=2.2 \times 10^{15}~M_{\odot}$ and actually consistent with it, especially if we also take into account the dispersion observed in the simulations ($15\%$). If this effect is real, this would point to a fundamental difference of form in the total mass profile at large radii, i.e. the real cluster dark matter density profile drops less steeply than the canonical r^-3 law. The observed discrepancy in the normalisation of the M-T relation at high density contrast would thus be due to a flaw in the numerical simulations for the dark matter component.

However, it is more likely that this level off of the observed profile is an artifact due to incomplete virialisation. If there is residual kinetic energy due to infall, the HE equation applied to the observed temperature profile would over-estimate the true mass. Such an effect is observed in the simulations of EMN96, although its expected magnitude is somewhat smaller that we observe, $\sim$$10\%$ at  $\delta=500$ on average (but a large scatter exists). A further indication that this is the correct explanation comes from a comparison with the polytropic temperature model. Let us assume that the best fit MQGSL model indeed reflects the true total mass distribution. The fact that the mass profile derived from the best fit polytropic model closely follows this profile down to  $\delta = 200$(see Fig. 13), indicates that this model is a true representation of the thermodynamic state of the gas if it was in HE up to there. The drop in temperature in the last bin ( $\delta> 550$), as compared to this model, would thus be an direct indication of incomplete thermalisation. We also recall that there is a sudden drop in the surface brightness profile at $\theta=7.8\hbox{$^\prime$ }$, corresponding to a density contrast of 450 (computed with the MQGSL model). This further supports our interpretation: we might actually be seeing the expected drop of the X-ray brightness beyond the edge of the virialised (and hot) part of the cluster. Finally, incomplete equipartition between the electrons and the ions at the border of the cluster (Chieze et al. 1998) could also contribute to the low (electronic) temperature observed.

The observed scaled mass profile depends on the cosmological model via the function h(z), used in the scaling, and the angular distance, $d_{\rm A}$, used to convert angular radius to physical radius. On the other hand the theoretical normalisation  $M_{10}(\delta)$ appears to be insensitive to cosmology. We thus examined if a better agreement with the theoretical normalisation is obtained for the currently most favoured $\Lambda$CDM model ( $\rm\Omega_{\rm m}=0.3, \Omega_\Lambda=0.7$). From Eqs. (12), (24) and (25), the derived $\delta$ value scales as $(d_{\rm A} h(z))^{-2}$ and  $\widetilde{M}$ as  $d_{\rm A} h(z)$, with $h^{2}(z)=\rm\Omega_{\rm m}(1+z)^{3} +\Omega_\Lambda$. For a $\Lambda$CDM model (open circles in Fig. 13), as compared to the SCDM model (filled circles), the data points are moved down and left along a line of slope 1/2 in the log-log plane, with $\delta$ multiplied by 1.137 and  $\widetilde{M}$ multiplied by 0.938. The translation is modest and its slope is similar to the slope of the scaled mass profile around $\delta=1000{-}500$, so the observed scaled mass profile remains essentially unchanged and the agreement with the theoretical curve is no better.

In summary the good agreement between the mass profile shape and the numerical simulations, measured for the first time in the whole virialised part of the cluster, suggest that the modelling of the Dark Matter component is correct. However, the offset in the normalisation of the M-T relation suggests that some physics is lacking in the modeling of the gas. Several groups have studied the effect of non gravitational physics, like pre-heating or cooling, on the M-Trelation (e.g. Loewenstein 2000; Bialek et al. 1999; Tozzi & Norman 2001; Babul et al. 2002; Thomas et al. 2002; Voit et al. 2002; Borgani et al. 2002). A detailed comparison with numerical simulations, which would require a statistically representative sample, is beyond the scope of this paper. We simply note that pure pre-heating seems to have a small effect of the M-T relation in the high temperature range of A1413 (Loewenstein 2000; Tozzi & Norman 2001; Babul et al. 2002) and that models including cooling (e.g. Thomas et al. 2002) seem to be more successful. We also emphasize that some care must be exercised when comparing theoretical predictions with these X-ray observations. The magnitude of the effect is not large as compared to the typical difference of $\sim$$50\%$ (Henry 2000) in the normalisation derived by various groups using purely adiabatic simulations. The dispersion of the relation, observed for both simulated ($\sim$$20\%$) and real clusters, require the use of statistically well controlled samples. Finally, a further ambiguity lies in the definition of the temperature. The temperature profile is not exactly isothermal (although our data suggest that the departure is small). Ideally we should compare the data from a given instrument using the spectral temperature estimated from simulation, after full modelling of the plasma emission folded with the instrument response. The study of Mathiesen & Evrard (2001) indicates that the spectral temperature could be an underestimate of the mass-weighted temperature, by about $20\%$. Note that this effect would worsen the discrepancy observed above. All these systematic effects have to be well controlled, if we want to confirm the departure from the self-similar scaling and identify the physical process responsible for it.

Figure 14: The gas mass fraction of A1413 as a function of overdensity $\delta$. Data points: gas mass fraction obtained using the total mass derived from the Monte-Carlo method. Line: same using the best fit MQGSL mass profile.

   
9.4 Gas distribution

From our determination of the central gas density we have calculated a cooling radius of $r_{\rm cool} = 0\hbox{$.\mkern-4mu^\prime$ }6$. Figure 3 shows that the gas density has begun to rise above the ${\beta }$-model fit at about twice this radius. This suggests that the main driver of the central peak in the gas distribution is the cusp in the dark matter profile, not the CF. It is likely that the cusp in the dark matter profile acts as a focus for the gas and the CF. This has implications for CF mass deposition rates $\dot{M}$deduced from excesses to the ${\beta }$-model, in that any application to this cluster would result in a gross overestimate of $\dot{M}$.

The ratio of the X-ray gas mass to the total gravitating mass is shown as a function of density contrast $\delta$ in Fig. 14. The $f_{\rm gas}$ rises slowly with increasing radius, up $\delta \sim 1000$. After this point, the $f_{\rm gas}$ either stabilises within the error if we use the total mass data points, or continues increasing beyond if we used the best fit MQGSL total mass profile. A conservative estimate of the gas mass fraction at $\delta \sim 500$ is $f_{\rm gas}=0.2\pm0.02$. This shows that the hot ICM is more extended than the dark matter distribution, as has been found in previous studies (e.g., David et al. 1995), and is also seen in numerical simulations, and expected from purely dynamical reasons (Chieze et al. 1997).

Assuming that the gas density profile follows the dark matter profile at large radii (within a factor of 2 between r₂₀₀/2 and 2r₂₀₀) and a polytropic equation of state, Komatsu & Seljak (2001) derived an analytical solution of the gas in HE in an NFW profile. They obtained $\gamma=1.15 +0.01(c-0.5)$, or  $\gamma=1.14$ for the concentration parameter c=5.4 derived for A1413, and a X-ray outer slope of $\beta\sim 0.65$ for a typical 6.5 keV cluster (their Fig. 14). The first assumption is roughly verified for our best fit profiles and not surprisingly, our derived values are in fair agreement with their model, although we obtain a slightly but significantly lower value of  $\gamma = 1.07\pm 0.01$ and a slightly higher ${\beta }$ value.

The value at $\delta=600$, corresponding roughly to the virialised part of the cluster, $f_{\rm gas}\simeq 0.2$, can be used to calculate the total mass density in the Universe, following the arguments that assume that the properties of clusters constitute a fair sample of those of the Universe as a whole (e.g., White et al. 1993). Assuming that the luminous baryonic mass in galaxies in A1413 is approximately one-fifth of the X-ray ICM mass (e.g., White et al. 1993), and neglecting other possible sources of baryonic dark matter, $\Omega_m = (\Omega_{\rm b}/1.2 f_{\rm gas})$, where $\Omega_{\rm b}$ is the mean baryon density. For $\Omega_{\rm b} h_{50}^2 = 0.0820$ (O'Meara et al. 2001, h₅₀ is the Hubble constant in units of 50 km s^-1 Mpc^-1), we obtain $\Omega_m = 0.34 h_{50}^{-0.5}$.