10 Conclusions

The main conclusions of this work may be summarised as follows:

1.

We have reported XMM-Newton observations of A1413, a relaxed galaxy cluster at z=0.143.

2.

In a 2D ${\beta }$ model analysis, we detect substructure to the south which does not appear to be interacting with the main cluster.

3.

Excluding the data from this region and all obvious point sources, we have measured the gas density and temperature profiles up to r₅₀₀ (corresponding to a density contrast  $\delta \sim 500$, with respect to the critical density at the redshift of the cluster). With the assumptions of HE and spherical symmetry, we have calculated the mass profile out to the same distance.

4.

The gas density profile is well described with a ${\beta }$-model beyond $\sim$ $250 h_{50}^{-1}~{\rm kpc}$. We further parameterise the inner regions with a modified version of the ${\beta }$-model (the KBB, Eq. (5)), which allows a more centrally peaked gas distribution.

5.

The temperature profile (excluding the inner point) is well described by a polytropic model with $\gamma = 1.07\pm 0.01$. The decline is modest: a decrease of $\sim$$20\%$ between 0.1 r₂₀₀ and  0.5 r₂₀₀.

6.

The mass profile, derived from the HE equation, is determined with an accuracy of about $\pm$$5\%$ up to r₆₀₀ and $\sim$$\pm 18\%$at r₅₀₀. It can be remarkably well described by a Moore et al. (1999) profile with a scale radius $r_{\rm s}= 845 \pm 43$kpc and concentration parameter $c = 2.6 \pm 0.1$. An NFW profile also gives an acceptable fit but describes less well the central regions. The c values we find are in good agreement with those expected from numerical simulations for a cluster of this mass. The Dark Matter modelling in these simulations is thus strongly supported by the excellent agreement between observed and simulated profiles.

7.

Beyond r₆₀₀, the observed temperature and derived mass profiles begin to depart systematically from, respectively, the polytropic description and the Moore et al. (1999) profile. There is also a sudden drop of the surface brightness profile at r₄₅₀. This suggests that the gas in these regions may not be in HE, and we may thus be seeing the outer edge of the virialized parts of this cluster.

8.

The offset in the normalisation of the $M_{\delta}{-}T$ relation, with respect to the simulations of Evrard et al. (1996) is now confirmed to be $\sim$$40\%$ across the entire radial range up to r₅₀₀ (i.e., in the virialised part of the cluster).

9.

The gas distribution is peaked primarily as a result of the cusp in the dark matter profile. The gas mass fraction increases with increasing radius, to reach $\sim$0.2 at r₅₀₀.

We are now in a position directly to confront simulations with observations. The results are encouraging (the obvious validity of the modelling of the Dark Matter distribution at large scale) but many questions remain. How peaked are dark matter profiles? What is the relationship between between central dark matter cusps and CFs? Why are some studies finding unrealistic values of the concentration parameter? What is the source of the discrepancy in the M-T relation?

The statistical errors on the observed quantities are now small enough so that we can determine in detail the intrinsic dispersion in cluster properties and systematic discrepancies with the classical self-similar model. To answer the above and other questions, a statistical sample of cluster properties would be of great help, preferably using Chandra to probe the central regions and XMM-Newton to determine properties at great distances from the cluster centre. Confrontation with numerical simulation is essential. The full range of observations, correlations between cluster properties, and detailed internal gas structure should be derived taking into account that they are viewed through a given instrument, so that we are able truly to compare like with like.

Acknowledgements

We thank the referee, J. Irwin, for insightful comments and suggestions which have improved the paper. We thank T. Ponman for providing the script used in the deprojection of the temperature profile, and D. Neumann for providing the Monte-Carlo code to derive the mass profile. We thank R. Teyssier and A. Refregier for useful discussions, and S. De Grandi for providing the BeppoSAX results shown in Fig. 12. We thank A. Evrard, M. Markevitch and T. Ponman for their comments on the manuscript. The present work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA).