Astron. Nachr. / AN 334, No. 4/5, 445 – 448 (2013) / DOI 10.1002/asna.201211876 On mass and shape of galaxy clusters by comparison of X-ray, Sunyaev- Zel’dovich effect, and gravitational lensing observations M. Sereno 1,2,⋆ , K. Umetsu 3 , S. Ettori 4,5 , and A. Baldi 6,4 1 Dipartimento di Fisica, Politecnico di Torino, corso Duca degli Abruzzi 24, I-10129 Torino, Italia 2 INFN, Sezione di Torino, via Pietro Giuria 1, I-10125, Torino, Italia 3 Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan 4 INAF, Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italia 5 INFN, Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italia 6 Dipartimento di Astronomia, Universit` a di Bologna, via Ranzani 1, I-40127, Bologna, Italia Received 2012 Jul 23, accepted 2012 Sep 18 Published online 2013 May 2 Key words galaxies: clusters: general – galaxies: clusters: individual: (Abell 1689) – galaxies: clusters: intracluster medium – methods: statistical – X-rays: galaxies: clusters The accurate determination of the mass of galaxy clusters is crucial to precision astronomy. We have often to deal with either biased measurements or precise determinations of peculiar quantities. A better understanding of intrinsic shapes plays a major role. We consider how the shape of the intracluster medium (ICM) can be inferred by combining X-ray photometry and spectroscopy with the measurement of the Sunyaev-Zel’dovich effect (SZe). Together with gravitational lensing (GL), we can get unbiased estimates of mass and concentration and probe the hydrodynamical properties of the cluster. c  2013 WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim 1 Introduction The determination of the mass of galaxy clusters is a clas- sic astronomical problem, crucial to understanding of cos- mic structure formation and evolution and precision cos- mology. Even if accurate measurements should rely on safe hypotheses, usual approaches assume either spherical sym- metry, which can bias high mass and concentration esti- mates for clusters elongated along the line of sight, or hy- drostatic equilibrium, which can underestimate the mass in the likely presence of non-thermal pressure (Rasia et al. 2012). The high concentrations derived for several strong- lensing clusters present a major inconsistency between the- oretical ΛCDM expectations and measurements. Triaxiality and orientation biases might be at the origin of this disagree- ment (Sereno et al. 2010a). Deprojecting a surface density map to infer the intrin- sic three dimensional shape is an under-constrained prob- lem. The first attempts were statistical approaches consist- ing in the inversion of the distribution of apparent shapes. Hubble (1926) first determined the relative frequencies with which galaxies of a given intrinsic ellipticity, oriented at random, are observed as having various apparent projected ellipticities. Similar methods were then applied to different classes of astronomical objects (Binggeli 1980; Binney & de Vaucouleurs 1981; Cooray 2000; Fasano & Vio 1991; Noerdlinger 1979; Plionis et al. 2004; Ryden 1996). With ⋆ Corresponding author: mauro.sereno@polito.it the exception of disc galaxies, prolate-like shapes appear to dominate all cosmic structure on a large scale. A multi-probe approach enables us to infer the shape of single objects. Since galaxy clusters are triaxial struc- tures (Jing & Suto 2002), they can be modelled as ellipsoids, which is the most logical step beyond the spherical symme- try hypothesis. However, even combining multi-wavelength observations, from X-ray through optical to radio bands, one can only constrain the elongation of the cluster along the line of sight (Sereno 2007). Ellipsoids map into ellipses (Stark 1977). The ellipticity and the orientation of the projected ellipses in the plane of the sky depend only on the intrinsic geometry and orienta- tion of the system. Here, we review some methods recently proposed to tackle the unbiased determination of mass and shape of galaxy clusters by combining multi-probe data. 2 Map inversion The basic idea to constrain the intrinsic shape is to measure the elongation of the cluster along the line of sight and to use it with its projected features on the plane of the sky. The elongation along the line of sight can be parametrized as e Δ , i.e. the ratio between the major axis of the projected ellipse in the plane of the sky and the size of the ellipsoid along the line of sight (Sereno 2007). If e Δ < 1, the cluster is more elongated along the line of sight than wide in the plane of the sky, i.e., the smaller e Δ , the larger the elongation along the line of sight. c  2013 WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim