The question of how to deproject observed cluster images is a prime example for so-called inverse problems , which often arise in astronomy. As pointed out by Lucy (1974, 1994) inverse problems in astronomical research reflect the fact that it is in general not possible to directly measure the quantities of interest due to the large distances between observers and studied objects. Furthermore, the theoretical understanding of the relevant physical phenomena is often so limited or the problem is so complex, that it is not possible to derive appropriate models from first principles.
At the same time the 3-dim. structure of rich galaxy images is particularly interesting as it impacts on the physical and cosmological interpretation of clusters of galaxies in general. The shape of a cluster is relevant for the combined analysis of Sunyaev-Zel'dovich (SZ), X-ray, and lensing data, e.g. when the Hubble constant is determined from SZ-and X-ray images (Gunn 1978; Silk & White 1978; Cavaliere et al. 1979; see Sulkanen 1999 for a first estimation of systematic errors introduced by galaxy cluster shapes) a galaxy cluster shape has to be assumed. The same is true for different cluster mass determinations as projection effects are, at least partly, suspected to account for the discrepancy of the different mass determinations from lensing, X-ray or dynamical mass estimates (Bartelmann & Steinmetz 1996). In addition, it is important for understanding the cluster galaxy orbit structure (The & White 1986; Merritt 1987) and also the baryon mass fraction (White & Frenk 1991; Böhringer et al. 1993).
Here we employ for the deprojection of clusters of galaxies an extension of the classical algorithm of Richardson (1972) and Lucy (1974). It has widely been used for the rectification of optical images. In addition, it has been applied to determine the 3-dim. stellar orbit structure in elliptical galaxies (e.g. Binney et al. 1990; Dehnen & Gerhard 1993, 1994). In order to optimize the results of the reconstruction we combine different data sets, in this case the lensing potential , the X-ray surface brightness and the Sunyaev-Zel'dovich effect. For more and more clusters these high quality data sets become available. The combination of multiple data sets allows one to exploit the different dependences of the various observable distributions on the gravitational potential along the line-of-sight (LOS hereafter).
The plan of the paper is as follows: For deriving this multiple-data Richardson-Lucy deprojection algorithm we first describe the general Richardson-Lucy (hereafter RL) approach in Sect. 2, then we specify a geometrical model for the cluster that is suitable for deriving a RL-type deprojection equation for the gravitational potential ; in Sect. 3 we discuss the dependencies of the three considered observable distributions, namely the lensing potential, the X-ray emissivity and the Sunyaev-Zel'dovich effect, on the gravitational potential. Finally, in Sect. 4 we show how the three above-mentioned observable distributions can be incorporated into this deprojection procedure. We also discuss strategies for implementing the algorithm into computer programs, and study their respective numerical stability. In Sect. 5 we assess the performance of the algorithm by applying it to synthetic cluster data from gas-dynamical simulations. In Sect. 6 we briefly show how the inclination angle can be recovered from the data using a minimization. In the end, we give an outlook on the suitability of the algorithm for practical applications to true observational data.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001