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Astron. Astrophys. 364, 377-390 (2000)

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7. Discussion

With the advent of an ever increasing number of clusters, which have several superb data sets including lensing, X-ray and SZ-images, it is an interesting question whether it is possible to obtain an improved reconstruction of the structure of individual clusters along the LOS by combining observational data from these different sources. In this paper we were able to device an algorithm based on the Richardson-Lucy deconvolution algorithm that uses lensing, X-ray and Sunyaev-Zel'dovich data to gain information about the 3-dim. structure of a cluster of galaxies. The first implementation of this multiple-data Richardson-Lucy (MDRL) algorithm was applied to synthetic clusters generated in gas-dynamical simulations, and we found it to work stably and efficiently:

Starting from earlier work of Binney et al. (1990) on deprojection of elliptical galaxies from photometric data, an integral kernel for an axisymmetric cluster model was derived and integral expressions for the n-th projected distribution [FORMULA] and the n-th iterative estimate [FORMULA] to the true potential [FORMULA] were obtained, that are suitable for numerical evaluation. In particular, we found that the simple LOS integral for [FORMULA] is easier to evaluate and numerically more stable as the expression given in Binney et al. (1990), and thus is to be preferred.

It was shown that the three observables of interest, the lensing potential [FORMULA], the X-ray luminosity [FORMULA], and the Sunyaev-Zel'dovich temperature decrement [FORMULA] can all be written as functionals of the gravitational potential [FORMULA] assuming an appropriate equation of state. The lensing potential [FORMULA] by itself is given directly as a simple LOS integral over [FORMULA], whereas [FORMULA] and [FORMULA] both have an exponential dependence on [FORMULA] mediated by the electron density within an isothermal hydrostatic intracluster gas model. [FORMULA] is here assumed to result in continuum bremsstrahlung of the gas distribution, and [FORMULA] is due to Compton scattering of CMB photons passing through the hot cluster gas.

The practical implementation of the MDRL algorithm into a computer program led to a few important observations. The evaluation of the n-th iterative estimate [FORMULA] requires integrations over ellipses on the projection plane, where the size and the relative location of the ellipse is determined by the current pair of [FORMULA] values. In the X-ray and SZ case the logarithmic dependence on the projected quantities requires the use of cut-off criteria to minimize the magnification of small deviations between observed and reconstructed projected data.

A first application of the MDRL method to a cluster from gas-dynamical simulations showed that the approach already works well for single data reconstructions from [FORMULA], [FORMULA], or [FORMULA]. The algorithm is very insensitive to the initial guess provided for [FORMULA]; qualitatively no significant difference in the reconstructed potentials obtained from a very simple constant value function or the realistic NFW model potential chosen as initial guesses were found. As expected theoretically, the X-ray and SZ reconstructions, which have their main contributions coming from the cluster core and are less affected by projection effects, give a better description of the core region, whereas the lensing potential that is only sensitive to the dark matter distribution, but more prone to projection effects, better reproduces the overall shape of the potential. Finally, we found that a combination of all three data sets within a multiple data reconstruction improves upon the single data results: the inner region is described as well as in the X-ray and SZ case, and the description for larger [FORMULA] radii lacking in the X-ray and SZ-case is compensated for by the lensing data.

We believe that this multiple data Richardson-Lucy reconstruction method will be a valuable and widely applicable tool.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001