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

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6. [FORMULA]-based determination of inclination angle i

Clearly in addition to a quantitative assessment of the quality of the reconstruction a systematic procedure for determining the input parameters for the MDRL algorithm is highly desirable.

As explained in Sect. 2.2 the MDRL-algorithm in its current formulation needs to be provided with the inclination angle i as input parameter and a choice for the weight factors [FORMULA], [FORMULA] and [FORMULA] for the three different contributions has to be made. Once all these parameters have been specified, the MDRL algorithm yields a reconstructed estimate [FORMULA] for the potential, which in turn determines the best estimates [FORMULA], [FORMULA], and [FORMULA] for the observed input distributions [FORMULA] (lensing potential), [FORMULA] (X-ray surface brightness), and [FORMULA] (Sunyaev-Zel'dovich temperature decrement).

The idea is now to minimize an appropriate [FORMULA] function, e.g.


to obtain the best parameters within the framework of the model assumptions underlying the presented MDRL algorithm. Here [FORMULA] is the number of grid points, with the i-th grid point given by a vector [FORMULA] in the xy-plane onto which the observables are projected. The desired quantitative assessment of the qualitity of the reconstruction could then be given in terms of a subsequent goodness-of-fit (GoF) evaluation, which would tell us how likely the reconstruction within this model is for the best set of parameters found before.

This idea is illustrated in the following using the cluster example from Sect. 5. We recall that the true 3-dimensional gravitational cluster potential [FORMULA] is observed under an angle of [FORMULA], and assume that we do not know the proper angle i in advance. For setting up the [FORMULA] statistics we compute a series of single data and multiple data reconstructions with various values for the angles i and the weight factors. We then use Eq. 32 to evaluate the corresponding [FORMULA] function, and obtain a [FORMULA] value for every set of input parameters.

The minimum over all the [FORMULA] values then indicates the best choice for the inclination angle i and the weight factors. According to our qualitative analysis from Sect. 5 we would expect that the inclination angle i indeed should be very close to the true value.

Work on the quantitative estimation of the inclination angle and the weight factors [FORMULA], [FORMULA] and [FORMULA] using the approach sketched above is currently in progress. Preliminary results indicate that with the simulation data available in this study the main contribution to the [FORMULA] for a fixed inclination angle i does not result from the mismatch between reconstructed potential and true potential, but from discretization effects due to the finite grid. The question, if the above sketched determination of the inclination angle i and the weight factors [FORMULA], [FORMULA] and [FORMULA] is feasible for realistic observational data sets of finite sizes thus requires further investigation.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001