## Appendix A: derivation of the kernel and formulation of the RL-schemeThe derivation of the kernel follows closely the derivation given in Binney et al. (1990) for the deprojection of elliptical galaxies. In order to derive the kernel we start with the following identity for , which we insert into Eq. 5 for the LOS integral over and obtain Next, we simplify by
substituting the coordinate transformation (4c) for provided that . Here and in the following we can restrict the discussion to the inclination angles in the range of , leading to non-zero, positive values for . Together with and the corresponding substitution 3, 2.2 for we arrive at Finally, integration of the second
-function over which is in the desired form of Eq. 6. We can identify the kernel and find that it is properly normalized: The second step 2b in the iterative RL-scheme reads in our case For this second integral the evaluation of the probability kernel
cannot be avoided. However, it is
possible to eliminate the -function
by again applying the same rule as before to
and by subsequently integrating over
For the quadratic equation which for a fixed pair of cluster coordinates describe an ellipse in the observer's sky. Since we obtain after integration over The integration over Utilizing that we arrive at our final result, namely a formulation for the second integral in the RL-scheme without -functions: © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |