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

Appendix A: derivation of the kernel and formulation of the RL-scheme

The 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 Z and applying a computational rule for -functions  ². In this case we have

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 z yields

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 y.

For the quadratic equation

we find the two roots

which for a fixed pair of cluster coordinates describe an ellipse in the observer's sky. Since

we obtain after integration over y

The integration over x in Eq. A.12 is limited to a finite range . Thus it is convenient to introduce a new variable t via

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
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