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

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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 [FORMULA] we start with the following identity for [FORMULA],

[EQUATION]

which we insert into Eq. 5 for the LOS integral over [FORMULA] and obtain

[EQUATION]

Next, we simplify [FORMULA] by substituting the coordinate transformation (4c) for Z and applying a computational rule for [FORMULA]-functions  2. In this case we have

[EQUATION]

provided that [FORMULA]. Here and in the following we can restrict the discussion to the inclination angles in the range of [FORMULA], leading to non-zero, positive values for [FORMULA]. Together with [FORMULA] and the corresponding substitution 3, 2.2 for [FORMULA] we arrive at

[EQUATION]

Finally, integration of the second [FORMULA]-function over z yields

[EQUATION]

which is in the desired form of Eq. 6. We can identify the kernel

[EQUATION]

and find that it is properly normalized:

[EQUATION]

The second step 2b in the iterative RL-scheme reads in our case

[EQUATION]

For this second integral the evaluation of the probability kernel [FORMULA] cannot be avoided. However, it is possible to eliminate the [FORMULA]-function by again applying the same rule as before to [FORMULA] and by subsequently integrating over y.

For the quadratic equation

[EQUATION]

we find the two roots

[EQUATION]

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

[EQUATION]

we obtain after integration over y

[EQUATION]

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

[EQUATION]

Utilizing that

[EQUATION]

we arrive at our final result, namely a formulation for the second integral in the RL-scheme without [FORMULA]-functions:

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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