Astron. Astrophys. 363, 415-424 (2000)

Appendix A: structure integral for the y parameter

The structure integral for the SZ effect is given by the expression

where is the density profile of the electrons, the central density of the cluster and l the maximum extension of the hot gas on the line of sight , in units of the core radius . With an ellipsoidal density profile, we obtain

where and l are given in units of the core radius . With the definition of a function G and transforming the variable of integration from to such that

we obtain after some algebra the structure integral

Finally, with a last change of variable,

we get

with

The structure integral turns out to be

where we introduced the Beta and the incomplete Beta functions defined by

and

with the Gamma and the incomplete Gamma-functions and , respectively.

-ray surface brightness

The structure integral for the X-ray surface brightness is given by

With the same transformations, as given above, we get

Appendix B: projection effects on the parameter

The rotation in the ()-plane around with an angle leads to the density profile:

The rotation angle is the angle between the major half axis of the rotated ellipse in the plane and the -axis. The line of sight is taken to be along the -axis.

To investigate the projection effects we will compare a rotated cluster with respect to the axis with its projection on the () plane, corresponding to the sky plane. We assume an infinite extension and an isothermal profile.

The rotated structure integral for the y parameter turns out to be

After some algebra and with the same kind of variable changes as in Appendix A, we get (assuming an infinite cluster extension for the structure integral):

with

and

On the other hand, the projection of this cluster on the observed sky plane in the infinite cluster extension case leads to the density profile

where is the maximum value that we get along the axis in units of

Moreover, and .

The projected structure integral for the y parameter turns out to be

with

-ray surface brightness

For the structure integral of the X-ray surface brightness we find in the case of the rotated cluster

and for the projected case

© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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