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

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3. Observables sensitive to the LOS-structure

As already stated in the introduction, the overall goal is to combine multiple data sets within the Richardson-Lucy algorithm to deproject cluster images. For observed distributions - denoted as [FORMULA] in the previous sections - we now discuss concrete observables, namely the weak lensing potential, the X-ray surface brightness and the SZ temperature decrement. For employing the Richardson-Lucy algorithm, it is important to connect the observables to the theoretical distribution [FORMULA]. We choose the gravitational potential [FORMULA] as theoretical distribution. In principle it is possible to choose the density [FORMULA] as theoretical distribution, but the gravitational potential [FORMULA] possesses better symmetry properties than the density [FORMULA]. Therefore substructure has less impact on the potential than on the density, thus better fulfilling the symmetry assumptions made deriving the kernel [FORMULA].

In the following we briefly discuss the connection between the respective observable and the gravitational potential [FORMULA] needed for the derivation and implementation of the MDRL-algorithm.

3.1. Lensing potential

Weak gravitational lensing emerged in the past couple of years as a tool to map the mass distribution of clusters of galaxies (e.g. Clowe et al. 2000 for one recent example). For a derivation and discussion of gravitational lensing theory cf. Schneider et al. (1992). Here we concentrate on the effective lensing potential [FORMULA], which can be written as the appropriately scaled, projected Newtonian potential of the lens

[EQUATION]

[FORMULA], [FORMULA], and [FORMULA] are the angular-diameter distances from the observer to the sources, from the observer to the lens, and from the lens to the sources. The scaled lensing potential [FORMULA] can be directly obtained from the observed data using a maximum likelihood approach (Bartelmann et al. 1996). The lensing potential is connected to the local properties of the lens, namely the convergence [FORMULA], and the shear [FORMULA] in terms of the second derivatives of [FORMULA]

[EQUATION]

[EQUATION]

[EQUATION]

where indices i following commas denote partial derivatives with respect to [FORMULA].  1

Hereafter, [FORMULA] shall exclusively denote the lensing potential. In Eq. 12 the dependence of the lensing potential on the 3-dim. gravitiational potential [FORMULA] is given as the LOS integral

[EQUATION]

With current observational techniques the lensing potential of clusters can be determined up to a radius of [FORMULA] Mpc from the center.

3.2. X-ray emissivity

Clusters of galaxies are powerful X-ray emitters with luminosities in the range of [FORMULA] erg s-1, making them the most luminous X-ray emitters in the sky. The X-ray emission in clusters is extended rather than point-like, and the X-ray spectra are best explained by thermal bremsstrahlung (free-free radiation) from the hot, dilute plasma with temperatures in the range [FORMULA] K and densities of [FORMULA] particles per cm3.

For the present purpose it is sufficient to include continuum emission only. Semiclassical derivations of free-free emission can be found in standard textbooks, e.g. in Rybicki & Lightman (1979) and in Shu (1991). The emissivity at a frequency [FORMULA] associated with electrons accelerated by ions of charge [FORMULA] in a plasma with temperature T is given by

[EQUATION]

where [FORMULA] and [FORMULA] are the number densities of ions and electrons, respectively. The Gaunt factor [FORMULA] corrects for quantum-mechanical effects and for the effect of distant collisions. It is a slowly varying function of frequency and temperature, and can be set to unity for nearly all frequencies and temperatures of practical interest. For a completely ionized gas mixture with a mass ratio of [FORMULA] hydrogen and [FORMULA] helium, i.e. a gas with a mean mass per particle [FORMULA] g, the thermal bremsstrahlung at position x in the energy range [FORMULA] is

[EQUATION]

where the electron density [FORMULA] in this case is given by

[EQUATION]

The observable X-ray surface brightness received at the 2-dim. position zeta is the LOS integral of the X-ray emissivity [FORMULA],

[EQUATION]

where the factor [FORMULA] accounts for the redshifting of the ratio between luminosity distance and angular diameter distance.

Assuming a hydrostatic gas distribution, it is possible to relate the observed X-ray surface brightness [FORMULA] to the 3-dim. gravitational potential [FORMULA] by the Euler equation

[EQUATION]

where the gas pressure P obeys the ideal equation of state

[EQUATION]

For this isothermal gas distribution, where the temperature T is independent of the position we obtain a dependence on the potential [FORMULA] of the form

[EQUATION]

We thus arrive at the following relationship between observed X-ray surface brightness [FORMULA] and the 3-dim. gravitational potential [FORMULA]:

[EQUATION]

3.3. Sunyaev-Zel'dovich effect

The inverse Compton scattering of the cosmic microwave background (CMB) radiation field off thermal electrons in clusters of galaxies on, which is called the Sunyaev-Zel'dovich effect (Sunyaev & Zel'dovich 1972, 1978, 1980), is one of the most important astrophysical processes in a low-energy environment, where only small energy transfers occur, with observable consequences. In essence, the Sunyaev-Zel'dovich (hereafter SZ) effect causes a perturbation of the spectrum of the CMB as its photons pass through the hot gas of clusters of galaxies. The SZ-effect is a very important cosmological probe, which can be used to study the evolution and structure of the Universe.

The frequency shift leads to an apparent deficit in intensity at low frequencies of the CMB spectrum, and an increase at higher frequencies meaning that the temperature of the CMB photons is lowered through the SZ-effect. Here we assume that the temperature decrement [FORMULA] at certain frequencies can be measured. The temperature decrement as a function of redshift, expressed in terms of the Rayleigh-Jeans brightness temperature ([FORMULA]), is given as

[EQUATION]

where y is the Comptonization parameter

[EQUATION]

and [FORMULA]. The first term of the componization parameter y describes the effect on energy transfer by a single electron, while the second term gives the probability. As opposed to the X-ray case which depends on [FORMULA], the SZ-effect depends on [FORMULA], which is proportional to the pressure P, thus providing additional constraints on the cluster. As in the X-ray case a hydrostatic gas distribution is assumed. Therefore the temperature decrement depends on the gravitational potential in the following way

[EQUATION]

It is worthwile noting that for both, the X-ray surface brightness [FORMULA] and the SZ-temperature decrement [FORMULA], the dependence on the quantity of interest, the 3-dim. gravitational potential [FORMULA], is exponential, requiring great care in the numerical implementation.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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