## 3. Observables sensitive to the LOS-structureAs 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 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 . We choose the gravitational potential as theoretical distribution. In principle it is possible to choose the density as theoretical distribution, but the gravitational potential possesses better symmetry properties than the density . Therefore substructure has less impact on the potential than on the density, thus better fulfilling the symmetry assumptions made deriving the kernel . In the following we briefly discuss the connection between the respective observable and the gravitational potential needed for the derivation and implementation of the MDRL-algorithm. ## 3.1. Lensing potentialWeak 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 , which can be written as the appropriately scaled, projected Newtonian potential of the lens , , and 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 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 , and the shear in terms of the second derivatives of where indices Hereafter, shall exclusively denote the lensing potential. In Eq. 12 the dependence of the lensing potential on the 3-dim. gravitiational potential is given as the LOS integral With current observational techniques the lensing potential of clusters can be determined up to a radius of Mpc from the center. ## 3.2. X-ray emissivityClusters of galaxies are powerful X-ray emitters with luminosities
in the range of erg s 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
associated with electrons
accelerated by ions of charge in a
plasma with temperature where and
are the number densities of ions and
electrons, respectively. The Gaunt factor
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
hydrogen and
helium, i.e. a gas with a mean mass
per particle g, the thermal
bremsstrahlung at position where the electron density in this case is given by The observable X-ray surface brightness received at the 2-dim.
position where the factor 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 to the 3-dim. gravitational potential by the Euler equation where the gas pressure For this isothermal gas distribution, where the temperature
We thus arrive at the following relationship between observed X-ray surface brightness and the 3-dim. gravitational potential : ## 3.3. Sunyaev-Zel'dovich effectThe 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 at certain frequencies can be measured. The temperature decrement as a function of redshift, expressed in terms of the Rayleigh-Jeans brightness temperature (), is given as where and . The first term of the
componization parameter It is worthwile noting that for both, the X-ray surface brightness and the SZ-temperature decrement , the dependence on the quantity of interest, the 3-dim. gravitational potential , is exponential, requiring great care in the numerical implementation. © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |