## 3. Cluster mass models and their lens propertiesIn this section, we summarize lens properties of three basic cluster mass distribution models. ## 3.1. The singular isothermal sphere (SIS) modelAlthough we do not use the SIS model, the SIS model is employed by HF97 and then we give its lens properties to compare with those of lens models we employed. The density profile of a cluster described by the SIS model is where is the velocity dispersion of the cluster. The lens equation of the SIS model thereby becomes where where [] is the angular diameter distance from the lens [the observer] to the source. If we assume all background sources circular, length-to-width ratio of a GLA equals to the ratio of the tangential stretching rate to the radial stretching rate or vise versa. (As one can see in the lens equation, the radial stretching rate of the SIS model is always unity.) The moduli of the eigenvalues of the Jacobian matrix of the lens mapping are and the length-to-width ratio of an arc is described as Following Wu & Hammer (1993), a giant arc is defined as the image which has where is the threshold value of length-to-width ratio for identifying a giant arc (Paper I). The position of arc to be giant has to fulfil the following inequalities: The cross section in the source plane for forming arcs with hence reads ## 3.2. The isothermal modelUnder the assumptions of spherical symmetry, isothermality, and the
hydrostatic equilibrium for the ICM distribution, we can calculate the
total gravitational mass within a radius where The density profile of the isothermal model is given by The lens equation of the isothermal model becomes where When , the lens always produces a single image of a source. On the other hand, the lens is able to form three images of a source when . The moduli of the eigenvalues of the Jacobian matrix of the lens mapping are and the length-to-width ratio of an arc is or ## 3.3. The universal dark matter halo profile modelNFW (1996, 1997) made series of studies on the density profiles of dark matter halos formed through a gravitational collapse in hierarchically clustering cold dark matter (CDM) cosmology using N-body simulations. They concluded that dark matter halo reached a density profiles with a universal shape that did not depend on their mass ranging from dwarf galaxy halos to those of rich clusters, nor on the power spectrum of initial fluctuations, nor on the cosmological parameters through dissipationless hierarchical clustering. They found that the universal density profile can be specified by giving two parameters: the halo mass and the halo characteristic (dimensionless) density : where is a scale radius related to the virial mass of the halo, and is the critical density: for cosmology. The universal dark halo profile becomes one parameter function in the spherical top-hat model (e.g. Eke et al. (1998), henceforth ENF98). The virial mass is defined to be the mass contained within the radius, , that enclose a density contrast . This density contrast depends on the value of and can be approximated by where for cosmology. This definition shows that the ratio of virial radius to scale radius, which is denoted by the `concentration' parameter: , uniquely related to by The structure of a halo mass is hence completely specified by a single parameter. The lens equation of the universal dark matter halo profile model is described as (Bartelmann 1996) where and ENF98 performed cosmological hydrodynamical and particle simulations to examine the evolution of X-ray emitting hot gas in clusters in a flat (), low-density () CDM cosmology. They showed that radial density profiles of gas in relaxed clusters are well described by the standard model. Table 3 in ENF98 enables us to calculate and from the X-ray data. They have shown that the values of most of their simulated clusters are around 0.79 if we exclude the cluster which is likely in dynamically non-equilibrium state. Parameters of the universal density profile is thereby specified with the observed core radius and the normalized central gas densities using the relations of and found in their result for . In
applying this model to our data, we had to remind that the best-fit
value got artificially lower as the
central surface brightness got closer to the background surface
brightness (e.g. Bartelmann & Steinmetz 1996). In general, the
central surface brightnesses of high redshift clusters are very low
and close to the background of For the ENF98-NFW model, one can evaluate its virial temperature : using
## 3.4. Total cross-sections to make giant arcsIn Fig. 1, we show total cross-sections of LF94 sample to make giant arcs of background galaxies assuming that the background source galaxies are circular. Dashed line, solid line, dot-dashed line, and long-dashed line respectively represent the total-cross section to make giant arcs calculated with the SIS model, the isothermal model, the ENF98-NFW model, and the MSS98-NFW model (we discuss this model in Sect. 5). A cross-section of a sample cluster to make giant arcs of background galaxies with a lens model is an area on the source plane in which background galaxies show their images with length-to-width ratio greater than or equal to 10 in the image plane. The total cross-section is calculated by summing up all the cross-sections of sample clusters. We calculated the total cross-section with the SIS model using relation used in HF97. Note that clusters MS 1333.3+1725,MS 1621.5+2640, and MS 2053.7-0449 are excluded in the calculation with the isothermal model, the ENF98-NFW model, and MSS98-NFW model (See Sect. 4.3). As shown in Fig. 1, the total cross-sections calculated with the ENF98-NFW model and that with the SIS model are comparable for all the source redshifts. Considerably larger total cross-section of the ENF98-NFW model than that of the isothermal model is due to their much higher . The main reason why HF97 could reproduce the observed number of GLAs in the LF94 sample is considerably high temperatures which they overestimated with inappropriate relation.
## 3.5. Properties of the giant luminous arcsIn Table 7, observational properties of GLAs found in the LF94
sample are summarized as follows; major axis length: the projected mass deduced from the isothermal model: and the projected mass deduced from the ENF98-NFW model: where , assuming source redshift .
© European Southern Observatory (ESO) 1999 Online publication: November 3, 1999 |