3. Cluster mass models and their lens properties
In this section, we summarize lens properties of three basic cluster mass distribution models.
3.1. The singular isothermal sphere (SIS) model
Although 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 y [x] is the angle between a source [an image] and the lens center in the unit of the Einstein ring radius of SIS model :
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 model
Under the assumptions of spherical symmetry, isothermality, and the hydrostatic equilibrium for the ICM distribution, we can calculate the total gravitational mass within a radius r of a cluster whose ICM spatial distribution is described with the standard model (Eq. 1):
where G is the gravitational constant, µ is the mean molecular weight, is the proton mass, and is a physical core radius; where is the angular diameter distance from the observer to the lens. We employ . Henceforth we call this model `isothermal model '.
The density profile of the isothermal model is given by
The lens equation of the isothermal model becomes
where y [x] is the angle between the lens center and the source [the image] in the unit of , and is called the lens parameter:
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
3.3. The universal dark matter halo profile model
NFW (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:
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
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 y (x) is the angle between the lens center and a source (an image) in the unit of the angular scale radius (), is a constant coefficient:
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
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 ROSAT HRI . It is hence likely that the best-fit values of shown in Table 2 are biased by this effect. To overcome this problem, we employed another model (henceforth `ENF98 model ') fitting in which was fixed to the median value of in Table 3 of ENF98; . In this procedure, we implicitly assumed that if the radial profile was resolved up to the virial radius, the should be . We list the ENF98 model fitting result in Table 3. We also list fluxes, luminosities and temperatures of the sample clusters calculated using the best-fit values of the ENF98 model fitting on the right hand side of each column in Table 4. We also list central electron number densities, central cooling times, ages of the universe at the redshift of each cluster, cooling radii, and mass-flow rates () for the sample clusters on the right hand side in each column in Table 5. On the left hand side of each column in Table 6, we list values of parameters of the NFW model (henceforth ENF98-NFW model ) derived from ROSAT HRI data using the procedure described above. The virial mass was evaluated using the equation (Makino et al. 1998)
For the ENF98-NFW model, one can evaluate its virial temperature :
using only the information of ICM spatial distribution; and . Some clusters have virial temperatures which are much higher than temperatures obtained by ASCA or obtained using the relation of AE98.
Table 6. Parameters of ENF98-NFW (and MSS98-) model.
3.4. Total cross-sections to make giant arcs
In 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 arcs
In Table 7, observational properties of GLAs found in the LF94 sample are summarized as follows; major axis length: l, length-to-width ratio: , distance from the cluster center: d, apparent magnitude: m, the projected mass within Einstein ring radii regarding :
the projected mass deduced from the isothermal model:
and the projected mass deduced from the ENF98-NFW model:
where , assuming source redshift .
Table 7. Properties of the giant luminous arcs from TABLE 1 in LF94.
© European Southern Observatory (ESO) 1999
Online publication: November 3, 1999