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Astron. Astrophys. 351, 413-432 (1999)

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5. Discussion

Although the ENF98-NFW model could marginally reproduce the observed number of GLAs in the LF94's sample, some clusters have unrealistic [FORMULA]. If ENF98-NFW model describes the truth and if it is this high virial temperature that describes the true depth of gravitational potential of clusters, the non-thermal pressure model (Loeb & Mao 1994) or the cooling flow model (Allen 1998) would be expected. However, as shown in Table 5, cooling times of all the clusters with [FORMULA] keV are longer than the ages of the universe at clusters' redshifts. This indicates that higher [FORMULA] obtained from the ENF98 [FORMULA] model fitting were not due to the existence of cooling flows. The high values of [FORMULA] are likely due to too large values of the scale radii [FORMULA]. The large values of scale radii ([FORMULA]) come from large X-ray core radii ([FORMULA]) and those who have large X-ray core radii show significant ellipticity. As NFW discussed, their profile came from the virialized system and those which have too high [FORMULA] hence would be regarded as in relaxing process.

For the comparison, we calculated a total cross-section to make giant arcs with another NFW type model in which temperatures are required to be those that are observed by ASCA or that are estimated using [FORMULA] relation of AE98 using the result of a theoretical work by Makino et al. (1998, henceforth MSS98). MSS98 computed the X-ray cluster gas density distribution in hydrostatic equilibrium from NFW model assuming isothermality of the ICM. MSS98 showed that the resulting distribution was well approximated by the standard [FORMULA] model. Their result gives relations of





and we could evaluate [FORMULA] and [FORMULA] (henceforth we call this model MSS98-NFW model ). We list the parameters of MSS98-NFW in Table 6 and we plotted the total cross-section to make giant arcs with MSS98-NFW model with long-dashed line in Fig. 1. As one can see in Fig. 1, the total cross-section to make giant arcs is considerably small and almost the same as the isothermal [FORMULA] model. This means the NFW model which is consistent with the ICM spatial and spectral data of sample clusters cannot reproduce the observed number of GLAs.

This is also the same to say that what is needed is just to make the sample clusters' temperatures much higher to reproduce the observed number of GLAs. However, such high temperatures are no more consistent with ASCA results or expected values from the [FORMULA] relation of AE98.

Systematic errors introduced by uncertainties in the background galaxy model we employed are summarized as follows.

As is in Paper I, the luminosity function was taken from Efstathiou et al. (1988). This luminosity function is in good agreement with the recent Las Campanas (Lin et al. 1996) and the Stromro-APM (Loveday et al. 1992) redshift surveys. These are called luminosity functions with lower normalization. On the other hand, the ESO Slice Project redshift survey (Zucca et al. 1997) gives higher normalization. Its amplitude is then higher, by a factor of [FORMULA] 1.6 at [FORMULA]. Luminosity functions with higher normalization thereby increase the number of GLAs, with rough estimation, by a factor of two. As noted above, we assumed no evolution of galaxy number density in the comoving volume for the luminosity function. On the other hand, HF97 showed that observed evolution of the luminosity function, which came from Canada France Redshift Survey (Lilly et al. 1995), increases the number of GLAs by a factor of several.

We investigated whether changing the type mixing ratio of background galaxies affected the GLA number by changing (E/S0, Sab, Sbc, Scd, Sdm)= (0.321, 0.281, 0.291, 0.045, 0.062) and (0.38, 0.16, 0.25, 0.10, 0.11) and then little difference was found.

As is in Paper I, the amplification factor was approximately constant over the whole area of an image and the validity of this assumption has already been confirmed in Paper I.

The intrinsic ellipticity of source galaxies could increase the number of GLAs by a factor of two as discussed in Paper I.

All these may affect on the expected number of GLAs by about an order of magnitude at most. Therefore even if uncertainties in the background galaxy model are taking into account, the main conclusion never change; the models consistent with the ICM spatial and spectral data of sample clusters, cannot reproduce the observed number of GLAs in the LF94's sample.

As noted in Sect. 4, recent observations give us new insights on galaxy surface brightness and size evolution up to [FORMULA] (Roche et al. 1998 and references therein). There is no size and luminosity evolution of elliptical galaxies at higher redshift (Roche et al. 1998). Spiral galaxies become smaller in size and brighter in surface brightness as redshift increases (Roche et al. 1998). Miralda-Escudè (1993b) and Paper I showed that the number of GLAs responds sensitively on the intrinsic size of galaxies and the number of GLAs decreases drastically when the intrinsic size of galaxies becomes smaller than the seeing FWHM. (See Fig. 3 in Miralda-Escudè 1993bor Fig. 4 in Paper I). We believe that this effect is stronger than that of surface brightness evolution because spirals seems not showing strong evolution in luminosity.

If the galaxy evolution history is drawn with the merger model which is currently popular (e.g. Kauffmann 1997; Bekki 1997; Bekki & Shioya 1997; Noguchi 1997), the galaxy evolution model we employed should be largely modified. Owing to merging-induced star formation, the merger model predicts the existence of temporarily very bright galaxies at various redshifts. Since the current galaxies are to be formed by aggregation of smaller building blocks in the merger model, the number density of source galaxies at high redshift is larger than the current galaxy number density. These two effects may thus increase the number of GLAs. Although the precise modeling of the merger history is required to quantify the effect, we can regard that these effects are included as re-normalization in the evolution of the galaxy luminosity function. The result obtained by HF97 thus provided a rough idea how this effect changes model prediction. On the other hand, as we discussed above, smaller size of the block-building galaxies at high redshift leads drastic decrease of the number of GLAs. Becoming larger in size by merging is competed by becoming less in number of galaxies by merging. On the arc statistics, the effect of being smaller intrinsically seems stronger than both being more luminous intrinsically and being numerous at high redshift.

Asada (1998) showed that the use of the Deyer-Roeder distance to take into account the inhomogeneity of matter distribution in the universe decreased the cross-section for forming GLAs for all the sets of ([FORMULA]). Therefore the appropriate application of angular diameter distance taking account of the inhomogeneity of the universe further decrease the number of GLAs.

Our calculation assumed that one GLA is generated from one single source galaxy. However, it may happen that two or more GLAs are generated from a single source galaxy. This means that the `true' expected numbers of GLAs exists between one times the expected numbers we calculated and two times of them under the spherically symmetric mass distribution models.

We close our discussion by this simple question: are all observed GLAs in LF94's sample really giant luminous arcs? LF94 discussed the possibility of mis-identification by elongated objects and claimed that 6 GLAs they found were really GLAs either because their widths were not spatially resolved, or because they presented a well-defined curvature. However, the giant luminous arcs in MS 1910.5+6736 and MS1008.1-1224 are located 67 and 47 arcsec away from cluster center respectively, which are unusually large values that would make the cluster extremely massive. Although, such mis-identification does not affect our result, it would seriously affect the GLA statistics which use the number of GLA on the whole sky extrapolating the GLA detecting rate in EMSS sample. Spectroscopic conformation of GLAs therefore is needed for all the GLAs in the sample.

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© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999