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

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4. The statistics of giant luminous arcs

In this section, we examine the expected numbers of GLAs predicted by the isothermal [FORMULA] model and the ENF98-NFW model using the numerical code described in detail in Paper I.

4.1. The theoretical model of galaxy evolution

As is in Paper I, prescriptions of properties of background galaxies are essentially the same as those in Yoshii & Takahara (1988) and Yoshii & Peterson (1991). We adopted the galaxy type mixing ratio of (E/S0, Sab, Sbc, Scd, Sdm)= (0.215, 0.185, 0.160, 0.275, 0.165) given by Pence (1976). The K and E corrections for each type were calculated using the type-dependent, present day spectral energy distribution (SED) updated by Yoshii & Peterson (1991) and the type-dependent galaxy luminosity evolution models by Arimoto & Yoshii (1986& 1987), except for the UV light of E/S0 galaxies. We adopted the UV-intermediate NGC3379 SED (henceforth `case I') and the UV-bright NGC4649 SED (henceforth `case II') as the SED for E/S0 galaxies. These models are reliable, especially at low redshift of [FORMULA] (Yoshii & Takahara 1988, Totani et al. 1997). Although recent observations of galaxies and their evolution at [FORMULA] (e.g. Roche et al. 1998and references therein) have considerably been improving our knowledge of them, these models are still in good agreement with such recent observations; e.g. data of star formation history taken by the Hubble Space Telescope (Madau et al. 1996) or galaxy number counts (Yoshii & Peterson (1991). We discuss some effects of the recent new knowledge for the galaxy evolution (e.g. a major merger of galaxies) on our GLA statistics bellow. The luminosity function of all galaxy types was assumed to be same and was taken from Efstathiou et al. (1988), namely the Schechter function with [FORMULA], [FORMULA] and [FORMULA] (h is the Hubble constant in the unit of [FORMULA]). The galaxy formation epoch was assumed to be [FORMULA]. The absolute magnitude in B band for each type of galaxies was converted into V band magnitude using the relation [FORMULA] where [FORMULA] for [FORMULA], -0.79 for Sab, -0.64 for Sbc, -0.56 for Scd and -0.46 for Sdm. No evolution in the comoving galaxy number density was assumed for the galaxy luminosity function. The luminosity profiles and the effective radii for each galaxies were modeled as the same way in Paper I. The intrinsic shape of the source galaxy image was assumed to be circular.

4.2. The detection condition and the arc identification scheme

The detection condition in the LF94 arc survey were taken into account as the same way in Paper I. The adopted GLA identification scheme was also same as described in Sect. 2.4 of Paper I. The length-to-width condition and the apparent magnitude condition were applied for the images smeared by the seeing and by the limiting surface brightness. The threshold value for the axis ratio and the apparent V magnitude respectively were set to be [FORMULA] and [FORMULA].

4.3. Numbers of giant luminous arcs

As noted above, the object MS 1333.3+1725 is not a cluster and thus was excluded from the cluster sample. Since the central surface brightness of cluster MS 1621.5+2640 is very close to the background, X-ray emission from only the central small portion was resolved by HRI (Morris et al. 1998). Because of this reason and poor photon statistics, the standard [FORMULA] model fitting gave unusual values as the best-fit result and it was impossible to estimate its errors. The ENF98 [FORMULA] model fitting for this cluster did not converge. The cluster MS 1621.5+2640 in which a GLA is found hence was excluded from the arc statistics sample in the current studies, following which the observed number of GLAs in the sample of 14 clusters becomes 5. Although two giant arcs were detected by the Hubble Space Telescope in clusters MS 1358.4+6245 (Franx et al. 1997) and MS 1512.4+3647 (Seitz et al. 1997), they are not extremely bright and hence do not enter the GLA statistics. X-ray emission from object MS 2053.7-0449 was not detected in spite of pointing observation with 5 ksec exposure time. The [FORMULA] upper limit on the X-ray flux of MS 2053.7-0449 was set. The upper limit on the temperature could be obtained from this upper limit by following procedure. First the temperature was assumed to be 6 keV and the upper limit of X-ray luminosity was calculated. Then using the [FORMULA] relation, the temperature value was reset and an upper limit of X-ray luminosity was re-calculated with this temperature. Iterating this procedure until the value of the temperature converged, the upper limit on temperature listed in Table 4 was obtained. The [FORMULA] value of MS2053.7-0449 was assumed to be [FORMULA]. A core radius of 10 arcsec for MS2053.7-0449 was also assumed because smaller core radius makes a cluster more efficient to make GLAs. Unfortunately, we could not calculate the central electron number density of MS2053.7-0449 and therefore MS2053.7-0449 was excluded from the GLA statistics with the ENF98-NFW model (and with MSS98-NFW when we calculate total cross-section to make giant arcs). The expected number of GLAs hence becomes 4 in 13 clusters in this case. The expected numbers of GLAs in LF94 sample, [FORMULA], were calculated summing up all the expected numbers of GLAs in each cluster, [FORMULA], which were described in detail in Paper I.

We list the result in Table 8. It shows that the isothermal [FORMULA] model consistent with current best X-ray data, cannot reproduce the observed large number of GLAs in the LF94's sample. On the other hand, the ENF98-NFW model marginally reproduces the observed number of GLAs in the LF94 sample. However, as one can see in Table 6, some clusters have extraordinary high [FORMULA] which are in disagreement with the temperatures measured by ASCA or estimated using the [FORMULA] relation of AE98. The dominant contribution to increasing the predicted number of GLAs is these high temperature clusters.


Table 8. Expected number of giant luminous arcs. Case I: E/S0 SED is UV-intermediate NGC3379 SED. Case II: E/S0 SED is UV-bright NGC4649 SED.
[FORMULA] See Sect. 4.3.

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© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999