5 Discussion

We compared our results with X-ray and virial analyses. From Fig. 4 of Lewis et al., we see that within 175 h^-1 kpc (1 arcmin) radius the mass inferred from X-ray emissivity is $M_{\rm X} \approx 6. \times 10^{13} ~h^{-1} ~M_{\odot}$ which is a factor of 2 lower than the minimum estimate from our shear analysis, $M_{\rm shear} = 1.24\pm0.28 \times 10^{14} ~h^{-1} \,M_{\odot}$ (CL = 90%).

The agreement is better on larger scales. Both estimates increase monotonically and reach $M_{\rm X}=1.82^{+0.34}_{-0.23} \times 10^{14} ~h^{-1} \,M_{\odot}$ and a lower limit of $M_{\rm shear}=2.3_{-0.8}^{+0.8} \times 10^{14} ~h^{-1} \, M_{\odot}$ (CL = 90%) at r = 350 h^-1 kpc (2 arcmin). At that radius, which is the limiting distance to which the X-ray data are reliable, the relative discrepancy of $\sim$20% is within the errors. However, even if we assume that the 20% difference is real and constant beyond r = 350 h^-1 kpc, the baryon fraction only changes from $f_{\rm b}=0.18$quoted by Lewis et al. (1999) to $f_{\rm b}=0.14$.

We compared the mass profile inferred from the shear analysis to several model mass profiles. These model curves have been plotted on the observed mass profile in Fig. 13. We note that the tightest constraint on the profiles occur at small radii. The error bars are too large to really discriminate between the models at large radii. For this reason and others described below we have plotted several models on the data and discussed them in some detail.

One drawback is the marginal detection ( $+1.3\sigma$) of shear in the outermost annulus. It may just be random fluctuation - sections of this annulus lie outside the image or on the masks and so it contains fewer galaxies than it otherwise would have; indeed its error-bar is a third again as much as that of its neighbour. On the other hand, this is perhaps an indication that the mass extends out beyond the edge of the field. If so, we will have a radius dependence to the mass underestimation ( $\propto R^2$) which is not expected to be significant at small radii but could be considerable at the outer points. However, we note that our models, which are basically constrained by the inner points, are not very different from the observed profile at large radii. So unless profiles in the real Universe are very different from those plotted in Fig. 13 the total mass (including the "missed'' fraction) should lie within the upper limits of the present error-bars. The second problem is the presence of the background cluster described previously in the depletion analysis. Clearly, its (unknown) contribution to the projected mass density, at $\sim$1 arcmin from the mass centroid, has to be subtracted before fitting a model profile.

In the upper two panels of Fig. 13 we have plotted Pseudo-isothermal sphere (PIS) models on the observed profile. For $z_{\rm lens} = 0.306$ the PIS profile is given by:

$$M(r)=1.28 \times 10^{14} ~h^{-1} \,{M}_{\odot} \left({\sigma_... ...^2 \left({r_{\rm c} \over 1'}\right) {x^2 \over \sqrt{1+x^2}}$$ (15)

where $r_{\rm c}$ is expressed in arc-minutes, $x=r/r_{\rm c}$, M(r) the mass within radius r and $\sigma_{\infty}$ the three-dimension velocity dispersion at infinity. Each panel contains 3 curves with the same velocity dispersion (top: 925 kms^-1; middle: 1000 kms^-1) but different core radii. The core radius values (top: 53, 70 and 88 h^-1 kpc; middle: 70, 88 and 106 h^-1 kpc) were selected to span the extent of the errorbar on the innermost data point where the effect of the core radius is expected to be most significant; indeed the convergence of the 3 models at large radii shows that core radius values have very little effect there. In the top panel ( $\sigma _{\infty } = 925$ kms^-1) the models follow the data at large radii and we see a hint of the expected background excess at 1 arcmin. On the otherhand the $\sigma _{\infty } = 1000$ kms^-1 models (middle panel) follow the inner mass points but lies above the observed profile in the outer regions; this may be a more appropriate model if we have underestimated the mass at large radii as discussed before. In general we could not fit both the rapid rise at low radii and the flatter profile at large radii satisfactorily with the same set of parameters. Thus depending on the magnitude of the correction to be applied to the observed profile we estimate a velocity dispersion of $\sigma _{\infty } = 925$-1000 kms^-1. More generally, one may cover the space occupied by the $1\sigma$ errorbars using $870<\sigma_{\infty}< 1010$ and $40<r_{\rm c}< 110~ h^{-1}$ kpc (lower $\sigma_{\infty}$ with smaller $r_{\rm c}$). The presence of strong-lensing features also indicates a small core radius.

Carlberg et al. (1996) measured $\sigma_{\infty} = 1054 \pm 104$ kms^-1 and our estimate, though somewhat smaller, is consistent with theirs. Their velocity dispersions (of MS 1008-1224 and other clusters) were in general considerably less than previous estimates. The agreement between our value and theirs suggests that their algorithm and prescriptions were reliable.

The universal profile (NFW) has been plotted on the observed profile in the bottom panel of Fig. 13. The NFW profile may be expressed for this cluster as:

$$M(x)=1. \times 10^{10} ~h^{-1} \,{M}_{\odot} \left({r_{\rm s} \over 1'}\right)^3 \delta_{\rm c} \ m(x)$$ (16)

where $x=r/r_{\rm s}$, m(x) is the dimensionless mass profile (Bartelmann 1996) and $\delta_{\rm c}=\rho_{\rm s}/\rho_{\rm c}$, where $\rho_{\rm c}$ is the critical density. The best fit gives $r_{\rm s}=140^{+212}_{-52}~h^{-1}$ kpc and $\delta_{\rm c}=3.52^{+5.0}_{-2.9} \times 10^4$ (CL = 90%). The error bars are too large for any definitive statement and it may be meaningful to talk of a model only for the average profile from many clusters.

Figure 17 shows the radial luminosity profile of cluster galaxies selected from the Colour-Magnitude plot.

Figure 17: Top panel: radial distribution of the total I-band luminosity of MS 1008-1224 galaxies located on the cluster sequence on the Colour-Magnitude plot. The solid line is the best fit straight line (see Sect. 5). Bottom panel: the observed Mass-to-Light ratio profile of MS 1008-1224 determined from weak-lensing mass and I-band luminosity (note: the M/L value scales with the Hubble factor h). The curves are model mass profiles divided by the linear fit to the observed luminosity profile : the solid line represents the NFW model plotted in bottom panel of Fig. 13 while the dotted lines represent the PIS models in the middle panel of 13 (lower core radius values make for flatter curves at small radii). The vertical bars represent $1\sigma$ errors.

The luminosity was computed assuming a no-evolution model and a K-correction appropriate for elliptical/S0 galaxies. This is reasonable since galaxies selected from the cluster sequence on a Colour-Magnitude diagram are mainly early-type systems comprising the brightest galaxies which contribute most of the luminosity of the cluster. The z = 0.31 K-correction value of 0.23 was obtained from the most recent Bruzual & Charlot (1993) models in I-band filter. At these magnitudes photometric errors were much larger than photon noise and so we assumed a constant error of $\delta_I = 0.1$ for all galaxies. This conservative estimate made allowance for the underestimation of the K-correction for late-type galaxies of the sample.

The light profile is remarkably linear. Hoesktra et al. (1998) found similar results for Cl1358+62. The best fit to the profile gave a slope of $3.89\pm0.09 \times 10^{11}~h^{-1}L_{{\odot}I}/$arcmin and a y-intercept of $-0.032\pm0.029\times 10^{12}~h^{-2} \,L_{{\odot}I}$ (CL = 90%) which is consistent with zero.

The radial profile of the mass-to-light ratio, M/L, is also shown in Fig. 17. At r = 350 h^-1 kpc (2 arcmin) from the cluster center, $(M/L)_I = (360\pm80)h$. Extrapolating the outermost data points provides a value of $(M/L)_I \sim 290~h\pm80$ at r = 700 h^-1 kpc (4 arcmin). This value must be scaled to a value appropriate for the r-band used in Carlberg et al. We find that our equivalent estimate of $(M/L)_r \approx 340\pm80~h$ is in good agreement with the value of ( $(M/L)_r = 312\pm84~h$ from the CNOC analysis (all errors $1\sigma$).

There is not much additional information (beyond that provided by the mass profile) to be had by fitting model profiles to this quantity. However, this plot brings out in a more obvious way the points we made when discussing the mass profile. The NFW model provides a better fit to the M/L profile than the PIS models because for a linear luminosity profile the M/L for the NFW model has the functional form ${\rm Log}(x)/x$ which has a maximum at some intermediate point. The strong constraint exercised by the innermost data point on the allowed core-radius values (PIS models : 40-110 h^-1 kpc) and the excess of mass on intermediate scales (for both NFW and PIS models) are also seen more clearly.

The origin of this excess at 1 arcmin radius may be due to the second cluster at z = 0.9 which increases the gravitational amplification and shear of galaxies at z > 1 and located within 1 arcmin of the mass centroid. From the depletion point of view the most distant galaxies are deflected twice which increases the depth and the angular size of the depleted area. From the gravitational shear point of view, the increase in distortion due to the second cluster could have been mistakenly ascribed to the stronger gravitational potential of MS 1008-1224. This could explain why the mass from the weak lensing analysis, and therefore the radial distribution of the mass-to-light ratio shown in Fig. 17, increases rapidly at small radii (r < 1 arcmin) despite a linear increase of the cluster luminosity. A similar effect is also discernable in the depletion which has a very steep growth curve.

The discrepancy between X-ray and lensing mass only appears on small scales. Also with our weak-lensing mass estimate it is only a factor of 2 which is significantly lower than the factor 3.7 obtained by Wu & Fang (1997) from the analysis of strong lensing features. The decrease of the discrepancy with radius seems to be a general trend which has already been reported (Athreya et al. 1999; Lewis 1999, see Mellier 1999 and references therein). It must be noted that in most of the studies reporting a discrepancy the comparison has been done between X-ray and strong-lensing (not weak-lensing) analyses.

Some of the discrepancy observed in MS 1008-1224 can be produced by the distant cluster behind it. However, such a projection effect, similar to those discussed by Reblinsky & Bartelmann (1999), cannot explain the factor of 2 discrepancy because (i) the distant cluster occupies only a small fraction of the lensed area (1 quadrant of the ISAAC field) and (ii) only background galaxies at z > 0.9 are magnified twice. An upper limit to the magnitude of its impact on the mass estimate is roughly the ratio

$${1 \over 4} \ {\left<{D_{\rm ls} \over D_{\rm os}}\right>^{-1... ...ht>^{-1}_{z_{\rm l}=0.3}} \times {n_{z>0.9} \over n_{z>0.3}} ,$$ (17)

where $n_{z>z_{\rm l}}$ is the fraction of lensed galaxies with redshift larger than $z_{\rm l}$ and the factor 1/4 is the fraction of the ISAAC area covered by the cluster. The photometric redshifts inferred with hyperz (Fig. 4) identified a large fraction of galaxies at z > 0.9 ($\sim$50% of the z > 0.4 sample). Therefore, we estimate that the ratio in Eq. (17) is at most 30%, in agreement with the prediction of Reblinsky & Bartelmann (1999). In fact, the mass reconstructions do not show any obvious clump of mass at the location of the second cluster and so the effect due to this second lens is probably much weaker than the above upper limit.

It is worth noting that apart from this distant cluster contamination by other projection effects are not visible at the center where photometric redshifts provide a good idea of the clustering along the line of sight. The ISAAC field encompasses the region where strong lensing features are visible and where the mass estimate from lensing exceeds the X-ray prediction. We find no evidence that biases like the ones proposed by Cen (1997) or Metzler (1999) are significant in the central region.

There is compelling evidence that the center of mass does not coincide with the cD galaxy:

• Lewis et al. (1999) reported that the X-ray centroid was 15 arcsec north of the cD galaxy (no error listed);
• our mass reconstruction shows that the centroid is offset 19^+22.5_-18.5 arcsec or alternatively at least 5 arcsec (both CL = 90%) north of the cD galaxy;
• our depletion analysis puts the mass centroid 10 arcsec north and 7 arcsec west of the cD (error could not be estimated).

Three independent techniques point to the offset in the same direction and of roughly the same magnitude which suggests that the cD is indeed located 10-20 arcsec to the south of the mass centroid. It may be associated with the lower clump seen in our high resolution mass maps.

The contours of isoluminosity and number density are clearly clumpy and extend northward of the cD galaxy, as do the contours in our mass maps and in the X-ray maps of Lewis et al. (1999) $\ldots$ all pointers toward a dynamically unstable and perhaps merging system. If so, the hot gas is unlikely to be in equilibrium. A merging process produces shocks and gas flows between clumps, such as those seen in Schindler & Müller's simulations (1993) or those reported by Kneib et al. (1996) and Neumann & Böhringer (1999) in the lensing cluster A2218.

Athreya et al. (1999) reported very similar trends in Abell 370: good agreement between X-ray and weak lensing mass estimates on large scales and a factor of 2 discrepancy near the centre. A370 is clearly composed of merging clumps and they ascribed the X-ray - lensing discrepancy to an oversimplified model of the hot gas in the inner regions. We suspect a similar case in the inner regions of MS 1008-1224. This, as suggested earlier by Miralda-Escudé & Babul (1995), explains the good agreement on large scales between the weak lensing, the X-ray and also the virial mass (see Lewis et al. 1999) and the apparent contradiction between X-ray and strong lensing.

We cannot rule out the possibility that the clumps in MS 1008-1224 are close to each other only in projection. The lensing signal due to a collinear collection of condensates would mimic that of an equivalent projected mass density lens but the X-ray mass estimate would be considerably lower. This would be a more appropriate explanation if the lensing mass missed by this analysis because of the small field of view is considerable.