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Astron. Astrophys. 361, 429-443 (2000)

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4. Results

4.1. Distribution of the various components

The results from our analysis for each cluster in our sample are summarised in Tables 3 to 6: in Tables 3 and 5 mass estimates at [FORMULA] are derived from the SLM with the [FORMULA]-[FORMULA] calibrations respectively given by EMN and BN (used to compute [FORMULA]). Tables 4 and 6 contain mean dynamical quantities over the sample at three different overdensities, [FORMULA], [FORMULA] and [FORMULA]. The same quantities are also given with mass estimates from the IHE model.

4.1.1. The binding mass

Mass profiles of the various components for a few clusters are displayed in Fig. 2, together with mass ratio profiles (right side). Fig. 3 shows baryon and gas fraction profiles for the whole sample. Quantities are plotted against the mean enclosed contrast density, which is the natural variable in the scaling model. A clear feature arising from Fig. 2 concerns the different behaviors of hydrostatic masses and total masses deduced from NFW's dark matter profile, normalised by the EMN [FORMULA]-[FORMULA] relationship: NFW profiles are more centrally concentrated, as could be foreseen from Eqs. 11 and 12, a property which is in agreement with the density profile of clusters inferred from lensing (Hammer 1991; Tyson et al. 1990). In the outer part, where the contrast density is smaller than a few [FORMULA], the shapes of the density profiles are quite similar, although some difference in the amplitude exists. In fact, profiles calibrated from the EMN [FORMULA]-[FORMULA] relationship tend to be systematically more massive than with the isothermal hydrostatic model, with a significant dispersion. The last column of Tables 3 and 5 gives the ratio between masses computed with both methods. The mean of masses estimated by the IHE [FORMULA]-model is significantly smaller than SLM masses (at [FORMULA]): [FORMULA] with EMN's normalization, and [FORMULA] with BN's normalization. Clearly, such a difference will translate into the baryon fraction estimates.

[FIGURE] Fig. 2. Mass and mass ratio profiles for a few objects. The meaning of line styles is as follows: Left panels: thick line: SLM mass; thin line: IHE mass; dashes: gas mass; dot-dashed line: stellar mass. Right panels: thick lines: baryon fraction (continuous), gas fraction (dashed) and stellar to total mass ratio (dot-dashed) in the SLM case (with EMN calibration); thin lines: same quantities for the IHE model; three-dots-dash: stellar to gas mass ratio.

[FIGURE] Fig. 3. Profiles of the baryon fraction and gas fraction as a function of mean overdensity for objects with the most reliable data. Left panels show these profiles in the case of the hydrostatic assumption and right panels for mass estimates derived from NFW's dark matter profile, with EMN normalization. Groups ([FORMULA]) are represented with dotted lines, cool clusters ([FORMULA]) with dashed lines and hot clusters with continuous lines. The group with a very steeply rising baryon fraction in the IHE case has [FORMULA].

4.1.2. The X-ray gas

Second, the distribution of gas is more spread out than that of dark matter, which results in steadily rising baryon fractions with radius (Fig. 4), as was already pointed out by numerous teams, among which Durret et al. (1994) and D95. NFW also recover this trend in their simulations. This fact makes the choice of the limiting radius an important matter. In particular, extrapolating masses to the virial radius (which is reached by the gas emission in only five clusters among our sample) could be very unsafe, especially for cool clusters, the gas of the most extended of our objects with [FORMULA] being detected only out to [FORMULA].

[FIGURE] Fig. 4. Average profiles for all clusters (groups included) with the most reliable data. Top: baryon fraction (continuous lines) and gas fraction (dashes) in the case of SLM mass estimates with the EMN normalization (thick lines) and hydrostatic masses (thin lines). Middle: mass to luminosity ratio for the whole sample (continuous lines), for King galaxy profiles only (dots) and for de Vaucouleurs profiles only (dash-dots), with the same convention as previously. Bottom: stellar mass to gas mass ratio, with the same line styles as for [FORMULA].

4.1.3. Mass to light ratio

The derived mean mass to blue luminosity ratio is shown in Fig. 4. As it can be seen, [FORMULA] remains remarkably constant from [FORMULA] to the outer parts of clusters, in the case of total masses derived from SLM as well as that of hydrostatic masses. Thus, the widely spread assumption that light traces mass is confirmed, at least at [FORMULA]. The influence of the choice of de Vaucouleurs galaxy density profiles as compared to King profiles is also clearly highlighted. In fact, in the core, dark matter is normally much more concentrated than galaxies, but using a de Vaucouleurs distribution, it turns out that the concentration factor is considerably lowered and even reversed in the case of hydrostatic masses. Mixing the two shapes of galaxy distribution in our sample, the result is an intermediate behavior.

4.2. The baryon fraction

We find that inside a same object, the gas and baryon fractions increase from the center to outer shells (Fig. 3 and Fig. 4), reflecting the fact that the distribution of gas is flatter than that of dark matter, a trend similar to what is found by D95. Secondly, an interesting feature can be noted from Fig. 3: the baryon fraction profiles versus density contrast are remarkably similar and seem to follow a regular behavior, consistent with a universal baryon fraction shape, even in the central part (although with a larger dispersion). This behavior appears more clearly when one is using the SLM model. This result is consistent with the baryon fraction following a scaling law as it has been already found for the emissivity profiles (Neumann & Arnaud 1999) and gas profiles (Vikhlinin et al. 1999).

Thirdly, the comparison of the graphs of Fig. 3 shows that the baryon fraction [FORMULA] estimated from the NFW profile normalized with the EMN [FORMULA]-[FORMULA] relationship is less dispersed at all contrast densities. This effect is asymmetric: the high baryon fractions [FORMULA] found with the IHE method disappear. The fact that [FORMULA] appears less dispersed has already been found by Evrard (1997) and Arnaud & Evrard (1999). However, our work indicates that this feature exists at any radius. We also plot in Fig. 5 the histograms of baryon fractions derived from both the IHE and SLM methods, at the virial radius [FORMULA] but also at [FORMULA], chosen because each object of the sample is detected in X-rays at least out to [FORMULA]. The comparison of the two indeed provides evidence for SLM masses to lead to more tightened baryon fractions than hydrostatic masses. At the virial radius, we found that the intrinsic dispersion is 50% with the IHE and 20% with SLM. This bears an important consequence for the interpretation of mass estimates as well as the interpretation of the baryon fraction. Clearly the fact that the baryon fraction is less dispersed in the SLM at all radii shows that this mass estimate is safer and that the IHE method provides less accurate mass estimates, even in the central region where hydrostatic equilibrium is expected to hold.

[FIGURE] Fig. 5. Histogram of baryon fractions at [FORMULA] and [FORMULA], with IHE masses in grey and SLM masses in black. The object at [FORMULA] is NGC 4261, which has the lowest X-ray slope [FORMULA].

4.2.1. Stellar to gas mass ratio [FORMULA]

Also shown in Fig. 4 is the mean [FORMULA] ratio as a function of overdensity, slowly going down after the central part. The galaxy density is indeed steeper than that of gas, decreasing in [FORMULA] with [FORMULA] instead of [FORMULA] for a typical value of [FORMULA], and the situation is even worse when a de Vaucouleurs profile is used for the galaxy distribution. Again, the latter contributes in a large amount to the steep decrease in the central regions, whereas there the ratio is flat with King profiles.

4.3. Numerical results

Average numerical results are presented in Tables 4 and 6. It is found that the mean baryon fraction using the SLM with the [FORMULA]-[FORMULA] normalization of EMN is 13.4% and the gas fraction 11.5% at [FORMULA] to be compared with hydrostatic results: respectively 19.2 and 17.0%. As expected, the two methods of mass estimation lead to different baryon (gas) fractions. This difference is not negligible ([FORMULA]) and is mainly due as already noted, to the difference between the IHE mass and the SLM mass. The IHE mass can be 50 to 60% lower with respect to the SLM mass (this is the case, for instance, of the groups HCG 62, NGC 2300 and NGC 4261). This difference between [FORMULA] and [FORMULA] increases when using the [FORMULA]-[FORMULA] normalization of BN (the mean baryon fraction being then 11.5 and the mean gas fraction 10.3%). Cirimele et al. (1997) found [FORMULA]% for their 12 clusters included in our sample (and 20% excluding A76), instead of our result of 19% (and 16%) using their parameters and the same hydrostatic [FORMULA]-model and their limiting radius (they choose a uniform [FORMULA]) and of [FORMULA] using the SLM method. The disagreement is due to the adopted stellar mass to light ratio ([FORMULA] 10.7 instead of our 3.2 value). From the results of D95, it comes out that their 7 clusters also have a mean baryon fraction of [FORMULA]. Thus, this is a confirmation of the divergence between hydrostatic [FORMULA]-model mass estimates and SLM's masses. From a sample of 26 clusters among which 7 hot and 3 cool clusters are in our sample, Arnaud & Evrard (1999) have made a similar analysis and derived in the frame of simulation-calibrated virial masses a mean gas fraction at [FORMULA] of [FORMULA]% in rough agreement with our value of 12%. If the comparison is restricted to hot cluster subsamples, the agreement is as good (they found 16% to be compared with our 14%), and also at [FORMULA]. A somewhat higher gas fraction ([FORMULA]) has been obtained recently by Mohr et al. (1999), as compared to ours, which is probably due to the difference in the normalization of the [FORMULA] relationship.

Another output from the present study is the mean total mass to blue luminosity ratio [FORMULA] at [FORMULA] (the hydrostatic assumption leading to [FORMULA]), groups and clusters of all temperatures put together. However, when looking in more detail at the three classes of groups (with [FORMULA]), cool clusters ([FORMULA]) and hot clusters, [FORMULA] (at [FORMULA]) goes from 200 to 270 and 340 respectively, with similar statistics (7 groups, 10 cool clusters and 8 hot clusters). Hence, we disagree with D95 who claim that the mass to light ratio is roughly constant from groups to rich clusters (using the group NGC 5044 which also belongs to our sample with [FORMULA], 2 cool clusters and 4 hot clusters, 3 of which are also in common with ours). It is worth noting that 2 of the 3 clusters in common have a low [FORMULA] in our analysis: 150 for A85 and 170 for A2063. This conclusion holds whatever the limiting radius: there is a factor of 1.7, 1.9 and 1.9 respectively between groups and hot clusters when examining [FORMULA] out to [FORMULA], [FORMULA] or [FORMULA].

As to the mean gas to stellar mass ratio [FORMULA], its values are summarised in Tables 4 and 6. We have computed this quantity to estimate the stellar contribution to the baryon fraction and to investigate any correlation with temperature, which will be discussed in the next section. Let us simply mention that our value for groups at [FORMULA] is in good agreement with the mean value [FORMULA] of Dell'Antonio et al. (1995) for 4 poor clusters, after correcting for the different [FORMULA] they have used.

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Online publication: October 2, 2000