THE NUCLEUS OF THE SAGITTARIUS DSPH GALAXY AND M54: A WINDOW ON THE PROCESS OF GALAXY NUCLEATION

From these data we constructed a final catalogue containing the radial velocity and the metallicity of 1152 stars in the innermost 15' × 10' of the Sgr galaxy, as displayed in the lower panel of Figure 3. The CMD of all these stars is shown in the upper panel of the same figure. The RGB sequences of the two systems are quite well separated in the CMD down to the faintest stars included in our sample (I ≃ 18). However, we also selected stars lying on the red clump of the Sgr,N population, at V − I ≃ 1.15 and 16.8 < I < 17.3: this feature overlaps with the RGB of M54, hence we will need some further information to establish the membership of these stars, in addition to their color, magnitude, and radial velocity. In particular, the metallicity would be very useful to disentangle the two populations in this range. M54 has [Fe/H] = −1.55 (Brown et al. 1999; Da Costa & Armandroff 1995), with a small dispersion (see Sarajedini & Layden 1995, and Section 3.4). In contrast, the metallicity distribution of the Sgr galaxy is dominated by a wide peak at [Fe/H] ≃ −0.4 (see M05b; Bonifacio et al. 2006; Bellazzini et al. 2006a, and references therein), extending from [Fe/H] ∼ −1.0 to [Fe/H] ∼ 0.0. Stars more metal poor than [Fe/H] ∼ −1.2 are quite rare and, presumably, they would pass their core-helium-burning phase as RR Lyrae or blue horizontal branch stars (Monaco et al. 2003): hence red clump stars of Sgr should have [Fe/H] ≳ −1.0, much more metal-rich than any M54 star.

As a first step for the selection of two samples representative of the Sgr,N and M54 populations, we introduced the photometric classification defined by the three curves overplotted on the CMD of Figure 3. Proceeding from left (blue) to right (red) we assigned the flag cmd in the following way: cmd = 1 to the stars enclosed between the first and second curve, as likely M54 members, cmd = 2 to the stars enclosed between the second and third curve, as likely Sgr,N members, and cmd = 0 to stars lying to the blue of the first curve or to the red of the third curve. A check a posteriori has shown that all the stars having cmd = 0 have radial velocities incompatible with being members of Sgr,N or M54, thus supporting the validity of our photometric classification.

For the coordinates of the centers of M54 and Sgr,N we adopted α₀ = 283.763750° and δ₀ = −30.478333° from Noyola & Gebhardt (2006), and we converted to Cartesian coordinates X, Y (in arcmin) projecting the equatorial coordinates of each star (α, δ) on the plane of sky as in van de Ven et al. (2006),

$$\begin{displaymath} X=-(10800/\pi)\cos (\delta)\sin (\alpha -\alpha _0) \end{displaymath}$$

$$\begin{displaymath} Y=(10800/\pi)[\sin (\delta)\cos(\delta _0)-\cos (\delta)\sin (\delta _0)\cos (\alpha -\alpha _0)], \end{displaymath}$$

with X increasing toward the west and Y increasing toward the north. Adopting the distance to Sgr and M54 measured by Monaco et al. (2004), 1 arcmin corresponds to 7.65 pc.

As there are several stars in common between the observational sets taken with different instruments and/or set-ups, we have the opportunity to check the consistency and the accuracy of our V_r and [Fe/H] estimates. In Figure 4 we show the comparison between V_r estimates from the various sources: V^F_f are the velocities obtained with FLAMES/fibers, V^D_h are from DEIMOS/holes, and V^D_s are from DEIMOS/slits. It can be readily appreciated that the consistency among the different sets of measures is excellent (i.e., ΔV_r ≃ 0.0 km s⁻¹). The typical accuracy, as measured from the rms of the V_r differences, is ⩽±2.0 km s⁻¹, which is satisfying for the present purpose. The actual uncertainty on the single measure should be a factor $\sqrt{2}$ smaller than the rms of ΔV_r (i.e. ≃±1.4 km s⁻¹), as the latter includes the uncertainties of both estimates, added in quadrature. Of the 414 velocity differences plotted in Figure 4, just ∼10 are significantly larger than ±3.0 km s⁻¹: some of these may be associated with binary system observed at different orbital phases (see Monaco et al. 2007).

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Figure 5 shows a comparison between [Fe/H] estimates. In this case, while the consistency between measures from FLAMES and DEIMOS/holes spectra is very good, [Fe/H] from DEIMOS/slits spectra are ∼0.08 dex larger, in average, than those from the other two sources. We were unable to find the cause of this mismatch and to preserve the homogeneity of the final merged dataset we corrected all the [Fe/H] values from DEIMOS/slits spectra by adding −0.08 to all of them. The overall accuracy is more than satisfying, with a typical rms of 0.1 dex, and also in this case the outliers are rare.

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Finally Figure 6 shows a comparison between our final V_r and [Fe/H] estimates and those obtained by M05b from high-resolution FLAMES-UVES@VLT spectroscopy, for the seven stars in common between these datasets. The agreement in the radial velocity is good: if the only outlier (at ΔV_r ∼ −10 km s⁻¹) is excluded the rms of the difference is just 2.3 km s⁻¹ and the average difference is zero.¹⁵ Therefore the reliability and the accuracy of our radial velocity scale is fully confirmed. The comparison of metallicities is more difficult to interpret and would have benefited by a larger number of stars in common between the two samples. For four of the seven stars, ranging from [Fe/H] ≃ −1.5 to [Fe/H] ≃ −0.2 the agreement is more than acceptable, given the associated uncertainties. The other three stars show a considerable Δ[Fe/H], but one of them was reported to have a tentative [Fe/H] estimate from M05b because its spectrum was affected by molecular bands making the analysis less reliable. In any case, none of the observed differences are so large as to cause us to erroneously classify a Sgr,N star as a M54 member, or vice versa, according to the selection criteria that are adopted below. Also, taking into account the high degree of self-consistency among different sets of measures shown in Figure 5, we conclude that our metallicity scale is sufficiently reliable and accurate for the purposes of the present analysis.

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The distribution of the radial velocity of all the observed stars as a function of their (projected) distance $(r=\sqrt{X^2+Y^2})$ from the center of the system is shown in Figure 7. Two very different populations can be identified in this plot: a broadly-distributed cloud of Galactic field stars showing a large dispersion around V_r ∼ 0 km s⁻¹, and an abundant low-dispersion population with a systemic V_r ∼ 141 km s⁻¹, typical of M54 and Sgr,N. As a first broad selection, and following Ibata et al. (1997), we retained as possible M54/Sgr,N members all the (832) stars having 100 km s⁻¹ ⩽V_r⩽ 180 km s⁻¹ (dashed lines in Figure 7). We then computed the mean and the dispersion of this sample: the ±3σ range around the global mean is enclosed within the two dotted lines. We do not reject the stars outside the 3σ range at this stage, since the velocity distribution is different at different radii (see below), and so a more refined choice is to exclude >3σ outliers in any considered radial bin. However, in Section 4 we will see that the adopted bin-by-bin 3σ rejection criterion will exclude all of these stars.

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Figure 8 shows that we expect some (limited) contamination from Milky Way stars, even in our V_r selected sample. The thick continuous line superposed on the observed histogram of the radial velocities zoomed on the Galactic component shows the excellent agreement between the observations and the predictions of the Robin et al. (2003) Galactic model once it has been shifted by −8.0 km s⁻¹ to match the observed peak and rescaled to minimize χ² for V_r < 100.0 km s⁻¹. The model suggests that up to ∼30 Milky Way stars may be present even in the relatively narrow V_r window we have adopted to select M54/Sgr,N stars. It is interesting to note that among the stars having 100 km s⁻¹ <V_r< 180 km s⁻¹ but lying outside of the 〈V_r〉 ± 3σ range, seven have V_r < 〈V_r〉 − 3σ and four have V_r > 〈V_r〉 + 3σ, in agreement with the trend with V_r predicted by the model. Figure 8 shows also that most of the galactic interlopers are expected to be giants, according the the Robin et al. (2003) Galactic model. However, it is likely that such stars should be, in general, much closer to us than Sgr: the model finds that the contaminating giants have a mean distance of 8.7 kpc (corresponding to Δ(M − m) = +2.4 mag, with respect to Sgr), 90% of them have D ⩽ 11 kpc and 99% of them have D ⩽ 14 kpc, i.e. much lower than D_Sgr = 26.3 kpc. For this reason the difference between their V magnitude and V_HB of Sgr must be a bad proxy for their gravity. The typical spectroscopic metallicity error incurred by overestimating the distance modulus of these galactic stars by 2.4 mag is Δ[Fe/H] = −0.58, so the stars will appear to be less metal-rich than they are in reality while their color and magnitude mimic those of genuine Sgr/M54 stars. As a consequence, the metallicity obtained for a galactic star from the calcium triplet (CaT) technique would normally be much different from what would be expected from the magnitude and color measurements. Therefore, there is some hope to exclude these galactic interlopers from our final samples, identifying them by their "odd" color—metallicity combination, as we will do in our final selection (Section 3.5). As the mean metallicity of contaminating giants is 〈[Fe/H]〉 = −0.75 and 90% of them have [Fe/H] < −0.33 it is quite likely that many of them would have a measured metallicity lower than the lower threshold we will assume for Sgr,N members ([Fe/H] ⩾ −0.8). On the other hand, those spuriously showing metallicities compatible with M54 will probably be too red to be selected as possible cluster members (Section 3.5 and Figure 3).

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Finally it is worth noting that the spectra of all the 843 stars that passed the selection in V_r have signal-to-noise ratio (per pixel) S/N>12; 300 of them have S/N>50 and 63 of them have S/N>100 (see Figure 9(d)).

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The metallicity distribution of the velocity selected stars is shown if Figure 9(a). The presence of two populations is very clear: there is a metal-poor peak around [Fe/H] ≃ −1.5 that must be dominated by M54 stars (Brown et al. 1999) and a broader distribution extending from [Fe/H] ≳ −1.0 to super-solar metallicity corresponding to the main population of Sgr,N (see M05b; Bellazzini et al. 2006a; Bonifacio et al. 2006, and references therein). As mentioned earlier, in the present context we will use the metallicity just as a further means to select samples of M54 and Sgr,N members that are as clean as possible, so as to allow further analysis of their kinematic properties. Therefore, we limit the discussion of the metallicity to the brief considerations listed below.

• 1.  
  We confirm that the bulk of Sgr stars have [Fe/H]> − 1.0, belonging to a broad distribution peaking around [Fe/H] = −0.4 and reaching super-solar metallicities (M05b; Bonifacio et al. 2006). Monaco et al. (2003) estimated that in the main body of the Sgr galaxy old metal-poor stars ([Fe/H] ≲ −1.2) should provide less than 12% of the whole stellar content. Several arguments suggest that this fraction should be even lower in the very central region considered here (see Alard 2001; Bellazzini et al. 2006b; Siegel et al. 2007, and references therein). However, we note that there are four stars having [Fe/H] < −2.0 in our sample (see Figure 9) that are probably too metal poor to be members of M54 and might be part of the metal-poor population of Sgr.
• 2.  
  The peak of the metallicity distribution corresponding to M54 is best fitted by a Gaussian curve having mean 〈[Fe/H]〉 = −1.45. This is slightly higher (∼+0.1) than what was found by Brown et al. (1999) and by Da Costa & Armandroff (1995), while it is in good agreement with Armandroff (1989). Note that any possible small shift in the zero point of our metallicity scale does not affect the accuracy of the metallicity ranking, that is ≃±0.1 and it is the relevant figure in the present context.
• 3.  
  Based on the analysis of the color—magnitude distribution of RGB stars, Sarajedini & Layden (1995) concluded that there is an intrinsic metallicity spread among M54 stars of σ_int([Fe/H]) = 0.16 dex. Using spectroscopic metallicities obtained with the CaT technique for five M54 members, Da Costa & Armandroff (1995) found further support for this hypothesis (but see Brown et al. 1999).¹⁶ Our much larger sample of CaT metallicities provides new supporting evidence for this possibility. The sample with −1.0 ⩽ [Fe/H] ⩽ −2.0 has an observed standard deviation of 0.17 dex, larger than what is expected from the statistical scatter (see Figure 5). Deconvolving the (internal) rms scatter (0.11 dex, see Figure 5) from the observed width of the distribution we obtain an estimate of the intrinsic spread of σ_int([Fe/H]) = 0.14 dex, in good agreement with previous estimates. It seems very unlikely that the width of the M54 peak is significantly contaminated by Sgr stars, as the width remains unchanged if we limit the sample to the innermost 3' where M54 should dominate the population mix. If finally confirmed by high-resolution spectroscopy of a large sample of stars, the purported metallicity spread would constitute another element of similarity between M54 and other large-size very bright clusters such as ω Cen, G1 and the like, which are also suspected of being remnant nuclei of disrupted galaxies (see Mackey & van den Bergh 2005; Federici et al. 2007; Catelan 2008; Maschenko & Sills 2005; Böker 2008, and references therein).

We think that the currently available metallicity estimates provide a metallicity ranking that is more than sufficient for the purposes of the present paper, i.e., (a) to discriminate M54 stars from Sgr,N stars in the magnitude range where the two population overlaps in the CMD (16.8 < I < 17.3), and (b) to exclude possible interlopers as stars having the "wrong" metallicity for their color. The details of the adopted selections are described in the following subsection.

The scheme of our finally adopted selections is illustrated in Figures 9(b) and (c). Our aim is to have a sample of M54 stars and a sample of Sgr,N stars as clean as possible from any kind of interlopers. The adopted criteria are the following:

• 1.  
  We accept as members of the M54 sample the stars having cmd = 1 and −1.8 ⩽ [Fe/H] < −1.1.
• 2.  
  We accept as members of the Sgr,N sample the stars having cmd = 2 and −0.8 ⩽ [Fe/H] ⩽ +0.2, or having cmd = 1 and −0.8 ⩽ [Fe/H] ⩽ +0.2 if 16.6 < I < 17.3 (Sgr RC stars superposed on the RGB of M54).

The metallicity separation avoids mixing between samples. The requirement that the cmd flag and the metallicity broadly agree in assigning the membership is likely to exclude most of the Galactic interlopers. In the region in which the RC of Sgr,N and the RGB of M54 overlap, the cmd flag does not provide any discrimination and we rely just on the metallicity. We have adopted quite conservative criteria to obtain very reliable samples of genuine Sgr,N and M54 stars. As a further check, all the results presented in the following about the kinematics of Sgr,N have been verified to hold also when subsamples in which stars in the RC region were excluded are adopted. The main properties of the selected samples are shown in Figure 10.

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With the selection described above we select a sample of 425 very likely M54 members and of 321 very likely Sgr,N members. In the process of clipping 3σ velocity outliers in any considered radial bin (which will be performed in Section 4), we will further exclude eight stars from the M54 sample and three stars from the Sgr,N sample. Hence the actual detailed analysis of the kinematics of the two systems will be performed on 417 stars for M54 and on 318 stars for Sgr,N.

In the upper panel of Figure 11 we show the distribution of the M54 stars in the V_r versus r plane. We have divided the sample into two sets of six and five independent radial bins of different size, respectively, in order to keep the number of stars per bin as high as possible while maintaining the highest degree of spatial resolution (the primary set of independent bins corresponds to the odd rows of Table 6, the secondary set corresponds to the even rows of the table). The primary bins are enclosed by the vertical lines. In each bin we computed the average V_r and the velocity dispersion σ, with their uncertainties (derived with the "jackknife" method, as above). An iterative 3σ clipping algorithm was applied bin by bin. As we proceeded from the innermost bin to the outer ones, any star rejected in a given bin by the clipping algorithm was excluded from the sample. The eight rejected stars are clearly indicated in the plot.

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The derived velocity dispersion profile is reported in the lower panel of Figure 11 (large filled pentagons, primary set, small filled pentagons, secondary set) and in Table 6. The table reports also an alternative estimate of σ obtained with the Gaussian maximum-likelihood method described by Walker et al. (2006), essentially equivalent to that adopted by Martin et al. (2007). It is remarkable that in all cases the estimates obtained with the two methods differ by much less than the reported uncertainties (in all cases by ⩽0.3 km s⁻¹). The profile is complemented with the central estimate obtained by Illingworth (1976) from integrated spectroscopy of the cluster core.¹⁷ The profile shows a steep decrease from σ = 14.2 km s⁻¹ at the center to σ ≃ 5.0 km s⁻¹ at r = 35. Then it begins to grow gently up to σ ≃ 9 km s⁻¹ in the last bin (r ≃ 7').

Figure 12 shows the velocity distribution and dispersion profile for the Sgr,N sample, with the same arrangement as in Figure 11 (see also Table 2). The absence of any obvious trend of velocity dispersion with radius allowed us to adopt a single set of bins of nearly uniform size. The dispersion in each bin has been estimated exactly as in the case of M54, described above. The two innermost stars of the sample (crosses) have been excluded from the analysis as they lie in a too poorly sampled radial region (r < 0.8') and may affect the computation of the mean radius of the first bin. However, the inclusion of these stars in the innermost bin changes the velocity dispersion estimate by ≃0.2 km s⁻¹.

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It is quite obvious from the inspection of Figure 12 that the velocity dispersion profile of Sgr,N is completely flat over the whole radial range explored by our data. The lower panel shows that the velocity dispersion of each radial bin is in good agreement with the velocity dispersion of the whole sample after the clipping of 3σ outliers, σ = 9.6 ± 0.4 km s⁻¹. The maximum observed difference among bins is 1.6 ± 1.4 km s⁻¹; the maximum difference between the dispersion in a given bin and the dispersion of the whole sample is −0.9 ± 1.0. Possibly, there may even be a weak tendency toward lower-velocity dispersions in the inner regions: in the range 00 ⩽ r < 50 we find σ = 9.2 ± 0.5 km s⁻¹ (180 stars), while in the range 50 ⩽ r < 90 we find σ = 10.1 ± 0.6 km s⁻¹ (138 stars); however, this difference is clearly not statistically significant (see Geha et al. 2002; Valluri et al. 2005, for some example of the variety of velocity dispersion profiles observed in dE,N).

It is rather clear that the nucleus of Sgr and the M54 cluster have different velocity dispersion profiles. M54 displays the typical behavior of a globular cluster, i.e., the system becomes increasingly kinematically hot toward its central regions while Sgr,N has a uniformly flat profile, though the SB profiles of the two systems are broadly similar. Figure 13 shows a comparison between the profiles in deeper detail. The inner growth of σ of M54 stars makes the two profiles overlap at r ∼ 1.5'; the innermost two points of the M54 profile are hotter than any Sgr,N point. The outermost point of the cluster profile does match the dispersion of Sgr,N: the velocity dispersion estimate here comes from the least populated bin as well as the most (possibly) prone to contamination by old and metal-poor stars belonging to Sgr (see below for a more detailed discussion). These are the reasons why the comparison between the two samples taken as a whole does not reveal any striking difference. However, in the intermediate radial range the M54 population is significantly colder than that of Sgr,N and the statistical significance of this result is very strong.

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Table 8 reports the results of non-parametric Kolmogorov–Smirnov tests performed on different radial ranges. In all the considered ranges the probability that the two samples are drawn from the same parent population is smaller than 2% and it is smaller than 0.2% in the 1.5' ⩽ r < 6.5' range. In the present case, however, the F statistic is the most appropriate mean to compare statistically the two samples (F = σ²_Sgr/σ²_M54, see Brandt 1970, and references therein), as the corresponding F test evaluates the probability that two samples of given F value are extracted from Gaussian distributions having the same σ. Table 9 shows that in our case this probability is lower than 0.3% in all the considered radial range and it is as small as < 0.01% in several intermediate ranges, including 1.5' ⩽ r < 6.5'. While such a large quantitative difference may not be so obviously apparent from the dispersion profile, it is clear at a first glance when looking at the comparisons in the X–V_r and in the R–V_r planes shown in Figure 14.

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Hence, the direct comparison of the velocity distributions of the two samples clearly establishes that the two populations have very different kinematic properties, indicating that the motion of M54 stars and that of Sgr,N stars are driven by different gravitational equilibria.

Moreover, Gilmore et al. (2007) clearly states that a velocity dispersion profile monotonically declining from the center outward "...is an unavoidable requirement for any mass-follows-light system..." In the present case, this means that while M54 behaves like an ordinary mass-follows-light "purely baryonic" self-gravitating star cluster (at least in its innermost region, containing most of its light/mass; see also Figure 14 and Section 4.2), the flat velocity dispersion profile of Sgr,N implies that the mass distribution driving the kinematics of the nucleus is significantly different from the distribution of the stars. In principle, a radial gradient in the velocity anisotropy affecting only one of the two systems may be invoked to explain the observed difference in the velocity dispersion profiles. As this would imply a correlation of the anisotropy variation with metallicity, we regard this hypothesis as rather ad hoc and we no longer discuss it in the following. We will re-consider the case in more detail in a future contribution (R. A. Ibata et al. 2008, in preparation).

Finally, the velocity dispersion profile of Sgr,N does not differ significantly from the overall profile of the Sgr galaxy, which is found to be flat out to large radii (Ibata et al. 1997), as typical of dwarf spheroidals (see Walker et al. 2006; Muñoz et al. 2008, and references therein). In Figure 15 we have combined the inner 0.8' ⩽ r ⩽ 9' profile obtained here, with estimates of the dispersion at different radii out to r ≃ 100' obtained from publicly available data from the literature (M05b; Ibata et al. 1997) in the same way as done for M54 and Sgr,N, above. Note that r ≃ 100' is less than half of the core radius of the whole Sgr galaxy (Majewski et al. 2003). All the observations are consistent with a constant σ in the considered range and, in particular, there is no apparent transition in the profile at the onset of the nuclear overdensity of stars (r ⩽ 10'). This fact strongly suggests that the inner kinematics of Sgr, including its nucleus, is dominated by the potential set by a distribution of unseen (dark) matter, not by M54 as in the scenario (b) described in Section 1.

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In Figure 16 the observed velocity dispersion profile of M54 is compared with the theoretical profiles of single-mass isotropic King (1966) models that have been proposed by various authors as best-fits to the SB profile of the cluster (see Table 4). The models proposed by Trager et al. (1995, continuous line) and by McLaughlin & van der Marel (2005, dashed line) provide a reasonable description of the data out to r ≲ 4', while there is a clear tendency of the data to be hotter than the models for r > 4'. The outermost observed point, at r ≃ 7' ≲ r_t, displays a velocity dispersion significantly higher than what is predicted even by the hottest model $[\sigma _{\rm obs}-\sigma _{KM}]_{r\sim 7^\prime }=5.8\pm 2.0$ km s⁻¹. There are two possible explanations for this unexpected rise of the velocity dispersion curve near the tidal radius of the cluster.

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First, while our selection criteria prevents any contamination of the Sgr,N sample by M54 stars, since cluster stars with [Fe/H] ⩾ −0.8 simply do not exist, it is well known that the Sgr galaxy does host a minority of stars with [Fe/H] ⩽ −1.0 (≲12%, Monaco et al. 2003), hence some degree of contamination of the M54 sample by Sgr stars is expected (see Section 3.1) and the less populated outer bins of the M54 profile are clearly the most sensitive to the effect of the inclusion of contaminants. Rough estimates indicate that the contamination by metal-poor Sgr stars should amount to less than 3% in the innermost 2', less than 10% for 2' ⩽ r < 4', but it can raise to ≲ 20% for 4' ⩽ r < 6' and be even larger for r > 6'. So, contamination by metal-poor Sgr stars seems a viable explanation for the rise of the dispersion profile of M54 such that it becomes similar to that of the Sgr,N sample in the outermost, and presumably most contaminated, bin.

Alternatively, it may be conceived that, during its spiraling toward the center of Sgr,N (see Section 5.2), M54 suffered some tidal harassment from the host galaxy (Sgr). In this case, some stars in the last bins would have been stripped from the outskirts of the cluster and heated up to the same dispersion of the surrounding field (see Muñoz et al. 2008, and references therein). In the lower left panel of Figure 14 the ±3σ dispersion profiles of the best-fitting King model for M54 (Trager et al. 1995) are superposed on the M54 sample in the R versus V_r plane: the good match between the shape of the bulk of the observed distribution and the model profile out to the tidal radius strongly suggests that most of the velocity outliers at large radii are likely not gravitationally bound to the cluster, independent of their origin (unrelated metal-poor Sgr stars or former M54 members that have been tidally stripped).

While we consider the "contamination" hypothesis as more likely, the "tidal" hypothesis is a very fascinating possibility and it cannot be dismissed with the data we have presently in hand (see also Section 4.3).

Turning back to the inner regions of the cluster, the main conclusion we can draw here is that the observed velocity dispersion profile is consistent with the expected kinematics of the King models that best fit the light distribution of the cluster. Hence, M54 has the kinematics of an ordinary self-gravitating globular cluster, at least in its innermost ∼25 pc, enclosing more than 90% of the cluster light/mass.

4.2.1. The Case of Sgr,N

In Figure 17 the observed velocity dispersion profile of Sgr,N is compared with various theoretical models. In the upper panel we compare King's models with r_c = 0.05' and various values of the concentration parameter C (the best-fit model for the SB profile corresponds to C = 1.9). All the theoretical profiles have been normalized at the central velocity dispersion of M54: this is a quite arbitrary choice, the family of profiles can be shifted up and down according to the preference of the reader. It is clear that, independent of the adopted normalization, none of the plotted profiles provides a satisfactory match to the observations. The dotted profiles, reported here for comparison, show the effects of factor ×2 changes in the adopted values of r_c and r_t. Even if such large values were compatible with the observations presented in Section 2 (and they are not), the corresponding profiles remain unable to reproduce the observed dispersions.

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One could be tempted to suggest that any King model with a sufficiently large scale to fit the whole surface profile of the Sgr galaxy (as the C = 0.90, r_c = 224' model by Majewski et al. 2003) would have a nearly flat dispersion profile on the relatively small radial range studied here (r ⩽ 9'), thus providing a reasonable fit to the observed profile. This is true, but these models would completely fail in reproducing the observed SB profile in the same range by more than 10 mag arcsec⁻² (see Figure 2), since they lack any nuclear overdensity at their center.

Hence, contrary to the case of M54, King models that fit the SB profile of Sgr,N appear unable to reproduce its kinematics: this further supports the conclusion that M54 and Sgr,N are systems of very different nature.¹⁸ In particular, as already noted, mass-follows-light models are incompatible with the observed velocity dispersion profile of Sgr,N (Gilmore et al. 2007).

In the lower panel of Figure 17 the observed profile is compared to those obtained from the N-body realization of a Navarro et al. (1996, hereafter NFW) model of the suite that is described in Section 5 (model NFW3, Table 10). The continuous line is the inner velocity dispersion profile of the NFW3 model at the beginning of the simulation; the dotted line is the profile at the end of the simulation, i.e. after the complete orbital decay of a model of M54 that is launched in orbit within the NFW3 halo. It is interesting to note that the theoretical profiles are fairly flat in the considered radial range, in good agreement with the data. It may be conceived that the (baryonic) nucleus of Sgr lies within the inner cusp of the NFW halo of dark matter that is presumed to embed the Sgr galaxy (Ibata & Lewis 1998; Majewski et al. 2003, and references therein). The potential in this inner region would be dominated by the DM cusp that will impose the flat dispersion curve to the observed Sgr,N stars. It does not seem necessary to assume a strict correlation between the typical size and/or density profile of the DM cusp/core and those of the embedded stellar nucleus: the DM overdensity would have simply provided the "local" minimum of the overall potential well to "attract" the largest density of the infalling gas that formed Sgr,N. It may be interesting to note that the NFW3 model, whose velocity dispersion profile is shown in the lower panel of Figure 17, encloses a mass of 3.2 × 10⁶M_☉ within the range of projected radii covered by our data (r_p ≃ 70 pc), and that the region in which its projected density profile is well fitted by a power-law, i.e. the inner cusp, has a size of r_p ≲ 100 pc, similar to Sgr,N.

Table 10. N-Body Simulations: Fundamental Parameters and Results

Name	Halo	Mhalo (M☉)	Nhalo	Mclus (M☉)	Nclus	X0 (kpc)	VY,0 (km s−1)	td (Gyr)	ttot (Gyr)	1	5
bh2_a	NFW1	2.4 × 108	105	2.0 × 106	1	2.0	4.0	0.37	0.84	0.86	0.67
bh2_b	NFW1	2.4 × 108	105	2.0 × 106	1	2.0	8.0	0.74	1.28	0.73	0.52
bh2_c	NFW1	2.4 × 108	105	2.0 × 106	1	2.0	12.0	1.28	1.58	0.56	0.45
bh2_d	NFW1	2.4 × 108	105	2.0 × 106	1	4.0	4.0	1.22	1.28	0.87	0.65
bh2_e	NFW1	2.4 × 108	105	2.0 × 106	1	4.0	8.0	2.90	3.00	0.68	0.65
bh1_a	NFW1	2.4 × 108	105	1.0 × 106	1	2.0	4.0	0.67	0.74	0.83	0.72
bh1_b	NFW1	2.4 × 108	105	1.0 × 106	1	2.0	8.0	1.19	1.34	0.72	0.58
bh1_N	NFW2	6.1 × 107	105	1.0 × 106	1	2.0	4.0	2.25	2.50	0.36	0.13
cl1.2_N	NFW2	6.1 × 107	105	1.2 × 106	104	2.0	4.0	...	2.50	...	...
bh15_a	NFW3	4.2 × 108	105	1.5 × 106	1	2.0	4.0	0.80	1.00	0.76	0.71
bh15_b	NFW3	4.2 × 108	105	1.5 × 106	1	2.0	8.0	1.25	1.70	0.61	0.44
bh15_c	NFW3	4.2 × 108	105	1.5 × 106	1	2.0	12.0	2.05	3.00	0.40	0.36
bh15_d	NFW3	4.2 × 108	105	1.5 × 106	1	3.0	8.0	2.10	2.42	0.60	0.52
cl15_b	NFW3	4.2 × 108	105	1.5 × 106	104	2.0	8.0	...	2.00	...	...
bh10_b	NFW3	4.2 × 108	105	1.0 × 106	1	2.0	8.0	1.60	2.05	0.61	0.49
bh15_p	NFW3	4.2 × 108	105	1.5 × 106	1	5.0	12.0	−a	6.00	0.25	0.15
bh02_a	NFW3	4.2 × 108	105	2.0 × 104	1	2.0	4.0	−b	6.00	0.75	0.75

Notes. NFW1: V_c = 30 km s⁻¹; NFW2: V_c = 15 km s⁻¹. NFW1, NFW2: a = 0.1 and b = 2.5. NFW3: V_c = 17 km s⁻¹; a = 0.5 and b = 15.0. t_d: time at which the orbit is decayed to a distance from the center r ⩽ 30 pc. t_tot: total time of the simulation. cl1.2_N: t_d cannot be defined as in the above cases since the cluster is no more self-bound at t ≳ 0.8 Gyr; at t ∼ 1.0 the remnant is approximately at the center of the host halo. The case of cl15_b is analogous. ^aAt the end of the simulation the BH has reached a distance of ∼2 kpc from the center of the halo. ^bThe orbit of the BH remain stable over the whole duration of the simulations.

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Unfortunately, we have no conclusive observational evidence in support of the above scenario. Note also that a DM halo with a core much larger than the nuclear scale would also display a flat dispersion curve in this region (as discussed above for the case of the C = 0.90, r_c = 224' King model); it is just the small-scale inner cusp as a possible seed for the formation of a barionic nucleus that can make the NFW option slightly more attractive.

A flat velocity dispersion profile may arise as a consequence of strong tidal disturbance, in mass-follow-light models of dwarf galaxies (Muñoz et al. 2008). While Sgr is clearly in course of tidal disruption, here we are dealing with the kinematics of the innermost ≃70 pc of the galaxy, that should (reasonably) be considered as untouched by the Galactic tides, since the tidal radius of Sgr is as large as a few thousands of parsecs (the formal King's major axis limiting radius is r_t ≳ 10 kpc, according to Majewski et al. 2003).

Finally, it may appear as a curious coincidence that M54 and Sgr,N have similar spatial scales, if they have independent origin. In this sense we can only note that the distributions of sizes and luminosities of globular clusters and stellar galactic nuclei largely overlap, so the observed similarity may not be particularly odd (see Federici et al. 2007; Böker 2008). Moreover, it should be noted that the core and half-light radii of M54 are ∼2 times larger than those of Sgr,N, clearly a non-negligible difference. Finally, while the best fitting King models have essentially the same C parameter, the two observed profiles appear to have significantly different slopes at large radii, where the profile of Sgr,N departs from the King model.

We searched for signals of rotation in the two samples, also taking into account the correction for perspective rotation, according to van de Ven et al. (2006) and taking the proper motion of Sgr/M54 from Dinescu et al. (2005); in the present case the absolute value of the maximum correction is ⩽1.0 km s⁻¹. Given a rotation of the coordinate axes by an angle θ

$$\begin{displaymath} \eta =X \cos (\theta)-Y \sin (\theta) \end{displaymath}$$

$$\begin{displaymath} \chi =X \sin (\theta)+Y \cos (\theta) \end{displaymath}$$

we tried all the possible values of θ in steps of 1°. For each adopted θ we computed the median velocity of stars with negative and positive η. The difference between the two median velocities is a robust measure of the amplitude of any systemic motion around an axis tilted by θ degrees from the χ axis, i.e. of the rotation of stars about this axis.

In some cases, ordered patterns of V_r as a function of η emerged (in particular for the M54 sample), but their amplitude was quite weak, typically lower than 2.0 km s⁻¹. While we can clearly exclude the presence of rotation signals stronger than this, we cannot exclude that such weak patterns are in fact real. In Figure 18 we show the best case for the M54 sample compared with the end product of one of the N-body simulations that are described in Section 5. We refrain from attaching any significance to such a weak pattern, in particular as it has an amplitude similar to the typical uncertainty of the individual velocity measures and is much weaker than the typical velocity dispersion of the sample. We just want to draw the attention of the reader to the fact that the orbital decay of a massive cluster within the parent dwarf galaxy seems to naturally produce these kinds of weak velocity gradients in the inner part of the sinking cluster at the end of the decay process (see Muñoz et al. 2008, for a deeper discussion of the effects of tidal disruption). However, it is quite clear that the detection at r ≳ 10' of stars having metallicity compatible with M54, possibly lower velocity dispersion than the surrounding metal-rich Sgr population, and, above all, having significant differences in mean systemic velocity compatible with a tidally induced rotation like that shown in the upper right panel of Figure 18, would be the final "smoking-gun" of the orbital decay of M54 to its current position. On the other hand, the failure to find such a component would not be sufficient to rule out the hypothesis as, for instance, the density of stars in the tidal tails is expected to be very low and the "rotation" signal may be greatly weakened by unfavorable orientations of the orbital plane with respect to the LOS (Muñoz et al. 2008).

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The simulations were performed with falcON, a fast and momentum-conserving tree-code (Dehnen 2000, 2002), within the NEMO environment (Teuben 1995). Gravity was softened with the kernel "P₂" (Dehnen 2000), with a softening length of 3 pc (except for two special cases, bh_15p and bh_02a, discussed in Section 5.2). The minimum time-step was 1.9 × 10⁻⁶ Gyr, and the tolerance parameter θ = 0.6.

To model the Sgr galaxy we adopted a truncated NFW model of the form

$$$ \rho (r) \propto {{{\rm sech}(r/b)}\over {r (r+a)^2}} \,. $$$

For the model NFW1, the parameters a = 0.1 kpc, b = 2.5 kpc, were selected, which together with a maximum circular speed of V_c = 30 km s⁻¹, gives a total mass of M_TOT = 2.4 × 10⁸ M_☉. This rather massive model was chosen to approximate to the sort of dense system suggested by Ibata & Lewis (1998) that would not be rapidly destroyed by Galactic tides. A second, lower-mass system (NFW2) with a = 0.1 kpc, b = 2.5 kpc, and V_c = 15 km s⁻¹ (M_TOT = 6.1 × 10⁷ M_☉) was also simulated. The motivation for this second model was that the central velocity dispersion in this case should be similar to the observed dispersion of Sgr (while keeping a and b identical to the first model). The NFW3 with a = 0.5 kpc, b = 15.0 kpc, and V_c = 17 km s⁻¹ was chosen to have both a size and a central velocity dispersion similar to the present-day main body of Sgr, irrespective of its robustness to Galactic tides.

In all cases the NFW halos were modeled with 10⁵ particles. M54 was modeled as a single massive particle of 1, 1.5, or 2 million solar masses, as if it were a black hole (bh simulations, as opposed to cl simulations, see Table 10 and Section 5.2). The particle is launched within the NFW halo from (x, y, z) = (2, 0, 0) kpc or (4, 0, 0) kpc¹⁹ (note that the core radius of Sgr is r_c ≃ 1.7 kpc Majewski et al. 2003), with velocities in the y direction in the range 4.0 km s⁻¹ ⩽V_y⩽ 12.0 km s⁻¹, while V_x = V_z = 0.0 km s⁻¹, hence the orbits of the M54 models lie in the x, y plane. As a mere convention to simplify the discussion, we consider the y direction coincident with the LOS between us and the center of Sgr, and the x and z directions as projected on the plane of the sky. When needed we will select particles according to their projected radius $r_p=\sqrt{x^2+z^2}$.

The initial conditions of the various simulations are reported in Table 10. The simulations were usually stopped when it was clear that the orbit of the M54 particle had reached a stable configuration at the center of the host halo(t_tot). The decay time (t_d) is defined as the epoch at which the 3D apocentric distance of the M54 particle become r_apo ⩽ 30 pc. Note that around this limit the adopted time step becomes too small with respect to the orbital period of the infalling massive particle, hence the evolution of the orbit cannot be followed further. To provide a quantitative idea of the evolution of the eccentricity of the orbit ( = (r_apo − r_peri)/(r_apo + r_peri)) we report the eccentricity at the first (₁) and fifth (₅) pericentric passages. The evolution of the orbital radius and of the average velocity of the M54 models is shown in Figure 19.

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From the inspection of Table 10 and Figure 19 we can draw the following conclusions:

• 1.  
  In spite of the wide range of initial conditions that has been explored, the orbit of all the M54 models decayed completely in less than 3 Gyr (5–15 orbits). At the end of our simulations M54 is always virtually at rest (mean velocity ≲1–2 km s⁻¹) at the very center of the host halo (within a few softening lengths). Hence the hypothesis that M54 has reached its presently observed status by dynamical-friction-driven orbital decay appears completely realistic and viable (within the limitations imposed by the resolution of our N-body experiments).
• 2.  
  Two special simulations (bh15_p and bh02_a) have been performed at a lower resolution (softening length of 10 pc) as the BH was not expected to reach the densest regions of the host halo (NFW3). bh15_p explores the case of a M54-like point mass decaying from somehow extreme initial conditions (large distance x₀ = 5.0 kpc and high velocity V_y,0 = 12.0 km s⁻¹). In six Gyr the orbit of the BH is sufficiently decayed to bring it within 2 kpc from the center of the halo. The other simulations indicate that from this point it will require less than 3 Gyr to reach the very center of the halo. From the bh15_p simulation we can deduce that for starting distances larger than 5.0 Kpc the times required for a complete decay begin to be comparable to the Hubble time, in good agreement with the results by M05a, and also that it is quite possible that M54 has reached the very center of Sgr only in recent times, if it was born sufficiently far away from the center. The analytical computations by M05a also indicated that clusters with a mass similar to the other GC residing in the main body of Sgr (Ter 7, Arp 2, Ter 8, M ⩽ 2. × 10⁴ M_☉) would have infinitely long decay times (i.e. stable orbits) if they born at r ≳ 1 kpc from the center of Sgr. As the closest of these clusters lie at ∼2.9 kpc from the center of Sgr, M05a concluded that all of them are on stable orbits. Simulation bh02_a is intended to verify this conclusion by letting a BH as massive as the most massive Sgr cluster (except M54, i.e. Arp 2, M ≲ 2. × 10⁴ M_☉, adopting the same (M/L)_V of M54, which seems appropriate given the age and metallicity of the cluster) evolve from a starting point x₀ = 2.0 kpc and a low initial velocity (V_y,0 = 4.0 km s⁻¹). After 6 Gyr no sign of decaying is noticed and the orbit is absolutely stable, in excellent agreement with the results of M05a.
• 3.  
  The effect of dynamical friction is to progressively decrease the radius, the period, and the eccentricity of the orbit. The mean velocity, on the other hand, appears to drop suddenly in the last phases of the decay, possibly when the point-mass particle reaches the dense central cusp of the host NFW halo. This may suggest that the presence of a central cusp may be a crucial ingredient to lead to the observed phase-space coincidence between M54 and Sgr,N, in particular if the recent results by Sánchez-Salchedo et al. (2006) and Read et al. (2006) are considered. This suggestion is worth being followed up with specific high-resolution simulations adopting different kinds of host galaxy halo models (such as King models, for instance). Recent studies indicate that dynamical friction is probably much less efficient in cored than in cusped structures (see Sánchez-Salchedo et al. 2006; Read et al. 2006; Goerdt et al. 2006).

To check the impact of our adoption of a point-mass model of M54, we repeated the simulation bh1_N and bh15_b adopting a 10⁴ particle King model resembling the real cluster as much as possible (W₀ = 8.0, r_t = 60 pc; simulations cl 1.2_N and cl 15_b of Table 10); in the case of cl1.2_N the mass was adjusted to M = 1.2 × 10⁶ M_☉ to have a central LOS velocity dispersion of σ₀ ≃ 15 km s⁻¹. All the final results are fully consistent with those obtained in simulations with a point-mass model for M54, hence for the purposes of the present study, we do not further distinguish between these two classes of models. To illustrate the typical behavior of these live M54 models in Figure 20 we show a series of snapshots of the evolution of the cluster model of simulation cl15_b. It is interesting to note that in this simulation (as well as in the cl1.2_N one) the cluster suffers strong tidal disruption from the host halo and the final relic sitting nearly at rest at the center of the halo is probably unbound, at odds with the real case. The very fact that the actual cluster is still bound may provide useful constraints on the mass distribution within the Sgr galaxy, which can probably be explored with a systematic suite of N-body simulations.

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If we take as demonstrated the above concluding remarks, we can ask what we have learned about the process of galaxy nucleation from the case studied here. Concerning the two main mechanisms that have been put forward in the literature, i.e., (a) formation of the nucleus by infall of globular cluster(s) to the center of the galaxy, or (b) in situ formation by accumulation of gas at the center of the potential well and its subsequent conversion into a stellar overdensity (see Section 1 and Grant et al. 2005; Côté et al. 2006), the main conclusion that can be drawn from the case of Sgr is that both channels are viable and actually both have been at work "simultaneously" in Sgr.

The present analysis has shown that a stellar nucleus made of the typical material that dominates the baryonic mass budget of Sgr is present in this galaxy, independent of M54, as it displays the same flat velocity dispersion profile as the whole core of Sgr, much different than that of the cluster.

In a likely scenario, the enriched gas from previous generations of Sgr stars accumulated at the bottom of the overall potential well of the galaxy, until star formation transformed it into a stellar nucleus whose SB is ≳100 times larger than in the surrounding Sgr core—a substructure within a larger galaxy. On the other hand, M54 is a (relatively) ordinary massive, old and metal-poor globular cluster. Independent of its birthplace within the early Sgr galaxy, its mass and the density of the surrounding medium of the host galaxy drove it to the bottom of the Sgr potential well by dynamical friction. During its trip to the densest central region of Sgr, the dense cluster managed to survive the tidal force of the host galaxy, hence it reached the present position as a (partially?) self-bound stellar system.

While the mass budget at the center of Sgr is probably dominated by the underlying DM halo, M54 dominates the overall light distribution: an observer taking photometry and/or spectra of the unresolved nucleus of Sgr from a distant galaxy (say, a galaxy in the Virgo cluster) would find that the object looks like a bright and blue globular cluster; the integrated velocity dispersion would not show any peculiar feature revealing the composite nature of the observed nucleus (see Figure 21). It would probably be impossible to disentangle the two systems from the integrated light.²⁰

Finally, the Sgr case seems to support the observed ubiquity of nuclei (see Section 1 and C06). The Sgr galaxy was able to form a sizable nucleus "twice" and with two different formation channels: if either of the two channels had not been viable for some reason, the galaxy would have ended up with a nucleus in any case. The fact that Sgr is the only case of galaxy with a clear nucleus among those classified as dwarf spheroidals in the Local Group may suggest that the progenitor of Sgr was in fact a brighter dE or disc galaxy that has been transformed into a dSph by the interaction with the Milky Way (Majewski et al. 2003; Mayer et al. 2007, M05b, and references therein)

The results presented in this paper suggest several interesting lines of research that we did not follow up for practical reasons. However, we feel that it is worth briefly mentioning some of them here, as a possible starting point for future studies.

• 1.  
  The results presented by Read et al. (2006), Goerdt et al. (2006), and Sánchez-Salchedo et al. (2006) suggest that the complete decay of a massive cluster to the very center of the host galaxy is much easier and faster if a central density cusp is present. It is even possible that a central cusp is actually required to bring a cluster down to the very bottom of the overall potential well. If this is the case, the position of M54 within Sgr,N would support the existence of a central cusp in actual DM halos, a point that has been questioned by several authors (see Sánchez-Salchedo et al. 2006, and references therein). The "complete infall" of a massive cluster may need a NFW cusp and, simultaneously, it may transform the cusp into a core by transferring orbital energy and momentum to the surrounding "medium", thus possibly providing a self-regulating mechanism that simultaneously prevents the further decay of other clusters (that, in general cases, would be quite difficult, given the expected decay times, see Hernandez & Gilmore 1998; Oh et al. 2000; Read et al. 2006; Sánchez-Salchedo et al. 2006; Goerdt et al. 2006, and references therein). In this context, it may also be worth recalling the work by Strigari et al. (2006), whose results militates against the presence of a large-size core in the Fornax dSph, and by Boylan-Kolchin & Ma (2004) on the resilience of cuspy halos in major mergers. These ideas seems worthy of detailed theoretical follow-up.
• 2.  
  It has been suggested several times that bright and metal-poor globular clusters may be of cosmological origin (see Brodie & Strader 2006; Maschenko & Sills 2005, and references therein). If this is the case, they may be very intimately linked to the earliest phases of formation of galaxies and there may have been plenty of opportunities for most of them to become the nuclei of some dwarf galaxy. The tidal stress that they suffered during their infall to the center of their host galaxies may be at the origin of the larger half-light sizes that are observed in those that have been suggested as possible nuclear remnants of ancient dwarf galaxies (Freeman & Bland-Hawthorn 2002; Federici et al. 2007; Mackey & van den Bergh 2005). This kind of scenario may be explored with dedicated N-body simulations, possibly including gas and stars. It would be interesting also to consider in detail the results presented here in relation with the scenario for the origin of globular clusters recently discussed by Böker (2008).
• 3.  
  All the analysis presented in this paper have been performed within the standard Newtonian gravitation theory and dynamics. It may also be worthwhile to consider the observational scenario that emerged from this study in the framework of Modified Newtonian Dynamics paradigms (MOND, see Milgrom 2008; Sanders & McGaugh 2002, and references therein), even if it may not necessarily be an ideal case. The transition between the ordinary Newtonian regime and MOND regime occurs around r ≃ 4.5'–6.5' (a range covered by our data), depending on the actual stellar mass of M54+Sgr,N.
• 4.  
  There is little doubt that the final fate of the Sgr galaxy will be its complete tidal disruption. Once the large-scale stellar body of Sgr will be completely dispersed into the Galactic halo, the final remnant of this (once) relatively large galaxy would be a faint nucleus embedding a bright globular cluster. An observer lacking any knowledge of the origin of this object would conclude that it is a very bright and peculiar globular cluster, dominated by a metal-poor population ([Fe/H] ∼ −1.5, possibly with some spread) but also including a small fraction (∼10%) of metal-rich stars (with average [Fe/H] ∼ −0.4). Also the abundance patterns would appear different: for instance, the metal-poor (M54) stars would appear as moderately α-enhanced (Brown et al. 1999), while metal-rich stars (Sgr,N) would have solar or subsolar [α/Fe] ratios (M05b). The radial velocities would reveal that the metal-rich and metal-poor stars have the same systemic velocity, but different velocity dispersion profiles and slightly different density profiles, possibly with some rotation in the metal-poor component. However, the dominance of metal-poor (M54) stars would be so high that there would be no hint of the presence of a dark matter component. The half-light radius of the system would appear slightly larger with respect to ordinary globulars (Mackey & van den Bergh 2005; Federici et al. 2007). Most of these likely characteristics of the future remnant of Sgr seem to have a counterpart in the widely studied and mysterious stellar system ω Centauri (see Norris et al. 1997; van Leeuwen et al. 2002; Pancino et al. 2000, 2002, 2003; van de Ven et al. 2006, and references therein), which has been proposed as the possible remnant of a nucleated dwarf elliptical since the work of Freeman (1993). While there are also noticeable differences between the two cases, the analogy seems very intriguing and potentially powerful.²¹ It is possible that at least some of the observational features of ω Centauri that appear so difficult to explain may find their natural place within a scenario like the one described above.