9 Summary and conclusion

We summarize our results below:

1.

The average period of RR Lyrae stars in Sgr places this galaxy in the long period tail of the Oosterhoff I group;

2.

We found 53 double-mode RR Lyraes (40 in Sgr and 13 in the Galaxy), and 13 RR Lyrae stars with two closely spaced pulsation frequencies (5 in Sgr and 8 in the Galaxy). The multi-periodic RR Lyraes in Sgr are the first such stars ever discovered in this galaxy whereas the 13 foreground RRds increase the number of known Galactic RRd stars to 18;

3.

The period and amplitude distributions of the RR Lyrae population in Sgr suggest an average metallicity of $\sim$-1.6 dex, with a contribution of a minor but perceptible population at ${\rm [Fe/H]}\lesssim-2.0$ dex. Furthermore, together with the period distribution of RRd stars, there seems to be a large metallicity spread within the RR Lyrae population;

4.

We find no significant variation of the period distribution over our field, suggesting a homogeneous population. There is therefore no evidence of a metallicity gradient in the RR Lyrae population. This result is in contradiction with claims of a metallicity gradient within the main body of Sgr, unless the RR Lyrae population is not directly related to the other populations;

5.

We find a striking similarity between the RR Lyrae populations in Sgr and the LMC. This similarity is based on comparisons of the period distributions, the period-amplitude diagram, and, to a lesser extent, the period distributions of RRd stars;

6.

We estimate the total number of RRab stars to be $\sim$4200 in the main body of Sgr. If Sgr has lost 50$\%$ of its mass since its formation, the total number of RRab stars associated with Sgr would be $\sim$8400. If RRab stars trace light, the above estimates would correspond to an integrated V-magnitude of -13.9 mag for the main body, and -14.7 mag for the whole system.

The stellar population of Sgr has a high metallicity dispersion. This was first suggested from interpretation of the Color-Magnitude diagram of Sgr (Marconi et al. 1998; Bellazzini et al. 1999a) and later confirmed by spectroscopic observations with values ranging from -1.4 dex up to solar abundance (Smecker-Hane & McWilliam 1999; Bonifacio 2000). It seems that the RR Lyrae population also presents a high dispersion of $\sim$1 dex around ${\rm [Fe/H]}\simeq -1.6$, suggesting that the early star-formation process in Sgr was complex. Sgr could thus have formed on a large scale with star bursts occuring in several gaseous clumps of different metallicities. Another scenario would be a continuous star formation with progressive metal enrichment.

The similarity of RR Lyrae populations between Sgr and the LMC is indicative of similar horizontal branch morphologies. This similarity implies that the parameter(s) driving the HB morphology are similar in both systems. Furthermore, since RR Lyraes represent the old metal-weak population of these systems, this result suggests that the LMC and Sgr formed at the same epoch and in a similar environment with respect to the metal abundance. It is thus tempting to speculate that Sgr and the LMC had a common progenitor. In this picture, Sgr could correspond to a piece of the LMC pulled out during a collision with the Galaxy. Such a scenario could help to explain how a galaxy with an old stellar population can be observed on such a low orbit without being completely disrupted through Galactic tides.

Acknowledgements

I am pleased to thank Christophe Alard for constant support and many useful suggestions during the elaboration of this paper.