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Appendix A: Effect of the binaries mass-ratio distribution on the mass functions

As already discussed in Sect. 5.2 the choice of a correct mass-ratio distribution of stars in MS-MS binary systems is crucial for a reliable estimate of the binary fraction. We have assumed in this paper that the binaries in NGC 2808 have flat distributions in f(q), consistent with the findings of Milone et al. (2011) in their study of MS-MS binaries in 59 GCs. In this Appendix, we investigate whether a different mass-ratio distribution could effect the main results of this paper.

Tout (1991) suggested that f(q) can be derived by randomly extracting secondary stars from the observed initial MF. The Hess diagram of a CMD made of binary systems only obtained by randomly extracting pair of stars from the MFs of NGC 2808 is plotted in Fig. A.1 and the corresponding mass ratios are displayed in the inset.

It should be noted that Sollima et al. (2007), on the basis of their study of the binaries population in 13 GCs, suggested that the mass-ratio distribution derived from the above procedure could significantly increase the fraction of binaries with low-mass secondaries and produce a significant overestimate of the binary fraction. Furthermore, as said, the Tout (1991) distribution is also in sharp disagreement with the observed distributions of binaries with q > 0.5 presented by Milone et al. (2011) for a sample of 59 GCs. While a precise measurement of f(q) for NGC 2808 binaries is beyond the purpose of this paper (and is not even possible with the given data set), we emphasise that in the case of binary stars formed by random associations between stars of different masses we expect a higher fraction of binaries with q  <  0.5 than can be found with the present data set.

	Fig. A.1 Hess diagram of a simulated CMD made of MS-MS binaries only. The adopted mass-ratio distribution is shown in the inset.
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Once we had fixed the f(q) shape, we followed the procedure of Sects. 5 and 6 to estimate the fraction of binaries and the MFs of the three MSs. For simplicity, we used BaSTI models only. As expected, we found an higher f^BIN = 0.09 ± 0.01 and obtained the MFs plotted in Fig. A.2. The three MFs are slightly steeper, but fully consistent within the uncertainties, with those presented in Sect. 6. In this case, we find that the rMS, the mMS, and the bMS MF have slopes of −1.36 ± 0.24, −0.96 ± 0.28, and −1.01 ± 0.34, respectively. These results, obtained for an “extreme” f(q) demonstrate that the adopted binaries mass-ratio distribution has a small effect on the obtained MFs and does not change the main conclusions of this paper.

	Fig. A.2 Mass function of the three MSs obtained by using the BaSTI mass-luminosity relations and assuming the mass-ratio distribution of MS-MS binaries shown in Fig. A.1.
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	Fig. B.1 Hess diagram of a simulated CMD made of MS-MS binaries only. In contrast to the similar plot of Fig. 11, in this case we assumed that both components of a binary systems come from the same MS.
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Appendix B: Effect of the binaries on the mass functions

In this work, we have assumed that both components of the binary systems have the same probability of belonging to any MS. As anticipated in the text, we explored a different extreme scenario to investigate how the membership of binaries to different MSs affects the derived MFs and the binary fraction. In this scenario, we assume that both the components, in every binary system, are taken from the same stellar population, i.e. from the same MS. The corresponding Hess diagram for a CMD made of binary systems only, is shown in Fig. B.1.

We adopted the same procedure described in Sects. 5 and 6 to estimate the fraction of binaries and the MFs of the three MSs. For simplicity, we used in this case only BaSTI models. The resulting value of f^BIN = 0.05 ± 0.01 is very close to the one obtained in Sect. 5.3 and the rMS, the mMS, and the bMS MF have similar slopes of −1.25 ± 0.22, −0.87 ± 0.25, and −0.91 ± 0.32, respectively, again similar to those of Sect. 6. Once again, these results suggest that the procedures adopted in this paper are insignificantly affected by the adopted scenario.

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