Issue 
A&A
Volume 540, April 2012



Article Number  A16  
Number of page(s)  41  
Section  Galactic structure, stellar clusters and populations  
DOI  https://doi.org/10.1051/00046361/201016384  
Published online  16 March 2012 
Online material
Description of the HST additional archive data sets used in this paper, other than those from GO10775.
Fraction of binaries with mass ratio q > 0.5, q > 0.6 and q > 0.7, and total fraction of binaries measured in different regions.
Collection of literature binary fraction estimates.
Appendix A: Reliability of the measured binary fraction
In this appendix we investigate whether the fraction of binaries with q > 0.5 that we measured with the procedure described in Sect. 4 are reliable or are affected by any systematic uncertainty due to the method we used. The basic idea of this test consists of simulating a number of CMDs with a given fraction of binaries, measuring the fraction of binaries in each of them, and comparing the added fraction of binaries with the measured ones.
Simulation of the CMD. We started by using artificial stars to simulate a CMD made of single stars following the procedure already described in Sect. 4.2. To simulate binary stars to be added to the simulated CMD we adopted the following procedure:

we selected a fraction of single stars equal to the fraction of binaries that we want to add to the CMD and derived their masses by using the Dotter et al. (2007) massluminosity relation. In our simulations we assumed the values of , 0.10, 0.30, and 0.50;

for each of them, we calculated the mass ℳ_{2} = q × ℳ_{1} of the secondary star and obtained the corresponding m_{F814W} magnitude. Its color was derived by the MSRL. For simplicity we assumed a flat massratio distribution;

finally, we summed up the F606W and F814W fluxes of the two components, calculated the corresponding magnitudes, added the corresponding photometric error, and replaced the original star in the CMD with this binary system.
As an example, in the upper panels of Fig. A.1 we show the artificial star CMD made of single stars only (left panel), and the CMD where we added a fraction of binaries (right panel), for the case of NGC 2298.
Simulation of the differential reddening. To probe how well the reddening correction works, we considered a simple model. The simulation of the differential reddening suffered by any single star is far from trivial as we have poor information on the structure of the interstellar medium between us and each GC. For simplicity, in this work we assumed that reddening variations are related to the positions (X, Y) of each stars by the following relations:
Here X_{MIN,MAX} and Y_{MIN,MAX} are the minimum and the maximum values of the coordinates X and Y, C_{1} is a free parameter that determines the maximum amplitude E(B − V) variation, and C_{2} governs the number of differential reddening peaks within the field of view.
Fig. A.1
Artificial stars CMD for NGC 2298 (upperleft) and simulated CMD with a fraction of 0.10 of binaries added (upperright). Bottom panels show the simulated CMD after we added differential reddening (left) and the simulated CMD after the correction for differential reddening (right). 

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Fig. A.2
Bottomleft: Map of differential reddening added to the simulated CMD of NGC 2298. The gray levels indicate the reddening variations as indicated in the upperright panel. Upperleft and bottomright panels show ΔE(B − V) as a function of the Y and X coordinate respectively for stars into 8 vertical and horizontal slices. 

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In this work, we used for each GC the value of C_{1} that ranges from 0.005 to 0.05 to account for the observed reddening variation in all the GCs, while we arbitrarily assumed three values of C_{2} = 3, 5, and 8 to reproduce three different finescales of differential reddening. As an example, in Fig. A.2 we show the map of differential reddening added to the simulated CMD of NGC 2298 that is obtained by assuming C_{1} = 0.025 and C_{2} = 5.
Fig. A.3
Difference between the measured fraction of binaries and the fraction of binaries in input as a function of the parameter C_{1} for four difference values of the input binary fraction. Black lines indicate the average difference. Red circles, gray triangles and black crosses indicate simulations with C_{2} = 3, 5, and 8 respectively. 

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Fig. A.4
Fractions of binaries per unit q measured in five massratio intervals as a function of q for all the simulated GCs. To compare the q distribution in simulated clusters with different fraction of binaries, we divided ν_{bin} by two times the fraction of binaries with q > 0.5. For clarity, black points have been randomly scattered around the corresponding q value. The means normalized binary fractions in each massratio bin are represented by red points with error bars, while the gray line is the best fitting line, whose slope is quoted in the inset. 

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Then, we have transformed the values of ΔE(B − V) corresponding to the position of each stars into ΔA_{F606W}, and ΔA_{F814W} and added these absorption variations to the F606W and the F814W magnitudes. The CMD obtained after we added differential reddening is shown in the bottom left panel of Fig. A.1 for NGC 2298. We applied to this simulated CMD the procedure to correct for differential reddening described in Sect. 3.1 and obtain the corrected CMD shown in the bottom right panel. For each of these binaryenhanced simulated CMD, we also generated a CMD made of artificial stars by following the approach described in Sect. 4.2. In our investigation we did not account for field stars. For each combination of the and C_{2} we have simulated 200 CMDs with random values of the C_{1}.
Measurements of the binary fraction. Finally, we used the procedure of Sect. 4 to measure the fraction of binaries with mass ratio q > 0.5 defined as:
where and are the numbers of stars in the regions A and B in the CMD, as defined in Fig. 11 in the binaryenhanced simulated CMD and and the numbers of stars in the same regions of the artificial stars CMD.
Fig. A.5
Fraction of binaries with q > 0.5 in three magnitude intervals as a function of Δm_{F814W} for all the simulated GCs. To compare the measured fraction of binaries in different clusters we have divided the measured binary fractions in each magnitude interval by the value of measured in the interval between 0.75 and 3.75 F814W magnitudes below the MS turn off. Red points with error bars are the means normalized binary fractions in each magnitude interval. The gray line with the quoted slope is the bestfitting leastsquares line. 

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Results are shown in Fig. A.3 where we plotted the difference between the measured and the input fraction of binaries versus the parameter C_{1} for four difference values of the input binary fraction. We found that, for input binary fraction of 0.05, 0.10 and 0.30, the average difference are negligible (<0.5%), as indicated by the the black lines and the numbers quoted in the inset. In the case of a large binary fraction () the measured fraction of binaries with q > 0.5 is systematically underestimated by ~0.03. Apparently our results do not depend on the value of the parameter C_{2}. Simulations with C_{2} = 3, 5, and 8 (indicated in Fig. A.3 with red circles, gray triangles, and black crosses, respectively) give indeed the same average differences. Our comparison between the fraction of binaries added to the simulated CMD and the measured ones demonstrate that the fraction of binaries determined in this work and listed in Table 2 are not affected by any significant systematic errors related to the procedure we adopted.
We have also determined the fraction of binaries in five massratio intervals by following the approach described in Sect. 5.1 for real stars. To this aim, we have divided the region B of the CMD defined in Sect. 11 into five subregions as illustrated in Fig. 22 for real stars. The size of each region is chosen in such a way that each of them covers a portion of the CMD with almost the same area. The resulting massratio distribution is shown in Fig. A.4, where we have plotted the fraction of binaries per unit q as a function of the mass ratio. As already done in the case of real stars, to compare the massratio distribution in simulated CMDs with different binary fraction, we have divided ν_{bin} by two times the measured fraction of binaries with q > 0.5. The best fitting gray line closely reproduce the flat massratio distribution in input with ν_{bin} = 1.
Finally we have measured in the simulated CMDs the fraction of binaries with q > 0.5 in three intervals [0.75,1.75], [1.75,2.75], and [2.75,3.75] F814W magnitudes below the MSTO. To do this we used the procedure already described in Sect. 5.4 for real stars and we have normalized the value measured in each magnitude bin by the fraction of binaries with q > 0.5 measured in the whole interval between 0.75 and 3.75 F814W magnitudes below the MSTO. Results are shown in
Fig. A.5 where we have plotted the normalized binary fractions as a function of Δm_{F814W}. The bestfitting gray line is nearly flat, and well reproduces the input magnitude distribution.
These tests demonstrate that both the massratio distribution determined in Sect. 5.1 for the 59 GCs studied in this work and shown in Fig. 25 as well as the binary fractions measured in different magnitude intervals in Sect. 5.4 are not biased by significant systematic errors related to the procedure we adopted.
Appendix B
Fig. B.1
Fraction of binaries with q > 0.5 in the core as a function of some parameters of their host GCs. Clockwise: ellipticity, central concentration, central velocity dispersion, logarithm of the central luminosity density, halfmass and core relaxation timescale, and metallicity. In each panel we quoted the Pearson correlation coefficient (r). PCC clusters are marked with red crosses and are not used to calculate r (see text for details). 

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Fig. B.2
As in Fig. B.1 for the r_{C − HM} sample. 

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Fig. B.3
As in Fig. B.1 for the r_{oHM} sample. 

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Fig. B.4
Upperleft: fraction of binaries with q > 0.5 in the core as a function of the absolute visual magnitude of the host GC. Dashed line is the best fitting straight line whose slope (s) and intercept (i) are quoted in the figure together with the Pearson correlation coefficient (r). PCC clusters are marked with red crosses and are not used to calculate neither the bestfitting line nor r. For completeness in the upperright panels we show the same plot for the fraction of binaries with q > 0.6, and q > 0.7. Lower panels: fraction of binaries with q > 0.5 in the r_{C − HM} (left) and r_{oHM} (right) sample as a function of M_{V}. 

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Fig. B.5
Fraction of binaries with q > 0.5 as a function of the BSS frequency in the core. PCC GCs are marked with red points. 

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Fig. B.6
Fraction of binaries with q > 0.5, q > 0.6, and q > 0.7 in the r_{C} region (upper panels) and fraction of binaries with q > 0.5 in the r_{C − HM} and r_{oHM} regions (bottom panels) as a function of the collisional parameter (Γ_{∗}). The adopted symbols are already defined in Fig. B.4. 

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Fig. B.7
As in Fig. B.6. In this case we used the encounter frequency adopted by Pooley & Hut (2006) in the approximation used for virialized systems. 

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Fig. B.8
Fraction of binaries with q > 0.5, q > 0.6, and q > 0.7 in the r_{C} region (upper panels) and fraction of binaries with q > 0.5 in the r_{C−HM} and r_{oHM} regions (bottom panels) as a function of the relative age measured by MarínFranch et al. (2009). The adopted symbols are already defined in Fig. B.4. 

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Fig. B.9
As in Fig. B.8 but in this case we used the age measures from Salaris & Weiss (2002) and De Angeli et al. (2005). 

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Fig. B.10
Fraction of binaries with q > 0.5 in the r_{C} sample for low density clusters (log (ρ_{0}) < 2.75) as a function of the relative age from MarínFranch et al. (2009) (left panel) an absolute age from Salaris & Weiss (2002) and De Angeli et al. (2005) (right panel). 

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Fig. B.11
Fraction of binaries with q > 0.5 as a function of the temperature of the hottest HB stars (bottom), the HB morphology index (middle), and the median color difference between the HB and the RGB (top). 

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© ESO, 2012
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