Relation between spectral indices and binary fractions in globular clusters
Authors: Zhang F. , Li L. , Han Z. , Gong X. 
Published: September 2023

Context. We study the relation between the known binary fraction and spectral absorption feature index to judge whether (and potentially which) spectral absorption feature indices are suitable for determining the binary fraction.
Aims. The determination of the binary fraction is important in studies of binary star formation, evolutionary population synthesis models, and other works. The number of binary stars is difficult to determine for nearly all stellar systems because the individual stars are need to be resolved photometrically or spectroscopically. By comparison, their integrated spectra or spectral absorption feature indices are relatively easy to obtain.
Methods. We used Galactic globular clusters (GCs) as our sample since they have relatively accurate binary fraction measurements and spectroscopic observations along the radial direction. When studying the relation between binary fractions and the spectral absorption feature index, we used three types of binary fractions: one with a mass ratio of q > 0.5 (f(q > 0.5)) and two with a total binary fraction (the methods of counting (f(tot)mf) and fitting (f(tot)mc)), calculated and obtained the equivalent widths or magnitudes of 46 spectral absorption feature indices at three spectral resolutions (FWHMLick/IDS, 5, and 15 Å). Since the regions for the binary fraction measurements (photometric) and the spectroscopic observations are different, we used the method of constructing the radial binary-fraction profile to get the binary fractions corresponding to the regions in the spectroscopic observations. The construction of the radial binary-fraction profile was obtained by using the python curve_fit module to fit the measured and analytic binary fraction values. The analytic value was expressed by taking advantage of the King surface-density profile and the assumed forms with respect to the radial binary-fraction profile (linear, quadratic, exponential, and Gaussian).
Results. We find that the low-resolution (15 Å) spectrum is not suitable for this study and the binary fraction type would affect the results: f(q > 0.5) and f(tot)mc exhibit better correlations with the spectral absorption feature index than f(tot)mf and the difference in metallicity would significantly affect the above relationship. Finally, to eliminate the effects of metallicity, age, and dynamical evolution, we only used those GCs with multiple spectra observed among different regions. We find that OIII-1, OIII-2, HγF, HδF, HγA, HδA, Hβ, Ca4455, C24668, and TiO1 indices have strong correlations with binary fraction. The two OIII indices are the most sensitive to the binary fraction, followed by four Balmer indices – the two narrower central bandpass Balmer indices (∼20 Å, F-definition) are more sensitive than the wider two (∼40 Å, A-definition) and, lastly, the Ca4455, C24668, and TiO1 indices. Using the binary fraction-sensitive spectral absorption feature indices in combination with the age- and metallicity-sensitive spectral absorption feature indices, we can determine the ages or metallicities first. Then we can go on to obtain the binary fractions for those stellar systems in which the individual stars cannot be resolved (e.g., dense or distant stellar systems). Furthermore, we are then able to carry out further studies of binary star formation and improve evolutionary population synthesis models.