Modeling Unresolved Binaries of Open Clusters in the Color-Magnitude Diagram. I. Method and Application of NGC 3532

The binary properties of open clusters place crucial constraints on star formation theory and clusters' dynamical evolution. We develop a comprehensive approach that models the color-magnitude diagram (CMD) of the cluster members as a mixture of single stars and photometric unresolved binaries. This method enables us to infer the binary properties, including the binary fractionf(b)and binary mass ratio distribution index gamma(q)when a power law is assumed, with high accuracy and precision, which was unfeasible in conventional methods. We employ a modified Gaussian process to determine the main-sequence ridgeline and its scatter from the observed CMD as model input. As a first example, we apply the method to the open cluster NGC 3532 with the Gaia DR2 photometry. For the cluster members within a magnitude range corresponding to FGK dwarfs, we obtainf(b) = 0.267 0.019 and gamma(q) = -0.10 0.22 for binaries with mass ratioq > 0.2. Thef(b)value is consistent with the previous work on NGC 3532 and smaller than that of field stars. The close-to-zero gamma(q)indicates that the mass ratios of binaries follow a nearly uniform distribution. For the first time, we unveil that the stars with smaller mass or in the inner region tend to have lowerf(b)and a more positive value of gamma(q)due to the lack of low mass ratio binaries. The clear dependences of binary properties on mass and radius are most likely caused by the internal dynamics. In this scheme, binaries with smaller primary mass or lower mass ratio have smaller binding energy; hence, they are more vulnerable to dynamical disruption, especially in the inner region where stars interact more frequently.