Modelling unresolved binaries of open clusters in color-magnitude diagram. I. method and application of NGC3532

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 the mixture of single stars and photometric unresolved binaries. This method enables us to infer the binary properties, including the binary fraction $f_\mathrm{b}$ and binary mass-ratio distribution index $\gamma_q$ when a power-law is assumed, with high accuracy and precision, which were unfeasible in conventional methods. We employ a modified Gaussian process to determine the main sequence ridge line and its scatter from the observed CMD as model input. As a first example, we apply the method to the open cluster NGC3532 with the Gaia DR2 photometry. For the cluster members within a magnitude range corresponding to FGK dwarfs, we obtain $f_\mathrm{b} = 0.267\pm0.019$ and $\gamma_q = - 0.10\pm0.22$ for binaries with mass ratio $q > 0.2$. The $f_\mathrm{b}$ value is consistent with the previous work on NGC3532 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 lower $f_\mathrm{b}$ and 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.