Smoking Gun of the Dynamical Processing of the Solar-type Field Binary
  Stars
  We investigate the binarity properties in field stars using more than 50\,000
main-sequence stars with stellar mass from 0.4 to 0.85\,$M_\odot$ observed by
LAMOST and {\emph Gaia} in the solar neighborhood. By adopting a power-law
shape of the mass-ratio distribution with power index of $\gamma$, we conduct a
hierarchical Bayesian model to derive the binary fraction ($f_{b}$) and
$\gamma$ for stellar populations with different metallicities and primary
masses ($m_1$). We find that $f_b$ is tightly anti-correlated with $\gamma$,
i.e. the populations with smaller binary fraction contains more binaries with
larger mass-ratio and vice versa.The high-$\gamma$ populations with
$\gamma>1.2$ have lower stellar mass and higher metallicity, while the
low-$\gamma$ populations with $\gamma<1.2$ have larger mass or lower
metallicity. The $f_b$ of the high-$\gamma$ group is anti-correlated with
[Fe/H] but flat with $m_1$.Meanwhile, the $f_b$ of the low-$\gamma$ group
displays clear correlation with $m_1$ but quite flat with [Fe/H]. The
substantial differences are likely due to the dynamical processing when the
binaries were in the embedded star clusters in their early days. The dynamical
processing tends to destroy binaries with smaller primary mass, smaller
mass-ratio, and wider separation. Consequently, the high-$\gamma$ group
containing smaller $m_1$ is more effectively influenced and hence contains less
binaries, many of which have larger mass-ratio and shorter period. However, the
low-$\gamma$ group is less affected by the dynamical processing due to their
larger $m_1$. These are evident that the dynamical processing does effectively
work and significantly reshape the present-day binary properties of field
stars.
