A Chandra Survey of Milky Way Globular Clusters II: Testing the Hills-Heggie Law

Binary-single and binary-binary encounters play a pivotal role in the evolution of star clusters, as they may lead to the disruption or hardening of binaries, a novel prediction of the Hills-Heggie law. Based on our recent {\it Chandra} survey of Galactic globular clusters (GCs), we revisit the role of stellar dynamical interactions in GCs, focusing on main-sequence (MS) binary encounters as a potential formation channel of the observed X-ray sources in GCs. We show that the cumulative X-ray luminosity ($L_X$), a proxy of the total number of X-ray-emitting binaries (primarily cataclysmic variables and coronally active binaries) in a given GC, is highly correlated with the MS binary encounter rate ($\Gamma_{b}$), as $L_X \propto \Gamma_{b}^{0.77\pm0.11}$. We further test the Hills-Heggie law against the {\it binary hardness ratio}, defined as the relative number of X-ray-emitting hard binaries to MS binaries and approximated by $L_{X}/(L_{K}f_{b})$, with $L_{K}$ being the GC K-band luminosity and $f_b$ the MS binary fraction. We demonstrate that the binary hardness ratio of most GCs is larger than that of the Solar neighborbood stars, and exhibits a positive correlation with the cluster specific encounter rate ($\gamma$), as $L_{X}/(L_{K}f_{b}) \propto \gamma^{0.65\pm0.12}$. We also find a strong correlation between the binary hardness ratio and cluster velocity dispersion ($\sigma$), with $L_{X}/(L_{K}f_{b}) \propto \sigma^{1.71\pm0.48}$, which is consistent with the Hills-Heggie law. We discuss the role of binary encounters in the context of the Nuclear Star Cluster, arguing that the X-ray-emitting, close binaries detected therein could have been predominatly formed in GCs that later inspiralled to the Galactic center.