Linking Stellar Coronal Activity and Rotation at 500 Myr: A Deep Chandra Observation of M37

Empirical calibrations of the stellar age-rotation-activity relation (ARAR) rely on observations of the co-eval populations of stars in open clusters. We used the Chandra X-ray Observatory to study M37, a 500-Myr-old open cluster that has been extensively surveyed for rotation periods ($P_{\rm rot}$). M37 was observed almost continuously for five days, for a total of 440.5 ksec, to measure stellar X-ray luminosities ($L_{\mathrm{X}}$), a proxy for coronal activity, across a wide range of masses. The cluster's membership catalog was revisited to calculate updated membership probabilities from photometric data and each star's distance to the cluster center. The result is a comprehensive sample of 1699 M37 members: 426 with $P_{\rm rot}$, 278 with X-ray detections, and 76 with both. We calculate Rossby numbers, $R_o = P_{\rm rot}/\tau$, where $\tau$ is the convective turnover time, and ratios of the X-ray-to-bolometric luminosity, $L_{\rm X}/L_{\rm bol}$, to minimize mass dependencies in our characterization of the rotation-coronal activity relation at 500 Myr. We find that fast rotators, for which $R_o<0.09\pm0.01$, show saturated levels of activity, with log($L_{\rm X}/L_{\rm bol}$)$=-3.06\pm0.04$. For $R_o\geq0.09\pm0.01$, activity is unsaturated and follows a power law of the form $R_o^\beta$, where $\beta$=$-2.03_{-0.14}^{+0.17}$. This is the largest sample available for analyzing the dependence of coronal emission on rotation for a single-aged population, covering stellar masses in the range 0.4$-$1.3 $M_{\odot}$, $P_{\rm rot}$ in the range 0.4$-$12.8 d, and $L_{\rm X}$ in the range 10$^{28.4-30.5}$ erg s$^{-1}$. Our results make M37 a new benchmark open cluster for calibrating the ARAR at ages of $\approx$500 Myr.