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The recent LMC-SMC collision: Timing and impact parameter constraints from comparison of Gaia LMC disk kinematics and N-body simulations

We present analysis of the proper-motion (PM) field of the red clump stars in the Large Magellanic Cloud (LMC) disk using the Gaia Early Data Release 3 catalog. Using a kinematic model based on old stars with 3D velocity measurements, we construct the residual PM field by subtracting the center-of-mass motion and internal rotation motion components. The residual PM field reveals asymmetric patterns, including larger residual PMs in the southern disk. Comparisons between the observed residual PM field with those of five numerical simulations of an LMC analog that is subject to the tidal fields of the Milky Way and the Small Magellanic Cloud (SMC) show that the present-day LMC is not in dynamical equilibrium. We find that both the observed level of disk heating (PM residual root-mean-square of 0.057$\pm$0.002 mas yr$^{-1}$) and kinematic asymmetry are not reproduced by Milky Way tides or if the SMC impact parameter is larger than the size of the LMC disk. This measured level of disk heating provides a novel and important method to validate numerical simulations of the LMC-SMC interaction history. Our results alone put constraints on an impact parameter $\lesssim$10 kpc and impact timing $<$250 Myr. When adopting the impact timing constraint of $\sim$140--160 Myr ago from previous studies, our results suggest that the most recent SMC encounter must have occurred with an impact parameter of $\sim$5 kpc. We also find consistent radial trends in the kinematically- and geometrically-derived disk inclination and line-of-node position angles, indicating a common origin.