Recent observations have constrained the orbit and structure of the Large Magellanic Cloud (LMC), implying a well-constrained pericentric passage about the Milky Way (MW) ~ 50 Myr ago. In this scenario, the LMCs gaseous disk has recently experienced maximal ram pressure stripping, suggesting the current extent of its HI disk directly probes the medium in which it is moving. From the observed stellar and HI distributions of the system we find evidence of a truncated gas profile along the windward ``leading edge of the LMC disk, despite a far more extended stellar component. We explore the implications of this ram pressure stripping signature, using both analytic prescriptions and full 3-dimensional hydrodynamic simulations of the LMC. Our simulations subject the system to a headwind whose velocity components correspond directly to the recent orbital history of the LMC. We vary the density of this headwind, using a variety of sampled parameters for a Beta-profile for a theoretical MW circumgalactic medium (CGM), comparing the resulting HI morphology directly to observations of the LMC HI and stellar components. This model can match the radial extent of the LMCs leading (windward) edge only in scenarios where the MW CGM density at pericentric passage is n(R = 48.2 +/- 5 kpc) = 1.1 (+.44/-.45) x 1e-4 cm^-3. The implied pericentric density proves insensitive to both the broader CGM structure and temperature profile, thus providing a model-independent constraint on the local gas density. This result imposes an important constraint on the density profile of the MWs CGM, and thus the total baryon content of the MW. From our work, assuming a Beta-profile valid to ~ Rvir, we infer a total diffuse CGM mass M(300 kpc) = 2.6 +/- 1.4 x 1e10 Msun or approximately 15% of a 1e12 Msun MWs baryonic mass budget.
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