We use Chandra and XMM observations of the globular clusters $omega$ Cen and NGC 6397 to measure the spectrum of their quiescent neutron stars (NSs), and thus to constrain the allowed ranges of mass and radius for each. We also use Hubble Space Telescope photometry of NGC 6397 to identify a potential optical companion to the quiescent NS, and find evidence that the companion lacks hydrogen. We carefully consider a number of systematic problems, and show that the choices of atmospheric composition, interstellar medium abundances, and cluster distances can have important effects on the inferred NS mass and radius. We find that for typical NS masses, the radii of both NSs are consistent with the 10-13 km range favored by recent nuclear physics experiments. This removes the evidence suggested by Guillot and collaborators for an unusually small NS radius, which relied upon the small inferred radius of the NGC 6397 NS.
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