Upper limits on the shock speeds in supernova remnants can be combined with post-shock temperatures to obtain upper limits on the ratio of cosmic ray to gas pressure (P_CR / P_G) behind the shocks. We constrain shock speeds from proper motions and distance estimates, and we derive temperatures from X-ray spectra. The shock waves are observed as faint H-alpha filaments stretching around the Cygnus Loop supernova remnant in two epochs of the Palomar Observatory Sky Survey (POSS) separated by 39.1 years. We measured proper motions of 18 non-radiative filaments and derived shock velocity limits based on a limit to the Cygnus Loop distance of 576 +/- 61 pc given by Blair et al. for a background star. The PSPC instrument on-board ROSAT observed the X-ray emission of the post-shock gas along the perimeter of the Cygnus Loop, and we measure post-shock electron temperature from spectral fits. Proper motions range from 2.7 arcseconds to 5.4 arcseconds over the POSS epochs and post-shock temperatures range from kT ~ 100-200 eV. Our analysis suggests a cosmic ray to post-shock gas pressure consistent with zero, and in some positions P_CR is formally smaller than zero. We conclude that the distance to the Cygnus Loop is close to the upper limit given by the distance to the background star and that either the electron temperatures are lower than those measured from ROSAT PSPC X-ray spectral fits or an additional heat input for the electrons, possibly due to thermal conduction, is required.
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