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Ground-based observatories across a wide range of wavelengths implement cryogenic cooling techniques to increase the sensitivity of instruments and enable low temperature detector technologies. Commercial pulse tube cryocoolers (PTCs) are frequently used to provide 40 K and 4 K stages as thermal shells in scientific instruments. However, PTC operation is dependent on gravity, giving rise to changes in cooling capacity over the operational tilt range of pointed telescopes. We present a study of the performance of a two stage PTC with a cooling capacity of 1.8 W at 4.2 K and 50 W at 45 K (Cryomech PT420-RM) from 0 - 55 ° away from vertical to probe capacity as a function of angle over a set of realistic thermal loading conditions. Our study provides a method to extract temperature estimates given predicted thermal loading conditions across the angular range sampled. We then discuss the design implications for current and future tilted cryogenic systems.