Ram pressure stripping can remove hot and cold gas from galaxies in the intracluster medium (ICM), as shown by observations of X-ray and HI galaxy wakes in nearby clusters of galaxies. However, ram pressure stripping, including pre-processing in group environments, does not remove all the hot coronal gas from cluster galaxies. Recent high-resolution Chandra observations have shown that $sim 1 - 4$ kpc extended, hot galactic coronae are ubiquitous in group and cluster galaxies. To better understand this result, we simulate ram pressure stripping of a cosmologically motivated population of galaxies in isolated group and cluster environments. The galaxies and the host group and cluster are composed of collisionless dark matter and hot gas initially in hydrostatic equilibrium with the galaxy and host potentials. We show that the rate at which gas is lost depends on the galactic and host halo mass. Using synthetic X-ray observations, we evaluate the detectability of stripped galactic coronae in real observations by stacking images on the known galaxy centers. We find that coronal emission should be detected within $sim 10$ arcsec, or $sim 5$ kpc up to $sim 2.3$ Gyr in the lowest (0.1 - 1.2 keV) energy band. Thus the presence of observed coronae in cluster galaxies significantly smaller than the hot X-ray halos of field galaxies indicates that at least some gas removal occurs within cluster environments for recently accreted galaxies. Finally, we evaluate the possibility that existing and future X-ray cluster catalogs can be used in combination with optical galaxy positions to detect galactic coronal emission via stacking analysis. We briefly discuss the effects of additional physical processes on coronal survival, and will address them in detail in future papers in this series.
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