The physical sizes of supernova remnants (SNRs) in a number of nearby galaxies follow an approximately linear cumulative distribution, contrary to what is expected for decelerating shock fronts. This has been attributed to selection effects, or to a majority of SNRs propagating in free expansion, at constant velocity, into a tenuous ambient medium. We compile a list of 77 known SNRs in the Magellanic Clouds (MCs), and argue that they are a fairly complete record of the SNe that have exploded over the last ~20kyr, with most now in the adiabatic, Sedov phase of their expansions. The roughly linear cumulative size distribution (uniform in a differential distribution) can result from the combination of a deceleration during this phase, a transition to a radiation-loss-dominated phase at a radius that depends on the local gas density, and a distribution of ambient densities varying roughly as rho^{-1}. This explanation is supported by the observed -1 power-law distributions of three independent tracers of density: HI column density, Halpha surface brightness, and star formation rate from resolved stellar populations. In this picture, the observed cutoff at r~30 pc in the SNR size distribution is due to a minimum in the mean ambient gas density in the regions where supernovae (SNe) explode. We show that M33 has a SNR size distribution similar to that of the MCs, suggesting these features, and their explanation, may be universal. In a companion paper (Maoz & Badenes 2010), we use our sample of SNRs as an effective SN survey to calculate the SN rate and delay time distribution in the MCs. The hypothesis that most SNRs are in free expansion, rather than in the Sedov phase of their evolution, would result in SN rates that are in strong conflict with independent measurements, and with basic stellar evolution theory.
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