The superbubble size distribution in the interstellar medium of galaxies

We use the standard, adiabatic shell evolution to predict the differential size distribution N(R) for populations of OB superbubbles in a uniform interstellar medium (ISM). Assuming that shell growth stalls upon pressure equilibrium with the ambient ISM, we derive N(R) for simple cases of superbubble creation rate and mechanical luminosity function (MLF). For constant creation and an MLF phi(L) proportional to L-beta, we find that N(R) proportional to R1-2 beta for R < R-e, and N(R) proportional to R4-5 beta for R > R-e, where the characteristic radius R-e similar to 1300 pc for typical ISM parameters. For R < R-e, N(R) is dominated by stalled objects, while for R > R-e it is dominated by growing objects. The relation N(R) proportional to R1-2 beta appears to be quite robust, and also results from momentum-conserving shell evolution. We predict a peak in N(R) corresponding to individual supernova remnants (SNRs), and suggest that the contribution of Type Ia SNRs should be apparent in the observed form of N(R). We present expressions for the porosity parameters, Q(2D) and Q(3D), derived from our analysis. Q(2D) is dominated by the largest superbubbles for beta<2 and individual SNRs for beta >2, whereas Q(3D) is normally dominated by the few largest shells. We examine evolutionary effects on the H II region luminosity function (H II LF), in order to estimate beta. We find that for a nebular luminosity fading with time t, L proportional to t(-eta), there is a minimum observed slope a(min) for the H II LFs. Empirical measurements all show a > a(min) therefore implying that usually we may take beta=a. We also find that if nebular luminosity is instantaneously extinguished at some given age, rather than continuously fading, no a(min) will be observed. Comparison with the largely complete HI hole catalogue for the SMC shows surprising agreement in the predicted and observed slope of N(R). This suggests that no other fundamental process is needed to explain the size distribution of shells in the SMC. Further comparison with largely incomplete H I data for M31, M33 and Holmberg II also shows agreement in the slopes, but perhaps hinting at systematic differences between spiral and Im galaxies. We estimate porosities that are substantially <1 for all of the galaxies except Holmberg II, for which we obtain values greater than or similar to 1. Most of these galaxies therefore may not be strongly dominated by a hot interstellar component. However, porosity results for the Galaxy remain inconclusive with the available data.