Narrow escape: how ionizing photons escape from disc galaxies

In this paper we calculate the escape fraction ($f_{\rm esc}$) of ionizing photons from starburst galaxies. Using 2-D axisymmetric hydrodynamic simulations, we study superbubbles created by overlapping supernovae in OB associations. We calculate the escape fraction of ionizing photons from the center of the disk along different angles through the superbubble and the gas disk. After convolving with the luminosity function of OB associations, we show that the ionizing photons escape within a cone of $\sim 40 ^\circ$, consistent with observations of nearby galaxies. The evolution of the escape fraction with time shows that it falls initially as cold gas is accumulated in a dense shell. After the shell crosses a few scale heights and fragments, the escape fraction through the polar regions rises again. The angle-averaged escape fraction cannot exceed $\sim [1- \cos (1 \, {\rm radian})] = 0.5$ from geometrical considerations (using the emission cone opening angle). We calculate the dependence of the time- and angle-averaged escape fraction on the mid-plane disk gas density (in the range $n_0=0.15-50$ cm $^{-3}$) and the disk scale height (between $z_0=10-600$ pc). We find that the escape fraction is related to the disk parameters (the mid-plane disk density and scale height) roughly so that $f_{\rm esc}^\alpha n_0^2 z_0^3$ (with $\alpha\approx 2.2$) is a constant. For disks with a given WNM temperature, massive disks have lower escape fraction than low mass galaxies. For Milky Way ISM parameters, we find $f_{\rm esc}\sim 5\%$, and it increases to $\approx 10\%$ for a galaxy ten times less massive. We discuss the possible effects of clumpiness of the ISM on the estimate of the escape fraction and the implications of our results for the reionization of the universe.