Supernova Feedback and the Hot Gas Filling Fraction of the Interstellar Medium

Supernovae (SN), the most energetic stellar feedback mechanism, are crucial for regulating the interstellar medium (ISM) and launching galactic winds. We explore how supernova remnants (SNRs) create a multiphase medium by performing 3D hydrodynamical simulations at various SN rates, $S$, and ISM average densities, $\bar{n}$. The evolution of a SNR in a self-consistently generated three-phase ISM is qualitatively different from that in a uniform or a two-phase warm/cold medium. By travelling faster and further in the low-density hot phase, the domain of a SNR increases by $>10^{2.5}$. Varying $\bar{n}$ and $S$, we find that a steady state can only be achieved when the hot gas volume fraction $f_{\rm{V,hot}}\lesssim 0.6 \pm 0.1 $. Above that level, overlapping SNRs render connecting topology of the hot gas, and the ISM is subjected to thermal runaway. Photoelectric heating (PEH) has a surprisingly strong impact on $f_{\rm{V,hot}}$. For $\bar{n}\gtrsim 3 \cm-3 $, a reasonable PEH rate is able to suppress the thermal runaway. Overall, we determine the critical SN rate for the onset of thermal runaway to be $S_{\rm{crit}} = 200 (\bar{n}/1\cm-3)^k (E_{\rm{SN}}/10^{51}\erg)^{-1} \kpc^{-3} \myr-1$, where $k = (1.2,2.7)$ for $\bar{n} \leq 1$ and $> 1\cm-3 $, respectively. We present a fitting formula of the ISM pressure $P(\bar{n}$, $S$), which can be used as an effective equation of state in cosmological simulations. Despite the 5 orders of magnitude span of $(\bar{n},S)$, the average Mach number varies little: $\mathcal{M} \approx \ 0.5\pm 0.2, \ 1.2\pm 0.3,\ 2.3\pm 0.9$ for the hot, warm and cold phases, respectively.