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.
