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Evolution of multiple supernova remnants

(Abridged) Heating of the interstellar medium by multiple supernovae (SNe) explosions is at the heart of producing galaxy-scale outflows. We use hydrodynamical simulations to study the efficiency of multiple SNe in heating the interstellar medium (ISM) and filling the volume with gas of high temperatures. We argue that it is important for SNe remnants to have a large filling factor {\it and} a large heating efficiency. For this, they have to be clustered in space and time, and keep exploding until the hot gas percolates through the whole region, in order to compensate for the radiative loss. In the case of a limited number of SNe, we find that although the filling factor can be large, the heating efficiency declines after reaching a large value. In the case of a continuous series of SNe, the hot gas ($T \ge 3 \times 10^6$ K) can percolate through the whole region after the total volume filling factor reaches a threshold of $\sim 0.3$. The efficiency of heating the gas to X-ray temperatures can be $\ge 0.1$ after this percolation epoch, which occurs after a period of $\approx 10$ Myr for a typical starburst SNe rate density of $ν_{\rm SN} \approx 10^{-9}$ pc$^{-3}$ yr$^{-1}$ and gas density of $n\approx 10$ cm$^{-3}$ in starburst nuclei regions. This matches the recent observations of a time delay of similar order between the onset of star formation and galactic outflows. The efficiency to heat gas up to X-ray temperatures ($\ge 10^{6.5}$ K) roughly scales as $ν_{\rm SN}^{0.2} n^{-0.6}$. For a typical SNe rate density and gas density in starburst nuclei, the heating efficiency is $\sim 0.15$, also consistent with previous interpretations from X-ray observations. We discuss the implications of our results with regard to observational diagnostics of ionic ratios and emission measures in starburst nuclei regions.