Energy Budget in Supernovae-Driven H I Shells

Abstract—Giant H I shells of several hundred parsecs and larger observed in nearby galaxies with a moderate inclination ($i \lesssim$ 50°) are formed by multiple supernova explosions in stellar clusters. To estimate the total energy of these supernovae the relation obtained for the evolution of a single supernova in a homogeneous medium is commonly used. Here we study uncertainties encountered in estimating the total energy using the quantities measured in observations, i.e. radius and velocity of a shell, gas density before shock front of a shell. We analyze these quantities gained from the “synthetic observations” of the data obtained in the 3D simulations of the shell driven by multiple supernovae in a stratified interstellar medium. We show that the value of the total energy can be overestimated as well as underestimated in several times and more using the relation for a single supernova. We discuss how the estimate of the total energy depends on the properties of a gas (density, metallicity), scale height of a disk, number of supernovae and when this estimate is applicable for interpreting observations of H I shell in nearby galaxies.

APPENDIX

Let us find from our numerical calculations the radius and the expansion velocity of an adiabatically evolving bubble. This will serve both for better understanding of the evolution of these quantities described in Section 3.3 and as a test of the code used here. Let us consider the “radius–velocity” relation for the bubble, which, according to the self-similar solution for the adiabatic case, has the form $v \sim {{r}^{{ - 3/2}}}$ for a single supernova and $v \sim {{r}^{{ - 2/3}}}$ for wind.

5.1. Single Supernova

Numerical simulations of the evolution of an isolated SN remnant in a uniform-density medium allow the radius and the expansion velocity to be determined using the method described at the beginning of Section 3.3. The “radius–velocity” relation derived follows quite closely the dynamics of adiabatic supernova remnant, $v \sim {{r}^{{ - 2/3}}}$ (Fig. A.1). The deviations at the beginning of the expansion are due to the small size of the bubble.

Fig. A.1.

The “radius–velocity” relation for a bubble evolving adiabatically in gas with uniform density n = 0.9 cm⁻³. The velocity is determined for a small region of the shell (see Section 3.3). The explosion energy is 10⁵² erg. The dashed line corresponds to the dependence $v\sim {{r}^{{ - 3/2}}}$. The color scale shows the age of the bubble in Myr.

5.2. Multiple Supernovae

Consider the evolution of a bubble produced by multiple SN explosions adiabatically expanding in a uniform medium. In two-dimensional slices of gas density and temperature (panels (a) and (b) Fig. A.2) the dense shell has close a spherical shape. However, the inner boundary of the shell exhibits certain features, which are the result of the interaction of sound waves and shocks coming from the hot cavity. This is particularly apparent in the distribution of the vertical velocity component ${{v}_{z}}$ (panel (c)). For the dense shell, these waves are shocks. In a single impact, they cannot significantly change the momentum of the whole shell, but they can produce some local perturbations in its thermal and dynamic structure. Therefore, for our method of computing the local shell expansion velocity (for a small part of the shell, such as in panel (d) Fig. A.2) these perturbations result in appreciable errors in the velocity determination. While the shell radius found from the numerical simulations follows the self-similar solution $r \sim {{t}^{{3/5}}}$ quite closely, the expansion velocity computed for a part of the shell using a small number of cells in the numerical grid is estimated inaccurately. Thus, the “radius–velocity” relation may not follow the self-similar solution for adiabatic wind, $v \sim {{r}^{{ - 2/3}}}$, and may even have non-monotonic portions (panel (e) Fig. A.2).

Fig. A.2.

Distribution of density (log n [cm⁻³]), temperature (log T [K]), and vertical velocity component (v_z in km s⁻¹) of gas in the plane passing through the center of a bubble evolving adiabatically in gas with constant density n = 0.9 cm⁻³ at 4 Myr. By this time, 30 SNe with the energy of 10⁵¹ erg have exploded in the central region of the bubble. Panel (d) shows the velocity distribution in a 20 pc-wide slice passing through the center of the bubble. Panel (e) shows the “radius–velocity” relation for this bubble throughout its evolution. The color scale shows the age of the bubble in Myr.