4. 3D numerical simulations
The expansion of gas layers around OB associations may be described as a blastwave propagating into the interstellar medium (Ostriker & McKee, 1988; Bisnovatyi-Kogan & Silich, 1995). A multi-supernova remnant follows an initial quasi-adiabatic stage, driven by the thermalized energy of the supernova ejecta. The energy from each new supernova is released within the remnant created by all previous stellar winds and explosions. Once the radiative losses become important a thin and cold shell forms at the outer edge of the cavity, which is filled by the hot medium. Due to its supersonic speed the shell continues to collect the ambient medium and it slows down to velocities comparable to the random motion of the interstellar medium.
Since the radius of the shell R is much larger than the thickness of it, the thin layer approximation considered by Sedov (1959) and developed by Kompaneets (1960) and Bisnovatyi-Kogan & Blinikov (1982) can be applied. This approximation has been adopted in two dimensional models of expanding shells by Tenorio-Tagle & Palou (1987), Mac Low & McCray (1988) and Palou et al. (1990) and compared with the 2D hydrodynamical simulations by Mac Low et al. (1989). These models have been further extended into three dimensions by Palou (1990, 1992), Ehlerova & Palou (1996) and by Silich et al. (1996).
Here, the 3D computer model of an expanding infinitesimally thin shell as described by Ehlerova & Palou (1996) is modified. Later, the results are compared with the Sedov solution.
4.1. Initial conditions
At an initial time we insert an initial energy into a spherical cavity of a small initial radius . The values of , and have to fulfill together with the given density of the ambient medium the relation given by Eq. (10). In Sedov's solution 19% of the initial SN energy is transformed into kinetic energy of the expanding shell and the rest into thermal energy of the hot bubble :
where is the initial volume corresponding to a sphere with the radius pc. The initial mass in this spherical shell is the total mass inside the initial volume. This defines together with the initial kinetic energy the initial expansion velocity of the shell .
4.2. Energy inputs from sequential supernovae
During their evolution, SN explosions supply energy to the bubble. The rate of energy input is given by , which we define in units of supernova events per years. We always take as the energy released by a single supernova a value erg. We also give the total number of supernovae which may deliver the energy from an OB association. When this number is reached the energy input stops.
In previous simulations (Ehlerova & Palou, 1996) with the abrupt injection of the energy, the pressure was computed using the adiabatic equation (). The temperature was then determined by the equation of state. There was no apparent connection between pressure and thermal energy.
In this paper, the internal thermal energy is evaluated at every timestep, and subsequently the pressure and temperature are derived. The internal energy of a bubble changes due to:
where is the increment of the bubble volume over the timestep .
where and are the values of the internal energy before and after one step in time.
4.3. Integration scheme
In the 3D numerical simulations, the shell is divided into layers and every layer into elements. The motion of each of the elements is solved numerically. The momentum conservation equation is given as
where are the mass, expansion velocity and the surface of an element, respectively, and are the pressures inside and outside of the bubble, and are the density and velocity of the ambient medium, and g is the gravitational acceleration.
After the expansion becomes subsonic, the mass m of an element is constant.
The momentum conservation equation (21) and mass conservation equation (22) are solved numerically together with the equations for the internal energy (20) using finite timesteps. An adaptive step-size control scheme is used, which will be described elsewhere. The main advantages of this scheme are the known accuracy of the integrated quantities and the savings of the CPU time, particularly in the subsonic stage where the timestep can be large without loosing the accuracy.
© European Southern Observatory (ESO) 1997
Online publication: March 24, 1998