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Astron. Astrophys. 335, 488-499 (1998)

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3. Expansion of supernova remnants and superbubbles

We restrict our attention to the turbulent motions generated by isolated SNs and by SBs. Each of these events drives an outward propagating shock wave in the ISM, which sweeps up the ambient interstellar gas and magnetic field lines into an expanding cold dense shell. In this study, we make the thin-shell approximation whereby we let all the swept-up gas and field lines be concentrated at the leading shock wave.

In the case of isolated SNs, the energy input is instantaneous. For SNI, we adopt the standard value of [FORMULA]  ergs, while for SNII, we adopt the somewhat larger value of [FORMULA]  ergs, to account for the contribution from the wind blown by the progenitor star (see Ferrière 1995). In the case of SBs, the energy input is continuous, as it is due to repeated SN explosions and to stellar winds; its time dependence was calculated by Ferrière (1995) and is given by the luminosity curve plotted in Fig. 7.

[FIGURE] Fig. 7. SB luminosity per SN progenitor as a function of time.

The set of equations governing the expansion of the shell produced by an isolated SN or by a SB is presented in Ferrière (1995). This set includes evolution equations for the radius of the shell, for its mass and momentum per unit solid angle, and for the energy of the interior cavity. The momentum equations contain neither the magnetic tension force, which is negligible in the thin-shell approximation, nor the Coriolis force, which has a minor effect on the shell expansion (even though it is crucial for the alpha-effect). The external parameters (density, pressure [FORMULA]) entering the mass and momentum equations are taken from our model ISM (Eqs. (6), (11) [FORMULA]) at the Galactic radius of the explosion site. This approach is justified by (1) the result that the horizontal dimension of a cavity remains small compared to the radial scale length of the ISM parameters (see Sect. 5), and (2) the argument that interstellar magnetic fields keep the different ISM phases sufficiently tied together that the ISM can be treated as a homogeneous medium (Elmegreen 1992). In the framework of the above approximations, a shell is axisymmetric about the vertical, and its radius, mass, and velocity are functions of time, [FORMULA], and polar angle, [FORMULA] (defined with respect to a vertical axis through the SN or SB center), only.

A shell keeps expanding until its velocity (normal to itself) drops to 68% of the local external signal speed:

[EQUATION]

(the proportionality factor obtained by Ferrière [1995] was updated such as to remain consistent with the new ISM parameters). At that time, denoted by [FORMULA], the shell merges with the background, and the swept field lines reconnect with the ambient magnetic field. For the following, it is important to bear in mind that [FORMULA] is a function of [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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