Dynamic Catastrophes in Stars

Abstract

(+) Chercheur Qualifie du FNRSThe invariance of the equations of linear stability of a physical system under a change of sign of the dependent variables implies that we cannot foresee, once an instability is detected, in which direction the system will evolve. In the context of radial dynamical stability this means that the linear analysis does not allow us to discriminate between a subsequent contraction or expansion. To fix up the arrow of the evolution beyond the onset of the instability, a nonlinear analysis is required.Recently mathematicians have developed a broad framework currently known as "catastrophe theory" (CT) which can be typified as a systematically precise equivalent to the analysis of a singular point of a vector field. Therefore we are entitled to expect that CT offers an ideal context for a nonlinear description of stellar stability. To take full advantage of CT however, we have to meet two conditions for our system : (a) it is a "gradient system", i.e. stable equilibrium states correspond to the minima of a potential ; (b) the total number of "effective" dependent variables ("behaviour variables" in Zeeman's 1977 terminology) must not exceed 2. At first sight these requirements seem so stringent that one might rightly wonder whether Nature has any such systems in store. To our surprise the problem of radial dynamical stability can actually be reformulated to obey both requirements. Under conservation of entropy the equilibrium states of a star satisfy a minimum energy theorem. The total energy which plays the part of the potential of CT (condition a) is given in standard notations by(V is here the specific volume, X stands for the composition profiles and r is the derivative of the radial position with respect to mass). The explicit form of V in terms of r and r flows from the conservation of mass. Any distribution r(m) that renders the functional E{r} minimum specifies a dynamically stable equilibrium state. Since r(m) stands in fact for an infinity of behaviour variables, namely the radial location at any mass point m, one has to devise a method to drastically reduce this number of variables to << 2 in order to -conform to condition (b). 501