Mathematical Principles of Theoretical Physics

418 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

whereξis as in (7.2.12). By (7.1.82) and (7.2.16)-(7.2.17), we can deduce that

(7.2.21) ξ( 1 ) = 0 , (i.e.ξ(r 0 ) = 0 ).

Thus, theσ-factor (7.2.11) follows from (7.2.16), (7.2.18) and (7.2.20).
5.Thermal Force and (7.2.21).The thermal expansion force acting on the stellar shell
(i.e. atr=r 0 ) can be deduced from (7.2.9) and (7.2.11) in the following (nondimensional)
form

(7.2.22) fT=σ 0 T, σ 0 =

c^2 r 03 β( 1 −δ)δ

2 κ^2

A (A> 0 ).

By 0<δ<1, we have

σ 0 > 0 (σ 0 =σ( 1 ),i.e.σ(r 0 )).

Hence, it follows that there is anε≥0 such that

(7.2.23) σ(r)> 0 , forε<r≤ 1.

The positiveness ofσ(r)in (7.2.23) shows that the thermal forcefTof (7.2.22) is an
outward expanding force. It is this power that causes the swell and the nebular matter spurt
of a red giant. We also remark that the temperature gradientAon the boundary is maintained
by the heat sourceQ.

7.2.3 Stellar interior circulation

Recapitulation of dynamic transition theory

First we briefly recall the dynamic transition theory developed by the authors in (Ma and Wang,
2013b) and the references therein. Many dissipative systems, both finite and infinite dimen-
sional, can be written in an abstract operator equation formas follows

(7.2.24)

du

dt

=Lλu+G(u,λ),

whereLλis a linear operator,Gis a nonlinear operator, andλis the control parameter.
It is clear thatu=0 is a stationary solution of (7.2.24). We say that (7.2.24) undergoes
a dynamic transition fromu=0 atλ=λ 1 ifu=0 is stable forλ<λ 1 , and unstable for
λ>λ 1. The dynamic transition of (7.2.24) depends on the linear eigenvalue problem:

Lλφ=β(λ)φ.

Mathematically this eigenvalue problem has eigenvaluesβk(λ)∈Csuch that

Reβ 1 (λ)>Reβ 2 (λ)>···.

The following are the main conclusions for the dynamic transition theory; see (Ma and Wang,
2013b) for details: