420 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
Based on the dynamic transition theory introduced early in this section, the stellar circu-
lation depends on the following three forces:
(7.2.28) Pr∆P,
1
κFGP, σT~k,where in general Pr∆Pprevents/slows-down the circulation.
By (7.2.14) we see thatFGPdepends on theδ-factor. The Sun’sδ-factor isδ⊙= 10 −^5 / 2 ,
and in general theδ-factors for stars are of the order:
(7.2.29)
δ∼ 10 −^8 for red giants,
δ∼ 10 −^5 for main-sequence stars,
δ∼ 10 −^3 for white dwarfs,
δ∼ 10 −^1 for neutron stars,
δ= 1 for black holes.Hence it is clear that all stars, except black holes, have smallδ-factors.
On the other hand, for a smallδ-factor, it follows from equations (7.1.79) and (7.1.80)
thatα,ψ,φhas the order
α∼ 1 +δ+η, η,η′∼δ^2 , α′∼δ,
ψ∼rδ, ψ′/ψ∼δ, φ′,φ′′∼δ.Hence, we deduce from (7.2.14) that
FGP∼δP for δ≪ 1.Thus, in view of (7.2.29) we conclude that the relativistic effectFGPis negligible on the
stellar interior motion for all stars except supernovae.
Hence the main driving force for stellar circulations is thethermal expansion force char-
acterized by theσ-factorσ 0 in (7.2.22). Due toδ≪ 1 ,σ 0 can be approximately given by
(7.2.30) σ 0 =
r^20 mGβ
κ^2A,
which plays the similar role as the Rayleigh number Re in the earth’s atmospheric circulation.
The valueσ 0 of (7.2.30) is large enough to generate thermal convections for main-sequence
stars and red giants.
We remark that the heat sourceQis caused not only by nuclear reactions, but also by the
pressure gradient, the density and the gravitational potential energy. Based on theσ-factor in
(7.2.30), we obtain the following physical conclusions:
1.Main-sequence stars.Based on the dynamic transition theory, by (7.2.23), we deduce
that there is a critical numberσc>0, independent of the parameterσ 0 in (7.2.30), such that
equations (7.2.26) undergo no dynamic transition ifσ 0 <σc, and a dynamic transition if
σ 0 >σc:
(7.2.31) σ 0 −σc
{
<0 there is no stellar circulation,
>0 there exists stellar circulation.