424 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
It implies that the forceFin (7.2.40) is explosive asδ→1. In addition, by (7.2.38) the first
eigenvalueβ 1 satisfies the following inequality
β 1 ≤ −k 1 +k 2√
Re+k 3
1 −δforδ→ 1.Thus we can deduce the following conclusions.
Physical Conclusion 7.11.Due to the explosive force F of (7.2.40), the stars withδ⋍
1 (δ= 1 is the black hole) has no the stellar atmospheres, which are erupted into the outer
space. However, as 1 −δ> 0 is small, the stellar atmospheres are in the turbulent convection
state,caused by the relatistic gravitational effect.
- Letlτandlrbe the convection scales in the horizontal direction and radial direction.
The ratio
r=lτ
lrdepends on the coefficient ratio of the horizontal components of the momentum(Pθ,Pφ)and
the radial momentum componentPrin (7.2.36), i.e. the ratio
(7.2.41) η=
δ− 1 −c 0 /ν
− 2 ( 1 −δ)−c 1 /ν+δ^2 / 2 ( 1 −δ)
.
As|η| ≪∞,γandηhave the qualitative relation
γ∝{
η−^1 forη> 0 ,
|η| forη< 0.By (7.2.41) we see that
η>0 andη−^1 ∼o( 1 ) forδ≪1 and largeν,
η<0 and|η| ≪ 1 forδ→ 1.Hence we deduce the following conclusions.
Physical Conclusion 7.12.The convection scale ratioγdepends onδforδ≪ 1 andδ→ 1 ,
and on the ratio c 0 /c 1 forδ< 1 and v≪ 1. Moreover,γpossesses the following properties
(7.2.42) γ=
{
o( 1 ) forδ≪ 1 and v large,
≪ 1 forδ→ 1.The sun atmosphere convections satisfy the relation (7.2.42). For the Sun,δ⋍^12 × 10 −^5.
The observations show that
the photosphere convection r.
= 1 ∼ 2 ,
the chromosphere convection r.
= 2 ∼ 3.
For the solar corona,v≪1, we need to know the ratioc 0 /c 1.