Mathematical Principles of Theoretical Physics

7.2. STARS 423

For stellar atmosphere circulations, the star radiusr 0 is much greater than the convection
heighth=r 1 −r 0. Hence, we can approximatively takeδas theδ-factor as defined in
(7.1.74).
Thus the eigenvaluesβof (7.2.36)-(7.2.37) depend onδand the Rayleigh number Re in
(7.1.74):
β=β(δ,Re), ( 0 <δ< 1 , 0 <Re).
For (7.2.36)-(7.2.37), the following conclusions hold true:

1. The differential operator∆in (7.2.36) is the Laplace-Beltrami operator, which is sym-
   metric. Hence, the differential operator in (7.2.36)-(7.2.37) is symmetric. It implies that all
   eigenvaluesβof (7.2.36)-(7.2.37) are real. Hence, the stellar atmospheric system (7.1.71)-
   (7.1.73) undergo the first dynamic transition to stationary solutions.


2. LetLbe the differential operator on the left-hand side of (7.2.36); then the first eigen-
   valueβ 1 satisfies

(7.2.38) β 1 =max
||u||= 1

1

2

〈Lu,u〉.

As theδ-factor of (7.1.74) is smaller than one, then by (7.2.38),β 1 satisfies the inequality

(7.2.39) β 1 ≤ −k 1 +k 2

√

Re,

for some constantsk 1 ,k 2 >0. By (7.1.74) we see that

Re→∞asm→∞,or asr 0 →∞,for(T 0 −T 1 )> 0.

By (7.2.39) we get
β 1 →+∞asm→∞,or asr 0 →∞.

In addition, it is known that

β 1 (k)→+∞(k≥ 2 )asβ 1 →+∞.

Hence based on the dynamic transition theory, we derive the following physical conclu-
sions.

Physical Conclusion 7.10.For the stars with smallδ≪ 1 ,their atmosphere have thermal
convections to occur. In particular for the stars with largemass and radius, the convections
are in turbulent state.

3. In the third equation of (7.2.36), the term

(7.2.40) F=

δ^2

2 ( 1 −δ)

1

r^2

Pr

represents the radially expanding force which is the relativistic gravitational effect. The force
(7.2.40) satisfies
F→+∞asδ→1 forPr> 0.