7.2. STARS 421
In particular,σchas the same order of magnitude as the first eigenvalueλ 1 of the the following
equations in the unit ballB 1 :
−Pr∆P+∇p=λ 1 P forx∈B 1 ⊂R^3 ,
(
Pr,∂Pθ
∂r,
∂Pφ
∂r)
=0 atr= 1 ,where∆Pis as in (7.2.27).
For the main-sequence stars, theσ-factors are much larger than the first eigenvalueλ 1 of
(7.2.32). For example, the Sun consists of hydrogen, and
r 0 = 7 × 108 m, m= 2 × 1030 kg, G= 6. 7 × 10 −^11 m^3 /kg·s^2.Using the average temperatureT= 106 K, the parameterκis given by
κ= 0. 18(
T
190 k) 1. 72
10 −^4 m^2 /s≃ 50 m^2 /s.With thermal expansion coefficientβin the orderβ∼ 10 −^4 /K, theσ-factor of (7.2.30) for
the Sun is about
(7.2.32) σ⊙∼ 1030 A [m/K].
Due to nuclear fusion, stars have a constant heat supply, which leads to a higher boundary
temperature gradientA. Referring to (7.2.32), we conclude that there are always interior
circulations and thermal motion in main-sequence stars andred giants, which has largeσ-
factors.
2.Red giants.The nuclear reaction of a red giant stops in its core, but doestake place
in the shell layer, which maintains a larger temperature gradientAon the boundary than
the main-sequence phase. Therefore, the greaterσ-factor makes the star to expand, and the
increasing radiusr 0 raises theσ-factor (7.2.30). The increasingly largerσ-factor provides a
huge power to hurl large quantities of gases into space at very high speed.
3.Neutron stars and pulsars.Neutron stars are different from other stars, which have
biggerδ-factors, higher rotationΩand lowerσ-factor (as the nuclear reaction stops). Instead
of (7.2.26) the dynamic equations governing neutron stars are
(7.2.33)
∂P
∂ τ+
1
ρ(P·∇)P=Pr∆P+1
κFGP− 2 ~Ω×P−∇p+σT~k,∂T
∂ τ+
1
ρ(P·∇)T= ̃∆T+Pr,divP= 0.
As the nuclear reaction ceases, the temperature gradientAtends to zero as timet→∞,
and consequently
(7.2.34) σ→0 as t→∞.
Based on the dynamic transition theory briefly recalled earlier in this section, we derive
from (7.2.33) and (7.2.34) the following assertions: