422 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
- By (7.2.34), neutron stars will eventually stop convection.
- Due to the high rotationΩ, the convection of (7.2.33) for the early neutron star is time
periodic, and its periodT is inversely proportional toΩ, and its convection intensity
Bis proportional to
√
σ−σc, i.e.(7.2.35)
periodT ≃C 1
Ω
,
intensityB≃C 2√
σ−σc,whereσcis the critical value of the transition, andc 1 ,c 2 are constants. The properties
of (7.2.35) explain that the early neutron stars are pulsars, and by (7.2.34) their pulsing
radiation intensities decay at the rate of√
σ−σcor√
kA−σc(kis a constant).7.2.4 Stellar atmospheric circulations
The model describing stellar atmosphere circulation without rotation is given by (7.1.71)-
(7.1.73), and the eigenvalue equations read as
(7.2.36)
Pr[
̃∆Pθ+^2
r^2∂Pr
∂ θ−
2cosθ
r^2 sin^2 θ∂Pφ
∂ φ−
Pθ
r^2 sinθ−
c 0
r^2 νPθ+
δ
2 r^2∂
∂r(rPθ)]
−
1
r∂ φ
∂ θ=βPθ,Pr[
̃∆Pφ+^2
r^2 sinθ∂Pr
∂ φ+
2cosθ
r^2 sin^2 θ∂Pθ
∂ φ−
Pφ
r^2 sin^2 θ−
c 0
r^2 ν
Pφ+
δ
2 r^2∂
∂r
(rPφ)]
−
1
rsinθ∂p
∂ φ
=βPφ,Pr[
̃∆Pr−^2
r^2( 1 −δ)(Pr+
∂Pθ
∂ θ+
cosθ
sinθPθ+1
sinθ∂Pφ
∂ φ)−
c 1
r^2 vPr+
δ^2
2 ( 1 −δ)1
r^2Pr−δ
2 r∂Pr
∂r+
1
r^2√
ReT]
−( 1 −δ)∂p
∂r=βPr,̃∆T+^1
r^2√
RePr=βT,divP= 0 ,with the boundary conditions
(7.2.37) T= 0 ,Pr= 0 ,
∂Pθ
∂r=
∂Pφ
∂r=0 atr=r 0 ,r 0 + 1 ,where
̃∆f=^1
r^2 sinθ∂
∂ θ(sinθ
∂f
∂ θ)+
1
r^2 sin^2 θ∂^2 f
∂ φ^2+
1 −δ
r^2∂
∂r(r^2
∂f
∂r),
andδ= 2 mG/c^2 rhis theδ-factor withr 0 <r<r 0 +1 being nondimensional.