7.4. GALAXIES 449
By the boundary conditions in (7.4.14), we obtain that
(7.4.16) β 1 =
rk 02 ζ 1 −rk 12 ζ 0
r 1 k^1 rk 02 −rk 01 rk 12, β 2 =rk 11 ζ 0 −rk 01 ζ 1
r 1 k^1 rk 02 −rk 01 rk 12.
Thus we derive the solution of (7.4.14) asP ̃φ, T ̃=T 0 +T^0 −T^1
r 1 −r 0r 1 (r 0
r− 1 ), p ̃=∫[P ̃ 2
φ
rρ−
ρG
r^2( 1 −βT ̃)Mr]
dr.Make the translation
Pr→Pr, Pφ→Pφ+P ̃φ, T→T+T ̃, p→p+ ̃p;then the equations (7.4.4) and boundary conditions (7.4.5) become
(7.4.17)
∂Pφ
∂ τ+
1
ρ
(P·∇)Pφ=ν∆Pφ−(P ̃φ
r+
dP ̃φ
dr
)Pr−1
r∂p
∂ φ,
∂Pr
∂ τ+
1
ρ(P·∇)Pr=ν∆Pr+2 P ̃φ
αrPφ+ρ βMrG
αr^2T−
1
α∂p
∂r,
∂T
∂ τ+
1
ρ(P·∇)T=κ∆ ̃T+
r 0 r 1
ρr^2γPr−1
ρrP ̃φ∂T
∂ φ,
divP= 0 ,
P= 0 , T=0 atr=r 0 ,r 1 ,wherer= (T 0 −T 1 )/(r 1 −r 0 ).
The eigenvalue equations of (7.4.17) are given by
(7.4.18)
−∆Pφ+1
ν(
P ̃φ
r+
dP ̃φ
dr)Pr+1
rν∂p
∂ φ=λPφ,−∆Pr−2 P ̃φ
α νrPφ−ρ βMrG
α νr^2T+
1
α ν∂p
∂r=λPr,−∆T+
1
ρrP ̃φ∂T
∂ φ−
r 0 r 1 γ
κ ρr^2Pr=λT,divP= 0 ,
P= 0 , T= 0 atr=r 0 ,r 1.The eigenvaluesλof (7.4.18) are discrete (not counting multiplicity):
λ 1 >λ 2 >···>λk>···, λk→ −∞ask→∞.The first eigenvalueλ 1 and first eigenfunctions(7.4.19) Φ= (Pφ^0 ,Pr^0 ,T^0 )