capítulo 3 modelo de uido para uxos zonais e modos...
TRANSCRIPT
♣ít♦
♦♦ ♦ ♣r ①♦s ③♦♥s
♠♦♦s úst♦s ♦és♦s
st ♣ít♦ t③♠♦s t♦r ♠♥t♦r♦♥â♠ ♠ ♠♦♦
♦s ♦s q ♥ s♦s ♣r í♦♥s ♣r ♦tr rqê♥ ♠♦♦s ♦és♦s
①s rqê♥s ♥â♠ sts ♠♦♦s ♣♦♥r♠♥t s♦rt♦s ♣♦r ❲♥s♦r
t ❬❪ é srt ♥ sçã♦ ♦ ♥str ♦ qír♦ ♦♠ r♦tçã♦ ♣♦♦ t♦r♦
t♥♦ ♦♠♦ s ♦ tr♦ s♥♦♦ ♣♦r ❱ s♦♥s ❬❪ ♦t♠♦s rçõs ♥tr ♦
r♥t t♠♣rtr r♦tçã♦ ♣♦♦ ♦♥sr♥♦ ♦♥trçã♦ ♦ ①♦ ♦r
♣r♦♥♥t ♦ r♥t r t♠♣rtr ♦t♠♦s ♥♦ r♠ s♦tér♠♦ é♠ s s
s♦çõs ♦rrs♣♦♥♥ts ♠♦♦s úst♦s ♦és♦s s ♦ ♠♦♦ úst♦ í♦♥ ❲s
♦rrçã♦ ♣r rqê♥ ♦s ①♦s ③♦♥s ❩s q é s♥sí à r♦tçã♦ ♣♦♦ ♠s
♥ã♦ à r♦tçã♦ t♦r♦ rst♦ ♦ ♣♦ r♥t♠♥t ❬❪ ♦♠ rçã♦ ♦ ♠♦♦
♦s ♦s ♣r♠r♠♥t st♠♦s ♦ t♦ ♥s♦tr♦♣ ♣rssã♦ í♦♥s trés
qçã♦ ♦çã♦ t♠♣♦r s♦s ♣r st t♦ q♥♦ ♦♥sr♦
♥ ♥â♠ ♦s s ♣r♦③ ♠ s♥sí r♥ç ♥♦ ♦r ♣r rqê♥ sts ❬
❪ P♦str♦r♠♥t ♥í♠♦s ♥st ♠♦♦ t♦s ♠♥ét♦s ♦s qs sã♦ ♣r♦♥♥ts
r♥ts t♠♣rtr í♦♥s ♥s s ♦♥çõs ♣r ♥st ♦s s
♦ sts r♥ts s qs ♦r♠ ♣s r♥t♠♥t ♠ ❬❪ sã♦ srts ♥
sçã♦ ♣rs♥t♠♦s ♥♦ ♥ ♦♠♦ ♣r♦♣♦st ♣r tr♦s tr♦s ♠ r sssã♦
s♦r t♦s tr♦♠♥ét♦s ♥♦s s st sssã♦ é t ♥tr♦ ♦ ♦♥t①t♦ t♦r
♦s ♦s
♦♦ ♠♥t♦r♦♥â♠
♦♠♦ ♣♦♥t♦ ♣rt ♣r st ♣ít♦ t③♠♦s t♦r ♦♥sr♥♦ ♦
♣s♠ ♦♠♦ s♥♦ ♦♠♣♦st♦ ♣♦r ♠ ú♥♦ í♦ q ♣♦r s ③ t♠ s ♥â♠ ♦r♥
♣s qçõs ♣rs♥ts ♥tr♦r♠♥t ♥ sçã♦
①♦ ♣r tr tr r♣t♠♦s ts qçõs ♣♦ré♠ rs♥t♥♦ ♦ í♥ Σ q
♥ s♦♠ s ♣rts qír♦ st♦♥ár ♣rtr ♣♥♥t ♦ t♠♣♦ s
r♥③s ♠r♦só♣s ♦ ♣s♠
EΣ + Σ ×Σ = 0,
ρΣdΣdt
+∇pΣ − Σ ×Σ = 0,
dpΣdt
+ γpΣ∇ · Σ + (γ − 1)∇ · qΣ = 0,
dρΣdt
+ ρ∇ · Σ = 0,
∇ · Σ = 0.
í♥ Σ é t③♦ ♣r s♠♣r ♥♦tçã♦ ♦ ♦♥tú♦ q s s ♣ós ♥r③
çã♦ s qçõs ♣♦r ♠♦ t♦r ♣rtrçõs st t♦r s r♥③s
♠r♦só♣s ♦ ♣s♠ pΣ ρΣ s ♦♠♣♦♥♥ts t♦rs EΣ Σ Σ Σ qΣ sã♦ ♦♥
srs ♦♠♦ s♥♦ ♦♠♣♦sts ♣♦r ♠ ♣rt st♦♥ár ♣♦r ♠ ♣q♥ ♣rtrçã♦
♠ ♠ó♦ ♣♥♥t ♦ t♠♣♦ ♦r♠ q
XΣ = XΣ(r, t) = X(r) + X(r)−iωt,|X||X| ≪ 1,
♦♥ XΣ r♣rs♥t qqr r♥③ ♠r♦só♣ ♦ ♠ ss ♦♠♣♦♥♥ts t♦rs
♦ ♣s♠ ♦t♠♦s t♠é♠ ♦ sí♠♦♦ ˜ ♣r ♥r s q♥ts ♣rtrrs
str♥♠♦s ♦ st♦ st sçã♦ ♦ s♦ ♣s♠s ♦♠ β = O(ε2) ♦♠ ♦
qír♦ ssô♥ |❱|2 ≪ c2s ♦r♠ q ♣rtrçõs ♦ ♠♣♦ ♠♥ét♦ = O(β)
♣♦♠ sr s♣r③s ♥ ♥ás ♣r♠r♦s r♠ô♥♦s m = ±1 rr♥ts ♦ ♥ú♠r♦
♣♦♦ ♥♦ ss♠ ♣♥s ♦ ♣♦t♥ tr♦stát♦ é ♦♥sr♦ ♠ ♥♦ss ♥ás ♠♦♦s
①s rqê♥s ♦ s EΣ = −∇Φ−∇Φ
qír♦ ♦♠ r♦tçã♦
♦♥sr♥♦ ♦r♠ vE ∼ vTi ♦r♠ q ♦ t♦ r ♠♥ét
♣♦♠ sr s♣r③♦s ♥ ♠ ♦ qír♦ é srt♦ ♣s qçõs
❱× = −∇Φ,
❱ ·∇ρ+ ρ∇ ·❱ = 0,
❱ ·∇p+ γp∇ ·❱+ (γ − 1)∇ · q = 0,
ρ❱ ·∇❱+∇p− × = 0,
♦♥
q =γ
γ − 1
p×∇T
eB2,
é ♣rt ♦♠♥♥t ♦ ①♦ ♦r ♥♦ s♦ ♥ã♦ ♦s♦♥ q sr ♦♥sr ♥♦
st♦ ❩ ♥ ♥stçã♦ t♦s s♦ ♣♦ r♥t t♠♣rtr
ss♠♠♦s q ♦ ♠♣♦ ♠♥ét♦ é s♠étr♦ ♠ rçã♦ ♦ â♥♦ t♦r♦ φ ♦r♠
q
= F∇φ+∇φ×∇Ψ, ∇Ψ ·∇φ = 0,
=∇×
µ0=
(R2∆∗Ψ∇φ−∇φ×∇F )
µ0, ∆∗Ψ = ∇ · (∇Ψ/R2),
♦♥♦r♠ ♠♦str♦ ♠
s qs s q
❱ =κ(Ψ)
ρ− Ω(Ψ)R2
∇φ, Ω =dΦ
dΨ,
♦♥ κ é ♠ ♥çã♦ ①♦ s♦♥ ♣♦ré♠ q stá rt♠♥t r♦♥ à r♦tçã♦
♣♦♦ qír♦ ♦♠ ssttçã♦ ❱ ♠ s q
κ
ρ ·∇p+ γp ·∇
(
κ
ρ
)
+ (γ − 1)∇ · q = 0
♣♦rt♥t♦ ♦srs q ♥ s♥ r♦tçã♦ ♣♦♦ κ = 0 ♦ ①♦ ♦r t♠
rê♥ ♥ ♦ s ♠ ♠é ♥ã♦ á tr♦ ♦r ♥tr s s♣rís ♠♥éts
rçã♦ ♥tr ♣rssã♦ ♥s t♠♣rtr p = ρT/mi ♣♦ sr ♦♥♥♥t♠♥t
①♣rss ♣r s♦ tr♦ ♦♠♦
·∇ρ
ρ− ·∇p
p+ ·∇T
T= 0.
♠ét♦♦ ér♦ ♣♦ q ♦s rst♦s ♥tr♦rs ♦s ♣ró①♠♦s ♦r♠ ♦t♦s é ♣r
s♥t♦ ♥♦ ♣ê♥ ♦ ♣r♥♣ ♦t♦ é r ♦t♥çã♦ s ①♣rssõs érs
♣r s ♦♠♣♦♥♥ts ∇φ ∇Ψ qçã♦ ♠♦♠♥t♦ s ♦♠♣♦♥♥ts ♥t♠♥t
♥s sã♦ ♦ts ♣♦ á♦ ♦ ♣r♦t♦ sr ∇φ ∇Ψ ♦♠ q ♣♦♠
sr ①♣rsss ♦♠♦
·∇[
F
(
1− µ0κ2
ρ
)
+ µ0κΩR2
]
= 0,
·∇(
κ2B2
2ρ2− Ω2R2
2
)
+ ·∇p
ρ= 0,
(
1− µ0κ2
ρ
)
∆∗Ψ+1
2
∇Ψ ·∇F 2
|∇Ψ|2 +µ0R
2
|∇Ψ|2∇Ψ ·∇p+µ0ρR
2
2×
[
∇Ψ
|∇Ψ|2 ·∇(
κ2
ρ2|∇Ψ|2R2
)
− ∇Ψ
R2·∇
(
κ2
ρ2
)
−(
Ω− κF
ρR2
)2∇Ψ ·∇R2
|∇Ψ|2]
= 0,
♦♥ ∆∗Ψ = R2∇ · (∇Ψ/R2) é ♦ ♦♣r♦r r♥♦
sr♥♦ q s ·∇f = 0 ♣r qqr ♥çã♦ sr f ♥♣♥♥t φ ♠♣
♠ f = f(Ψ) ♦♥s q s♦♠♥t ♥ sê♥ r♦tçã♦ ♣♦♦ κ = 0 ♦r♦ ♦♠
♥tã♦ F = F (Ψ) ♥ ♥st ♠s♠♦ ♦♥t①t♦ s ♦♥srr♠♦s ♦ s♦ r♦tçã♦
①s♠♥t t♦r♦ ♦r♦ ♦♠ · ∇p = ρΩ · ∇R2/2 ♥trt♥t♦ ♦♠♦
·∇R2 6= 0 ♦♥s q p ♥ã♦ ♣♦ sr ♠ ♥çã♦ ①♦ ♦ ♦♥trár♦ ♦ q ♦♦rr ♠
♣s♠s s♠ r♦tçã♦ ♥♦s qs p = p(Ψ)
♣ró①♠♦ ♣ss♦ é t③çã♦ t♦r ♣rtrçã♦ ♣r rs♦r s qs
♦s s♠♦s ♥♦ ♠ét♦♦ ♣rs♥t♦ ♥ ❬❪ ♥ q s r♥③s qír♦ sã♦
♦♠♣♦sts ♥ ♦r♠ Q = Q0(Ψ) +Q1(Ψ, θ) ♦♠ |Q1/Q0| ≪ 1 ♦♥ Q r♣rs♥t p ρ T
♦ F ♥♠♦s ♥tã♦ ♣♦r ♦♥♥♥ r♥③
∆Q =( ·∇Q1)/Q0
( ·∇R2)/R20
.
rqê♥ ♥r r♦tçã♦ ♣♦♦ t♦r♦ é ♣♦r
ΩP = ∇θ ·❱ =κF
ρqR2, ΩT = ∇φ ·❱ = qΩP − Ω,
♦♥ q é ♦ t♦r sr♥ç q é ♥♦ ♣♦r
q = q(Ψ) =∇φ ·∇θ · =
F
JR2, J = ∇θ · (∇φ×∇Ψ).
P♦r ♦♥♥♥ ♥s qçõs q s s♠ ♥tr♦③♠♦s s s♥ts ♥çõs
MP =qΩP0R0
cs, MT =
ΩT0R0
cs, Mt =
R0
ecs
dT0
dΨ, c2s =
γp0ρ0
,
ΩP0 =κF0
ρ0qR20
, ΩT0 = qΩP0 − Ω, B0 =µ0ρ0c
2sR
20
F 20
∼ β.
q sã♦ rts ♦s ♥ú♠r♦s ♣♦♦ t♦r♦ tér♠♦ ♦ ♣râ♠tr♦ β
♣rtr ♦ á♦ rê♥ ♦ ①♦ ♦r
∇ · q = Mt
[
1−∆F +∆p − (1 +Rρ −RF +RR2)∆T
(γ − 1)F0/R0
]
·∇R2
R20
ρ0c3s,
RF =T0
F0
dF0/dΨ
dT0/dΨ, Rρ =
T0
ρ0
dρ0/dΨ
dT0/dΨ, RR2 =
T0
R20
∇Ψ ·∇R2
∇Ψ ·∇T0,
q é t♦ ♠ ♣♦♠♦s rsrr ♦ sst♠ s♥t ♦r♠
∆ρ −∆p +∆T = 0
(1− B0M2P )∆F + B0M
2P∆ρ = B0MP (MT −MP ),
M2P∆F −M2
P∆ρ +∆p
γ=
M2T
2−MPMT +M2
P ,
Mt∆F +MP∆ρ − (MP /γ +Mt)∆p + (1 +Rρ −RF +RR2)Mt∆T = Mt.
r♥t qçã♦ rr♥♦ ♠♦ ♣♦♠♦s rsr ♦♠♦
∆∗Ψ+
[B0R2
γR20
(1 +Rρ) +RF
]
F 20
T0
dT0
dΨ+ T (κ,Ω,Ψ),
♦♥ T = O(B20F0/LT ) é ♦ tr♠♦ ♣r♦♥♥t r♦tçã♦ qír♦ ♦ q ♣♦ sr ♣r♦①
♠♦ ♣♦r
T ≈ −B0M2P∆
∗Ψ+
[
∇Ψ ·∇p1∇Ψ ·∇p0
B0R2
γR20
(1 +Rρ)+
(
∇Ψ ·∇F1
∇Ψ ·∇F0− F1
F0
)
RF +B0
2
( |∇Ψ|2F 20
M2PRΨ2 −M2
T
)]
F 20
T0
dT0
dΨ,
RΨ2 =T0
|∇Ψ|4∇Ψ ·∇(|∇Ψ|2)
dT0/dΨ∼ T0
|∇Ψ|2∂|∇Ψ|2/∂ΨdT0/dΨ
.
♠♥♦s q ♦♦rr ♠ ♦rt ③♠♥t♦ r ♦ ♠♣♦ ♠♥ét♦ ♣♦♦ ♦ s s
∂2Ψ/∂r2 ≫ (∂Ψ/∂r)2 é ♦♥③♥t ♦♠ r t♦♠s ♠ r♠s ① ♣rssã♦
β ∼ ε2 st♠r s r♥③s ♣rs♥ts ♠ s♥t ♦r♠
B0 ∼ ε2, ∆∗Ψ ∼ B0F 20
T0
dT0
dΨ∼
√B0F0
LT,
1
LT=
1
T0
∂T0
∂r
♦ q ♠♣ ♠ RF ∼ B0 Rρ ≈ η−1 ∼ 1 ♦♥ η = Lρ/LT Lρ = ρ−10 ∂ρ0/∂r ♦♠ rçã♦
♦ tr♠♦ RR2 ♥♦ ♠ ♣r st♠t s ♦r♠ r♥③ ♦♥sr♠♦s
t♦♠s sçã♦ rr ♦♠♦ ♦ ♣♦r ①♠♣♦ ♦r♠ q ∂Ψ/∂θ ≪ r∂Ψ/∂r
♥ ♥st ♦♥t①t♦ q♥♦ LT ≤ r ♦ s q♥♦ á ♠ ♦♥srá r♥t r
t♠♣rtr ♥♦ t♦♠ ♦ q é t♦t♠♥t ríst♦ ♥ ♣rát s q
RR2 =T0
R20
∂R2/∂Ψ
dT0/dΨ≈ 2
LT
R0cos θ ∼ ε ≪ 1.
♥t♦ ♠♦r ♦r r♥t t♠♣rtr ♠s stá s t♦r♥ ♣r♦①♠çã♦ ♦
q ③ s♠♣ ♦ s♥♦♠♥t♦ ♠ ♠♦♦ ♥ít♦
♦tçã♦ t♦r♦
Pr ♦ s♦ ♣rtr r♦tçã♦ ♣r♠♥t t♦r♦ MP = 0 ♦♥sr♥♦ s ♣r♦①
♠çõs ♠♥♦♥s ♠ ♦ sst♠ ♦♠♣♦st♦ ♣s qçõs ♣rs♥t
s♥t s♦çã♦
∆F = 0, ∆p =γ
2M2
T , ∆ρ = ∆p −∆T
♦♠ rçã♦ ♥ás q é ♥ssár♦ tr ♠ ♠♥t s qs
q ♣r♠t♠ ♦♥r q ∇ · q = 0 q♥♦ ♥ã♦ á r♦tçã♦ ♣♦♦ κ = 0 P♦ré♠
♦r♦ ♦♠ st♦ só ♦♦rr ♠ ♦s s♦s ∆T = (1 + ∆p)/(1 + Rρ) ♦ Mt = 0
♣r♠r♦ s♦ ♠♣r q ♥♦ ♠t s♠ r♦tçã♦ qír♦ MT → 0 t♥t♦ t♠♣r
tr q♥t♦ ♥s qír♦ ♣♥r♠ ♦rt♠♥t ♦♠ ♣♦sçã♦ ♣♦♦ ♣♦s
∆ρ = −∆T = −(1 +Rρ)−1 ∼ 1 ♠ s♦r♦ ♦♠ ♦ qír♦ s♠ r♦tçã♦ ♥♦ q∆p = ∆ρ = ∆T = 0
❬❪ s♥♦ s♦ ♥trtt♥t♦ ♠♣ q ♣♦ ♠♥♦s ♠ ♣r♠r ♦r♠ t♠♣rtr é
♦♥st♥t ♠ s♣rís ♠♥éts r♥ts ♦r♦ ♦♠ ♦ q t♠é♠ ♥ã♦ ♦♦rr
♠ t♦♠s t♠♣rtr é ♠á①♠ ♥♦ ♥tr♦ ♥ ♥ ♦r ❯♠ ♦r♠ ♦♥r
st ♥♦♥sstê♥ é ss♠r q Mt ∝ MP ♦ ♦r♠ q♥t q r♦tçã♦ ♣♦♦
qír♦ é ♠ ♦♥sqê♥ rt ①stê♥ r♥ts rs t♠♣rtr
P♦rt♥t♦ ♥st ♠♦♦ ♦♥í♠♦s q ♥ã♦ ①stê♥ r♦tçã♦ ♣♦♦ só é ♣♦ssí ♦
♠♥t s ss♦ ♦♦rrr ♠ tr♠♥ ♣♦sçã♦ r á ♠ ♥çã♦ r q ♥st
♣♦sçã♦ ♦♦rr ♠ ♣r ♣♥♦ ♥♦ ♣r t♠♣rtr
s s♥ts r♠s ♣rtr ♥trss ♣♦♠ sr ♦♥sr♦s ♥st s♦
• át♦ st s♦ q♥t S = pρ−γ q r♣rs♥t ♥tr♦♣ ♦ sst♠
é ♠ ♥çã♦ ①♦ ♦r♠ q rçã♦ ∆p − γ∆(S)ρ = 0 s r s♦çã♦
♦rrs♣♦♥♥t st r♠ é
∆p =γ
2M2
T , ∆(S)ρ =
1
2M2
T , ∆(S)T = (γ − 1)M2
T .
• s♦tér♠♦ rtr③♦ ♣♦r sr ♦ r♠ ♠s ríst♦ ♦♦rr q♥♦ ∆(T )T = 0 ♦
q ♠♣ ♥ s♦çã♦
∆(T )ρ = ∆p.
• s♦♠étr♦ st r♠ rtr③♦ ♣♦r ∆(V )ρ = 0 ♠♦r ♥ã♦ s ♦♠♠ ♠ ①
♣r♠♥t♦s t♠ rt ♠♣♦rtâ♥ ♣♦r sr ♦ ú♥♦ r♠ rtríst♦ ❩s ♥stás
♦♥♦r♠ ♦ ♠s ♥t s♦çã♦ ♦rrs♣♦♥♥t é
∆(V )T = ∆p.
♦tçã♦ ♣♦♦ t♦r♦
♦♠ rs♦çã♦ ♦ sst♠ ♦♥sr♥♦ B0 ∼ ε2 ≪ 1 Rρ ≈ 1/η M2P,T ≪ 1
♦r♠ q ∆F = O(B0M2P,T ) ♣♦ sr s♣r③♦ ♦t♠♦s s♥t s♦çã♦
∆ρ =N∆
D∆
[
1 +
(
1
N∆− γ
η
)
Mt
MP
]
,
∆p = γN∆
D∆
[
1 +
(
M2P
N∆− η + 1
η
)
Mt
MP
]
,
∆T = (γ − 1)N∆
D∆
[
1−(
1− γM2P
(γ − 1)N∆+
γ
γ − 1
)
Mt
MP
]
,
♦♥
N∆ =M2
T
2+MP (MP −MT ), D∆ = 1−M2
P − η + 1
η
Mt
MP+
γ
ηMPMt.
ss♠ ♦♠♦ ♥♦ s♦ r♦tçã♦ ①s♠♥t t♦r♦ ♥st s♦ t♠é♠ é ♦♥♥♥t
♥sr ♦s três r♠s ♣r♥♣s ♠♥♦♥♦s ♥tr♦r♠♥t
• át♦ ♦♥srs ♥st r♠ M (S)t = 0 ♦ q rst ♠
∆(S)p = γ∆(S)
ρ , ∆(S)T = (γ − 1)∆(S)
ρ , ∆(S)ρ =
N∆
D(S)∆
, D(S)∆ = 1−M2
P .
• s♦tér♠♦ s s♦çõs sã♦ ♦ts ♣ ssttçã♦ ∆T = 0 ♠
♦r♠ q ♣r MP ≥ 0
M(T )t =
(γ − 1)MPN∆
1 + γ(N∆ −M2P )
> 0.
• s♦♠étr♦ ♦r♠ ♥á♦ ♦ r♠ ♥tr♦r ♣rtr ♦♥çã♦ ∆ρ = 0 ♣r
MP ≥ 0 ♦té♠s
M(V )t =
−MPN∆
1− (γ/η)N∆< 0.
Pr ♦ t♦♠ ♦r♦ ♦♠ r♥t rtór♦ ❬❪ ♠♦str♠♦s ♥ r ♦
♣r r r♦tçã♦ qír♦ ♦t♦ ①♣r♠♥t♠♥t ♣rtr st rá♦ ♣♦♠♦s
st♠r ♦s ♦rs MP MT ♦♠ ♦ ♥tt♦ ♦tr ♠ st♠t ♣r rqê♥ ♦s
s ❲s ❩s
♥trss♥t ♦srr ♦ q ♦♦rr ♥♦ ♠t MT → 0 ♦ s ♦r♦ ♦♠ r
♣ró①♠♦ r = 0.7a st ♠t ♦srs q
M(V )t = −M3
P , M(S)t = 0, M
(T )t = (γ − 1)M3
P .
♦♥sr♥♦ ♥♠♥t t♦♠s sçã♦ rr t r③ã♦ s♣t♦ é ♣♦ssí
♥♦♥trr s r♥③s qír♦ Pr ♠ r♥③ ♥ér Q s♠étr ♠ rçã♦ φ
s ♥çã♦ ∆Q q
·∇Q = ∆QQ0 ·∇R2
R20
♣♦ sr s♥♦♦ ♦♥sr♥♦ Ψ ≈ Ψ(r) ♦ s ≈ F (r)R−1φ + (Rr)−1(dΨ/dr)θ
r rá♦ ♦ ♣r r ♦ r♦tçã♦ ♣♦♦ tr♦ t♦r♦♥ ♦♠♦ ♥çã♦ ♣♦sçã♦ r ♥♦r♠③ r/a ♥♦ t♦♠ srçã♦ st rá♦ ♦ ①trí♦ ♣t♦ ❬❪
st ♥tã♦ ssttçã♦ ♠ s♥t qçã♦ ♥trá
∂Q
∂θ= −2ε∆QQ0 sin θ +O(ε2Q)
s♦çã♦ ♣r♦①♠ tr♠♥ Q = Q(r, θ)
Q(r, θ) = Q0(r) + 2ε∆Q(r)Q0(r) cos θ.
♣rtr ♣♥ê♥ ♣♦♦ s q♥ts qír♦
♣♦♠ sr tr♠♥s ♦ s
ρ = ρ0(1 + 2ε∆ρ cos θ), p = p0(1 + 2ε∆p cos θ),
T = T0(1 + 2ε∆T cos θ) =mic
2s
γ[1 + 2ε(∆p −∆ρ) cos θ],
❱ = VP θ + VT φ, VP = ΩP r, VT = ΩTR,
VP ≈ ε
qMP cs, VT = (MT +∆V ε cos θ)cs, ∆V = MT − 2MP (1 + ∆ρ).
st♠ qçõs ♣rtrs rçã♦ s
♣rsã♦
♦♥sr♥♦ ♦r ♣rtrçõs t♠♣♦rs ♦s ♠♦♦s ♦sçã♦ ①s rqê♥s
♥♦ ♣s♠ sã♦ ♦t♦s ♣rtr rs♦çã♦ ♦ s♥t sst♠
ρ0∂v‖
∂t+∇‖p+ F‖ = 0,
∂(ρ+ R)
∂t+ ρ0∇ · = 0
∂(p+ P )
∂t+ γp0∇ · = 0
♦♥
= E + v‖, E =×∇Φ
B,
é ♦ ♣rtr ♣r♦♥♥t r E × ♦♠♣♦♥♥t ♣r s tr♠♦s
F‖ R P sã♦ s ♦♥trçõs r♦tçã♦ qír♦ ♥♦s ♦r♠ ♦♥♥♥t
♣♦r
F‖ = ρ0( : ∇❱+ ❱ : ∇) + ρ❱ : ∇❱,
∂R
∂t= ❱ ·∇ρ+ ·∇ρ0 + ρ∇ ·❱,
∂P
∂t= ❱ ·∇p+ ·∇p0 + γp∇ ·❱+ (γ − 1)∇ · q,
s ♥♦ ♣ê♥ st á♦ ♠♦s ♠ ♦♥t ♣♥s ♦s tr♠♦s ♦♠♥♥ts
♦♠ rçã♦ ♦ t♦r ε = r/R0 ≪ 1 q sã♦ ♦ ♦♥trçã♦ ♦s ♣r♠r♦s r♠♦♥♦s
Pr ♦t♥çã♦ rçã♦ s♣rsã♦ é ♥ssár t③çã♦ qçã♦ ♦ ♠♦♠♥t♦
♥r③
ρ∂
∂t+∇p− ×+ = 0, = ρ(❱ ·∇+ ·∇❱) + ρ❱ ·∇❱,
q q♥♦ ♠t♣ t♦r♠♥t ♣♦r rst ♥ ①♣rssã♦ ♥ít ♣r ♥s
♦rr♥t
=j‖
B+
ρ
B2× ∂
∂t+
B2×∇p+
B2× .
rçã♦ s♣rsã♦ é ♣r♦♥♥t ♦♥çã♦ qs♥tr ♦ ♣s♠ q ♣♦
sr ①♣rss ♣ qçã♦ ∇ · = 0 ♠t♦♦♦ ♥ít ♣rã♦ é s ♥♦ á♦
♠é t qçã♦ s♦r ♠ s♣rí ♠♥ét P♦♠♦s r D t♦♠♥♦ ♠é
♦♠ rçã♦ ♦ ♦♠
D =
∫
V dV∇ · ∫
V dV= 0, dV = (R0 + r cos θ)rdrdθdφ,
trés ♦ t♦r♠ rê♥ ss ♦t♠♦s
D =
∫
S · d∫
V dV= 0, d = (R0 + r cos θ)rdθdφr.
①♦s ③♦♥s ❩s ♠♦♦s úst♦s ♦és♦s
s
sr sr♠♦s ♦ ♠♦♦ ♠s s♠♣s ♣r ①♣♦rr ♥â♠ ás s ♦s
çõs tr♦státs ♦♥s ♦♠♦ s st ♣rt s♦♥sr♠♦s r♦tçã♦ q
ír♦ ♣♦r ♠♦t♦s át♦s ♦♠ ♥ ♥t③r ♦ ♠â♥s♠♦ ís♦ ♦r♠çã♦
♦s s ♥♠♥t t③♠♦s ssttçã♦ F‖ = P = R = 0 ♠ ♦♠♦
♣♥s ♦s ♣r♠r♦s r♠ô♥♦s s♠♣♥♠ ♠ ♣♣ r♥t ♥ ♥â♠ ás ♦s
s tr♦stát♦s ❬❪ ♦♥sr♠♦s s♦çõs ♦r♠X = Xs sin θ + Xc cos θ ♣r s ♣r
trçõs ♠s ♠ s trt♥♦ ♠ ♥ás ♥r X ∝ −iωt ♦r♠ q sst
tçã♦ ∂/∂t → −iω ♠ ♣♦ sr ♠♣r
tr♠♦∇· tê♠ s ①♣rssã♦ s♥♦ ♥♦ ♣ê♥ ♦r♦ ♦♠ s qs
♣♦ sr srt♦ ♥ ♦r♠
∇ · = −2ωE sin θ + k‖∂v‖
∂θ, ωE =
irΦ0
B0R0=
i
2
eΦ0
Tirρiωi, ωi =
vTi
R0.
tr♠♦ é ssttí♦ ♠ rst♥♦ ♥ rçã♦ ♥tr p v‖
p = iρ0c2s
(
−2ωE
ωsin θ −
k‖
ω
∂v‖
∂θ
)
,
q ♣♦r s ③ é ssttí ♠ ♦♥sq♥t♠♥t s♥t qçã♦ r♥
♣r v‖ ♠ θ é ♦t
(
1 +k2‖c
2s
ω2
∂2
∂θ2
)
v‖ = 2k‖c
2s
ω2ωE cos θ.
s♦çã♦ ♦rrs♣♦♥♥t
v‖ =2k‖c
2s
ω2 − k2‖c2s
ωE cos θ,
q♥♦ ♥sr ♠ ♥ ♦♠♣t♠♥t ♦ tr♠♦ ∇ · q ♣ós sr ssttí♦ ♠
♦♠♣t ♦ ♦♥♥t♦ s♦çõs ♦♠
∇ · = − 2ω2
ω2 − k2‖c2s
ωE sin θ,
ρ = iρ0
(
2ω
ω2 − k2‖c2s
)
ωE sin θ, p = ρc2s.
♦ ♥sr♠♦s s qçõs é ♣♦ssí ①trr s ♦♥sõs ♠
♣♦rt♥ts Pr♠r♠♥t s♦çã♦ ω = 0 ♥ã♦ é ♠ s♦çã♦ tr ♣♦s ♣r st s♦
v‖ = −2ωE cos θ/k‖ 6= 0 ♦♥♦r♠ ①♣♦ ♠s ♥t st s♦çã♦ ♦rrs♣♦♥ ♦s ①♦s
③♦♥s ♥♦♠♣rss ♦ ♣s♠ ♦r♦ ♦♠ ♠ ♦♠♦ sê♥ ♦r
r♥ts ♠♥éts ♣♦s p = 0 é ♠ rtríst ♥♠♥t sts ①♦s st♦♥ár♦s
s♥ rtríst ♠♣♦rt♥t é ♦♠ rçã♦ ♦ t♦r sr♥ç ♦t q ♣r q → ∞
v‖ → 0 ♥♦ s♦ s ω 6= 0 v‖ → ∞ ♣r ❩s ♣♦s k‖ = 1/qR0
♥trss♥t ♦srr t♠é♠ ♦ q ♦♦rr s ωE = 0 ♦ s ♥ sê♥ ♦ ♠♣♦
étr♦ ♦r♦ ♦♠ qçã♦ rê♥ ♦ é ♣r♦♣♦r♦♥ à rçã♦
♦ ♣r ♦♠ rçã♦ ♦ â♥♦ ♣♦♦ θ ♦ q ♥③ ♠ ♣rtrçã♦ ♥
♣rssã♦ ♦r♦ ♦♠ ♦t♥♦ ♦ ♠s♠♦ ♣r♦♠♥t♦ ♦té♠s ♠ qçã♦ s♠r
à q
(
1−k2‖c
2s
ω2
)
v‖ = 0
q ♣♦ss s s♦çõs ♣r♠r tr v‖ = ρ = p = 0 ♣♦rt♥t♦ ♥ã♦ ♠♣♦rt♥t
s♥ ω2 = k2‖c2s q ♦rrs♣♦♥ ♦♥s ústs ♦t q s♥ s♦çã♦ ♥ã♦ ♣r♠t
tr♠♥çã♦ s ♣rtrçõs v‖ ρ p ♥st ♠♦♦ s♠♣s
♦rr♥t ♣rtr é ♦♠♣♦st ♣♦r s ♣rts ♥♠♥ts ♣r sts ♠♦♦s ♦♥
trçã♦ ♥r ♠♥ét s ①♣rssõs ♥íts ♣r ss ♦♠♣♦♥♥ts rs
sã♦
jr =
(
ρ
B2× ∂
∂t
)
· r ≈ iR0
B0ρ0ωωE ,
j♣r =
(
B2×∇p
)
· r ≈−1
εB0R0
∂p
∂θ(1 + ε cos θ),
♦t q ♠ ♠♥t♠♦s ♦ tr♠♦ ε cos θ q é ♣r♦♥♥t B ≈ B0(1− ε cos θ)
♣♦s st tr♠♦ é r♥t ♥♦ á♦ ♠é ♠ ♠ s♣rí ♠♥ét
♣rtr ♦ s♥♦♠♥t♦ ♠♦str♦ ♥♦ ♣ê♥ rst rçã♦
s♣rsã♦
D = −i2R0ρ0rB0
(
1 +ips
ρ0ωωER20
)
ωωE = KD(0) = 0,
♦♥ K = −2iR0ρ0ωE/rB0 é ♠ tr♠♦ ♠♣♦rt♥t ♥♦ st♦ t♦♠♦♦s ♦ ♦♥tí♥♦
qçã♦
D(0) = ω
[ω2 − (2c2s/R20 + k2‖c
2s)
ω2 − k2‖c2s
]
= 0,
♦r♥ s s♦çõs ♣r srqê♥s ❩s s
ω❩ = 0, ω2 =
(
2 +1
q2
)
c2sR2
0
.
r ♣r ❩s ♦♠♦ ♥ã♦ ♦r♠ ♦♥sr♦s tr♠♦s ♦r♠ s♣r♦r ♠ ♣r♥í♣♦
s♦çã♦ é ♠♦r srt ♣♦r ω❩ ≈ 0
♦t q ♠ ♦r♠ ♦♠♥♥t á t♠é♠ ♠ ♦♠♣♦♥♥t ♣♦♦ ♦rr♥t ♠
♥ét ①♣rssã♦ é
j♣θ =irpB0
.
❯t③♥♦ ♦♥sr♥♦ r ≫ r−1 ♦t♠♦s ♠ rçã♦ s♣rsã♦ ♦♠♦ ♦r♠
tr♥t à q
∇ · ≈ ir jr − 2j♣θsin θ
R0+ k‖
∂j‖
∂θ= 0,
q q♥♦ s♥♦ r♠♥t rst ♠
−ρ0R0rωE
B0ω
(
1− 2c2s/R20
ω2 − k2‖c2s
)
− ρ0R0rωE
B0
2ωc2s/R20
ω2 − k2‖c2s
cos(2θ) + k‖∂j‖
∂θ= 0.
♦♠♦ ♣r qqr θ qçã♦ sr stst ♦ tr♠♦ ♦♥t♦ ♥♦ ♣r♠r♦
♣rê♥ts ss qçã♦ s ♥r rst♥♦ ss♠ ♥s s♦çõs ♠♦strs ♥ qçã♦
❯♠ ♥t♠ ♦ s♦ é ♦t♥çã♦ ♦rr♥t ♣r
j()‖ =
√
2q2 + 1
4
ρ0R0
B0rωE sin(2θ), j
❩‖ = 0,
q s ♠♦str ♣♥♥t s♥♦s r♠ô♥♦s r♣rs♥t♦s ♣♦ tr♠♦ sin(2θ) ♦t
q ♣r♥♣♠♥t ♥♦ ♠t q ≫ 1 ♦♥trçã♦ ♦rr♥t ♣r j‖ ∝ q é s♥t
st♥♦ ♠ ♣r♥í♣♦ ♦♥srr t♦s tr♦♠♥ét♦s ♣♦s j‖ = ·∇× é♠ ss♦
♠ ♠t♦s ①♣r♠♥t♦s s sã♦ tt♦s trés ♥ás s♥s r♠ô♥s
♦r♠ q ♦rr♥t ♣r ♣rtr s♠♣♥ ♠ ♣♣ ♠♣♦rt♥t ♥st t♣♦
♦sçã♦
sr ♠ srçã♦ s♠♣ ♦ ♠♥s♠♦ ís♦ ♥♦♦ ♥s ♦sçõs ♣rs♥ts
♥♦s s é ♣rs♥t Pr s♠♣r s ①♣rssõs ♦ r♦í♥♦ ó♦ st ♠♥s♠♦
♦♥sr♠♦s ♦ ♠t q → ∞ ♦ s ωGAM =√2cs/R0 ♣♦♠♦s q ♥♠♥t ♠ t = 0
①st ♠ ♠♣♦ étr♦ ♠á①♠♦ q é ♦r♠ E = ωEB0R0r ♦♥ ωE = |ωE | cos(ωt)
|ωE | =1
2
e|Φ0|Ti
rρiωi, Φ0 = Φ0(r, t),
♦♥sr♠♦s rρi > 0 ♣♦r s♠♣ s ♣rtís ♦ ♣s♠ ♥♥s ♣♦r st
♠♣♦ étr♦ ♠ ♦♠♦ ♣♦ ♠♣♦ ♠♥ét♦ t♦r♦ qír♦ B ≈ B0(1 − ε cos θ)
s♦r♠ ♠ ♠♦♠♥t♦ r t♦ t♣♦ E × ♦ q ♣r♦③ ♠ ①♦ ♣♦♦ ♦♠♣rssí
q é ♦r♠
E = |ωE |R0(1 + ε cos θ) cos(ωt)θ, v‖ ≈ 0,
♦ s ♥t♥s r♥t ♥♦s ♦s ♠♣♦ ♦rt ♠♣♦ r♦
♦♥♦r♠ str r ♠ ♦rrê♥ st r♥ç ♥t♥s ♦ ♣s♠ é
♦♠♣r♠♦ ♥ r③ã♦
∇ · = −2|ωE | sin θ cos(ωt),
♠♣ q rρi < 0 ♣♦s ♣♥ê♥ r Φ ♦♥sq♥t♠♥t s rr sã♦ s♦♥s ♠ ♣r♥í♣♦
♦
♦ q ♦s♦♥ ♠ ♣rtrçã♦ ♥ ♥s ♦♥sq♥t♠♥t ♥ ♣rssã♦
p =√2|ωE |ρ0csR0 sin θ sin(ωt),
♦♠ ♦ ♠♦♠♥t♦ r E × ♦ ♣s♠ sr ♠ ♦rr♥t ♥r q é r
♣r♦①♠♠♥t ♦♥st♥ q t♥ ♥r ♦ ♠♣♦ étr♦ ♥ ♣♦ tr♥s♣♦rt
r ♣♦st ♣r ♦r s♣rí ♠♥ét rrê♥ ♥trt♥t♦ ♠ ♦rr♥ ♦
r♥t ♣♦♦ ♣rssã♦ s ♣ ♣rtrçã♦ st sr t♠é♠ ♠ ♦rr♥t
♠♥ét q ♠ tr♠♥s ♣♦sçõs s♣r ♦rt♠♥t ♣r♠r t♠♣♦ ♠ q
é ♠á①♠ ♠♣t ♦rr♥t r t♦t ①♣rssã♦ ♥ít s ♦rr♥ts ♥r
♠♥ét sã♦ ♠♦str♦s rs♣t♠♥t ♥ r ♥ qçã♦ ①♦
jr =√2ρ0csB0
|ωE | sin(ωt), j♣r = −jr
(
1
2+
1
εcos θ +
1
2cos(2θ)
)
.
♠ ♠é ♥st ♠♦♠♥t♦ é ♠á①♠♦ ♦ tr♥s♣♦rt rs ♣♦sts ♣r ♦r s♣rí
♠♥ét ♠ r♥ ♥ r ♦ q ♥ ♦ ♠♣♦ étr♦ r ♦♥sq♥t♠♥t
♦ r E × ♥trt♥t♦ ♦ ♥ér í♦♥s à ♦rr♥t ♠♥t
♥ ♣rs♥ts ♦ ♠♣♦ étr♦ ♥rt s s♥t♦ ♠ t = π/ωGAM ♦ r é
♠①♠ ♥♦ s♥t♦ ♥t♣♦♦ ♦♥♦r♠ str r ♠ t = 3π/2ωGAM ♦ ♠♣♦
étr♦ é ♥♦ ♥♦♠♥t ♦rr♥t é ♠á①♠ ♣♦ré♠ ♥♦ s♥t♦ ♦rá ♦ tr♥s♣♦rt
r ♣♦st ♣r s♣rí ♠♥ét ♠ qstã♦ ♦♥♦r♠ r ♥♠♥t
♠ t = 2π/ωGAM ♥â♠ srt ♠ s r♣t ❯♠ ♥stçã♦ ①♣r♠♥t t♥t♦
♦ ♦r ♥s ♣rtr ♦♠♦ s ♣♦sçã♦ ♣♦♦ ♠á①♠♦ ♦r s♦t♦ é
♣rs♥t ♣♦r rä♠r♥ t ❬❪
♦ s♦ ♦s ❩s ♥â♠ é ♦♥sr♠♥t ♠s s♠♣s ♦ s ♦♠♣♦rtr ♦r♠
♦♠♣rssí ♦ ♠ ①♦ rt♦r♥♦ ♥ rçã♦ ♣r
v‖ = −2qωER0 cos θ, ωE = |ωE |
♦ ♣s♠ ♥ã♦ ♣r♠t ♣rtrçõs ♥s ♣rssã♦ ♠ ♦♥sqê♥ ♣♥s ♠ ①♦
st♦♥ár♦ ♣♦♦ ♦tr♦ t♦r♦ ♥♦r♠♠♥t ♠♣t ♠ ♠♦r q ♦ ♣r♠r♦
♣♦♠ ♦①str ♦r♠♠♥t ♦♠♣♦♥♥t ♣♦♦ sts ①♦s ♣♦ss♠ s♠♥t♦
r ♥rt♥♦ s♥t♦ ♦♠ ♣♦sçã♦ r ♠ ♠ ♥tr♦ s♣ ♦rrs♣♦♥♥t ♦
♦♠♣r♠♥t♦ ♦♥ r st s♠♥t♦ ♣r♠t ♦ ♦♥tr♦ trê♥ s ♣♦r
♦♥s r ❬❪
(HFS)
(LFS)
(HFS)
(LFS)
(HFS)
(LFS)
(HFS)
(LFS)
R0
r
θ
vE = E×BB2
∇ · vE = −2vE · κ ∝ sin θ cos(ωGAM t)
p ∝∫
dt∇ · vE
Er ∝ cos(ωGAM t)
BTBT
κ
κ = b · ∇b
Superfıcies magneticas
a) Instante inicial t = 0
Er > 0 → max.
vE > 0
jr = 0
c) Instante t = π/ωGAM
Er < 0 → min.
vE < 0
jr = 0
BTBT
κ
κ
BTBT
jprjpr
= 0
p max
p min
b) Instante t = π/2ωGAM
Er = 0
∂Er
∂t< 0
vE = 0
|jr| → max.
BTBT
jprjpr= 0
d) Instante t = 3π/2ωGAM
Er = 0
∂Er
∂t> 0
vE = 0
|jr| → max.
p min
p max
r ♥â♠ ♠♦♦s úst♦s ♦és♦s s ♠ t♦♠s
t♦ r♦tçã♦ ♥♦s s ❩s
P♦ t♦ ♦ sst♠ sr ♥r ♣♦♠♦s srr s q♥ts ♣rtrs
♦♠♦ ♦♠♥çõs s ♦♥trçõs ♥♠♥ts t♦r♦ ♣♦♦ P ♦r♦
♦♠ ♦r♠
X = X(0) + X(T ) + X(P ),
♦♥ X(0) é s♦çã♦ ♦t q♥♦ MP = MT = 0 X(T ) é ♦♥trçã♦ t♦r♦ q♥♦ s
♦♥sr ♣♥s r♦tçã♦ t♦r♦ ♦ X(P ) é ♦♥trçã♦ ♣♦♦ sst♠♦s ♥♦ ♥t♥t♦
q q♥♦ ♦s ♦s t♣♦s r♦tçã♦ sã♦ ♦♥sr♦s ♥♦s tr♠♦s ∆ρ ∆p ♦♥t♦s ♠ X(T )
X(P ) é ♥ssár♦ ♦♥srr MP 6= 0
♣rt rst♥t st sçã♦ ss♠ ♦♠♦ ♥♦ ♣ê♥ ♦♥sr♠♦s ♥♦r♠③çã♦
Ω =ω
k‖cs, ΩE =
ωE
k‖cs,
♥st ♣ê♥ ♦t♠♦s rçã♦ s♣rsã♦ q é ♠♦str sr
2ΩE
Ω2 − 1(D() +D() +D(P)) = 0,
♦♥
D() =Ω
2q2(−Ω2 + 2q2 + 1),
D() =M2
T
Ω
[(
1 +1
2
∆V
MT+
1
γ
∆p
M2T
+1
2∆ρ
)
Ω2 +1
2
(
∆p
γ−∆ρ
)]
,
D(P) =N p
+1(P)
D+1(P)
− N p−1
(P)
D−1(P)
+MT
[N v+1
(P)
D+1(P)
− N v−1
(P)
D−1(P)
+MT
2
(N ρ+1
(P)
D+1(P)
− N ρ−1
(P)
D−1(P)
)]
,
D±1(P) ≈ (MP ∓ Ω)(Ω + 1∓MP )(Ω− 1∓MP ) + [2γ(Ω∓MP )
2 − 1]Mt.
♥ts ♣r♦ssr ♦♠ ♦ s♥♦♠♥t♦ ér♦ ♦s ①t♥s♦s t
s sã♦ ♣rs♥t♦s ♥♦ ♣ê♥ é ♦♥♥♥t r s s♥rs ♠ D(P) ♣r
ss♦ ♦♥srs q Mt ∼ M3P ♦r♠ t♦r♥r ♣♦ssí ♣♦r ♠♦ ♣r♦①♠çõs rs♦r
♥t♠♥t D±1(P) = 0 s ♦rs s s♥rs ♦♥sr♥♦ MP ≥ 0 sã♦ ♠♦str♦s
r♠♥t ♥ r
♥♠♥t ♣rs♥t♠♦s sr rçã♦ ♥ ♣r♦♥♥t ♦ s♥♦♠♥t♦ ér♦
r ♥rs ♦ ♥♦♠♥♦r D(P) ♣r MP ≥ 0
D() = Ω
(
− Ω2
2q2+ 1 +
1
2q2
)
,
D() =M2
T
Ω
[
2(1 +M2P )
(
1− MP
MT+
1
2
M2P
M2T
)
+
(
1
4− MP
MT
)
M2T+
(
1
2− MP
MT+
M2P
M2T
)
Mt
MP
]
Ω2 − 1
2
Mt
MP
D(P) =MP
(Ω2 − 1)5
4∑
k=0
K2k+1Ω2k+1,
♦♥ ♦s ♦♥ts K2k+1 = K2k+1(MP ,MT ,Mt) sã♦ ♠♦str♦s ♥♦ ♣ê♥ rs♦çã♦
♥ít t♥♦ ♠ st s ①♣rssõs ♠s é t t ♠♥t s♥t
♣r♦①♠çã♦ ss♠♣tót
• ♠♦ úst♦ ♦és♦ Ω ≫ 1
• ♠♦ s♦♥♦r♦ í♦♥ ❲ Ω ∼ 1
• ①♦s ③♦♥s❩ Ω ∼ MP ≪ 1
♦ ♣r♠r♦ ♥♦ trr♦ s♦ ♦ ♣♦♥ô♠♦ tê♠ s r r③♦ q♥♦ ♦ s♥♦♠♦s
♠ ♠ sér ♣♦tê♥ ♠ Ω ♦♥srr♠♦s ♣♥s ♦s três tr♠♦s ♠s ♦♠♥♥ts
s♥♦ s♦ ♣♦ sr ♥s♦ ♦ ss♠r♠♦s s♦çõs ♦r♠ Ω2 ≈ 1 +O(M2P ) ♠♦♦
q ♦ ♥♦♠♥♦r t♦r♥s ♣q♥♦ ♣♦rt♥t♦ ♣♦♠♦s ♦♥srr D(P) ≈ 0
♦t♥♦ ss♠ s♦çã♦ ♥♦ r♠♦ s♦♥♦r♦
sr ♥s♠♦s s♣r♠♥t ♦ s♦ ♦♠ r♦tçã♦ ♣♥s t♦r♦ ♦ s♦ ♠ q
r♦tçã♦ s s♥♦ ♠ ♠s s rçõs
t♦ r♦tçã♦ t♦r♦
♦♠ sssttçã♦ MP = 0 ♠ ♦t♠♦s ♣♥s s s♦çõs
ω2GAM
c2s/R20
= 2 +1
q2+ 4M2
T +
(
2q2∆ρ
M2T
+1
2
)
M4T
2q2 + 1,
ω2ZF
c2s/R20
=
(
∆ρ −∆p
γ
)
M2T
2q2 + 1, ∆p = γ
M2T
2,
q ♦rrs♣♦♥♠ rs♣t♠♥t ❩ t ♦s ♦rs s rqê♥s
rts sts ♠♦♦s sã♦ ♠♦strs ♥♦s três r♠s ♠s ♠♣♦rt♥ts át♦ s♦tér♠♦
s♦♠étr♦
t♦ r♦tçã♦ ♣♦♦ t♦r♦
sr ♦♥sr♠♦s ♦s r♠s át♦ s♦tér♠♦ ♥ ♥ás ♦ t♦ r♦tçã♦
♣♦♦ t♦r♦ ♥♦s s ❩s ♦ r♠♦ ♦és♦ ♥♦ r♠♦ úst♦ í♦♥s s ♦rr
s♣♦♥♥ts rqê♥s sã♦ ♦♠♥s ♥sts ♦s r♠s ♣r q ≫ 1 ♣♦♠ sr ♣r♦①♠s
♣♦r
ω2GAM
c2s/R20
≈ 2 +1
q2+M2
P + (MP − 2MT )2,
♦♠♣rçã♦ ♥tr ♦s qr♦s s rqê♥s ♥♦r♠③s ♣♦r cs/R0♦s s ♦s ❩s ♥♦s r♠s s♦♠étr♦ át♦ s♦tér♠♦
♠ R20ω
2GAM/c2s ❩ R2
0ω2ZF/c
2s
s♦♠étr♦ 2 +1
q2+ 4M2
T +M4
T
4q2 + 2− M4
T
4q2 + 2
át♦ 2 +1
q2+ 4M2
T +M4
T
20
s♦tér♠♦ 2 +1
q2+ 4M2
T + (2γq2 + 1)M4
T
4q2 + 2(γ − 1)
M4T
4q2 + 2
ω2SW
c2s/R20
≈ 1
q2+
(3MP − 4MT )
q2MP .
♠ s trt♥♦ ❩s ♥♦ r♠ át♦ rqê♥ ♥ã♦ s tr ♦ ♦♥trár♦ ♦ q
♦♦rr ♥♦ r♠ s♦tér♠♦ ♥♦ q ♦ ♦ t♦ ♦ ①♦ ♦r q
ω2ZF
c2s/R20
≈ M2P
q2.
①♣rssã♦ é ♣r♦①♠ á ♣♥s ♥♦ ♠t q ≫ 1 M2P ≪ 1 M4
T ≪ M2P
st♦r♠♥t ♦s rst♦s ♠♦str♦s ♠ ♥♦s ♦ ♦ t♦
r♦tçã♦ ♣♦♦ ♦r♠ ♦ts ♣r♠r♠♥t ♣♦r ❱ s♦♥s t ❬❪ ♦♥sr♥♦ ♦
r♠ át♦ ♣rtr ♦ st♦ st tr♦ ♦♥sr♥♦ ♦ t♦ ①♦ ♦r
♥♦ r♠ s♦tér♠♦ ♦t♠♦s ♦rrçã♦ ♦s ①♦s ③♦♥s ❬❪ á ♥ ♦ r♠ s♦♠étr♦
sr ♥s♦ ♦ q ♣rt♥♠♦s ③r ♠ ♠ tr♦ tr♦
sssã♦ s♦r ♦ í♥ át♦
♥ts ♦ ♥í♦ ♣ró①♠ sçã♦ é ♦♥♥♥t ①♣rssr ωGAM ♠ tr♠♦s ♦
tér♠ í♦♥s st ♦♥♥ê♥ s ♦ ♥tt♦ ♦♠♣rr t♦r ♠ í♦ ♦♠
t♦r ♦s ♦s ♦s rst♦s ♦♥♠ ♦♠ t♦r ♥ét q♥♦ ♦ ♠♦rt♠♥t♦
♥ ♥ã♦ é ♦ ♠ ♦♥t st ♦r♠ ♦♥♦r♠ rqê♥ ♦s s ♣♦
sr ①♣rss ♦♠♦
ω2GAM =
(
2 +1
q2
)
γp0ρ0
= γ
(
1 +1
2q2
)(
1 +Te
Ti
)
v2Ti
R20
,
♦♥ s rçõs p0 ≈ n0(Ti + Te) ρ0 ≈ n0mi ♦r♠ t③s t♦r ♠ ♦ ♥ã♦
♦♥sr r♥ç ♥tr ♦s í♥s át♦s γ í♦♥s étr♦♥ t♦ ♦♥♦r♠
t♦r ♥ét s♣♦sçã♦ ♠s rst ♣r ♣s♠s t♦♠ é γi = 5/3 ≈ 1, 7 γe = 1
st sr♣â♥ ♦rs s à r♥ r♥ç ♥tr ♠ss í♦♥s étr♦♥s
♦r♠ q ♣♦r ♣rs♥tr♠ ♥ér ♠t♦ ♠♥♦r ♦s étr♦♥s sã♦ ♣③s r♣♠♥t
♥trr♠ ♠ qír♦ tér♠♦ ♥tr s st ♦r♠ ♣r t♦s ♦♠♣rçã♦ ♥tr s s
t♦rs é ♦♥♥♥t t③r ssttçã♦
γ → γ(♦rrt♦) =γi + γeTe/Ti
1 + Te/Ti,
♦♥ ♣r Te = Ti γ(♦rrt♦) ≈ 1, 3 < 5/3 ≈ 1, 7 r♣rs♥t♥♦ ♠ rr♦ ♣r♦①♠♠♥t
25%
♣ró①♠ sçã♦ é♠ rr♠♦s ♠ rçã♦ ♠s ♣rs ♣r rqê♥ ♦s s
♦♠ rçã♦ ♦ í♥ át♦ ♦♥sr♠♦s t♠é♠ ♦ t♦ ♥s♦tr♦♣ ♣rssã♦ ♦
s p⊥ 6= p‖ st t♦ rst ♠ ♠ ♠♥t♦ ♦ í♥ át♦ t♦ ♣r í♦♥s
γi = 5/3 → γ(t♦)i = 7/4 Pr s♠♣r ♦ ♠♦♦ ♥♦s rstr♥♠♦s ♦ ♠t q → ∞
P♦ré♠ ♥♦ ♣ró①♠♦ ♣ít♦ ♥♦ q trt♠♦s rs♣t♦ t♦r ♥ét ♦♥sr♠♦s ♦r
rçõs O(q−2) ♥ rqê♥ ♦s s
♦♦ ♦s ♦s ♦♠ s♦s ♣r
st sçã♦ ♣rt♠♦s ♦ sst♠ ♣r srr ♣s♠s ♥♦ q t♦s
r♥ts ♥s t♠♣rtr ♠ sr ♦♥sr♦s ♣♦ré♠ ♥ã♦ ♠♦s ♠
♦♥t ♥♦ qír♦ r♦tçã♦ ♥♠ ①♦ ♦r ♦r♦ ♦♠ ♠♦♦ ♣rs♥t♦ ♠ ❬❪
st ♦r♠ t sst♠ é ♦♠♣♦st♦ ♣s s♥ts qçõs
∂ni
∂t+∇ · (n0i) = 0,
∂pi∂t
+ i ·∇p0i + γp0i∇ · i = 0,
∂π‖i∂t
+ p0i
[
−2i ·∇ lnB − (γi − 1)∇ · i]
= 0,
min0∂i∂t
+∇pi +∇ · π‖i− en0(E + i ×) = 0,
men0∂e∂t
+∇pe + en0(E + e ×) = 0,
∇ · (i + e) = 0.
sr ♦♠♦s ♥♦s ♦t♦s át♦ ♦ ♣rs♥t ♠♦♦ ♦r♠ q ♥♠♥t ♥ã♦
♠♦s ♠ ♦♥t t♦s r♥ts ♥s t♠♣rtr ♦♥t♦ ts t♦s
sã♦ ♦♥sr♦s ♣♦str♦r♠♥t ♥ ♥st ♣ít♦
t♦ ♥s♦tr♦♣ ♣rssã♦ ♥♦s s
♥♠♥t ♣rtr ♦ s♥♦♠♥t♦ ér♦ ♥♦ ♠t q ≫ 1
♦♥sr♥♦ i ≈ E
∂ni
∂t− 2n0E ·∇ lnB = 0,
∂pi∂t
− 2γip0i E ·∇ lnB = 0,
∂π‖i
∂t− 2(2− γi)p0i E ·∇ lnB = 0.
♦t♠♦s s s♥ts rçõs
ni±1= ± i
2
ωω
eΦ0
Tin0, pi = γiTini, π‖i = (2− γi)Tini.
Pr étr♦♥s ♥â♠ é ♦♥sr♠♥t r♥t ♣♦s sts ♦ s ♣q♥
♥ér sã♦ ♦♥sr♦s ♥♦ r♠ át♦ s♦tér♠♦ st ♦r♠ ♦♠♦ me ≪ mi
♣rtr ♦t♠♦s ♦♠♣♦♥♥t ♣r qçã♦ ♠♦♠♥t♦
∇‖pe + en0E‖ = 0, E‖ = −∇‖Φ,
q q♥♦ t③ ♠ ♦♥♥t♦ ♦♠ qçõs s♠rs ♣♦ré♠ ♣r
étr♦♥s ♦r♥ rçõs s♠rs às ♦ts ♠
pe = Tene, ne±1=
en0
TeΦ±1, Te = 0.
♠♣♦rt♥t tr ♠ ♠♥t q ♦ ♦♥trár♦ v‖i ♠s♠♦ ♥♦ ♠t q ≫ 1 e ♥ã♦ ♣♦ sr
s♣r③♦ ♥♦r♠çã♦ s♦r ♦ ♣r í♦♥s étr♦♥s ♣♦♠ sr ♦ts s
qçõs q ♥ã♦ ♠♥♦♥♠♦s ♠ ♣♦ré♠ st é ♠ t♠ ♣r tr♦s tr♦s Pr
♣rs♥t ♥ás é ♠♣♦rt♥t é ♦srr q ♦ t♦ q γe = 1 ♦♥♦r♠
étr♦♥s ♥ã♦ ♦♥tr♠ ♣r ♥s♦tr♦♣ ♣rssã♦ π‖e ≈ 0
♦♥çã♦ qs♥tr e(ni − ne) = 0 ♦t♠♦s
Φ±1 = ±iτe2
ωω
Φ0, τe =Te
Ti,
♦ ♥ ♦r♠ tr♦♥♦♠étr
Φs = τe(ωi/ω)rρiΦ0, Φc = 0, ωi =vTi
R0.
♦t q ♠ t③♠♦s ssttçã♦ ω = rρiωi q t♠ ♣♦r ♥tt♦ ♠♦strr q
Φs ∼ rρiΦ0 ♦♥ ♦ ♦♥♦ st ts ♦♥sr♠♦s rρi ≪ 1
♣rtr ♦♥♦r♠ ♠♦str♦ ♥tr♦r♠♥t ♦té♠s ♥s
♦rr♥t
⊥α = α + pα + πα+ Eα,
♦♥
i =min0
B× dE
dt, pα =
×∇pαB
, α = i, e, πi=×∇ · πi
B,
sã♦ s ♦♥trçõs ♠♣♦rt♥ts q ♠ sr s ♣r ♦t♥çã♦ rçã♦ s♣r
sã♦ ♦t q rr♥t ♦ ♠♦♠♥t♦ r E× á ♠ ♥♠♥t♦ ♣♦s Ei+ Ee
= 0
♦♠ rçã♦ ♦s étr♦♥s ♦♥trçã♦ ♦rr♥t ♥r é ♣q♥ ♦ s e = (me/mi)i
♣♦♥♦ sr s♣r③ t♠é♠ πe≈ 0 st ♦r♠ ♣♥s s ♦♥trçõs ♠♥♦♥s
♠ sã♦ ♠♣♦rt♥ts ♣r ♦ á♦ ♥s ♦rr♥t t♦t
⊥ =∑
α=i,e
⊥α.
♦ ♣r♦r♠♦s ♦r♠ s♠r ♦ ♣r♦♠♥t♦ ♦t♦ ♥ sçã♦ ♣rtr q
♦t♠♦s rçã♦ s♣rsã♦
eΦ0
Tirρiω +
(
pisn0Ti
+pesn0Ti
+1
4
π‖isn0Ti
)
ωi = 0,
♦ s♥♦♠♥t♦ é ♣r♦♥♥t ♦s s♥ts rst♦s
JIr = −rρ2i2
eΦ0
Tien0ω, Jpr + Jπ‖r = −ρi
2
ωi
ε
e
Ti
[
∂
∂θ
(
p−π‖
2
)
+ 3επ‖ sin θ
]
.
♥♠♥t ♦♠ ssttçã♦ ♠ ♦t♠♦s rçã♦
2rρ2i
[
ω − ω2i
ω
(
γi + γeτe +2− γi
4
)]
eΦ0
Ti= 0
♣rtr st rqê♥ ♦s s
ω2
v2Ti/R2
0
= γi + γeτe +2− γi
4= γ
()i + γeτe,
♦♥ γ()i = 3γi/4 + 1/2 é ♦ í♥ át♦ t♦ ♣r í♦♥s
♦♥sr♥♦ γi = 5/3 í♦♥s ♥♦ r♠ ♦ γe = 1 étr♦♥s ♥♦ r♠ át♦
s♦tér♠♦ s q γ()i = 7/4 ♦♥sq♥t♠♥t s q
ω =
(
7
4+
Te
Ti
)1/2 vTi
R0,
♦♥♦r♠ ♦sr♦ ♥tr♦r♠♥t ❬❪ srs q ♦ t♦ ♥s♦tr♦♣ ♣rssã♦
í♦♥s ♣rs♥t ♥♦ tr♠♦ π‖i r♣rs♥t t♦r♠♥t ♠ ♣q♥♦ ♠♥t♦ ♥ rqê♥ ♦s
st ♠♥t♦ é ♣r♦①♠♠♥t ♣r♦①♠♠♥t 3, 0% ♣r τe = 1 ♣r
τe ≫ 1 ♦ t♦ é ♥ ♠♥♦r ♣ró①♠♦ 1, 7% ♦♥sr♥♦ γ = γ(♦rrt♦) ♦♥♦r♠
t♦s ♠♥ét♦s ♥♦s
sr ♦♥sr♠♦s t♦s ♠♥ét♦s ♦ t♦s r ♥♦s ♠♦♦s Pr
s♠♣r s ①♣rssõs ♦♥sr♠♦s s ♥í♦ s ssttçõs γi = 5/3 γe = 1 t♦s
r sã♦ ♣r♦♥♥ts tr♠♦s ts ♦♠♦ E ·∇n0 E ·∇Ti0 ♦ s ♦♦rr♠ ♦
r♥ts rs ♥s t♠♣rtr qír♦ ♦♠♣rs ♦♠ s qs
s qçõs sr♠ rs♦s ♥st s♦ ♦r ♣rs♥t♠ tr♠♦s ♦♥s
∂ni
∂t− 2n0E ·∇ lnB + E ·∇n0 = 0,
3
2
∂pi∂t
− 5p0i E ·∇ lnB +3
2E ·∇p0i = 0,
♦ q ♥ã♦ ♦♦rr ♦♠ q ♦çã♦ s♦s ♣r q ♣r♠♥ ♥tr
s♦çã♦ ♣r ♥s ♣rssã♦ ♣rtrs í♦♥s ♥st s♦
ni±1=
(
± i
2
ωω
Φ0 ∓ω∗i
ωΦ±1
)
en0
Ti,
pi±1=
(
±5
3
i
2
ωω
Φ0 ∓ (1 + ηi)ω∗i
ωΦ±1
)
en0
♣♦♠ sr ♦♥trsts ♦♠ ♦s rst♦s ♣rs♥t♦s ♠ ♦♥ ♦srs q ♦s
tr♠♦s ♦♥s ♠ sã♦ ♣r♦♥♥ts r♥ts ♥s t♠♣rtr
qír♦ s tr♠♦s ♥♦s ♦♠♦ ω∗i = Ti/erBLN ω∗e = Te/erBLN ♦♥ L−1N = dn0/dr
sã♦ ♦♥♦s ♦♠♦ rqê♥s r í♦♥s étr♦♥s rs♣t♠♥t ♠é♠
é ♦♠♠ ♥♦♥trr ♥ trtr ár rqê♥ ♠♥ét q ♥♦ s♦ í♦♥s é
♥ ♦♠♦ ω∗pi = (1 + ηi)ω∗i ♦♥ ηi = LN/LTi
L−1Ti
= dTi/dr
♥â♠ étr♦♥s ♥ã♦ s tr ♣ ♣rs♥ç t♦s ♠♥ét♦s tr♦stát♦s
♣♦ré♠ q♥♦ ♦♥sr♠♦s t♦s tr♦♠♥ét♦s ♦♥♦r♠ st♦ ♠ ♦ r♥t
t♠♣rtr étr♦♥s s♠♣♥ ♠ ♣♣ ♥♠♥t ♥st ♥â♠
♦♠♥t ♦♥sr♠♦s ♦♥çã♦ qs♥tr ni = ne ♣r ♦tr rçã♦
♥tr ♦s r♠ô♥♦s ♦ ♣♦tê♥ tr♦stát♦
Φ±1 = ± i
2
τeωω ± ω∗e
Φ0,
♦r♠ q ♥ ♣rs♥ç t♦s ♠♥ét♦s s ♦♠♣♦♥♥ts s♥♦ ♦ss♥♦ ♥ã♦ ♥
♥ ♣rs♥ç t♦s ♠♥ét♦s ♦ ♣♦t♥ tr♦stát♦ sã♦ s ♣♦r
Φs =τeωiω
ω2 − ω2∗e
rρiΦ0, Φc = −iτeωiω∗e
ω2 − ω2∗e
rρiΦ0 = −iω∗e
ωΦs.
♥♦♠♥t ♦ s♦ ♥tr♦r s♠ t♦ ♠♥ét♦ ♦ tr♠♦ ♣r♥♣ ♣r ♦ s♥
♦♠♥t♦ ér♦ é ♦♠♣♦♥♥t sin θ q♥t p+ π‖/4 ♦ á♦ ♦r♥
(
p+π‖
4
)
s
= −ωω
(
7
4+
τeω2 + (1 + ηi)ω
2∗e
ω2 − ω2∗e
)
en0Φ0.
♦♥♦r♠ ♦ ♣r♦♠♥t♦ ♥tr♦r♦r♠♥t ♣rs♥t♦ ♦ á♦ ♠é ♠ ♠ s♣r
í ♠♥ét s ♦♠♣♦♥♥t r ♥s ♦rr♥t ♥r ♠♥ét ♦r♥
rçã♦ s♣rsã♦ q é ♠ qçã♦ qrát ♠ ω2 ♦♠ s♦çõs
ω2± =
1
2
(
ω2 + ω2
∗e ±√
(ω2 + ω2
∗e)2 + (4ηi − 3)ω2
∗eω2i
)
,
♦♥ ω2 = (7/4 + τe)ω
2i ♠s♠ ♦r♠ ♦♠♦ ♥♦ ♥tr♦r♠♥t
sts s♦çõs q ♦t♠♦s s qs ♦r♠ ♣s ♠ ❬❪ ♣♦♠ tr ss ①♣rssõs
s♠♣s s ♣r♦①♠s ♥♦ ♠t ω∗e ≪ ωi
ω2+ = ω2
+1 + τe + ηi7/4 + τe
ω2∗e ω2
− =3/4− ηi7/4 + τe
ω2∗e
srs q r♥ts ♥s t♠♣rtr s♠ ♠ ♠♥t♦ ♥ rqê♥
♦s q é ♣r♦♣♦r♦♥ à rqê♥ r étr♦♥s Pr ηi = 0, 75 s♥
s♦çã♦ ♣♦ss rqê♥ ♣ró①♠ à ♦s ❩ ♦ q ♣♦ s♠♣♥r ♠ ♣♣ ♠♣♦rt♥t
♥ ♥â♠ q ♦r♥ trê♥ ♦♥s r ♦ à ♥trçã♦ ♥ã♦ ♥r ♥tr
sts s rqê♥s ♥t♦ ηi > 0, 75 st ♠♦♦ ♣rê ♠ ♥st ♣♦ssí
♦♥r q á rs ♥çõs q r♥ts t♠♣rtr ♦♥ t♥♠ sst③r
♦ ♣s♠ ♦ ♣ss♦ q r♥ts ♥s ♦♥tr♠ ♣r st③á♦ ♦r♦ ♦♠
♥ás ♦ ♦r ηi ♥ s♦çã♦ ♥t
sssã♦ s♦r s tr♦♠♥ét♦
♦♠ ♦ ♥tt♦ ♣rs♥tr ♦♣çõs ♣r ♣r♠♦r♠♥t♦ ♦s ♠♦♦s ♣r ♦s s
t♠♦s sr ♦ t♦ s♦ ♣♦ ♠♣♦ ♠♥ét♦ ♣rtr♦ ♣r♣♥r ♦ ♠♣♦
♠♥ét♦ qír♦ s t♦s sã♦ srt♦s ♣♦ ♣♦t♥ t♦r ♣r♦ A‖ ♦r♠
q ♦s ♠♣♦s étr♦ ♠♥ét♦ ♣rtr♦s sã♦ ♦ ♣♦r
E = −∇Φ−∂A‖
∂t, = ∇× (A‖)
♥s ♦rr♥t ♣r ♣♦ sr r♦♥ ♦♠ ♦ ♣♦t♥ t♦r ♣♦r ♠♦ ♦
s♦ ♠♣r
(∇× ) · = µ0J‖ =⇒ J‖ =2rµ0
A‖,
♦♥ t③♠♦s s rçõs ♠é♠ é út r♦♥r st ♥s ♦♠
♦ ♦ s
J‖ = J‖i + J‖e, J‖α = eαn0v‖α,
♦r♠ q é ♥ssár♦ tr♠♥r ♦♠♣♦♥♥t ♣r ♦ í♦♥s étr♦♥s
♣r r♦♥r A‖ ♦♠ Φ
♣rssã♦ ♦♥sq♥t♠♥t ♥s étr♦♥s sã♦ ♦ts ♣rtr ♦♠♣♦♥♥t
♣r qçã♦ ♠♦♠♥t♦ ♣♦ré♠ é ♥ssár♦ ♦♥srr ♦♥trçã♦
♥st á♦ qçã♦ rst♥t ♥tã♦
∇‖pe + ∇‖pe0 + en0E‖ = 0
♦♥ ∇‖ = (/B) · ∇ é ♠ ♦♣r♦r ①♣rssã♦ é ♠♦str ♠ ♥♦ ♣ê♥
E‖ = −∇‖Φ + iωA‖ é ♦♠♣♦♥♥t ♣r ♦ ♠♣♦ étr♦ ♠♦str♦ ♠ ♦t
q ♦ ♦ s♥♦ tr♠♦ q só st ♣rs♥t ♥♦ s♦ tr♦♠♥ét♦ ♦♥
q sr ηe = LN/LTe ♣♦rt♥t♦ é ♣r♦á q ♥â♠ étr♦♥s s♠♣♥
♠ ♣♣ ♠♣♦rt♥t ♥♦ s♦ tr♦♠♥ét♦ st s♦ ♦ ♦ r♥t t♠♣rtr
étr♦♥
♦♠ rçã♦ ♥â♠ í♦♥s ♠ ♥trs s r♥③s π‖i pi ♦♠ rçã♦ ni
s ♥tr♦r♠♥t ♥trt♥t♦ ♥♦ s♦ ♠♥ét♦ é ♥ssár♦ ♦♥srr s ♦s
♣rs étr♦♥s í♦♥s ♥s ss rs♣ts qçõs ♦♥t♥
♦♠ ♦ ♣r♦ss♠♥t♦ ♦s á♦s ♣r♦♥♥ts s qçõs ♦♥çõs srts ♠
srrá ♦ ♠♣♦rt♥t tr♠♦ K2⊥ = k2‖
2rλ
2De
c2/ω2 ♠♥s♦♥ ♦♥ λDe=
√
ε0Te/n0e2 é ♦
♦♠♣r♠♥t♦ ② ♣r étr♦♥s ♠t ♣r♠♥t tr♦stát♦ é ♦t♦ ♦♥sr♥♦
K⊥ → ∞ ♣♦ré♠ ♣♦r ♦tr♦ ♦ q♥♦ K⊥ < 1 t♦s tr♦♠♥ét♦s ♣ss♠ sr
♠♣♦rt♥ts ♥â♠ ♦s st qstã♦ é st ♦r♠ ♠s r ♠ ❬❪ ♦♥
♦ ♣râ♠tr♦ K⊥ ♦ ♥♦ ♠ ❬❪ ♣rt♥♦ qçã♦ ♥ét r é ♠♦str♦ q
♦ ♠♦♦ ♣♦♦ m = 2 é ♠♣♦rt♥t ♥♦ st♦ t♦s tr♦♠♥ét♦s ♥♦s
♠ár♦ sssã♦
st ♣ít♦ ♣rtr t♦r ♦ ♠♦♦ ♦s ♦s ♥♦ ♥s♦tr♦♣
♣rssã♦ ♣rtr í♦♥s é ♦♥sr ♦t♠♦s ①♣rssõs ♥íts ♣r três ♠♣♦r
t♥ts r♠♦s ①s rqê♥s ①♦s ③♦♥s úst♦ í♦♥s úst♦ ♦és♦
st♥çã♦ ♦r♠ r♥③ s rqê♥s ♣rt♥♥ts sts r♠♦s ♣♦ tr ♣çõs
♠♣♦rt♥ts s ♦♠♣rs s ①♣rssõs ♥íts ♦♠ ♦rs ①♣r♠♥ts s rs♣ts
rqê♥s ♦ ♣ss♦ q ♠s ♣çõs ♣♦ss♠ ♦t♦s ♥óst♦s ts ♦♠♦ ♦tr
♦ ♣r r ♦ t♦r sr♥ç q(r) t♠♣rtr T (r) ♦trs s r♦♥♠ ♣r
♥ás st ①♦s ③♦♥s ♠♦♦s úst♦s ♥tr s ♦♥çõs ♠ q
♦♦rr♠ ♥sts ♣♦ r tr rçã♦ ♦ ♦♥♥♠♥t♦ s♦ ♣♦ tr♥s
♣♦rt ♥ô♠♦ ①♦s ③♦♥s ♠♦♦s úst♦s❩s ❲s s sã♦ ♣③s r③r
trê♥ s ♣♦r ♦♥s r ♣♦r ♠♦ ♠ ♣r♦ss♦ t♦♦r♥③çã♦ q
♦♦rr ♥♦ ♣s♠ ♦ q ♥ ♥ã♦ é ♠t♦ ♠ ♦♠♣r♥♦ ❬ ❪ ♠s ♣♦ss ♠ ♦rt
♠♣t♦ ♥ ár ♦♥tr♦ sã♦ ♥r ♣rtís ♣♦rt♥t♦ ♦tr rqê♥
sts ♠♦♦s s ♦♥çõs ♥st q♥♦ ♦tr♦s t♦s ♦ ♣s♠ sã♦ ♦♥sr♦s
é ♠♣♦rt♥t
♥♠♥t ♣rt♥♦ s qçõs ♥st♠♦s ♦ qír♦ ♦♠ r♦tçã♦
♣♦♦ t♦r♦ st ♥stçã♦ ♦♥stt♠♦s q ♦ r♥t r t♠♣rtr
♦♥sq♥t♠♥t ♦ ①♦ ♦r qír♦ ♣♦r s♦ stã♦ r♦♥♦s à r♦tçã♦
♣♦♦ ♥trt♥t♦ ♥♦ r♠ át♦ ♥♦ q ♥ã♦ á ①♦ ♦r ♦♠♥t é ♣♦ssí
♥♦♥trr ♠ qír♦ ♦♠ r♦tçã♦ ♣♦♦ ♥ã♦ ♥ ♥rsã♦ s♥t♦ ♦ r♥t
t♠♣rtr ♦ ♣s♠ t stçã♦ é ♣♦ssí ♣♥s ♦♠♥t ♣♦s t♠♣rtr é ♠♦r
♥♦ ♥tr♦ ♦ q ♥ ♦r ♥r q ♦ ♠ ♠♥ç r♠ ♦ r♠ át♦
♣r ♦ s♦♠étr♦ ♦ q ♣♦r sr ♥st ♥♦s ❩s t♠♥t♦ st st♦
q s ♥♦ ♣♦ tr♦ ❱ P ♥ ❬❪ é ♠ s ♣r♦♣♦sts ♣r tr♦s tr♦s
Pró①♠♦ à rã♦ r = 0.7a ♦sr♠♦s q ♦ r♥t r t♠♣rtr é ♣r♦♣♦r♦♥
♦ ♦ ♦ ♣♦♦ Mt ∝ M3P ❯t③♥♦ ♦ ♠♦♦ t♦r
♦s ♦s ♥♦ qír♦ é ♣♦ssí ♣♦ ♠♥♦s ♦r♠ ♣r♦①♠ ♦tr ♦ ♣r r
t♠♣rtr í♦♥s ♦ q ♦ ♣♦♥t♦ st ①♣r♠♥t é ♦♠♣♦ sr t♦ Pr ss♦
é ♥ssár♦ tr ♥♦r♠çõs s♦r ♦ ♣r r ♦ r♦tçã♦ ♣♦♦ t♦r♦ ♦
q ♦ ♦t♦ ①♣r♠♥t♠♥t ♣r ♦ t♦♠ ❬❪
Pr ♦t♥çã♦ s rqê♥s ♦rr♥ts ♣rtrçõs tr♦stát s♥♦♠♦s
♠ ♠ét♦♦ trt♦ ♣r ♠ qír♦ r♠ rtrár♦ st ♠ét♦♦ é s♥♦♦ ♠ três
t♣s ♦♥sts ss ♥s s♥ts ♦♥çõs qír♦ ♠ r♦tçã♦ MP = MT = 0
♦♠ r♦tçã♦ ♥♠♥t t♦r♦ MP = 0 MT 6= 0 ♥♠♥t ♦♠ r♦tçã♦ ♣♦♦
t♦r♦ MP 6= 0 MT 6= 0 st ♠ét♦♦ é stá ♣♦ t♦ q ♦ sst♠ sr r
s♦♦ é ♥r ♣♦rt♥t♦ ♦ ♣r♥í♣♦ s♣r♣♦sçã♦ ♣♦ sr ♣♦ ♠♦tçã♦ ♣r
st ♠ét♦♦ é ♣r♦♥♥t ♦ st♦ r③♦ ♣♦r r♦♠♦♦♣♦♦s ❬❪ rs♣t♦
♥①stê♥ qír♦ ♦♠ r♦tçã♦ ♥♠♥t ♣♦♦ ♦♠ rçã♦ st t♠ á ♥
qstõs ♠ rt♦ ♣♦s ♦r♦ ♦♠ ♥ás r ♦srs q ♠ r ≈ 0.7a
♦ ♦r ♦ ♣♦♦ é ♣ró①♠♦ ♦ ♠á①♠♦ ♦ t♦r♦ s ♥ ♥st
♣♦sçã♦ st rã♦ t♠é♠ ♦♦rr ♥rsã♦ s♥t♦ r♦tçã♦ t♦r♦ ♦ q ♥ ♥ã♦
é ♠ ♦♠♣r♥ ♦ ♣♦♥t♦ st tór♦ ♠s ♣♦ tr ♠ ♦rt ♠♣t♦ ♥ ♦r♠çã♦
rrr tr♥s♣♦rt ❬❪ ♦♥sq♥t♠♥t ♥♦ tr♥s♣♦rt tr♥t♦ ♦ ♥tr♦ ♦♥
♣s♠ rsst é ♠t♦ ① ♦ ♣s♠ ♣♦ sr ♦♥sr♦ ♥ã♦♦s♦♥ ♣♦ré♠
♦♥♦r♠ ♥♦s ♣r♦①♠♠♦s ♦r ♦♥ ♣s♠ st s t♦r♥ ♦s♦♥ ♣♦rt♥t♦ ♦
st♦ ♥st rã♦ rqr ♠ ♣r♥í♣♦ ♠ ♠♦♦ ♦ ♠s r♥♥t ♣③ ♥r
s♦s rsst ♦♥trçõs ♦s♦♥s ♣r ♦ ①♦ ♦r
trés ♦ st♦ ♥â♠ ♠♦♦s ♦és♦s ①s rqê♥s ♥ sçã♦
♦sr♠♦s q á três rqê♥s tí♣s ♦rrs♣♦♥♥ts ❩s ω ∼ 0 ❲s ω ∼ vTi/qR0
s ω ∼ 2vTi/R0 t♣♦ ♠♦♦ ss♦♦ ♠ sts rqê♥s é ♠♣♦rt♥t
♣♦rq sr ♦ ♣r♦ss♦ ís♦ ♥♦♦ ♣r♠r♦ ❩ ♦♦rr q♥♦ ♦ ♣s♠ rs♣♦♥
♠♥r ♥♦♠♣rssí à ♣rtrçã♦ tr♦stát ♠ ♦♥trst ♦♠ ♦s ♦tr♦s ♦s t♣♦s
rtr③♦s ♣♦r ♦♠♣rss ♦ ♣s♠ ♥s s♦♠ ❲s só ♣♦♠ ♦♦rrr ♠ s
stçõs ♥ sê♥ ♣rtrçõs tr♦státs Φ0 = const. q♥♦ á r♦tçã♦ ♣♦♦
qír♦ ♥ ♥ã♦ á ♥ trtr ♠ ♦♠♣r♥sã♦ t s♦r s r③õs íss
♣r ♦ ♦♠♣♦rt♠♥t♦ ♦ ♣s♠ ♠ rçã♦ às ♣rtrçõs tr♦státs ♦ tr♦♠♥éts
♦♠♦ ♥ã♦ ♦♦rr ♠ tr♥sçã♦ s ♥tr ♦s ♦rs s rqê♥s sts três t♣♦s ♠♦♦s
♦♦rr ♥s ♣s ♥♦ s♣tr♦ rqê♥ ♣♦rt♥t♦ rçã♦ ♥tr ♠♦♦s ♦és♦s
♠♦♦s ♥♦s ❬❪ é ♠ ♠♣♦rt♥t ár ♥stçã♦ ♣r♥♣♠♥t ♥♦ q s rrr
♥óst♦s ♠ s♣ ♣r ♦t♥çã♦ ♦ ♣r r q(r) ♠ss t ♥♦ ♣s♠
♦ ♥r ♦♠♣♦♥♥t ♦ ①♦ ♦r ♦rr♥t ♦ r♥t r t♠♣rtr
♥ qçã♦ ♦♥srçã♦ ♥r ♦t♠♦s ♦rrçã♦ ♥ rqê♥ ♦s ❩s q ♥
♣rs♥ç r♦tçã♦ ♣♦♦ ss♠ ♦r ♥ã♦ ♥♦ ♣r♦♣♦r♦♥ MP /q ❬❪ st ♠♣♦rt♥t
rst♦ q ♦t♠♦s ♣♦ ♠ ①♣rssã♦ ♥ít t ♣r rqê♥
♦s ❩s é ♠ s ♦♥trçõs st ts
t♦ ♥s♦tr♦♣ ♣rssã♦ í♦♥s ♦ ♦♥sr♦ ♥♦ ♦♥t①t♦ t♦r ♦s í♦s
♥st ♣ít♦ ♦♠ s♣♦sçã♦ q ♣rssã♦ ♦ ♦♥♦ s ♥s ♠♣♦ ♣r♣♥r
sts ♣r í♦♥s sã♦ r♥ts ♦t♠♦s ♦ í♥ át♦ t♦ γ(t♦) = 7/4 ♣r
rqê♥ ♦s s ♥trt♥t♦ ♦rrçã♦ ♥s♦tr♦♣ ♣rssã♦ ô♥ ♥ã♦ é ♠t♦
s♥t ♦r♠ ♥♦ ♠á①♠♦ t♦ rá♣ tr♠③çã♦ étr♦♥s ♦ s
♦ s ♣q♥ ♠ss sts s ♦♠♣♦rt♠ s♦tr♠♠♥t t♠♥t ③♥♦
♦♠ q t♠♣rtr étr♦♥s s t♦r♥ ♦♥st♥t ♠ ♠ s♣rí ♠♥ét ♣♦r
st r③ã♦ ♦ t♦ ♥s♦tr♦♣ étr♦♥s ♣♦ sr s♣r③♦ Pr í♦♥s ♥♦ ♥t♥t♦ ♣♦
t♦ sts ♣♦ssír♠ ♥ér ♠t♦ ♠♦r s sã♦ ♥♣③s ♥trr♠ ♠ qír♦ tér♠♦
♥♦ t♠♣♦ rtríst♦ ♠♦♦s ♦és♦s
♦♠ rçã♦ t♦s ♠♥ét♦s ♦ s t♦s s♦s ♣♦r r♥ts rs
♥s t♠♣rtr í♦♥s ♦s qs ①♣rss♠♦s ♣♦r ♠♦ ♦s ♦♠♣r♠♥t♦s r
tríst♦s LN LTi rs♣t♠♥t ♦t♠♦s ♦s s tr♦stát♦s ♣r♠r♦ ♣rs♥t
♠ ♠♥t♦ rqê♥ ♦ ♣rs♥ç r♥t ♥s ♠ ♦♥trst q♥♦
á ♦rts r♥ts t♠♣rtr s♣♠♥t ηi = LN/LTi> 3/4 ♦ s♥♦ ♠♦♦
s t♦r♥ ♥stá ♥ã♦ ♦stór♦ t① rs♠♥t♦ st ♥st é ♣r♦♣♦r♦♥
rqê♥ r étr♦♥s ω∗e = Te/erBLN ♣♦rt♥t♦ trts ♠ t♦ ♦ r♦
r♠♦r ♥t♦ ♣♦ré♠ ♦♠ rçã♦ ♦s í♦♥s st rst♦ ♣♦ r♥t♠♥t ❬❪
t♠é♠ é ♠ s ♣r♥♣s ♦♥trçõs st ts ♦ ♣ít♦ s♥t ♠♦str♠♦s q
♠ rõs t♦r sr♥ç ♠♥♦r st ♥st é s♣r♠
sçã♦ ♠♦str♠♦s q r♥ts t♠♣rtr r♦tçã♦ ♣♦♦ stã♦ r♦♥♦s
st ♦r♠ ♦♥srr t♦s ♠♥ét♦s ♦ t♦s r ♦ ♥és r♦tçã♦
qír♦ ♣♦ sr ♠ tr♥t ♦♥♥♥t ♣r ♥str st s st
♦♥♥ê♥ rs ♥ ♠♦r t③çã♦ ♦ ♠♦♦ ♥ét♦ ♦ q sr ♥tr
♦tr♦s t♦s ①s♦s t♦r ♥ét ♦ ♠♦rt♠♥t♦ ♥ ♦ q é st♦ ♥♦
♣ró①♠♦ ♣ít♦
♦♥srr t♦s tr♦♠♥ét♦s t♦s ♠♥ét♦s s♠t♥♠♥t ♣♦ srr
qstõs ♠♣♦rt♥ts ♥tr s sr♣â♥ ♥♦ ♦r ①s rqê♥s ♦ts ①♣r
♠♥t♠♥t ♦♠ ♦s ♦rs tór♦s s rqê♥s ♠♦♦s ♥♦s s é♠ ss♦
♦ t♦ ♦ r♥t t♠♣rtr étr♦♥s ♠ ♠♦♦s ♦és♦s stá ♦ ♣rtr
çõs ♠♥éts ♣r♣♥rs srts ♣ ♦♠♣♦♥♥t ♣r ♦ ♣♦t♥ t♦r A‖
Pr ♥stçã♦ s tr♦♠♥ét♦s ❬❪ é ♥ssár♦ ♦♥srr s♥♦s r♠ô♥♦s
m = ±2 ♥♦s ♠♦♦s ♣♦♦s ♦s qs s♠♣♥♠ ♠ ♣♣ ♠♣♦rt♥t ♣♦r ♦♥trr♠
♣r ♦rr♥t ♣r j‖ ❬❪
♥r s♠t♥♠♥t r♦tçã♦ qír♦ t♦s ♠♥ét♦s tr♦♠♥ét♦s ♥♦
st♦ s é ♠ ♣r♦♣♦st ♦r ♦ s♦♣♦ st ts ♣♦r r á♦s ♠t♦ ①t♥s♦s
♣♦ré♠ ♣rt♥♠♦s r ♥t st st♦ ♠ tr♦ tr♦ ♣rtr ♠t♦♦♦
srt ♥st ts ♠é♠ ♣rt♥♠♦s ♦♥srr ①♦ ♦r ♥ t♦r ♦s ♦s ♣r
♥sr ♠♦♦s rqê♥s ♠♥♦rs ♠ s♣ ①♦s ③♦♥s ❩s