♣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