Universidade de São PauloInstituto de Física

Investigação cinética de modos geodésicos debaixas frequências em plasmas magnetizados

Reneé Jordashe Franco Sgalla

Orientador: Prof. Dr. Artour G. Elfimov Co-Orientador: Prof. Dr. Ricardo Magnus Osório Galvão

Tese de doutorado apresentada ao Instituto de Físicapara a obtenção do título de Doutor em Ciências

Banca Examinadora:

Prof. Dr. Artour G. Elfimov (IF – USP )Prof. Dr. Ivan Cunha Nascimento (IF – USP)Prof. Dr. Ricardo Luiz Viana (UFPR)Prof. Dr. Luiz Fernando Ziebell (UFRGS)Prof. Dr. Francisco Eugenio Mendonça da Silveira (UFABC)

São Paulo2014

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FICHA CATALOGRÁFICA

Preparada pelo Serviço de Biblioteca e Informação

do Instituto de Física da Universidade de São Paulo

Sgalla, Reneé Jordashe Franco Investigação cinética de modos geodésicos de baixas frequências em plasmas magnetizados. São Paulo, 2014. Tese (Doutorado) – Universidade de São Paulo. Instituto de Física. Depto. de Física Aplicada. Orientador: Prof. Dr. Artour G. Elfimov Área de Concentração: Física Unitermos: 1. Física de plasmas; 2. Fusão nuclear; 3. Teoria cinética; 4. Magnetohidrodinâmica; 5. Turbulência eletrostática. USP/IF/SBI-057/2014

tór

♦ st ts à ♠♥ qr s♣♦s ♦st q s♠♣r ♠ ♣♦♦ ♠ ♠♥s

ts ê♠s à ♠♥ ♥ ♦st q♠ s♣r♦ ♦♥sr ①r ♠

r♥ç ♦♥♠♥t♦s ♣r t♦ ♦s ♠s ♣s ♠♦ r r♥♦

q♠ ♦ t♦s s ♠♥s ♦♥qsts ♣ss♦s ♣r♦ss♦♥s

r♠♥t♦s

r③çã♦ st tr♦ ♥ã♦ ♦ rt♦ s♦♠♥t ♦ ♠ s♦rç♦ ♠♥ çã♦ ♠s

t♠é♠ ♦♦rçã♦ ♣ss♦s ♥sttçõs às qs ♦ s♥r♦s r♠♥t♦s

rç♦ ♦ ♦♥s♦ ♦♥ s♥♦♠♥t♦ ♥tí♦ ♥♦ó♦ Pq ♣♦

♣♦♦ ♥♥r♦ ♣♦r ♠♦ ♦s ♦t♦r♦ P♥♦ q ♠ ♦ ♦♥♦ r♥t

♦ ♣rs♥t ♣r♦r♠ ♦t♦r♠♥t♦ rr♦ ♣♦♦ ♠ ♣r♦♣♦r♦♥♦ é♠ ♦♥çõs

♥♥rs ss♥s ♠♦tçã♦ s♥t ♣r ♣♦r r st tr♦ ♥t ♠ ♦♥

sqê♥ st ♦s ♣ró①♠♦s q ♣r♦♠♥t s srã♦

♠ s♣ ♠♥ qr s♣♦s ♦st ♦ ♠ ♠s s♥r♦ r♦♥

♠♥t♦ ♣♦ ♣♦♦ ♠♦♦♥ q s ♦ ♥ ♦ ♠ ♠str♦ t♠ s♦ ♥st♠á

♣r sr ♥t ♠ ♠♥ rrr ê♠ ♠é♠ r♦♥ç♦ ♣rst ♣♦r

q♥t♦ à rsã♦ r♠t stíst srt st ts

rç♦ t♠é♠ ♠♥ ♥ ♦st ♣ ♠♦tçã♦ ♣r ♦♥t♥r

♥t ♣r♥♣♠♥t ♥♦s ♠♦♠♥t♦s ♠s ís q ♥ã♦ ♦r♠ ♣♦♦s

ã♦ ♣♦r sqr rr t♠é♠ ♠♥ ♠ã r r♥♦ ♣♦s

♦s r♥♦ q t ♦♠ ♥ q ♠ ♣r♠tr♠ r ♠ ♦ ♣rt

♦ ♠ t♠♣♦ ♣r ♦ ♠ ♦t♦r♦ ♠s♠ ♦r♠ rç♦ ♦ ♠ ♦ ♣ ♠♦

♣ ♠♥ çã♦ ♦r♠çã♦ q rt♠♥t t ♥♥ ♥♦ ♠ ♦t♦r♦

♣ s r♥t ♠♥s ♥s ♦♠ ♥tt♦s ê♠♦s

♦ ♠ ♦r♥t♦r Pr♦ r rt♦r r♦r ♠♦ q rs♦ ♠s á♦s

s♠♣r st ♣rs♥t ♣r sssõs s♦r ís t♦♠s ♦ ♦ ♣rt r③çã♦

st tr♦ s ♦♥trçõs ♦♠ rçã♦ às rss ♦rrçõs st ts ♣♦r ♦r♥s

♦r♠ ss♥s ♣r t♥r ♦r♠ t

❯♠ r♥ ♣rt ♦ ♦♥♠♥t♦ q qr ♦♠ rçã♦ à ♣çã♦ ♦ ♠♦♦ r♦

♥ét♦ t♦r ♦s ♦s ♦ st♦ ♠♦♦s ♦és♦s ♠ t♦♠s ♦ ♦ Pr♦ r

♥r ♠♦②♦ ♦♠ q♠ t ♦ ♣r③r trr ♠ ♠ ♦♦rçã♦ q rst♦ ♥

♠♦r ♣rt ♦s ♣ít♦s st ts ♠ tr♦s ♣♦s ♣rs♥t♦s ♠ ♦♥rê♥s

é♠ ss♦ ♦ ♣rt s rrê♥s ♦rás st ts ♦r♠ s r♦♠♥çã♦

♦ Pr♦ r r♦ ♥s sór♦ ã♦ rç♦ ♣♦ rs♦ ís Ps♠s

♠♥str♦ ♥♦ s♠str ♦ q ♠ ♦ ♠♦rr ♠s ♦♥♠♥t♦s ♠ ís

♣s♠ ♠ ♠♥r r ♠é♠ rç♦ s sssõs s♦r r♦tçã♦ ♣s♠

s qs ♦♥tr♠ t♠é♠ ♦♠ ♣rt♣çã♦ ♦ Pr♦ r rt♦r ♠♦

♦ ♠ ♦ r♦ ♦r é♠ ♦s ①♥ts és s♠ çúr ♣ós ♠♦ç♦ ♠ s

s rç♦ ♣s rtírs sssõs s♦r qír♦ ♦tr♦s ss♥t♦s ♠♣♦rt♥ts

s♦r ♦♥♠♥t♦s r♦♥tr ♠ ís ♣s♠ rç♦ t♠é♠ ♦

♣rst ♥ çã♦ ♠s s rs st ts

s♠♦

♦ à s ♠♣♦rtâ♥ ♠ trê♥ s ♣♦r ♦♥s r à ♣çã♦ ♦♠

♣r♦♣óst♦s ♠ ♥óst♦s ♣s♠ ♥stçã♦ ①♦s ③♦♥s ❩ ♠♦♦s úst♦s

♦és♦s t♠ trí♦ st♥t t♥çã♦ ♥ trtr ♠ ís ♣s♠s st

ts ♣r♠r♠♥t ♦♥sr♠♦s t♦s qír♦ ♦♠ r♦tçã♦ ♣♦♦ t♦r♦ ♥sts

♠♦♦s ♣♦str♦r♠♥t ♥st♠♦s t♦s ♠♥ét♦s ♠ ♣rtr ♠ ♠♦♦

♦s ♦ ♥♦ q ♥í♠♦s s♦s ♣r í♦♥s ♥ ♣rt ♥ ♦♥sr♠♦s

♠♦rt♠♥t♦ ♥ t♦s ♠♥ét♦s s♠t♥♠♥t ♥♦ st♦ ♣♦ré♠

♣rtr ♦ ♠♦♦ r♦♥ét♦ t♦s ♠♥ét♦s sã♦ s♦s ♣♦r tr♠♦s q ♥♦♠

r♥ts ♥s t♠♣rtr ♣r♦♥♥ts ♥çã♦ ①♥ qír♦

♦♣♠♥t♦ ♥tr ♦s r♠ô♥♦s ♣♦♦s m = ±1 s rs rs q♥ts

♠r♦só♣s ♦ ♣s♠ é rs♣♦♥sá ♣♦ ♠♥t♦ ♥♦ ♦r rqê♥ ♥♦ t

rqê♥ ♣ ♥st ♥♦ ① rqê♥ st t♣♦ ♥st q é

♣r♦♣♦r♦♥ à rqê♥ ♠♥ét étr♦♥s à r③ã♦ ♥tr ♦s r♥ts t♠♣rtr

♥s é ♠s ♣r♦♣♥s♦ ♦♦rrr ♠ ♣♦sçõs rs ♠ q ♦ t♦r sr♥ç é t♦

♦♦s ♦és♦s sã♦ r♠♥t ♠♦rt♦s ♦ ♠ ♠â♥s♠♦ ♥ã♦ ♦s♦♥ ♦♥♦

♣♦r ♠♦rt♠♥t♦ ♥ ♦ q é s♦ ♣ ♥trçã♦ ♥tr ♦♥ tr♦stát

♣rtís rrs í♦♥s ♥♦ s♦ t① ♠♦rt♠♥t♦ é ♠♦r ♣ró①♠♦ ♦ ♥tr♦

♦♥ ♣s♠ ♦♥ ♦ t♦r sr♥ç ss♠ ♦rs ♠s ①♦s

qír♦ ♦♠ r♦tçã♦ ♦ ♥st♦ ♠ três r♠s ♦♠ rçã♦ às s♣rís

♠♥éts s♦tér♠♦ át♦ s♦♠étr♦ ♦ ♦sr♦ q ♦ r♥t t♠♣rtr

♣♦ss s♥t♦ ♦♣♦st♦ ♠ rçã♦ à ♦ r♦tçã♦ ♣♦♦ ♣♥s ♥♦ r♠ s♦♠étr♦

♦ ♦♥srr qír♦ ♦♠ r♦tçã♦ s♣rís ♠♥éts s♦tér♠s ♥r ①♦ ♦r

♥ qçã♦ ♥r ♦sr♠♦s q ❩ ♣rs♥t♠ rqê♥ ♥ã♦♥ q é ♣r♦♣♦r♦♥

à ♦ r♦tçã♦ ♣♦♦ ♥rs♠♥t ♣r♦♣♦r♦♥ ♦ t♦r sr♥ç

♦♠♦ rçõs trs rsst♠♦s q é ♠♣♦rt♥t ♦♥srr t♦s tr♦♠♥ét♦s s

tr t♦♠♦♦s ♦és♦s ♥r ♦ t♦ ♣rtís ♣rs♦♥s ♣r ♦ s♥♦♠♥t♦

ís ❩ s♥♦♠♥t♦ ♥rá t♥t♦ ár tr♥s♣♦rt ♠ t♦♠s

♦♠♦ ár ♥óst♦s ♥ q ♦t♥çã♦ ♦ ♣r r t♠♣rtr í♦♥s ♦

t♦r sr♥ç é ♠ ♦s ♦t♦s st ár ♠ ♥♦♦ t♣♦ ♥óst♦ ♦♥♦ ♦♠♦

s♣tr♦s♦♣ ♠ ♠♦♦s úst♦s ♦és♦s stá s♥♦ s♥♦♦ s♦ ♥♦ st♦

t♦♠♦♦s

Prs s ♦♦s úst♦s ♦és♦s t♦s ♠♥ét♦s t♦s

r ♠♦rt♠♥t♦ ♥ ①♦s ③♦♥s ❩ s♣tr♦s♦♣ ♠

strt

t♦ t ♠♣♦rt♥t r♦ ♥ rt tr♥ ♥ ♣♣t♦♥s ♦r ♣s♠ ♥♦st

♣r♣♦ss t ♥stt♦♥ ♦ ③♦♥ ♦s ❩ ♥ ss♦t ♦s ♦st ♠♦s

s rs♥ ♠ tt♥t♦♥ ♥ t ♣s♠ ♣②ss trtr ♥ ts tss rst ♦♥sr q

r♠ ♣♦♦ ♥ t♦r♦ r♦tt♦♥ ts ♦♥ ts ♠♦s s♥ t ♠♦ t♥

♥stt ♠♥t ts ♦♥ s♥ t♦ ♠♦ tt ♥s ♣r ♦♥

s♦st② ♥ ♥ t ♥ st♣ ♥ ♦t ♥ ♠♣♥ ♥ ♠♥t ts ♦♥ t

st② ♦ t♥ t r♠♦r ♦ t ②r♦♥t ♠♦ ② ♠♥t ts ♠♥

t ♥st② ♥ t♠♣rtr r r♥ts tr♠s ♦♠♥ r♦♠ t qr♠ ①♥

strt♦♥ ♥t♦♥ ts s ② t ♦♣♥ t♥ t m = ±1 ♣♦♦ r

♠♦♥s ♥ t r rts ♦ qr♠ ♠r♦s♦♣ q♥tts r rs♣♦♥s ♦r ♥

♥rs ♥ t rq♥② ♦ t rq♥② ♥ ♦r ♥ ♥stt② ♥ t ♦

rq♥② s ♥stt② s ♣r♦♣♦rt♦♥ t♦ t tr♦♥ rt rq♥② ♥ t

rt♦ t♥ ♦♥ t♠♣rtr ♥ ♥st② r♥ts r ♠♦r ② t♦ ♦r ♥ r ♣♦st♦♥s

r t st② t♦r s ❲ ♦sr tt ♦s ♠♦s r s♦② ♠♣ ② ♦

s♦♥s ♠♥s♠ ♥♦♥ s ♥ ♠♣♥ s s ② t ♣rt ♥trt♦♥

t♥ t tr♦stt ♣♦t♥t ♥ t í♦♥s s ♠♣♥ s ♥♥ ♥r t ♥tr ♦

t ♣s♠ ♦♠♥ r t st② t♦r s ♦r s

qr♠ t ♣s♠ r♦tt♦♥ r ♥stt ♥ tr r♠s rr♥ t

♠♥t srs s♦tr♠ t ♥ s♦♠tr t s ♦♥ tt t t♠♣rtr r♥t

s ♦♣♣♦st rt♦♥s ♦♠♣r t♦ t ♣♦♦ r♦tt♦♥ ♦♥② ♦r t s♦♠tr r♠ ②

♦♥sr♥ qr♠ r♦tt♦♥ t s♦tr♠ ♠♥t srs ♥ ♥♥ t ①

♦sr tt ❩ s ♥♦♥③r♦ rq♥② s ♣r♦♣♦rt♦♥ t♦ t ♣♦♦ ♦t② ♥

t ♥rs ♦ t st② t♦r

♦r tr rt♦♥s ♣♦♥t ♦t tt tr♦♠♥t ts ♦s ♥♠♦s ♥

tr♣♣ ♣rts ♣②ss s♦ ♠♣♦rt♥t ♦r t ♦♣♠♥t ♦ t ❩ ♥ ♣②ss

tr ♥ t r ♦ ♥♦♠♦s tr♥s♣♦rt s ② rt tr♥ ♦r ♦r ♥♦st

♣r♣♦ss ♦r ♦t♥♥ t r ♣r♦ ♦ t ♦♥ t♠♣rtr ♥ t st② t♦r ♥ ts

r ♥ ♥ ♦ ♥♦st ♥♦♥ s ♦s ♦st ♠♦ s♣tr♦s♦♣② s ♥ ♦♣♥

s ♦♥ t st② ♦ ♥♠♦s

②♦rs ♦s ♦st ♦s ♠♥t ts rt ts ♥

♠♣♥ ③♦♥ ♦s ❩ s♣tr♦s♦♣②

♠ár♦

♥tr♦çã♦

♥r ♣r trs rçõs

♦♥♥♠♥t♦ ♦ ♣s♠ ♥♦ t♦♠

♣r♥í♣♦ ♦♥♥♠♥t♦ ♠♥ét♦

rçã♦ ♦ ♦♥♥♠♥t♦ ♠ t♦♠s

♦tçã♦ ♣r ♦ st♦ ♠♦♦s ♦és♦s rst♦s ♦t♦s

rqê♥ ♦s ♠♦♦s úst♦s ♦és♦s s

r♥③çã♦ st ts

♣ít♦s

♣ê♥s

ís t♦♠s

♦♠♣r♠♥t♦s t♠♣♦s rtríst♦s ♦ ♣s♠

♠♣♦ ♠♥ét♦ qír♦ ♥♦ t♦♠

sã♦ tr♥s♣♦rt ♠ t♦♠s

♦♠♥t♦ ♣rtís ♦ ♦ ♥tr♦

♦r ♥ét

♥ás qçã♦ ♦t③♠♥♥

qçã♦ r♦♥ét

qçã♦ ♥ét r

♦r ♦s

♦r ♦s ♦s s qçõs r♥s

♦r ♠♥t♦r♦♥â♠

♦♦ ♦ ♣r ①♦s ③♦♥s ♠♦♦s úst♦s ♦és♦s

♦♦ ♠♥t♦r♦♥â♠

qír♦ ♦♠ r♦tçã♦

♦tçã♦ t♦r♦

♦tçã♦ ♣♦♦ t♦r♦

st♠ qçõs ♣rtrs rçã♦ s♣rsã♦

①♦s ③♦♥s ❩ ♠♦♦s úst♦s ♦és♦s

t♦ r♦tçã♦ ♥♦s ❩

t♦ r♦tçã♦ t♦r♦

t♦ r♦tçã♦ ♣♦♦ t♦r♦

sssã♦ s♦r ♦ í♥ át♦

♦♦ ♦s ♦s ♦♠ s♦s ♣r

t♦ ♥s♦tr♦♣ ♣rssã♦ ♥♦s

t♦s ♠♥ét♦s ♥♦s

sssã♦ s♦r tr♦♠♥ét♦

♠ár♦ sssã♦

♥stçã♦ ♠♦♦s úst♦s ♦és♦s ♣♦ ♠♦♦ r♦♥ét♦

st♦ ♣rtr ♦ ♠♦♦ r♦♥ét♦

♠t ♦ ♦♠ k‖vTi= 0 q → ∞

♠t ♦ ♦♠ k‖vTi♥t♦ q ≫ 1

ss♣çã♦ ♥ ♠ ω > k‖vTi

sssã♦ s♦r ♣çõs ♦ ♠♦♦ r♦♥ét♦ ♥ ♦r♠ ♠s r

t♦s ♠♥ét♦s ♠♦rt♠♥t♦ ♥ ♠

♦çõs ♥♦ ♠t ♦

t♦ ♥ét♦ ♠ ♠♦♦s ♦és♦s ♠♦rt♠♥t♦ ♥

♠ár♦ sssã♦

♦♥sõs rçõs trs

♦♦ ♦s

♦♦ r♦♥ét♦

Pr♦♣♦sts ♣r tr♦s tr♦s

♦♥st♥ts ♣râ♠tr♦s ♦

♦♥st♥ts ís

Prâ♠tr♦s ♦ ♦

r♥s r♥③ rqê♥s ♦s rtrísts ♦

♥ts rçõs t♦rs

♥ts t♦rs

♥ts t♦r♠s ♥♠♥ts

♥ts ♥♦♥♦ ♦ ♦♣r♦r ∇

r♥t r♥t ♦t♦♥ ♣♥♦ ♠ ♦♦r♥s í♥rs

r♥t r♥t ♦t♦♥ ♠ ♦♦r♥s qst♦r♦s

rs rs♦rs ♠ ♦♦r♥s í♥rs

rs rs♦rs ♠ ♦♦r♥s qst♦r♦s

t♥çã♦ s ①♣rssõs ♥íts rr♥ts à ♥ás qír♦ ♦♠ r♦

tçã♦

çõs ♥♦♥♦

çõs ♣r ❱

á♦ ∇ · q qír♦

rçã♦ ór♠s ss ♥♦ ♣ít♦

çõs ♣r

á♦ rê♥ π q

çõs ♣r ♦s

çõs ♣r ♥s ♦rr♥t

qçã♦ ♦çã♦ π‖

♣r♦①♠çã♦ ♣r t♦♠s s♣rís ♠♥éts ♦♥ê♥trs

♠♣♦ ♠♥ét♦ qír♦

♠♣♦ ♠♥ét♦ ♣rtr♦

❱♦ ♥s ♦rr♥t

♦çã♦ trt s qçõs ♣rtrs

qçõs ♥s s♦çã♦ qír♦

á♦ F‖ R P

r♠♦s ♦♥çã♦ rs ♥rs

á♦ F‖ Fθ

á♦ R

á♦ ∇ · q

á♦ P

♦çã♦ s♠ r♦tçã♦ ♣r♠r trçã♦

♦çã♦ ♦♠ r♦tçã♦ t♦r♦ s♥ trçã♦

♦tçã♦ ♣♦♦ t♦r♦ trr trçã♦

çã♦ s♣rsã♦

á♦ ♥trs ♥çã♦ strçã♦

çõs ♥♦♥♦ strçã♦ ♠①♥

á♦ s ♥trs ♥ ♣r♦①♠çã♦ ♦

♥çã♦ s♣rsã♦ ♣s♠

á♦ s ♥trs ♦♠ t♦s ♥ét♦s

t♥çã♦ ♦ ♠t ♦ ♣rtr s ♥trs ♦♠ t♦s ♥ét♦s

Prt♣çã♦ ♠ ♥t♦s ♥tí♦s

rs♦s ♥tr♥♦♥s

Pr♦çã♦ ♦rá

♦♥rê♥s ♥♦♥tr♦s ♥tí♦s

rê♥s ♦rás

❱

st rs

sq♠ ♦t♥çã♦ ♥r ♣♦r ♠♦ sã♦ tr♠♦♥r ♦♥tr♦ ♠

♠ tr♦ rt♦r sã♦ s♦ ♥♦ t♦♠

sq♠ ♠ t♦♠

Pr r ♦ r♦tçã♦ qír♦ ♣♦♦ sqr t♦r♦

rt ♥♦ t♦♠s

♥â♠ ♠♦♦s úst♦s ♦és♦s ♠ t♦♠s

♥rs ♦ ♥♦♠♥♦r D(P) ♣r MP ≥ 0

❱

st s

rqê♥s ♥♦r♠③s ♠ r♥ts r♠s qír♦ ♦♠ r♦tçã♦ t♦r♦

rqê♥s tí♣s ♥♦r♠③s ♣♦r vTi/R0 r♦♥s t♦s ♦és♦s

úst♦s í♦♥s ♠♥ét♦s

♦♥st♥ts íss ♣rt♥♥ts st ts

Pr♥♣s ♣râ♠tr♦s ♦s t♦♠s

❱♦r ♦r♠ r♥③ rqê♥ rçã♦ ♦ tér♠ ♦

r♦ r♠♦r ♣r ♦ t♦♠

❱

♣ít♦

♥tr♦çã♦

st ♥tr♦çã♦ ♥♠♥t st♠♦s s♣t♦s s♦r qstã♦ ♣r♦çã♦ ♥r

♣r ♦ ♦♥s♠♦ ♠♥♦ ♥ ♥st ♦♥t①t♦ ♣rs♥t♠♦s ♣r♦♣♦st sã♦ tr♠♦♥r

♦♥tr♦ ♦♠♦ ♠♦ tr♥t♦ ♣r s♣rr ♠♦r ♦s ♣r♦♠s ár ♥r

♣r♥í♣♦ ♦♥♥♠♥t♦ ♦ ♣s♠ ♥♦ t♦♠ ♥s ♦s ♣r♥♣s s♦s ís

t♦♠s sã♦ ♦s ♠ s

trr sçã♦ ♥ q s ♣r♥♣s rrê♥s r♦♥s ♦ t♠ st ts ♦r♠

♥srs ♣rs♥t♠♦s ♠♦tçã♦ ♣r st tr♦ q t♠ ♦♠♦ ♣r♥♣ ♦t♦ ♥s

tr ♠♦♦s úst♦s ♦és♦s ♦ s ♠♣♦rtâ♥ ♥ ár trê♥

♠s r♥t♠♥t ♥ ár ♥óst♦s ♣♦r str♠ ♣rs♥ts ♥♦ r♠ ♦♥♥♠♥t♦

♠♦r♦ ♠♦♦s ①s rqê♥s ♠ t♦♠s ♠ s♣ tê♠ s♦ ♦

♥t♥s s♣çã♦

♥♠♥t ♦r♠ ♦♠♦ st ts stá ♦r♥③ ♦s ss♥t♦s q trt ♣ít♦

♦ ♦♥tú♦ ♦s ♣ê♥s sã♦ ♣rs♥t♦s ♦r♠ rs♠ ♥ út♠ sçã♦

♥r ♣r trs rçõs

á ♠s és trás q♥♦ ♦ q♠♥t♦ ♦ ♥ã♦ r ♦♥sr♦ ♠ ♣r♦♠

♠ ♣♦t♥ é♠ ss♦ s rsrs ♥trs ♣tró♦ rã♦ r♠ ♦♥srs r♥t

♦s ♦st ♦s

♥r s♥t ♣♦r ♠ ♦♥♦ ♣rí♦♦ ♠ét♦♦s tr♥t♦s ♣r♦çã♦ ♥r ts

♦♠♦ ♥r s♦r ó ♦tér♠ t ♥ã♦ t♥♠ t♥t ♦rç ♣r ♣rr ♥ ♣rát

t♠♥t ♥trt♥t♦ ♠ç ♣♦ssís ♠♥çs ♥ts ♥♦ ♣♥t ♣r♦♦s

♣r♥♣♠♥t ♣♦ q♠♥t♦ ♦ t♠ s♦ ♦♥sr ♠ ♣r♦♣çã♦ ♣r ♥tsts

♠♥tsts ♥♦ ♠♥♦ ♥tr♦ ♠s s♥s sss③ rrs♦s ♥trs ♦ ♠♥t♦

♦ ♦♥s♠♦ ♥r ♣r♥♣♠♥t ♦ ♦ s♥♦♠♥t♦ t♥♦ó♦ ♦ ♠♥t♦

♣♦♣çã♦ ♠♥ ♠♦str♠s ③ ♠s ♥ts ♦r♠ q s ♣♦r ♦r♠s

tr♥ts ♣r♦③r ♥r ♦♠ ♠ ♠í♥♠♦ ♠♣t♦ ♠♥t ♠ ♥♥♦ ♦rç ♠

rs♦s ♣íss ♥s ♥ ♠í

♦♥t♦ é ♣♦ssí ♦♥sr♥♦ ♠s st♠ts q s ts ♦r♠s ♥r

♠♣ r♥♦ás ♥ã♦ ♣♦ss♠ s♣rr ♠♥ s ♣♦ ♠♥t♦ ♦ ♦♥s♠♦ ♥r

q rt♠♥t ♦♦rrrá ♥s ♣ró①♠s és P♦r st r③ã♦ t♦r♥s ♠♣rs♥í ♦ s♥

♦♠♥t♦ ♥♦♦s ♠♥s♠♦s ♣r♦çã♦ ♥r ♠ ♣rtr ♥r ♣♦r ♠♦

sã♦ tr♠♦♥r ♦♥tr♦ q é ♠ s♦çã♦ ♠ ♣♦t♥ ♣r ♦ ♣r♦♠ st♦ s

♦ t♦ q é♠ ①♦ ♦ ♠♣t♦ ♠♥t s♦ ♦♥♦ ♣r③♦ ♦ st♦ ♦t♥çã♦

♦s ♦♠stís sr♠ t③♦s ♥ rçã♦ sã♦ ♣♦rá sr r③♦ r♥ q♥t

♥r q ♣♦rá sr ♣r♦③ ♦r♠ sst♥tá é ♠ ♥t♠ s ♥t♦ às

qstõs s♦r ♣r♦çã♦ rsí♦s r♦t♦s ♦ rs♦ ♥ts ♥rs ♦♥sr♦s

s♠ râ♥ q♥♦ s trt rt♦rs ssã♦ ♥r ♦ t♠♣♦ r♠③♥♠♥t♦

sts rsí♦s ♣r q ♥ã♦ s♠ ♥♦s ♦ ♠♦ ♠♥t é ♠ ♠♥♦r ♥♦ s♦ sã♦

é♠ ss♦ ♦ ♠♣t♦ ♠ ♠♣r♦á ♥t ♥ã♦ s st♥r ♣r♦♣♦rçõs ♠♦rs ♦ q

♦ ♦ ♥stçã♦ ♦ rt♦r ♦ ♦♥trár♦ ♦ q ♦♦rr ♥♦ ♣ss♦ ♦♠ rt♦rs ssã♦

♦♠ s ♥sts r♠♥t♦s ❬ ❪ rts q ♥r sã♦ é s♦çã♦ ♠s ③ ♥♦

q s rr à ♣r♦çã♦ r♥ q♥t ♥r ♦r♠ sst♥tá ♦♠ ♠ ♠í♥♠♦

♠♣t♦ ♠♥t ♥trt♥t♦ á ♠t♦s s♦s t♥t♦ ♥ ís ♦♠♦ ♥ ♥♥r ♥ã♦

s♣r♦s ♥ q ♠♣♠ ♦t♥çã♦ ♥r ♣♦r ♠♦ sã♦ ♦♠ ♦ ♦t♦

s♣rr sts s♦s ♣sqss ♠ rs♦s s♣♦st♦s ♦♥♥♠♥t♦ ♣s♠ ê♠ s♥♦

r③s ♦ ♦♥♦ ♦ t♠♣♦ t♠♥t rts q ♦ t♦♠ ♥♥t♦ ♣♦r ♠♠

♥♦♠ t♦♠ é ♣r♦♥♥t t♦r♦♥② ♠r ♠♥t♥② ts q ♠ rss♦ s♥

r♦ ❬❪ ♥ é é ♦ s♣♦st♦ ♠s ♣r♦♣♥s♦ sr t③♦ ♥ ♣r♠r

s♥ ♥r sã♦ ♦♠s ♦③♦s ♠ rss ♣rts ♦ ♣♥t ❬❪ ê♠ s♥♦

♦♥strí♦s ♦ ♦♥♦ és ♣ós s♥ rr ♠♥ ♦♠ ♥ ♣r♦♠♦r

♣sqs ♠ ís ♣s♠s ♦♠ ♦ ♦t♦ s♣rr ♦s s♦s q ♠♣♠ ♦t♥çã♦

♥r ♣♦r ♠♦ sã♦

♣r♦ss♦ ♣r ♦t♥çã♦ ♥r étr ♣♦r ♠♦ sã♦ ♥r str♦ ♥ r

♦♥sst ♠ ♣r♦③r ♦♥♥r ♠ ♣s♠ ♦♥sttí♦ ♣♦r tér♦ rít♦ sót♦♣♦s

♦ r♦ê♥♦ ♠♥t♥♦♦ tr♠♥♦s ♦rs t♠♣rtr ♥s ♠s ♥

ss♠ ♠♥r stá ♦r♠ q ♣♦ss ♦♦rrr ♦♠ rqê♥ s♥t rçã♦

+ → 4 ❱+ ♥ ❱.

♠ ♠ s ♥ ♦ s♥♦♠♥t♦ rt♦rs sã♦ ♣♦r sr ♠ s rçõs ♥

rs ♠s s♠♣s s r③r rçã♦ ①♣rss ♠ sr t③ ♦ ♥és ♦trs

♠s ♥ts q ♥trt♥t♦ ♥sst♠ t♠♣rtrs ♠s s

êtr♦♥s ♥rét♦s ♣r♦♥♥ts ♦ ♣s♠ ♦ t♥r♠ ♠ ít♦ q

rst s ♣rs ♥tr♥s ♦ t♦♠ tr♥srr♠ s ♥r ♥ ♦r♠ ♦r ♠

tçã♦ á ♦rr♥t ♦♠ ♣r♦çã♦ ♣♦r t ♣rssã♦ ♦♠♦ ♦♥sqê♥ st

♣r♦ss♦ st ♣♦r sr ♥tã♦ t③♦ ♣r ♦♥r ♠ r♦r tr q ♣♦r s

③ str r étr s s ♥çã♦ ♠ ♥♦ t♦♠ s♣♦st

♠ ♠ ♠♥t é ♦r♥r át♦♠♦s trít♦ ♦ ♣s♠ ♦r♦ ♦♠ rçã♦

6+ ♥ → 4+ + ❱.

♠♥r ♦ ♥st ♣r♦ss♦ árs t♣s tr♥s♦r♠çã♦ ♥r ♦ rst♦

íq♦ sr tr♥s♦r♠r ♥r ♥r ♦♥t ♥♦s át♦♠♦s tér♦ trít♦ ♠ ♥r

étr r♥♦ ♦♠♦ rsí♦ st tr♥s♦r♠çã♦ ás é♦ q ♣♦r sr r♦ ♦

♠♦ ♠♥t s♠ ♦ ♣rí③♦ ♥♥♠ ♠♣t♦ ♠♥t ♠ ③ q st ás é ♥rt

stá ♣sr ♥♦r♠ q♥t ♥r q ♣♦r sr ♦♥s ♦♠ ①s

â♠r t♦r♦ ♥♦ ♣♦r ♦♥s ♠♥éts

q♥ts tér♦ ít♦ ♥♥ts ♥ ♥tr③ ♦ rs♦ ♥ts ♥rs é

①♦ ♦ ♠♣t♦ ♠♥t ♣rt♠♥t ♥①st

r sq♠ ♦t♥çã♦ ♥r ♣♦r ♠♦ sã♦ tr♠♦♥r ♦♥tr♦♠ ♠ tr♦ rt♦r sã♦ s♦ ♥♦ t♦♠s ♣tçã♦ ♣rtr s rs ♦r♥s ♣r♦♥♥ts s s♥ts ♦♥tstt♣trr♠♥sst♥s♦♥P♥t♥①♣♣ ss♦ ♠ tt♣♥♦s♦♦♠q♠q♠♥rss♦ ♠

♠♦r s♦ ♥♦ ♥t♥t♦ ♦♥sst ♠ ♦♥sr ♠♥tr ♦ ♣s♠ ♦♥♥♦ ♥s ♦♥çõs

♥ssárs ♣r q rçõs sã♦ ♣♦ss♠ ♦♦rrr rss ♥sts ♣r♦ss♦s

ss♣çã♦ q ♦♦rr♠ r♥t ♦ ♣r♦ss♦ ♦♥♥♠♥t♦ ♦ ♣s♠ ♠♣♠ q st

♦♥♥♠♥t♦ r t♠♣♦ s♥t ♣r q ♦ t♦♠ ♣♦ss sr s♦ ♦♠♦ ♠ rt♦r ♥r

♠ ♣rtr á ♦ ♠♥s♠♦ tr♥s♣♦rt ♥♦ q ♣rtís ♥r ♦ ♣s♠ sã♦

♣rs r♣♠♥t r♥♦ ss♠ ♦ ♦♥♥♠♥t♦

♦♥♥♠♥t♦ ♦ ♣s♠ ♥♦ t♦♠

♣r♥í♣♦ ♦♥♥♠♥t♦ ♠♥ét♦

♣s♠ ♣r♠♥ ♦♥♥♦ ♥♦ t♦♠ q♥♦ ♦♦rr ♦ qír♦ ♥tr ♦rç

♦ r♥t ♣rssã♦ ♥ét ♦rç ♠♥ét ♦r♦ ♦♠ qçã♦

∇p = ×,

♦♥ p é ♣rssã♦ ♥ét ♦ ♣s♠ = µ−10 ∇ × é ♦rr♥t ♣r♦③ ♣♦ ♠♣♦

♠♥ét♦ ♣♦ r♥t ♣rssã♦ ♣r♠r ♦rç à ♣rssã♦ ♦ ♣s♠ é ♠

♦♥sqê♥ ♥tr ♦ ♦♠♣♦rt♠♥t♦ ás ♣rs♥t♦ ♣♦ ♣s♠ ♦ q é ♦rr♥t

s rq♥ts ♦sõs q stã♦ sts s ♣rtís q ♦♠♣õ♠ st ♠ ♦♣♦sçã♦ st

♦rç ♠♥ét q ♥♦ s♥t♦ ♦♥♥r ♦ ♣s♠ é ♣r♦③ ♣ ♥trçã♦ ♥tr ♦

♠♣♦ ♠♥ét♦ ♦rr♥t q ♣r♦rr ♦ ♣s♠

♦ ♣rár♦ q s s ♦ ♠♥s♠♦ ♦♥♥♠♥t♦ ♦ ♣s♠ ♣rtr rçã♦

st ♠ ♦♠♦ ♦s ♣r♥♣s ♠♣♦s ♠♥ét♦s ♦rr♥ts s♣♦st♦s ♠s ♠♣♦rt♥ts

♦s qs t♠é♠ sã♦ ♠♦str♦s ♥ r sã♦ srt♦s

♠ ♠ ♣r♠r♦ ♠♦♠♥t♦ ♦ ♣s♠ é ♣r♦③♦ ♣rtr ♦♥③çã♦ ♦ ás ♥tr♦ ♥♦r

♠♠♥t r♦ê♥♦ ♦ tér♦ ♦♥t♦ ♥ â♠r á♦ q ♦♦rr ♦ à t t♥sã♦

♥③ ♣ tr♥srê♥ ♥r ♦♥t ♠ ♠ ♦♥♥t♦ ♣t♦rs t ♣♦tê♥

♣r ♦ ♥r♦♠♥t♦ ♥tr ♦t q ♦ t♦♠ ♦♠♦ ♠ tr♥s♦r♠♦r ♦♥ ♦♥

♣s♠ ♠ ♦r♠ ♥ t ♦♠♦ ♥r♦♠♥t♦ s♥ár♦ ♣ós ♦♥③çã♦ ♦ ás ♦♦rr

♥tã♦ ♠ q rs ♥ rsst st ♠♦♠♥t♦ ♠ q ♦ ♣s♠ é ♣r♦③♦ ♣s

s♥♦ rr ♥tã♦ ♠ ♥t♥s ♦rr♥t ♥♦ ♣s♠ ♦rr♥t ♣s♠ Ip ♣s♠

♦♠♠ t♠é♠ t③r ♦r♠çã♦ ♠ tr♠♦s ♣rssã♦ ♠♥ét B2/2µ0

♠♣♦ ♠♥ét♦

♣♦♦ BP

♦♥t♦r♦

♦♥rt

♠♣♦ ♠♥ét♦t♦r♦ BT

♦rr♥t ♣s♠ Ip ♦

♦rr♥tt♦r♦ JT

♦rr♥t♣♦♦ JP

♦♥ ♥tr ♠♣♦ ♠♥ét♦ ♥tr

aθR0

♦♥

♣s♠

r sq♠ ♠ t♦♠

é ♥tã♦ q♦ t♠♣rtrs strs ♦♥çã♦ ♥s♣♥sá ♣r q rçõs sã♦

♥r ♦♠♦ ♠♦str ♥ q ♣♦ss♠ ♦♦rrr ♦♠ rqê♥ st t♣ ♦ ♣r♦

ss♦ ♣r ♦♥tr ♦ t♦ ♦rt ♣rssã♦ ♥ét q ♠♥t ♦♠ t♠♣rtr ♦rç♥♦

♦ ♣s♠ s ①♣♥r t♥♦ ♣♦rt♥t♦ ♦ ♦♥♥♠♥t♦ ♦rç ♠♥ét s♠♣♥

♠ ♣♣ ss♥ st ♦rç sr ♠ ♦rrê♥ ♥trçã♦ ♥tr s ♦rr♥ts q r♠

♥♦ ♣s♠ ♦ ♠♣♦ ♠♥ét♦ ♥ ♣rs♥t Pr q ♦♥♥♠♥t♦ é ♥ssár♦ q

st ♠♣♦ ♠♥ét♦ ♣rs♥t s ♦♠♣♦♥♥ts ♠ ♥ rçã♦ t♦r♦ BT ♣r♦③

①tr♥♠♥t ♣♦ ♥r♦♠♥t♦ t♦r♦ ♦tr ♥ rçã♦ ♣♦♦ BP r ♣ ♣ró♣r

♦rr♥t ♣s♠ é♠ ♦rr♥t ♣s♠ ♥ rçã♦ t♦r♦ ♦ ♦♠♣♦rt♠♥t♦ ♦t♦

♦ ♣s♠ ♣r♠t ♦ sr♠♥t♦ ♠ ♦tr ♦rr♥t ♥st ♣♦ré♠ ♥ rçã♦ ♣♦♦ st

♦rr♥t sr ♠ ♦♥sqê♥ r ♠♥ét ♦ à ①stê♥ ♠ r♥t

r ♥s P♦r ♠ á ♥ ♠ ♦tr♦ t♦ ♥sá ♦ ♦♥♥♠♥t♦ ♦ q

st t s ♦♥s rts ♦♥ ♣s♠ t♥ ♠♥tr s r♦ ♠♦r

♠ rçã♦ à â♠r á♦ s♠r ♦ q ♦♦rr ♠ ♠ ♥ ♦rr♥t ♠rs♦ ♠ ♠

♠♣♦ ♠♥ét♦ ❬❪ st ♣r♦♠ ♣♦ré♠ ♣♦ sr ♦rr♦ ♣♦r ♠♦ ♠ ♦rr♥t

♦♥tr♦á q trss s ♦♥s rts s qs ♦r♠ ♠♣♠♥ts ♦♠ ♦ ♦t♦

rr ♠ ♠♣♦ ♠♥ét♦ rt ♣③ ♦♥tr♦r ♣♦sçã♦ ♦ ♣s♠

trtr á ♥ú♠rs rrê♥s rs♣t♦ t♦r t♦♠s ♥s qs ts

♠♣♦rt♥ts ♦ ♣♦♥t♦ st ís♦ s♦r ♦ ♣r♦ss♦ ♣♦ q ♦ ♣s♠ é ♦♥♥♦ ♠ t♦

♠s sã♦ ♣rs♥t♦s ❯♠ ①♣♦sçã♦ át s♠♣s qtt q trt ♥ã♦ s♦♠♥t

ís t♦♠s ♠s t♠é♠ ♠ts qstõs r♦♥s à ár ♥r ♣♦ sr

♥♦♥tr ♠ ❬❪ Pr ♠ st♦ ♠s ♣r♦♥♦ q ♥♦ ts ér♦s ♠s

s rrê♥s ♠s tr♦♥s ❬ ❪ sã♦ r♦♠♥s

rçã♦ ♦ ♦♥♥♠♥t♦ ♠ t♦♠s

♣ós ♦ ♦♥♥♠♥t♦ ♦ ♣s♠ ♠ ♠ ♣r♠r♦ ♠♦♠♥t♦ á ♥ ♦tr♦s ♦stá♦s ♦s

qs ♠♥♦♥♠♦s ♣♥s ♦s sr♠ ♥♦s ♣r q s ♣♦ss t♦r♥r ♠ r

♣♦rt♥t♦ ♣r♦♣♦r♦♥r ♠ ♥♦r♠ ♥ç♦ ♥tí♦ à ♣♦ss ♦t♥çã♦ ♥r ♣♦r

♠♦ sã♦ ♥r

Pr♠r♠♥t á s ♦♥srr ♠♣♦ss ♦ t♦♠ r ♦♠♦ tr♥s♦r♠♦r

♣♦r ♠t♦ t♠♣♦ ♦ q ♠♥tr ♦rr♥t ♣s♠ ss♥ ♣r ♦ ♦♥♥♠♥t♦ ♣♦s

♣r ss♦ ♦ ♦r t♥sã♦ ♣ ♦ ♥r♦♠♥t♦ ♥tr tr q rsr ♦♥st♥t♠♥t

♣♦r ♠ ♦♥♦ ♣rí♦♦ ♦ q é ♠♣♦ssí ♦ ♣♦♥t♦ st ♣rát♦ ♥trt♥t♦ é ss♥

♣rs♥ç ♦rr♥t ♣s♠ ♣r ♦ ♦♥♥♠♥t♦ ♦ q rqr ♠ s♦çã♦ tr♥t ♣r

♦ ♣r♦♠ ♦♥stt ♠ s ♥s ♣sqs ♠ ♣s♠s t♦♠s ♦ s♥♦♠♥t♦

♠♥s♠♦s ♣③s ♠♥tr ♦rr♥t ♣s♠ qr ♦ ♣s♠ s ♠♥s♠♦s

s s♠ ♥ ①tçã♦ ♦♥s ♥♦ ♣s♠ ♥ tr♥srê♥ ♣rtís ♥trs st

♣♦r ♠♦ s♣♦st♦s ①rs ♠ ♣rtr ♦♠ s♦rt ♦♥s é♥ ❬❪

s t③çã♦ ♥ rçã♦ ♦rr♥t ♥♦ q♠♥t♦ ♦ ♣s♠ ♠ sst♠s ♦♥♥

♠♥t♦ ♠♥ét♦ s t♦r♥♦ ♠ ár ♣sqs t ♠ ♠t♦s t♦♠s ♠ s♣ ♥♦

❬❪

❯♠ ♦tr♦ r♥ ♦stá♦ ♣r ár ♣s♠s sã♦ é ♦ tr♥s♣♦rt ♣rtís

♥r ♦r q ♦♦rr ♥♦ ♣s♠ r♥♦ r♣♠♥t ♦ ♦♥♥♠♥t♦ s♣ ê♥s

sr ♦ tr♥s♣♦rt ♥ô♠♦ ♦ tr♥t♦ ♦ q é ♠t♦ ♠♦r ♦ q ♦ tr♥s♣♦rt

áss♦ q ♦♦rr ♠ ss ♥tr♦s ♥trt♥t♦ ♦♠ s♦rt ♠ ♥♦♦ r♠

♦♥♥♠♥t♦ t♠é♠ ♦♥♦ ♦♠♦ ♠♦♦ ❬❪ ♦ ♠ s♥t ♦♥trçã♦

♣r ♦ s♥♦♠♥t♦ t♦♠s ♠ s♣ ♦ ❬❪ ♦ ♠♦♦ ♦r♠s ♠ ♦rt

r♥t ♣rssã♦ ♠ rt rã♦ ♦ ♣s♠ t♠é♠ ♦♥♦ ♦♠♦ rrr tr♥s♣♦rt

♣♦s ♦ tr♥s♣♦rt tr♥t♦ é s♥t♠♥t r③♦ ♥st rã♦ ♦♠ rçã♦ ♦

♣r♦ss♦ trê♥ ♥♦ ♣s♠ q ♦♥tr s♥t♠♥t ♣r ♦ tr♥s♣♦rt ♥ô♠♦

s♦rs ♦♦rrê♥ tr♠♥♦s ♠♦♦s ♥♦ ♣s♠ ♦s ♠♦♦s úst♦s ♦és♦s

t♠é♠ ♦♥♦s ♣♦r ①♦s ③♦♥s ❩ t rqê♥ ♣♦ré♠ rqê♥

♠ r♦r rqê♥ rçã♦ í♦♥s q sã♦ ♣③s s♣r♠r ♠ t♣♦ s♣

trê♥ trê♥ ♦♥s r ❲ ①tçã♦ ♥tçã♦ ①♣r♠♥t

♠ t♦♠s ♠ ♦♠♦ ♦♠♣r♥sã♦ ♦ ♣r♦ss♦ t♦♦r♥③çã♦ rs♣♦♥sá

♣ s♣rssã♦ trê♥ t♠ s♦ ♦ ♥t♥s ♣sqs tór ①♣r♠♥t ❬❪

♦tçã♦ ♣r ♦ st♦ ♠♦♦s ♦és♦s

rst♦s ♦t♦s

♦♦s ♦és♦s ①s rqê♥s ♠ s♣ tê♠ s♦ ♦ ♥t♥s ♥s

tçã♦ tór ①♣r♠♥t ♥ã♦ s♦♠♥t ♦ ♦ s ♣♣ ♥ s♣rssã♦ tr♥s♣♦rt

tr♥t♦ ♠ t♦♠s ❬ ❪ ♠s t♠é♠ ♦ s rçã♦ ♦♠ t♦♠♦♦s

♦♠ é♥ rés♥ ♦③♦ ♥♦ ♥sttt♦ ís ❯♥rs ã♦ P♦ ❯P ♥♦ rs

♦♥♥♠♥t ♥tr♥t♦♥ r♠♦♥r ①♣r♠♥t t♦r é ♦ ♣r♠r♦ rt♦r sã♦ ♠ ♦♥s

trçã♦ ♥ s♦ ♥ t♥♦♦ t♦♠s ♦③♦ ♠ rs ♥ r♥ç st s♥♦♣r♦t♦ ♣r ♣r♦③r ❲ ♣♦tê♥

r♥s♣♦rt rrrrt tr♥

é♥ ♥③♦s ♣ ♣rssã♦ ❬❪ ♦srçã♦ t ♠♥t♦r♦♥â♠

♦ ♠♦♦s ♦és♦s ♣♦ tr t♠é♠ ♣çõs ♥ósts s♣♠♥t

♥♦ q s rr à s♣tr♦s♦♣ ♠ ❬❪ st♦ é ♥♠♥t ♠♣♦rtâ♥

♥ ♥stçã♦ trê♥ ♥♦ ♥ rçã♦ trê♥ ♠ s♣tr♦s♦♣

♣r ♥♦str ♦ t♦r sr♥ç q ❬ ❪ ♦ q s rr à rçã♦ ①♣r♠♥t

♦s ♠♣♦rt♥ts ①♣r♠♥t♦s ♠ r♥ts t♦♠s ❬❪ ♥ã♦ só ♦♥r♠r♠ s

①stê♥ ♠s t♠é♠ rr♠ s♣t♦s rs ♦♠ rçã♦ s ♦③çã♦ ♥t♥s

♣r♥♣ ♠♦tçã♦ st ts rs ♥♦ ♦t♦ ♦♥trr ♥ q ♥rt♠♥t

♦♠ rst♦s tór♦s qtt♦s q♥ttt♦s q ♠ ♥t♥r ♠♦r ♦ ♦♠♣♦r

t♠♥t♦ ♠♦♦s ①s rqê♥s ♦ ♠♥s♠♦ s♣rssã♦ trê♥ ♣♦r ♦♥s

r ♠ t♦♠s ♦♥♠♥t♦ ís♦ s♦r st ♠♥s♠♦ ♠ ♦♠♦ ♦ st♦

tr♥s♣♦rt ♥ô♠♦ ♣rt♥♠ ♠ ♦♥♥t♦ s♦s ♥tí♦s sr♠ s♣r♦s ♣r ♦

s♥♦♠♥t♦ ♦ ♣r♠r♦ rt♦r sã♦ ♥r ♣♦r ss♦ ♣♦♠ s♠♣♥r ♠ ♣♣

♠♣♦rtâ♥ ♦♥ô♠ s♦ ♦ó ♠ s ♦ ♥♦ tr♦

st s♥t♦ ♣♦r ♦♥srr♠♦s ♠ t♠ ♠t♦ ♠♣♦rt♥t ♥st♠♦s rs♦s t♦s

♥♦s ❬❪ ♦s qs ♣♦ t♦ str♠ rt♠♥t ♦s à ❲ tê♠ s♦ ♦

♥t♥s ♣sqs tór ①♣r♠♥t ♠ t♦♠s ①♦ ♠ r sssã♦ s♦r

sts t♦s ♦s ♣r♥♣s rst♦s ♦rr♥ts q ♦t♠♦s

• ♦tçã♦ qír♦ s ♣ ♣rs♥ç ♠ ♠♣♦ étr♦ r qír♦

r♥ts t♠♣rtr ♥s t♠ s♦ ♥st ♣r♥♣♠♥t ♦ s

♦♦rrê♥ r♥t ♦r♠çã♦ rrr tr♥s♣♦rt ♥♦ r♠ ♦♥♥♠♥t♦

♠♦r♦ ♦ ♠♦♦ ❬❪ ♥st♠♦s ♥ê♥ r♦tçã♦ ♣♦♦ t♦r♦

♥ rqê♥ ♦s ♠♦♦s ♣ít♦ ♣rtr ♦ ♠♦♦ rsst

♥ ♣r ♠ ♦ ♦r♠ ♦♥sr♦s três t♣♦s qír♦ át♦

s♦tér♠♦ s♦♠étr♦ ♦ s♦ór♦ ♦s qs sã♦ srt♦s ①♦

qír♦ át♦ P♦r str ♥♦ r♠ rtríst♦ ♣r♦♣çã♦ ♦♥

s s♦♠ ♥ã♦ s♦♠♥t ♠ ♣s♠ ♠s t♠é♠ ♠ ♦s ♥tr♦s ss♠ ♦♠♦

té♥ ♥♠♦s♥t♦②r♦②♥♠s

s♣r♦ ♥♥ rt♠♥t rqê♥ ♦s ♥♦ s♥t♦ ♠♥tr

st ♦♥♦r♠ q ♥ã♦ ♠♣♦rt♥♦ ♦ s♥t♦ r♦tçã♦ ♥♦ ①st

♠ ①♦ ♣♦♦ qír♦ á ♠ s♥ s♦çã♦ q ♦rrs♣♦♥ ♦ r♠♦

s♦♥♦r♦ í♦♥s rqê♥ ♥♦r♠③ ♦r♠ ♦ ♥rs♦ ♦ t♦r s

r♥ç q−1 ♠ rçã♦ à rqê♥ ♦s s ♥♦♥tr ♥♦ ♦r ♥tr♠ár♦

♥tr ♦ r♠♦ ♦s ♦s ❩ st út♠♦ r♠♦ rqê♥ ♥ ♥♦r♠♠♥t

♥ã♦ t♠ s rqê♥ tr ①s♠♥t ♣r st qír♦ ♦ q ♥

q ♦ ♣♦♥t♦ st tór♦ ♦ ①♦ ♦r q s♠♣♥ ♠ ♣♣ ♠♣♦rt♥t

♥♦ ♠♥s♠♦ ♦r♠çã♦ ♦s ❩

qír♦ s♦tér♠♦ ♣r♠♥t♦ ❩ stás ♣♦ré♠ rqê♥ ♥t

q♥♦ á r♦tçã♦ ♣♦♦ é ♣r♥♣ ♦♥sqê♥ st t♣♦ qír♦ ♥♦

q ①♦ ♦r ♣r♣♥r ♦ ♦ r♥t r t♠♣rtr

sr ♦♥sr♦

qír♦ s♦♠étr♦ ♦ s♦ór♦ Pr♠t ♣rs♥ç ①♦s ③♦♥s ❩

♥stás ♠s♠♦ ♣r ♦ s♦ r♦tçã♦ ①s♠♥t t♦r♦ ❬❪ ♦ q

rt ♦r♠ ♦♥r♠ rçã♦ sts ♦♠ ①♦s ♥♦♠♣rssís ♥♦ ♣s♠ ❯♠

s ♥♦çõs st ts ♦ ♥str ♦ t♦ ♦ qír♦ s♦♠étr♦ ♦♠ r♦tçã♦

♣♦♦ ♥♦s ❩

♠ rs♠♦ ♦ ♦♥srr♠♦s r♥ts qír♦s ♦♠ r♦tçã♦ ♥♦♥tr♠♦s três s♦

çõs ♦♠ ♦rs rqê♥s st♥t♦s rtríst♦s ♥ô♠♥♦s ís♦s r♥ts

q ♦♦rr♠ ♥♦ ♣s♠ s ♥ô♠♥♦s ♦rr♠ rtrísts ♣rs ♦ ♣s♠

♦ t♦♠ s qs stã♦ r♦♥s à rtr ♦és ♦ ♠♣♦ ♠♥ét♦ ♥♦

t♦♠ à ♣r♦♣çã♦ ♦♥s tí♣ ♦s ♦ ♠♣♦rt♥t ♣râ♠tr♦ ♦ t♦♠

♦♥♦ ♦♠♦ t♦r sr♥ç à ♦♥çã♦ ♦♠♣rss ♦ ♥♦♠♣rss

q ♦ ♣s♠ ss♠

• t♦s ♠♥ét♦ ♦ t♦s rs t♦s s♦s ♣♦r r♥ts

♥s t♠♣rtr qír♦ ♠ ♠♦♦s ♦és♦s ①s rqê♥s

rt ts

♦r♠ ♥st♦s ♥st ♣rt ♣ít♦s ♦ss♦s ♦t♦s sã♦ ♣rs♥tr ♦s

♠♥s♠♦s ís♦s ♥♦♦s ♥s ♦sçõs ♦s ①♣♦r ♦ ♦♥tú♦ ♦r♠ ♠s

s♠♣s ♦♠♣r♥sí ♣♦ssí ♥r t♦s ♥ét♦s Pr t♥r ts ♦t♦s

♦r♠ t③♦s ♦s ♠♦♦s ♦ ♠♦♦ ♦s ♦ ♠♦♦ r♦♥ét♦ ♦r♠ q

♦♥sstê♥ ís ♣r♦♣♦r♦♥ ♣♦ ♠♦♦ ♦ ♠♦r ♥r ♦ ♠♦♦

♥ét♦ ♦♥tr♠ ♣r ♥♦ss ♠t ♦ ♦♥srr ♦ t♦ r ♠♥ét

♦ t♦ r♦tçã♦ qír♦ ♥♦s ♣♠♦s ♦srr q sts t♦s stã♦

r♦♥♦s s♣t♦s rt♦s sss ♦s ♠♦♦s ♦s rst♦s ♦rr♥ts s

s♦ sã♦ ♠♦str♦s sr

♦♦ ♦s ♦s st ♠♦♦ é r♥t ♦ ♠♦♦ ♦♥

sr♦ ♥tr♦r♠♥t ♣♦r s r③õs ♣r♠r♦ ♦s ♦s ♦s rtríst♦s

í♦♥s étr♦♥s sã♦ ♦♥sr♦s ♠ r♠s st♥t♦s s♥♦ ♦ à

♥trçã♦ ♥tr s ♣rtís ♦ ♣s♠ ♦ ♠♣♦ ♠♥ét♦ ♠r♦só♣♦ ♣rs♥t

♥st r♥ç ♥tr ♣rssã♦ ♣r ♣r♣♥r ♦♠ rçã♦ ♦ ♠♣♦

♠♥ét♦ qír♦ sr ♦♥sr ♦♠ ♥sã♦ ♦ t♥s♦r s♦

s ♣r ❬❪ st ♦♥srçã♦ ♣r♠t ♦tr rst♦s ♦♥③♥t ♦♠

t♦r ♥ét ❬❪ q ♠ ♦♥srçã♦ ♦ t♦ ♥s♦tr♦♣ ♣rssã♦

♠♦r ♦r♠ ♥ã♦ tã♦ r ❯♠ ♦r♠ tr♥t ♣③ srr ♦ t♦

♥s♦tr♦♣ é trés ♦ ♦r♠s♠♦ ❬❪ ♠ ♦r s♠♣ ♦

ts♠♦ ♥ ①♣♦sçã♦ ♦ ♣ít♦ ♦♥sr♠♦s ♦ ♠t q → ∞ ♥t③♥♦

ss♠ ♣♥s s qstõs ♦ t♦ ♦s r♥ts ♥s t♠♣rtr

♥♦s ♠♦♦s ♦rrs♣♦♥♥ts ♦ r♠♦ ♦és♦ ♦ r♠♦ ♠s ①

rqê♥ ❩ ♦ à ♥s♦tr♦♣ ♣rssã♦ ♣rtr ♦rrçã♦ ♣r ♦

♦♥t át♦ t♦ í♦♥s ♥ ①♣rssã♦ ♣r rqê♥ ♦s

♦♦ r♦♥ét♦ ♠ ♦♠♦ ♠t♦♦♦ ss♠ ♦♠♦ qqr ♠♦♦

♥ét♦ ♦tr ♥çã♦ strçã♦ s ♣rtís q ♦♠♣õ♠ ♦ ♣s♠ ♣rtr

rs♦çã♦ qçã♦ ❱s♦ qçã♦ ♥ét s♠ ♦ tr♠♦ ♦sõs ❬❪

P♦ré♠ s♣♠♥t ♣r st ♠♦♦ st qçã♦ é s♥♦ ♥ts sr

rs♦ ♣rtr t♦r r♦♥ét ❬❪ q é ♣rs♥t ♥♦ ♣ít♦

♣ ♥ ♦r♠ st ♠♦♦ ♥♦ ♣ít♦ ♦♠ t③çã♦ ss ♠♦♦

♣r ♥r t♦s ♠♥ét♦s ♠ ♠♦♦s ①s rqê♥s ♦t♠♦s três

s♦çõs s♠rs às ♦ts ♣ ♥♦r♣♦rçã♦ r♦tçã♦ qír♦ ♥♦ ♠♦♦

♣ít♦ ♦r♠ q ♣r♦♣♦♠♦s ♥st ts ♠ tr♥t

♠ ♣r♥í♣♦ ♣r ♦ st♦ t♦s r♦tçã♦ qír♦ ♠ ♠♦♦s

①s rqê♥s q♥♦ st st♦ ♦r ♦♠♣①♦ ♠ ♠♦♦s ♥ét♦s s♦çã♦

♥♦ r♠♦ s♦♥♦r♦ ♦t q♥♦ s ♦♥sr q ♥t♦ ss♠ ♦♠♦ ♦ ♠♦rt♠♥t♦

♥ ♦s ♠♦♦s ♣rt♥♥ts ♦s três r♠♦s ♦r♠ ♦t♦s ♥♦ ♣ít♦

♥tr s ♣r♥♣s ♦♥sqê♥s ss ♣♦ ♠♦rt♠♥t♦ ♥ stá

♦r♠çã♦ ♥çã♦ strçã♦ í♦♥s ♠♥çã♦ ♦ ♣♦t♥ tr♦stát♦

♣rtr♦ ♦♠ ♦ t♠♣♦ ♠♣t♥♦ ♥t♠♥t ♥♦ ♦♥♥♠♥t♦ ♦♠ ①t♥

çã♦ ♦s s♦rt♦ ♣♦r ♥ ❬❪ ♦ ♠♥s♠♦ ♠♦rt♠♥t♦

♦ rs♠♥t♦ ①♣♦♥♥ ♠ ♦♥ tr♦♠♥ét ♠s♠♦ ♠ ♣s♠s ♥ã♦

♦s♦♥s t♠é♠ ♦♥♦ ♦♠♦ ♠♦rt♠♥t♦ ♥ é ♠ ♦♥sqê♥

♥trçã♦ ♦♥♣rtí q ♦♦rr ♥♦ ♣s♠ ♦ à ♣rs♥ç ♣rtís

♦♠ ♦s ♣ró①♠s às ♦ s ♦♥

rqê♥ ♦s ♠♦♦s úst♦s ♦és♦s s

♣r♠r ①♣rssã♦ ♥ít ♣r rqê♥ ♦s ❬❪ ♦t ♣♦r ❲♥s♦r t

♠ ♦ s♥♦ ♣rtr ♦ ♠♦♦ ♥♦ q s ♦♥sr ♦ ♣s♠

♦♠♦ ♠ ♦ ú♥♦ í♥ át♦ γ = 5/3 ♣♦ sr srt ♦♠♦

ω2♠ =

(

2 +1

q2

)

γT

mi,

♦♥ T = Ti + Te é t♠♣rtr ♦ ♣s♠ Ti Te sã♦ rs♣t♠♥t t♠♣rtr

í♦♥s étr♦♥s mi é ♠ss ♦s í♦♥s ♥♦ s♦ ♣s♠ r♦ê♥♦

P♦str♦r♠♥t ♠ st♦s r♦♥â♠♦s ❬❪ sts ♠♦♦s t♠é♠ ♦r♠ ♥♦♥tr♦s

♣ós ♠s és ♣rtr t♦r ♥ét ❬ ❪ ♦♥sr♥♦ t♦s ♦♥s

①♣♦ ♦ ♦♥♦ st ts ①♣rssã♦ ♥ét ♦t ♣r rqê♥ ♦s ♦

ω2♠ = 2

(

γi + γe

Te

Ti+O(q−2)

)

Ti

mi,

♦♥ γi = 7/4 é ♦ í♥ át♦ t♦ ♣r í♦♥s γe = 1 é ♦ í♥ át♦ ♣r

étr♦♥s ❯♠ rçã♦ tr♥t ❬❪ ♣rtr t♦r ♦s ♦s ♦♥sr♥♦ í♦♥s

♥♦ r♠ ♦ γi = 5/3 étr♦♥s ♥♦ r♠ át♦ s♦tér♠♦ ♦♠ γe = 1

♠♦str♦ sr ♣♦ssí r♦rr ♦ rst♦ ♥ét♦ q ♣rtr t♦r ♦s ♦s

♠ t rst♦ r♥ç ♥tr ♦s ♦♥ts át♦s t♦s í♦♥s étr♦♥s s

à ♥♦r♠ r♥ç ♥tr s ♠sss sts s s♣és q ③ ♦♠ q rs♣♦st

♦s étr♦♥s às ♣rtrçõs s ♠t ♥q♥t♦ ♦s í♦♥s ♣♦r rs♣♦♥r♠ ♠s ♥t♠♥t

♠ st♦s ♦ t♦ ♥♦♠♦♥ ♣rssã♦ ♦ ♣rs♥ç ♦ ♠♣♦ ♠♥ét♦

t♦ ♥s♦tr♦♣ ♣rssã♦ ♠ qstã♦ ♣♦ sr srt♦ ♣♦r ♠♦ ♦ t♥s♦r s♦s

♣r π‖ ♥s qçõs ♦ ❬❪ ♦♥♦r♠ ①♣♦ ♥♦s ♣ít♦s

r♥③çã♦ st ts

♣ít♦s

st ts é ♦♠♣♦st ♣♦r ♣ít♦s ♣ê♥s ♦s ss♥t♦s sã♦ s♥t♠♥t srt♦s

sr ♦ ♣ít♦ ♦s ♣r♥♣s ♠♦♦s ís♦s ♣r♦♥♥ts t♦r ♦s t♦r

♥ét ♣s ♣s♠s ♠♥t③♦s sã♦ rs♦s rqê♥s ♦♠♣r♠♥t♦s rt

ríst♦s ♥♠♥ts ♣râ♠tr♦s ♣s♠s ♠ t♦♠s sã♦ ♥♦s ♣♦r ♠ ♠ r

rs♠♦ s♦r t♦r tr♥s♣♦rt ♠ t♦♠s é ♣rs♥t♦ ♥r ♦ ♦♥tú♦

q ♣r s♣ ♦s ♣ró①♠♦s ♣ít♦s é ♠ rtríst st ♣ít♦ q t♠

♣♦r ♦t♦ rsã♦ ♦♥t♦s ♥♠♥ts ♠♣♦rt♥ts ♠ s ♣rtr ♦ ♠♦♦

♦ ♠♦♦ ♦s ♦s ♥♦ ♣ít♦ ♣rs♥t♠♦s ♦ st♦ ♠♦♦s

úst♦s ♦és♦s ①♦s ③♦♥s ❩ ♦tçã♦ qír♦ t♦s ♠♥ét♦s

sã♦ ♦♥sr♦s ♥ss ♣ít♦ á ♥stçã♦ ♥ét ♣r♥♣ t♠ st ts t♦s

♠♥ét♦s ♥ê♥ ♦ ♠♦rt♠♥t♦ ♥ ♥♦s sã♦ ♦ ♦♥tú♦ ♦ ♣ít♦

P♦r ♠ ♥♦ út♠♦ ♣ít♦ ♣rs♥t♠♦s s ♦♥sõs ♥tís st ts ♣r♦♣♦sts

♣r ♦♥t♥çã♦ ♣rs♥t ♥ ♣sqs rs ♣r♦t♦s ♣r tr♦s tr♦s

♣ê♥s

♣ê♥ q ♦ s♥♦ ♣r ♥s ♦♥st ♦♥t♠ s ♦♥st♥ts ♥♠♥ts

ís ♣rt♥♥ts st ts ♠s s ♣r♥♣s r♥③s t♦♠s ♥♦ss♦

♥trss ♥ts ♠♣♦rt♥ts çõs ①♣rssõs úts ♦♥♦s

á♦s ér♦s t③♦s ♣r♥♣♠♥t ♥♦ ♣ít♦ sã♦ ♣rs♥t♦s ♥♦s ♣ê♥s

r♥t ♦ ♣ít♦ s ♦ ♣ê♥ q ♣rs♥t çã♦ ♦♥s ①♣rssõs

♦ á♦ ♥trs r♦♥s à ♥çã♦ ①♥ à ♥çã♦ s♣rsã♦ ♣s♠

♥♠♥t ♥♦ ♣ê♥ ♥t♦s ♦♥rê♥s ♥♦♥tr♦s s♦s t ♦♦rçõs

♣çõs r③♦s r♥t ♦ ♣rí♦♦ ♣ósrçã♦ str♦ ♦t♦r♦ ♥ ár

ís ♣s♠s sã♦ rs♠♠♥t srt♦s

♣ít♦

ís t♦♠s

st ♣ít♦ st♥s à sssã♦ ♥s ♦s tó♣♦s ís t♦♠s ♦ ♦♥

tú♦ srá ♣♦ ♦s ♠♦♦s s♣í♦s q trt♠♦s ♥♦s ♣ró①♠♦s ♣ít♦s ♣rs♥t

♠♦s ♠ t①t♦ rrê♥ rátr ♠s r ♥♦ s♥t♦ ♠ q s t♦rs q trts

á ♠ sts ♥ ár ís ♣s♠s sã♦ ♣③s srr ♥ú♠r♦s ♣r♦ss♦s

♠♥s♠♦s ís♦s q ♦♦rr♠ ♠ ♣s♠s ❯♠ srçã♦ s ♥çõs r♥③s ♥

♠♥ts ♦ ♣s♠ ts ♦♠♦ ♦♠♣r♠♥t♦s t♠♣♦s rtríst♦s ♦♠♦ ♣♦♥t♦ ♣rt

♣r st ♣ít♦ ♥t♠ t♠s ♥♠♥t ♠♣♦rtâ♥ ♣r ♦♠♣r♥sã♦ st

ts ♦s qs s rr♠ strtr ♦ ♠♣♦ ♠♥ét♦ ♦ ♠♥s♠♦ tr♥s♣♦rt

♣rtís ♥r ♠ t♦♠s t♦r ♥ét ♣rtís rrs ♦r♠

çã♦ ♠r♦só♣ ♦s srt ♠ tr♠♦s ♠♦♦s t③♦s ♣♦str♦r♠♥t st

♣ít♦ sã♦ ♣rs♥t♦s ♠ s

P♦r qstõs s♠♣ ♦♥sr♠♦s ♣♥s ♣s♠s r♦ê♥♦ ♥ã♦ ♥♦ ♠

♦♥t rçõs sã♦ ♦r♠ q Z = 1 é ♦t♦ ♦ ♦♥♦ t♦ ts ♣s♠ é

♦♠♣♦st♦ ♣♦r ♣♥s ♦s s ♦s s♣és ♣rtís ♥①♦ss ♣♦r α = i, e

í♦♥s étr♦♥s ♠é♠ ♦♠♦ é ♦♠♠ ♠ r♥ ♣rt trtr ♠ ís ♣s♠s

♦t♠♦s ♣rát s♣r♠r ♦♥st♥t ♦t③♠♥♥ k ♦ ♦r é ♠♦str♦ ♥♦ ♣ê♥

♦r♠ q ssttçã♦ kT → T ♦ t③ ♦ ♦♥♦ st ts

♦♠♣r♠♥t♦s t♠♣♦s rtríst♦s ♦ ♣s♠

sr st♠♦s ♥♠♦s ♥s ♦s ♦♠♣r♠♥t♦s t♠♣♦s rtríst♦s ♦ ss

♥rs♦s s rqê♥s rtrísts ♣rs♥ts ♠ ♣s♠s t♦♠ ❯♠ sssã♦ ♠s

t ♦s ss♥t♦s trt♦s q é ♣rs♥t ♣♦r ♥ ❬❪

♥♠♦s ♥♦ss sssã♦ ♣ rqê♥ ♦trô♥ q ♣♦ sr ♥ ♦♠♦

ωcα =eB

mα,

♦♥ e é r ♠♥tr mα é ♠ss ♣rtí ♦ t♣♦ α B é ♦ ♠♣♦ ♠♥ét♦

rqê♥ ♦trô♥ é ♠ ♠ ♠é r♣③ ♦ ♠♦♠♥t♦ s ♣rtís ♠ t♦r♥♦

s ♥s ♦rç ♠♣♦rt♥t ♥♦tr q ♦♠♦ me ≪ mi ωce = (mi/me)ωci ≫ ωci

♦t♥♦ ♦♥♣çã♦ ♦s í♦♥s étr♦♥s rtr③♦s ♣♦r t♠♣rtrs

♣ró♣rs Ti Te é ♦♥♥♥t ♥r ♦ tér♠

v2Tα=

2Tα

mα,

♦♥ vTe ≫ vTi ♣♦s ♥♦r♠♠♥t Ti ∼ Te vTe = (Te/Ti)(mi/me)vTi

♦ tér♠ rqê♥ ♦trô♥ s r♦♥♠ ♣♦r ♠♦ ♠ r♥③

♦♥ ♦♠♦ r♦ rçã♦ ♦ r♦ r♠♦r q r♣rs♥t ♦ ♦♠♣r♠♥t♦ r

tríst♦ ♦s r♦s s órts ♣rtís ♠ t♦r♥♦ s ♥s ♦rç ♦ ♥s ①♦ ♦

q é ♥♦ ♦♠♦

ρα =vTα

ωcα

.

♦♠♦ ρe ∼√

me/miρi ≪ ρi ♦ t♦ r♦ r♠♦r ♥t♦ ♣r étr♦♥s ♥ã♦

s♠♣♥ ♠ ♣♣ ♠♣♦rt♥t ♠ ♠♦♦s ①s rqê♥s ♣♦r st r③ã♦ ♦♥sr♠♦s

♥s t♦rs ♦t♠ ♥çã♦ tr♥t ωcα = eαB/mα ♦r♠ q ♣r étr♦♥s ee =−e rqê♥ ♦trô♥ t♦r♥s ♥t ♦ ♦♣tr♠♦s ♣ ♦♥♥çã♦ rqê♥s s♠♣r♣♦sts rsst♠♦s ♥trt♥t♦ ♥ss ♠♦r t♥çã♦ ♥♦s á♦s ér♦s

♥♠♦s t♠♣rtr ♠ tr♠♦s ♥r ♥ét ♠é Kα = mαv2

Tα

/2 = Tα ♣♦ré♠ ♥çã♦ v2Tα

= Tα/mα t♠é♠ é st♥t ♠♣r ♥ trtr♥t r♠♦r s

ρe ∼ 0 ♥♦s ♣ít♦s

é♠ ♦ tér♠ á s ♦trs ♦s ♣rtr ♥trss ♣r ♦ st♦

♠♦♦s ①s rqê♥s ♠ ♣s♠s ♠♥t③♦s ♣r♠r s ♣♥♥t

♥s ♣rtís ni ≈ ne é ♦ é♥

cA ŠBõ0nimi

,

♣rs♥t ♥♦ st♦ ♦♥s é♥ ❲ ❬❪ q ♣♦ss ♠♣♦rt♥ts ♣çõs ♠ ♣s♠s

♥tr s tr♠♥çã♦ ♦♠tr ♦ ♠♣♦ ♠♥ét♦ ❬❪ sts t♣♦s ♦♥ q

♣♦ss♠ ♠ ♠♣ ♠ ssçã♦ sr♠ ♦ ♣rtrçõs ♦ ♠♣♦ ♠♥ét♦

q t♠é♠ sã♦ rrs sss ♥ trtr ♦♠♦ t♥sã♦ ♦♠♣rssã♦ ♦ t♦rçã♦ s

♥s ♦rç s♥ ♦ ♥trss q é st♥t t③ ♦ ♦♥♦ st ts

é ♦ s♦♠ ♥♦ ♣s♠ q s r♦♥ à t♠♣rtr ♥s ρ ♣rssã♦ p

♣♦r

cs ≈√

γiTi + γeTe

mi=

√

γp

ρ∼ vTi

,

♦♥ γi γe sã♦ ♦s ♦♥ts át♦s í♦♥s étr♦♥s ♦ ♥tr③ ♠ss

sts ♣rtís ♦s ♦rs ♠s r③♦ás ♣r sts ♦♥ts sã♦ γi = 5/3 γe = 1

s♦ γ ♦ í♥ át♦ t♦t é t♦ ♥tr♦ ♦ ♦♥t①t♦ t♦r ♠ ♦ q é

st ♠s ♥t ♠

r③ã♦ qrát ♥tr s ♦s s♦♠ é♥ ♣♦ss ♠s♠ ♦r♠

r♥③ ♠ ♠♣♦rt♥t ♣râ♠tr♦ ♣r ♦ ♦♥♥♠♥t♦ ♣s♠s ♠ t♦♠s ♦ t♦r

t q é ♥♦ ♦♠♦ r③ã♦ ♥tr s ♣rssõs ♥ét p ♠♥étB2/2µ0 ♦ s

c2sc2A

∼ β

2γ, β =

2µ0p

B2.

♠ ♠t♦s ♠♦♦s ♦♠♦ ♦s q ♣rs♥t♠♦s ♥st ts ♦♥srs r♠s ①

♣rssã♦ rtr③♦ ♣♦r β = O(ε2) ♦♥ ε = r/R0 é r③ã♦ ♥tr ♣♦sçã♦ r ♦ r♦

é♥ ❲s

♠♦r ♦ t♦♠ ♠ ts r♠s ♥♦ ♠♦♦ é stá s♣r③r ♣rtr

çõs ♠♥éts ♥♦ st♦ ♠♦♦s ①s rqê♥s ❬ ❪ ♠é♠ é ♣rt♥♥t

♦♥srr ♥st t♣♦ st♦ ♦♥çã♦ qs♥tr q s ♣ ♥ô♠♥♦s

♦♠♣r♠♥t♦s rtríst♦s ♠t♦ ♠♦rs ♦ q ♦ ♦♠♣r♠♥t♦ ② q ♣♦

sr ♥♦ ♦♠♦

λ2Dα

=ε0Tα

nαe2.

♠ ♦♠♣r♠♥t♦s ♠♥♦rs ♦ q λDα ♦♦rr♠ ♦sçõs étr♦♥sí♦♥s ♠ rs♣♦st à

♣rs♥ç ♠♣♦ étr♦ ♦ ♣s♠ ① sr ♥tr♦ ♦♠♥t rqê♥ sts

♦sçõs é ♦♥ ♦♠♦ rqê♥ ♣s♠ ♥st ♦♥t①t♦ é ♥ ♦♠♦

ω2pα =

nαe2

ε0mα.

♦t q λDαωpα ∼ vTα ♦r♠ q ♣r rqê♥s ω ∼ vTi/R0 ♦♠♦ λDα ≪ R0 ω ≪ ωpα

♥st s♦ ♣s ♦♥çã♦ qs♥tr

sr ♠ r sssã♦ s♦r ♦s ♣r♥♣s t♠♣♦s rqê♥s rtríst♦ss

rr♥ts ♣r♦ss♦s ♦sõs é ♣rs♥t ♠ ♦sõs ♦♦♠♥s ♦♥♦r♠ ♦s ♠♦♦s

t③♦s ♠ ❬❪ ♣r♦♥♥ts t♦r ♥ét ss s♣rs q rqê♥

♦sõs étr♦♥í♦♥ s ν ∼ niσvTe ♦♥ σ = πb2 é sçã♦ ♦q tr♥srs b é ♦

♣râ♠tr♦ ♠♣t♦ ♠♦r ♥ tr♠♥çã♦ st rqê♥ rs ♥♦ á♦

b ♦♠♦ rrê♥ ♦t♠♦s ♣r ♦ t♠♣♦ rtrst♦ ♦sã♦ í♦♥í♦♥ étr♦♥

í♦♥ ♦s ♦rs ♠♦str♦s ♠ ❬❪ q q♥♦ ♦♥sr ♦♥çã♦ qs♥tr

ni ≈ ne = n sã♦ ♦s rs♣t♠♥t ♣♦r

τ =12π3/2ε20ln Λe4n

m1/2i T

3/2i , τ =

1√2

m1/2e

m1/2i

T3/2e

T3/2i

τ.

sts ①♣rssõs Λ ∼ nλ3D é ♦ ♥ú♠r♦ ♠é♦ ♣rtís ♥tr♦ ♠ sr ②

ln Λ é ♠ tr♠♦ ♦♥♦ ♦♠♦ ♦rt♠♦ ♦♦♠♥♦ ❬ ❪ ♦ ♦r ♥♠ér♦ s

♥♦♥tr ♥tr

st ♦♥t①t♦ ♦ ♠♦r ♦s t♠♣♦s rtríst♦s é ♦ ♣rí♦♦ ♦sã♦ í♦♥étr♦♥

τie ∼ τq ♦♥

τq =mi

2meτ,

♥tr♦ ♦ q ♦♦rr ♦ qír♦ tér♠♦ ♥tr í♦♥s étr♦♥s ❬❪ sr q ①st ♠

rrq ♥tr s rqê♥s ♦sã♦ ν ≪ ν ≪ ν

♠♣♦ ♠♥ét♦ qír♦ ♥♦ t♦♠

♦ t♦♠ ♦ ♠♣♦ ♠♥ét♦ qír♦ ♣♦ sr r♣rs♥t♦ ♣♦r ❬❪

= F∇φ+∇φ×∇Ψ,

♦♥ F Ψ sã♦ ♥çõs r♦♥s à ♦♠♣♦♥♥t t♦r♦ φ ♦ ♠♣♦ ♠♥ét♦ ♦ ①♦

♠♥ét♦ ♣♦♦ θ rs♣t♠♥t ♦t q ♦♥çã♦ ∇ · = 0 é t♦♠t♠♥t

stst ♣♦r

♥♦ ♥ã♦ á r♦tçã♦ qír♦ ❱ = 0 F = F (Ψ) p = p(Ψ) ♦♥çã♦ st

♦ ♣s♠ ♣♦ sr srt ♦♠♦

× = ∇p,

♦♥ = µ−10 ∇ × é ♥s ♦rr♥t ♥♦ ♣s♠ ♦r♠ q♥t ♣♦

sr r♣rs♥t ♣ qçã♦ rr♥♦ ❬❪

∆∗Ψ+ µ0R2 dp

dΨ+

1

2

dF

dΨ= 0, ∆∗Ψ = R2

∇ ·(

∇Ψ

R2

)

,

♥ q ∆∗ é ♠ ♦♣r♦r í♣t♦ ♦♥♦ ♦♠♦ ♦♣r♦r r♥♦ ❬❪

r③ã♦ ♥tr ♦ r♦ ♠♦r ♦ r♦ ♠♥♦r ♦ t♦♠ é ♠♣♦rt♥t ♣râ♠tr♦ ♦♥♦

♦♠♦ r③ã♦ s♣t♦

A =R0

a≤ 1

ε, ε =

r

R0.

♠ t♦♠s sçã♦ rr t r③ã♦ s♣t♦ ε ≪ 1 ♣♥ê♥ ♦♠ ♣♦sçã♦

♣♦♦ θ Ψ ♣♦ sr tr♠♥ ♥t♠♥t trés ♠s ♥ã♦ ♣♥ê♥

r ❬❪ ♦r♠ ♣r♦①♠ ♦té♠s q Ψ(r, θ) ≈ Ψ0(r)[1 + (∆s(r)/R0) cos θ] ♦♥

∆s(r) ♦ s♦♠♥t♦ r♥♦ é ♠ ♠ ♦ q♥t♦ s s♣rís ♠♥éts

s s♦♠ ♠ rçã♦ ♦ ♥tr♦ ♦♥ ♣s♠

♣rs♥ç r♦tçã♦ qír♦ ❬❪ s♦ q ♦♥sr♠♦s ♥♦ ♣ít♦ ♦ tr♠♦

ρ(❱ · ∇)❱ sr ♦♥♦ ♦ ♦ rt♦ ♦ q rst ♥ qçã♦ r

r♥♦ ♠♦ ❬❪

(

1− µ0κ2

ρ

)

∆∗Ψ+ µ0R2

(

dpdΨ

)

R

+ 12

(

dF 2

dΨ

)

R

− µ0κ∇Ψ ·∇(

κρ

)

+

µ0ρ2

ddΨ

(

κ2

ρ2|∇Ψ|2

)

R

= 0,

♦♥ κ = κ(Ψ) é ♠ ♥çã♦ ①♦ ♣r♦♣♦r♦♥ à ♦ r♦tçã♦ ♣♦♦ ♦ í♥ R

♥ q s rs ♦♠ rçã♦ Ψ ♠ sr s R ♦♥st♥t ♥ã♦ r♠♦s

♠ ♦♥t ♥çã♦ ♦ ♣r ♣rtís ♦ ♣s♠ ♦ qír♦ ♦r♦ ♦♠

s qçõs ♦♥t♥ ♠ ♣♦ sr ①♣rss ♦♠♦

❱ =κ

ρ− dΦ

dΨR2

∇φ,

♦♥ Φ = Φ(Ψ) é ♦ ♣♦t♥ tr♦stát♦ qír♦

rçã♦ ♥tr ♦♠♣♦♥♥t ♣♦♦ t♦r♦ ♦ ♠♣♦ ♠♥ét♦ é srt ♣♦r

♠ ♣râ♠tr♦ ♠♣♠♥t ♣rs♥t ♠ ♠t♦s ♠♦♦s ♠♣♦rt♥ts ♦ t♦♠ ♦ t♦r

sr♥ç q ♣♦ sr ♥♦ ♦♠♦ ❬❪

q = q(Ψ) = ·∇φ

·∇θ=

∫

dθdφ

dθ.

st ♣râ♠tr♦ é ♠ ♠ s ♥s ♦rç ♥♦ ♣s♠ stá rt♠♥t

♦ à st ♦ ♣s♠ q rqr q ♦ ♠t rsr♥♦ ❬❪

q > 1 s stst♦ ♣♦ ♠♥♦s ♥♦ ♥tr♦ ♦♥ ♣s♠ ♠ t♦♠s ♦ t♦r

sr♥ç ♦st♠ sr ♠♦r ♥ ♦r q ∼ 3 ♦ té ♠s♠♦ q ∼ 5 ♦ q ♥♦ ♥tr♦ q ∼ 1

❯♠ ♦tr r♥③ ♠♣♦rt♥t ♣r st rt♠♥t r♦♥ st ♣râ♠tr♦

é ♦ s♠♥t♦ ♠♥ét♦

s(r) =r

q

dq

dr,

q ♠♥s♦♥♠♥t ①♣rss rçã♦ q ♦♠ ♣♦sçã♦ r ♠ st r♥③

t♠ ♠♣♦rtâ♥ ♥♠♥t ♥ ár ♥óst♦s ♣r tr♠♥çã♦ ♦ ♣r r

q

♦ st♦ ♥ít♦ ♠♦♦s ①s rqê♥s ss♠ ♦♠♦ ♠ ♠♦♦s ♥♦áss♦s

é ♣♦ssí ♠ ♠t♦s s♦s t③r ♣r♦①♠çã♦

=B

1 + ε cos θ

(

ε

q(r)θ + φ

)

,

♦♥ θ φ sã♦ rs♦rs ♥ rçã♦ ♣♦♦ t♦r♦ rs♣t♠♥t ♠ ♠♦♦s ♦s

♦♠♦ é ♦ s♦ st ts ♠ ♠t♦s s♦s ♣♦♠♦s s♦♥srr ♦ t♦ ③♠♥t♦

♠♥ét♦

♦t q ♦ ♠♣♦ ♠♥ét♦ ♠♦str♦ ♠ é s♠étr♦ ♦♠ rçã♦ φ ♠s ♥ã♦ ♦♠

rçã♦ ♦ â♥♦ ♣♦♦ θ ♦r♠ q ♦ ♠♣♦ ♠♥ét♦ é r♠♥t ♠♦r ♥ ♣rt

♥tr♥ ♦ t♦♠ ♦ q ♥ ♣rt ①tr♥ ♣sr ♣q♥ st r♥ç

∆B/B ∼ ε s♠♣♥ ♠ ♠♣t♦ s♥t♦ ♥♦s ♦rs ♦s ♦♥ts tr♥s♣♦rt

♦ st♦ tr♥s♣♦rt ♦s ♦♥ts áss♦s tr♥s♣♦rt ♣r♦♥♥ts t♦r ss

♥tr♦s ♣s♠s ♠♥t③♦s ♠ sst♠s í♥r♦s ♠ sr sstt♦s ♣♦s ♦

♥ts ♥♦áss♦s q sã♦ ♠t♦ ♠♦rs ♠ ♠t♦s s♦s ♣♦♠ té r sr ♠ ♦r♠

♠♥t s♣r♦r ❯♠ r sssã♦ s♦r tr♥s♣♦rt ♠ t♦♠s é ♣rs♥t ♥

s♦

sçã♦ s♥t

sã♦ tr♥s♣♦rt ♠ t♦♠s

❯♠ ♦s s♦s ♠s ♠♣♦rt♥ts ís ♣s♠ ♦♥♥♦s ♠♥t♠♥t é ♦

r③r ♣r ♣rtís ♥r ♠ t♦♠s ♦♠ st ♥ ♦r♠ s♥♦s

t♦rs tr♥s♣♦rt q ♦♥sst♠ ss♥♠♥t ♠ tr♠♥r ♦s ♦♥ts D κ r

r♥ts ♦s ①♦s ♣rtís ♦r rs♣t♠♥t ♦s qs ♣♥♠ r♥ts

♥s t♠♣rtr ❬❪ ♦ s

Γ ≈ −D∇⊥n q ≈ −κ∇⊥T.

♠ D é ♦ ♦♥t sã♦ κ é ♦♥t tér♠ s s ♦♥srçã♦

♣rtís ♥r ♣♦♠ sr ♥♥s ♦♠♦

∂n

∂t+∇ · Γ = S♣rt,

3

2

∂T

∂t+∇ · q = S,

♦♥ S♣rt S r♣rs♥t♠ ♦♥ts ①tr♥s ♣rtís ♦r s ①♦s Γ q sã♦ q♥

ts ♠r♦só♣s ♦ ♣s♠ q ①♣r♠♥t♠♥t sã♦ ♠♦s ♣♦r ♠♦ ♥óst♦s

t♦r♠♥t ♣♦♠ sr st♠♦s Pr ♦ ♦♥t sã♦ ♣♦r ①♠♣♦ trés

♥ás ♠♥s♦♥ ♣♦♠♦s st♠r D

D ∼ ν(∆r)2,

♦♥ ν é rqê♥ ♦sõs ∆r é ♦ ♦♠♣r♠♥t♦ rtríst♦

t♦r tr♥s♣♦rt áss♦ q s ♣ sst♠s ♦♠tr í♥r é s

t ♠ ts ♣♦r s ❬❪ á ♥♦ t♦♠ ♦ ♦ t♦ ♦♠tr st ♥♦

♠♣♦ ♠♥ét♦ qír♦ ♥ã♦ á s♠tr ♣♦♦ t♦ st ♦ q ♥♦s rr♠♦s ♦♠♦

t♦ ♥♦áss♦ st ♦r♠ t♦r tr♥s♣♦rt ♥♦áss♦ ❬ ❪ sr ♣

♥st s♦ ♦ s♦ í♥r♦ ♦ r♦ r♠♦r ρ r♣rs♥t ♦ ♦♠♣r♠♥t♦ rtríst♦

♣r q♥tçã♦ sã♦ ♣r♣♥r ♦ s

D⊥ ∼ νρ2, κ⊥ ∼ nνρ2.

♠ ♦♥tr♣rt ♦ r ♦♦♠♦çã♦ s ♣rtís ♦ ♦♥♦ ♦ ♠♣♦ ♠♥ét♦

♠t ♣♥s ♣♦r ♦sõs é ♣♦ssí st♠r ♦ ♦♥t ♦♥t ♦r ♣r♦

♦♠♦

κ‖ ∼ nνλ2 ∼ ω2c

ν2κ⊥,

♦♥ λ = vTα/ν é ♦ r ♠♥♦ ♠é♦ ♦♠♦ ωc/ν ≫ 1 ♣r ♣s♠s ♠♥t③♦s é

♣♦ssí ♦♥r q ♦ ♦r s ♥ ♠t♦ ♠s ♠♥t ♦ ♦♥♦ ♦ ♠♣♦ ♠♥ét♦

q♥♦ ♦ ♣s♠ s t♦r♥ ♠s ♦s♦♥ ♦♥t ♣r ♠♥ ♥q♥t♦ q

♦♥t ♣r♣♥r ♠♥t ③♥♦ ♦♠ q ♦ ♣s♠ t♥ ♣rr ♠s ♥r

s ♣rár♦s ♥tr♦rs ♥ã♦ ♠ ♠ ♦♥t ss♠tr ♣♦♦ ♦ ♠♣♦ ♠♥ét♦ ♦

q ♦ t♦r♥ ♠s ♥t♥s♦ ♥♦ ♦ ♥tr♥♦ ♦♥ ♣s♠ ♦ q ♥♦ ♦ ①tr♥♦

ss ss♠tr t♠ ♥ê♥ ♥♦ ♠♦♠♥t♦ ♦ ♥tr♦ s ♣rtís ♦♥♦r♠

srt♦ qtt♠♥t ♠ ❬❪ q♥ttt♠♥t ♠ ❬❪ ♠ ♦♥sqê♥ ♠s ♣r

tís ♦ ♣r é rt♠♥t ① ♥ã♦ ♦♥s♠ ♥r rrr ♦ ♣♦ç♦

♠♥ét♦ ∆B/B ∼ ε ♠ ♦♥sqê♥ rt♦r♥♠ ♣♦ré♠ ♠ ♦tr s♣rí ♠♥ét

rst♦ ss ♣r♦ss♦ q s r♣t ♣♦r ♥s ♦s é q sss ♣rtís ♣rs♥t♠

órts rrrs ♦♥s ♦♠♦ órts ♥♥ ♦♠ ♠ s♦♠♥t♦ t♦ st♠♦

♣♦r ∆r ∼ (q/√ε)ρ ≫ ρ Pr ♦ á♦ ♦s ♦♥ts tr♥s♣♦rt é ♥ssár♦ r ♠

♦♥t ♥ã♦ s♦♠♥t ♦ ♠♦♠♥t♦ s ♣rtís t♥♦ s s♦♠♥t♦ rtríst♦ ♠s

t♠é♠ s rqê♥ t ♦sõs

ss♥♠♥t á três r♠s ♥♠♥ts sr ♦♥sr♦

• Prsütr st r♠ t♠é♠ ♦♥♦ ♦♠♦ r♠ ♦s♦♥ ♦ r♠

s ♦ ♠♣♦ ♠♦r♦ ♦ ♠♥♦r ♠♣♦

♦s ♦ t♠♣♦ ♥ssár♦ ♣r q s ♣rtís ♣♦ss♠ ♦♠♣tr ♠ órt é ♠♦r

♦ q ♦ t♠♣♦ ♦sã♦ ♦r♠ q s órts s ♣rtís sã♦ ♦♥st♥t♠♥t

♥trr♦♠♣s ♣♦r ♦sõs st ♦r♠ st r♠ é srt♦ ♣ ♦♥çã♦ ν/ωtr ≫ 1

♦♥ ωtr = vT /qR0 é rqê♥ rçã♦ st r♠ sr ♠ ♦r

♦♥ ♣s♠ ♦ ♦♥t sã♦ ♣r♣♥r ♣♦ sr st♠♦ ♦♠♦ ❬❪

D(PS)⊥ ∼ q2νρ2,

♦ s ♦r♠ q2 ♠♦r ♦ q ♦ s♣r♦ ♣ t♦r áss

• Pt st r♠ ε3/2 ≪ ν/ωtr ≪ 1 s qçõs ♦ ♥ã♦ s ♣♠ s♥♦

♥ssár♦ ♦ s♦ qçã♦ ♥ét r rts ♠ ♦♥çã♦ ♥tr♠ár

♥tr ♦ r♠ Prsütr ♦ r♠ ♥♥ q s ♣ ♦ ♥tr♦ ♦♥

♣s♠ t♠ ♦♠♦ ♦♥t sã♦ ①♣rssã♦ ❬❪

D(P )⊥ ∼ ωtr

νq2νρ2,

♦ s t ♦♥t é ♠ ♠♦r ♦ q ♥♦ s♦ ♥tr♦r ♣♦s ωtr/ν ≫ 1

• ♥♥ ♦ r♠ ♥ã♦ ♦s♦♥ s ♣rtís ♠ ♠é ♣♦ss♠ t♠♣♦ s

♥t ♣r ♦♠♣tr ss órts ♥ts ♦r♠ ♦♠ ♦trs ♥trt♥t♦ ♦ ♦

r♥ ♦♠♣r♠♥t♦ rtríst♦ sts órts ♦sã♦ ♦♦rr ♦r ss s♣rís

♠♥éts ♦r♠ ♦ q rrt ♠ r♥ ♦♥trçã♦ ♣r ♦ tr♥s♣♦rt r

rqê♥ ♦sã♦ é srt ♣♦r ν/ωtr ≪ ε3/2 st♠t ♣r ♦ ♦♥t

sã♦ rst ♠ ❬❪

D(B)⊥ =

q2

ε3/2νρ2,

♣♦rt♥t♦ ♦ t♦r ♠t♣t♦ q2/ε3/2 ♠ rçã♦ ♦ ♦r áss♦ ♣r ♦ ♦♥t

sã♦ ③ ♦♠ st ♣♦ss sr té ♠ ♦r♠ ♠♥t ♠♦r ♠♦♦ sr

♦t♦ t♠é♠ s s ♥ qçã♦ ♥ét r ♦r♠ q ♦ ♠♦♦ ♦

t♠é♠ ♥ã♦ s ♣ ♥st r♠

♠♣♦rt♥ts tr♦s ♣♦s s♦r tr♥s♣♦rt ♥♦áss♦ ❬ ❪ té ♠s♠♦

r♦s q trt♠ ♦ ss♥t♦ ♦♠ st♥t rq③ ts ❬❪ tê♠ ♥ ♦r♥r

♠ ♦ ♦♠♣r♥sã♦ s♦r t♦s ♥♦áss♦s ss ♠♣t♦s ♥♦ ♦♥♥♠♥t♦ ♣s♠

♥trt♥t♦ ♠ rõs ♦♠♥s ♣♦r ♣r♦ss♦s tr♥t♦s ♦ ♦♥t sã♦ é ♥

♠♦r ♦ q ♦s srt♦s ♣♦s ♠♦♦s ♥♦áss♦s st s♦ ♦♦rr ♦ q ♠♠♦s

sã♦ ♦♠ ❬❪ ♥ ár tr♥s♣♦rt tr♥t♦ ♦ tr♥s♣♦rt ♥ô♠♦ sssã♦

r♥♥t s♦r st t♣♦ tr♥s♣♦rt é t ♣♦r s ❬❪ ♦♠♦ ♦♥t♥çã♦ ss

tr♦s ♥s ❬ ❪

♠♦r ♥ã♦ s ♦ ♦♦ st ts trtr s♦r tr♥s♣♦rt tr♥t♦ ♠ s♠♣s s

t♠t ♦r♦ ♦♠ ❬❪ ♣r ♦ ♦♥t sã♦ ♥ô♠ é út tít♦ ♦♠

♣rçã♦ ♦♠ ♦s ♦♥ts ♥♦áss♦s Pr étr♦♥s D⊥e = (∆r)2/τ ♣♦♠♦s st♠r

∆r/τ ∼ vE ∼ Φ/∆rB eΦ/T = k ♦♥ vE é ♦ r ♥♠♥t E ×

♣♦rt♥t♦ q

D(♦♠)⊥e = k

T

eB∼ ωcρ

2.

♦t q ♦♠♦ ωc ≫ q2ν/ε3/2 D(♦♠)⊥e ≫ D

(B)⊥e

st♦r♠♥t t ♦♥t ♦ srt♦ ♦♠♦D(♦♠)e = T/16eB ♦♥ r③ã♦ ♣r ♦ t♦r

té ♦ ♣r♠♥ ♦sr ❬❪ rts ♠ í ♣r♦♠ ♥ã♦♥r tr♠♥çã♦

k s♥♦ q k < 1 ❬❪ Pr ♦♥r st sçã♦ ♦sr♠♦s q ♦ ♦r ♦ ♦♥t

sã♦ ♥ô♠ ♣r étr♦♥s ① ♦ ♦r áss♦ ♠ ♣r♦①♠♠♥t kωceτ ≫ 1 ♣♦r

ss♦ ♠♣r rçã♦ ♦ ♦♥♥♠♥t♦ ♦ ♦ tr♥s♣♦rt ♥ô♠♦ é ♦♥sr♦ ♠

♦s ♠♦rs s♦s ís t♦♠s

♦♠♥t♦ ♣rtís ♦ ♦ ♥tr♦

Prtís rrs ♠rss ♠ ♠ ♠♣♦ tr♦♠♥ét♦ ♦♠♦ s q ♦♠♣ô♠ ♦

♣s♠ ♠ s♠ts à çã♦ ♦rç ♦r♥t③ ♦r♦ ♦♠ qçã♦ sr q

①♠♣ ♥â♠ í♦♥s

d

dt=

e

m

[

E(r, t) + ×(r, t)

]

,

♦♥ = dr/dt é ♦ sts ♣rtís q stã♦ ♦③s ♥ ♣♦sçã♦ r ♦t

q ♥st qçã♦ ♠ ♦♠♦ ♥s ♣ró①♠s st sçã♦ ♦♠t♠♦s ♦ í♥ α ♦ q sr

s♥t♥♦

♦ ♠ ♣♦ sr ①♣rss ♥ ♦r♠

= v‖+ ⊥, ⊥ = v⊥(cos γ1 − sin γ2),

♦♥ γ = − tan−1( · 1/ · 2) é ♦ â♥♦ rçã♦ 1 2 ♦r♠♠ ♥st ♦r♠ ♠

s ♦rt♦♥♦r♠ ♦♥♥♦♥♠♥t ♦r♥t ♥ q = /B

♦r♠ s♠r ♣♦sçã♦ s ♣rtís ♠ ♠ ♣s♠ ♠♥t③♦ ♣♦ sr ①♣rss

♦♠♦

r = rg + ρ, ρ =× ⊥

ωc,

♦♥ ρ é ♦ r♦ r♠♦r t♦r rg é ♣♦sçã♦ ♦ ♥tr♦ ♦ ♣♦♥t♦ ♥tr

órt ♣r♦①♠♠♥t rr s ♣rtís ♥♦ss♦ ♣r♥♣ ♥trss ♥st sçã♦ stá ♥

♦t♥çã♦ ♦ ♦ ♥tr♦ q é ♦t r♥♦ ♦♠ rçã♦ ♦ t♠♣♦

♠ s t♦♠♥♦ ♠é ♠ rçã♦ ♦ â♥♦ rçã♦ Pr ♠é ♠ r♥③

♥ér X tr♠♦s ♥çã♦

〈X〉 = 1

2π

∫ 2π

0dγX.

♦ ♦srr q 〈〉 = v‖ 〈ρ〉 = 0 ♦té♠s

g =

⟨

drgdt

⟩

= v‖−⟨

dρ

dt

⟩

♦♥ ♦ á♦ 〈dρ/dt〉 q é rt♠♥t ♦♥♦ ♣♦ sr st♦ ♠ ❬❪

♥ts ♣rs♥tr♠♦s ♦ ♦ ♥tr♦ é ♦♥♥♥t ♥r ♦ ♠♦♠♥t♦

♠♥ét♦ ♥r ♠ ♣rtí

µα =mαv

2⊥

2B Eα = eαΦ+ µαB +

mαv2‖

2,

q ♠ ♣r♠r ♦r♠ ♠ ρ/L ♦♥ L r♣rs♥t ♥ér♠♥t ♦ ♦♠♣r♠♥t♦ rtríst♦

♦ r♥t qqr q♥t ♠r♦só♣ ♦ ♣s♠ sã♦ ♦♥st♥ts ♠♦♠♥t♦

♠s♠ ♦r♠ q ♦ ♠♦♠♥t♦ ♥ô♥♦ ♣r♦ ❬❪

♠ ♣r♠r ♦r♠ ♠ δρ = ρi/L ♦ ♦ ♥tr♦ ♣♦ sr ①♣rss ♦♠♦

gα = v‖α+ + Bα + κα,

♦♥

=E ×

B2,

é r E × q ♣♦ sr ♦r♠ δ0ρ ss♠ ♦♠♦ v‖ ♦ ♦r♠ δ1ρ rt

♦♥♦r♠ st♦ ♥ ♣ró①♠ sçã♦ st r ♣♦ss ♦ ♠s♠♦ s♥t♦ ♣r í♦♥s étr♦♥s

♣♦s ♥♣♥ r ♣rtí r ♠♥ét

Bα =µα

eα×∇ lnB,

sr ♦ à ♥♦♠♦♥s ♦ ♠♣♦ ♠♥ét♦ ♣♦r str r♦♥ ♦ ♠♦♠♥t♦

♦trô♥♦ ♣♦ss s♥t♦s ♦♣♦st♦s ♣r rs ♣♦st ♥t ♥♠♥t

κ =eα|eα|

v2‖ωcα

× κ,

q t♠é♠ t♠ s s♥t♦ ♠♦♠♥t♦ ♣♥♥t r ♣rtí é r rs

t♥ rtr ♦ ♠♣♦ ♠♥ét♦ κ = ( ·∇)

♦ r s ♦s r ♣♦♠ sr ①♣rsss ♥ ♦r♠

α =1

mα

α ×

B2,

♦♥ α r♣rs♥t s rss ♦rçs q ♠ ♥ ♣rtí ♦ s étr ♠♥ét

rtr t

♦r ♥ét

❯♠ ♦s ♦t♦s t♦r ♥ét é tr♠♥r ♥çã♦ strçã♦ fα ♣r s♣é

♣rtí ♣♦s ♣rtr ♦ á♦ ♠♦♠♥t♦s st ♥çã♦ st♦ é ♥tr ♦♠ rçã♦

às ♦♦r♥s ♦ ♥çã♦ strçã♦ ♠t♣ ♣♦r ♣♦tê♥s ♦

♦té♠s r♥③s ♠r♦só♣s ♦ ♣s♠ s qs ♣♦♠ sr ♦♠♣rs ♦♠ ♦rs

①♣r♠♥ts s qçõs ♦ ♦ts ♣rtr st ♠t♦♦♦ ♦♥♦r♠ st♦

♠ sr♠ ♠♣♦rt♥ts s íss ♥♦ q s rr ♦♥srçã♦ ♠♥srás

♠r♦só♣♦s ♦ ♣s♠

♠ ♣r♥í♣♦ ♦♥srs q ♥çã♦ strçã♦ é ♦r♠ fα = fα(t, r,) ♣♦ré♠ ♥♦

♠♦♦ r♦♥ét♦ t③♦ ♥♦ ♣ít♦ ♦r♠ f (g)α = f

(g)α (t, rg, µ, E , γ) é ♠s ♦♥♥♥t

út♠ ♦r♠ é t③ ♥ rçã♦ qçã♦ r♦♥ét ♠♦str ♥ sçã♦

♥trt♥t♦ ♣r st♠r ♦r♥s r♥③ tr♠♦s qçã♦ ♦t③♠♥♥ t③♠♦s

♣r♠r ♦r♠

♥ás qçã♦ ♦t③♠♥♥

♠ ♦r♠ r s qçõs ♥éts ♣♦♠ sr ①♣rsss ♦♠♦

df

dt=

6∑

i=0

dxidt

∂f

∂xi= C(f),

♦♥ f = f(x0, x1, x2, x3, x4, x5, x6) é ♥çã♦ strçã♦ ♣rtís ♥♦ s♣ç♦ s

srt♦ ♣s rás xi x0 = t C(f) é ♦ tr♠♦ ♦sõs qçã♦ ♦t③♠♥♥

♦t♥çã♦ ♣rtr s ♠s rs s ♥tr♣rtçã♦ ís sã♦ ♣rs♥ts ♦r♠

r r♥♥t ♠ ❬❪ é ♦ s♦ ♣rtr ♠ q x1, x2, x3 sã♦ s ♦♦r♥s s♣s

♦③çã♦ ♥st♥tâ♥ ♣rtí x4, x5, x6 sã♦ s ♦♠♣♦♥♥ts ♦ st st

qçã♦ q ♥♦r♠♠♥t é t③ ♥ ♦t♥çã♦ ♦ t♥s♦r étr♦ ❬❪ é ①♣rss ♦♠♦

∂fα∂t

+ · ∂fα∂r

+ α · ∂fα∂

= Cα(f),

♦♥ α é rçã♦ ♣rtí t♦ t♣♦ α ♦♠ ♦ ♦③ ♠ r q ♣r ♣s♠s

♦rtór♦ é ♠ srt ♣ ①♣rssã♦

α =eαmα

(E + ×).

♦ ♦♥t①t♦ ís t♦♠s ♦ ♣s♠ é ♦♥sr♦ ♠♥t③♦ q♥♦ ρi/LB ≪ 1

♦♥ LB é ♦ ♦♠♣r♠♥t♦ rtríst♦ rr♥t ♦ r♥t ♦ ♠♣♦ ♠♥ét♦ qí

r♦ ♥♦ ♣s♠ é♠ ss♦ ♦♥sr♠♦s ♥st ts ♥ô♠♥♦s ①s rqê♥s q♥♦

♦♠♣rs ♦♠ rqê♥ í♦tr♦♥ ♠ts ♥ ♥ás st sçã♦ q ♦ ♦♠♣r

♠♥t♦ ♦♥ ♣r♣♥r λ⊥ s ♣rtrçõs ♣♦ss♠ sr ♦r♠ ♦ r♦ r♠♦r

rtríst♦ ♦♥ts ♥sts ❬❪

s ♥♦r♠çõs ♦ ♣rár♦ ♥tr♦r ♣♦♠ sr ①♣rsss s♥t ♦r♠

δρ =ρiL

≪ 1, δk = k⊥ρe ≪ 1, δω =ωt

ωci

≪ 1

♦♥ L é ♦ ♠♦r ♦s ♦♠♣r♠♥t♦s rtríst♦s rr♥ts ♦s r♥ts q♥ts

♠r♦só♣s qír♦ ωtr ≤ vTe/L é rqê♥ rçã♦ r♦♥ rçã♦

t♠♣♦r ♥çã♦ strçã♦ ♦♥sr♠♦s ♥ q rqê♥ ♦sã♦ é ♣q♥

s ♦♠♣r ♦♠ rqê♥ ♦trô♥ ν/ωc ∼ δρ ≪ 1 t♦ ♣r s♠♣r ♥

rçã♦ qçã♦ r♦♥ét ♥ã♦ ♠♦s ♠ ♦♥t ♦ ♦♣r♦r ♦sã♦

♥ts str s♦r qçã♦ ♥ét r qçã♦ r♦♥ét s

qs sã♦ r♠♥t t③ ♥♦ st♦ ♠♦♦s ①s rqê♥s ❬❪ é ♦♥♥♥t

st♠r ♦r♠ r♥③ ♦s tr♠♦s qçã♦ ♦t③♠♥♥ ♦♥♦r♠ ♦ ③♠♦s sr

∂f

∂t∼ δρωcf,

e

m(×) · ∂f

∂∼ ωcf, C(f) ∼ νf ∼ δρωc,

e

mE‖ · ∂f

∂∼

E‖/B

vTωcf ∼ δρωcf

e

m⊥ · ∂f

∂∼ vE

vTωcf.

♠ , vE = |E ×|/B2 E‖/B ∼ δρvTi♣ró①♠♦ ♦ qír♦ ❬❪

♥t♦ à ♥ás ♦ tr♠♦ ♦♥t♦ · ∇f ♦♥srs q ♥çã♦ strçã♦ é

♦♠♣♦st ♣♦r s ♣rts f ∼ F + G ♥q♥t♦ q ♣r♠r r s♠♥t ♥♦ s

♣ç♦ |∇F | ∼ F/L s♥ ♦♥sst ♣rtrçõs ♣q♥♦s ♦♠♣r♠♥t♦s ♦♥

|∇G| ∼ G/λ ♦♥ ♦♥çã♦ λ ∼ ρi é ♣♦ssí ♦r♠♠♥t s qçõs ♥éts sã♦

rs♦s ♦r♠ ♣rtrt ♦♥sr♥♦ q G ∼ δkF ♦r♠ q

·∇f ∼ ωF + ωcG ∼ (δρ + δk)ωcf,

♦♥sq♥t♠♥t á ♦s ♣râ♠tr♦s ♥♣♥♥ts sr ♥s♦ δρ ≪ 1 δk ≪ 1

♥ás ♥ três s♦s st♥t ♥trss ♣r t♦♠s

• r♠ r ♦♥srs δk = 0 vE ∼ δρvTi ♦r♠ q ≈ v‖ ♥♦r♠

♠♥t t③s qçã♦ ♥ét r ♦♠♦ ♣♦♥t♦ ♣rt ♣r ♠♦♦s

st t♣♦ ♦r♠ é r♠♥t ♣♦ ♣r srr ♣r♦ss♦s tr♥s♣♦rt rs♦s

t♣♦s ♥sts

• r♠ st s♦ δk = 0 ♠s vE ∼ vTi ♦ s ≈ v‖+ E t♦r

♦s sçã♦ q é st♥t r ♦ ♣♦♥t♦ st ís♦ ♣♦ sr t③

♠ ♥ô♠♥♦s q ♥♦♠ st t♣♦ ♦r♠ ts ♦♠♦ ♦♥ts ♥sts

rt ♥t qt♦♥②r♦♥t qt♦♥

• r♠ rçã♦ ♦ s♦ ♠ q ♠♦r vE ∼ δρvTi♠ts ♣♦ss

♣rtrçõs ♦♠ r♥s rçõs s♣s (δk ∼ δρ) qçã♦ r♦♥ét

sr t③ ♥st s♦

qçã♦ r♦♥ét

sr s♦s ♥s rs ❬ ❪ ♣rs♥t♠♦s ♦s ♣r♥♣s ♣ss♦s ♣r ♦t♥çã♦

qçã♦ r♦♥ét q é t③ ♥♦ ♣ít♦

♦♥♥♥t ♦♥srr ♠♥ç rás (t, r,) → (t, rg, µ, Eα, γ) ♠ rçã♦ às

rás qçã♦ ♦t③♠♥♥ ♦r♠ q q ♣♦ss sr srt♦ ♦♠♦

∂f

∂t+

drgdt

· ∂f

∂rg+

dµ

dt

∂f

∂µ+

dEαdt

∂f

∂Eα+

dγ

dt

∂f

∂γ= 0.

♣r♦♠♥t♦ ♣r ♦t♥çã♦ qçã♦ r♦♥ét ♦♥sst ♠ r ♠é

♦♠ rçã♦ γ ♦♥sr♥♦ ♣r st á♦ q rg = rg(r,) µ = µ(rg,) Eα = Eα(rg,)

γ = γ(rg,)

Pr ♦ ♥s♥♦♠♥t♦ ♥ít♦ ♥s ♣ró①♠s t♣s ♦ts ♣r♦①♠çã♦ ♦♥ ♦♠

rçã♦ à ♦♦r♥ ♦ ♥tr♦

❳(r) = ❳g(rg)i⊥·r,

♦♥ ❳ r♣rs♥t ♦r♠ ♥ér qqr t♦r ♦ t♦♠r ♠é ♦t♠♦s

〈❳(r)〉 = J0(k⊥ρ)❳g(rg)i⊥·rg ,

♦♥ Jn(x) é ♥çã♦ ss ♦r♠ n r♠♥t♦ x ♠ ♦ t③

rçã♦⟨

i⊥·ρ⟩ = J0(k⊥ρ) ❬ ❪ ♣rtr ♦té♠s ❬❪

⟨

drgdt

⟩

= g +

[

J0(k⊥ρ)(Φ− v‖A‖) + 2J1(k⊥ρ)k⊥ρ

µe B‖

]

i⊥×B ,

⟨

dµdt

⟩

≈ 0,

⟨

dEdt

⟩

= ∂∂t

[

J0(k⊥ρ)e(Φ− v‖A‖) + 2J1(k⊥ρ)k⊥ρ µB‖

]

,

♦♥ ♥♦tçã♦ X = X(rg, t) ♦♠ X = Φ, A‖, B‖ ♥t s ♣rtrçõs

♥çã♦ strçã♦ ♣♦ sr ♦♠♣♦st ♠ três ♣rts

fα = Fα + Gα + G(γ)α ,

♦♥ Fα é ♦♥trçã♦ qír♦ Gα + G(γ)α = (O(δρ)+O(δk))Fα é ♣rtrçã♦ G(γ)

α é

♣rt ♣♥♥t γ ♠ ♣r♠r ♦r♠ ♠ δρ δk ♦ á♦ ♠é rst

♥ qçã♦ r♦♥ét

(

∂∂t + gα ·∇

)

gα =

eα

(

∂Fα

∂Eα∂∂t +

×∇Fα

mαωcα· i⊥

)[

J0(k⊥ρα)

(

Φ− v‖A‖

)

+ 2J1(k⊥ρα)k⊥ρα

µα

eαB‖

]

.

♦♥ ♦s r♥ts sã♦ ♦s ♥s ♦♦r♥s ♦ ♥tr♦ rg ♠ t♦♠s ♣r♦①

♠çã♦ B‖ ≈ 0 ♣♦ sr ♦♥sr ♥ ♠♦r ♦s ♠♦♦s ❬❪ t♦ é ♦♥sr ♥♦

♣ít♦

♠♣♦rt♥t ♦srr q gα ♥ã♦ r♣rs♥t ♥tr♠♥t ♣rt ♣rtr ♥çã♦

strçã♦ q é ♦t ♣ ①♣♥sã♦ fα ♠ t♦r♥♦ ♥r qír♦ Eα0 =

eαΦ + mαv2/2 ♦♥ Φ é ♦ ♣♦t♥ tr♦stát♦ qír♦ Eα = Eα0 + Eα1 é ♥r

t♦t Eα1 = eαΦ é ♣rtrçã♦ ♥r ♦♥♦r♠ qçã♦ s q

fα ≈ Fα(Eα0) + Eα1∂FMα

∂Eα|Eα=Eα0

+ G(γ)α ,

♦♥ G(γ)α é ♦♥trçã♦ ♣♥♥t γ ♣r♦♥♥t qçã♦ r♦♥ét ♦

s

fα = Fα + fα, fα = eαΦ∂FMα

∂Eα+ gα

i⊥·ρα .

♦ ♣ít♦ t③♠♦s ssttçã♦ ∇ → i(⊥ + k‖) ♣r rs♦r ♦♥

⊥ ≈ rr + θ1

r

∂

∂θ, k‖ =

1

qR0

(

∂

∂θ+ q

∂

∂φ

)

.

♦♥♦r♠ ♠♦str♦ ♥♦ ♣ít♦ st ♣♦t♥ ①st s s♦♠♥t s á r♦tçã♦ qír♦

qçã♦ ♥ét r

rts ♠ qçã♦ q ♣r ♣r♥♣♠♥t ♠ st♦s s♦r tr♥s♣♦rt ♥♦áss♦

♠s t♠é♠ ♣♦ sr t③ ♥ ♥stçã♦ ♠♦♦s úst♦s ♦és♦s ♦♠♦

♠ ❬❪ ♣♦r ①♠♣♦ st qçã♦ q ♣♦ sr ♦t ♦r♠ rrs ❬❪ trés

♦ ♣r♦ss♦ r♦♠é ♣♦ sr srt ♦♠♦

∂fα∂t

+ gα ·∇fα +

(

eαgα · E + µα∂B

∂t

)

∂fα∂Kα

= 0,

♦♥ Kα = Eα − eαΦ é ♥r ♥ét ♣rtís ♦ t♣♦ α E ≈ −∇Φ− (∂A‖/∂t) é ♦

♠♣♦ étr♦ ♣rtr♦ ❯♠ rçã♦ ♠s át st qçã♦ s ♥♦ tr♦

♦r♥ ③t♥ ❬❪ ♣♦ sr ♥♦♥tr ♠ ❬❪

♦r ♦s

t♦r ♦s ♠t♦♦♦ ♣r ♦t♥çã♦ ss qçõs ♣rtr qçã♦

♦t③♠♥♥ sã♦ ♣rs♥t♦s ♥st sçã♦ ♣rs♥t♠♦s t♦r ♦s ♦s t♦r

♠♥t♦r♦♥â♠ ♠s t③s ♥♦ ♣ít♦

♦r ♦s ♦s s qçõs r♥s

á♦ ♠♦♠♥t♦s qçã♦ ♦t③♠♥♥ st♦ é ♥tr t qçã♦

♠t♣ ♣♦r ♣♦tê♥s ♦♠♥çõs t♦rs ♦ ♥♦ s♣ç♦ ♦s

♣r♠t ♦t♥çã♦ s qçõs ♦s ♦ qçõs r♥s s qçõs

♦s sr♠ ♦çã♦ t♠♣♦r ♠♣♦rt♥ts q♥ts ♠r♦só♣s ♦ ♣s♠

q sã♦ sts sr

Pr♠r♠♥t ♦ á♦ ♦ ♠♦♠♥t♦ ♦r♠ ♥ ♥çã♦ strçã♦ rst ♥

♥s ♣rtís

nα = nα(r, t) =

∫

fα(r,, t)d3,

q q♥♦ ♦♠♥ ♦♠ s rs ♦s r♥ts t♣♦s ♣rtís ♣r♠t ♦t♥çã♦

♥s r

ρc =∑

α

eαnα.

♠♦♠♥t♦ ♣r♠r ♦r♠ ♦r♥ ♦ ①♦ ♣rtís nαα ♦ q ♣r♠t ♥r

♦ ♦ ♦ ♦ t♣♦ α

α =1

nα

∫

fα(r,, t)d3,

♦r♠ s♠r à ♥s r ♥s ♦rr♥t é ♦t

=∑

α

eαnαα.

♥♦ ♦ ♥♦ rr♥ ♦ ♦ ♦ s♥♦ ♠♦♠♥t♦ ♦r♥ ♦ t♥s♦r ♣rssã♦

q ♣♦ sr ♦♠♣♦st♦ ♥ ♣rssã♦ sr ♥ét pα ♥♦ t♥s♦r s♦s π

♦♥♦r♠ ♠♦str♦ ①♦

♣α = pα+ πα =

∫

mα(− α)(− α)fα(r,, t)d3.

t♥s♦r s♦s é ♥♦r♠♠♥t ♦ ♠ três ♣rts

πα = π‖α + πα + π⊥α,

♥♦♠♥s s♦s ♣r r♦s♦s s♦s ♣r♣♥r

♥♠♥t t♠é♠ ♦ ♥♦ rr♥ ♦ ♦ ♦ ♣ró①♠♦ út♠♦ ♠♦♠♥t♦ fα q

♦♥sr♠♦s ♥st ts ♦r♥ ♦ ①♦ ♦r

qα =

∫

1

2mα[(− α) · (− α)](− α)fα(r,, t)d

3.

♦♠♥t♦s ♥çã♦ strçã♦ ♦r♥s ♠s t ♥ã♦ t♠ t♥t ♠♣♦rtâ♥ ♦ ♣♦♥t♦

st ís♦ ♣♥s ér♦ ♣♦r st r③ã♦ ♦ s ♣♦ t③çã♦ ♥♦s ♠♦♦s ♠s

♠♣♦rt♥ts ♥ã♦ ♦s ♠♦str♠♦s ♥st ts sr ♣rs♥t♠♦s s q♥ts ss♣ts

q sã♦ s ♣rtr ♠♦♠♥t♦s ♦ ♦♣r♦r ♦sõs Cα(f) s s q♥ts

♠♦r ♥trss ♣r t♦♠s sã♦ ♦rç rçã♦ ♦ tr♠♦ tr♥srê♥

♦r ♥♦s rs♣t♠♥t ♣♦r

α =

∫

mααCα(f)d3,

Qα =

∫

1

2mα(− α) · (− α)Cα(f)d

3.

s qçõs r♥s ♦ qçõs ♦s ♦s

rts s s qçõs ♦ts ♣rtr ♦ á♦ ♠♦♠♥t♦s qçã♦ ♦t③♠♥♥

sr♠ ♠♣♦rt♥ts s ís ♦♥srçã♦ ♠ss ♠♦♠♥t♦ ♥r ❬❪

sr ♣♦st♠♦s ts qçõs ♦t♥çã♦ é ♠♦str ♦r♠ r t ♥s

s ❬ ❪

♥♠♥t ♦♥sr♠♦s s qçõs q sr♠ ♦♥srçã♦ ♠ss ♠♦♠♥t♦

♥r rs♣t♠♥t

dαnα

dt+ nα∇ · α = 0,

mαnαdααdt

+∇pα +∇ · πα − eαnα(E + α ×) = α,

dαpαdt

+ γpα∇ · α + (γ − 1)(πα : ∇α +∇ · qα) = (γ − 1)Qα,

♦♥ dα/dt = ∂/∂t+ α ·∇ é r ♦♥t ♦ r ♠tr γ é ♦ ♦♥t

P♦ss♦♥ ♦ ♦♥t át♦

sst♠ ♦♠♣♦st♦ ♣s qçõs é ♥♦♠♣t♦ ♣♦s ♣r rs♦ê♦ sã♦

♥ssárs ♥♦r♠çõs s♦r ♦ ♦♣r♦r ♦sõs rs♣♦♥sás ♣♦s tr♠♦s α Qα é♠

s qçõs ♦çã♦ t♠♣♦r ♣r π q ♣sr q ♠ ♠t♦s ♠♦♦s s ♣♦ssí

s♦♥srr ts r♥③s ♦♠♦ ♥♦ ♠♦♦ ♠♥t♦r♦♥â♠ á s♦s ♠

q é ♥ssár♦ ♦ á♦ ♦s ♣ró①♠♦s ♠♦♠♥t♦ qçã♦ ♦t③♠♥♥ ♣r ♦tr ts

r♥③s ♦ ♣ít♦ ♦♥sr♠♦s ♦ t♦ ♥s♦tr♦♣ ♣rssã♦ srt♦ ♣♦ t♥s♦r

s♦s ♣r π‖ st t♥s♦r é ♦ trés qçã♦ ♦çã♦ π

dπα

dt + (∇ · α)πα + [πα ·∇α + (πα ·∇α)T − (γ − 1)(πα : ∇α)]−

ωcα(πα) + p[∇α + (∇α)T − (γ − 1)(∇ · α)]+

(1− 1/γ)[∇qα + (∇qα)T − (γ − 1)(∇ · qα)] +∇ · τ = πα ,

q ♦ ♦t ♣r♠r♠♥t ♠ ❬❪ ♥♦ ♦♥t①t♦ ss ♥tr♦s ♣♦str♦r♠♥t ♣

t ♣r ♣çõs ♠ ís ♣s♠s ❬❪ st qçã♦ ♦ í♥ T ♠ s♦rsrt♦

r♣rs♥t tr♥s♣♦st ♠tr③ q s ♦té♠ ♥ r♣rs♥tçã♦ ♦ tr♠♦ q st í♥ s

rr ♥ ♦r♠ ♠tr τ πα sã♦ t♥s♦rs ♣r♦♥♥ts ♠♦♠♥t♦s ♠s t ♦r♠

♥çã♦ strçã♦ ♦ ♦♣r♦r ♦sõs ♦s qs ♥ã♦ sã♦ ♦♥sr♦s ♥st ts

é ♠ ♦♣r♦r q ♦r♦ ♦♠ ♥çã♦ ♠ ❬❪ sts③ s s♥ts ♣r♦♣rs

(A) = A× − ×A

−1

(A) = 14

[×A · (+ 3)] + [×A · (+ 3)]T

,

♣r qqr t♥s♦r s♠étr♦ A t♥s♦r s♦s ♣r í♦♥s ♦ q ♦♥sr♠♦s

♥♦ ♣ít♦ é ♦ ♣ s♥t ①♣rssã♦

π‖i =π‖i

γi − 1

(

− 1

3

)

♦r ♠♥t♦r♦♥â♠

Pr str ♦ s♦ s qçõs é ♥ssár♦ ♦♥srr ♥s ♦♠♣r♠♥

t♦s t♠♣♦s rtríst♦s ♠♣♦rt♥ts ♦♥♦r♠ ♠♦str♦s sr

L ∼ a, τ =a

vTi

, ω ∼ 1

τ, λα = vTαταα

♠ L é ♦ ♦♠♣r♠♥t♦ rtríst♦ rt♦ r♥ts q♥ts ♠r♦s

ó♣s ω é rqê♥ ss♦ ♠♦♦s λα é ♦ r ♠♥♦ ♠é♦ ♦ q

♣♥ ♦ tér♠ ♦ t♠♣♦ ♦sõs ♣rtís ♠s♠♦ t♣♦

♣sr q t♦r ♦ ♣♦♥t♦ st tór♦ s ♣q s♦♠♥t ♥s s♥ts

r♥stâ♥s

(

mi

me

)1/2ων

≪ 1 ♣s♠ t♠♥t ♦s♦♥,

ρia ≪ 1 r♦ r♠♦r ♠t♦ ♣q♥♦,(

ρia

)2(

mi

me

)1/2ν

ω≪ 1 ♣s♠ ① rsst,

q rr♠♥t ♣rt♥♠ à r ♣s♠s t♦♠ ♦ s s♦ ♠ ♥ú♠r♦s ♠♦♦s q

♦♠ ts r♥stâ♥s r♣r♦③ rst♦s ♦♥③♥ts ♦♠ ①♣r♠♥t♦s ❬❪ ♣♦rt♥t♦

♠♦r t♦r s ♠ t♦r rt♠♥t s♠♣s ♣♦ss ♥ú♠rs

♣çõs ♠♣♦rt♥ts ❬❪

♦r♠♠♥t ♥ t♦r ♦♥srs q ♦ ♣s♠ é ♠ ♦ ♥s

♠ss ρ ♥s r ρc q ♥♦ s♦ s♦ t♦♠s é ♣rt♠♥t ♥ ♣rs

sã♦ p t♠♣rtr T ♦rr♥t sts r♥③s q rtr③♠ ss ♦ rs♣t♠

tr♠♥s rçõs ♦♠ s r♥③s ♣rt♥♥ts ♦s ♦s í♦♥s étr♦♥s s qs

sã♦ ♠♦strs sr

ρ =∑

α

mαnα ≈ mini, ni = n ≈ ne, me ≪ mi

ρc =∑

α

eαnα = e(ni − ne) ≈ 0,

=

∑

αmαnααρ

≈ i,

=∑

α

eαnαα ≈ ne(i − e),

p =∑

α

pα = niTi(1 + τe), τe =Te

Ti.

♦t q ♦rrs♣♦♥ê♥ ♥rs rr♥t às ♦s í♦♥s étr♦♥s ♣♦ sr

srt ♥ ♦r♠

i ≈ e ≈ −

en.

rsst ♦ ♣s♠ q é ♠ ♠♣♦rt♥t ♣râ♠tr♦ ♥ t♦r ♣♥

♣r♥♣♠♥t rqê♥ ♦sõs étr♦♥í♦♥ ♥♦r♠♠♥t é ♥ ♦♠♦ ❬❪

η =meνe2n

.

♦♠♥çõs ♥rs s qçõs ♣♦♥rs ♣♦r r♥③s rtrísts

í♦♥s ♦ étr♦♥s ♦s ts ér♦s ♣♦♠ sr ♥♦♥tr♦s ♠ ❬ ❪ ♣r♠t♠

♦tr ♦ ♦♥♥t♦ qçõs q sã♦ ♣rs♥ts sts s♣r♠♥t ♥♦s

♣rár♦s q s s♠

Pr♠r♠♥t ♦♥sr♠♦s qçã♦ rr♥t ♦♥srçã♦ ♠ss

dρ

dt+ ρ∇ · = 0,

qçã♦ ♥á♦ ♠s ♣r ♦♥srçã♦ r étr é

∂ρc∂t

+∇ · = 0.

♠♦r ♣rt ♦s ♣r♦ss♦s ♠ ♦♥çã♦ qs♥tr s ♣ ♦r♠

q ρc ≈ 0 ♦♥sq♥t♠♥t

∇ · = 0.

qçã♦ ♦♥srçã♦ ♠♦♠♥t♦ ♣♦ sr srt s♥t ♦r♠

ρd

dt+∇p− ×+∇ · (πi + πe)− ρcE +

me

e2n[∇ · ()− ·∇ lnn] = 0,

♥ q ♥s ♦♠♥tár♦s sã♦ ♣rt♥♥ts ♣r str s♠♣çõs q ♣r♠t♠ ♣á

t♠♥t ♠ ♠♦♦s Pr♠r♠♥t ♠s♠♦ s♠ ♦♥srr r③♦á ♣r♦①♠çã♦

|i − e| = |/en| ≪ || ss♠♥♦ q ts tr♠♦s sã♦ ♠s♠ ♦r♠ ♦srs q ♠

♦s ♦s út♠♦s tr♠♦s sã♦ s♣r③ís ♦♠ rçã♦ ♦ ♣r♠r♦ ♣♦r ♠ t♦r me/mi

st♦ q |ρd/dt| ∼ ρv2/a |me∇ · ()/e2n| ∼ (me/mi)ρv2/a ♠♥♦s q ♦♦rr ♠ ♦rt

r♥t ♥s |∇ lnn| ∼ (mi/me)/a ♦ q r♠♥t ♥ã♦ ♦♦rr ♠ ①♣r♠♥t♦s

♦♠ rçã♦ ♦ t♥s♦r s♦s ♦ tr♠♦ ♦♠♥♥t é s♦s ♣r ♣♦s

ωcατ ≫ 1 ♣r ♣s♠s ♠♥t③♦s í♦♥s st♦ q πe‖ ∼ (me/mi)1/2πi‖ ♦♠♣rçã♦

|∇ · πi|/|∇p| ∼ ω/νii ♠ q |πi‖| ∼ piω/ν st s♣r③r ♦s t♦s s♦s

♠ ♣s♠s t♠♥t ♦s♦♥s P♦r ♠ ♠ s trt♥♦ ♠♦♦s ①s rqê♥s

♦♥çã♦ qs♥tr ♣♦ sr ♦♥sr ♦♠ s ♥sts r♠♥t♦s q

♣♦ sr ♣r♦①♠ ♣r s♥t ♦r♠

ρd

dt+∇p− × = 0.

♥♦♠♥t ♣rtr ♦té♠s ♠ ♥r③

E + × = 1en(×−∇pe −∇ · πe +e)+

me

e2n

[

∂∂t +∇ · (+ − /en)

]

.

st qçã♦ ♣r♠r♠♥t ♣♦♠♦s ♦♠♣rr ♦ tr♠♦ ×/en ♦♠ ♦s tr♠♦s

♥tr ♦ts ♦s qs ♦s três ♣r♠r♦s sã♦ ♠s♠ ♦r♠ ♦ qrt♦ é ♠t♦ ♠♥♦r

♦ q sts ♣♦s /en ≪ ♠ ③ q β−1ρe/L ≪ 1 ♠s♠♦ ♠ sst♠s ①

♣rssã♦ β ∼ ε2 st ♦r♠ |me(∂/∂t)/e2n|/|×/en| ∼ ω/ωce = ρe/a ≪ 1 ♦ q

♠♦str q ♦s tr♠♦s ♥tr ♦ts ♣♦♠ sr s♣r③♦s P♦rt♥t♦ ♥ás ♦r s

rstr♥ ♦ tr♠♦ ×/en ♦ tr♠♦ ♠ét♦ ∇pe/en ♦ tr♠♦ rçã♦

e/en s ♦s ♣r♠r♦s sã♦ ♠s♠ ♦r♠ ♣♦s ∇pe ∼ × ♥♦ qír♦ ♣♦♠ sr

s♣r③♦s ♥ ♦r♠ vE ∼ vTi ♣♦s |∇pe/en|/|×| ∼ ρi/a ≪ 1 s♠♦ ♥ ♦r♠

r vE ∼ δρvTi ♦ tr♠♦ ×−∇pe ♣♦ sr ♠t♦ ♣q♥♦ ♦r♠ q s♣r③r

♦s tr♠♦s ♠♥ét♦ s♠t♥♠♥t ♣♦ sr stá ♦♠♦ Ti ∼ Te ♦

tr♠♦ ♣r♦♥♥t s♦s étr♦♥s ∇ · πe t♠é♠ ♣♦ sr s♦♥sr♦ st

♥♠♥t ♠ ♦ tr♠♦ rçã♦ sr ♥s♦ ♦ q ♣♦ sr ①♣rss♦ ♦♠♦

e

en= η(0, 51‖ + ⊥)−

1

2

Te

TivTi

Bρi

(

∇‖ lnTe +3

2

×∇ lnTe

ωceτ

)

.

♠ ♦s tr♠♦s ♠ ♣rê♥tss ♣♦♠ sr s♣r③♦s q♥♦ ♦♠♣r♦s ♦♠ ×

♣♦s s sã♦ ♦r♠ ρi/λe ∼ ν/ωci ≪ 1 (ρi/a)/ωceτ ≪ 1 rs♣t♠♥t

♠ ♦♠ rsst ♣♦ ♥tã♦ sr ♣r♦①♠ ♣r

E + × = η(0, 51‖ + ⊥),

♣♦ré♠ ♣r ♥ô♠♥♦s q ♥♦♠ t♠♣♦s rtríst♦s ♥r♦rs ♦ t♠♣♦ sã♦ ♦

♠♣♦ ♠♥ét♦ τ = µ0a2/η ♦ t♦ rsst ♣♦ sr s♣r③♦ st ♦♥çã♦ é

stst ♠ ♠t♦s s♦ ♣♦s |η|/|×| ∼ (ν/ω)(ρ2e/a

2) ♦ s só é ♦ ♠

♥ô♠♥♦s rqê♥s ♠t♦ ①s ♠ ♣s♠s t♠♥t ♦s♦♥s s♦ st♦ ♥ã♦ ♦♦rr

♠ ♣♦ sr ♣r♦①♠ ♣♦r

E + × = 0.

♥♠♥t rst ♥sr út♠ qçã♦

dpdt + γp∇ · − ·(∇pe−γpe∇ ln ρ)

en +

(γ − 1)

[

(πi + πe) : ∇+∇ · (qi + qe)− πe : ∇(/en)− ·e

en

]

= 0,

q ♦rrs♣♦♥ à ♦♥srçã♦ ♥r ♠ ♦♥çõs ♥♦r♠s γpe∇ ln ρ ∼ ∇pe ♣r

t♦♠s ♦♠♦ | ·∇pe/en|/|dp/dt| ∼ ρi/a ≪ 1 sts s♣r③r ♦ tr♠♦ q ♥♦

♦s ♣rê♥tss ♥ ♣r♠r ♥ ♦ ♦♥srr♠♦s q πe ≪ πi /en ≪ ♥♦

♠♥t à ♥ás ♦♠♣rçã♦ (|πi : ∇|/|dp/dt| ∼ ω/ν ♥♦s ♦♥r q ♦

tr♠♦ s♦s ♣♦ sr s♣r③♦ ♠ ♣s♠s t♠♥t ♦s♦♥s ♦♠ rçã♦ à ♥á

s ♦♠♣♦♥♥t ♣r ♦ ①♦ ♦r q‖α ≈ −κ‖α∇‖Tα ♦♠♦ κ‖i/κ‖e ∼ (me/mi)1/2 ≪ 1

|∇ · qe|/|dp/dt| ∼ ν/ω ♣♦♠♦s s♣r③r t ♦♠♣♦♥♥t ♠ ♥ô♠♥♦s ♦♥srá

rqê♥ ♠ r♠s ♥ã♦♦s♦♥s st tr♦ ♦♥sr♠♦s ♣♥s ♣s♠s s♠

rsst ♦r♠ q ♦rç rçã♦ ♣♦ sr s♣r③ ♠ ♦♠ rçã♦ ♦

①♦ ♦r ♣♥s ♦♠♣♦♥♥t

q× =p

eB2×∇T,

q é s♥sí ♦ r♥t t♠♣rtr é t③ ♥ qçã♦ ♥r q s r③

dp

dt+ γp∇ · + (γ − 1)∇ · q× = 0.

s qs ♦♥stt♠ ♠ sst♠ qçõs srs

♦♠ rás r♣rs♥ts ♣♦ ♦♥♥t♦ r♥③s srs ρ p T t♦rs

E st ♦r♠ tr♠♥çã♦ st sst♠ rqr ♠s qçõs ♥♣♥♥ts s

♠ ♥ês st ♦♠♣♦♥♥t é ♦♥ ♦♠♦ r♦ss t ①

sts qçõs sã♦ s ♦♠♣♦♥♥ts t♦rs s s♥ts qçõs ①

∇×E = −∂

∂t,

∇× = µ0.

♦t q qçã♦ ① q ①♣rss sê♥ ♠♦♥♦♣♦♦ ♠♥ét♦

∇ · = 0,

♥ã♦ ♣♦ sr ♦♥sr ♠ qçã♦ ♥♣♥♥t ss♠ ♦♠♦ ∇ ·E = 0 ♣♦s ♣♦

sr ♦t ♣♦ á♦ ♦ r♥t ♦s ♦s ♦s ♣♦ s♦ ♥t

♦ ♥t♥t♦ qçã♦ ♣♦ss ♠♣♦rtâ♥ ♥♠♥t ♠ ís ♣s♠s ♣r

tr♠♥çã♦ ♦ ♠♣♦ ♠♥ét♦ qír♦ ♣♦s st ♦♥çõs érs t♦rs

♣r ♦ á♦ ❬❪ ♦♥♦r♠ st♦ ♠

út♠ qçã♦ ♥ssár ♣r ♦♠♣tr ♦ sst♠ srt♦ ♠ é rçã♦ ♥tr

♥s ♣rssã♦ t♠♣rtr q ♦♥♦r♠ s ♥çõs ♥tr♦rs s q♥ts ♠

r♦só♣s ♣r ♠ ♦ ♣♦ sr ①♣rss ♦♠♦

p ≈ ρ

miT.

♠ ♠t♦s s♦s ss♠ ♦♠♦ ♥♦ ♣rs♥t tr♦ st ts é ♦♥♥♥t ①♣rssr E

♠ tr♠♦s ♣♦t♥s ♦ s

E = −∇Φ− ∂A

∂t, = ∇×A.

♣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 ♦ ♠♦♦ úst♦ í♦♥ ❲

♦rrçã♦ ♣r rqê♥ ♦s ①♦s ③♦♥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 ♣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 ♦

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 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♥♦ ♦ 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 ♥r③çã♦

s qçõs ♣♦r ♠♦ t♦r ♣rtrçõs st t♦r s r♥③s ♠r♦s

ó♣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 ♣♥s ♣r♠r♦s r♠ô♥♦s m = ±1 ♥♦ 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 ♥ sê♥ 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+ 12∇Ψ·∇F 2

|∇Ψ|2 + µ0R2

|∇Ψ|2∇Ψ ·∇p+ µ0ρR2

2 ×[

∇Ψ|∇Ψ|2 ·∇

(

κ2

ρ2|∇Ψ|2R2

)

− ∇ΨR2 ·∇

(

κ2

ρ2

)

−(

Ω− κFρR2

)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♦ ♦ ♦rrçã♦ ♦rç

♥trí ♦ à r♦tçã♦ ♦ ♦♥trár♦ ♦ q ♦♦rr ♠ ♣s♠s s♠ r♦tçã♦ ♦♥ 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 ♦♥♥ê♥ 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 ♦♥♥ê♥ ♥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

γR2

0

(1 +Rρ)+(

∇Ψ·∇F1

∇Ψ·∇F0− F1

F0

)

RF + B0

2

(

|∇Ψ|2F 2

0

M2PRΨ2 −M2

T

)]

F 2

0

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♦ à ♥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♠♣rtr 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♦ ♥trt♥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♦ ♦♠ ❩ ♥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♠ ♣rs♥t♦ ♥♦ r♥t rtór♦ ❬❪ ♦ st♦

r♦tçã♦ rs ♠ ♣s♠s ♦ s♦ ♥♦ tr♦ r♦ t ♠ ❬❪

♠♦str♠♦s ♥ r ♦ ♣r r ♦ r♦tçã♦ qír♦ ♦t♦ ①♣r

♠♥t♠♥t ♥st t♦♠ ♣rtr st rá♦ ♣♦♠♦s st♠r ♦s ♦rs MP MT

♦♠ ♦ ♥tt♦ r rqê♥ ♦s ❲ ❩ sr q ♦ ♦r ♦

r♦tçã♦ ♣♦♦ ♣ró①♠♦ ♦r ♦♥ ♣s♠ ♣rst♦ ♣ t♦r ♥♦áss ♣r

♣s♠s ♦s♦♥s stá ♠ s♦r♦ ♦♠ ♦ rst♦ ♦t♦ ①♣r♠♥t♠♥t ♥♦

♥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

Posicao radial (cm) Posicao radial (cm)

0 5 10 15 200 5 10 15 20

0

1

2

3

4

5

6

7

8

-25

-20

-15

-10

-5

0

5

10

Velocidad

etoroidal

(km/s)

Velocidad

epoloidal

(km/s)

1 - Experimental2 - Neoclassica

r Pr r ♦ r♦tçã♦ qír♦ ♣♦♦ sqr t♦r♦ rt ♥♦ t♦♠s Pr ♦ s♦ r♦tçã♦ ♣♦♦ ♦ ♣r ①♣r♠♥t ♦ ♦♠♣r♦ ♦♠ ♦ ♣r tór♦ ♦t♦ trés t♦r ♥♦ássst r é ♠ ♣tçã♦ s rs ♦ tr♦ ♦r♥ r♦ t ❬❪

s ♥çã♦ ∆Q q

·∇Q = ∆QQ0 ·∇R2

R20

♣♦ sr s♥♦♦ ♦♥sr♥♦ Ψ ≈ Ψ(r) ♦ s ≈ F (r)R−1φ + (Rr)−1(dΨ/dr)θ

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) = mic2s

γ [1 + 2ε(∆p −∆ρ) cos θ],

❱ = VP θ + VT φ, VP = ΩP r, VT = ΩTR,

VP ≈ εqMP 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 s r♦tçã♦ qír♦ ♥♦s ♦r♠ ♦♥♥♥t

♣♦r

F‖ = ρ0( : ∇❱+ ❱ : ∇) + ρ❱ : ∇❱,

∂R

∂t= ❱ ·∇ρ+ ·∇ρ0 + ρ∇ ·❱,

∂P

∂t= ❱ ·∇p+ ·∇p0 + γp∇ ·❱+ (γ − 1)∇ · q,

s ♠ st á♦ ♠♦s ♠ ♦♥t ♣♥s ♦s tr♠♦s ♦

♠♥♥ts ♦♠ rçã♦ ♦ t♦r ε = r/R0 ≪ 1 q sã♦ ♦s à ♦♥trçã♦ ♦s ♣r♠r♦s

r♠ô♥♦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 úst♦s ♦és♦s

sr sr♠♦s ♦ ♠♦♦ ♠s s♠♣s ♣r ①♣♦rr ♥â♠ ás s ♦sçõs

tr♦státs ♦♥s ♦♠♦ st ♣rt s♦♥sr♠♦s r♦tçã♦ qír♦ ♣♦r

♠♦t♦s át♦s ♦♠ ♥ ♥t③r ♦ ♠â♥s♠♦ ís♦ ♦r♠çã♦ ♦s

♥♠♥t t③♠♦s ssttçã♦ F‖ = P = R = 0 ♠ ♦♠♦ ♣♥s ♦s

♣r♠r♦s r♠ô♥♦s s♠♣♥♠ ♠ ♣♣ r♥t ♥ ♥â♠ ás ♦s tr♦s

tát♦s ❬❪ ♦♥sr♠♦s s♦çõs ♦r♠X = Xs sin θ + Xc cos θ ♣r s ♣rtrçõs

♠s ♠ s trt♥♦ ♠ ♥ás ♥r X ∝ −iωt ♦r♠ q ssttçã♦ ∂/∂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♥ ♦rr♥

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♦ ω 6= 0 v‖ → ∞ ♣r ❩ ♣♦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♦ 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 s rqê♥s ♦s ❩

ω❩ = 0, ω2 =

(

2 +1

q2

)

c2sR2

0

.

r ♣r ❩ ♦♠♦ ♥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♥tss 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 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 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

♠♣ q rρi < 0 ♣♦s ♣♥ê♥ r Φ ♦♥sq♥t♠♥t s rr sã♦ s♦♥s ♠ ♣r♥í♣♦

♠♣♦ étr♦ ♠ ♦♠♦ ♣♦ ♠♣♦ ♠♥ét♦ t♦r♦ qír♦ B ≈ B0(1 − ε cos θ)

s♦r♠ ♠ ♠♦♠♥t♦ r ♦ 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 ♦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♥t q t♥ ♥r ♦ ♠♣♦ étr♦ ♥ ♣♦ tr♥s♣♦rt

r ♣♦st ♣r ♦r s♣rí ♠♥ét rrê♥ ♥trt♥t♦ ♠ ♦rrê♥ ♦

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

♦

(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 ♠ t♦♠s

♦ r E × ♥trt♥t♦ ♦ ♥ér í♦♥s à ♦rr♥t ♠♥ét

♥ ♣rs♥ts ♦ ♠♣♦ étr♦ ♥rt s s♥t♦ ♠ t = π/ωGAM ♦ r é

♠á①♠ ♥♦ 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 ❩ ♥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 ❬❪

t♦ r♦tçã♦ ♥♦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,

♥ss ♣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♦

D() = Ω

(

− Ω2

2q2+ 1 +

1

2q2

)

,

r ♥rs ♦ ♥♦♠♥♦r D(P) ♣r MP ≥ 0

D() =M2

T

Ω

[

2(1 +M2P )

(

1− MP

MT+ 1

2M2

P

M2

T

)

+

(

14 − MP

MT

)

M2T+

(

12 − MP

MT+

M2

P

M2

T

)

MtMP

]

Ω2 − 12MtMP

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 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 ♦♥sr♥♦ ♣♥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♦

♦♠ ssttçã♦ 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 ❩ ♦ r♠♦ ♦és♦ ♥♦ r♠♦ úst♦ í♦♥s s ♦rrs

♣♦♥♥ts rqê♥s sã♦ ♦♠♥s ♥sts ♦s r♠s ♣r q ≫ 1 ♣♦♠ sr ♣r♦①♠s

♦♠♣rçã♦ ♥tr ♦s qr♦s s rqê♥s ♥♦r♠③s ♣♦r cs/R0♦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

♣♦r

ω2GAM

c2s/R20

≈ 2 +1

q2+M2

P + (MP − 2MT )2,

ω2SW

c2s/R20

≈ 1

q2+

(3MP − 4MT )

q2MP .

♠ s trt♥♦ ❩ ♥♦ 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 ♠ ♦ ♦♠

t♦r ♦s ♦s st ♦r♠ ♦♥♦r♠ rqê♥ ♦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

♦♠ rçã♦ ♦ í♥ át♦ ♦♥sr♠♦s t♠é♠ ♦ t♦ ♥s♦tr♦♣ ♣rs

sã♦ ♦ 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 ♦rrçõs

O(q−2) ♥ rqê♥ ♦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♦s ♦ ♣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

♥♠♥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♦ ♣♦ré♠ ♥♦r♠çõs s♦r ♦ ♣r í♦♥s étr♦♥s srã♦ ♦♥s

rs ♣♥s ♠ tr♦s tr♦s ♣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

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 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∗iω

Φ±1

)

en0

Ti,

pi±1=

(

±5

3

i

2

ωω

Φ0 ∓ (1 + ηi)ω∗iω

Φ±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∗erρiΦ0, Φc = −i

τeωiω∗eω2 − ω2∗e

rρiΦ0 = −iω∗eω

Φs.

♥♦♠♥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♠♥t ♣rs♥t♦ ♦ á♦ ♠é ♠ ♠ s♣rí

♠♥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∗eω

2i

)

,

♦♥ 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 ♥♦ ηi > 0, 75 st ♠♦♦ ♣rê ♠ ♥st ♣♦ssí

♦♥r q á rs ♥çõs q r♥ts t♠♣rtr ô♥ t♥♠ sst③r

♦ ♣s♠ ♦ ♣ss♦ q r♥ts ♥s ♦♥tr♠ ♣r st③á♦ ♦r♦ ♦♠

♥ás ♦ ♦r ηi ♥ s♦çã♦ ♥t

sssã♦ s♦r 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♦

♦♠ 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 q♥♦

♥s♦tr♦♣ ♣rssã♦ ♣rtr í♦♥s é ♦♥sr ♦t♠♦s ①♣rssõs ♥íts ♣r

três ♠♣♦rt♥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 s♦♥♦r♦s ♠♦♦s úst♦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 ❩ 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 rs♦♦ 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 ❩ ω ∼ 0 ❲ ω ∼ vTi/qR0

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♦♠ ❲ 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 rr

♥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 ❩ 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 ❩ é ♠ s ♦♥trçõs st ts

t♦ ♥s♦tr♦♣ ♣rssã♦ í♦♥s ♦ ♦♥sr♦ ♥♦ ♦♥t①t♦ t♦r ♦s ♦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 ♥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 rt

ríst♦s LN LTi rs♣t♠♥t ♦t♠♦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♦♥ à

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 ♣♦ sr 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 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 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♦ 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♦ 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 ❩

♣ít♦

♥stçã♦ ♠♦♦s úst♦s

♦és♦s ♣♦ ♠♦♦

r♦♥ét♦

♥♦ ♠ st ♦ q ♦ ①♣♦st♦ ♥♦s ♣ít♦s ♥♦s qs sssã♦ s♦r ♦ ♠♦♦

r♦♥ét♦ ♣çã♦ ♦ ♠♦♦ ♦s ♦ st♦ t♦s ♠♥ét♦s ♥♦s ♠♦♦s

úst♦s ♦és♦s sã♦ ♣rs♥t♦s ♥st♠♦s ♥st ♣ít♦ t♦s ♠♥ét

♦s ♠♦rt♠♥t♦ ♥ ♥♦s t③♥♦ t♦r r♦♥ét

st♦ ♣rtr ♦ ♠♦♦ r♦♥ét♦

st sçã♦ ♣r tr ♦♠♣r♥sã♦ ♦ ♠♦♦ r♦♥ét♦ ♣♦ à ♥â♠

♣♦r s♠♣ ♥ã♦ ♠♦s ♠ ♦♥t r♥ts ♥s t♠♣rtr

♥♠♥t ♦♥sr♠♦s ♥çã♦ strçã♦ fα q ♦♥♦r♠ é r♣rs♥t

♣ ①♣rssã♦

fα = eαΦ∂FMα

∂Eα+ gα

i⊥·ρα , ρα =⊥ ×

ωcα

,

♦♥ gα t♠ s ♥â♠ ♦r♥ ♣ qçã♦ r♦♥ét q q♥♦ s♣r

③♦s ♣rtrçõs ♦ ♠♣♦ ♠♥ét♦ ≈ 0 r♥ts ♥s t♠♣rtr

∇Fα ≈ 0 ♣♦ sr srt ♦♠♦

∂gα∂t

+ (gα ·∇)gα = −eα∂FMα

∂Eα∂Φ

∂tJ0(k⊥ρα).

♦ ♦ ♥tr♦ ♦♥♦r♠ q♥♦ ♥ã♦ á r♦tçã♦ qír♦

Φ0 = 0 é ♦t ♣ ①♣rssã♦

gα = v‖+

ωcα

×(

v2⊥2∇ lnB + v2‖κ

)

,

q ♠ r♠s ① ♣rssã♦ ♦ s q♥♦ β = O(ε2) ♣r t♦♠s t r③ã♦

s♣t♦ ε ≪ 1 ♣♦ sr ♣r♦①♠ ♣♦r

gα ≈ v‖− 1

ωcαR0

(

v2⊥2

+ v2‖

)

(sin θr + cos θθ).

Pr s♥♦r r♠♥t ♦t♠♦s s s♥ts ssttçõs

∂

∂t→ −iω, ∇ = i,

∂FMα

∂Eα= −FMα

Tα

♦♥ FMα é ♥çã♦ ①♥ q ♠ tr♠♦s ♥r ♣rtí Eα é r♣rs♥t

♣♦r = rkr + k‖ s ♦♣r♦rs r♥s ♥t♦s ♣♦r ˆ q ♦♥stt♠

s ♦♠♣♦♥♥ts t♦rs sã♦ ♥♦s ♣♦r

kr = −i ∂∂r , k‖ = · (θkθ + φkφ) = −ik‖

(

∂∂θ + q ∂

∂φ

)

,

kθ = −ikθ∂∂θ , kφ ≈ 1

R0

∂∂φ , kθ =

1r , k‖ =

1qR0

.

♠s♠ ♦r♠ ♦♠♦ ♥♦ ♣ít♦ ♦♥sr♠♦s ♦♥çã♦ r ≫ kθ ♦♥③♥t ♦♠

♦r♠ ♦ q st ♣r♦①♠çã♦ k⊥ ≈ r ♥♦ r♠♥t♦ ♥çã♦ ss

J0(k⊥ρα) ♠ ♥tã♦ q q ♣♦ sr ♣r♦①♠ ♣r

[

ω − k‖v‖ +

(

v2⊥2

+ v2‖

)

sin θ

ωcαR0r

]

gα = ωeαTα

J0(rv⊥/ωcα)ΦFMα .

♦♠ rçã♦ s rás θ φ ♥çã♦ gα ♣♦ sr ①♣♥ ♠ sér ♦rr

gα =∑

m,n

g(α)mn(r)i(mθ−nφ),

♦♥ m n sã♦ rs♣t♠♥t ♦s ♠♦♦s ♣♦♦ t♦r♦ Pr n = 0 ♦r♠

q q ♣♦ sr srt ♦♠♦

∞∑

m=−∞

[

iΩα

2g(α)m−1 + (1−mΩtrα)g

(α)m − i

Ωα

2g(α)m+1 − FMαJ0α

eαTΦm

]

imθ = 0,

♦♥ J0α = J0(rρα) g(α)m = g

(α)m0(r)

Ωα =

(

1

2

v2⊥v2Tα

+v2‖v2Tα

)

ωαω

, ω = rρiωi, ω = −τeω, ωi =vTi

R0,

Ωtrα =v‖vTα

ωtrαω

, ωtri = k‖vTi= ωi/q, ωtre =

√

τe(mi/me)ωtri ≫ ωtri.

♠ ωtrα é rqê♥ rçã♦ q stá ss♦ ♦ ♠♦♠♥t♦ ♦ ♦♥♦ s

♥s ♠♣♦ ♠♥ét♦ ♦♠♣r♠♥t♦ ♦♥ rt♦ ♦ ♠♦♠♥t♦ ♣r♦ ♣♦ sr

r♣rs♥t♦ ♣ ①♣rssã♦ λ‖ = 2πqR0/|m − nq| ♦♥ ♣♦ sr ♥♦t♦ q λ‖ → ∞ ♥s

s♣rís ♠♥éts r♦♥s ♠ q q = m/n ♣rís ♠♥éts r♦♥s sã♦ qs

♠ q s ♥s ♠♣♦ s ♠ s♦r sí ♠s♠s q r♦♥ ♣♦r sr♠ rtr③s ♣♦r

♥sts s rss♦♥â♥s q ♥s ♦♦rr♠ t♠ r ♠♣♦rtâ♥ ♠ rs♦s

♥ô♠♥♦s ♠ ís t♦♠s

Pr rs♦r ♠ ♠ ♣r♠r ♣r♦①♠çã♦ ♣♦♠♦s ♦♥srr gm = 0 ♣r |m| ≥ 2

♦ s ♥♦ ♠ ♦♥t ♣♥s ♦ t♦ ♣r♠r♦s r♠ô♥♦s ♦♠ ♦ s♦ st ♣r♦①♠

çã♦ q é á ♣r rρi ≪ 1 ♦t♠♦s s♦çã♦

g(α)0 =

(J0αeα/Tα)FMα

1− Ω2trα − Ω2

α/2

[

(1− Ω2trα)Φ0 − i

Ωα

2(Φ1 − Φ−1)− i

ΩαΩtrα

2(Φ1 + Φ−1)

]

,

g(α)±1 =

1

1∓ Ωtrα

(

±iΩα

2g(α)0 + J0α

eαTΦ±1FMα

)

,

♥ q ♦ ♥♦♠♥♦r ♣♦ sr ♣r♦①♠♦ ♣♦r

1

1− Ω2trα − Ω2

α/2=

1

1− Ω2trα

+1

2

Ω2α

(1− Ω2trα)

2+O(Ω4

α).

♦r♠ tr♦♥♦♠étr s qs ♣rtr ♦ s♦ s rçõs g(α)s = −i(g(α)1 − g

(α)−1 )

g(α)c = g

(α)1 + g

(α)−1 ♣♦♠ sr ①♣rsss ♦♠♦

g(α)0 =

J0αeα/Tα

1− Ω2trα

[(

1 +Ω2α

2− Ω2

trα

)

Φ0 +Ωα

2Φs + i

ΩαΩtrα

2Φc

]

FMα ,

g(α)s =J0αeα/Tα

1− Ω2trα

(

ΩαΦ0 + Φs + iΩtrαΦc

)

FMα ,

g(α)c =J0αeα/Tα

1− Ω2trα

[

−iΩαΩtrαΦ0 − iΩtrαΦs + Φc

]

FMα .

♦ ♦♥♦ st ts ♦♥sr♠♦s st ♠♦♦ ♣r ♦t♥çã♦ ♠♦♦s rqê♥s ♥♦

♥tr♦ ωtri < ω ≪ ωtre ã♦ ♠♦s ♠ ♦♥t ♦ t♦ ♥t♦ ♠ss étr♦♥ ♦ s

me/mi ≈ 0 ♦r♠ q ωtre → ∞ sr ♥st♠♦s ♦s t♦s ♦ ♠♦♠♥t♦ ♣r♦

s ♣rtís ♦s qs stã♦ r♦♥♦s ♦♠ ♦ ♦r rqê♥ ωtri ♠t♦♦♦

q ♦t♠♦s ♣r ♦ st♦ st ♠♦♠♥t♦ s ♠ ♦r♠ rt ♦♠

rçã♦ ♦ s♥♦♠♥t♦ ér♦ st ♦r♠ ♣r♠r♠♥t ♦♥sr♠♦s q → ∞ ♦

s ωtri = 0 ♥trt♥t♦ ♦♥çã♦ mi/qme → ∞ sr ♦♥sr s♠t♥♠♥t ♦

q s tr③ ♣r ωtre → ∞ ♠ ♠ s♥♦ ♣ss♦ ωtri/ω ≪ 1 é ♦♥sr♦ ss♠♥♦ q

ω é r ♦ q ♥♦s ♣r♠t ♦tr ♦rrçõs O(q−2) ♣r rqê♥ ♦s s♠♦ ♦♠

st ♦♥srçã♦ ♥ ♥ã♦ str♠♦s ♥♦ ♠ ♦♥t ♠ ♠♣♦rt♥t t♦ ss♣çã♦

♥ã♦ ♦s♦♥ ♦♥ ♣♦r ♠♦rt♠♥t♦ ♥ ❬❪ ♦ q é ♥♦ ♥♦ ♥ st

sçã♦ ♥ trr t♣ st t♣ ω é st♦ ♦♠♦ ♠ rqê♥ ♦♠♣① ♠ ♣r♥í♣♦

♠ tr♠♦s ♥çã♦ s♣rsã♦ ♣s♠ ❬❪ Z(x) ♦ rst♦ ♣r ωtri ∼ ω é ♦t♦

♠t ♦ ♦♠ k‖vTi= 0 q → ∞

♥ts ♥r ♦ á♦ ér♦ ♣rtr ♣r♦①♠çã♦ k‖vTi= 0 st♠♦s rs♣t♦

♥s ♣rtís q é t③ ♥ ♦t♥çã♦ rçã♦ s♣rsã♦ ♥s

♣rtís ♣♦r s ③ é ♦t ♣ ♥trçã♦ ♥♦ s♣ç♦ ♦s ♥çã♦

strçã♦ ♦ s

nα =⟨

fα

⟩

= n(C)α + n(G)

α ,

♦♥ ♣r ♠ r♥③ ♥ér ♦r♠ Xα = Xα(t, r, v⊥, v‖, γ) ♥♠♦s 〈〉 ♦♠♦

X♠r♦só♣♦α (r, t) = 〈X〉 =

∫

vd3vX♣rtí

α , d3v = dγdv‖dv⊥v⊥.

♠ n(C)α n

(G)α r♣rs♥t♠ rs♣t♠♥t s ♦♥trçõs í♥r s♠étr ♠

rçã♦ θ ♦és s♥sí rtr ♦és ♦ ♠♣♦ ♠♥ét♦ ♥ã♦ s♠étr ♠

rçã♦ θ ♣r ♥s ♣rtís ♦r♦ ♦♠ s q

n(C) = eαΦ

⟨

∂FMα

∂Eα

⟩

= −eαΦ

TαFMα , n(G) =

⟨

gαi⊥·ρα

⟩

.

Pr étr♦♥s ♦♠♦ ωtre → ∞ Ωtr → ∞ ♣♦ sr ♦♥sr♦ ♠

♦r♠ q

g(e)0 =

−eΦ0

TeFMe g(e)s = g(e)c = 0,

♦♥sq♥t♠♥t

n(C)e =

en0

Te(Φ0 + Φs sin θ + Φc cos θ), n(G)

e ≈ −en0

TeΦ0.

♦t q 〈FMe〉 = 〈FMi〉 = n0

♦♠ rçã♦ ♦♥trçã♦ ♦és t③♠♦s rçã♦

〈exp(−i⊥ · ρe〉 = J0(k⊥ρe) ≈ 1 ♣♦s ♣♦♠♦s s♣r③r ♦ t♦ r♦ r♠♦r ♥t♦

st rst♦ é ♠t♦ s ♦srr♠♦s q ♥tr ♦ ♣r♠r♦ ♠♦♠♥t♦ ♥çã♦ strçã♦é ♥s ♣rtís n0 s ♦♥srr♠♦s ♦♥çã♦ qs♥tr ni = ne = n0

♣r étr♦♥s

♦♥trçã♦ í♥r ♣r í♦♥s é ♥♦ étr♦♥s ♦ s ♥s t♠ ♠

rs♣♦st ♦t③♠♥♥

n(C)i

n(C)e

= −τe, τe =Te

Ti.

♦ ♥t♥t♦ ♦♠ rçã♦ à ♦♥trçã♦ ♦és rs♣♦st é ♦♠♣t♠♥t r♥t ♦♥

sr♠♦s sr ♦ s♦ ωtri/ω = 0 ♦♥sq♥t♠♥t Ωtr = 0 ♦r♦ ♦♠

♦r♠ q ♦ rst♦ ♣r ni(G) ♣♦ sr ①♣rss♦ ♦♠♦

n(G)i =

en0

Ti

[(

I(i)0 +

I(i)2

2

)

Φ0 +I(i)0

2Φs

]

+

[

I(i)1 Φ0 + I

(i)0 Φs

]

sin θ + I(i)0 Φc cos θ

,

♦♥ s ♥trs I(i)n =

⟨

J20i(Ω/ω)

n⟩

n = 0, 1, 2 sã♦ s ♠ ♣♦♠ sr ♣r♦

①♠s ♣♦r I(i)0 = 1 − 2rρ

2i /2 I

(i)1 = rρi/Ω I

(i)2 = (7/4)2rρ

2i /Ω

2 ♠ ♦t q

♥♦♠♥t ♦ ♣ít♦ ♥tr♦r ♦♥sr♠♦s rqê♥ ♥♦r♠③ ♠s ♥

♦r♠ r♥t q ♦ s

Ω =ω

ωi=

ωR0

vTi

.

♣♦rt♥t♦ q ♦ á♦ ♥s ♣rtr t♦t í♦♥s étr♦♥s ♠ ♣r♠r

♦r♠ rst ♠

ni

en0/Ti= 1

2

[

(−Ω2 + 7/4)2rρ

2

i

Ω2 Φ0 +rρiΩ Φs

]

+rρiΩ Φ0 sin θ,

ne

en0/Ti= τi(Φs sin θ + Φc cos θ), τi =

Ti

Te.

Pr ♦tr rqê♥ ♦s ♠♣r♠♦s ♦♥çã♦ qs♥tr

e(ni − ne) = 0,

♥ q ssttí♠♦s ♦s ♦rs s ♥ss ♣rtrs í♦♥s étr♦♥s ♠♦str♦s ♠

♦ q rst ♥ qçã♦

1

2

[(

−Ω2 +7

4

)

2rρ2i Φ0 +ΩrρiΦs

]

+

(

ΩrρiΦ0 − τiΩ2Φs

)

sin θ − τiΩ2Φc cos θ = 0,

q é á ♣r Ω 6= 0 é stst ♣r θ rtrár♦ s s♦♠♥t s

Φc = 0, Φs = τerρiΩ

Φ0,

(

−Ω2 +7

4+ τe

)

2rρ2i Φ0 = 0.

♦t q sr♠ rçã♦ Φm = O(mr ρmi )Φ0 q ♣♦ sr t③ ♠

♠♦♦s ♠s r♥♥ts út♠ rçã♦ ♠ rst ♥ rqê♥ ♦s

ω = ΩvTi

R0, Ω2

=7

4+

Te

Ti.

q ♦♥♦r ♦♠ ♦ ♦r ♦t♦ ♣ t♦r ♦s ♦s ♦♠ s♦s ♣r ♦♥♦r♠

sr ♦♥sr♥♦ k‖vTi♥t♦ ♣♦ré♠ k‖vTi

≪ ω ♦t♠♦s ♦rrçõs O(q−2) ♣r

rqê♥ ♠♦str ♠

♠t ♦ ♦♠ k‖vTi♥t♦ q ≫ 1

♦ ♠t ♦ t③♠♦s s♥t ①♣♥sã♦ ♠ sér

1

1− Ω2tr

= 1 + Ω2tr +O(Ω4

tr),

♠ ♦ s♣r③r tr♠♦s O(Ω3tr) rçã♦ s♣rsã♦ srt ♥

♦r♠ ♠tr

e2n0

Ti

(

1 sin θ cos θ

)

R00 R0s R0c

Rs0 Rss Rsc

Rc0 Rcs Rcc

Φ0

Φs

Φc

= 0,

♦♥ ♦s ♠♥t♦s ♠tr③ ♥tr sã♦ ♥♦s ♣s s♥ts ①♣rssõs

R00 = I(i)00 − 1 + 1

2(I(i)20 + I

(i)22 ) =

12

[

−Ω2 + 74 + 23

81

q2Ω2

]

2rρ2

i

Ω2 ,

R0s =12Rs0 =

12(I

(i)10 + I

(i)12 ) =

12

(

1 + 1q2Ω2

)

rρiΩ ,

Rss = Rcc = I(i)02 − τi =

12q2Ω2 − τi,

R0c = −12Rc0 =

i2I

(i)11 = 0, Rsc = −Rcs = iI

(i)01 = 0,

♦♥ s ♥trs ♦r♠ I(i)ab =

⟨

J20iΩ

aΩ

btr

⟩

♦♠ a, b = 0, 1, 2 sã♦ ♦♠♣ts ♠ Pr

q ♥ã♦ s♦çã♦ tr ♠♣õs q ♦ tr♠♥♥t ♠tr③ ♥tr s

♥♦ ♦ s

R2ss

(

R00 − 2R2

0s

Rss

)

= 0.

♦♥çã♦ Rss = 0 ♦t♠♦s rqê♥ sr ♥♦ r♠♦ úst♦ í♦♥s

ωsw0 = ΩsvTi

R0= k‖cse, Ω2

s =τe2q2

,

♦♥ cse =√

Te/mi é ♦ ♦ s♦♠ ♥♦ ♠t í♦♥s r♦s Ti ≪ Te ♣♦s c2s =√

(γiTi + γeTe)/mi γe = 1

Pr Rss 6= 0 s q

(

−Ω2 +7

4+ τi

23

4

Ω2s

Ω2+

τe + 4Ω2s/Ω

2

1− Ω2s/Ω

2

)

2rρ2i

Ω2= 0.

♦t q ♥♦ ♠t Ωs → 0 q → ∞ s♦çã♦ ♠♦str ♠ é ♦t ♦♥sr♥♦

rρi 6= 0 s s♦çõs ♠ Ω2 sã♦ ♦ts ♣rtr s♥t qçã♦ ú

Ω6 − (Ω2 +Ω2

s)Ω4 −

(

23

4τi +

9

4

)

Ω2sΩ

2 +23

4τiΩ

4s = 0, τi =

Ti

Te.

P♦ t♦ ♥ã♦ r♠♦s ♠ ♦♥t r♥ts ♥s t♠♣rtr ♦♥ts ♥s

ts t♦s ♥ét♦s ♥st sçã♦ s ú♥s s♦çõs ♦♠ s♥♦ ís♦

sã♦ s ♣♦sts sts s♦çõs ♣♦♠ sr tr♠♥s ♥t♠♥t ♦r♠ ♣r♦①♠

♣♦s s ♣♦ss♠ ♦r♥s r♥③s st♥ts Ω1 ∼ 1 Ω2 ∼ Ωs ♦♥ Ωs ≪ 1 Pr

tr♠♥çã♦ ♣r♠r ♣♦♠♦s s♣r③r ♦ út♠♦ tr♠♦ ♦ ♦ sqr♦ ♠ q é

O(Ω4s) ♦♠ rçã♦ à s♥ s♦çã♦ ♣♦♠♦s s♣r③r Ω6 ♠ ♦t♥♦ t♠é♠

♠ qçã♦ qrát ♦ ♣r♠r♦ s♦ qçã♦ sr rs♦ é ♣♦r

Ω4 − (Ω2 +Ω2

s)Ω2 −

(

23

4τi +

9

4

)

Ω2s = 0.

s s♦çõs

Ω2 =Ω2

2

[

1 +

(

7

4+ τe

)

Ω2s

Ω2

]

[

1±√

1 +(23τi + 9)Ω2

s/Ω2

(1 + Ω2s/Ω

2)

2

]

ú♥ ♦♠ s♥♦ ís♦ q r♣rs♥t ♦♠ ♦rrçõs ♠ q é s♦çã♦ ♣♦st

q ♣♦ sr ♣r♦①♠ q♥♦ Ωs/Ω ≪ 1 ♦ s

ωGAM =

(

Ω2 +

Ω2s

Ω2

)1/2 vTi

R0+O(q−4)

vTi

R0,

♦♥

Ωs =

(

23

4τi + 4 + τe

)1/2

Ωs.

♠t♥♦ Ω ∼ Ωs q ♣♦ sr ♣r♦①♠ ♣r

Ω4 +

(

23

4τi +

9

4

)

Ω2s

Ω2

Ω2 − 23

4τiΩ2s

Ω2

Ω2s = 0,

s♦çã♦ á q r♣rs♥t ❲s é ♣♦r

ωSW =

[√

1 +4(7 + 4τe)τe

23(1 + 9τe/23)2− 1

]1/2(

23

8τi +

9

8

)1/2Ωs

Ω

vTi

R0.

♦t q ♥♦ ♠t í♦♥s r♦s τe → ∞ Ω❲ → 1/q ♠ ♦r♦ ♦♠ t♦r

s ♦♥sr♦ γe = 1 ♦ ♥t♥t♦ ♣r τi ♥t♦ Ω❲ < 1/q ♦r♠ q ♥♦ s♣tr♦

rqê♥ st s♦çã♦ stá ①♦ rqê♥ úst ♦♥s

♦♠ ssttçã♦ ♦s ♦rs Ω Ωs Ωs ♠ ♦t♠♦s ♦r♠ ♥

rqê♥

ω =

[(

7

4+

Te

Ti

)

+1

q2

(

23

8+ 2

Te

Ti+

1

2

T 2e

T 2i

)(

7

4+

Te

Ti

)−1]1/2 vTi

R0,

q stá ♦r♦ ♦♠ ♦ rst♦ ♦t♦ ♥tr♦r♠♥t ♠ ❬❪ ♥sts s s qs

rs♣t♠♥t

♦ ♠t í♦♥s r♦s Ti ≪ Te s♦çã♦ ♣rs♥t ♠ ♦♥r ♣r ♦ ♦r

♦t♦ ♣♦ ♠♦♦ ❬❪ s ♦♥srr♠♦s γe = 1 ♦r♦ ♦♠

st ♦srçã♦ ♦ t ♥tr♦r♠♥t ♣♦r ♠♦②♦ ❬❪ ♥st s♦ rqê♥

♣♦ sr ♣r♦①♠ ♣♦r

ω ≈(

2 +1

q2

)1/2 cseR0

, cse =

√

Te

mi.

♣rtr s q

Φc = 0, Φs = −2R0s

RssΦ0 =

τeΩ2 + 2Ω2

s

Ω2 − Ω2s

rρiΩ

Φ0,

♦♥ ♣♦ sr ♥♦t♦ q Ω = Ωs ss♠ ♦♠♦ ♠ r♣rs♥t ♠ s♥r

ss♣çã♦ ♥ ♠ ω > k‖vTi

Pr ♦rs ①♦s q ①♣♥sã♦ ♠ sér ♠ ♥ã♦ ♣♦ sr t③ ♦r♠

q ♦ s♥♦r s qçõs ♦s rst♦s ♣♥rã♦ ♥çã♦ s♣rsã♦ ♣s♠ ❬❪

q é ♥ ♦♠♦

Z(ζ) =1√π

∫ ∞

−∞dx

−x2

x− ζ,

♦♥ x é r ζ ♦♠♣①♦ ♦♠ s ♣rt ♠♥ár Im(ζ) ♣♦♥♦ ss♠r ♦rs ♣♦st♦

♥t♦ ♦ ♥♦ ♠♥t ♦ ♠ét♦♦ ♣r♦♦♥♠♥t♦ ♥ít♦ Pr ♦t♥çã♦ rst♦s

♥ít♦s ♦♥srs |Im(ζ)| ≪ |Re(ζ)| ♦r♠ q st ♠♦♦ s ♣ ①s♠♥t

♦ st♦ ♠♦♦s r♠♥t ♥stás ♦ ♠♦rt♦s

á♦ ♦s tr♠♦s ♠tr③ ♥tr ♠ tr♠♦s ♥çã♦ s♣rsã♦ rst

♠

R00 = L(i)0 − 1 + 1

2L(i)2 − L(i)

02 = −12

32ζ

2α + ζ4α +

(

12ζα + ζ3α + ζ5α

)

[Z(ζα)− i√π−ζ2i ]

2rρ2

i

Ω2 ,

R0s =12Rs0 =

12L

(i)1 = −1

2

ζ2i +

(

12ζi + ζ3i

)

[Z(ζi)− i√π−ζ2i ]

rρiΩ ,

Rss = Rcc = L(i)0 − 1− τi = −

1 + τi + ζi[Z(ζi)− i√π−ζ2i ]

,

R0c = −12Rc0 =

i2L

(i)11 ∝ −ζ2i , Rsc = −Rcs = iL(i)

01 ∝ −ζ2i ,

♦♥ ζi = qΩ Ω = ΩR + iΓ ♠ t③♠♦s s s♥ts ♥çõs

L(α)a =

⟨

J20αΩ

aα

1− Ω2trα

⟩

L(α)ab =

⟨

J20αΩ

aαΩ

btrα

1− Ω2trα

⟩

, a, b = 0, s, c,

♦s á♦s sã♦ t♦s ♠

rçã♦ s♣rsã♦

R2ss

[(

1 +R2

sc

R2ss

)

R00 − 2(R2

0s −R20c)

Rss− 4

R0sR0cRsc

R2ss

]

= 0,

é ♦t ♣rtr ♦♥ ♦s ♦rs Rab sã♦ ♦s ♦♥sr♥♦s

♦r♠ ♣r♦①♠ Z(ζi) ♦♥♦r♠ ♣r ζi ≫ 1 ♦♠ st ♣r♦♠♥t♦ ♣rtr

ssttçã♦ ♦s tr♠♦s ♦♠♥♥ts ♠

R00 =12

[

−Ω2 +

(

74 + 23

81ζ2i

+ i(σ − 1)√πζ5i

−ζ2i + i(σ − 1)√πζ3i

−ζ2i

)]

2rρ2

i

Ω2 ,

R0s =12

(

1 + 1ζ2i

+ i(σ − 1)√πζ3i

−ζ2i

)

rρiΩ , Rss = −

(

τi − 12ζ2i

+ i(σ − 1)√πζi

−ζ2i

)

.

♠ ♥ã♦ ♥♦ ♠ ♦♥t tr♠♦s ♦r♠ −2ζ2i rst ♦r♠ ♥

−Ω2 +Ω2 +

Ω2s

Ω2+ i(σ − 1)

[

1 + 2(2 + τi)Ω2s

Ω2

]√πq5Ω5−q2Ω2

= 0.

♦t q ♣r σ = 1 rt♦♠♠♦s ♦ ♠t ♦ ♦s rst♦s ♦r♠ ♣rs♥t♦s ♥tr

♦r♠♥t ♦♠ s♦ ♦rrt σ = 2 q r♣rs♥t ♦ s♦ ♠ q ♦♦rr ♠♦rt♠♥t♦

♥ ♦t♠♦s rçã♦ ♦rrt

r③ã♦ ís ♣r q sts ♠♦♦s s♠ ♠♦rt♦s é s ♥♦ t♦ q ♦ ♥tr♥♦

♥tr ♦♠ rçã♦ ♦ ♣r é ♠ ♥çã♦ ss♥ ♣♦rt♥t♦ ♦ ♥ú♠r♦

♣rtís ♦♠ ♦s ♥r♦rs à ♦ s ♦♥ vph q stá ♦③

♥ strçã♦ é ♠♦r ♦ q ♦ ♥ú♠r♦ ♣rtís ♦♠ ♦s s♣r♦rs

vph ♠ ♦♥sqê♥ á ♠s ♣rtís q r♠ ♥r ♦♥ ♦ q ♣rtís

q ♠ ♥r ♦r♠ q ♦♠ ♣r ♥r ♦♥ é ♠♦rt ♦

♥trçã♦ ♦♥♣rtí ♠é ♦ t♦ ♥ ❬❪

st sçã♦ ♣♦ t♦ str♠♦s ♥trss♦s ♥♦s ♦rs rqê♥s t①s ♠♦r

t♠♥t♦ s♦♠♥t ♠ ♣r♠r ♦r♠ ♦s qs sã♦ ♣rs♥t♦s ♥♥♦ ♣ró①♠ ♦r♠

♥ sçã♦ ♦♥sr♥♦ Ω = ΩR + iΓ ♦♠ Γ < 0 |Γ| ≪ |ΩR| qçã♦ sr rs♦

♣♦ sr ♣r♦①♠ ♣♦r

D(Ω) = F(Ω) + iK(Ω) = 0,

F ≈ Ω2 − (7/4 + τe), K ≈ √πq5Ω5−q2Ω2

.

Pr rs♦r t③♠♦s ♠ ♣r♦♠♥t♦ trt♦ s♦ ♥ ①♣♥sã♦ ♠ sér

②♦r ♣r Ω = ΩR ♠ ♣r♠r ♦r♠ ♥♠♥t ♦♥sr♠♦s ♣r♦①♠çã♦ ♠ sér

D(Ω) ≈ F(ΩR) + iK(ΩR) + iΓ

(

∂F∂Ω

+ i∂K∂Ω

)∣

∣

∣

∣

Ω=ΩR

= 0.

P♦str♦r♠♥t s♦♥♦ ♣rt r ♠♥ár ♠ ♦♥ ♠s ♠ s ♥r

♦t♠♦s s rçõs

F (ΩR) = 0,

Γ = − K(ΩR)

F ′(ΩR), F ′ =

∂F∂Ω

,

♥s qs s♣r③♠♦s ♦ tr♠♦ −ΓK′(ΩR) ♠ ♣♦r sr ♠t♦ ♣q♥♦ út♠♦ ♣ss♦

♦♥sst ♠ rs♦r ♣r ♦tr ΩR ♥♠♥t sstt♦ ♠ ♦ q ♣r♠t

♦t♥çã♦ Γ

♠ ♠ ♣r♠r ♣r♦①♠çã♦ s q

ΩR ≈√

7

4+ τe, Γ ≈ −

√π

2q5Ω4

R−q2Ω2

R .

♣♦ssí r ♦ ♠♦♦ ♣♦r ♠♦ ♦ á♦ ♥♠ér♦ Γ/ΩR Pr q = 1 τe = 3 ♦

s ♣ró①♠♦ ♦ ♥tr♦ ♦♥ ♣s♠ |Γ|/ΩR ≈ 0.17 ♦♥♦r♠ ♥♦s ♣r♦①♠♠♦s

♦r st ♦r ♠♥ rst♠♥t P♦r ①♠♣♦ s♣♦♥♦ q = 2, τe = 1 |Γ|/ΩR ∼ 10−4

♦♥í♠♦s ♣♦rt♥t♦ q ♦ ♠♦rt♠♥t♦ é ♠♣♦rt♥t ♣♥s ♣r q ①♦ ♦

q ♥♦r♠♠♥t ♦♦rr ♥♦ ♥tr♦ ♦♥ ♣s♠ ♣rtr st ♠♦♦ ♦ t♦ ♦

♠♦rt♠♥t♦ ♥ ♣r í♦♥s é t♦r♠♥t srt♦ ♦♠ st♥t ♣rsã♦ rst♦

q ♦t♠♦s ♣r Γ ♦♥♦r ♦♠ ♦ ♦t♦ ♠ ❬❪

sssã♦ s♦r ♣çõs ♦ ♠♦♦ r♦♥ét♦

♥ ♦r♠ ♠s r

sr ♣rs♥t♠♦s ♠ sssã♦ ♠s t s♦r ♦ ♠♦♦ r♦♥ét♦ ♥♦ q

♦♥sr♠♦s ♦ r♥t r ♥çã♦ ①♥ ♦ ♣♦t♥ t♦r ♣r♦ ♣rtr♦

A‖ ♦ q sr ♣rtrçõs ♠♥éts ♣r♣♥rs st ♦♥t①t♦ qçã♦

r♦♥ét sr s♥♦ é ①♣rss ♣♦r

∂gα∂t

+ (gα ·∇)gα = eα

(

−∂FMα

∂Eα∂

∂t+×∇FMα

mαωcα

· i⊥)

J0α(Φ− v‖A‖),

♦ ♥és rsã♦ ♣rs♥t ♠ ♥trt♥t♦ fα ♠ tr♠♦s gα ♣r♠♥ ♥tr

♦r♦ ♦♠ é ①♣rss ♦♠♦

fα = eαΦ∂FMα

∂Eα+ gα

i⊥·ρα .

♦♠ t③çã♦ ♠ rst q ♣r qqr m n sr stst

s♥t qçã♦

−iΩα

2g(α)m−1,n + [1− (m− nq)Ωtrα]g

(α)m,n + i

Ωα

2g(α)m+1,n = (1−mΩ∗)Ψ

(α)m,n,

♦♥

Ψ(α)m,n = J0α(Φm,n − v‖A‖m,n)

eαTα

FMα ,

Ω∗α =eαe

ω∗αω

[

1 + ηα

(

v2⊥v2Tα

+v2‖v2Tα

− 3

2

)]

, ω∗α =Tα

eBrLN∼ ρi/LN

r/R0

vTi

R0.

rçã♦ ♥tr A‖ Φ é ♦t ♣ ♠♣èr ♦ s ♦♥sr♥♦ r ≫ r−1 rst

q

J‖ =∑

α=i,e

eα

⟨

v‖fα⟩

≈ 2rµ0

A‖,

♦♥ fα = fα(Φ, A‖) ♥♦ s♦ ♠s r

qçã♦ ♠ ♦♥♥t♦ ♦♠ rçã♦ é ♦ ♣♦♥t♦ ♣rt ♣r ♥stçã♦

♥ú♠r♦s t♣♦s ♠♦♦s ê♥♦s ♦és♦s ss ♦rrs♣♦♥♥ts t♦♠♦♦s ♦♠

sts qçõs ♣♦♠♦s trtr rs♦s t♦s ts ♦♠♦ t♦s tr♦♠♥ét♦s A‖ 6= 0

t♦s r ω∗α 6= 0 ♠♦rt♠♥t♦ ♥ t ♦ ♥t♥t♦ s♥♦ ♦ ♦♦

st ts ♦♥sr♠♦s ♣♥s t♦s r ♠♦rt♠♥t♦ ♥ ♦s qs sã♦

♣rs♥t♦s ♥ ♣ró①♠ sçã♦ ♠ tr♦s tr♦s ♣rt♥♠♦s s♥♦r ♦s ♦tr♦s tó♣♦s

♠♥♦♥♦s

t♦s ♠♥ét♦s ♠♦rt♠♥t♦ ♥

♠

♦♥srr♠♦s ♣♥s ♣r♠r♦s r♠ô♥♦s ♥♦ r♠ tr♦stát♦ ♥♥♦ r♥ts

t♠♣rtr ♥s ♣r♦♥♥ts ♦ r♥t ♥çã♦ strçã♦ qír♦

s s♦çõs s rs♠♠

g(α)0 =

J0αeαFMα/Tα

1−Ω2trα−Ω2

α/2

[

iΩα2 (1− Ωtrα)(1 + Ω∗α)Φ−1+

(1− Ω2trα)Φ0 − iΩα2 (1 + Ωtrα)(1− Ω∗α)Φ1

]

,

g(α)±1 =

1

1∓ Ωtrα

[

±iΩα

2g(α)0 + J0α

eαTα

FMα(1∓ Ω∗α)Φ±1

]

,

♦ ♦r♠ tr♥t ♠ ♦♠♣♦♥♥ts s♥♦s ♦ss♥♦s

gα0 =eαFMα

TαJ0α

[(

1 + 12

Ω2

α

1−Ω2trα

)

Φ0 − 12

(

Ωα1−Ω2

trα− ΩαΩtrαΩ∗α

1−Ω2trα

)

Φs−

i2

(

ΩαΩtrα1−Ω2

trα− ΩαΩ∗α

1−Ω2trα

)

Φc

]

,

gαs =eαFMα

TαJ0α

[

− Ωα1−Ω2

trαΦ0 +

(

11−Ω2

trα− ΩtrαΩ∗α

1−Ω2trα

)

Φs+

i

(

Ωtrα1−Ω2

trα− Ω∗α

1−Ω2trα

)

Φc

]

gαc =eαFMα

TαJ0α

[

iΩαΩtrα1−Ω2

trαΦ0 − i

(

Ωtrα1−Ω2

trα− Ω∗α

1−Ω2trα

)

Φs+(

11−Ω2

trα− ΩtrαΩ∗α

1−Ω2trα

)

Φc

]

.

♣rtr ♥trçã♦ ♥♦ s♣ç♦ ♦s ♦t♠♦s ♦s ♠♥t♦s

♠tr③ ♥tr ♦r♦ ♦♠ r♣rs♥tçã♦ ♠tr ♠ st á♦ q♥♦

t♦ ♠ ts rst ♥s s♥ts ①♣rssõs

R00 = L(i)0 + 1

2L(i)2 − 1, Rss = Rcc = L(i)

0 − L(i)011 − 1− τi,

R0s = −12L

(i)1 + 1

2L(i)111, Rs0 = −L(i)

1 , Rsc = −Rcs = iL(i)01 − iL(i)

001,

R0c = − i2L

(i)11 + i

2L(i)101, Rc0 = iL(i)

11 ,

q ♦♠ ♦ s♦ ♦ s♣rr ♣rts rs F ♠♥árs K ♣♦♠ sr s♥♦

s s♥t ♦r♠

R(F)00 = 1

2

(

−Ω2 + 74 + 23

8q2Ω2 + 98q4Ω4

)

2rρ2

i

Ω2 ,

R(K)00 = −i

√π2

(

1 + 1q2Ω2 + 1

2q4Ω4

)

2rρ2

i

Ω2 q5Ω5−q2Ω2

,

R(F)0s = −1

2

(

1 + 1q2Ω2 + 9

4q4Ω4

)

rρiΩ +

R(K)0s = i

√π2

[

1 + 12q2Ω2 −

(

ηiq2Ω2 + 1 + 1+ηi/2

2q2Ω2

)

Ω∗i

]

rρiΩ q3Ω3−q2Ω2

R(F)0c = −

√π2

[

1q2Ω2 + 1

q4Ω4 −(

ηi +1

q2Ω2 + 1+ηi/22q4Ω4

)

Ω∗i

]

rρiΩ q5Ω5−q2Ω2

,

R(K)0c = i

2

(

1 + ηi +1+2ηiq2Ω2 + 9+3ηi/4

4q4Ω4

)

rρiΩ Ω∗i,

R(F)s0 = −

(

1 +1

q2Ω2+

9

4q4Ω4

)

rρiΩ

, R(K)s0 = i

√π

(

1 +1

q2Ω2

)

rρiΩ

q3Ω3−q2Ω2

R(F)ss = −τi +

1

2q2Ω2+

3

4q4Ω4, R(K)

ss = −i√π

[

− 1

q2Ω2+

(

ηi +1− ηi2q2Ω2

)

Ω∗i

]

q3Ω3−q2Ω2

,

R(F)c0 =

√π

(

1 +1

2q2Ω2

)

rρiΩ

q3Ω3−q2Ω2

,

♦♥ Rab = R(F)ab +R(K)

ab a, b → 0, s, c

♠ ♦r♠ ♦♠♥♥t s♦♥sr♥♦ tr♠♦s qrát♦s ♥çã♦ ①♣♦♥♥ ♦ tr

♠♥♥t ♠tr③ ♥tr ♠ ♦ q ♣♦ sr ♣r♦①♠♦ ♣♦r

Rss(RssR00 −R0sRs0 −R0cRc0) = 0,

♦ sr s♥♦♦ rst ♥ s♥t rçã♦ s♣rsã♦

F(Ω) + iK(Ω) = 0, F(Ω) ≈ ∑3j=0C

(F)2j Ω2j , K(Ω) ≈ √

πΩ5∑4

j=0C(K)j Ωj−q2Ω2

,

C(F)0 = (Ω2

s +Ω2s)Ω

2s, C

(F)2 = 2Ω2

Ω2s +Ωs +Ω2

, C(F)4 = −(Ω2

+ 2Ω2s +Ω2

∗e),

C(F)6 = 1, C

(K)0 = (3ηi − 2)ηiΩ∗iΩ∗eΩ2

s, C(K)1 = (1 + τe)(3ηi − 2)Ω∗iΩ2

s,

C(K)2 = 2(1 + τi)Ω

2s + (ηi − 1)Ω2

∗e, C(K)3 = −1, C

(K)4 = 1.

s rqê♥s t③s ♠ sã♦ ♥s ♥ t q t♠é♠ ♠♦str ♦s ♦rs

♣r♦①♠♦s sts rqê♥s ♥♦ ♥tr♦ ♥ ♦r ♦♥ ♣s♠

rqê♥s tí♣s ♥♦r♠③s ♣♦r vTi/R0 r♦♥s t♦s

♦és♦s úst♦s í♦♥s ♠♥ét♦s

rqê♥ ①♣rssã♦ ♥ít τe = 1 q = 3.5 ♦r τe = 3 q = 2 ♥tr♦

Ω2

7

4+ τe

Ω2s

τe2q2

2.04× 10−2 1.88× 10−1

Ω2s

(

23

4τi + 4 + τe

)

Ω2s 2.19× 10−1

Ω2 (1 + τe + ηi)Ω

2∗e (1 + 0.5ηi)

ρ2i /L2N

R20/r

2(2 + 0.5ηi)

ρ2i /L2N

R20/r

2

Ω2

(

3

4− ηi

)

Ω2∗e

Ω2s

(

15

2τi +

9

4

)

Ω4s

Ω2s τi

(

η2i +9

2ηi −

17

4

)

Ω2∗e

♦çõs ♥♦ ♠t ♦

♠t ♦ é ♦t♦ ♦♥sr♥♦ −q2Ω2 → 0 ♠ ♦ s trés rs♦çã♦

F(Ω) = 0 P♦r s trtr ♠ qçã♦ ú ♣♦rt♥t♦ í sr s♦♦♥

♥t♠♥t t③♠♦s ♦r♠ ♣r♦①♠ st qçã♦ q ♦r♥ s s♦çõs

ss♥tóts ♦♥sr♥♦ q s três s♦çõs ♣♦ss♠ s s♥ts ♦r♥s r♥③ Ω ∼

Ω ∼ 1 Ω ∼ Ωs ∼ δ ≪ 1 Ω ∼ Ω ∼ δ ≪ 1

• Pr♠r s♦çã♦ P♦r s trtr rqê♥ ♦r ♠s t♦ Ω ∼ 1

♣♦♠♦s s♣r③r C(F)0 = O(δ4) ♦r♠ q F = 0 t♦r♥s ♠ qçã♦ qrát

♠ Ω2 s♦çã♦ ♣♦st ♦r♥ ♦♠♦ s♦çã♦ rqê♥ ♦s ♦rr ♣♦

t♦r sr♥ç (q) ♣♦r t♦s ♠♥ét♦s

Ω2 = Ω2

+Ω2s

Ω2

+Ω2

Ω2

• ♥ s♦çã♦ ❲s st s♦ ♦♠♦ s♦çã♦ é Ω ∼ δ ♦ tr♠♦ Ω6 ♣♦ sr

s♣r③♦ ♠ ♠t♥♦ ♦♥çã♦ Ωs ≫ Ω∗e ♦t♠♦s ss♥t♦t♠♥t

rqê♥ ❲s ♦rr ♣♦r t♦s ♠♥ét♦s

Ω2❲ =

(

1 +7

4τ2i

)

Ω2s −

[

η2i +

(

1

2+ τe

)

ηi −3

4

(

5

3+ τe

)]

Ω2∗i

• rr s♦çã♦ t♦ ♠♥ét♦ st s♦çã♦ ♦rrs♣♦♥ ♦ s♦ ♠ q

Ω∗e ≫ Ωs ♥♦s r♠s ♠ q ηi ≫ 3/4 ♦ ηi ≪ 3/4 ♣♦ sr ♣r♦①♠ ♣♦r

Ω2 =

Ω2

Ω2

+ fdia(ηi, τe)Ω2s

Ω2

,

♦♥

f(ηi, τe) =

(

ηi −3

4

)−1[

Ω4η

2i +

(

Ω4 +

Ω2

2− 29

16

)

ηi −(

3

4Ω4 +

5

4Ω2 −

87

64

)]

.

♦♠ rçã♦ ♦s ♠ts ss♥tót♦s f á ♦s s♦s sr ♦♥sr♦s

♠ r♦ r♥t t♠♣rtr ηi ≪ 1 s♦çã♦ é stá

♣♦ sr ♣r♦①♠ ♣♦r

Ω =3

4

Ω2∗e

Ω2

+

(

Ω4 +

5

3Ω2 −

29

16

)

Ω2s

Ω2

♠ ♦rt r♥t t♠♣rtr ηi ≫ 1 st s♦ s rtr③

♣♦r sr ♥stá ♥ã♦ ♦stór♦ s ♦♦rrr ♦rts r♥ts ♥s ♦ s

s Ω∗e &√2Ω/2q t① rs♠♥t♦ st ♥st ♣♦ sr ♣r♦①♠

♣♦r

Γ♥st =√ηi

(

Ω2∗e

Ω2

− τiΩ2s

)1/2

t♦ ♥ét♦ ♠ ♠♦♦s ♦és♦s ♠♦rt♠♥t♦

♥

Pr ♦ á♦ t① ♠♦rt♠♥t♦ s s♦çõs ♠♦strs ♠

t③♠♦s s ①♣rssõs ♦ q rst ♥♦s s♥ts ♦rs

Γ = −q5Ω4

√π2

[

1 +

(

5Ω2 − 9

2 + 29/16Ω2

)

Ω2

s

Ω2

+ ηi

Ω∗eΩ

+(

1 + ηi − Ω2

Ω2

)

Ω2∗e

Ω2

]

−q2Ω2

,

Γ❲ = −qΩ2s

√π2

[

τe − 34 + τi + g

(1)sw∗

Ω∗eΩs

− g(2)sw∗q

Ω2∗e

Ω2

s

]

−q2Ω2

❲ ,

g(1)sw∗ =

(

ηi2 − 1

)

τe + 3− 54ηi +

17ηi−578 τi, g

(2)sw∗ =

ηi−14 τe +

34η

2i − 3

2ηi + 1,

Γ = −(1 + τe)

√π

2

Ω4∗e

Ω4

[(

3

4

Ω∗eΩ2

)2

− 3

2

(

Ω2 −

√3Ω −

55

24− 87

32Ω2

)

τiΩ2s

Ω2

]

−q2Ω2

.

♠ár♦ sssã♦

st ♣ít♦ ♠♦str♠♦s ss♥♠♥t ♠♣♦rtâ♥ ♣çã♦ ♦ ♠♦♦ r♦♥é

t♦ ♦ st♦ ♠♦♦s ①s rqê♥s ♣rtr st ♠♦♦ ♥♠♥t ♣rs♥

t♦ ♥♦ ♣ít♦ ♥ú♠r♦s ♥ô♠♥♦s q ♦♦rr♠ ♠ ♠♦♦s ①s rqê♥s ♣♦♠

sr ♥s♦s ♥tr s t♦s ♠♥ét♦s ♠♦rt♠♥t♦ ♥ t♦s tr♦♠

♥ét♦s t♦s ♣rtís ♣rs♦♥s t♦♠♦♦s r♦♥♦s ♠♦♦s ♥♦s

♠♦r stq st ♠♦♦ ♥♦ ♥t♥♦ s à ♣♦ss ♦tr t①

♠♦rt♠♥t♦ ♥ã♦ ♦s♦♥ ♠♦rt♠♥t♦ ♥ ♥str t♦s ♣rtís

♣rs♦♥s ♠ ♠♦♦s ♦és♦s ♦ q ♥ã♦ é ♣♦ssí ♣rtr ♠♦♦s ♦s ♣♦r s

trtr t♦s ♣r♠♥t ♥ét♦s

♥t♦ à ♠t♦♦♦ ♦t ♥st ♣ít♦ ♦♠ ♥ r t③çã♦ ♦

♠♦♦ r♦♥ét♦ ♠♦s st ♣ít♦ ♠ t♣s Pr♠r♠♥t ♦♥sr♠♦s k‖vTi= 0

k‖vTe → ∞ ∇FMα = 0 ♦ q t♦r♥ ♦ á♦ ♥ít♦ ♦♥sr♠♥t ♠s s♠♣s

♣r♠t ♦tr ♦s tr♠♦s ss♥s rqê♥ ♦♠ sts ♦♥srçõs é ♣♦ssí

♥t♥r ♠s ♠♥t ♥â♠ ás ♣♦ ♠♦♦ r♦♥ét♦ ♠ s

♦♥sr♥♦ k‖vTi♥t♦ ♣♦ré♠ ♥ã♦ ♥♦ ♠ ♦♥t ♥ ♦ ♠♦rt♠♥t♦ ♥ ♦

q é ♠♣♦rt♥t ♣r ①♦s ♦rs q é♠ ♦rrçõs ♦r♠ q−2 ♣r rqê♥

♦s t♠é♠ ♦t♠♦s s♦çã♦ ♣r SWs sr♠♦s ♥st ♣rt q rqê♥

ω =√

Te/miq2R20 r♣rs♥t ♠ s♥r ♣♦s ♦s ♥♦♠♥♦rs rçã♦ s♣rsã♦

♦♠♣♦♥♥t s♥♦ ♦ ♣♦t♥ tr♦stát♦ s ♥♠ ♥st rqê♥ ♠♦rt♠♥t♦

♥ é ♥tã♦ ♦♥sr♦ ♦r♠ ♣r♦①♠ ♦♠ ♦t♦s át♦s ♥ ssçã♦

sçã♦ st♠♦s r♠♥t ♦r♠ ♠s r qçã♦ r♦♥ét q ♣r♠t

♥ás t♦s ♠♥ét♦s é♠ ♦tr♦s t♦s ♠ ♠♦♦s ♦és♦s ♥♦s

♦s qs ♣rt♥♠♦s str ♠ tr♦s tr♦s út♠ sçã♦ ♦t♠♦s s ①♣rssõs

♥íts ♣r rqê♥ ❲s ♥♦ ♠ ♦♥t ♦rrçõs ss♥tóts ♦rr♥ts

r♥ts rs ♥s t♠♣rtr í♦♥s ♦ s ∇FMα 6= 0 é ♦♥sr♦

①s ♠♦rt♠♥t♦ ♦rrs♣♦♥♥ts sts rqê♥s s qs sã♦ ♠♣♦rt♥ts q♥♦

k‖vTi. vTi

/R0 t♠é♠ ♦r♠ s ♥st sçã♦

s ①♣rssõs ♥íts ♣r rqê♥ ♦s q ♦t♠♦s ♦ ♦♥srr ♠♦rt

♠♥t♦ ♥ ♦♥♦r♠ ♦♠ ♦s ♦rs ♦t♦s ♠ tr♦s ♥tr♦rs ❬ ❪

♣♥s ♥♦ ♠t í♦♥s r♦s Ti ≪ Te ♦♥sr♥♦ étr♦♥s ♥♦ r♠ át♦ s♦tér♠♦

γe = 1 ♦ q é r③♦á ♣r t♦♠s ❬❪ ①♣rssã♦ ♥ét ♣r rqê♥

♦♥♦r ♦♠ ①♣rssã♦ ♦t ♣♦ ♠♦♦ ❬❪ ♦♥s ♣♦rt♥t♦ q

♥â♠ étr♦♥s ♥ã♦ é ♥♥ ♣♦r t♦s ♥ét♦s ♣♦ sr ♠ srt ♣ t♦r

♦s ♦rrçã♦ ♦ ♦ t♦ ♥s♦tr♦♣ ♣rssã♦ ♣rtr ♣r í♦♥s ♦♠

rçã♦ às ♦♠♣♦♥♥ts ♣r♣♥r ♣r é ♦r♠ ♣♦rt♥t♦ ♥ã♦ s♠♣♥

♠ ♣♣ ♠t♦ s♥t♦ ♥♦ ♦r rqê♥ ♦s

s ①♣rssõs ♥íts ♣r rqê♥ ❲s ♦r♠ ♥ss ♥♦s ♠ts

ss♥tót♦s ♠♦r ♥trss ♠ s♦ s♦çã♦ q ♦rrs♣♦♥ ♥ ♣rs♥ç

t♦s ♠♥ét♦s ♦r♦ ♦♠ t♠ s rqê♥ ♠♥t ♦ ♦ r♥t

♥s t♠♣rtr st t♦ ♣♦ sr t③♦ ♠ ♦♥♥t♦ ♦♠ ♥stçã♦

♦tr♦s t♦s ♠ ♣r ①♣r ♣♦ss♠♥t ♥tr ♥s t♣♦s ♠♦♦s

tt♦s r♥t♠♥t ♥♦ ❬❪ ♦ ♠t Te ≫ Ti 1 ≪ q ≪ qmax q ♥♦r♠♠♥t

♦♦rr ♠ ♣♦sçõs rt♠♥t ♣ró①♠s ♦ ♥tr♦ ♦♥ ♣s♠ ①♣rssã♦

♣♦ sr t③ ♣r ♦ á♦ ♣r♦①♠♦ rqê♥ ❲s s q ♥ã♦ ♦♦rr♠

♦rts r♥ts ♥s ♦ ①tr♠♦ ♦♣♦st♦ q♥♦ q ∼ qmax á ♦rts r♥ts

♥s s♦çã♦ ♦rrs♣♦♥♥t ❲s é t ♣♦r t♦s ♠♥ét♦s ♥st s♦

①♣rssã♦ sr t③ ♣r ♦ á♦ rqê♥ Pr ♦ s♦ s♣í♦ ♠ q

ηi ≫ 3/4 s♦çã♦ é ♠♦str ♠ sts ♠♦♦s ❲s s t♦r♥♠ ♥stás ♥ ♣rs♥ç

♦rts r♥ts ♥s t♠♣rtr ♣♦ssí ♦♥r q st st

t♣♦ ♠♦♦ ♦és♦ ♣♥ ♥ã♦ s♦♠♥t rçã♦ ♥tr ♦s r♥ts ♥s

t♠♣rtr í♦♥s ♠s t♠é♠ rçã♦ ♥tr ♦ t♦r sr♥ç ♦ r♥t

♥s st út♠ ♦♥çã♦ ♣♦ sr t③ ♠ ①♣r♠♥t♦s ♣r ♦t♥çã♦ ♦ ♣r

r q ♣rtr ♠s rqê♥ st t♣♦ ♠♦♦

♠ tr♦s tr♦s ♣rt♥♠♦s str t♦s tr♦♠♥ét♦s s♦s ♣♦r ♣rtr

çõs ♦ ♠♣♦ ♠♥ét♦ ♣r♣♥r ♠ rçã♦ ♦ ♠♣♦ ♠♥ét♦ qír♦

♦♥♦r♠ ♥st♦ ♥tr♦r♠♥t ♠ ❬❪ rts ♠ ♠♦♦ ♦s á♦s sã♦ ♠s

①t♥s♦s ♣r♥♣♠♥t ♦ à ♥ss ♥r ♠♦♦s t♦r♦s n 6= 0 ♠♦♦s ♣♦♦

s m = ±2 ♦♥♦r♠ st♦ ♠ ❬❪ ♥trss♥t ♦♥srr s♠t♥♠♥t t♦s

tr♦♠♥ét♦s t♦s ♠♥ét♦s ♠ s t♥ é♥ ♥♠♦s

♦♥♦r♠ st♦ ♠ ❬❪ ♣♦s ♦trs ♦rrçõs ♣r rqê♥ ♠♦♦s ♦és♦s sã♦

♥ssárs ♣r ♦♥r♠çã♦ ①♣r♠♥t sts ♠♦♦s ❬❪ é♠ ss♦ ♦ ♦♥srr t♦s

tr♦♠♥ét♦s ♦s qs stã♦ r♦♥♦s à ♥â♠ étr♦♥s ♦ t♦ r♥ts

t♠♣rtr étr♦♥s ♣♦ sr q♥t♦ ♦♥♦r♠ st♦ ♥♦ ♣ít♦

♣ít♦

♦♥sõs rçõs trs

st ♣ít♦ ♣rs♥t♠♦s ♣r♠r♠♥t ♠ sssã♦ s♦r ♦s ♠♦♦s ♦

r♦♥ét♦ ♦♥sõs s♦r ♦s rst♦s ♦t♦s q♥t♦ à r♥ê♥ sts

♠♦♦s q♥t♦ à ♣çã♦ sts rtt♦s ♥♦ t♦♠ ♠ s ♦♣çõs ♣r ♦ s♥

♦♠♥t♦ ♠♦♦s ♠s r♥♥ts sã♦ sts ♣r♦♣♦sts ♣r tr♦s tr♦s sã♦

♣rs♥ts

♦♦ ♦s

♦ ♠♦♦ ♦s ♣rt♠♦s s qçõs r♥s q sr♠ ♦çã♦ s

r♥③s ♠r♦só♣s ♦ ♣s♠ ♣rssã♦ p s♦s ♣r π‖ ♥s n

♦ s qs sã♦ ♠s ♣♦r ♠♦ ♥óst♦s ♥♦ t♦♠ ♣r str qír♦

♦♠ r♦tçã♦ ♣♦♦ t♦r♦ ♠♦♦s ♦és♦s ①s rqê♥s ♥ ♣rs♥ç

r♦tçã♦ qír♦ r♥ts ♥s t♠♣rtr P♦ t♦ trtrs ♠

♠♦♦ ♠s ssí ♣r ♦♠♣r♥sã♦ ís ♦s ♥ô♠♥♦s ♥♦♦s ♣♦r ♣r♠tr ♠

trt♠♥t♦ ♠s s♠♣s ♠♦♦s ♥ã♦ ♥rs t♦r ♦s ♣♦ss ♠♣♦rt♥ts ♣

çõs ♠ ís ♣s♠s s ♠♦♦s ♦♥sr♦s ♠t♦♦♦ t③ ♦r♠ s♦s

♣r♥♣♠♥t ♠ ♦s tr♦s ♥tr♦rs ❬ ❪

♦ ♦♥srr♠♦s qír♦ ♦♠ r♦tçã♦ ♣♦♦ t♦r♦ ♦ tr♠♦ ①♦ ♦r ♥♦

♠♦♦ ♦t♠♦s ♦rrçõs ♥íts ♠ tr♠♦s ♦ ♥ú♠r♦ ♣♦♦

t♦r♦ ♣r rqê♥ ①♦s ③♦♥s ❩s ♦♥s s♦♠ ❲s ♠♦♦s úst♦s

♦és♦s s sr♠♦s ♠ ♠♥t♦ ♥♦ ♦r rqê♥ sts ♠♦♦s q é

♦ à r♦tçã♦ ♦ ♣s♠ ♥♦ á ♣rtrçõs tr♦státs ♥♦ ♣s♠ Φ 6= 0 ♥♦

♦♥t①t♦ t♦r ❲s só ♦♦rr♠ q♥♦ á r♦tçã♦ ♣♦♦ qír♦

♥trt♥t♦ ♣rtr ♦ ♠♦♦ r♦♥ét♦ ♠♦str♠♦s q á ❲s ♠s♠♦ q♥♦ r♦tçã♦

♣♦♦ ♥ã♦ é ♦♥sr

♦ ♦♥trár♦ ♦ q ♦♦rr ♦♠ s ❲s q sã♦ ♠♦♦s úst♦s ♣♦rt♥t♦ ♦♠♣rs

sís ♦srs q ❩s ♠♦♦s ♥♦♠♣rssís sã♦ ♦rt♠♥t ♥♥♦s ♣ ♦♠♣♦♥♥t

♥♦r♠ ♣r♣♥r ♦ ♠♣♦ ♠♥ét♦ qír♦ ♥♦r♠ às s♣rís ♠♥éts

♦ ①♦ ♦r q ♣♥ ♦ r♥t r t♠♣rtr ♦ ♦♥srr♠♦s ①♦

♦r ♥s qçõs ♦t♠♦s ①♣rssã♦ ♥ít ♣r rqê♥ ♦s ❩s

q t♦r♥s ♥ã♦ ♥ ♦ ♦ t♦ ♦ ①♦ ♦r r♦tçã♦ ♣♦♦ qír♦ ❬❪

♦ ♦srr♠♦s q ♥♦ r♠ s♦♠étr♦ ♦♠ r♦tçã♦ ♥♠♥t t♦r♦ ♦s ❩s sã♦ ♥s

tás ❬❪ ♦sr♠♦s ♥ss ♦♥srr t♠é♠ r♦tçã♦ ♣♦♦ ♥st r♠ ♦

q ♣♦ t♦ r ♦♥ts ①t♥ss r ♦ s♦♣♦ st ts ♣rt♥♠♦s ③r ♠

♠ tr♦ tr♦ ♥ás ♦s rst♦ ♣r♦③♦s ♣♦r st tr♦ ♠ ♦♥♥t♦ ♦♠

♥ás ♦ ♠♦♦ ♦s ♦s ♥♦ q ♦sr♠♦s q r♥ts t♠♣rtr sã♦

rs♣♦♥sás ♣♦r ♠♦♦s ♦és♦s ♥stás é ♣r♦♠ss♦r ♣♦s ♦♠♦ r♦tçã♦ ♣♦♦ stá

r♦♥ ♦♠ r♥ts t♠♣rtr ♣♦rí♠♦s ♦tr ♥♦r♠çõs ♦♥s s♦r ♦

♣r r t♠♣rtr í♦♥s é♠ ♥t♥r ♠♦r s ♦♥çõs st

❩s ❲s s

♦♠ rçã♦ ♥ás ♦ qír♦ ♦♥sr♠♦s três t♣♦s qír♦ át♦

s♦tér♠♦ s♦♠étr♦ ♣rtr ♦ ♣r r ♦ r♦tçã♦ ♣♦♦ t♦r♦

♦t♦ ①♣r♠♥t♠♥t ♥♦ t♦♠ ❬❪ ♦srs q ♠ ♠♥ç r♠

qír♦ ♦♠ ♣♦sçã♦ r ♦ s♦tér♠♦ ♣r ♦ s♦♠étr♦ ♠♣r ♠ ♠ ♠♥ç

s♥t♦ ♦ r♥t t♠♣rtr ♦ ♣s♠ ♦♥sr♥♦ q ♥ã♦ ♥rsã♦ s♥t♦

♦ ♠♣♦ ♠♥ét♦ ♣♦♦ ♦♠ s ♥♦ rst♦ ①♣r♠♥t ♦t♦ ♣r r♦tçã♦

♣♦♦ ♥♦ ♦ r♠ át♦ ♦srs q ♦ r♥t t♠♣rtr é ♥♦

♣♦rt♥t♦ st r♠ é ♣♦ssí ♣♥s ♦♠♥t ♣♦s t♠♣rtr rs ♦ ♥tr♦ ♣r

♦r ♦♥ ♣s♠ ♦ q s rr ♥rsã♦ s♥t♦ ♦ r♦tçã♦

t♦r♦ ♥ ♣♦sçã♦ r ≈ 0.7a ♦ q ♥ ♥ã♦ é ♠ ♦♠♣r♥♦ ♦ ♣♦♥t♦ st tór♦ é

♥ssár♦ ♦♥srr ♦tr♦s tr♠♦s ♥♦ ♠♦♦ ♥♦ ♠♦♦ ♦s ♦s sts

tr♠♦s sã♦ ♦ ♦sõs ♣♦s ♣ró①♠♦ à ♦r ♦♥ ♣s♠ ♦ ♣s♠ s t♦r♥ ♠s

♦s♦♥ ♣♦rt♥t♦ rsst♦

t♦s ♠♥ét♦s ♣r♦♥♥ts r♥ts rs ♥s t♠♣rtr

♦r♠ ♦♥sr♦s ♥♦ ♠♦♦ ♦s ♦s ♥♦ q í♦♥s sã♦ ♦♥sr♦s ♥♦ r♠

♦s át♦ étr♦♥s ♦ s ♣q♥ ♠ss sã♦ ♦♥sr♦s s♠t♥♠♥t

♥♦ r♠ át♦ s♦tér♠♦ ♦ à ♥♦r♠ r♥ç ♠ss ♥tr sts ♣rtís

é ♦r♥t ♦♥srr s♦s ♣r ♣♥s ♣r í♦♥s ♦ q q r ♠ ♦♥t ♦

t♦ ♥s♦tr♦♣ ♣rssã♦ ♦♠♣♦♥♥t ♣r♣♥r é r♥t ♦♠♣♦♥♥t ♣r

♣r ♣rssã♦ ♣rtr í♦♥s ss♠♥♦ ♦ r♥t t♠♣rtr ♥ ♠s♠ rçã♦ ♦

r♥t ♥s ♦ q é ♠s r③♦á ♦♠ rçã♦ ♦ q s ♦sr ♠ ①♣r♠♥t♦s

♦♥í♠♦s q ♥st ♦s s é ♣r♦♥♥t ♦ r♥t t♠♣rtr q ♦

r♥t ♥s rt ♦r♠ t♥ st③r ♦ ♣s♠ s ♣rs♠♥t q♥♦

L−1Ti

> 3L−1N /4 ♦♦rr♠ ♠♦♦s ♥stás ❬❪ ♥st ♦rrçã♦ rqê♥ ♦s

s ♦ t♦s ♠♥ét♦s ♦ t♦s r sã♦ ♣r♦♣♦r♦♥s à rqê♥

♠♥ét étr♦♥s ω∗e = Te/eBrLN st t♠ stá r♦♥♦ ♦ st♦ ♦♥s

r ❲ ❬❪ ♠♦♦s r♥t t♠♣rtr í♦♥s ❬❪

s ♣r♥♣s ♦♥sõs ♣r♦♥♥ts ♦ ♣ít♦ s♦r ♠♦♦s ♦s ♣♦♠ sr

rs♠s ♦♠♦

• ♦ rtr ♦és ♦ ♠♣♦ ♠♥ét♦ qír♦ ♥♦ t♦♠ ♦ q r

rt ♥ qr s♠tr ♣♦♦ ♦ sst♠ sr♠ ♠♦♦s ♦és♦s ♥♦ ♣s♠ q

♣♦r s♠♣♥r♠ ♠ ♣♣ ♠♣♦rt♥t ♥♦ ♦♥tr♦ trê♥ ♣♦r tr♠ ♣

çõs ♥ ♦t♥çã♦ ♦ ♣r r ♣râ♠tr♦s ♦ ♣s♠ sã♦ ♥♦s ♠ ♥t♥s

♥stçã♦ tór ①♣r♠♥t ♥♦r♠çõs s♦r s rqê♥s sts ♠♦♦s ♣♦

♠ sr t③s t♥t♦ ♣r str ♦s ♥ô♠♥♦s ís♦s ♥♦♦s ♦♠♦ ♣r ♣çõs

rt ❲s♦♥ ♠♣rtr r♥t

♠ ♥óst♦s

• ♥s♦tr♦♣ ♣rssã♦ ô♥ srt ♣♦ t♥s♦r s♦s ♣r π‖ tr

♦ ♦r ♦ ♦♥t át♦ t♦ γ(t♦) 5/3 ♣r 7/4 ③♥♦ ♦♠ q

rqê♥ s t♥ ♠ ♣q♥♦ ♠♥t♦ ♠♥♦r ♦ q

• t♦ r♦tçã♦ qír♦ ♦s r♥ts t♠♣rtr ♥s ♣r♦③r♠

♠ ♠♥t♦ rqê♥ ♥♦ r♠♦ ♠♦r rqê♥ ♦s s ♣♦ sr t③♦

♣r t♥tr ①♣r ♥tr ♠♦♦s r♥t♠♥t ♠♦s ♥♦ t♦♠ ❬❪

• ♥stçã♦ ♦ t♦ r♦tçã♦ qír♦ t♦s ♠♥ét♦s ♠ ❩s r

qr q ♦ ①♦ ♦r s ♦ ♠ ♦♥t ♣♦rt♥t♦ r♥ts t♠♣rtr

s♠♣♥♠ ♠ ♣♣ ♥♠♥t ♥ ♥â♠ ①♦s ③♦♥s

• ♥t♦ ♦t♥çã♦ ①♣rssõs ♥íts ♣r rqê♥ ♠♦♦s ♦és♦s q♥♦

r♦tçã♦ qír♦ ♦ t♦s ♠♥ét♦s sã♦ ♦♥sr♦s ♦♠♦ ♦ st♦ q

ír♦ ♦♠ r♦tçã♦ ♠ r♥ts r♠s ♦s rst♦s ①♣r♠♥ts ♦t♦s

♠ t♦♠s ♣♦♠ sr t③♦s ♠ ♦♥♥t♦ ♣r ♦t♥çã♦ ♦s ♣rs rs t♠

♣rtr í♦♥s ♦ t♦r sr♥ç ♦ q t♠ ♠♣ ♣çã♦ ♣r ♦r♥tçã♦

tr♦s ①♣r♠♥t♦s ♥♦

• r♥t t♠♣rtr í♦♥s é ♦♥t ♥r rs♣♦♥sá ♣ ♥st

s ♦♥çã♦ ♣r st ♥st q é ♣r♦♣♦r♦♥ à rqê♥ ♠

♥ét étr♦♥s ω∗e = Te/eBrLN t♠é♠ ♣♥ ♦ r♥t ♥s ♠

rõs ♠ q q é ♠t♦ r♥ ♣♦ sr ①♣rss ♦r♠ ss♥tót q → ∞ ♣♦r

ηi = (∂ lnTi/∂r)/(∂ lnn0/∂r) > 3/4 ♣♦rt♥t♦ ♦srs q ♦ r♥t ♥s

♦♥tr ♣r st s

♦♦ r♦♥ét♦

♦ ♣♦♥t♦ st q♥ttt♦ t③çã♦ ♦ ♠♦♦ r♦♥ét♦ ♥♦ ♣ít♦ t ♦s

♦t♦s ♦tr ①♣rssõs ♥íts ♣r s rqê♥s s ❲s ♥ ♣rs♥ç t♦s

♠♥ét♦s ♦♥sr♥♦ s♠t♥♠♥t tr♠♦s O(q−2) ♥str ♦ ♠♦rt♠♥t♦

♥ sts ♠♦♦s q sr ♦ ♥trçã♦ ♦♥♣rtí ♣♦r ss♦ rqr ♠

trt♠♥t♦ ♥ét♦ P♦r qstõs áts té ♠s♠♦ ♣♦rq ♠♦♦s ♥ét♦s ♣rs♥t♠

♠♦r ♦♠♣r♥sã♦ ís s ♦♠♣r♦s ♦♠ ♠♦♦s ♦s sr♠♦s

s♣r♠♥t t♦ ♥ts ♦♥srá♦ ♠ ♦♥♥t♦ st♦ ♦ ♠♦rt♠♥t♦

♥ ♠♦♦s ♦és♦s ♣rs♥t♦ ♥st ts ♦ s♦ ♥ ❬❪ ♦s rst♦s

stã♦ ♠ ♦r♦ ♦♠ ♦s ♥♦ss♦s

Pr tr♠♥r t① ♠♦rt♠♥t♦ ♥ã♦ ♦s♦♥ ♦s ♠♦♦s ♦és♦s ♠ qs

tã♦ ♦♥♦r♠ str♦ ♥ sçã♦ t③♠♦s ♠ ♠ét♦♦ trt♦ ♥♦ q ♣r♠r♠♥t

♥♦ ♠t ♦ ♦ s ♥ã♦ ♥♦ ♠ ♦♥t ♦ ♠♦rt♠♥t♦ ♥ ♦t♠♦s ♠

qçã♦ ú s♦çã♦ ♦r♥ s rqê♥s sts ♠♦♦s P♦str♦r♠♥t sts s♦

çõs sã♦ ♦♥srs ♥s ①♣rssõs ♥éts s qs sã♦ ①♣♥s ♠ sér ②♦r

♦♥sr♥♦ ♥st ①♣♥sã♦ t① ♠♦rt♠♥t♦ ♥ ♦♠♦ ♣râ♠tr♦ ①♦

♦r Pr s♦çã♦ ♥ít qçã♦ ú ♣rt♥♦ ♦ ♦♥♠♥t♦ ♣ré♦ ♦t♦

♣♦ ♠♦♦ ♦s ♦s ♦r♠ r♥③ s s♦çõs s♣r③♠♦s

tr♠♦s ♠♥♦r ♦r♠ ♣r ♣r♦①♠á ♣♦r ♠ qçã♦ qrát t♠♦s ss♠

s três rqê♥s ♥♦s ♠ts ss♠♣tót♦s q ♦rrs♣♦♥♠ rs♣t♠♥t s

t rqê♥ ❲s s ①s rqê♥s ♥stás s♦ rts ♦♥çõs s ①

♣rssõs ♥íts sã♦ ♠♦strs ♠ rs♣t♠♥t ❬❪ ♦♠ rçã♦

♦ ♠♦♦ ♥stá ♦ ♠t ♠♦r ♥trss é q ∼ qmax L−1Ti

≫ L−1N Ti ≪ Te ♦sr♠♦s

q ♦♥♦r♠ ♠♦str♦ ♠ ♦ r♥t t♠♣rtr ô♥ ♦ ♦ r♥t

♥s ♦♥stt♠ ♦♥ts ♣r ♥st s ♠ rõs ♦♥ ♦ t♦r sr♥ç

é t♦ ♦ s ♥♦r♠♠♥t ♥ ♦r ♦♥ ♣s♠ st ♥♦r♠çã♦ ♣♦ sr ①♣♦r

①♣r♠♥t♠♥t ♣rtr ♠s sts rqê♥s ♣♦♠♦s ♦tr ♠ rçã♦ ♥tr ♦

♦r q LN ♦ ♦r ηi = LN/LTi ♦ s ♦♠♦ LN ♣♦ sr ♦t♦ ①♣r♠♥t

♠♥t ♠♥r ♠s s♠♣s ♣♦♠♦s tr♠♥r ♦ ♣r r t♠♣rtr í♦♥s

q ♠s♠ ♦r♠ ♦♠♦ t① ♠♦rt♠♥t♦ ♥ ♣♥ q tr♠♥çã♦

♠♣t ♦ ♣♦t♥ tr♦stát♦ ss♦♦ sts ♠♦♦s ♠ ♥çã♦ ♦ t♠♣♦ ♣♦ sr

t③ ♣r tr♠♥çã♦ ♦ ♣r r q

é♠ ♣çõs ♠ ♥óst♦s ♥ tr♠♥çã♦ ♦ ♣r r q(r) Ti(r)

①♣rssã♦ ♥ét ♣r rqê♥ s ♦t ♥♦ ♣ít♦ ♣♦ sr t③ ♥ ♦♠

♣rçã♦ ♥tçã♦ ♠♦♦s ♦sr♦s ♥♦ ❬❪ ♦♠ ♠s ♣rsã♦ ♣♦s ♠♦s

♠ ♦♥t tr♠♦s O(q−2) ♦♠ rçã♦ t① ♠♦rt♠♥t♦ st ♣♦ tr ♣çõs ♠

♥ô♠♥♦s tr♥s♣♦rt ♣♦s é ♣r♦á q ♠ rçã♦ ♥tr t① ♠♦rt♠♥t♦ ♦

♣♦t♥ tr♦stát♦ ♦ ♠♥t♦ ♦ tr♥s♣♦rt

Pr♦♣♦sts ♣r tr♦s tr♦s

Prt♥♠♦s r ♦♥t♥ à ♥ ♣sqs ♥ ♥st ts ❯♠ ♦s tó♣♦s

♥ã♦ ♠♥♦♥♦ ♥tr♦r♠♥t ♦ q ♣rt♥♠♦s ♥str é ♥ê♥ ♠ s ❲s

❩s ♣♦♣çã♦ ♣rtís ♣rs♦♥s ♥♦ ♣s♠ t♦ ss♠tr ♦ ♠♣♦

♠♥ét♦ ♠ ♣rtís ♦ ♣r é rt♠♥t ① ③ ♦♠ q sts

s♠ ♣rs♦♥s ♦ ♠♦♠♥t♦s r ♠♥ét sts ♣rtís sr♠

♠♦♠♥t♦s s♣s ♦♥♦s ♦♠♦ órts ♥♥ ❬❪ q ♣♦r s ③ sã♦ ♥rt♠♥t

rs♣♦♥sás ♣♦ tr♥s♣♦rt ♥♦áss♦ ❬ ❪ ♦ q ♥ q ♠ ♠♥♦s ♥t♥s ♦

q ♦ tr♥s♣♦rt ♥ô♠♦ r ♦ ♦♥♥♠♥t♦ ♦ ♣s♠ P♦r ♦tr♦ ♦ á t♠é♠

♣rtís ♦♠ ♦ ♣r rt♠♥t t s ♣rtís ♣ss♥ts ♦ r♥ts

q ♦♥s♠ ♥r ♦ ♣♦ç♦ ♠♥ét♦ ♣♦r ♥ã♦ srr♠ órts tã♦ s♣ç♦ss ♦♥tr♠

♠♥♦s ♦♠ ♦ tr♥s♣♦rt Pr ♠♦♦s ①s rqê♥s t♣♠♥t ♠♥♦r ♦ q

rqê♥ rçã♦ í♦♥s ωtri =√

2Ti/mi/qR0 ♦♥ Ti mi sã♦ t♠♣rtr

♠ss ♦s í♦♥s R0 é ♦ r♦ ♠♦r ♦ t♦♠ ♥â♠ í♦♥s étr♦♥s ♣rs♦♥♦s

sã♦ r♥ts ♣♦rt♥t♦ ♠ sr ♦s ♠ ♦♥srçã♦ ♥♦ ♠♦♦ ♣r srr ts

♠♦♦s ❬❪ t♦ ♣rtís ♣rs♦♥s ♥♦ ♦♥tí♥♦ ♦♥s é♥ ss

❲ ① rqê♥ ♦ ♥st♦ ♥t♠♥t ♠ ❬❪ t③♥♦ ♦ ♠♦♦ r♦

♥ét♦ ♦r♦ ♦♠ ❬❪ ♣r q s ♣♦ssí ♠ trt♠♥t♦ ♥ít♦ é ♥ssár♦ q

st♥çã♦ ♥tr ♣rtís t♦t♠♥t ♣rs♦♥s ♣rtís t♦t♠♥t r♥ts

sr♠ ♦♥srs ♥♦ ♠♦♦ s ♣r♥♣s rrê♥s ♣r ♦ st♦ rçã♦ ♥tr ♠♦♦s

r é♥ ❲s

♦és♦s ♥â♠ ♣rtís ♣rs♦♥s q r♦♠♥♠♦s sã♦ ❬❪ Prt

t♦r ♥♦ ♥st st♦ é st ♦r♠ ♠s ssí ♠ ❬❪

♠ ♠ ♠♦♠♥t♦ ♥ ♦s ss♥t♦s trt♦s ♥st ts ♣♦♠ sr ①♣♥♦s ♠ ♦s

r♠♦s ♦ st♦ s tr♦♠♥ét♦s ♥ ♣rs♥ç t♦s ♠♥ét♦s ♥st

çã♦ t♦♠♦♦s úst♦s ♦és♦s ♥♦ ♥ ♣sqs st ts ♣rt♥♠♦s

♥str ♠ ♣r♦ sts ♦s ss♥t♦s ♣♦s ♦♥srá♦s ♠ ♦♥♥t♦ s♦ ♥s

s ❬ ❪ ♦ t♦ ♦ r♥t t♠♣rtr étr♦♥s ♠ ♠♦♦s ♦és♦s ①s

rqê♥s srt♦ ♣ q♥t ηe = LN/LTe ♣♦ sr q♥t♦ q♥♦ s ♦♥sr

♣rtrçã♦ ♣r ♦ ♣♦t♥ t♦r A‖ 6= 0 st st♦ ♦♥srr ♥ê♥

s♥♦s r♠ô♥♦s ♣♦♦s m = ±2 é ss♥ ♦r♦ ♦♠ ❬❪ ♣♦s t♦s tr♦

♠♥ét♦s t♠ ♣r♥♣♠♥t ♠ étr♦♥s q ♦♥stt♠ ♦rr♥t ♣r q ♣♦ss

♣♥ê♥ ♠ sin(2θ) ♠s♠ ♦r♠ ♦ st♦ t♦♠♦♦s ♦és♦s ① q s♠

♦s ♠ ♦♥ts tr♠♦s r♦♥♦s m = ±2 t♦♠♦♦s ♦és♦s ♦♦rr♠ ♠ rõs

♦♥ rqê♥ s ♦♥tí♥♦ é ♣rt♠♥t ♦♥st♥t ♦♠ ♣♦sçã♦ r ♣♦r

♣r♠tr ♦tr ♥♦r♠çõs s♦r ♦ ♣♦t♥ tr♦stát♦ ♣rtr♦ Φ(r, t) ♣♦r tr ♣

çõs ♥ósts s♣tr♦s♦♣ ♠ ❬ ❪ ♠ s♣ ♣r ♦tr ♦ ♣r r

t♠♣rtr í♦♥s ♣♦ss ♠ ♠♣♦ ♥trss ♥ ár sã♦ ♥r

á ♥ ♥ú♠rs qstõs ♠ rt♦ q♥t♦ ♦s ♠ ♣rtr ♦tr♦s t♦s

♦♥srçã♦ tr♠♦s ♦r♠ s♣r♦r ♠ ♣♦♠ r srr ♠s

sts qstõs ♥sã♦ tr♠♦s O(4rρ4i ) ♣♦r ①♠♣♦ é♠ ♥ssár ♣r ♦

st♦ t♦♠♦♦s ♦és♦s ♣r♠t tr♠♥çã♦ ♣♥ê♥ r rqê♥

♦s ❬❪ ♦ ♠♦♦ í♦s sss tr♠♦s ♦r♠ s♣r♦r ♣♦♠ sr ♥í♦s s

♦♥sr♦s r♦s♦s ①♦ ♦r ❬ ❪

♣ê♥

♦♥st♥ts ♣râ♠tr♦s ♦

♦♥st♥ts ís

♦♥st♥ts íss ♣rt♥♥ts st ts

í♠♦♦ r♥③ ís ❱♦r ♥♠ér♦ ❯♥s ♦ ♦r♠ r♥③

me ss r♣♦s♦ ♦ étr♦♥ 10−30

mp ss r♣♦s♦ ♦ ♣rót♦♥ 10−27

e r étr ♠♥tr 10−19

ε0 Pr♠ss ♥♦ á♦ 10−11 ♠−1

µ0 Pr♠ ♥♦ á♦ 10−6 ♠−1

h ♦♥st♥t P♥ 10−26 s

k ♦♥st♥t ♦t③♠♥♥ 10−23 −1

Prâ♠tr♦s ♦ ♦

Pr♥♣s ♣râ♠tr♦s ♦s t♦♠s

Prâ♠tr♦ í♠♦♦ ❯♥s

♦ ♠♦r R0 ♠

♦ ♠♥♦r a ♠

♠♣♦ ♠♥ét♦ t♦r♦ BT

♦rr♥t ♣s♠ IP

t♦r sr♥ç qa(q95)

♥s étr♦♥s ne ♥tr♦ a ♠é 1019♠−3

♠♣rtr étr♦♥s Te ♥tr♦ a ♠é ❱

r♥s r♥③ rqê♥s ♦s

rtrísts ♦

❱♦r ♦r♠ r♥③ rqê♥ rçã♦ ♦ tér♠ ♦ r♦ r♠♦r ♣r ♦ t♦♠

ωci 108 rs vTi5.4× 104 ♠s ρi 4.5× 10−4 ♠

ωce 2.2× 1011 rs vTe2.3× 106 ♠s ρe 1.0× 10−5 ♠

♣ê♥

♥ts rçõs t♦rs

st ♣ê♥ ♣rs♥t♠♦s ♥ts rçõs t♦rs ♥♠♥t ♠♣♦rtâ♥

♣r ♦s á♦s ♦ ♣ê♥ ♦ ♣ít♦ s rçõs ♠ts ③s ♥♦♠ ♦ ♦♣r♦r

r♥t ∇ q ♥ ♠♦r ♦s s♦s é ♦♥♥♥t♠♥t srt♦ ♠ tr♠♦s ♦♦r♥s

qst♦r♦s s rçõs ♥ts q ♣rs♥ts ♣♦♠ sr ♥♦♥trs ♠ ❬

❪

♥ts t♦rs

(A×) ·C = (×C) ·A = (C ×A) ·

A× (×C) = (A ·C)− (A ·)C

(A×) · (C ×D) = (A ·C)( ·D)− (A ·D)( ·C)

♥ts t♦r♠s ♥♠♥ts

∇ · (∇×A) = 0, ∇× (∇f) = 0

∫

V∇ ·AdV =

∮

SA · dS,

∫

S(∇×A) · dS =

∮

lA · dl

♥ts ♥♦♥♦ ♦ ♦♣r♦r ∇

∇ · (fA) = f∇ ·A+A ·∇f

∇× (fA) = f∇×A+∇f ×A

∇ · (A×) = (∇×A) ·− (∇×) ·A

∇× (A×) = (∇ ·)A− (∇ ·A)+ ( ·∇)A− (A ·∇)

A× (∇×) +× (∇×A) = ∇(A ·)− (A ·∇)− ( ·∇)A

∇× (∇×A) = ∇(∇ ·A)−∇2A

∇ · (A) = (∇ ·A)+ (A ·∇)

r♥t r♥t ♦t♦♥ ♣♥♦

♠ ♦♦r♥s í♥rs

∇Ψ =∂Ψ

∂RR +

1

R

∂Ψ

∂ϕϕ +

∂Ψ

∂ZZ , R = R0 + r cos θ

∇ · =1

R

∂(RBR)

∂R+

1

R

∂Bϕ

∂ϕ+

∂BZ

∂Z

∇× =

[

1

R

∂BZ

∂ϕ− ∂Bϕ

∂Z

]

R +

[

∂BR

∂Z− ∂BZ

∂R

]

ϕ +1

R

[

∂(RBϕ)

∂R− ∂BR

∂ϕ

]

Z

∇2Ψ =1

R

∂

∂R

(

R∂Ψ

∂R

)

+1

R2

∂2Ψ

∂ϕ2+

∂2Ψ

∂Z2

r♥t r♥t ♦t♦♥ ♠ ♦♦r♥

s qst♦r♦s

∇Ψ =∂Ψ

∂rr +

1

r

∂Ψ

∂θθ +

1

R

∂Ψ

∂φφ,

∇ · =1

Rr

[

∂

∂r(RrBr) +

∂

∂θ(RBθ) + r

∂Bφ

∂φ

]

∇× =

1

R

[

(

1

r

∂(RBφ)

∂θ− ∂Bθ

∂φ

)

r +

(

∂Br

∂φ− ∂(RBφ)

∂r

)

θ +R

r

(

∂(rBθ)

∂r− ∂Br

∂θ

)

φ

]

.

∇2Ψ =1

Rr

[

∂

∂r

(

Rr∂Ψ

∂r

)

+∂

∂θ

(

R

r

∂Ψ

∂θ

)

+∂

∂φ

(

r

R

∂Ψ

∂φ

)]

rs rs♦rs ♠ ♦♦r♥s í♥rs

∂R∂R

=∂ϕ∂R

=∂R∂R

= 0,∂Z∂ϕ

= 0,∂R∂Z

=∂ϕ∂Z

=∂Z∂Z

= 0,

∂R∂ϕ

= ϕ,∂ϕ∂ϕ

= −R.

rs rs♦rs ♠ ♦♦r♥s qst♦r♦s

∂r∂r

=∂θ∂r

=∂φ∂r

= 0,∂φ∂θ

= 0,

∂r∂θ

= θ,∂θ∂θ

= −r,

∂r∂φ

= − cos θφ,∂θ∂φ

= − sin θφ,∂φ∂φ

= − cos θr + sin θθ.

♦♥♥♥t ♦srr s♥t rçã♦ ♥tr ♦s sst♠s ♦♦r♥s ♣rs♥t♦s ♠

R = cos θr − sin θθ, ϕ = −φ, Z = sin θr + cos θθ.

♣ê♥

t♥çã♦ s ①♣rssõs ♥íts

rr♥ts à ♥ás qír♦ ♦♠

r♦tçã♦

♣r♥♣ ♥tt♦ st ♣ê♥ ♠♦strr ♠t♦♦♦ ♣r ♦t♥çã♦ s qçõs

q sr♠ ♦ qír♦ ♦♠ r♦tçã♦ ♣♦♦ t♦r♦

çõs ♥♦♥♦

♥♠♥t ♣rtr ♦ ♠♣♦ ♠♥ét♦ qír♦ ❬❪

= F∇φ+∇φ×∇Ψ,

♥s ♦rr♥t ss♦♦ ♣♦ sr ♦t ♣♦ s♦ s ♥ts

rst♥♦ ♠

= µ−10 ∇× = µ−1

0 [−∇φ×∇F + (∇ ·∇Ψ)∇φ+ (∇Ψ ·∇)∇φ− (∇φ ·∇)∇Ψ],

Pr ♦ s♥♦♠♥t♦ ér♦ ♣♦♠♦s srr ♦s ♦s út♠♦s tr♠♦s ♦ ♦

rt♦ ♠ ♦♦r♥s í♥rs ♦r♦ ♦♠ ♦♥sr♥♦ ♣r ss♦ φ = −ϕ ♦

s ♦r♠ ①♣ít s q

(∇Ψ ·∇)∇φ− (∇φ ·∇)∇Ψ =

∂Ψ∂R

∂∂R

(

−ϕR

)

+ 1R2

∂∂ϕ

(

∂Ψ∂R R

)

= 2R2

∂Ψ∂R ϕ = R2

[

∇

(

1R2

)

·∇Ψ

]

∇φ.

♦♠ ♦ s♦ t♦r♥s ♦♥♥♥t ①♣rssr ♥s ♦rr♥t ♠ tr♠♦s

♦ ♦♣r♦r r♥♦ ∆∗Ψ = ∇ · (∇Ψ/R2) ♦r♠ q

= µ−10 (R2∆∗Ψ∇φ−∇φ×∇F ).

♣rtr s ①♣rssõs ♥ít ♣r ♣rs♥ts ♠ ♦té♠s

∇φ× = −∇ΨR2 , (∇φ×∇Ψ)× = F

R2∇Ψ,

∇Ψ× = |∇Ψ|2∇φ− F (∇φ×∇Ψ) = B2R2∇φ− F, B2 = F 2+|∇Ψ|2

R2 ,

∇φ× (∇×) = ∇FR2 , (∇φ×∇Ψ)× (∇×) = −(∆∗Ψ)∇Ψ− ( ·∇F )∇φ,

∇Ψ× (∇×) = −R2∆∗Ψ(∇φ×∇Ψ)− (∇Ψ ·∇F )∇φ,

♦♥ ♦r♠ t③s s ♥ts ♦♠ rçã♦ s♠tr ③♠t ∇φ · ∇f = 0

♣r qqr ♥çã♦ f ♠

tr♠♦ ♦ ♦rç ♠♥ét é ♦ ①♦ t③♥♦

× = − 1

µ0R2

[

(∆∗Ψ)∇Ψ+1

2∇F 2 − ( ·∇F )R2

∇φ

]

,

çõs ♣r ❱

Pr♦♣rs ♠♣♦rt♥ts ♦ qír♦ ♣♦♠ sr tr♠♥s ♣rtr

♠ qçã♦ ♦♥t♥

E +❱× = 0,

∇ · (ρ❱) = 0,

♦♥ ♥ ♣r♠r s♦♥sr♠♦s ♦ t♦ ♠♥ét♦ ♦♥③♥t ♦♠ ♦r♠

s q ❱ = ❱′ + C ♦♠ ❱′ ⊥ é ♠ s♦çã♦ ♣♦ssí ♦♥ ❱′ ♣♦ sr

tr♠♥♦ ♣rtr ♦ ♣r♦t♦ /B2 ♦♠

❱′ =E ×

B2=

F∇φ×∇Φ− (∇Ψ ·∇Φ)∇φ

B2.

♦♠♦ E · = −∇Φ · (∇φ ×∇Ψ) = 0 ∇Φ ·∇Ψ = 0 ♣♦r s♠tr ③♠t ♦♥s q

E stá ♥ rçã♦ ∇Ψ ♦ ♦r♠ q♥t

E = −Ω∇Ψ, Ω = Ω(Ψ) =dΦ

dΨ,

♦ q ♣r♠t s♥♦r ♦r♦ ♦♠

❱′ = −ΩR2∇φ+

FΩ

B2,

♦♥sq♥t♠♥t ♣♦♠♦s ①♣rssr ♦ ❱ ♦♠♦

❱ = C ′− ΩR2∇φ, C ′ = C +

FΩ

B2.

st q♥♦ ssttí ♠ ♦♠ ♦ ①í♦ ∇ · = 0 à s♥t

qçã♦

∇ · (ρC)−∇ · (ρΩR2∇φ) = ·∇(ρC) = 0,

q ♣♦ sr tr③ ♣r C = κ(Ψ)/ρ ♣♦s ·∇f = 0 ♠♣ q f = f(Ψ) ♣r qqr

♥çã♦ f s♠étr ♠ φ ♣♦rt♥t♦ tr♠♥ ♦ qír♦ ♠ ♥çã♦ s

r♥③s κ(Ψ) Ω(Ψ) F Ψ

❱ =κ

ρ− ΩR2

∇φ,

♦♥ κ = κ(Ψ) é ♠ ♥çã♦ rtrár ①♦ rt♠♥t r♦♥♥ ♦♠ ♦

♣♦♦

♠s♠ ♦r♠ ♦♠♦ ♦r♠ ♦t♦s é ♦♥♥♥t ♦tr rçõs ♥♦♥♦

♦ ♣r♦t♦ t♦r ♦♠

∇φ×❱ = − κρR2∇Ψ, ×❱ = −Ω∇Ψ,

∇Ψ×❱ = κ|∇Ψ|2ρ ∇φ+

(

ΩR2 − κFρ

)

(∇φ×∇Ψ)

♣ró①♠ t♣ é ♦ á♦ ♦rç ♥trí ♦r♦s ♦ r♦tçã♦ ♦ ♣s♠

♥ qçã♦ ♠♦♠♥t♦ Pr st á♦ é ♦♥♥♥t t③r s♥t ♥t

❱ ·∇❱ = ∇

(

V 2

2

)

−❱× (∇×❱),

♦t ♣rtr q é ♠s r Pr tr ①♣rssõs ♠t♦ ①t♥ss é ♦♥♥♥t

r s ♦♠♣♦♥♥ts ♥ rçã♦ ♠ t♦r rrtár♦ ❯ ♦r♦ ♦♠

♥t

❯❱ : ∇❱ = ❯ · (❱ ·∇❱) = ❯ ·∇(

V 2

2

)

− (❯×❱) · (∇×❱).

♣r♦♥♥t ♦ s♦ st t♦r rrtár♦ trr♠♦s ❯ = ∇φ ❯ = ❯ = ∇Ψ

♣r ♦tr s qçõs

Pr ♦ s♥♦♠♥t♦ é ♥ssár♦ ♣r♠r♠♥t tr ♦s s♥ts á♦s

V 2

2=

1

2

κ2B2

ρ2+

1

2Ω2R2 − κFΩ

ρ,

∇×❱ =

[

κ

ρ∆∗Ψ+∇Ψ ·∇

(

κ

ρ

)]

∇φ+∇φ×∇

(

ΩR2 − κF

ρ

)

,

♦♥ ♦ s♥♦ ♦♥srá ①t♥sã♦ ér ♦ t♦ ♣rtr rçã♦ s

♥ts

♦t♠♦s s ♦♠♣♦♥♥ts ♦rç ♥r ♦ à r♦tçã♦

ρ∇φ❱ : ∇❱ =1

R2 ·∇

(

κ2F

ρ− κΩR2

)

,

ρ❱ : ∇❱ = ρ ·∇(

κ2B2

2ρ2− Ω2R2

2

)

,

ρ∇Ψ❱ : ∇❱ = |∇Ψ|2R2

[

−κ2

ρ ∆∗Ψ− ρ∇Ψ ·∇(

κ2

2ρ2

)

+ ρ|∇Ψ|2∇Ψ ·∇

(

κ2

ρ2|∇Ψ|2

)]

+(

κFΩρ − ΩR2

2 − κ2B2

2ρ2

)

∇Ψ ·∇R2,

s qs ♣r ♦t♥çã♦ s qs ♠ sr s♦♠s strís rs♣t♠♥t

às ♦♠♣♦♥♥ts ♦ r♥t ♣rssã♦ ♦rç ♠♥ét ♦♥♥♥t ♣r st á♦

♣♦rt♥t♦ t③r ♦s s♥ts rst♦s

∇φ · (×) = ·∇F

µ0R2,

∇Ψ · (×) = −|∇Ψ|2µ0R2

(

∆∗Ψ+1

2

∇Ψ ·∇F 2

|∇Ψ|2)

,

♦t♦s ♣rtr s rrs qçõs ♣ós ♦ s♥♦♠♥t♦ ér♦ s ♦♠♣♦

♥♥ts qçã♦ ♠♦♠♥t♦

ρ❯❱ : ∇❱+❯ ·∇p−❯ · (×) = 0, ❯ = ∇φ,,∇Ψ,

♣♦♠ sr ①♣rsss ♥ ♦r♠

• ♦♠♣♦♥♥t ∇φ

·∇[

F

(

1− µ0κ2

ρ

)

+ µ0κΩR2

]

= 0.

• ♦♠♣♦♥♥t

·∇(

κ2B2

2ρ2− Ω2R2

2

)

+ ·∇p

ρ= 0.

• ♦♠♣♦♥♥t ∇Ψ

(

1− µ0κ2

ρ

)

∆∗Ψ+ µ0R2∇Ψ·∇p

|∇Ψ|2 + 12∇Ψ·∇F 2

|∇Ψ|2 − µ0ρ2 ∇Ψ ·∇

(

κ2

ρ2

)

+

µ0ρ2|∇Ψ|2∇Ψ ·∇

(

κ2

ρ2|∇Ψ|2

)

+ µ0R2

|∇Ψ|2

(

κFΩρ − ΩR2

2 − κ2B2

2ρ2

)

∇Ψ ·∇R2.

á♦ ∇ · q qír♦

Pr r ♦ sst♠ é ♥ssár♦ ♦ á♦ ∇ · q ♣rs♥t ♥ q q ♣♦ sr

t♦ ♣rtr ♣çã♦ ♥t ♥ qçã♦ ♥çã♦ ①♦ ♦r

♦ q rst ♠

∇ · q =γ

γ − 1

[

p

eB2∇ · (×∇T ) + (×∇T ) ·∇

(

p

eB2

)]

.

♦ q s rr ♦ ♣r♠r♦ tr♠♦ ♥tr ♦ts ♦ s♦ ♣r♠t ♦tr

rçã♦ ♣r♦①♠

∇ · (×∇T ) = (∇φ×∇T ) ·∇F ≈ dT0

dΨ ·∇F − dF0

dΨ ·∇T,

♦♥ é♠ ♣çã♦ t♦r ♣rtrçã♦ t♠♣♦r ♦ t③ ♥ út♠

♣ss♠ q ♠♣r ♣r♦①♠çã♦ |T1(Ψ, θ)| ≪ T0(Ψ) |F1(Ψ, θ)| ≪ |F0(Ψ)|

♦r♠ s♠r ♦ s♥♦ tr♠♦ ♥tr ♦ts ♠ ♣♦ sr s♥♦♦ ♦r♦

♦♠ ①♣rssã♦

(×∇T ) ·∇(

peB2

)

≈ F0

e

[

dT0

dΨ ·∇(

pB2

)

− ddΨ

(

pB2

)

·∇T

]

≈p0R2

0

eF0

dT0

dΨ

[

·∇pp0

− 2·∇FF0

+ ·∇R2

R2

0

]

−[

1p0

dp0dΨ − 2

F0

dF0

dΨ + 1R2

0

dR2

dΨ

]

·∇T

,

♦♥ ♦ ♦♥sr♦ q |∇Ψ| ≪ |F | | ·∇(|∇Ψ|)| ≪ | ·∇F | d|∇Ψ|/dΨ ≪ |dF/dΨ|

♦♠ s ♥çõs ♥tr♦③s ♣s qs s qçõs

♣♦♠ sr ①♣rsss ♥ ♦r♠

peB2∇ · (×∇T ) = Mt(∆F −RF∆T )

p0csF0

·∇R2

R0, (×∇T ) ·∇

(

peB2

)

=

Mt

[

1− 2∆F +∆p −(

1 +Rρ − 2RF + T0

R2

0

∂R2/∂ΨdT0/dΨ

)

∆T

]

p0csF0

·∇R2

R0.

♠ rs♠♦ ♦ ss♠r q s r♥③s qír♦ sã♦ ♦r♠ X = X0(Ψ)+ZX1(Ψ, θ)

♦♠ |X1| ≪ |X0| ♦ s♣r③r♠♦s ♦ tr♠♦ |∇Ψ|2/F 20 ss rs r♥ts ♣♠♦s

s♥♦r ①♣rssã♦ rê♥ ♦ ①♦ ♦r ♦tr ♦ s♥t rst♦

∇ · q = Mt

[

1−∆F +∆p −(

1 +Rρ −RF +T0

R20

∂R2/∂Ψ

dT0/dΨ

)

∆T

]

γp0csR0

(γ − 1)F0

·∇R2

R20

,

♦ q é t③♦ ♠

♣ê♥

rçã♦ ór♠s ss ♥♦

♣ít♦

①♣rssõs érs ♣r ♦ r♥t ♦ ♦ t♥s♦r s♦s ♣r

♥s ♦rr♥t qçã♦ ♦çã♦ ♦ t♥s♦r s♦s ♣r ♠s

st♥t t③s ♥♦ ♣ít♦ sã♦ ♦ts ♥st ♣ê♥ í♥ α = i, e s q♥ts

♠r♦só♣s ♦ ♣s♠ é s♣r♠♦ ♥s ①♣rssõs ♣r s♠♣r ♥♦tçã♦ ♦♥t♦ s

♠ r s♥t♥♦s s rst♦s q ♣rs♥t♦s sã♦ á♦s ♠ s ♠♦r ♣r

sst♠s ① ♣rssã♦ β ∼ ε2

çõs ♣r

Pr ♦t♥çã♦ trs rçõs é ♦♥♥♥t s♣rr s ♦♠♣♦♥♥ts ♣r ♣r

♣♥r ♦♠ rçã♦ = /B ♦ ♦♣r♦r ∇ ♦ s

∇‖ = ·∇ ∇⊥ = ∇− ∇‖,

♦r♠ q ♣rtr ♦té♠s

∇ · = ∇‖ lnB, ∇× = µ0

B+ ×∇ lnB ≈ ×∇ lnB,

s ♦♥srr♠♦s J‖ ∼ J⊥ ♣♦s ∇p ≈ × ♦♥sq♥t♠♥t

|µ0⊥/B||∇× | ∼ β, · (∇× ) = µ0

J‖B

∼ β

L.

❯t③♥♦ ♦té♠s s♥t ♣r♦①♠çã♦

κ = ∇‖ = −× (∇× ) ≈ ∇⊥ lnB,

♣r ♦ t♦r rtr ♦ ♠♣♦ ♠♥ét♦ κ

Pr ♦s á♦s ♣ró①♠ sçã♦ é ♦♥♥♥t ♥r = f×∇g ♦ á♦ ♦

r♥t t♦ ♣rtr rst ♠

∇ · ≈ ·∇ ln f.

♠ ♣rtr ♣r f = B−1 ∇ · = (× κ) ·∇g ♥st s♦ ♦té♠s

D =×∇g

B, ∇ · D = −2D ·∇ lnB.

á♦ rê♥ π q

♥♠♦s st sçã♦ ♣rs♥t♥♦ ♥çã♦ s♦s ♣r

π‖ =3

2π‖

(

− 1

3

)

,

♦ á♦ s r♥t ♦t♦ ♣rtr s rçõs ♣rs♥ts ♥

sçã♦ ♥tr♦r rst ♠

∇ · π‖ =3

2

[

(∇‖ lnB + κ)π‖ + ∇‖π‖

]

− 1

2∇π‖.

q é ♠ t♦r rtrár♦ ♠s ♥t t③♦ ♣r r♣rs♥tr t♦t q D ♣♦ss ♦r♠ ♠ ♦ r

♦t t♠é♠ q

·∇ · π‖ =3

2π‖∇‖ lnB +∇‖π‖,

×∇ · π‖ =3

2π‖(× κ)− 1

2×∇⊥π‖,

t③♥♦

∇ · (×∇ · π‖) = (× κ) ·∇π‖.

çõs ♣r ♦s

❯t③♥♦ ♦t♠♦s ♠ ♣r♠r♦ s♥♦♠♥t♦ ér♦ ♣r ♦ r♥t

s ♣r♥♣s ♦s r ♠ ♦s st s♥♦♠♥t♦ ♦♥sr♠♦s ♦r♠

vE ∼ vTi ♥ã♦ ♠♦s ♠ ♦♥t ♦ tr♠♦ r♦s♦s πg s♦s ♣r♣♥

r π⊥ ♦s qs sã♦ ♦r♠ s♣r♦r ♠ ρi/L sr ♣rs♥t♠♦s ts rst♦s

E =×∇Φ

B, p =

×∇p

enB, π =

×∇ · πenB

, I ≈

ωc× dE

dt,

∇ · (nE) = E ·∇n− 2nE ·∇ lnB,

∇ · (np) = −2np ·∇ lnB,

∇ · (nπ) = nπ ·∇ lnB,

∇ · (nI) = I ·∇n− nI ·∇ lnB +1

ωc∇ ·

(

× dEdt

)

∇ · (v‖) = ∇‖v‖ − v‖∇‖ lnB.

çõs ♣r ♥s ♦rr♥t

♦r♠ s♠r à ssçã♦ ♥tr♦r ♦t♠♦s s ①♣rssõs s ♣r♥♣s ♦♠♣♦♥♥ts

♥s ♦rr♥t ss rs♣t♦s r♥ts

p =×∇p

B, π =

×∇ · π‖B

, ≈mn

B× dE

dt,

∇ · p = −2p ·∇ lnB,

∇ · π = π ·∇ lnB,

∇ · ≈ ·∇ ln

(

n

B

)

− mn

B

[

∂

∂t∇ ·

(

∇⊥ΦB

)

+∇ · (× E ·∇E)

]

.

qçã♦ ♦çã♦ π‖

Pr♠r♠♥t ♦sr♠♦s q ♦ tr♠♦ Tijk = i · [(j ·∇)k] i, j, k = 0, 1, 2 sts③

♥r♠♥t s s♥ts rçõs

Tijk = −Tikj + i · [(∇× j)× k] + i · [(∇× k)× j ],

T00k = −T0k0 − Tk00, 0 = .

♦♠♦ · d/dt = 0 · ∂/∂t = 0 · κ = 0 sq q

· ddt

= ·(

∂

∂t+ (⊥ ·∇)+ v‖κ

)

= ⊥ : ∇ = 0

♦♥sq♥t♠♥t ♦♠♦ ⊥ = v⊥11 + v⊥22

v⊥1T010 + v⊥2T020 = 0.

st ♦r♠ ♦r♦ ♦♠ ♦ ♥♦tr q ♥ ♦r♠ ♠tr s

♦♠♣♦♥♥ts ∇ ♦ tr♠♦ : ∇ ♣♦♠ sr ①♣rss♦s rs♣t♠♥t ♦♠♦

Mij = (i ·∇)vj +

2∑

m=0

vmTijm,∑

i,j

biMijbj = b0M00b0 = M00

♦♥ v0 = v‖ v1 = v⊥1 v2 = v⊥2 ♦té♠s rçã♦

: (∇)T = : ∇ = ∇‖v‖ − ⊥ ·∇ lnB.

♦♥sr♠♦s ♦r qçã♦ ♦çã♦ ♦ t♥s♦r s♦s ❬❪

dπdt + π∇ · +

[

π ·∇+ (π ·∇)T − (γ − 1)(π : ∇)

]

+ ωc(× π − π × )+[

p∇+ p(∇)T − (γ − 1)p∇ · ]

+ γ−1γ

[

∇q+ (∇q)T − (γ − 1)∇ · q]

+∇ · τ = 0,

♦♥ τ é ♠ ♦s ♣ró①♠♦s ♠♦♠♥t♦s ♥çã♦ strçã♦ q é ♦♥sr♦ ♥♦ ♥st

♦♥t①t♦ ♦♥sr♠♦s ♦ s♦ ♥ã♦ ♦s♦♥

♦♥sr♥♦ t♠é♠ ♣♥s ♦ t♦ s♦s ♣r ♦ s π ≈ π‖ = π‖(− /3)

♦r♦ ♦♠ ♦té♠s

:dπ

dt=

dπ‖dt

− d

dt· ( · π)− ·

(

d

dt· π

)

=dπ‖dt

.

st ♦r♠ ♥♦ r♠ ♥r ♣r ♦ s♦ át♦ q = 0 ♠ q ♥ã♦ á r♦tçã♦

qír♦ ♣rtr s q

dπ‖dt

+ p

[

2∇‖v‖ − 2⊥ ·∇ lnB − (γ − 1)∇ · ]

= 0.

♣r♦①♠çã♦ ♣r t♦♠s s♣rís ♠

♥éts ♦♥ê♥trs

♠♣♦ ♠♥ét♦ qír♦

♠ t♦♠s s♣rís ♠♥éts ♣r♦①♠♠♥t ♦♥ê♥trs ♦♥♦r♠ srt♦

♥♦ ♣ít♦ ♦ ♠♣♦ ♠♥ét♦ qír♦ ♣♦ sr ♣r♦①♠♦ ♣♦r

= B, B ≈ B0(1− ε cos θ), =ε

qθ + φ ≈ φ, ε ≪ 1.

♦♥♥♥t♠♥t r♣rs♥t♦ ♥♦ sst♠ ♦♦r♥s qst♦r♦s (r, θ, φ) ♦♥s

q♥t♠♥t ♦♥sr♥♦ ♣♥s tr♠♦s ♦♠♥♥ts ♦ s♥♦♠♥t♦ ér♦ ♦s ♦♣r

♦rs ∇‖ ∇⊥ ♦ t♦r rtr κ rst ♠

∇‖ = k‖

(

∂

∂θ+ q

∂

∂φ

)

, ∇⊥ = r∂

∂r+ θkθ

∂

∂θ, κ = − R

R0,

♦♥ k‖ = 1/qR0 kθ = 1/r ♦r♦ ♦♠ R = cos θr− sin θθ ♠é♠ ♦srs

q ♣r q ≫ 1

∇× ≈ × κ = − ZR0

, ∇ · ∼ · (∇× ) = O(εk‖) ∼ 0,

♦♥ Z = sin θr + cos θθ

♠♣♦ ♠♥ét♦ ♣rtr♦

r♠♦s sr ♠s rçõs ♣r ♦ ♠♣♦ ♠♥ét♦ ♣rtr♦ ♦♥sr♥♦ q

= ⊥ = ∇× (A‖) ♦♥ é ♦ ♣♦t♥ t♦r ♣♦rt♥t♦ q

= −×∇A‖ + A‖(× κ) ≈ −1

r

∂A‖∂θ

r + irA‖θ

♣rtr ♦té♠s

∇‖ =

B·∇ ≈ 1

rB

(

irA‖∂

∂θ−

∂A‖∂θ

∂

∂r

)

,

∇× ≈ irk‖∂A‖∂θ

r + 2rA‖.

❱♦ ♥s ♦rr♥t

♣rtr s rçõs

v2Ti=

2Ti

mi, ρi =

vTi

ωci

,1

B=

vTiρi2

e

Ti, ρi/r ≪ rρi ≪ 1

♦♥sr♥♦ ♦ ♣♦tê♥ tr♦stát♦ ♣rtr♦ Φ ♠ ♦♠♦ ♣rssã♦ (p s♦s

♣r π‖ t③♠♦s ♣r ♦tr

E = ωER0

(

θ + iρi/r

rρi

∂ln Φ

∂θr

)

, ωE =irΦB

=irρi2

eΦ

Ti

vTi

R0,

p =i

2

e

TiωR0

(

pθ + iρi/r

rρi

∂p

∂θr

)

, ω = rρivTi

R0

π = − i

4

e

TiωR0

(

π‖iθ + iρi/r

rρi

∂π‖i∂θ

r

)

,

I = ienωER0ω

ωcr.

♣ê♥

♦çã♦ trt s qçõs

♣rtrs

qçõs ♥s s♦çã♦ qír♦

st ♣ê♥ ♣rs♥t♠♦s ♠ ♠ét♦♦ trt♦ ♣r rs♦r ♦ sst♠ ♦♠♣♦st♦ ♣s

qs s qs sã♦ r♣ts sr ♣r tr tr

ρ0∂v‖∂t

+∇‖p+ F‖ = 0,

∂(ρ+ R)

∂t+ ρ0∇ · = 0,

∂(p+ P )

∂t+ γp0∇ · = 0,

= E + v‖, E ≈ ωER0(1 + ε cos θ)θ,

F‖ = ρ0( : ∇❱+ ❱ : ∇) + ρ❱ : ∇❱,

R =

∫

(❱ ·∇ρ+ ·∇ρ+ ρ∇ ·❱)dt,

P =

∫

(❱ ·∇p+ ·∇p0 + γp∇ ·❱+ (γ − 1)∇ · q)dt.

Pr s ♦tr rçã♦ s♣rsã♦ é ♥ssár♦ r ♦♠♣♦♥♥t ♣♦♦ ♦rç

♦ s

Fθ = ρ0(θ : ∇❱+ θ❱ : ∇) + ρθ❱ : ∇❱.

s ①♣rssõs érs ♣r s r♥③s ♥s ♣♦r sã♦ ♦ts ♣rtr ♦

qír♦ q é srt♦ ♣♦r

ρ = ρ0(1 + 2ε∆ρ cos θ),

p = p0(1 + 2ε∆p cos θ),

❱ = VP θ + VT φ,

VP ≈ ε

qMP cs, VT = (MT +∆V ε cos θ)cs, ∆V = MT − 2(1 + ∆ρ)MP .

♠ét♦♦ trt♦ q t③♠♦s ♣r rs♦r s qs ♥s rás v‖ ρ p

♣r ♦t♥çã♦ rçã♦ s♣rsã♦ é st♦ ♣♦r s trtr ♠ ♠♦♦ ♥r ♥♦

q ♦ ♣r♥í♣♦ s♣r♣♦sçã♦ s ♣ ♠ét♦♦ ♦♥sst ♠ ♦♠♣♦r s q♥ts

♣rtrs ♥ ♦r♠

X = X() + X() + X(P),

♦♥ P ♥♠ s ♦♥trçõs ♣r s q♥ts ♣rtrs ♦ ♥â♠

s♠ r♦tçã♦ ♦♠ r♦tçã♦ ①s♠♥t t♦r♦ ♦♠ r♦tçã♦ ♣♦♦ t♦r♦ rs♣t

♠♥t ♣r♦♠ ♥ é ♥tã♦ ♦ ♠ três ♣rts Pr♠r♠♥t s♠ ♥r

r♦tçã♦ qír♦ ♥♦♥trs s♦çã♦ ♠s s♠♣s P♦str♦r♠♥t ♣rtr st s♦

çã♦ tr♠♥s s♦çã♦ ♣r♦♥♥t r♦tçã♦ t♦r♦ ♥♠♥t ♦ ♥r r♦tçã♦

♣♦♦ ♥♦♥trs s♦çã♦ ♦♠♣t

á♦ F‖ R P

r♠♦s ♦♥çã♦ rs ♥rs

á♦ ♦s tr♠♦s ♦♥t♦s t♦s ♣rtr s qs rst

♥s s♥ts rçõs

❱P ·∇ = MPk‖cs∂

∂θ, E ·∇ =

ωE

ε

∂

∂θ,

❱T ·∇ = qMTk‖cs∂

∂φ, v‖ ·∇ =

v‖R0

∂

∂φ.

é♠ ♦s tr♠♦s ♦♥t♦s t♠é♠ é ♥ssár♦ ♣r ♦ rst♥t ♦s á♦s st sçã♦

♦tr s rs s ♦s ♦♠ rçã♦ ♦s â♥♦s ♣♦♦ t♦r♦ q ♦r♦

♦♠ sã♦

∂❱P

∂θ= −ε

qMP csr,

∂❱T

∂θ= −ε∆V cs sin θφ,

∂❱P

∂φ≈ 0,

∂❱T

∂φ≈ MT cs(− cos θr + sin θθ)

∂E∂θ

= −ωER0r,∂(v‖)

∂θ=

∂v‖∂θ

φ

∂E∂φ

= −ωER0 sin θφ,∂(v‖)

∂φ= v‖(− cos θr + sin θθ).

á♦ F‖ Fθ

♦♠ ssttçã♦ ❱ = 0 ❱ = ❱T ❱ = ❱P ♠ ♦té♠s

F()‖ = 0,

F()‖ = ρ0(❱T : ∇E + E : ∇❱T ) = −ρ0ωEcs(MT +∆V ) sin θ,

F(P)‖ ≈ ρ0❱P : ∇(v‖) = MPρ0k‖cs

∂v‖∂θ

,

♦r♠ s♠r ♦♠ rçã♦ Fθ s q

F()θ = 0, F

()θ = ρ0[θ❱T : ∇(v‖) + v‖θ : ∇❱T ] + (θ❱T : ∇❱T )ρ =

qMTk‖cs(2ρ0v‖ +MT csρ) sin θ, F(P)θ = 0.

á♦ R

♦r♦ ♦♠ ♣♦♠♦s s♣r③r ♦ tr♠♦

∇ ·❱ = ·∇(

κ

ρ

)

≈ MPk‖csρ0∂ρ−1

∂θ≈ −2εMP∆ρk‖cs sin θ = O(εMP∆ρε),

q ♥ã♦ ♦♥tr ♠ ♣r♠r ♦r♠ ♣r ♦ á♦ R ♦ q é rst ♠

R() = 0,

R() =

∫

dtE ·∇ρ = −2i∆ρωE

ωρ0 sin θ,

R(P) =

∫

dt❱P ·∇ρ = iMP

k‖csω

∂ρ

∂θ.

á♦ ∇ · q

trés ①♣rssã♦ ♣r ♦ ①♦ ♦r t♦t

qΣ =γ

γ − 1

pΣeB2

(×∇TΣ), pΣ = p+ p, TΣ = T + T ,

♦té♠s ♦ ①♦ ♦r ♣rtr♦ ♠ ♣r♠r ♦r♠

q =p

pq+

γ

γ − 1

p

eB2(×∇T ),

♦♥sq♥t♠♥t

∇ · q = ∇ ·(

p

pq

)

+γ

γ − 1

[

(×∇T ) ·∇(

p

eB2

)

+p

eB2(∇×) ·∇T

]

.

♦♠♦ ∇ · q = O(εMtp0k‖cs) é ♠ tr♠♦ s♥ ♦r♠ s q

∇ ·(

p

pq

)

≈ q ·∇(

p

p

)

≈ γ

γ − 1

R20

eF0 ·∇p ≈ γ

γ − 1Mtk‖cs

∂p

∂θ.

♦r♠ s♠r ♦s ♦tr♦s tr♠♦s ♣♦♠ sr s♥♦♦s rst♥♦ ♥s rçõs

(×∇T ) ·∇(

p

eB2

)

≈ R20

eF0

dp

dΨ ·∇T = Mtk‖csp0

∂

∂θ

(

T

T0

)

∼ ∇ ·(

p

pq

)

,

p

eB2(∇×) ·∇T ≈ p0R

20

eF 20

dF0

dΨ ·∇T = RFMtk‖cs

∂

∂θ

(

T

T0

)

= O(B0)∇ ·(

p

pq

)

.

s♦ rçã♦ ♣r♦①♠ ♥tr ♣rssã♦ ♥s t♠♣rtr ♣rtrs

p ≈ p0T

T0+ p0

ρ

ρ0,

♣r♠t ♦♠♥r s ①♣rssõs ♠ ♦tr ①♣rssã♦ ♥ ♣r ∇ · q

∇ · q =Mtk‖csγ − 1

(

2γ∂p

∂θ− c2s

∂ρ

∂θ

)

.

á♦ P

♦ ss♠r♠♦s q limMP→0Mt = 0 ♦té♠s ♥♦♠♥t á♦ R q

P () =

∫

dtE ·∇p = −2i∆p

γ

ωE

ωρ0c

2s sin θ

P (P) =

∫

dt[❱P ·∇p+ (γ − 1)∇ · q] = ik‖csω

[

(MP + 2γMt)∂p

∂θ−Mtc

2s

∂ρ

∂θ

]

.

♦çã♦ s♠ r♦tçã♦ ♣r♠r trçã♦

♦♠ s ssttçõs F‖ = R = P = 0 = () = E + v()‖ ♠ ♦ s♦

♥♦r♠③çã♦

Ω =ω

k‖cs, ΩE =

ωE

k‖cs,

♦♥♦r♠ ①♣t♦ ♥ sçã♦ ♦té♠s

∇ · ()k‖cs

= −2ΩE sin θ +∂

∂θ

v()‖cs

,

v()‖ = v

()‖c cos θ, v

()‖c =

2ΩE

Ω2 − 1cs,

ρ()s

ρ0=

p()s

ρ0c2s= iΩ

v()‖ccs

, ρ()c = p()c = 0.

♦çã♦ ♦♠ r♦tçã♦ t♦r♦ s♥ trçã♦

s♦♥sr♥♦ ♠ s♦çã♦ ♦t ♠ ♥ sçã♦ ♥tr♦r ♦t♠♦s ♦ sst♠

−iΩv()‖cs

+∂

∂θ

p()

ρ0c2s+

F()‖

ρ0k‖c2s= 0,

−iΩ

(

ρ()

ρ0+

R()

ρ0

)

+∂

∂θ

v()‖cs

= 0,

−iΩ

(

p()

ρ0c2s+

P ()

ρ0c2s

)

+∂

∂θ

v()‖cs

= 0,

q ♣rs♥t ♦♠♦ s♦çã♦

v()‖ =

i

2

Ω

ΩE

(

∂

∂θ

P ()

ρ0c2s−

F()‖

ρ0k‖c2s

)

v()‖c = v

()‖s sin θ + v

()‖c cos θ,

ρ() = −[

1

2ΩE

(

P ()

c2s+

∂

∂θ

F()‖

k‖c2s

) v()‖ccs

+ R()

]

= ρ()s sin θ + ρ()c cos θ,

p() =1

2ΩE

(

Ω2P () +∂

∂θ

F()‖k‖

) v()‖ccs

= p()s sin θ + p()c cos θ.

♥t ♦ s♦ ♦s rst♦s ♥tr♦rs ♠ s ♥♠♥t q

v()‖s = iΩ

(MT +∆V )

2v()‖c , v

()‖c =

∆p

γv()‖c ,

ρ()s =i

Ω

[

∆p

γ+ (Ω2 − 1)∆ρ

] v()‖ccs

ρ0, ρ()c =(MT +∆V )

2

v()‖ccs

ρ0,

p()s = iΩ∆p

γρ0csv

()‖c , p()c = ρ()c c2s.

♦tçã♦ ♣♦♦ t♦r♦ trr trçã♦

Pr♠r♠♥t sstts ♠ ♣r ♦tr

F(P)‖

ρ0k‖c2s= MP

∂

∂θ

v(P)‖cs

+MP

[

−(

1 +∆p

γ

)

sin θ +iΩ

2(MT +∆V ) cos θ

] v()‖ccs

.

R(P)

ρ0=

iMP

Ω

∂

∂θ

ρ(P)

ρ0− MP

Ω2

i

2Ω(MT +∆V ) sin θ +

[

Ω2 +∆p

γ+ (Ω2 − 1)∆ρ

]

cos θ

v()‖ccs

,

P (P)

ρ0c2s= i

MP

Ω

∂

∂θ

[(

1 + 2γMt

MP

)

p(P)

ρ0c2s− Mt

MP

ρ(P)

ρ0

]

+MP

Ω

i

[

1 + (2γ − 1)Mt

MP

]

sin θ+

[

Ω∆p

γ+

(

2γΩ− 1

Ω

)

Mt

MP+ 2Ω∆p

Mt

MP+

(

1

Ω− Ω

)

∆ρMt

MP

]

cos θ

v()‖ccs

,

q ♣♦r s ③ sã♦ ♥sr♦s ♥♦ sst♠

v(P)‖ =

iΩ

Ω2 − 1

(

∂

∂θ

P (P)

ρ0cs−

F(P)‖

ρ0k‖cs

)

,

ρ(P) = −R(P) − 1

Ω2 − 1

(

P (P)

c2s+

∂

∂θ

F(P)‖

k‖c2s

)

,

p(P) = − 1

Ω2 − 1

(

Ω2P (P) +∂

∂θ

F(P)‖k‖

)

,

s qçõs sã♦ s♠rs às qs

Pr rs♦r st sst♠ ♦ ♦ ♦♣♠♥t♦ s qçõs é ♦♥♥♥t t③r

♦r♠çã♦ ①♣♦♥♥ ♦ ♥és tr♦♥♦♠étr ♦r♦ ♦♠ s rçõs

X = Xs sin θ + Xc cos θ = X+1iθ + X−1

−iθ,

Xs = i(X+1 − X−1), Xc = X+1 + X−1, X±1 =1

2(Xc ∓ iXs).

♦r♠ ①♣♦♥♥ é ♦t ♣♦r s r③õs Pr♠r♠♥t ♦ à ♣rt ♥♦ á♦

r ♦ s ∂/∂θ → im m = ±1 s♥ é ♦ ♣♦ss ♥ás

♥ ♦s r♠ô♥♦s m = 1,−1 q ♦r♠ ①♣♦♥♥ ♣r♦♣♦r♦♥

st ♦♥çõs ♦r♦ ♦♠ t③♥♦ s q

F(P)‖±1

ρ0k‖cs= ±iMP

v(P)‖±1

cs+

i

2MP

[

±(MT +∆V )

2Ω + 1− ∆p

γ

] v()‖ccs

,

R(P)±1

ρ0= ∓MP

Ω

ρ(P)±ρ0

− MP

2

[

1 + ∆ρ ±1

2

(MT +∆V )

Ω+

(∆p/γ −∆ρ)

Ω2

] v()‖ccs

,

P(P)±1

ρ0c2s= ∓MP

Ω

[(

1 + 2γMt

MP

)

p(P)±1

ρ0c2s− Mt

MP

ρ(P)±1

ρ0

]

−

MP

2

1 +∆p

γ+ (2γ − 1)

Mt

MP± 1

2

[

1 + (2γ − 1)Mt

MP

]

(MT +∆V )

Ω

v()‖ccs

,

♦♠ ssttçã♦ ♠ ♦t♠♦s ♠ sst♠

6× 6 q ♣♦ sr r♣rs♥t♦ ♥ ♦r♠ ♠tr

C(±)11 C12 C13

C21 C(±)22 C(±)

23

C31 C(±)32 C(±)

33

v(P)‖±1/cs

ρ(P)±1 /ρ0

p(P)±1 /ρ0c

2s

=1

2

MP v()‖c

Ω2 − 1

K(±)v

K(±)ρ

K(±)p

,

♦♥

C(±)11 = 1± C31Ω

Ω2−1, C12 = Mt

Ω2−1, C13 = C21 + 2γC12,

C21 = −C31Ω2−1

, C(±)22 = 1±

(

C12+C31Ω

)

, C(±)23 = ∓C13

Ω ,

C31 = −MP , C(±)32 = ±C12Ω, C(±)

33 = 1∓ C13Ω,

K(±)v = ± (MT+∆V )

2 Ω2 +

[

1− ∆p

γ ±(

1 +∆p

γ + (2γ − 1)MtMP

)]

Ω+

(MT+∆V )2

(

1 + (2γ − 1)MtMP

)

,

K(±)ρ = (1 +∆ρ)Ω

2 + (1± 1) (MT+∆V )2 Ω±

(

1− ∆p

γ

)

+ 2

(

∆p

γ −∆ρ

)

+

(2γ − 1)MtMP

± (2γ − 1) (MT+∆V )2

MtMP

1Ω +

(

∆ρ − ∆p

γ

)

1Ω2 ,

K(±)p =

(

1 +∆p

γ + (2γ − 1)MtMP

)

Ω2 + (MT+∆V )2

(

1± 1± (2γ − 1)MtMP

)

Ω

±(

1− ∆p

γ

)

.

s♦çã♦ st sst♠ ♣♦ sr r♣rs♥t ♣s s♥ts rçõs

v(P)‖±1

cs=

N v±1

(P)

D±1(P)

,ρ(P)±1

ρ0=

N ρ±1

(P)

D±1(P)

,p(P)±1

ρ0c2s=

N p±1

(P)

D±1(P)

,

♦♥

D±1(P) ≈ (MP ∓ Ω)(Ω + 1∓MP )(Ω− 1∓MP ) + [2γ(Ω∓MP )

2 − 1]Mt,

N v±1

(P) = MP∑3

k=0 C(v)k,±1Ω

k, C(v)0,±1 =

MP

2

(

−MP + MT

2 + MP∆V

2

)

,

C(v)1,±1 = ∓1

2

(1 +M2P )

(MT+∆V )2 − 3MP

(

1 +∆p

γ

)

+

[

(2γ − 1) (MT+∆V )2MP

− 4γ

]

Mt

,

C(v)2,±1 = −

(

1 +∆p

γ

)

+ (MT+∆V )2 MP −

(

γ − 12

)

MtMP

, C(v)3,±1 = ∓ (MT+∆V )

4 ,

N ρ±1

(P)= MP

Ω

∑4k=0 C

(ρ)k,±1Ω

k, C(ρ)0,±1 = ∓1

2

(

∆ρ − ∆p

γ

)

,

C(ρ)1,±1 =

MP

2

[

1− 2∆ρ + 3∆p

γ −MP(MT+∆V )

2

]

+

[

1− (2γ − 1) (MT+∆V )4MP

]

Mt,

C(ρ)2,±1 = ∓1

2

[

1− 2∆ρ + 3∆p

γ − 32MP (MT +∆V ) +M2

P + (2γ − 1)MtMP

]

,

C(ρ)3,±1 = −MT+∆V

2 + (1 +∆ρ)MP + γMt, C(ρ)4,±1 = ∓1

2(1 + ∆ρ),

N p±1

(P)= MP

∑3k=0 C

(ρ)k,±1Ω

k, C(p)0,±1 =

MP

2

[

1 +∆p

γ − (MT+∆V )2 MP

]

+ Mt2 ,

C(p)1,±1 = ∓1

2

(

1 +∆p

γ − 32(MT +∆V )MP +M2

P

)

,

C(p)2,±1 =

(

1 +∆p

γ

)

MP − (MT+∆V )2 +

[

2γ − 12 − (2γ − 1) (MT+∆V )

4MP

]

Mt,

C(p)3,±1 = ∓1

2

(

1 +∆p

γ + (2γ − 1)MtMP

)

.

çã♦ s♣rsã♦

Pr ♦t♥çã♦ rçã♦ s♣rsã♦ é ♥ssár♦ ♥♦r qçã♦ ♦ ♠♦♠♥t♦

ρ∂

∂t+∇p− ×+ = 0, = ρ(❱ ·∇+ ·∇❱) + ρ❱ ·∇❱,

q q♥♦ ♠t♣ t♦r♠♥t ♣♦r ♣r♠t ♦t♥çã♦ 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 ♦r♠ s♠r ♣♦♠♦s t♦♠r ♠é

qçã♦ ♠♥♦♥ ♦♠ rçã♦ ♠ ♦♠ rtrár♦ ♣s♠ ♦ s

D =

∫

V dV∇ · ∫

V dV= 0, dV = (R0 + r cos θ)rdrdθdφ,

á♦ ♦ ♥♠r♦r D ♠ é t♦ ♣rtr ♦ s♦ ♦ t♦r♠ rê♥

ss ♦r♠ q

D =

∮

S · d∫

V dV= 0, d = (R0 + r cos θ)rdθdφr.

D() ≈ K(r)

2iπ

[

−iΩE

q2Ω

∫ 2π

0dθ + 2

∫ 2π

0dθ

∂

∂θ

(

p()

ρ0c2s

)

cos θ

]

,

D() ≈ K(r)

2iπ

[

2

∫ 2π

0dθ

∂

∂θ

(

p()

ρ0c2s

)

cos θ +1

q

∫ 2π

0dθ

F()θ

ρ0k‖c2s

]

,

D(P) ≈ K(r)

2iπ

[

2

∫ 2π

0dθ

∂

∂θ

(

p(P)

ρ0c2s

)

cos θ +1

q

∫ 2π

0dθ

F(P)θ

ρ0k‖c2s

]

,

♦♥ K(r) ≈ 2iγp0(r)/rF0(r) ♦♠ ssttçã♦ ♦s rst♦s ♥tr♦rs ♦t♦s ♥st sçã♦

♥s ①♣rssõs ♦t♠♦s ♦s rst♦s ♠♦str♦s ♠

♣ê♥

á♦ ♥trs ♥çã♦

strçã♦

çõs ♥♦♥♦ strçã♦ ♠①♥

♠ ♣r♦♠s ♥♦♥♦ ♥çã♦ strçã♦ ①♥ q é ♥ ♣♦r

FMα =n0

π3/2v3Tα

exp

(

−v2⊥ + v2‖v2Tα

)

,

é ♦♠♠ ♣rr ♥trs ♦ t♣♦

I(a, b) =

∫ 2π

0dγ

∫ ∞

−∞dv‖

∫ ∞

0dv⊥v⊥

(

v⊥vTα

)a( v‖vTα

)b

FMα , a, b ≥ 0

q ♣♦♠ sr s♠♣s ♣rtr s ♠♥çs rá x = v⊥/vTα y = v‖/vTα

I(a, b) =2n0√πI⊥(a)I‖(b).

s s♦çõs ♣r I⊥(a) I‖(b) ♦r♠ ♥ér sã♦ s ♣♦r

I⊥(n) =∫ ∞

0xn+1−x2

dx =

2−1(n/2)! ♣r n ♣r

2−(n+1)/2n!!√π ♣r n ♠♣r

,

I‖(n) =∫ ∞

−∞xn−αx2

dx =

√π ♣r n = 0

0 ♣r n ♠♣r

2−n/2(n− 1)!!√π ♣r n ≥ 2 n ♣r

,

♣rtr ♥r ♦ tér♠ ♣rtís ♦ t♣♦ α

Eα = eαΦ+1

2mαv

2, v2Tα=

2Tα

mα

♦♥ v2 = v2⊥ + v2‖ ♣♦♠♦s ①♣rssr ♥çã♦ ①♥ ♦♠♦

FMα =n0m

3/2α

(2πTα)3/2exp

(

−Eα − eαΦ

Tα

)

.

♣♦rt♥t♦ q

∂FMs

∂Eα= −FMs

Tα,

∇FMα =

[

∇ lnn0 +

(Eα − eΦ

Tα− 3

2

)

∇ lnTα +e∇Φ

Tα

]

FMα ,

Pr ♦ s♦ ♣rtr ♠ q Φ = 0 n0 ≈ n0(r) Tα ≈ Tα(r) ♦ q ♦♥sr♠♦s ♥st ts

q ♣♦ sr s♠♣ rst♥♦ ♠

∇FMα ≈ rn0

LN

[

1 + ηα

(

v2⊥v2Tα

+v2‖v2Tα

− 3

2

)]

FMα ,

♦♥

L−1N =

∂lnn0

∂r, L−1

Tα=

∂lnTα

∂r ηα =

LN

LT.

á♦ s ♥trs ♥ ♣r♦①♠çã♦ ♦

♣rtr ♥tr ♥♦ s♣ç♦ ♦s r♥③ rtrár Xα = Xα(r, v⊥, v‖, γ)

q é ♥♦t ♣♦r

〈Xα〉 =1

n0

∫

vd3vFMαXα =

1

π

∫ 2π

0dγ

∫ ∞

0dv⊥v⊥

−v2⊥/v2Tα

v2Tα

1√π

∫ ∞

−∞dv‖

−v2‖/v2

Tα

vTα

Xα,

♥st sçã♦ ♠♦str♠♦s ♦s rst♦s s s♥ts q♥ts

I(α)a =⟨

J20αΩ

aα

⟩

, I(α)ab =

⟨

J20αΩ

aαΩ

btrα

⟩

, I(α)abc =

⟨

J20αΩ

aαΩ

btrαΩ

c∗α⟩

,

♦♥

J0α = J0(rv⊥/ωcα) ≈ 1− 1

2

v2⊥v2Tα

2rρ2α +

3

32

v4⊥v4Tα

4rρ4α +O(6rρ

6α),

Ωα =ωαω

(

1

2

v2⊥v2Tα

+v2‖v2Tα

)

, Ωtrα =ωtrαω

v‖vTα

, Ω∗α =ω∗αω

[

1 + ηα

(

1

2

v2⊥v2Tα

+v2‖v2Tα

)]

.

Pr ♦ á♦ s ♥trs ♠ ♦sr♥♦ q Ωα Ωtrα Ω∗α sã♦ ♥♣♥♥ts γ

é ♦♥♥♥t ♣r♠r♠♥t ♥trr ♥st rá ♣♦str♦r♠♥t ♠ v⊥ ♣♦r ♠ ♠ v‖

♣rtr s ♠♥çs rás ♥tr♦③s ♥tr♦r♠♥t v⊥/vTα = x

v‖/vTα = y ♣r rρi ≪ 1 ♠ tr♠♦s s rqê♥s ♥♦r♠③s Ω = ωR0/vTi Ω∗α =

ω∗αR0/vTi sã♦ ♦t♦s ♦s s♥ts rst♦s

I(α)0 =

⟨

J20α

⟩

= 1− 12

2rρ

2α + 3

164rρ

4α,

I(α)1 =

⟨

J20αΩα

⟩

≈(

1− 34

2rρ

2α

)

rραΩ ,

I(α)2 =

⟨

J20αΩ

2α

⟩

≈(

74 − 13

8 2rρ

2α

)

2rρ2α

Ω2 ,

I(α)02 =

⟨

J20αΩ

2trα

⟩

=

(

12 − 1

42rρ

2α

)

1q2Ω2 ,

I(α)12 =

⟨

J20αΩαΩ

2trα

⟩

=

(

1− 58

2rρ

2α

)

rραq2Ω3 ,

I(α)22 =

⟨

J20αΩ

2αΩ

2trα

⟩

=

(

238 − 23

162rρ

2α

)

2rρ2α

q2Ω4 ,

I(α)001 =

⟨

J20αΩ∗α

⟩

=

[

1− (1+ηα)2 2rρ

2α

]

Ω∗αΩ ,

I(α)101 =

⟨

J20αΩαΩ∗α

⟩

=

[

1 + ηα − 34(1 + 2ηα)2rρ

2α

]

Ω∗αrραΩ2 ,

I(α)121 =

⟨

J20αΩαΩ

2trαΩ∗α

⟩

=

[

1 + 2ηα − 58(1 + 3ηα)2rρ

2α

]

Ω∗αrραq2Ω4 ,

I(α)021 =

⟨

J20α

Ω2trαΩ∗α

ω3

⟩

=

[

12(1 + ηα)− 1

4(1 + 2ηα)2rρ2α

]

Ω∗αq2Ω3 ,

I(α)abc =

⟨

J20αΩ

aαΩ

btrαΩ

c∗α⟩

= 0, s b ♦r ♠♣r.

♥çã♦ s♣rsã♦ ♣s♠

Pr ♥çã♦ s♣rsã♦ ♣s♠

Z(ζ) =1√π

∫ ∞

−∞dx

−x2

x− ζ, Im(ζ) > 0,

sã♦ ststs s s♥ts ♣r♦♣rs

Z(−ζ) = 2i√π−ζ2 − Z(ζ),

dZ

dζ= −2[1 + ζZ(ζ)].

|Im(ζ)| ≪ 1 ♠♥t ♦ ♣r♦♦♥♠♥t♦ ♥ít♦ ♣r ♥r ♦ s♦ Im < 0 s s♥ts

♣r♦①♠çõs ss♥tóts ♣♦♠ sr ts

Z(ζ) ≈ i√π−ζ2 − 2ζ +

4

3ζ3 +O(ζ5),

♣r |ζ| ≪ 1

Z(ζ) ≈ iσ√π−ζ2 −

[

1

ζ+

1

2ζ3+

3

4ζ5+

15

8ζ7+O(ζ−9)

]

, σ =

0 s Im(ζ) > 0

1 s Im(ζ) = 0

2 s Im(ζ) < 0

,

♣r |ζ| ≫ 1

Pr s♠♣r ♥♦tçã♦ ♥♠♦s Z(k) = dkZ/dζk Pr ♦s á♦s q s s♠ é

♦♥♥♥t ♣rtr r s s♥ts rs

Z(1) = −2− 2ζZ,

Z(2) = 4ζ + (−2 + 4ζ2)Z,

Z(3) = 8− 8ζ2 + (12ζ − 8ζ3)Z,

Z(4) = −40ζ + 16ζ3 + (12− 48ζ2 + 16ζ4)Z,

Z(5) = −64 + 144ζ2 − 32ζ4 + (−120ζ + 160ζ3 − 32ζ5)Z,

♦♠ rçã♦ à s♥t ♥çã♦ r♦♥ Z(ζ)

Zn(ζ) =1√π

∫ ∞

−∞dζ

xn−x2

x− ζ, n ≥ 0,

t③♥♦ ♦t♠♦s ♦s s♥ts rst♦s

Z0(ζ) = Z(ζ),

Z1(ζ) = −12Z

(1) = 1 + ζZ(ζ),

Z2(ζ) =14 [2Z + Z(2)] = ζ + ζ2Z(ζ),

Z3(ζ) = −18 [6Z

(1) + Z(3)] = 12 + ζ2 + ζ3Z(ζ),

Z4(ζ) =116 [12Z + 12Z(2) + Z(4)] = 1

2ζ + ζ3 + ζ4Z(ζ),

Z5(ζ) = − 132 [60Z

(1)(ζ) + 20Z(3)(ζ) + Z(5)(ζ)] = 34 + 1

2ζ2 + ζ4 + ζ5Z(ζ),

♥♠♥t é ♦♥♥♥t ♥r s♥t ♥çã♦ r♥ç

Dn(ζα) = Zn(−ζα)− Zn(ζα), ζα =ω

ωtrα,

q ♣ t③çã♦ rst ♥♦s s♥ts ♦rs

D0(ζα) = 2[i√π−ζ2α − Z(ζα)],

D1(ζα) = −2ζαi√π−ζ2α ,

D2(ζα) = −2ζα + ζ2αD0(ζα),

D3(ζα) = −2ζ3αi√π−ζ2α ,

D4(ζα) = −ζα − 2ζ3α + ζ4αD0(ζα),

D5(ζα) = −2ζ5αi√π−ζ2α .

á♦ s ♥trs ♦♠ t♦s ♥ét♦s

♥♠♥t ♣rs♥t♠♦s s ♥trs ♥éts

L(α)a =

⟨

J20αΩ

aα

1− Ω2trα

⟩

, L(α)ab =

⟨

J20αΩ

aαΩ

btrα

1− Ω2trα

⟩

, L(α)abc =

⟨

J20αΩ

aαΩ

btrαΩ

c∗α

1− Ω2trα

⟩

,

♦♥ ♣r s á♦ é ♦♥♥♥t ♦srr q

1

1− Ω2trα/ω

2=

ζα2

(

1

v‖/vTα + ζα− 1

v‖/vTα − ζα

)

, ζα =ω

ωtrα,

♣♦s st ♦srçã♦ ♥♦s ♣r♠t ♦tr s♥t rçã♦

1√π

∫ ∞

−∞

dv‖vTα

(v‖/vTα)n−v2

‖/v2

Tα

1− Ω2trα

=ζα2Dn(ζα).

♣♦rt♥t♦ q

L(α)0 = ζα

2 D0(ζα)

(

1− 12

2rρ

2α

)

, L(α)1 = ωα

ωζα2

[

12D0(ζα) +D2(ζα)

]

,

L(α)2 =

ω2

α

ω2

ζα2

[

12D0(ζα) +D2(ζα) +D4(ζα)

]

,

L(α)01 = 1

2D1(ζα),

L(α)02 = 1

2ζαD2(ζα),

L(α)11 = ωα

ω12

[

12D1(ζα) +D3(ζα)

]

,

L(α)001 =

ω∗αω

ζα2

[(

1− ηα2

)

D0(ζα) + ηαD2(ζα)

]

,

L(α)101 =

ωαω∗α

ω2

ζα2

[(

12 + ηα

4

)

D0(ζα) +D2(ζα) + ηαD4(ζα)

]

,

L(α)011 =

ω∗αω

12

[(

1− ηα2

)

D1(ζα) + ηαD3(ζα)

]

,

L(α)111 =

ωαω∗α

ω2

12

[(

12 + ηα

4

)

D1(ζα) +D3(ζα) + ηαD5(ζα)

]

,

♥♠♥t ♦♠ ssttçã♦ ♦s ♦rs ♠♦str♦s ♠ rst

L(α)0 = −ζα[Z(ζα)− i

√π−ζ2α ]

(

1− 2rρ2α

2

)

,

L(α)1 = −

ζ2α +

(

12ζα + ζ3α

)

[

Z(ζα)− i√π−ζ2α

]

rραΩ ,

L(α)2 = −

32ζ

2α + ζ4α +

(

12ζα + ζ3α + ζ5α

)

[

Z(ζα)− i√π−ζ2α

]

2rρ2α

Ω2 , L(α)01 = −ζαi

√π−ζ2α ,

L(α)02 = −[1 + ζαZ(ζα)]

(

1− 2rρ2α

2

)

, L(α)11 = −

(

12ζα + ζ3α

)

rραΩ

√π−ζ2α ,

L(α)001 = −

ηαζ2α +

[(

1− 12ηα

)

ζα + ηαζ3α

]

[

Z(ζα)− i√π−ζ2α

]

Ω∗αΩ ,

L(α)101 = −

(

1 + 12ηα

)

ζ2α + ηαζ4α +

[(

12 + 1

4ηα

)

ζα + ζ3α + ηαζ5α

]

[

Z(ζα)− i√π−ζ2α

]

Ω∗αrραΩ2 ,

L(α)011 = −

[(

12 − 1

2ηα

)

ζα + ηαζ3α

]

Ω∗αΩ i

√π−ζ2α ,

L(α)111 = −

[(

12 + 1

4ηα

)

ζα + ζ3α + ηαζ5α

]

Ω∗αrραΩ2 i

√π−ζ2α .

t♥çã♦ ♦ ♠t ♦ ♣rtr s ♥trs

♦♠ t♦s ♥ét♦s

♦ ♠t |ζi| → ∞ ♦ tr♠♦ −ζ2i ♣♦ sr s♣r③♦ ♦ t③r ♦ ♠t ss♠♣tót♦

Z(ζi) ♠♦str♦ ♠ ♦t♠♦s ♣rtr ♦ ♠t ♦

L(i)0 ≈ I

(i)0 + I

(i)02 ≈

(

1− 12

2rρ

2i

)(

1 + 12q2Ω2

)

, L(i)1 = I

(i)1 + I

(i)12 =

(

1 + 1q2Ω2

)

rρiΩ ,

L(i)2 = I

(i)2 + I

(i)22 =

(

74 + 23

81

q2Ω2

)

2rρ2

i

Ω2 , L(i)11 = I

(i)11 = 0,

L(i)101 = I

(i)101 + I

(i)121 =

[

1 + ηi +1+2ηiq2Ω2

]

Ω∗irρiΩ2 , L(i)

111 = I(i)111 = 0, L(i)

011 = I(i)011 = 0,

L(i)001 = I

(i)001 + I

(i)021 =

[

1 + 1+ηi2q2Ω2 − 1

2

(

1 + ηi +1+2ηi2q2Ω2

)

2rρ2i

]

Ω∗iΩ ,

♣ê♥

Prt♣çã♦ ♠ ♥t♦s ♥tí♦s

rs♦s ♥tr♥♦♥s

• t r♦s ♥s ♠♠r ♦♦ ♦♥ Ps♠ P②ss ♥ s♦♥ ♥

❬❪ st rs♦ rt rçã♦ q ♦♦rr st♠r♦ ♠

♦♥♥ ♥ ♠♥ ♦♥sst ♥ú♠r♦s s♠♥ár♦s s♦r ♦s ♣r♥♣s tó♣♦s

rs♣t♦ ís Ps♠ sã♦ r ♥t♦r♦♥â♠ ♦r ♥ét

q♠♥t♦ r♥s♣♦rt t ♦♠ ê♥s ♠ ♣çõs ♣r ♦ s s♠♥ár♦s

q ♦r♠ ♠♥str♦s ♣♦r ♣sqs♦rs ♣r♦ss♦rs s♣sts ♠ ár ♦r♠

♣♦s ♥♦ ♦r♠t♦ rt♦ r ♠ ❬❪

Pr♦çã♦ ♦rá

• r ts ♦♥ ♦s ♦st ♠♦s ♦ t♦ ♠ ♣♦ ♥ rsã♦

♥ ♠ ❬❪ st tr♦ ♦ ♠ r♥ ♣rt rst♦ ♦♦rçã♦ ♦♠

Pr♦ r ♠♦②♦ ♣rt♥♥t ❯♥rs st♥ é s ♣r

♣r♦♣♦sts tr♦s tr♦s q ♣rt♥♠♦s r③r

• ♦tt♦♥ t ♦♥ ♦s ♥ ③♦♥ ♦ ♠♦s ♥ t♦♠ ♣s♠s t

s♦tr♠ ♠♥t srs t♠ st tr♦ ♦ ♦ ♦ ♣sqs ♦ ♠s

tr♦ ♣r♥t ♦ ♣rs♥t ♦t♦r♦ ❬❪ ♦ ♥í♦ st ♦t♦r♦ ♦r♠ ♥

♦♥sr♥♦ ①♦ ♦r qír♦ ♦♠ s♣rís ♠♥éts s♦tér♠s st tr

♦ ♦ ♣♦ ♠ ❬❪

♦♥rê♥s ♥♦♥tr♦s ♥tí♦s

• t r♦♣♥ P②s ♦t② ♦♥r♥ ♦♥ Ps♠ P②ss ❬❪ rts

♠ ♦♥rê♥ ♠ ís ♣s♠s q ♦♦rr ♦ ♥ ♥

â♥ ♣rs♥t♠♦s ♥st ♦♥rê♥ ♠ ♣♦str ♥tt♦ ♦♠♦ ♠♥t ts

♥ ♥ ♠♣♥ ♦♥ ♦s ♦st ♠♦s ♠ rt♦ ♥♦ t♠♥♦ ♠á①♠♦ ♣r

♠t♦ ♣á♥s ❬❪ st tr♦ é s ♣r ♦ ♣tít♦ st ts ♣rtr

♣rt♥♠♦s ♥♦ tr♦ s♠tr ♠ rt♦ ♦♠ ♠♦r t♠♥t♦ ér♦ ♠

♠ rst ♥tr♥♦♥ q ♦ t♣♦ tr♦

• ♥♦♥tr♦ rsr♦ ís Ps♠s ❬❪ ♦ ♠ ♥t♦ ♥tí♦ ♥

♦♥ q ♦♦rr ③♠r♦ ♥ ❯♥rs rsí ❯♥ ♠

rsí st ♥t♦ ♣rs♥t♠♦s ♥ ♦r♠ ♣♦str ♠t♦♦♦ ♣r ♠

st♦ s♦r t♦♠♦♦s úst♦s ♦és♦s ♣çã♦ ♠♦r ♠♣t♦ ♠t♦

stá ♥ tr♠♥çã♦ ♦ ♣r r t♠♣rtr í♦♥s ♠ t♦♠s ♣♦r ♠♦

♠ ♥♦♦ t♣♦ ♥óst♦ ♠ s♥♦♠♥t♦ s♣tr♦s♦♣ ♦♠ rts

♠ t♠ q ♣rt♥♠♦s s♥♦r ♠♦r ♠ ♠ tr♦ tr♦ ♥♦ ♥t♥t♦ ♦s

♦♥t♦s ♥s ♠t♦♦♦ ♣r ♦ s♥♦♠♥t♦ st tr♦ ♦r♠ ♣rs♥

t♦s ♥♦ ♣♦str ♥♦ rs♠♦ ♥tt♦s ♦s ♦st ♥♠♦s ♥ t ♣rs♥ ♦

♠♥t ts P ❬❪

rê♥s ♦rás

❬❪ tt♣tr♦r ss♦ ♠ ♦

❬❪ t r♦s ♥s ♠♠r ♦♦ ♦♥ Ps♠ ♥ s♦♥ ♥r② P②ss

s♦♥ ♥

❬❪ ♠♠ ♥ r♦ ♦r② ♦ ♠♥t tr♠♦♥r rt♦r ♥ Ps♠

♣②ss ♥ t ♣r♦♠ ♦ ♦♥tr♦ tr♠♦♥r rt♦♥s t ②

♦♥t♦ ♣s ② Pr♠♦♥ Prss ♦♥♦♥ ❨♦r

❬❪ ❲ss♦♥ ♦♠s r♥♦♥ Prss ①♦r

❬❪ P rr Ps♠ ♣②ss ♥ s♦♥ ♥r② ♠r ❯♥rst②

Prss ❨♦r

❬❪ ♥ ♥ ♥s♣♥s trt ♦ s♦♥ ♣♦r ♥ s t ♣♥t

♣r♥r

❬❪ ❲t ♦r② ♦ t♦♠ ♣s♠ ♦rt♦♥ ♠str♠

❬❪ é♥ ①st♥ ♦ tr♦♠♥t②r♦②♥♠s s tr

❬❪ ♠♦ sss ③♦ rs♥♦ rs♦

P♦t♣♥♦ ♥ ❱ s②♣♥ é♥ t♥ ♥ rr♥t r ♥②ss ♥

♠♥t③ ♣s♠ strtrs r③ P②s

❬❪ ❲♥r ♥ t ♠ ♦ ♠♣r♦ ♦♥♥♠♥t ♥ t ♥ ♥tr

♠t srs ♦ t ❳ t♦♠ rt♦r P②s tt

❬❪ ♠♦♥ t♦ t♦ ♥ ♠ ❩♦♥ ♦s ♥ ♣s♠ r

Ps♠ P②s ♦♥tr♦ s♦♥

❬❪ ♦s♥t ♥ ♥t♦♥ P♦♦ ♦ r♥ ② ♦♥♠♣rtr

r♥t tr♥ ♥ t♦♠s P②s tt

❬❪ t♦ ts ♥ t♦ ①tt♦♥ ♦ ♦s ♦st ♠♦ ♥

t♦r♦ ♣s♠s Ps♠ P②s ♦♥tr♦ s♦♥

❬❪ r♥ trt ❲ ❲ r♥ ♦♥ r♥

♦ ②♦r ♥ ♦♠♣s♦♥ ♦ é♥ ♠♦s ♦r② ♥

①♣r♠♥t P②s s

❬❪ ❲ ❲ r♥ trt ♥ r♥ srt♦♥ ♦ t

♥ é♥ ♥♠♦s ♥ t t♦♠ P②s tt

❬❪ ❲ ❲ r♥ s♦ r♦♣♦ ♥ ♥ ❩♥ ♥

♠s ❲t s t t♥ é♥ ♥♠♦ P②s Ps♠s

❬❪ ❲ ❲ r♥ s ♣②ss ♦ é♥ ♥stts r♥ ② ♥rt ♣r

ts ♥ t♦r♦② ♦♥♥ ♣s♠s P②s Ps♠s

❬❪ ❩♦♥ ♥ ♥ ♥t♦r♦ ♥t t♦r② ♦ ♦rq♥② é♥ ♠♦s

♥ t♦♠s Ps♠ P②s ♦♥tr♦ s♦♥

❬❪ s♦ st r♣♦ r r③♠♥ ♦♥r tr

♥ts♥♥ ♥ ❲ s♣tr♦s♦♣② Ps♠ P②s ♦♥tr♦ s♦♥

❬❪ r♣♦ st ♣r ♦r s♦ s tr

♥ts♥♥ ♥ ❱♦♥ r♠♥♥ s♣tr♦s♦♣② tr♦ tt♥

t♦r♦ é♥ ♥♠♦s ♥ é♥ ss P②s tt

❬❪ ♠♦ ♥t ♦♥ t ♦♥ ♦s ♦st ♥ ♠♦s ♥ t♦♠s

P②s Ps♠s

❬❪ ♦♥② r♦str ♦tt ♥ ts rq♥② s♥ ♥ ♦

③t♦♥ ♦ ♦s ♦st ♠♦s ♥ ❳ ❯♣r Ps♠ P②s ♦♥tr♦

s♦♥

❬❪ s r ♦ ③♦♥ ♦ ①♣r♠♥ts r s♦♥

❬❪ ♣t ♦♥ ♦ssr ❲ r ♥ ♦

trtr ♥ s♥ ♣r♦♣rts ♦ t ♦s ♦st ♠♦ Ps♠ P②s

♦♥tr♦ s♦♥

❬❪ ❱ ♥♦ ❱ ❱rs♦ s rs♥ ❱ ♦③♥

r♣♥ ②s♥♦ ❱ r♥ ❱ Pr♦ ♥ ❱ ♦t♦

❱ ❯♠ts ❯r③ ❱♥ ♦st ♥ ❩♠ ♥stt♦♥ ♦

♦s ♦st ♠♦ ♦st♦♥s ♥ t t♦♠ Ps♠ P②s ♦♥tr♦

s♦♥

❬❪ ❲♥s♦r ♦♥s♦♥ ♥ s♦♥ ♦s ♦st s ♥ ②r♦♠

♥t s②st♠s P②s s

❬❪ ❱ P ♥ ❱ s♦♥s ♥ ♠♦②♦ ♦s ♦st ♠♦s ♥ ③♦♥

♦s ♥ t♦r♦② r♦tt♥ t♦♠ ♣s♠s P②s tt

❬❪ r♥s r♥s♣♦rt ♣r♦sss ♥ ♣s♠ ♥ s ♦ Ps♠ P②ss

t ② ♦♥t♦ ♣s ② ♦♥st♥ts r ❨♦r

❬❪ ❱ ❨s♠♥♦ ♠♦♥ ❱ ♦♦s ♥ ♠♦②♦

Pt r♠ ②♥♠s ♦ t r①t♦♥ ♦ ♣♦♦ r♦tt♦♥ ♥ t♦♠

♣s♠s P②s Ps♠s

❬❪ ③t♥ ♥ ss Ps♠ ♦♥♥♠♥t s♦♥❲s② P

s♥ ♦♠♣♥②

❬❪ tt♥♦rt ♥♠♥ts ♦ Ps♠ P②ss ❨♦r Pr♠♦♥ Prss

❬❪ tt♦ ❲ ♥ ♥ ♥ ♥r③ ②r♦♥ts Ps♠ P②s

❬❪ ♥ ♥ t rt♦♥ ♦ t tr♦♥ ♣s♠ P②s ❯

❬❪ ♦s rt ♥stts st♦rt♥ t ♠♥t srs ♦

♦♠t②♣ t♦r♦ s②st♠s s♦♥

❬❪ ♠♦ é♥ ♦♥t♥♠ ♦r♠t♦♥ ② ♥t ♦s t ♥ r♦t

t♥ t♦♠ ♣s♠s P②s Ps♠s

❬❪ ♠♦②♦ ❳ rt tt♦ ♥ tt♥ t♣ ♣♦r③t♦♥ ♦

♦s rtr ♥ ♠♦s P②ss ttrs

❬❪ ♠♦②♦ ②r♦s♦s ♦rs ♥ ♦s♦♥ss ♣s♠ t t♠♣rtr

r♥ts ♥♥ P②s

❬❪ ♥ ♥tr♦t♦♥ t♦ ♣s♠ ♣②ss ♥ ♦♥tr♦ s♦♥ ♣r♥r

♥

❬❪ ♦♦ ♥ P♦ts Pr♥♣s ♦ ♥t♦②r♦②♥♠s ♠

r

❬❪ s s♣t ♦ ♥♦♠♦s tr♥s♣♦rt ♦rt♦♥ ♠str♠

❬❪ t♦ r♦tçã♦ ♥♦s ①♦s ③♦♥s ♠♦♦s úst♦s ♦és♦s

strs tss ♥sttt ♦ P②ss ❯♥rst② ♦ ã♦ P♦ ❯P r③

❬❪ r♥♦ Prt ♥trt♦♥s ♥ ② ♦♥③ ♣s♠ ♥ s ♦

Ps♠ P②ss t ② ♦♥t♦ ♣s ② ♦♥st♥ts

r ❨♦r

❬❪ ❱ ♥ ♦♦♠ ♦s♦♥s ♥ ② ♦♥③ ♣s♠ ♥ s ♦ Ps♠

P②ss ② ♦♥t♦ ♣s ② ♦♥st♥ts r

❨♦r

❬❪ ♥r ♥ ♠r ♦s♦♥ r♥s♣♦rt ♥ ♥t③ Ps♠s

♠r ♦♥♦r♣s ♦♥ Ps♠ P②ss

❬❪ ♦st♦♥ ♥ tr♦r ♥tr♦t♦♥ t♦ ♣s♠ ♣②ss P

Ps♥ t

❬❪ P rr ♥t♦②r♦②♥♠s P♥♠ Prss ❨♦r

♦♥♦♥

❬❪ r ♥ ♥②r♦♠♥t qr ♥ ♦rr s ♥ Pr♦♥s

♦ t ♥ ❯ ♦♥ ♦♥ t P ❯ss ♦ t♦♠ ♥r② ♥

❬❪ ❱ r♥♦ Ps♠ qr♠ ♥ ♠♥t ♥ s ♦ Ps♠

P②ss ② ♦♥t♦ ♣s ② ♦♥st♥ts r

❨♦r

❬❪ s ♥ Prr♥ ①t s♦t♦♥s ♦ t stt♦♥r② qt♦♥s

♦r r♦tt♥ t♦r♦ ♣s♠ Ps♠ P②s

❬❪ ❱ s♦♥s ♥ t ♦s ♦st ♠♦s ♥ ③♦♥ ♦s ♥ r♦tt♥

rs♣trt♦ t♦♠ ♣s♠s Ps♠ P②s ♦♥tr♦ s♦♥

❬❪ ❱ s♦♥s ♥ P♦③♥②♦ ❨ r♥s♣♦rt rrr s rt♦♥ ♦ t

qr♠ ♦ t♦♠ ♣s♠ Ps♠ P②s ♣

❬❪ rs ♥ ♥stt② ♦ ♣♥ t ♦♥t♥

♠♥t Pr♦ ♦ ♦♥♦♥ r

❬❪ ❱ r♥♦ ♥ t stt② ♦ ②♥r s♦s ♦♥t♦r ♥ ♠♥t

❯r ♥r②

❬❪ s r♥s♣♦rt ♣r♦sss ♥ ♣s♠ ♦rt♦♥ ♠str♠

❬❪ s r♥s♣♦rt ♣r♦sss ♥ ♣s♠ ♦rt♦♥ ♠str♠

❬❪ ♥t♦♥ ♥ ③t♥ ♦r② ♦ ♣s♠ tr♥s♣♦rt ♥ t♦r♦ ♦♥♥

♠♥t s②st♠s ♦ P②s

❬❪ P rs♠♥ ♥ ♠r ♦ss tr♥s♣♦rt ♦ ♠ts♣s t♦r♦

♣s♠ ♥ r♦s ♦s♦♥t② r♠s P②s s

❬❪ ♥ ❩ ♦r② ♦ ♥♦ss s♦♥ ♥ s ♦

Ps♠ P②ss t ② ♦♥t♦ ♣s ② ♦♥st♥ts

r ❨♦r

❬❪ P rs♠♥ ♥ ♠r ♦ss tr♥s♣♦rt ♦ ♠♣rts ♥ t♦♠

♣s♠s s♦♥

❬❪ ♦♠ rtrsts ♦ tr srs ♥ ♠♥t s

t ② tr ♥ ❲r♥ ♣s r ❨♦r

❬❪ ♦r♦③♦ ♥ ♦♦ ♦t♦♥ ♦ r ♣rts ♥ tr♦♠♥t

s t ② ♦♥t♦ ♣s ② ♦♥st♥ts r

❨♦r

❬❪ tt♣♣s♣r⑦r♥②r♦❴s♥♦♣ ss♦ ♠ ♦

❬❪ ③t♥ ♥ ❲r♦ r♠♦r ♦ Ps♠ P②ss

❲st ♦r

❬❪ ♠♦②♦ ②♥ ♥ ❳ rt ♥t t♦r② ♦ tr♦♠♥t

♦s ♦st ♠♦s Ps♠ P②s ♦♥tr♦ s♦♥

❬❪ ♦s ♥ r♣♦ t♥ t♠♣rtrr♥t ♥

♠♦s ♥ t♦♠s ♥t t♦r② Ps♠ P②s ♣♦rts

❬❪ ❩♦♥ ♥ ♥ strtrs ♥ ♥♦♥♥r ①tt♦♥ ♦ ♦s

♦st ♠♦s r♦♣②s tt

❬❪ ❨ ♥ ❩ ♥ ♦r② ♦ ♥ t♦r♦t②♥ sr é♥

♥♠♦ ♥ t♦♠s P②s s

❬❪ ❩♦ tr♦♠♥t ♦s ♦st ♠♦s ♥ t♦♠ ♣s♠s P②s

Ps♠s

❬❪ ③t♥ rs rt♦♥ ♦ rt♥t qt♦♥ Ps♠ P②s

❬❪ r ♥ t ♥t t♦r② ♦ rr ss ♦♠♠♥t♦♥s ♦♥ Pr ♥

♣♣ t♠ts

❬❪ r ♦r♦ ♦♥t♥♠♥t ♦ Ps♠ P②s s

❬❪ P ❲♥ ♥ ♥ ♥t ②r ♠♦♠♥t sr♣t♦♥ ♦ ♣s♠s

♣♠♥♥s♦ ♣♣r♦ P②s s

❬❪ ♦s ♥ ❱ s②♣♥ r♥s♣♦rt qt♦♥s ♦ ♣s♠ ♥ r♥r

♠♥t tr Ps♠ P②s

❬❪ s ③t♥ ♥ ♦rrs♦♥ ♥r③ r ♠♦

t ♥t ♦♥②r♦rs ts P②ss ♦ s

❬❪ ♠♦②♦ ❳ rt ♥ ♦r ♥ t ♣r ♠♦♠♥t♠ ♥

♥ ♦ ♣rssr ♣s♠s t ♥ ♥♦♠♦♥♦s ♠♥t s♦♥

❬❪ ♠♦ ♦ ♥ ♦tt♦♥ t ♦♥ ♦s

♥ ③♦♥ ♦ ♠♦s ♥ t♦♠ ♣s♠s t s♦tr♠ ♠♥t srs

Ps♠ P②s ♦♥tr♦ s♦♥

❬❪ ♠♦②♦ ♠♦ ♥ sr rt ts ♦♥ ♦s

♦st ♠♦s P②s tt

❬❪ ã♦ ❲ qr♦ s s ❱ rt

♠♦ ③♦♥♦ ♦♥s r♠♥♦ ❩ ♠rãs♦

r♦♥♠♦ ❨ ③♥ts♦ ♥rq s♠♥t♦ Prs

P P P P s ♦♥ ♦ ❲ P á

♥ r♦ ❱ ♦♦r♦ ♠♦r ♦rs ♦♥

♣♦rt ♦♥ r♥t rsts ♦t♥ ♥ t t♥ ♠r♥ ❲♦rs♦♣ ♦♥

Ps♠ P②ss P

❬❪ r♦ s♠♥t♦ ❱ s②♣♥ ♥ ã♦ Ps♠ rs

r♦tt♦♥ ♥ t t♦♠ s♦♥

❬❪ r ♦s ♦r r♦ ♦♥s♦♥

P♥s r♣♦ ♥ ♦♥trt♦rs ①♣♥t♦♥ ♦ t ♥

r♣♥ ♠♦ s♦♥

❬❪ rä♠r♥ ♦t♦ ❩♠♠r♠♥♥ ♥ ♠ ♦st

♠♦s ♥ t rt♦♥ t♦ ♥st② tt♦♥s P②s tt

❬❪ ♠♦②♦ ②♥ ♥ ❳ rt tr♦♠♥t ts ♦♥ ♦s

♦st ♥ t♥ é♥ ♥♠♦s s♦♥

❬❪ r♦♠♦♦♣♦♦s ❲t③♥r ♥ ss♦ ♥ ♥♦♥①st♥ ♦ t♦♠

qr t ♣r② ♣♦♦ ♦ P②s Ps♠s

❬❪ r ♥ ♦♥t Ps♠ s♣rs♦♥ ♥t♦♥ ♠

Prss ❨♦r

❬❪ ♠ ♥ ❲t♥ ♦s♦♥ss ♠♣♥ ♦ ♦s ♦st ♠♦s

Ps♠ P②s

❬❪ P ♦♠♠♥t♦♥ t r♦s

❬❪ ❨ ③♥ts♦ s♠♥t♦ r♦ ❩ ♠rãs♦

s ã♦ r♦ ♦♥ ♦ ♠♦

③♦♥♦ ❲ P á ❯sr ♥ ❱ ♥♦ P r②③♥

Prs ♦♥s♦ P s r♠ r♠♥♦ r②♥♥

♦r♥♥ss r♥ ♦♠♠ ♦rs ❱ ♥t♥

P ♥ P rt str♦ ❱♦r♦②♦ ③♥ts ❱ ♦s♥♦

♦ s♠♥t♦ ♦♥ ♥ ♠t③r ♦♥st♥ ♦rrt♦♥s ♥

s♥ ①♣r♠♥ts s♦♥

❬❪ ❲ ♦rt♦♥ rt s ♥ tr♥s♣♦rt ♦ P②s

❬❪ ♠♦ ♥ ♠♦②♦ ♠♥t ts ♥ ♥

♠♣♥ ♦♥ ♦s ♦st ♠♦s t P ♦♥r♥ ♦♥ Ps♠ P②ss

P

❬❪ ❩♦♥ ♥ ♦tr♥♦ rtt r♥ sr♦ ❱ Pr♦ ♦♥

♥ ♦♥trt♦rs rq♥② s♦♥s t t♦rt ♥tr

♣rtt♦♥ ♦ ①♣r♠♥t ♦srt♦♥s s♦♥

❬❪ r ♥ ♥tr ♠♣♥ ♥ r ♦ ♦rq♥② ♠♦s ♥ t♦♠

♣s♠s s♦♥

❬❪ r r♠ rr♥ ❱ ♦♥ ss♥r ♥tr rs

r♥♦③ s ♥ ❳ ❯♣r ♠ ♥t é♥ ♥♠♦s

t ❳ ❯♣r Ps♠ P②s ♦♥tr♦ s♦♥

❬❪ r♦s ♥ ❩♦♥ ts ♦ tr♣♣ ♣rt ②♥♠s ♦♥ t strt

rs ♦ ♦rq♥② sr é♥ ♦♥t♥♦s s♣tr♠ Ps♠ P②s ♦♥tr♦

s♦♥

❬❪ rs♦ ♠♦ ❱ s②♣♥ ③♦ ♥ sss

♣r ♣r♠tt② ♦ ♠♥t③ t♦r♦ ♣s♠s t ♣t ♠♥t

srs ③♦s♦ P②s

❬❪ rr ♥ ♠♦ ♣r♠ttt② t♥s♦r ♥ ♦♥ rs♦

♥♥ ts ♦♥ ss♣t♦♥ ♥ t♦r♦ ♣s♠s ③♦s♦ P②s

❬❪ rs♦ ♠♦ ③♦ ♥ sss ♦♦♠

sttr♥ t ♦♥ tr♣♣ ♣rts ♦♥rs♦♥♥ ss♣t♦♥ ♥ ♠♥

t③ t♦r♦ ♣s♠s P②s Ps♠s

❬❪ ❩♦ ❩♦♥ ♦ ♠♦s ♥ t♦♠ ♣s♠ t ♦♠♥♥t② ♣♦♦ ♠♥

♦s P②s Ps♠s

❬❪ t♦ t♦ ♠♦♥ s ❨ ❲tr ❨ s♠ ♥

②♠ ♦s ♦st ♥♠♦s Ps♠ ♥ s♦♥ sr ♣

♦♠♠♥t♦♥

❬❪ t♦ t♦ s s ♦ ♥ ❨ s♠♦s ♦st

♠♦ s♣tr♦s♦♣② Ps♠ P②s ♦♥tr♦ s♦♥

❬❪ P♦ts ♠♦②♦ ♥ r♦s ♥t♦②r♦②♥♠ qt♦♥s ♦r

♣s♠s t ♥tr♠♦rrs ts Ps♠ P②s

❬❪ ❲ sr ❲ t♦♥ ♥ ♥ ♦t ① ♦♦r♥ts

♥ ♠♥t strtr t♦ ♥♠♥t t♦♦ ♦ ♣s♠ t♦r②

♣r♥r❱r r♥ r

❬❪ Ps♠ ♦r♠r② ♣♣♦rt ② ♦ sr

♦rt♦r②

❬❪ tt♣r♦s♠♥s♥t ss♦ ♠ ♦

❬❪ tt♣♣st♦ ss♦ ♠ ♦

❬❪ tt♣ss♦rr⑦♣① ss♦ ♠ ♦

❬❪ tt♣sss♦rr♥t♦s♣♣r♦r♠st❴tr♦

s♣ssstr ss♦ ♠ ♦