. Plasma Resonance Yadiat ion Martin D.Altschuler Yale l inivers i t y Observatory l iew fiaven, Connecticut GPO PRICE s CFSTl PRICE(S) Hard copy Microfiche 11853 Julv85 ',N66 32303' I 0 UCCESSlON NUMBER1 ITHRU) B I https://ntrs.nasa.gov/search.jsp?R=19660023013 2020-03-16T21:11:03+00:00Z -2- The s p e c t r a l c h a ~ - a ~ t c r i s t i c s of e lec t roEagnet ic r a d i a t i o n i n t h e neighborhood of t h e plasma frequency are der ived. The l i n e a r i z e d hydromagnetic equat ions are used t o f i n d t h e se l f - cons i s t en t e l e c t r i c f i e l d of a tes t 2 a r t i c l e i n t h e p l a s m . From t h i s e l e c t r i c f i e l d t h e acce ie ra t ion of an e l e c t r o n and t h e r a d i a t i o n emitted i n t h i s process a r e d e t e m i n e d for a s i n g l e encounter e l e c t r o n - test p a r t i c l e . %e s n e c t m of t h e r a d i a t i o n emi t t ed by t h e e l e c t r o n s surrounding t h e t e s t ? a r t i c l e is found by i n t e g r a t i n g t h e e a i s s i o n s p e c t m of a s ing le exxounter over a s u i t a b l e range of Lipact p a r m e t e r s . dhen t h e r e l a t i v e speed of t h e t e s t p a r t i c l e i s less than t n e thermal speed of t h e plasma e l e c t r o n s , t h e s p e c t m agrees wi th c a l c u l a t i o n s based on a s p h e r i c a l Debye p o t e n t i a l f o r t h e t e s t p a r t i c l e . For a tes t g a r t i c l e s2eed g r e a t e r tkian t h e t h e m a l e l e c t r o n s2eed a resonance a r i s e s near t h e e l e c t r o n glasna frequency. The energy erai t ted a t t h i s resonance i s c a l c u l a t e d and i s found t o increase with t h e t es t p z r t i c l e speed, whereas t h e energy emi t ted i n the neighboring cont inurn i iecreases. - 3 - k t l e a s t two of the nos t i n t e r e s t i n g con-therrzal r a d i o e v i s s i o n s frosl t h e ST&?, rL~-ic>J~, t h e +,r*>- 11 ~ 7 5 C - 7 - h T T T ,.----+- --- ------l-* +L----l-A -n . - V b U J y b Lli L V C L L V D , U I L LuvlsrullLJ L # . l W U ~ l l b VI as n a n i f e s t a t i o n s of resonance r a d i a t i o n by the coronal e l e c t r o n s a t t h e i r l o c a l plasma frequenc:J. Xnereas t;ee I1 events a r e a p y r e n t l y connected x i t h t h e passage of a snock f r o n t through t h e corona, t ype I11 b u r s t s a r e caused o j clouds of ? a r t i c l e s s e n e r a l l y tra.vell i i12 outdaras with nea r ly r e l e t L v i s t i c s-;eec. F, xo t ion of clovlis of caw-ed > a - t i c l e s t-rough t b e corona and the e x c i t a t i o n of e l e c t r o n :2lasma o s c i l l a t i o n s by t h e i r Jassase sugzests t h 3 t a t l e a s t a c -ua l i ta t ive uzderstanain, of t h e raCizi,ion g a t t e r n caz be obta ined by cons ider ing t o e electromagnet ic f i e l d s induced 3y a " tes t p a r t i c l e " o r chzrge q i n a ;lama. Zie nurpos? of t n i s r e 3 o r t i s t o aum?arize t h e ?evelo?=iont of such a theory . T 5 e extens ion of t h e s e r e s u l t s t o e n s e m l e s of , a r t i c l e s a u , i n :>ar t icu lar , t h e i r %A-?l ica i ion t o ty2e ZII a u r s t s 2re Ciscusse5 i n E suoseruent ?a?er. i n orcier t o f i n d t h e se l f - cons i s t en t e l e c t r i c and - . a p e t i c f i e l d s , 5 and 3 , 1 a r i s i n g f r o 3 t h e passage of a t e s t ? a r t i c l e tnrouqn 2 full;. i o n i z e a ?as , one can use the s y s t e n of l i n e a r i z e d hydronagnetic equat ions . Tor a t e s t p a r t i c l e moving throuqh t h e p l a s m wi th uniform v e l o c i t y i n t h e a3sence of an e x t e r n a l magnetic f i e l d , t h e se l f - cons i s t en t f i e l d s 3 and E both decay a t l eas t as fas t as trle inve r se square of tae d i s t a n c e ? roc t h e t e s t y a r t i c l e . Clonaet:uentlJ-, t i e s u r f i c e i c t e g r x l of t h e 7oyntin; v e c t o r througn a s p e r e enclosin;; t n e t e s t p a r t i c l e irill q p r o a c h zero as t n e r z d i u s of t h e s2here becones i n f i n i t e . ;io e lec t roxa -ne t i c r a d i a t i o n arises t h e r e f o r e f r o 3 t n e .. . - - _- - 4 - se l f - cons i s t en t f i e l d of a uniformly moving t e s t p a r t i c l e i n t n e absence of an a29liea z q n e t i c f i e l d . Blectromagnetic r a d i a t i o n i s emitted, noirever, wcenever t h e t es t p a r t i c l e C ~ J i i i d e S W i t h mi S l S C t r G I i iii its V i C i I i i t g . iv'e V i 1 1 be c O : l C e r r l e d d i t h t h e e lec t romagnet ic r a d i a t i o n e m i t t e d by those e l e c t r o n s which are acce le ra t ed by t h e se l f - cons i s t en t f i e l d s of t h e t e s t p a r t i c l e . I n t h e n o n - r e l a t i v i s t i c s i t u a t i o n whicn we w i l l treat exc lus ive ly t h e con t r i3u t ion of t h e se l f - cons i s t en t m g n e t i c f i e l d toward t h e a c c e l e r a t i o n of e l e c t r o n s i s ne-;l i ; ible, and x i 1 1 3e oznitted i n t:ie ca l cu la t ions t o follo-4. Fu r tne r , t n e t r e a t s e n t vi11 be l i s l i t e d t o t n e r a d i o f r e i aency region :There t h e s t ra i ;h t - l ine asproximation i s va l i i ! . 2 Tne spectrum of t h e r e s u l t i n g e l ec t ronagne t i c r a d i a t i o n has been c a l c u l a t e d up till nox us ing t h e agpox ima t ion t h a t t h e t es t p a r t i c l e f i e l d i s s p h e r i c a l l y syllunetric, i n p a r t i c u l a r , t h a t it can be represented by an unshie lded Coulozb p o t e n t i a l 2 , or a Coulomt, 2 o t e n t i a l c u t of f a t t h e Debye d is tance3 . It i s w e l l i aovn , however, t ha t t h e f i e l d s of a t e s t ? a r t i c l e i n a f u l l y ion ized plaszm do not have s i sner iea l syametry a t t e s t ? a r t i c l e speeds n e a r o r above t h e t n e m a l speed of t h e plasma e l e c t r o n ~ ~ * 5 , ~ , 7 3 ~ . Tie r a d i a t i o n f i e l d w i l l b e c a l c u l a t e d i n t h e d i p o l e approxipiation. The d ipo le e w r e s s i o n s are obviously valiC: f o r e lec t ron- ion i n t e r a c t i o n s , however, t hey are equa1l;r app l i cab le t o e l ec t ron -e l ec t ron i n t e r a c t i o n s i f one of the 2 a r t i c i s a t i n g s y s t e n s i s r e s t r i c t e d i n i t s r?.otion by o t h e r f i e l d s . I n ? a r t i c u l a r , t h e e l e c t r o n s of a 3ebye cloud i n t e r a c t wi th a "free" e l e c t r o n accordin2 t o t n e S ipo le ru l e s . '=iiis r e s u l t i s well-known f r o 3 k i n e t i c tneory 9 . Put i n s i 9 2 l i f i e d t e r n s , it i s r equ i r ed t h a t t e s t p a r t i c l e sTeed m d d i r e c t i o n i s e s s e n t i a l l y unchanged during t h e i n t e r a c t i o n T i t h t h e r a d i a t i n q e l e c t r o n . - 5 - IC t h e frane work of t h e kqrdrosa~net ic exuz t iocs , t3e v e l o c i t y % i s t r i m t i o n of t h e plasma e l e c t r o n s i s rep lacea by t h e nean t h e r n a l ( r o o t aean square) speed. The consexuent l o s s of i n f o m a t i o n is compensated i n t ha t an e x p l i c i t ss>,i%ion e a r 22 o5tair;ed f ~ r the eiitlre eiectruiili*-;rletic spectrum., wnereas t n e I apj?roach based on k i n e t i c t heo ry requires ex tens ive machizle c a l c u l a t i o n s .9 ,lo *11¶12 Following a b r i e f review of t h e r a d i a t i o n equat ions , we d i scuss t h e hyciroaagnetic eguat ions snd &r ive t h e se l f - cons i s t en t e l e c t r i c f ie ld . .{e t i e n find t h e r s i i i a t i o n s r e c t r a i n t h e case of a subsonic an6 o r a suFer- I I 1 s on i c t e s t p a r t i c 1 e. Tne i n s t a n t m e o u s 2ower emi t ted i n t h e fom of e l e c t r o n a g n e t i c r a d i a t i o n by zn acce le ra t ed cherge e i s rT3ex-e a(t> i s t n e ins tan taneous zcce le ra t ion of t h e cmr:e. Zq.(l) i s v a l i d under n o n - r e l a t i v i s t i c conai t ions . T!;e tota.1 ener5y e E i t t e d b;: t n e cnarge reads +aJ E = (2e2/3c3) I 1a(t)I2 d t . ,OD From Tarseva l ' s theorem re w i s tne angular frequent:;, a( t ) and a( w ) are a Four ie r - t ransfomL rsair: I += c l ( t ) = 1 / J ( 2 a ) J a(&) exr,(-iwtf z t . -03 1' inus, +W E = I ?,21.>, 31 I ,' \ i L C / -JL J 1 ]L(iAl) j 2 d&. i 'J I ,OD Defin inf t h e s p e c t r a l i n t e n s i t y a ( @ ) , t h n t i s , t h e energy emi t ted pe r u n i t bani v i a t h , by t h e r e l a t i o n +m m J ;(t) Qt = J :(w) do , 0 ,m ana us ing S ~ S . ( 2 ) anci (61, we o5tai.n ( 7 ) T h e express ion of Q ( w ) of Eq. (8) is n m e r i c a l l y equiiralent t o t h e 7 r e fe rence (2 ) i n s p i t e of t h e f a c t t h a t Eq. ( 7 ) does not have a f a c t o r o f t 7 on t ae r i g h t . Tiis fol loirs because t h e Four i e r t ransforms of EP-S. ( 4 ) and ( 5 ) a r e def ined u i f f e r e n t l g fro3 those of r e fe rence (2). of w rn? i:ius t o c a l c u l a t e t h e s 2 e c t r w of t h e r a c i s t i o n err i t te i 3;' an e lec t ror , it i s necessery t o f i n d t h e acce le ra t ion a(w). Tnis w i l l be c a r r i e d out i n t h e f o l l o u i n g sec t ions . L e t a tes t ? a r t i c l e of sGeed u an6 charse q Zove i n t ae d i r e c t i o n of t h e ;)ositive x-axis. "he acce le ra t ion cf zn e lec3ron cf char,-e -e m c ! miss =? a t 9 o s i t i o n r and t i r te t due t o tne f i e l d Z(r,t) of t h e t es t p a r t i c l e i s a ( r , t ) = - ( e / m ) E ( r , t ) . ( 9 ) .. ' .' - 7 - If during t h e i n t e r a c t i o n , i .e., during a ti:lie of t h e order 2 ~ / u m e r e b i s t h e i s p e c t pa rane te r , t h e e l ec t ron i s not d i sp laced s i g n i f i c m t l y f ro= i t s i n i t i a l p o s i t i o n , r i s appoxiziateljr cons tan t , and This s ta tement i s e s s e n t i a l l y the well-known s t r a i g h t - l i n e a2.;troximation 1 . A t any f i x e a value of r, and a l s o +m E ( r , t ) = 1/ ( 2 ~ ) ~ 1111 Z ( k , u ) e x p [ i ( k * r - u t ) ] dk d w , -W where E(k,w) is t h e space and time Fourier t ransform of E ( r , t ) . E q s . (11) and (12) then y i e l d 2rom T l Sqs. ( s ) , (lo), and (13) ve ob tz in t h e e l ec t roxegne t i c r a d i a t i o n snectrum ::(a) 2rovided xe knov E(k ,u) . This quan t i ty w i l l be deduced fro? t h e nydro- n sgne t i c equat ions. X e de f ine a " r a d i a t i o n p robao i l i t y" w i t h dinensioR [ ~ ( o ) ] = a r e 8 x x energy/frequency as t h e i n t e g r a l of 3, over a range of impact parameters: The n m b e r of e l e c t r o n s i i i t h srkich t h e t e s t ? a r t i c l e i n t e r e c t s per u n i t t i x e i s u n urqer turbed plasma. The electrornagnetic energy r a d i a t e d p e r u n i t t i n e i n 21rb 6b, i f no i s t h e n m b e r of e l e c t r o n s per u n i t vohme of 0 ~ - 8 - t h e frequency i n t e r v a l w, w+do , by t h e 2lasma e l e c t r o n s between t h e inFac t F a r a e t e r s b and 5 i s t h e r e f o r e 1 2 b2 52 P ( w ) = 2nnou I Q(b,w,u) bdb = nou ~ ( o ? ) ] . "1 bl A s b approaches zero, t h e s t r a i g h t - l i n e approxiaa t ion fails . A more complete t rea tment shows t h a t t i le lower l i m i t of t h e i n t e g r a t i o n may 3e c u t o f f a t 2 I n t n e rest f r a z e of t h e test ; ?a r t i c l e , an e l e c t r o n at t h i s ir?pzct .sarmeter xould be a e f l e c t e d by 90'. Tne l i n e a r i z e d hydromagnetic equat ions , neg lec t ing d a p i n g and e x t e r n a l f i e l d s , bu t includiLg t h e t e s t p a r t i c l e i n t h e form of a Dirac d e l t a func t ion are596 c VXG = -an /a t , c vxx = az/at - 4ren v + 4rqu G(r-ut) , V*E: = -4ren + 4 v q G(r-ut) , av/at = -(e/m)E - (V2/no) Vn, 0 0 , V*E = a d a t + no vov = 0, I -&ere Z i s t h e e l e c t r i c f i e l a a r i s i n z bot:? from t h e t e s t p a r t i c l e of charze q and t h e 2 e r t u r b a t i o n n i n t h e e l e c t r o n n w b e r d e n s i i y [c . f . Eq. (19>]; 3 i s t h e magnetic f i e l d a r i s i n g f r o 3 t h e cnarge motions; v i s the p e r t u r b z t i o n - 9 - i n t h e w e r a g e plasma e l e c t r o n ve loc i ty ; n i n t h e abser-ce of a ge r tu rba t ion ; 'J i s t h e t h e r r a l speed cf t h e 2 lzsma e l e c t r o n s , i . e . , 3 V 2 / 2 = 3K"/2 i n t h e case of a 'hxvell d i s t r i b u t i o n ; n i s t h e mass of t h e e l ec t ron . It i s a s smed t h a t t h e 2 o s i t i v e charges f o r a a uniform s t a t i o n a r y background of charge dens i ty +en . These equat ions have been used by o t h e r moving p o i n t charge i n a p l a s n a and t o d i scuss c o l l e c t i v e p l a s m o s c i l l a t i o n s . 2;s. (17) t i iroush ( 2 3 ) c o n s t i t u t e t e n equat ions f o r t h e t e n unknows 5, 5, v, and n , and i n t h i s sense ere se l f - cons i s t en t . 11.(22) cari 3e der ived Fro3 23s . (18) and (l?), v n i l e Zq. ( 2 : ) serves as i n i t i a l condi t ion Pcr 12. (17). i s t h e n m b e r dens i ty o? e l e c t r o n s 0 0 t o f i n d the charge d e n s i t y d i s t r i b u t i o n about a - Following a method used by "ajuxdar 5 , w e Four i e r t ransFox2 E i s . (17) tkrough (20 ) i n space zr~0 time, us inc Taese t e n l i n e a r a l g e b r a i c eTaatio:ia are x o - ~ solved si;:ultzneously f o r t n e e l e c t r i c f i e l d , w i tn t h e r e s u l t .- - 10 - ,. 2 3 * _. i s ne3liC;iDle i r i t;iz non-relat i - ke no% s e t up &I? ortho-zonal c o o r d i r a t e system wi th the x-axis Taral le l t o u 2nd t h e u n i t vec to r s 2,, Po, 20. The t o t a l e l e c t r i c field, viz., through (31) i n t o (13). ile i l l u s t r a t e t he xethod by solvinp f o r cox2oner;t Zn is found i n a aiBiler zanner . (r,a); t h e -1 z t‘sing 2qs.(l3) ( 2 3 ) , -$e o b t a i n t s e i n t e g r a l +m ~~ - 11 - t h e i n t e g r a t i o n over k is r e a d i l g pe r fomed t o g ive X + (ky2+k,2)V2]-1] exp{i f (w/u)x + k + kzz]) dk dkz . YY ( 3 5 ) Y To c a l c u l a t e t h e f i e l d at t h e p o s i t i o n of an e l e c t r o n , we can ZssuTe without 10s; of g e n e r a l i t y t h z t t n e e l ec t ron i s a t p o s i t i o n (g,b,o), t h a t i s , a t i q z c t p a r m e t e r b. Senhrat ing S i n t o s--, y- , and z -coqonen t s , we czn w r i t e 1 L l x -, ( b , w ) = - [ 2 q i w / ( 2 ~ ) ~ ' ~ u ~ ] I , (36) uhere ( 3 9 ) + (kr2 + kZ2)V2]-') exF(ibk ) dk &kZ . y Y T h i s i n t - eg ra l has d i f f e r e n t s o l u t i o n s f o r u < V (subsonic t es t p a r t i c l e ) an6 f o r u > v (supersonic test p a r t i c l e ) . ?or t h e subsonic case , a contour i n t e g r a t i o n in tile c o q l e x k --s1 z - r e a d i l y y i e l d s t h e r e s u l t [ c . f . A??endix A ] : e I " - . .- - 12 - Severa l remarks should be made about t h e e l e c t r i c f i e l d s of Eqs.(40) through (46 ) : i;e, both p m e r a t o r 2nd Aenc?ninztcr ~ - w r m c k l zerc? ir? tce 1. ;i3e?, & -+ -rr f i e l d s Six, E l y Y Epy 9 L ' z b p i t a l ' s %le shoxs, however, t h a t t h e f i e l d s 1 . - 13 - are f i n i t e and continuous a t o = w . l e l 2. b2/u1 2 >> u2/c2, t n e n o n - r e l e t i v i s t i c a22roxinlatior u2/c2 << 2 e l inci icates t h a t 5, = Ju, a i r tl;erefore 1z2j << 1 ~ ~ 1 . 3. h e n &/be2 << u*/c2 c < 1, w e can assme t h a t w -+ '3, i n which -1x - n * -f 0; Z + 0. '???us, w e case + 0 ; E + v'(2/7~) (qu,/uv) K ~ ( ~ U ~ / V ) ; A2x ca.n neg lec t t he f i e l d conqonents IT2 1 f o r a l l f requencies i n t h e non-rela- t i v i s t i c case. I 1Y 27 2 I T J, r.'iq7 CTS. (40) through (43) f o r t h e t o t a l e l e c t r i c f i e l d , t oge the r v i t n 22s. (6) an6 (131, -de o t t a i a t h e s -nec t rw of tie e l e c t r o . ; a p e t i c r x i i s t i o n e-iitted b;- a s i r igle e l e c t r o n a t i-.Gact p a r a x t e r b f r m 3 subsonic test $ a r t i c l e i n a pl- asna : vhere 6 i s aefirrea by ZQ. (43). 1 I n t h e dbsence of 2lasm, we = 9, m a wkicn i s t h e well-?mmn exgression fo r t h e electronisgnet ic r a i i i a t ion s p e c t r m of ac e lec t ron- ion c o l l i s i o n , i f one i d e n t i f i e s t h e char> w ) , t h e s p e c t r m of ~ q . (42) -* Jecones e which i s i u e n t i c a l with Zq. (49). Thus, t n e s p e c t r m of a subsonic c o l l i s i o n s i n t n e p e s e n c e of plasm should converge a t high f requencies t o tne s p e c t r m i n t h e absence of plasma, providing t n a t u2/V2 << 1. 5. S igh f r e q u e n c y l i n i t f o r test p a r t i c l e w i t h U=V (idath 1): When t h e t es t p a r t i c l e speed equals t h e xean t : i ema l e l e c t r o n speed V, t h e hTgh frequency - limit nf rc. (48) becoxes >( o > W e ,D,u=v~ = ( 8e2/3sc 3 ) ( ae/n) 2 ~ - 4 x I [ w I ” : ~ ( ~ ~ / ~ ) - (we2/w);i 0 (bUle/v) 1 2 + [ & I ~ ( ~ ~ / U ) - (we31w2)i;l(bwe/ir)12j* ( 5 4 ) i 1 - 15 - aut c lear ly l i r n wKl(bwfV) << lin (y3 /u2) Iil(bwe/V), w- W-tOD s i n c e t h e asy-qtotic foms of K and K 8re 0 1 %quation (54 ) i s then ( 5 6 ) Tnus, a t u=V t h e high frequency t a l l of the spectrurn decaus as 1/u2, wnereas for u V,and. i s a@n nez1ecte.i i n our n o n - r e l a t i v i s t i c t r e 3 t z e c t . ?ne s2ec t rwi of Lie e1ectroTs;netic r a d i a t i o n e-Titted by an e l e c t r o n i n t n e vici::17;;- of a sTJ2ersoni.c t e s t S z r t i c l e i s de te rz ined b x r ;qs. ( 3 6 ) tnrog;,: ( 3 2 ) . 'or tliose fre;uencies that setisf:: t h e i n e q u a l i t y - - 18 - t h e s p e c t r a l i n t e n s i t y d i s t r i b u t i o n Q(o,b,u) is given by Eqs. (48) ana (43). The q u a l i t a t i v e d i f f e rence t o t h e previous case i s t h a t s i n c e nox u V it i s 2 o s s i b l e f o r p t o -De zero. 3 u t i n t h e l i m i t of s x a l l ar,gment T!IUS, a( w,b,u) -+ m when 6, -+ 3 , o r equ iva len t ly , when w -t w . For t h i s reason ire w i l l c a l l w t h e resonant frequency. r r Ofi t h e o the r hand, i f c12 < 0 , t h a t i s , (79) o r equ iva len t ly , t hen fro3 Lqs. (36) tnrough ( 3 9 ) t h e e l e c t r i c f i e l d s becorne 1c.f. .1?2eridix 31 I - - 19 - and Jo, Ti, Yo, and Y1 sre Ijessel func t ions i n convent ional no ta t ion . Zsing ~ q s . (10) , ( 7 3 ) , (74), a d (75) i n ( 3 , w e ob ta in f o r t h e s n e c t r m Tiis exgress ion i s v a l i d f o r c12 2 0 , o r equ iva len t ly w G(u,b,u) goes t o infinit:: as w -t w t h e resonant frequency w i l l be discussed i n t h e next s ec t ion . w . :?e note t h z t r - %e na tu re of t h i s d i s c o n t i n u i t y a t r’ For supersonic t e s t p e r t i c l e s , ye found i n t n e l as t s e c t i o n t h a t as w approacnes t h e resonarit frequency w whether f r o n h ighe r o r lower f requencies , t n e s s e c t r a l i n t e n s i t y ;(w,?~,u) approaches i n f i n i t y . ”his f b l l o w s because, on one hand, o t h e r hand, c1 + 0 i n Eq. (76) and thus t h e term Y 2(br, ) -+ m. r’ + 0 i n Eq. (48) and t h e t e rn u 2Ko(bc1) -f =, and, on t h e 5 e 0 1 Althougn t h e s p e c t r a l i n t e n s i t y ‘3, i s i n f i n i t e a t t h e resonance w t h e in t e - r’ p a l I;(w,b,u) d w i s f i n i t e over t h e e n t i r e frequency range. ”at i s , t h e r e sonmce a t w = w i s an i n t e g r a b l e d i s c o n t i i u i t y . 3e fo re ve c o x p t e an do9roxinate express ion f o r t h i s i n t e g r a l , we f i r s t would l i k e t o SliOW t h a t t h e d i s c o n t i c u i t y i s s p m e t r i c witn r e s2ec t t o t h e resonant frequency. r - 20 - Thus, for w = w , w e have r ami i n the vicinit:,. of w be comes approxirriat e l ; r bu t Jus t above i t , t h e sgectr-A? of :(u,-D,u) r' 51 J u s t Leiov t h e resonant frequency, we conaicer t h e lizit of s za l l in -,q. ( L 2 ) aLci r ep lace I ~ ~ ( - D < ~ ) -cy - ~ r i ( y ~ < / 2 ) .ma - r :', ( 3 ~ ~ ) b:r I / s ~ ~ . -!:e 1 - r e a z l t i n g exzress ion i s i d e n t i c a l t o Zq. (79 ) exces t t h a t t h t n i r 4 t e r n is x i s s i n z . Since t n i s t e r l i n trie neighjorhoo5 of t h e resonance nro.ser i s n e s l i g i b l e i n comparison wi th t h e rist t e n , ve conc1;tde t h a t t h e resonance indeed i s symnet r ica l w i th respec t t o w. I n o r a e r t o c o m p t e t h e t o t a l energy m i t t e l : i n t h e resonance, ue cons iae r only t h e cioninsnt t e rn , v i z . , ane i n t e g r a t e over w fro:; t h e resonance G: a q u a l i t a t i v e d i scuss ion nay oe f ixed such t h a t t o sole value L; + E -$:Xch, fGr r r y5c 2 = 1, if w = w, + E, 1 2. - 21 - (21) that i s S ince ur+ E Iln2( YbG1/2) Uw = (V2/wry2b2) [u2/(u2-V2) 1 , w r ye f i n o w i t a w f roE sq. ( 6 8 ) r .- i:ence, t'ne energy emi t ted i n t h e resonance i s indeed f i n i t e . It inc rezses wi th inc reas ing t es t p a r t i c l e speed, s t z r t i n ; f roa zero a t u = L'. Xe s h a l l cove back on t h e dependence on the i n p a c t Daraaeter 2 resen t ly . I n s e r t i n g the spec t run ( 7 6 ) i n t o 2%. ( 1 4 ) and u s i n 3 t n e r e s u l t s of Agpendix L" f o r t h e eva lua t ion of t h e i s t e g r a l s , r e o b t a i n f o r t h e r a d i a t i o n :;rob a b i l i t y : :. - 22 - Tne r a d i a t i o n p r o b a b i l i t y diverges as b approacnes zero jecause of t h e 0 t e r m ~ ~ ~ ( j i 1; t h i s behavior i s also t r x e i n t l ie subsonic case 2nd arises sizrgly because t h e p o t e n t i a l of tne t e s t p a r t i c l e i nc reases i n d e f i n i t e l y as w e approach t h e t e s t 2 a r t i c l e . The r a d i a t i o n p r o b a b i l i t y a l s o d iverges as t h e upper l i m i t bm a p p o a c h e s inf in@. This behavior arises because t h e test p a r t i c l e is assumed t o have been i n t h e plasma fo r an i n f i n i t e t i m e s o t'nat tne d i s c o n t i n u i t i e s of t h e 2ach cone sre a l s o i n f i n i t e . 1 - - vte can again c a l c u l a t e t h e t o t a l r a d i a t i o n emi t ted i n t h e resonance by t n e average ~ l a s m e lec t ron . Again, w e cons ider only t h e resonsnt t e n , v iz . , I n a r r i v i n g a t Eq. ( 8 6 ) , terns of o r 6 e r Yo(b,G1), as w e l l as a l l t e r n s m u l t i p l i e d wi th bo2 , such as b,2Y02(bo<1), e t c . , have been neglec ted . The terms c o n t a i n i n t ~ ~ ( 5 5 ) and y1i(5< ) cance l a t t h e resonance f o r upper and lower l i n i t of i n t e g r a t i o n . 1 1 If :?re again r ep lace t h e Bessel func t ion by l n ( y b < /2 ) and i n t e g r a t e 3 1 over w f roz w to w + E, r e c a l l i n s timt r r we f i n d f o r t h e t o t a l r a d i a t i o n a t t h e resonance This express ion may be coEpared with Z q . (84) wnich, upon i n t e g r a t i o n over b , l e a d s t o t h e s z e r e s u l t except f o r z f zc t s r lz(5 '"0) thzt i s ne i n t e r e s t f o r our p re sen t purposes. 2' - 23 - 12. co7cLiT3Ic)::s The 5Fdrozlagnetic equat ions n m e been used t o f i n d t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d of a t e s t p s r t i e l e . I n an i n t e r a c t i o n wi th such a t e s t p a r t i c l e e l e c t r o n s are a c c e l e r a t e d and e m i t e l ec t ronagne t i c r a d i a t i o n according t o t h e s p e c t r a l d i s t r i b u t i o n of a. Eq. (48j vhen t h e tes t ? a r t i c l e i s subsonic , b. Sc,. (48) when t h e t es t p a r t i c l e i s sugersofiic ana t h e f requencies consideret? are less than u o f ~ q . (G) , r c. Eq. (76) when t h e t e s t >?er t ic le i s s u p r s o n i c a17d t h e frec3;lencies considered are g r e a t e r t h m w . The energy r a d i a t e d at, t h e resonance frequency i s given by Ea_. (84). r Tine t o t a l power r a d i a t e d by t h e average e l e c t r o n as def ined i n Sq. (15) has t h e s p e c t r a l d i s t r i b u t i o n of a. Eq. (60) -&en t n e t es t p a r t i c l e i s subsonic , 5. 3q. (85) x3en t h e t es t ? a r t i c l e Is supersonic . Tie t o t a l enery;' emi t t ed a t t h e resonznce hy e l e c t r o n s wi tk in a 3e3;-e d i s t ance t:?e t e s t I indeb tea t o Prof . Lu6viS ? s t e r of t h e Yele UciversiQr fibservator:; f o r suzges t ions , c r i t i c i s ; . s , and his con t inua l encourageaent. - 24 - 4-32 i s to bi? evaluate2 for t h e case v?im w < w / J ( l - V /u ) 5 (it e r i n t h e s q e r s o n i c case. To i n t e g r a t e over the variable k r r e def ine z and 'Then t h e i n t e g r a t i o n over t h e v a r i a b l e kZ i s .- . - 25 - Tne i n t e g r a t i o n over dkTT car, now 4ve coll;.leted u s i n s the s tan3ard Tntegrc ls f o r the n o d i f i e 6 3 e s s e l func t ions v i t h t h e r e s u l t .- I - I - . * - 26 - 52 = w2(1/’J2 - 1 /u2) - w 9\72 > 0. e The i n t e g r a t i o n over t h e va r i ab le k i s then n Since t h e r e are t w o p o l e s on t h e real axis, we spec i fy t h e contour t o 3e t h a t for outgoing waves. The roo t on t h e p o s i t i v e s i d e of t h e r e a l axis i s included when t h e contour e n c i r c l e s t h e lower ha l f plane. ?,om t h e r e s idue theore= fo l lows :de coraplete t h e i n t e g r a t i o n using t h e e q r e s s i o n f o r t h e Hankel func t ion . Thus, where 5 i s def ined by ~ q . (75 ) . F i n a l l y i n s e r t i n 5 t h i s r e s u l t i n t o E q s . ( 3 6 ) throuzh (39) w e have Tie i n d e f i n i t e i n t e g r a l s used t o obtain tne r a d i a t i o n p - o b a j i i i t i e s Stre listed here f o r convenience. IxKo2(.x) dx = x2[Ko2(ax) - ;52(ax) ] /2 /xK12(ax) dx = x2[K12(ax) - KO2(=) - (2/ax)Ko(ax)Kl(ax)]/2 0 . -1 -1 IXK,[ a~ )KO( cx j ex = X( a 2 - e ) -[-do( cx j irl( ax j + c ~ ~ i j?:,( cx j j I X K ~ ( ~ X ) K ~ ( C X ) dx = x(a2-c2) JxJo2(ax) dx = x2[Sc2(ax) + J1 2(=41/2 ~ x ~ ~ z ( a x ) ii = x2['io2(ax) + ~ ~ 2 ( a x ) ] / 2 IxJ:(ax) dx = x2[Jo2(ax ) + JI2(ax) - (2 / ax ) Jo (ax ) J l ( ax ) ] /2 [-aK 0 ax)^ 1 ( c x ) + ci: 0 ( c . y . ) ~ ~ ( a x ) ] ixY:( ax) dx = x2[Yo2( ax) + Y1 2(a) - (2/ax)Yo(2x)Yl(aX)l/2 /ao( cx)Yo( ax) dx = x( a2+c2)-I[ -cyo( ax)K1( cx) + a0( cx)Yl( ax) 1 I X K ~ ( CX)Y,( a) dx = -x( a2+cz) [ C K ~ ( cx)vl( ax) + aye( a x ) ~ , ( c x ) I -1 .. e c . - 28 - 1. S. K. ;:ajmdar, Proc. ~ h y s , SOC. (London) E, 657 (1960) 2. L, ~ s t e r , Re%-. :*IO&. Thys. 33, 525 (i9C;i) 3. L, Oster, Phys. Fluids 1, 263 (1364) 4, A. A, Vlasov, i4any P a r t i c l e Theory and Its A ~ ~ l i c a t i o n s t o Plasma. Sa&. T r a n s l . Gordon and Breach h b l i s h e r s (1361) 5. S . $1. ::aJu?aar, Tech. 3ep. 431, MIT 3es. Lab. of X e c t r o n i c s (1352) 5 . 1. 2. Zonen, ?kys. 3ev. 1.23 711 (1961) 7. 3. Pines and D. Boiur:, "hys, liev. 8 5 , 338 (1952) 3. 2. Gemin and 3. Guernsey, Te-hn. yep. ~1-32-0263 Boein; Sc ien t . qes. -? - Labora tor ies (1963) 9. T . Birmingham, 3. Dawson, and C. gbeman, Phys. Fluids 8, 297 (1965) -

Plasma resonance radiation

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