Theoretical Analysis of the Electron Spiral Toroid Concept

Transcription

1 NASA/CR Thtcal Analyss f th Elctn Spal Td Cncpt Jan-Luc Camb and Davd A. Mchltt MSE Tchnlgy Applcatns, Inc., Butt, Mntana Dcmb 000

2 Th NASA STI Pgam Offc... n Pfl Snc ts fundng, NASA has bn ddcatd t th advancmnt f anautcs and spac scnc. Th NASA Scntfc and Tchncal Infmatn STI Pgam Offc plays a ky pat n hlpng NASA mantan ths mptant l. Th NASA STI Pgam Offc s patd by Langly Rsach Cnt, th lad cnt f NASA s scntfc and tchncal nfmatn. Th NASA STI Pgam Offc pvds accss t th NASA STI Databas, th lagst cllctn f anautcal and spac scnc STI n th wld. Th Pgam Offc s als NASA s nsttutnal mchansm f dssmnatng th sults f ts sach and dvlpmnt actvts. Ths sults a publshd by NASA n th NASA STI Rpt Ss, whch ncluds th fllwng pt typs: TECHNICAL PUBLICATION. Rpts f cmpltd sach a maj sgnfcant phas f sach that psnt th sults f NASA pgams and nclud xtnsv data thtcal analyss. Includs cmplatns f sgnfcant scntfc and tchncal data and nfmatn dmd t b f cntnung fnc valu. NASA cuntpat f p-vwd fmal pfssnal paps, but havng lss stngnt lmtatns n manuscpt lngth and xtnt f gaphc psntatns. TECHNICAL MEMORANDUM. Scntfc and tchncal fndngs that a plmnay f spcalzd ntst,.g., quck las pts, wkng paps, and bblgaphs that cntan mnmal anntatn. Ds nt cntan xtnsv analyss. CONTRACTOR REPORT. Scntfc and tchncal fndngs by NASA-spnsd cntacts and gants. CONFERENCE PUBLICATION. Cllctd paps fm scntfc and tchncal cnfncs, sympsa, smnas, th mtngs spnsd c-spnsd by NASA. SPECIAL PUBLICATION. Scntfc, tchncal, hstcal nfmatn fm NASA pgams, pjcts, and mssns, ftn cncnd wth subjcts havng substantal publc ntst. TECHNICAL TRANSLATION. Englshlanguag tanslatns f fgn scntfc and tchncal matal ptnnt t NASA s mssn. Spcalzd svcs that cmplmnt th STI Pgam Offc s dvs ffngs nclud catng custm thsau, buldng custmzd databass, ganzng and publshng sach sults... vn pvdng vds. F m nfmatn abut th NASA STI Pgam Offc, s th fllwng: Accss th NASA STI Pgam Hm Pag at E-mal yu qustn va th Intnt t hlp@st.nasa.gv Fax yu qustn t th NASA STI Hlp Dsk at Phn th NASA STI Hlp Dsk at Wt t: NASA STI Hlp Dsk NASA Cnt f ASpac Infmatn 7 Standad Dv Hanv, MD

3 NASA/CR Thtcal Analyss f th Elctn Spal Td Cncpt Jan-Luc Camb and Davd A. Mchltt MSE Tchnlgy Applcatns, Inc., Butt, Mntana Natnal Anautcs and Spac Admnstatn Langly Rsach Cnt Ppad f Langly Rsach Cnt Hamptn, Vgna und Puchas Od L-887 Dcmb 000

4 Acknwldgmnts Wk was cnductd thugh th U.S. Dpatmnt f Engy Natnal Engy Tchnlgy Labaty at th Wstn Envnmntal Tchnlgy Offc und DOE Cntact Numb DE-AC-96EW Th us f tadmaks nams f manufactus n th pt s f accuat ptng and ds nt cnsttut an ffcal ndsmnt, th xpssd mpld, f such pducts manufactus by th Natnal Anautcs and Spac Admnstatn. Avalabl fm: NASA Cnt f ASpac Infmatn CASI Natnal Tchncal Infmatn Svc NTIS 7 Standad Dv 585 Pt Ryal Rad Hanv, MD Spngfld, VA

5 Cntnts Sctn Pag Fgus... v Tabls... v Excutv Summay.... Intductn.... Fundamntal Dynamcs Th EST Cnfguatn Stablty Pblms Th EPS Thtcal Mdl Scalng Ppts Engy Stag and Lftm Expmntal Obsvatns Cnclusns Rfncs... 35

6 Fgus Pag : Schmatc f EST plasma... : Schmatc f cdnat tansfmatn. Th z-dctn s ut f plan. Th td s appxmatd by an nfntly lng cylnd : Numcal slutn f Equatns and 4 lctns nly f th chc f paamts: α.6, fˆ.0, b /. 0 Cas # n Tabl : Numcal slutn lctns nly f th chc f paamts: 3 ˆ 4 α 0, f 0, b /.0 s Cas # n Tabl : Nmalzd vlcty f Cas #3 cmpa wth Fgu 4 f [] : Nmalzd cycltn fquncy f Cas #3 cmpa wth Fgu 5 f [] : Nmalzd vlcty f Cas #4 f Tabl symbls a plttd vy 5 gd pnts...0 8: Nmalzd cycltn fquncy f Cas #4 f Tabl symbls plttd f vy 5 gd pnts...0 9: Cmpasn btwn th tw tms f Equatn 30 dtmnng xtnt f physcal slutn f th n vlcty. Rsults f Cas #3 f Tabl... 0: Cmpasn btwn th tw tms f Equatn 30 dtmnng xtnt f physcal slutn f th n vlcty. Rsults f Cas #4 f Tabl... : Scalng latnshp btwn cuplng paamt α and chag paamt f ˆ f cnstant magntc fld...9 : Scalng latnshp f cnstant lctn ngy...9 3: Scalng f lctn ngy vsus shll thcknss...0 4: Scalng f magntc fld wth sz at. Bth data and appxmat ft shwn...0 5: Schmatc f kntcs at th EST ntfac...8 Tabls : Lst f cass studd by Chn n []... 7 : Cmpasn f u numcal sults wth ths f Chn n [] : EST paamts lstd n []... 4: Typcal stuctual paamts satsfyng ngtc valus n Tabl 3... v

7 Excutv Summay Ths pt dscbs th analyss by MSE Tchnlgy Applcatns, Inc. MSE f th Elctn Spal Td EST cncpt bng dvlpd by Elctn Pw Systms Inc. EPS. Ths analyss was cnductd f th Natnal Anautcs and Spac Admnstatn s Langly Rsach Cnt NASA-LaRC. Th EST s dscbd by EPS as a plasma shll f tdal shap, n whch lctns a n bts aund th tdal axs. Th cunt pducd by th lctn mtn gnats a tdal magntc fld, whch s cnfnd wthn th tus shll. It s clamd that ths plasma stuctu s vy stabl and can st vast amunts f ngy, pmaly as a magntc fld. Th mst dtald dscptn s fund n a thtcal study f th EST cncpt pfmd by th Massachustts Insttut f Tchnlgy MIT and publshd by EPS, hn lstd as fnc []. EPS als cntly publshd th Phas I pt f a NASA Insttut f Advancd Cncpts NIAC awad, n whch sm f th spcfc clams abut lftm and ngy dnsty stag w mad, dntd hn as fnc []. Th psnt analyss was gnally basd n ths and anth dcumnt dscbng th EST cncpt, whch was als publshd by EPS [3]. Hwv, dssmnatn f an aly daft f ths pt ld t a futh xchang f nfmatn btwn MSE and EPS [4]; th las f tw nw dcumnts [5,6]; and multpl vsns f th mdl by EPS. Rfnc [5] s actually a vsd vsn f [3], and th scnd dcumnt [6] s an xpmntal pt f th Dfns Spcal Wapns Agncy DSWA. Th analyss f ths addtnal matal s ncludd n th fnal vsn f ths pt. Fllwng th gnal dscptn f th EST, th psnt analyss s manly basd n th cldflud quatns f th plasma shll cupld t th slf-gnatd magntc fld. Th dynamcal quatns f th cld lctn flud, as wll as th numcal slutns, a fund t match ths psntd n []. Hwv, ths quatns nly psnt an ncmplt subst f th plasma dynamcs, snc th n mtn was gnd. It s asly sn that th n flud s unstabl, du t th pulsng lctstatc fld qud f lctn cnfnmnt. A dtald analyss wth th numcal pcdu cnfmd ths sults. Th EST cncpt qus an xcss f pstv chags, and th ptntal wll thus catd s spnsbl f cnfnng th lctns as wll as th magntc fld std wthn th tus. Althugh vaus scnas f n cnfnmnt w thn suggstd [4,8] by EPS and MIT, nn f thm s plausbl. In fact, t s dmnstatd n ths pt that ths pblm s fundamntal and cannt b slvd wthn th cntxt f th ppsd EST cnfguatn. Sval pblms w als fund wth th sults psntd n [] gadng th physcal chaactstcs f a hgh-ngy EST, whch w fund t b ncnsstnt wth th scalng ppts f th numcal slutns. Th ats f ngy lsss thugh cllsns and cycltn mssn w als xamnd and fund t b n by sval ds f magntud. Futhm, th analyss by EPS f th xpmntal data [6] ds nt pvd pf f an EST, as dfnd n []. It s dmnstatd n ths pt that th clams f abslut stablty and lag ngy stag capacts f th EST cncpt hav nt bn substantatd. Hwv, th s undnabl xpmntal vdnc f sm typ f plasma stuctus whs ppts and chaactstcs stll man t b dtmnd and whch culd ptntally hav applcatns f spac ppulsn. Hwv, m alstc thtcal mdls must fst b dvlpd t xplan th xstnc and ppts bf applcatns f ntst t NASA can b assssd and dvlpd.

8 . Intductn Ths pt dscbs th analyss by MSE Tchnlgy Applcatns, Inc. MSE f th Elctn Spal Td EST cncpt bng dvlpd by Elctn Pw Systms Inc. EPS. Ths analyss was cnductd f th Natnal Anautcs and Spac Admnstatn s Langly Rsach Cnt NASA-LaRC. Th cmpany cvd a Phas I awad fm th NASA Insttut f Advancd Cncpts NIAC gant numb Th nfmatn gadng th EST cncpt s smwhat sktchy, snc th a n p-vwd publcatns. Ou fst analyss f th cncpt was cmpltd n Nvmb 999, and was basd n th lmtd ltatu avalabl at th tm; pmaly a thtcal dscptn f th cncpt by C. Chn [], th NIAC Phas I pt [], as wll as an EPS dcumnt dscbng th aspcts f th cncpt [3]. Aft th las f an al vsn f ths pt and a dscussn wth EPS nvstgats [4], futh nfmatn was mad avalabl [5,6] and subsquntly vwd. Fnally, a mtng tk plac wth EPS and th Massachustts Insttut f Tchnlgy MTI [7] dung whch th latst, nnpublshd nfmatn [8,9] was mad avalabl. Nt als that fncs [3,5,8] a dffnt vsns f a smla dcumnt; th vsns w mad fllwng th sults f th psnt analyss. In th gnal publshd dcumnts [-8], t s clamd that a stabl plasma, capabl f stng vast amunts f ngy, can b catd wthut th nd f xtnal cnfnmnt.., f ESTs. Th s n xact dfntn f stabl, but ndcatns fm th avalabl ltatu [,3] suggst lftms f th d f yas. Snc plasmas n labats a ntusly sht-lvd ntts, ths clam s qut spctacula. Oth clams a smla n scp. 6 F xampl, th EST spcfc ngs culd b f th d f 6 0 J/kg. By cmpasn, cmbustn f kg f hydgn wth xygn wuld gnat. 0 8 J, whch s quvalnt t th lbatn f appxmatly.5 V/atm. T ach th clamd lvl f spcfc ngy, ach atm wuld hav t b abl t st 650 MV f ngy appxmatly 60% f th st-mass f th ptn tslf! T dat, nly antmatt s knwn t hav ths lvl f ngy dnsty. Th std ngy wuld pmaly b n th fm f a cnfnd magntc fld, f th d f 6,000 Tsla. Ths wuld lad t an ntnal pssu f blln atmsphs, yt n xtnal cnfnmnt mthd s mntnd. Th plasma n th EST s bascally cntand wthn a tdal shll, wth th lctns at hgh vlcty n th pldal dctn.., n bt aund th cntln f th tus. Th lctn mtn gnats a magntc fld alng th tdal dctn, whch fms th gat ccl f adus R T. A schmatc f th cncpt s shwn n Fgu blw takn fm []. Fgu : Schmatc f EST plasma.

9 & & & & & & & Th plasma s assumd t b stngly cupld. Th analyss f plasma dynamcs n [] s basd nly n th cld-flud mdl, whch s ctanly vald f stngly cupld plasmas. It s suffcnt t say that thmal ffcts a assumd t b nnxstnt nglgbl, and thf, thy a nt cnsdd n [] n th psnt analyss.. Fundamntal Dynamcs Th td f nn adus b and ut adus R T, shwn n Fgu, can b appxmatd by an nfntly lng cylnd. Th xamnatn f th stablty ppts f th td can thn b ducd t a tw-dmnsnal pblm. As dscbd n [], th plasma s cnsdd t b wthn a tdal shll, btwn th bunday adus b and an nn bunday. Th plasma s thf cntand n a cylndcal shll. Ths cnfguatn s dntcal t th n dscbd n []. Cnsd als th tansfmatn fm an thgnal cdnat systm xˆ, yˆ, zˆ nt a cylndcal cdnat systm ˆ, ϕ ˆ, zˆ, as shwn n Fgu. Th asscatd tansfmatn matx R ϕ dscbs a tatn by an angl ϕ, such that: u x u csϕ snϕ u Rϕ, u y uϕ snϕ csϕ uϕ u u x u csϕ snϕ x Rϕ, uϕ u y snϕ csϕ u y and x Rϕ 3 y ϕ y ϕ x b Fgu : Schmatc f cdnat tansfmatn. Th z-dctn s ut f plan. Th td s appxmatd by an nfntly lng cylnd. Th quatn f mtn f a cld, ncmpssbl plasma flud s: u t α + Zαα u u E + u B α α m α α 4 In th tus, th ẑ dctn bcms θˆ, th tdal dctn. 3

10 4 wh, α s th ndx f th plasma cmpnnt lctns and ns. Th dynamcal Equatn 4 can thn b xpssd n th cylndcal cdnat systm, usng th fact that th magntc fld has a cmpnnt n th ẑ dctn nly: z t B u E m Z u u u u ϕ α α ϕ a z t B u E m Z u u u u u + + ϕ α α ϕ ϕ ϕ 5-b Snc n s ntstd n stady qulbum cndtns, th tm-dvatvs and th adal vlcty a assumd t b dntcally z. Th qulbum cnfguatn bng symmtc, w als hav 0 ϕ E. Th nly vlcty s n th ϕˆ dctn, th nly lctc fld s n th ˆ dctn nly, and th magntc fld s n th ẑ dctn. Thf, n can dp th spatal ndx f ths quantts. Assumng a sngl spcs f ns f mass m, w thn hav: B u E m Z u B u E m u Th lctc fld s gvn by Pssn s quatn,..: n n Z E ε 7 At ths stag, t s usful t ntduc dnsty scals n n, and bng all th spatal dpndnc nt nmalzd functns, f f, such that: ; f n n Z f n n and f f f 8 A spatal scal s als ntducd as th mn adus f th td, b and a nmalzd adal vaabl dfnd as: 0 ; ρ ρ b 9 Intducng als th plasma fquncs p p ω ω, : p p m n Z m n ε ω ε ω ; 9 Pssn s quatn bcms: 0 ' ' ' ρ ρ ρ ρ ρ ω ρ Φ f d m E b p 0 Th slutns f th quatns f mtn thn bcm: Φ ± Φ + ± / 4 / 4 ρ ρ ω ω ω ρ ρ ρ ω ω ω ρ p c c p c c u u wh th dfntns f th cycltn fquncs hav bn usd:

11 ω c B δ m Fnally, th magntc fld s gvn by Amp s law: Z m ; ω c δω c wth m n f u n f u B µ 3 Assumng that th plasma s naly nutal, n can st n n, wth any spac-chag ffcts ncludd n th dstbutn functns f, f. In that cas, th vlutn f th magntc fld bcms: ω p ω c f u f u 4 c Equatns 0, and 4, cmbnd wth th dfntn, fm a clsd systm and can thby b slvd. 3. Th EST Cnfguatn Accdng t f. [], th EST s chaactzd by an lctn dnsty functn f th fm: / f b f ρ 5 0 thws Th ns fm a backgund dnsty wth th sam spatal dpndnc as th lctns, albt wth a slght latv xcss fˆ, such that: f ˆ ρ f f ρ 6 In that cas, and ρ / b b b ' ' ' ˆ Φ ρ dρ ρ f ρ f ρ Φ ρ ρ fˆ Althugh th s n th xplct nfmatn gadng th ns n th EST mdl, t s assumd that thy pvd a fxd backgund snc th n quatn f mtn s nt bng cnsdd n []. It shuld als b pntd ut that th s a smwhat cnfusng statmnt n [], wh th paamts f a lag-scal EST a lstd pag f f. []. Althugh th ttal chag s 86 culmbs, th systm s calld chag-nutal. Evn thugh th latv dvatns fm nutalty wuld appa small, th dnsts cnsdd n ths ntns EST cnfguatns lad t a lag amunt f lctstatc ngy. In fact, t s pcsly ths lctstatc chag that s spnsbl f cntanmnt f th plasma and stag f th ntns 6,000 Tsla! magntc fld. At suffcntly lag dstancs fm th EST, th lctc fld gvn by th EST can b appxmatd by th z th -d tm n th multpla xpansn: 7 8 Q E 9 4πε R 7 F xampl, th last thtcal cas cnsdd n [] pag 6 uss a dvatn ˆ f

12 whch f th 86 Culmbs qutd n [] gvs a fld f E.7 0 V / m at m dstanc. It s nt knwn hw th dvlps f th EST ntndd t pvnt lctcal dschags fm spntanusly ccung btwn th EST and ts suundngs, gvn ths lvl f fld ntnsty. Nvthlss, th thtcal analyss f th EST cncpt, as dscbd n [], can b cntnud. It s usful t ntduc futh nmalzatn paamts, such that: u u v ; v 0-a ω p ω p and ωc ωc Ωˆ c ; Ωˆ c 0-b ω p ω p Instad f nmalzng th dstanc t th bunday f th td Equatn 9, n can scal th dstanc wth spct t th nn adus f th lctn shll, as n []: ρ /. In that cas, th lctn vlcty s gvn by: ρ Ωˆ v ± Ω + 4 ˆ c c f ρ ρ Intducng th cuplng paamt: ω p α c and snc th ns a fxd, th quatn f th magntc fld can smply b wttn as: Ω ˆ c α v 3 ρ ρ and thf: Ωˆ c α Ωˆ ± Ω + ˆ 4 c c f 4 ρ ρ ρ Equatns and 4 dscb th vlutn f th systm and a cmpltly dntcal t Equatns 3 and 4 f. []. Thf, th dynamcal systm f th EST as dscbd n [] has bn cvd. Ths s a nnlna systm, whch qus a numcal ntgatn pcdu. Thf, a smpl numcal pcdu f th slutn f th systm, 4 was wttn, and numcal slutns w btand f th sam paamts as ths gvn n []. F xampl, th nmalzd slutn f α. 6, f ˆ. 0, and b /. 0 s shwn n Fgu 3. Nt that th bunday adus b usd h s quvalnt t th b adus usd n []. Th slutn psntd n Fgu 3 s cmpltly dntcal t th cspndng Fgu f f. []. All th cass cmputd n [] a lstd n Tabl and hav bn cmputd wth u numcal ntgatn cd. Ths slutns a shwn n Fgus 3 thugh 8 and a dntcal wth th sults psntd n []. Hwv, th a n fgus cspndng t th last cas #4 n []. Spcal attntn has bn gvn t th slutn accuacy, and th cunt numcal schm uss a sttchd gd na th dgs f th plasma shll, such that a smth slutn s guaantd n that gn. Usng th paamts lstd n Tabl, th physcal slutns can b btand fm th nmalzd numcal slutns. Of spcal ntst a th pak lctn vlcty at th ut dg f th tdal shll, th pak magntc fld at th nn dg f th shll, and th ttal 6

13 numb f lctns and ns n th shll. Th latt can b appxmatly btand fm th lctn dnsty and th vlum f th tdal shll, nglctng th vaatn f th dnsty n / wthn th thn shll b < <. Snc and α a gvn, th plasma fquncy can b btand fm Equatn and th lctn dnsty fm th dfntn f th plasma fquncy 9. Th vlum f a thn tus R > > s appxmatly gvn by: T π RT b VlT 5 and snc th plasma s cntand wthn th shll dlmtd by, b, ts vlum s: b Vl p π RT b 6 And thf, w appxmatly hav: αc m ε N Vl p A m accuat valu f th ttal numb f lctns dnsty pfl dung th numcal ntgatn. Ths s accmplshd n MSE s cd. / 7 N can b btand by ntgatng th Th sults f all cass cnsdd by Chn n [] a shwn n Tabl. Nt that th vlcts and magntc flds can b th pstv ngatv, dpndng n th sgn f th slutn takn n Equatns and 4. Only th abslut magntuds a lstd n Tabl. Wth a fw xcptns, th cunt sults cmpa vy wll wth Chn s. Th dscpancy n N f th fst tw cass s bvusly an accuntng n []. Ou sults a cnsstnt wth th appxmatn 7. Anth dscpancy s n th fld valu f Cas #4. Snc th nmalzd slutn was nt shwn n [], t s dffcult t assss whth th ls n Chn s numcal slutn n th scalng. Snc th lctn vlcts ag vy wll f ths cas and th numcal accuacy s btt f th fld than f th vlcty pfl th vlcty gadnt bcms vy lag na th bunday at lag α, thus ncsstatng a hghly sttchd gd, n can asnably assum that Chn s numcal slutn s als accuat. Thf, t s assumd that th s a smpl scalng n []. As wll b dscussd blw, u valus f th magntc fld a sn t by stct scalng laws, thus addng valdty t th cunt sults. Cas # α f- / m R T m Tabl : Lst f cass studd by Chn n []. Cas U m/s B θ T N Chn cunt Chn cunt Chn cunt Tabl : Cmpasn f u numcal sults wth ths f Chn n []. It wuld appa that th ttal numb f patcls n [] hav bn nusly dvdd by n ths tw cass. Th cct numb can asly b btand fm th tus vlum and th lctn plasma fquncy, usng Equatn. Ntc th dscpancy n magntc fld n ths cas, whl th vlcty s n gd agmnt. 7

14 U /ω p, ω c /ω p v + v - Ω c - Ω c / Fgu 3: Numcal slutn f Equatns and 4 lctns nly f th chc f paamts: α.6, fˆ.0, /. 0 Cas # n Tabl. b U /ω p, ω c /ω p v + v - Ω c - Ω c / Fgu 4: Numcal slutn lctns nly f th chc f paamts: 3 ˆ 4 α 0, f 0, /.0 s Cas # n Tabl. b 8

15 U /ω p / Fgu 5: Nmalzd vlcty f Cas #3 cmpa wth Fgu 4 f [] ω c /ω p / 0.00 Fgu 6: Nmalzd cycltn fquncy f Cas #3 cmpa wth Fgu 5 f []. 9

16 U /ω p / Fgu 7: Nmalzd vlcty f Cas #4 f Tabl symbls a plttd vy 5 gd pnts ω c /ω p / 0.00 Fgu 8: Nmalzd cycltn fquncy f Cas #4 f Tabl symbls plttd f vy 5 gd pnts. 0

17 4. Stablty Pblms Bth th dynamcal quatns and th numcal slutns f f. [] hav bn cvd, whch wuld sm t bng sm valdty t th EST cncpt. Hwv, t s cucal t xamn th dynamcal bhav f th cmplt systm f quatns ncludng th ns! A ky assumptn baly mntnd n [] s that th ns fm a fxd backgund. Hwv, snc th magntc fld vanshs at th bunday f th td, th s n cnfnng fc bng appld t th n flud na th sufac. Snc th td s als pstvly chagd th lctstatc fc pvds th cnfnng fc f th lctns, th s a stng pulsn pushng th ns utwad. Thf, th s n mchansm f kpng th ns nsd th td. Thy wuld vy apdly vapat, thby lavng th td abslutly chag-nutal. Ths pcss wuld als lmnat th cnfnng fc f th lctns, and th whl td wuld thn apdly xpand, unabl t cntan th magntc pssu. T lk at th n dynamcs, cnsd Equatns 5-a,b. F n ntal azmuthal.., pldal vlcty u ϕ 0, and snc E > 0 f ˆ > f lctn cnfnmnt, Equatn 5-a claly ylds t u > 0, and th n gas s apdly xpandng. Th fxd n backgund assumptn s thf cmpltly nvald f n-mtn tm scals. On culd thn agu that a stabl slutn culd stll b achvd, but wth th ns n mtn. It s cla that as ns gan a adal vlcty, th Lntz fc wll als act t pvd thm wth an azmuthal vlcty s Equatn 5-b. In that cas, n must xamn th stabl slutn f th n flud wth an azmuthal vlcty. Th n slutn has alady bn dscbd n Equatn. Usng th sam nmalzatn 0-a,b, n btans f th sam dnsty dstbutn: ρ Ωˆ v ± Ω 4 ˆ c c f 8 ρ ρ Usng, ths can als b wttn as: ρ Ωˆ v ± Ω 4 ˆ δ c c f 9 δ ρ ρ A stabl slutn s pssbl f th dscmnant mans pstv,.., f: ˆ Ωc Ψ 0 30 wth Ψ 4 fˆ 3 δ Obvusly, snc th fld vanshs at b, Ωˆ c 0 and th cndtn cannt b satsfd. In fact, th wll b a gn f fnt xtnt na th ut bunday, wh ths cndtn cannt b satsfd. Th sz f ths gn dpnds n th paamts α and f ˆ. Fgus 9 and 0 shw plts f bth ˆ Ω c and Ψ f Cass #3 and #4 f Tabl. Th allwd gn wh a physcal slutn can b fund f th ns s dtmnd by th dstanc btwn th gn and th pnt at whch th tw cuvs css ach th. F Cas #3 Fgu 9, t appas that mst f th plasma n th shll wuld b unstabl t n vapatn. F Cas #4, ths gn s much small. On culd agu that f th ght chc f paamts, th unstabl gn wuld b ducd t a skn. Hwv, ths ds nt slv th pblm snc th ns wuld always b mttd fm th sufac f th EST. As th latv chag at fˆ dps, th lctstatc cnfnng fc wuld als dp, and th tus wuld apdly xpand und th pssu f ts wn magntc fld. As th EST sufac ncass dung xpansn, s wuld th at f n lss.

18 Ω c ψ 0.0 Ω c, ψ / Fgu 9: Cmpasn btwn th tw tms f Equatn 30 dtmnng xtnt f physcal slutn f th n vlcty. Rsults f Cas #3 f Tabl Ω c, ψ Ω c ψ Fgu 0: Cmpasn btwn th tw tms f Equatn 30 dtmnng xtnt f physcal slutn f th n vlcty. Rsults f Cas #4 f Tabl. /

19 Th pcss s slf-acclatng and wuld vntually lad t th xplsv lbatn f th std ngy. Hwv, th a stll futh pblms that pvnt a stabl slutn. As th tm bcms pstv and a physcal slutn f th ns can b fund, th dynamcs a n lng gvnd by Equatns and 4. In fact, th full xpssn f th cunt must b usd ncludng th n vlcty and th vlutn f th magntc fld s nw gvn by Equatn 4, ts nmalzd quvalnt. Slvng f th cmplt systm, t s fund that n physcal slutn can b btand. Dtald xamnatn f th sults vals that th pblm has nthng t d wth numcal s. It s fst ncssay t pnt ut an mptant aspct f th dynamcs f th pblm. Th a tw pssbl slutns f th lctn vlcty, dpndng n th sgn + chsn n Equatn. If th + sgn s chsn and v > 0, th fld taks n ngatv valus Ω ˆ + c < 0. If th - sgn s chsn and v < 0, th fld s pstv Ωˆ > 0 H c. Ths can asly b sn n Fgus 3 and 4. In th cas, th dynamcs a such that th abslut valu f th lctn vlcty s gvn by a dffnc btwn tw tms: Ωˆ c and th squa t tm f Equatn. Thf, usng th dfntn 3, n can wt: ρ Ψ Ωˆ δ v c + 3 Ωˆ c Hwv, at th cssv pnt wh th n flud can hav a physcal slutn, Ωˆ c Ψ, and thf: ρ Ωˆ ρ v c [ + δ ] δ Ωˆ c 33 4 n that gn. On th th hand, th n vlcty s appxmatly: ρ v δ Ωˆ c 34 snc th squa-t tm n Equatn 9 vanshs at th cssv pnt. Thf, at th cssv, th n vlcty s appxmatly twc th magntud f th lctn vlcty and wth th ppst sgn. Th at f vlutn f th magntc fld mmdatly changs sgn, and th magntud f th fld stats dcasng as n mvs twads th nn bunday. Ths als mpls that th dscmnant gvn n Equatn 30 mmdatly cass t b pstv, and th n flud has n physcal slutn. Ths bhav s xactly cnfmd by a caful xamnatn f th numcal ntgatn pcdu as n attmpts t fnd a slutn that ncluds th n flud. Thf, vn f th paamts a chsn t cncntat th dynamcs na th ut dg f th shll, th cmplt dynamcs val that n stabl slutn can b fund. It s als usful t stmat th fld qud t nfc a stabl slutn f th n flud. Ths culd b stmatd by qung that 0 at th bunday b. F xampl, takng Cas #3 f Tabl, n fnds that th must b a magntc fld at th ut bunday b. 0 n th d f: B xt 0. 9 T. Ths s cmpaabl t th fld std nsd th EST f ths cas. Thf, vn f n w t pvd an xtnal cnfnmnt fld, ts valu wuld nd t b f th sam d as th n std nsd th td. As pntd ut by Chn [4], th dscussn f n stablty can b avdd altgth by nvkng th Val thm [0], pp. 7-74, whch stats that a plasma cannt b slf-cnfnd. Hwv, w blv that th xamnatn f th dynamcal quatns lucdats th gns f th slf- 3

20 cnfnmnt pblm btt than mayb smply qutng a thm. At th sam tm, u analyss ndcats that an xtnally cnfnng fld wuld hav t b f th sam d as th std fld. Althugh ths may appa bvus fm ngtc cnsdatns, th sult s mptant bcaus t can b gnalzd t any mthd f cnfnmnt ncludng xtnal mchancal pssu. Th magntc ngy std nsd th tus s n dct latn wth th lctstatc ptntal ngy gnatd by th xcss chag f th ns usd t cnfn th lctns. In tun, t cnfn th ns wuld qu an ngy dnsty.., a pssu f at last th sam d f magntud. In th ntductn f f. [], t s mntnd that th mdl s vald nly f tm scals sht n cmpasn wth th n mtn tm scal.g. nvs n plasma fquncy; th pblm f n cnfnmnt s thf nt addssd. Ths s th nly mntn f ths mptant tm-scal lmtatn n all th pvdd dcumntatn. On th th hand, sval EPS dcumnts [,3,5,8] stat that th physcs f th EST w dvd and vfd, th EST was stabl wth n xtnal fld qud f cnfnmnt, and th was n knwn bvus nmal ccunc that wll lad t nstablty. Th can b n ambguty n th dfntn f stabl plasma snc th dscbd applcatns qu lftms f many hus, ctanly much gat than th n mtn tm scal. F xampl, fncs [3,5] xplctly mntn an EST lftm f 0 6 hus [3,5] n a vacuum nvnmnt. It s thn cla that th f EST, as dscbd n [], cannt achv lng-tm stablty. Futh dscussns wth Chn [4] cnfmd that mchancal pssu f sm st s qud f n cnfnmnt. Th alzatn f a hgh-ngy EST n a vacuum nvnmnt s thf cmpltly unfasbl. In a fllw-n dscussn [4], EPS ffcals mpld that th fncs w actually f a patal vacuum nly. Ths s nt a vald agumnt, snc th tm vacuum can b usd t dscb any nvnmnt wh th pssu s much lw than atmsphc and thf, nglgbl cmpad t th ntnal pssu gas-dynamc lctmagntc f th hgh-ngy EST. At any at, t s xtmly dffcult t magn hw hgh-ngy ESTs can b cnfnd by a mchancal pssu many ds f magntud small than th std magntc ngy. Th xampl dscbd n [3,5] mntns an EST wth 00 MJ f std ngy, cspndng t an ngy dnsty f appxmatly 0 J/m 3.., a magntc pssu f mlln atmsphs. Th xampl n [] cspnds t ntnal magntc pssus f blln atmsphs, yt n shuld xpct ths hgh-ngy EST cnfguatns culd b abslutly stabl n na-vacuum cndtns! T fnd a mchansm that can achv ths xtadnay fat wuld qu damatc changs n th fundamntal laws f physcs. Th pblm f n cnfnmnt wll b futh dscussd n Sctn Th EPS Thtcal Mdl Bf dscussng futh th mplcatns f th cld-flud mdl, t s wth nvstgatng anth thtcal mdl f th EST ppsd by EPS, as dscbd n [5,8]. Ths s an attmpt at cnstuctng a dsct mdl usng a summatn v a fnt numb f patcls. Th ns and lctns sd n spaat shlls th ns fm th nn shll, cmpsd f a lag numb f lps f adus and a lag numb f patcls alng ach lp. Th spacng btwn lctns n a lp s d, and thf, th numb f lctns n a lp s: π N lp 35 d Th numb f lps n th td s appxmatly R > > : T 4

21 πrt N lp kd 36 wh kd s th spacng btwn lps, wth k. Th ttal numb f lctns s thn: 4π RT N N lp N lp kd 37 Nt that th EST sufac aa s als gvn by: Aa 4π R T. 38 Th EPS nvstgats fst cmput th fc actng n an lctn n a sngl lp. Snc th dstanc btwn lctns s gvn by th chd d sn δθ / δθ, wh th angula dstanc s f cus δθ π / N lp, th adal cmpnnt f th lctstatc pulsn fc fm all lctns n a lp s thn: N lp / sn nδθ / F 4πε n d n 39 wh d n sn nδθ /. Th EPS nvstgats thn us th appxmatn f small angls t wt th fnal answ as: F πε d Stctly spakng, ths s a dvgng ss and a ptntal pblm f lag N lp. EPS stmats th summatn t b 5, a ath abtay chc. Usng ths small fnt valu f ths summatn, EPS thn xpsss th fc f pulsn du t all th lctns n th lp as: t F C 4πε d 4 wh C s a cnstant. Nt that ths s th fc actng n a sngl lctn, and that ths sult s btand n th lmt f lag N lp. EPS thn uss ths xpssn n th balanc f fcs. Hwv, ths s a sus f sval asns. Fst, th xpssn f th fc tslf 4 s wng, bcaus th tuncatn f th summatn t small numbs s nvald. Th summatn n 40 s lgathmcally dvgnt, and th xpssn f th fc shuld thn nclud a tm lg / d, whch s nt nglgbl. Scnd, and vn m mptantly, ths xpssn ds nt tak nt accunt th cntbutn fm nghbng lps. Snc th dstanc btwn lps s assumd by EPS t b f th sam d as th dstanc btwn lctns n a lp k d, wth k, ths cntbutn s fa fm nglgbl and s, n fact, f th sam d. Th cct valuatn f th fc s shwn lat n ths sctn, but an stmatn f ths cntbutn can b gvn blw. T nclud th lctns n nghbng shlls, n can plac ach lctn n th lp by a tub xtndng n th ẑ dctn th maj adus f th td. Ths tub has a damt d, and th ffctv dnsty f lctns n ths tub s π d kd / 4. Th slutn f Pssn s quatn f ths chag cnfguatn ylds th fllwng fld stngth at a dstanc d fm th tub: 4 E 4 πε k d d 5

22 Usng ths valu f th adal fc n an lctn and summng v all tubs alng th lp: F πε lp 8 k d N n / sn nδθ / d Ths tm, th sum can b valuatd snc th sn n δθ / cancl ut. Th summatn bcms: N lp / N lp π n 4 4d Th fc n th lctn thn bhavs as d, nt as d and s thf much lag. Th s mad by EPS a thn futh cmpundd whn cmputng th fc du t th magntc fld. Ths s xpssd n [5,8] as: vµ N Fm v B 45 πrt wh v / d s th cunt n a sngl lp, and RT s th adus f th td. Th s sm cnfusn as t th natu f ths fld. In [5,8], th magntc fld s takn as an ntatng fld causd by th ac cunt usd t cat th EST. In that cas, t s nt cla why th ac cunt shuld als b th sam as th lp cunt, unlss all th ac cunt gs nt th td. In that cas, th magntc fld catd by th cunt lp shuld b: µ B 46 π and th td adus R T shuld nt appa n ths xpssns. Th s als cnfusn abut th manng f th numb N appang n 45. Ths shuld actually b th numb f lps: πrt Nlps 47 kd but n fact, EPS uss th pduct N N lp N lp n Equatn 45. Th sultng xpssn n [5,8] ylds th fc actng n all lctns n a lp nstad f n ach lctn. Th cmpasn wth th lctstatc fcs Equatns 4 43 s thf cmpltly nvald. Th pp way t valuat ths fcs can nw b dscbd n th sam lmt f lag N. Assum that th lctns fm a cylndcal shll f thcknss d. Intgatng Pssn s quatn 7 f th lctn cmpnnt nly, th lctc fld actng n ach lctn bcms: n E d 48 ε wh n s f cus th lctn dnsty n th shll. Usng th ttal numb f lctns 37 and th shll vlum π dπr, n can asly av at th fnal xpssn: T E and ε k d F n ε k d Ntc that ths sult s ndpndnt f th assumd thcknss. Th fld du t th n shll can b cmputd n a smla fashn and addd lna suppstn t yld th ttal lctstatc fc n an lctn. Th magntc fld nsd th td s appxmatd by a slndal fld: dnl B µ µ 50 dx k d 6

23 7 : d k d k B ε µ v v 5 Nt that B B z. Th magntc fc n ach lctn s thn: c d k d k F M v v ε µ 5 Nt that th at f magntc t lctstatc fcs s / c v, as xpctd. Ths xpssns can nw b usd t valuat th balanc f fcs, and th cntbutn f th ns can b sn t b fˆ tms th fc du t th lctns 49. Th balanc f fcs f th lctns s: v v m B d k f + ˆ ε 53 Anth applcatn cnssts f xamnng th stablty f th n shll nsd th lctn td. By symmty, th s n lctnc cntbutn t th lctstatc fld, and th balanc f fcs F n +F t F mag n ach n f pssbl ducs t th fllwng qualty: B m d k f v v + ˆ ε 54 wh d d s th n spacng. Lt us nw ntduc th n Lam adus: L c m d B m v v v ε 55 wh w hav agan usd Equatn 5 f th magntc fld. Th cntfugal psud-fc n Equatn 54 can nw b bught nt th fm: c d k m L v v v ε 56 Th balanc f fcs f th ns 54 thn mpls: ˆ c d k d k f L v v ε ε 57 Bth th cntfugal and magntc fcs a f th d f / c v v and can b nglctd cmpad t th lctstatc fc. Thf, th s n stabl physcal slutn f th n shll. Of cus, ths was als dmnstatd n Sctn 4. It s als ntstng t xamn th cnsquncs n a mchancal cnfnmnt schm. Supps that th n stablty s pvdd by a mchancal gas cntanmnt vssl pssu. In that cas, th cnfnng fc appld t ach n must balanc th lctstatc pulsn. Th balanc f fcs thn bcms: ˆ n cnf F B m d k f + + v v ε 58 Nglctng th tms f d / c v v, n can thn xpss th qud cnfnmnt fc f th nt EST:

24 ˆ 4π RT F cnf EST F cnf n N n f ε kd d 59 wh th ttal numb f ns s fˆ tms th ttal numb f lctns, gvn by Equatn 37. Th ttal cnfnmnt fc can b dvdd by th aa 38 t gv th cnfnmnt pssu: Pcnf ˆ f ε k d d 60 Ths xpssn can b manpulatd futh. Usng 5, th lctn spacng can b xpssd as a functn f th std magntc fld: v kd µ B 6 In th appxmatn f naly dntcal spacng d d, w thf hav: v B ε k d d ε µ µ 6 Th cnfnmnt pssu bcms: Pcnf c B fˆ v µ 63 W can als xpss th at f spds as a at f kntc t st mass f th lctn: v E 64 c mc and th fnal xpssn f th cnfnng pssu bcms: 5 P ˆ cnf f Pmag 65 E [ kv ] Nt that f ˆ and unlss th lctns a ultalatvstc n whch cas ths analyss must b mdfd, th qud cnfnmnt pssu nds t b sval tms lag than th magntc fld pssu nsd th tus. Thf, vn f n culd nvsn applyng a cnfnmnt fc n th ns, th qud magntud f ths fc s such that th cncpt f ngy stag.., a lag ntnal magntc fld s cmpltly mpactcal. Th latnshp 65 s vy usful bcaus f ts gnalty and ndpndnc wth spct t th EST dmnsns. W cnclud that th thtcal mdl ppsd by EPS s svly flawd n many aspcts and ds nt pvd a stabl slutn. Th nly cct thtcal mdl at ths stag s th cldflud mdl, whch s dscbd n [] and als n th psnt analyss. Hwv, Chn s mdl s ncmplt, snc th n mtn has nt bn cnsdd. M cmplx dscptns f th plasma suggstd by Chn.g. phas-spac appach wuld nt chang th fundamntal ppts f th mdl, and wuld thf nt slv th pblm f n cnfnmnt. F that, a cmpltly dffnt physcal mdl may b ncssay. 6. Scalng Ppts Althugh t has bn dtmnd that th EST as dscbd anywh n th ltatu [-8] s nt a stabl stuctu, th a th clams t b valuatd. It s fst ncssay t xamn th It s nt cla hw EPS ppss t apply a mchancal.g. cntanmnt vssl gas pssu fc t th nn shll f th ns wthut affctng th ut shll f th lctns. As s shwn n Sctn 7, t s nt pssbl t pvnt cllsns btwn lctns and th patcls f put nt cntact. 8

25 scalng ppts f th stuctu, whch wll hlp n th xamnatn f th clams f []. Th ncmplt slutns, fund wth nly th lctn dynamcs, a analyzd as functn f th stuctual paamts α and f ˆ. Th dmnsns f a lag-scal EST a chsn as fllws: 0.5 m, /.05, R.5 m b T It s thf a scald-up vsn f Cass #3 and #4 f Tabl. A ss f numcal slutns can thn b btand f vaus valus f α and f ˆ and chaactzd as a functn f th maxmum magntc fld B max at th nn bunday and maxmum lctn kntc ngy E,max at th ut bunday b. Th sults a shwn n Fgu f a cnstant maxmum fld, chsn t b appxmatly 6,000 Tsla. Th sults f a cnstant maxmum lctn ngy chsn t b.5 MV sc a shwn n Fgu. f B max 6,000 T α x 0 Fgu : Scalng latnshp btwn cuplng paamt α and chag paamt f ˆ f cnstant magntc fld. f- x E,max.5 MV α x 0 Fgu : Scalng latnshp f cnstant lctn ngy. 9

26 On shuld pnt ut that ths chc f lctn ngy s cmpltly abtay and s nly ntndd t dmnstat th bvus scalng latnshp shwn n Fgu. A lag lctn ngy allws n t cv a wd ang f valus f th α and fˆ paamts. Anth chc, such as th 0 kv usd blw, wuld gv a ln paalll t th n shwn n Fgu. Th physcal answs f th magntc fld and lctn ngy wuld als dpnd n anth stuctual paamt, th latv shll thcknss.., th at b /. F fxd valus f th 8 cuplng and chag paamts ˆ α 0, f 0, th maxmum fld and ngy a cmputd f sval valus f th at b /, and th sults a plttd n Fgus 3 and α0, f -0-8 E,max MV / - Fgu 3: Scalng f lctn ngy vsus shll thcknss α0, f -0-8 B max.09x0 5 ln / 0000 B max T / Fgu 4: Scalng f magntc fld wth sz at. Bth data and appxmat ft shwn. 0

27 It appas thn that th lctn ngy scals xactly wth th nmalzd shll thcknss b /. Th scalng bhav f th magntc fld s m cmplx, and th fmula shwn n Fgu 4 s an appxmatn. Snc th pak lctn ngy s btand at th ut dg wh th fld vanshs, a smpl scalng latnshp can b btand f th vlcty. Fm, w hav at th bunday: fˆ / b v 66 b cnvtng t physcal vaabls and xpssng th ngy E v, w fnally btan: / ˆ b f / m E,max α 67 whch s cnfmd by bth Fgus and 3. Cmbnng all th data, th fllwng appxmat scalng latnshps can b btand: ˆ 5, max b α f E B max b ln α 0 ˆ 8 0 f / kv Tsla Althugh th lctn ngs n th EST can b qut hgh, t s assumd that th tmpatu s vy lw.., th lctns a mnngtc. In that cas, th cuplng paamt can b lag: 4 [ π n] / 3 Γ > 70, wth R 70 3 kt 4πε R Ths mpls that T K and psnts anth ngnng challng. F xampl, fs. [,3] dscb EST fmatn wth an njctd lctn bam. It s nt cla hw EPS ppss t gnat an lctn bam and captu t n a shll wth a latv ngy spad f lss than 0-8. Th s als sm cnfusn gadng th EST pductn mthd; n [5] an lctc ac s mntnd nstad. Th ac shuld hav an vn hgh ngy spad Engy Stag and Lftm Th EST paamts lstd n [] a dffnt fm th cass studd abv and n [], and snc th s sm mssng nfmatn, n cannt xtact th xact lvant quantts. Nvthlss, th ght ds f magntud can b btand. Th lvant paamts a lstd n Tabl 3. Damt, maj, T : 0.63 m Damt, mn, : 0. m Elctn ngy: 0 kv Ttal Mass: kg Magntc Fld: 6,000 Tsla Engy Lss, Radatn:.6 J/sc Engy Lss, Cllsns: J/sc Ttal Engy:.9 0 J Tabl 3: EST paamts lstd n [].

28 It s nt cla hw ths valus hav bn btand, and th a sm ncnsstncs. Fst, th lst mntns damts, whl symbls f adus a usd. Assumng a hydgn plasma, th ttal mass gvs a ttal numb f ns and lctns: N N. 0. Snc th s n nfmatn cncnng th shll thcknss, t s ncssay t assum a valu. Chsng b. 05, th plasma vlum s f th d f m, and th lctn dnsty s 3 thf: m. Th plasma fquncy s thn.3 0 s, and th n cspndng cuplng cnstant assumng th nn adus s 0. m must b α.6 0. Th plasma fquncy als cspnds t an quvalnt fld f m ω p / 3 T. Ths s n makd cntast wth th clamd valu f 6,000 T gvn n []. Althugh sm numcal slutns nt shwn hav a pak valu f th nmalzd cycltn fquncy Ωˆ c >, thy hav all bn fund t b f th sam d. Ths s wh th scalng latns can b usful. Invtng 68,69, n fnds: / ˆ max f E kv ln / Bmax T / α 5 Bmax T / Emax kv [ ln ] / Cnsdng tw typcal valus f th latv shll thcknss,.0 and.05, and usng th clamd valus f E max and B max n Tabl 3, n fnds: / α f Tabl 4: Typcal stuctual paamts satsfyng ngtc valus n Tabl 3. ω p 8 A sach n paamt spac n th cas /. 05 yldd th bst numcal slutn.., 5 6 n wth E, max 0kV and B max 6,000 T f α.9 0 and ˆ f 5 0, vy cls t th valus pdctd by th scalng laws. Ths valus f th cuplng cnstant a 8 ds f magntud gat than th n dvd fm th mass f th systm! Futhm, ths valus 9 3 cspnd t an lctn dnsty n 6 0 m. Ths s hgh than sld-stat dnsts th numb dnsty f sld cpp s 8 0 8, whl t s f wat! Any n th scalng latns 6,63 wuld sd n th scalng f th magntc fld wth spct t th shll sz. As sn n Tabl 4, th ffct f sz s lmtd. On must thn cnclud that th valus f magntc fld, ngy, and mass gvn n [] a ncmpatbl wth ach th, as wll as wth th numcal slutns. Th nxt stp n th analyss s t xamn th at f cllsns n th systm. Snc th lctns and ns a nt spaatd n th mdl dscbd n [], th lctns wll ls ngy thugh Culmb cllsns wth th ns. Th nly way ths cllsns culd b pvntd s f th ns w mvng at th sam vlcty, n whch cas th wuld b n cunt gnatd, as wll as n magntc fld. An xpssn f th at f ngy lss s fund n [], p.4: It s mphaszd h that th pblm s nt s much wth th abslut scal f th dnsty, althugh astnshng n ts wn ght, but wth th sv ncnsstncy f th paamts.

29 4 de Z b ln Λ n H x, m / m 73 dt 4πε m wh E s th ngy f th lctn, b / / m kt, and ln Λ s th Culmb lgathm. F lctn-n cllsns: ln Λ 3.3 ln n + ln T 74 Th functn H s dfnd by: / H x, µ f x / x π + µ xp x 75 / m E wth x b u 76 mt wth E and T xpssd n th sam unts. F th paamts lstd n Tabl 3 and assumng a 5 hydgn plasma at T K, w hav x F lag valus f x, H / x. Usng th 3 lw valu f dnsty dvd fm th mass n.7 0 and ln Λ 7. 35, th fnal sult s: de 0 J/s 7000 kv/s 77 dt Ths s th at f ngy lss f ach lctn. T fnd ut th at f ngy lss f th nt tus, t s ncssay t multply by th ttal numb f lctns, whch s appxmatly N. 0. Thus, th ttal at f ngy lss s appxmatly: cl E EST f 0 9 J/s th nt EST 78 Expssn 77 mpls that t taks appxmatly.4 msc f ach lctn t ls ts ngy. Ths cspnds t 300 bts aund th mn ccl. Althugh th man-f path s lag cmpad t th dmnsns f th systm and th plasma can b cnsdd cllsnlss, th nglct f cllsns s vald n th tm-scals chaactstc f th plasma dynamcs. Hwv, f n lks at lng-tm stablty, th abslut at f cllsns must b cnsdd. Anth way t valuat th at f cllsns s t lk at th Ruthfd scattng css-sctn,.., th sngl bnay cllsn f a dflctn angl f 90 m. F xampl, usng xpssn 6..5 f f. [] 9-0 f f. [0], th css-sctn f such an vnt s appxmatly σ m. Ths lads t a fquncy f vnts 0 5 s -. Snc nly a factn δ f th lctn ngy s tansfd dung ach such cllsn, th tm qud f cmplt laxatn f th lctn ngy bcms 0 - sc. Hwv, at ths plasma cndtns, th mpact paamt f 90 dflctns p 90 s vy small. Th at f lag t small angl cllsns s thn [0]: p << 79 λd and th xpssn 73 f multpl cllsns s m apppat. Thf, n xpcts th at f ngy lss t b gvn by Equatn 78, but t culd b much lag f th dnsty w t b cnsstnt wth th scalng latnshps btand fm th numcal slutns. Nt that EPS clams a at f 0-9 J/s f th nt EST, a dffnc f 8 ds f magntud! T mphasz th cnsquncs f ths, cnsd th fllwng. Assumng th at f ngy lss s ndd gvn by th valu n Tabl, th at f ngy lss p lctn bcms: de J / s 80 dt f th nt systm. 3

30 Ths at wuld b btand f a numb dnsty f th d f 0 3 m -3. Snc th at scals lnaly wth th numb dnsty f ns and f th numb dnsty was f th sam d as th numb dnsty f cpp, th at f ngy lss p lctn wuld b: de J/s.3 0 kv/s 8 dt It wuld thf tak sc f th 0 kv lctn t ls ts ngy, whch mpls that such an lctn bam wuld b abl t g thugh km f sld cpp wthut stppng! On culd agu that th lss-at by cllsns gvn by Equatn 73 s a lcal quantty and shuld b valuatd thugh vlum ntgal, ath than usng glbal and maxmum valus f th dnsty and lctn ngy. Althugh th dnsty vas as /, th ang f vaatn s small snc b /. 05, thby pvdng at mst a 0% chang. Hwv, th vaatn f th lctn ngy s much m apd and culd mak a dffnc. Thf, th numcal ntgatn pcdu was mdfd t nclud a vlum ntgatn f th cllsnal lsss. On shuld pnt ut fst that f m / m 000, th functn Hx has a maxmum na x 3 and can b appxmatd as / x bynd that. Usng lcal valus x 3 nly n th ntgatn, n fnds that th vlum ntgatn gvs a sult wthn 5% f th sult 78 btand fm glbal and pak valus. Usng x 0 valus nly n th ntgatn gvs sults lag by n d f magntud. Bcaus th Culmb cllsn css-sctn ncass as th lctn ngy dcass, usng th pak valu f lctn ngy n 73 f a glbal stmat ylds a valu that undstmats th actual ats f ngy lsss by cllsns wth ns. It s thf cla that th valu clamd n [] s unphyscal n a damatc way f th ns and lctns a c-lcatd, as ndcatd by th thtcal mdl f th EST f []. It was suggstd by EPS that th EST sufac shuld b cnsdd as a stngly cupld systm wth cllctv bhav, and that patcl-patcl cllsns d nt apply. Hwv, ths suggstn s at dds wth th fundamntal physcs nvlvd. It s tu that th lctns n th cllctv stat d nt clld wth ach th, and f th ns als fm a stngly cupld plasma, n-n cllsns a als mvd. Hwv, th ns and lctns tgth d nt fm a stngly cupld plasma! Th cndtn 70 f stng cuplng s nt satsfd f th lctn-n systm, bcaus th latv dffnc n ngy 0 kv btwn ns and lctns s fa t lag. Thf, hgh-ngy lctn-n cllsns dfntly ccu. Th stuatn can b btt vsualzd wth th fllwng analgy. Cnsd a mtallc sld wth a cnductn band an nc cystal. Th systm can b thught f as an n cmpnnt n a lattc stngly cupld plasma wth a nutalzng backgund f lctns. Cllsns btwn ns d nt ccu n ths systm; nly cllctv xctatns n ths cas, phnns a natually psnt. Yt, f a 0 kv atmc bam s dctd at th sld, n s guaantd t hav a lt f ndvdual patcl cllsns wth a sgnfcant amunt f ns jctd fm th sld. Th cas f th EST s smla. Ins and lctns wuld spctvly fm lattcs at na sld dnsty, but mpactng ach th at hgh vlcts. Bcaus th latv kntc ngy btwn th tw cmpnnts s lag, patcl cllsns fquntly ccu and a cctly dscbd by Equatns 73 thugh 8. Th adatv lsss shuld als b cnsdd. Th at f cycltn mssn can b btand fm f. [] agan p. 68: 7 3 P 6. 0 n T B W/m 8 c kv Althugh sld cpp s nt a plasma p s, th Ruthfd css-sctn wuld stll apply at ths hgh ngs. Stctly spakng, snc th lctns a mn-ngtc, n culd us th adatd pw mttd by a sngl lctn, usng classcal fmulas such as 4.3 f [3]. Hwv, ths ds nt affct th dscussn. 4

31 H, T E, and ths gvs a at f ttal ngy lss by cycltn mssn f th EST: 5 B J/s 83 E cycl EST If th fld s 6,000 T as clamd, th at f ngy lss by cycltn mssn s: cycl EST 4 E.8 0 J/s 84 agan a dffnc f 4 ds f magntud wth th at clamd n [] s Tabl 3. Hwv, ths wuld b a gss vstmat f th mttd pw, snc th hghst lctn vlcts a n a gn na th ut dg f th plasma shll wh th magntc fld s small. T cctly valuat th cycltn adatv pw, th numcal ntgatn schm was mdfd t nclud an ntgatn f th cycltn pw. A slutn f th dynamcal quatns was chsn such that th maxmum fld was 6,000 Tsla and th maxmum lctn ngy 0 kv, f th sam sz as th EST cnfguatn gvn n Tabl 3. Of cus, th cspndng plasma dnsts and ttal numb f lctns w much hgh than ths dvd fm th EST mass, fllwng th appxmat scalng latnshps dscussd abv. Th ttal mttd pw f ths slutn was: cycl 7 E 4 0 J/s 85 EST whl Equatn 8, usng th maxmum valus glbally, wuld gv 0 J/s. Thf, th s a lag ductn n mttd pw whn th clatn btwn lctn ngy and magntc fld s takn nt accunt. Nvthlss, th slutn gvs a cycltn pw much lag than th n qutd n []. Hwv, t wuld bcm f th sam d f th sults w scald t match th dnsty btand fm th EST mass. Thf, th cycltn pw mttd may b cct, but th qutd magntc fld valu s n sus dsagmnt wth th numcal slutns and th scalng laws. Th ttal ngy can b valuatd as th magntc fld tms th ttal vlum f th tus,..: B E EST π b RT 86 µ Assumng B 6,000 T, th ttal ngy s J. Evn f th szs gvn n Tabl 3 w th actual damts nstad f ad, th ngy wuld b 7 0 J. Ths s stll lag than th valu mntnd n [] and lstd n Tabl 3, yt anth ncnsstncy n th clamd sults. It can b asly vfd that th kntc ngy f th lctns s nglgbl cmpad t th magntc ngy. If th ats f ngy lsss gvn n [] w usd, th EST wuld hav a lftm f 50,000 yas! Usng th lss at fm Equatn 78, th EST wuld last abut 7000 sc, stll an xtadnay achvmnt n tms f plasma stablty. Hwv, th tu cllsnal lftm s gvn by th lctn ngy lftm tslf, whch accdng t Equatn 77, s nly n th d f a mllscnd. Thf, t s cla that cllsns btwn lctns and ns wuld pvnt th EST t achv th vy lng clamd lftms. Of cus, ths s a pblm f th lctns and ns a bth lcatd n th sam gns, whch s a ky assumptn n th mdl dscbd n []. Snc ths mdl was patdly mntnd as an ndpndnt dscptn f th physcs f th EST, t s thn asnabl t assum that t s th cunt stat-f-th at n th thtcal dscptn f th EST. In fact, as mntnd n Sctn 4 gadng th n dynamcs, th s n cla dscptn f th n ppts anywh ls n th ltatu [-3]. Th nw vsn f a dcumnt [5] xplctly mntns th spaatn f ns and lctns. Ths s a nw addtn, mad aft a plmnay vsn f th psnt pt was mad avalabl t EPS. Spaatn f ns and lctns s pssbl n sm cass. As pntd ut by Swad [,3], stngly cupld n-cmpnnt plasmas can ganz thmslvs n shll-lk stuctus, as 5

32 dmnstatd by Glbt t al [4] f an n plasma n a Pnnng tap.., cnfnd by an stng xtnal magntc fld. Stngly cupld plasmas a nw cmmnly catd n th labaty, althugh always n th psnc f cnfnng fld cnfguatns such as Pnnng and Malmbg taps. Stng cuplng s ndd a ky assumptn f th EST mdl, thby mplyng that th ns and lctns fm tw dstnct cmpnnts, ach n at lw tmpatu. In fact, th cuplng cndtn Γ > 70 f EST s typcal f a sld-lk plasma phas, cspndng t Culmb cystal stuctus [5] ath than flud-lk shlls, and t s pbably m stngnt than ncssay. Du t th lag mass dffnc btwn ns and lctns, n culd thn xpct n and lctn shlls t b spaatd. Hwv, n n has yt alzd such a tw-cmpnnt systm. Futhm, th alzatn f stngly cupld plasma cnfguatns s ath dlcat and cuntly qus clng t vy lw tmpatus n th psnc f xtnal cnfnng flds. T xpct that such stuctus can b fmd fm hgh-ngy bams lctc acs s yt anth fmdabl clam that dmands xtnsv pf. Fnally, th n shll wuld als b fundamntally unstabl, snc th systm s glbally chagd wth th latv xcss f ns. Thf, th n shll wuld nt man spaat fm th lctns, and agan, th cllsns btwn lctns and ns wuld b unavdabl. Th n cnfnmnt pblm mans a ky ssu wth mptant cnsquncs. F xampl, supps that th ns sd n a shll nt t th lctn shll. If mchancal pssu s qud t cnfn th ns, as nw suggstd by EPS [4], th must b a sgnfcant numb dnsty f nutals at th dg f th n shll and thf wthn th lctn shll. Thn, th cllsns btwn lctns and nutals bcm nvtabl, and at ths ang f lctn ngs 0 kv, th pvus classcal analyss mans lvant. Thf, xcssv cllsnal ngy dsspatn wth bth ns and nutals, mans an mptant pblm that pvnts any pssblty f lng lftms f any EST cnfguatns. Aft mntnng th nd f a cnfnng fc pvdd by an xtnal pssu, EPS agan mdfd th thtcal mdl and statd [4] that th s a fxd n backgund. Th latt wuld b pvdd by an nzatn pcss f th nutal patcls dffusng nsd th EST. In that cas, lctn-nutal cllsns must b cnsdd. Th ang data [6] f 0 kv lctns n a spcfs a fgu f 9.7 MV.cm /g fm whch n can dv th lctn-nutal lastc cllsn css-sctn t b f th d f 0-0 m. If th nutal dnsty w f th sam d as th n dnsty n th EST, th lctn lftm wuld b 0-8 sc. If nstad, nly a patal pssu f µt s psnt, th lftm wuld b appxmatly 50 msc. Ths s stll sgnfcantly lss than th 0 6 hus clamd n []. At ths pnt, t bcms mptant t valuat th actual nutal patal pssu that wuld b qud t mantan th EST n ths scna. Ths was alady dn n Sctn 5, thby ladng t a latnshp btwn th xtnal pssu and th magntc fld pssu, Equatn 65. Anth appach s nw usd, and agan, th cas f hgh-ngy EST Tabl 3 s xamnd. Th lctstatc cnfnmnt f th lctns fm th xcss nc chag s spnsbl f th psumd gnatn f th lag magntc flds nsd th EST 6,000 T n ths cas. Fm ngtc cnsdatns, f th fld ngy s much lag than th kntc ngy f th plasma, n shuld xpct a balanc btwn th magntc fld ngy dnsty B / µ and th ngy dnsty pssu f th lctstatc fld ε E /. As th fm dcass na th dg, th latt ncass. Ths ndcats that th lctc fld at th dg f th shll s f th d f 5 0 V/m. Anth valuatn cms fm th ttal chag f th EST, qutd n [] t b 86 Culmbs. Assumng a cylndcal shap, th fld wuld b: It s nt a dvd cndtn, as mpld n sm f th EST ltatu. 6

33 E Q πr / T V / m πb Ths tw appxmatns a n faly cls agmnt. On can thn cnsd th stady-stat dynamcs f th ns na th dg f th shll. Th magntc fld vanshs n that gn, and th lctc fld pvds a lag acclatn, whl th cllsns wth th nutals pvd a fctnal fc,..: Gv E v ν nv G[ m wh x s th dstanc alng th n path. Snc th cllsn fquncy btwn ns and nutals als dpnds n th n vlcty ν n N nσ nv, an appxmat slutn can b fund f th fm: v E m x / Λ Λ 89 wh Λ N nσ n s th man f-path. Nt that as th magntc fld vanshs twads th dg f th shll, th Lam adus bcms lag, and th cllsn man f-path bcms th lvant lngth scal. Th css-sctn s typcally f th d f 0-0 m, and f th gas pssu s µt at 300 K, th man f path s,500 mts. Th ns wuld thf stam away fm th EST at xtmly hgh vlcts wthut stppng, thby nvaldatng th cncpt f patal vacuum cnfnmnt. If th gas dnsty s that f th psumd n backgund nsd th EST, th man f path bcms 300 µm, whch s a small factn f th shll thcknss. It wuld appa thn that ths s th ang f nutal dnsty qud. A m accuat dtmnatn s shwn blw. It s ntstng t als pnt ut that th vlcty gan dung ths dstanc s accdng t 89, abut m/s! Ths valu s appxmatly th sam as th n btand fm th E B dft vlcty, usng pak valus f bth flds. It was pvusly mntnd hw ngtc cnsdatns dctat that th nutal gas pssu, n d t cnfn th ns, shuld b f th sam d as th lctstatc ngy dnsty n d t cnfn th ns;.., f th d f atmsphs f th EST dscbd n [] and Tabl 3. A smla sult can nw b btand fm kntc cnsdatns. Equlbum s achvd whn th mmntum fluxs at th EST ntfac can b achvd. Th ns nsd th EST a acclatd t hgh vlcty fm th lctc fld, and th vlcty at th ntfac can b btand fm 89 usng an avag v a man f path. Th psntatv dstanc s thf Λ / Λ / 3, and th avag n vlcty at th EST bunday s: E v Λ 3 90 m L Th n mmntum flux acss th ntfac s N m v v and must b balancd by th cspndng flux fm th nutal backgund s Fgu 5,..: n / n x n n N m v v N m v N kt 9 wh th statstcal avag v v th nutal patcl dstbutn has bn usd. Usng 87, th qud nutal pssu thn bcms Pa, 900 mlln atmsphs; agan f th sam d as th magntc fld pssu. Evn usng th Lam adus f pak magntc fld ds nt actually alt th cnclusns. 7