Magnetic Shielding of the Acceleration Channel Walls in a Long-Life Hall Thruster

Transcription

1 46th AIAA/ASME/SAE/ASEE Jont Propulson Confrnc & Exhbt July 2010, Nashvll, TN AIAA Magntc Shldng of th Acclraton Channl Walls n a Long-Lf Hall Thrustr Ioanns G. Mkllds, * Ira Katz, Rchard R. Hofr, Dan M. Gobl, Jt Propulson Laboratory, Calforna Insttut of Tchnology, Pasadna, CA, and Krst d Grys, ** Alx Mathrs Arojt, Rdmond, WA In a Qualfcaton Lf Tst (QLT) of th BPT-4000 Hall thrustr that rcntly accumulatd >10,000 h t was found that th roson of th acclraton channl practcally stoppd aftr ~5,600 h. Numrcal smulatons of ths thrustr usng a 2-D axsymmtrc, magntc fld-algnd-msh (MFAM) plasma solvr rval that th procss that ld to ths sgnfcant rducton of th roson was multfactd. It s found that whn th channl rcdd from ts arly-n-lf gomtry to ts stady-stat confguraton svral changs n th nar-wall plasma and shath wr nducd by th magntc fld that, collctvly, consttutd an ffctv shldng of th walls from any sgnfcant on bombardmnt. Bcaus all such changs n th bhavor of th onzd gas nar th rodng surfacs wr causd by th topology of th magntc fld thr, w trm ths procss magntc shldng. A 0,1,2 = fttng coffcnts for lctron xctaton nrgy loss B = magntc nducton fld C 0,1,2,3 = fttng coffcnts for th H&W shath c 0,1,2,3 = fttng coffcnts for f θ(k) E = lctrc fld = lctron charg, C Ĩ = nlastc on drag forc dnsty I b = total on bam currnt I = on spcs currnt f = nutral vlocty dstrbuton functon f θ(k) = fttng functons for th angl (nrgy) dpndnc of th sputtrng yld j = lctron currnt dnsty j = on currnt dnsty K T = fttng coffcnt for f K K 0 /2=frst Maxwllan cross-ovr Nomnclatur Q T = thrmal hatng of lctrons q = on charg (Z) p () = on (lctron) prssur, Pa 1 Grk Symbols α= spatally varyng multplr for ν α βˆ = magntc nducton fld unt vctor A, V =computatonal ara and volum ncrmnts φ =shath potntal drop at channl walls ε = roson rat ε 0 = prmttvty n vacuum ζ s =onzaton potntal of spcs s η = total or ffctv lctrcal rsstvty η b = currnt utlzaton ffcncy η = lctron-on lctrcal rsstvty η m = mass utlzaton ffcncy θ = on ncdnc angl * Mmbr of th Tchncal Staff, Elctrc Propulson Group, 4800 Oak Grov Drv, Pasadna, CA, 91109, Mal Stop , Assocat Fllow AIAA. Group Suprvsor, Elctrc Propulson Group, 4800 Oak Grov Drv, Pasadna, CA, 91109, Mal Stop , Snor Mmbr AIAA. Mmbr of th Tchncal Staff, Elctrc Propulson Group, 4800 Oak Grov Drv, Pasadna, CA, 91109, Mal Stop , Snor Mmbr AIAA. Scton Staff, Thrmal and Propulson Engnrng Scton, 4800 Oak Grov Drv, Pasadna, CA, 91109, Mal Stop , Assocat Fllow AIAA. ** Managr, Elctrc Propulson Systms, P.O. Box 97009, Rdmond, WA, , Mmbr AIAA. Program Managr, Elctrc Propulson Systms, P.O. Box 97009, Rdmond, WA, , Mmbr AIAA. Copyrght 2010 by th, Inc. Th U.S. Govrnmnt has a royalty-fr lcns to xrcs all rghts undr th copyrght clamd hrn for Go

2 K = on kntc nrgy k B = Boltzmann s constant L=acclraton channl lngth at BOL R=radus of acclraton channl cntrln at BOL R () = lastc on (lctron) drag forc dnsty S = scondary lctron yld t = tm T ()(β) = on (lctron) (on spcs β) tmpratur u = lctron drft vlocty u B = Bohm spd (2k B T /m ) ½ u = on drft vlocty u β = drft vlocty of on spcs β v = partcl vlocty Y=sputtrng yld Z = charg stat M β = Mach numbr, v /(k B T β /m β ) ½ m () = mass of on (lctron) m& =bam (anod) mass flow rat b(a) n& = lctron-mpact onzaton rat nˆ = normal unt vctor n () = on (lctron) numbr dnsty n β = numbr dnsty on spcs β r,z=radal, axal dmnsons κ = lctron thrmal conductvty Λ β= coulomb logarthm for -β par λ mfp = man fr path for (classcal) collsons ν α = non-classcal collson frquncy ν B = Bohm collson frquncy ν = total collson frquncy ν = lctron-on (coulomb) collson frquncy ν n = lctron-nutral collson frquncy ~ ν = lctron-nutral onzaton collson frquncy n ν w = lctron-wall collson frquncy ν β = coulomb collson frquncy for -β on par Φ s = lctron nrgy loss from nlastc collsons wth spcs s ϕ=non-dmnsonal xctaton nrgy loss φ = plasma potntal φ T = thrmalzd plasma potntal χ= frst conjugat harmonc functon Ψ=Mach numbr functon for -β coulomb collsons ψ= scond conjugat harmonc functon ω c = lctron cyclotron frquncy Ω = lctron Hall paramtr I. Introducton all thrustrs provd an attractv combnaton of thrust and spcfc mpuls for a varty of nar-arth H mssons. In many cass ths systms allow for sgnfcant rductons n propllant mass and ovrall systm cost compard to convntonal chmcal propulson. Th rang of thrust and spcfc mpuls attanabl by ths thrustrs maks thm applcabl also to a varty of NASA scnc mssons. Scnc mssons howvr, such as thos consdrd by NASA s Dscovry Program for xampl, rqur typcally wdr throttlng and longr thrustr throughput compard to nar-arth applcatons. 1,2 A crtcal prcvd rsk about th applcaton of Hall thrustrs to scnc mssons s thr throughput capablty. Spcfcally, thr ar two major war procsss known to xst n Hall thrustrs that can lmt thr applcablty to NASA mssons, roson of th acclraton channl and roson of th hollow cathod. Multpl approachs ar currntly bng dmonstratd that addrss ths challng. NASA s Scnc Msson Drctorat In-Spac Propulson Tchnology Program has bn supportng snc md-2000 th dvlopmnt of a low-cost, long-lf, hgh voltag, Hall acclrator (HVHAc) at th NASA John Glnn Rsarch Cntr (GRC). To addrss th rsk on throughput capablty, ths thrustr ncorporats an nnovatv dscharg channl rplacmnt tchnology as a mans of xtndng ts lf. 3,4 Th frst laboratory HVHAc thrustr bult and tstd, th NASA 77- M, 5 was dsgnd and manufacturd by Arojt. In 2007, NASA GRC tamd wth Arojt to dsgn and fabrcat a flght-lk HVHAc ngnrng modl thrustr that ncorporatd th channl rplacmnt lf-xtndng nnovaton. In 2008 th NASA-103M.XL 6 had accumulatd >4,700 h at 700 V n a war tst prformd at NASA GRC. 7 Mor rcntly, Arojt and Lockhd Martn Spac Systms (LMMS) Company succssfully xtndd th QLT of th qualfcaton modl 4.5-kW BPT-4000 byond 10,400 h. 8 Th BPT-4000 has fxd nsulators and a magntc dsgn for hgh ffcncy and long lf. Post-tst assssmnt of th data showd no masurabl roson of th acclraton channl walls from 5,600 h to 10,400 h ndcatng that th thrustr rachd a zro roson confguraton. Ths confguraton wll b rfrrd to hrnaftr as th stady-stat confguraton. Th QLT rsults hav, potntally, brakthrough mplcatons for NASA. Thy mply that, f proprly dsgnd, th srvc lf of Hall thrustrs can b xtndd to (or xcd) that of on thrustrs thrby rtrng th prcvd rsk assocatd wth thr throughput capablty. Th dtald physcs that ld to ths sgnfcant rducton of th roson n th BPT-4000 was not dntfd upon th concluson of th QLT. As part of thr dvlopmnt of Hall thrustrs, Arojt cratd a sm-mprcal 2-D roson modl and th company rports xcllnt agrmnt wth masurd roson profls. 8 Whl th 2-D roson 2

3 modl dos not captur th dtald physcs, th QLT showd that th Arojt dsgn 9 provdd th bass for ffcnt opraton and low stady-stat roson. Th QLT has sgnfcantly xcdd th rqurmnts for most commrcal or mltary mssons. 10 Bcaus many NASA mssons rqur long opratonal tms, hgh throughput, and a wd rang of opratng condtons, a rgorous undrstandng of th roson physcs must b attand. Such undrstandng s mportant bcaus t must b dmonstratd unambguously that ground tst obsrvatons wr not anomalous and that thrustr lf projctons basd on mprcal scalngs and/or sm-mprcal modls alon wll b vald (or not) for th wd throttlng rqurmnts of NASA scnc mssons. Also, t would rduc any rsk prcvd by msson tams and would allow for probablstc lf analyss. To mt ths nd th Jt Propulson Laboratory (JPL) has bn supportng an advancd modlng and smulaton actvty for Hall thrustrs. Th modlng actvty at JPL complmnts a lf modlng program on on thrustrs and hollow cathods that has bn ongong for svral yars now at th Laboratory. Rgardng th rcnt roson trnds n th BPT-4000, t was rcognzd arly at JPL that to fully undrstand such physcs on must account, at mnmum, for th two-dmnsonal (2-D) dstrbuton of th lctrc fld nar th rodng surfacs, th shath physcs thr, and th local topology of th magntc fld. To accomplsh ths, t s rqurd usually that th soluton to an xtnsv systm of govrnng laws for th Hall thrustr plasma n two or thr dmnsons s obtand. Thrfor, th actvty mployd ntally HPHall (Hybrd-Partcl-n-Cll Hall), a 2-D plasma solvr for Hall thrustrs that was dvlopd orgnally by Ff and Martínz-Sánchz 11 and latr upgradd to HPHall-2 by Parra and Ahdo. 12 Addtonal algorthm advancmnts ncludng a nw roson sub-modl wr also compltd at JPL. 13,14,15 Th HPHall-2 smulatons xposd a lmtaton of th cod that prohbtd ts applcaton to xplan th roson physcs n th stady-stat confguraton of th BPT Th lmtaton was du to a fundamntal assumpton that formd th bass for ts dvlopmnt, th so-calld quas-on-dmnsonal (quas-1d) approxmaton for lctrons, 11 that dos not prmt th numrcal smulaton of th thrustr plasma n th spcfc magntc fld topology xposd by th roson of th BPT-4000 channl. Th mportanc of undrstandng th roson physcs motvatd th dvlopmnt of a nw Hall thrustr plasma solvr at JPL, dubbd Hall2D, 16 that s not lmtd by th quas-1d approxmaton. Hall2D s a 2-D computatonal modl of th partally-onzd gas n Hall thrustrs that mploys th full vctor form of Ohm s law wth no assumptons rgardng th rat of lctron transport n th paralll and prpndcular drctons of th magntc fld. Th modl s a dscndant of OrCa2D, a 2-D computatonal modl of lctrc propulson hollow cathods that mploys a mx of mplct and xplct algorthms to solv numrcally th consrvaton laws for th partally-onzd gas n ths dvcs. 17,18 Excssv numrcal dffuson du to th larg dsparty of th transport coffcnts n th two drctons s vadd n Hall2D by solvng th quatons n a computatonal msh that s algnd wth th magntc fld. To th bst of our knowldg, Hall2D s th frst computatonal modl of Hall thrustrs to mploy MFAM, and s a capablty that was largly motvatd by th nd to assss th lf capablty of Hall thrustrs n complcatd magntc fld topologs. Hall2D has bn usd to smulat th BPT-4000 wth th goal to dntfy th physcs that ld to th roson trnds obsrvd durng th Arojt/LMMS QLT and our fndngs ar prsntd n ths papr. Th papr s organzd as follows. Scton II provds an ovrvw of Hall2D and dscrbs rcnt physcs upgrads that hav bn mplmntd n th cod. A mor dtald dscrpton of th cod has bn provdd n Rf. 16. Scton III dscusss comparsons btwn numrcal smulaton rsults and a varty of plasma and prformanc masurmnts obtand at JPL, n two dffrnt gomtrcal confguratons of th acclraton channl. W shall b rfrrng to ths confguratons hrnaftr as th 1200-h and stady-stat channl gomtrs. Th two gomtrs mulat closly th rodd channl walls aftr ~1200 h of opraton and th zro-roson confguraton, rspctvly. W shall b rfrrng also to th bgnnng-of-lf (BOL) confguraton, whch for th purposs of ths papr wll rprsnt th nomnal gomtrcal dsgn of ths thrustr at zro hours of opraton. In Scton IV w compar computd roson rats btwn th 1200-h and stady-stat gomtrs, xplan th basc physcs that producd thm, and conclud wth an assssmnt of our plasma modl uncrtants on th computd roson rats. Th Hall2D numrcal smulatons at JPL rval that svral changs wr nducd n th plasma as portons of th magntc fld, burd nto th dlctrc matral n th BOL dsgn of ths thrustr, bcam xposd to th plasma as th acclraton channl rcdd from ts orgnal gomtry to ts stady-stat confguraton. Collctvly, th nducd changs of th plasma proprts consttutd an ffctv shldng of th channl walls from on bombardmnt. Bcaus all such changs n th bhavor of th plasma nar th rodng surfacs hav bn found to b drvn by th magntud and topology of th magntc fld thr, w call ths war-rducng mchansm magntc shldng. 3

4 II. Th 2-D Axsymmtrc MFAM Cod Hall2D A. Gnral dscrpton of th cod and motvaton bhnd ts dvlopmnt Bcaus th fundamntal prncpl bhnd th acclraton of ons n Hall thrustrs s basd on opratng th acclrator at hgh lctron Hall paramtr (Ω >100), th dffuson of mass and hat for th lctron flow n th drcton paralll to th magntc fld s much gratr (by ~Ω 2 ) than that n th prpndcular drcton for most of th channl rgon. Ths lads to th qupotntalzaton and sothrmalzaton of th lns of forc 19 : stramlns of th magntc vctor fld ar, approxmatly, also lns of constant thrmalzd potntal φ T φ- T ln(n ) and constant lctron tmpratur. Numrcally, ths may allow for th soluton of th plasma potntal and lctron tmpratur n a (quadrlatral) computatonal cll that s boundd by two adjacnt lns of forc rathr than on wth arbtrary dmnsons. Th so-calld quas-1d approxmaton formd th bass for th dvlopmnt of a numbr of 2-D computatonal modls of Hall thrustrs n xstnc today such as HPHall. HPHall uss a Partcl-n- Cll (PIC) - Drct Smulaton Mont Carlo mthod for ons n 2-D axsymmtrc gomtry, and was dvlopd by Ff and Martínz-Sánchz n th lat 90s. 11 Snc thn t has provdd nvaluabl nsght nto Hall thrustr physcs at varous nsttutons of acadma, ndustry and govrnmnt. A dsrd computatonal capablty for Hall thrustrs at JPL s th assssmnt of roson of th channl walls nar rgons of th thrustr that may contan complx magntc fld topologs. For xampl, t s possbl that xcssv roson of th acclraton channl may xpos magntc fld arrangmnts n whch a ln of forc bgns and nds at th sam surfac rathr than travrsng th channl. In such rgons th quas-1d approxmaton cannot b usd to smplfy th numrcal approach for th soluton of th lctron quatons. A fully-2d lctron solvr s ncssary n such cass. A man challng howvr wth such solvrs s xcssv numrcal dffuson that s causd by th larg dsparty of th transport coffcnts paralll and prpndcular to th magntc fld. Ths s vadd by dscrtzatng and solvng th quatons on a computatonal msh that s algnd wth th magntc fld. MFAM s a long-standng computatonal approach for smulatng hghly ansotropc plasmas (.g. s Rfs. 20, 21, 22, 23, 24) and s th approach followd n Hall2D. Hall2D has bn undr dvlopmnt at JPL snc lat It s a physcs-basd computatonal modl of th partally-onzd gas producd by Hall thrustrs that mploys th full vctor form of Ohm s law wth no assumptons rgardng th rat of lctron transport n th paralll and prpndcular drctons of th magntc fld. Th consrvaton quatons, numrcal mthodology and prlmnary smulaton rsults hav bn prsntd n dtal n Rf. 16. Hr w provd a brf ovrvw of th cod and dscrb physcs and numrcal upgrads that hav bn mad to th cod snc our 2009 rport. For convnnc and clarty w shall rfr to th vrson of Hall2D dscrbd n Rf. 16 as th 09 vrson. cathod to plat plum boundary cathod boundary horzontal far plum boundary anod boundary dlctrc wall boundary xtnt of th HPHall computatonal rgon vrtcal far plum boundary dlctrc wall boundary cathod boundary axs of symmtry Fgur 1. Schmatc of th computatonal rgon and namng convntons for th boundary condtons (BC) n th MFAM cod Hall2D. Although thr ar many smlarts wth othr hybrd-basd cods, thr ar also svral dstnctv faturs of Hall2D both n th physcs and th numrcal approach. Such faturs hav bn ncorporatd n th cod aftr many lssons larnd from Hall thrustr smulatons ovr th last dcad or so. Du to th wd us of HPHall both 4

5 at JPL and othr nsttutons t may b most nstructv to th communty to outln th major dstnctons btwn Hall2D and HPHall. Thr ar four: 1) Dscrtzaton of all consrvaton laws on MFAM allowng for th assssmnt of roson n rgons wth complx magntc fld topologs. Th MFAM also allows for th slf-consstnt smulaton of th plasma n th nar-anod rgon; unlk HPHall, thr s no magntc fld stramln n ths rgon of Hall2D upstram of whch th consrvaton laws for th plasma ar not solvd slf-consstntly. 2) Numrcal soluton of th complt vctor form of Ohm s law, accountng for lctron transport paralll and prpndcular to th magntc fld. 3) Numrcal soluton of th havy-spcs consrvaton quatons that ar not basd on dscrt-partcl mthods (lmnatng th nhrnt statstcal fluctuatons assocatd wth such mthods) (a) ons ar tratd as an sothrmal, cold (rlatv to th lctrons) flud accountng for th drag forc and on-prssur gradnt, (b) atomc spcs (nutrals) ar tratd as part of a collsonlss gas and thr voluton n th computatonal doman s computd usng ln-of-sght formulatons that accounts for onzaton. 4) Larg computatonal doman that xtnds svral tms th thrustr channl lngth n th axal drcton, and ncompasss th cathod boundary and th axs of symmtry. Th largr physcal doman allows us to ncorporat th hollow cathod boundary slf-consstntly. Also, although byond th scop of ths papr, th largr physcal doman wll allow us n th futur to follow th lctron flow from th cathod to th channl avodng th modfcaton of th transport coffcnts and Hall paramtr that s commonly mposd n rgons of th far plum. A schmatc of th Hall2D physcal doman wth namng convntons for th varous boundars ar provdd n Fgur 1. Th typcal xtnt of th computatonal rgon n an HPHall smulaton s also shown for comparson. B. Physcs, numrcal approach and rcnt augmntatons n Hall2D 1) Ions Th voluton of ons n Hall2D s computd usng a hydrodynamc approach basd on th assumpton that th on gas consttuts an sothrmal, cold (rlatv to th lctrons) flud. Th justfcaton and mprcal support 25 for th on-flud approxmaton has bn dscussd n Rf. 16. Th momntum and on contnuty quatons for th ons account for multpl-onzaton collsons, ncludng trply-chargd ons. Prsntly, Hall2D accounts for th followng ractons: X 2 + X, + X 3 + X, + X 4 + X X 2 + X, + X 3 + X + X X X + X X + X, X + X ++ X X. +++ (II-1) Th racton that ylds trply-chargd ons from an lctron collson wth a sngly-chargd on was rcntly ncludd n th 09 vrson of th cod, and s basd on cross-sctonal data provdd n Rf. 26. Th momntum quaton (Eq. (II-2)) ncluds th drag forc on ons, mposd on thm du to thr lastc and nlastc collsons wth othr havy spcs that may b prsnt n th smulaton doman, and th on prssur-gradnt forc: Du nm Dt ~ = q n E p + R + I (II-2) whr R and Ĩ rprsnt th lastc and nlastc contrbutons to th transport of on momntum, rspctvly. For R, th 09 vrson of th cod accountd only for th drag du to collsons wth nutrals. Th cod has bn updatd to nclud contrbutons from coulomb collsons of ons of dffrnt charg stats Z: whr, R coulomb ( u u ) ν β n m (II-3) 5 β

6 ν β = 2ν β,0 ( M ) x ( M ) ( Z Z ) 3 π Ψ β Ψ 3 2 2M β Ψ β 2 x = ( x) xp( y ) dy. π 0 ν β,0 2 β 4 n lnλ β β 3/ 2 B 2 2 ( π) 3/ ε ( k T ) 0 m (II-4) Th on consrvaton laws ar closd wth condtons spcfd at all boundars n Fgur 1. At th anod and dlctrc-wall boundars th Bohm condton s prscrbd for th spd wth whch th ons xt th physcal doman (.. at ntry to th shath). At th plum boundars th ons ar allowd to flow out of th systm frly (gradnts of th two vlocty componnts ar st to zro). Rflcton boundary condtons ar st at th axs of symmtry. Hall2D solvs numrcally th non-consrvatv form of th on momntum quaton usng a frst-ordr upwnd schm for th vlocty fld. Th quaton s marchd forward n tm xplctly. Th vlocts ar dfnd at th vrtcs of ach quadrlatral computatonal cll. Blnar ntrpolaton s usd to dfn forcs at vrtcs from thr prmtv valus at th cll dgs and to dfn scalar quantts at vrtcs from thr prmtv valus at th cll cntrs. 1) Elctrons Th lctron populaton n Hall2D s tratd as a flud. Th lctron momntum quaton n th absnc of th vscous trms s gvn n vctor form by Du nm Dt ( E + u B) p + R = n (II-5) whr th frcton forc R /n for lctrons s rlatd to th ntgral of th collson trm n th lctron dstrbuton functon and th lctron momntum. In th absnc of hgh-frquncy lctron dynamcs th lctron nrta may b nglctd. Thn on obtans th vctor form of Ohm s law as follows: 2 p ( 1+ Ω ) j + η p E j, (II-6) = η + η j E = η j n n xprssd hr n th fram of rfrnc of th magntc fld (wth and dnotng paralll and prpndcular componnts rspctvly). Equatons (II-6) mply th approxmaton u >>u (n addton to u >>u n ) and thus R - n m ν u wth th total collson frquncy ν ncludng th contrbutons from collsons of lctrons wth all othr spcs. It has also bn suggstd that th dffuson of lctrons n Hall thrustrs s nhancd n a non-classcal mannr,.g. by plasma turbulnc. Many attmpts to captur ths nhancmnt n numrcal smulatons wth HPHall and smlar cods hav bn mad through th us of an ffctv collson frquncy, whch w trm hr ν α. Ff and Martínz-Sánchz proposd orgnally 11 that ν α n Hall thrustrs may b basd on Bohm s 1/B scalng for th crossfld moblty, 27 and usd a coffcnt α to nhanc th total collson frquncy for th lctrons whl rtanng th proportonalty of ν α wth ω c. In ths papr w hav mposd th gnralzd functon ν = α(r,z) ω, α c (II-7) allowng α to vary spatally vrywhr n th smulaton rgon. As w shall show latr n th papr numrcal smulatons wth Hall2D that wr gudd largly by plasma data suggst larg dvatons of α from a constant valu and thrfor lttl to no corrlaton of ν α wth ω c. Thus, n ths papr w rfran from any suggstons that Bohm physcs prsst anywhr n th channl or n th plum rgon of th thrustr untl strong vdnc to th contrary mrgs. Durng thr azmuthal drft lctrons colld wth walls also and ths was proposd (orgnally by Morozov 19 ) to b on mor procss that affcts th transport of lctrons n th acclraton channl. In numrcal smulatons of Hall thrustrs ths addtonal transport mchansm has bn accountd for through th addton of anothr ffctv 6

7 collson frquncy, ν w. Bcaus Ohm s law n computatonal modls that mak us of th quas-1d assumpton (lk HPHall) s solvd btwn adjacnt fld lns, an avrag valu s usd commonly n ths modls. That s, th two stramlns and th two boundars at th nnr and outr walls of th acclraton channl that bound a sngl computatonal cll for th dtrmnaton of th currnt dnsty across magntc fld lns, also dfn th volum and surfac aras of that lmnt. Ths ar thn usd to dtrmn th avrag wall collson frquncy n th acclraton channl. 11 Hall2D s not lmtd by th quas-1d approxmaton and th collson of th lctrons wth th walls s accountd for at th collson st,.. th walls. Spcfcally, th wall collson frquncy s non-zro only at th computatonal clls that shar at last on dg wth th dlctrc walls of th Hall thrustr, that s: ν w S A = u. B (II-8) 1 S V In Eq. (II-8) A s th surfac ara of rvoluton assocatd wth th boundary dg of th computatonal cll and V s th volum of th computatonal cll. Th scondary lctron yld s dnotd by S. In Scton III-A w provd comparsons of th collson frquncs at th channl cntrln and along th wall boundars. Accountng thn for all transport mchansms th lctrcal rsstvty and th Hall paramtr ar dtrmnd as follows: mν B η = Ω = 2 (II-9) n n η whr th total lctron collson frquncy ν s dfnd as ν ν + ν + ν + ν. n w α (II-10) Th frst two trms on th rght of Eq. (II-10) ar th classcal (total) lctron-on and lctron-nutral collson frquncs. Th lctron nrgy consrvaton law s solvd for th lctron tmpratur (xprssd n V) and s gvn by 3 T n 2 t 5 3 T = E j + T j + κ T T j Φs + Q. (II-11) 2 2 s Fgur 2. Hall2D has bn updatd wth lctron nrgy losss by xctaton and wth shath BCs for th convctv lctron hat flux at th anod, allowng for mor accurat solutons of th lctron tmpratur n th nar-anod rgon. Lft: Only onzaton losss and Drchlt BCs for T =1 V as mplmntd n th 09 vrson of Hall2D. Rght: Wth slf-consstnt xctaton losss and shath BCs at th anod as rvsd n th prsnt vrson of Hall2D. Th contours and boxd valus dpct lctron tmpratur n V. Th last trm on th rght rprsnts th nrgy xchang pr unt tm btwn lctrons and th havy spcs 28 du to dvatons from thrmal qulbrum, and s proportonal to n (m /m)ν (T -T ) for ons and n (m /m)ν n (T -T n ) 7

8 for nutrals. In Hall thrustrs t s usually a small contrbuton to th total lctron nrgy. Inlastc nrgy losss ar accountd for by th followng trms: 3 + Z Φ = ( ϕ )( ζ) + ( ζ) + + ( ζ) ( ζ) + s 1 n& n n& n ZT n& T n& 2 + T (II-12) s Z= whr ϕ ϕ ζ ~ ~ aζ 1 & (II-13) ( + Z ) + Z A ζ /T + A n = n ( n a ) + + n = + = A xp n aζn 1 n 2 s s ζ s and a=<σv > for th mpact onzaton btwn lctrons and spcs s. Elctron xctaton losss ar ncludd n th frst trm of Eq. (II-12). Prsntly, ths losss account only for ground-stat transtons. Th xctaton loss rat s a trm that has bn addd n th prsnt vrson of Hall2D and s basd on a ft to a soluton drvd by Dugan, t al. 29 wth coffcnts A 0 =0.6, A 1 =0.304 and A 2 =1 (s also Rf. 11 for th fttng functon). Anothr addton that has bn mad n th prsnt vrson of th cod s rlatd to th anod BC for th convctv hat flux, j T. Th 09 vrson ncorporatd Drchlt BCs for th tmpratur at th anod boundary. Th prsnt vrson mploys shath BCs assumng an lctron-rpllng shath at th lctrod. Such BCs hav bn dscrbd numrous tms lswhr (.g. s Rf. 30). Th rvsd BCs no longr dpnd on a spcfcaton of th plasma tmpratur at th anod boundary, whch n prvous smulatons wth Hall2D practcally dtrmnd th soluton of T n ths (nar-anod) rgon. Fgur 2 compars th soluton bfor (lft) and aftr (rght) th mplmntaton of xctaton nrgy losss and shath BCs n th BPT-4000 acclraton channl. Th quatons for th lctrons ar closd wth BCs at all surfacs n Fgur 2. For all dlctrc-wall boundars a zro-currnt condton s mposd, j +j =0. At th anod a Drchlt condton spcfs drctly th voltag at ts dscharg valu. For th smulaton cass that ar prsntd n ths papr ths valu s 300 V. A Drchlt condton s also mposd at th cathod wth a valu of 10 V. For th lctron nrgy BCs th convctv hat loss follows th formulatons of Hobbs and Wsson 31 for th potntal drop n a shath wth scondary lctron msson. Th scondary lctron yld S must b provdd for th spcfc matral that s bng smulatd. Th nrgy quaton s solvd n a sm-mplct fashon; th thrmal conducton trm s mplct whras all othr trms ar valuatd at th prvous tm-stp. Th numrcal approach followd n Hall2D for th transport of lctrons s to solv th lctron quatons n th fram of rfrnc of th magntc fld, n two dmnsons. Spcfcally, th plasma potntal s solvd by combnng th quaton for currnt consrvaton and Ohm s law nto on quaton and dscrtzng on a computatonal msh that s algnd wth th magntc fld lns. Numrcal dffuson s rducd by assumng that cll dgs ar xactly thr paralll or prpndcular to th magntc fld lns (Fgur 3 mddl). Th accuracy of th soluton s thn dpndnt upon th xtnt of th spatal dvatons of th msh from th tru lns of constant potntal and stram functons χ and ψ. Hr, χ and ψ ar th commonly-usd st of conjugat harmonc functons satsfyng th Cauchy-Rmann condtons for th radal and axal componnts of th magntc fld. A st of such lns n th vcnty of th acclraton channl of a 6-kW laboratory Hall thrustr ar shown n Fgur 3 lft. Th corrspondng MFAM for ths thrustr s shown n Fgur 3 rght. Th quaton for th plasma potntal quaton s solvd mplctly. φ βˆ nˆ ( j βˆ ) βˆ βˆ ( βˆ j) j rˆ ẑ Fgur 3. Lft: A st of lns of constant stram functon (ψ) n blu (stramlns of th magntc fld) ovrlad by lns of constant potntal functon (χ) n rd, n th vcnty of th acclraton channl n a 6-kW Hall thrustr. Mddl: Each dg of a computatonal cll n Hall2D s closly algnd wth thr a χ-ln or a ψ-ln. Rght: corrspondng fnt-lmnt computatonal msh. 8

9 2) Nutrals Th nutral gas n most Hall thrustrs s n th fr-molcul rgm. A nw algorthm that s not basd on dscrt-partcl statstcs was dvlopd for Hall2D to dtrmn th dnsty of nutrals. 32,33 Th algorthm taks advantag of th fact that almost all nutral partcls n ths rarfd mdum procd along straght-ln, constantvlocty trajctors untl thy ar thr onzd, strk a wall, or lav th physcal doman. Th algorthm assums that th partcl vlocty dstrbuton functon for nutrals mttd from a gvn surfac rmans unchangd xcpt for a scal factor that rflcts th loss of nutrals to onzaton. Thn w solv for th nutral gas dnsty by ntgratng forward n tm th lnar Boltzmann quaton n th absnc of any forcs on th partcls: f t + v f = ν ~ f. (II-14) n Th sourcs of nutrals ar gas nlts and sotropc, thrmally-accommodatd gas molculs manatng from chambr surfacs ncludng rcombnd ons. Compard wth th PIC mthod, commonly usd n many plasma smulaton cods lk HPHall, ths algorthm achvs qut and smooth solutons as shown by th comparsons n Fgur 4. Th fgur compars th nutral gas dnsty n a 6-kW laboratory Hall thrustr. It s notd that th HPHall soluton shown n Fgur 4 s avragd ovr thousands of cycls whras th soluton from th Hall2D algorthm s nstantanous. Hall 2D n (m -3 ) n (m -3 ) 5.0E E E E E+19 HPHall 1.0E E E E E E E+18 4.E+19 R R Nutral dnsty (m -3 ) 3.E+19 2.E+19 1.E+19 Hall 2D HPHall Z Z 0.E E E E E E E-04 Numrcal smulaton tm (sc) (s) Fgur 4. Lft & Mddl: Channl and nar plum nutral gas dnsts calculatd by th nw algorthm n Hall 2D compard wth tm-avragd rsults of HPHall for a 6-kW laboratory Hall thrustr. Rght: Tm-dpndnt nutral gas dnsts calculatd by th nw nutral gas algorthm n Hall2D compard wth th PIC Mont Carlo calculatons from HPHall. Th fgur plots th soluton at th computatonal cll outlnd n th contourd plots (lft & mddl). III. Comparsons of Numrcal Smulaton Rsults and Masurmnts In 2009 th BPT-4000 was opratd n a larg vacuum faclty at JPL to valuat th thrustr s prformanc at opratng condtons of ntrst to NASA scnc mssons and to provd n stu plasma masurmnts for th valdaton of Hall2D. In ths scton w prsnt comparsons btwn smulaton rsults producd by th most rcnt vrson of Hall2D and plasma masurmnts at varous locatons n th plum of th thrustr. Th plasma dagnostcs and a mor dtald dscrpton of th masurmnts ar provdd n Rf. 34. In Scton III-A w dscrb th channl gomtrs smulatd, and prsnt our comparsons for th lctron tmpratur and plasma potntal along varous axal and radal slcs n th thrustr plum. W also prsnt comparsons wth th masurd prformanc. Scton III-A concluds wth a prsntaton of slctd 2-D rsults and xplanatons of th sgnfcanc of th computd trnds nar th walls. In Scton III-B w dscuss th comparsons btwn thory and xprmnt n gratr dtal and dntfy aras n our thortcal fforts that wll b th focus of futur nvstgatons. A. Comparsons n th 1200-h and stady-stat channl gomtrs of th BPT-4000 W hav mployd Hall2D to smulat th plasma and roson n th BPT-4000 wth th goal of undrstandng th mchansm(s) that ld to th rducd war rat durng th Arojt/LMMS QLT. Our approach has bn to smulat two dffrnt channl gomtrs, on mulatng opraton arly n th lf of th thrustr and on lat n lf, and thn compar th roson rats. Throughout ths and subsqunt sctons w shall b rfrrng to thr dffrnt confguratons of th BPT-4000 acclraton channl that w hav constructd for our numrcal smulatons: (1) bgnnng-of-lf (BOL) gomtry, (2) 1200-h gomtry and (3) stady-stat gomtry. Th 1200-h 9

10 gomtry rsmbls closly th shap of th channl walls aftr 1200 h of opraton of th thrustr n th Arojt/LMMS QLT. 8 Th stady-stat gomtry mulats closly th shap of th channl aftr 5,600 h; byond ths tm rcsson of th walls by roson had practcally casd. Th thr dffrnt channl gomtrs and rfrnc scal lngths ar shown n Fgur 5. Dspt mor than two dcads and numrous nvstgatons n th Untd Stats and othr wstrn countrs, th tru physcs of th dffuson of lctron hat and partcl flux across magntc fld lns n convntonal Hall thrustrs rman lusv. Bcaus th soluton to ths long-standng problm s byond th scop of ths ffort our approach hr has bn to sk spatal varaton(s) of th non-classcal collson frquncy ν α (r,z) and Hall paramtr Ω (r,z) basd on th plasma masurmnts. Our approach s smlar to that followd by Haglaar, t al. 35,36 Ths approach allows us, frst, to obtan a soluton for th plasma proprts n th acclraton channl that w may thn us to dtrmn roson of th acclraton channl and, scond, to dntfy rgons whr th largst dvatons from classcal transport occur. Th sgnfcanc of th lattr s that th rsults proms to gud furthr nvstgatons of lctron transport physcs and subsqunt mprovmnts of Hall2D s physcs modls. Cathod Acclraton channl Ext plan L Stady-stat channl gomtry 1200-hr channl gomtry BOL channl gomtry Channl cntrln R r Thrustr cntrln z Fgur 5. Schmatc of th BPT-4000 acclraton channl showng th 1200-h and stady-stat gomtrs as usd n th Hall2D numrcal smulatons. Also shown for rfrnc s th BOL gomtry. Th smulatons usd a msh that was algnd wth th magntc fld n BPT W prsnt blow four smulaton cass, thr for th 1200-h gomtry and on for th stady-stat gomtry. Each cas s assocatd wth dffrnt functons ν α (r,z) and Ω (r,z). Both functons for th thr cass n th 1200-h gomtry ar plottd n Fgur 6 along th channl cntrln. Also plottd n Fgur 6 lft and n Fgur 7 ar th rmanng collson frquncs comprsng ν (s Eq. (II-10)). W not n Fgur 7 that nar th dlctrc, lctron collsons wth walls can domnat th total collson frquncy for a larg xtnt of th channl. Th larg drop of ν w byond z/l for th nnr channl wall compard to th outr wall sn n Fgur 7 s assocatd wth th largr valus of th magntc fld nar th nnr surfac, whch n turn yld largr ν α. Ths largr valus ar mor clarly dpctd n th 2-D contour plots of Fgur 8 comparng ν α (lft) and ν (rght) for cas 1. Also, w pont out that for all smulaton cass wth dffrnt profls of ν α, Ω (r,z) was dtrmnd slfconsstntly as th rato B /m ν nsd th channl but t was assgnd profls that dvatd from B /m ν n th plum rgon of th thrustr. Wth th prsnt physcs of Hall2D ths assgnmnt was found ncssary to produc th bst possbl agrmnt wth both th masurd plasma proprts and thrustr prformanc. In th 1200-h cass, Ω (r,z) was n ssnc rlaxd to zro byond spcfc magntc fld stramlns n th nar-plum, analogously to th approach followd n othr Hall thrustr smulaton cods. For xampl, n HPHall th Hall paramtr s st qual to zro byond th magntc fld stramln that s connctd to th cathod, 11 whch also srvs as th 10

11 ffctv plum boundary of th cod s computatonal doman. Th stramln byond whch ths cutoff was mposd n Hall2D corrsponds to z/l=1.31 along th channl cntrln for cas 2 and z/l=1.69 for cass 1 and 3. Bcaus th computatonal doman of Hall2D xtnds svral tms th channl sz downstram of th thrustr xt w wr n a poston to quantfy th mposd dvaton of Ω from ts slf-consstnt valu. Ths s shown for cas 1 n Fgur 6 (rght) by th two black lns. W fnd for ths cas that a dvaton of svral ordrs of magntud n th far plum was rqurd. Th othr two cass wr found to xhbt dvatons of smlar ordr of magntud. Thrfor, th far plum rmans a rgon of lusv physcs that ar currntly not capturd by thr Hall2D or othr SOA smulaton cods lk HPHall. In lght of Hall2D s xtndd plum rgon howvr, and snc mor than on profls of ν α and Ω (r,z) hav bn found to produc rsults that ar wthn or clos to th xprmntal uncrtanty of th plasma masurmnts and th obsrvd thrust, dscharg currnt and voltag, w sought profls that dd not rlax Ω (r,z) compltly n th plum rgon. W dscrb on such cas latr n ths scton as t was part of th stady-stat channl gomtry smulatons. Also, w dscuss th ovrall mplcatons of all th mposd profls n gratr dtal n Scton III-B. Fgur 6. Numrcal smulaton rsults along th channl cntrln of th 1200-h gomtry (cass 1-3 of th smulatons). Each cas corrsponds to dffrnt profls of ν α and Ω as mposd n th smulatons. Lft: rlvant frquncs. Rght: Hall paramtr Ω. All cass n ths 1200-h gomtry mpos a cutoff of Ω n th plum rgon byond a spcfd locaton downstram of th channl xt. Cas 1 ( B /m ν ) plots Ω that corrsponds to th cas-1 profl of ν α wthout a cutoff, showng th locaton of th magntc fld stramln along th channl cntrln whr th cutoff s mposd. Path along outr channl wall Path along nnr channl wall Fgur 7. Contrbutons of lctron collsons wth th channl walls to th total collson frquncy ν. Th collson frquncy ν w (s Eq. (II-8)) s mplmntd n Hall2D at th computatonal clls adjacnt to th dlctrc wall boundars, as ndcatd n th schmatc to th lft. Ths allows us to account for th nar-wall conductvty n ths thrustr. Th rght plot shows th rato of ν w ovr th total collson frquncy ν (s Eq. (II-10)), whch ncluds ν α, along th outr and nnr boundars nsd th acclraton channl. 11

12 Th smulaton solutons of th lctron tmpratur and plasma potntal for all thr cass along th channl cntrln n th 1200-h gomtry ar compard wth th plasma masurmnts n Fgur 9. Dspt th sgnfcant dffrncs btwn th mposd frquncy ν α, all thr cass ar found to yld rsults that ar wthn or clos to th xprmntal uncrtanty of th plasma masurmnts. Rgardng thrustr prformanc and rlatd ntgratd paramtrs th 1200-h smulaton rsults from all thr cass ar also wthn 3% of th masurd thrust (s Tabl 1). Th most notcabl dscrpancs ar assocatd wth th thrustr currnts. Spcfcally, w fnd a consstnt ovr prdcton of th doubly-chargd on currnt fracton and an undr prdcton of th sngly-chargd fracton. Ths and othr dscrpancs btwn thory and xprmnt ar dscussd furthr n Scton III-B. Th bam currnt, mass utlzaton and currnt utlzaton ffcncs n Tabl 1 hav bn computd usng Eqs. (III-1) as follows: I b = 3 Z+ I 3 Z m + & b Z= 1 I ηm = Z= 1 m& A m& A ν α (s -1 ) / Z / m Ib ηb I d ν (s -1 ) (III-1) Fgur 8. Contours of th non-classcal collson frquncy ν α (lft), mplmntd n th numrcal smulatons of th BPT-4000 for cas 1, compard to th total collson frquncy ν (rght). ν accounts for classcal coulomb collsons btwn lctrons and ons, lctron-nutral collsons and nar-wall collsons. Fgur 9. Comparsons btwn numrcal smulaton rsults and axal plasma masurmnts obtand along th channl cntrln (r/r=1) n th 1200-h gomtry of th BPT Lft: lctron tmpratur. Rght: plasma potntal. Radal comparsons btwn th computd rsults and plasma masurmnts hav also bn prformd. Thy ar shown n Fgur 10 for thr axal locatons n th 1200-h gomtry. Th top-lft fgur shows contour plots of th plasma potntal for cas 1; th thr radal slcs along whch th comparsons ar mad ar also plottd as dashd lns. Th top-rght fgur s th comparson closst to th acclraton channl, z/l=1.113, followd by z/l=2.088 (bottom-lft) and z/l=4.715 (bottom-rght). Th smulaton prdcts wll th plasma potntal wthn approxmatly a channl hght n front of th acclraton channl but th agrmnt s found to dmnsh at gratr angls rlatv to th thrust vctor. Ths dscrpancy s dscussd furthr n Scton III-B. 12

13 Fgur 10. Comparsons btwn numrcal smulaton rsults and radal masurmnts of th plasma potntal. Th comparsons ar mad at dffrnt locatons downstram of th channl xt n th nar-plum of th BPT-4000 opratng wth th 1200-h channl gomtry. Top-lft: computd contours of th plasma potntal for cas 1 showng radal slcs along whch th comparsons btwn thory and xprmnt hav bn prformd. Top-rght, bottom-lft and bottom-rght: comparsons at z/l=1.113, and 4.715, rspctvly. Tabl 1. Prformanc comparsons btwn numrcal smulaton rsults and masurmnts prformd at JPL 34 n th 1200-h and stady-stat channl gomtrs of th BPT Channl gomtry 1200-h Stady-stat Exprmnt vs. Thory Exprmnt Thory (Cas 1) Thory (Cas 2) Thory (Cas 3) 13 Exprmnt Thory (Cas 4) Anod (mass) flow rat (mg/s) Dscharg voltag (V) Dscharg currnt (A) Thrust (mn) Bam currnt, I b (A) X + currnt fracton, I + /I b X 2+ currnt fracton, I 2+ /I b X 3+ currnt fracton, I 3+ /I b X 4+ currnt fracton, I 4+ /I b NA NA NA NA Mass utlzaton, η m Currnt utlzaton, η b NA=not accountd for n th smulaton.

14 Smlar comparsons btwn thory and xprmnt ar prsntd blow for th stady-stat channl gomtry. Fgur 11 lft plots th rlvant collson frquncs along th channl cntrln ncludng th mposd profl for ν α ; w shall rfr to ths smulaton as cas 4. By utlzng th xtndd computatonal rgon of Hall2D w attmptd n ths smulaton gomtry to fnd profls of th collson frquncy ν α and Ω that producd closly th plasma and prformanc masurmnts whl rlaxng th strngnt cutoff of Ω mposd byond a gvn magntc fld stramln n th 1200-h cass. Th mposd profl for Ω that bst rproducs th xprmntal obsrvatons s plottd n Fgur 11 rght as th sold ln, and s compard on th sam plot wth th slf-consstnt profl (dashdottd ln). W fnd that ths doubl-humpd dstrbuton rsmbls qualtatvly th slf-consstnt valu of Ω whl rducng sgnfcantly th dscrpancy btwn thm n th plum rgon compard to th 1200-h cass. Th axal comparsons btwn thory and xprmnt that rsult from ths mposd profls ar dpctd n Fgur 12. Fgur 11. Numrcal smulaton rsults along th channl cntrln n th stady-stat gomtry (cas 4 of th smulatons). Lft: rlvant frquncs. Rght: Hall paramtr (sold ln). Th dash-dottd ln corrsponds to th Hall paramtr as computd drctly from th total collson frquncy ν. A comparson of th two profls quantfs th magntud of th modfcaton ndd n th plum rgon to achv corrlaton wth th plasma masurmnts. Fgur 12. Comparsons btwn numrcal smulaton rsults and axal plasma masurmnts obtand along th channl cntrln (r/r=1) n th stady-stat gomtry of th BPT Lft: lctron tmpratur. Rght: plasma potntal. Mthod 1: drctly from Langmur prob tracs. Mthod 2: drvd from mssv prob masurmnts. 14

15 Th radal comparsons at two dffrnt axal locatons downstram of th channl xt, z/l=1.188 and 1.525, ar plottd n Fgur 13. Th comparson n Fgur 13 lft s of partcular ntrst snc both thory and xprmnt dsplay a non-monotonc profl of th lctron tmpratur wth radus. Th sgnfcanc of ths trnd s that such non-monotoncty s xpctd at ths proxmty to th channl xt du to th magntc fld topology n th rgon and th sothrmalzaton of th fld lns. It s also obsrvd that th numrcal smulaton prdcts a profl that s vry smlar, both qualtatvly and quanttatvly, to th masurmnt but at a dffrnt locaton rlatv to th channl. Th spatal dscrpancy s of th ordr of a fw mllmtrs and may b causd by dffrncs btwn th magntc fld modl usd n th smulatons and th ral fld appld n th thrustr. Also, cumulatvly, probthrustr msalgnmnts ar stmatd to b of th ordr of 1-2 mm. Fgur 13. Comparsons btwn numrcal smulaton rsults and radal masurmnts of th lctron tmpratur obtand at two dffrnt locatons, z/l=1.188 (lft) and z/l=1.525 (rght), downstram of th acclraton channl n th stady-stat gomtry of th BPT Th lft and rght vrtcal lns on ach plot ndcat th radal locatons of th nnr and outr walls of th channl s cylndrcal scton, rspctvly. Contour plots n th two gomtrs ar shown n Fgur 14. Th ffct of th qupotntalzaton of th lns of forc 19 s shown clarly n Fgur 14 top. Th plot compars computd plasma potntal contours n th 1200-h and stady-stat gomtrs. For xampl, w fnd only a 30-V rducton (275 to 245 V) along th outr dvrgng wall of th stady-stat confguraton compard to sgnfcant drop of 286 V n th 1200-h gomtry. Thus, acclraton of ons along th dvrgng wall n th 1200-h s consdrably hghr compard to th stady-stat gomtry. Ths hghr acclraton, n turn, ncrass th on kntc nrgy nto th shath. Th sam prncpl that lads to th qupotntalzaton of th lns of forc s rsponsbl also for thr sothrmalzaton. Thn, snc lns of forc ar narly sothrmal n th acclraton channl, th ln that grazs th dvrgng wall and procds almost paralll to t s assocatd also wth a low valu of th lctron tmpratur snc such ln xtnds dp nto th acclraton channl whr th lctrons ar consdrably coolr (s also lft of Fgur 9 and Fgur 12). For th BPT-4000 channl ths s shown n Fgur 14 mddl comparng contours of th lctron tmpratur n th 1200-h and stady-stat gomtrs. Th comparson shows a rducton n th maxmum tmpratur, as much as svn tms along th dvrgng walls. Bcaus th lctron tmpratur was rducd sgnfcantly along th dvrgng walls n th stady-stat gomtry w also obtand a rducton n th shath drop across ths surfacs. A thrd ffct sn n th stady-stat gomtry s largly lnkd to th plasma potntal profl. Spcfcally, bcaus th componnt of th lctrc fld paralll to th dvrgng scton of th dlctrc s margnalzd n th stady-stat gomtry, th acclraton of ons s largly axal and away from th dvrgng wall. Ths s shown by th rprsntatv vlocty stramlns of sngly-chargd ons ovrlad on th lctron numbr dnsty contours n Fgur 14 bottom. Thus, th radal xpanson of th plasma byond th cylndrcal scton of th acclrator s rducd compard to that n th 1200-h gomtry, and th rgon nar th dvrgng wall of th stady-stat gomtry s populatd by a much lowr numbr of chargd partcls. Th lctron numbr dnsty n th two gomtrs n Fgur 14 bottom shows mor than on ordr of magntud lss dnsty n th stady-stat gomtry. Th sgnfcanc of ths rducton on roson s that th total flux to th wall s also rducd. 15

16 φ=286 V φ=0 V φ=245 V φ=275 V φ(t )=33.4 V φ(t )=19 V φ(t )=16 V φ(t )=9.8 V n = m -3 n = m -3 n = m -3 n = m Fgur 14. Comparson of th numrcal smulaton rsults n th 1200-h (lft) and stady-stat (rght) channl gomtrs. Top: Plasma potntal. Mddl: Elctron tmpratur. Bottom: Elctron numbr dnsty ovrlad by rprsntatv trajctors of sngly-chargd ons. 16

17 B. Dscusson on th comparsons btwn thory and xprmnt Th wd rang of comparsons btwn thrustr masurmnts and numrcal smulatons that hav bn prformd hr stablsh th stat of th valdaton of Hall2D and dntfy clarly aras n Hall2D s physcs modls that rqur furthr mprovmnt. Th most vdnt nd for bttr undrstandng s assocatd wth th fact that a dffrnt profl of th collson frquncy ν α must b prscrbd n dffrnt channl gomtrs and/or opratng condtons, whch mpls that Hall2D s not yt a fully-prdctv cod for Hall thrustrs. Ths has bn a long-standng challng n th numrcal smulaton of Hall thrustrs that s mor than two dcads old. Howvr, Hall2D s now at a stat of dvlopmnt that allows us to addrss ths challng drctly. Th xtndd computatonal rgon and th MFAM ar capablts that wr spcfcally dvlopd to allow us, n part, to ntrrogat mchansms prtnnt to lctrons transport (such as larg-scal lctron currnt paths to th acclraton channl) that cannot b capturd by othr SOA smulaton cods. As a prcursor to a mor dtald ntrrogaton on ths subjct w prformd a srs of snstvty smulatons wth dffrnt profls of th collson frquncy ν α to dntfy rgon(s) that dvat most sgnfcantly from classcal transport. Th ntnt was to sk functonal dpndncs of ths non-classcal frquncy hr, as gudd by th masurmnts, whch could lad us to frst-prncpls formulatons of th tru transport physcs. Our snstvty smulatons wth dffrnt profls n th 1200-h gomtry show that dffrnt functonal forms of ν α can produc smlar solutons for th plasma and thrustr prformanc. Ths prsnts a challng n our qust of th tru lctron transport physcs bcaus dffrnt spatal varatons of ν α, all yldng rsults that ar thr wthn or clos to th xprmntal uncrtanty wth th plum plasma masurmnts, thrust and dscharg charactrstcs of th thrustr, mak corrlatons wth tru frquncs ambguous. For xampl, by comparson to th lctron cyclotron frquncy or th Bohm (ν B =ω c /16) collson frquncy (also plottd n Fgur 6 and Fgur 11), w fnd no functonal smlarts wth th mposd ν α that span both th acclraton channl and th nar plum. Notd s th comparson of th frquncs nsd th channl: although ν α for cas 3 s drctly proportonal to ω c ( B), cass 1 and 2 fall fastr wth dcrasng magntc fld nsd th channl. Outsd th channl w fnd ν α to dffr sgnfcantly from ν B n pak magntud and spatal varaton, n both th 1200-h and stady-stat channl gomtrs. Thus, although lctron collsons othr than thos assocatd wth classcal partcl ntractons such as collsons wth th thrustr walls may ndd lad to dstnctv transport trnds nsd and outsd th acclraton channl, w fnd no corrlatons btwn ν α and ν B, and ar unabl to dntfy dstnct rgons of ths problm that can b lnkd unambguously wth dffrnt fractons of ν B. Thrfor, w rfran from makng any suggstons of th xstnc of Bohm physcs as ths would mply spcfc dpndncs of transport physcs on th magntc fld. Th rsults from our snstvty smulatons on th modfcaton of th Hall paramtr (s lft of Fgur 6 and Fgur 11) n th plum rgons suggst that procsss hr cannot b uncoupld from th numrcal smulatons of th acclraton channl; but such procsss rman not wll undrstood. Th couplng of th physcs n th acclraton channl and plum rgons was a concluson of our prvous work (s Rf. 16) and rmans unchangd basd on th rsults of ths ffort. Th radal comparsons btwn thory and xprmnt n th h gomtry (s Fgur 10) dntfy clarly rgons whr th bhavor of th plasma rmans lusv. Spcfcally, rfrrng to Fgur 15, w fnd that although th mposd profls of ν α and Ω lad to good agrmnt wth th plasma potntal masurmnt nar th channl cntrln, th agrmnt bgns to dmnsh away from t n th radal drcton. In ths hgh-angl rgons of th thrustr plum th smulaton rsults smply rflct th xpctd varaton of th plasma potntal thr as th lowdnsty on bam convrgs onto th thrustr cntrln. Th radal gradnt of th plasma potntal computd by Hall2D and dpctd n Fgur 10 s, thrfor, not surprsng. Howvr, th masurmnt (also shown n Fgur 10) suggsts a dffrnt trnd wth radus. Ths trnd s most vdnt n th z/l=1.113 and locatons and 0<r/R<0.75,.. n th nnr rgons of th plum closst to th thrustr walls. Of partcular ntrst s th nglgbl radal lctrc fld masurd at th thrustr cntrln, whch mpls th prsnc of low-nrgy plasma. Ths plasma s not capturd by th smulaton. Usng th masurd valus of th plasma potntal and lctron tmpratur 34 for z/l=1.113, and th computd φ,n, T 17 Slc wth radal masurmnts Thrustr plum φ φ 0,n0, T ( n / ) 0 + Tln n0 Thrustr cntrln Fgur 15. Schmatc for th stmaton of th plasma dnsty nar th thrustr cntrln basd on masurmnts of th plasma potntal and smulaton rsults at th channl cntrln. φ Magntc fld stramln

18 valus of th plasma dnsty n th thrustr bam w may stmat th dnsty on would xpct to hav n ths rgons of dscrpancy btwn thory and xprmnt. As a rprsntatv locaton w choos (r/r, z/l)=(0.4, 1.113) whr th computd and masurd valus dffr by approxmatly 20 V. Rfrrng to Fgur 15, snc th plasma potntal along fld lns s approxmatly φ φ 0 +T ln(n /n 0 ), w can dntfy a magntc fld stramln that contans ths pont and also crosss th channl cntrln. At th channl-cntrln pont w us th computd valus for th plasma potntal, lctron numbr dnsty and tmpratur, φ 0, n 0 and T rspctvly, to stmat n as follows: n n φ φ xp m =. 0 T (III-2) Ths valu s approxmatly two ordrs of magntud hghr than ~ m -3 as computd by Hall2D at ths locaton. Low-nrgy charg-xchang ons ar a possbl sourc for ths dscrpancy. Prsntly, such ons ar not tratd as a dstnct spcs n Hall2D and as a rsult thr concntraton may b undrstmatd sgnfcantly n ths rgons of th thrustr. Emsson from th walls that ar not prsntly capturd by th classcal wall modls n Hall2D, and/or nhancd onzaton not accountd for by th nlastc modls n Hall2D ar othr canddats that may b rsponsbl for th dscrpancy. Th natur of th msson charactrstcs and rlatd shath physcs has bn a subjct of nvstgaton for svral yars [.g. s Rfs. 37, 38, 39, 40 and 41] and s an ara w plan to rvst n th nar futur snc such physcs may affct our smulaton rsults n th plum rgons of th thrustrs but also th transport of lctrons nsd th acclraton channl. For xampl, nhancd msson of low-nrgy lctrons from walls may altr sgnfcantly th shath and, n turn, th lctron nrgy balanc nsd th channl. Wth th prsnt physcs modls n Hall2D th cod appars to yld hghr lctron tmpratur nsd th BPT channl than th valus xpctd basd, for xampl, on th on currnt fractons (s Tabl 1). Th currnt fractons ar currntly ovr prdctd by Hall2D for doubly-chargd ons and undr prdctd for sngly-chargd ons. Th on-on-on comparsons wth th BPT-4000 masurmnts at JPL hav dntfd aras of mprovmnt n th Hall2D physcs smulaton capablts but hav also allowd us to obtan plasma solutons for th two channl gomtrs that now prmt dtald assssmnts of th roson rats n ths thrustr. For xampl, n th channl and nar fld plum, w hav found valus of an ffctv collson frquncy such that, usng th slf-consstnt valu of th Hall paramtr, Hall2D modl rsults ar comparabl to th masurmnts. Howvr, for all cass (Fgurs 7 and 11) t was ncssary to rduc sgnfcantly or lmnat ntrly th Hall paramtr n th far plum n ordr to kp calculatd potntals and lctron tmpraturs as low n th far plum (z/l >3) as th masurmnts. In th far plum, th lctron flow nutralzs th on bam currnt. As dscussd n Katz, t al., 42 most of th lctrons that lav th hollow cathod procd to nutralz th on bam currnt and only a small fracton of thm ntr th channl. Prsntly, th Hall2D far plum BCs mpos currnt nutralty. Ths local BC rqurs lctrons lav th computatonal rgon at th sam locaton as th ons. Snc th ons ar not magntzd many lctrons ar forcd across fld lns to nutralz th bam. Ths currnt gnrats a substantal lctrc fld bcaus t s multpld by th Hall paramtr squard. On possblty s that th path for ths lctrons xtnds far outsd th Hall2D computatonal grd and bam currnt nutralzaton occurs mtrs downstram of th thrustr xt plan. If so, rgons n th Hall2D plum that prdct prsntly fnt lctron currnt dnsty could sustan nglgbl flow of lctrons, whch would margnalz th contrbutons of th rsstv lctrc fld n ths rgons. Currnt nutralzaton far down stram of th channl xt s consstnt wth publshd data showng that hollow cathod couplng mprovd whn th cathod was far from th thrustr. 43 W plan to nvstgat ths n th futur by usng a much largr computatonal rgon and a slf-consstnt Hall paramtr. IV. Eroson Calculatons and th Basc Physcs of th Plasma at th Channl Walls A. Background and coffcnts for th roson rats Th sputtrng roson rat du to on bombardmnt s gvn by, ε = j Y (IV-1) whr th ncdnt on currnt dnsty prpndcular to th channl wall j s dpndnt on th on numbr dnsty and th on vlocty at th wall. Th sputtrng yld of th channl matral Y s a functon of th on mpact nrgy 18