ATheoretical Analysis of Vacuum Arc Thruster Performance

Transcription

1 AThortical Analysis of Vacuum Arc Thrustr Prformanc Jams E. Polk, Mik Skrak and John K. Zimr Jt Propulsion Laboratory, M/S , 48 Oak Grov Driv, Pasadna, CA 9119 (818) , Jochn Schin, Nianshng Qi and Robrt Bindr Alamda Applid Scincs Corp, 2235 Polvorosa Av, San Landro, CA (51) , André Andrs Lawrnc Brkly National Laboratory, 1 Cyclotron Road, Brkly, CA 9472 (51) , aandrs@lbl.gov IEPC In vacuum arc dischargs th currnt is conductd through vapor vaporatd from th cathod surfac. In ths dvics vry dns, highly ionizd plasmas can b cratd from any mtallic or conducting solid usd as th cathod. This papr dscribs thortical modls of prformanc for svral thrustr configurations which us vacuum arc plasma sourcs. This analysis suggsts that thrustrs using vacuum arc sourcs can b opratd fficintly with a rang of propllant options that givs grat flxibility in spcific impuls. In addition, th fficincy of plasma production in ths dvics appars to b largly indpndnt of scal bcaus th mtal vapor is ionizd within a fw microns of th cathod lctron mission sits, so this approach is wll-suitd for micropropulsion. Introduction Vacuum arcs, or dischargs burning in mtal vapor libratd from th cathod into an intrlctrod gap initially at vacuum, produc high vlocity, highly ionizd plasma flows which can b xploitd for propulsion applications. On cathod surfacs which ar too cold to support bulk thrmionic mission, currnt continuity across th mtal-vacuum intrfac is maintaind through on or mor highly mobil, luminous spots. Although th bulk cathod tmpratur is rlativly low, th local tmpratur in ths spots is wll ovr th boiling point of th cathod matrial and lctrons ar mittd by a combination of thrmal and fild mission. Ths mission sits Copyright c 21 by th California Institut of Tchnology. Publishd by th Elctric Rockt Propulsion Socity with prmission. Prsntd at th 27th Intrnational Elctric Propulsion Confrnc, Pasadna, CA, Octobr, 21. ar initiatd at locations whr thr ar local microprotrusions or dilctric inclusions which caus local nhancmnt of th applid lctric fild. Fild mission in ths rgions causs xplosiv vaporization of th microprotrusion or inclusion du to vry rapid Joul hating. A combination of Joul hating and ion bombardmnt hating sustains th tmpraturs rquird to mit lctrons and vaporiz cathod matrial. Th loss of cathod matrial causs th formation of a tiny cratr on th surfac. As th cratr diamtr grows, th powr dposition by ohmic hating and ion bombardmnt dcrass. Evntually th tmpratur drops to th point whr it is no longr possibl to sustain lctron and vapor mission and th sit xtinguishs. Th charactristic sit liftim appars to b on th ordr of a fw tns of nanosconds [1, 2, 3, 4, 5]. Th rsulting cratrs hav a diamtr that is typically only 1-1 microns [6, 7], although gross mlting may rsult in largr struc- 1

2 POLK: VACUUM ARC THRUSTER PERFORMANCE 2 turs [8, 9, 1, 11, 12]. Th xtinction of an mission sit is gnrally followd by th ignition of a nw sit at a narby microprotrusion, oftn apparntly slf-gnratd by th moltn mtal flows from th prvious sit. Th luminous spot thrfor appars to mov ovr th cathod surfac. Extraordinary conditions ar achivd in th cathod spots. Currnt dnsitis on th ordr of 1 8 A/cm 2 [13, 14] in th mission sit and hat fluxs of W/cm 2 [15] produc rapid vaporization and ionization of th cathod matrial. Plasma dnsitis in th nar-cathod rgion rach cm 3 [5, 2], narly th dnsity of th solid mtal. Th plasma is gnrally almost 1 prcnt ionizd, oftn with multipl charg stats [16]. What is truly rmarkabl is how asy it is to gnrat ths conditions in cold cathod arc dischargs. Ths xtrm nvironmnts lad to vigorous acclration of th mtal vapor plasma away from th cathod spot, and vlocitis achivd in th xpanding plasma plum ar typically on th ordr of 1 4 m/s [17, 18]. Vacuum arc-gnratd plasmas can b usd in svral diffrnt typs of propulsion dvics. Th plasma plums producd in cathod spots ar highly dirctional, and can b usd to produc thrust dirctly. Vacuum arcs may also b usd as plasma sourcs in ion acclrators such as ion or Hall thrustrs. Th focus of this papr is on application of vacuum arc dischargs in thrmal Vacuum Arc Thrustrs (VAT s) and in lctrostatic Vacuum Arc Ion Thrustrs (VAIT s). Th uniqu physical conditions achivd in vacuum arcs offr svral potntial advantags in ths dvics. A highly ionizd plasma is gnratd vry fficintly in cathod spot opration. Bcaus th ionization procss occurs within tns of microns of th mission sit, th plasma sourc is inhrntly scalabl to vry small sizs for micropropulsion applications. No magntic fild is rquird for an fficint discharg, unlik lctron bombardmnt ion ngins. Vacuum arc dischargs can b opratd in pulss with no sacrific in plasma production fficincy, so th duty cycl can b varid to match th ngin powr to that availabl from th spaccraft. This can nabl th us of high spcific impuls lctric thrustrs for powr-constraind microspaccraft. Finally, bcaus th propllant is providd by th consumabl cathod, no gas fd systm is ncssary. This not only rducs mass and volum, but liminats th nd for low lak-rat valvs. It is vry difficult to achiv low lak rats in microfabricatd valvs, so this is a significant advantag ovr convntional ion ngins for microspaccraft applications. Vacuum arc dischargs xhibit crtain rgularitis in thir bhavior which allow simpl, smimpirical modls of thrustr prformanc. Th purpos of th modls dvlopd in this papr is to provid guidanc in choosing cathod matrials and xplor th prformanc potntial of svral implmntationsofvacuum arc thrustrs. W willfirst dscrib th prformanc modls and thn discuss th prformanc charactristics of a numbr of candidat propllants. ASmi-Empirical Prformanc Modl Vacuum Arc Plasma Sourcs Mass is rodd from cathod spots in th form of mtal vapor ions, droplts or macroparticls, and nutral vapor [19, 2, 21], although th majority of th nutral vapor appars to b vaporatd from th macroparticls in flight [2, 22]. For sufficintly low valus of nrgy dpositd in th cathod, th total rosion rat ṁ t scals with th arc discharg currnt J d [23, 24, 25], ṁ t = E r J d, (1) Th rosion rat E r is constant in this rgim bcaus th mass loss occurs primarily within singl, isolatd mission sits. Highr currnt lvls ar accommodatd by mor mission sits. Abov a crtain thrshold currnt or during long pulss th tmpratur filds of individual mission sits may ovrlap and caus gross mlting. Undr ths conditions th droplt rosion rat may incras dramatically [26, 27, 28, 29, 3, 31, 32, 33, 34]. Th ion componnt of th mass flux can b dscribd as a currnt J i,whichisth sum of th currnts J Z associatd with th fluxs of ions in various charg stats Z. Exprimnts ovr a wid rang of conditions show that th ion currnt in th spot

3 POLK: VACUUM ARC THRUSTER PERFORMANCE 3 plasma plum can b xprssd as a narly constant fraction f i of th discharg currnt ranging from.7 to.1 [23, 26, 31]. W can thrfor writ th ion flow in th cathod spot plasma as φ J i = Z J Z = f i J d (2) l and th ion mass flow rat as ṁ i = Z J Z M i Z = f ij d M i Z 1, (3) da 1 θ whr is th charg on an lctron, M i is th mass of th ion, Z 1 rprsnts th man invrs charg stat, Z 1 = f Z Z. (4) Z and f Z is th ratio of th currnt du to a singl charg stat to th total ion currnt, f Z = J Z J i. (5) This charg stat distribution (CSD) is assumd to b constant for a givn cathod matrial. An quilibrium composition of multiply chargd ions is cratd in th hot, high dnsity mtal vapor plasma nar th mission zon and frzs at som point in th plum from th cathod spot as th rcombination rat drops du to plasma xpansion and cooling [35]. Masurmnts in pulsd dischargs show that th CSD for a crtain matrial changs ovr th first 1 µs ofthdischarg and thn bcoms rlativly constant [36]. Th CSD dos not vary significantly with discharg currnt ovr a rang of 5-12 A [4, 37], but may b influncd by applid magntic filds and highr discharg currnts [38, 39]. Th fraction of th total mass loss that occurs in th form of ions is givn by th xprssion F i = f im i Z 1 E r. (6) Th plasma is acclratd to high vlocitis within a fw hundrd microns of th cathod spot by gasdynamic (lctron prssur gradint and lctronion friction) forcs and possibly by lctrostatic forcs cratd by a potntial hump that forms abov Figur 1: Gomtry usd to dfin ion currnt distribution. Currnt Dnsity Distribution Cosin Distribution Exponntial Distribution, k= Angl from Surfac Normal (dg) Figur 2: Normalizd angular ion currnt distribution for cosin and xponntial functions. th mission sit [4]. Exprimntal masurmnts of th ion currnt dnsity in th plum xpanding from th cathod rgion in vacuum arcs suggst that it follows a cosin [2, 41, 42] or xponntial distribution [4]. Th gomtry usd in dfining ths distributions is shown in Fig. (1). For a cosin distribution in polar coordinats th currnt dnsity at a radius l and angl φ dfind from th surfac normal du to mass gnratd in ara da 1 on th cathod surfac is j ip (l, φ) = j ic cos φda 1 πl 2 (7) whr j ic is th ion currnt flux from th cathod surfac. This function normalizd to a valu of j ic da 1 =1is plottd in Fig. (2). Th cosin distribution is naturally truncatd at

4 POLK: VACUUM ARC THRUSTER PERFORMANCE 4 φ =9dgrs, but ion currnt is oftn obsrvd at highr angls [4]. This ion backflow is obviously not gnratd at th cathod surfac but is th rsult of forcs acting on th plasma plum away from th lctrod structur. Th xponntial distribution assums that th currnt drops xponntially with th solid angl subtndd by th polar angl φ. This distribution oftn mor accuratly capturs th ion currnt dnsity at high angls, and as shown in Fig. (2), prmits ion currnt backflow. This distribution is givn by anod cathod SCR gat puls arc PFN + - cathod insulator with conducting surfac vacuum spot initiation conducting layr insulator j ip (l, φ) = 2j ic da 1 πl 2 k rf(2π/k) xp( [2π(1 cos φ)] 2 /k 2 ), (8) Figur 3: Schmatic of a vacuum arc thrustr. whr k is a constant which dtrmins th sprad in th distribution. With th assumptions discussd abov, th ion currnt dnsity at th cathod surfac can b writtn as j ic = f i j d.thionmass flow rat at th point (l, φ) du to rosion at da 1 is givn by ṁ ip (l, φ) = M ij ip Z 1. (9) This rlationship rlis on th assumption that th CSD is indpndnt of angl, so that j Z /j ip = f Z vrywhr in th plum. Som angular variation in CSD du to lctric filds in th plum and charg xchang with nutral atoms vaporatd from macroparticls at high angls is obsrvd [43], but ths ffcts ar not xpctd to significantly affct th calculatd prformanc, howvr. Th ion vlocity u i is assumd to b a constant for all ion charg stats of a givn matrial. This is justifid by xprimntal data which gnrally show that th vlocity dos not vary significantly with currnt or angl [18]. A wak variation with charg stat is obsrvd [44]. Th discharg voltag V d is also assumd to b constant for a givn cathod matrial. This is also justifid by xprimnt, which shows that th burning voltag for a givn matrial is dominatd by th cathod fall [45]. Th voltag dos vary wakly with anod gomtry [46] and significant voltag drops can occur in currnt lads or th cathod matrial itslf [45]. Figur 4: Photograph of a laboratory modl vacuum arc thrustr. Vacuum Arc Thrustr Prformanc In th vacuum arc thrustr (VAT) th xpanding cathod spot plasma is usd to produc thrust. Th gnral configuration of a thrustr dvlopd by Alamda Applid Scincs Corporation [47] is shown in Fig. (3). Figur (4) shows a photograph of avat with a titanium cathod and molybdnum anod that was built and tstd at JPL. This dsign consists of a cntral rod-shapd cathod sparatd from acoaxial anod by a thin-walld insulating tub. Th fac of th cathod and insulator may in gnral b rcssd from th anod fac. Th consumabl cathod rod would hav to b fd into th discharg rgion as matrial is rodd from it. Inductivly or

5 POLK: VACUUM ARC THRUSTER PERFORMANCE 5 Anod Surfac or Grid Plan da 2 r 2 Cathod Surfac l φ r 1 θ2 L θ1 da 1 Figur 5: Gomtry usd to calculat th momntum flux through th anod plan. capacitivly drivn pulsd dischargs ar ignitd in this configuration by what is calld triggrlss opration [48]. A thin conducting film on th surfac of th insulator producs a high but finit impdnc btwn th cathod and anod. Whn a rlativly low voltag of svral hundrd volts is applid btwn th lctrods brakdown occurs at vry small gaps or flaws inth thin mtal film. Ths tiny dischargs produc nough mtal vapor to initiat th main discharg in th gap. Mtal vapor and droplts rodd from th cathod rplnish th thin conducting layr on th insulator. Th thrust for this configuration can b drivd by intgrating th momntum flux through th aprtur in th anod plan. Th gomtry usd to calculat th thrust is shown in Fig. (5). Th diffrntial thrust du to ion flux from ara da 1 on th cathod surfac through ara da 2 at th anod xit plan is givn by dt = dṁ ip (l, φ)(u i cos φ)(cosφda 2 ) = M ij ip Z 1 u i cos 2 φda 2. (1) Intgrating this ovr th cathod surfac and th anod xit plan yilds th xprssion T = M if i J d u i Z 1 C t (L, r a ), (11) which rprsnts th thrust that would b achivd if all of th rodd ion flux xitd th ngin with avlocity u i dirctd along th axis multiplid by a thrust corrction factor C t which accounts for plum divrgnc and dposition of rodd cathod matrial on th anod intrior walls. This xprssion also assums that th discharg currnt dnsity is uniform ovr th cathod surfac, j d = J d /πr 2 c, whr r c is th cathod radius. In any singl discharg this assumption is violatd bcaus th currnt is highly nonuniform, concntratd in individual cathod spots. This approach will giv th corrct avrag thrust ovr many dischargs if th rosion rat of th cathod is uniform, as it must b in a good thrustr dsign. Th thrust corrction factor for a cosin distribution is givn by intgrating ovr th cathod sourc plan to dtrmin th total momntum flux through a point on th anod xit plan and thn intgrating that rsult ovr th anod xit ara. This yilds C t = L3 π 2 2π 2π ra 1 r 1 r 2 dr 1 dr 2 dθ 1 dθ 2 l 5, (12) whr l =[L 2 + r r2 2 2r 1r 2 cos(θ 1 θ 2 )] 1/2. Th thrustr gomtry paramtrs ar nondimnsionalizd by th cathod radius, so L = L/r c whr L is th cathod lngth, r 1 = r 1 /r c and r 2 = r 2 /r c, whr r 1 and r 2 ar th radii in th cathod and anod plans and r a = r a /r c,whr r a is th anod xit radius. Th angls in th cathod and anod plans ar givn by θ 1 and θ 2.Th corrsponding thrust corrction factor for th xponntial distribution is whr C t = 2π 2π ra 1 2L 2 π 3/2 k rf(2π/k) I 1 (13) xp( [2π(1 L/l)]2 k 2 )da 1 da 2 I 1 = l 4 (14) and da 1 da 2 = r 1 r 2 dr 1 dr 2 dθ 1 dθ 2. Ths intgrals wr valuatd numrically for a rang of lctrod gomtry paramtrs L and r a and plottd in Fig. (6). For th xponntial distribution a sprad factor k =4.5was chosn, basd on masurmnts mad with a coppr vacuum arc discharg at 1 A [4]. Th maximum thrust is achivd whn th anod and cathod surfacs ar coplanar, so th anod dos not shadow th plasma flux. With this gomtry, th thrust cofficint is.67 for a cosin distribution and.64 for an xponntial distribution

6 POLK: VACUUM ARC THRUSTER PERFORMANCE 6.7 Momntum Flux Intgral, C t Cosin Distribution Exponntial Distribution, k= Anod Lngth, L / r c Anod Radius, r a / r c Figur 6: Thrust factor which corrcts for plum divrgnc and loss of plasma to anod walls. with k =4.5. This maximum thrust cofficint rprsnts th thrust loss du to plum divrgnc alon. Th spcific impuls for a VAT is givn by th xprssion I sp = T ṁ t g = M if i u i C t Z 1 E r g = F iu i C t. (15) g Th powr consumd in this simpl dvic is just th discharg powr, P = J d V d.thtotal fficincy can thrfor b xprssd as η = T 2 2ṁ t P = M i 2f i 2u2 i C2 t ( Z 1 ) 2 2 2, (16) E r V d and th thrust-to-powr ratio is T/P = M if i u i C t Z 1. (17) V d Th thrust of a VAT scals with th ion mass M i, ion currnt fraction f i,plasma vlocity u i and th charg stat distribution factor Z 1,which ar fundamntally cathod matrial proprtis. It also scals with th discharg currnt J d,which is limitd to valus gratr than th chopping currnt (th minimum valu rquird to sustain a vacuum arc discharg) and lss than th thrshold for gross mlting. Th thrust also dpnds on th thrust cofficint C t,which argus for an lctrod gomtry in which th anod dos not intrcpt th cathod spot plasma, such as flush anod and cathod facs. Howvr, othr considrations may affct this conclusion. Th discharg voltag will dpnd to som xtnt on lctrod gomtry and will likly b highr if th anod surfac is not in good contact with th low impdnc plasma. Bcaus th droplt flux from th cathod is pakd at high angls (i.. along th cathod surfac) [19, 31, 21], most of th droplts will b capturd on th intrior anod wall in gomtris with a rcssd cathod. Som loss in thrust and fficincy may b tolrabl if it rsults in lss mass dposition on th spaccraft. Th spcific impuls also dpnds on th matrial proprtis and th gomtry, but varis invrsly with th cathod rosion rat E r. For a fixd ion currnt fraction f i,alowr total rosion rat implis lowr droplt rosion, a mass flux from th consumabl cathod which dos not contribut significantly to th thrust. Th I sp is indpndnt of th discharg currnt bcaus th plasma acclration occurs in individual cathod spots for low currnts. Incrasing th discharg currnt changs th numbr of activ mission sits, but dos not influnc th fundamntal acclration mchanism. At highr currnt lvls

7 POLK: VACUUM ARC THRUSTER PERFORMANCE 7 Cathod Anod Mtal Vapor Plasma Ion Bam cussd abov. Th bam currnt is givn by th fraction of th total arc ion currnt which is xtractd by th ion optics, J b = f i J d C j φ g. (18) Discharg Supply Bam Supply Acclrator - Grid Supply + - Ion Optics Nutraliz Figur 7: Schmatic of a vacuum arc ion thrustr. intractions btwn spots may lad to additional lctromagntic or lctrothrmal acclration, but this will likly occur undr conditions which also lad to gross mlting and intolrabl droplt rosion. Th total fficincy scals with th squars of th thrust paramtrs, invrsly with rosion rat and discharg voltag and is indpndnt of discharg currnt. This undrscors th importanc of dsigns which minimiz voltag drops in th currnt lads, cathod structur and intrlctrod plasma. For many powr-limitd micropropulsion applications th thrust-to-powr ratio is a critical paramtr. To maximiz th T/P, cathod matrial with high valus of th paramtrs M i, f i, u i and low CSDs should b chosn and th lctrod gomtry should b dsignd to maximiz th paramtr C t /V d. Vacuum Arc Ion Thrustr Prformanc In Vacuum Arc Ion Thrustrs (VAIT s) a dvic lik th VAT is usd to supply a plasma which is thn acclratd lctrostatically with ion optics. A schmatic of this implmntation with two grid ion optics is shown in Fig. (7), and Fig. (8) shows a photograph of a laboratory modl thrustr built by JPL and Alamda Applid Scincs Corporation. VAIT s ar mor complx than th VAT discussd abov, but offr much highr prformanc potntial bcaus th ion acclration procss is dcoupld from th plasma production. Th opration of th plasma sourc for th VAIT is assumd to b lik that dscribd by th modl dis- Th fraction of th total ion currnt f i J d intrcptd by th grids is givn by th paramtr C j and th fraction of that acclratd through th ion optics is givn by th grid transparncy φ g.th currnt fraction incidnt on th grids is dtrmind by intgrating th currnt flux from th cathod ovr th grid ara, similar to th procdur usd abov to dtrmin th momntum flux intgral. For a cosin distribution this yilds C j = L2 π 2 2π 2π rg 1 r 1 r 2 dr 1 dr 2 dθ 1 dθ 2 l 4. (19) For anxponntial distribution th optics currnt is whr C j = 2π 2π rg 1 2L π 3/2 k rf(2π/k) I 2 (2) xp( [2π(1 L/l)]2 )da k I 2 = 2 1 da 2 l 3. (21) and, as abov, da 1 da 2 = r 1 r 2 dr 1 dr 2 dθ 1 dθ 2. In ths xprssions, L isth distanc btwnth cathod and th scrn grid, L = L/r c, r g is th radius of th ion optics activ bam ara and r g = r g /r c. Ths intgrals ar plottd in Fig. (9) for a rang of cathod-grid gomtry paramtrs. As with th VAT, ths rlationships ar not valid for a singl discharg with a nonuniform distribution of currnt on th cathod surfac, but yild th corrct valus avragd ovr many dischargs if th cathod rods uniformly. Th grid transparncy φ g is assumd to b qual to th physical opn ara fraction of th scrn grid. Ion optics with a quiscnt upstram plasma typically xhibit an ffctiv transparncy gratr than th opn ara fraction bcaus th convx shath upstram of th scrn grid xtracts ions from an ara largr than th aprtur ara. In vacuum arc sourcs th plasma typically approachs th grid with a vlocity gratr than th Bohm vlocity, so

8 POLK: VACUUM ARC THRUSTER PERFORMANCE 8 Figur 8: Photograph of a laboratory modl vacuum arc ion thrustr. 1. Grid Radius, r g / r c Currnt Flux Intgral, C j Cosin Distribution Exponntial Distribution, k = Cathod-to-Grid Spacing, L/r c Figur 9: Gomtry factor which accounts for ion currnt to surfacs othr than th ion optics.

9 POLK: VACUUM ARC THRUSTER PERFORMANCE 9 th ion trajctoris will b lss affctd by th convx quipotntial lins nar th ntranc to th scrn grid aprtur and th ffctiv transparncy will b clos to th physical opn ara fraction. Equation (18) assums that th bam currnt xtraction is not limitd by optics focusing ffcts. If w modl th ion optics as a plan diod, th maximum ion currnt dnsity that can b xtractd from th upstram plasma is givn by th Child-Langmuir law [49], j max b = 4ɛ ( ) 2 1/2 V 3/2 9 Z 1/2 M i t l 2. (22) In this xprssion, ɛ is th prmittivity of fr spac, l is th ffctiv gap ovr which th ions ar acclratd, Z 1/2 = Z f Z/Z 1/2,andthtotal acclrating voltag btwn th grids V t is th diffrnc btwn th bam voltag V b and th acclrator grid voltag V a. In rality, th currnt flow btwn th grids is always at th spac charg limit; th plasma boundary (upstram shath dg) movs, varying th ffctiv acclration lngth l,sothatthbam currnt dnsity givn in Eq. (22) is consistnt with th flux of ions toward th scrn grid aprtur. As th upstram plasma flux incrass, th plasma boundary movs into th scrn grid aprtur, and at a crtain valu of th upstram dnsity th shath no longr proprly focuss th ions through th acclrator grid aprtur. This prvanc limit for circular aprturs is givn approximatly by Eq. (22) if l = [lg 2 + d 2 s/4] 1/2,whr l g is th gap btwn th grids and d s is th scrn grid hol diamtr [49]. For a VAIT,thpak ion currnt dnsity at th scrn grid must not xcd th valu givn by Eq. (22) to avoid dirct ion impingmnt on th acclrator grid. This placs a limit on th maximum discharg currnt (which controls th total plasma production rat and thrfor th ion currnt dnsity) that can b usd for a givn gomtry. Th pak ion currnt dnsity at th scrn grid dpnds on th distribution of discharg currnt on th cathod surfac and th angular distribution of plasmaflowing from th cathod spots. W will considr two limiting cass for ach of th angular distributions considrd abov. Th worst cas condition occurs whn th discharg currnt is blow th spotsplitting currnt and a singl cathod spot carris th ntir discharg currnt. Th pak ion currnt dnsity thn occurs dirctly abov this spot. Th othr limiting cas occurs whn th discharg currnt is much highr than th spot splitting currnt and many spots ar simultanously activ. In this limit th discharg currnt dnsity is approximatd as a uniform distribution ovr th ntir cathod surfac. Th pak ion currnt dnsity at th scrn grid will thn occur along th cntrlin of th thrustr. In all cass th pak ion currnt dnsity at th scrn grid is givn by an xprssion of th form, j max ip = f ij d C j (L) πr 2 c (23) whr Cj is a gomtry factor which dpnds on L, th ratio of th distanc btwn th cathod and th grids and th cathod radius. For a cosin ion currnt dnsity distribution th gomtry paramtr for opration with a singl spot (low discharg currnts) is C j = 1 L 2. (24) For opration with many cathod spots (high discharg currnts), th pak ion currnt dnsity at th grids scals with th factor C j = 1 L (25) Foranxponntial distribution, singl spot opration yilds a pak ion currnt dnsity gomtry paramtr of Cj 2 π = L 2 k rf(2π/k). (26) Formultipl spot opration, th pak ion currnt dnsity scals with C j = 4 π L k rf(2π/k) 1 xp( [2π(1 L/l)]2 k 2 )r 1 dr 1 l 3/2. (27) Equating ths xprssions to th spac charglimitd currnt dnsity dfind by Eq. (22) and solv-

10 POLK: VACUUM ARC THRUSTER PERFORMANCE 1 ing for J d givs xprssions for th maximum discharg currnt, J max d = 4πrc 2 ɛ ( ) 2 1/2 V 3/2 9f i Cj Z 1/2 M i t l 2. (28) It may b possibl to xtract mor bam currnt by oprating at discharg currnts abov ths limits, but som fraction of th acclrator grid upstram surfac will xprinc dirct ion impingmnt, lading to unaccptably high sputtr rosion rats. Th thrust for a givn mass flow rat in th bam ṁ ib and xhaust vlocity u is rlatd to th bam voltag V b by th xprssion ( ) 2Mi V 1/2 b T = ṁ ib u = f i J d C j φ g Z 1/2. (29) Th maximum thrust is obtaind whn th sourc is opratd at th maximum discharg currnt, b V 3/2 t 9Cj l2 T max = 16πr2 c ɛ C j φ g V 1/2 Th spcific impuls is givn by I sp = f ic j φ g E r g = F ic j φ g g ( ) 2Mi V 1/2 b Z 1/2 ( 2Vb M i ) 1/2 Z 1/2 Z 1. (3) (31) Th total powr is assumd to b P = J b V b + J d V d, although this nglcts th powr rquird for a nutralizr cathod. With this assumption, th total fficincy can b xprssd as η = η u Z 1/2 2 (ɛ B /V b +1) Z 1, (32) whr th propllant utilization fficincy is η u = J Z M i C j φ g = M if i C j φ g Z 1 Zṁ Z t E r = F i C j φ g (33) and th bam ion production cost is dfind as ɛ B = J dv d J b = V d f i C j φ g (34) Finally, th thrust-to-powr ratio can b writtn as T/P = (2M iv b /) 1/2 Z 1/2 V b /J d + ɛ B (35) Ths quations dmonstrat what cathod matrial proprtis, thrustr dsign paramtrs and oprating conditions influnc VAIT prformanc. Th VAIT thrust dpnds on th matrial proprtis M i, f i and th charg stat distribution factor Z 1/2. Th dpndnc on th CSD is somwhat wakr in th VAIT compard to th VAT, which scals with Z 1.Ththrustalso dpnds on th ngin dsign through th paramtrs C j and φ g.maximizing th fraction of ion currnt capturd by th grids drivs dsigns toward clos spacing btwn th cathod and th grids and a larg grid radius compard to th cathod radius, as Fig. (9) shows. Howvr, this dsign approach may xcd th currnt xtraction capabilitis of th ion optics. As Eq. (3) shows, th maximum thrust is obtaind for th optimum combination of th gomtry paramtrs rc 2 C j φ g /Cj l2. Thr is an intrsting tradoff btwn th compting paramtrs C j and Cj. Th apparanc of th grid transparncy φ g shows that prformanc scals with opn ara fraction. In convntional gas discharg ion ngins, prformanc improvs with dcrasing acclrator grid opn ara fraction bcaus this hlps prvnt th loss of nutral propllant atoms. In th VAIT, howvr, on is not drivn to this bcaus thr ar vry fw nutrals and thos that do xist in th cathod spot plum hav a high probability of condnsing on th thrustr surfacs thy contact. Th thrust also dpnds on th oprational paramtrs J d and V b,asinconvntional ion ngins. Thr is gratr flxibility to choos th spcific impuls with a VAIT compard to a VAT bcaus th ion vlocity dpnds on th bam voltag, not th cathod spot plasma xpansion vlocity, as Eq. (31) shows. Th I sp also dpnds on fficint gnration of ions with low droplt production through th paramtr F i and ffctiv us of th ion flux through th ngin dsign paramtr C j φ g. Th total thrustr fficincy scals with th propllant fficincy and CSD and invrsly with th ratio of bam ion production cost and bam voltag. High propllant fficincy dpnds on choosing

11 POLK: VACUUM ARC THRUSTER PERFORMANCE 11 cathod matrials with high ion currnt fraction and low droplt rosion (high valus of F i )and propr acclrator dsign (high C j φ g ). Highr fficincy is achivd by using propllants with low avrag charg stats. Clarly th ion production cost must b minimizd by choosing matrials with a low burning voltag and high ion currnt fraction and good thrustr gomtry dsign. Highr bam voltags can b usd to compnsat for high ion production costs, although this will incras th I sp and dcras th thrust-to-powr ratio for a givn discharg currnt, as Eq. (35) shows. Ths rlationships suggst th following prscription for dsigning vacuum arc ion thrustrs to mt crtain prformanc spcifications. Th cathod matrial must b chosn for high ion currnt fraction, low droplt rosion rat, low avrag charg stat and good ignition rliability. Th bam voltag is dtrmind by th rquird spcific impuls. Th ion optics dsign involvs choosing an acclrator grid voltag which prvnts lctron backstraming from th nutralizr and a grid spacing which is sufficint to stand off th total acclrating voltag. Th grid opn ara fraction should b maximizd, but must satisfy structural and ion bamlt focusing rquirmnts. For opration at th maximum achivabl thrust, th gomtric paramtr rc 2 C j /Cj must b maximizd. Thrust lvls lowr than th maximum can b obtaind by varying th discharg currnt, and undr ths conditions th ngin prformanc is optimizd by maximizing th paramtr C j. Th discharg currnt and duty cycl must b chosn to giv th rquird avrag thrust lvl, but must not rsult in cathod hat inputs which caus gross mlting. Modl Prdictions Matrial and Arc Proprtis Proprtis for various potntial cathod matrials usd to stimat th prformanc of vacuum arc thrustrs and vacuum arc ion ngins ar summarizd in Tabl (1). Th currnt CSDs wr calculatd from particl charg stat distributions and avrag charg stats masurd in 1 A pulsd dischargs [5]. Th CSD was sampld at about 15 µsaftr arc initiation; othr masurmnts show that th avrag charg stat is highr for th about th first 1 µs ofapulsd discharg [36]. Th paramtrs Z 1 and Z 1/2 wr calculatd from th valus of f Z in th tabl. Most of th rosion data ar from wight loss masurmnts aftr squncs of 25 µs pulss at 1 A [25], although som ar from 8 25 A dischargs of svral sconds duration [23] or 17 3 A arcs of 5sconds duration [17]. Th ion mass fraction F i was calculatd from th rosion rat data assuming an ion currnt fraction f i =.1. As discussd abov, this is typically obsrvd in vacuum arc xprimnts, but is apparntly not consistnt with som of th rosion rat data. Th ion mass fraction for chromium, gadolinium, tantalum and tungstn xcds unity, suggsting ithr that th rosion rat masurmnts ar too low or that th ion currnt fraction is lss than.1. Th impact of this on calculatd prformanc is discussd blow. Th discharg voltag data ar from xprimnts with 25 µs pulsd dischargs at 3 A [45]. Th voltag drops through th lads and th cathod matrial hav bn subtractd, so ths data rprsnt th arc burning voltag for an anod configuration which is similar to that in VAT s or VAIT s. Th ion vlocity masurmnts wr sampld from 25 µs long, 1 2 A dischargs [18, 51]. Prdictd Thrustr Prformanc Th quations dvlopd abov wr usd with th matrial proprtis for ths cathod matrials to calculat potntial VAT and VAIT prformanc. Th rsults for th VAT shown in Tabl 2 assum a discharg currnt of 1 A and a momntum flux intgral C t =.666. This is th maximum thrust corrction factor for an xponntial or cosin distribution, but might b achivabl in configurations with th anod and cathod facs flush. Th thrust scals linarly with th discharg currnt, but th othr prformanc paramtrs ar indpndnt of th assumd valu. Th maximum spcific impuls and total fficincy prdictd for thos spcis for which w hav all th matrial proprtis ar plottd in Fig. (1). Th cathod matrials that appar to hav th high-

12 POLK: VACUUM ARC THRUSTER PERFORMANCE 12 st prformanc potntial in a VAT ar magnsium, chromium, yttrium, tantalum and tungstn, with fficincis ranging from.7 to.12 and spcific impulss of 86 to 166 s. Ths matrials offr th bst prformanc bcaus of a combination of high ion vlocity and high ion fraction. Som of ths prdictions ar artificially high bcaus of th unralistically high valu of ion mass fraction calculatd from th xprimntal rosion rats and an ion currnt fraction of.1. Ths valus ar not slf-consistnt, ithr bcaus th rosion rats ar too low or th ion currnt fractions ar too high. Th maximum achivabl valu of th ion mass fraction is unity, so th ffct of incrasing th rosion rat or dcrasing th ion currnt fraction to gt F i =1was xplord. As shown in Tabl 3, this would rduc th spcific impuls by 1 2 s and th fficincy by.2.4. This prformanc is still comptitiv with othr tchnologis such as pulsd plasma thrustrs that offr vry small, controlld impuls bits, and th inhrnt scalability of vacuum arc-basd concpts may provid uniqu advantags. Th prformanc prdictd for VAIT s is shown in Tabl 2 and in Fig. (11). For ths calculations w assumd a discharg currnt of 1 A, a bam voltag of 1 V, a currnt flux intgral C j =.8 and a grid opn ara fraction of.8. Ths valus of C j and φ g ar high, but probably achivabl in acarfully dsignd ngin. As notd abov, this approach also nglcts th powr rquird for a nutralizr cathod, although this would b justifid if a low powr nutralizr such as a fild mission array cathod wr usd with th ion sourc [52]. All of th prformanc paramtrs dpnd on th discharg currnt or bam voltag, so ths rsults rprsnt a point dsign which illustrats th impact of cathod matrial and th prformanc potntial. Th bst matrials appar to b chromium, coppr, yttrium, gadolinium, tantalum and tungstn, with fficincis of and spcific impulss at this bam voltag of s. Th impact of varying f i or E r to gt a ralistic uppr bound on th ion mass fraction F i is shown in Tabl (3). This rducs th spcific impuls by 4 15 s and th fficincy by.4.2. This suggsts that th pak fficincis for th bst matrials would b on th ordr of.45 for a bam voltag of 1 V. Th prformanc could, of cours, b improvd with highr bam voltags, as Fig. (12) shows. In this figur th total fficincy as a function of spcific impuls is plottd for th bst cathod matrials. Th bam voltag rangs from 5 to 5 V ovr ths curvs and for Cr, Gd, Ta and W th ion mass fraction F i was assumd to b qual to on. As ths curvs show, th fficincy xcds.5 at spcific impulss of 3 to 9 s for th diffrnt matrials. Bcaus th ion gnration occurs in a lngth scal on th ordr of 1 µmin th cathod spots, ths rsults ar largly indpndnt of th ngin siz. This thrfor appars to b an xcllnt approach for miniaturizing ion ngins. Th prformanc of convntional lctron bombardmnt ngins suffrs as thy ar scald to small sizs bcaus th incrasd surfac-to-volum ratio lads to lss fficint plasma production [53]. Conclusions Th modls dscribd in this papr show how th prformanc of vacuum arc thrustrs and vacuum arc ion ngins scals with cathod matrial, ngin gomtry and oprating paramtrs. Ths modls wr usd with publishd valus of matrial proprtis and optimistic stimats of paramtrs such as th ion currnt fraction and th currnt and momntum flux intgrals to dtrmin uppr bounds on th prformanc potntial of VAT s and VAIT s. Th modl rsults suggst that vacuum arc thrustr configurations may b capabl of oprating at up to 9 prcnt total fficincy at a spcific impuls of 9 13 s. Th advantags of this configuration ar that it is vry simpl, rquirs no gas fd systm, and can b opratd with a vry simpl, low mass powr procssing unit [47]. Th modl shows that vacuum arc ion thrustrs ar capabl of much highr prformanc bcaus th ion acclration is dcoupld from th plasma production procss. Efficincis of 5 6 prcnt for spcific impulss of 3 9 s should b achivabl in carfully dsignd ngins. This is a vry xciting rsult, bcaus this prformanc should b largly indpndnt of ngin scal. Vry small, high prformanc VAIT s rquiring no magnts and no gas fd systms and oprating at vry low avrag powr lvls (in low duty cycl pulsd opration)

13 POLK: VACUUM ARC THRUSTER PERFORMANCE Mg Cr Y W Total Effiicincy Ti Ta Spcific Impuls (s) Atomic Mass (AMU) 15 Efficincy Spcific Impuls Figur 1: Prdictd prformanc of vacuum arc thrustrs for various cathod matrials. 2 3 Total Efficincy Cr Cu Y Gd Ta W Spcific Impuls (s). 5 1 Atomic Mass (AMU) 15 Efficincy Spcific Impuls Figur 11: Prdictd prformanc of vacuum arc ion thrustrs for various cathod matrials. 2

14 POLK: VACUUM ARC THRUSTER PERFORMANCE 14 Total Efficincy Spcific Impuls (s) Chromium, F i = 1 Coppr Yttrium Gadolinium, F i = 1 Tantalum, F i = 1 Tungstn, F i = Figur 12: Variation of VAIT fficincy with spcific impuls for bam voltags ranging from 5 to 5 V. could b an idal propulsion systm for powr- and mass-constraind microspaccraft. Currnt work is focusd on xprimntally validating ths modls with dirct thrust and lctrical masurmnts and furthr charactrizing th modl input paramtrs. Prliminary thrust masurmnts for a VAT with a titanium cathod yildd a thrustto-powr ratio of 2.2 µn/w [47, 54]. Using th gomtry of th thrustr and th masurd discharg voltag, th modl dscribd hr prdicts a prformanc of 3.5 µn/w. This is rasonably good agrmnt, givn th uncrtaintis in th lctrod gomtry [47]. Additional thrust masurmnts using th thrustr shown in Fig. (4) with diffrnt cathod matrials, lctrod gomtris and discharg currnts will b prformd to furthr validat th modl. Masurmnts of th lctrical paramtrs and dirct thrust masurmnts for th VAIT shown in Fig. (8) with various matrials, gomtris and oprating conditions will b usd to vrify th VAIT modl prdictions. In addition, vacuum arc plasma sourcs ar bing charactrizd in a spcial ultra-high vacuum facility. Th purpos of ths xprimnts is to vrify th modl assumptions and masur th modl input paramtrs such as rosion rats, ion currnt fractions, CSDs, angular ion currnt dnsity distributions, ion vlocitis and discharg voltags [55]. Additional nginring challngs will hav to b ovrcom to raliz th potntial of vacuum arc plasma sourcs. Arc ignition rliability with th triggrlss mthod is an important issu, although prliminary tsts dmonstrating up to 1 million dischargs ar ncouraging [48]. Uniform rosion of th cathod must b achivd and a rliabl mthod of dispnsing th consumabl cathod must b dvlopd. Th thrat of spaccraft contamination by condnsibl mtal propllants has to b addrssd. Finally, adquat ngin liftim must b dmonstratd, which will rquir a thorough undrstanding of potntial failur mods such as insulator and grid rosion and lctrical shorts from mtal vapor or macroparticl dposition. Ths ar crtainly challnging issus, but th prdictd prformanc and scalability of vacuum arc-basd thrustr concpts indicat that th ffort is justifid. Th unusual conditions achivd in vacuum arc dischargs provid uniqu advantags for advancd propulsion systms. Acknowldgmnts Th rsarch dscribd in this papr was conductd in part at th Jt Propulsion Laboratory, California Institut of Tchnology, and was sponsord by th National Aronautics and Spac Administration.

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18 POLK: VACUUM ARC THRUSTER PERFORMANCE 18 Mass Ion Charg Stat Distribution [5] Erosion Ion Discharg Ion Spcis M i Rat a E r Frac- Voltag [45] Vlocity b (AMU) f 1 f 2 f 3 f 4 f 5 f 6 Z 1 Z 1/2 (µg/c) tion F i V d (V) u i (m/s) Li C [23] Mg Al [51] Si Ca Sc Ti [51] V Cr Mn F Co Ni Cu Zn [17] G Sr Y Zr Nb Mo Pd Ag [17] [51] Cd [17] In Sn Ba La C Pr Nd Sm Gd Dy Ho Er Yb Hf Ta [51] W [51] Ir Pt Au Pb Bi Th U a Erosion rats ar from rfrnc [25] unlss othrwis notd. b Vlocitis ar from rfrnc [18] unlss othrwis notd. Tabl 1: Matrial and Arc Proprtis for Potntial Cathod Matrials.

19 POLK: VACUUM ARC THRUSTER PERFORMANCE 19 Mass Vacuum Arc Thrustr Vacuum Arc Ion Thrustr Flow Total Thrust-to- Propllant Ion Produc- Total Thrust-to- Spcis Rat, ṁ t Thrust I sp Effici- Powr T/P Thrust I sp Efficincy tion Cost Effici- Powr T/P (mg/s) T (mn) (s) ncy η (µn/w) T (mn) (s) η u ɛ B (V/ion) ncy η (µn/w) Li C Mg Al Si Ca Sc Ti V Cr Mn F Co Ni Cu Zn G Sr Y Zr Nb Mo Pd Ag Cd In Sn Ba La C Pr Nd Sm Gd Dy Ho Er Yb Hf Ta W Ir Pt Au Pb Bi Th U Tabl 2: Prdictd Prformanc of Vacuum Arc Thrustrs and Vacuum Arc Ion Thrustrs.