《航天推进 Space Propulsion》（英文版）Lecture 16: Ion Engine Performance. Brophy’s Theory DEFINITIONS:

16.522, Space Propulsion Prof. Manuel martinez-sanchez Lecture 16: Ion Engine performance. Brophy's Theory DEFINITIONS: JB=Beam ion(and neutralizer electron current) Je=cathode emitted current ]=ion current to cathode- otential surfaces D=current through disch power supply Op=total ion production rate Jia=lon current to anode Jacc=lon current intercepted by accel. grid Current balances JD=JE+Jc+JB+J and also JD=P+JE -, Jp=JB +Jc +Jia Jacc(ion balance) Useful Power JB(VB+vD) Total Power JBVB+JDVD +JaccvB+ heaters Energy cost r beam i Total P - P. p-JB Vo Jace vB+ PH PH =JEVO JD-JB=JE +Jc+Ja JEVD+PHp JVo+J f Tac lvB+ vo)(plasma ion cost) Lecture 16 Prof. manuel martinez- Sanchez

16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 16: Ion Engine Performance. Brophy’s Theory DEFINITIONS: JB=Beam ion (and neutralizer electron current) JE=cathode emitted current Jc=ion current to cathode- potential surfaces JD=current through disch. power supply JP=total ion production rate Jia=Ion current to anode Jacc=Ion current intercepted by accel. grid Current Balances: JD = JE + Jc + JB + Jacc and also JD = JP + JE − Jia JP = JB + JC + Jia + Jacc (ion balance) B Useful Power = (V J + VD ) B ' Total Power = V J + V J B B D D + accV J B + PHeaters Energy cost ' B P. Total - P. Useful (JD − )V J D + JaccVB + PH per beam ion ε B = = JB JB PH = V J E c JD − JB = JE + Jc + Jacc ' ε B = ⎜ ⎜ D E + PH ⎞ ⎟ ⎟ JP V J D + Jacc (V + VD ⎛ V J ) + c B ⎝ JP ⎠ JB JB ε p ε ' ε B = p + fc VD + facc (V + VD ) (plasma ion cost) fB fB fB B 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 1 of 8

JeVo +P here 8,f=c, f acc and Ep= )p More definitions: U+= ionization energy per ion U= excitation energy of level j Jj= excitation rate(total) JLp= loss rate of primary electrons Em mean energy of Maxwellian electron group Discharge Energy Balance JLpVo p+JE-Jia-JLp) JEVo P Define s Then (VD-Em)LpJE-Jia PH/Jp fe em++ P JE VI JE PvD PoPo Jevo Jp E 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 2 of 8

V J + PH where fB = JB ,fc = Jc , facc = Jacc and ε p = D E J J J J p p p p More definitions: U+ = ionization energy per ion Uj = excitation energy of level j Jj = excitation rate (total) JLP = loss rate of primary electrons ε m = mean energy of Maxwellian electron group Discharge Energy Balance U+ j + ∑ j J P j J U + P DLp J V J + ε m ( pJ + EJ P ia J − J − ) Lp J = p D E J V J = p ε − p H J P Define o ε = U+ j + ∑ j J p j J U . Then − P H J P + p ε = oε + ε m + (V D − ε m ) E Lp J J p E J J − p ia J J ε m p m⎜ ⎜ ⎝ ⎛ + ε ε D H V − P / pJ ⎟ ⎟ ⎠ ⎞ Use p ε D H V − P P / J p ⎢ ⎢ ⎣ ⎡ ε 1 − D mD V V − ε E Lp J J − D m V ε ⎥ ⎥ ⎦ ⎤ = o ε + ε m − (V D − ε m ) E Lp J J D p H V J P − p ia J J ε m − ε m D p H V J P + D p H V J P + p H J P 1 − E Lp J J − D m V ε ⎜ ⎜ ⎝ ⎛ 1 − E Lp J J ⎟ ⎟ ⎠ ⎞ 1 − ⎜ ⎜ ⎝ ⎛ 1 − E Lp J J ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ 1 − D m V ε ⎟ ⎟ ⎠ ⎞ 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 2 of 8

Eo + em Jp).PH J VD NOTE: If we write PH=EVC, the equation for a, becomes Eo +Emll Jia Survival equation for Primary Electrons Jue =e-rotnine, where atot=(o,+Exc/primaries.Also, te is the path length for a JE primary electon before it would be captured by the anode, if it did not collide with a neutral before before that. This path lenght is that of the electrons helical path around one of the magnetic lines of force created by the confinement magnets The neutral density is related to the flow rate by no (o= grid system transparency for neutrals; nu utilization efficiency) 4m(1-nukole 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 3 of 8

1 JLp − ε m JLp ε m + ⎛ ⎜ ⎜ ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎞ ⎟ ⎟ ε m − ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ PH VD Jia ε p = ε o m + ε − J − 1 − 1 − ε m J ⎝ E V D ⎠ Jp ⎝ V J V D E D ⎠ p ⎞ ⎟ ⎟ ε m J ⎛ 1 + ⎛ ⎜ ⎜ ⎞ ⎟ ⎟ Lp PH 1 Jε ⎜ ⎜ ⎠⎝ - − V J ⎝ p ⎠ p Jia o m 1 Jp PH J J Lp 1 m p 1 JE VD + ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ε − − ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ + ε − ε ⎛ ⎜ ⎜ ⎝ ε p = NOTE: If we write PH = V J the equation for ε p c E , becomes ⎛ ⎜ ⎜ ⎞ ⎟ ⎟ Jia ε o m + ε 1 − J ⎛ ⎞ ⎟ ⎟ ⎝ p ⎠ VC ⎜ ⎜ ⎞ ⎝ ⎟ ⎟ ⎛ ⎜ ⎜ ε p = ⎝ 1 − 1 + J ⎞ V ⎟ ⎟ V ⎛ ε m ⎜ ⎜ Lp D ⎠ J 1 − E ⎠⎝ D ⎠ Survival Equation for Primary Electrons JLp σ σ ) tot exc primaries. primary electon before it would be captured by the anode, if it did not collide with a neutral before before that. This path lenght is that of the electron’s helical path around one of the magnetic lines of force created by the confinement magnets. The neutral density is related to the flow rate by (σ −σ A totn e n = e , where = + Also, Ae is the path length for a + JE Γn cn 4 η 4 u m n (1 − ) (φ = grid system transparency for n = = A m g i cnφ φ neutrals; ηu = utilization efficiency) φ m (1 η u c A m i n g J ⎡ 4 − )σ A ⎤ LP J e = exp ⎢− ⎥ E ⎢⎣ ⎥⎦ 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 3 of 8

1-exp eo +5m( 11+ve 4 m 1 The quantity Co is a measure of the confinement effectivenss for primary electrons (better for long electron path (e, small grid open area Ago ) If Co>o0, the energy cost per ion, Ep, tends to the limit ep, which then represents the cost per ion primary losses Calculation of Primary /Secondary population Ratio Primaries are endowed initially with an energy Vo, and, if they did not escape, would all thermalize eventually to an energy Em. The rate at which they disappear in that case is simply the rate of ionization or excitation by primaries(a primary is assumed to become a secondary-Maxwellian after one ionization or one excitation). So, the net energy input rate per unit volume due to injection of primaries is(without escape nnnpUpOrDXvo-Er (or =o. + exd This energy is used by the primaries and their secondary progenie"to (a) Produce ionization by primaries. Per ionization event, this uses U++ Em, since the new electron created has energy s Total p.u. volume n,,(VD )u++6m) (b)Excite atom, by primaries Total energy rate p.u. volume n," exc(voJexc ( a shorthand for npUp∑oy) (c) Produce ionization by secondaries(Mawellian). The rate p.u. volume nn fm(cko. (cAr cdc=n, cema 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez

ε* p ε p = 1 − exp[− m c  (1 − η )] o n [ε + ε m (1 − fia )]⎜ ⎛ 1 + Vc ⎟ ⎞ o ⎜ ⎝ VD ⎠ ⎟ Co 4 σ T Ae ε* where p = , and = ε m c A m n g i φ 1 − VD The quantity Co is a measure of the confinement effectivenss for primary electrons (better for long electron path Ae , small grid open area Ag φ ). If Co → ∞ , the energy cost per ion, ε p , tends to the limit ε * p , which then represents the cost per ion no primary losses. Calculation of Primary/Secondary Population Ratio Primaries are endowed initially with an energy VD, and, if they did not escape, would all thermalize eventually, to an energy ε m . The rate at which they disappear in that case is simply the rate of ionization or excitation by primaries (a primary is assumed to become a secondary-Maxwellian - after one ionization or one excitation). So, the net energy input rate per unit volume due to injection of primaries is (without escape) υ σ T (V )(V − ε m ) (σ = σ + + σ exc n n ) n p p D D T This energy is used by the primaries and their secondary “progenie” to (a) Produce ionization by primaries. Per ionization event, this uses U+ + Em, since the new electron created has energy ε m . Total p.u. volume n n υ σ T (V )(U + + ε m ) n p p D (b) Excite atom, by primaries Total energy rate p.u. volume n n υ σ exc ( ) p p n U v o exc (a shorthand for υ ∑σ jU j n ) p p j (c) Produce ionization by secondaries (Mawellian). The rate p.u. volume is ∞ 2 nn ∫ m ( )c c f σ + ( ) c 4π c dc ≡ n e n c n m σ + o 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 4 of 8

where a,=nfm(ck o,(chmc2dc a 8 kTe nm=m ncdc The ionization cross-section o (c)in zero below c Using a Maxwellian form for fm(c), we find easily Ge=euo, udu and the energy spent by secondaries in ionization(p u. time and volume) is then n,nmCeo, (u++em) Similarly, the energy spent in excitation is nnnmCedexcUexc The energy balance is therefore(dividing by n throughout) PuPa. (Wo-Em)=npuplo, (vD Nu++6m)+Gexc(VD Vexd +nmcb. (u++Em)+excUexc) This can be solved for - p U++e+Exc 0. exc -9)-+- which is a function of Te for a fixed Vo. Hence n is also a function of Te. This is because, gi excOM(1- Lecture 16 Prof. manuel martinez- Sanchez Page 5 of 8

∞ 2 where σ c c σ ( )4π dc c + = 1 fm ( ) + c c n e ∫ m o ∞ 8 kTe 2 and ce = , nm = ∫ fm 4π dc c π me o The ionization cross-section σ t (c) in zero below c + = 2eU + . Using a Maxwellian me form for fm ( ) c , we find easily ∞ σ t = ∫ e−u u σ + ( ) ⎛ E ⎞ du u ⎜ ⎜u = ⎟ ⎝ kTe ⎟ ⎠ +u and the energy spent by secondaries in ionization (p.u. time and volume) is then c n n eσ + (U + + ε m ) n m Similarly, the energy spent in excitation is n e m c n n σ excUexc The energy balance is therefore (dividing by nn throughout) υ σ (V υ σ (V )(U + + ε m ) + σ ( D )U V exc ] + c n e [σ (U + + ε m ) + σ excUexc n ] p p + D − ε m ) = n p p [ + D exc m + np This can be solved for : nm U + + ε m + Uexc σ exc np = ce σ + nm (V ) − σ (V ) σ exc (V ) υ p D + D D (V − ε m ) σ T K (U + + ε m D ) − σ + σ + σ + * which is a function of Te for a fixed VD. Hence ε p is also a function of Te. This is because, given ε* p ε p = 1 − exp[− m C  (1 − η )] o u 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 5 of 8

cn is seen to be the energy per ion created if no primary electrons were to escape (E,+o). The expression for ep(neglecting ion capture by the screen, fia=0, and heating power, V=o)was 1 Eo=U++U nuPlEx(o) and this does depend on Te and"(Te). Substituting the expression for pfound above, and simplifying, we obtain NOTE: An intermediate expression for En(still containing -p), which will be useful later, is n Calculation of utilization efficienc d Ceo, r JB=fBJ e D/+n ↑m(1-n) 4 v(np2,)+na)4-n)=2m But also n,=nm+np and JB=en, 0.61DB A, i 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 6 of 8

ε * is seen to be the energy per ion created if no primary electrons were to escape p * (ε o → ∞). The expression for ε (neglecting ion capture by the screen, f p ia = o , and heating power, Vc = o ) was pε * = D m mo V − + ε εε 1 ; oε = U+ + p exc J J U exc or ε o = + U + U exc ( ) ( ) + + + υ σ υ σ Dp p Dexc p p Vn Vn σ + σ e m exce m c n c n and this does depend on Te and ( ) e m p T n n . Substituting the expression for m p n n found above, and simplifying, we obtain * σ (V σ + (U + + ε m ) + σ excUexc ε p = V T D D ) [σ σ (V ) − σ (V )σ ]U + σ σ T (V )(V − ε m ) exc + B exc D + exc + D D n * p NOTE: An intermediate expression for ε p (still containing ), which will be useful nm later, is ⎛σ T ⎞ ⎜ ⎟ * VD ⎝ ⎜ σ + ⎠ ⎟ ε p = nm ce σ + 1 + n υ σ (VD ) p p + Calculation of utilization efficiency  Jp = ( n Vd υ σ (V ) + c n eσ + )n JB = J f = e )( p p + D m n p B η um e mi ↑ m (1 − η ) 4 u cnφAgmi ( η u V(n υ σ (V ) + c n eσ + ) 1 4 − η u ) p p + D m = cnφAg fB But also n+ = nm + np and JB = n e + 61.0 υ A φi g B 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 6 of 8

Agoi 1+2 UpD)o(D) Divide Cec+ G,4(1-mn)e0.6A响 e0.61B49 o,D) 1 4JBfBvcea, 1+ Vp or(vo)np Up 0.15e68421+ np JBfBVou, Vor(vi So: Given VD, JB, Walls >Cn Geometry: fB,8,i,V,Ag Gas U*,U ELM Can conclude P(Te)Ep(Te)>nu(Te)NOTE: m=i2B so real parameter is m not Then given also le( magnetic geometry)andm→Co→5p→EB Summary Geometry and Gas perating Additional Magentic Field Properties Parameters arameter p,向,V,4g E),exc(E DrB e 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 7 of 8

JB so nm + np = e υBAg φi np υ p (VD ) σ + (VD ) 1 + 1 4 − η ) = ( n e 6.0 Ag φ i Divide: c V e σ nm ce σ + cn φAg + η u B 1 + np f J B nm 2 e 61.0 υ Ag φφ cn ⎛ ⎜1 + np ⎞ ⎟ ⎟ 1 − η u 1 = B i ⎜ nm ⎠ = η 1 + y u + D 4 c V f J e σ + ⎜ ⎜ ⎛ 1 + np ⎝ υ p σ (V )⎞ ⎟ = y → η u B B ⎝ nm ce σ + ⎠ ⎟ VD σ T (V D ) np υ p ∗ ε P σ + nm ce ⎛ n ⎞ * 2 15. 0 e ε υ Ag φφicn ⎜ ⎜1 + p ⎟ p B n ⎟ ⎝ m ⎠ y = V f J D υ pV σ (V ) ⎛ ⎜ np ⎞ ⎟ B B T D ⎜ ⎟ ⎝ nm ⎠ So: Given VD , JB ,Twalls → cn Geometry: fB ,φ,φi ,V , Ag Gas (U + , Uexc ,σ + ( ) E ,σ (E ), M ) exc n * Can conclude p ( ) → ε ( ) → η ( ) NOTE: m = mi JB Te p Te u Te  so real parameter is m , nm e ηn not JB . Then given also A  e (magnetic geometry) and m → Co → ε p → ε B Summary Inputs: Geometry and Gas Operating Additional Magentic Field Properties Parameters Parameter fB , fc ,... U + , Uexc VD , JB φ,φ i , V , Ag σ + ( ) E E ,σ exc ( ) Te → ε m Tw → cn Ae M 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 7 of 8

U. +s+Uw. -exc nm Up (D-5m)-a ne up o. o) 0.15epB4n1 r Ny miB C ) 1-exp[-C i(I-n +VD 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 8 of 8

U+ + ε m + Uexc σ exc np ce σ = nm p υ + VD (V − ε m)σ ( ) Uexc D σ + ↓ ne υ p σ (VD ) + ε T = VD ⎜ ⎜ ⎛σ T ⎞ ⎟ ⎟ * nm ce σ + ⎝ σ + ⎠VD 1 + np υ p σ + (VD ) nm ce σ + ↓ Y = ( ) n n VVV f J n n cAe m p DTp D B B m e n i gB p ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 15.0 1 + 2* συ ε φφυ ; η u = 1 + Y 1 → m  = u Bi J e m η C o = ( ) φ σ n g i DT cA m 4 AeV → p ε [ ( )] o u p mC η ε −−− = 1exp1 *  ↓ Bε = B p f ε + B c f f V D 16.522, Space Propulsion Lecture 16 Prof. Manuel Martinez-Sanchez Page 8 of 8