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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 87, NO. A2, PAGES 661-669, FEBRUARY 1, 1982 Adiabatic ChargedParticleMotion in Rapidly Rotating Magnetospheres T. G. NORTHROP Laboratory for HighEnergyAstrophysics, NASA/Goddard Space FlightCenter,Greenbelt, Maryland20771 Centerfor Astrophysics andSpaceSciences, Universityof California,SanDiego,LaJolla, California92093 T. J. BIRMINGHAM Laboratoryfor ExtraterrestrialPhysics,NASA/GoddardSpaceFlight Center,Greenbelt,Maryland20771 When the cE x BIB2 drift velocityis largeenoughto exceedthe particlegyro velocity,adiabatic theory becomesmore complicated and less useful: more complicated becausethere are five drifts in additionto the E x B, gradient,and line curvaturedriftsand threemoretermsin the parallelequationof motion; lessusefulbecausethe secondand third invariantsJ and ß are no longer valid in mirror geometry due to the rapid drift acrossfield lines that destroysany semblanceof periodicity in the bouncemotion. (Approximate periodicity is necessaryto have an adiabatic invariant.) But the specialcase of a rapidly rotatingrigidmagnetic field,withEll -- 0, isanexception. If thefieldat anytimeismerelya rotationofthat at an earlier time, the drift and parallel equationssimplify,and even better, the secondinvariant is again valid. The drift equationnow has two rather than five additionalterms--a Coriolisdrift and a centrifugal drift. Just as in a slowly or nonrotating mirror system,the secondinvariant now providesa (rapidly rotating) drift shell to which the guiding center is confined.The particle kinetic energyin the rotating frame,minusthecentrifugal potential,is an exactconstantof motionin a rigidrotator.Thisconstant is wellknownbutdoesnotbyitselfputlimitsonradialmotiontowardor awayfromtherotationaxis,hence it doesnot limit particleenergychanges.But the drift shelldoeslimit and make periodicany radial excursions,and this is why previous studiesof particle motion, in particular examplesof rigid rotators, haveshownthat particlesexperienceno steadyenergygain or loss. drifts. If one has presentin addition the other five drifts, the proofcannotbe carriedthrough.The samedifficultyappearsin The theory of adiabatic particle motion in electromagnetic differentguiseif onetriesto carry out thoseproofsthat employ fieldsis well developedthrough first order in gyroradius[e.g., Hamiltonian theory [Kruskal, 1962]. Alfv•n, 1950; Northrop, 1963; Roederer, 1970] and to a far Becausethe invariantsare suchpowerfultools, it is importlesserextentthroughsecondorder [NorthropandRome,1978]. ant to know whether the invariance of d (and •) can be salAdiabatic theory givesthe equationsof motion of the guiding vaged in specialsituations.An applicationwould be to the center and the approximate constants(magneticmoment M, low-energysulfur ions in Jupiter'sIo plasmatorus, for which longitudinalinvariant J, and flux invariant •) of that motion. us•>%,whereusis thecomponent of thecorotational velocity These invariants place strong constraints on the particle perpendicularto the magneticfield line. At Io us is 74 km/s, motion in mirror geometry.Furthermore,the invarianceof M while Io movesat 17 km/s, giving a sulfurion picked up cold and J is crucial in deriving diffusion equations for trapped fromIo a gyrovelocityv• = 57 km/s.Severalauthors[Siscoe, particles in planetary magnetospheres[Birmingham et al., 1977; Goertz and Thomsen,1979] have assumedthe invariance 1967]. of J duringdiffusionof plasmainjectedby Io and otherJovian 1. INTRODUCTION All proofsof the invarianceof J requirethat the magnitude moons. of theelectric fielddriftvelocity us-- cE x B/B2 besmallcom- In this paperwe will studynonrelativisticparticlesonly and paredto theparticlegyrovelocity%. In thissituation,theusual deriveadiabaticequationsof motion as seenby an observerin threeguidingcenterdrifts(us,magneticfield gradient,and field the corotatingreferenceframe.We will showthat J is in fact linecurvature) aretheonlylargeones.However,whenus•>%, stillconserved evenif us•>%, providedtheelectricandmagthe other five drifts besides the curvature drift that arise from the guidingcenteraccelerationbecomecomparableto the curvature (and gradient) drifts and must be included [Northrop, 1963,equations 1.13and 1.17].Furthermore, whenus•>%, J and ß are generally no longer adiabatic invariants, and the usual diffusionequation cannot be justified.The reasonthat J is no longer conservedis that the guidingcentervelocitycomponent perpendicularto the magneticfield B is comparableto or greater than the componentparallel to B, and after one mirror bounceperiod the guiding centerdoes not return anywhere near the mirror point it left a period earlier. In direct proofs of the invariance of d [Northrop and Teller, 1960; Northrop, 1963] one showsdirectlythat the drifts conserveJ, assumingthe presenceof only the gradient,curvature,and us at any time is the sameas that at an earliertime but rotated about an axis through an arbitrary angle (and translated if desired).The electricfield is assumedto have the corotational value [Birminghamand Northrop, 1979, equation28] E =(r x 12) x B/c,where12istherotationalangularvelocityof therigid rotator and r is the vector to the observationpoint from an origin on the rotating axis (seeFigure 1). The rotator may be oblique.Evenif the rotatingmagneticfieldpatternshouldhave an intrinsic axis of symmetry(suchas a dipole has), this axis neednot be alignedwith 12.(SeeHonesandBergeson [ 1965] for an exampleof this geometry.)But an intrinsicsymmetryaxis need not even exist for what follows below. The usual adiabatic requirementsare still needed,i.e., that the gyroradiusbe small Copyright¸ 1982by the AmericanGeophysical Union. Paper number 1A1550. 0148-0227/82/001A- 1550501.00 netic fields a-re those of a 'rigid rotator.' A rigid rotator is definedasa timedependentmagneticfieldconfiguration, which 661 662 NORTHROPAND BIRMINGHAM: CHARGEDPARTICLEMOTION IN MAGNETOSPHERES Let w = v - f• x r, so that w is the particlevelocityseenby the observerriding with the rotator. Then dv/dt = dw/dt + d/dr (f• x r). Let the rate of change of a vector as seen by the corotating observerbe denoted by (d/dt)c. Then [Goldstein, 1950, equation 4.105], (d/dt)c= (d/dO- f•x. Application of this to both dw/dtand d/dt(f• x r) gives dt- + 2• x w+ • x (• x r) (2) Because• x (• x r)= -V•p2• 2, wherep is the distance (Figure 1) from the rotation axis to the particleposition,(1) becomes dwc = -2• xw+V•p2• 2+-m (r,t) e +- Fig. 1. Depictionin crosssectionof an obliquemagneticrotator. The magneticfield neednot have an axis of symmetry.Here r is the vectorpositionof a particlewith respectto the origin 0 and p is its mE (w + • x r) x •r, t) (3) E still beingas observedfrom the nonrotatingframe.Equation (3) is quite general,requiringonly that f• be time and space independent.For a rigid rotator [Birminghamand Northrop, 1979], E = (r x n). x B/c= -(•n/c)$ x B vectordisplacement fromthe spinaxis11;• is in the azimuthal directionwith respectto where• istheunitazimuthal vectoraboutthef• axis.For this comparedto the scalesize of the magneticfield and that the magneticfield changeslowlyover the gyroperiod(fl <<co,the electricfield,(3) becomes e 2mc dwc =V«p2f•2 +__ wX B(r) + mc gyrofrequency). We will showin this paper that the rapidly rotatingrigid e fI (4) rotatorisonespecial casewhereuE• vgandyetJ isconserved.In the rotating frame the electricfield vanishes,and B is time Another(trivial)specialcasewhereJ is conserved in spiteof the independentas indicated.Equation (4) containsboth centrifu- factthatuE• vgcanbe constructed asfollows. Takea static gal and Coriolisforces.Dotting (4) with w gives magneticmirror field with E = 0, whereJ is conserved by the drifts. Then look at this arrangementfrom a referenceframe translating with respectto the static frame at a velocity ld --- (W2 __p2f•2)___ 0 2 dt (5) /•2/2_ pfl¾. 4 is an exactcon>>%.An observer in thistranslating framesees ue>• %,andyet so that H =-«(w2-- p2f12)___ stantof the particlemotion.Here «p2f12is the centrifugal the J of the particleis still conserved, beinga scalarpropertyof the particle. The rigid rotator is not sucha trivial specialcase,becausean observerin the corotatingframe seesCoriolis and centrifugal drifts,eventhoughhe seesno electricfielddrift. One mustshow that thesetwo additional drifts do not destroythe invarianceof J. The motion as observedin the rotating systemis almost periodic,sothe invarianceof J canbe suspected. In the followingwe derive the guidingcenterequationsof motion in a rigid rotator and prove the invarianceof J. A by-productof the work will be a clearerunderstanding of why rigid rotators of arbitrary geometryneverproducelong-term changes in trappedparticlekineticenergy.Otherstudiesof this potential,and the particleenergyseenby the rotating observer is larger when the particle findsitself further from the rotation axis.But (5) doesnot placeany constrainton how far from the rotation axis the particle can find itself; hence it places no constrainton energy.So far this is all well known. What is new is that the secondinvariant J still holds and placesa constraint onwheretheparticlegoes,henceonp•'fl•'andconsequently on w •' and v•'. Since (clr/clt)c = w, equation (4) is formally identical to the equation of motion of a particle in a static electricpotential -(m/2e)p•'fl•' and a staticmagneticfield •'-= B + (2mc/e)f•. Theelectric fieldin thisanalogyis,q = (m/e)V(p•'fl2/2). Because • is a constant, ,q and •' satisfy the Maxwell equations V that a magnetizedsphererotating about an axis not aligned x ,q = -(l/c) c•/Ot = 0 and V ß •' = 0, both of which are with the dipole producesno secularenergychanges.Birming- repeatedlyusedin adiabatictheory. If there exists an electric field 6E in addition to that which hamand Northrop [1968] extendedthis resultto an arbitrarily produces corotation,it can be addedto the right sideof equamovingmagnetwith the sameresult.Both papersassumed tion (4). Of course,H is modified: if 6E is derivablefrom a E ßB = 0, as we will here,but here we will drop the assumption potential (6E = -V•), e(•/m) must be added to maintain its thatue<• v•madein bothofthosepapers. constancy;if V x 6E•: 0, there is no longer an 'energy conmatter have been made. Hones and Bergeson[1965] proved 2. stant' H. PARTICLE MOTION IN THE ROTATING FRAME The chargedparticleequationof motion in the nonrotating frame is, in the usual notation dv dt e - m e E(r, t) + • mc v x B(r, t) (1) The formal analogybetween(1) and (4) permitsapplication of all heretoforedevelopedadiabatic theory to particle dynamics in the rotating frame, as describedby (4). Because8 is proportionalto the adiabaticparameterE -- m/e,only the small electricfield form of adiabatic theory is required,and this fact NORTHROP AND BIRMINGHAM: CHARGED PARTICLE MOTION IN MAGNETOSPHERES providesmuch simplificationcomparedto working from the point of view of the non-rotatingframe with its large E and attendant more complexformalism. By making useof (4), the Vlasov equationcan be derivedfor the rotating frame. Let f(r, w, t) be the particle distribution functionin (w,r) space.By conservationof particles 663 analogy.The guiding center velocity in an inertial systemis (whenue,• %)thefamiliarexpression t,•( Mc me2• ßV•l) +0(g2) (11) t,•v,+ • x --cE+ • VB e e 0'-•+V•-(fw)+ V,,ß k,dtJ½ =0 (6) wheret,•= B/Bandvii= t,• ß•. From (4), V,, ß (dw/dt)½= 0. Furthermore,V• ß w = 0. Thus (6) becomesa Vlasovequationin the rotating world: Lets bedistance alonga B line.Then t,• ß Vt,• = 0•j/0s.•We cautionthat (11) is not itself valid from the point of view of the non-rotating observerin our presentproblem,because it assumes thatug• %, whereas just the reverseis true for him. But (11) is the expressionwhose analog will lead to the rotating frame equation (15) and to equation(17) that is valid for the nonrotating observer.We can Of(w, r,t)+w. Vf+[V• x•] ßV,f=0(7) -lp2f• 2+-ew Ot mc Application of the analogy to the usual Vlasov equation also yields(7) directly. Anisotropiesof the low energyparticlesin rigid rotatorsmay be analyzedin therotatingsystemby thetechniques of BirminghamandNorthrop[1979Jand of Northropand Thomsen[1980J and then convertedto any other frame of referencesuchas the spacecraftframe [seeNorthrop, 1976J.This procedurecircumventscarryingout adiabaticsolutionsof the Vlasovequationin drawthe analogyfromaug • v• equationbecause the analog electricfield o• in the rotating systemleadsto a centrifugaldrift velocity(seeequation(16))whichis muchlessthan theparticle velocityw solong as•2 • co. By the analogy,the guidingcentervelocity•½ seenby an observerin the corotating frame is the nonrotating spacecraft frame,whereug• vo.If ug• vo, many terms that were drop•d by the above authorsas small would have to • retained,therebygreatly expandingan alreadyextensiveanalysis. In the remainder of this paper we usethe analogy repeatedly to obtain equationsvalid for the corotatingobserver. 3. T• + Mc e +-mc e . where/•= •/• Ol• 1+ (12) anda isdistan• alonglinesoftheanalogfield •. Through first order in gyroradius,the last term of (12) may GUIDING CEN•R EQUATIONOF MOTION In a nonrotatingframe the differentialequation of motion of the guiding center located at R is [Northrop, 1963, equation 1.123 e + t•x I M • = 5 E(R,t)+ • • x •R, t)--• VB(R, t)+ 0(•) (8) justascorrectly bewrittenwiththesubstitution of• for• and B for • because•=B+2(mc/e)•=B+•), and •= •/• = • + 2•/w + 0(•2) = • + 0(•), where • = x •l). The error made is of 0(•2), becausethe expression is bracketsis already explicitly of 0(•), sinceM/e and • are both proportional to m/e • •. The first term of (12) becomes,with substitution for•, where • • m/e and 0(•) means terms of that order which have not been written down; superposeddots stand for total time derivatives.The analog of (8) is .. Rc=•+--R e •- ( •=• ß •l +2 gl + gl+ (13) where• = eB/mc• •-•. When(13)is expanded throughfirst M cxg---Vg+0(•) (9) order in e, one finds ß = 0, + where• m(dR/dt)•and• m(d2R/dt2)•. Conversion of (9)to u, + the actualfieldsis made by the definitionsof • and •. whereull• • ß•. We haveusedthefactthat e • = (2/w•z ß•z = 0(e2),because aswill • seenbelowin (16),•z isproportional to w-l, whichis itselffirstorderin •c = V}P2n2+-mc •c x B(R) The guiding•nter velocityin the rotating frame then is (with M ' + 2Rcx n-- m VB + 0(•) (10) • = m/eV}p2fl2) M, the magneticmoment, is a scalar property of the particle and not frame dependent. 4. THE DRIFT VELOCI• k•= ;•u,+•••x (-•mc +•Mc e V•1 e VB p2f12 mc2O• • 2mcx • +•u•i • +•u,• e The drift velocityas seenby a corotatingobservermay be derived in the following three ways (in increasingorder of difficulty): (1) by solving(10)for Rcz,the component of •c perpendicularto B, by crossing(10) with B; (2) by using the usualdrift velocityvalid for ue• % plusfor analogy;(3) by startingwith the generaleight-termdrift expression[Northrop, 1963,equation1.17]validwhenue• v• andsubstituting intoit E(r, t) and B(r, t) for the rigid rotator. We have carried out all three and give the secondone here to again illustrate useof the e + 0(••) • •u, + •z (15) where•z isthedriftvelocity in therotating frame. • m • •l ( •1p2f12 M =•x -V +•VB +uH•+2u H•x• +0(•2) (16) 664 NORTHROPAND BIRMINGHAM' CHARGEDPARTICLEMOTION IN MAGNETOSPHERES The drift velocity (16) showsthe expectedgradient and line for the magneticmoment of a particle and holds that number curvaturedrifts, plus a centrifugaland Coriolis drift, the latter fixed throughoutthe calculation,so that he is in reality using being proportional to the first power of the parallel velocity. M0 + eMx. But to determinethe number to be used,givensay Theguiding center velocity • asseen bya nonrotating observerthe initial particler and v, onemustusethe completeexpression is•c plusthecorotational velocity fl x R = for M0 + eMx. We now continue to work with the small uE inertial frame equation (18) beforetaking its rotating frame analog.The rate of change of theparallelspeed is d/dt(•'xß•1),andthisdiffers from (18) by thelastformbecause E = -p• •1 ßd•'l/dt= (•1•1 ß•. + •;{_1_) ' d•'l/dt x B/c.Theparallelvelocity soon bythenonrotating observer isVim • Jl ßR = ull+ (• x R) ß•1 = U)m + •g' •. In deriving(16) we have gonethrough the followingroutine: (1) written the desiredadiabatic equation for a nonrotating = ßa,/at = ß ßa,/as + where•_ is theusualthreeterm(E x B, gradient, andcurvature) drift velocity.One findsthat x O&,/as) system for thecasewhereuf • %; (2)writtendowntheanalog •1 ßd&l/dt= --(•1 ß•{/co)(&l by the prescription B•, E• •, •1• •, s• •, Vml•Uml, ß(VMB/m-- eE/m)+ 0(e2) R• R•; (3)converted theanalogequation to actualquantities and with the useof (18), seen by the rotating observerby meansof •=Jl + 2•ffm + O(e•),• = B + 2(mc/e)•, d(t'1 ' •1[) Mo + sM1 c3B e = B(1 + 2• ß •l/m)+ 0(e•),all of whichare easilyderived; and (4) finally, if desired,convertedthe rotating frameequation -- -- dr to anequation applicable in thenonrotating frameby +p• and R•=R-2•xR-•x(•xR). This latter m (19) '•ss -l-m (D [k•l XOS / V , ---- --[0(82)(20) comes from using twice the relationship between total time derivativesof vectors,d/dt = (d/dt)• + •x. The four steps(1•4) are illustratedby equations(11), (12), (15), and (17), respectively.It is important not to conMsean expressionlike (11), from which the analogis drawn, with (17), which appliesto the real situation seenby the nonrotating Equation (20) and its drift velocitycompanion(11) are the two equations that determineguiding center motion in ordinary, small uE,inertial referenceframe situations.We again caution, just as after (11), that (20) is not itself valid from the point of view of the nonrotatingobserverin our presentproblembut is observer. It is alsoimportantto distinguish amongUml(guidingthe expressionwhoseanalog leads to the rotating frame equacenter velocity parallel to B seen in the rotating system), tion (27) that we want. Equation (30) is the one valid from the vii(guiding centervelocityparallelto B asseenfromthenonro- point of view of the nonrotatingobserver.The rotating frame tatingsystem), and•. • (guiding centervelocity in thero- analogof (20) is tating systemparallel to •, a 'parallel velodty' for which we have not introduced a symbol). Possibleconfusionmay also arisebetween Uml andds/dtor between • ßR• andda/dt.The surfacesof constants or • are not necessarily perpendicularto B or •, respectively.The necessaryand su•cient condition that a set of surfacesnormal to a divergence-freefield suchas the magneticfield evenexistis that B ß V x B = 0 [Newcomb, 1958, 1980]. That is, there can be no cu•ent parallel to B. And even if that condition should be Mlfilled, the orthogonal sur- d(• . flc)=- Mo+meMx c3•+-e e Z m 2) (21) mc (• flc)l• xc3l•/c•a ( M•$.)+0(e e•' In (21) and in other equationsbelow, d/dt doesnot carry the subscriptc becausethe time derivativeof a scalaris the samein either reference frame, so the subscript is not needed. -- V(«p2fi2) are now substituted facesare not constants surfaces,exceptin highly symmetric • -- B + (2rnc/e)f•and e$ø/rn geometry,to which we do not wish to confineourselves.So in into (21) to get the parallelequationof motion alongthe B lines generalAs • •, u]] • ds/dr,•d b' Re • d•/dt. In fact, in the rotating frame. The following are neededand can easily ds/dt= Uml + R• ßVs+ 0(e:)andsimilarly forda/dt,sothat be verified: ds/dtisthesameasumm onlyif theorderedriftsareignored. 5. T• /• = •1 + 2f•_ + 0(e2) (22) PARALLELEQUATIONOF MOTION The componentof (8) parallelto B is e • ßR =-•m M OB ßE- --•+ m Os •=B 0½2) 1+ (•8) The reader will notice that (18) is written as if correctthrough 0(e),whereas(8) is correct only up to 0(e).Northrop and Rome /•. V•-- Oa- Os + 2• 1ß•)+0(e2) ß VBq- /• øV«p2•'• 2 -" I0•'•2/• øVp -" •'•2b" ßp [1978] provedthat whenE = 0 all of the 0(e)termsomittedin (23) q-0(S 2) (24) (25) (8) are perpendicularto B, providedM is the sumM 0 + eM 1 of where p is as in Figure 1. The left Side of (21) must also be thelowestordermagnetic momentM0 = mv•Z/2Bandthefirst correctioneM1 [Kruskal•1958;Northrop,1963;equation3.87]. converted: It is plausiblethat E shouldenterasin (18):without introducing any unanticipatedterms,but, to be safe,we haveverified(18) by the method of Kruskal [ 1962]. So (18) is correctthrough 0(e).In a numerical calculation, forexample, onedetermines a number at • gx+• '•c ---dt (•1 ø•'c)"[-0(g2) (26) NORTHROP ANDBIRMINGHAM'CHARGEDPARTICLEMOTIONIN MAGNETOSPHERES the lastequalityagainarisingfrom the factthat 665 roradius.It is the point on the rotating field line where the guiding centerwillremainprovided it isputtherewithu, - 0 andprovidedthe drift velocityfica_is ignored. By (27),the Thelasttermin (21)is of ordere because ofm/e,in itscoefficient centrifugal equatoron a field line is where0/0S(p2•2/2)-andmaytherefore have/•,•, anda replaced byt,•, B, ands, (M/m) OB/&. For M = 0, it is where the field line has no = = 0½ respectively.Then component along p and so is a maximum distance from the rotation axis. dul• dt - --•MOB •+• • •_p2f12 6. TI• LONGITUDINAL INVARIANT In an inertialsystem,whenue,• %, the secondor longitudinal invariant is definedas J= • Pllas= m• vllas 2MB•x ' -Z (31) K isdefined as(mvll2)/2 + MB + (27)If forstaticE andBfields, + where4• is the electrostatic potential,thenby (20),l•/rn = 0 whereM = M0 + eM,.The right-handsideof (27)containsthe + 0(e), so that in 'ordinary' (i.e., nonrotating referenceframe expectedmirror and centrifugalforces,aswell astwo termsthat are firstorderin gyroradius. andue• %)adiabatic theory,orcanbeequallywelldefinedfor staticfieldsby Throughlowestorder,(27) has the energyintegralfor the guidingcentervelocityalongB of d rn (K- MB eck) ds (32) m Kcm• (Ull_ p2f12) q_ MB= const (28)The analogsin the rotating frame of (31) and (32) are consistent with the exactintegral(5), MB beingthe gyration energy. Or moreexplicitly, l(c/rn= 0 + 0(•). /m =•([• ' fic)da =•da[• (•- M•) qI,'•2p2] 1/2(33) The parallel equation of motion actually valid from the where •/rn -- «(• ß•[c)2 + M•/rn - «p2f•2. From (21), viewpoint of thenonrotating observer, whosees ue•>v0,may •/rn = 0 + 0(e).In (33)theintegral isnowalongthelinesof beobtained from(27)by theadditionofd/dt(pf•. •,•) to it, not B. because vii= Ullq_pf• ß•'x.Wegivethefinalexpressions only anddo not includetheproofs. At this point one may correctly invoke any of the usual proofs [Northrop and Teller, 1960; Northrop, 1963] of the invarianceof J to assertthat o• is also conserved,in the sense that dt + pfl• ß(•tc•_ ßVL'0+ 0(e2) (29) 2) (Zdo•> m =0+0(8 (34) where anglebrackets meanthebounce average •--x • da(/•ß •tc)-x( ), •- being thebounce period • de(t;ß•c)-•. o•/rnis and consequentlyconservedover drift time scales. Supposeinstead one would like to use as invariant in the + pn$ ß 2Mc glz0• ß Jc= m• dsg'•ßP,c= m• dSUll mc eB + rotatingsystemthemoreconventionalintegralalongthe actual magneticfieldB. The questionis,doesthequantity 2 co fl•_ ß VB +--'co V«p2•'• 2 Ill ß 0• u c•s w t'• x 0• c•s (35) satisfy((d/dt)Jc/m)=0 + 0(e2)?In (35),the integralis now alongB linesandanglebrackets meanTc- • •[dsUll-• ( ), Tc beingthe bounceperiod.One might surmisethat the answeris 'yes,' forthereason thatullin (35)and/•ßflc in (33)differin 0(E 2)only,since/•= • + 2fix/toand2(fl•_/to) ßfic= 2(fl•_/co) ßfica_ = 0(e2) by (16).Thissurmise is correct, but because of ßv(mmB--«p2fl2) q_O(E 2) (30) geometricquestionsthat might arise concerningdifferences betweenda and ds and the end points of integration,we now where •[c•_ isgiven by(16), and uL.=vii- p,.•ß•1' The fact that v, = ull+ Pfl4•' • meansthat a guiding outlinea directproof of the invarianceof Jc. The proof follows center could haveparallel velbcity vu= pfl• ßg,•,asseen from alongthe linesof Northrop[1963]. A crucialpoint turnsout to to Ull, whichreverses thenonrotating system, andhaveu, = 0. It wouldthenremain bethattheCoriolisdriftis proportional at thesamegeometricpointon thefieldline(if the slowerdrifts signbetweenthe two directionsof integrationalong the field theroundtripintegration • ds.... Thatis, areignored) because thenby(17)•t = g,avll+ ue= pfl•, and linein performing oftheform• dsullnh(s)vanishes if nisodd(Coriolis thereis corotation.From thepointof viewof thenonrotating anintegral observer,any point fixedon a fieldline movesparallelto the drift) but doesnot vanishfor n even(linecurvatureand gradilineatPfl4J' g'•.However, it isnotnecessary thata particle's v, ent(drifts).Jc/mmay bewritten as equalPfl4'' •a. The parallelvelocities u, and v, are governed byinitial6onditions fortheparticle considered plusthe m appropriatedifferentialequation(27)or (30). The centrifugal equator[Cummings et al., 1980]is the point on a fieldline wheredull/dtvanishes, to zero orderin gy- Jc(•, fl,Kc, M)= •ds ]1/2 666 NORTHROP AND BIRMINGHAM' CHARGED PARTICLE MOTION IN MAGNETOSPHERES where Kc is definedin (28), and where M = M0 + eMx. Here 0•(r,t) and fi(r, t) are the usual Euler potentials[Northrop and Teller, 1960; Stern, 1970] constanton a field line and suchthat B = V0•x V/•, with the vector potential A = 0N/•. Even for a givenfield, 0•and/• are nonunique.Of the possiblechoicesfor the rigid rotator,we selectthemsothat the •, • labelsof a given rotating field line do not changewith time. Then time doesnot appear explicitly in the argument of B in (36). The rate of changeof Jc/mis finds 1 dKc mdt - /•, being constanton a field line, are changedonly by the drift velocity.By an analysisquite similar to that givenby Northrop [1963], an analysiswhich will not be repeatedhere, one can derive that the drift velocity(16) causes0•and/• to changeat ratesgivenby • e ullfl . • + O(e 2) (42) Consequently, (l•c/rn)= 0 + 0(e2)because Rc isproportional to an odd powerof ull. Thuswith somemodification of the usualdirectproof of the invarianceof J in a nonrotating,small uE situation,we have shown that the bounceaverageof (37) vanishesherealsoin the rapidlyrotatingframe: dt m-m + ••+••+ •tJ (37) dJc 1[•c•+•X •Jc•Jc •Jc •Jc•Jc] Now •$c/Ot= 0 and2•/-- d/(dtXMo+ eM•) = 0(e2),and0•and 2Mc dJc)=0q-0(I• 2) (43) Therefore,both o• and dc are conservedby the first order in gyroradiusguiding center motion. The invarianceof Jc prescribesa shell on which the guiding centerslowly drifts in the fashionsofamiliar for static,smalluE,nonrotatingsystems. The only differenceis that the drift shell is now rapidly rotating about When (36) is used to find a drift shell,Kc should be held e d• constant. Thisis because by (42)(/•c) = 0 on the drift time scale. Therigidrotatorisa fortunate special casein thisrespect, rnc dt --•-• --«P2f12 +2ullB +0re) (38) e dO mc dt +• (7_- +2ull B' +0(e) the fieldsbeingstaticin the corotatingframe,eventhoughtime dependentto the nonrotating observer.Ordinarily when fields are time dependent,the appropriateK is not a constantas the particle drifts, even though the time dependenceand ue are smallenoughto conservethe secondinvariant d. In suchcases K mustbe followedin time by integratingthe relationship[see Northrop, 1963,equation 3.62](/•) = -(l/T) t•d/t•t. The right-hand side of (42) equals -(2Mc/e) d(fl ß •'l)/dt + 0(e2)because ullc9/•s = d/dr+ 0(e),thetimederivative being where R(0•,fi, s) is the guiding center position. The bounce that seenby an observermoving with the guidingcenter.Thus m- • d/dt(Kc + 2MmcII ßg,t/e) = 0 + 0(e2), and averages are 2 __«•O2["•2 + MB/m + 2(Mc/e) 11 ß r,x is conservedby the total guiding centermotionfie (parallelvelocity plusdrifts),whereas K, is constantonly if the drifts are ignored.In finding a drift shell (&)--•1•ds Ul• - e• •• (M2_«p2f12) +0(e2) &= mc•d••l (39) from (36), K, + (2Mmc/e) fl ß •,• may be usedin placeof K,, l •d• mc •ds • (7 _}p2n2) 7. where T is the bounceperiodalongB. By comparisonwith (36), (40) eT • PARTICLE ENERGIZATION Previousdirect calculations[Hones and Bergeson,1965; Birminghamand Northrop, 1968] of particleenergizationin rigid rotators have never revealeda systematicenergychange.The (&)- eTc •Jc + 0(e2) 00 = and is in fact •-0½2) Hones-Bergeson paperinvestigated thespecial caseof a dipole: field rotating about an fl axis oblique to but intersectingthe dipole axis.The Birmingham-Northroppaper generalizedthat paper to arbitrary fieldsmoving in an arbitrary fashion.Both papers assumed thatthemotionwassoslowthatue<•%.In the In deriving(39) by bounceaveraging(38),the integral presentwork we have not made the restrictiveassumptionthat the electric field is small. Let the exact constant of the motion • asuII- • (d/dt)(u II • ' OR/Off) vanished because ds/ull= dt to lowestorderin e, andUllvanishesat the mirror points.Furthermore, the Coriolis term (last term in (38)) bounceaveragedto zero, being an exampleof an odd n integral. «(w•' - p•'fl•) bedenoted byH. Gyroaveraging gives q-0(e 2) {-•}=H +•p2(R)f•2 (44) Finally,wemustexamine/•canditsbounce average. From where as a consequenceof the gyroaveraging,p is now at the the definition(28) of Kc 1dK c- ull•dull dB dtd-«p2•2 mdt q-M rn dt guiding centerpositionR rather than at the particle position. Thebounce average of(44)isH = •YF+ 0(t•2). (41) wheredB/dt=(•xUll+ fla_) ßVB and similarlyfor (d/dt) «•02[r•2' •[c_l_ is givenby (16)anddull/dtby (27).Withthese substitutionsinto (41) and straightforwardvectoralgebra,one ( -•- ) = H+ (--•) + 0(e 2) (45) From the point of viewof the corotatingobserver,the particle'skineticenergychangesperiodicallyin accordancewith (44) NORTHROP AND BIRMINGHAM: CHARGED PARTICLE MOTION IN MAGNETOSPHERES 667 as the particle movesrapidly back and forth along field lines, therebychangingp:T•2/2 periodically.Superposed on this periodicity is a longer term changecausedby the slower drift around the drift shell,the shellbeingdefinedby the invariance of Jc. The bounce averagedenergy(45) changeson the drift t=0 time scale,in response to changing(p:T•:/2). However,after one drift period, the guiding center returns to its original (p2f12/2),henceto itsoriginal({w2}/2). Thustherecanbeno long term changein particle energyover many periodsof fl or over many drift periodsaround the rapidly rotating drift shell. This conclusionholds even for completely asymmetric rigid rotators. Hones and Bergeson[1965] could have had their rotation axis skewto the dipole axis,and there still would have beenno long-termenergychanges.So an off-centerdipole,such asJupiter's,causesno long-termenergization,at leastinsofaras Jupitermeetsthe criteria of a rigid rotator. From the point of view of the nonrotatingobserveror spacecraft, the energypicture is a little different,but the conclusion that no long-term energy changesoccur remains valid. The velocityseenby the nonrotatingobserveris v = w + flx r = w + pfl• 4 Fig. 2. Crosssectionof a magneticdipole 'tumbling end for end,' i.e., with its spin axis 11 out of the page and its magneticaxis vertical. Shown schematicallyat the right are the variations in centrifugal energy ,rr2fF, Fermienergy UllP,O•b ß•a,andtheirsum {v2/2} - H asa guiding center traverses a complete bounce trajectory. accelerationon field lineslying in the plane of fl and the dipole because •p.•x = 0 on theselines.A combination of tilt and field line sweepbackout of a plane producesmore complicated patternsthan in Figure 2. 8. THE THIRD INVARIANT The third or flux invariant q>is conservedhere, since the drift and so the magneticflux through them is trivially constant.The third invariant +•-- +pfl$ßfic+0(e 2) 4 (46) shellsare static in shapeas observedfrom the rotating frame Squaringand gyroaveragingleadsto = 2 offers no useful information here. If the rotating magnetic field pattern is not quite rigid but deformsslowly compared to the drift time (not compared to a period of fl), it seemsvirtually certainthat the secondand third (47) Withthesubstitution of{w2/2}from(44),(47),becomes invariants contribute {-•-}--H+p2f•2+pf• 4ß•{c q0(E 2) (48) p and• in (47)and(48)beingat theguiding center position R, would be conserved and the invariance nontrivial information. One would of q> would then have the picture of a rapidly rotating drift shell which deforms very slowly and about which the particle drifts on an intermediate time scale. and•c = • Ull+ •c_•asin (15).If thedriftfic_ • isignored in (48) and just the rapid motion along the field line studied,one has H + p2f•2q_ul] pf•, •1 q_0(E) (49) 9. SUMMARY AND CONCLUSIONS We have investigatedsomeaspectsof adiabatictheory in the regime where the E x B drift velocity is comparablewith or larger than the gyro velocity.Previouswork showsthat such particlesundergofive drifts in addition to the familiar E x B, gradient,and line curvaturedrifts,have three more termsin the parallel equation of motion, and possessneither bounceJ nor Figure 2 showsschematicallyhow the energy changesoccurring during the rapid oscillationalong the field line would look for the caseof a dipole rotating about an axisperpendicu- flux q> adiabatic invariants. The absence of conserved J and q> lar to the dipole(i.e.,90ø dipole tilt). Figure 2 appliesto a field arisesfrom the fact that a large E x B drift in generaldestroys line lying in the planeperpendicularto fl. Time is taken aszero the near periodicityof the bounceand drift motions. at the northern mirror point. The energy gains and losses The caseof the rapidly rotating rigid magneticfield configurillustrated by the {v2/2}-- H curvein Figure2 arecaused by ation is an exceptionto theseconclusions,becausea corotating Fermi acceleration(typesA and B, seeNorthrop [1963], equa- particle then has nearly periodic motion. The equation of tion1.36)contained in uIIpflqbßg,xandbythecentrifugal effect motion of a chargedparticle in a rapidly rotating asymmetric p2f•2. The Fermi acceleration term agreeswith Honesand magnetosphere hasa formalidentity to the particleequationof Bergeson[1965], who do not howeverhave the centrifugalterm motion in nonrotating static magneticand electric fields. Bebecauseof their assumptionof small E. For a field line in the cause of this similarity, all previously developed adiabatic plane definedby fl and the dipole,Figure 2 is simplerbecause theory of guiding center motion can be used to produce ana4• ßg'x- 0 everywhereon it and thereis no Fermi acceleration, logousguidingcenterequationsin the rotating referenceframe. only the centrifugaleffectwhich itself vanishesat the equator. Using this analogyis an easyway to obtain the guiding center As time progresses, the guidingcenterdrifts around its 'end- drift velocity and parallel differential equation of motion. The wise' rotating drift shell. The shell will not be axisymmetric drift velocity(16) in the rotating frame consistsof the expected aboutthedipolebecause of thep2f•2termin (36).Herep2f•2 field gradient and line curvature drifts plus centrifugal and dependsnot only on dipolelatitudebut alsoon the field line.In Coriolis force induced drifts. The parallel equation of motion Figure2,p2f•2issmallerat a givendipolelatitudefor fieldlines (27) containsmirror and centrifugalforcesalong the field line to lying in the plane definedby fl and the dipolethan for thosein zero order in gyroradiusplustwo first order terms. The particle equation of motion in the rotating frame has an the planeof the paper.This leadsto an asymmetricdrift surface exact constant (5) of the motion equal to the particle kinetic in the axisymmetricdipolefield. If the tilt is lessthan 90ø, e.g., 10ø as at Jupiter, the energy energyin the rotating frame plus a centrifugalpotentialenergy. changesare qualitativelyasin Figure 2. Again thereis no Fermi If the particle movesfurther from the rotation axis,its rotating 668 NORTHROPAND BIRMINGHAM.'CHARGEDPARTICLEMOTION IN MAGNETOSPHERES i i NORTHROPAND BIRMINGHAM:CHARGEDPARTICLEMOTION IN MAGNETOSPHERES frame kinetic energy increases.This constant of the motion doesnot by itselfput any limits on the particle energyincrease or decrease.However,we havealsoshownin this paper that the secondinvariant is conservedin the rotating frame by the four drifts in (16), so that a particle slowly drifts around on its rapidly rotating drift shell and returns to its original field line. Thus thereis no long-term energychange.Any energychangeis 669 qb electrostatic potential; T• bounceperiodof guidingcenter; { },< ) gyro, bounceaverage,respectively,of the enH gc periodicon thebounceanddrifttimescales. Thissame' state- closedquantity; total (kinetic + potential) particle energy/m in the rotating system; constant of the parallel (to B) guiding center motion; • ment also holdsfor energychangesas observedby a nonrotating observer.This is in brief why all previousexplicit calculations of particle energizationin rigid rotators have failed to show any seculareffect. constant of the parallel (to •) guiding center motion. Finally, a Vlasov equation holds in the rotating system, Acknowledgments.We thank A. W. Schardtand M. F. Thomsenfor againbecauseof the analogybetweenequations(1) and (4). their many provocativequestionswhich led to our study of particle Table 1 summarizesthe most important equationsfrom the motion in rotating systems. The Editor thanks M. F. Thomsen for her assistancein evaluating point of view of both rotating and nonrotating observers,while this paper. the notation list definesthe notation. Taken together,thesetwo serve as a referencethat can be used without delving into the details of the text. REFERENCES NOTATION Alfv•n, H., CosmicalElectrodynamics, Clarendon, Oxford, 1950. Birmingham, T. J., and T. G. Northrop, Charged particle energetizationby an arbitrarily moving magnet,d. Geophys.Res., 73, 83, 1968. v magneticand electricfields; analog magneticand electricfields; a/o, particle positionvectorfrom origin (locatedon rotation axis particle velocity; v0 particle gyrovelocity; fl p angularrotation velocityof rigid rotator; distancefrom rotation axis; unit vector in azimuthal direction about ro- •p tation axis fl; w particle velocity inrotating frame- v - pfl•; guidingcenterposition; guiding center velocity seenfrom nonrotating frame; guiding center velocity seen from rotating UE E M 1958. guiding center acceleration seen in rotating Northrop, T. G., The Adiabatic Motion of ChargedParticles, John Wiley, New York, 1963. Northrop, T. 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Fiilthammar, Charged particlediffusionby violation of the third adiabaticinvariant, Phys. charge on particle (negative for negative charge); m/e; gyrofrequencyeB/mc (negative for negative charge); sum Mo + eM• of first two terms of magnetic moment series; J½ second(longitudinaladiabatic invariant for particlein a rigid rotator; analog secondadiabatic invariant; third or flux invariant; Livermore, Calif., Nov. 12, 1980. 1960. Northrop, T. G., and M. F. Thomsen, Theory of scan plane flux anisotropies,J. Geophys.Res.,85, 5719, 1980. Roederer, J. G., Dynamics of GeomagneticallyTrapped Radiation, Springer-Verlag,New York, 1970. Siscoe,G. L., On the equatorial confinementand velocityspacedistribution of satellite ions in Jupiter'smagnetosphere,J. Geophys. Res., 82, 1641, 1977. Stern,D. P., Euler potentials,Am. J. Phys.,38, 494, 1970. (ReceivedJune 15, 1981; revised October 2, 1981; acceptedOctober 5, 1981.)