Download Adiabatic Charged Particle Motion in Rapidly Rotating

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motion; lessusefulbecausethe secondand third invariantsJ and ß are no longer valid in mirror geometry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. G., Angular distribution of particle fluxes in rotating
systems,J. Geophys.Res.,81, 5150, 1976.
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to higherorder,Phys.Fluids,21,384, 1978.
Northrop, T. G., and E. Teller, Stability of the adiabaticmotion of
chargedparticlesin the earth'smagneticfield, Phys.Rev., 117, 215,
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co
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component
offieperpendicular
to•,•;
frame;
S, O'
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charge on particle (negative for negative
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m/e;
gyrofrequencyeB/mc (negative for negative
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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;
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1960.
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(ReceivedJune 15, 1981;
revised October 2, 1981;
acceptedOctober 5, 1981.)