Download Propagation of cosmic rays in the solar wind

REVIEWSOF GEOPHYSICS
ANDSPACEPHYSICS,VOL. 9, NO. 1, FEBRUARY
1971
Propagation of Cosmic Rays in the Solar Wind
J. I•. JOKIPII•
PhysicsDepartment
California Institute o[ Technology
Pasadena,California 91109
This paper presents a coherent exposition of the modem statistical theory of the
transport of fast charged particles (cosmic rays) in the solar wind. Observations are
discussedonly as they illustrate the phenomena under discussion.A brief introductory
section surveys the historical development of the theory. The dominant effect on the
motion of cosmic rays in the solar wind is the interplanetary magnetic field, which is
irregular and which is therefore best treated statistically, using random functions. The
magnetic irregularities scatter the cosmic rays in pitch angle, so that to a good approximation the cosmic rays diffuse through the irregular magnetic field. Using a statistical
analysis of the equations of motion, one may relate the diffusion tensor to the power
spectrum of the magnetic field, which is in principle measurable. The resulting general transport theory relates the motion of cosmic rays, statistically, to the solar-wind
velocity and magnetic field. Application of the theory both to the modulation of galactic
cosmicrays by the solar wind and to the propagation of solar cosmic rays is discussed
in detail. It is concluded that the present theory explains the principal phenomena
quite well. Future theoretical work will probably be devoted to obtaining better solutions of the equations, to obtaining better values of the parameters, and to studying
higher-order or more subtle effects.
CONTENTS
1.
Introduction
2.
3.
Historical Background........................................................
Statistical Description of the Interplanetary Magnetic Field ......................
A.
4.
................................................................
Introduction
.............................................................
28
32
32
B. The Average Magnetic Field ...............................................
C. The Fluctuating Component...............................................
32
33
D.
36
Observations
of the Fluctuations
...........................................
Statistical Description of Particle Motion .......................................
A.
Introduction
.............................................................
B. The Fokker-Planck Equation in Pitch Angle and Position.....................
C.
5.
6.
28
The General Diffusion
Tensor ..............................................
39
39
40
44
D. ApproximateExpressionsfor the Fokker-Planck Coefficients...................
E. Relation to Magnetic-Field Observationsin Space............................
46
48
Diffusive Motion of Fast Charged Particles in the Solar Wind .....................
Modulation of Galactic CosmicRays by the Sun.................................
51
56
A. Variation in Density or OmnidirectionalIntensity ............................
56
B.
63
Anisotropies.............................................................
C. Energy Balanceof GalacticCosmicRays....................................
Alfred P. Sloan Foundation
Fellow.
27
67
28
7.
8.
J.R. JOKIPII
Diffusive Propagationof Solar CosmicRays.....................................
69
A. The Density or OmnidirectionalIntensity ...................................
B. Anisotropies.............................................................
69
75
General
Discussion ...........................................................
Appendix A.
Appendix B.
Appendix C.
Appendix D.
Transformation of Power Spectra...................................
StochasticEquations from Liouville's Equation .......................
Random Walk of Magnetic Lines of Force............................
List of Frequently Used Symbols....................................
77
78
79
80
81
1. INTRODUCTION
Cosmic-ray astrophysicsis concernedwith two major problems' to understand the basic physicsunderlying the origin and accelerationof the cosmic-ray
particles,and to relate the subsequentpropagationof the particlesto the characteristics of the media through which they pass. Cosmic rays are conveniently
divided into two species.Galactic cosmicrays come from outside •he solar system, and solar cosmicrays are producedby the sun. Very little still is known
concerningthe origin and accelerationof most primary cosmicrays, and indeed,
we are not yet completelysure just what are the sourcesof galactic cosmicrays.
However, over the past twenty years considerableprogresshas been made in
our understandingof the physics of cosmic-ray propagation in inters•ellar and
interplanetary media. Becauseof the convolutedpaths traversed by cosmic-ray
particles and their consequentsensitivity to the ambient electric and magnetic
fields, the study of cosmic-raypropagationhas come to be a fascinating and
rewarding discipline, which casts light on many fundamental problems in
astrophysics.It is this aspect of cosmicrays that is the topic of this review.
In largepart, this recentprogresshas beenmadepossibleby the opportunity,
throughthe useof spaceprobes,to observethe interplsnetarymediumdirectly,
so as to permit detailed comparisonof theory and observation.It is now possible to relate quantitatively many observedfeaturesof cosmic-raypropagation
in the solarsystemto directly observedpropertiesof the interplanetarymedium,
which is sometimescalledthe solar wind to indicatethat it flowssupersonically
outward from the sun. It is the purposeof this review to present a coherent
expositionof the theory of cosmic-raypropagationin the solar wind. The.present
restrictionto the solar wind is made becauseour understandingof the solar wind
is highly developed,and only there is detailed comparisonbetweentheory and
experimentpossible.Most of the conceptsand equationsdiscussed
in this paper
are also applicableto the interstellar medium.
This discussion
is primarily devotedto.an expositionof the theory of cosmicray transportas appliedto the solar wind. Observa'tions
will be discussed
only
as they illustrate the basicphenomenaunder consideration.
Previousreviewstouchingon the theory of cosmic-raytransport were written by Dorman [1962], Morrison [1961], Parker [1963], Webber [1963], Quenby
[1967], and Parker [1969].
2. HISTORICAL
BACKGROUND
The modernapproachto cosmic-raytransportis fundamentallystatistical.
The particlespropagatein an irregular interplanetary or interstellar magnetic
COSMIC.RAY PROPAGATION
IN THE SOLAR WIND
29
field, and their motion is describedstatistically;the randommagneticirregularities
scatter the particlesmuch as in the ordinary kinetic theory of gases.
The modem, statistical approach to cosmic-ray propagation was first suggestedby Fermi [1949]in his profoundlyinfluential work on the accelerationof
cosmicrays, in which the famousFermi mechanismwas introduced.Fermi was
interested in the accelerationof particles as they were scattered by randomly
moving magneticirregularitiesor clouds,and he did not explicitly considerthe
propagationproblem;but implicit in his work is the conceptof a scatteringtime
and the consequentrandom walk of particles.Very shortly after the publication
of Fermi's work, papersby Cocconi[1951] and Terletskiiand Logunov[1951] appeared in which the random walk through an irregular magnetic field was explicitly discussed.Specifically,if U(r, t, T) is the number density of particles
having energy T and speedw at positionr and time t, then the evolution of U
was assumedto be governedby a diffusionequation.If X is the mean step length
and Ix•uI << u, the averagemotionof the particlesis givenby a flux of particles
I* = -«XwVU
= -gVU
(la)
Conservation of particles then leads to the equation
•U
0t
- --V.(•VU)
•- S
(lb)
where S representsany sourcesof particles. Here K is a phenomenologicaldiffusion coefficientdeterminedin someunspecifiedway by the magnetic irregularities. Terletskii and Logunov (1951) noted that equation i is an example of a
Fokker-Planck equation.
In these early discussionsit is implicit that energy changesoccur much
more slowly than direction changes(or scatterings).This is a reasonableassumption becauseof the small magnitude of the electric field in a highly conducting
astrophysicMplasma, which permits neglect of energy change of fast particles
in • first approximation.
After these original works, a seriesof papers appeared in which •he basic
conceptswere •pplied to cosmic-raytransport in the galaxy [Terletskii and
Logunov,1952; Morrison et .al., 1954]. It is interestingto note that the postulated
scatteringhas as a consequence
a high degreeof isotropyof the cosmicradiation,
in agreementwith observation.At this time the isotropywas the only datum that
supported,to • limited extent, the point of view that particles diffused.There
was no real evidencethat equation i or any related equation was an adequate
approximation,nor was it possibleto deriveequationi as a limiting form of the
particle equationsof motion.Diffusionof cosmicrays was an ad hoc hypothesis,
based on intuition.
The first observational
evidence that
diffusion was in fact
•n
excellent
approximationto cosmic-raymotionwasput forth by Meyer et al. [1956]. In this
important p•per, a study of the February 23, 1956, solar-flare event established
that the observedtime-intensity profile could be accountedfor quantitatively by
equationI with appropriateparameters.Much of this very early discussionis
quite close•o presentideas.In particular, the ide• that the exponentialdecay
30
J.R. JOKIPII
requires diffusion with an absorbingboundary at about 5 AU is in agreement
with some more recent suggestions,as discussedbelow in section 6. However,
these authors also postulated a field-free inner cavity, which is not consistent
with more recent observations.
About this same time it was becomingclear that the sun 'modulates'the
galactic cosmic-ray intensity in antiphase with solar activity. This effect was
first reportedby Forbu.sh[1954] and is now known to consistof a cyclic, elevenyear, solar-cycle variation upon which are superposedshorter-term irregular
fluctuations (called Forbush decreases). This modulation is strongest at low
energies.Initially, a number of authors consideredthe possibility of geocentric
modulation [Nagashima, 1951; Parker, 1956], but it finally was establishedthat
the modulation was heliocentrieand affected the cosmic-ray intensity throughout the inner solar system. It remained to determine the mechanismresponsible
for the modulation. Various models involving static heliocentric electric or magnetic fields were proposed [Freier and Waddington, 1965], but none was satisfactory. Morrison [1956] proposedthat certain featuresof the transient modulations could be explained in terms of diffusion in tangled magnetic fields carried
out by plasma ejecta from solar outbursts.In his picture, the outward-moving
irregular magnetic fields carried with the plasma initially sweep the galactic
particles out of the inner solar system,after which the particles gradually diffuse
back,up to their original level.This is essentiallythe modernview of the transient
Forbush decreasesand, as we will see shortly, is about as closeto the present
picture as was possiblewithout the conceptof a continuoussolar wind.
The real breakthrough in our understandingof cosmicrays in the solar system camewith the conceptof a continuoussolar wind. The modern conceptof a
continuoussolar wind goes back to the considerationof the physics of comet
tails by Biermann [1951, 1953, 1957] and was cast in essentially its modern
form by Parker [1958a.]. The interestedreader is referred to the review by
Dessler [1967] for an informative and readable summary of the development
of the solar-wind concept.Axford [1968] and Parker [1969] have recently published reviewsof the detailed physicsof the solar wind. The point here is that
the continuouspresenceof solar-wind plasma in interplanetary spaceprovides
a natural vehicle for many of the cosmic-ray effects discussedabove. The interplanetary diffusionpostulatedby Meyer et al. [1956] is a result of scattering
of chargedparticlesby irregularitiesin the interplanetarymagneticfield, which
is carried out with the solar wind. The basic interpretation of Forbush decreases
suggested
by Morrison [1956], and describedabove,can be seento fit naturally
into this over-all solar-windpicture. Finally, as Parker [1958b] was quick to
point out, the continuoussolar wind provides a compellinginterpretation of
the quasi-steady,eleven-yearmodulation of galactic cosmicrays. Parker noted
that the irregular magnetic field frozen into the solar wind will tend to conveer
particlesradially outwardat the solar-windspeedV•. This will set up an outward gradientof cosmicrays that causesparticlesto diffuseinward.In a steady
state,therefore,the outwardconvection
flux UV•omustbe balancedby..an equal
in.warddiffusion
flux -K OU/OrlThus,if the systemis approximately
spherically
symmetricand if U• (T) is the interstellarnumberdensityof cosmicrays, as a
COSMIC. RAY PROPAGATION
IN THE SOLAR WIND
31
functionof kinetic energy,T, reachedat an outerboundaryr - D, onefindsupon
integrating the flux balance equation
(OU/Or) =
(2)
that
U(r,
T)=U•(T)
exp
(- DV•dr)
K
(3)
Althoughwe will find that this simplepicture is incomplete,the various features
of the eleven-year variation are qualitatively explained by equation 3. For
example, as the parameters Vw, D, or g changewith the solar cycle one can
understandthe basic features of the eleven-year cyclic variation. Similarly, the
daily variation observedby neutronmonitors [Sarathai and Kane, 1953; and
Rao et al., 1963] is due to the corotarionof the interplanetary magneticfield
with the sun [Ahluwalia and Dessler,1962].
With the introductionand acceptanceof the solar wind conceptand with
the recognitionthat fast chargedparticles (particleswith velocitiesmuch greater
than the Alfv•n velocity) diffusethrough the interplanetary magnetic field, the
basic qualitative features of cosmic-raytransport in the interplanetary medium
were established.The years from 1963 to the present have been a period of considerablerefinement and modification of this over-all picture, but the basic idea
remains as discussedabove. The fundamental improvementsin our understanding include the following' The nonzero divergenceof the solar-wind velocity
leads to adiabatic energy change as was pointed out by Parker [1965] and
Gleeso• and Axford [1967]. ($inqer et al. [1962] discussedadiabatic deceleration in a slightly different context.) That diffusion is probably anisotropic,since
a consequence
of the presenceof a nonfluctuatingpart (average part) of the
magnetic field was pointed out by Axford [1965a] and Parker [1965]. The
subtleties involved in the various diffusion-related anisotropiesand their relation
to energychangewere revealed [Gleesonand Axford, 1967; Jokipii and Parker,
1967, 1970]. Finally it was shownthat the cosmic-raydiffusiontensor may be
obtainedfrom the magnetic-fieldpower spectrum[Jokipii, 1966, 1967; Hasselman
and Wibberentz,1968; Roelof, 1968]. This step at last establishedthat the diffusionof cosmicrays, as indicatedin equation1, couldbe derivedas an appropriate
statistical limit of the particle equationsof motion.It has also provedto be very
important in utilizing direct space-probeobservationsto compareobservations
of the solar wind and cosmicrays. These major stepshave been accompaniedby
a large numberof papersconcernedwith detailed applicationto observations.
A• the risk of being overoptimistic,one may say tha• the basic principles
underlyingthe transportof cosmicrays in the solar systemare now well established.The principal phenomenaare reasonablywell understood.Future theoretical work will be devoted to obtaining more accurate solutions of the com-
plex differentialequations,to better evaluationof the relevant parameters,and
to the explorationof higher-orderor more subtleeffectsassociatedwith fluctuations,anisotropies,
and lack of sphericalsymmetry.
J. R. JOKIPII
32
.
A.
STATISTICAL
DESCRIPTION
MAGNETIC
OF THE
FIELD
INTERPLANETARY
Introduction
Consider first the problem of specifying the interplanetary magnetic field,
which is the dominant factor determining the motion of cosmic-rayparticles in
the solar wind. The electric field may be neglectedbecauseof the high conductivity
of the plasma and becausethe Alfv•n velocity is small comparedwith the particle
velocity, as is shown below in section4. Collisionswith particles, of course,are
completelynegligiblebecauseof the low density in the solar wind. To calculate
the particle trajectoriesit is necessaryfirst to specifyquantitatively the magnetic
field, and this is difficult. A completespecificationof the magneticfield as a function of position and time, although in principle possible,is not desired,for computing the individual particle trajectoriesin the field would be beyond our capability. Instead, we describethe magneticfield statistically,in the senseof random
function theory [Yaglom, 1962]. In such a scheme,specificationof a hierarchy of
correlation functions constitutes a complete statistical specificationof the field.
That is, we consideraveragesover an ensembleof identical systemsand denote
suchaveragesby the symbol(). If B•(r, t) is the magneticvector, the specification
of the correlation functions,
t•,
...
t3 = (B,(rl, t)Bi(r,
...
t3 )
(4)
for all n, as a function of r•, t• ... r•, t•, constitutesour descriptionof the field. For
example,for n = 1 onehassimply {B•(r, t)•, whichis the averagemagneticvector,
and so on. This approachis basicallythe sameas that usedto describethe irregular
velocity field in ordinary turbulence, and the reader is referred to the book by
Batchelor[1960] for an excellent and readable summary of turbulence theory. The
goal is to describethe motion of cosmic-rayparticles statistically in terms of the
field correlationsdefinedin equation 4. Before doing this, let us considerin some
detail the structure of the interplanetary magnetic field.
B.
The Average Magnetic Field
First considerthe average interplanetary magnetic field. Direct observations
[Ness et al., 1964] show that although the interplanetary magnetic field at a
given point fluctuatesconsiderably,the averageat the orbit of earth over a time
scaleof one day is very closeto the idealized spiral predictedby Parker [1958a].
Althoughavailableobservations
coveronly the heliocentricrangebetweenroughly
0.8 and 1.5 AU, it is assumedhere that the average magnetic field at any point
r, 0, • in the solarwind is adequatelyrepresentedby Parker'sspiral
= (B(rs, O,r)- r9s/Vw))(rs'/r
= 0
(5)
(B•) = (Br(rs, O,r) - r9s/Vw))(rs29ssin O/Vwr)
where rs is the radius of the sun, 9s is its angular veloeit:y,and 0 = 0 along the
rot•at•ionaxis. In Figure 1, [his average field in the solar equatorial plane is
schematicallyrepresentedby the dashedline. The angle g, = tan-x (r9s/V,o.
COSMIC.RAY PROPAGATION
IN THE SOLAR WIND
33
The variousconstantsare suchtha• a• earth'sorbi• the averagefield magnitude
is ~ 5 • (1 • = 10-• gauss),andthe spiral is inclinedabou•50ø to •he solarradius
vector. It is no•ed that the interplanetary field appearsin sectors [Ness and
Wilcox,1965], centeredon •he sun,in which•he averagespiral field is directed
either in •oward the sun or out away from the sun,separatedby rather abrupt
changesin signof <Bd and <B,>in equation5. In the following,all averageswill
apply over a given sector, and each average will be assumed•o be essentially
independentof time within each sector.
C. The Fluctuating ComTonent
The observeddeviationsfrom the ideal spiral within a given magnetic
sectorare a manifestationof the irregularitiesin the magnetic field that are so
important to cosmicrays. It is these irregularitiesthat scatter the cosmic-ray
particlesin pitch angle to make them isotropicand causetheir diffusion.The
problemis to obtain a quantitative descriptionof the fluctuations.
To a goodapproximationthe observedfluctuations,in time, at a stationary
observercan be regardedas due to a pattern that is stationary in the frame
of the plasma.beingconveered
past the observerby the solar wind. That is, the
temporal fluctuationsare essentiallydue to the motion of spatial fluctuations
at the solar-windvelocity ¾•.
Onemay understandthis last point mostintuitively by notingthat the solar-
wind plasmaflowsout from the sun at a speedof some300-400 km/sec. The
relevan•fluctuationsare either hydromagneticwaves or quasi-staticstructures,
sothat, relativeto the plasma,their speedVvasatisfiesthe inequality
V• < (V,•2+ V•2)TM
<• 50 km/se½•t 1 AU
(6)
where Va - B/(4•rp)•/a --• 40 km/sec is a quantity with dimensionsof speed
called the Alfv•n speed,and Vs is the soundspeed,which usually is some30
km/sec.Sincethe speedVvais so muchlessthan the bulk plasmaspeed,it may
be neglected,and the irregularitiesmay be taken to be approximatelystationary
in the frame of the wind. Thus, the irregular variation of the field with distance
may be related to the observedvariation with time. This result is discussedmore
formally, from the point of view of correlationfunctionsand power spectra,in
Appendix A.
It should be noted also that the smallnessof the Alfv•n velocity also helps
insurethat the electricfield is small in the frame of the solar wind, and thus it
may be neglectedin treating particle motion. We will come back to this point
later.
The resultingview of the interplanetarymagneticfield, proiectedinto the
solarequatorialplane,is indicatedschematicallyin Figure 1. One has irregularly
fluctuatinglines of force being drawn out from the sun by the solar wind, being
braidedand twistedby turbulentmotionsin the solarplasma.The linesof force
are tangled and intertwinedbut, on the average,lie along Parker's Archimedean
spiral, indicatedby the dashedline.
34
J.R. JOKIPII
Fig. 1. Schematic view of the interplanetaw magnetic field projected into
solar equatorial plane. Rotation of the
sun at angular speed 9, results in indicated spiral average field. • .-- tan-•
(9,r/V,o) is the angle between the average field and the radius vector from the
sun. Note that magnetic lines of force do
not actually cross,but are braided and
intertwined
in three dimensions.
To describethe fluctuations quantitatively, one makes use of the higherorder correlationsdefined in equation 4. First define the fluctuating part of the
magnetic field
Bx(r, t) = B(r, t) -- 03>
(7)
Of primary interest is the two-point, two-time correlationof B•
(•,(r•,
tOB•,(r,.,t,)) = R,,(r•, r,.,
t,)
(s)
We saw above that the average field within a sector was invariant with respect
to time; we make the approximation that all means have this invariance, so that
R•(t•, t•) dependsonlyon the differenceIt• - t•.I. The characteristic
of a two-point
correlationfunctionis that it is a maximumfor Ir• -- r•.I = 0, It• -- t•.I -- 0, and
goesto zero for large arguments.The characteristicscales•of the function R• are
calledthe correlationlength L and correlationtime r•, respectively;R• is sketched
schematicallyin Figure 2. Observationally, one finds that in the solar wind L is
much lessthan 1 AU, in which caseit is possibleto assume
R,,(r•,
r,.,t•,t•)'•'R•,•(r•
-- r• t•
' -
t•)
(9)
=
Herethe(r) superscript
indicates
the(slow)
variation
of thecharacter
of the
fluctuationswith heliocentri½radius r. In the languageof random function theory,
equation 9 states that the fluctuationsare approximatelyhomogeneous
or stationary random functions with respectto the arguments r and t. Note that the definition of R•(i•, •) is such tha• the mean-squarefluctuation in the magnetic field is
given by
(Bx"(r)>
= •]] R,• (0, 0)
(10)
COSMIC-RAY PROPAGATION IN THE SOLAR WIND
35
The readeris again referredto the book by Batchelor[1960]for a thoroughdiscussion of the propertiesof the two-point, two-time correlationfunction of a divergence-free
vectorfield.Such• discussion
is beyondthe scopeof this p•per.
In what followsthe fluctuationsare treated locally as if all statistical properties are independentof r and t, in what is effectivelya 'quasi-stationary'approximation. After the calculation of local parameters is carried out, the mean values
are allowedto vary slowlywith r and t. A morerigoroustreatment of this variation
of the meanis possible,usingstructurefunctions,as outlinedby Tatarski [1961],
but the results are the same as those obtained in tkis more heuristic treatment.
The (r) superscriptwill be suppressed
below,with the understandingthat all local
quantitiesmay vary slowlywith r and t.
It is often desirableto considerthe Fourier transform of equation 9
P,i(k,
o•')
=f:•da• d•'
R,i(•,
•')e
i(k't-•'•)
(11)
which is usually termed the 'power spectrum'of B•. For illustration, in Figure 2
are displayedschematicallytypical functions,R•i(•j, •) and P•(k, o•').
In general,it is true that the higher-ordermomentsdefinedin equation 4
will be requiredfor a completedescription;but work up until nowhasconcentrated
on effectsonly to secondorder in the fluctuatingfield B• so that a consideration
of R•i or P•i is sufficient.It will turn out belowthat, to a reasonablefirst approximation, the scatteringand diffusionof fast chargedparticlescan be expressedin
I
I
Rxx(0,0,r])
(a)
Fig. 2. Schematicillustration of the general form of the two-point correlation
function (a) and power spectrum (b) for
a typica.1random magneticfield. The
scale L is the correlation length of the
fluctuations.
Pxx(0,0,k)
(b)
36
J.R. JOKIPII
terms of the averagemagneticfield and the magnetic-fieldpower spectrum,both
of which are observablequantities.
D.
Observations
o• theFluctuations
Before proceedingto the theory of particle transport, it is desirableto consider
the practicalproblemof obtainingR•i(t:, •) (or P•(k, o•))from spacecraftobservations in the solar wind. The point is that it is not possibleat present (nor for the
foreseeablefuture) to obtain precisemeasuresof the interplanetarymagneticfield
B as a function of r and t. The presentlyavailable data consistof the magnetic
vector B as a function of time at an essentiallystationary spacecraft.From these,
the mean field may be subtracted, so that one has the fluctuating field B•(r, t)
measuredat a given point. Now, becausethe Alfv•n velocity is so small, as discussedabove in section3B, the field fluctuationsare essentiallyfrozen into the
plasma so that the time dependenceof B• at a given position gives information
concerningthe spatial variations. Define B•'(r, t) as the fluctuating field in the
frame of the solar wind. Clearly, if the fluctuationsare frozen into the solar wind,
which is movingat velocity V•, onehas in the movingframe r' -- r - V•(t - to),
so that
B(r,
B,'[r - Vo(t-
to), to]
(12)
wherethe prime here refersto quantitiesin the movingframe. More precisely,the
relevant correlation functions in the fixed and moving frames are related by
Rii(•-- O,q')_____
(Bli(ri,
t)Bli(ri,
t -]-•-))
=
(13)
0)
In practice, the averagesappearingin equation 13 are obtained by averaging
over time at a spacecraft.This relation is discussed
more rigorouslyin Appendix
A where i5 is shownthat equation13 followsif the only important fluctuations
presentare thosefor which the phasevelocity ,,,'/k • V,•. Equation 13 may be
written more convenientlyas follows.Let P•j(/) be the observedtemporalpower
spectrumas a function of frequency•:
Pi•(D -
(Bii(ro,t)Bli(ro,t -]- 7'))e
-2wil'v
d•-
=f;•R•i(O,
•-)e
-2•i•
d•Then, if • is along the wind velocity vector, equation 14 can be written
i(2•.f•._•/¾w)
P,•(•)_____
• I f:•R,•'(O,
O,•'•,O)e
d•'s
(15)
Equation 15 states that the observedfrequency power spectrum is, to a good
approximation,the instantaneousone-dimensionalwave-numberspectrumalong
5hedirectionof the solar-windvelocity. Clearly, becausethis is only one-dimensional, we do no5 obtain complete information concerningthe spectrum from
observationsat a givenpoin5in space.Further assumptions
are usually necessary
COSMIC-RAY PROPAGATION IN THE SOLAR WIND
37
in discussingany actual problem. For example,i• may be assumed•ha• •he
fluctuationsare s•a•is•ically isotropic,so •ha• •he form of •he correlationtensor
is invarian• under ro•a•ions.In •his case,specificationof •he one-dimensional
spectrum(equation15) is sufficien••o specify•he complete•ensor.Alternatively,
one may assume•ha• •he fluctuationsdepend on essentiallyonly one direction
in space.Thesevariousapproximationswill be discussedla•er, in connectionwi•h
the discussionof particle mo•ion.
Illustrated in Figure 3 is a typical frequency power spectrumfor •he component of the interplanetary magnetic field normal to the solar equatorial plane.
The spectrumwas computedfrom data obtained on the spacecraf• Mariner 4
in late 1964 near the orbit of earth [Jokipii and Coleman, 1968]. Indicated on
the lower ordinate is the corresponding
wave number k = 2,r//V•o, for the onedimensionalspatial spectrumof the fluctuations,as given in equation 15, assuming a wind velocity V• = 350 km/sec. This spectrumhas the characteristicwave
number k• _ 6 x 10-•a cm-•, changingto an approximatepower law k-•/a at
higher frequencies.Of course,the value of k, is somewhatuncertain and is only
correct to about a factor of 3. The basic propertiesof Fourier transforms then
state that the correlationlengthL is roughly 1/k, -- 2 x 10• cm, or 0.01 AU.
Thus, the assumptionthat L is much smaller than I AU is borne out. It will
turn out that the most important effect on cosmic-ray particles comes from
the resonantwave numbersk • 1/r•, wherer, is the averagecyclotronradius,
so that the wave numbersgiven in Figure 3 are the onesrelevant to cosmicrays
having kinetic energiesfrom a few Mev to several Gev, as indicated on the lower
scale. The correspondingspectra of the other two componentsof the magnetic
field are basically similar to the spectrum shown,with the power in the radial
componentbeing somewhat smaller.
Power spectrareportedby other authors [Coleman, 1966; Siscoeet al., 1968;
$ari and Ness,1969; Quenbyand Sear, 1970] bear out thesegeneralcharacteristics
of the spectrum, although they differ in detail. There is evidencethat the amplitude of the spectrum,at the higher wave numbers (~10 -•x - 10•ø), fluctuates
by roughly a factor of 3 on a day-to-day basis and that the spectrumis steeper
in 1967 comparedwith 1964-1965. The observedspectra will be discussedagain
below in connectionwith the computation of the cosmic-ray diffusion tensor in
interplanetary space.The observedvariations in the spectraimply that the diffusion tensor varies with time.
It is important to rememberthat even completeknowledgeof the threedimensionalpower spectrum may not be sufficientto specify completely the
nature of the fluctuations.For example,the fluctuationscould consistprimarily
of discontinuous
changesin the field and still have nearly the samepower spectrum as a systemhaving more gradual changesin the field. However, the discussion concerningthe relative importance of waves or discontinuitiesin the solar
wind [Siscoeet al., 1968; Burlaqa, 1969; Belcheret al., 1969] is not relevant to
the cosmic-rayproblem except as it affects the assumptionsrelating the onedimensionalspectrumto the full spectrum,as discussedabove.
Finally, note that the electric field in the plasma may be treated in this
same way, but it will be shownin the next sectionthat it is not necessaryto
38
J.R. JOKIPII
i0 7
MARl NER
4
29 NOV- $0 DEC, 1964
i0 6
B8
TOTPWR=4.3 7'2
io4
LLI
• IO3
ß
IO 2
i0 •
10-6
10-7
10-13
10-5
FREQUENCY
10-12
10-4
(Hz)
iO-II
10-3
i0-10
10-2
10-9
WAVE NUMBERk (cm-I)
I
IOO
I
$o
I
IO
I
2.
I
.5
I
.I
I
.Ol
RE SONANT PROTON ENERGY (GeV)
Fig. 3. Powerspectrumof the componentof the interplanetarymagnetic
field normal to the solar equatorialplane, observedon Mariner 4. Middle
ordinategiveswave numberrelatedto the observedfrequencyif the solar-
wind velocity V• -- 350 km/sec,and the lower ordinategivesthe proton
energy correspondingto the resonantwave number /• -- 1/r•, where r• is
the cyclotronradiusin a 5-• averagemagneticfield. One • •
[Jol•ipii and Coleman, 1968].
10-5 gauss
COSMIC-RAY PROPAGATION IN THE SOLAR WIND
39
considerthe electric field explicitly in computing the motion of fast charged
particles. The magnetic-field fluctuations are the dominant effect on particle
motionin the frame of the plasmaif the particle velocity is large.
4. STATISTICAL
A.
DESCRIPTION
OF
PARTICLE
MOTION
Introduction
Now considerthe motion of fast chargedparticles in an irregular magnetic
field such as that describedin section 3. As in the case of the magnetic field,
the goal is not to obtain a completedescriptionof the trajectory of each particle,
but instead to find a statistical equation governingthe evolution of the particle
distribution function. The behavior of a large number of particles will be assumed
to followthe probability distributionof a singleparticle. The presentapproachis
very similar to the ordinary random walk problem, exceptthat here we are tracing the random walk of a particle trajectory under the influenceof an irregularly
fluctuating magnetic field. It will turn out that the desired descriptionof particle
motion can be obtained in terms of observablestatistical properties of the magnetic field, such as its mean value and power spectrum.The end result will be
an expressionfor the particle diffusion coefi%ientin terms of the observedmagnetic field.
The goal of this calculation can perhapsbe stated in more familiar terms as
follows. If the magnetic field were uniform, the particle would travel in a
helical orbit along the field. The irregularities perturb this orbit and ca.use,
amongother things, a scatteringin pitch angle. If the irregularitieswere.all of a
given shape •;B(x), but occurred,say, with random sign, one could compute an
'elementary'scatteringAt•(< 1, and t•hencomputethe net changeafter N scatter-
ings(A,t•) _• N-•/a [Ate[,
because
of the randomsign.Thisthenleadsto a relaxation of the angular distributiontoward isotropyin a characteristictime •c, which
can be computedfrom A.t•.Hence the diffusioncoefficientK • w•c/3 can be
obtained.But in actuality the irregularitiesdo not all have the sameshape,and
a more general treatment is necessary.
The outline of the present approach is as follows. The particle equation of
motion is considered,and the evolution of the probability distributionin pitch
angle and position is determined.This is carried out from the point of view of
a Fokker-Planck equation,but it is alsoshownthat essentiallythe sameequation
arises from a considerationof Liouville's equation. The resulting equation is
usually too detailed for practical use. However, it turns out that in most cases
of interest the scattering causesa rapid relaxation to isotropy, so that the
pitch-angle distribution differs from isotropy by only a small amount. For this
situation the particle density (averaged over pitch angle) satisfiesthe diffusion
equation i with the diffusiontensorexpressiblein terms of the magnetic field. In
this approximation,then, one regainsthe diffusionequation,which has been of
such use in describing cosmic-ray transport, except that the diffusion tensor is
expressiblein terms of observedmagnetic-fieldparameters.
Consider,then, the propagationof a chargedparticle in an irregularly
fluctuatingturbulent plasma.As discussedabove the irregularitiesare, at least
40
J.R.
JOKIPII
in par[, hydromagneticwaves propagating in the plasma and therefore have
electric as well as magnetic fields associatedwith them. The following argument
showsthat the electric fields may be neglected in a good first approximation.
Most simply, the point is that the magnitudeof the electric field is E ~ V•B/c,
where V• is the Alfv•n velocity and c is the velocity of light. Hence the magnitude of the ratio of the electric force to the magnetic force is [Joki•ii, 1966]
F• • qwB/•
qE • qV•B/e•
Fs
qwB/c V•
w
(16)
Particles satisfying the inequality w • V• will be called fast chargedparticles;
for them the effect of magnetic fluctuationsdominatesthe effect of electric fields.
Sturrock and Hall [1967] made the relation in equation 16 more explici• and
showedtha• for wavesof frequency•' and wave number k
Fs- Iwx(kxE)[•kw
(17)
which is essentially•he same as (16) for Alfv6n waves. The neglec• of •he electric field for fas• particles or cosmicrays simplifiesthe analysis considerably
becausethen the effect of the magneticfield is only •o changethe direction of
motion without changingthe energy.
B.
The Fokker-Planck Equation in,Pitch Angle and Position
Supposea particle of velocity w and mass7•mo propagatesin •he magnetic
fieldB(r), where7• = (1 - w•/c•)-•/•. The equationof motionis
dw_
dt
-
y•moC
- w o(r)
(18)
with • = qB/ymmoC.
The goalis to find • d•erenfial equationgoverning•he distribution of p•rticles subject •o equation 18, in terms of •he correlationfunctionsof
the ma•etic field defined in equation 4. One c•n e•sily see that • general description would involve the entire infinite f•ly
of correlations•nd is thus impractical. To avoid this d•culty, one •pproach is •o define
and
(19)
o(r) = o(r) -
•nd •o assume(w•/wo• • 1 so•h• •he orbi• is only slighfiyperturbedin a cohereneelength of •he field. Then only •he lowest-ordercorrelationsof o• need be
retained •o obtain • reasonableapproximation•o •he particle motion [Jokipii,
A• this poin• one m•y proceed in one of •wo disfinc• directions to find •he
equationfor •he particledistribution.The ofi•n•l •ppro•ch [dokipii, 1966,1968a;
Hasselmannand Wibberentz,1968] proceedsby meansof Fokker-Planck coefficients
in • manner firs• used by •t•rrock [1965]in • d•eren• problem. More recently
Hall and•t•rrock [1967],Dol•inovand Topty•in [1968],and Rodof[1968]pointedou•
•h• •he sameequationscouldbe obtainedfrom a moregeneral•pproachby means
of •Liou•lle's equation.(Seealsovery recentwork by Kl•mas and •andri [1970]).
COSMIC.RAY PROPAGATION IN THE SOLAR WIND
41
In fact, Hall and Sturrockextendedthis techniqueto includeelectricfields and
energy changes.Appendix B sketchesthe derivation by means of Liouville's
equation.In this review, the Fokker-Planckapproachof Jokipii [1966] will be
discussed
becauseof its comparativealgebraicsimplicity. One first notesthat the
unperturbed particle trajectory in the uniform field •0o•.is the usual helix determinedby its instantaneous
positionand velocity. It is assumedthat the average
cyclotronfrequency•0ois large comparedwith any other frequencies,so that all
quantitiesmay be averagedover the phaseof gyration. Hence, the orbit is com-
pletelycharacterized
by the pitchangie0 = cos-x w./w, and positionat a given
velocity w.
It provesconvenientto definethe complexvariables
x+ =x+iy
w+ = w•+iw•
in the plane normal to the averagemagneticfield. The unperturbedor zero-order
orbit is then given by
z• = Zo+ W•ot
(20)
x.•, = X.o •- i(w.oe-•øt/O•o)
(21)
2 = W2 = constant.
with Iw.01
•' •- W•o
Now,thefluctuating
field• causes
perturba-
tions in this zero-orderorbit. The orbit parametersx, y, z, W•oexecutea random
walk under the influenceof •. Define/z = w•/w. Then let
n(r, •, t) dr d•
(22)
be the probability of findinga particle in r to r •- dr, • to • •- d• at time t. Hence n
is a probability density that may be identifiedwith the measureddensityin position and pitch angle. If the orbit changescausedby the random field •x are small
in a correlation time of the fluctuations as seenby a particle, then the evolution
of n is governedby a Fokker-Planckequationas outlinedby Chandrasekhar[1943].
That is, the evolution of n is causedby a succession
of small, random increments,
and the particlesmay be regardedas random-walking,or diffusing,in pitch angie
and position.The processis a straightforwardgeneralizationto morevariablesof the
ordinary particle diffusiondue to a spatial random walk, as given in equation 1.
If r, • is replacedby the four parametersX•, then the Fokker-Planck equation
reads formally
On
0•'2L
Ot- - •
i=• • 0 r(AX•)
k
n] +•1•
•=,OX•
*
At n
n] (23)
The problemis to evaluatethe variousFokker-Planckcoefficients(AX •')/At,
i
etc., appearingin equation 23.
To calculate these coefficientsone considersthe perturbations about the
42
J.R. JOKIPII
unperturbedorbi• givenin equations20 •nd 21. Se•
z(t) --z,,(t) q-z•(t) q-z•(t) q- ...
x+(t) :
+
+
(24)
+ ...
(25)
wherez•(t) andx+,(t) arelinearin (o,andz,.andx+, are of secondorder.Substituting
equations24 and 25 into the equationof motion (18) one obtains,to firs• order
in
i
x
• iCOot
• -- --• [co+(z,,,
x+.)W+o
e -- co+*(z.,
x+.)W+oe
-'•ø•]
(26)
•+ • q- iwox+
• -- iWzoOJ+
(z., x+.) -- iw+oe-'•øtW,z(Z.,
x+.)
(27)
and so on, where •he superscrip•* indicates complexconjugate.
Consider firs• •he sca•ering in pitch angle. From •he definition of • and
equation26, onehas immediately
1
=
--- -- 4w
•.
'•ø• - oJ+*
(z., x+.) +
dz'([oJ+(z.,
x+.)W+o*e'
dz
W+oe
-'•ø•]
ß[w+(z.,x+.)W+o*e
'•' -- w+*(z.,x+.)W+oe-'•*•']> (28)
co•ec• •o secondorderin •,. Inspectionof (28) reveals•ha• in general
is expressiblein •er• of •he co,elations of •. If it is rememberedtha• the •wopoint correlationsgo •o zero a• •st•nces •eater •han the correlationlength L,
one seestha• if it is assumedthat At is l•rge compared•th L/w, equation 28
migh•simplify.It is a simplema•terto showthat in sucha case((A•)•> is proportional to At. To evaluate((A•)') explicitly,considera specialcase.Sincewo•.
pic• ou• the o•y characteristicdirection,it is reasonabletha• the fluctuations
are statistically axially symmetric abou• the z axis, in which casethe correlation
function of •x must be of the form [Batchelor,1946]
R•(•, •, •) = (wx•(x,y, z)wx•(x• v, Y ß •, z • •))
av
•+b av• av•
+dv
=Lan•
] avf
af•• b af•• df
• dv af• • df a• • • b •c •
2d•
(29)
where a, b, and c are even functionsof o = (• + •)x/• and •, and d is even in
o and odd in •. Carryingout •he opergfionsindicg•edin equg•ion28, one finds
thg5upondefining• = W•o(•' - •) and
p,(r) =
-
'
(30)
1/2
----2(1 too'•
•
1-- cos--r
CO0
COSMIC-RAY PROPAGATION
IN THE SOLAR WIND
43
then
((A•)
•) i- • foa,f,oo(a,-•)
=
dr
d•'
p•(•')](1
2 •ate',
ß{b[•',
p•(•)]e
-•ør/•
+(1-2•o•)w
(31)
or, if W•o•t > L for most particles of interest
{(A•)
•) 1-•({
q_
(1-•')w"
_
2•Oo
ate',
p•(•)](1
(32)
Equation32 is the generalexpression
for the Fokker-Planckcoefficient
if the fluctuationsare statisticallyaxially symmetric.I5 givesthe rate of relaxation toward isotropy.
By followingthis samegeneralprocedure,
onemay computethe remaining
Fokker-P!anekcoefficients.
However,somework may be circumvented
by making useof Liouville'stheoremand symmetryproperties.One neednot compute
(•)/•t, sinceLiouville's•heorems•a•es•ha• •he steady-s•a•econfiguration
is
isotropic.Hence(5•)/st and ((5•)•)5t mustbe relatedby [Jokipii, 1966]
I 0• (<(•u)•> 0 [(•u}
I 0 (<(•u)•>
(33)
whichis zerofor an isotropicdistribution.This resul•alsoarisesin the alternate
approachusingLiouville's[heorem.Similarly,•he symme• abou• •he z axis
leads •o
<a•>/at = <as>/at = o
(Ax Ay)/At = 0
(35)
The equationfor x+•(t) may be integrated,
precisely
as wasdonefor z•(t) •o
obtain
((ax)•)
<(ay)•)
At
At
- 2•o• I•1•
'
+ (1 - •)w•R•[r, o•(D]e
'
+2i•(1•')w
•{a[•,
p•(•)]•
•o
Considering
(•z)/•t and (•z•)/•t, onefindsfur[her•ha[
(38)
(•'>/at
= o(at)
.(39)
44
J.R. JOKIPII
HereO(At)meansthat the righ•sideof equation39 goesto zeroas At • 0. Thus,
the full Fokker-Planckequationin •his approximationbecomes
with the Fokker-Planckcoefficients
given by equations32 and 37. This is a
complete
solution
to theproblem,
in the limit that (.o•?)•02
• 1, sothat theorbits
are onlyslightlyperturbed
in a coherence
lengthof the field.The reasonfor this
conditionis that if the orbit changes
are largein a coherence
length,thenuseoœ
the Fokker-Planckequation23 is not justified.Thereis onepoint,in the analysis
that,requiresfurthercomment.
In goingfromequation31 to 32, it wasassumed
that W•oAt= izwAt>>L; but clearlythis breaksdownat tz -- 0 (0 = 90ø),and
somecautionmust be exercised
in interpretingthe resultsnear 90ø pitch angle.
For oneinterpretation,
the readeris referredto the paperby Noerdlinger[1968].
Otherwise,
the discussion
and conclusions
are straightforward,
and the goalof
describing
particlemotionin termsof the field correlations
hasbeenrealized.
The alternateapproach,by meansof the Liouville theorem,also leadsto
equation40 in this limit. This derivation,dueprincipallyto Hall and Sturrock
[1967], is sketchedin AppendixB.
C.
T,he General Diffusion Tensor
Equation40 containsa great deal of informationconcerning
the particle
distribution,includingthe pitch-angledistributionand its evolution.Quite often
the scatteringin pitch angleis rapid comparedto otherratesof change,sothat
n is nearly isotropic.If this is the case,onemay approximateequation40 by a
diffusionequation.One way of obtainingthe diffusionequationis as follows
[Jokipii,1968a].n(r, t•, t) may be expandedin Legendrepolynomials
n(r,tz,t) = •1 U(r,t)q-• n,(r,t)P,(tz)
(41)
whereU is the pa•ic]e density (or probabilitydensity) averagedover pitch
anglea• a givenenergyT. A• thispoin•i• is necessary
to assume
tha• the scattering is suchtha• n•+• decaysfasterthan n• (the n• mus•all decaydueto the scat•ering). Then, •o lowestorder•or slowvariations,
he,
t)
][U +
wi•h n• •< U. Substitutingequation42 into the Fokker-Planckequation40 and
integratingover• from• - -1 •o • - 1 yields
at + s Oz-2Lax• + oy• j
Subs[i[uting
again,multiplyingby >, andintegratingagainyields
wO
Uoz
n,
((au•))
d•
(44)
And,finally,combining
equalions
43 and44 yieldsthe diffusion
equalionfor U
COSMIC-RAY PROPAGATION IN THE SOLAR WIND
45
Ot- • --•Oz • •X[_Ox
•' • Oy
•'J
where
•,1- 9
At d•
At'•)
1fo
•(Ax
dt•
'•' = •
(46)
(47)
A different
derivation
of theparalleldiffusion
coe•cient,
•, in terms
((A•)"/At wasfirstputforthby Jokipii[1966]andsubsequently
emphasized
by
Hasselmann
and Wibberentz[1968].In this derivationthe Legendre
expansion
is not used.Insteadit is assumed
that On/Or• vw (On/Oz)in equation40. By
carryingthroughappropriate
manipulations
whilestill regarding
the anisotropy
to be small, one arrives at
= w•
•'
(• -- •) d• d•'
(48)
Thisis •he sameas (46) onlyin ce•ain cases.
However,in •he actualcases
discussed
below,•hed•erencebetween
(46)and(48)isless•hano•heruncertainties;
•hus,•hedifference
is no•a• presen•
of practical
importance.
Onedifficulty•h
•helatterapproach
is •ha• •hemajorassumption
of neglecting
On/Or•o find•he
dependence
onu breaks
do• badlynear90øpitchangle.On•heo•herhand,it has
notbeenproved•ha••heLegendre
coefficients
decayasassumed
in •hefirstderivation.In •he present
paper,•heform•afionleading•o equations
46 and47 •11
be used.
A• thispointit is desirable
to generalize
•he resultslightly.The aboveconclusions
wereobtained
by assuming
a statistically
uniformmediumin which
average
quantifies
areindependen•
of position
(e.g.,(•) = Wo•,e•c.).However,
discussed
in section
3, themeanvaluesin thesolarwindvaryslowlywithposition.
The present
conclusions
concerning
scattering
andd•usionarestillvalidif •he
lengthscales
overwhich•he means
vary arelargecompared
•h •he scalefor
sca•efing(mean-free-path
X).Thiswill be•hecaseif X<<i AU, whichis•ruefor
particles
up •o about100Gevenergy.
Consideration
is restricted
•o thiscase.If
•he average
ma•e[ic fieldvariesM•h position,
i• waspointedou• by Jokipii
[1969b]
andJokipiiandParker[1970]
•ha[•hereareadditional
•erms•ha• mustbe
included
in •heaverage
particlemo•ion.
Therearedriftsdue•o •hecubalureand
gradien•
of •he average
ma•efie field,and•hereis a furthereffec•caused
by
particle
density
gradien•
normal•o •heaverage
field.Foranisotropic
pi•changle
distribution,Jokipii andParker[1970]•ve •he additionalflux as
• • *•møw•c
(Bx•U)
3ZeB •
(49)
Theseauthors
pointedoutthat thisadditional
fluxmaybe included
by adding
antisymmetric
part[o thediffusion
tensor.In a framewiththe z axisalongthe
local averagemagneticfield, one has
46
J.R.
JOKIPII
1
,,,,=
Bo
wr•Imol •'
(50)
Here the termsBo/IBolchangesignif z • --z. This formwasfirstwritten downby
Parker [1965] and is implicit in the work of Axiord [1965a].One can write the
diffusionflux of particlesin the form
F, = --K,i(OU/Oxi)
(51)
Conservationof particlesis then expressedby
ot- •, ""3-i7•
(52)
I• is interesting•o no•e tha6 the antisymmetricpart of • doesno• contribute
to equation 52 if Bo is independen•of position. Equations 50, 51, and 52 eons6i•u•ea completespecificationof particle motion in 6he diffusionapproximation
and in •he res• frame of •he plasma.In general,•he antisymmetricpar• of r•
has beenneglectedin •he literature; bu•, as shownby Jokipii [1969b], the effects
of 6heseterms can be important, pa•icularly in connectionwi•h solar-flare
particles.
D.
ApproximateExpressions
for the Fokker-Planck Coefficients
Although the Fokker-Planck coefficientscan be obtained in principle from
equations32 and 37, in practice the full expressions
are rather complex,and
further approximations
are necessary.
We considertwo approximations.
1. First, supposethat the fluctuatingfield • dependsonly on z and is independen•of x and y. Then i• is clear•ha• •he correlation•ensordefinedin equation
29 mus• nobdependon • and •. Hence, a = d = 0, and b and c are independen•
of •. Then equations32 and 37 •ake on •he relatively simpleform
(a•")
(ay") _
I
[2f'w"b(•)
q-(1-- g•)
w•'
R•(•')e- ,•o•/•]
d•'
(54)
2•o• 1•1w •
Thesemay be expressed
directly in terms of the one-dimensional
spatial power
spectrumof the fluctuating part of B(z)
(55)
Theseresultsare implicit in the work of Jokipii [1966] and were clearlystated
by Hasselmannand Wibberentz[1968].
COSMIC-RAY PROPAGATION IN THE SOLAR WIND
47
I• is of interest here [o discussthe physical interpretation of equations55
and 56. Equation 54 statesthat particlesare scatteredby thosefluctuationsthat
are resonantwith the particle'sgyromotionabout the averagemagneticfield Bo.
The resonantwave number is readily seento be k - 1/•r, - •o/•w. (In this
regard seealso work by Parker [1964a].) In equation56 one seesthat in addition to a resonantscatteringterm there is a nonresonantterm P•(k - 0), which
contributes[o ((Ax)•)/At. This may be shown[Jokipii, 1966;Jokipii and Parker,
1968b,1969a] •o representthe tendencyof particles[o follow the meanderingor
randomwalk of the magneticlines of force as they travel down the z axis. This
effect is physically quite novel and is discussedmore fully in Appendix C. It
turns out that for low-ener• particles, the nonresonant term in equation 56
dominatesthe transportof particles'normal to the averagemagneticfield in the
solar wind.
2. Second,supposethat the fluctuatingfield is statisticallyisotropic.Then
c - 0, and a and b dependonly on r - (p• + •)•/2 [see,e.g.,Batchelor,1960].
In this caseit is convenientto considerthe two limits, r, - W/•o >> L, and r, <<
L, which will be called the large and small cyclotron radius limits, respectively.
The large cyclotronradius limit is readily obtained directly from equations32
and 37, following the procedure of Jokipii [1967]. In this limit •1•(•) •
[(1 - •)/•]
• over the relevant range of integration,sincea(r) and b(r) are
zero for r • L << r,. Thus, one obtainsfor r, >> L
((Au)•)
At •
(1 - u•) Z•e•
At -
w •"mo•
[(1- u•)P,,(0)
+ u•P•(0)]
w
At - 2Bo
•[1+ 5u
•- 4u]P•(0)
(57)
(58)
making use of the fact tha• the power spectrumis flat at small wave numbers.
The limit r, << L is more diffieul• to obtain. Consider firs• •he scattering in
pitch angleas in equation32. One expectsthe second•erm [o be small compared
wi•h the first, so
{Au
•)• I --
dr
(59)
•)l/•]e_i•or/•w
b[(r• + p•
which can be expandedin terms of
P•(k) = 2
b(D coskr ar
to yield
((Au)
•) 1- u
dkP•(k) arcos
At - JulW 7m2mo2C
•
UW
whereJ,(x) is •he usualBesselfunctionof order l. Now, •he firs• •erm in •he
48
J.R.
JOKIPII
bracketsyields preciselyequation55, and the remainingterms are of higher
orderin ro.However,oneis not yet justifiedin neglectingthesehigher-orderterms.
Careful examinationsuggeststhat if P•(/c) falls off less rapidly than /•-•' at
I• • o•o/W,the remainingterms are small and alternate in sign. It is therefore
concludedthat for such spectra the first term in brackets is a reasonablefirst
approximationfor ro << L. Similar argumentslead to equation56 for the transverse random walk. The errors involved in truncating the seriesare difficult to
estimate,but they are almost certainly less than presentobservationalerrors,
particularly since it is not likely that the fluctuationsare preciselyisotropic.
The effectof the truncation is to neglectvariations in the directionnormal to the
average field becauseof the smallnessof the cyclotron radius. Hence, in this
small cyclotronradiusapproximationwe regainthe cylindricalresults,equations
55 and 56.
E.
Relation to Magnetic-Field Observationsin Space
It is the goal of this calculationto use observedmagnetic-fieldpower spectra
to estimatethe cosmic-raydiffusiontensor.Above it was shownthat, under certain plausible assumptions,the Fokker-Planck coefficientscould be written
directly in terms of one-dimensionalspectraoœthe fluctuatingpart oœthe magnetic field. But in section3 it was shownthat the observedfrequencyspectrum
at a stationary spacecraœtis essentially a one-dimensionalspectrum along the
solar-winddirection.Hence,onemight hopeto expressthe particlemotiondirectly
in terms of the observedspectrum.For example,this would be possiblei• the
fluctuationswerestatisticallyisotropic,in which casethe one-dimensional
spectra
are the samein all directions.To the extent that the approximationsare valid,
one could substitutethe observedspectra directly into equations55 and 56. I•
the fluctuationswere a functiononly of z, the distancealongthe field, then one
needsthe spectrumalongthe z direction,and a small correctionwould be required.
In actual •act, the fluctuationsare not likely to have either symmetry, and
in view of the uncertaintiesconcerningthe form o• the fluctuations,it will be
assumedhere that expressions(55) and (56) are correctfor particleswith r• •< L,
and (57) and (58) for r• >> L, with the relevant powerspectrabeingthe observed
spectra. This gives as good an estimate of the Fokker-Planck coe•cients as is
possibleconsideringthe obse•ational uncertainties. It should be kept in mind
that an improvedunderstandingo• the correlationtensormay eventuallyrequire
a more careful distinction betweenthe various approximationsto the complete
equations.Thus the observedspectrumV•P•(/=
V•k/2•) will be substituted•or
P•(k) in the expressionsfor the Fokker-Planck coe•cien•.
Before computingthe diffusiontensor •rom particular spectra,considersome
generalrelations.First, if P•(k) = Ak-• whereA is a constant,then from equation
55
((A•)'>
i •--w•••
At
--
Z•e
• A•w
moc
2
2
2
(61)
•o•e •hat if • > 1, as is •he casefor mos• observedspectra,the ra•e of sca•ering
in pi•ch angle goes•o zero for 90ø pi•ch angle (• • 0). •owever, as discussed
COSMIC-RAY PROPAGATION
IN THE SOLAR WIND
49
previously,the formalismmust be appliedwith cautionnear 90ø, and there are
probablyhigher-ordertermsthat give somescatteringat 90ø. Nonetheless,
scatteringin thesecasesis probablysmallbut not zeronear 90ø. By usingequation
46 one finds in this case
K,=
cla(a-2)IBo"e"-2R29A
(62)
whereR is the particlemagneticrigidity,R = 7mmowc/Z,
and/• = w/c. It should
be mentionedthat for ((A•2))/At of the form givenin equation61, this form
for K, followsfromequation48 onlyif a - 1. Equation62 shouldbe a reasonable
approximationif I < a < 2. At high rigidities,both equations55 and 57
indicate
that
• ___
R2•/P=(k = 0)
(63)
Consider
nextthe generalformof theperpendicular
diffusion
at lowrigidities.
I-Iereequation56 is taken as givinga reasonable
approximation.
It is composed
of twoparts.At lowenergies,
theresonant
termis verysmallcompared
withthe
nonresonant
one,because
the powerspectrum
falls off sharplytowardhighfrequencies.Thus, we expectthat for ra • L
i w•.P=(k= O)
Kz•---•Bo
(64)
The corresponding
form at large cyclotronradii is
i • •.P=(k= 0)
•x -----•Bo
(65)
Finally, considerthe specificvalue of the diffusiontensorobtainedfor magnetic-fieldpower spectraobservedin late 1964-early 1965 by magnetometerson
boardthe Mariner 4 spacecraft.
Thesepowerspectrawereanalyzedin somedetail
by Jokipiiand Coleman[1968] and subsequently
by Jokipii and Parker [1969a].
Shownin Figure 3 is the spectrumobtainedin late 1964, at a heliocentricradius
of approximatelyI AU. Jokipii and Colemanalsostudieda corresponding
spectrum obtained near 1.5 AU for data obtained in early 1965. The values of KII
were computedfrom equations55 and 46 and are shown in Figure 4 as a.
functionof particle rigidity. These estimatesshouldbe taken as correctto per-
haps-+50%. It is immediatelyapparentthat within the expectederrorsthereis
no variation of •11between1 AU and 1.5AU.
For particlesbelowroughlyI Gv rigidity,corresponding
to ra • L, rl•hasthe
analytic form
•11----5 X 10•R•/• cm•/sec
with a. smooth transition
(66)
to
Ki•- 1.5 X 10•R2• cm•/sec
at higherenergies,whereR is expressed
in units of Gv. Note that the R •/• de-
Fig. 4. The parallel diffusion coefficient
Klldividedby particlevelocity/9 plotted
versus rigidity R, computed from the
powerspectraof Figure 1. K•//9is only a
function of particle rigidity. The change
in slope at R ~ 2 Gv is due to a transition from particle cyclotron radius ro
less than the correlation length L to
ro • L [Jol•ipii and Coleman, 1968].
pendencefollowsdirectly from equation62 with a power-lawspectrumof the
form Alc -a/•'.
The valueof •ñ for rigiditieslessthan about.i Gv is determined
principally
by the randomwalk of the magneticlinesof force,as measuredby the powerat
zerowavenumber.Jolcipiiand Par•er [1969a] estimatedP•(/c - 0) by noting
that P• (/c)musthavezeroslopeat lowfrequencies,
andsoextrapolated
the spectrum shownin Figure3 to /c - 0. They deducedthat belowabouti Gv rigidity
Kl -----2 X 102• cmZ/sec
(67)
Kñ hasroughlythe sameform,within a factorof 2, at higherenergies.
It is interestingthat the ratio Kii/•ñ --• 2.5R•/• is not largeand decreases
with
decreasing
particlerigidity, becomingapproximatelyequalto unity at R --• 0.15.
Diffusionof thesecosmicrays at the orbit of earth is effectivelyisotropic,and
even at rigidities0 (1 Gv) the ratio •,/•ñ is not as large as has sometimesbeen
assumedlAxford, 1965b;Parker, 1965].
It must be rememberedthat these expressions
are valid only if the orbit
changeis smallin a correlationlengthof the fluctuations.In practice,this requires
that the scatteringmean-free-pathX satisfy
k = 3•11/w• L
(68)
By substitutingequation 66 into 68, one finds that the present expressions
are not reliable belowroughly R = 0.1 Gv.
These determinationsof the diffusiontensor,valid for the period aroundlast
solar minimum,constitutethe best values available at the presenttime. In the
following
discussion
of selected
applications
of the transport
theory,thesevalues
will be usedextensively.However, it shouldbe remembered[Siscoeet al., 1968;
Sari and Ness, 1969] that the observedspectrumvaries with time and position.
Hence the diffusiontensor varies. For example, the spectra of Sari and Ness
indicatea k-2 dependence
on wave numberand a muchsmalleramplitudein 1966,
COSMIC-RAY PROPAGATION IN THE SOLAR WIND
51
suggesting
tha• •11
•/• and tha• diffusionis much faster at certaintimes. I• would
bevery desirable
to monitorthe powerspectracontinuously,
to followthevariation in the diffusion,
but thisis not yet beingdone,andonlya smallnumberof
spectra are available.
Onefurtherpointis relevant.A numberof authors[Burlaga,1969;$iscoeet
al., 1968;Sari and Ness,1969] havesuggested
that a substantial
part of the
observedpoweris due to tangentialdiscontinuities
beingconvected
past the
spacecraft.These discontinuities
would probablybe relatively inefficien•in
scatteringlow-energyparticles,and hencetheir contributionto the observed
spectrumshouldnot be includedin Kij.This is because
the field linesare parallel
to the tangentialdiscontinuityand a gyratingparticlewouldrarely encounter
the discontinuity.
In effect,this wouldmeanthat the assumptions
usedabove
to obtainKijfrom the observations
are invalid. However,this questionhas not
yet beenresolved,and it will not be pursuedfurther here.
Finally, the possibility that the small electric fields associatedwith the
plasmaacceleratethe cosmicrays shouldbe pointedout. This effectis the usual
second-order
Fermi acceleration[Fermi, 1949]. Itowever,Parker [1965] has
concluded
that suchFermi acceleration
effectscanbe neglected
exceptperhaps
at low energies.
Seealsothe discussion
by Hasselmann
and Wibberentz[1968].
Fermi acceleration will not be considered further.
5. DIFFUSIVE
MOTION
OF FAST
IN
SOLAR
TYIE
CYIARGED
PARTICLES
WIND
In the previoussectionthe equationsof motionof a chargedparticlein a
statisticallyspecified
magneticfieldwereconsidered,
and the diffusionapproximation was derivedin the coordinateframe at rest with respectto the local
plasma.The diffusionlimit in this framegivesriseto a net flux of particlesthat
is relatedto the densityU (X•', t, T') by
F•(X•', t, T') = -•,•(OU/OX•')
(69)
with • •s givenin equation50 above.The p•rticle kineticenergyin the plasma
frame is denotedby T'. Associated
with this flux is anisotropyamplitude
$,(X•', t, T') = 3F,/Uw
(70)
in the frame of referenceof the plasma (movingwith the solarwind). Note that
[a•! • I forthediffusion
approximation
to bevalid.
The purposeof this sectionis to write downthe differentialequationsfor the
density and flux of cosmicrays in a form suitable for applicationin the solar
wind.Equations69 and 70 are correc5
to first orderin [a[in the frameof the
solar plasma; but observationsare carried out, and boundary conditionsare
applied, in a coordinateframe effectivelyat rest with respectto the sun (the
orbital motionof the earth or spacecraf5
is negligible).The particle energyis
not constant in this rest frame. The transformation
between the two frames
is not trivial, and somecare is required in carrying it out.
The mostgeneralderivationpresentlyavailableis due to Jokipii a•d Parker
[1970]. Firs• they write downthe differentialequationexpressing
conservation
52
J.R. JOKIPII
of particles,still in •he frame a[ res[ with respeelto •he plasma.The particle
densi[y U at• a given energy T may change because[he divergenceof F• is
nonzero or becausea nonzero divergenceof the plasma veloei[y ¾•ocausescompressionor expansionof the cosmicrays. Associatedwi[h [he latter efteel are
both a changein numberdensi[yand an adiabatic energychange[Sinqeret al.,
1962;Parker, 19•5, 1966]. More precisely,•o lowes[orderin 8•,[he motionof •he
plasmaa[ velocity V• givesriseto a flux V•U, whosedivergencemay be nonzero.
A[ [he sametime [his expansion(or compression)of [he plasma leads• a cooling (or heating) of •he cosmicrays a[ a ra[e
dT'/dt = -- [a(T')/3] (V.V•)T'
(71)
where a(T •) = (T • • 2To)/(T' + To), and To = mo• is •he rest-energyof •he
cosmic-ray particle. PossibleFermi acceleration,which may also contribute to
•he energy change•erm, is probably no5 an importan5 effec5as discussedabove
and will no• be considered further.
The differential equation for the rate of changeof •he particle density U
is •hen given by summing the rates of changedue to these various effects.One
obtains
Ot-
OX,' OT'
U + OX,'•
(72)
still in the frame movingwith the plasma.Equations69, 70, and 72 constitutea
complete
description,
to firs•orderin IS[,of •hemo•ionof fas•charged
pa•icles.
I• is usually desiredto obtain 5hecorresponding
equationsin a frame a5 res•
Wi•h respec••o •he sun, in which case we wish to •ransform •o •he coordinates
(X• = X•' + V•t), keepingtermsto firstorderin V•/c. In thistransformation,
of
course,T' also changes,but neglectthis for a momentand still regardT' as the
energy definedin the rest frame of the plasma. Then equation 72 transforms
into the Fokker-Planckequationfirst obtainedby Parker [1965],
Ot • OX,(UV•,)-- •7•
U +•
,,
(73)
andthe particleflux a• a givenT' is now V•U - •,•OU/OX•.
Now considerthe transformationof the energyT' [o •he energyT measured
in' the res• frame. This transformationis complicatedby •he fae• that T for a
givenparticleoscillatesby •V•wx relativisticmassas •he particlegyratesabou•
the movingmagneticfield or is scatteredback and forth along•he field, as
pointedout by Jokipii and Parker [1967]. If • is the angleof •he instantaneous
particlevelocityrelative•o •he wind veloeky,in •he movingframe,[hen[o firs•
order in V•/c
T = T,+
[T,(T,
+
cos
0'
The goal is to obtain the density U and •he flux F• in the res[ frame. Now consider the 6ransformationof 6he particle density U(X•, t, T'). I• is s•eien• [o
consider•he isotropic par• of •he distribution because•he eorree[ionsare of
COSMIC-RAY PROPAGATION
IN THE SOLAR WIND
53
O(Vw/w) or smallerso that small anisotropies
O(V/w) will have corrections
O(¾wu/w").It may be established
that U is unchangedfor a smoothenergy
spectrumas follows.An isotropicdistributionof particlesall havingthe energy
T' in the movingframehasa distribution(1/2)sin t/dO' overthe anglet/. Hence
thedistribution
q•(T, T•) of theseparticlesoverT is givenby
•(T, T') dT = (1/2)sin t•' dO'
(75)
upon normalizingto one particle, or
1
•(T,T')= 2(V•/c)[T'(T'
•- 2To)]
•/'ø
(76)
uponusingequation74. But O' cannotbe greaterthan • or lessthan zero, so
from (74) it followsthat ½is nonzerofor
V• (T'q-2To)]
•/z (77)
T'_ __
V•[T'(T'
q-2To)]
•/2<:T<:T'q--•-IT'
and zero otherwise.
Suppose
nowthat the localenergydistribution
in themovingframeis •(T').
The energydistribution½(T) in the fixedframeis•hen
•(•) =
a•' •(•, •')•(•') = • •(•/•)••
• •o)•.]
(78)
whereA is the lowestenergyT• that can transformto T, and B is the highest.
From equation74 onefindsthat
A = T- V,.[T(T+ 2To)]
•/•'
B = T q-V._•_•
[T(Tq-2To)]
•/•'
Now specializeto •he casewhere(I,(T') is a sufficiently
smoothfunctionof T• that
it may be expanded
in a Taylor seriesaboutthe energyT. More precisely,define
the parametere - (V•,/c)OIn U/O In T andassumee << 1. Then it followsdirectly
from equation78 that [Jokipiiand Parker, 1967]
•(T) = •(T')[1 q-O(e•)]
(79)
so that to order V•o/w for whichthe diffusionapproximationis valid, a smooth
spectrumat an interiorpoin5is unchanged
by the transformation.
It tums ou5
tha5 near boundariesthere are indeed correctionsto the order of V•/w, but the
questionof boundaryconditions
will be considered
later. It is concluded,
therefore,that the densityU at an interiorpointis unchanged
upontransformation
to
the fixed frame if the energy spectrum is smooth. Thus, if U(X•, t, T) is the
cosmic-raydensityin the fixed frame,it alsosatisfiesthe Fokker-Planckequation of Parker [1965]
'
o(ou)
0•= -v.(v•)+•1[(v.v2•o(••)] +y•
•,,•
Ot
(so)
54
J.R. JOKIPII
This invarianceof U may also be establishedmore elegantlyby considering
invariancepropertiesof •he densityin phasespace,œollowing
•he discussion
of
Forman [ 1970a].
Next, consider•he •ransforma•ionof •he flux F• or aniso•ropy.$•.This is moss
simply obtainedfrom a considerationof •he Comp•on-Ge•ing effec• [Compton
and Getting, 1935; Gleesonand Axford, 1968a; Forman, 1970a]. One proceeds
by noting •ha• •he aniso•ropyin •he frame oœ•he wind, given by equations69
and 70, is very small comparedto uni•y. Transformingto •he fixed coordinate
frame introduces another small aniso•ropy due •o •he Comp•on-Ge•ing effec•
on 5he density U (X•, t, T). This additional aniso•ropyis given by
01nT
3
V•o
I2_a(T)
O
In(Uw)]
•ie-o•- w
(81)
whichagainis 0 (V•/W) (( 1.The corresponding
fluxis (zt,/3)$c-a•U.Application of the Comp•on-Ge•fingformula also requires a smoothenergy spectrum
for which .• <( 1. The ne• flux in •he fixed frame is simply •he sum oœ•hese•wo
•erms,sinceany cross•ermsare oœ•he orderof ,•' or higher.Thus,to first orderin
Vw/w onehasthe result,firs• presented
by Jokipii andParker [1970]
OU
V•o 0
•5,(X,,
t,T)= ¾•,U- •,,OX, • OT[,•(T)TU]
(Se)
with •n •ssociated•nisotropy •t• = 3F•/wU. Equations 80 •nd 82 provide •
completedescriptionof particle•ransportin the solarwind, correctto terms of
the order of V,./w, provided that sca•Leringis sufficientto drive •he •ngular
distributionnearly isotropicand provided that the energy spectrumis smooth
enough.
At this point it shouldbe noted tha• the basicequations80 •nd 82, for •he
specialcaseof isotropicdiffusion• = • •t• andsphericalsymmetryabou••he sun,
were derived from • very differentpoin• of view by Gleesonand Axford [1967].
ßhey beginwith a Boltzmannequationfor the particledistributionf(r, w, 0, t)
where cos0 = w.r/wr, with magneticscatteringrepresenLed
by a collisionoperator. They write Bol•zmann'sequationin the form
Of
ot
+ •ocos0•0]_w
r sin0
•Of
= (Of)
• •o•,•o•
(83)
and expand• in sphericalharmonics,keepingLetresof lowesborder in Vw/w.
This procedureresults,aϥera seriesof mathematicalmanipulations,in the
above-menLioned
specialcasesof (80) and (82). In this paperan explici•equaLion
/or •he flux oœcosmicrays in the solarwind was discussed
for •he firs• time. This
generalprocedurewas subsequently
generalized
by Gleeson[1969] Loincludean
averagemagneticfield, which causesthe diffusionLo be anisoLropic.
However,
•his approachappearsto require•hat •ñ/•, -----1/(1 + •o";t2w"), which is nob
true in general(seesection4). The readeris referredto the originalpapersfor
s,ninterestingalternatederivationoœthe basicequationsin a lessgeneralform.
I• remains to discussthe general problem oœboundary condiLions.Usually,
one'willbesolving
equation
80/or thecosmic-ray
densityU as•hequantityof
COSMIC-RAY PROPAGATION IN THE SOLAR WIND
55
primaryinterest,and it will be necessary
to specifya linearcombination
of U
andOU/Orat the boundaries
of thesystem.
Onenaturalboundaryfor cosmic
rays
in the solarwindis the interfacewith the interstellar
medium.Sincevery little
is knownaboutthis interface,varioussimplifyingassumptions
are used.It has
oftenbeenassumed
that thereis a sharpboundary
at a heliocentric
radiusr - D,
beyondwhichthereis little scattering,
andthe diffusion
theorydoesnot apply.
Beyondr -- D the densityU takeson its interstellarvalue U•o(T). Intuitively,
one might expectthat the appropriateboundaryconditionis that at r - D,
U(xi, t, T) - U•(T). Closeexamination
[JokipiiandParker,1967]showsthis
to be correctonly to zerothorderin Vw/w, andthat thereare corrections
to the
first orderin Vw/w. The analysisis lengthyand complexand will not be detailed
here.Essentiallywhat happens
is that particlesenterthe windmovingcounterto
the wind.Hence,by equation74 their energyin the frameof the wind is greater
thanwhenthey entered.This produces
a change
to the orderof V,•/w in U at the
boundary.Othercomplications
includethe familiar effectin diffusiontheorythat
sharpboundaries
arewherethe diffusionapproximation
breaksdown.Jokipiiand
Parker [1967] give the resultthat if the systemis sphericallysymmetricand
K•j- K•;•j,then the properboundaryconditionfor nonrelativisticparticlesis
U(D,t,T)
i 2 • q_• D
4dU•ø
U•(T)
[1--•4(-•)1- õ
dT(T--•) (84)
In view of otheruncertainties,
the approximation
of lettingU take on the value
U• (T) at theboundary
hasbeenquitegoodenough
upto thepresent.
Nonetheless,
the above correctionsmust be included in a calculation correct to the order of
Anotherapproachis to let r•j graduallybecomelarge with increasingr,
requiringU to approachU• (T) at largedistances.
In additionto the outerboundary,thereis the boundaryat the sun.When
discussing
galacticparticlesit is usuallysuiticient
to requirethat U remainfinite
as r approaches
zero;in effectonetreatsthe sunas a point.However,in the
discussion
of solarparticlesonemustbe careful.It hasbeengenerallyassumed
in the literaturethat thereis a unit impulsivesourceat the sunandthat the sun
is treatedas a point, so that againonly thosesolutionsthat are finite at the
originarerelevant.Theseassumptions
avoidtheproblem
of theproperboundary
conditionsat the sun,and we will seein section7 that in the caseof solar cosmic
rays this may be important.The problemof properboundaryconditions
at the
sunhas not yet beensolved.
Finally,consider
brieflythe casewherethe energyspectrum
variesstrongly
with energysothat .e• 1. This situationoccurs,
for example,if onewishesto
consider
the evolutionof the distribution
of particleswith a givenenergy,or
perhapsoneparticle,introducedat the sunor at the interfacewith the interstellar
medium.It was shownby Jokipii and Parker [1970] that in sucha caseone
encountersa complex,integro-differentialequation in the coordinateframe at
56
:I. R. JOKIPII
rest with respec5•o the sun.They pointedout that in this caseit is often easier
to work in the frame movingwith the wind.
6. MODULATION
OF
GALACTIC
COSMIC
RAYS
BY
TI-IE
SUN
One of the two major applicationsof the basictransporttheory developed
in this paper is to the modulation of the galactic cosmic-rayintensity by •he
solar wind. The study of this phenomenon
hastwo complementaryaspectsin that
it helpsus to understandbetter the plasma dynamicsof the solar wind, whereas
at the sametime it leadsto a better knowledgeof the unmodulated,interstellar
spectrum,which is important in understandingthe origin of theseparticles.Our
understanding
of the modulationproblemis still imperfect,and a detailedsolution
is still unavailable.But at presentit appearsthat the basictransport equations
80 and 82 representmost of the underlying physicsadequately,and a fuller
understandingonly awaits more sophisticatedsolutionswith better knowledge
of the boundaryconditionsand parameters.
A.
Variation in Density or Omn,idirectionalIntensity
The modulation phenomenoncomprisesa number of effects,including •he
eleven-yearvariation of the cosmic-rayintensityin antiphasewith solaractivity,
quiet-time anisotropiesof various sorts, and transient effects such as Forbush
decreasesbehind shock œronts.Perhaps the most studied of these effects is the
quiet-time eleven-yearvariation with its associatedanisotropies,and it is appropriate to begin with a study of this effect. The basic observationalfact is that
the intensi•y of cosmicrays between about 10 Mev and 10 Gev energy, in the
absenceof solar-flare effects,varies inversely with solar activity, with the effect
being strongerat lower energiesas illustrated in Figure 5. The qualitative interpre•ation of the modulation in terms of a balance between inward diffusion and
outward convectionin the solar wind has already been sketchedin the Introduction. Now considerhow this modulationmay be understoodmore quantitatively
in terms of the full transport equation 80.
The basic geometry is again as sketchedin Figure 1. The galactic cosmicray intensity is reduced in the inner solar system by the convection-diffusion
effect mentionedabove. Beyond some distanceD the convectionis no longer
effective, and the cosmic-ray intensity attains a maximum value that is
generally taken to be the full interstellar intensity. This distanceD is called the
boundary of the modulating region. It may be associatedwith the slowingdown
of the solar wind so that the convectionvelocity goesto zero or, perhapsmore
reasonably,the wavesthat scatterthe cosmicrays damp away [Jokip•i and Davis,
1969], and •, -• •. The physicsof this boundaryis still very poorlyunderstood,
and D will be treated as a parameterin the analysis.The parameters•,, V•, etc.,
within the boundaryD, and perhapsD itself, vary with the solar cycle,giving
rise to the varying modulation that is observed.Various lines of evidenceto be
mentionedbelow suggestthat perhapsD • 3-5 AU.
Now considerhow this modulation may be understoodin terms of the full
transport equation 80. To illustrate the basic effect, first idealize to the case
of a sphericallysymmetricsolarwind with constantwindvelocityV• and assume
COSMIC-RAY PROPAGATION
IN THE SOLAR WIND
57
I i I
June- Sept. 1965
IMP
111
o.•
Oct.-
Nov. 1964
IMP
0.07
TI
0.05
Dec.1963 - May 1964._
E=
0.05
IMP I
0.02
I I!II
O.Ol
5
I0
JI J I I I I ,•
20
30
50
70
I00
Kinetic Enerqy per Nucleon(Mev/nucleon}
Fig. 5. I)ifferen•ial energy spectra for primary helium during •hree time
intervals near minimum solar activity. These satellite measurements were
made in interplanetary space over continuous time intervals and do not
include periods when solar-associated particle events are present. It is
possible that some of the particles at ~ 10 Mev/nucleon and lower are
solar particles that are present more or less continuously [Gloeckler a•d
Jokipii, 1966].
•ha• U(X•, t, T) is also independen•of angle abou• the sun. Then equation 80
reduces •o
Ot- r•1O(r2Krr-•rr
q-2V•
3r O(aTU)_i O
OU
OU)
where •rr is a linear combinationof •ll and •z, and the antisymmetric part of
58
J.R.
JOKIPII
• has beenassumednegligible.I[ is usualin •he literature to considerequation
85 with • = •&•. The presentform representsa slight generalizationto include
anisotropicdiffusion.The relevantgeometryis illustratedin Figure 1; • =
[an4(r•s/V•) is taken to be the angle betweenthe radius vector from the sun
and the outwarddirectionalongthe magneticfield.The form of • in a coordinate
system oriented along the average magnetic field is given in equation 51. Applicationof the transformationmatrix that rotatesto a frame in whicha principle
axisis along•he radiusvectoryields
•
= • cos• • + • sin• •
(86)
Of course,if U dependedon other spatial variables,other components
of • would
appear. Now note that, followingthe discussionin section4, •y has the general
magnitude10•ø - 1• • em¾see,V• • 4 x 10* era/see,and the gradientscales
are of •he order of 0.1-1 AU. Thus, the terms on the right of equation85 are of
the order of 10-%whereasthe time scale associatedwith the eleven-year variation is some10s seconds.Hence the time-derivative may be neglectedin equation
86, and one computesthe eleven-yearvariation in a quasi-static approximation.
It is still di•eult to solve equation 85 analytically, and only a few approximate solutionsare available. For example, if • = Kr•T • with b • 1, and the
variation o• •(T) with T is neglected,the general solution may be shown to be
of the form [seeJokipii, 1967; Fisk, 1969; Fisk and Ax/ord, 1969]
U(r,
T)=fffd•e-(ir+•)
TM
ßg(•)•F•
• 1+
'
'1 -• +h(•)
2b;(1-2b)•rV..
(1-
b)S
b+l'bl 2b
rV•.3}
ß•F•••(1+ i•)+ b-I '(1-2b)%
•
(87)
wherethe f•ctions g(•) and h (•) are to be chosento fi• the boundaryconditions,
and • is definedby
2/(1 - b + õaa)
Fisk and Ax•ord [1969] have been able to invert equation87 to find solutions
in closedform if U (T) approaches
AT-* at largevaluesof r. Clearly, this solution
is sufficientlycomplexthat only limiting forms are of much use. For example,
if b < 1, h (•) = 0 if the solutionis [o be finite at the origin.Then, the asymptotic
form for •Fx (a, b; x) in the limit that x becomeslarge yields, if g(•) goesto zero
for large a,
1 ou
---•u Or
I
(1-
v
b)v •
• [1 -]- o(•-•)]
for
2
rV•) 1
(88)
COSMIC.RAY PROPAGATION IN THE SOLAR WIND
59
(F•s• and Axford [1969] have presenteda thoroughdiscussion
of theseresults
usingtheir exact solutions.)Usingthis large • limit, Par•er [1965] studiedin
some detail the behavior of the probability distributionof a single particle
injected at r - D for b = 0. He showedthat for D V•/•, - 5, nonrelativistic
particlestha• reach r = 0 have lost nearly 90% of their ener•. This effec• was
also studied by Goldste• et al. [1970]. Thus the particles observedat earth
having energiesof 10 to 50 Mev correspondto particleswith roughly 100-Mev
energyin interstellar space.
Another approximationthat is of considerableinterestis the limi• rV•/•
1, in whichcasethe solutionmay be expanded
in ascending
powersof r'V•/•,. It is
not necessaryhereto assumeV• independen•or r. The quasi-staticform of equation 80 is written
r1
•Or
•(r•V•)] •(aTU)
r•Or(r•V•U)
=0
0(r•• OU
) +•-1••0
0
(89)
Suppose•ha5 a• somelarge value of r = D, beyondthe region of effectivemodulation, U takes on the value U• (T). Following an approachsimilar to •ha5 used
by Gleesonand Axford [1968b], le• U(r, T) have the form
a(r, T) = U•(T)[1 + a•(r, T) + a•(r, T) + ...]
where Ux(r, T) is of firs• order in rV•/r•, U• of secondorder, etc. Clearly, U•,
U•, e•c., mus• go to zero a• r = D. By substitutingthis form into equation88 one
obtains,to firs5orderin rV•/r•,
r• Or(r •/
+ 1•
OU,•
[O(r•V•)•
O(aTU•)- 1O(r•V•U•)= 0
(90)
from which i• is trivial to obtain the equation
OU•V••i 3U•
10OT(aTU•)• = •-
• Or
Here C is an integrabionconstanbbhabis se5equal to zero • correspondto an
absenceof sourcesor sinks a• the sun. A secondintegration,with the condition
•ha5 Ux(D, T) = 0, yields the derivedsolutionto firs• order in
U(r,
o
-
at'
.
This can be cas5into a more useful form by definingthe parameter7(T) =
-0 In (j)/O In (T), which is the effective power-law index of the differential
inbensityspecbrumj - Uw/•. Equabion91a bhenbecomes
U(r
T)•U•(T)•i-(2•aY)
fOV•drt• (9lb)
'
--
3
Krr
The radial gradient in this limi5 is, kom equations91a' and 91b
I OU
_U Or
2•ayV•
3
60
J.R.
JOKIPII
I• may be easily shownfrom equation94 •ha• •he flux ff• is zero go this order
in rV•/•.
(The •u•hor is indebted•o W. I. Axford•or suggestions
•h• led •o clarification
of the •bove deriwfion.)
Equations91 •nd 92 •re • reasonable•pproxim•fion for DV,o/• • 1. By
•king D •___
3 AU •s • reasonable
wlue as suggested
• •he beginningof this section, one finds •h• •his is true if
•
• 1.5 X 10• cm•'/sec
for V• -----4 x 10• cm/sec.From •he discussion
in section4 •his appears•o be
•rue for particlesof grea•er•han roughly I Gv rigidity.
Equations 91 and 92 illustrate, for a fairly realistic situation, •he expected
behavior of •he galactic cosmic-raydensity under •he influenceof •he solar
wind. The cosmic-ray in•ensi•y is depressedbelow i•s in•ers•ellar value by •he
sweepingou• of cosmicrays by •he magneticirregularitiesand by •he adiabatic
coolingof particles in •he divergingwind. We shall seebelow •ha• also associated
wi•h •his modulation are small aniso•ropiesin •he angular distributions.
I• is interesting •o con•ras• equations 91 and 92 wi•h •he corresponding
results in •he absenceof energy change,outlined in •he Introduction. One finds
tha• the two resultshave the samebasicform in the limi• of smallrV,o,•, but tha•
in general if • • 2.5 as is observed,the inclusion of adiabatic decelerafion
increases •he modulation.
A differen•and useful approximationin •he limi• of small rV,o,• was presentedby Gleesonand Axeoral [1968c] and discussedfurther by Fisk et al.
[1969].They alsoconsider
•he caseof sphericalsymmetryand assume
isotropic
diffusion• = •.
In •his case•hey no•e•ha• equations80 and 82 can be written
in the form
.
O(r•5)
ri• Or
V•
O•OT(aTU)
3 Or
(93)
0 (aTU)
5 = V•U-• OU
Or V•.
30T
(94)
where ff is now in •he radial direction. They no•e •ha• • is negli•ble a• small v•lues
of DV/•; •hus, in •his limit onemay set •he left sideof equation94 equal •o zero.
(The interestedreader m•y verify •his by evaluation• from equations91a and
92 andnoting•h•t it is of secondorderin rV•/•.) Using•he •o•l particleener• W
•s •he energyvariable, equation 94 becomes
O
U
To•)
•/•0 •w(w-
=o
where To is the pa•icle res• ener•.
Writing • in the generalfo•
whereB = w/c andR = 7Mwf[Ze[is •he pa•icle rigidiW,equation95 canbe
COSMIC.RAY PROPAGATION IN THE SOLAR WIND
61
integratedto give for the differential intensity
4•r j(r, W)
c (ZeR)•
U(r, W)
W(W•-
Toe)
•/•
=H
(W':-- To2)
'/e--
•-•(-•,
t)ds,
(96)
Here H(x, t) is an arbitrary function to be determinedfrom boundary conditions.
One introducesthe quantities
; dx
r3•(x,
V•(x
t)
t)
-- To2)
1/2
dW'
r(w,
z,t)= o(w,:
ek(r,t) =
and •he inversefunction•(•, z, t) such•ha•, given (•, z, t) •hen W - •/(•, z, t).
Defining a quan•i•y
q•(r, W, z) = •(•' + ek,z) -- W
and,requiring•ha½j(D, W) - j• (W), i½is a simplema6½er
•o show•ha½equation
99 Cakeson ½hesugges½ive
form
j(r,W) _ j(•, W+ q,)
(97)
W•- Toe - (W + q•)•-- Toe
This is directly analogous[o Liouville'stheoremif q• is taken [o be a kind of
'potential energy.' In the particular case • -- gx(r)R•c, Gleesonand Axford
[1968c]pointou[ tha[ß - IZel (k(r, t) and• becomes
somewha[
analogous
to
an electrostaticpotential.
Two points shouldbe made concerningthis solution.First, note that it may
be trivially generalizedto include anisotropicdiffusion by replacing • by • as
definedin equation86. Second,althoughit is not immediatelyapparentthat the
two solutions (91) and (97) are very similar, a shor[ calculation demonstrates
their equivalencefor weak modulation.
For the casethat rVw/• << 1, the parameter q• can be shownto take on the
form
•
W '
-• dr'
Hence ß << W and equation 97 can be written
(Te+ 2TTo)j•(W)
1+••
j(r,W)• T•+2TTo
+2q•(T
+To)
•j•(W) 1 -- •7 -- •-
+ •o'
(98)
62
J.R.
JOKIPII
Finally, usingthe definition of •, one obtains
j(r,
W)=
j•(W)I1
_(2-[-a'r)
f•V•dr,l
3
(99)
This is preciselyequation91b, if i• is remembered
tha• • may be replacedby
in •he presen•formulationof •he problem.I• thusappearstha• the •wo solutions
are equivalent.
I• appears•ha• •he Liouville4ypesolutionis valid overa slightly
largerrangeof parameters
because
i• is basedon a strongerassumption,
andtha•
i• •hereforehas grea• utility in discussing
observations.
Considernow briefly a comparison
wi•h selectedobservations.
Fisk et
[1969] presentedcurvescomparingthe resultsobtainedusing•he presen•solution, numericalcalculations,
and observations.
Theseare illustratedin Figure6.
Similarly, in Figure 7 is presentedthe observedradial gradien•betweenearth
I0O-
FORCE FI ELD
UNMODULATED
SPECTRUM
NUMERICAL
SOLUTION
SIMPLE
CONVECTION-
DIFFUSION
SOLUTION
>.. io-2_
Z
Z
O'
lb2 KINETIC
ENERGY
I$
(MeV/Nucleon)
Fig. 6. A comparisonbetween the numerical solution, the approximate solution (equation 97)
and observed data presented by Fisk et al. [1969]. The data are for 1965 [Gloeckler and
Jokipii, 1967], and the parameters have been chosenso that the theoretical results agree with
the data.
COSMIC-RAY PROPAGATION
IN THE SOLAR WIND
I
Fig. 7. Computed and observed radial
gradient between earth and Mars in 19641965. The data are those of O'Gallagher
[1967], and the theoretical value was
computedfrom equation 92 with the value
of K•j deduced in section 4. The dashed
line at lower energiesindicates the theoretical prediction, although the assumptions in the approximation 92 probably
break
down below
about
R
---- I
I
I
I I I I
I
63
I
I
[
I
[
t't-
_
_
-
COMPUTED_•
Gv
[Jokipii and Coleman, 1968].
I
.I
I
I i lllll
I
RIGIDITY R (GV)
and Mars as measuredon Mariner 4 and reported by O'Gallagher [1967] and
O'Gallagherand Simpson[1967]. The agreementwith the predictedgradient,
using equation 92 and the diffusioncoefficientof Jokipii and Coleman [1968], is
clearly excellent and suggeststhat theory and observation are in good accord.
Recent observational evidenceobtained from meteorites supports O'Gallagher's
result at energies_• 400 Mev [Fireman and Spannagel,1970; Forman, 1971].
However,it shouldbe mentionedthat Krimigis [1968, 1969] and Anderson[1968]
reportedvalues of the gradient from Mariner 4 data that disagreewith that of
O'Gallagher,and which are also inconsistentwith each other. If it is established
that the gradientis indeedas low as reportedby Krimigis or Anderson,for particle energiesof a few hundredMev or higher, then some of the assumptions
relating •u to the powerspectrummust be incorrect.
The above approximatesolutions,someexact casesconsideredby Fisk and
Axford [1969], and a few unpublishednumericalcalculationsof Fisk [1969],
constitutethe presentlyavailable solutionsfor the numberdensity U (or omnidirectional intensity j - wU/4-k).
B.
Anisotropies
Now consider another effect associatedwith the eleven-year variation. In
addition to decreasingthe omnidirectional intensity of cosmic rays, the solar
wind producessmall anisotropiesin the angular distributionof cosmicrays. For
example,there is observedan azimuthal anisotropyof cosmicrays, in the direction of solar rotation, at right anglesto the solar radius vector. This anisotropy
causes the famous diurnal variation
in neutron monitors at the surface of the
earth, as the rotation of the earth sweepsthe acceptanceconesof the detectors
acrossthe sky. The magnitude of the anisotropy is approximately 0.4% at
energies~5 Gev for protonsand less than 0.1% at energies~10 Mev [Rao
et al., 1967]. A secondanisotropyis a small, outward-directedflux of particles
a• ~10-Mev energy[Rao et al., 1967].
64
J.R.
JOKIPII
First considerthe interpretation of the azimuthal anisotropy.Basically, the
presenceof the azimuthal anisotropycan be understood[Ahluwalia and Dessler,
1962; Parker, 1964b, 19'65;Axford, 1965a] as a straightforwardconsequence
of
the spiral interplanetary magnetic field. Particles gyrate about the magneticfield
that corotareswith the sun. Unless the particles are to be forced out of the
system by this rotation, they must move azimuthally at essentially the rotational velocity rgs. Two effects complicate the issue. First, the particles are
scattered by magnetic irregularities, and second,possible gradients out of the
ecliptic may give rise to azimuthal fluxesthat contribute to the observedanisotropy. Thesetwo effectsare consideredseparately.
A fundamental point, made first by Stern [1964], is, that in the absenceof
scattering (in which casethe electric and magnetic fields are static), Liouville's
theorem rules out anisotropies. The combined magnetic field and electric field
(E = --V• x B) are such that a density gradient is set up normal to the sun's
equatorial plane. The magnitude of this gradient in the absenceof scattering is
precisely of the right magnitude that the associatedflux exactly cancelsthe flux
due to corotarion. For details of this calculation see Parker [1964b]. Now, the
observedflux at relativistic energiesis about ] of the 0.6% expectedfor full corotation. That is, corotarion implies a velocity, at radius r, of rgs, which yields an
anisotropyby the Compton-Getting effect
-
w
(2+
(100)
____-0.6%
upon setting .a -- i for relativistic particles and using 7 - 2.5. This is larger
than observedby a factor of 1.5.
Parker [1964b] arguedthat perhapsscatteringacrossheliocentriclatitude
was sufficientto relax the gradient implied by Liouville's theorem by about «
so that the resultinganisotropywas roughly « oœthe full corotaiionalanisotropy,
or 0.4%. One would then presumablyinterpret •he lack oœan azimuthal anisot-
ropy for 10-Mev particles (where •ot ~ 5%) as an indication oœvery little
scattering.However, as first pointed out by M. A. Forman (personalcommunication, 1968) the implied latitude gradient oœ10-Mev protons is enormous,and
one beginsto doubt this model.
An alternate interpretationwas suggested
by Jokipii and Parker [1969a].
They pointedou• tha• (a) the magnitudeoœthe perpendiculardiffusioncoefficient
•ñ as determinedby the magnetic-fieldpower spectrumis large enoughthat
gradientsnormal •o •he ecliptic are probably small, and •hat (b) using the
deriveddiffusiontensoronemay accountfor the observedazimuthalanisotropy
in terms of radial gradientsalone. To seethis, considerequation 82 for the flux
in the azimuthal direction. Let x be a Cartesian coordinate normal to the radius
vectorr, in the directionoœsolarrotation.Then, if U dependsonly on r, onehas
ff• = -•(OU/Or)
But, usingthe appropriaterotationmatrix to transformfrom the principalaxes
COSMIC.RAY PROPAGATION IN THE SOLAR WIND
65
alongthe magneticfield,anddefining• asthe anglebetweenthe field andradius
vector, one obtains
5• = cos• sin •(•11- Ki)(OU/Or)
(101)
Oneseesimmediatelythat if diffusionis isotropic(KiI= Kñ), the flux and hence
the anisotropygo to zero.As pointedout by Jokipii and Parker [1969a], the
form of the diffusiontensorimplied by magnetic-fieldpower spectrain 1965
suggests
that indeed•11• •ñ for protonswith a kinetic energyof some10 Mev.
This, then,may be the interpre[ationof the vanishinganisotropyat low energies.
Considernow quantitatively the situation at rela[ivistic energieswhere the
solution(92) is applicable.SubstitutingOUfOrfrom equation92 into equation101
andexpressing
•rr in termso• •11,•ñ and•, onehas
•_- (KlI--K.)tan•
+am)
Kii-]- Kx tan2 • V•U(2
3
(102)
with an associatedanisotropyamplitude from equation70,
•az-(Kll
-- Kj_)tan
• V•(2-]-am)
- Kii•-K•tan 2• w
(103)
Againif •ñ/•11-• 1, •az• O.Considerfor a.momentthe oppositelimit •ñ•11-• 0.
One quickly obtains
•az.O= tan •(V•/w)(2 -]- am)
(104)
But by equation86 tan • = r9s/V•, so that we regain the result in (100); i.e.,
•az --- •rot;if Kx/Kii-• 0 providedthat U dependsonly on r. However, it has been
shownabove that in this limit of Kx/Kii• 0 one might expectgradientsnormal to
the ecliptic that contribute to $a,. As pointed out by Jokipii and Parker [1969a],
the powerspectrumof the interplanetary magneticfield suggeststhat Ki/K• • 0.15
for protonswith energiesof 5 Gev in late 1964. With this large value of Kx/K•it
followsthat the gradientnormal to the plane of the eclipticis small [Parker, 1967],
and (103) gives
(2/3)az,O
(105)
= 0.4%
Thus, the observedvalues of •ñ/•11 yield a simple picture of the azimuthal
anisotropythat doesnot invoke density gradientsout of the ecliptic.
Consider next the radial anisotropy. Physically, there have been two basic
causesput forth for a radial anisotropy.In a sphericallysymmetricsystem,the
fact that adiabatic decelerationproducesor absorbsparticles in certain energy
ranges,causingsourcesor sinks, leads to a• anisotropythat may be seenas follows.For sphericalsymmetry,equation80 can be written
O(r,5•)= V•
O'aT(aTU)
r1
• Or
3 Or
(106)
J. R. JOKIPII
66
which integratesimmediately
30o OT(aTU)
do
(107)
in •he absence of sources or sinks a• the sun. This shows •ha• •he radial flux de-
pends only on conditions between •he poin• of observation and •he sun. As an
illustration of •he magnitudeof 5• expected,Jokipii and Parker [1968a]evaluated
(107) under the assumption•ha• •he radial gradien• (1/U)(dU/dr) measuredby
O'Gallagher[1967] ex•ended all •he way •o •he sun. Their derived aniso•ropy is
shownin Figure 8. This aniso•ropyis much larger •han •hag which is observed;
•hus, •he gradien• near •he sun is probably substantially less•han •ha• measured
by O'Gallagher. Ig is no• implied •ha• •he aniso•ropyand gradien• observations
are in conflict.SeealsoForman [1968]and Fisk and Ax•ord [1970]for a discussion
of •his problem.
A secondpossible cause of a radial anisotropy is s•reaming of particles
because of a lack of spherical symmetry in the solar wind [Parker, 1964b;
Ax•ord, 1965a; Sarabhai and Subramanian, 1966; Jokipii and Parker, 1968a].
For example,if the wind is slowerat high latitudes,then the cosmic-rayintensity
will be higher at high latitudes than in the solar equatorial plane. The particles
will
then tend to diffuse across heliocentrie
latitude
and then
outward
in the
equatorial plane, as illustrated in Figure 9a. Alternatively, of course,the intensity may be lower at high latitudes, and the streamingwill then be inward in
the equatorialplane. The magnitudesof theseeffectsdependon the value of Kñ
as well as conditionsout of the ecliptic plane, and quantitative estimates are
difficult.Parker [1969] has suggested
that the associatedradial anisotropiesmay
reasonablybe as largeas 0.5%.
Also associatedwith a latitude-dependentcosmic-rayintensity is a possible
second-harmonic
anisotropy(proportionalto P2(cos0) ), as discussed
by $arabhai
and $ubramanyan[1966] andLietti and Quenby[1968]. If the intensityis higher
on both sides of the equatorial plane, an observerin this plane would see an
enhancedintensity in a directionnormal to the magneticfield and parallel to
the equatorial plane. I• appearstha• the observed0.05% amplitude of this
7.5 %
Fig. 8. A plot of the radial anisotropy
a, for a sphericallysymmetric cosmic-ray
distribution, based on equation 107 and
assumingfor purposesof illustration that
the gradient (1/U)(OU/Or) of O'Gallagher [1967] extends to the sun [Jokipii
5.O%
•r 2.5%
0
-2.5%]
I
20
I
50
T (MeV)
and Parker, 1968a].
I
I00
COSMIC.RAY
PROPAGATION
IN THE SOLAR WIND
67
high,,.•.•/high
•
._
high
low
high
.,• high
high
= high
high
Fig. 9. A schematic
illustrationof the cosmic-ray
anisotropyif (a) the cosmic-ray
intensity
werelessin the equatorialplanethan elsewhereand (b) greaterin the equatorialplane than
elsewhere [after Parker, 1969].
anisotropyat 10 Gev can be explainedwith nominalgradientsnormalto the
ecliptic [Lietti and Quenby,1968].
C. Energy Balance o[ Galactic CosmicRays
As a final aspectof the modulation of galactic cosmicrays by the sun, consider the energy exchangebetweenthe solar wind and galactic cosmicrays. This
problem was recognizedquite early, in the considerationof the interaction of
the solar wind with the interstellar medium [Parker, 1958a], when it was recognizedthat the cosmicrays would help to stop the solar wind. A completediscussionof the energy balance problem involves some subtleties and was first
consideredin detail by Jokipii and Parker [1967].
The basic point is that galactic cosmicrays enter the solar system while
moving counter to the solar wind and leave while moving with the wind. The
basic geometryis illustrated in Figure 10. Thus, in the absenceof further energy
changewhile in the wind the particles experiencean energy gain of the order
of mV•w. More precisely,averaging over isotropic (over 2•r steradian) entering
and leaving angular distributions,one finds that the average particle gains
(AT) _____
mV,.w
(108)
However, while •he particle is random-walking in •he interplanetary magne•
field, i• is continually losing energy becauseof adiabatic deceleration a• a rate
given by equation 71. The result is •ha• most particles gain energy, although
•hose•ha• penetrate deepin•o •he solarsystem(the oneswe see)losea substantial
fraction of •heir energy. To understand•his las• poin• more clearly, consider•he
following argument. The average particle traverses a path similar to tha• indicated in Figure 10 and spendsa short time of the order of k/w in the solar wind,
where k is the scatteringmean free path. SinceV.V• •-- 2V/D at the boundary,it
losesaltogether •he amount of energy AT•o• ,.., (dT/dt)k/w --_• mwV•(k/D),
which is much smaller•han •he gain by the factor k/D. Hence the •ypical particle
gainsenergy.
68
j.R.
JOKIPII
Fig. 10. Sketch of a typical cosmic-ray
particle interacting with a modulating
regionthat endsat a radiusr -- D.
But there are a few particlesthat by chancepenetratedeeply into the solar
system.Thosethat reach the inner solar systemtake of the order of D2/A2 steps,
requiringa time of the order of D2/wX. They thus losea fraction of their energy
• D Vw/K,where K • (l/3)wX. SinceD Vw/• may be of the order of unity, it
is clear that for these few particles the adiabatic decelerationfar exceedsthe
small gain mVwW.
The result is that the solar wind does a net amount of work on the cosmic
rays, so that the cosmicrays tend to slow down the wind. It may be shown
[Jokipii and Parker, 1967], by consideringthe entry and departure of particles
from a small region of space,that the solar wind doeswork on the cosmicrays
at a rate per unit volume
dQ/dt -- Vw dP•/dr
(100)
where P• is the cosmic-raypressure
P• = õ
U(T,r)'•mow
•'dT
This work must be includedin a completedynamical equation for the solar wind,
as pointed out by Axford [1965a]. Axford and Newman [1965] and Sousk and
Lenchek [1969] have consideredin somedetail the integration of the solar-wind
equations, incorporating the effect of the cosmic-ray pressure P•. Sousk and
Lenchek pointed out that Axford and Newman neglecteda term involving energy
change,thus incorrectlyestimatingthe effect of the cosmicrays. Souskand
Lenchek find that the effec5 is important if the effective modulation boundary
D is much greater than 4 AU. Jokipii [1968d] has emphasizedthat the energy
gained by cosmicrays from the solar wind might resul5 in acceleration of lowenergy particles,to give rise to an effective'source'of cosmicrays at the solar-
COSMIC.RAY
PROPAGATION
IN THE SOLAR WIND
69
wind boundary.Little enoughis known about the parametersto know whether
suchaccelerationis important.
A number of o•her effectsthat are no• consideredhere in deSallalso are pax4
of the modulationprocess.The sbudyof the phaselag betweensolar indicesand
•ime variaSionsleads •o esbimabes
of D [Simpson, 1963; Charakhchyan and
Charakhchyan,1968; Dotman and Dotman, 1967; Simpsonaad Wang, 1970].
Also Jokipii [1969] has nobedbha5there shouldbe characberisbic
frequenciesin
the short-term (period of days) •ime variations of the cosmic-ray in•ensiSy
causedby a sharp boundary a•' r = D. Tt•ese differen5analyses,together wibh
inferencesfrom solar-flare propagation (seenex5section), sugges•tha• the effective depth of •he modulating region is some 3-7 AU. Parker [1968] has discussedbhe role of •he field-line random walk (see Appendix C) in admibting
galacSiccosmicrays to the inner solar sysbem.Finally, 5he shorb-bermForbush
decreasesshouldbe menbioned.The theory of this effec• has progressedlibtle
beyondParker'soriginaldiscussion[Parker, 1963].
7. DIFFUSIVE
PROPAGATION
OF
SOLAR
COSMIC
RAYS
The secondmajor application of •he cosmic-ray transpor5 bheory in bhe
solar systemis •he transpor• of energeticparticles acceleraSedat the sun and
injecSedin•o •he in•erl•lanetarymedium.The variety of phenomenais such•hat
the presen5discussionis resbrictedto problemsillustrating diffusivebranspot5in
the solar wind. The modern diffusionapproachto this problem goesback to bhe
paper of Meyer et al. [1956], which has been discussedin •he Introduction. The
explanaSionof a solar cosmic-rayevent is basicallymore complex•han •he modulation problem because•he full, time-dependen•form of •he •ranspor• equations
80 and 82 mus• be used. As in •he case of modulation, mos• abSentionhas been
concentratedon •he •ime dependenceof the density or omnidirectionalintensi•y,
and we begin by addressingthis problem.The importan5problem of anisobropies
will be discussedlaber.A recen• review has also beenprepared by Axfo•d [1970].
A.
The Density or OmnidirectionalIntensity
The sbandardapproach bo bhe solution of bhe branspor•equabion80 for
solarcosmic
r.ayshasbeen•o.assume
•ha.••he particlesare released
impulsively
into •he solar wind a• a poin5 on the sun. Thus, bhe initial condibionis bha•
U (x•, t, T) is a delSafunction in posiSiona5 time t = 0. One •hen imposesboundary conditionsto complebespecificationof bhe problem, a.sdiscusseda• the end
of section4. Usually it is suiTicien••o se• U equal to zero a• the outer boundary,
if bhis is sharp, or to require bhabU -• 0 as r -• • if bhe boundary is gradual.
The problem of bhe inner boundary,ab the sun, remains.This lat•er boundary
condiSionhas no5yet beenproperly •rea•ed in the literature.
As in bhecaseof the modulabionproblem,i• is usual to considerpropagabion
in a sphericallysymmetricsolar wind. This has beendone in various approxima5ions by Parker [1963, 1965], Krimigis [1965], Axford [1965b], Fibich and
Abraham [1965], Shishov [1966], Burlaga [1967], Feit [1969], Fisk and Axford
[1968], and Forman [1970b]. I5 is interesting 5ha• in each of •hese papers
(excep• for •hab of Forman, who did not considerthe boundary condibions),the
70
J.R. JOKIPII
inner boundaryconditionis that U is finite at r = 0, eventhoughin mos• of
the later papersi• is clear that diffusionis not taking place in the sun; thus,
r - 0 is irrelevant.The readeris alsoreferredto a paper by Axford [1965b]. In
the following,the solutionsof Burlagawill be discussed
in detail and the effec•
of usingmore physicallyrelevant boundaryconditionswill be examined.
Considerthen the calculationof Burlaga [1967] that illustratesthe basic
featureof the behaviorbf solarcosmicrays. Burlagastartswith the simple
anisotropic
diffusionequationfor the densityU, whichis proportional
to the
intensity j -- wU/4,r
Ot
O
U-- Ox•
0(••i O•_•xUi)
(110)
whichmaybe obtained
from•he full transpor•
equation
80 by neglecting
all
termsinvolvingthe wind velocityV•. The justificationsfor this neglec•wereno•
discussed
by Burlaga,and i• is worthwhile
to consider
herethe conditions
under
whichequation110migh•bevalid.I• appears•ha• if • is largeenough,
diffusion
dominatesthe convectionand energy change,in which ease equation 110 is
adequate.Order-of-magnitude
considerations
indicate•ha• sincethe length
scalesare of the orderof r or smaller,•he neglectedtermsare small if
•,, >> -•arV•(y - 1)
(111)
where7 is asdefinedin section6. The evaluationof •, in section4 thenindicates
that Burlaga'ssolutionis a reasonablefirs5 approximationfor protonswith
energies
in excess
of a few hundredMev. For thesesolarparticles,then,diffusion
dominates,and convectionand energychangeare small effects.The simple
diffusionequation110should•ot be considered
applicablefor solarprotonsbelow
roughly100-Mevenergy,or othercaseswhereequation111is violated[e.g.,Li•
et al., 1968; Lanzerotti, 1969].
One next assumestha• negligible error is introducedby setting the spiral
angleq•-- 0, sotha• • - •11andthe diffusion
in latitudeand.longitude
is governedby •ñ. Further,take •11to be independen•
of r and •ñ ecr•. Note tha•
assuming
.•ñ ecra hasrecentlybeengivensomesuppor•by considerations
of
field-linerandomwalk [Jokipii and Parker, 1969]. Equation 110 becomes
OU
0r,OU I 0 (1- •')• +r•'sin
•00,
ot-. •,ñ
_ •x•
(112)
wherev - cos0 and0 and • are angularvariables.The initial condillonis [ha•
air=0
U(r,•, t: O,T) - Uo(T)
2•ro•$(r- ro)•(• - 1)
(113)
corresponding
to an impulsivepointsourceat r - to, 0 - 0. Definethe dimensionless variables
•--- t•/r •'
• = (•/•)•/•' •r
(114)
and note that with a point sourceat • - 0, U cannotdependon •. With the
COSMIC.RAY PROPAGATION IN THE SOLAR WIND
71
assumedform for K•j,equation111 separates.Le•
U(p, •, 7.,T) -- R(p, 7.)O(•, 7.)
(115)
with the energy dependenceretained implicitly. Equation 112 then reducesto
two equations.The R equationis
OR O"'R
-!-2OR
07. -- Op:: p Op
with the initial
(116)
condition
(\r K•y
/
R(p, O) = Uo •
-
2wpo
2
(117)
and
00
07'
0I(ll •:)0•1for0<•<1
(118)
with
0(•, 0)-
$(•-
1)
(119)
First considerthe angular equation 118. Burlaga showedthat the solution
could be written
20(•, •) = Qo(•)= •'• (2n-[- l"e-n(n+•)•P
;
f•
(120)
•mO
where P,•) is the usual Legendrepolynomial. Similarly, the solution to the
radial equation 116 can be obtained by a Laplace •ransform and one finds
where •he allowedvalues of B are specifiedby the boundaryconditions.
The boundary conditionsare to be applied at •he inner and outer boundaries
of the system,and hencerefer •o the functionR(o, •). At •he outer boundary
r • D(o • or) •he requiremen•is •hat R(or, •) • O. With regard to the inner
bounda•, Burlaga followsthe generalpracticeof requiringR (0, •) to be finite,
eventhoughit is quite apparentthat equation110 doesnot apply inside•he sun,
and one doesnot care what happensa• • - 0. Here, we first give •he solution
using the boundary conditionapplied by Burlaga, and then illustrate the uncertainty by applyinga different,physicallymorereasonable,bounda• condition.
Thus, if R is to be finite wheno • 0, the parameterB in equation121 is
zero.Then the conditiontha• R (•, •) • 0 specifies
B, and onefinds
U(p,
•, v,T)= Uø(•(•x/r•")s/:
• e-(•"•=>"
s•x p• • ,s•
2wppop•
•
Q•(•)(122)
Of particularinterestis the asymptotic
time dependence
of U. If •=•/•= • 1
andpo• p•, equation122takeson the form
U(r,O,t, T)• Uo(T)
2rD• s•
e
(123)
72
J.R. JOKIPII
Similarly, for •r/t•) • << 1, Burtagafinds
Illustrated in Figure 11 is a typical flare even[ observedon [he Deep River
neutronmonitor,togetherwith a theoreticalfit. Clearly, the form of [he solution
agreesquite well with the observations.
Burlaga findsD ~ 2.5 AU.
Now go back [o the questionof boundary conditionsat the sun. Clearly, a
generallinear boundaryconditionis [hat
EU q- F(OU/Or)].... = 0
for
r > 0
(124)
The above, with F = 0, corresponds
to a perfectly absorbingsun, and E = 0
15
NOVEMBER
DEEP
RIVER
,,,
to = 0213
uJ
• I00
•o = 16ø
1960
EVENT
NEUTRONS
UT
-r:: -X
i--
-
,:5.,
"-;...:.:an..:
.
• 10II
THEORETICAL
CURVE/••'•
I
0
3
6
9
TIME
12
15
18
21
(HOURS)
Fig. 11. Theoretical fit, using equation 122, to the Deep River neutron
monitor data for the November 15, 1960, event. 0ois the angle between the
flare and the foot of the average magnetic field line passingthrough the
point of observation [Burlaga, 1967].
COSMIC.RAY PROPAGATION IN THE SOLAR WIND
73
corresponds
to a perœectly
reflectingsun. Certainly a perœectlyabsorbingsun is
not tenable unlessthe impulsivesourceis situated at r > rs, œorotherwise
the particleswould be immediatelyabsorbed,and none would propagate•o 1
AU. I• is likely tha• the actual situation is mixed, in which neither E nor F is
zero.Bu• i• may no• be •ar wrong•o assumea perfectly reflectingboundarycondition. Sincethe strong macetic fields at the solar surface decreaserapidly
outward, particles would tend to mirror and be reflectedfrom the sun. In any
event, choosingonly solutionsthat are finite at the origin is manifestly incorrect.
To see the effect of choosingphysically more meaningfulboundary conditions, considerthe effect of imposingthe conditionin equation 124 at radius ro.
First, it is evidentthat requiringU (p -- p•) -- 0 leadsto
=; •A• inly_ po
L'• poa
Then, subs•i•u•in• in•o equation 1•4 leads •o
--•po•Sn•+Fsn•+•p•_ gO= 0
•hich serves•o specify•he allowed•slues of •. I• is cles• already • •his s•s•e
•hs• •he solution•hs• diverges• •he origin mus• in •ene•sl be •e•sined, so •hs•
•he correc6solution may diffe• considerably from equation 1•. •oweve•,
appears•hs• if •o/(•, - po) << 1, •he lon•-•e•m limi• is reasonablywell spproxims•ed by equation 1•3. This problem of boundary conditions was discussed
b•iefiy by Fei• [1969]. The questionof prope• boundary conditionsis 6herefore
of considerablepractical importance. One should •e•srd wi•h caution any solution •hs• i•ores •he questionof prope• boundary conditions• •he sun, exeep6
insofar as i6 may •ive • 6u•li•a•i•e indication of •he nature of •he solution.
Thus, for example,•he observedexponentialdecay is accurately •iven by equation 1•3.
I• may be argued •hs• •he observedexponentialdecay is • s•ron• ar•men•
for •he existenceof • •els•ively sharp bounds• • some3-5 AU, •here •he
•erin• mesn-f•ee-ps•h becomeslarge. The s•umen• is no• conclusivesince sl•ernative causesof an exponential decay • high energieshave no• been excluded.
In 6he li•h6 of •he precedin• discussion,•hen, i6 appears•hs• simple anisotropic diffusion•ives • reasonableapproximation•o •he behsvio• of high-energy
particles, excep• •hs• •he boundary condition • •he sun has been improperly
•res•ed. Fu•he•more, proper •res•men• of low-energy particles (p•o•ons of •100
Mev or lower) •equires considered.
ion of •he complete •mnspor• equation 80.
Unfortunately, snsl•ic solutionsof •his equation unde• realistic circumstances
are no• ye6 available. An analytic solutionunde• somewha6•es•ric•ive conditions
was presentedby Fis• a•d •]ord [1968].
They consider•he casexo m x8o,•i•h x(r) • xor.Wi•h •his form for xo and
6he assumptionof sphericalsymmetry, equation80 becomes,upon ne•lec•in• •he
ener• dependence
of .•
OU
Ot+ •V.•0½•U) 4V.
3 r oT
0(TU)=••o0(r•O•)
(127)
74
J.B.
JOKIPII
whichis separableif the particlesare injectedwith a powerlaw energyspectrum.
Fisk and Axford imposethe initial condition
U(r, t, T) - AT -•' $(r - rs)/r•
(128)
with U (r, T, t) -) 0 as r --) • and U (0, T, t) is finite. Now, we haveseenabove,
in discussing
•he high-energyapproximation,tha• this latter boundarycondition
is suspect,and caution must be used with regard to this solution.Nonetheless,
•he solutionshouldbe a5 leas•qualitativelycorrect.Fisk and Axford find
U(r,t,T)- •otr
AT-•
(--ri
sr xrs/
exp[--(r -}-ro)/got]I•[2(rro)•/•/got]
wi•h
• = [(2 + V•/•o)•+ 16V•(u- 1)/3•o]
(129)
whereI, (4) is the modifiedBesselfunctionof the firs• kind. They find •hat •here
is a residualanisotropya• largetimestha• is rela•ed•o the energylossdue to
adiabaticenergychange.Fisk andAxfordmakeno at•emp••o comparethis solution wi•h observation;
this is probablyrealisticsinceobserved
decaysare usually
exponential,not the power law in time as suggested
by equation129.
Available solutions,•hen, do no• give a realistic descriptionof •he behavior
of solarparticlesat lowerenergies.
Althoughit doesno• appear•o be fruitful •o
derivediffusioncoefficients
from •he solar-flareparticleda•a to comparewi•h the
resultsof section4, moreexac•solutionsshouldhelp •o completeour understanding.
Recenfiy Lin [1969] has presentedevidencetha• 40-key electronsocca-
sionallypropagateou• from the sunwithou•undergoing
appreciable
scattering.
This givesa new aspec••o the transpor•a• low energies,
whichas ye• is poorly
understood.
Burlaga [1970] has a•temp•edto interpretsomeof •heseeventsin
terms of somedifferen•assumptions
concerningsca•ering near the sun. These
problemsare no• yet properlyunderstood
and will no• be discussed
here; they
are mentionedonly •o guide•he reader•o somecurren•problems.
Finally, i• shouldbe pointedou• tha• in recen•years i• has beenobserved
•ha• the sunalsoemitsparticlescontinuously
for longperiodsof time, from certain locationson the sun,givingrise •o recurringcoro•a•ingstreamsof energetic
solarparticles[Fan et al., 1968]. I• appearstha• the basicfeaturesof •he phenomena can be understoodas a logical extensionof •he above discussionwith
impulsive emissionreplacedby continuousemission.No full solutionrelevan• •o
•his situationhasbeenpublished.
Jolcipiiand Parker [1968b,1969a]pointedou•
that the relativelybroadextentof the streamsof ~l-Mev protonsreportedby
Fan et al. couldbe understood
in termsof the randomwalk of magneticlinesof
force,as measuredby the powerspectrumat zerowavenumber(seealsoAppendix 3). The contributionof this randomwalk to the perpendiculardiffusioncoefficienthas alreadybeendiscussed
in section4. The pictureof Jokipii and Parker
[1969a] is that the bundleof linesof forcethrougha point at the sunis braided
and twisted so that at a distance r the lines are distributed in a Gaussian with a
half-widthdetermined
by the powerat zerowavenumber.The low-energy
par-
COSMIC.RAY PROPAGATION IN THE SOLAR WIND
75
ticlesthen trace ou• this distribution.The predictedhalf-width usingthe spectrum in Figure 3 is about 0.15 radian, to be comparedwith the observedwidth of
0.28 radian. $okipii and Parker concludetha½•he random walk of lines of force
is a dominan•œactor
in producing•he angulardistribution.
Anotherremarkablefeatureof the corotatingstreamswasreportedby Anderson [1969]. I-Ie observesboth 40-key electronsand 1-Mev protonsin the streams
and findsthat the electronstypically leadprotonsby up •o severalhours.That is,
the centerof the electrondistributionleads•he cen•erof the protondistributionby
this amount.Anderson[1969] suggested
that this separationmay be a manifestation of gradien•drift occurringin magneticloopsnear the sun.Jokipii [1969b]
has pointed out tha• curvature and gradient drifts in the spiral interplanetary
magnetic field would be expectedto produce such a separation if the diffusion
coefficien½
for 1-Mev protonsis oœthe orderof 10•ø cme/sec,whichis a reasonable
value. I• is an interestingpoint in this analysis tha• the combinationof both
curvature and gradien½drifts in the solar wind is suchtha½the electronstend to
lead the protons for both sensesof the spiral magnetic field. This is illustrated
in Figure 12.
B.
Anisotropies
The secondaspec•of the transpor• of solar cosmicrays relevan• •o the diffusion theory is their anisotropy.The observedanisotropiesmainly refer to
impulsiveeventsand are of two kinds, the initial anisotropyassociatedwith the
injectionand the equilibriumanisotropythat se•sin later in the evenL
Considerfirst the initial, or nonequilibrium,anisotropy.This is very difficul•
to handletheoretically,and very little has beenaccomplished.
It is not properly
treated in the diffusionapproximation.The basic effect is that particles arriving
from a flare arrive firs5along the averagemagneticfield, and then gradually one
observesparticlescomingfrom the sidesand finally from the backwarddirection.
The initial anisotropy may be quite large and relaxes to an 'equilibrium' value
with a time scaleof the orderof one hour. This phenomenon
was studiedby McSOLAR
ROTATION
(].
b.
%
EQUATOR
+
+
+
+
\
%
\
sou,c
+•
I
+
+
+
+
+
+
+/
+
+
+
+
+
+
ß
ß
ß
ß
ß
ß
ß
ß
ß
©.,sou,c
ß
ß X.
EQUATOR
/
•'/SOURCE
+
+
+
/
(• SOURCE
+
+
ß
/
/
ß
/
/
Fig. 12. The dotted lines illustrate schematically the motion of a proton due to curvature
and gradient drifts in the spiral interplanetary magnetic field. The magnetic field is directed
toward the sun in part (a) and away from the sun in part (b). In either case, one of the
sourcesis in the northern and the other in the southern hemisphere.The drift of low energy
electronsis negligible,so that as the sun rotates, the electronswill lead the protons.
76
J.R.
JOKIPII
Cracken [1962] at relativisticenergiesand morerecentlywas studiedat energies
~ 10 Mev by McCracken et al. [1967].
Intuitively,
onemightexpect
thattherelaxation
timerr andmean-free-pat•
Xll = 3 Kii/w are related by
rrw = Xi•
(130)
as suggested
by McCracken et al. [1967]. However, it appearsthat the observed
relaxation time at low energiesis muchtoo long[o be accountedfor in this manner. That is, the expectedscatteringmean-free-path,based both on a powerspectral analysis as in section4 or on a diffusionfit to the observedprofile, is
much smaller •han r•w. At presentthis discrepancyis not well understood.One
possibleinterpretationwas suggestedby Jokipii [1968b] and is based on the
analysisof particle trajectoriesgiven in section4. The point is [ha• t.herate of
scatteringin pitch angle,given at low energiesby equation55, varies with pitch
angle.If the powerspectrumfalls off with increasingwave numberas k-a/a,then
i• followsdirectly from equation55 that the rate of scattering((Atz)2)/At is small
for t• near zero,corresponding
to 90ø pitch angle.Particlestraveling out from the
sunthereforeexperiencedifficultyin beingscatteredthrough90ø pitch angle,and
the relaxation time may be anomalously long. Jokipii [1968b] presented a
detailed analysisthat indicated that this interpretation, in terms of a variation
of the rate of scatteringwith pitch angle, is consistentwith observation.
The equilibriumanisotropyis better understoodthan the initial anisotropy,
since it falls within the province of the transport equations80 and 82. In partieular, since the initial, large anisotropy has relaxed, the flux equation 82 is
valid, and the radial anisotropyis given by
& - 3V•
V• OT
0 (aTU) wU
3Kr•
OU
w - 3wU
Ox•
or, if v:
(131)
-O In (wU)/O In T,
w
wU
&=(2+a•)V•
3 [ OU
1
(132)
Forman (1970b) has publishedan analysisof •he equilibrium aniso•ropy,based
on equation 132. She finds •ha• •he low-energyda•a of McCracke• et al. [1967]
and Rao et al. [1969] are consis•en•wi•h •he major contributions•o 8• coming
from only •he firs• •erm in equation 132, which suggests•ha• •he gradientsare
small during •he equilibriumphaseof •heseevents.The agreemen•between6he
predictedaniso•ropyusing•he firs• par• of equation132 and observationis shown
in Figure 13. A completeunderstandingof •his aniso•ropywould involve knowing
why 6he gradient contribution•o 8• is so small, and •his requiresa solutionfor U.
Nonetheless,•he discussionin •erms of equation 131 indicates•ha• •he observed
phenomenonfits very well within •he basic•ransport•heory.
More recenfiy, Gleesonet al. [1970] have discussed
•he s•eady-s•a•egradien•
and aniso•ropy of presumably solar cosmicrays on •he basis of equation 132.
They also find good agreemen•between•heory and observation.
A large number of other phenomenaassociatedwith solar flares have been
COSMIC.RAY PROPAGATION
,
IN THE SOLAR WIND
,
I
77
,
I
i
i
IN,'•10
i
RADIAL
ANISOTROPY
EVENTS
SOLAR
PARTICLES
IN 1966 (McCRACKEN,ETAL)
MeV
>" 20
MAR. 20 I-4
o
ß
ß JULY II
o
z
i,i
I0
ß
ß
JAN
Z•.1/•JAN.
19
SEPT.28 H
: ' MAR.26
0
o-
-
0
'
I
I0
CONVECTIVE
I
!
20
ANISOTROPY,%
Fig. 13. Comparisonbetween equilibrium anisotropiesobservedby McCracken et al. [1967] and Rao et al. [1969.] and the effect calculated from
the first term in equation 132 [from Forman, 1970b].
reportedin the literature. However, the ones discussedabove illustrate mos•
clearlythoseaspects
of the problemrelevantto the presentview of cosmic-ray
transpor•in the solarwind.With the possible
exception
of Lin's [1969] observationsof scanter-free
propagation
of ~40-key electrons,
mos•propagation
effects
appearto fit very well into •he basictransporttheory.Hopefully,the next few
yearswill seebetterand morerealisticsolutions
of the differentialequations
to
provide a better test of these ideas.
8. GENERAL
DISCUSSION
The foregoingdiscussion
has presenteda thoroughaccountof the modern
statisticaltheoryof cosmic-ray
propagation.
As shownin sections
6 and 7, this
theoryhas had very goodsuccess
in explainingthe observedfeaturesof cosmicray transportin the solarwind. The basicobservedphenomena,
suchas the
,
eleven-year
solarmodulation
of galactic
particles,
Forbush
decreases,
thetimeintensityprofileof solarcosmic-ray
events,andtheobserved
fluxesor anisotropies
of bothsolarand galacticparticles,appearto be well understood.
Futureprogresswill probablycenteron obtainingbetter solutionsto the equations,
on
obtainingmore accuratevaluesof the variousparameterssuchas the diffusion
78
J.R. JOKIPII
tensoror solar-windvelocity,and on the explorationof higher-order
or more
subtle effects.
It shouldbe emphasized,
however,the transportequations
80 and82,together
with the determinationof the diffusiontensorin termsof the spectrumof magnetic fluctuations,are valid in many placesother than the solar wind. In discussing
the scatteringof trappedparticlesin the earth'smagnetosphere,
equations
similarto equation55 have beenused [Kenneland Petschek,1966]. Further,the
transportof cosmicraysin the galaxyhasbeendiscussed
usingpreciselythe same
equationsthat are usedin the solarwind [Wentzel, 1969; Kulsrud and Pierce,
1969; Earl and Lenchek,1969; and Jokipii and Parker, 1969b].One effectthat
is not important in the solar wind, but that must be consideredin the interstellarmedium,is the fact that the cosmicrays have sufti½ient
energydensityto
affect the magneticspectrum [Lerche, 19.67].One must in generaluse a selfconsistentcalculationto obtain the magneticspectrum.Once the spectrumis
determined,the interactionof the particleswith the interstellarmagneticfield
is preciselyas outlined here.
The techniquesdiscussed
in this paper, then, are applicablewhereverfast
chargedparticles (with speedsw >> V•) propagatein an irregular,rarefied
plasma.This includesmany situationsin astrophysics
and spacephysics.
APPENDIX
A. TRANSFORMATION
OF POWER
SPECTRA
Considerthe relationbetweenpowerspectraobservedin fixed and moving
frames. Let B•'(r, t) be the fluctuating field observedin a coordinateframe fixed
relative to the plasma.Then, in this frame
e,,,(t:,
=
+ t:, t +
(A1)
(A2)
Now supposethat the magneticfield is observedby a stationaryobserver,past
whom the plasma is convectedat a constant (nonrelativistic)velocity V.. This
observerseesa fluctuating magneticfield
s(r, t) = n'(r - vt,
and a correlation
t)
tensor
R,•(t:,
r) = (B•,(r
+ •, t + r)B•(r,
t))
= R,i'(t•- V,,,r,r)
(A3)
'
But this stationary observerobservesonly at a fixed point, so that in fact
O, and he observes
R,•(O, r) = R,,'(-V•o,
r)
(A4)
The Fourier •ransformof R•(O, •-) with respec•to • is the observedfrequency
power spectrum
COSMIC.RAY PROPAGATION IN THE SOLAR WIND
P•(I')
=f;=,
R,•(O,
r)e
-"'•'•
dr
-- f;•R•
i' (- V•r,
•)e-2riIr dr
79
(•)
Usingthe f•ct that R, •' is the invemeFou•er transformof P•'(k, •'), and letting
k, be the componentof k along V•, one finds the relation between the obse•ed
spect•m and that in the plasma frame
Equation A6 is exact.
Now make use of the fact that for any waves of interest in the solar wind,
V• is muchlargerthan the phasevelocity•'/[k I. If Vo" is the maximumvalue
o• •71kl, thenP•s(k,g) is smallfor o,'/Ik[ • Vo'. Hence,wherethe integrand
in
(A6) is nonzero,onehas (2=/-
•')/V•
• •VVo". Sinceit is knownthat V• >
Vo" • Va, one has 2=]/V• > •VV•, and equationA6 becomes
P,•(I)• 8•aV• '"
dk•dka
dw'
P•' k•,ka,ks= V•'•'
(A7)
Bu• the figh• side of (A7) is reco•zed to be (1/V•) •imes the one-•mensional
wave number spect•m along V•; that is
P,•(•••
R,•'(0,
0,•, O)e
+'(•'"•-'•'
d•
(AS)
Equations
A• andA8 contain
•hedesired
result.
APPENDIX
B. STOCHASTIC
EQUATIONS
LIOUVILLE'S
EQUATION
FROM
Hall and Sturrock[1967] and Roelo• [1968] have pointed out that the basic
Fokker-Planck equation can be derived from Liouville's equation.This approach
will be sketchedin this Appendix.Considerthe one-particleprobability distribution n(r, w, t) that satisfiesthe equation
On
On
0Wq-w.Vn q-(wx •o).•-•= 0
(B1)
Again,let • = •o + •, with (•) = 0. Similarly,let n = no + n•, with (n•) = 0.
The ensembleaverageof equationB1 is
ot + w.Vno+ [w• •o]'ow
= - ((w• •).T•/
Ono
Ono
On•
(B2)
Subtracting(B2) from (B1) yields an equationfor n•
On•
On•
ot + w.Vn•+ (wx •o)'ow
= -(w x •)' Ono
ow
where the term [(w x •).(0n•/Ow)]
(B3)
is neglectedbecauseit is of higher order.
80
]. R. ]OKIPII
The solutionto equation B3 can be written
Ono
(wx •0'•-•.]•
dt•
n•(•,w,t) = --
(B4)
where the integrand is to be evaluated along an unperturbed particle trajectory,
which is the characteristicof the differentialequation.One may substituteequation B4 into B2 to obtain the equation governingthe evolution of the average
distribution no
0t + w.Vno+ [(wx ,•o)]'¾•
- <(wx ,•).•¾
(wx ,•).-•_• •t'
(BS)
First; note that; the right side of (B5) containsonly two-point correlationsof e,.
If the orbit is substantially unchangedin s correlationtime of the fluctuations,
the lower limit of the integration may be taken to --•,, and equation B5 takes
on the Fokker-Planck form. This result was shownby Hall and Sturrocto[1970]
t;o lead to the equationsgiven in section4 if e is independentof time. They also
generalizedtheseresultsto incorporatevarying electricfields.
The reader is referred to Klimas and Sandri [1970]for sn alternate view of
this problem.
APPENDIX
C. RANDOM
WALK
OF
MAGNETIC
LINES
OF
FORCE
It; is possibleto obtain an interestingphysical interpretation of the transverse power at zero wave number that appearsin equation 56. This nonresonant;
term can be shownto representthe random walk of the magnetic lines of force,
as firs• pointed ou• by Joripit [1966] and subsequenfiyelaboratedby Jokipi{
and Par}er [1969a].
The lines of force of •he magnetic field B (r) are defined by the family of
solutions•o •he first-order equations
dx
dy_ dz
• =• - •
(c1)
Supposeagain that
B•r) = Bo•, •- B•r)
(C2)
Thentheequations
for a fieldlineareto firstorderin 1B•[/Bo
Ax= x - Xo= •I fo
b'B•(Xo,
yo,
z')(/z'
(C3)
•y = y - yo= •! I •'B•(Xo,
yo,
z9•z'
(C4)
One may readily verify •hag •he ensembleaverages<Ax>and <Ay>are zero. In
•erms of •he correlation•ensorof B•, onehas
<(ax)•>
=•
•'
•r<•(Xo,
•o,•')•(Xo,
yo,
•' + r)> (c5)
wi•h similar equationsfor <Aye>and <AxAy>.Now suppose•ha• Az >> L, where
COSMIC-RAYPROPAGATION IN THE SOLAR WIND
81
L is the correlation
lengthof thefluctuations.
Thenthelimitsonthe • integration
canbe taken from -•
to + •, and onehas
((•x)
•)-- •z•o
• (o,o,•)6•
(c•)
But the term in bracketsis just the one-dimensional
spectrumP•(k) evaluated
at k -- O. Thus
((•x)•')
= •z?•(•
•o=
• o)
(c•)
with similarequations
for ((•y)•') and ((/•x/•y)). The coefficients
((Ax)2)//•z,etc.,
are in fact the appropriateFokker-Planckcoefficients
describing
the evolutionof
fieldlinesas a functionof distancealongthe averagemagneticfield.The distribution of thoselinespassingthroughthe pointXo,yo,Zois a Gaussiandistribution
inx and y atz - Zo+ zx.
It is easyto seethat low-energyparticletravelingalongthe z axiswill tend
to followthisrandom
walk.Sincetheirspeed
in the z direction
is Igl w, they
random walk in the x direction at a rate
((•x)•
- I•l•o•x•
At
Az
(cs)
Pxx(k = O)
Bo•
whichis preciselythe nonresonant
term in equation56.
APPENDIX
a, b, c, d
B
Bi
D
D. LIST
OF FREQUENTLY
USED
SYMBOLS
Functions defining structure of correlation tensor.
Equation 29.
Magnetic-field vector.
Fluctuating magneticfield.
Speedof light.
I-Ieliocentricradius of boundary of modulating
!
ff,(r, T, t)
region.
Frequency.
Cosmic-rayflux in frame at rest relative to the sun.
Equation 82.
j(r, T, t)
Differentialomnidirectional
intensityper unit solid
k
Wave
n(r, tz, t)
Density, or probabilitydensity, of particlesas a
functionof r, •, t. Equation 2.
Observedtemporalpowerspectrumtensor.
•ngle, j = Uw/4•r.
number.
Particle rest mass.
P,•(•)
Legendrepolynomial of the order of l.
r =
x, y, z
r, O, 4•
Position vector.
Sphericalpolar coordinatescentered on sun.
82
J. R. JOKIPII
Cyclotron radius.
Radius of the sun = 6.96 X 10'ø cm.
we
Particle magneticrigidity, R = •mcw/Z.
Two-point, two-time correlationtensor of magnetic
field as function of spatial and temporallagsi•,•.
Time.
t
Particle kinetic energy.
Cosmic-raynumber density or probability density,
averaged over angle.
Density of galactic cosmicrays beyond the region
U(r, t, T)
U•(T)
of effective solar modulation.
VA
Vs
w
Alfv•n speed.
Sound speed.
Solar-wind velocity, speed
Particle velocity, speed.
Total particle energy.
z = q/e
Particlechargein unitsof elementarycharge.
V•, V•
W• W
a(T)= (T
q-2moc
•)
(T q- moc
•)
Energy chargeparameter.Equation 71.
Separationconstant.Equation 121.
(1 -- w•ic•)-•/2.
v(T) = -01nj/01n
T
Effective index of powerclaw energy spectrum.
(Equation91b.)Also1 • - 10-5 gauss.
•C-G,i
•, = 35i/Uw
K.L• KII
k = 3Ku/w
tz -- Wz/W
p
p•
• = tan-• r•s/V•
-- qB/'ymoc
Anisotropyassociatedwith Compton-Gettingeffect.
Equation 81.
Cosmic-ray anisotropy associated with flux ff•.
(V•/c) 0 in U/O in T.
Cosmic-rayscalardiffusioncoefficient.
Cosmic-raydiffusiontensor.Equations50 and 51.
Cosmic-raydiffusioncoefficientsperpendicularand
parallel to the averagemagneticfield.
Scatteringmean free path.
Cosineof pitch angle.Equation 27.
Dimensionlessdistanceparameter. Equation 89.
Orbit parameter.Equation 30.
Effectivepotential. Equation 97.
Angle between average magnetic field and heliocenttic radius vector. Equation 5.
Vector frequency associatedwith magnetic field.
Fluctuating part of •(r).
Average cyclotronfrequency.
Rotationfrequencyof the sun: 2.9 X 10-• sec-•.
Ensembleaverageoperation.
Acknowledgments.
This work has benefitedfrom helpfulsuggestions
by W. I. Axford,
Leverett Davis,Jr., and E. N. Parker,for whichI am grateful.
COSMIC.RAY
PROPAGATION
IN THE SOLAR WIND
$3
Work on this review has been supported,in part, by the National Aeronautics and Space
Administration under grant NGR-04-002-160.
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