VOL. 76, NO.

10 3OURNAL OF GEOPHYSICAL RESEARCH APRIL 1, 1971

Low-Energy
Cosmic
RaysnearEarth
L. J. GLEESON

Monash University, Clayton, Victoria, Australia

S. M. I•RIMIGIS

Applied Physics Laboratory, The Johns Hopkins University

Silver Spring, Maryland 20910

W. I. AXFORD

University o/California at San Diego, La Jolla 92037

Observationsof protons with kinetic energy ),0.31 Mev in interplanetary space during
quiet times made from the spacecraftExplorer 33 showthat there is an anisotropyof ~15%,
the maximum flux being directed approximately radially outward from the sun. This observa-
tion, and the correspondingradial gradient results reported by S. M. Krimigis, are shown
to be in very good agreement with those expected from a model in which the observed
protons are convectedoutward at the solar wind speed.It is concludedthat theseparticles
have come from an interior region and that the most probable source is the sun, but the
possibilitythat they are of galacticorigin,althoughremote,cannotbe positivelyexcluded.

Recent measurements of the steady-state tronswith energies

in the range1-20 Mev [e.g.,
energetic particle flux in interplanetary space Simnett and McDonald, 1969; Sirenerret ed.,
suggest that the sunis an importantsourcethat 1969], if they are of solaroriginand hencehave
contributescontinuouslyto the interplanetary a large negative gradient (-•-250%/AU).
particlepopulation.Observations of the gradient Ho.wever, a careful calculation shows that al-
of 0.3-Mev protons during 1967 [Krimigis, thoughtheseelectronsmay have an important
1970] have been interpreted to show that the effectthey cannotwholly accountfor the nega-
sun is the dominant contributor of suchprotons tive gradient,and hencethe radial gradientof
to the interplanetary particle flux. Kinsey protonsmust be negativefor someenergies at
[1970], from a study of the interplanetary pro- least (S. M. Krimigis and D. Venkatesan,pri-
ton and a-particle spectraat I AU, has inde- vate communication,1970). Vernovet ed.[1970]
pendentlysuggested that the interplanetaryflux independentlyrepo.rteda negativegradientof
consistsof two componentsin the 4- to 80- --13%/AU observedduring 1967 for protons
Mev/nucleonrarge in kinetic energy,the vari- of kineticenergyT • 30 Mev. In contrastto
able componentbelow 10 Mev being of so!ar these results, O'Gallagher [1967] finds large,
origin. The componentbelow 10 Mev has also positive gradients' 187%/AU and 145%/AU
been discussedin detail by Fan et ed..[1968, for protonsin the range 30 to 60 and 70 to 100
1970]. Mev, respectively.The subject of the present
At higher energies,Krimigis and Venkatesan paperistheregionT • 10Mev.
[1969] suggested that the sun is an important The particleflux for T • 10 Mewis transient
sourceof 50-Mev protonseven at the time of in nature becauseof the large number of solar
solar minimum, on the basis of an apparent particle events. However, there are quiet pe-
negativegradientfor suchprotonsof --14%/ riods in which steady-stateconditionsappear to
AU during1965.It is possiblethat the negative prevail. The radial gradient observation of
gradientis in part due to interplanetaryelec- Krimigis [1970] at T • 0.3 Mev was made
during the quiet period August 27-31, 1967.
Copyright ¸ 1971 by the American GeophysicalUnion. In this paper we present observationaldata
2228
 Low-ENERGY CosMIC RAYS NEAR EARTH 2229

0.5
on the steady-stateanisotropyof the T • 0.3
Mev protonsreferred to above and proceedto
developa model that is in accordancewith both
the gradientand anisotropyobservations.The
model assumes that convection is the dominant
0.4 MARINER
5
transport mechanismand is consistentwith par-
ticlesoriginatingat the sun.It hasbeenpointed
out by several authors [e.g., Parker, 1965;
Quenby, 1966, 1967; Fisk and Ax•ord, 1968;
0.3 \ J=
•,o. 2
Jo(-•
)-0.9
Forman, 1968, 1970b, 1971; Ax[ord, 1970] that
energeticparticles releasedfrom solar flares are • 0.1 EXPLORER
33
likely to undergo substantialdecelerationif the
diffusionvelocity is comparableto or lessthan 0.07

the solar wind speed.Here we considersteady

emissionand use resultsthat are valid asymp- o.o5/ I I I I I I I I I
0.1 0.2 0.4 0.6 1.0
totically in order to make comparisonswith
T (MeV)
observations.
The observations described below have been Fig. 1. Integral intensity spectrumobservedon
Mariner 5 at r -- 0.84 AU (data points), together
obtained with intercalibrated solid-state detec-
tors from the University of Iowa experiments with the exponential and power-law functions
on the Venus-bound Mariner
used to approximate it in the analysis. A single
5 and the earth-
point observedon Explorer 33 at 1.01 AU is also
orbiting Explorer 33 spacecraft.The detectors included.
on both spacecrafthave been describedin detail
elsewhere[Krimigis, 1970].
day, sincemost sectorcountingrates are within
OBSERVATIONS
two standard deviationsof each other. By aver-
Figure i showsJ(r, T), the mean integral aging over the entire 5-day period,however,it
directionalintensityin the kinetic energyrange is possibleto obtain an amplitudefor the anisot-
0.32 < T < 1.0 Mev at Mariner 5 for the ropy with reasonableaccuracy(~20%), and to
periodAugust27-31, 1967,whenthe heliocentric establishthe direction of the anisotropy.
radius r was 0.84 AU. At this radius, the func- The total number of countsin a given sector
tion J(r, T) can be adequatelyapproximatedby over the 5-day period is ,,730 (not including
backgroundcounts), i.e., ~4% statisticsif a
J-- Jo(T/To)-ø'9 or J-' Joe -r/r' (1) Poissondistribution is assumed.The anisotropy
where To and Jo are constants and T• -- 0.66 can be computedin two ways:
Mev. Of thesetwo forms,the exponentialrepre- a. Assuming that the directional counting
bentsthe data slightlymore accuratelythan the rate at an angle• to the sun-spacecraft line is
power-lawfunction.Both are usedlater in order of the form
to illustrate the rather critical dependenceof the
Compton-Getting factor on the form of the + B cos (2)
function used.
Figure 2 showsthe corresponding
directional with A, B, and •o constants,we find •o -- 160ø
--+ 20ø and B/A -- 0.156 ---+0.03 are the best
countingrate due to protonswith T • 0.31
values to represent the data. Thus, to within
Mev as measuredon Explorer 33 in the vicinity
of earth. The observations are divided into four the accuracythat the data warrant, the maxi-
mum flux comes from the direction of the sun
sectorspointing in the solar, antisolar,ecliptic
and the radial anisotropyis 16 --+ 3%.
north, and eclipticsouthdirections,respectively.
b. The anisotropyis often definedas
As can be seen from the figure, in all but one
of the four days of interest (239-243) the sun-
= ++
ward sectorhas the highestcountingrate. Be-
causeof inadequatestatistics,one cannotmake where J+, J- are the directionalintensitiesfrom
a statementabout the anisotropyon any given the solar and antisolar directions, respectively.
2230 GLEESON•KRIMIGIS, AND AXFORD
0.07

0.06

0.05

I I
I I
0.04 •1 k' SECTOR
3 r---'-I-1" -..-, ANTISUN
0.03 --4--'I -"'"•"-'i II SOUTH
-- , -- - ,,,
0.02
'-'-!•'c•-:-•.
• --.--IV
NORTH
...... •o-•••-• ....

0,03
i•1•
0
235 240 245 250
Day of Year 1967

Fig.2. Directional
dependence
of the integralfluxfor T _• 0.31Mev observed
on Ex'
plorer 33 at r -- 1.01 AU. The anisotropyduring the period August 27-31 (days 239-243)
in a plane approximatelyperpendicularto the ecliptic, is estimated to be 16% and radial.

Using this method, one finds that • --- 0.14 ñ THE MODEL
0.03 (i.e., 14 ñ 3%), the maximumflux coming A model that fits the observations well is
again from the direction of the sun. obtained when we take into account convection
One notes that with either method of com-
and energy-loss processesdue to the scattering
putation the anisotropy is approximately the of the particles by irregularitiesin the inter-
same. The errors quoted in b are maximum
planetarymagneticfields.The generalsteady-
errors, computedfrom the expression
state continuity and momentumequationsthat
a• at/ apply are

1 O V O•'
with a similar expressionfor method a. The r
• Or (•'s) + 3 aT Or
(•"v) = 0 (4)
anisotropyin the north-southdirectionis statis-
tically insignificant. S(•, •") = C(•, •) V V -- • OVlO• (5)
During this period, the mean integral inten- where .
sity for protons with T _• 0.31 Mev measured
on Explorer 33 at r -- 1.009 AU was 0.11 ñ
0.01 protons (cm' secster)-•, and that measured C(r,T) = I 3• U
• aT
o (,•'v) (6)
on Mariner 5 at r -- 0.84 AU was 0.2 _+ 0.01
protons (cm' sec ster)-• [Krimigis, 1970]. The is the Compton-Gettingfactor [Gleesonand
radial gradient of the mean integral intensity Axford,1968;Forman,1970a].Here a spheri-
cally symmetric model is assumedwith r the
(l/J) (OJ/Or) is then given approximatelyfrom
thesetwo measurements by heliocentricradius; U(r, T) is the differential
number density,S(r, T) is the differentialcur-
2 z(•, •') - •(•,, •') rent density,V(r) is the solarwindspeed,• is
[ -[ the diffusioncoefficient,.
and a = (T -I- 2Eo)/
(T + Eo), whereEo is the rest energy.For
(0.11 -- 0.20)
nonrelativisticparticles,a •_2, and in the sub-
0.085(0.11 -1- 0.20) sequent analysis we will assumeit to be con-
= - a•o 4- 80%/av (a) stant. However, we will usuallyleave it in the
 LOW-ENERGY COSMIC RAYs NEAR EARTH 2231
formulasas a in order to keep the approxima- where v is the particle speed [Gleeson and
tion in view. Axlord, 1967].
The Fokker-Planckequation,which governs If the mean integral intensity of the form
the number density, followson eliminatingS Jo(T/To) • is used in (1), the associateddif-
betweenequations4 and 5 [Parker, 1965; Glee- ferential numberdensityis
son and Ax•ord, 1967; Jokipii and Parker,
1967]: U(r, T) = Uo(T/To)
-('+s/•' (10)
and thus, from (6), the Compton-Gettingfactor
is
r Orr•VU-
C(r, T) = (4 -]- 2g)/3 (11)
1 o
(7) The differential anisotropy (9) now becomes

•(r, T) = (4 + 2•)V/v (12)

A simplified form of this model, which is --

useful in treating the present observations,is The integral anisotropy•(T) for kinetic energies
obtained by assumingthat the particles have •_ T is determinedin generalfrom
difficulty in diffusing through the irregular
magnetic field, and thus are convectively
transportedby the solarwind. This is equivalent
(13)
to assuming that KOU/Orcanbe neglected in (5),
which is the caseif K<< CV•, where• is a char- Since j(T) = vU/4•. and U(T) is given by
acteristic length defined by OU/Or-- U/•. (10), in the presentcasethe integralanisotropy
From the observedgradient (3) and the spectra corresponding to (12) is readily foundto be
(1), we find that • • 5 X 107km and C • 2;
hence,takingV -- 400 km/sec,we requirethat
• <( 4 X 10TM
cm2/secif the diffusionterm is to = (•_]_
«) -; -- (•_]_
«)•(T) (14)
be negligible.Formally, the approximate equa- In the presentcase,• = 0.9, so that C = 1.93
tions to be usedare obtainedby putting • - 0 and
in equations5 and 7.
Note that by neglectingdiffusioneffectsWe •(T) = 2.7 X 10-•V/T •/•= 0.64•(T) (15)
avoid someof the obviousshortcomingsof the where V is in kilometers per secondand T is
model describedby equations4-7, such as its in •llion electron volts. Taking V = 400 •
failure to take the interplanetary magnetic sec-• during this period (A. Lazarus, private
field properly into account. If the approximate- communication,1970), the differential anisot-
equations are locally valid (which requires ropy is found to range from 29% at T = 0.31
only that • ((CV•), they must be a very good Mev to 12% at T = 2 Mev. The integral
approximation to the actual situation if the solar anisotropyfor T = 0.31 Mev is 18.5%, which
wind is essentiallyradial. is in good agreementwith the observedvalue
In the followingsections,we examinethe pre- of 16 • 3%.
dictionsof the radial anisotropy and gradient If the exponentiMform of the mean integral
that follow from these approximateequations intensity given in (1) is chosen,the Compton-
and comparethem with the observations. Getting factor is givenby
The anisotropy. First we note that from
equation5 with • -- 0 = +
From (9) and (15), we find for the exponential
= vu (8) formthat
so that the radial •differential anisotropy is
givenby
li(T)= 2 q-\-•/ j (T,/2m),/:•
(17)
where m is the massof a proton. This function
T) - ss _ sc , _v ( has a minimum at T -- T• -- 0.66 Mev, where
 2232 GLEESON,KRIMIGIS, AND AXFORD
/j -- 14.3%if V -- 400km sec-x.The anisotropy If the power-law spectrumis adopted at
•(T) varies by only 5% of its value between Mariner 5, C is given by (11) and is inde-
T -- 0.5T• and T -- 2T•, so that we can expect pendentof energy,so that the gradients-ofthe
the integralanisotropyto be closeto 14%. Thus mean differential and integral intensitiesare
again we have good agreementwith the obser- equalandaregivenby (19).Thus,takingr --
vations,eventhoughthe energydependence of 0.9 AU and C ----1.93,we obtainG ------430%/
•(T) is quite different. AU, in reasonableagreementwith the observa-
It shouldbe noted that this comparisonhas tions.
been made betweenan anisotropyobservedat If the exponentialspectrumis adopted at
r -- 1.01 AU and that predictedat r -- 0.84 AU Mariner 5, then C(r, T) is not constantand
and hence is somewhat approximate. A more the gradient of the mean differentialintensity
precise analysis describedbriefly in the last at r -- 0.9 AU varies from --220%/AU at
section of this paper showsthe error involved T ----0.31 Mev to, for example,--370%/AU at
to be negligible. T -- 1.0 Mev. The radial gradient of the
The radial gradient. Using the model de- mean integral intensity (21) in this case be-
scribed earlier, it follows directly from the comesspecifically

•I OJ
0-•--;2I4
• q-•2(Ta/T•)e
-ya/y•--
e-•-•/•_e
•-i-•o/• .J

Fokker-Planck equation 7 with K -- 0 that Thus with r -- 0.9 AU and T• -- 0.31 Mev
the gradient of the mean differential density is the gradient is predictedto be --365%/AU,
which is in excellentagreementwith the ob-
lOU I o served value (3).
T) = 'u (r v) y) When the exponentialspectrum(1) is used,
C(r, T) increasesfrom 0.98 at T -- 0.31 Mev
(18)
to 2 at T -- 1.2 Mev; in contrast,the constant
With V -- constant,this becomes value1.93is obtainedfrom the power-lawspec-
trum. This appreciabledifferencearisesbecause
the expressionfor C(r, T) containsa second
(19) derivativeof the integralspectrumfunctionwith
respectto T and is sensitiveto the preciseform
This gradientis alsoequalto the gradientof the chosenfor the spectrumfunction.However,the
mean differentialintensity,since
integralanisotropies andintegralgradients pre-
dictedfrom the spectralforms (1) do not differ
lOj I OU substantially.We attribute this to the fact that
(20)
j Or U Or the predictions,which are comparedwith the
observations, are integral valueswhosederiva-
However,it is not usuallyequalto the gradient tion entailsreversingthe differentiation
process
of the mean integralintensity,which is usedto derive C(r, T) from the integralspec-
trum.

10J DISCUSSION OF THE OBSERVATIONS

AND MODEL
J Or
--
r •
In settingup a modelto explainthe gradients
and anisotropiesof low-energyprotons in
0.31 _< T _< 10 Mev, which have beenobserved
in the vicinity of earth, we have assumedthat
when the differential gradients are given by the particlesare soimpededin theirmotionby
(19) and particles are observed in T, _< irregularitiesin the interplanetarymagnetic
T<T•. field that they are effectivelyprevented from
 LOW-ENER(•Y CosMxc RAYs NEAR EARTH 2233

diffusing and thus are convectedalong with form of the spectralfunctions(1) at r -- 0.84
the solar wind. This assumptionis valid if AU is retainedat r -- 1.01 AU, exceptthat
]KaU/ar]• CVU; its consequences
havebeen
analyzed by putting K -- 0 in the equations T• -* (1.01/0.84)4/8T• -- 0.41 Mev (24)
governingthe differential number density and Thus the discussionin terms of a power-law
the differentialcurrent density.The goodagree- spectrum is unmodified and that in terms of
ment between the observed values of the radial
the exponentialspectrumis slightlymodifiedin
intensity gradientand the anisotropyand those that an anisotropyof 17% is predictedinstead
predictedfrom the integral intensity spectrum of 14%. Sincethis is to be comparedwith an
givessomesupportto the model,at leastin the observedvalue of 16%, the goodagreementis
range0.8 AU _• r _• 1.0 AU. unaltered.
With K -- 0, it is alsopossibleto integratethe In making a more precisecomparisonin the
equationsof the model, and this has been 'done case of the radial gradient of the mean inten-
in a companionpaper [Gleeson,1971], where sity, we note that the relationship23 enablesus
the implicationsare analyzed in detail. The to predict the integral spectrumat Explorer 33
solutionpredictsU and $ as functionsof (r, T); (r -- 1.01 AU) from that at Mariner 5 (r --
it alsoshowsthat the particlesloseenergydue to 0.84 AU) as shownin Figure 3. At Mariner 5,
adiabaticdeceleration as they are convectedout- the flux above T -- 0.31 Mev was observed to
ward, the kinetic energydecreasing by a factor be 0.2 proton/(cm" sec ster); at Explorer 33,
•2.5 betweenr -- 0.5 AU to earth and by a according to this model, the flux of particles
factor •500 if the particlesare convectedfrom with 0.31 Mev (Figure 3) is 0.115 proton/(cm •
the outer corona.However, it is unlikely that sec ster), in excellentagreementwith the 0.11
convectivetransport (i.e. K ((CV•) will prevail proton/(cm• sec ster) observed.Of course,the
all the way to the sun (for a further discussion corresponding averagegradient of --345%/AU
of this point, seeGleeson [1971]andAxford[1970]). is in agreementwith (3). In view of the error
In investigatingthe validity of the model,we bars in Figure 3, this agreementmay be for-
have usedonly relationshipsthat comedirectly
from the partial differentialequations4, 5, 6, 0.4
without actually solvingtheseequations.Thus
the validity establishedis local to the point of 0.3
observations.This procedureis instructive and MARINrER 5OBSERVATIONS
= 0.84 AU
has the advantage of simplicity, but can be
criticizedin the presentwork, first, becausethe 0.2

anisotropyobservedat r -- 1.01 AU was com-

pared with that predicted from a spectrum
observedat r -- 0.84 AU and, second,because
an average gradient betweentwo points (r -- 0.1
0.84 AU, 1.01 AU) was comparedwith a gra-
dient predicted from the spectrum at r -- 0.84 OBSERVAT . ON
0.07 EXPLORER 33
AU. The inaccuraciesshould,however,be ex-
pected to be small, with the validity of the
0.05
model still established.A more precise com- PREDICTED SPECTRUM

parisoncan be made by utilizing the solutionof ATEXPLORER 33, r = 1.01AU/ __

the equations4, 5, 6, in a form that predicts
J(r, T) in terms of J(a, T) under convective 0.03
0.1 '0.2 ' 0.4 0.6 1.0
conditionswith V -- constant [Gleeson,1971;
T (MeV)
Fisk and Ax•ord, 1969]'
Fig. 3. The integral intensity spectrum at the
•(•, •) -- (•a/•)•[a, (•/a)•Y] (•S) position of Explorer 33 (r -- 1.01 AU) predicted
from that observed on Mariner 5 at r -- 0.84 AU.
In realringthe comparison
of the predicted The integral intensity actually observed on Ex-
and' observedanisotropiesmore precise,it is plorer 33 for T •_ 0.31 Mev is shown for com-
sufficientto note that, accordingto (23), the parison.
2234 GLEESON,KRIMIGIS, AND AXFORD
tuitous, but it provides further evidence in T < 10 Mev [Forman, 1968, 1970b]; thus it
supportof the model proposed.By using (23), may play a major role in many aspectsof the
we can go directly to the observationsfor our propagation of low-energy cosmic rays. The
predictionsand avoid having to dependupon -main conclusions of this paper are that direct
representationof the data by Continuousfunc- accessof galacticcosmic-rayprotonswith T • 1
tions (1); clearly this techniqueis to be pre- Mev is not possible; and that the observed
ferred. particlesare comingfrom regionscloserto. the
It is, of course,not sufficientto showonly that sun. The observations are consistent with the
the predictions of the model are consistent solar origin of the particles,and this appearsto
with observations;in addition, we must consider be the most likely source.However, in view of
also the validity of our assumptionthat • • the energy changesnoted above, it is possible
CVP = 4 X 10•øcm• sec-1. Since we are concerned that the particles observedcould have pene-
with radial diffusionin a sphericallysymmetric trated near the sun with higher energy, lost
solar wind, we require that •11•- • cosec•x •( someenergythere, and are being convectedout-
8 X 10•ø cm• sec-1, where •11is the diffusion ward again.
coefficientparallel to the interplanetarymagnetic
field and x: is the 'gardenhose'angle (x • 45ø). Acknowledgments. We wish to thank Drs. J. A.
Jokipii and Coleman[1968] find, on the basis Van Allen and T. P. Armstrong for their efforts
of measurements of the power spectrum of in making the University of Iowa experiments on
Mariner 5 and Explorer 33 a success,and for
interplanetary magnetic field fluctuations, that helpful discussionsabout the data.
The experiment on Mariner 5 was supported by
t•x• 5 X 10•'l(•T•)
:!/•' cm•'sec-• (25) subcontract 950613 with the Jet Propulsion Lab-
oratory and the Office of Naval Research contract
whereT is in gevand• - v/c. Jokipiiand Cole- NOw 1509(06). The Explorer 33 experiment was
man emphasizethat the theory on which this supported by NASA under contract NAS 5-9076.
result is based begins to be questionablefor Data analysisat the Applied PhysicsLaboratory,
T <• 3 Mev. However, if we extrapolate the Johns Hopkins University, was supported in part
result to T • 0.3 Mev to obtain a roughestimate, by NASA under task I of contract NOw 62-0604-c.
Work at the University of California, San Diego,
we find that gll•" 7 X 1018cm• sec-•. Klimas and and Monash University was supported by NASA
Sandri [1970] have given an improved version of under grant NGR-05-0094)81.
the theory which suggeststhat this value should
be reducedby a factor of about 2.7. One cannot
Associate Editor W. R. Webber wishes to thank
make use of observations of solar energetic
J. J. Quenbyand E. C. Roelof for their assistance
particle events to obtain estimates for the in evalua.
ting this paper.
diffusion coefficient, since the usual diffusion
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