Page 1
THE MODEL DEPENDENCE OF SOLAR ENERGETIC PARTICLE MEAN FREE
PATHS UNDER WEAK SCATTERING
G. Qin,1M. Zhang,1J. R. Dwyer,1H. K. Rassoul,1and G. M. Mason2
Receivv ed 2004 May 24; accepted 2005 March 9
ABSTRACT
The mean free path is widely used to measure the level of solar energetic particles’ diffusive transport. We model a
solarenergeticparticle event observedbyWindSTEPat 0.31–0.62 MeVnucleon?1,bysolvingthefocused transport
equation using the Markov stochastic process theory. With different functions of the pitch angle diffusion coefficient
D??,weobtaindifferentparallelmeanfreepathsforthesameevent.Weshowthatthedifferentvaluesofthemeanfree
path are due to the highanisotropy of thesolar energetic particles. This makes it problematic to use just themean free
path to describe the strength of the solar energetic particle scattering, because the mean free path is only defined for a
nearly isotropic distribution. Instead, a more complete function of pitch angle diffusion coefficient is needed.
Subject headingg s: diffusion — interplanetary medium — solar wind
1. INTRODUCTION
Pitch-angle scattering by magnetic fluctuations is an essential
transport mechanism of solar energetic particles (SEPs). The
particles’ parallel mean free path can be derived from the pitch
anglediffusioncoefficient(thequasi-lineartheory[QLT],Jokipii
1966; Hasselmann & Wibberenz 1968; Earl 1974) for the parti-
cleswithanearlyisotropicpitch-angledistribution.Parker(1965)
obtained a diffusion equation of particle transport for applica-
tions such as the study of cosmic-ray modulation. However,
considering SEPs, the focused transport equation is needed be-
cause the particles experience strong magnetic focusing near
the sun (Parker 1963; Roelof 1969; Ng & Reames 1994). To
describe the SEPs’ scattering by the magnetic fluctuations, the
focused transport equation can be solved to model the observed
intensityandanisotropyprofiles.Sinceitisverydifficulttosolve
the equation analytically, numerical solutions have been pro-
vided (e.g., Ng & Wong 1979; Schlu ¨ter 1985; Ruffolo 1991;
Kocharov et al. 1998). Kocharov et al. (1998) found different in-
tensity profiles for isotropic and anisotropic pitch-angle scatter-
ing under a nominally similar mean free path. Note that previous
authors have not studied how the different pitch angle diffusion
models used in their numerical methods affect the simulation re-
sults by modeling observations.
However, it is known that under some conditions the SEPs
have high anisotropy during the initial phase of a solar event
(e.g., Schulze et al. 1977; Mason et al. 1989; Torsti et al. 2004).
From the numerical calculations of the focused transport equa-
tion, Mason et al. (1989) showed that the particles’ anisotropy
increases dramatically as kkincreases. They got large values of
the parallel mean free paths (k1 AU) for some impulsive SEP
eventsbyfittingthenumericalsolutionsofthetransportequation
tothespacecraftobservations.Torstietal.(2004)reportedthatin
some events when SEPs propagate under weak scattering con-
ditions, very large parallel mean free paths, kk? 10 AU, are
obtained from fitstothese events.With test-particleorbit-tracing
simulations and theoretical calculations, Qin et al. (2004) showed
that the strong magnetic focusing effect can make all the pitch-
angle distribution of SEPs become a beamlike distribution im-
mediately after they are released near the Sun.
In this article we model an impulsive SEP event observed by
Wind STEP (von Rosenvinge et al. 1995) with a focused trans-
port equation, which is solved by a method of Markov stochas-
tic process simulation (Zhang 1999, 2000). We obtain mean free
paths by fitting numerical results to the spacecraft data. The
STEP instrument is a time-of-flight mass spectrometer, capable
of measuring ions from several tens of keV nucleon?1up to sev-
eral MeV nucleon?1. The data are measured in eight azimuthal
bins. In order to calculate pitch angles, Wind MFI magnetic field
data (Lepping et al. 1995) are used. For the 0.5 MeV nucleon?1
SEP data we study we can compute the Compton-Getting cor-
rections (Dwyer et al. 1997) for the event-averaged pitch angle
distribution function using Wind SWE (Ogilvie et al.1995) so-
lar wind velocity data, but we have difficulty computing the
Compton-Getting corrections for the anisotropy time profile be-
cause of the low time resolution of the data. Furthermore, be-
cause the particle speed is much greater than the solar wind
speed, the Compton-Getting corrections we get are smaller than
the data’s error bar, so in this work we ignore the Compton-
Getting corrections. We concentrate on the investigation of how
the pitch angle diffusion models can affect the determination of
the mean free path. We obtain significantly different mean free
paths from the different models of the pitch angle diffusion co-
efficients.Furthermore,wediscussthevalidityofusingthemean
free path to describe the transport of SEP events for which their
anisotropy is high. Note that we get similar results for different
species and energy, but as an example, in this paper we only
show the results from 0.31–0.62 MeV nucleon?1He.
2. APPROACH AND METHODS
We model the transport of the Wind STEP energetic solar
particle events (von Rosenvinge et al. 1995) with the numerical
simulations using the Markov stochastic process theory (Zhang
1999,2000).Inourmodel,thefocusedtransportequationinsolar
wind frame can be written as (Roelof 1969)
@f
@tþ ?v@f
@z?v(1 ? ?2)
2B
@B
@z
@f
@??@
@?
D??@f
@?
??
¼ q(z;?;t);
ð1Þ
1Departmentof PhysicsandSpaceSciences,FloridaInstituteof Technology,
150 West University Boulevard, Melbourne, FL 32901; gqin@fit.edu, mzhang@
pss.fit.edu, jdwyer@fit.edu, rassoul@pss.fit.edu.
2Department of Physics, University of Maryland, College Park, MD 20742;
gmmason@umd.edu.
562
The Astrophysical Journal, 627:562–566, 2005 July 1
# 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.
Page 2
where B is the averaged background interplanetary magnetic
field from the standard Parker spiral (Parker 1958) with the in-
terplanetary magnetic field (IMF) path length to Earth about
1.15 AU for Vsw¼ 400 km s?1, and z is the coordinate along the
magnetic field spiral. In this work we ignore the particles’ per-
pendicular diffusion and adiabatic cooling effects to simplify the
condition so that we can concentrate on the effects of pitch an-
glediffusion models. Inaddition,wechoose a low-energy,short,
impulsive event in a case with a large mean free path to help
illustrate the effect of model dependence, and since the duration
time is short, the adiabatic cooling effects are also smaller. Large
particle fluxes in SEP events may generate IMF turbulences so
that weak scattering only occurs in small events (Mason et al.
1989). For the complete formula, we should also include these
perpendicular diffusion and adiabatic cooling effects (Skilling
1971; Qin et al. 2004). In another paper being prepared we will
study the effect of the adiabatic cooling.
If particles are in the isotropic pitch-angle distribution, the par-
allelmeanfreepathkkcanbewrittenas(Jokipii1966;Hasselmann
& Wibberenz 1968,1970; Earl 1974)
kk¼3v
8
Zþ1
?1
d?
1 ? ?2
D??
ðÞ2
;
ð2Þ
or the radial mean free path can be defined as
kr? kkcos2 ;
ð3Þ
where is the angle between radial distance and interplanetary
magnetic field.
In order to evaluate equations (1) or (2), the knowledge of the
pitch angle diffusion coefficient as a function of ?, D??(?), is
essential. In the QLT (Jokipii 1966) D??can be written as
D??¼1 ? ?2
? j jv
Z2e2Vsw
?2m2
0c2Pxx
Vsw
2??v
??
;
ð4Þ
wherePxx(Vsw/2??v)istheobservedpowerspectrum.Ininertial
range
Pxx/ ?q;
ð5Þ
where q > 1 is the slope of the power spectrum. We can further
assume particles have constant radial mean free path, kr, prop-
agating through the heliosphere (Bieber et al. 1994). But other
theories suggest different forms of the pitch angle diffusion co-
efficient.HerewefollowBeeck&Wibberenz(1986)touseanad
hoc model of pitch angle diffusion coefficients
Dr
??? D??=cos2 ¼ D0
? j jq?1þ h
??
1 ? ?2
??;
ð6Þ
where D0is a constant indicating the magnetic field fluctuation
level. The constant h is introduced to help particles transport
through ? ¼ 0 in simulation. A number of theories have been
discussed to address particle scattering through ? ¼ 0 in QLT,
e.g., a higher order of corrections to QLTregarding the dissipa-
tion range of magnetic fluctuation spectra (Coroniti et al. 1982;
Denskatetal.1983),wavepropagationeffects(Schlickeiser1988),
and the effects of dynamical turbulence (Bieber & Matthaeus
1991)andthermalwavedamping(Achatzetal.1993;Schlickeiser
& Achatz 1993). If nonlinear effects are considered, the transport
through ? ¼ 0 is governed by the level of magnetic fluctuations,
i.e., dB/B (e.g., Goldstein 1976; Jones et al. 1978; Kaiser et al.
1978;Qin2002).Recently,Dro ¨ge(2003)developedadynamical
QLT model, which has been successfully used to reproduce
details of observed particle events, implies resonance broaden-
ingeffectsthatcanbe related tosolarwindparameters (magnetic
fieldstrength,density,andtemperature).Theparameterhcouldbe
usedas anapproximation ofthe above modelsand also berelated
to solar wind observables. If the magnetic spectrum index q ¼ 1,
it is reduced to the isotropic scattering model. For any given
model with certain q and h, the level of the diffusion can be ad-
justed by changing the value of D0, and thus we can get the best
fit of numerical simulations of the transport equation (1) to the
observational data. With the value of D0for the best fit we go
furthertocalculatethemeanfreepathfromequations(2)and(3).
This mean free path is called the best-fit mean free path of the
observational data.
In this work we use different models of the pitch angle dif-
fusioncoefficienttofitanimpulsiveSEPeventobservedon1998
May 16 by the Wind STEP instrument. At first, we fix the con-
stant q ¼ 5/3 as the Kolmogorov spectrum and vary h to obtain
models with different scattering levels at ? ¼ 0. The two models
weconsiderarewithh ¼ 0:2and0.01.InFigure1thedottedand
dashed lines of the top panel shows Dr
models with h ¼ 0:2 and 0.01, respectively. From the figure we
canseethatinbothmodelsDr
??reachesitsmaximumvalueatthe
values of ? around ?0.5, which is similar to the QLT model
(h ¼ 0). But in contrast to the QLT models, in these two models
Dr
?? is not zero at ? ¼ 0 and particles can be scattered through
90?. Note that with the h ¼ 0:2 model, Dr
value around ? ¼ 0. In x 4 we also show the model dependence
of radial mean free path with a series of simulations with dif-
ferent models (h, q).
??versus ? from the
??has a much larger
3. NUMERICAL METHODS AND RESULTS
In order to solve the focused transport equation (1) we use the
Markov stochastic process theory (Zhang 1999, 2000). If we in-
troduce a virtual distribution function g(?;z;t) ? A(z)f (?;z;t),
Fig. 1.—Comparison between two models with h ¼ 0:2 (heavy dotted line)
and h ¼ 0:01 (heavy dashed line) in arbitrary units. Top: D??vs. ?. Bottom:
g(?) ? (1 ? ?2)2/D??vs. ?. The two horizontal lines are the averaged value of
(1/2)R1
?1g(?)d? for h ¼ 0:2 (dotted line) and h ¼ 0:01 (dashed line) models.
MODEL DEPENDENCE OF SEP MEAN FREE PATHS
563
Page 3
where A(z) / 1/B(z) is the magnetic flux tube area, the focused
transport equation (1) could be written as
@g
@t¼ ?@
@z(?vg) ?@
þ@2
@?
?þ q0(z;?;t);
v
2L
1 ? ?2
??þ@D??
@?
??
g
@?2D??g
?
ð7Þ
where the virtual source q0¼ Aq and the focusing length L is
defined by
1
L¼ ?1
B
dB
dz¼1
A
dA
dz:
ð8Þ
We write the equation in this form because we want to simulate
the stochastic particle transport in a time-forward manner. The
corresponding time-forward Ito stochastic differential equa-
tions may be written as (see Gardiner 1983; Zhang 2000)
dz(t) ¼ ?vdt;
d?(t) ¼
ð9Þ
ð10Þ
ffiffiffiffiffiffiffiffiffiffiffi
2D??
p
dW?(t) þ
v
2L
1 ? ?2
??þ@D??
@?
??
dt;
where W?(t) is a Wiener process (e.g., Gardiner 1983). The
probability density in the random variable space of fz;?g is
proportional to the virtual particle distribution function g, which
can be used to further obtain the particle distribution function f
with f ¼ g/A.
With the two pitch angle diffusion models (h ¼ 0:2 and 0.01)
with q ¼ 5/3, in x 2 we made the numerical calculation to solve
the focused transport equation (1) using the above method. A
particle source, which has a rectangular type of release time dis-
tributionfunctionwith?t ¼ 0:03daysandisotropicinitialpitch-
angle distribution, is placed at r0¼ 0:05 AU and approximately
at 2:30 UTon 1998 May 16. For the short, impulsive, SEP event
westudy,itisappropriatetoassumeanearlypointsource,which
can be confirmed by fitting the anisotropy profiles. The release
times match the flare in the GOES-8 solar X-ray flux profiles
whose peak was at 2:00 UT on 1998 May 16.
As a comparison between spacecraft data and simulations
with different models, Figure 2 shows time profiles of intensity
and anisotropy. Note the anisotropy is in the spacecraft frame.
Thediamonds(toppanel)andsolidline(bottompanel)arefrom
the Wind STEP 15 minute average data of 0.31–0.62 MeV nu-
cleon?1He. The dotted and dashed lines are from simulations
with kr¼ 1:2 AU, h ¼ 0:2 and kr¼ 2:2 AU, h ¼ 0:01, respec-
tively. The vertical lines indicate the slightly different releasing
times, 2:29:45.6 and 2:39:50.4 UT on 1998 May 16, for simu-
lations with kr¼ 1:2 and 2.2 AU, respectively.
The top panel of Figure 2 shows particle intensity versus time.
Note that in the intensity profile of the spacecraft data there are
short-timescale (3 hr) variations; we assume the ‘‘dropouts’’ are
caused by the convection of magnetic flux tubes (Mazur et al.
2000). Here we discarded the three biggest dropouts from the
spacecraft data ( filled diamonds) so that we fit intensity data ex-
cluding them. In the top panel for different diffusion models
we show the best-fit simulations, kr¼ 1:2 AU for h ¼ 0:2 and
kr¼ 2:2AUforh ¼ 0:01.Wehavesimulations(notshown)with
small alternations, e.g., ?0.2 AU, that produce unacceptable fits.
The bottom panel of Figure 2 shows the particles’ anisotropy
measuredinspacecraftframeversustime.Thedottedanddashed
lines are from the simulations of best fit in the top panel with
model parameter h ¼ 0:2 and 2.2 AU, respectively. We can see
before 11 hr (UT) when the particles’ intensity is not too low,
both of the profiles of anisotropy of best-fit simulations fit the
data well.
Fig. 2.—Comparison between Wind STEP 15 minute average data of 0.31–
0.62 MeV nucleon?1He (top: diamonds, bottom: solid line) and simulations
(dotted and dashed lines with model parameter h ¼ 0:2 and 0.01, respectively).
Top: Particle intensity vs. time. Note that the three dropouts are shown as filled
diamonds. Bottom: Particle first-order anisotropy measured in spacecraft frame
vs. time.
Fig. 3.—Particlecumulativepitchangledistributionfunctionatthepositionof
the satellite for 0.31–0.62 MeV nucleon?1He during the period 6:00–15:36 UT
on 1998 May 16. The solid line is from the Wind STEP observations with eight
sectors, and the dotted and dashed lines are from the simulations.
QIN ET AL.564Vol. 627
Page 4
From the two panels of Figure 2, for h ¼ 0:2 and 0.01 we
obtain mean free paths of kr¼ 1:2 and 2.2 AU, respectively.
Figure 3 shows the event-averaged particles’ pitch angle dis-
tribution function from spacecraft observation and simulations
detected at the position of the satellite for the SEP event shown
in Figure 2 during the period 6:00–15:36 UT on 1998 May 16.
We can see that for both of the best-fit simulations with different
pitch angle diffusion coefficient models, kr¼ 1:2 AU with h ¼
0:2andkr¼ 2:2AUwithh ¼ 0:01,theparticleshaveverylarge
anisotropy.Bothofthesimulation resultsoftheanisotropyagree
withspacecraftdata,andthedifferencebetweentheresultsofthe
two simulations is not large enough to decide which model of
pitch-angle diffusion to be used to compare to the observational
data.
4. DISCUSSION
Mean free paths have been widely used to measure the SEPs’
pitch-angle diffusion. However, we get different values of the
mean free path from the different models of the pitch angle dif-
fusion coefficient D??. In fact, for the definition of equation (2)
to be valid, it is required that the particles are nearly isotropic in
thepitch-angledistribution,butfromFigure3wecanseethatthe
results from the Wind STEP observation and simulations show
the particles are not at all in an isotropic distribution. Actually,
for the particles in the anisotropic distribution, the effects of D??
ontheparticletransportequation(2)isweightedbytheparticles’
pitchangledistributionfunction,butinthedefinitionofthemean
freepathinequation(2)thecontributionofdifferentpitchangles
is weighted equally.
We can rewrite the definition of the radial mean free path as
kr¼ 3v=8
ðÞ
Z1
?1
g(?)d?;
ð11Þ
where g(?) ¼ (1 ? ?2)2cos2 /D??¼ (1 ? ?2)2/Dr
tom panel of Figure 1 shows g(?) versus ? from the two best-fit
simulations kr¼ 1:2 AU with h ¼ 0:2 and kr¼ 2:2 AU with
h ¼ 0:01. The two thin horizontal lines indicate the results of
(1/2)R1
kr,andthedifferencebetweentheradialmeanfreepathsfromthe
two simulations are mainly from the difference of the integration
of g(?) in this range of ?. In the simulation with model h ¼ 0:2,
particlesscattermoreefficientlythrough? ¼ 0andshouldreturn
from the antisunward direction easier. By comparing the pitch
angle distribution function of models h ¼ 0:2 and 0.01 (Fig. 3)
we can see that there are more particles with ? < 0 for the sim-
ulation with model h ¼ 0:2. However, for both of the two
models, because the particles are highly anisotropic and more
likely to be nearly parallel to the averaged magnetic field, the
amountofparticleswith? < 0isonlyaverysmallpercentageof
the total particles and the time profile of the particles’ intensity
and anisotropy are mainly determined by the particles nearly
paralleltothemagneticfield.Sincethestrengthofdiffusivescat-
tering of the particles, described by D??¼ Dr
values of ? contributes more significantly to the intensity and
anisotropy time profile, generally, for each model, we can adjust
the constant D0to make D??in proper value in a higher range of
? to simultaneously fit simulations to the spacecraft data of the
timeprofileoffluxandanisotropy.Therefore,inhigh-anisotropy
conditions particle simulations have different best-fit mean free
path krdescribed by equation (11).
??. The bot-
?1g(?)d?forthetwomodels.Fromthefigurewecansee
thatwhen j?jT1,g(?) contributessignificantly totheresultsof
??cos2 , at higher
Furthermore, under the condition qT2 or h30, with ? 3
0, and from the definition of the model of pitch angle diffusion
coefficients, equation (4) can be written as
@D??=@?
D??
¼(q ? 1)?q?21 ? ?2
ðÞ ? 2 ?q?1þ h
Þ 1 ? ?2
ðÞ?
?q?1þ h
ððÞ
? ?
2?
1 ? ?2:
ð12Þ
This means that for higher ranges of ?, D??has the same value
for different models fitting the same spacecraft data. Thus, we
assumetheadhocmodelofthepitchanglediffusioncoefficient,
equation (4), is
D??¼ D0
0D00
0
? j jq?1þh
??
1 ? ?2
??;
ð13Þ
where D00
models ?030, so D??has the same value in a higher ? range.
We can insert equation (13) into equations (2) and (3) to get a
theoretical radial mean free path of each model with only one
model-independentfreeparameter,D0
from the fitting results of one simulation to the spacecraft data
with any model.
Inadditiontothetwosimulationswithdifferentmodelsshown
above, we obtain a series of radial mean free paths for different
models(h,q)byfittingsimulationstothespacecraftdata.Figure4
shows the model dependence of the radial mean free path kr. The
toppanelshowskrversush,andthebottompanelshowskrversus
q. Diamondsand asterisks are fittings of simulations tospacecraft
data. As a reference, theoretical calculation of radial mean free
paths with the assumption of equation (13) is also shown as solid
and dotted lines for different models. From the figure we can see
the fitting results agree with the theoretical calculation very well,
except as q ! 2 in the bottom panel when the conditional equa-
tion (12) is invalid. This result demonstrates that under the
conditional equation (12) in impulsive SEP events with high
anisotropy we only need to fit simulation for one model to the
0¼ 1/½(?q?1
0
þ h)(1 ? ?2
0)? with a constant for different
0,whichcanbedetermined
Fig. 4.—Model dependence of radial mean free path kr. Top: krvs. h. Bot-
tom: kr vs. q. Solid and dotted lines are theoretical model calculations for
different parameters. Diamonds and asterisks are fittings of simulations to
spacecraft data for different parameters.
MODEL DEPENDENCE OF SEP MEAN FREE PATHS
565No. 1, 2005
Page 5
spacecraft data and we can obtain radial mean free paths for any
model by simple theoretical calculations.
FromthetoppanelofFigure4wecanseethatwhenwefixthe
parameter q, the fitted krdecreases from its maximum value to
the minimum one as h increases from the limit h ! 0 to hk1.
On the other hand, the bottom panel shows that when q ¼ 1, all
themodelswithdifferentharereducedtotheisotropicscattering
one with the minimum value of fitted mean free path kr. Fur-
thermore, krincreases from its minimum to maximum as q in-
creases from q ¼ 1 to qk2. In addition, from the figure we can
see that the model dependence of q is more significant if h is
smaller,unlessqk2,andthatthemodeldependenceofhismore
significant if q is larger.
From the above discussions we conclude that for highly an-
isotropicparticlesthevalueofthepitchanglediffusioncoefficient
D??in the higher range of ? contributes more to the particles’
intensity and anisotropy time profile, which can be obtained from
the numerical solution of the focused transport equation. There-
fore, the mean free path obtained from equations (2) or (3) is not
an adequate physical quantity to describe the transport of SEPs.
Furthermore, since in such conditions the values of the mean free
path obtained from the best fit of the numerical simulations are
different with different pitch angle diffusion coefficients models,
we cannot even use the parallel mean free path kkor krdefined
with equations (2) or (3), respectively, to indicate the strength of
the particles’scattering by the magnetic fluctuations. Instead, we
have to use the full definition of the pitch angle diffusion coeffi-
cient D??. However, it is a challenge to determine D?? from
theoretical study or observation.
For example, there are puzzles of some SEP events with ab-
normallylargemeanfreepathscomparedtotheQLTresultsfrom
magnetic field power spectra (e.g., Tan & Mason 1993). These
might, in part, relate to the difficulty of the lack of knowledge of
D??for the large anisotropic case. Generally, we cannot easily
get the accurate D??function to numerically solve the focused
transport equation. Therefore, in high-anisotropy cases, if we
practically still report the value of the large mean free path as a
crude measure of the relative scattering conditions between dif-
ferent SEP events or different species, at least the model of dif-
fusion coefficients used should also be shown.
The authors benefited from the Wind SWE data provided by
K.W.Ogilvie,R.B.Torbert,andA.J.LazarusandtheWindMFI
data provided by R. P. Lepping. This work was supported partly
by NASA grants NAG5-11036, NAG5-13514, NAG5-11921,
NSF SHINE grant 0203252, and JPL contract 1240373.
REFERENCES
Achatz, U., Dro ¨ge, W., Schlickeiser, R., & Wibberenz, G. 1993, J. Geophys.
Res., 98, 13261
Beeck, J., & Wibberenz, G. 1986, ApJ, 311, 437
Bieber, J. W., & Matthaeus, W. H. 1991, Proc. 22nd Int. Cosmic Ray Conf.,
Vol. 3, ed. M. Cawley et al. (Dublin: Dublin Inst. Adv. Stud.), 248
Bieber, J. W., Matthaeus, W. H., Smith, C. W., Wanner, W., Kallenrode, M.-B.,
& Wibberenz, G. 1994, ApJ, 420, 294
Coroniti, F. V., Kennel, C. F., Scarf, F. L., & Smith, E. J. 1982, J. Geophys.
Res., 87, 6029
Denskat, K. U., Beinroth, H. J., & Neubauer, F. M. 1983, J. Geophys., 54, 60
Dro ¨ge, W. 2003, ApJ, 589, 1027
Dwyer, J. R., Mason, G. M., Mazur, J. E., Jokipii, J. R., von Rosenvinge, T. T.,
& Lepping, R. P. 1997, ApJ, 490, L115
Earl, J. A. 1974, ApJ, 193, 231
Gardiner, C. W. 1983, Handbook of Stochastic Methods for Physics, Chem-
istry, and the Natural Sciences (Berlin: Springer)
Goldstein, M. L. 1976, ApJ, 204, 900
Hasselmann, K., & Wibberenz, G. 1968, Z. Geophys., 34, 353
———. 1970, ApJ, 162, 1049
Jokipii, J. R. 1966, ApJ, 146, 480
Jones, F., Birmingham, T. J., & Kaiser, T. B. 1978, Phys. Fluids, 21, 347
Kaiser, T. B., Birmingham, T. J., & Jones, F. 1978, Phys. Fluids, 21, 361
Kocharov,L.,Vainio,R.,Kovaltsov,G.A.,&Torsti,J.1998,Sol.Phys.,182,195
Lepping, R. P., et al. 1995, Space Sci. Rev., 71, 207
Mason, G. M., Ng, C. K., Klecker, B., & Green, G. 1989, ApJ, 339, 529
Mazur, J. E., Mason, G. M., Dwyer, J. R., Giacalone, J., Jokipii, J. R., & Stone,
E. C. 2000, ApJ, 532, L79
Ng, C. K., & Reames, C. K. 1994, ApJ, 424, 1032
Ng, C. K., & Wong, K. Y. 1979, Proc. 16th Internat. Cosmic Ray Conf., Vol. 5,
ed. S. Miyake (Tokyo: Univ. Tokyo Press), 252
Ogilvie, K. W., et al. 1995, Space Sci. Rev., 71, 55
Parker, E. N. 1958, ApJ, 128, 664
———. 1963, Interplanetary Dynamical Processes (New York: Interscience)
———. 1965, Planet. Space Sci., 13, 9
Qin, G. 2002, Ph.D. thesis, Univ. Delaware
Qin, G., Zhang, M., Dwyer, J. R., & Rassoul, H. K. 2004, ApJ, 609, 1076
Roelof, E. C. 1969, in Lectures in High Energy Astrophysics, ed. H. O¨gelman
& J. R. Wayland (NASA SP-199; Washington: NASA), 111
Ruffolo, D. 1991, ApJ, 382, 688
Schlickeiser, R. 1988, J. Geophys. Res., 93, 2725
Schlickeiser, R., & Achatz, U. 1993, J. Plasma. Phys., 49, 63
Schlu ¨ter, W. 1985, Ph.D. thesis, Univ. Kiel
Schulze, B. M., Richter, A. K., & Wibberenz, G. 1977, Solar Phys., 54, 207
Skilling, J. 1971, ApJ, 170, 265
Tan, L. C., & Mason, G. M. 1993, ApJ, 409, L29
Torsti, J., Riihonen, E., & Kocharov, L. 2004, ApJ, 600, L83
von Rosenvinge, T. T., et al. 1995, Space Sci. Rev., 71, 155
Zhang, M. 1999, ApJ, 513, 409
———. 2000, ApJ, 541, 428
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