Researchers discuss the radial dependence of the peak flux and the fluence of solar flare produced energetic particles under the assumption that they propagate diffusively in the heliosphere

Hamilton, Douglas C.

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THE RADIAL DEPENDENCE OF THE SOLAR ENERGETIC PARTICLE FLUX 
D. C. HAMILTON, Dept. of Physics and Astronomy/Institute for 
Physical Science and Technology, University of Maryland, 
College Park, MD 20742 USA 
ABSTRACT 
We discuss the radial dependence of the peak flux and the fluence 
of solar flare produced energetic particles under the assumption that 
they propagate diffusively in the heliosphere. 
MODEL 
There is considerable evidence that in many cases the propagation 
of solar energetic particles can be described by Parker's spherically 
symmetric transport equation which includes the effects of diffusion, 
convection, and adiabatic energy loss in the expanding solar wind 
(Parker, 1965) . 
where U(r,T,t) - differential number density 
V - solar wind speed 
K - radial diffusion coefficient 
T = particle kinetic energy 
a - (T + 2m C')/(T + m c2> 
r 
The omnidirectional flux is then 
j - VU/4R 
where v - particle speed. 
Equation (1) has generally been solved under the assumptions that 
b K = K r  
r 0 
where K - diffusion coefficient at 1 AU and 
0 
r - radial distance in AU. 
( 3 )  
When the differential number density is expressed as a power law 
in kinetic energy, U - U T-',the explicit energy dependence of (1) can 
be eliminated. Even with these assumptions, analytic solutions to (1) 
have been found only for special cases of the radial dependence of K : r 
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https://ntrs.nasa.gov/search.jsp?R=19890019090 2020-03-20T02:00:31+00:00Z
b - 1 (Fisk and Axford, 1968) and b - 0 (Lupton and Stone, 1973). 
Therefore (1) is usually solved numerically (Webb and Quenby, 1973; Ng 
and Gleeson, 1975; Hamiiton, 1977; Zwickl and Webber, 
of numerical solutions to (1) are shown in Fig. 1. 
IO', , , , I I I I I I I 
10. - 
t 10)- 
- - 
102-  
c 
IO' - 
1977)-. Examples 
V.0 103 hf.1 
Fig. 1. Intensity-time profiles at (a) 1 AU and (b)z 5 AU from a 
numerical solution of (1) for K -1.5 x loz1 cm /s and b-0.5, 
values typical for 10 MeV protons. The three cumes 
correspond to diffusion only ( b o ) ,  diffusion + convection 
(V400 h / s ,  7=l.O), and diffusion + convection + adiabatic 
deceleration (V-400 km/s, 7-3.5). (From Hamilton, 1981). 
If the effects of convection and energy loss are removed from (1) 
(by setting V - 0, for example), then a pure diffusion equation 
results, and a solution has been given by Parker (1963) for any value 
of the radial index b. For the case of 3-dimensional space, 
87 
3 
2-b - 
1 r exp ( - u(r,t> a ( - )2-b 
0 (2-b)2K t 
1 
K t  (5) 
Of interest are the radial dependences of the time to maximum 
intensity, the maximum flux, and the fluence. 
2-b r 
tmax - 3K (2-b) 
- 3  a r  j m a x  
- (b+l) F a r  
(independent of K or b) (7) 
(f hence) 
These analytic solutions to the pure diffusion equation are 
useful for comparison with numerical solutions to the complete 
transport equation, but generally they do not agree well with 
observations of solar energetic particles at moderate energies (< 100 
MeV). At these energies the effects of convection and energy loss 
become important. Including these additional terms reduces tmax and. 
produces a more rapid decrease of j and F with increasing r. max 
OBSERVATIONS 
Observations of solar energetic particles beyond 1 AU have been 
made with the Pioneer 10/11 and Voyager 1/2 spacecraft. Observations 
inside 1 AU have been reported from Helios 1/2. We review here the 
two studies which used simultaneous observations at two or more radial 
distances to deduce radial propagation parameters. 
The power law index b of the radial diffusion coefficient has 
been determined for several events. Hamilton (1977) analyzed 11-67 
MeV protons in solar particle events covering the radial range 1-6 AU. 
His values for b ranged from 0.3 to 0.5 with the trend towards the 
smaller value at larger radial distances. Beeck et al. (1987) studied 
protons and heavier ions over a somewhat lower energy range (0 .4 -27  
MeV/nuc) for two particle events covering the radial range 0.65-1.9 
AU. Their value for b ranged from 0.5 to 0.7. In both of these 
at two or three radial studies, b was deduced by fitting t 
distances. 
To summarize these results, a value of b - 0.5 f 0.2 covers all 
six events studied, with a trend to vary from b - 0.7 near 1 AU to 
b = 0.3 near 5 AU. Other observations at larger radial distances 
max 
88  
indicate a further reduction to b fs  0 or somewhat less by 10 AU 
(Webber and Goeman, 1979). This variation of b with r of course 
implies that K is not really a simple power law in r. Nevertheless, 
it is a useful approximation over limited radial ranges. 
1.5 for b - 0.5 in the pure diffusion 
approximation, and this is close to what is observed. The radial 
dependence of t is only slightly weaker than this (Hamilton, 1977). 
The solution to Equation (7) predicts an r-3 dependence for j 
(l), on the other hand, gives an re3.' dfzfndence for parameters 
typical for 30-67 MeV protons and r-3*3 to r for 11-20 MeV protons 
depending on the spectral index 7. The more rapid decrease of 
with r results largely from energy loss and thus is more important jmax 
at lower energies and for larger values of 7. 
Hamilton(l977) also studied two events for which the maximum flux 
fell off more rapidly with radial distance (r-3'8 to r-'.'). These 
events were observed in solar wind rarefaction regions in w h p h  the 
flux tube cross section increases more rapidly with r than a r as is 
appropriate for 3-dimensional isotropic space. Parker (1963) has 
shown this more rapid flux tube expansion results in a more rapid 
decrease of jmax with increasing r. 
To my knowledge, io observations of the radial dependence of the 
particle fluence have been reported. To make an estimate we may be 
guided once again by the pure diffusion approximation. Equation (8) 
would then suggest a r-lm5 dependence for b - 0.5. Including effects 
of convection and energy Joss causes a more rapid decrease. A 
resonable estimate is F a r- 
r 
From (6), we expect tmax a r 
max 
max . 
to r-2*5. 
APPLICABILITY OF RESULTS 
The model discussed above applies to a situation of spherical 
symmetry with istropic diffusion. Ng and Gleeson (1971) have shown, 
however, that the model (Eq. 1) applies anywhere within the flux tube 
connected to the flare site even though diffusion parallel to the 
interplanetary magnetic field is much more rapid than that 
perpendicular to it. 
The particle events selected for the two studies cited above were 
some of the very few observed when two or more spacecraft at different 
radial distances have simultaneously been well-connected to a flare 
site. In most events, the rapid particle intensity decrease away from 
the best connected field line causes large flux differences at 
separated spacecraft in addition to any radial dependence. Thus the 
radial dependences in j and F cited above will rarely be observed 
in individual events except for fortuitously located spacecraft. On 
the other hand, these predictions may be useful on a statistical basis 
in extrapolating from the large data base collected at 1 AU. 
Finally, we note that there are particle events in which there is 
very little interplanetary scattering, particularly inside of 1 AU 
max 
8 9  
(e.g. Bieber et al., 1980). These "scatter free" events are not 
described by (1). However for most of the events in the 1-100 
MeV energy range, (1) generally appears to be a good approximation. 
This work was supported in part by NASA under subcontract JHU 
601620 between The Johns Hopkins University and the University of 
Maryland. 
REFERENCES 
Beeck, J . ,  G.M. Mason, D.C. Hamilton, G. Wibberenz, H. Kunow, D. 
Hovestadt, and B. Klecker 1987. A multispacecraft study of the 
injection and transport of solar energetic particles. 
Astrophys. J .  322:1052. 
Wibberenz 1980. Interplanetary pitch angle scattering and 
coronal transport of solar energetic particles: new information 
from Helios. J. Geophys. Res. 85:2313. 
F i s k ,  L.A., and W.I. Axford 1968. Effects of energy changes on solar 
cosmic rays. J. Geophys. Res. 73:4396. 
Hamilton, D.C. 1977. The radial transport of energetic solar flare 
particles from 1 to 6 AU. J. Geophys. Res. 82:2157. 
Hamilton, D.C. 1981. Dynamics of solar cosmic ray events: processes 
at large heliocentric distances (+1 AU). Adv. Space Res. 1:25. 
Lupton, J.E., and E.C. Stone 1973. Solar flare particle propagation: 
Comparison of a new analytic solution with spacecraft 
measurements. J. Geophys. Res. 78:1007. 
Ng, C.K., and L . J .  Gleeson 1971. The propagation of solar cosmic ray 
bursts. Solar Phys. 20:166. 
Ng, C.K., and L.J. Gleeson 1975. Propagation of solar-flare cosmic 
rays along corotating interplanetary flux-tubes. Solar Phys. 
43:475. 
Parker, E.N. 1963. Interplanetary Dynamical Processes. (New York: 
Interscience), pp. 214-219. 
Parker, E.N. 1965. The passage of energetic charged particles 
through interplanetary space. Planet. Space Sci. 13:9. 
Webber, W.R., and R.A. Goeman 1979. Solar proton propagation 
characteristics out to 16 AU. Proc. 16th Intl. Cosmic Ray Conf. 
5:334. 
Webb, S., and J.J. Quenby 1973. A comparison of theoretical and 
experimental estimates of the solar proton diffusion coefficient 
during three flare events. Solar Phys. 29:477. 
Zwickl, R.D., and W. R. Webber 1977. Solar particle propagation from 
1 to 5 AU. Solar Phys. 54:457. 
Bieber, J .W. ,  J.A. Earl, G. Green, H. Kunow, R. Mtiller-Mellin, and G. 


The radial dependence of the solar energetic particle flux

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