E N E R G E T I C S O L A R P A R T I C L E E V E N T S IN A

S T R E A M - S T R U C T U R E D S O L A R W I N D

M. S C H O L E R

Max-Planck-Institut fiir Physik und Astrophysik, Institut far extraterr. Physik, 8046 Garching, F.R.G.

G. M O R F I L L

Max-Planck-Institut fiir Kernphysik, 69 Heidelberg, F.R.G.

and

A. K. R I C H T E R

Max-Planck-Institut fiir Aeronomie, 3411 Lindau, F.R.G.

(Received 27 November , 1978; in revised form 30 July, 1979)

Abstract. Theoretical considerations lead to a solar cosmic ray diffusion coefficient which varies with heliolongitude in a s t ream-st ructured solar wind. By solving numerically the t ime dependent convection- diffusion equat ion for the particle transport we investigate the effect of the azimuthal variation of the diffusion coefficient on intensity-time profiles as seen by a stationary observer. Depending on the position of the observer relative to the solar wind stream at the time of flare occurrence, completely different intensity-time profiles will be observed. When the spacecraft is at the time of the flare occurrence right at the leading edge of a solar wind stream, the large mean free path leads to rapid s teepening of the initial phase of the intensity profile. The longitudinally decreasing mean free path ~ 1 day in front of the leading edge will lead to intensity-time profiles similar to long-t ime injection events if the event occurs before the stationary observer enters the flux tubes with the decreasing diffusion coefficient.

1. Introduction

Interplanetary observations of energetic solar particle events in the MeV energy range usually show complicated intensity time structures which are associated with changes in the solar wind and magnetic field parameters. The most frequent and prominent variation of interplanetary parameters is due to the solar wind stream structure, i.e. the dependence of the solar wind velocity on heliolongitude. As the solar wind corotates over the spacecraft, the spacecraft is connected to different longitudes at the Sun and will sample the longitudinal particle intensity profile in the corona. Any change in the solar wind velocity will lead to corresponding changes in the connection longitude and will therefore have an effect on the intensity-time profile measured at the satellite (e.g. Roelof and Krimigis, 1977). This description assumes that there is no dependence of the interplanetary propagation on longitude. However, in a recent paper by Morrill et al. (1979) it was calculated that the interplanetary mean free path for energetic particles can vary by as much as a factor five from the leading edge to the trailing edge of solar wind streams. As such a stream corotates over the spacecraft, the detectors will sample flux tubes with different interplanetary propagation conditions and this will lead to a change in the measured intensity-time profile even if there is no longitudinal intensity variation in the corona.

Solar Physics 64 (1979) 391-401 . 0 0 3 8 - 0 9 3 8 / 7 9 / 0 6 4 2 - 0 3 9 1 5 0 1 . 6 5 . Copyright 0 1979 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

392 M . S C H O L E R E T A L .

The theoretical implications of coronal propagation and release on the intensity and anisotropy profiles as seen by a stationary observer have been discussed in detail by Ng and Gleeson (1976). In this paper we will use the calculated and measured longitudinal variation of those interplanetary parameters which influence energetic particle propagation in order to calculate intensity time profiles as seen by a stationary spacecraft over which the solar wind is corotating. Although the coronal intensity structure is undoubtedly an important factor in determining the intensity time profile we wish to emphasize the importance of varying propagation conditions from one flux tube to the next, which has been neglected in the past.

In the next chapter we first summarize briefly the calculations of Morrill et al.

(1979), which have led to the result that the diffusion coefficient is large near the leading edge of high speed solar wind streams. With this azimuthally dependent diffusion coefficient we solve then numerically the time dependent convection- diffusion equation. In Section 3 we will present several examples, indicating how the intensity time profile can vary depending on the position of the spacecraft relative to the solar wind stream at the time of the flare.

2. The Cosmic Ray Diffusion Coefficient and the Solution of the Convection- Diffusion Equation

A detailed derivation of the radial and azimuthal dependence of the cosmic ray diffusion coefficient is given in Morrill et al. (1979). In this section we first summarize the physical assumptions underlying the calculations and present the results.

Morrill et al. (1979) applied a superposed epoch analysis to the Mariner 5 plasma and magnetic field observations of 13 corotating high speed solar wind streams. This yielded the average azimuthal distribution of all relevant parameters of the back- ground interplanetary medium and in particular of the superimposed Alfv6n waves at the spacecraft position. The interplanetary magnetic field fluctuations were identified as Alfv6nic ones by calculating the absolute values of the correlation coefficients between the radial components of both the interplanetary magnetic field and the solar wind velocity (e.g. Belcher and Davis, 1971). Assuming that the solar wind is an ideal, isotropic one-fluid MHD plasma expanding stationarily in a frame of reference corotating with the Sun all parameters of the background interplanetary medium and the fluctuations are then 'mapped' to other radial distances (Richter, 1975). Since the direction of the wave number vector k of the Alfv6n waves cannot be inferred from observations based on the variance method or from observations of the directional anisotropy (Richter, 1978), the most plausible assumption was therefore to assume all Alfv6n waves purely outwardly propagating with an isotropic k-vector distribution at a distance of R0 = 20Rs (solar radii). Ro corresponds roughly to the Alfv6nic point.

The pitch angle diffusion coefficient was then calculated using standard quasi- linear theory of wave-particle interaction (Jokipii, 1966; Hasselmann and Wib- berenz, 1968) with the usual nonlinear corrections at large pitch angles (Jones et al.,

ENERGETIC S O L A R PARTICLE EVENTS IN A STREAM-STRUCTURED S O L A R WIND 3 9 3

1973; V61k, 1975) and the contribution of compressional waves added (Lee and V61k, 1976; Morrill et al., 1976). Finally, the spatial diffusion coefficient KLI (parallel to the mean magnetic field) was calculated by including the effect of 'guiding' fluctuations, i.e. the deviation of the local magnetic field from the long term Archimedian spiral field (Morrill et al., 1976).

The calculations of the diffusion coefficient were done for 100 MeV protons. According to Zwickl and Webber (1977, 1978) the interplanetary scattering mean free path seems to be independent of particle rigidity. The diffusion coefficient was therefore scaled down to lower energies, assuming to be proportional to particle velocity only. However, in ths paper we are not so much concerned with the absolute value of the diffusion coefficient or the rigidity dependence as with the azimuthal and radial dependence in interplanetary space. Some results of the calculations are shown in Figures 1 and 2. Figure 1 shows in the top panel the solar wind velocity from

I ' I 1 ' I

llJ t/1 _-

.~ 45o >,- I.--

2 ~- ~00 r'-n

rY

_J cz 350 m

102o J i i J 0 30 60 90

~) [ deg . ]

Fig. 1. T h e so l a r w i n d ve loc i ty f r o m a s u p e r p o s e d e p o c h ana lys i s of M a r i n e r 5 p l a s m a o b s e r v a t i o n s of 13

c o r o t a t i n g s o l a r w i n d s t r e a m s ( top) a n d the r ad i a l d i f fus ion coeff ic ient of 1 M e V p r o t o n s (bo t tom) as a f u n c t i o n of so l a r l o n g i t u d e a t 1 A U .

394 M . S C I q O L E R E T A L .

the superposed epoch analysis at 1 A U and in the bot tom panel the radial diffusion coefficient as a function of solar longitude. There is a considerable increase of the diffusion coefficient near the leading edge of the solar wind stream. The reason for

this is due to the fact that the diffusion coefficient depends strongly on the angle of the wave number vector k relative to the background magnetic field: The particles' resonant frequency is proport ional to the inverse of the cosine of this angle.

Increasing this angle will increase the resonant frequency, this means that less power is available to scatter the particles so that the pitch angle diffusion coefficient decreases and hence the spatial diffusion coefficient increases. The high radial diffusion coefficient near the leading edge of the stream is caused by the refraction of the k-vector in the azimuthal solar wind velocity gradient.

Figure 2 shows the radial dependence of the diffusion coefficient at the leading edge of the stream and in the high speed region. The diffusion coefficient increases

towards the Sun with a power law, has a minimum at - 0.3-0.4 AU, increases to

1-2 A U and stays constant thereafter. The increase of the diffusion coefficient towards the Sun is due to the fact that the k-vectors start off being isotopic at R0, and that consequently only part of the wave power is able to scatter cosmic rays. As the

k-vectors are focussed more and more into the radial direction (which is also the direction of mean magnetic field), the diffusion coefficient decreases. At large distance, however, the field is almost azimuthal whereas the k-vectors are radial, so that the diffusion coefficient is large again. The radial dependence of the diffusion

coefficient can be approximated by

K = K1 r -t~ + K 2 [ 1 - e x p ( - r / r o ) ] . (1)

It should be noted that the diffusion coefficient shown in Figure 2 is the diffusion

Fig. 2.

U

E

1022 ! I I I I I I I I I I I I

Leading Edge \

10 21

Region

1 0 2 0 . . . . . . . . ~ . . . . 0.1 1 5

r [ A U ]

The radial dependence of the radial diffusion coefficient (for 1 MeV protons) in the high speed region and in the leading edge of the solar wind stream.

E N E R G E T I C S O L A R P A R T I C L E E V E N T S IN A S T R E A M - S T R U C T U R E D S O L A R W I N D 395

coefficient along a flux tube, i.e. at constant solar wind speed and not at constant

helio- longitude. The coefficients Kx, K2,/3, and r0 in (1) vary from one flux tube to the

next and we will write them as functions of heliolongitude & at 1 AU. A flux tube passing through a certain & at i A U will of course be at other heliolongitudes at other distances r.

In order to calculate intensity t ime profiles of impulsive solar flare events we would have to solve the particle t ransport equation in four dimensions, i.e. in energy space, radial distance, azimuth and time. This is a formidable task and instead we rely on the following procedure: We assume that within each 'flux tube ' the propagation of cosmic rays can be described approximately by the spherically symmetric diffusion equation for the omnidirectional intensity U(r, T, t) as a function of radial distance r and energy T,

oU 1 0 { oU 2 ] 4 V O Ot r20r r2Kr--~-r - r V U J + ~ r - ~ TU, (2)

with V being the solar wind velocity. We used the word flux tube in a somewhat loose

way and rather mean a certain part of the solar wind stream in which the velocity changes only slightly. As the solar wind corotates over the spacecraft, the spaceraft enters different flux tubes, where the transport coefficients are different and we have

consequently different solutions to Equation (2). However , having constructed solutions to the transport Equation (2) for a fine grid in heliolongitudes &, i.e. for a number of corotating points at 1 AU, we can compose the intensity profile at a fixed observation point. This procedure is similar to the one employed b y N g and Gleeson

(1976) in order to investigate the effect of coronal propagat ion of flare particles on intensity profiles at a fixed observation point. The numerical method in order to solve the time dependent diffusion-convection Equation (2) is described in Scholer (1977). As can be seen from Figure 2 the mean free path of 1 MeV protons at ~ 0.2 A U is

large (between 0.1 and 0.2 AU) and comparable with r. This means that at small radial distances the mean free path is of about the same order of magnitude as the characteristic length over which the magnetic field changes and adiabatic focussing neglected in Equation (2) will become important (Earl, 1976).

3. Results

The results of our calculations are presented in Figures 3 to 6. The upper panel of

each figure shows the intensity versus time profile for 1 MeV protons, the lower panel

shows the corresponding radial anisotropy in the solar wind frame. The insert in the lower panel shows the azimuthal profile of the solar wind stream at 1 AU. The t = 0 hr mark gives the position of the satellite relative to the solar wind s t ream at the occurrence of the flare, the t = 96 hr shows the position of the spacecraft relative to the solar wind s t ream after four days. All calculations were per formed with a mean free path independent of energy, a T -2 injection spectrum at the Sun, and in Figures 3 to 5 the injection at the Sun was assumed to be momentaneous , i.e. we used as

3 9 6 M. S C H O L E R E T A L .

f I I I I I I

6-1NJECTION { j ~ T -2)

X = CONST.

:::3~X ~ ~ . t / -. T = 1MeV

._J i i

kO / ~ "- ... o

10C / I I I I I I t

\ .c_ . t=Oh

0 30 60 90

ra >'- ~ qb [deg] 0 '~

I--- ", c~

z

I I I I I I I

0 2# #8 62 96 TIME (h)

Fig. 3. The intensity-time profile (top) and the radial anisotropy (bottom) of 1 MeV protons as observed at 1 AU. The insert shows the solar wind stream at 1 AU and'the position of the observer at the time of flare occurrence relative to the stream. The solid curves are calculated under the assumption that the solar wind stream corotates over the observer, the dashed curves give the time profiles for a corotating observer.

in i t ia l c o n d i t i o n U = Uo fo r r = ro a t t = to a n d a t o t a l l y r e f l e c t i n g S u n OU/Or = 0 at ro

fo r l a t e r t i m e s . T h e d a s h e d c u r v e s in F i g u r e s 3 to 5 s h o w t h e i n t e n s i t y as we l l as t h e

a n i s o t r o p y p ro f i l e in t h e ca se w h e n t h e d i f fu s ion coe f f i c i en t is i n d e p e n d e n t o f

h e l i o l o n g i t u d e , o r in o t h e r w o r d s w h e n t h e s p a c e c r a f t a l w ay s s t ays in t h e s a m e flux

E N E R G E T I C S O L A R P A R T I C L E E V E N T S I N A S T R E A M - S T R U C T U R E D S O L A R W I N D 397

tube marked by t = 0 in the insert. This is actually the assumption which has always

been made in the past when intensity profiles have been calculated in order to fit solar flare particle events (e.g. Hamiton, 1977; Zwickl and Webber, 1977). Figures 3 to 5 were calculated for different positions of the spacecraft relative to the solar wind stream at the time of flare occurrence (t = 0). From each individual figure it can be seen that there is a considerable difference in the intensity time profile when the azimuthal variation of the diffusion coefficient is taken into account (i.e. the solar wind corotates over the spacecraft), compared to the case when the diffusion coefficient is independent of longitude (dashed profile). The radial anisotropy, however, shows almost no significant difference between the two cases. In addition, the profiles (solid lines) can be very different depending on the satellite position relative to the stream at the flare occurrence (e.g. fast rise, slow decay, and vice versa).

Let us discuss Figures 3 to 6 in some more detail. In Figure 3 the spacecraft is in the minimum of the low speed stream at t = 0 (flare), where the diffusion coefficient is small; we observed therefore a relatively slow rise in particle intensity to the time of maximum. However, as the spacecraft enters the leading edge of the high speed stream the diffusion coefficient rises, resulting in a steepening of the initial phase of the particle intensity profile. The late phase is characterized by a faster decay due to the enhanced convection in the high speed stream. If, as is the case in Figure 4, the occurrence of the flare is only somewhat earlier with the respect to the stream, the diffusion coefficient decreases in the beginning by about a factor 2 and this results in a slower rise to time of maximum as compared to the longitude independent case. In Figure 5 the spacecraft is initially (at t = 0) in a flux tube with a large diffusion coefficient, resulting in a steep rise in particle intensity. As the spacecraft enters the high speed stream the diffusion coefficient drops, resulting in a gradual turn-over to the time of maximum. The important thing to notice is that fitting only part of an event, say the time to maximum, the initial rise phase or the late decay phase can lead to different values of the derived diffusion coefficient. The diffusion coefficient generally increases with particle rigidity; the time to maximum flux therefore decreases with increasing rigidity. Fitting e.g. the decay phase of low and high rigidity particles implies fitting the diffusion coefficients in different parts of the inter- planetary medium (different times and helio-longitudes). Thus the rigidity depen- dence of the mean free path inferred from such events, where solar wind streams occur, may be inaccurate.

The solid curves in Figure 6 are identical to the solid curves in Figure 3. Although this profile is calculated for a delta-function type injection at the Sun, most observers would assign this event to long time injection events. We were indeed able to reproduce this intensity time profile quite well with an injection of the form

I(t) exp { - t/r}

with an e-folding time z of 10 hr and an azimuthally independent diffusion coefficient

398

A

X

d LL

(.D 0 d

100

>...

0 t'r- I,-.- 0 s z

Fig. 4.

M . S C H O L E R E T A L .

I I I I I

BO

O i

0

O-INJECTION ( j - T -2)

X = CONST. iii1~/~"~ T = 1 MeV

1 / ~ - - - .

I/ " - .2-- - , y " I I o I t I I

/ \

' ~ ~ t'=Oh

\ \ ~ & 3'0 6'0 9b \~kk, ~ [deg ]

k

I I I I I I I

24 48 62 96

TIME (h)

Same as Figure 3, except that the observer is at a different position relative to the stream.

corresponding to a flux tube near the maximum velocity of the solar wind stream. Only by measuring the anisotropy (bottom panel of Figure 6), is it possible to distinguish between a delta-function and long-time injection, The importance of long-time injection profiles for high anisotropies late in solar flare events has been discussed in detail by Schulze et al. (1977).

E N E R G E T I C S O L A R P A R T I C L E E V E N T S IN A S T R E A M - S T R U C T U R E D S O L A R W I N D 399

A

X E3 ,._1 LL

L9 0 ,._J

10(

o'-t 5(

o I3E I--- O tf) Z

< 0

I I I I I I I

i / / - ' 'N 5-INJECTION (jNT -2) ~ \ . ?S COeNvST

I I I I I I I

U

/', ~ / \ / " >~0c / t ='96h

\~, ~ /t

,'o ,'o eg]

- - - - - - I I I I I I I -- . . . . . 0 2/+ /+8 62 96

TIME (h) Fig. 5. Same as Figure 4.

4. Summary

F r o m m e a s u r e m e n t s at 1 A U and f rom theoret ica l considera t ions involving Alfv6n wave p ropaga t i on and wave-par t ic le in teract ion Morrill et al. (1979) found a distinct radial and az imuthal d e p e n d e n c e of the energet ic part icle diffusion coefficient in the inner hel iosphere . The implicat ions of the radial d e p e n d e n c e have not been exp lored

400 M . S C H O L E R E T A L .

r 3 ] r ] r I 5 - FUNCTION INJECTION, J OBSERVED THROUGH STREAM I INJECTION ~ EXP ( - t /~ ) J c :1Oh, OBSERVED IN SAME

~ ~ FLUX TUBE

>- n 0 n-

O s

Z <

100

50

0

0

', /t [ / T'oj. wS~ ' ~'\ ~ AO0

~ 350 \ \ -~ I t'=Oh

~ - - • .i I i • . L ~

2L~ 48 62 %

T I M E (h )

Fig. 6. Solid curves are the same as those of Figure 3, however, the dashed profiles were calculated for a corotating observer near the maximum of the solar wind speed and using an injection profile with an

e-folding time of 10 hr.

here. Us ing the m e t h o d of fitting intens i ty - t ime profi les of solar flare particles,

prev ious authors found either a constant m e a n free path (Zwick l and Webber , 1977)

or a m e a n free path increasing with radial distance according to a p o w e r law (Hami l ton , 1977) . The latter is not surprising, s ince their ansatz for K, w h e n solving the d i f fus ion-convect ion equat ion, is.a p o w e r law in r in the first place. B o t h these

results m a y not be in conflict with each other or with the calculat ions of Morrill et aI.

(1978) s ince H a m i l t o n (1977) restricted his analysis to 'dwell' regions, i .e. t imes of decreas ing solar wind speed, whereas Z w i c k l and W e b b e r (1977) m a d e no se lect ion .

ENERGETIC SOLAR PARTICLE EVENTS IN A STREAM-STRUCTURED SOLAR WIND 401

We emphasize in this paper the importance of the azimuthal variation of the diffusion coefficient on intensity-time profiles as seen by a stationary observer over which the solar wind corotates. Pure diffusion theory in an infinite scattering medium predicts the time to maximum flux to be proportional to the inverse of the diffusion coefficient. Thus changes in the diffusion coefficient by only a factor of two in neighboring flux tubes will lead to considerable differences in the time-intensity profiles in these flux tubes and an observer over which these flux tubes are corotating will observe a distorted profile which resembles neither of the individual profiles. The actual profile depends critically on the position of the observer at the time of the flare occurrence relative to the gradient in solar wind velocity, since this gradient is the determining factor for the diffusion coefficient. It has been shown, that the azimuthal dependence of the diffusion coefficient can lead to intensity-time profiles obtained with m o m e n t a n e o u s injection at the Sun, which are indistinguishable from long time injection events. Only anisotropy measurements can differentiate between the two. In o u r - m o d e l calculations we used results obtained from a superposed epoch analysis; each individual stream will, of course, have its own characteristics. The purpose of this note was to point out that any longitudinal change in the solar wind velocity can, in addition to a shift in the solar connect ion longitude, lead to distortions of intensity-time profiles as seen by a stationary observer via a longi- tudinal dependence of the radial diffusion coefficient.

References

Belcher, J. W. and Davis, L. Jr.: 1971, Jr. Geophys. Res. 76, 3534. Earl, J. A.: 1976, Astrophys. J. 205, 900. Hasselmann, K. and Wibberenz, G.: 1968, J. Geophys. 34, 353. Hamilton, D. C.: 1977, Z Geophys. Res. 82, 2157. Jokipii, J. R.: 1966, Astrophys. J. 146, 480. Jones, F. C., Kaiser, T. B., and Birmingham, T. J.: 1973, Phys. Rev. Letters 31, 485. King, J. H.: 1977, National Space Science Data Center A, 77-04. Lee, M. A. and Volk, H. J.: 1975, Astrophys. Z 198, 485. Morrill, G. E., V61k, H. J., and Lee, M. A.: 1976, J. Geophys. Res. 81, 5841. Morrill, G. E. Richter, A. K., and Scholer, M.: 1979, Geophys. Res. 85, 1505. Newkirk, G. Jr. and Wentzel, D. G.: 1978, J. Geophys. Res. 83, 2009. Ng, C. K. and Gleeson, L. J.: 1976, SolarPhys. 46, 347. Reid, G. C.: 1964, J. Geophys. Res. 69, 2659. Richter, A. K.: 1975, Astrophys. Space Sci. 36, 383. Richter, A. K.: 1978, J. Geophys. Res., submitted. Roelof, E. C. and Krimigis, S. M.: 1977, in M. A. Shea, D. F. Smart, and S. T. Wu (eds.), Study of

Travelling Interplanetary Phenomena, D. Reidel Publ. Co., Dordrecht, Holland, p. 343. Scholer, M.: 1977, Planetary Space Sci. 25, 1081. Schulze, B. M., Richter, A. K., and Wibberenz, G.: 1977, SolarPhys. 54, 207. Van Hollebeke, M. A. I., Ma Sung, L. S., and McDonald, F. B.: 1975, SolarPhys. 41, 189. Volk, H. J.: 1975, Rev. Geophys. Space Phys. 13, 547. Zwickl, R. D. and Webber, W. R.: 1977, SolarPhys. 54, 457. Zwickl, R. D. and Webber, W. R.: 1978, J. Geophys. Res. 83, 1157.