Adv. 5~ace Rae. Vol.1, pp.93—96. 0273—1177/81/0201—0093$05.OO/O© COSPAR, 1981. Printed in Great Britain.

SHOCK ACCELERATION OFENERGETIC PARTICLES INCOROTATING INTERACTIONREGIONS

L. A. Fisk and M. A. Lee

SpaceScienceCenter/DepartmentofPhysics,UniversityofNewHampshire,Durham, NewHampshire,USA

ABSTRACT

A simple shock model for the acceleration of energetic particles in corotatinginteraction regions (CIR) in the solar wind is presented Particles are acceler-ated at the forward and reverse shocks which bound the CIWby being compressedbetween the shock fronts and magnetic irregularities upstream from the shocks, orby being compressed between upstream irregularities and those downstream from theshocks. Particles also suffer adiabatic deceleration in the expanding solar wind,an effect not included in previous shock models for acceleration in CIR’s. Themodel is able to account for the observed exponential spectra at earth, the observedbehavior of the spectra with radial distance, the observed radial gradients in theintensity, and the observed differences in the intensity and spectra at the forwardand reverse shocks.

INTRODUCTION

It is well established that energetic particles (energies ‘~l MeV/nucleon) are accel-erated in association with corotating interaction regions (CIR’s) in the solar wind(e.g., McDonald et cii. [1]; Barnes and Simpson [2]; Van Hollebeke eb al. [3]). Pos-sible sites for the acceleration are the forward and reverse shocks which bound theCIR, as was suggested by Barnes and Simpson [2] and discussed using model calcula-tions by, e.g., Palmer and Gosling [4]. It is the purpose of this paper to pointout that a deficiency with current models is that they do not include the effectsof adiabatic deceleration in the expanding solar wind. We present here a simpleshock acceleration model which includes adiabatic deceleration, and which is thenable to account for the major features of the observations.

The accelerated energetic particles are seen in two locations. A broad increasein the intensity of ~l MeV/nucleon particles occurs in the high—speed solar windstream. This increase extends from thereverse shock back to the orbit of earth,and is seen as far inward towards the Sun as “,O.4 AU (Van Hollebeke, et al. [5]).An increase in energetic particles is also seen in front of the forward shock. How-ever, this increase generally involves fewer particles than the increasebehind thereverse shock, has a softer spectrum, and does not appear to extend back to earth(Barnes and Simpson [2]; Van Hollebeke et al. [3]). The particle intensity is alsoobserved to decrease substantially towards the center of the CIR (Barnes andSimpson [2]).

93

94 L.A. Fisk and M.A. Lee

The University of Maryland/Max—Planck Institute group report that the spectra atearth in several events are well fit if they are plotted as a distribution func-tion that is an exponential in velocity independent of particle species (Gloeckler,Hovestadt and Fisk [6]). This result holds over a wide energy range, from “0.15 —

8 MeV/nucleon, and for all major species from protons through iron. The GoddardSpace Flight Center group obtains similar spectra (Van Hollebeke et al. [5]). Thee—folding velocity of the spectra~of particles in the high—speed stream is alsoobserved to be essentially independent of radial distance. The gradient in theintensity of the particles in the high—speed stream is observed to be steepest inthe inner solar system (“350//AU from “’0 4 — 1 AU), and then decreases scmewhatbeyond earth (“’100%/AU) (Van Hollebeke et al. [5]).

A SHOCKMODEL

Particles can gain substantial energy at the forward and reverse shocks which bound

the CIR’s by being compressed between the shock front and the magnetic irregulari-ties in the solar wind upstream from the shock, or by being compressed between theupstream irregularities and those downstream from the shock.. Such a shock—accel-eration model is of course not a new idea. It has been applied to the earth’s bowshock by, e.g., Jokipii [7], to flare—induced shocks by, e.g., Fisk [8], and toCIR’s by, e.g., Palmer and Gosling [4]. However, a deficiency with current shockmodels for CIR’s is that they neglect the effects of adiabatic deceleration in theexpanding solar wind. In the case of, e.g., the earth’s bow shock or flare—inducedshocks, this neglect is not significant. However, for CIR’s where the particleincrease extends over several AU, the effects of adiabatic deceleration are impor-tant, and have the consequence that the resulting spectra can be exponential,rather than the power laws which are common to this mechanism.

In the corotating frame, the equation for the energetic particle distribution func-tion, f, in a given magnetic flux tube is (Parker [9])

1 ~ 2 3f 1 3 (‘2 ~f)--~—~-~— (r V)v-~--=—-

2--~-- ~r K~J-V-i-- (1)

where r is heliocentric radial distance, v is particle speed, and K is the radialdiffusion coefficient. It is assumed that the particles are nonrelativistic andmove only along and not across magnetic field lines. V is the speed of the solarwind in the inertial frame fixed with respect to the sun.

The effects of the shock are described by means of a boundary condition on thesolution to equation 1. We note that for a given magnetic flux tube the productof the particle differential streaming S and the cross—sectional area A of theflux tube must be conserved across the shock front. The appropriate boundary con-dition at a shock located at r = r5 is then

Vv3f 3f Vv3f—————K——-—6—-——, at r=r (2)

3 3v 3r 3 3v S

where 6 a (V’2 + 02r2)½ (V2 + çr2i~ B(B’)1. Here, V, 0 and B are, respec-

tively, the solar wind speed, the angular velocity of the sun and the magneticfield strength upstream from the shock; V’ and B’ refer to conditions downstream.We have assumed, for, simplicity in equation 2, that there is substantial parti-cle scattering in the CIR so that particles are transported downstream from theshock primarily by convection. We also assume that f is continuous across theshock front. .

Shock Acceleration of Energetic Particles 95

AN APPROXIMATE SOLUTION

The diffusion coefficient for low—energy particles is observed to be of the formKvg(r), where g(r) is a function of radial distance only (Zwickl and Webber 110]).Since on the RES of equation 1, then, the first term.dominates the second forenergies ~l MeV/nucleon, we consider an asymptotic expansion of equation 1 in largev. We take, for.example, the case in which K=K

0vr upstream from the shocks. Theexpansion of equations 1 and 2 to order exp[h(r)v

1] then yields

281(1—8) + V/(K v) —31(1—8) 6 8vf = [~_J ° v exp [~vcl_~)2j (3)

Clearly, the solution in equation 3 can account for many of the observed featuresof energetic particles upstream from the CIR~s.. The leading dependence of the dis-tribution function on particle velocity is an exponential which is independent ofparticle species. The e—foldIng velocity [V(l—8)2/(6K

08)] is independent of radialdistance. Further, the radial gradient of the intensity, (l/f)’(3f13r), varies asl/r and thus is steepest in the inner solar system.

The solution in equation 3 can also yield a steeper spectrum at the forward shockthan at the reverse, again as is observed. There are several quantities in thee—folding velocity in equation 3 which should be roughly equal for the two shocks.For shock interactions at large radial distances, 8 in equation 2 is 6 B/B’, whichis observed to be a similar ratio for the two shocks (Smith and Wolfe [11]). AlsoK0 could be similar for the high— and low—speed streams. However, the e—foldingvelocity varies directly as the upstream solar wind speed V which for the forwardshock is, of course, substantially slower.

We also find that the solution for f in the CIR, which corresponds to the upstreamsolution in equation 3, is

- ‘-3/cL-8) fr )2/(l-6) 6 Ko 8v 1~2I3f — v exp vu_8>2 ~rj

A spacecraft such as Pioneer 10 or 11 passes through a CIR at roughly constantradial distance. However, the spacecraft will intersect many different streamlinesin the CIR, each with their own value of r

5. The streamlines near the center ofthe CIR will have the smallest values of r5. Clearly, the solution in equation 4predicts that there chould be a substantial reduction in the intensity towards thecenter of the CIR, again as is observed.

NUMERICAL RESULTS

For a detailed comparison between theory and observation, we consider full numericalsolutions to equation 1 subject to equation 2, which can be obtained by usingthe numerical techniques that were developed by Fisk [12]. Shown on the left sideof Figure 1 are the results of a numerical solution to equation 1 for conditionsupstream from the reverse shock. The diffusion coefficient is assumed to be of theform K=K0vr and the mean, free path at r=1AU is taken to be l.6~lO

12cm. Thesolar wind speed is V= 800 km/sec since this is the high—speed stream. The shock isplaced at r

5=4AU; 8 =1/3. The calculated curves are normalized to fit the datafor the June 27, 1974 corotating event, which was seen at earth by Gloeckler,. I-love—stadt, and Fisk [6]. Clearly, the calculated curve at r=1AU provides an excellentfit to the data The radial gradients in f implied in Figure 1 are also in reason-able agreement with the observations At the higher velocities f increases by abouta factor of ten between 1 and 3 AU which corresponds to a gradient of ~l00//AU Asimilar increase in f occurs between 0 4 and 1 AU or a gradient ‘~350//AU

96 . L.A. Fisk and M.A. Lee

I I

~ \ June 27, 1974 Spectrum 01 Shockl0o-~ -

AU - \~verse Shock -

~Z04~~\I

? 6~- V Oxygen - - -0 Iron Forward Shock

a6 ._________________________________0.0 0.5 1.0 15 2.0 25 0.0 0.5 10 1.5 20 25 3.0\Mlocity , v, (MeV/nuc) /2

Fig.l The results of a numerical solution to equation 1 for theconditions and parameters discussed in the text. In the left panelis shown the omnidirectional distribution function f at various radialdIstances upstream from the reverse shock, Plotted for comparisonare the observed values for £ at earth for the June 27, 1974 event,as measured by Gloeckler, Hovestadt, and Fisk [6]. In the right panelare the calculated spectra at the forward and reverse shocks.

Shown on the right side of Figure 1 are the spectrum at the reverse shock (ther=4AU curve), and a possible spectrum at the forward shock. The latter spectrum iscalculated by assumIng that K 1S the same in the high— and slow—speed flows and that

8 (~B/B’) is the same for the two shocks. However, the upstream solar wind speedfor the forward shock is taken to be V=400 km/sec. Clearly, the spectrum at theforward shock is substantially steeper than it is at the reverse shock, again as isrequired by the observations. The numerical results plotted in Figure 1 are well

fit by the approximate solution in equation 3 for velocities v~l (MeV/nuc)h/2.

The research was supported, in part, by NASA grant NSG7411.

References

1. F. B. McDonald, B. J. Teegarden, J. H. Trainor, T. T. von Rosenvinge, andW. R. Webber, Ap. J. (Letters) 203, Ll49 (1976).

2. C. W. Barnes and J. A. Simpson, ~ (Letters) 210, L91 (1976).3. M. A. I. Van Holiebeke, F. B. McDonald, J. H. Trainor and T.T. von Rosenvinge,

Proc. Solar Wind Conf. 4, Lecture Notes in Physics, Springer—Verlag, Berlin,in press.

4. 1. D. Palmer and J. T. Gosling, J. Geophys. Res. 83, 2037 (1978).5 M A I Van Hollebeke, F B McDonald J H Trainor and T T von Rosenvinge,

J Geophys Res 83 4723 (1978)6. G. Gloeckler, D, Hovestadt and L. A. Fisk, Ap. J. (Letters), 230, L191, 1979.7. J. R. JokIpii, Ap. J. 143, 961 (1966).8. L. A. Fisk, J. Geop)~ys. .Res. 76, 1662’ (1971).9. E. N. Parker, Planet. Space Sci. 13, 9 (1965).

10. R. D. Zwlckl and W. R. Webber, Solar phys, 54, 457 (1977).11. E. J. Smith and J. H. Wolfe, Study of Traveling Interplanetaty Phenomena, ed.

by 14. A. Shea, D. F. Smart and S. T. Wu, p. 227, D. ReidelPublishiagCo., 1977.12. L. A. Fisk, J. Geophys. Res. 76, 221 �1971),