Cosmic-ray gradients in the outer heliosphere

Adv. Space Res. Vol. 13, No.6, ~,. (6)257—(6)260, 1993 Printed in Great Britain. All rights reserved.

0273—1177~3$24.00 Copyright © 1993 COSPAR

COSMIC-RAY GRADIENTS IN THE OUTER HELIOSPHERE J. KOta and J. R. Jokipii Lunar and Planetary Laboratory, University ofArizona, Tucson, AZ 85721,

U.S.A.

ABSTRACT We report on a study of the expected spatial variation of the galactic cosmic-ray intensity in the vicinity of the solar-wind termination shock. Model simulations predict that the radial gradients change abruptly at the shock, and that the nature of the effect varies significantly with particle energy. At low energies, the radial gradient changes abruptly from a lower value inside the shock to a higher value outside, whereas at high energies, the higher value of the gradient is inside the shock. This effect, which is a consequence of the matching conditions at the shock and is closely related to diffusive shock acceleration, is qualitatively the same for both heliospheric magnetic polarity states and remains much the same in one-dimensional, two-dimensional and three-dimensional models. Hence drifts do not change the nature of this phenomenon, although they change it quantitatively. The effect may prove to be an important diagnostic tool for the study of the termination of the solar wind. INTRODUCTION The modulation of cosmic rays by the solar wind involves a number of distinct physical effects diffusion, convection, guiding-center drifts and energy change. The energy change can be either cooling in the expanding solar wind or acceleration at shocks or compression regions. The termination shock of the solar wind is thought to be responsible for the acceleration of the anomalous cosmic rays /1,2/. —

On the other hand, discussions of the modulation of galactic cosmic rays have, for the most part, neglected the presence of a termination shock. Initially, this was because this seemed to be too distant to be of interest. More-recently, this was because the commonly-used numerical technique did not allow for the necessary particle acceleration at the shock. In our most-recent simulation codes /3,4/ acceleration and drift at the terminal shock (as well as at other shocks) are included. Webb, Forman and Axford /5/ considered aspects of this problem analytically in a onedimensional model for the special case of a diffusion coefficient, and pointed out that similar effects must be considered in the problem of the modulation of galactic cosmic rays. Jokipii and Merényi /6/ considered the effect of the terminal shock on modulation in two-dimensional numerical simulations, and concluded that the effects were not of much importance for observations carried out in the inner heliosphere. However, recent emphasis on observations near the terminal shock has caused us to reconsider this problem, concentrating on modulation near the termination shock. We find a very interesting, and possibly important effect which may be an important diagnostic tool for the in-situ study of the termination shock. JASR 13:6-R

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J. K&a and J. R. Jokipii

MODEL CALCULATIONS AND THEIR INTERPRETATION Our heliospheric solar wind and magnetic field model is that used previously and described elsewhere (see /3,4,7/). Here, we report on simpler 2- and 1-dimensional models. The complex velocity and magnetic field structure of our most-recent full, three-dimensional simulations would slightly modify the picture (see /7/). We consider the modulation of protons and use parameter sets which we have used in the past and which yield simulations in good agreement with observations. The parallel and perpendicular diffusion coefficients used here are and

= ~

1cj

= IlK

(1)

11

where B is the magnetic field, BE is its value at 1 A.U.,2s~. ~j is usually set equal to .01 .05 and, if the rigidity P is expressed in GV, ~cc~ 1.5 x 1022 cm In Figures 1 and 2 we show the radial variation of cosmic-ray intensity for two simple models azimuthally-symmetric and spherically symmetric. In each case we see that the radial gradient, ôf/ôr, changes abruptly at the shock. Moreover, the nature of the abrupt change varies considerably with particle energy. The nature of the phenomenon, for a very broad range of parameters, is summarized as follows: at low energies the radial gradient increases abruptly at the shock whereas at high energies it decreases. —

—

1.2 :‘

I:

1.2

~

I

~1

~

Ii

0

ill

Iii

20

40

Iii

60

ii

80

0

radius [au]

Iii

20

iii

40

iii

60

80

radius [au]

Fig. 1. 27-day averages of cosmic-ray proton intensities at 100 latitude and normalized to be unity at the outer boundary, as obtained from a 2-dimensional model calculation for both epochs A < 0 (a) and A > 0 (b). The termination shock is placed at 60 a.u. Note that the gradient changes abruptly at the shock: it turns upward at low energies and downward at high energies. The effect is not a consequence of drifts, nor can it be due to differing modulation parameters upstream and downstream of the shock, since the modulation parameter ~ remains constant across the shock in our simulations. We find that a straightforward consideration of the matching conditions across leads to a simple interpretation. Consider first the one-dimensional, spherically symmetric case. The jump condition across the shock prescribes that both the cosmic ray phase space density, f, and the the normal component of the net flux, S~,must be the same on both sides of the shock +

f~=0

and

SnI

+

0

(2)

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Cosmic-Ray Gradients

with S~being the radial component S~in our case

(3)

SrCfVw+Krr~

These jump conditions represent the basis of the theory of diffusive shock acceleration (see e.g., Axford /8/). Here C = —Olnf/91np is the Compton-Getting factor, which is obviously continuous across the shock. If er = Sr/f denotes the bulk speed of cosmic rays, then clearly er is continuous across the shock, and the ratio of the logarithmic gradients, (g,. = rnnf/i9r ) can be expressed as gr2

—

gri

-

(V

2/K2)(C er/vw,2) (Vi/~i)(C~r/Vw,i)~

4

Relations (3) and (4) have some straightforward implications. First, if convection and diffusion happen to exactly balance each other, i.e. ~r = 0, we have the force-field approximation, which is valid only for high-energy particles. If, further, both V,~,and K change by the same factor across the shock, we have the same gradient on both sides of the shock. However, even for this simple approximation the radial gradient changes discontinuously if V~/k changes. For the sake of simplicity, assume that V~/K remains constant across the shock. Then a simple connection can be found between the sign of the radial streaming and the sense of the change of radial gradient. If the net streaming is directed outward (~r > 0), then the gradient decreases across the shock. In the case of an inward streaming, on the other hand, the radial gradient will increase increase upon crossing the shock. Assuming an inward streaming and a positive Compton-Getting factor the maximum change of the gradient is limited to the compression ratio, V~,1/V~,2,which would occur at C = 0. An even larger jump of the radial cosmic-ray gradient can be obtained if the Compton-Getting factor happens to be negative. In our model calculations we find that cosmic-ray gradients appear to turn upward at the shock at low energies, where the diffusion coefficient becomes considerably smaller than rV~. The force-field approximation breaks down for this case. We also find, that the Compton-Getting factor is, indeed, very small or even negative for these cases. 1.2

1-~

:1 I

—(a)

1

I

I

I

1.2 I

with shock

-

~L2•5~MeV75Me15Me!

ii

20

i -(b) without

I I

I

shock

~~25~~35MeV1

0 0

I

40

0

iii

60

radius [au]

80

iii

0

20

40

60

80

radius [auj

Fig. 2. Radial variation of the cosmic-ray density at various energies as obtained from a one-dimensional model calculation with a termination shock included (a). For comparison, also shown are the results obtained with the same parameters but without a shock (b).

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J. K6ta and J. R. Jokipu

As for the energy dependence, Figure 2 indicates that the radial gradient turns sharply upward at the shock below about 100 MeV, which corresponds to rV~/K~ 1. If we take a smaller value of KU then this phenomenon starts at correspondingly higher energies. In a two- or three-dimensional model, the jump condition yields a less transparent relation. The underlying basic physics, however, remains the same: at the lowest energies, where the radial diffusion coefficient becomes small (smaller than V~r),the Compton-Getting factor becomes small or even negative, and the radial gradient sharply increases at the termination shock. At high energies, on the other hand, the radial diffusion tends to be smaller beyond the shock (Figure 1). As a result of acceleration, the cosmic-ray intensity at the shock may even exceed the interstellar intensity, leading to a negative radial gradient behind the shock. At low latitudes, this might be expected for the qA < 0 case, when particles drift toward the neutral sheet during the acceleration. We note that the role of drift manifests itself in the differences between Figures la and lb. Furthermore, we also notice that a thicker subsonic region, does not only reduce the cosmic-ray density inside the shock but will result in a smaller gradient as well (see /7/). SUMMARY AND CONCLUSION The standard cosmic-ray transport theory implies a sudden change in the radial gradient of galactic cosmic rays at the termination shock. This effect is primarily due to the change in radial wind velocity at the shock, and is present in models ranging from our most-sophisticated three-dimensional, time-dependent models to simple one-dimensional, time-independent models which neglect drift. The nature of the change in radial gradient varies greatly with particle energy, so that at low energies the gradient is significantly smaller inside the shock than outside, whereas at high energies the reverse is true. At high energies the gradient outside the shock can be negative. This change in radial gradient may provide a further useful diagnostic of the solar-wind termination shock. Furthermore, this effect may be related to the effective “modulation barrier” in the outer heliosphere discussed by Webber and Lockwood /9/, and by Potgieter, le Roux, and Burger /10/. Acknowledgement. This research was supported, in part by the National Science Foundation under Grant ATM-892215l and the National Aeronautics and Space Administration under Grant NAGW-2549. REFERENCES 1. M.E. Pesses, J.R. Jokipii, and D. Eichler, Ap. J. 246, L85 (1981). 2. J.R. Jokipii, in: Physics of the Outer Heliosphere, ed. S. Grzedzielski and D.E. Page, Pergamon, 1990, p.169. 3. J. Kóta, in: Physics of the Outer Heliosphere, ed. S. Grzedzielski and D.E. Page, Pergamon, 1990, p.119. 4. J. Kdta and J.R. Jokipii, Geophys. Res. Lett. 18, No. 10, 1797 (1991). 5. G.M. Webb, M.A. Forman and W.I. Axford, Ap. J. 298, 684 (1985). 6. J.R. Jokipii and E. Merényi, unpublished manuscript (1986). 7. J.R. Jokipii, J. Kóta, and E. Merényi Ap. J., in press (1993). 8. W.I. Axford, in: Proc. 10th Texas Symp. on Relativistic Astrophysics, ed. R. Ramaty and F. C. Jones (Ann. N Y Acad. Sci., 375), 1980 p.297. 9. W.R. Webber and J.A. Lockwood, Ap. J. 317, 534 (1987). 10. M.S. Potgieter, J.A. le Roux and R.A. Burger, Proc. t~0thInt’l Cosmic Rag Con!., Moscow, 3, 291 (1987).