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THE ASTROPHYSICAL JOURNAL, 477 : L115–L118, 1997 March 10 q 1997. The American Astronomical Society. All rights reserved. Printed in U.S. A.

THE INFLUENCE OF PICKUP, ANOMALOUS, AND GALACTIC COSMIC-RAY PROTONS ON THE STRUCTURE OF THE HELIOSPHERIC SHOCK: A SELF-CONSISTENT APPROACH JAKOBUS A.

LE

ROUX

AND

HORST FICHTNER

Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742-2431 Received 1996 August 15; accepted 1996 December 26

ABSTRACT A self-consistent time-dependent model is used to study the modification of the heliospheric shock in the upwind direction under the influence of three suprathermal particle populations, namely, pickup, anomalous, and Galactic cosmic-ray protons. By ensuring that the resulting modulated cosmic-ray proton spectra are consistent with those observed by the Voyager and Pioneer spacecraft during the solar activity minimum in 1987, two alternative modifications of the heliospheric shock can be identified. While the first is characterized by a low injection efficiency of pickup protons into the process of diffusive shock acceleration and is mainly determined by those particles, the second, resulting from a high injection efficiency, is clearly dominated by anomalous cosmic rays. Subject headings: acceleration of particles — cosmic rays — shock waves — solar wind energetic particle populations (Zank et al. 1993; Lee, Shapiro, & Sagdeev 1996; le Roux et al. 1996), we study their simultaneous influence on the structure of the HS describing the energetic particles with the CR transport equation. We demonstrate that, on the basis of the currently available data, the HS structure cannot be clarified unambiguously, but that there are at least two different alternatives consistent with observations obtained so far.

1. INTRODUCTION

The shock transition terminating the supersonic solar wind (SW), the so-called heliospheric shock (HS), has received increasing attention during the last few years for several reasons. First, the deep space probes Pioneer and Voyager are entering the outer heliospheric region where the HS is supposedly located, and it is of importance to have some expectation of how it might show up in the data (e.g., Barnes 1993; Suess 1993; Paularena et al. 1996). An indication that Pioneer 10 and Vo yager 1, both located beyond a heliocentric distance of 160 AU, might in fact be relatively close to the HS is given by the detection of anomalous hydrogen by these spacecraft (Christian, Cummings, & Stone 1995; McDonald, Lukasiak, & Webber 1995; Stone, Cummings, & Webber 1996). Second, the HS is a key element in structuring the global heliosphere, which is currently the subject of extensive numerical modeling (e.g., Baranov & Malama 1993; Karmesin, Liewer, & Brackbill 1995; Linde et al. 1996; Pauls & Zank 1996; Ratkiewicz et al. 1996). Third, the notion that beyond 120 AU the pressure of pickup ions (PUIs) might be much larger than the thermal pressure of the SW (Isenberg 1986) or the magnetic field pressure (Whang, Burlaga, & Ness 1995) drives some interest in its influence on the dynamics of the outer heliosphere (e.g., Fahr & Fichtner 1995) concerning the location and modification of the HS (Zank, Webb, & Donohue 1993; Lee 1996). And fourth, the properties of anomalous cosmic rays (ACRs), probably produced at the HS, give rise to the question of how the HS structure, determining the diffusive shock acceleration process, looks in detail (e.g., Lee 1996; le Roux, Potgieter, & Ptushin 1996). We report here briefly about the first results of a new time-dependent model that, in a generalization of earlier approaches, takes into account the self-consistent interaction of the thermal SW plasma (including PUIs) with two cosmicray (CR) populations. While previously the influence of PUIs, ACRs, and galactic cosmic rays (GCRs) on the structure of the HS have been studied separately (e.g., Ko, Jokipii, & Webb 1988; Lee & Axford 1988; le Roux & Ptuskin 1995a, 1995b) or, in a simplified approach, for combinations of some or all of the

2. MODEL EQUATIONS

The CR protons with a momentum p . mpu measured in the SW frame, with mp denoting the proton rest mass and u(r, t) the SW velocity at heliocentric distance r and time t, both measured in the observer’s frame, are described by the fundamental CR transport equation (Parker 1965):

S D

1 ­ 2 ­f ­f ­f ­f 1 ­ r 2k 2 2 1u 2 ~r u! p 5 h Qpi . ­t ­r r 2 ­r ­r 3r ­ r ­p

(1)

Here f 5 f (r, p, t) is the omnidirectional CR distribution function and k 5 k (r, p, t) 5 k icos2C 1 k 'sin2 C is the radial diffusion coefficient, with k i 5 k i(r, p, t) 5 (3.8 3 1022 cm2 s21 )(w/c)[P/(1 GV)](5 nT/B) and k ' 5 k '(r, p, t) 5 ek i(r, p, t) characterizing diffusion parallel and perpendicular to the heliospheric magnetic field B 5 B(r, t) with a winding angle C 5 C(r, t) 5 tan21(Vr/u); V is the angular velocity of the Sun. Reproducing the GCR spectrum requires e 5 0.015 (le Roux et al. 1996). The time dependence of the SW velocity u implies a time dependence of C and B and, in turn, of the level of turbulence upstream of the HS. The particle velocity and rigidity are given by w and P, respectively. Equation (1) is also used to describe the simultaneous transport of pickup protons for p # mpu, but with k 5 0. This means that pickup protons are convected and adiabatically cooled in the upstream SW and adiabatically heated across the HS. Those pickup protons that cross the threshold, assumed to be at p 5 mpu, undergo diffusive shock acceleration because of heating at the HS and are considered as ACRs. Pickup protons are introduced via v, d the source term Qpi 5 Qpi(r, p, t) 5 nHn ce d ( p 2 mpu), deL115

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scribing pickup protons produced by charge exchange with an u (r) 5 u(cm s21 )nSW(cm23 )[7.5 2 2.1 ionization frequency n ce log u(km s21 )] 3 10215 s21 upstream of the HS (e.g., Fite, Smith, & Stebbings 1962; Isenberg 1986), with nSW denoting d (r) 5 1.23wrms the number density of SW protons and n ce (cm s21 )nSW(cm23 )[7.5 2 2.1 log 1.23wrms(km s21 )] 3 10215 s21 downstream of the HS (e.g., Lee 1996), with wrms the root mean square velocity of SW protons. The density of interstellar hydrogen in the heliosphere is calculated according to nH(r) 5 0.077 exp(23.74 AU/r) cm23 following the cold gas model (Gloeckler et al. 1993). The use of this expression restricts the validity of the model computations to the upwind direction of the heliosphere. The contribution due to photoionization, with a frequency n ph(r) 5 (9 3 1028/r 2 ) s21, is included in equation (1). The injection efficiency h represents the fraction of those pickup protons that, by adiabatic heating across the HS, attain p . mpu. This modification of the actual injection rate is needed to simulate observed modulated proton spectra. The given absolute values of the various quantities and parameters are not the only possible choices, but are standard or typical. Since we are aware of the many simplifications of the outlined treatment of the CR transport, we explicitly note that particle drifts are neglected and that the use of equation (1) for p . mpu instead of p .. mpu demands some caution (e.g., Gombosi et al. 1993). The velocity profile u(r, t) is self-consistently computed from a system of time-dependent equations describing a spherically symmetric one-fluid SW (including PUIs) in the presence of CRs:

­p 1 ­ 2 ˆ ph , 52 2 ~r m! 1 Q ­t r ­r

S D

(2)

­m m2 ­ ~ pth 1 pcr! m 1 ­ ˆ ce , r2 2 52 2 2 Q ­t r ­r r ­r r

F

G

F

m2

r2

G

FIG. 1.—Differential intensity jT for pickup, ACR, and GCR protons as a function of kinetic energy T for different heliocentric distances in the ecliptic plane. The different injection efficiencies in (a) h 5 0.0003 and (b) h 5 0.9 manifest themselves in the different flux levels of the PUI spectra visible at energies less than 11026 GeV. The filled circles represent the 1987 proton data detected by Voyager 2 and Pioneer 10 at 24 and 42 AU, respectively (McDonald et al. 1995).

(3)

S D

m ­ pcr ­e m a ­ m 1 ­ r 2 ~e 1 pth! 2 r2 52 2 1 2 ­t r ­r r r ­r r ­r r 2 0.5

Vol. 477

2 ˆ ce , H~rsh 2 r! 1 w rms H~r 2 rsh! Q

(4)

where r 5 r (r, t) and pth 5 pth(r, t) are the mass density and thermal pressure of the gas mixture consisting of electrons, SW, and pickup protons, m(r, t) 5 r (r, t) u(r, t) is the mass flux, and e(r, t) 5 pth(r, t)y( g 2 1 1 m2(r, t)/2 r (r, t) is the total energy with a polytropic index g 5 5/3 for the thermal gas; pcr 5 (4p/3) * p3wf dp is the CR pressure. The source and loss terms are determined by the production rates of pickup protons resulting from photoionization of, and ˆ ph,ce /mp 5 charge exchange with, interstellar hydrogen Qph,ce 5 Q nph,ce(r)nH(r). Following Lee (1996), the terms were derived under the assumption that upstream of the HS, the energy density loss of the SW and Qce depend mainly on u, and downstream mainly on wrms; hence the appearance of Heaviside functions H in equation (4). The function a 5 a(r, t) 5 (4 p /3)mpu 3 Ekin f (r, mpu, t) results from the transfer of PUIs with p # mpu from the thermal to the suprathermal particle population (Zank et al. 1993) across the threshold p 5 mpu by adiabatic compression and heating. The parabolic transport equation (1) describes both ACRs resulting from the injection and diffusive shock acceleration of PUIs at the HS and GCRs incorporated by prescribing an interstellar spectrum (see, e.g., McDonald et al. 1995) at the

outer boundary at 120 AU. It is solved by using a combination of the implicit Crank-Nicholson method for spatial diffusion and the explicit monotonic transport scheme for convection and adiabatic energy changes. For the system of hyperbolic fluid equations (2), (3), and (4) describing the thermal gas mixture, solved with a Riemann algorithm (LeVeque 1994), standard SW conditions at the inner boundary ri 5 1 AU are used (ui 5 400 km s21, r i 5 5 mp cm23, Ti 5 105 K). At the outer boundary ro 5 120 AU, a constant downstream density ( ­ r / ­ ru r 5 ro 5 0), a mass flux decreasing proportional to 1/r 2 [ ­ (mr 2 )/ ­ ru r 5 ro 5 0], and a thermal pressure equal to the local interstellar pressure ( p 5 pth,LISM 5 1 eV cm23 ) are assumed. Note that absolute values are not prescribed for either the density or the mass flux at r 5 ro. Their actual values result from the integration. 3. RESULTS

For h, the injection efficiency of pickup protons into the process of diffusive acceleration at the HS, we found one high and one low value of the free parameter h resulting in CR flux levels consistent with Pioneer and Voyager observations during 1987. Figure 1 shows the differential intensity jT 5 p 2f of CRs as a function of kinetic energy T for various heliocentric distances for these solutions, (a) h 5 0.0003 and (b) h 5 0.9. Since the injection efficiency as defined above denotes only a fraction of those PUIs with w . u, the actual number of injected particles represents a smaller fraction of the total PUI population than is indicated by h. The percentage of pickup protons at the HS having velocities greater than u as a

No. 2, 1997

STRUCTURE OF THE HELIOSPHERIC SHOCK

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TABLE 1 HS PARAMETER

Case

Energetic Particles

1. . . 2. . . 3. . . 4. . . 5. . .

No PUIs or CRs PUIs, no CRs PUIs and GCRs PUIs, GCRs, and ACRs PUIs, GCRs, and ACRs

FOR DIFFERENT CASES

h

rsh (AU)

s

L (AU)

Du/ui (%)

0 0 0 0.0003 0.9

80.8 73.7 71.8 74.3 73.8

4.0 3.5 3.5 3.4 1.5

0 0 37 34 12

0 12.8 17.3 19.8 45.0

NOTES.—h 5 PUI injection efficiency, rsh 5 heliocentric distance, s 5 subshock compression ratio, L 5 extent of precursor, Du/ui 5 total SW deceleration (ui 5 400 km s21 ).

consequence of adiabatic heating is found to be 98% and 92% for (a) and (b), respectively. For h 5 0.0003 (h 5 0.9) for (a) ([b]), we find that 0.03% (83%) of all PUIs are diffusively accelerated. From an analytical estimate employing an upstream PUI distribution derived by Vasyliunas & Siscoe (1976) in combination with the self-consistently determined SW deceleration, one obtains (a) 0.02% and (b) 15% for the actual injection rate. These numbers demonstrate not only that the values obtained numerically represent a tendency of the algorithm to accelerate particles too efficiently (Hawley, Smarr, & Wilson 1984), but also that, in order to reproduce the observed spectrum with a high-injection case, the actual injection fraction has to be increased. Such an increase could be achieved by the inclusion of a preacceleration mechanism for PUIs (e.g., Chalov & Fahr 1996; Fichtner et al. 1996; Lee et al. 1996; Zank et al. 1996). While the modulated spectra for distances smaller than 60 AU are basically identical, there are differences farther out due to the different HS structure. The parameters describing this structure are listed in Table 1 for both injection cases (4 and 5) along with those for three noninjection cases (1–3) serving as reference solutions. Case 1 corresponds to a SW expansion unaffected by the presence of PUIs and CRs. Consequently, the HS is strong, with a compression ratio s 5 4.0, and there is no deceleration upstream. The mere presence of neutral particles and, subsequently, PUIs, i.e., case 2 (dashed lines, Figs. 2a and 2b), decelerates the SW (12.8%). A higher temperature of the gas mixture in the outer heliosphere causes its Mach number to decrease from 141 to 4.3; thus the compression ratio decreases to s 5 3.5. Because of the lower SW ram pressure, the HS moves in from 80.8 to 73.7 AU. Reference case 3 also includes GCRs. The effect is twofold: besides further reducing the heliocentric distance of the HS to 71.8 AU as a consequence of the increased external pressure ( pth,LISM 1 pGCR ), GCRs enhance the SW deceleration. Compared with case 2, the deceleration is increased in a region of 137 AU upstream by 14.5%, resulting in a total increase in deceleration of 17.3%. In the following, we refer to this region of deceleration in excess of that due to PUIs (case 2) as the precursor. If such a precursor exists, the shock transition is referred to as a subshock. The extended precursor of case 3 can be attributed to the high effective diffusion length ( k /u) of GCRs. The results obtained for these reference cases correspond to those reported by other authors (e.g., Ko et al. 1988, Lee & Axford 1988, Lee 1996). We now turn to the situation where all three energetic particle populations are present simultaneously. Figure 2 shows the HS structure resulting from (a) low and (b) high injection. According to Grzedzielski & Ziemkiewicz

FIG. 2.—Structure of the HS in the presence of PUIs, GCRs, and ACRs for injection efficiency of (a) h 5 0.0003 (case 4) and (b) h 5 0.9 (case 5) compared with the solution without CRs (case 2). The actual values of the shock position, subshock compression ratio, and extent of the precursor are given in Table 1.

(1990) and Lee (1996), the ACRs should push the HS away from the sun. Our results confirm this finding (see Table 1). While the effects of GCRs and ACRs on the subshock location nearly compensate each other, so that the location is close to that of the pure PUI case 2, the other parameters defining the HS structure are distinctly different. For the low-injection case 4 the compression ratio is slightly lower than for case 3, the extent of the precursor is somewhat shorter (134 AU), and the deceleration of the SW is more pronounced (19.8%). Even so, the precursor is determined mainly by GCRs and the subshock by PUIs. For case 5, the SW deceleration is 45.0% and both the compression ratio (s 5 1.5) and the precursor extent (112 AU) are significantly reduced, indicating that ACRs, with a relatively small effective diffusion length, dominate the overall structure of the HS. Note, however, that for such a weak HS it is not easy to determine the exact compression ratio. Our low-injection case differs from the finding by Stone et al. (1996), who found a compression ratio of 2.63 H 0.14 for a GCR-modified HS. For a comparison with our result, s 5 3.5, one has to keep in mind that the influence of anomalous helium is not yet included and that it might reduce the compression ratio further. The difference between the two cases can be further illustrated with a comparison of the combined ACR and GCR pressure with the SW ram and thermal pressure (Fig. 3). For h 5 0.0003, the contribution from ACRs is negligible and the CR pressure profile is rather flat. Also, the CR pressure is everywhere smaller than that of the thermal plasma; its ram pressure dominates upstream, its thermal pressure downstream. For h 5 0.9, however, the acceleration of ACRs results in a significant CR pressure at and downstream of the

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LE ROUX & FICHTNER These effects occur because the steeper spectral gradient ­jT/­T at the HS (see Fig. 1) implies a larger Compton-Getting factor C 5 [2 2 (T 1 2E0)/(T 1 E0) ­ln jT/­ln T]/3, where E0 is the particle’s rest energy. As a consequence of both effects, case 5 gives flux levels similar to those of case 4 inside 160 AU. At distances smaller than 160 AU, it is difficult to distinguish between the low- and the high-injection case (see Figs. 1, 2, and 3). Thus, an observational discrimination between the two alternatives (before a spacecraft encounters the HS) can only be made in that part of the precursor that is close to the HS. While Pioneer 10 will soon run out of power and cease to function, and Voyager 2 is currently located at only 151 AU, Voyager 1 (located at 165 AU) should soon be able to discriminate between the two alternatives on the basis of the spatial gradients gr. These CR observations, however, cannot be supplemented by plasma measurements before the year 2000, when Voyager 2 will reach distances greater than 160 AU. 4. SUMMARY

FIG. 3.—CR pressure (solid line) compared to the ram (dashed line) and the thermal pressure (dash-dotted line) for (a) h 5 0.0003 and (b) h 5 0.9. All pressures are in eV cm23.

HS. The strong modification of the HS shown in Figure 2b and given in Table 1 can be understood in view of the large CR pressure gradient close to the HS seen in Figure 3b. This gradient forces the SW to decelerate strongly. It is also evident why both the high- and the low-injection case result in similar flux levels for distances smaller than 160 AU: the production of ACRs leads to a pressure buildup close to, at, and beyond the HS, but not very far upstream. If only a small fraction of the PUI population is injected, the subshock strength remains relatively high (s 5 3.4) and the acceleration remains sufficiently efficient to produce the observed flux levels. If, on the other hand, a larger fraction of PUIs gets injected, the HS is strongly modified (s 5 1.5) and its acceleration efficiency reduced, accompanied by stronger modulation, i.e., larger radial gradients gr 5 Cu/ k , particularly beyond 160 AU.

With a new time-dependent and self-consistent model we have studied the influence of pickup, anomalous, and Galactic cosmic-ray protons on the structure of the heliospheric shock. We found two different modifications of the HS that result in cosmic-ray flux levels and solar wind plasma properties compatible with observational data obtained by Pioneer and Vo yager in the outer heliosphere. The first alternative assumes a low injection efficiency of pickup protons into the process of diffusive acceleration at the heliospheric shock. For this case the strength of the subshock and the structure of the precursor remain dominated by pickup and Galactic cosmic-ray protons, respectively. In contrast, the second alternative, with high injection efficiency, represents a shock structure mainly determined by anomalous cosmic-ray protons. Although in this case significantly more particles are injected, their subsequent acceleration is rather inefficient because the subshock is weak as consequence of a steep cosmic-ray pressure gradient on its upstream side. Because of stronger modulation near the shock, the modulated spectrum and properties of the thermal plasma are almost indistinguishable from those of the lowinjection case for distances smaller than 160 AU. Consequently, only data obtained beyond 160 AU can clarify the actual structure of the heliospheric shock. Such cosmic-ray measurements are currently being made by Vo yager 1. They will soon (in the year 2000) be supplemented with in situ plasma observations to be made with Voyager 2.

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