NASA Technical Reports Server (NTRS) 19990008887: Direct Acceleration of Pickup Ions at The Solar Wind Termination Shock: The Production of Anomalous Cosmic Rays

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DIRECT ACCELERATION OF PICKUP IONS
AT THE SOLAR WIND TERMINATION SHOCK:
THE PRODUCTION OF ANOMALOUS COSMIC RAYS
Donald C. Ellison
Department of Physics
North Carolina State University
Raleigh, NC 27695, U.S.A.
Frank C. Jones and Matthew G. Baring
Laboratory for High Energy Astrophysics
NASA Goddard Space Flight Center
Greenbelt, MD 20771, U.S.A.
Accepted for publication in the Astrophysical Journal.

THE ASTROPHYSICAL JOURNAl,, INPRESS
Preprint typeset using D,TEX style emulateapj
DIRECT ACCELERATION OF PICKUP IONS AT THE SOLAR WIND
TERMINATION SHOCK: THE PRODUCTION OF ANOMALOUS COSMIC RAYS
DONALD C. ELLISON
Department of Physics, North Carolina State University,
Box 8202, Raleigh NC 27695, U.S.A.
don_ellis onOncsu.edu
FRANK C. JONES _z MATTHEW G. BARING l
Laboratory forHigh Energy Astrophysics,
NASA Goddard Space Flight Center, Greenbelt, MD 20771, U.S.A.
]rank.c.jonesOgsfc.nasa.gou, baringOlheavx.gs]c.nasa.gov
The Astrophysical Journal, in press
ABSTRACT
We have modeled the injection and acceleration of pickup ions at the solar wind termination shock
and investigated the parameters needed to produce the observed Anomalous Cosmic Ray (ACR) fluxes.
A non-linear Monte Carlo technique was employed, which in effect solves the Boltzmann equation and
is not restricted to near-isotropic particle distribution functions. This technique models the injection of
thermal and pickup ions, the acceleration of these ions, and the determination of the shock structure
under the influence of the accelerated ions. The essential effects of injection are treated in a mostly self-
consistent manner, including effects from shock obliquity, cross-field diffusion, and pitch-angle scattering.
Using recent determinations of pickup ion densities, we are able to match the absolute flux of hydrogen
in the ACRs by assuming that pickup ion scattering mean free paths, at the termination shock, are
much less than an AU and that modestly strong cross-field diffusion occurs. Simultaneously, we match
the flux ratios He+/H + or O+/H + to within a factor ,-, 5. If the conditions of strong scattering apply,
no pre-termination-shock injection phase is required and the injection and acceleration of pickup ions
at the termination shock is totally analogous to the injection and acceleration of ions at highly oblique
interplanetary shocks recently observed by the Ulysses spacecraft. The fact that ACR fluxes can be
modeled with standard shock assumptions suggests that the much-discussed "injection problem" for
highly oblique shocks stems from incomplete (either mathematical or computer) modeling of these shocks
rather than from any actual difficulty shocks may have in injecting and accelerating thermal or quasi-
thermal particles.
Subject headings: Cosmic rays: general -- particle acceleration -- shock waves -- diffusion --
interplanetary medium -- termination shock
1. INTRODUCTION For input at the termination shock, we use a standard
expression for the shape of the isotropic pickup ion phase-
It is believed that Anomalous Cosmic Rays (ACRs) orig-
space distribution based on the derivation of Vasyliunas &
inate as interstellar pickup ions (Fisk, Kozlovsky, & Ra-
Siscoe (1976) (e.g. Gloeckler et al. 1993, 1994; le Roux,
maty 1974) which are accelerated at the solar wind termi-
Potgieter, & Ptuskin 1996), and normalize this to the val-
nation shock (Pesses, Jokipii, _ Eichler 1981). Such ions
ues reported by Cummings & Stone (1996) for the inter-
originate as neutrals that are swept into the solar system
stellar ion flux in the heliosphere (see also Stone et al.
from the external interstellar medium, and subsequently
1996; Isenberg 1997). We use the Cummings & Stone
ionized by the solar UV flux or by charge exchange with
fluxes, even though more recent values have been reported
solar wind ions. Recent observations of pickup ions by the
(e.g. Gloeckler 1996; Gloeckler, Fisk, & Geiss 1997), so
Ulysses spacecraft (e.g. Gloeckler et al. 1993) adds to the
we can make a direct comparison with their results. In
indirect evidence for this scenario, which by now has be-
addition, since important parameters are uncertain at the
come quite compelling. However, one essential element of
termination shock, we perform a limited parameter sur-
the process, namely how pickup ions are first injected into
vey but always find that we can easily match the observed
the acceleration mechanism, has engendered controversy.
flux of H+,by varying the strength of scattering. For typ-
We show here that standard and well-tested assumptions
ical cases, we require that All _ 5-10 r0, where All is the
of diffusive (also called first-order Fermi) shock accelera-
scattering mean free path parallel to the mean magnetic
tion allow the direct injection and acceleration of pickup
field and ra is the ion gyroradius. This length scale of
ions without a pre-injection stage. We have employed
diffusion parallel to the field seems fairly typical of that
our Monte Carlo simulation code (e.g. Ellison, Baring,
inferred in the vicinity of planetary bow shocks (Ellison,
& Jones 1996) to study the physical parameters that the
MSbius, & Paschmann 1990), interplanetary shocks (Bar-
solar wind termination shock must have in order to pro-
ing et al. 1997), supernova shocks (Achterberg, Blandford,
duce the observed ACR fluxes.
& Reynolds 1994), and that found in hybrid simulations
of quasi-parallel shocks (e.g. Giacalone, et al. 1993) but is
1Universities Space Research Association
1
ELLISONJ,ONES&, BARING
muchlessthanthatfoundforthe
undisturbed interplane- planetary shock would, of course, make the diffusive shock
tary medium (e.g. Forman, Jokipii, & Owens 1974; Palmer accelera_ ion of these particles to ACR energies impossible,
1982; Moussas et al. 1992; Bieber et al. 1994; Gloeckler, since so:ne particles must be able to diffuse back to the
Fisk, & Geiss 1997). If the turbulence we postulate for shock fr(_m the downstream region in order to obtain MeV
pickup ions is, in fact, present, it implies that the termi- energies.
nation shock generates fairly strong, local magnetic field Our nonlinear shock acceleration model calculates the
turbulence as has long been observed or inferred at other full distribution functions of the various ion species at the
collisionless shocks (e.g. Lee 1982; 1983 and references shock in,:luding effects from the shock smoothing produced
therein). We are somewhat less successful in matching by the back-reaction of accelerated particles on the solar
the ACR flux ratios, He+/H + and O+/H +, seeing less en- wind flow. The three most abundant ACR species, H+,
hancement based on mass/charge than reported by Cum- He+ ,an.J O+, are included self-consistently in the deter-
mings & Stone (1996). We do, however, match the ratios minatio_t of the shock structure. Since our Monte Carlo
to within a factor of ,-, 5, a relatively small difference given technique has not yet been generalized to spherical geom-
the uncertainties of extrapolating flux densities to the ter- etry, we are forced to assume that the termination shock
mination shock and the possibility that species-dependent is plane. However, the most important process we inves-
heating or pre-acceleration could occur in the solar wind tigate, the injection of pickup ions, occurs locally and will
before pickup ions reach the termination shock. not be _riously affected by this approximation. In addi-
Regardless of any uncertainty about flux ratios, that tion, this implementation of the Monte Carlo simulation
fact that we can model the absolute hydrogen flux with does not treat solar modulation in a complete fashion, nor
no pre-acceleration is in clear contradiction with the con- does it include adiabatic losses; we anticipate including
clusions of most previous work addressing pickup ion in- these in future work. For now, we artificially mimic the ef-
jection at the termination shock. For the most part, pre- fects of ",_liabatic losses and truncate acceleration by plac-
vious work has argued that the highly oblique termina- ing a frec escape boundary upstream from the plane shock.
tion shock would not be able to accelerate pickup ions We also neglect the presence of galactic cosmic rays on the
directly. It was postulated, for example, that some inde- shock structure, relying on the estimate of Fisk (1996) that
pendent pre-acceleration phase, perhaps at interplanetary galactic cosmic rays will not produce pressure gradients
shocks (e.g. Jokipii & Giacalone 1996) or by second-order strong eaough to smooth the termination shock.
Fermi acceleration off Alfv6n waves (e.g. Isenberg 1986; The most important model parameter we require is
Bogdan, Lee, & Schneider 1991; Fichtner et al. 1996), r/= Aii/rg, the ratio of the particle mean free path paral-
or transit-time damping of magnetosonic waves (e.g. Fisk lel to the magnetic field to the particle's gyroradius, and
1976; Schwadron, Fisk, & Gloeckler 1996), or shock 'surf- this is c_osen to match observed spectral intensities. For a
ing' (e.g. Lee et al. 1996; Zank et al. 1996) was necessary given r/, our model gives the absolute normalization of all
before the pickup ions encountered the termination shock spectra _t the shock. Unfortunately, the absolute inten-
and underwent their final acceleration to ACR energies. sities at,.' strongly dependent on the pickup ion densities
It has also been suggested that the termination shock was at the termination shock which are uncertain. Our deter-
not quasi-perpendicular for a substantial fraction of the mination of the relative intensities of different ion species,
time (e.g. Liewer, Rath, & Goldstein 1995; Chalov & Fahr however, is not influenced in any important way by the
1996b) thus allowing injection at times when the shock was absolute normalization or by small changes in r/, although
less oblique. Furthermore, Chalov & Fahr (1996b) and Le changes in Mach number can affect relative intensities as
Roux, Potgieter, & Ptuskin (1996) suggest that reflected we show below. In partial support of the findings of Cum-
pickup ions from an already energized population serve mings & Stone (1996), we see evidence for an acceleration
as seed particles for Fermi acceleration. While it is cer- efficienc:, that increases with A/Q (mass number to charge
tainly possible that some pre-acceleration may occur or number's. The actual values that we obtain, however, are
that the shock is highly variable, our results indicate that less than those inferred by Cummings & Stone. Similar
the termination shock seems fully capable of injecting and A/Q enhancement effects have been reported for diffuse
accelerating pickup ions directly in a single step if stan- ions observed at the quasi-parallel Earth bow shock (Elli-
dard diffusive shock acceleration assumptions are made son, MSbius, & Paschmann 1990).
and if the self-generated turbulence is as strong as rou-
2. MODEL
tinely assumed in virtually all other astrophysical shocks
which accelerate particles. Since diffusive shock accelera- The Itlonte Carlo technique we use here has been de-
tion predictions have been tested extensively and success- scribed _n Ellison, Baring, & Jones (1996) and we refer
fully at directly observable shocks in the inner heliosphere the reader to that paper for complete details. Briefly, we
(and less directly at shocks outside the heliosphere), we see have de_ eloped a technique for calculating the structure of
no physical reason why the termination shock should act a plane, steady-state, collisionless shock of arbitrary obliq-
differently, i.e. should be incapable of generating sufficient uity and arbitrary sonic and Alfv_n Mach numbers greater
turbulence, or why standard shock assumptions shouldn't than one. We include the injection and acceleration of ions
apply (e.g. Drury 1983; Jones & Ellison 1991). We note directly from the background plasma and assume that,
that claims of extremely weak scattering of pickup ions with th_ exception of pickup ions, no ad hoc population of
(All ,,-AU) seem to be based on modeling of the quiet in- superth_ rmal seed particles is present. The model assumes
terplanetary medium (Gloeckler, Fisk, & Geiss 1997; Fisk that the background plasma, including accelerated parti-
et al. 1997; MSbius et al. 1998). Convincing evidence that cles, an, 1magnetic fields are dynamically important and
the mean free paths of pickup ions with energies much less their effects are included in determining the shock struc-
than ACR energies are ,-, AU immediately behind an inter- ture. TI _emost important difference between the code we
PICKUP IONS AT THE TERMINATION SHOCK
employ here and that described in Ell±son, Baring, _: Jones clear that a plane shock assumption is valid for the curved
is that we are no longer restricted to subluminal shocks, termination shock. However, the curvature of the termina-
i.e. shock geometries where the de Hoffmann-Teller (H-T) tion shock will only be important if the diffusion length of
speed is less than the speed of light (note, however, that particles is comparable to the shock radius, at which point
our application in this paper to the termination shock still high energy particles tend to leak away from the shock and
focuses on subluminal shocks). We no longer move par- the acceleration ceases. Otherwise the shock appears pla-
ticles by transforming into the H-T frame, a frame where nar to the accelerating particles. Adiabatic losses in the
the u x B electric field disappears. Instead, we move par- expanding solar wind are more of a concern, since these
ticles in the normal incidence frame and explicitly include losses can be shown to set the maximum energy particles
effects of the u × B field in translating the particles, al- obtain (Jones, in preparation). For our work here, we
lowing us to model shocks of arbitrary obliquity. Apart parameterize the maximum energy obtainable by placing
from this generalization, and the injection of pickup ions, a free escape boundary at a distance upstream from the
the code used here is essentially identical to that described shock. The distance is chosen to give maximum energies
in Ell±son, Baring, & Jones (1996). _, 100 MeV, typical of ACRs. We find (we believe coinci-
The most basic assumption we make is that the compli- dentally) that the diffusion lengths of the highest energy
cated plasma physics can be described by a simple scat- particles along the magnetic field are comparable to the
tering relation for individual particles, i.e. , pole-to-equator distance. Since our models generally have
_¢± << tell, the maximum scalelength perpendicular to the
1 termination shock (i.e. in the radial direction) is much
All - _Trg or _Cll= _orgv , (1) shorter, of the order of a few AU.
Another important simplification is that we do not in-
where v is the particle's speed in the local frame, rg -- clude a cross-shock, charge separation potential in our
pc/(QeB) is the gyroradius of a particle of momentum p model. A cross-shock potential should exist and such a
and charge Qe, Allis the mean free path parallel to the lo- potential may have some effect on injection. We leave this
cal magnetic field, Xllis the diffusion coefficient parallel to generalization to later work.
the local magnetic field, and r/is a model parameter which Once a satisfactory model for oblique shocks and shock
characterizes the strength of scattering and the importance acceleration is developed, it becomes clear that the ma-
of cross-field diffusion. We assume _ is a constant inde- jor problem with modeling a given source is the array of
pendent of particle energy, particle species, and position parameters that are required. Oblique shocks are compli-
relative to the shock. It's clear from equation (1), that as a cated and we do not see how they can be modeled self-
particle convects, it will on average scatter after moving a consistently without a large number of both environmen-
distance )'11along the magnetic field. We assume the parti- tal (e.g. Mach numbers, shock speed, size, obliquity, etc.)
cles pitch-angle scatter elastically and isotropically in the and model (e.g. _/,type of scattering assumed, cross-shock
local plasma frame regardless of their energy. After each potential, etc.) parameters. While simplifications and a
pitch-angle scattering (which occurs every 6t << All/V) a reduction in the number of parameters can be made in
new direction is obtained for a particle's velocity vector some circumstances, the most extreme and useful being
and a new gyrocenter is calculated. After some number of the assumption of a plane-parallel shock (eBn = 0 ev-
scatterings, a particle's pitch angle will deviate by ,_ 90° erywhere, where OBn is the angle between the magnetic
from it's original direction and it will be gyrating around field and the shock normal), they cannot be made if the
a field line within 2rg of the one the particle was circling obliquity and particle acceleration are important, i.e. if
originally. Such cross field diffusion is an integral part the accelerated particles are numerous enough so that a
of diffusive acceleration at oblique shocks (e.g. Jokipii test-particle solution is unrealistic. In this case, all of the
1987; Ostrowski 1988) and Ell±son, Baring, & Jones (1995) parameters become important and must be included.
showed that the scheme we employ here and in Ell±son,
Baring, & Jones (1996) for cross-field diffusion, together
3. RESULTS
with the assumption contained in equation (1), is equiva-
lent to a kinetic theory description of diffusion (e.g. Ax- The upstream parameters required for a solution are:
ford 1965; Forman, Jokipii, & Owens 1974; Jones 1990), the shock speed, Vsk = Vsw (in the solar wind frame), the
where the diffusion coefficients perpendicular to (x±) and magnetic field strength, B, the obliquity, OBn, the temper-
parallel to (_11) the mean field direction are related via ature, T, the number densities of the various thermal ion
to± = _11/(1 + T/2). The parameter ,7in equation (1) then species, ni, and the number densities of the various pickup
clearly determines the "strength" of the scattering and ions, npu, all of which are ambient upstream conditions
when _ _- 1, x± --, telland particles diffuse across the mag- and can, in principle, be determined by observations (we
netic field as quickly as they move along it (the so-called assume that the termination shock is stationary so that
Bohm limit). The properties of highly oblique and quasi- the shock speed equals the solar wind speed, Vsw). Here,
parallel shocks tend to merge when the scattering is strong "upstream" means far enough in front of the shock so that
(i.e. ,7<< 10). backstreaming energetic particles do not influence the flow
We simplify our model of the termination shock in sev- parameters. We also require our model parameter, _/,and
eral important ways, namely we assume that the shock is our scattering assumption, equation (1), which depend on
plane and in a steady-state. While the steady-state as- the highly complex plasma interactions that occur in the
sumption is sensible for the termination shock unless it is shock environs. In principle, these could be determined by
undergoing some form of perturbation on time scales short comparison with observations of space plasma shocks or 3-
compared to the acceleration time of the ACRs, it is less D plasma simulations; the prescription in equation (1) is
ELLISON,
JONES, & BAitING
a simple and transparent way to model these plasma pro- to their greater conductivity; large scale averages for elec-
cesses.
tron temperatures are presented by Phillips et hi. (1995).
In addition to all of the above, we must also define the We further assume a fixed value for OBn, repeating that
size of the acceleration region by setting the distance (in our's is a plane shock model and can only describe a shock
units of mean free paths), dFEB, between the upstream with a constant far upstream obliquity. We note that,
free escape boundary (FEB) and the shock. Accelerated while dFEB << DIS in our models, the size of our system
particles that diffuse upstream of the FEB are removed along the field lines, dFEB tan OBn, is comparable to DTS
from the system, producing a high energy turnover in the since OEn _ 90°. We do not model the range of magnetic
spectrum and giving a crude approximation of adiabatic field geometries around a spherical shock.
losses. We emphasize that this complexity and array of
parameters is intrinsic to oblique shocks and must be in- 3.2. Pickup Ion Contribution to the Sonic Mach Number
cluded in any realistic model.
Pickup ions contribute to the sonic Mach number Ms
through both their mass loading of the solar wind, and
3.1. Parameters at the Termination Shock
also their velocity dispersion relative to the mean speed
We use a simple model to relate values for solar wind of the wind. This latter component is crucial to the de-
parameters at the termination shock to those at 1 AU. termination of Ms at large distances from the sun, where
Assuming that the solar wind speed remains constant in adiabatic cooling of the solar wind has diminished its pres-
its passage to the outer heliosphere, the density of a solar sure below that of the pickup ions. The sound speed in
wind ion species at the termination shock is the solar wind frame is
(1AU "__
cs = -- , for P 0cp'r _ ]fl-_ , (5)
ni,TS = _,DTs] ni,AU , (2)
the magnetic field at the termination shock is for a gas of one species (e.g. protons or helium ions),
where 7 is the ratio of specific heats for that species, P is
(1AU_ the pressure, p is the mass density, and Y is the volume.
BTS -" \DTs] BAU , (3) If the shvck speed in the solar wind frame is Vsk (_ _w),
then the sonic Mach number is
and the temperature of an ion species at the termination
/..-
shock is _ V,k _ /d V,k
Ms (6)
Cs V_ _ '
( 1AU_ 2("?sw-1)
Tt, IS = \D'_TS] Tt, AU , (4)
where the pressure, P = nm(v2)/d = p(v2)/d, is expressed
in terms of the mean of the square (i.e. dispersion) of the
where DTS is the distance to the termination shock, the
particle speeds v (measured in the solar wind frame). Here
subscript "AU" indicates values at Earth, and the sub- d is the dimensionality of the system. For a phase space
script "TS" indicates values at the termination shock. We
speed distribution f(v) of non-relativistic particles,
have assumed that the solar wind flux per solid angle is
jo /j:
conserved, and that the magnetic field strength decreases
as r-1 in the tightly-wound Archimedean "Parker" spiral (,?) : dv v2 (7)
(Parker 1958); this field is dominated by the tangential
component, while the radial component drops off as 1/r 2.
Also, the temperature is determined by adiabatic expan- For a monoenergetic pickup ion injection distribution
sion of the wind, i.e. V'rsw-lT -- constant, where V is a f(v) = _(v- Vsw), we have (v2) = V,_ (= V_), while
volume element and %w is the ratio of specific heats for the for thermal solar wind particles with exp[-mv2/(2kT)],
solar wind. We take %w = 5/3 and assume that the termi- the fam liar result (v2) = 3kT/m for a non-relativistic
nation shock is at 85 AU in all that follows. If, for example, Maxwellian emerges.
we take values at 1AU of np,AU = 8cm -_, BAU = 5x 10-s To accommodate the two-component population of so-
G, Tp, AU = 2 x 105 K, and Vsw = 500 km s-1, we have lar wind and pickup ions (denoted by subscripts "sw"
for the termination shock parameters: np,WS = 1.1 x 10-3 and "pl:," respectively), the speed of sound must by
cm -3, BTS = 5.9 x 10-7 G, Tp, IS = 535 K, and, for the modified. Since pressures and densities add linearly, i.e.
Mach numbers, Ms _- 130 and ./VIA --_ 13 (Ads is the P = Psw + Ppu and p = Psw+ Ppu, then the adiabatic laws
sonic Mach number and ¢k4a is the Alfv_n Mach num- of
ber). For this example, we have neglected pickup ions and
ion species other than protons. The addition of pickup PewV 3'_w-1 = consts and PpuV "¢pu-I = constp (8)
ions will lower Ms dramatically. We assume here and
can be used to derive
elsewhere that the electron and proton temperatures are
equal, and that all ions have the same temperature per nu- OP OP dV
2
cleon. This equality is used for its expediency and can, of Cs -_
cop OV dp
course, be relaxed if data shows otherwise. In reality the (9)
electron component of the solar wind is somewhat hotter (oP.,, oP , v
than the protons (e.g. see Baring et hi. 1997), perhaps due = +-zrr-.,.= ,
Psw + Ppu
PICKUP IONS AT THE TERMINATION SHOCK 5
where %w (_/pu) is the ratio of specific heats for the solar & Siscoe (1976):
wind (pickup ions). This exhibits an intuitive property,
namely that the pressure terms can be added in the numer-
ator and densities can be added in the denominator, imi- /pu(r,v) = _3 uo¢ v n(r, v) O(V_w - v)
tating the situation for the spring constant and the mass (a2)
in a harmonic oscillator. In general, 3'pu ¢ %w, and we ob-
where v isthe particle speed, r is the radial distance (of the
serve that if the pickup ions maintain the two-dimensional
termination shock in this application) from the sun along
ring distribution of their injection, as -/= (dpu + 2)/dpu,
the line pointing toward the nose of the termination shock,
then one would obtain _/pu= 2 for the pickup ion 7. Other- and uoo ---20 km s-1 is the velocity of the Sun relative to
wise, if the pickup ions are isotropized, as will be assumed
the local interstellar medium. Here A = _r_/uoo is the
later in the paper, then 3'pu --- 5/3 = %w. It follows, that
characteristic ionization distance for interstellar neutrals,
if dpu and (v2)pu are the dimensionality and mean square where uE is the frequency of ionization at the Earth, i.e.
speed, respectively, of the pickup ions, then
at a radial distance of rE = 1 AU from the Sun. While
A is written in terms of quantities measured at 1 AU, it
is actually independent of radius due to the 1/r 2 decline
.Ads = Vsk 3(5nn_sww+knTp/um) + dpud_+u 2 nnspwu+(Vn2p)upu } -I/2 in the solar wind density (which is involved in charge ex-
change with the interstellar neutrals) and the ionizing so-
(10)
lar UV flux. The values we adopt for the ionization fre-
At the termination shock, the solar wind is very cold, so
quencies of various ionic species at Earth are taken from
that the pickup ion component dominates the pressure, the determinations at solar minimum of Rucinski, Fahr,
primarily because the pickup ion abundance is significant
& Grzedzielski (1993), and were those used by le Roux,
(for protons, beyond around 5 AU, the pickup ion density Potgieter, & Ptuskin (1996), namely _ = 5 x 10-7 s-1,
drops off roughly as 1/r since the accumulated injection 6.7 × 10-8 s-x, and 5 × 10-7 s-1 for hydrogen, helium,
of pick up ions scales more or less as r, which is diluted by
and oxygen, respectively. These values differ significantly
the spherical expansion factor 1/r2; this contrasts the so-
(at least for hydrogen and helium) from the earlier values
lar wind, whose density scales purely as the 1/r _ dilution quoted by Vasyliunas & Siscoe (1976) and fall below the
factor). In this case, mean ionization frequencies recorded over the entire solar
cycle by factors of around 1.5 (e.g. Rucinski et al. 1996).
The Heaviside step function O(Vsw - v) is unity for non-
l v.i (11) negative arguments and zero otherwise, so that it cuts the
Ms = dp_ + 2 np_ (v2)p_ distribution off at the solar wind speed, Vsw. The factor
n(r, v) in Equation (12) is given in terms of the neutral
density in the interstellar medium, n_, by:
It follows that the dependence of the sonic Mach num-
ber on the dimensionality of the pickup ions is con-
veniently very weak, and that pickup ion abundances
npu/(n_ + npu) exceeding around 1%, limit the Mach n(r,v)=_x(l+x)2exp{-(A)(_) -3[2 II+2xl } '
number to around ten. As one moves from 1AU towards
(x3)
the termination shock, adiabatic cooling of the solar wind with
forces Ms to increase slowly (_ r) until the pickup ions
dominate the pressure and the Mach number saturates at = - , (14)
the above value. r
In all that follows, we take dpu = 3 and calculate (V2)pu and
directly from the injected pickup distributions assuming
rp(O) = 2GMo(I_- 1)/u_ . (15)
they are isotropic in the local frame.
In these expressions, G is the gravitational constant, M o is
the mass of the Sun, and/J is the ratio of the solar radiation
3.3. Adiabatic Evolution of the Pickup Ion Distribution
pressure to the solar gravitational force. Hence rp (0) is the
As the solar wind expands in its progression to the outer negative (for # _<1) of the radius where the gravitational
heliosphere, it cools as does the pickup ion distribution. potential (suitably modified for radiation pressure) equals
The pickup ions are, however, continually injected at rates the kinetic energy of the interstellar neutrals, i.e. approxi-
depending on their distance from the sun, so the determi- mately where gravitational deflection of neutrals becomes
nation of their distribution at the termination shock is important. Note that, except for the step function, the
non-trivial. The calculation of the injection rates and re- particle speed v and the radial distance r in Eqs. (12)-
suiting distribution function depends on details such as (13) always appear in the combination ri -----(v/Vsw) 312 r,
the radial variations of the ionizing solar UV flux and so- which isjust the radius of injection of pickup ions that adi-
lar wind density, and the gravitational focusing of inter- abatically cool to speed v at radius r. Therefore, if n(r, v)
stellar neutrals in the inner heliosphere. We use a stan- is expressed as a function of ri, n represents the density
dard expression for the pickup ion phase-space distribu- of neutrals at the radius ri of injection. These character-
tion, fpu (r, v), in the solar wind frame, in the nose region of istics of adiabatic evolution of the pickup ion distribution
the termination shock (e.g. Gloeckler et al. 1993, 1994; le were established by Vasyliunas & Siscoe (1976), who also
Roux, Potgieter, & Ptuskin 1996), for a three-dimensional presented results for two-dimensional pick-up ion popula-
isotropic population, based on the derivation of Vasyliunas tions.
ELLISON, JONES, & BAILING
Except for minor changes in notation, the above expres- noo(He) = 0.0153 cm -3. These differences are relatively
sions are taken directly from le Roux, Potgieter, & Ptuskin small an( we use the Cummings and Stone values to allow
(1996), and following them, we use p = 0.7 for hydrogen for a direct comparison. The values are listed in Table 1
and p = 0 for helium and oxygen to generate the pickup under Model I along with corresponding solar wind vMues
ion distributions. For our comparisons with the ACR ob- at the Earth, for densities, temperatures, and the esti-
servations presented in Cummings & Stone (1996), we nor- mated magnetic field strength. We assume OBn = 89°,
malize the pickup distributions generated with the above inject the thermal and pickup ions with far upstream (i.e.
equations to the densities estimated by Cummings and -x >> r/rgl, where rgl -= mpV6kc/e) phase space distribu-
Stone. tions as shown in Figure 1, and use _ = 14, chosen to give
a good fi', to the observed ACR intensities, as will become
evident shortly.
_ H+ Model I
-2O TABLE 1
He 2+ TS at 85 AU PARAMETERS FOR TERMINATION SHOCK MODELS
6" /
E
u -22 "r='l'L_ _" "1_ Parameters a Model I Model II Model III
_3 "lt.t] Thermal solar wind
V,k [kin s-i] 500 500 360
:1.,-,_-,...,_....,. -lll t T/nuc = 5,.35 K eBn 89° 800 87°
>,
•"_ -24 0s+ _, ;J BTS [G] 5.9x 10-7 5.9x 10-7 8x 10-7
"Ltt,j
c'-
14 35 5
a
Hydrogen
u -26 LiJ Pickup H+
np,AU icm -3]b 8 8 2.5
np,TS [cm -3] 1.1 x 10-3 1.1 x 10-3 3.44 X 10-4
ID -2a npu [cm-3]c 2.0x 10-4 2.0x 10-4 2.43x 10-4
1-- Helium
13_
"t_. l nHe.AU [cm -3] b 0.4 0.4 0.12
0
-'J --30 ' ,...,.,.•._.. r',_,,...,,.04, nHe.TS [cm -3] 5.5 x 10-s 5.5 x 10-s 1.72 x 10-s
J" ''l,... pu lore-ale 4.6 x 10-s 4.6 x 10-6 5.65 X 10-6
L'J. "x,_. nile
Oxygen
-52 ........ I ........ , ........ nO,AU :cm-3] b 8 X10-3 8 x 10-3 2.5 x 10-3
10 -3 0.01 0.1
nO,TS [cm -3] 1.1 x 10-s 1.1 x 10-6 3.44 x 10-z
Velocity/Vsw [Solar wind frame] npou [cm-3]c 1.1 x 10-v 1.1 x 10-v 1.34 x 10-v
24s 5.2 5.2 3.1
FIG. 1.-- Upstream Phase-space densities for pickup ions ex-
J_4A 15.4 15.4 5.4
pected at 85 AU calculated using equations (12) and (13) with
noo(H) = 0.077, noo(He) = 0.01, noo(O) = 9.7 x 10-s cm -3. The r 3.54 3.54 2.77
velocity is in units of the solar wind speed, Vsw. The three thermal
ion species are all injected with atemperature per nucleon of 535 K dFEb [A0]d 274 1100 4030
with charge states: H+, He2+, and Os+ . The pickup ions have a dFEE [AU] d 2.3 23 6.3
charge state of +1. The flat nature of the He+ distribution relative
A0 [AU] 8.3 x 10-3 0.02 1.6 × 10-3
to those of Hand 0 reflects its much longer ionization length.
2dll/(_rRtk) 0.98 0.97 0.91
3.4. Direct Acceleration of Anomalous Cosmic Rays at NOTS.-- (a) All models assume that the shock is at 85AU
the Termination Shock and that the temperature per nucleon at 1 AU is 2 x 105 K
and is ,,_500 K at the termination shock for all ion species. In
We now present a model for the acceleration of anoma- all cases, the electron temperature is set equal to the proton
lous cosmic ray H+ , He + , and O+ . Using the observations temperat are. (b) The solar wind values assumed for Models
ofGeiss et al. (1994) and the model ofVasyliunas & Siscoe I and 1I _ive the ratios, np,AU/nXe,AU/nO,AU = 1/0.05/0.001,
(1976) just discussed, Cummings & Stone (1996) estimate which at, close to normal solar wind values (e.g. Ipavich et al.
the following pickup ion fluxes at the nose of the helio- 1988 and references therein). (c) The pickup ion values we use
sphere: Fp" __ 1.0 x 104 cm -2 s-1, F_Iu _--230 cm -2 s-1, for Modets Iand II axe from Cummings & Stone (1996) who give
and Fpu = 5.3 cm -_ s-1. Again assuming that the solar the ratios: 5.3/228/10240 for pickup O/He/H number fluxes
wind speed is constant and equal to 500 km s-1 , the pickup (in units of cm -2 s-l). For V,k = 500 km s-1, these give the
number densities shown. The pickup ion densities for Model III
densities at the termination shock are then: nppU_ 2x 10-4
are set t( give the same He+/H+_ _ 43 and O+/H+_ 1800 ratios
cm -3, nilepu ,--,, 4.6 x 10-6 cm -3, and nPou = 1.1 x 10-7
as Model s Iand II. (d) These values of dFEB, listed both in units
cm -3. These values correspond to noo(H) = 0.13 cm -a,
of A0 an( AU, are chosen to yield proton turnover energies near
noo(He) = 0.02 cm -3, and noo(O) = 7 x 10-5 cm -3,
150 MeV. The parameter A0 = r/rgl is calculated using BTS.
somewhat different from the values assumed by le Roux,
Potgieter, & Ptuskin (1996). More recently, Gloeckler,
Fisk, & Geiss (1997) report noo(H) = 0.115 cm -3 and
PICKUP IONS AT THE TERMINATION SHOCK
magnetic field goes smoothly from OBn -" 890 far up-
stream to OBn --" 89.720 downstream. The addition of
the pickup ions has caused the sonic Mach number to de-
Model I crease substantially from the example we gave above, i.e.
0.8
here ]Vis = 5.2 versus fl4s = 130 without pickup ions (see
®Sn "=89°
-_ 0.6 Table 1).
"0---14 Note that even though dFEn, which is measured along
= 0.4 r = 3.54 the shock normal, may be a small fraction of the shock
radius, the high obliquity means that ions will move much
0.2 greater distances along the shock face. Setting the distance
parallel to the shock face in our plane-shock approxima-
0
tion to be dll --, dFEB tan OBn, we require for consistency
1.4
that particles stream no more than the pole-to-equator
distance, i.e.
1.2
dll < tiP, k/2, (16)
or,
rrRsk
0-
0.8 dFEB _ 2tanOBn ' (17)
0.6 Clearly, the quasi-spherical geometry of the termination
shock renders the effective value of dll somewhat (but not
1.4 much) less than the bound in equation (16). The quan-
tity, 2dll/(rP_k ) is listed in Table 1 and for this example
D 1.2 (Model I), 2dli/(_r/_k ) __ 1.1.
In Figure 3we show the model spectra, calculated at the
E 1 termination shock, along with Voyager 1 (V1) data mea-
sured (well within the termination shock) during 1994 on
• 0.8 days 157-313 (Cummings & Stone 1996; see also Christian,
¢-
Cummings, & Stone 1995). The value of q has been cho-
0.6 sen to obtain general agreement with the normalization of
the ACR proton observations, but there has been no other
-2 -1 0
adjustment of normalization in the top panel. Smaller val-
Linear Distance, x [V rgl]
ues of _/(i.e. stronger scattering) would yield higher model
intensities at ACR energies (i.e. in conflict with the data)
FIG. 2.-- Determination of shock structure by iteration. The top
and larger values would yield lower model intensities. Note
panel shows the x-component of the flow speed, ux(x), the middle
panel shows the xx-component of momentum flux, and the bottom that the intensity of the ACR H+ peak at ,-- 50 MeV is
panel shows the energy flux, all normalized to far upstream values. ,-, 10 orders of magnitude below the H+ pickup bump at
In each panel, the first and last iterations are shown as dashed lines
,-_ 2 keV; only a tiny fraction of pickup ions need to be
and solid lines, respectively. The energy and momentum fluxes are
conserved throughout the shock to within 1%. accelerated to energies above ,_ 10 MeV to account for the
observed ACR fluxes. The value of 7/= 14 used to obtain
The self-consistent shock profile is shown as a solid line the "fit" should be regarded only as a rough indication of
in the top panel of Figure 2, along with the zx-component the true value, given the sensitivity of the ACR flux to the
of the momentum flux and the energy flux in the lower two pick-up ion abundance and the shape of their distribution.
panels. In each panel, the dashed line is the test-particle In examining Figure 3 it must be remembered that the
quantity obtained with the discontinuous shock and the V1 observations were made at an average radial location
solid line is the value obtained after the self-consistent of 57 AU and show the effects of solar modulation. Our
smooth shock structure has been found. For a complete model spectra, on the other hand, are calculated at the
description of how the shock structure is determined, see termination shock and do not include modulation. Note
Ellison, Baring, & Jones (1996). The important point is that our shock acceleration simulation does generate low
that, even for an obliquity of 89°', the injection and ac- energy "modulation-like" depletions in upstream popula-
celeration of thermal and pickup ions is efficient enough tions (e.g. Baring, Ellison, & Jones 1994; Ellison, Baring,
to cause some departures from momentum and energy & Jones 1996) due to inefficient convection against the
conservation in the discontinuous shock; the downstream fluid flow, much like the model of Lee (1982). However,
fluxes rise to a factor of > 1.1 above the far upstream for the termination shock models of this paper, such deple-
values. Even though the shock smoothing is quite small tions appear relatively close to the shock, on scales << DTS,
(Figure 2uses a linear distance scale and the portion of the due to our incomplete modeling of particle convection and
shock shown is a small fraction of the size set by dF_8), diffusion in the complex geometry of the heliosphere. The
it is necessary to conserve momentum and energy fluxes. heavy solid line in Figure 3 is the estimate of Cummings
Our self-consistent, smooth shock solution conserves all & Stone (1996) for the power law H+ spectrum at the ter-
fluxes, including the zz-component of momentum and uni- mination shock and, as mentioned above, we have chosen
formity of the tangential electric field (not shown), across r/to approximately match this intensity (fine tuning of 7/
the shock. The angle between the shock normal and the would give a more precise match).
ELLISONJ,ONES,
& BASING
I ! ! I we would have toincrease,relativetoH,theHe+ intensity
"_ 4 by a factor,-,4,and theO + intensitya factor_.12.This
Model I
adjustmenthas been done inthebottom panelofFigure3
> (9 = 89 ° toshow thattheshapes oftheobserved ACR spectraare
-O_ 2 matched exceedingwellby our model above the modula-
L _7= 14
tionturfo,ver. In particular,our singleparameter dFEB,
-qg) simultaneouslygivesa good match to the cutoffforall
I 0 threespecies.
O \ Consideringtheuncertaintiesinvolvedinestimatingthe
mI '\ variousl_arametersneeded attheterminationshock,such
¢NE -2 \\ as the sclarwind speed and the pickup iondensities,we
O \\ believethematch indicatedinFigure 3 isacceptable.
v
t_
-O5 -4 \ I I I I I ," I 2q
C&S
O
¢_
-6 \
x"
Vl \ (gen= 700, r/ = 25 ..."" /' / j
ta_
O Observotions 0 s=>==,==_ -I
0 o
E
...'" /° /
-2
-I-0 61 o;...,.,,
¢=
.£
o -3
He+/H + Adj: 4 • - t_
O
O+/H + Adj: 12 o
t.)
-8 o --4 :; .//, / / H+ E)=8g o
, , , , , \, / Ix. &z
0
-2 -1 0 1 2 3 4 5 6
o --5 : / /_/ _ He* "r}=14
_J
Loglo Energy per Nucleon, [keV//A]
FIo. 3.-- Comparison of Voyager 1 observations of ACR H, -6
He, and O (made during 1994/157-313 when V1 was at an average
radiallocation of,,,57AU) toModel Ispectra calculated attheter-
mination shock. The model spectra have an absolute normalization -7 J 1 I , I I I
determined by the injection parameters, i.e. np,TS Vsw = 5.5 x 104 0 1 2 3 4 5 6
cm -2 s-l for the protons and corresponding values for the He and
O. The value of q has been chosen to give a general fit to the in- Loglo Energy per Nucleon [keY/A]
tensities of the observed ACR's. The sharp thermal peaks show the
relatively cold solar wind ions that have not yet thermalized. As
FIG. 4.--- Acceleration time in years versus energy per nucleon
the observation position is moved downstream, these peaks broaden.
for the H÷ (solid llne), He+ (dashed line), and O+ (dot-dashed)
Note that the H thermal peak intensity is ,-_ 11 orders of magni-
produced bt Model I. These curves axe calculated from the analytic
tude above the observed ACR intensity. The heavy solid line is the
result give_ in Ellison, Baring, & 3ones (1995). The upper limits on
Cummings & Stone (1996) estimate for the ACR proton intensity at
the O acceleration time from electron stripping at 10 and 20 MeV/A
the termination shock. In the bottom panel, we have individually
are from Adams & Leising (1991). The 70°, _ = 25 curve is included
adjusted the normalizations to match the ACFt observations. The
to indicate what shock parameters are necessary to encroach upon
relative adjustments for He + and O+ are labeled.
the experiv rental upper limits.
There are several points to consider. First, the limits on
the maximum ACR energy are such that the Cummings 3.4.1. Acceleration Time
& Stone extrapolation extends into the exponential cutoff.
An important constraint on the production of ACRs
Second, even though the shock model we are using has
comes from the charge stripping rate; clearly consider-
a compression ratio of r = 3.54, well above that inferred
ations of ACR generation are simplified when charge-
by Cummings & Stone, the spectral slope in the very lim-
stripping timescales exceed those of acceleration. Such
ited energy range (i.e. above the modulation turnover)
ionization is relevant to species heavier than He (whose
provided by the data is reasonably well fit. Because of
stripping -,imescales are long), in this case oxygen. Adams
the limited energy range and the spectral cutoff, it may
& Leising (1991) showed that 10 MeV/A singly charged
not be possible to meaningfully constrain the termination oxygen ions will be further stripped, in conflict with ob-
shock Mach number by extrapolating ACR observations servations, if they propagate more than -_ 0.2 pc in the
made well inside the heliosphere back to the termination
local interstellar medium. Jokipii (1992) showed how this
shock as done by Cummings & Stone (1996). Third, the
relates to the acceleration rates of various mechanisms,
ACR data clearly show that the observed He+/H + ratio is
and concblded that first-order Fermi acceleration at highly
greater than our model predicts. This is also true for the oblique sh3cks is the only mechanism fast enough to satisfy
O+/H + ratio and in order to match the observed fluxes, this limit Our results are in agreement with this assess-