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Modulation of anomalous protons: Effects of
different solar wind speed profiles in the
heliosheath
U. W. Langner
1
Institut fu¨r Theoretische Physik IV, Ruhr-Universita¨t Bochum, Bochum, Germany
M. S. Potgieter
Unit for Space Physics, School of Physics, North-West University, Potchefstroom, South Africa
H. Fichtner and T. Borrmann
Institut fu¨r Theoretische Physik IV, Ruhr-Universita¨t Bochum, Bochum, Germany
Received 11 February 2005; revised 20 October 2005; accepted 3 November 2005; published 21 January 2006.
[1] Two termination shock acceleration modulation models are used to study the
modulation of anomalous protons, in particular the effects of different scenarios for global
solar wind speed (V) variations in the heliosheath. The first numerical model simulates
a symmetric heliosphere and the second simulates an asymmetric heliosphere with
respect to the Sun. The modulation differences between these models are illustrated and
discussed. The geometry of the heliosphere in the latter model is deduced from a
time-dependent three-dimensional hydrodynamic model of the heliosphere which provides
the different scenarios for the V-profiles in the heliosheath. The modulation models include
the solar wind termination shock, global drifts, adiabatic energy changes, diffusion,
convection, and a heliosheath. The anomalous protons are kinetically described using the
Parker transport equation. A solar wind speed decreasing stronger than the generally
assumed V
/ 1/r
2
dependence, with r the radial distance from the Sun, is studied as well as
an extreme scenario with V
/ r
2
. The stronger decrease produces a compressive flow in the
heliosheath which results in additional acceleration of anomalous protons in the
heliosheath. The solutions are shown for solar minimum and moderate maximum
modulation conditions for both heliospheric magnetic field polarity cycles. Significant
modulation differences are found to occur between these different scenarios for V in the
heliosheath. If the stronger than V
/ 1/r
2
scenarios in the heliosheath are real, the
anomalous intensities should increase beyond the TS, which should be measurable by the
Voyager 1 spacecraft in the near future.
Citation: Langner, U. W., M. S. Potgieter, H. Fichtner, and T. Borrmann (2006), Modulation of anomalous protons: Effects of
different solar wind speed profiles in the heliosheath, J. Geophys. Res., 111, A01106, doi:10.1029/2005JA011066.
1. Introduction
[2] The two Voyager spacecraft have been measuring the
spectra of anomalous protons for several years in the outer
heliosphere [e.g., Webber and Lockw ood, 2004]. These
particles seem absent at the Earth’s orbit [Christian et al.,
1988, 1995; Cummings and Stone, 1998]. The origin of
anomalous protons is generally assumed to be interstellar
neutral hydrogen atoms that are ionized by charge exchange
with the solar wind protons relatively close to the Sun.
These ionized particles are then ‘‘picked up’’ by the solar
wind and because they have significantly higher random
velocities than the solar wind protons, they are more easily
accelerated to energies at which diffusive shock acceleration
by the termination shock (TS) becomes effective [Pesses et
al.,1981].Sincetheirdiscovery[Garcia-Munoz et al.,
1973a, 1973b] they also have been the subject of numerous
theoretical papers, studying their acceleration and propaga-
tion in the heliosphere with numerical models [e.g., Jokipii,
1986; Jokipii , 1990; Potgieter and Moraal, 1988; Steenberg
and M oraal, 1996; Sreenivasan and Fichtner, 2001;
Langner and Potgieter, 2004b]. These particles are also
the subject of study in this paper.
[
3] The interest in the large-scale structure and geometry
of the heliosphere and the transport of cosmic rays within
this configuration is not limited to an understanding of
anomalous protons in the outer heliosphere but has strongly
grown since the approach of the TS by the Voyager 1
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, A01106, doi:10.1029/2005JA011066, 2006
1
Also at Unit for Space Physics, School of Physics, North-West
University, Potchefstroom, South Africa.
Copyright 2006 by the American Geophysical Union.
0148-0227/06/2005JA011066
A01106 1of14
spacecraft [e.g., Webber et al., 2001; Krimigis et al., 2003;
McDonald et al., 2003; Stone and Cummings, 2003]. For
recent reviews on the structural shape of the heliosphere, see
Zank [1999], Fichtner [2001] and Florinski et al. [2004a].
In this work a two-dimensional (2-D) TS modulation model
is used to study the acceleration of the anomalous protons at
the TS in an asymmetrically bounded heliosphere [see
Langner and Potgieter, 2005; Langner et al., 2005] and
their subsequent propagation into the heliosheath and to-
ward the Sun. This model was further extended [Langner et
al., 2005] to include different scenarios for the global solar
wind speed ( V) in the heliosheath for galactic protons, since
it became apparent that the change of V in the heliosheath
may be different than the previously assumed V / 1/r
2
dependence [e.g., Baranov and Malama, 1993; Pauls et al.,
1995; le Roux and Fichtner, 1997; Fahr et al., 2000;
Florinski et al., 2004b].
[
4] The effect of different scenarios for V in the helio-
sheath on anomalous protons is now studied in this work
with a 2-D TS model, using as input the different solar wind
speed profiles and the shape of the heliosphere from a three-
dimensional (3 -D) hydrodynam ic model develope d by
Borrmann [2005] [see also Borrmann and Fichtner,
2005]. This is done because a reasonable treatment is
possible only if a hydrodynamic approach is used in
accordance with the particle propagation approach, thus
establishing a simplified ‘‘hybrid model.’’ These approaches
must be used together because a hydrodynamic approach
relies on a momentum-averaged diffusion coefficient,
resulting in the problem being overdetermined [Florinski
et al., 2003]. A fluid approach also does not yield particle
spectra that can be compared to available observations to
validate assumptions, whereas the particle approach, on the
other hand, cannot predict the geometry of the heliosphere,
simply relying in this regard on informed assumptions.
Therefore a time-dependent 3-D hydrodynamic model of
the dynamics of the heliosphere is used to provide the
different scenarios for the solar wind speed profiles in the
heliosheath, for the heliospheric nose and tail regions. So
far, no comprehensive hybrid model exists that effectively
combines these effects in a self-consistent treatment of the
transport of CRs [Fichtner, 2005], although first steps were
made by Ferreira et al. [2004] and Ferreira and Scherer
[2004] for galactic electrons and by Scherer and Ferreira
[2005] for protons. The solutions resulting from the com-
bination of the results of these models are therefore a step
closer to a more realistic self-consistent modulation model,
which provides reasonable scenarios for the modulation of
anomalous protons in the heliosphere, which, to our knowl-
edge, has not been done before.
[
5] The TS model is first used to study the modulation
differences caused in a symmetrica l heliosphere by the
different profiles of V in the heliosheath which are (1) V
decrease as V / 1/r
2
,(2)V decrease as V / 1/r
8
, and (3) V
increase as V / r
2
. Although some of these scenarios are
unphysical for a symmetrical heliosphere, they were used
for an empirical study in order to quantify the effects of the
different V profiles. This study is then repeat ed for an
asymmetrical bound heliosphere for two scenario s: (1) a
scenario where V decrease as V / 1/r
2
at all latitudes in the
heliosheath and (2) a scenario where V change over latitude
from a decrease of V / 1/r
8
in the heliospheric nose region
to an increase of V / r
2
in the tail. This change over latitude
of the radial component of V and its strong decrease in speed
in the heliospheric nose region are a combination of flow
stagnation, charge exchange, and the structure of the helio-
sheath due to the s ubsonic solar wind flo w there. In
particular, the role of the rV in the modulation of
anomalous protons will be investigated as the global profile
of V changes in the heliosheath. The solutions are shown for
solar minimum and moderate maximum conditions for both
polarity cycles of the heliospheric magnetic field. The
heliopause is assumed to separate the solar and interstellar
plasmas and is considered the outer modulation boundary
with the heliosheath defined as the region between the
heliopause and the TS. For this work, the intensities of
anomalous protons are forced to approach zero at the
heliopause for numerical reasons and as a first approxima-
tion, although the ano malous particles could, in reality,
diffuse into the interstellar medium. There is a reasonable
consensus that the TS should be in the vicinity of (90 ± 5)
AU [e.g., Stone and Cummings, 2001] in the heliospheric
nose direction in which the heliosphere is moving (the TS
was recently crossed at 94 AU by the Voyager 1 spacecraft),
although over a solar cycle the TS may move outward and
inward [e.g., Steinol fson, 1994; Karmesin et al.,1995;
Baranov and Zaitsev, 1998; Whang et al.,2004].The
position of the heliopause is less certain; 30–50 AU
beyond the TS in the nose direction, while in the tail
direction the heliosphere is most probably an open structure,
causing a considerable asymmetry with respect to the Sun
[e.g., Scherer and Fahr, 2003; Zank and Mu¨ller, 2003;
Ferreira et al., 2004; Scherer and Ferreira, 2005]. The
positions computed with the hydrodynamic model could
not be used directly in the asymmetric TS model because
of computer power constraints. However, the solutions
presented here do establish a reasonable presentation of
modulation in an asymmetric heliosphere and give a good
representation of what modulation effects are to be expected
if these changes in V would occur in the heliosheath.
2. Numerical Models
2.1. Kinetic Model
[
6] The modulation model is based on the numerical
solution of the time-dependent cosmic ray transport equa-
tion [Parker, 1965]:
@f
@t
¼V þhN
D
iðÞrf þr K
S
rfðÞþ
1
3
rVðÞ
@f
@ ln p
þ J
source
; ð1Þ
where f(r, p, t) represents the omnidirectional cosmic ray
distribution function, p is the particle momentum, r is
position, and t is time, with V(r, q)=V(r, q)e
r
the solar wind
velocity. Terms on the right-hand side represent convection,
gradient and curvature drifts, diffusion, adiabatic energy
changes caused by the rV, and a source function,
respectively. In this work the role of the rV in equation
(1) will be investigated as the global profile of V changes in
the heliosheath. The tensor K
S
consists of a parallel
diffusion coefficient (k
k
) and of perpendicular diffusion
coefficients (k
?
). The averaged guiding center drift velocity
for a nearly isotropic cosmic ray distribution is given by
A01106 LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH
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A01106
hn
D
i = r(k
T
e
B
), with e
B
= B/B
m
, where B
m
is the
magnitude of the modified background heliospheric
magnetic field (HMF) as described below; here k
T
(sometimes indicated as k
A
) is the diffusion coefficient
specified by the antisymmetric elements of the generalized
diffusion tensor K, which describes gradient and curvature
drifts in the large-scale HMF. The function J
source
represents in this work a source of anomalous protons,
which are injected at the TS position as a delta function at
an energy of 2.0 MeV. Although this injection energy
seems high, it was shown previously that solutions of this
particular model is independent of this energy, as long as it
is well below the spectral ‘‘cutoff’’ energy for anomalous
protons [Langner, 2004].
[
7] The TS model combines diffusive shock acceleration,
drifts, and a heliosheath, neglecting any azimuthal depen-
dence. This two-dimensional TS modeling approach was
described originally by Jokipii [1986] for a symmetrical
heliosphere and has been applied numerous times, recently
by, e.g., Ferreira et al. [2001], Potgieter and Ferreira
[2002], Langner et al. [2003], and Langner and Potgieter
[2004a, 2004b]. The HMF was assumed to have a basic
Parkerian geometry in the equatorial plane but was modified
in the polar regions similar to the approach of Jokipii and
Ko´ta [1989].
2.2. Diffusion Tensor
[
8] The basic expressions for the diffusion coefficients
k
k
, k
?
, and k
T
were given by Burger et al. [2000] for a
steady-state model, except for minor changes caused by
the introduction of the TS in this model as described by
Langner et al. [2003]. Of importance is that perpendicular
diffusion is assumed to enhance toward the poles [e.g.,
Potgieter, 1996]. These diffusion coefficients were used
previously with success for a symmetrical model for
various species of particles and are therefore not changed
for this work. They are optimal for modeling without
solar maximum effects, e.g., global merged interaction
regions. For a complete description of these diffusion
coefficients and details of the model, see Langner et al.
[2003], Langner [2004], and Langne r and Potgieter
[2004a]. For a description of the asymmetric boundary
configuration of this model, see also Langner and
Potgieter [2005].
2.3. Numerical Representation of an Asymmetrical
Heliosphere
[
9] For the first part of the study, the locations of the
heliopause and the TS in the symmetrica l model are
assumedtobeatr
HP
= 120 AU and r
s
= 90 AU,
respectively. For the asymmetrical model, equation (1) is
derived by using the coordinate transformation:
u ¼ rx y sin qðÞ; v ¼ q; w ¼ p; ð2Þ
f(r, q, p) is then transformed to g(u, v, w) with x and y
variables that can be changed in order to give the desired
locations of the TS and heliopause so that equation (1) in
polar coordinates becomes
@g
@t
¼ a
0
@
2
g
@u
2
þ b
0
@
2
g
@v
2
þ c
0
@g
@u
þ d
0
@g
@v
þ e
0
@g
@ ln w
þ j
0
@
2
g
@u@v
; ð3Þ
where the primed variables are the transformed coefficients.
The exact formulations of the coefficients in equation (3) as
well as the difference formulae were given in detail by
Haasbroek [1997] [see also Haasbroek and Potgieter,
1997]. The numerical grid was chosen to range from q =
0 to 360. The equatorial plane of the heliosphere is then at
q =90 (nose direction) and at q = 270 (tail direction),
respectively. For the asymmetrical model the different
positions of the heliopause (range of u in equation (2)) are
then assumed to be at r
HP
= 120 AU in the equato rial region
in the nose direction of the heliosphere, r
HP
= 140 AU at the
poles, and r
HP
= 180 AU in the equatorial region in the tail
direction of the heliosphere, respectively. The TS positions
are given in section 2.5.
2.4. Hydrodynamical Model
[
10] The flow fields in the heliosheath were computed
with a hydrodynamical model. This model [Borrmann,
2005; Borrmann and Fichtner, 2005] is a 3-D two-fluid
model with the solar wind and the interstellar plasma
forming one quasi-neutral electron-proton fluid and with
the interstellar neutral gas, which is treated as a pure atomic
hydrogen fluid. With the additional often-made assumption
that magnetic fields do not have a direct dynamical influ-
ence [see, e.g., Fahr et al., 2000; Zank and Mu¨ller, 2003],
the basic equations of this two-fluid model take the follow-
ing form for the rest frame of the Sun:
@
t
r
i
¼div r
i
u
i
ðÞ
þ Q
r
i
ð4Þ
@
t
u
i
¼
1
r
i
grad p
i
u
i
gradðÞu
i
þ Q
m;i
u
i
Q
r
i
= r
i
ð5Þ
@
t
E
i
¼div E
i
þ p
i
ðÞu
i
ðÞþQ
e;i
þ u
i
Q
m;i
u
2
i
Q
r
i
= 2 ð6Þ
with the mass density r
i
, velocity u
i
, and the total energy
density E
i
=
1
2
r
i
u
i
2
+ e
i
, where e
i
denotes the internal energy.
The source terms Q
ri
, Q
m,i
, and Q
e,i
contain the mutual
interactions between all species and read explicitly (at the
example of species i 6¼ j):
Q
r;i
¼ n
i
j
r
j
n
j
i
r
i
ð7Þ
Q
m;i
¼ n
i
j
r
j
u
j
n
j
i
r
i
u
i
ð8Þ
Q
e;i
¼ n
i
j
e
j
n
j
i
e
i
þ
1
2
n
i
j
r
j
u
j
u
i
2
; ð9Þ
where the n
j
i
are the ionization frequen cies describing the
processes of charge exchange, photoionization, and elec-
tron-impact ioniz ation. For their explicit values, s ee
Rucinski et al. [1996]. Mos t important is the charge
exchange for protons and hydrogen, which is modeled
according to Holzer [1972], who gave an approximation
valid for the supersonic region of the solar wind. While this
expression has been used for the heliosheath as well, a
refined treatment should employ the result obtained by
A01106 LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH
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A01106
Holzer and Banks [1969], who derived an expression for the
case that the thermal speeds are higher than the streaming
speeds. Terms related to the gravitational potential are
omitted because the inner boundary of our computational
domain is a sphere with radius 10 AU and at this and greater
heliocentric distances gravitational effects are negligible.
[
11] For the present study, only the HD solution for the
solar wind plasma in the heliosheath between the solar wind
termination shock and the heliopause is of interest. Such a
solution was presented as a contour plot of the density by
Langner et al. [2005] and, e.g., McComas et al. [2004,
Figure 6], which shows a typical result of a computation of
the interaction of a nonspherically symmetric solar wind
with latitude-independent mass flow, a speed varying from
400 km s
1
in the equatorial plane to 800 km s
1
above the
solar poles, a number density of 8.3 cm
3
in the equatorial
plane, and a temperature of 5 10
4
K at the Earth orbit. The
interstellar medium moves with 26 km s
1
relative to the
Sun and has a number density of 0.1 cm
3
and a temper-
ature of 8000 K.
[
12] The velocity profiles in the upwind and the down-
wind direction are given in the lower left and right panels of
Figure 1, respectively (open circles). These profiles show
the expected behavior: in the upwind direction a strong
decrease of the velocity is followed by a ramp-like structure
towards the heliopause (compare, e.g., with the results
shown in Figure 2 of Pauls et al. [1995]), and in the
downwind direction there is a moderate increase of the
velocity in the direction of the heliotail. This, at a first
glance, counterintuitive result is, however, well-known [see,
e.g., Zank et al., 1996; Pauls and Zank, 1997; Florinski et
al., 2003]: it is due to a charge-exchange-induced narrowing
of the heliotail that as a consequence of the continuity of the
mass flow, results in a local acceleration. While we cannot
Figure 1. Three different solar wind speed (V) profiles (top panels) as a function of radial distance
from 1 to 120 AU in the equatorial plane. In the middle panels, the corresponding divergence of the
solar wind speed is shown. The right panels show the same as the left panels but enlarged for the
heliosheath region from 85 AU to 120 AU. The bottom panels show V as used in the kinetic asymmetric
TS model as fitted to the solutions for the solar wind speed profiles as obtained with the three-
dimensional (3-D) HD model in the heliospheric nose and tail regions in the heliosheath. Note that the
TS is in the bottom panels (for the 3-D HD model) in a different location than for the top and middle
panels (for the 2-D asymmetric TS model).
A01106 LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH
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A01106
exclude the possibility that the narrowing of the tail is
somewhat overestimated in our model due to the heliotail
boundary conditions used, note, however, that the acceler-
ation effects remain small.
2.5. Implementation of the Fluid-Model Results Into
the Kinetic Model
[
13] The intensities of anomalous protons are assumed to
be zero in the interstellar medium for both models. The
position of the TS in the asymmetrical model, where the
anomalous protons are injected at all latitudes, is assumed at
r
s
= 90 AU in the equatorial nose region, r
s
=95AUatthe
poles, and r
s
= 100 AU in the equatorial tail region of the
heliosphere, respectively. The compression ratio, s = 3.2,
and a shock precursor scale length of L = 1.2 AU [see also
Langner et al., 2003] are used. This means that up to the
shock, the solar wind speed V decreases by 0.5 s starting at
L, then abruptly as a step function to the downstream value,
in total V/s. Beyond the TS, V changes as 1/r
2
,1/r
8
, and r
2
,
respectively, up to r
HP
for the symmetrical model. For
the asymmetrical model, scenarios were investigated where
V / 1/r
2
in the heliosheath at all latitudes and then where
V / 1/r
8
in the heliosheath in the heliospheric nose region
changing to V / r
2
in the tail region. It must be noted
that the modulation model assumes spherical geometry
and a purely radial solar wind velocity. This means that
some flow divergence is already built in. Using velocity
profiles from the hydrodynamic model in this case leads
to a significant over- or underestimation (depending on
whether velocity is decreasing fast er or slower than 1/r
2
)
of the actual rV because of the extra r
2
expansion
factor. In particular, in a 3-D HD model flow slowdown
toward the stagnation point near the nose is accompanied
by an overexpansion (the flow is spreading into the wings
of the heliosheath) and the actual divergence is larger
than in the spherical model with V / 1/r
8
. Conversely,
the tail flow is essentially one-dimensional and its accel-
eration as r
2
would lead to an overestimation of rV by
the spherical model. Basically, the heliosheath flow is
nearly incompressible (and hence has zero divergence) all
by itself and the effects that cause deviations from this are
mostly related to charge exchange with neutral hydrogen.
This implies that the difference between the extreme cases,
which is already small, will be further reduced.
[
14] Concerning the global latitudinal dependence of
the solar wind speed, it is assumed that V changes from
400 km s
1
in the equatorial plane (q =90, q = 270 )
to 800 km s
1
in the polar regions for solar minimum
conditions [Phillips et al., 1995] and that V has no clear
latitudinal dependence for moderate solar maximum con-
ditions as was observed by Ulysses [McComas et al.,
2000]. (Note that q is counted from 0 to 360 in order
include the downwind heliosphere for q 180 .) The
factor of 2.0 increase for solar minimum conditions
happens in the heliosphere for 60 q, q 300, and
for 240 q 120. For moderate solar maximum
conditions it is assumed to be on average 400 km s
1
in
the whole heliosphere. A modified version of the current
Figure 2. Solutions of a symmetric model of the heliosphere in the nose region (q =90) for anomalous
protons during solar minimum conditions (a =10)intheA > 0 (top panels) and A < 0 (bottom panels)
polarity cycles for different solar wind speed scenarios in the heliosheath. (left) Energy spectra at radial
distances of 1 AU, 60 AU, and the TS position (90 AU in these cases). (right) Differential intensities as a
function of radial distance at energies of 16 MeV and 200 MeV, respectively. The TS position is indicated
in the right panels with a vertical line. Black lines represent the reference solutions with V / 1/r
2
in the
heliosheath, thick gray (red in online version) lines for V / 1/r
8
, and dark gray (blue in online version)
lines for V / r
2
.
A01106 LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH
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A01106
sheet model of Hattingh and Burger [1995], which emu-
lates the waviness of the current sheet in two spatial
dimensions, was used [see also Langner, 2004]. The current
sheet ‘‘tilt angles,’’ a, as calculated by J. T. Hoeksema of
Wilcox Solar Observatory (available at http://wso.stanford.
edu), were assumed to represent moderate solar maximum
modulation conditions with a =75 and solar minimum
conditions with a =10 during A > 0 (e.g., 1990 – 2001)
and A < 0 (e.g., 1980–1990) polarity cycles. This is
accompanied by a change in V with changing solar activity
and an increase in the values of perpendicular diffusion
where the latter implies decreasing drift with increasing
solar activity [see also Ferreira et al., 2004].
[
15] The rV determines the energy losses and gains of
charged particles in the heliosphere, e.g., rV < 0 implies
further accelera tion of particles in the heliosheath, although
clearly not to the same extent as at the TS where its value
becomes significantly negative; rV > 0 in the heliosheath
implies adiabatic deceleration of particles, the same as
normally occurring inside the TS where it gets larger with
decreasing radial distances to produce the characteristic E
1
spectral slopes for protons at Earth. For an incompressible
fluid approach, rV = 0 with V / 1/r
2
so that neither
energy losses nor gains for charged particles can then play a
role in the heliosheath, which is the simplest assumption as
used before [Langner and Potgieter, 2004a, 2005].
3. Modeling Results and Discussion
[16] In Figure 1 the different V profiles and the resulting
rV in the equatorial plane as a function of radial distance
are shown. The bottom panels of Figure 1 also show the
fits for V as used in the kinetic asymmetric TS model to
the solutions for the solar wind speed profiles as obtained
with the 3-D HD model in the heliospheric nose and tail
regions in the heliosheath. From these fits it is clear that
the increase in the solar wind speed (as discussed in the
previous section) will probably occur for distances in the
heliosheath for the heliospheric tail region >350 AU,
after an initial strong decrease. The strong decrease of V in
the nose region (V / 1/r
8
) is necessitated by the fact that
the solar wind speed must become zero when the helio-
pause is reached. While principally in agreement with (part
of) the numerically computed velocity profiles, the power
law fits shown in the top panels of Figure 1, which were
used in the asymmetric TS model, must therefore be
considered extreme cases.
[
17] The first results illustrate the effect of the different
V-profiles used in the heliosheath region on the modulation
of anomalous protons for a symmetrical model of the
heliosphere (with the Sun in the center) by showing in
Figures 2 and 3 differential intensities for the solutions in
the nose region (q =90), for solar minimum (a =10) and
moderate solar maximum modulation conditions (a =75)
for both polarity cycles. The energy spectra are shown at
radial distances of 1 AU, 60 AU, and the TS position (r
s
=
90 AU in these cases) and the differential intensities as a
function of radial distance at energies of 16 MeV and
200 MeV, respectively. These solutions represent what
happens in the nose region of the heliosphere. Although
these scenarios for the solar wind profiles are strictly
speaking unrealistic for a symmetrical heliosphere, it is
shown here for comparative purposes only. The dominant
feature in the spectra for anomalous protons is the well-
reported ‘‘acceleration cutoff’’ in energy [ Potgieter and
Moraal, 1988; Moraal et al., 1999; Potgieter and Langner,
2004, 2005]. This ‘‘cutoff’’ clearly moves significantly to
higher energies when V is assumed to decrease stronger
than 1/r
2
in the heliosheath. The opposite happens when V
increases in the heliosheath. This shift in the ‘‘cutoff’’ in the
Figure 3. Similar to Figure 2 but for moderate solar maximum modulation conditions (a =75).
A01106 LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH
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A01106
spectra is probably caused by the values of the diffusion
coefficient just outside the TS. For V / 1/r
8
k
rr
just outside
the TS will be smaller than for the V / 1/r
2
case, therefore
reducing the rate of escape for higher-energy par ticles,
which lead to the shift in the ‘‘cutoff’’ to higher energies.
The opposite will happen for the V / r
2
case. The
acceleration process, as measured by the energy where
the ‘‘cutoff’’ occur, is even more effective for the A <0
cycle bec ause s hock-drift is in t he same direction as
perpendicular diffusion in the theta direction, therefore also
reducing the escape rate for the particles in this cycle. When
V / 1/r
8
in the heliosheath , additional acceleration of the
anomalous protons occurs in the heliosheath, evident from
the increasing intensities as a function of radial distance in
this region. When V / r
2
in the heliosheath, additional
deceleration (adiabatic cooling) of particles occur in this
region, resulting in the decreasing intensities as a function of
radial distance in the heliosheath. It must be noted that the
sharp decrease in intensities in front of the heliopause is
caused by the assumed boundary conditions which sets the
solutions to zero at the heliopause for all the cases because it
is assumed that there are no sources or sinks of anomalous
protons at the heliopause, which might be an oversimplifi-
cation. The effects of the different solar wind profiles in the
heliosheath on the spectra are therefore only a product of the
conditions in the heliosheath region and just downstream of
the TS. These factors have a considerable effect on the
anomalous proton spectra inside the TS; they differ for the
A > 0 cycle at solar minimum at Earth for the V / 1/r
2
and V / r
2
scenarios in the heliosheath by more than a
factor of 10 at 1 GeV and more than a factor of 100 for the
V / 1/r
2
and V / 1/r
8
scenarios. These differences are
Figure 4. Solutions of the asymmetric model for anomalous protons in the heliospheric nose ((top)
q =90), for the Voyager 1 trajectory latitude ((middle) q =55), and for the heliospheric tail ((bottom)
q = 270) regions. This is done for solar minimum conditions (a =10)intheA > 0 polarity cycle
for different solar wind speed scenarios in the heliosheath. (left) Energy spectra at radial distances of
1 AU, 60 AU, and at the TS position (90 AU in the nose region, 95 AU at the poles, and 100 AU in
the tail region). (right) Differential intensities as a function of radial distance at energies of 16 MeV and
200 MeV, respectively. The TS position is indicated in the right panels with a vertical line. Black lines
are for solutions where V / 1/r
2
in the heliosheath for all latitudes, gray (red in online version) lines
are solutions where V changes from V / 1/r
8
in the nose region of the heliosheath to V / r
2
in the tail
region of the heliosheath.
A01106 LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH
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A01106
enhanced for the A < 0 cycle. The differences which occur
inside the TS with decreasing radial distances are a product
of the global modulation effects. The different V-profiles
obviously cause differences in the intensities at the TS at
different latitudes, which in turn have an effect on the global
modulation, as shown in Figures 2 and 3. The differences
are also caused by the harder spectrum at the TS for the V /
1/r
8
scenario, caused by the shift in the ‘‘cutoff’’ to higher
energies, which in turn led to a supplement for the low-
energy population through adiabatic cooling of the high-
energy particles. If the stronger than V / 1/r
2
scenario in the
heliosheath would be real, the anomalous intensities should
increase beyond the TS, which should then be measurable
by the Voyager 1 spacecraft in the near future.
[
18] For moderate solar maximum modulation shown in
Figure 3 the ‘‘cutoff’’ in the spectra moves to even higher
energies, as noted before by Potgieter and Langner [2003]
which again should be a measurable feature in order to
distinguish between these scenarios and to establish which
one is closer to reality.
[
19] In Figures 4 to 9 the differences in modulation are
illustrated that occur for the asymmetrical TS model of the
heliosphere when V / 1/r
2
in the heliosheath at all latitudes,
compared to when V changes from V / 1/r
8
in the helio-
spheric nose region (q =90)toV / r
2
in the tail region (q =
270) over latitude in the heliosheath. As mentioned, the
latter V-profile was calculated with the hydrodynamic model
and is therefore not arbitrary. It must, however, be noted that
this is one of a series of possible profiles computed for V
and that these profiles can be highly dynamic and time-
dependent. The solar wind profiles used for the heliosheath
in the asymmetric TS model are therefore only used here to
show the maximum effects of a variable solar wind speed.
[
20] Solutions are shown in Figures 4 to 7 for the helio-
spheric nose (q =90), the Voyager 1 trajectory latitude (q =
55), and the tail (q = 270) regions of the heliosphere for
solar minimum conditions (a =10 )intheA > 0 cycle
(Figure 4), for solar minimum conditions in the A < 0 cycle
(Figure 5), for moderate solar maximum conditions (a =
75)intheA > 0 cycle (Figure 6), and for moderate solar
maximum conditions in the A < 0 cycle (Figure 7). This is
done for the two different V scenarios for the asymmetrical
model in the heliosheath. Energy spectra are shown at radial
distances of 1 AU, 60 AU, and the TS position (90 AU in
the nose region, 95 AU at the polar regions, and 100 AU in
the tail region) and the differential intensities as a function
of radial distance at energies of 16 MeV and 200 MeV,
respectively.
Figure 5. Similar to Figure 4 but for the A < 0 polarit y cycle at solar minimum modulation conditions
(a =10).
A01106 LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH
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[21] C omparing Figures 4 to 7, several modulation
aspects can be noted that will not be discussed again here
in detail, e.g., the difference between A > 0 and A <0
cycles, etc. The emphasis is rather on the effects of the
different V-scenarios in the heliosheath. For a discussion of
other modulation effects relevant to the anomalous compo-
nent, see Langner and Potgieter [2004a, 2004b]. It follows
from these figures that the modulation differences caused by
the different V-scenarios are the largest for the A < 0 polarity
cycle. The intensities are consistently less for the scenario
where V / 1/r
2
in the heliosheath for all latitudes than for
V / 1/r
8
! r
2
in the heliosheath. The differences are for
most cases less in the tail region than in the nose region.
The intensities for the V-scenarios differ markedly in the
heliosheath as a function of radial distance for 16 and
200 MeV particles, especially in the nose regions of the
heliosphere. Since the anomalous component dominates at
these energies, observations from the Voyager 1 spacecraft
should give clear indications which of the presented
scenarios are indeed more reasonable.
[
22] The corresponding anomalous proton intensities at
16 MeV and 200 MeV are shown as contour plots in the
meridional plane in Figure 8, with V / 1/r
2
for all polar
angles in the heliosheath and in Figure 9 with V / 1/r
8
! r
2
in the heliosheath. This is done again for the A > 0 and A <0
cycles and for solar minimum and maximum. The intensi-
ties in the heliosheath at these energies differ markedly for
the two scenarios. Interesting features of the asymmetric
model follow from this comparison. For the V / 1/r
2
scenario for both magnetic cycles during solar minimum,
the 16 MeV intensities in the heliosheath are moderately
latitude-dependent, with the highest intensities in the equa-
torial region, which get less latitude-dependent for moderate
maximum solar activity. Apart from the heliosheath being
wider in the tail region, not much modulation difference
occurs between the tail and nose. This scenario predicts
that there should be an abundance of these anomalous
particles in most of the heliosheath at almost all latitudes.
At 200 MeV the latitude dependence gets significantly
stronger, especially with a =10 during the A < 0 cycle.
With a =75 the intensity distribution in the heliosheath
becomes more uniform in latitude despite a change in
polarity. For the second scenario, on the other hand, this
picture of the heliosheath changes to one where the
intensities are much less in the tail region and differently
distributed with the highest intensity clearly in the nose
region of the heliosphere in all modulation situations. The
spectra at the TS position are also altered the most in the
Figure 6. Similar to Figure 4 but for the A > 0 polarity cycle at moderate solar maximum conditions
(a =75).
A01106 LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH
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A01106
heliospheric nose region as shown in Figures 4 to 7. For
this scenario the clear abundance of anomalous particles
may occur only in the nose direction of the heliosphere,
fortunately also in the directions the two Voyager space-
craft are moving. The asymmetrical modulation features
are significantly enhanced by this approach.
4. Summary and Conclusions
[23] A two-dimensional asymmetric TS model is used
to study the effects on the modulation of anomalous
protons of different solar wind speed profiles and conse-
quently different profiles for the rV in the heliosheath.
This is done for both polarity cycles and with modulation
changes from solar minimum to moderate solar maximum
conditions. First, several V scenarios in the heliosheath
were studied with a symmetrical heliosphere, with the
Sun placed in the center. Although some of the V
scenarios are unphysical for a symmetrical heliosphere,
it were done as an empirical study in order to quantify
the effects of the different V profiles. Second, two
different scenarios for V in the heliosheath were studied
with an asymmetrical model of the heliosphere. For the
first scenario for V it was assumed that the solar wind is
incompressible in this region, resulting in V / 1/r
2
in this
region. For the second scenario a three-dimensional time-
dependent hydrodynamic model was used to calculate
possible spatial dependencies for V in the heliosheath.
An approxim ation of this calculation, where V is changing
from V / 1/r
8
in the heliospheric nose region to V / r
2
in the heliospheric tail region over latitude, was used as
input for the asymmetric TS model.
[
24] The following were found for the solutions of the
symmetrical model: (1) The cutoff in energy spectra move
Figure 7. Similar to Figure 4 but for the A < 0 polarity cycle at moderate solar maximum conditions
(a =75).
Figure 8. Contour intensities for anomalous protons at (top) 16 MeV and (bottom) 200 MeV for an asymmetrical
heliosphere and a scenario where V / 1/r
2
in the heliosheath for all polar angles, for solar minimum and moderate solar
maximum conditions, and the (left) A > 0 and (right) A < 0 polarity cycles. Note that the legends of the contours correspond
to the exponent of the base 10 on a logarithmic scale and that the scaling of the legends differ for the 16 MeV and 200 MeV
plots. The TS and the heliopause are indicated in the plots by black lines.
A01106 LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH
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Figure 8
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11 of 14
A01106
Figure 9. Similar to Figure 8, but for a scenario where V changes from V / 1/r
8
in the heliosheath nose
to V / r
2
in the heliosheath tail.
A01106 LANGNER ET AL.: MODULATION OF ANOMALOUS PROTONS IN THE HELIOSHEATH
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significantly to higher energies when V is assumed to
decrease stronger than 1/r
2
in the heliosheath. (2) The
opposite occurs when V is assumed to increase in the
heliosheath. (3) When V / 1/r
8
in the heliosheath, addi-
tional acceleration of the anomalous protons takes place in
the heliosheath region, resulting in altered accelerated
spectra at the TS. (4) When V / r
2
in the heliosheath,
additional deceleration of particles occur in this region,
resulting in decreasing intensities in the heliosheath as a
function of radial distance. (5) These factors cause the
spectra at Earth for the V / 1/r
2
and V / r
2
scenarios to
differ by more than a factor of 10 for the A > 0 cycle at solar
minimum conditions at 1 GeV and by more than a factor of
100 for the V / 1/r
2
and V / 1/r
8
scenarios.
[
25] For the solutions of the asymmetrical model, with a
scenario where the solar wind speed profile in the helio-
sheath was kept constant at V / 1/ r
2
for all polar angles and
a scenario where it changes from V / 1/r
8
in the helio-
spheric nose region to V / r
2
in the heliospheric tail region,
it was found that (1) modulation differences between the
two V-scenarios are the largest for the A < 0 polarity cycle.
(2) For V / 1/r
2
at all latitudes the intensities are less than
for V / 1/r
8
! r
2
in the heliosheath. These differences are
also less in the tail region of the heliosphere than in the
nose region. (3) The intensities as a function of radial
distance for 16 and 200 MeV particles differ markedly in
the heliosheath for the two V-scenarios, especially in the
nose region. (4) The mild dependence of the 16 MeV
intensities in the heliosheath as a function of polar angle
for the V / 1/r
2
scenario for solar minimum and moderate
maximum conditions shifted to one where the intensities
are significantly higher in the nose region only for the V /
1/r
8
! r
2
scenario. (5) The highest intensities for the V /
1/r
2
scenario occurred in the equatorial regions with little
difference between the nose and tail regions, despite the
heliosheath being much wider in the tail region.
[
26] It is concluded that the Voyager 1 spacecraft is in an
advantageous position so that f uture measurements of
anomalous protons in the heliosheath should provide us
with a modulation pict ure of which solar wind speed
profile is most realistic in this region; the predicted
modulation differences are large enough to be observable.
From the asymm etrical model it was found that the
effectiveness of particle acceleration at the TS, as seen
by the higher intensities at the TS, can be altered by
changing the V-profile in the heliosheath over latitude,
especially in the A < 0 cycle, but not significantly.
Similarly, this can also happen for the symmetrical model
with a V profile that decreases stronger than the previously
assumed 1/r
2
dependence because particles then are accel-
erated to significantly higher energies at the TS and in the
heliosheath. The V / 1/r
2
scenario for all latitudes predicts
that there should be an abundance of anomalous particles
in the heliosheath at almost all latitudes. For the scenario
where V / 1/r
8
! r
2
, thus a strongly varying V with
latitude in the heliosheath from the nose to the tail, the
abundance of anomalous particles may occur only in the
nose direction of the heliosphere.
[
27] This modeling work, which includes an asymmetri-
cal heliospheric structure and a change over latitude for the
radial solar wind speed profile in the heliosheath, gives
insight into the modulation of anomalous protons that may
occur at the TS and beyond in the heliosheath, which is
presently been probed by the Voyager 1 spacecraft.
[
28] Acknowledgments. We thank the SA National Research Foun-
dation (NRF) and the Deutsche Forschungs Gemeinschaft (DFG) for partial
financial support under the bilateral DFG/NRF agreement, GUN 2049412,
and DFG SCHL201/14-1/14-2/14-3. UWL also wishes to thank the NRF
for partial financial support during his postdoctoral research in Germany
and the Claude Leon Foundation for financial support during his postdoc-
toral research in South Africa.
[29] Shadia Rifai Habbal thanks Eric R. Christian and another referee
for their assistance in evaluating this paper.
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T. Borrmann, H. Fichtner, and U. W. Langner, Institut fu¨r Theoretische
Physik IV, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany.
(ul@tp4.rub.de)
M. S. Potgieter, Unit for Space Physics, School of Physics, North-West
University, Potchefstroom 2520, South Africa.
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