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Galactic cosmic rays in the global heliosphere: an
axisymmetric model
V. Florinski , G. P. Zank and N. V. Pogorelov†
Institute of Geophysics and Planetary Physics, University of California, Riverside
†
Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow
Abstract. We present a new axisymmetric model of the heliosphere that includes the three principal particle species (plasma
with magnetic field, interstellar neutral atoms and galactic cosmic rays). An important improvement over the previous models
is the kinetic description for the cosmic rays instead of the usual fluid approach. We compute the cosmic-ray diffusion
coefficients from the quasi-linear theory assuming a constant power spectrum of the fluctuations in both the solar wind and the
interstellar medium. Our model predicts small cosmic rays gradients implying little modification of the plasma flow, except,
possibly, in the heliosheath region.
INTRODUCTION
gion 1 population only) and effects of the presence of the
neutrals on the cosmic rays discussed. Since the model is
two-dimensional, we are constrained to a plane through
the IW axis at an angle φ to the equator. We compare our
results with the fluid models and provide data to validate
their choice of momentum-averaged parameters.
It is widely recognized that galactic cosmic rays (GCR)
are capable of changing the flow pattern of the solar wind
and the surrounding local interstellar medium (LISM)
provided the particles’ coupling to the plasma is sufficiently strong. While the interstellar cosmic-ray population spectra are reasonably well constrained [1], large
uncertainties still exist in our knowledge of diffusion coefficients that ultimately determine the cosmic-ray pressure gradients. Currently, a fluid approach is most widely
used to describe the GCR in the context of a global
heliospheric model (GHM) [2, 3]. The deficiency of
this description is the necessity to impose apriori the
momentum-averaged diffusion coefficient as well as the
average cosmic-ray “adiabatic index”. Clearly this approach suffers from the fact that the average quantities
are themselves dependent on the cosmic-ray gradients.
In this paper we overcome the problems inherent in
fluid models by introducing a kinetic cosmic-ray GHM.
Our model can be seen as an extension of the model
[4] to include the region beyond the heliopause (HP).
The model, in a sense, bridges the gap between selfconsistent GHMs and the cosmic-ray modulation models.
A major feature of our model is the introduction of the
3-dimensional heliospheric magnetic field in a kinematic
approximation. This gives a more accurate representation of the field and the diffusion coefficients in the heliosheath and the heliotail than by using a Parker’s spiral
field as was done in [2]. The current model is axisymmetric with respect to the interstellar wind (IW) direction.
Neutral atoms and charge exchange are also included (re-
MODEL DESCRIPTION
We use the MHD model of the plasma described by
the usual set of conservation laws with charge exchange
terms and CR pressure gradients included. Due to axial
symmetry of the model, only the interstellar field is included in the MHD equations (see, e.g., [5]). However,
we include the heliospheric field kinematically by assuming Parker’s spiral field at the inner boundary and following its evolution in time and space under assumption
that the solar wind velocity is axially symmetric, i.e., the
velocity vector (u,0,w) satisfies ∂ u ∂ φ ∂ w ∂ φ 0. In
the cylindrical coordinate system the equations are
∂ Br
∂t
∂ Bz
∂t
∂ wB∂r z uBz
∂ wBr uBz
∂r
∂ Bφ
∂t
∂ uBφ ∂r
1 ∂ rBr ∂ Bz
∂z
r ∂r
u
(1)
1 ∂ rBr ∂ Bz
w
∂z
r ∂r
wBr r uBz
∂ wBφ CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
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∂z
0
(2)
(3)
TABLE 1. Diffusion parameters
heliospheric
interstellar
0 01 [7]
4 5 1011 cm [7]
0.002 [9, 7]
0 01 0 1 [8]
1017 1018 cm [8]
1.0
10
This description is valid for any φ , however the field
reversal at the neutral sheet cannot be taken into account
properly. We ignore the j B force produced by this
field because the latter will necessarily destroy the axial
symmetry.
The model includes the neutral particles and charge
exchange using the Pauls–Zank–Williams formalism [6].
Only interstellar neutrals are considered, while energetic
neutral atoms produced in the solar wind are currently
ignored. Both MHD and neutral equations are solved
numerically on a polar grid lying in the half-plane φ
const using the TVD Lax–Friedrichs scheme [5].
We describe the galactic cosmic-ray population with
the usual Parker–Skilling transport equation. Drift velocities are not included because they contain a component
perpendicular to the half-plane. We also note that diffusion in the direction perpendicular to the plane is necessarily ignored, which is a serious limitation of the axial
symmetry. To get a better idea about the amount of CR
modulation, one needs to consider the angle-averaged
cosmic-ray distribution computed for a range of φ angles. We plan to include such analysis in a future publication.
We compute the diffusion coefficients following [7] by
assuming that a fully developed energy-inertial turbulent
spectrum exists in both the solar wind and the interstellar
medium. The diffusion coefficients are
κ
1 3 2 3
27 rg lc v 7
35 A2
27
5
rg
lc
3
1
α A2 κ 10
10
10
-5
-6
-7
-8
-9
-10
10 10
100
r, AU
FIGURE 1. Heliospheric fields Br and Bφ for θ
1000
00
(solid), 900 (dashed) and 1800 (dotted lines). Bφ is the larger
of the two at low latitudes, but smaller at 900 .
Our current model uses the results obtained in [10]
showing that the turbulence level decays slower than in
the WKB approximation because pickup ions and stream
interaction create additional amounts of turbulence. We
therefore approximate both A 2 and lc by their respective
values at the inner boundary, located at 10 AU. While this
approximation is rather crude in the heliosheath region,
we must defer model refinement until sufficient improvements are made to the turbulence evolution theory.
Finally, we use the Jokipii–Kota modified field [11] to
suppress diffusion in the regions where the field is very
small. We specify a modified field B m in the φ direction
described by an equation similar to (3) with B m Br 02
at 1 AU.
(4)
RESULTS
where lc is the correlation length (turbulence outer scale)
and A2 δ B2 B2 for slab turbulence, and
κ
-4
10
B, Gs
A2
lc
κ κ
10
In this paper we present results for the single value of the
azimuthal angle φ 0, i.e., the meridional plane. The
simulation domain was 400 100 170 grid points in
the r, θ and p directions, respectively. The momentum
domain corresponds to proton kinetic energies between
10 MeV and 10 GeV. For the typical interstellar spectrum
[1] this includes about 80% of the total pressure.
Figure 1 shows all three components of the heliospheric magnetic field at different angles relative to the
IW axis (the LISM and outer heliosheath field is not
shown). One important feature of this plot is the existence of a region with a very large magnetic field in the
vicinity of the heliopause. This field is found to considerably reduce the ability of the lower energy cosmic rays
to penetrate into the heliosphere creating a “modulation
barrier”.
(5)
where α is a constant. The last expression corresponds to
the second (“modified QLT”) model of [7] and predicts
a relatively large perpendicular diffusion coefficient. The
various parameters used in computing the diffusion coefficients are summarized in Table 1. The interstellar diffusion coefficient computed according to (5) is several
orders of magnitude larger than the heliospheric. This is
consistent with the notion that the outer scale of the turbulence in the LISM is considerably larger than in the
Solar system. Because the direction of the LISM magnetic field is poorly known, we use isotropic diffusion in
this region.
645
10
27
2 -1
10
κ, cm s
30 MeV
26
400
25
x
10
200
10
24
10
10
23
0
-800
-600
-400
z
-200
0
200
22
500 MeV
400
21
10 10
1000
x
100
r, AU
200
FIGURE 2. 1 GeV proton diffusion coefficients for θ 00
(solid), 900 (dashed) and 1800 (dotted lines). κrr and κθ θ are
shown, κrr is larger at small solar distances.
0
-800
-600
-400
z
-200
0
200
FIGURE 4. Same as Figure 3, but with the neutrals included.
30 MeV
400
x
distribution inside the termination shock shows the typical lobe structure characteristic of the modulation models [12]. There is an enhancement in 500 MeV intensity
in the heliotail region which is caused by re-acceleration
due to slowing down of the heliotail flow and the contraction of the tail caused by charge exchange.
Modulated proton spectra are shown in Figure 5 together with the available solar-minimum observational
data. The computed spectra are seen to be in agreement
with the observations. The crossover is visible in the
heliotail region indicating GCR re-acceleration. Despite
significant (up to 50%) reduction in the size of the modulation cavity, GCR intensity in the presence of neutral
atoms differs by no more than 5% at 10 AU.
Finally, we would like to compare our momentumdependent diffusion coefficients with the κ i j used by
fluid models. The latter can be computed according to
200
0
-800
-600
-400
z
-200
0
200
500 MeV
x
400
200
0
-800
-600
-400
z
-200
0
200
FIGURE 3. GCR phase space density contours at 30 MeV
(top) and 500 MeV (bottom panel). Contour lines are drawn
with 5% intensity decrements.
κi j
We show 1 GeV proton diffusion coefficients in Figure
2. Note that κ is very large in the LISM and causes no
modulation even at the lowest energy on the scales of
the problem. However, the barrier is evident in the inner
heliosheath.
Figure 3 shows the phase space density contours of
cosmic ray protons at two different energies for the noneutral case, while Figure 4 plots the same quantity for
the case when neutral atoms were included. It can be
seen that low energy particles are mostly filtered by the
small diffusion region in the inner heliosheath. Particle
4π
3
∂ Pc
∂xj
1
κi j
∂f 3
p vd p
∂xj
(6)
where Pc is the cosmic-ray pressure. The average diffusion coefficient was found to correspond approximately
to κ 1 GeV , which is considerably larger than that used
in fluid models [3], when large GCR effects were obtained. We therefore expect that galactic cosmic rays
would not change the structure of the termination shock,
but merely produce a wide and shallow precursor.
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experience additional acceleration in the convergent flow
(∇ u 0).
To summarize, the most important result of our model
is that modulation in the heliosheath cannot be ignored.
In fact, the amount of attenuation experienced by particles with T 1 GeV is greater in the heliosheath than
in the solar wind. One implication of this is the existence of large gradients in Pc in this region, which may
result in the solar wind being strongly mediated by the
cosmic rays in the upwind heliosheath region. We also
found that, despite a large reduction in the size of the heliosphere when the neutral hydrogen is included in the
model, GCR intensities differ only by 5% in the inner
heliosphere for these two cases. We have also found that
GCR upwind-downwind asymmetries are too small at
small heliocentric distances to provide a useful tool for
detecting the outer heliospheric structures.
For the future, we plan to improve the model by including a better turbulence description. In particular, the
model [10] needs to be extended to 3 dimensions and
the restriction on the Alfvén speed being small removed.
One should also consider including turbulence production from pickup ions, whose density can be directly
computed from the charge exchange rate. This extra
source of turbulence may reduce diffusion in the outer
heliosheath causing additional attenuation of GCR intensity.
2
1
-2 -1
-1
J, m s ster MeV
-1
10
10
10
10
0
-1
-2
10
1
10
2
T, MeV
10
3
10
4
FIGURE 5. Modulated proton spectra in the heliosheath at
1 1rs , θ 0 (dashed line), in the crosswind direction at 1 1rs ,
θ 90Æ (dash-double dotted), in the heliotail at 800 AU, θ
180Æ (dotted) for the case with H atoms. The 10 AU spectrum
is plotted with a solid line and the no-neutral 10 AU spectrum
is shown with a dash-dotted line for comparison. The input
(galactic) spectrum at 1000 AU is also shown with a solid line.
Experimental data from BESS (squares) and IMP8 (circles) are
shown for comparison.
This work was supported, in part, by the NASA grant
NAG5-11621, the NSF grant ATM-0296114 and the
RFBR grant 02-01-0948.
DISCUSSION AND FUTURE WORK
We believe the new model of the CR-modified heliosphere to be a useful extension of the current fluid
GHMs. Our results indicate that cosmic rays are not coupled strongly to the plasma in the heliosphere and the
surrounding interstellar medium, except in the inner heliosheath, where the average diffusion coefficient is 2-3
orders of magnitude below the heliospheric values. We
find that while high-energy (1 GeV and higher) particles
can penetrate into the heliosphere quite easily, lowerenergy cosmic rays are stopped by the modulation barrier.
Due to space constraints we do not show Pc , however
the latter was found to vary little throughout the heliosphere. We find only 0.07 eV/cm 3 decrease in Pc between
the outer and the inner boundaries, which is insignificant
when compared with the wind dynamic pressure of 0.65
eV/cm3 at the termination shock. Galactic cosmic rays
are therefore not likely to modify the shock significantly.
We also note that neutral atoms may have an important effect on the cosmic-ray propagation in the heliotail
region. When neutrals are present, the heliosphere contracts due to reduction in the momentum density of the
solar wind as well as cooling and decelerating caused by
charge exchange in the heliotail [6]. In this case, GCR
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