2D STOCHASTIC SIMULATION MODEL OF COSMIC RAY MODULATION:
COMPARISON WITH EXPERIMENTAL DATA
P.Bobik(1) (3), M.Gervasi(1) (4), D.Grandi(1) (4), P.G.Rancoita(1), I.G.Usoskin(2)
(1)
INFN sezione di Milano, P.zza delle Science 3, 20126 Milano, Italy, piergiorgio.rancoita@mib.infn.it
(2)
Sodankyla Geophysical Observatory, University of Oulu, Finland, ilya.usoskin@oulu.fi
(3)
Institute of Experimental Physics, Watsonova 47, 04053 Kosice, Slovak Republic, bobikp@kosice.upjs.sk
(4)
Universitá di Milano-Bicocca, P.zza della Scienza 3, 20126 Milano, Italy, massimo.gervasi@mib.infn.it
ABSTRACT
We developed a 2D stochastic simulation model of
heliospheric propagation of galactic cosmic rays. The
model solves numerically the transport equation of
particles in the heliosphere. In the calculation we use
also drift effects which are included through analytical
effective drift velocities. We estimated the cosmic rays
spectrum at 1AU using this model formalism. The
calculated spectra are compared with other models
(CREME96) and with experimental data (IMP8 and
AMS) for positive (A>0) and negative (A<0) solar
periods.
1. INTRODUCTION
In the last years models of heliospheric propagation of
galactic cosmic rays (GCR) are continuously
developing. Models begin to use new structure of
heliospheric magnetic field [1] which has been
developed after Ulysses measurements of small
latitudinal gradient of cosmic rays intensity in the inner
heliosphere. One of most recent is a model with mixed
Fisk and Parker fields, which is a pure Fisk heliospheric
magnetic field at mid-latitudes but changes to a Parker
field in the solar equatorial plane and above the solar
poles [2]. These models also became time dependent
due to the measurements connected to drift effects in the
heliosphere [3]. Since a few years we have developed a
1D heliospheric model using the stochastic simulation
approach [4] and [5]. Now we have developed a two
dimensional (radius and latitude) drift model of GCR
propagation in the heliosphere based on the same Monte
Carlo approach. The model is dependent on the charge
sign of particles due to curvature, gradient and current
sheet drifts. It is time dependent too due to the variation
of the measured values of the solar wind velocity in the
ecliptic plane (V) and the tilt angle ().
U 1  2
U
 2 (r K rr
)
t r r
r
1

U
(1)
 2
( K sin 
)
r sin  

1  (r 2V ) 
1 
 2
(TU )  2 (r 2VU )
3r
r T
r r
where  is the heliolatitude, U is the cosmic ray number
density per unit interval of kinetic energy T (per
nucleon), r is radial distance and V solar wind velocity.
 = (T+2T0)/(T+T0) and T0 is proton’s rest energy. The
first two terms of equation (1) describe diffusion of
GCR in the heliosphere, the third term adiabatic energy
losses and the last one convection by outgoing solar
wind. Using a stochastic simulation technique, particle
co-ordinates variation in a small time step t results:
1 d (r 2 K rr )
t  V t  Rg 2 K rr t
r2
dr
d 
2 K 
2 K
 
 1    2  t  Rg 2.(1   ) 2 t
d 
r 
r
r 
(2)
Where r is the radial variation,  = cos is the
latitudinal variation of the particle, Rg is the Gaussian
distributed random number with unit variance, t is the
time step of calculation. Radial diffusion coefficient is
Krr  K cos2   K .sin 2 
(3)
where  is the angle between radial and magnetic field
directions [6]. The latitudinal coefficient is
K  K 
(4)
where parallel and perpendicular coefficients are
2. MONTE CARLO 2D MODEL WITH DRIFT
EFFECTS
Propagation of GCR across the heliosphere is described
by Fokker–Planck equation. 2D heliospheric stochastic
simulation is based on equation for GCR transport in the
heliosphere (without drift terms) [6] and [7]:
K  K 0  K P ( P)
K   ( K  )0 K 
B
3B
(5)
(6)
where K0= 2 x 1022 cm2s-1 [8], β is the particle velocity
(in unit of the light speed), Kp(P) take into accounts the
dependence on rigidity (in GV), (K)0 is the ratio among
parallel and perpendicular diffusion coefficient, B ( 5
nT) is the value of heliospheric magnetic field at the
Earth orbit, and B is the Parker field [9].
We use the Parker model of heliospheric magnetic field
because only for this model an analytical solution for
drift velocities exists. For the Fisk model there are
difficulties with the evaluation of diffusion tensor and
with the analytical solution for drift velocity. Moreover
solutions of the transport equation in the Fisk field
model can become unstable easier than in the Parker
field case.
In our model a solar wind speed V() as function of the
heliolatitude  has been used [10]:
V = V0 (1 + sin ), km s-1, for 0o <  < 60o
V = 750 km s-1, for 60o <  < 90o
(7)
Where V0 is the velocity of solar wind in ecliptic plane.
The heliosphere is considered to be a sphere with radius
equal 100AU. The model does not include an effect of
the termination shock. Drift effects are included through
analytical effective drift velocities [11]. The average
drift velocity is
vd =  (/3B)
(8)
Where  is the speed ratio,  is the CR particle’s
rigidity and B is the magnetic field. In the Parker spiral
field, gradient, curvature and drift along the neutral
sheet are added to the previous formulas (2) to calculate
position variation of the particle during a time step t:


rd  r  vg  vdns t

 vq t  
 d  cos  arccos(  )+ arctg 

 r 

(9)
Where rd is the radial variation with drift effect, d is
the latitudinal variation of the particle, vg is the velocity
of gradient drift, vdns is the velocity of neutral sheet drift
and vθ is the velocity of curvature drift. Both vg and vdns
are directed along er, while vθ is directed along eθ in
spherical coordinates.
We use Burger’s model [12] as Local Interstellar
Spectrum of protons (LIS hereafter). The heliospheric
propagation model has been optimized by fine tuning
the parameters Kp(P) and (K)0 of the diffusion tensor.
We performed long time consuming simulations and
compared results with the available experimental data.
In particular we are interested to the flux spectrum at
1AU and as a reference data set we used both the IMP8
measurements and Creme96 model. Moreover we
concentrated on periods with low-medium solar activity
and opposite solar field polarity.
Finally we get the best combination of these two
parameters of the model producing spectra and
comparing them with the data set at 1AU for both
positive and negative solar field periods. Regarding the
ratio between perpendicular and parallel diffusion
coefficients (K)0 we obtain as best value (K)0 = 0.025.
This is the same value reported by Giacalone et al. [13].
For the dependence of the diffusion tensor on rigidity
we have found similar but not the same results obtained
in [13]. Our simulations produce the best spectra at 1AU
for Kp(P)  P0.78, which corresponds to a slope  2 – q,
where q is the slope of the high energy part of LIS.
The heliospheric propagation model has been optimised
mainly for periods of minimum solar activity and
negative polarity. As a matter of fact negative polarity
periods are more sensitive to parameters variation. Not
only (K)0 and Kp(P), but also variations of other
parameters, like the tilt angle, induce very important
changes in the modulated spectrum at 1AU.
3. EXPERIMENTAL DATA BASE
We selected two periods in 1987 and 1998 during solar
minima in two opposite solar cycles. For these periods
we have experimental data to compare to the model and
to estimate the charge drift effect. In our model different
periods of solar activity are uniquely characterized by
the measured value of the tilt angles  and solar wind
velocities V in ecliptic plane. Experimental values of the
tilt angle  during years 1987 and 1998 are (see the web
page http://quake.stanford.edu/~wso/wso.html):
1987 : A < 0,   3o – 21o
1998 : A > 0,   16o – 50o
We considered average values for both these periods:
for 1987 we choose the value  = 10o; for 1998  = 30o
- 40o. We also consider as solar wind speed in ecliptic
plane at 1AU the value V = 400 kms-1 for both periods.
We compared results of the simulations with
measurements of IMP8-GME experiment and IMP8CRNC experiment and with Creme96, a model of GCR
in heliosphere. We selected measurements from IMP8
satellite apparatus, in particular from the Goddard
Medium Energy (GME) experiment (see the web page
http://spdf.gsfc.nasa.gov/imp8_GME/GME_home.htm)
and from the Cosmic Ray Nuclear Composition
(CRNC) (see the web page http://ulysses.sr.unh.edu/
WWW/Simpson/imp8.htm). These experiments cover
positive and negative solar periods from year 1973 to
year 2002 and energy range from 0.5 MeV to 500 MeV
for protons. We also used the model Creme96 [14] (see
the web page http://crsp3.nrl.navy.mil/creme96/) as a
smooth reference spectrum, even if it is not reproducing
completely the experimental data.
4. RESULTS
In Fig. 1 results from the present 2D drift model
spectrum for the year 1987 ( = 10o) are compared with
IMP8 measurements and Creme 96 spectrum. In this
period, i.e. minimum solar activity and negative
polarity, a good agreement of the model results with
experimental data is obtained. Besides both
experimental data and the 2D model are different from
Creme96 model, but difference is still inside the 25%
error bar of the Creme96 specrtum.
In Fig. 2 calculations based on the present 2D drift
model spectrum for June 1998 ( = 30o) are compared
with IMP8 measurements and Creme 96 spectrum. In
this period, i.e. minimum solar activity and positive
polarity, a good agreement among the model results,
experimental data and Creme96 model is obtained for
energies higher than 100MeV. For lower energies the
measured spectrum and Creme96 model are
systematically higher respect to our model. This is
probably due to an additional component in the
measured flux, which can be ascribed to anomalous
cosmic rays: both measurements and Creme96 model
take into account this extra component. Besides we need
to remember that this period is not a really quite solar
activity period, as it is confirmed by the measured value
of the tilt angle ( = 30o).
Fig. 1. Proton differential flux at 1AU. Comparison of
our 2D drift model with IMP8 measurements and
Creme96 model; year 1987, A  0 and  = 10o.
Fig. 3. Proton differential flux at 1AU. Comparison of
our 2D drift model with AMS-01 measurements and
Creme96 model; year 1998, A  0,  =30o and 40o.
Fig. 2. Proton differential flux at 1AU. Comparison of
our 2D drift model with IMP8 measurements and
Creme96 model; year 1998, A  0 and  = 30o.
In Fig. 3 we compare our simulated spectra with AMS01 data. As we can see the best value of the tilt angle is
intermediate between 30o and 40o.
The dependence of simulated spectra at 1AU on the tilt
angle during positive polarity periods is shown in Fig. 4.
It is evident the depletion of galactic protons at low
energies by increasing the tilt angle. This trend is still
more evident for negative polarity periods.
the model is able to produce spectra at 1AU which
reproduce experimental measurements quite well.
Agreement is better for the negative solar period
explored (1987) when the influence of drift effect is
stronger and the solar activity is lower than in the
opposite solar cycle. As a result we obtain that proton
fluxes at low energy strongly depend on heliospheric
current sheet position, i.e. on the tilt angle.
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Fig. 5. Proton flux at 1AU vs tilt angle for particles at
energy equal to 5.0, 3.1, 2.0, 1.2, 0.8, and 0.5GeV
during a positive polarity period (A  0).
The flux depletion of galactic protons is still more
evident if we plot the intensity vs the tilt angle for
several energy bins. From Fig. 5 we can se as the
modulation is more effective at lower energy while it
disappears at higher energy.
CONCLUSION
We developed a 2D stochastic simulation model of
heliospheric propagation which includes charge drifts
effects. For both, positive and negative solar periods,
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