Download Abstract - Istituto Nazionale di Fisica Nucleare

Solar modulation model with reentrant particles
P.Bobika, K. Kudelaa, M. Boschinib, D. Grandib, M. Gervasib, P.G.Rancoitab
a
Institute of Experimental Physics SAS, Slovak Republic, Watsonova 47, 040 01 Košice, Slovak Republic
Istituto Nazionale di Fisica Nucleare, Piazza della Scienza 3, 20 126 Milano, Italy
b
Abstract
We developed a one dimensional model of particle transport in the heliosphere. As opposite to widely
used models, we apply a method where a quasi-particle is traced back in time. The model gives us the
possibility to work on the possible existence of reentrant particles in the heliosphere that can be hardly
solved by the traditional forward tracking method. Particles escape from the heliosphere and may reenter
back. We estimate how these particles affect the modulation process in the heliosphere. Presented here are
the results for different values of particles mean free path in the interstellar space and for different
interstellar magnetic field values.
1. Introduction
The problem of transport and distribution of Galactic Cosmic Rays in the heliosphere and interstellar
space is described by the Fokker-Planck equation (FPE hereafter) (Parker 1965, Chandrasekhar, S. 1943).
∂f 1 ∂
∂f
∂f
= 2 r r2K
−V
∂
∂t r
∂r
∂r
1 ∂ 2
p∂f
r V
2
3∂p
r ∂r
(1)
Where f is the distribution function, r is the distance from the Sun, V is the velocity of radially directed
solar wind, K is the diffusion coefficient and p is particle momentum.
Many methods to solve the FPE were introduced in the last decades (Fisk, 1971, Jokipii and Owens,
1975). One of the commonly used is the solutions of the FPE by the Monte Carlo method based on Ito's
proof that a set of stochastic differential equations is equivalent to the FPE (Krülls and Achterberg, 1994).
The time reversal process is also possible (Kota, 1977, Yamada et al., 1998).
Stochastic differential equations equivalent to the FPE are
dr = V
2K
∂K
dt
2 K dW r
r
∂r
2 pV
p ∂V
dp= −
dt−
dt
3 r
3 ∂r
(2)
The Solution of the FPE back in time is much faster than the classical forward in time approach. This
is crucial for the problem of reentrant particles.
Particles of Galactic Cosmic rays from interstellar space penetrating the heliosphere are affected by
outgoing solar wind and lose energy adiabatically. Particles outside the heliosphere diffuse in the
interstellar space without any energetic losses. It may happen that a particle escaping the heliosphere reenter
later back again. This means that not every particle entering the heliosphere has the same characteristics of
local interstellar spectrum particles. Some of them are particles that were already modulated inside the
heliosphere. Almost in all present models of GCR modulation in the heliosphere, particles escaping the
heliosphere are no longer considered. This do not take into account the possibility of reentrant particles.
Using a back-tracing model we can evaluate the effect of possible reentrant particles to the modulated
spectrum inside the heliosphere. A particle injected at 1AU is followed inside the heliosphere and after
crossing its outer border also in the interstellar space. Here it experiences just simple diffusion with a
diffusion coefficient different from that one inside the heliosphere. Particles in interstellar space are
1
followed in a test domain with one parsec radius from Sun and killed when crossing this border because at
this distance the probability to reenter back in the heliosphere is negligible.
Let us note that instead of speaking about a particle for these calculation it is more appropriate to use
the term quasi-particle because what is reconstructed in the model is not a real particle but it is more the
particle gyration center trajectory in the heliospheric magnetic field between magnetic field inhomogenities
than the particle trajectory itself.
2. Method
Inside the heliosphere, when we do not take in account the acceleration of particles by termination
shock, (it means in a model without termination shock) equations (2) became
2K
dr = V
dt
2 K dW r
r
dp= −
2 pV
3 r
(3)
dt
Where the diffusion coefficient is K = .K0.P. In this article we use K0 = 5.1022 cm2 sec-1. P is particle
rigidity in GV and V = 400 km s-1 .
As is stated in Yamada et al. (1998) one possibility about how to proceed is to inject particles at 1AU
as a set of delta functions of energies for which we evaluate the flux at 1AU. Because in many cases we
need to compare model results with experimental data we applied different injection. Particles are injected
uniformly with energy at 1AU. Particle with injection energy T0 during its way back in time in the
heliosphere can gain energy (reversal process to adiabatic losses). After injection the particle is followed
back in time inside the heliosphere to the moment when it crosses the heliosphere border at 100AU. Kinetic
energy T1 of the particle at the moment of first heliopause crossing is recorded. The set of stochastic
differential equations (3) in the interstellar space where V = 0 km s-1 became
dr =
2K
dt
r
2 K dW r
(4)
dp= 0
Because the probability that a particle reenter back to the heliosphere should depend on the particle
mean free path in interstellar space, we use a diffusion coefficient in interstellar space constructed in
following way (Muraishi et al., 2005).
K IS = K B~ 3.3 x 10
22
Z
−1
E
GeV
B
1 G
−1
cm2 sec− 1
(5)
Where
is the ratio of the mean free path of the particle to the Larmor radius, KB is the Bohm
diffusion coefficient KB = Ec/(3ZeB) where E is total energy, Z is the atomic number and B is the magnetic
field intensity in the interstellar space.
is a parameter of simulation in interstellar space. We used a set of
values
= 10, 100, 1000. Magnetic field in the interstellar space is the second parameter of the simulation
outside the heliosphere. We assume locally a constant value of the interstellar magnetic field in the test
domain (sphere with radius one parsec from the Sun). For model calculations we choosed the values B =
0.1, 1, 10 G. We scale the time step dt in the interstellar space as equal to
2
2
dt=
(6)
2K
to have the mean free path
equal to the diffusion step in the interstellar space. Finally when particle
crosses the border 1 parsec we record its kinetic energy T2. In this moment we evaluate two flux values.
First for modulated spectrum at 1AU not affected by reentrant particles, and second affected by reentrant
particles. The number of particles from local interstellar spectrum (Burger et al., 2000) for energy T1 is
added to the energy bin belonging to initial energy T0 for the spectrum not affected by reentrant particles.
Same process for energy T2 is done to evaluate a spectrum taking into account reentrant particles. Then,
next particle is injected and the process is repeated for a time long enough to obtained stable statistics.
Let us note that all calculations presented in this paper are made for protons.
3. Results and Interpretation
To check the effect of reentrant particles to modulated spectrum at 1AU we evaluate both spectra
(with, and without reentrant particles effect) for a set of parameters
and B described in Table 1.
In Table 1. is presented the ratio of number of reentrant particles to the total number of particles in the
modulated spectrum at 1AU (particle integral flux from 48 MeV to 100 GeV). We call reentrant particles
those ones modulated in heliosphere at least two times. From the table we can see that the number of
reentrant particles in modulated spectrum at 1AU increases with increasing intensity of interstellar magnetic
field B at fixed . This happens because with increasing the interstellar magnetic field we decrease
particle Larmor radius and at fixed
we decrease the mean free path
of the particle in the interstellar
space. After escaping the heliosphere, at shorter
particle can reenter back with higher probability
because will remain closer to the border of heliosphere. We can see same results for increasing the value of
with fixed interstellar magnetic field B that increases the particle mean free path and decreases the
number of reentrant particles in the modulated spectrum.
Influence of reentrant particles to different energies of modulated spectra is presented on figures 1. 2.
and 3. Situation for
= 10 and B = 1
G is showed at figure 1. At upper panel is the spectrum containing
reentrant particles (S1) evaluated for mentioned
and B compared with the spectrum without reentrant
particles (S2) both calculated for same parameters in the heliosphere. Bottom panel of figure 1. shows ratio
between both spectra (S1/S2). Figure 2. and 3. present results for B = 1
While for a
G and
= 100 and
= 1000.
= 10 we have a difference between spectra for energies in a range 100MeV to 1GeV in order
of tenths percent, for
2%.
= 100 in same energy range it is a percents and for
= 1000 it became less than
Figure 4. illustrates how in average heliosphere modulation depend from reentrant particles at
in
different magnetic fields B. Because B is fixed at 0.1, 1 and 10 G it is a dependency from the mean free
path. In figure panels are presented ratios of injection energy at 1AU to particle energy when escapes from
the test domain. Because the model describes the modulation process back in time, physically figure 4.
shows how is the ratio of particle energy coming first time (T2) to heliosphere to energy of the same
particle registered at 1AU after modulation in heliosphere (T0). Particle registered at 1AU with energy 0.5
GeV is, without taking into account reentrant particles, a particle entering to the heliosphere with energy ~ 1
GeV from interstellar space with B = 0.1
G. For a stronger interstellar magnetic field, we present B = 1,
10 G we must have particles with higher average energy when enter for the first time inside the
heliosphere to be registered at 1AU with energy 0.5 GeV. If we let particle diffuse in interstellar space and
reenter back into the heliosphere we can see that with decreasing mean free path in interstellar space we
increase the modulation at 1AU. By decreasing the mean free path we increase the average energy loss of
3
particles registered at 1AU. The reason is transparent. Reentrant particle lose adiabatically more energy
because the additional period inside the heliosphere. Same effects we register for all energies, the difference
is how strong this effect is. As we can expect the effect decreases with increasing the particle energy.
Conclusions
We estimated the presence of reentrant heliospheric particles at 1AU. We show that the influence of
reentrant particles to the energy spectrum modulation inside heliosphere depends on the particle mean free
path in the interstellar space. If interstellar magnetic field is close to value 1 G, mean free path of particle
in the interstellar space should be less than one thousands of particle Larmor radius to have effect which
should be considered in heliospheric modulation process. The effect of reentrant particles increases with
decreasing the mean free path of particles in the interstellar space.
Since it is still not clear how exactly is the mean free path of particles in the interstellar space, the
possibility of non negligible influence of reentrant particles to solar modulation is still to be investigated.
Acknowledgments
This work was supported by the Slovak Research and Development Agency under the contract No.
APVV-51-053805.
References
Burger, R. A., Potgieter, M. S., Heber, B., Rigidity dependence of cosmic ray proton latitudinal
gradients measured by the Ulysses spacecraft: Implications for the diffusion tensor, Journal of Geophysical
Research, 105, A12, 27447-27456, 2000.
Chandrasekhar, S., Rev. Mod. Phys.,15, 1-89, 1943
Fisk, L. A., Solar modulation of galactic cosmic rays, 2, J. Geophys. Res., 76, 221-226, 1971
Jokipii, J. R., Owens, A. J., Implications of observed charge states of low-energy solar cosmic rays,
Journal of Geophysical Research, 80, 1209-1212., 1975.
Krülls, W. M.; Achterberg, A., Computation of cosmic-ray acceleration by Ito's stochastic differential
equations, Astron. Astrophys. 286, 314-327, 1994.
Kóta, J. , Energy loss in the solar system and modulation of cosmic radiation, Proc. of 15 ICRC, 11,
186, 1977.
Muraishi H., Yanagita S., Yoshida T., Galactic modulation of extragalactic cosmic rays: Possible
origin of the knee in the cosmic ray spectrum, Progress of Theoretical Physics, Vol. 113, No. 4, pp. 721731., 2005.
Parker, E. N., The passage of energetic charged particles through interplanetary space, Planetary and
Space Science, 13, 9-49, 1965.
Yamada Y., Yanagita, S., Yoshida, T.,, A stochastic view of the solar modulation phenomena of
cosmic rays, Geophysical Research Letters, vol. 25, 13, 2353-2356, 1998.
Figures captions.
Figure. 1. Upper panel show a modulated spectra at 1AU for
= 10 and B = 1 G together with a
local interstellar spectrum (solid line). Dotted line with triangles denote a spectrum without reentrant
particles, solid line with diamonds denote spectrum with a reentrant particles. Bottom panel of figure show
4
a ratio between both spectra (with reentrant/without reentrant).
Figure. 2. Ratio S1/S2 between modulated spectra at 1AU evaluated with (S1) and without (S2) taking
reentrant particles into account for
= 100 and B = 1 G.
Figure. 3. Ratio S1 / S2 between modulated spectra at 1AU evaluated with (S1) and without (S2) taking
reentrant particles into account for
= 1000 and B = 1 G.
Figure 4. Ratio of energy of particle coming first time (T2) to heliosphere to energy of this same
particle registered at 1AU after modulation in heliosphere (T0).
Table 1. Ratio of the reentrant to total number of the particle in the modulated spectra at 1AU
B = 0.1
G
B=1
G
B = 10
G
= 10
7.53 %
29.27 %
38.30 %
= 100
0.44 %
7.31 %
30.02 %
= 1000
-
0.21 %
6.32 %
= 10000
-
-
0.90 %
5