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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