Monte Carlo diffusion of CR from the Gould Belt

The physical parameters concerning the Gould Belt as deduced in Ref. 48, are reported in Table 1. The total energy is such as to produce results comparable with the observations and the kinematic age is the same as in Ref. 48. In order to obtain Etot = 6× 10⁵¹ erg with t^burst= 0.015 × 10⁷yr, we have inserted N^∗= 2000. The time necessary to cross the Earth orbit, that lies 104 pc away from the Belt center, turns out to be 0.078 × 10⁷ yr that means 2.52 × 10⁷ yr from now. The 2D cut at z = 0 of the superbubble can be visualized in Fig. 20. Our model gives a radial velocity at z = 0, Vtheo= 3.67 km s⁻¹. The influence of the Galactic rotation on the direction and modulus of the field of originally radial velocity the transformation (95) is applied, see Fig. 21. A comparison should be done with Figs. 5 and 9 in Ref. 48.

Fig. 20. Rhombi represent the circular section, the stars the rotation-distorted section and the big star the Sun. The Galaxy direction of rotation is also shown. The parameters aretage= 2.6×10⁷yr,

∆t = 0.001 × 10⁷ yr,t^burst₇ = 0.015, N^∗= 2000,zOB= 0 pc, andE51= 1.

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Fig. 21. The stars represents the rotation-distorted section of the Gould Belt and the big star the Sun. The velocity field of the expansion modified by the shear velocity is mapped. The Galaxy direction of rotation is also shown.

Fig. 22. Concentration of CR diffusing away from the Gould Belt at the x − y (z = 0) plane.

λ1 = 5.51, IP = 10000, ITMAX = 200, tD= 7204 yr andctr= 0.5. Assuming Z = 1, H−6= 1 we have (Bohm diffusion)E15= 5.08.

The results of the streaming of CR from the complex 3D advancing surface of the Gould Belt can be visualized through a 2D grid representing the concentration of CR along the Galactic plane x−y (z = 0), see Fig. 22, or through a perpendicular plane x − z (y = 0), see Fig. 23.

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Fig. 23. The same as Fig. 23 but in thex − z (y = 0) plane.

The variation of CR due to the presence of the Gould Belt are also analyzed in Subsec. 3.2 in Ref. 7.

5. Conclusions

We have studied the propagation of CR with the aid of mathematical diffusion in 1D, 2D and 3D. Due to the nature of the problem, diffusion from the Galactic plane, the first results can be obtained in 1D. The 1D diffusion has also been analyzed in the case of a variable diffusion coefficient: which in our astrophysical case consists in a magnetic field function of the altitude z from the Galactic plane. These ana-lytical results allow to test the Monte Carlo diffusion. The physical explanation is the diffusion through the Larmor radius which means that the diffusion coefficient depends on energy. The calculation of the transit times of CR explains the absence of the isotope¹⁰Be. Further on the theoretical and Monte Carlo setup of the Levy flights allows to increase the applications of the random walk. For example from an observed profile of CR function of distance from the Galactic plane is possible to evaluate the concavity: the 1D Levy flights predict profiles concave up and the 1D regular random walk predicts no concavity. The observations with EGRET in the gamma range, see Fig. 3 in Ref. 46, provide profiles that are concave up and there-fore the Levy flights with the shape regulating parameter α can be an explanation of the observed profiles, see Fig. 7.

The CR diffusion in a superbubble environment is then performed by evaluating the maximum energies that can be extracted (≈ 10¹⁷eV) and performing a Monte Carlo simulation of the propagation of CR from SB disposed on a spiral-arm net-work. According to the numerical results, the concentration of CR depends on the distance from the nearest SB and from the chosen energy.

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This fact favors the local origin of CR⁴⁹ that comes out from an analysis of the results of KASCADE data.⁵⁰ From an astrophysical point of view, local origin means to model the diffusion from the nearest SNR or SB. Like a practical example we now explore the case of the Gould Belt, the nearest SB, which has an average distance of 300≈ pc from us. The concentration of CR at the source of this SB is, according to formula (18):

Cm≈ C(300)300 pc ρZ

≈ 277 × C(300)H−6Z

E15 , (97)

where C(300) is the measured concentration on Earth at the given energy. It is interesting to point out that the crossing time (25 My ago) of the expanding Gould Belt with the Earth with a consequent increases in the concentration of CR can explain one event of the fossil diversity cycle that has a periodicity of 62 My.⁵¹ The gamma-ray emissivity is different in the various regions of the galaxy,⁵²and in order to explain this fact the theoretical map of gamma-ray from the Galactic plane has been produced. CR with energies greater than 10¹⁸ eV, Ultra-High Energy, are thought to be of extragalactic origin^27,53–57 and therefore the diffusion from extragalactic sources through the intergalactic medium is demanded to a future investigation.

Acknowledgments

I thank M. Ferraro for many fruitful discussions.

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