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New Insights into the Acceleration and Transport of Cosmic Rays in the Galaxy or Some Simple Considerations J. R. Jokipii University of Arizona Presented at the Aspen Workshop on Cosmic-Ray Physics Aspen, Colorado, April 18, 2007

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Acknowledgements to The New Yorker

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Outline Background/Introduction Simple Transport Approximations Observed Lifetime and Anisotropies ) Problems A possible resolution?

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The observed quiet-time cosmic-ray spectrum

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The observed energy spectra of cosmic rays are remarkably similar everywhere they are observed. The Galaxy The Sun

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Probably nearly all cosmic rays are due to diffusive shock acceleration

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Shocks in the heliosphere are also sources of energetic charged particles.

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Motion in an irregular magnetic field is sensitive to initial conditions (chaotic): This demonstrates the importance of small-scale turbulence.

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The Parker Transport Equation: Where the drift velocity due to the large scale curvature and gradient of the average magnetic field is: ) Diffusion ) Convection w. plasma ) Grad & Curvature Drift ) Energy change – electric field ) Source

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The associated anisotropy is obtained from the diffusive streaming flux S i = - ij f / x i + (U i /3) p f / p or bulk velocity S i /w, which then gives the anisotropy i = 3 S i /w here w is the particle speed. One can generally estimate the anisotropy as ¼ /L, where is the mean free path and L is the macroscopic scale.

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The turbulent electromagnetic field is described statistically. In the quasilinear approximation, the scattering rate / P B [1/(r c cos p) )]. Notice also the large-scale field-line meandering.

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We never find the classical condition ? = k /(1+ 2 2 ) which would give a much smaller ratio. Test-Particle Simulations using synthesized Kolmogorov turbulence (Gicalone and Jokipii, Ap. J. 1999 + 1 point)

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A very simple and reasonably successful picture of cosmic rays in the galaxy has evolved. Regard the galaxy as a box into which the cosmic rays are injected and from which they escape. Replace all of the diffusive transport and geometry complications by an effective loss rate which balances the acceleration and injection.

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Note that the cosmic rays escape predominantly across the average magnetic field. For a more detailed discussion in terms of Loss across the galactic field, see Jokipii in “Interstellar Turbulence” ed by Franco, Cambridge, 1999.

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Transport and Loss in the Galaxy The transport equation is sometimes simplified to the very simple and basic equation f / t ¼ 0 ¼ - f / L + Q or f = L Q where f is the distribution function (dj/dT = p 2 f, where p is the momentum of the particle), ¼ L 2 / ? and Q is the source of particles. For relativistic particles pc = T. Primary cosmic rays are accelerated from ambient material, presumably at supernova blast waves In this case Q p is a power law: Q p (T) / T -(2-2.3)

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The characteristic loss time L can be determined from secondary nuclei, produced from collisions (spallation) with ambient gas. Since, at high energies, the spallation approximately conserves energy per nucleon, we have the source of secondaries Q s / f p Then we have f s = L Q s / L f p or f s /f p / L This ratio is observed to vary as ¼ T -. 6 at T ¼ 1-10 GeV. Extrapolated to high Energies, this give problems. Observations show that L ¼ 20 Myr at GeV energies, or some 300 yr at 10 18 eV!

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We may inquire as to how large the perpendicular diffusion coefficient must be to yield. L ¼ L 2 / ? to be ¼ 2 £ 10 7 yrs Setting L equal to a characteristic scale normal to the disk of some 500 pc yields ? ¼ 4 £ 10 27 cm 2 /sec, which is quite large. A typically quoted value for k of the order of or less than 10 29 cm 2 /sec, in which case the ratio of perpendicular to parallel diffusion is about 4%. These all seem reasonable.

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Anisotropies Strictly speaking we should not do anisotropies in the leaky box model. Nonetheless, simple considerable lead to reasonable anisotropies at GeV energies. In the diffusion approximation (the Parker equation), we can write for the anisotropy c ¼ 3(L/ L ) or ¼ 3 L/( L c) ¼ 10 -4 relative to the local plasma, which is not unreasonable.

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At TeV energies. , relative to the local interstellar medium is < ¼3 £ 10 -4 From Amenomori, et al, Science, 2006

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BUT, what happens at high energies? We must remember that observations mandate that L scale as T.6. This gives ¼ 1 at 10 18 eV Observations give < ¼ 5% (Sokolsky, private communication, 2007. The theoretical scaling of L as T.33 for Kolmogorov turbulence is barely acceptable at about 5%.

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What can we do? The above arguments are quite basic. Perhaps the answer is to consider more- realistic geometries. We are observing near the center of the galactic disk. In this case, the gradients and hence the anisotropies can be much smaller.

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X

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Results of a simple 1-dimensional model which illustrates the point

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Even more-complicated scenarios are possible. Results of a model calculation with multiple sources.

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Conclusions Simple considerations based on observations lead to untenable conclusions regarding the anisotropy high-energy cosmic rays. Perhaps we must go to more-complicated models such as that illustrated here or those of Strong and Moskalenko. Can we find observational tests for these ideas?

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