Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-11T21:27:26.905Z Has data issue: false hasContentIssue false

A monoenergetic-source solution of the steady-state cosmic-ray equation of transport

Published online by Cambridge University Press:  17 February 2009

G. M. Webb
Affiliation:
Monash University, Clayton, Victoria 3168
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A monoenergetic point-source solution of the steady-state cosmic-ray equation of transport for cosmic-rays in the interplanetary region in which monoenergetic particles are released isotropically and continuously from a fixed heliocentric position is derived by Lie Theory. A spherically-symmetric model of the propagation region is assumed incorporating anisotropic diffusion with a diffusion tensor symmetric about the radial direction, and the solar wind velocity is radial and of constant speed V. Because of the point release the solution is non-spherically-symmetric.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1977

References

REFERENCES

[1]Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions (Dover, New York, 1965).Google Scholar
[2]Bluman, G. W., “Applications of the general similarity solution of the heat equation to boundary value problems”, Quart. J. Appl. Math. 31 (1974), 403.Google Scholar
[3]Bluman, G. W. and Cole, J. D., “The general similarity solution of the heat equation”, J. Math, and Mechanics 18 (1969), 1025.Google Scholar
[4]Bluman, G. W. and Cole, J. D., Similarity methods for differential equations (A.M.S. publications 13) (Springer-Verlag, New York, 1974).CrossRefGoogle Scholar
[5]Dolginov, A. Z. and Toptygin, I. N., “Diffusion of cosmic particles in the interplanetary medium”, Geomag. and Aeron. 7 (1967), 785.Google Scholar
[6]Dolginov, A. Z. and Toptygin, I. N., “Cosmic-rays in the interplanetary magnetic fields”, Icarus 8 (1968), 54.Google Scholar
[7]Fisk, L. A., “Behaviour of cosmic-rays in the interplanetary medium” (Ph.D. Thesis, University of California, San Diego, 1969).Google Scholar
[8]Fisk, L. A. and Axford, W. I., “Solar modulation of galactic cosmic-rays, 1”, J. Geophys. Res. 74 (1969), 4973.Google Scholar
[9]Gleeson, L. J. and Axford, W. I., “Cosmic-rays in the interplanetary medium”, Astrophys. J. Lett. 149 (1967), L115.Google Scholar
[10]Gleeson, L. J. and Axford, W. I., “The solar radial gradient of galactic cosmic-rays”, Canad. J. of Physics 46 (1968), S937.Google Scholar
[11]Gleeson, L. J. and Axford, W. I., “Solar modulation of galactic cosmic-rays”, Astrophys. J. 154 (1968), 1011.CrossRefGoogle Scholar
[12]Gleeson, L. J. and Urch, I. H., “A study of the force-field equation for the propagation of galactic cosmic-rays”, Astrophys. and Space Sci. 25 (1973), 387.Google Scholar
[13]Gleeson, L. J. and Webb, G. M., “Modulation and spectral redistribution of galactic. cosmic-rays”, Proc. 14th Int. Conf. on Cosmic-rays, Munich 3 (1975), 893.Google Scholar
[14]Harrison, B. K. and Estabrook, F. B., ‘Geometric approach to invariance groups and solution of partial differential systems”, J. Math. Phys. 12 (1971), 653.Google Scholar
[15]Jokipii, J. R. and Parker, E. N., “On the convection, diffusion and adiabatic deceleration of cosmic-rays in the solar wind”, Astrophys. J. 172 (1970), 319.Google Scholar
[16]Lie, S., “Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen”, Arch. Math. 6 (1881), 328.Google Scholar
[17]Ovsjannikov, L. V., Gruppovye svoystva differentsialny uravneni, Novosibirsk (1962) (“Group properties of differential equations”, translated by G. W. Bluman, 1967).Google Scholar
[18]Parker, E. N., “The passage of energetic charged particles through interplanetary spacePlanet. Space. Sci. 13 (1965), 9.CrossRefGoogle Scholar
[19]Parker, E. N., “The effect of adiabatic deceleration on the cosmic-ray spectrum in the solar system”, Planet. Space Sci. 14 (1966), 371.Google Scholar
[20]Sneddon, I. N., Special functions of mathematical physics and chemistry (Oliver and Boyd, Edinburgh, 1961).Google Scholar
[21]Toptygin, I. N., “Direct and inverse problem of cosmic-ray propagation in interplanetary space”, Geomag. and Aeron. 13 (1973), 181.Google Scholar
[22]Urch, I. H., “Theoretical studies of the interplanetary medium” (Ph.D. Thesis, University of Adelaide, 1971).Google Scholar
[23]Webb, G. M., “Steady-state cosmic-ray propagation in interplanetary space” (Ph.D. Thesis, University of Tasmania, Hobart, 1976).Google Scholar
[24]Webb, G. M., “Similarity solutions of the steady-state cosmic-ray equation of transport”, J. Aust. Math. Soc., Series B 19 (1976), 432.Google Scholar
[25]Webb, G. M. and Gleeson, L. J., “Monoenergetic-source solutions of the steady-state cosmic-ray equation of transport”, Proc. 13th Int. Conf. on Cosmic-rays, Denver University of Denver, Conf. Papers, 5, 1973, 3253).Google Scholar
[26]Webb, G. M. and Gleeson, L. J., “Green's theorem and Green's functions for the steady-state cosmic-ray equation of transport”, Astrophys. and Space Sci. 50 (1977), 205.CrossRefGoogle Scholar
[27]Webb, G. M. and Gleeson, L. J., “Green's formula for the cosmic-ray equation of transport”, Proc. 15th Int. Conf. on Cosmic-Rays, Bulgaria, 3 (1977), 12.Google Scholar