COSMIC RAYS, PROPAGATION J.R. JOKIPII Essential to an understanding of most areas of high-energy astrophysics is an understanding of the physics of the acceleration and propagation of energetic charged particles or cosmic rays. Since in all cases of interest the energy change and spatial transport are closely coupled, they will both be referred to as parts of a general transport process. In all cases of interest the ambient thermal plasma is sufficiently rarefied that collisions with particles are extremely rare. Moreover, the effects of gravity on the energetic particles are negligible. The number of cosmic ray particles (and hence their mass and momentum) is much less than that of the background plasma. Hence, the transport is governed by the effects of ambient electric and magnetic fields, determined by the thermal plasma. Collisions need only be considered in cases where the production of secondary particles by the infrequent nuclear interactions are of interest. This latter aspect of the problem is considered briefly later in this entry. The acceleration and transport of charged particles in astrophysics may then be described quite succinctly by the equation of motion for a particle of charge q, momentum p, and velocity w *************************************** where the electric and magnetic fields E and B are determined by the state of the background plasma, and where c is the speed of light. In any given situation, the basic problem is to call upon our knowledge of the structure and dynamics of the plasma to determine the electric and magnetic fields to be used in Eq.(1) and to determine the initial and boundary conditions for the accelerated particles. In nearly all cases of interest, the background plasma satisfies the hydromagnetic approximation, in which the electric field is determined by the magnetic field and the fluid flow velocity relative to the observer by the relation ********. Hence the statement that the particle transport is governed by the magnetic field. Because the equation is impossible to solve in general, a number of approximations have been utilized. Beginning with the seminal work of Enrico Fermi, these all recognize the fact that the plasma and magnetic field are in general irregular and turbulent, so that a statistical treatment is required. The most important and generally used approximation is the diffusion approximation. This is based chiefly on the observation that energetic particles in many important cases are observed to have a very nearly isotropic pitch angle distribution relative to the local plasma. This isotropy is a consequence of the "scattering" of the particles by small-scale, irregular fluctuations in the ambient magnetic field, and may be derived from a statistical analysis of the equation of motion. Space does not permit a discussion of this scattering theory here. In many situations the scattering is sufficiently rapid that the distribution of cosmic rays may be taken to be nearly isotropic, the anisotropies (in the local fluid frame) being generated by spatial gradients of the density. The basic, physical effects to be included are (a) diffusion, which describes the random walk of the cosmic rays through the random magnetic field irregularities; (b) convection of the cosmic rays with the background fluid flow; (c) drift of the cosmic ray guiding centers caused by the gradient and curvature of the large-scale magnetic field; (d) energy change of the cosmic rays due to the expansion or compression of the background fluid and its magnetic field; and (e) acceleration due to possible random motions of the magnetic irregularities (denoted second-order Fermi acceleration). The basic transport equation, which combines all these effects, may be written (with names of the various physical where *(*****) is the omnidirectional particle distribution as a function of position r, momentum magnitude p, and time t. U is the background convection velocity, *** is the diffusion tensor, which may be obtained from the spectrum of magnetic irregularities in the turbulence, the drift velocity ** is determined by the large-scale magnetic field **=(******)***(****), Dpp is the rate of acceleration due to the random motion of the magnetic irregularities (relative to U), and Q is the local source function. Another way of viewing Eq.(2), which shows that it should be in fact a reasonable approximation in widely varied circumstances, is to note from Liouville's theorem that a homogeneous, isotropic distribution in an arbitrary static magnetic field is in a steady state. Equation (2) essentially describes the first-order consequences of gradients in the distribution function and plasma flow velocity. Hence, even if we do not know some of the transport coefficients (such as K**) accurately, the general form of Eq.(2) may be adequate for many purposes. Equation (2) contains nearly all of the cosmic ray acceleration and transport mechanisms discussed in recent years and will be the basis of the present discussion. Examples include the diffusion of cosmic rays in the interstellar gas or in supernova remnants, the basic theory of acceleration at hydromagnetic shocks or the venerable second-order Fermi acceleration. Of course, in many cases, some of the terms in Eq. (2) may be omitted to simplify the analysis. PARAMETERS The most important applications of Eq.(2) have been to the solar wind and to the interstellar medium. Before considering detailed modeling, it is useful to consider the general magnitude of the basic parameters, which have been found to give reasonable agreement with observations. The typical cosmic ray is a proton with an energy of 1 GeV. For such a particle we expect Clearly, we expect diffusion, convection, and drift to play significant roles in the solar wind, whereas drift may clearly be neglected for the bulk of the cosmic rays in the Galaxy. APPLICATIONS AND OBSERVATIONS Solar Wind The most sophisticated application of Eq.(2) has been to the transport of galactic cosmic rays in the solar wind, which is subject to in situ observational testing. Here the solar system is regarded as residing in a constant, isotropic bath of galactic cosmic rays. The radially outflowing solar wind acts to partially exclude these particles from the inner solar system, "modulating" the cosmic ray intensity with a basic 11-yr period in antiphase with the sunspot cycle (see Fig.1). In this case, all the terms in the equation except for the D** term play important roles. Indeed, the full form of Eq.(2) (without the D** term) was first written down by Eugene N. Parker in response to the challenges presented by solar modulation. For particle energies greater than approximately 1 MeV, the D** term is small and may be safely neglected. Hence the solar modulation may be regarded as a balance between the inward random walk or diffusion, the outward convection by the solar wind, gradient and curvature drifts caused by the large-scale structure of the interplanetary magnetic field, and the adiabatic cooling due to the radial expansion of the solar wind. Sophisticated three-dimensional numerical solutions have been obtained, utilizing the presently accepted picture of the solar wind and its entrained magnetic field, (the heliosphere). In this picture, the solar wind flows out to some termination radius (not yet known, but certainly greater than 50 AU). At this boundary the cosmic ray intensity takes on its interstellar value. The heliospheric model is quite accurate in the solar equatorial regions, where there are many direct observations, but the values of the parameters such as ambient magnetic field and flow velocity in the polar regions are quite uncertain. This uncertainty notwithstanding, the calculated properties of the cosmic rays (radial and latitudinal gradients, energy spectrum, and time variations) are in basic agreement with Earth-based and spacecraft observations of cosmic rays. In summary, the modulation of cosmic rays by the solar wind provides a reasonably complete verification of the basic transport equation, with the exception of the Fermi acceleration term, which is unimportant in this case. Interstellar Medium The observed energy spectrum of galactic cosmic rays is shown in Fig. 2. The part of the spectrum corresponding to energies between *** and **** eV is thought to originate in supernova explosions in the Galaxy, and to arrive at the Earth after propagating for some *** years in the interstellar medium. The cosmic ray transport in the irregular magnetic field, governed by the basic transport equation, establishes a quasisteady equilibrium where the sources are very nearly balanced by losses. The ultra-high-energy particles (energies **** eV) have a different origin and may come from outside the Galaxy. Only those galactic particles with energies between about *** and **** eV are considered, to avoid the part of the spectrum that is seriously distorted by the solar wind. These "unmodulated" galactic cosmic rays are observed to be highly isotropic in arrival directions (with a relative anisotropy ********** at ****** eV). Furthermore, evidence from observed * rays and synchrotron emission indicates that they are uniformly distributed throughout the galactic disk. Finally, observations of unstable isotopes in meteorites (caused by nuclear interactions of impacting cosmic rays) show that the intensity of these cosmic rays has been constant to within about 50%, at the solar system, for approximately the last *** yr. The energy spectrum in interstellar space is apparently very smooth. Althuogh we could use Eq.(2) with appropriate parameters and boundary conditions, it is adequate to illustrate the propagation of these cosmic rays and their confinement to the Galaxy in terms of a simpler model. The galaxy is taken to be a leaky box containing cosmic rays of species * with a density **(*). The loss of these particles is described by a mean leakage time *(which is related to the diffusion coefficient * by *******, where L is a characteristic scale of the galactic confinement region). If there is a source of particles **(**), conservation of particles is then described by the equation *******************************. (3) Observations show that the mean trapping time * for cosmic rays in the Galaxy is about a few times *** yr, much less than galactic time scales, so the present distribution of cosmic rays reflects a steady state between sources and losses, and we have the simple results from Eq.(3): ******************************* (4) The equilibrium energy spectrum depends on both the source spectrum and the energy dependence of the transport. In particular, because the leakage time decreases with increasing energy, the spectrum is steepened by the loss process. The observed secondary cosmic ray nuclei as a function of energy per nucleon provide valuable constraints on the transport in the Galaxy. Secondaries are produced by spallation of heavier cosmic ray nuclei by the rare collisions with the ambient interstellar gas particles. Observations show that for energies above roughly 5 GeV per nucleon, the ratio of secondaries to primaries decreases monotonically with increasing energy. One observes, approximately, for the ratio of the secondaries to the primaries from which they have been created, as a function of energy T, ******************************** (5) Above 2 GeV per nuleon. This is generally interpreted to be generally interpreted to be a result of an energy-dependent transport and loss from the Galaxy. Because the cross sections for nuclear interactions at energies ** GeV are roughly independent of energy, the dependence given in Eq.(5) would be produced if the leakage time were approximately proportional to *** above ***** GeV. Galactic cosmic rays are currently believed to be accelerated by collisionless shock waves. Subject only to a few quite reasonable restrictions, which essentially amount to assuming a planar shock on scales of the particle gyroradius, and the validity of the basic transport equation (2), one finds the spectrum of accelerated particles above ***** GeV per nucleon to be ***************************, (6) Where *=0 if the shock is strong, and increases as the shock strength decreases. This shape is independent of the particle propagation parameters, insofar as the basic assumptions are satisfied. Quite naturally, then, from Eq.(4) we would expect the observed ******** primary nucleon energy spectrum from this source if ******** (moderately strong shock), and a leakage time corresponding to that obtained above from the secondary to primary ratio. We note that although this source spectrum is located at the shock fronts, over the *****-yr lifetime of a typical cosmic ray particle, one expects that much of the interstellar medium will be traversed by shocks. Hence, it is reasonable to assume a smooth source distribution in the disk. STATUS The current diffusive theory of cosmic ray propagation, summarized in Eq.(2), has proved to be successful in explaining many observed features of cosmic rays in the Galaxy and the solar system. It may be used with confidence in other situations as well. Additional Reading Axford, W.I.(1981). Acceleration of cosmic rays by shock waves. In proceedings of the 17th International Cosmic Ray Conference 12, p.155. Cesarsky, C.J.(1980). Cosmic-ray confinement in the Galaxy. Ann. Rev. Astron. Ap.18 289. Drury, L.(1983). An introduction to the theory of diffusive shock acceleration of energetic particles in tenuous plasmas. Rep. Prog. Phys.46 973. Jokipii, J.R.(1971). Propagation of cosmic rays in the solar wind. Rev. Geophys. Space Phys.9 27 Linsley, J.(1980). Very high energy cosmic rays. In IAU Symposium 94, Origin of Cosmic Rays, G. Setti, G. Spada, and R.A. Wolfendale, eds. Reidel, Dordrecht, p.53. Meyer, P.(1969). Cosmic rays in the Galaxy. Ann. Rev. Astron. Ap.7 1. Toptygin, I.N.(1985). Cosmic Rays in Interplanetary Magnetic Fields. Reidel, Dordrecht. Volk, H.J.(1976). Cosmic-ray propagation in interplanetary space. Rev. Geophys. Space Phys.13 547. See also Cosmic Rays, Acceleration; Heliosphere; Interplanetary Medium, Shock Waves and Traveling Magnetic Phenomena; Supemova Remnants, Evolution and Interaction with the Interstellar Medium.