Some Physics

Cosmic rays are amongst the less-well understood phenomena in the universe. Neither their origin, distribution, lifetime or acceleration mechanism are undisputed. Simpson (1983) gives a detailed overview of the observational background of the cosmic ray problem - but to fully understand and interpret the observations, models have to take into account the origin of the cosmic-ray particles (i.e. their source-composition and energy distribution), the acceleration mechanism (including possible mass- and energy-dependencies of the acceleration), the propagation of these particles through the interstellar medium (including the interaction of the particles with this medium and with the galactic magnetic field), and, to a lesser extent, the propagation of cosmic-ray particles through the outer heliosphere and the earth's magnetic field (and, possibly, residual traces of earth's atmosphere - depending on the location of the chosen detector) and have to explain the isotopic composition of the observed particles, its absolute and differential energy-flux (i.e. spectrum) and the lack of anisotropies in the observed particle distribution.

A relatively simple model that can account for all the above, is the so-called 'leaky box' model, in which the particles are injected into a confinement volume of uniform density (usually associated with the galactic disk) by sources, that are uniformly distributed within that volume. These 'primaries' have then certain probabilities (mainly determined by the energy-dependent nuclear interaction cross-sections of the encountered interstellar gas) to produce so-called "secondaries", i.e. particles that to not originate in the cosmic ray source itself, but are created in the propagation volume (mainly by spallation/fragmentation, in which nuclei of mass A hit protons to produce nuclei below A).

The particles in the "box" also undergo ionization energy loss and have a certain probability of being stopped entirely - thereby supplying a fraction (or maybe all) of their energy to the interstellar medium. They may also escape from it with an energy-dependent probability whenever they hit one of the "walls" of the volume, or may decay radioactively if they happen to be unstable isotopes.

This (intuitively naive) diffusive model can account for all observed effects, since it encompasses quite a number of free parameters that allow it to be fit to the observed parameters of the cosmic ray flux: The elemental and isotopic composition of the source, the spectral distribution of particles leaving the source, the total amount of matter encountered by the particles as a function of energy, the typical path-length of a particle and the distribution of path-lengths around this mean (if leakage is purely determined by a fixed probability on every encounter with the "walls" of the volume, the path-length distribution will be exponential). See e.g. Protheroe et. al. (1981), Garcia-Munoz et. al. (1987) or Engelman et. al. (1990).

The limitations of the model are grave, though: not only does it not allow one to draw conclusions on the traversed volume, but since it is a homogeneous equilibrium model, it does not yield information about the distribution of the sources within that volume, and to the extent that differences in the source spectrum and/or elemental distribution can be masked by difference in path-length distribution or grammage of encountered mass, is even only of limited use in the determination of type of the source.

Several modifications of the model exist, that try to introduce a more physical view onto the matter, like models in which the average density of the encountered matter in the source (supernova remnants) is different from that in the average medium (e.g Cowsik and Wilson, (1973)), leading to differences in the path-length distribution, or models that include distributed re-acceleration of the particles in the traversed medium (e.g. Silberberg et. al. (1983) or Simon et. al. (1986, 1996)). These refinements, however, lead to only subtly different predictions for the observed composition and/or spectrum of the particles that impinge upon the earth's atmosphere, which are beyond the current limits of observation and theoretical background (interaction cross-sections etc.), and are therefore not really required by the model.

More physical approaches to the problem usually first take the non-homogeneity of the traversed matter into account. If the cosmic-ray particles spend an appreciable fraction of their lifetime in the galactic halo, the encountered grammage will be smaller, and the distribution of elements (primaries and secondaries) will be different depending on the location: assuming that the primaries are produced/accelerated in the galactic disk (where the density of probable energy sources is highest) the density of radioactive nuclides will be higher close to the disk. This is also true due to the fact that radioactive secondaries will be produced with higher probability in the denser disk than in the halo (e.g. Prischep and Ptuskin (1975) or Ginzburg et. al. (1980)).

Generally, these models treat the different traversed volumes by assigning different diffusion coefficients to them, models that extend far from the disk may also different magnetic fields (the confinement mechanism for charged particles) into account.

Models that assume the cosmic rays to be in hydrostatic equilibrium, for example, can make predictions on the pressure (and therefore distribution) of the particles, which in turn can be used to derive more physically meaningful parameters from the observed data (e.g Jones, (1979)).

It is obvious, that the average age (and thereby path-length for a given velocity) derived from a non-homogeneous model will be necessarily longer than one that assumes that the density of radioactive nuclides is the same everywhere as it is in Earth's direct neighborhood in the disk.

References:

Cowsik, Wilson, Proc. 14th ICRC, 1, 500, 1973
Engelman et. al., Astr. Ap., 233, 96, 1990
Garcia-Munoz, Simpson, Guzik, Wefel, Margolis, Ap. J. Supp., 64, 269, 1987
Ginzburg, Khazan, Ptuskin, Astr. Sp. Sci., 68, 295, 1980
Jones, Ap. J., 229, 747, 1979
Prischep, Ptuskin, Astr. Sp. Sci., 32, 265, 1975
Protheroe, Ormes, Comstock, Ap. J., 247, 362, 1981
Silberberg, Tsao, Letaw, Shapiro, Phys. Rev. Letters, 51, 1217, 1983
Simon, Heinrich, Mathis, Ap. J., 300, 32, 1986
Simon, Heinbach, Ap. J., 456, 519, 1996
Simpson, Ann. Rev. Nuc. Part. Sci., 33, 323, 1983