A Possible Approach to Three-dimensional Cosmic-Ray Propagation in the Galaxy. III. Unstable Nuclei

We extend our three-dimensional propagation model for the stable cosmic rays (CRs) in the Galaxy to that for unstable CRs such as ¹⁰Be and ²⁶Al, including the energy gain due to reacceleration and energy loss due to ionization, where the latter is approximated by the cutoff procedure. We assume that there is no boundary in both the radial spread of the disk and the latitudinal spread of the halo, and that the three critical parameters, the diffusion coefficient D, the gas density n, and the CR source density Q, have exponential-type gradients in both Galactic longitude and latitude with scale heights (r_D,r_n,r_Q) for the former and (z_D,z_n,z_Q) for the latter. It is possible to apply the weighted-slab technique to the transport equation, even including the energy change process, as well as the decay effect, if the diffusion coefficient is separable in position r and rigidity R, i.e., D(rR)=D(r)(v/c)R^α. Combining this with the first-pole approximation in the path-length distribution, we can obtain the analytical solution rather easily. We compare our numerical results with experimental data on ¹⁰Be/ ⁹Be, ²⁶Al/²⁷Al, and so on and obtain a diffusion coefficient D_solar=1.17×10²⁸ cm² s^-1 at E=1 GeV nucleon^-1 and a gas density n_solar=0.832 H atoms cm^-3 at the solar system with scale heights z_D=2.5 kpc and z_n=250 pc. We found the residence time of CRs in the Galaxy to be 3.79×10⁷ yr at E=1 GeV nucleon^-1 and 1.14×10⁸ yr at E=100 MeV nucleon^-1.