Numerical Models for Cosmic Ray Propagation and Gamma Ray Production

An extensive program for the calculation of galactic cosmic-ray propagation has been developed. Primary and secondary nucleons, primary and secondary electrons, and secondary positrons are included. The basic spatial propagation mechanisms are (momentum-dependent) diffusion, convection, while in momentum space energy loss and diffusive reacceleration are treated. Fragmentation and energy losses are computed using realistic distributions for the interstellar gas and radiation fields. INTRODUCTION The main motivation for developing this code (Strong and Moskalenko 1997) is the prediction of diffuse Galactic gamma rays for comparison with data from the CGRO instruments EGRET, COMPTEL and OSSE. This is a development of the work described in Strong and Youssefi (1995). More generally the idea is to develop a model which self-consistently reproduces observational data of many kinds related to cosmic-ray origin and propagation: direct measurements of nuclei, electrons and positrons, gamma rays, and synchrotron radiation. These data provide many independent constraints on any model and our approach is able to take advantage of this since it must be consistent with all types of observation. We emphasize also the use of realistic astrophysical input (e.g. for the gas distribution) as well as theoretical developments (e.g. reacceleration). The code is sufficiently general that new physical effects can be introduced as required. The basic procedure is first to obtain a set of propagation parameters which reproduce the cosmic ray B=C ratio, and the spectrum of secondary positrons; the same propagation conditions are then applied to primary electrons. Gamma-ray and synchrotron emission are then evaluated. Models both with and without reacceleration are considered. DESCRIPTION OF THE MODEL The models are three dimensional with cylindrical symmetry in the Galaxy, the basic coordinates being (R; z; p) where R is Galactocentric radius, z is the distance from the Galactic plane and p is the total particle momentum. The numerical solution of the transport equation is based on a CrankNicholson implicit second-order scheme. In the models the propagation region is bounded by z = zh beyond which free escape is assumed. A value zh = 3 kpc has been adopted since this is within the range which is consistent with studies of 10Be=Be and synchrotron radiation. For a given zh the diffusion coefficient as a function of momentum is determined by B=C for the case of no reacceleration; with reacceleration on the other hand it is the reacceleration strength (related to the Alfven speed vA) which is determined by B=C. Reacceleration provides a natural mechanism to reproduce the B=C ratio without an ad-hoc form for the diffusion coefficient (e.g., Heinbach and Simon 1995, Seo and Ptuskin 1994). The spatial diffusion coefficient for the case without reacceleration is D = βD0 below rigidity ρ0, βD0(ρ=ρ0)δ above rigidity ρ0, where β is the particle speed. The spatial diffusion coefficent with reacceleration assumes a Kolmogorov spectrum of weak MHD turbulence so D = βD0(ρ=ρ0)δ with δ= 1=3 for all rigidities. For this case the momentum-space diffusion coefficient is related to the spatial coefficient (Seo and Ptuskin 1994). The injection spectrum of nucleons is assumed to be a power law in momentum for the different species, dq(p)=d p ∝ p γ for the injected density, corresponding to an injected flux dF(Ek)=dEk ∝ p γ where Ek is the kinetic energy per nucleon. The entire calculation is performed with momentum as the kinematic variable. The interstellar hydrogen distribution uses HI and CO surveys and information on the ionized component; the Helium fraction of the gas is taken as 0.11 by number. The interstellar radiation field for inverse Compton losses is based on stellar population models and IRAS and COBE data, plus the cosmic microwave background. The magnetic field is assumed to have the form 6 e (jzj=5kpc) (R=20kpc) μG. Energy losses for electrons by ionization, Coulomb, bremsstrahlung, inverse Compton and synchrotron are included, and for nucleons by ionization and Coulomb interactions. The distribution of cosmic-ray sources is chosen to reproduce the cosmic-ray distribution determined by analysis of EGRET gamma-ray data (Strong and Mattox 1996). The bremsstrahlung and inverse Compton gamma rays are computed selfconsistently from the gas and radiation fields used for the propagation. The π0-decay gamma rays are calculated directly from the proton and Helium spectra using the method of Dermer (1986). The secondary nucleon and secondary e source functions are computed from the propagated primary distribution and the gas distribution, and the anisotropic distribution of e in the μ system was taken into account. ILLUSTRATIVE RESULTS Some results obtained are shown in the Figures. Fig. 1: Secondary nucleons. The energy dependence of the B=C ratio can be reproduced with D0 = 2: