Propagation of cosmic-ray nucleons in the Galaxy

TL;DR

This study develops a comprehensive 3D numerical model for Galactic cosmic-ray propagation, incorporating diffusion, convection, and diffusive reacceleration, as well as energy losses and realistic gas and radiation fields. It calibrates models against B/C and 10Be/9Be ratios and gamma-ray gradients, using a Crank–Nicolson scheme with operator splitting to solve the transport equation for nucleons, electrons, and positrons. The results favor reacceleration models with halo heights of 4–12 kpc and a constraint of dV/dz < 7 km s⁻¹ kpc⁻¹, while diffusion/convection models without a diffusion-break fail to reproduce the data; Be-10/Be-9 and gamma-ray gradients also point to a broader cosmic-ray source distribution than the standard SNR profile. The work provides a publicly available, physically detailed framework for predicting cosmic-ray populations and associated gamma-ray and synchrotron signatures across the Galaxy.

Abstract

We describe a method for the numerical computation of the propagation of primary and secondary nucleons, primary electrons, and secondary positrons and electrons. Fragmentation and energy losses are computed using realistic distributions for the interstellar gas and radiation fields, and diffusive reacceleration is also incorporated. The models are adjusted to agree with the observed cosmic-ray B/C and 10Be/9Be ratios. Models with diffusion and convection do not account well for the observed energy dependence of B/C, while models with reacceleration reproduce this easily. The height of the halo propagation region is determined, using recent 10Be/9Be measurements, as >4 kpc for diffusion/convection models and 4-12 kpc for reacceleration models. For convection models we set an upper limit on the velocity gradient of dV/dz < 7 km/s/kpc. The radial distribution of cosmic-ray sources required is broader than current estimates of the SNR distribution for all halo sizes. Full details of the numerical method used to solve the cosmic-ray propagation equation are given.

Figures (7)

• Figure 1: The 3-D distribution of $^{12}C$ and $^{10,11}B$ at 515 MeV/nucleon for reacceleration model with $z_h$ = 10 kpc, for $v_A$ = 20 km s$^{-1}$. Parameters: see model 10500 in Table \ref{['table2']}.
• Figure 2: $B/C$ ratio for diffusion/convection models without break in diffusion coefficient, for $dV/dz$ = 0 (solid lines), 5 (dotted lines), and 10 km s$^{-1}$ kpc$^{-1}$ (dashed lines). The cases shown are (a) $z_h$ = 1 kpc, (b) $z_h$ = 3 kpc, (c) $z_h$ = 10 kpc. Solid lines: interstellar ratio, shaded area: modulated to 300 -- 500 MV. Data: vertical bars: HEAO-3, Voyager (Webber96), filled circles: Ulysses (DuVernois96: $\Phi$ = 600, 840, 1080 MV). Parameters as in Table \ref{['table1']}.
• Figure 3: $B/C$ ratio for diffusion/convection models with break in diffusion coefficient, for $dV/dz$ = 0 (solid lines), 5 (dotted lines), and 10 km s$^{-1}$ kpc$^{-1}$ (dashed lines). The cases shown are (a) $z_h$ = 1 kpc, (b) $z_h$ = 5 kpc, (c) $z_h$ = 20 kpc. Lower lines: interstellar ratio; upper lines: modulated to 500 MV. Parameters as in Table \ref{['table1']}. Data: as Figure \ref{['fig3']}.
• Figure 4: $^{10}Be/\,^9Be$ ratio for diffusion/convection models, for $dV/dz$ = 0 (solid lines), 5 (dotted lines), and 10 km s$^{-1}$ kpc$^{-1}$ (dashed lines). The cases shown are (a) $z_h$ = 1 kpc, (b) $z_h$ = 5 kpc, (c) $z_h$ = 20 kpc. Data points from Lukasiak et al. (1994a) (Voyager-1,2: square, IMP-7/8: open circle, ISEE-3: triangle) and Connell (1997) (Ulysses): filled circle. Parameters as in Table \ref{['table1']}.
• Figure 5: Predicted $^{10}Be/\,^9Be$ ratio as function of (a) $z_h$ for $dV/dz$ = 0, 5, 10 km s$^{-1}$ kpc$^{-1}$, (b) $dV/dz$ for $z_h = 1 - 20$ kpc at 525 MeV/nucleon corresponding to the mean interstellar value for the Ulysses data (Connell98); the Ulysses experimental limits are shown as horizontal dashed lines. The shaded regions show the parameter ranges allowed by the data.