Implementation of CR Energy SPectrum (CRESP) algorithm in PIERNIK MHD code. II. Propagation of Primary and Secondary nuclei in a magneto-hydrodynamical environment

Abstract

We developed a new model for the production and propagation of spectrally resolved primary and secondary Cosmic Ray (CR) nuclei elements within the framework of the Cosmic Ray Energy Spectrum (CRESP) module of the PIERNIK MHD code. We extend the algorithm to several CR nuclei and demonstrate our code's capability to model primary and secondary CR species simultaneously. Primary C, N, and O are accelerated in supernova (SN) remnants. The spallation collisions of the primary nuclei against the thermal ISM protons lead to secondary Li, Be, and B products. All the CR species evolve according to the momentum-dependent Fokker-Planck equations that are dynamically coupled to the MHD system of equations governing the evolution of the ISM. We demonstrate the operation of this system in the gravity stratified box reproducing the Milky Way conditions in the Sun's local environment. We perform a parameter study by investigating the impact of the SN rate, the CR parallel diffusion coefficient $D_\parallel$, and the rigidity-dependent diffusion coefficient power index $δ$. A novel result of our investigation is that the secondary-to-primary flux ratio \BtoC increases with increasing diffusion coefficient, due to the weaker vertical magnetic field resulting from CR buoyancy effects. Moreover, a higher SN rate leads to lower values of \BtoC because of stronger winds and the shorter residence time of primary CR particles in dense disk regions.

Figures (12)

• Figure 1: Example of $^{12}$C injection differential kinetic energy density spectrum $\mathrm{d}e/\mathrm{d}p$ at $t=0$ from a one cell simulation test. In the first and fourth bins (non-relativistic and trans-relativistic), the spectrum follows $p^{2 + s - q}\approx p^{-2.1}$. Above $p/m_pc=A_C$ ($A_C=12$ for Carbon), the ultra-relativistic bins exhibit a flatter power law $p^{-1.1}$.
• Figure 2: CR differential kinetic energy density spectrum for $^{12}$C after one step in a test simulation with primary $^{12}$C and secondary $^{11}$B only. The red curve is the primary injection spectrum of $^{12}$C (the amplitude is changed to compare both curves), as seen in Figure (\ref{['fig.1']}). We observe the differences between primaries and secondaries below $p/m_p c\leq A_C$, showing how the momentum-dependent spallation rate changes the slope. A power-law fitting method applied to a few bins estimates the slope of the $^{11}$B spectrum (blue curve). The corresponding curves and slopes are displayed. As expected, the slope of $^{11}$B and $^{12}$C differ by one for $p/m_p c \leq A_C$ and are equal for $p/m_p c \geq A_C$.
• Figure 3: Snapshots of the ISM density evolution (left) and vertical velocity evolution (right) in the stratified box over $500 \; \mathrm{Myr}$ in the $y-z$ plane at $x = 0$ for the model case A1 with SN rate $=80\;\mathrm{kpc}^{-2}\mathrm{Myr}^{-1}$, $D_{\parallel}^0 = 3\times 10^{28} \mathrm{cm}^2\mathrm{s}^{-1}$. The ISM presents CR-driven buoyancy structures, generating outflows in the vertical direction. The vertical velocity snapshots indicate the presence of outflows, meaning that advection for CRs is present.
• Figure 4: Top: snapshot of the gas density within the section $z \in [0,2]\,\mathrm{kpc}$ for A1 run at $t=100\;\mathrm{Myr}$, where magnetic fields vectors are displayed. Bottom: same snapshot as top, but at $t=500\;\mathrm{Myr}$.
• Figure 5: Snapshots of the stratified box from A1 (top row) and A3 (bottom row) simulations at $t = 500\; \mathrm{Myr}$ with, from left to right: gas density, CR protons energy density, the vertical velocity of the gas, vertical magnetic field, and four CR $^{12}$C bins at middle-valued momentum $p = 6.90\;\mathrm{GeV} \, \mathrm{c}^{-1}$, $p = 7.01\times10^{1}\;\mathrm{GeV} \, \mathrm{c}^{-1}$, $p = 7.12\times10^{2}\;\mathrm{GeV} \,\mathrm{c}^{-1}$ and $p = 7.24\times10^{3}\;\mathrm{GeV} \,\mathrm{c}^{-1}$. We observe a correlation between the magnetic field structure, outflows of gas, and outflows of CR protons and Carbon. Carbon at relativistic energies is more spread in the vertical direction, showing the action of rigidity-dependent diffusion.