RAMSES-MCR: A consistent multi-group treatment of cosmic rays physics in momentum-space with the RAMSES code

Abstract

Cosmic rays (CRs) are known to play a key role in many astrophysical environments: they can modify shock dynamics, influence the thermochemistry and the ionization of the interstellar medium, regulate galaxy mass content by driving galactic winds, and be released by jets from active galactic nuclei. They also serve as important observational tracers through $γ$-ray emission, radio synchrotron, and secondary particle production. Since CR particles follow power-law distributions in momentum space spanning many decades in energy, and because diffusion and radiative losses further shape these spectra, it is crucial to model spectrally resolved CRs in numerical simulations and to assess the impact of this modeling on gas dynamics and observational signatures. We present a consistent multi-group spectral method in momentum space for CR protons called RAMSES-MCR in the adaptive mesh refinement code RAMSES, based on the two-moment formalism that evolves both CR energy and number density in momentum space, together with their associated flux. The modeled CR processes include advection, anisotropic/isotropic diffusion, streaming instability, Coulomb and hadronic losses, adiabatic changes, and feedback onto the gas. We also show that the method can be naturally extended to CR electrons (e.g. including synchrotron losses) and generalized to multiple CR species. The implementation is validated against a suite of standard multi-dimensional tests. We finally apply RAMSES-MCR to the three-dimensional expansion of a supernova remnant including CRs with anisotropic diffusion and energy losses, and demonstrate how CR energy redistributes in a momentum-dependent manner and modifies the gas momentum during the snowplough phase.

Figures (15)

• Figure 1: Illustration of the spectral discretization. Each momentum bin $i$ can be characterized by the parameters $(f_{i-1/2},q_i)$. The distribution function is not necessarily continuous at the interfaces.
• Figure 2: CR spectrum evolution cooled by Coulomb and hadronic losses, with $N_{\rm c}=5$ bins, an initial distribution function $f\propto p^{-4.5}$ and $n_{\rm e}=n_{\rm N}=1\,\rm cm^{-3}$. The top panel shows the evolution in time of the spectrum. The blue dashed curve shows the spectrum for $N_{\rm c}=15$ at $t=100 \, \rm Myr$ and the black dashed line correspond to the expected shape of the spectrum considering the radiative processes separately. We find that our test agrees well with the analytic prediction. In the bottom panel, we show the time evolution of the total CR pressure, the pressures of the low-momentum and high-momentum CR components, and the gas pressure, which illustrate the transfer of energy from CRs to the thermal plasma.
• Figure 3: CR spectrum evolution in case of injection and radiative losses, with $N_{\rm c}=10$ bins and $n_{\rm e}=n_{\rm N}=1\,\rm cm^{-3}$. The first panel shows the time evolution of the spectrum with an injection slope $q_{\rm inj}=4$. The black dashed line correspond to the analytic solution of equation \ref{['eq:steadystate_solution']}. The second panel shows the error of the numerical solution compared to the analytic result using the $L^2$ norm.
• Figure 4: Static test with adiabatic compression cycles using $\boldsymbol{\nabla}.\boldsymbol{u} =-1\,\rm Myr^{-1}$. The panels show the relative ratio of the distribution function at a given time with respect to the analytical distribution function $f_{\rm ana}$ for $q_{\rm ini}=4.5$. The dots indicate the value of the ratio at the center of the bin. We present the solution for three spectral time steps and three time integration schemes.
• Figure 5: Spatial diffusion in one-dimension of the CR energy densities $e_{{\rm c},i}$ at $t=1\,\rm Myr$ for each momentum bin, assuming that diffusion scales as $\kappa(p)\propto p^{0.5}$. The green curves show the total CR energy density (sum over all bins). All CR quantities are normalized to their initial values at the peak of the gaussian. The two lower panels show the CR distribution function at two different positions $x=500\,\rm pc$ and $x=750\,\rm pc$ (middle and bottom panels respectively). Solid lines correspond to the spectrally-averaged diffusion coefficients ("with correction"), while dashed lines assume $\kappa^n_{{\rm c},i}=\kappa^e_{{\rm c},i}$ ("without correction").