A model for cosmic rays

high as_∼ 5_×1019_{eV. Higher energy cosmic ray particles are susceptible to a larger range of inelastic}

processes, and therefore are subject to faster dissipation by means of inelastic collisions.

4.2.2.5. Coulomb losses

While moving through a gaseous medium, charged particles do not only lose energy by actual collision events, but also are subject to the influence of the electromagnetic field created by the ambient matter. They continuously exchange momentum and energy with the surrounding ions and electrons mediated by the Coulomb force. In a purely thermal plasma, this interaction is statistically balanced, i.e. particles on average gain as much energy as they lose, provided the momentum spectrum has the well-known thermal equilibrium shape.

However, particles that move with an extreme energy compared to the thermal population find them- selves losing energy to the charged matter that they pass by, and gradually will have their momenta reduced to thermal levels. This effect is most efficient for cosmic ray particles at low velocities. Highly relativistic particles react slower to the thermalization process due to their shorter interaction time with individual ions of the background matter. It should be noted that not only an ionized plasma causes this kind of energy loss for cosmic ray particles. Even in a neutral gas, the negatively charged electrons can induce this effect, albeit at a largely reduced efficiency, while the positive ions are rendered ineffective due to atomic charge shielding effects.

4.2.3. A model for cosmic rays

The large range of different physical effects acting on the particles of the cosmic ray population make a detailed inclusion in simulations difficult, if not impossible. In principle, a fully general treatment would

have to evolve the full momentum distribution function of cosmic ray particles for every fluid element. The high dimensionality of this problem makes this impossible in practice, given the performance of even the newest generation of supercomputers. In addition, cosmic ray particles of different type and momentum are subject to different physical dissipation processes, adding further complications for a numerical treatment of the problem.

As a result, simulation approaches to cosmic ray physics have so far been restricted to post-processing analysis of outputs produced by ordinary hydrodynamic simulations, based on empiric relations to model cosmic ray populations in a post-hoc fashion (e.g.Miniati 2001,Miniati et al. 2001). A significant dis- advantage of this approach is that it cannot account for the mutual dynamical influence between thermal and relativistic gas components.

To make fully self-consistent hydrodynamic simulations possible that for the first time include the dynamical effects that a cosmic ray population may cause, we introduce a simplified model for cosmic ray physics. In this model, we represent the cosmic ray population existing in each fluid mass element of the simulation in a way that captures as many physical properties of the cosmic ray population as possible while it on the other hand induces only a moderate computational cost. We aim for a model that

Cosmic ray feedback in hydrodynamical simulations of galaxy formation

is accurate for the in-situ computation of gas-dynamical effects of the cosmic rays, and favor this aspect over an accurate representation of the detailed spectral distribution of the population. For this reason, we will invoke a simplified momentum distribution function of cosmic ray particles, guided by adiabatic invariants and the fundamental principles of energy and particle-number conservation.

4.2.3.1. CR population

Protons contribute the largest mass fraction to the cosmic ray population. Yet,α-particles and heavier ions make up a sizeable part of the cosmic rays, with abundances exceeding those found in thermal in- terstellar and intergalactic medium. However, taking the heavy ions explicitly into account in our cosmic ray model would require either a self-consistent or phenomenologically motivated enrichment model that resolves the different nuclei in the acceleration mechanisms. Also, there would be substantial additional

computational cost created by the need to iterate through all particle families in every simulation step. Instead, the model we present here restricts itself to the effects of protons andα-particles, where the

information on the latter is absorbed into the proton treatment. For all processes presented, this is a reasonable approximation. In hadronic interactions, the binding energy of theαnuclei of a few MeV can be taken to be of minor importance when considering the kinetic energies of the particles on the GeV scale. On the other hand, for Coulomb cooling, due to the proportionality of the process’ efficiency to

the square of the ion’s charge, the effect on the four nucleons of theαparticle each is equal to the energy loss that an individual proton would feel.

4.2.3.2. Confinement

Cosmic ray particles do not travel freely through space. Rather, they are subject to the strong Lorenz force of the magnetic field. It keeps cosmic ray particles of energies less than _∼ 3_×1018eV tightly bound in our Galaxy. In fact, the magnetic field that is frozen into a hot plasma can strongly couple the cosmic ray population to the baryonic gas, such that the two fluids effectively move together. In the following, we picture the cosmic ray population to be confined to its Lagrangian fluid element by a magnetic field, even though the latter is not explicitly included in the simulation formalism. Note that we also neglect the energy density and pressure that is in principle associated with the magnetic field component.

Irregularities in the magnetic field can scatter CR particles such that they escape from their field lines. The magnetic confinement is hence not perfect. To account for this effect, we include a formalism for diffusive transport of cosmic ray particles between adjacent gas mass elements.

4.2.3.3. Fixed spectral shape

In our Galaxy, it is found that cosmic rays follow a power-law spectrum with a spectral index ofα_≃2.75 for particle energies below the “knee” at_∼ 4_×1015_{eV. Most of the energy of the cosmic ray spectrum}