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Cosmic rays (CRs) are elementary particles and atomic nuclei that have been accelerated by very high energies by magnetised shocks. They pervade interstellar space, and though they are much less numerous than the background sea of thermal particles that makes up the bulk of the interstellar medium, they carry a significant amount of energy. This makes them potentially important for the dynamics of interstellar gas. Because CRs are charged particles, they can be deflected by magnetic fields, and interactions between CRs and the magnetic fields that thread interstellar plasma provide a mechanism for CRs and the plasma to exert forces on one another. However, because magnetic interactions that can take place between CRs and the background plasma around them are complex and poorly understood, the exact nature of CR transport remains an unsolved problem.

For this reason, astrophysicists who seek to model CRs often resort to ad hoc empirical prescriptions, for example assuming that CR transport can be approximated as a simple diffusion process. This simple assumption works surprisingly well when it comes to reproducing the observable signatures of CRs, for example gamma-ray emission produced when CR protons collide with protons in the interstellar medium. However, this is an empirical description only, and one at odds with plasma physics models, which suggest that for the CRs that carry most of the energy budget – protons with energies of order a GeV – the dominant mode of CR motion should be a process known as streaming, whereby CRs move along magnetic field lines locked in resonance with Alfvén waves. In this picture, CRs do not diffuse, they slide along magnetic field lines like beads on a wire.

In Sampson et al. (2023), ANU honours student Matt Sampson led a team of other ANU researchers, including PhD student James Beattie, in developing models that go a long way toward reconciling the empirical diffusion and theoretically-motivated streaming pictures. The key insight that motivated the research is that, while microscopically CRs move along field lines like beads on a wire, if the field lines themselves are in turbulent motion, the net effect may look something like diffusion. To test this hypothesis, we carried out a suite of 240 simulations of magnetohydrodynamic turbulence coupled to calculations of the trajectories of CRs through that turbulence. We then analysed the statistical properties of the spatial distribution of CRs after their launch (Figure 1) to investigate whether the effects indeed looked like diffusion, and to characterise the diffusion rate as a function of the fundamental properties of the turbulence.

We found a surprising result: while the statistical properties of the CR distribution produced by propagation through the turbulence look in some ways like diffusion, they are better described by superdiffusion. Superdiffusion differs from diffusion in that both can be thought of as the result of particles carrying out a random walk, but with ordinary diffusion the steps of that walk are of fixed size, or are drawn from a distribution like a Gaussian with small tails. By contrast, superdiffusion corresponds to a case where particles walk randomly, but the steps they talk are occasionally very large, so that the distribution of step sizes is heavy-tailed. CRs seems to undergo this sort of heavy-tailed random walk, rather than the classical random walk as assumed in simple models.

Our simulation suite enabled us to calculate the statistical properties of CR superdiffusion and how they depend on the fundamental properties of the plasma through which the CRs move. This will enable future modelling of CR transport using a physically-motivated rather than an ad hoc model. The realisation that CR transport is better described by superdiffusion than diffusion also helps resolve some outstanding problems in existing models. Previous diffusion-based models encountered the problem that it was not possible to reproduce all the existing observations using a single diffusion rate; observations on small sales focused on individual sources of CRs seemed to require a lower diffusion rate than observations that are sensitive to the average diffusion rate of the whole Galaxy. Superdiffusion naturally removes this problem, because one way of viewing superdiffusion is as diffusion where there are different diffusion rates on different length scales – in a superdiffusive system, particles travel long distances faster than would be expected for pure diffusion, but travel short distances more slowly. This is exactly what the observational constraints seem to demand, and thus our results enable us to reproduce all the observations with a single superdiffusive model.

Mark Krumholz

Reference: “Turbulent diffusion of streaming cosmic rays in compressible, partially ionized plasma”, Sampson, M. L., Beattie, J. R., Krumholz, M. R., Crocker, R. M., Federrath, C., & Seta, A. 2023, Monthly Notices of the Royal Astronomical Society, 519, 1503

Figure 1: Snapshots from a few simulations in our simulation suite. Each panel represents a different simulation, with the different simulations characterised by the sonic Mach number M and the Alfvénic Mach number MA0; colours indicate the local Alfvén speed vA in the plasma, with blue corresponding to places where it is lower than the mean Alfvén speed in the simulation vA0 and red to places where it is higher. The superimposed great trails show the trajectories of sample CRs through the plasma, with darker green corresponding to the most recently-injected CRs and lighter green to CRs that have been traveling through the plasma for a longer time.