Can cosmic rays gain a second boost behind a supernova shock?

What happens to energetic particles after a supernova shock has accelerated them? They enter the downstream region: the expanding, shocked gas behind the advancing front. There, moving magnetic disturbances can continue exchanging energy with them. The question is whether this slower process has enough energy, and enough time, to matter.

In an arXiv preprint, Antonio Capanema and Pasquale Blasi investigate that question by following energetic protons and magnetic turbulence together. Their central finding is conditional: turbulence generated ahead of the shock and carried through it barely changes the particles’ energy distribution. A sustained additional supply behind the shock, however, can produce appreciable changes.

A slower process with more time

Diffusive shock acceleration energizes particles through repeated crossings of a shock front. A different mechanism can operate downstream, where particles scatter from moving magnetic disturbances. Individual encounters can increase or decrease a particle’s energy, while the ensemble can gain energy overall. This is stochastic acceleration, also called second-order Fermi acceleration.

It is generally slower than shock acceleration. But particles may remain downstream much longer than they spend undergoing acceleration at the front. That longer exposure could let a modest acceleration rate accumulate into a significant change.

There is an essential constraint: the particles draw their extra energy from the disturbances that scatter them. As particles gain energy, those waves weaken through damping. Calculating acceleration while holding the turbulence fixed can therefore exaggerate the effect and potentially violate energy conservation.

Following both sides of the exchange

The researchers numerically solved coupled transport equations for the proton momentum distribution and the distribution of turbulent energy across spatial scales. These equations account for particles gaining energy, the associated wave damping, and a turbulent cascade that transfers energy toward smaller scales, where it can ultimately dissipate.

They tested two common prescriptions for that cascade, known as Kolmogorov and Kraichnan turbulence. First, they used a closed-box model to isolate the competing processes. They then modeled expanding supernova remnants as 100 spherical shells, each representing gas that crossed the shock at a different time.

The shell calculations included dilution and particle momentum losses caused by expansion. They considered both a Type Ia remnant expanding into uniform gas and a Type II remnant expanding into a stellar wind.

Inherited turbulence is not enough

When the only turbulence came from the upstream Bell instability—a process driven by energetic particles ahead of the shock—the downstream acceleration was negligible in the modeled conditions. The available turbulent energy was limited, and cascading removed it before it could substantially reshape the proton distribution.

Fresh downstream turbulence changed the outcome. The authors did not simulate a particular generation mechanism. Instead, they assumed a source operating continuously throughout the downstream gas, converting a prescribed fraction of its available energy into turbulence.

Their estimates indicate that substantial reshaping becomes possible when this turbulence-injection efficiency is comparable to or greater than the efficiency assigned to energetic particles. Damping still matters: it can remove the small-scale waves needed to accelerate low-momentum particles, shifting the changes toward higher momenta. In some modeled shells, the long downstream residence time even allowed changes near the upper momentum cutoff inherited from shock acceleration.

A possible gamma-ray signature

The authors also calculated gamma rays produced when energetic protons collide with gas protons, creating neutral pions that decay into photons. In a benchmark 1,000-year-old Type II remnant, downstream acceleration increased the predicted gamma-ray output by up to roughly 50% between about 1 and 100 gigaelectronvolts, compared with the corresponding calculation without stochastic acceleration. It also changed the spectral shape.

That is a model prediction, not an observed enhancement or a universal percentage. Its size depends on the turbulence prescription, injection strength, particle population and remnant evolution.

Other limitations remain important. The shells exchange neither particles nor waves, and the model neglects particle escape. The assumed turbulence source may be less realistic than localized or intermittent stirring. The acceleration prescription also extends a theory developed for small magnetic fluctuations into a strongly turbulent regime whose validity remains under investigation.

The work is a preprint, not an established observational explanation. Its useful lesson is that downstream acceleration cannot be assessed from an acceleration timescale alone: the turbulence’s energy supply, depletion and evolution must be followed too.