Shock when a supernova explodes

Question

I have been reading that when supernovae explode, they produce (Shockwaves) which act as cosmic accelerators.

I do not understand, what is meant with "shock" (partially because I do not study in English), or how it is produced when the supernova explodes, could someone elaborate?

I would also be interested in the mechanism of how these shock "accelerate" particles? As well as if good resources are linked in which this subject is explained well.

Answer 1

A shock is when a fluid flow moves much faster than the speed of sound of the medium. With supernovae, the material that was once the star (e.g., hydrogen, helium, etc) is pushed outwards with such great force that it travels at a few percent of the speed of light! As the speed of sound of the interstellar medium (ISM) is about 10 km/s, a velocity of ~30,000 km/s is definitely going generate a shock.

As the supernova ejecta expands, it begins sweeping up material from the ISM, which does slow it down, though over the course of many thousands of years. This is the material that is accelerated via what is called diffusive shock acceleration (arXiv link) or, equivalently, second-order Fermi mechanism and become the 'cosmic rays' we observe regularly.

The general idea of the mechanism is that at the shock front, a particle (proton, alpha, etc) from the ambient ISM scatters off the tangled magnetic field at the shock interface and can repeatedly cross the shock front. From the jump conditions between the two states, a particle can gain a very small energy, $E_{i+1}=(1+\alpha)E_{i}$ where $\alpha\propto u_1-u_2$ where the $u_i$ are the velocity of the 'upstream' (1) and 'downstream' (2) regions.
Now the interesting thing is that this gain of energy occurs when the particle crosses from either direction!

Thus, after $k$ crossings, the particle has gained an energy of $$E_k=(1+\alpha)^kE_0.$$ After each crossing, the particle has a return probability of, $P_\text{return}=1-\rho_1u_2/\rho_2$. Then after $k$ crossings, the number of particles remaining would be $$N_k=N_0P_\text{return}^k\equiv N_0\left(1-P_\text{escape}\right)^k.$$ Since the $k$ in both the previous two equations are the same, we can solve for it to find that, $$\frac{\log(N_k/N_0)}{1-P_\text{escape}}=\frac{\log(E_k/E_0)}{\log(1+\alpha)}\implies N(E)\sim N_0\left(\frac{E}{E_0}\right)^s.$$ This is consistent with the power-law distribution we observe in cosmic rays (see Matthiae 2010, for instance).

When modeling this, one utilizes the aforementioned diffusive shock acceleration framework in which a particle distribution, $f(\mathbf{x},\,\mathsf{p},\,t)$ with $\mathsf{p}$ the particle momenta, evolves as, $$\frac{\partial f}{\partial t}+\mathbf{u}\cdot\nabla f=\nabla\cdot\left(D\nabla f\right)+\frac{1}{3}\left(\nabla\cdot\mathbf{u}\right)\mathsf{p}\frac{\partial f}{\partial\mathsf{p}}+\mathcal{S}$$ where $\mathbf{u}$ is the fluid velocity and $\mathcal{S}$ the source gain/loss term. We can integrate this distribution to obtain a pressure term, $$p_\text{cr}(\mathbf{x}\,t)\sim\int\,\frac{\mathsf{p}^4}{\sqrt{1+\mathsf{p}^2}}f(\mathbf{x},\,\mathsf{p},\,t)\,\mathrm{d}\mathsf{p}.$$ This pressure must be accounted for in the dynamic evolution of the supernova remnant: \begin{align} \partial_t\rho+\nabla\cdot(\rho\mathbf{u}) &= 0 \\ \partial_t(\rho\mathbf{u})+\nabla\cdot\left(\rho\mathbf{u}\mathbf{u}+p_\text{total}\mathbf{I}\right) &= -\nabla p_\text{cr} \\ \partial_tE+\nabla\cdot\left(\mathbf{u}\left(E+p_\text{total}\right)\right)&=-\mathbf{u}\cdot\nabla p_\text{cr} \end{align} (though one probably should include the magnetic fields in the above hydrodynamic equations).

For a good treatment, Kirk's Plasma Astrophysics covers this well. At research-level, pretty much anything by Tom Jones, Don Ellison, Pasquale Blasi or Damiano Caprioli on the subject is going to be what you're looking for.