# Maximum Energy

#### Hillas Plot

The equation of the maximum energy from shock acceleration can be rewritten as:![$$E\_{max} \leq 10^{18} {\rm eV}\\; Z \\;\beta\_s \left(\frac{R}{\rm{kpc}}\right)\left(\frac{B}{\mu\rm{G}}\right)$$](https://render.githubusercontent.com/render/math?math=E_%7Bmax%7D%20%5Cleq%2010%5E%7B18%7D%20%7B%5Crm%20eV%7D%5C%3B%20Z%20%5C%3B%5Cbeta_s%20%5Cleft%28%5Cfrac%7BR%7D%7B%5Crm%7Bkpc%7D%7D%5Cright%29%5Cleft%28%5Cfrac%7BB%7D%7B%5Cmu%5Crm%7BG%7D%7D%5Cright%29\&mode=display)

where ![$\beta\_s$](https://render.githubusercontent.com/render/math?math=%5Cbeta_s\&mode=inline) is the shock velocity, ![$B$](https://render.githubusercontent.com/render/math?math=B\&mode=inline) the magnetic field strength, and  ![$R = u\_1 t\_{age}$](https://render.githubusercontent.com/render/math?math=R%20%3D%20u_1%20t_%7Bage%7D\&mode=inline) is the radius of the expanding shockwave, or in other words the size of the acceleration reguin. In 1984 Hillas arrived to a similar conclusion just by doing a back-of-an-envelope assumption that in order for it to accelerate CR particles to high energies in which he impossed that a condition where the size of the acceleration region must be at least twice the Larmor radius. The plot showing possible sources in the parameter space ![$B$](https://render.githubusercontent.com/render/math?math=B\&mode=inline) vs ![$R$](https://render.githubusercontent.com/render/math?math=R\&mode=inline) is usually referred as the Hillas' plot. For relativistic shockwaves (![$\beta\_s \sim 1$](https://render.githubusercontent.com/render/math?math=%5Cbeta_s%20%5Csim%201\&mode=inline)) many sources are able to accelerate protons up to ![$10^{20}$](https://render.githubusercontent.com/render/math?math=10%5E%7B20%7D\&mode=inline) eV, however for slower shockwaves (![$\beta\_s \sim 1/300$](https://render.githubusercontent.com/render/math?math=%5Cbeta_s%20%5Csim%201%2F300\&mode=inline)) the number of source candidates is strongly reduced.![](https://github.com/zemrude/PHYS-F-467/raw/12f2d10ab5e7b0128f70a9bcf8407ece300332c6/images/Hillas_plot.png)