How to calculate speed of cosmic ray particles from kinetic energy

In high energy and astroparticle physics energies for cosmic ray particles are given in GeV/n. But how much is that in term of speed? First of all we must remember that these energies are kinetic energies (k).
The total energy  E of a particle is the sum of its kinetic energy k and its mass m:

E = m + k

with c=1 and energy and mass measured with the same unit.
In special relativity:

E = g m

where g is the Lorentz factor that is equal to:

 \gamma =  \frac{1}{\sqrt{1-\beta^2}}

and \beta =  \frac{v}{c} is the ratio between the speed of a particle and the speed of light c.

Remember that the mass of a proton is m=0.938 GeV.

Lorentz factor is then given by:

image

or

image

and b is then

image

Assuming that mass is linear with number of nucleons in the nucleus, the same calculation applies to any ion using the kinetic energy per nucleon. We see that for energies bigger than 2 GeV/n particles travel almost at light speed (> 95 %).

k g b
100 KeV 1.000107 0.0146
1 MeV 1.001066 0.04614
10 MeV 1.010661 0.14486
100 MeV 1.106610 0.42825
1 GeV 2.066098 0.87507
2 GeV 3.132196 0.94767
5 GeV 6.330490 0.98744
10 GeV 11.66098 0.99632
100 GeV 107.6098 0.99996

image

 

 

This post was inspired by Protoni quasi veloci come la luce and Protoni quasi veloci come la luce: soluzione

It was very useful for tomorrow challenge.

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