Mathematical description: charged particle orbit in a magnetic field; magnetic rigidity

The Lorentz force exerted by the magnetic field \( \vec{B} \) on a particle with charge \( q \) , mass \(m\) and velocity \( \vec{\upsilon} \) is $$ \frac{d\vec{\upsilon}}{dt} = \frac{q}{\gamma m} \vec{\upsilon} \times \vec{B} ; . $$

\( \gamma \) is the Lorentz factor, i.e. the ratio of the energy to the rest energy \( m c^2 \). Since the acceleration is perpendicular to both the magnetic field vector and the velocity vector, the orbit is a circle (+ a uniform motion along the magnetic field). Integration of the equation of motion yields, for a uniform magnetic field that is constant in time, $$ \vec{\upsilon} = \vec{r} \times \frac{q \vec{B}}{\gamma m} = \vec{r} \times \vec{\Omega}_c ; . $$

The angular frequency of the circular motion is \( \Omega_c = \frac{|q| |\vec{B}|}{\gamma m} \) . The radius of the circular orbit (cyclotron radius or Larmor radius) is hence $$ r_c = \frac{\upsilon}{\Omega_c} = \frac{\gamma m \upsilon}{|q| |\vec{B}|} = \frac{\beta \gamma m c}{|q| |\vec{B}|} =\sqrt{\gamma^2-1} \frac{m c}{|q| |\vec{B}|} ; . $$

Here \( \beta \) is the ratio of the particle speed to the speed of light, and \( \gamma = \frac{1}{\sqrt{1-\beta^2}} \) . If the magnetic field is given in nanoTesla (nT), the cyclotron radius per unit charge is $$ r_c = 3.1 \times 10^9 \left(\frac{B}{1 ; \mathrm{nT} }\right)^{-1} \sqrt{\gamma^2-1} ; \mathrm{m}. $$

The cyclotron radius decreases with increasing charge

• a natural consequence of the fact that the Lorentz force is proportional to charge. We can express it in a charge-independent way by introducing the magnetic rigidity: since the momentum of the particle is \( p=\gamma m \upsilon \) , $$ r_c = \frac{\gamma m \upsilon}{|q| |\vec{B}|} = \frac{p}{|q| |\vec{B}|} = \frac{R}{c|\vec{B}|} , $$

where \( R:=\frac{pc}{|q|} \) is called the magnetic rigidity. This quantity measures the cyclotron radius in a given magnetic field, and is therefore an indicator of the sensitivity of the particle, whatever its charge or mass, to the magnetic field. The trajectory of a charged particle is the more strongly curved by the magnetic field, the lower its magnetic rigidity. If the magnetic rigidity is given in giga-Volt (GV), as is typical for cosmic rays detected by neutron monitors, the cyclotron radius is $$ r_c = 3.3 \times 10^9 \left(\frac{B}{1 ; \rm nT}\right)^{-1} \left(\frac{R}{1; \mathrm{GV} }\right) ; \mathrm{m}. $$

Some examples of cyclotron radii at the Sun, near Earth and at the surface of the Earth (R_E: radius of the Earth; energy and speed are quoted for a proton of the given rigidity):