# 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 (RE: radius of the Earth; energy and speed are quoted for a proton of the given rigidity):

 Magnetic Kinetic Speed Cyclotron radius rigidity energy [% of c] Corona (10 mT) Interplanetary 1 AU (5 nT) Earth (surface; 30 μT) 1 GV 0.43 GeV 73% 330 m 6.6x108 m ≈ 100 RE 1.1x105 m ≈ 0.017 RE 5 GV 4.1 GeV 98% 1.65 km 3.3x109 m ≈ 520 RE 5.5x105 m ≈ 0.086 RE 20 GV 19.1 GeV 99.8% 6.60 km 1.3x1010 m ≈ 2100 RE 2.2x106 m ≈ 0.34 RE