Effective potential energy for relativistic particles in the field of inclined rotating magnetized sphere

Abstract

The dynamics of a charged relativistic particle in electromagnetic field of a rotating magnetized celestial body with the magnetic axis inclined to the axis of rotation is studied. The covariant Lagrangian function in the rotating reference frame is found. Effective potential energy is defined on the base of the first integral of motion. The structure of the equipotential surfaces for a relativistic charged particle is studied and depicted for different values of the dipole moment. It is shown that there are trapping regions for the particles of definite energies.

Appendix: Stationary points of the relativistic potential energy

The power of potential formulation of the problem is the possibility to find the “potential valleys” where the charged particles can be trapped. And the slope of the “valley” shows the force exerted on the particle. Having this in mind, we find the stationary points of the potential energy, i.e. the points satisfying the set of equations:

$$\frac{\partial V}{\partial q_i}=0, $$

where q _i =ρ,θ,ψ. This gives a system of three equations

$$\begin{aligned} &\frac{\rho^3\sin\theta}{\sqrt{1-\rho^2\sin^2\theta}}+\frac {N_\perp\cos\theta}{\sqrt{1+\rho^2}}\bigl(\cos\eta+ \rho^3\sin\eta\bigr) \\ &\quad {}-N_\parallel\sin\theta=0, \end{aligned}$$

(A.1)

$$\begin{aligned} &\frac{\rho^3\sin2\theta}{\sqrt{1-\rho^2\sin^2\theta }}-2N_\perp\cos2\theta\sqrt{1+ \rho^2}\cos\eta \\ &\quad {}+2N_\parallel\sin2\theta=0, \end{aligned}$$

(A.2)

$$\begin{aligned} &\sin2\theta\sin\eta=0. \end{aligned}$$

(A.3)

Equation (A.3) has two solutions:

$$\begin{aligned} & (\mathrm{i})\quad \theta=\frac{\pi n}{2}, \quad n\in Z \end{aligned}$$

(A.4)

$$\begin{aligned} & (\mathrm{ii})\quad \eta=0, \pi. \end{aligned}$$

(A.5)

Solution (i)

The stationary points on the axis θ=0,π can exist provided that ∂V/∂ρ=0 and ∂V/∂θ=0 for any ψ, which is not the case as one can see in Eqs. (A.1)–(A.2). As to the equatorial plane $\theta=\frac{\pi}{2}$, Eqs. (A.1)–(A.2) have the next solutions:

$$\begin{aligned} &\rho^2=\frac{N_\parallel^ {2/3}}{2^{1/3}} \Biggl[\sqrt[3]{1+\sqrt {1+\frac{4N_\parallel^2}{27}}}+\sqrt[3]{1-\sqrt{1+\frac{4N_\parallel^2}{27}}} \Biggr]; \\ & \quad\eta=0,\ \pi. \end{aligned}$$

(A.6)

Coordinate ρ in Eq. (A.6) increases monotone as N_∥ increases, and asymptotically approaches the value of unity as N_∥→∞. For small N_∥ it takes the value $\rho \approx N_{\parallel}^{1/3}$.

Solution (ii)

Substituting η=0,π into Eqs. (A.1) and (A.2) we obtain two equations for ρ and θ of the stationary points:

$$\begin{aligned} &\frac{\rho^3\sin\theta}{\sqrt{1-\rho^2\sin^2\theta}}+\frac {\varepsilon N_\perp\cos\theta}{\sqrt{1+\rho^2}} -N_\parallel\sin \theta=0, \end{aligned}$$

(A.7)

$$\begin{aligned} &\frac{\rho^3\sin2\theta}{\sqrt{1-\rho^2\sin^2\theta }}-2\varepsilon N_\perp\cos2\theta \sqrt{1+\rho^2} +2N_\parallel\sin2\theta=0, \end{aligned}$$

(A.8)

where ε=1 for η=0 and ε=−1 for η=π. Solution of these equations gives the lines at which the stationary points are lying

$$\begin{aligned} \operatorname{tg}\theta=-\varepsilon\frac{3\cot\alpha+q\sqrt{9\cot ^2\alpha+8+4\rho^2}}{2\sqrt{1+\rho^2}}, \end{aligned}$$

(A.9)

and equation for coordinate ρ at these lines:

$$\begin{aligned} & \rho^6Q^2\bigl[4+4\rho^2+Q^2 \bigr]^2 \\ &\quad {}-N^2\sin^2\alpha\bigl[4+4 \rho^2+\bigl(1-\rho^2\bigr)Q^2\bigr] [2+Q\cot \alpha]^2=0, \end{aligned}$$

(A.10)

where $Q=3\cot\alpha+q\sqrt{9\cot^{2}\alpha+8+4\rho^{2}}$ and q=±1 is the sign of the particle charge. If N≪1 and ρ≪1 these equations transform to Eqs. (36) and (37) of Paper I. The lines given by Eq. (A.9) are plotted in Fig. 7 for α=π/3.