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New method to determine proton trajectories in the equatorial plane of a dipole magnetic field
- Damaschin Ioanoviciu^{1}Email author
- Received: 18 July 2014
- Accepted: 3 March 2015
- Published: 14 March 2015
Abstract
A parametric description of proton trajectories in the equatorial plane of Earth’s dipole magnetic field has been derived. The exact expression of the angular coordinate contains an integral to be performed numerically. The radial coordinate results from the initial conditions by basic mathematical operations and by using trigonometric functions. With the approximate angular coordinate formula, applicable for a wide variety of cases of protons trapped in Earth’s radiation belts, no numerical integration is needed. The results of exact and approximate expressions were compared for a specific case and small differences were found.
Keywords
- Proton orbits
- Van Allen belts
- Trapped protons
- Dipole magnetic field
- Stability condition
Background
The differential equations of the motion of charged particles in a magnetic dipole field, as approximation of the Earth’s magnetic field, were derived already by Stőrmer (1907). Their solutions were obtained only numerically in an attempt to explain the behaviour of aurora borealis. The interest for charged particles trapped inside the dipole magnetic field increased by the discovery of Van Allen belts. A deep discussion of this topic can be found in Elliot (1963). Alfven (1950) developed an approximation splitting particle motion into guiding centre motion and gyration. The method has been enriched by Northrop (1961). Detailed studies of the motion of charged particles in the meridian plane of magnetic dipole fields are due to Markellos et al. (1978a) and (Markellos et al. 1978b).
The specific case of charged particle motion in the equatorial plane of the dipole magnet allowed De Vogelaere (1950) to obtain general periodic solutions of a Hill type differential equation from two numerically found particular solutions. Dragt (1964) in a comprehensive paper, describes a power series solution originating from a double power series solution of Stőrmer (1955). Graef and Kusaka (1938) obtained solutions for the motion in that plane in terms of elliptic integrals of first kind. These include a step of numerical calculation, no matter if the elliptic integral is obtained by a subroutine of the trajectory determining programme or if its value is picked up from tables.
Here parametric expressions of the proton coordinates in the equatorial plane were derived. Both exact radial and approximated angular coordinate expressions are strictly speaking, analytical formulas including only basic arithmetic operations and trigonometric functions. A prescribed accuracy for the angular coordinate is obtained from an exact formula by numerical integration, performed with the desired degree of precision.
Results and discussion
Radial coordinate-exact expression
“Point” means derivative with respect to the time.
with μ the Earth’s magnetic dipole moment.
Next we use Richardson’s (1947) parameter ψ, the angle between the velocity and the theta velocity component, by the substitution v_{θ} = v_{0}cosψ, earlier applied to describe ion paths in “wedge” magnetic fields and ion optical studies, see also Ioanoviciu (1989).
Only the solution with “plus” sign offers physically acceptable values.
From the above expressions we obtain the maxima and minima of the proton radial coordinate by derivation with respect to ψ. As ρ_{i} is a function ρ_{i}(cosψ) the derivative is
for cosψ = −1 ,
for cosψ = 1
Here the index “i” has been removed.
Now ρ = r/r_{C}, η = μq/(p_{r}r_{C} ^{2}) with r_{C} the radial distance for ψ_{i} = π/2, p_{r} the relativistic particle momentum \( {p}_r={m}_0{v}_0/\sqrt{1-{\beta}^2}, \)
We can use the simplified expressions by looking for an equivalent equation connecting r_{i} and η_{i} to r_{C} and η. By equating the maximum and the minimum of the equation (12), (13) to those of eq. (15), (16) we obtain the following equivalence:
Angular exact coordinate expression
Approximating proton coordinates
Let’s estimate the η values for the Van Allen belts. For the Earth, dipole moment μ_{E} = 7.906×10^{15} T.m^{3}, equatorial radius R_{E} = 6.378×10^{6}m , the protons of 10÷100 MeV, moving inside the inner belt concentrated around 1.5R_{E,} have η = 188.536 to 58.247 (Hess 1962; Fiandrini et al. 2004).
For the protons of 1 MeV located between 2.5 and 8R_{E} (second belt) the values of η=215.147 and 21.01 while for those of 0.065 MeV η=844.087 and 82.43.
Application to a specific case
First loop of a 60 MeV energy proton with the basic radial distance
ρ | θ in radians | ( θ-θ _{ app } )/θ % |
---|---|---|
1.000000 | 0 | |
0.99674778 | 2.046283×10^{−3} | 0.1213 |
0.991148 | 5.138577×10^{−3} | 0.0476 |
0.987785 | 1.048920×10^{−2} | 0.0209 |
0.987394 | 1.203098×10^{−2} | 0.0173 |
0.990058 | 6.310101×10^{−3} | 0.0381 |
0.995200 | 2.626267×10^{−2} | 0.0050 |
1.001655 | 2.707027×10^{−2} | 0.0048 |
1.007859 | 2.462010×10^{−2} | 0.0052 |
1.0122026 | 1.941538×10^{−2} | 0.0055 |
1.013494 | 1.272709×10^{−2} | 0.0045 |
1.011368 | 6.208492×10^{−3} | 0.0021 |
1.006424 | 2.552444×10^{−2} | 0.0050 |
1.000000 | 1.635483×10^{−3} | 0.1150 |
Conclusion
The coordinates of the protons moving in the Earth’s equatorial plane were derived as functions of a parameter. The use of the exact expressions assumes only basic operations and trigonometric functions to be involved for the radial coordinate calculation, while for the angular coordinate a numerical integration is necessary. Combining the exact radial formula with the approximated angular coordinate expression, an entirely analytic set of formulas has been obtained. The accuracy of this description has been shown on an illustrative example. The amazing simplicity of the coordinate expressions suggests possible developments by accounting for field perturbations.
Declarations
Authors’ Affiliations
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Copyright
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