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The dynamics of these particles is the The perpendicular component of the electric field will main focus of this paper. move particles with an overall drift velocity, known as This paper is organized as follows: Section II intro- the E-cross-B-drift, which is perpendicular to both field duces the relativistic equation of motion for a particle vectors: in an electric and magnetic field and describes the cy- E⊥ × B clotron, bounce and drift motions. It also shows some vE = . (4) typical particle trajectories under the dipolar magnetic B2 field, approximating the Earth’s field. Section III intro- The particle will move with with the velocity v , plus the duces the concept of adiabatic invariants and derives the E cyclotron motion described above. The drift velocity vE first adiabatic invariant associated with the particle mo- is independent of particle mass and charge. Therefore, tion. Section IV gives the approximate equations of mo- in an inertial frame moving with vE, the E-cross-B-drift tion for the guiding center of a particle, obtained by aver- will vanish for all types of particles. aging out the cyclotron motion. Section V presents and For the remainder of this paper we take E = 0, that derives the second and third invariants associated with is, the acceleration due to the electric field is not taken the bounce and drift motions, respectively. Section VI into consideration. This is not because electric fields are lists some exercises building on the concepts described unimportant; on the contrary, they play an important in the paper. Two of these exercises describe non-dipole role in the complex dynamics of plasmas. The first reason fields that are used for modeling different regions of the for leaving out electric field effects is described above: magnetosphere. If the field is uniform and constant, we can transform to another frame that cancels it. Even if the field is II. PARTICLE TRAJECTORY IN DIPOLAR nonuniform and time-dependent, electric drifts can be MAGNETIC FIELD vectorially added to magnetic drifts in order to obtain the overall drift. Drift velocities due to different fields are independent. The motion of a particle with charge q and mass m in The second reason is the need for simplicity; a static E B an electric field and magnetic field is described by magnetic field provides sufficient real-life context for the the Newton-Lorentz equation: discussion of guiding-center and adiabatic invariant con- d(γmv) cepts in general-purpose lectures. The final reason is that = qE(r)+ qv × B(r). (1) dt this paper focuses on the region occupied by radiation belts, and in this region the magnetic term of (1) is the Here γ = (1 − v2/c2)−(1/2) is the relativistic factor and v dominant force.15 is the particle speed. Now we consider motion under the influence of a mag- Suppose that E = 0. Then, because of the cross prod- netic dipole. The field Bdip(r) of a magnetic dipole with uct, the acceleration of the particle is perpendicular to moment vector M at location r is given by:14 the velocity at all times, so the speed of the particle (and µ0 the factor γ) remains constant. Further suppose that the Bdip(r)= [3(M · ˆr)ˆr − M] , (5) magnetic field is uniform. Then, particles move on he- 4πr3 lices parallel to the field vector. The circular part of this where r = xxˆ + yyˆ + zˆz, r = |r| and ˆr = r/r. For motion is called the “cyclotron motion” or the “gyromo- Earth, we take M = −Mzˆ, antiparallel to the z-axis, tion”. The “cyclotron frequency” Ω and the “cyclotron because the magnetic north pole is near the geographic radius” ρ are respectively given by south pole.16 qB At the magnetic equator (x =1Re, y = z = 0) the field Ω= (2) −5 γm strength is measured to be B0 = 3.07 × 10 T. Substi- γmv tution shows that µ M/4π = B R3. Then in Cartesian ρ = ⊥ , (3) 0 0 e qB coordinates, the field is given by: where B = |B| is the uniform field strength and v⊥ is B R3 B = − 0 e 3xzxˆ +3yzyˆ + (2z2 − x2 − y2)ˆz . (6) the component of the velocity perpendicular to the field dip r5 vector. If there are not any other forces, the parallel com- ponent of the velocity remains constant. The combined Figure 2 shows trajectories of two protons with 10MeV motion traces a helix.14 kinetic energy, a typical energy for radiation belts.13 If the electric field is not zero, we can write it as The trajectories are calculated with the SciPy module 17 E = E⊥ + E||, where E|| is parallel to B, and E⊥ is using the Python language. One proton is started at perpendicular to it. If E|| 6= 0, particles accelerate with (2Re, 0, 0) and the other at (4Re, 0, 0). Both start with qE||/m along the field line and they are rapidly removed an equatorial pitch angle (angle between the velocity ◦ from the region. Therefore, the existence of a trapped and field vectors) αeq = 30 so that vy0 = v sin αeq, plasma population implies that the parallel electric field vz0 = v cos αeq and vx0 = 0. Both are followed for 120 must be negligible. seconds. 3
5 the drift motion period τd are approximately given as: 4
3 R0 c 3/4 τb ≈ 0.117 1 − 0.4635(sin αeq) (7) Re v 2 3 h i 2πqB0Re 1 1 0.62 1 τd ≈ 2 1 − (sin αeq) , (8)
] mv R0 3 e 0
y [R where R0 is the equatorial distance to the guiding line −1 and αeq is the equatorial pitch angle. Both approxima- −2 tions have an error of about 0.5%.
−3
−4 III. THE FIRST ADIABATIC INVARIANT −5 0 5 x [R ] e If the parameters of an oscillating system are varied very slowly compared to the frequency of oscillations, the system possesses an “adiabatic invariant”, a quan- tity that remains approximately constant. In the Hamil- tonian formalism, the adiabatic invariant is the same as the action variable:18
J = p(E) dq, (9) −5 H=E 1 I ] e 0 where q, p are canonical variables and the integral is eval-
z [R 0 uated over one cycle of motion satisfying H(p,q,t)= E. −1 The integral should be evaluated at “frozen time”, that x [R ] is, the slowly varying parameter is considered constant −4 e −2 0 5 during the integration cycle. 2 4 y [R ] There are three separate periodic motions of a charged e particle in a dipole-like magnetic field. This means there FIG. 2. (Color online) Trajectories of two 10MeV protons in are three adiabatic invariants for the particle’s motion. the Earth’s dipole field. The dipole moment is in the −ˆz di- The canonical momentum for a charged particle in a rection. Both panels show the same trajectories from different magnetic field with vector potential A is γmv + qA. To viewing angles. obtain the first adiabatic invariant J1, we integrate the canonical momentum over one cycle of the cyclotron or- bit:
J1 = (γmv + qA) · dℓ. (10) The motion is again basically helical, but the nonuni- I formity of the field introduces two additional modes of motion on large spatial and temporal scales. These are Here dℓ is the line element of the particle trajectory. called “the bounce motion” and “the drift motion”. Even though the path does not exactly close, we eval- uate the integral as if it does. Stokes’ Theorem states The bounce motion proceeds along the field line that that: goes through the helix (the “guiding line”). The motion slows down as it moves toward locations with a stronger A · dℓ = ∇× A · dσ, (11) magnetic field, reflecting back at “mirror points”. The I Z bounce motion is much slower than the cyclotron motion. where the right-hand side integral is taken over the sur- face bounded by the closed path of the left-hand side The drift motion takes the particle across field lines integral. Using Stokes’ theorem with B = ∇× A, the (perpendicular to the bounce motion). In general, drift integral J1 takes the form: motion is faster at larger distances, as observed in Fig- 2 ure 2. Particles in dipole-like fields are trapped on closed J1 =2πγmρv⊥ − qπρ B (12) “drift shells” as long as they are not disturbed by colli- πγ2m2v2 = ⊥ . (13) sions or interactions with EM waves. The drift motion is qB much slower than the bounce motion. The second equation follows from substituting ρ from (3). Under a dipolar field, the bounce motion period τb and It is assumed that the cycle is sufficiently small so that B 4 is considered uniform over the area bounded by the gyro- field does not change significantly within a cyclotron ra- motion. This assumption is essential for the existence of dius. This condition can be expressed as adiabatic invariants. The area element dσ is antiparallel B to B because of the sense of gyration of the particles; ρ ≪ (15) hence the negative sign of the second term. |∇B| Customarily, one takes The magnetic moment is an adiabatic invariant under γ2mv2 this condition. µ = ⊥ (14) Northrop19 and Walt5 give detailed derivations of the 2B equations of guiding-center motion. In order to derive the as the first invariant, which differs from J1 only in some acceleration of the guiding center, the particle position r constant factors. The parameter µ is called the “mag- is substituted with: netic moment” because it is equal to the magnetic mo- ment of the current generated by the particle moving on r = R + ρ (16) this circular path. where R is the position of the guiding center. The vector The adiabatic invariance of µ explains the existence of ρ lies on the plane perpendicular to the field, oscillates mirror points. As the particle moves along the field line with the cyclotron frequency, and its length is equal to toward points with larger B, the perpendicular speed v ⊥ the cyclotron radius. must increase in order to keep µ constant. However, v ⊥ Assuming that the cyclotron radius is much smaller can be at most v, the constant total speed. The particle than the length scale of the field, we can expand B(r) stops at the point where B = B = γ2mv2/2µ and falls m around R to first order in a Taylor series: back (see Section IV). The left panel of Figure 3 shows the magnetic moment B(r) ≈ B(R) + (ρ · ∇)B (17) µ for the two protons shown in Figure 2. The values are oscillating with the local cyclotron frequency because This expansion is substituted into the Newton-Lorenz instantaneous values of v⊥ and B are used. The actual equation (1) and the equation is averaged over a cycle, adiabatic invariant is the average of these oscillations and eliminating rapidly oscillating terms containing ρ and its it is constant in time. derivatives. The resulting acceleration of the guiding cen- The top right panel shows the oscillations of µ vs. time ter is given by: for a shorter interval. Comparison with the z vs. time plot below, it can be seen that these oscillations are cor- q qρ2Ω R¨ = R˙ × B(R) − ∇B(R) (18) related with the bounce motion. Oscillations of µ have a γm 2γm small amplitude near the mirror point because there the parallel motion slows down and the overall motion be- Taking the dot product of both sides of the equation comes more adiabatic. Near the equatorial plane (z = 0) with bˆ, the local magnetic field direction, will yield the parallel motion is fastest, the motion is less adiabatic, equation of motion along the field line. The first term and µ oscillates with a larger amplitude. becomes identically zero because it is perpendicular to The proper way of calculating the first invariant would the field vector. Then: remove all oscillations: After following the full trajectory, 2 dv|| qρ Ω find the times {ti} where vx = 0 (or vy = 0) by search- = − bˆ · ∇B(R) (19) ing along discrete path points and by interpolating. The dt 2γm difference between any successive time points is half a gyroperiod. Then take the average of µ over that time where v|| is the speed along the field line. Replacing 2 2 interval using the path points. Repeating this procedure qρ Ω/(2γm) = µ/(γ m) and defining s as the distance for all time intervals, we get a constant set of µ values along the field line, this equation can be written as: (apart from numerical errors). dv d2s ∂ µB(s) || = = − (20) dt dt2 ∂s γ2m
IV. THE GUIDING-CENTER EQUATIONS Here B(s) is the field strength along the field line. The factors µ and γ can be taken inside the derivative be- The “guiding center” is the geometric center of the cause they are constants. This expression shows that the cyclotron motion. If the magnetic field is uniform the quantity µB(s)/(γ2m) acts like a potential energy in the guiding center moves with constant velocity parallel to parallel direction. The negative sign indicates that the the field line. In nonuniform field geometries, there is a parallel motion is accelerated toward regions with smaller sideways drift in addition to the motion along the field field strength. lines, as seen in the dipole example in Figure 2. The motion of the guiding center perpendicular to field Calculation of the guiding center motion requires that lines can be determined by taking the cross product of the motion is helical in the smallest scale, and that the Eq. (18) with the field direction vector. The resulting 5
FIG. 3. (Color online) Left: The instantaneous values of the magnetic moment for the particle orbits shown in Fig. 2. Lower curve is for the proton starting at 2 Re distance, upper curve for the proton starting at 4 Re. The white horizontal line shows the average value. Top right: A close-up view of the instantaneous magnetic moment up to time 6s. Bottom right: The z-coordinate up to time 6s. equation is then iterated to obtain an approximate solu- need to resolve the cyclotron motion. This reduces the tion for the drift velocity across field lines. cumulative error, as well as the total computation time. Figure 4 shows the solution of the guiding-center equa- dR γm ⊥ = (v2 + v2 ) bˆ × ∇B (21) tions for the same protons shown in Figure 2 under a dt 2qB2 || dipolar magnetic field (Python source code provided in Supplement).17 This drift velocity is actually the sum of two separate It should be noted that the guiding-center equations drift velocities: The gradient drift that arises from the are approximate because only terms first order in cy- nonuniformity of the magnetic field, and the curvature clotron radius are used in their derivation. For parti- drift that occurs because the field lines are curved. For cles with larger cyclotron radii (higher kinetic energies), an example of pure gradient drift motion, see Exercise 2 there may be a noticeable difference between guiding- in Section VI. center and full-particle trajectories (see Exercise 5 in Sec- Equation (21) shows that electrons and ions drift in tion VI). opposite directions. This creates a net current around the Earth, called “the ring current”. Gradient and curvature drifts are the only drifts seen V. THE SECOND AND THIRD ADIABATIC in static magnetic fields. External electric fields, external INVARIANTS forces such as gravity, and time dependent fields create additional drift velocities.3 Combining these, we obtain the following equations of The second adiabatic invariant is associated with the motion for the guiding center: bounce motion, and it is calculated by integrating the canonical momentum over a path along the guiding field dR γmv2 v2 line: || bˆ ∇ bˆ = 2 1+ 2 × B + v|| (22) dt 2qB v ! J = (γmv + qA) · ds, (24) dv µ 2 || = − bˆ · ∇B (23) I dt γ2m where ds is the line element along the field line. The These equations are more complicated than the simple adiabatic integrals are evaluated in a “frozen” system: It Newton-Lorentz equation, and they require computing is assumed that the drift is stopped, so the motion moves bˆ · ∇B and bˆ × ∇B at each integration step. Still, they back and forth along a single guiding field line. have the advantage that we can follow the overall mo- Using Stokes’ theorem, the second term can be con- tion with relatively large time steps because we do not verted to an integral over a surface bounded by the 6
3
2
1 ] e 0 y [R −1
−2
−3
−5 0 5 x [R ] e FIG. 5. The second invariant values in time, calculated using the guiding-center trajectories in Fig. 4. Lower curve is for the proton starting at 2 Re distance, upper curve for the proton starting at 4 Re.
where Bm is the field strength at the mirror point. Sub- 2 2 2 stituting v⊥ = v − v|| and solving for v|| gives 1
] −5 e 0 sm2 B(s)
z [R J2 =2 γmv 1 − ds ≡ 2γmvI. (28) −1 B 0 Zsm1 s m
The integral J2 is an adiabatic invariant in general (even −2 x [R ] 0 2 5 e if there are electric fields or slow time-dependent fields). y [R ] If the speed is constant, I can be used as an adiabatic e invariant. The integral I depends only on the magnetic field, not on the particle velocity, so it can be used to FIG. 4. (Color online) Guiding-center trajectories for the par- compute the drift path using the field geometry only (see ticles shown in Figure 2. The cyclotron motion is averaged exercises 8-10 in Section VI). out. Figure 5 shows that the value of the second invariant I, evaluated using the guiding-center trajectories shown in Figure 4, stays constant in time. The integral is evaluated bounce path not using the definition of I in Eq. (28), but using the dynamical form qA · ds = q ∇× A · dσ = q B · dσ = 0 (25) I Z Z 1 1 2 I = v||ds = v dt, (29) which is zero because the bounce motion goes along the v v || Z Z same path in both parts of the cycle so that the enclosed area vanishes. Then, the second adiabatic invariant can where the integral is evaluated over a half period. The be written as limits of the integrals are determined by interpolation be- tween two points where the parallel speed changes sign. sm2 The values do not oscillate because the adiabatic invari- J2 =2 γmv||ds, (26) sm1 ant is calculated as an average over a cycle. Z The drift path is found by averaging the bounce mo- where s is the path length along the field line, and sm1, tion. In a dipolar field all drift paths are circular due sm2 are locations of the mirror points where the particle to the symmetry of the field. The third invariant, associ- comes back. ated with the drift motion, is defined as an integral along At the mirror point the parallel speed v|| vanishes so the drift path: that v⊥ = v. From the invariance of the magnetic mo- ment µ it follows that J3 = (γmv + qA) · dℓ, (30) γ2mv2 γ2mv2 I µ = ⊥ = , (27) 2B(s) 2Bm where dℓ is a line element on the drift path. 7
This can be written as 2. Gradient drift. Consider a magnetic field given as B = (Ax + B0)ˆz. The field has a gradient in the x- −1 J3 = γmvddℓ + q B · dσ. (31) direction, but no curvature. Set A = 1T · m and I Z B0 = 1T. Follow the trajectory of a particle with In the first term vd, the drift speed, is the magnitude mass m = 1kg and q = 1C initialized with velocity of the expression given in Eq. (21). The second term is v = 1m · s−1yˆ at the origin. Note that the sideways obtained by using Stokes’ theorem as above. drift arises from the fact that the cyclotron radius An order of-magnitude comparison shows that the first is smaller at stronger fields. term of J3 can be neglected because it is much smaller than the second term: From Eq. (21) the order of mag- 3. Equatorial particles. Consider a particle in a nitude of the drift speed can be written as dipolar magnetic field, located at the equatorial mv2 B plane (z = 0) with zero parallel speed. As the field vd ∼ , (32) strength is minimum at the equator with respect 2qB2 R to the field line, there is no parallel acceleration where B is the typical field strength at the drift path and and the particle stays on the equatorial plane at all R is the typical distance from the origin. Similarly, from times. Using the dipole model, follow an equato- Eq. (3), the cyclotron radius has the order of magnitude rial particle and verify that the center of the motion mv stays on a contour of constant B, as implied by the ρ ∼ . (33) qB conservation of the first adiabatic invariant. Then, the order-of-magnitude ratio of the terms in 4. Explore the drift motion. Run the programs in Eq. (31) is the supplement to trace protons and electrons using mv 2πR m2v2 ρ 2 the dipole model. Initialize particles with different d ∼ ∼ . (34) qBπR2 q2B2R2 R energies, starting positions and pitch angles. Verify that electrons and protons drift in opposite direc- According to the adiabaticity condition Eq. (15), ρ/R tions, and electrons have a much smaller cyclotron must be very small. Therefore the first term of Eq. (31) radius than protons with the same kinetic energy. is ignored and we have Estimate the periods of bounce and drift motions and compare them with Eqs. (7, 8). J3 = qΦ, (35) where Φ is the magnetic flux through the drift path. The 5. Accuracy of the guiding-center approxima- third adiabatic invariant is useful as a conservation law tion. Simulate the full particle and guiding cen- when the magnetosphere changes slowly, i.e., over longer ter trajectories with the same initial conditions and time scales compared to the drift period. plot them together. Shift the initial position of the The use of three invariants gives more accurate results particle properly so that the guiding center runs for the motion of particles over long periods. Numerical through the middle of the helix. solution of equations of motion are less accurate because of accumulated numerical errors. Roederer6 discusses Repeat with protons with 1keV, 10keV, 100keV in detail how drift shells can be constructed geometri- and 1MeV kinetic energies. At higher energies, the cally using the invariants (see Exercise 10 in Sec. VI). guiding-center trajectory lags behind the full par- Furthermore, as three invariants uniquely specify a drift ticle because the omitted high-order terms become shell, the invariants themselves can be used as dynami- more significant as the cyclotron radius increases. cal variables when investigating the diffusion of trapped Different numerical methods. particles.20 6. Solve the full particle and guiding center equations using differ- ent numerical schemes,21,22 such as Verlet, Euler- VI. EXERCISES AND ASSIGNMENTS Cromer, Runge-Kutta and Bulirsch-Stoer. Verify the accuracy of the solution by checking the con- servation of kinetic energy and adiabatic invariants. This section lists some further programming exer- cises with varying difficulty. The code given in the 7. Field line tracing. Plot the magnetic dipole field supplement17 can be modified to solve some of the ex- line starting at position (x ,y ,z ). For any vector ercises. 0 0 0 field u(r), a field line can be traced by solving the 1. Uniform magnetic field. Follow charged parti- differential equation cles under a uniform magnetic field B = Bzˆ where B = 1T. Verify that the particles follow helices dr u = (36) with cyclotron radius and frequency as given in ds |u| Eqs. (2, 3). Experiment with particles with dif- ferent mass and charge values. where s is the arclength along the field line. 8
8. Compute I. Compute the second invariant I (Eq. 28) under a dipolar field for a guiding cen- ter starting at position (x0,y0, 0) and an equato- rial pitch angle αeq. The integral should be taken 4 along a field line, which can be traced as described above. From the first adiabatic invariant one finds 2 Bm = B(x0,y0, 0)/ sin (αeq) and the limits of the 2
integral are found by solving B(sm)= Bm. ] e 0 9. Second invariant along the drift path. Pro- z [R duce a guiding-center trajectory under the dipole field. By interpolation, determine the points −2 (xi,yi, 0) where the trajectory crosses the z = 0 plane. Compute the second invariant I at these −4 equatorial points and plot. Verify that the values −8 −6 are constant as shown in Fig. 5. −4 −2 6 0 2 8 10. Drift path tracing using the second invari- 4 6 x [R ] y [R ] e ant. Pick a starting location (x0,y0, 0) and mir- e ror field Bm, and evaluate the second invariant I(x0,y0,Bm) as described above. Compute the gra- FIG. 6. A proton with 200keV kinetic energy, initialized ◦ dient ∇I = ∂xIxˆ + ∂yIyˆ numerically using central on the right edge at (7Re, 7Re, 0) with 60 pitch angle in a differences: double-dipole field. 1 ∂ I ≈ [I(x + δ, y ,B ) − I(x − δ, y ,B )] (37) x 2δ 0 0 m 0 0 m than 1 so that the magnetopause becomes curved. 1 Also, the two dipoles can be tilted by equal and ∂ I ≈ [I(x ,y + δ, B ) − I(x ,y − δ, B )] ,(38) y 2δ 0 0 m 0 0 m opposite angles with respect to the Sun-Earth line (x-axis), to simulate the fact that the dipole mo- where δ is a small number (e.g. 0.01Re). ment of the Earth is tilted. The second invariant is constant along the drift (a) Starting at various latitudes, plot the mag- path, so for a finite step (∆x, ∆y), it holds that netic field lines of Eq. (39) on the x − z plane. ∂xI∆x+∂yI∆y = 0. Use this relation to trace suc- Observe the compression of field lines on the cessive steps along the drift shell. This method is dayside and extension on the nightside. Note more accurate than following a particle or a guiding that no field line crosses the x = 10Re plane. center. Multiply the image dipole term by 1.5 and re- 11. The double-dipole model.23 The dipole ceases peat. to be a good approximation for the magnetic field (b) Follow several guiding-center trajectories of the Earth as we go farther in space. The double- starting position x between −7Re and −10Re, dipole model, although unrealistic, introduces a y = z = 0, and pitch angles between 30◦ day-night asymmetry that vaguely mimics the de- and 60◦ (smaller pitch angle creates a longer formation of the magnetosphere by the solar wind. bounce motion). With small pitch angles, the It can be used to capture some basic features of particle should come closer to Earth on the particle dynamics in the outer magnetosphere, if day side. Repeat with a pitch angle of 90◦. only qualitatively. Now the particle goes away from the Earth on The model has one dipole (Earth) at the origin, the dayside. Explain these observations using pointing in the negative z-direction, and an image the conservation of first and second adiabatic invariants. dipole at x = 20Re. If both dipoles are identical, the magnetic field is given by: (c) The double-dipole field can break the second invariant for some trajectories. The reason B(x,y,z)= Bdip(x,y,z)+ Bdip(x − 20Re,y,z) (39) of this breaking is that the field strength has a local maximum on the dayside around the where Bdip(x,y,z) is given by Eq. (6). equatorial plane. Particles with sufficiently The domains of each dipole are separated by the small mirror fields are diverted to one side of plane x = 10Re. This plane simulates the magne- the equatorial plane because they cannot over- topause, the boundary between the magnetosphere come this field maximum, as seen in Fig. 6. and the solar wind. For slightly better realism, the Start an electron guiding center at position image dipole can be multiplied by a factor larger x0 = −10Re, y0 = z0 = 0 with kinetic energy 9
K = 1MeV with an equatorial pitch angle 70◦ and follow its guiding center for 1000 seconds with time step 0.01. On the dayside the tra- jectory will temporarily move above or below the equatorial plane. Using the method used (b) 1.5 in Fig. 5, plot the second invariant I versus time. The second invariant will be constant 1.0 (a) between breaking points, but its value will dif- fer from the initial value. The reason is that z 0.5 near the breaking points bounce motion slows 0 40 down and the adiabaticity condition does not 20 hold. However, the first invariant is not bro- −0.5 0 10 ken. 5 −20 0 −5 −40 x This phenomenon, named drift-shell y −10 bifurcation,23 can be one of the causes of particle diffusion in the magnetosphere.
12. Magnetotail current sheet. On the tail region of the magnetosphere, magnetic field lines are heavily (c) stretched, and a sheet of current is flowing through them.24 The field in the magnetotail can be repre- sented by the simple form: 1 6
z z 0 B = B xˆ + B zˆ if |z| < 1 0 d n 4 −1 B0 2 = xˆ + Bnˆz otherwise (40) d 0 −2 0 −4 x In this problem we set B0 = 10, Bn = 1 and y −6 −2 d = 0.2. The field lines trace parabolas on the x − z plane, which can be seen by integrating the FIG. 7. (Color online) Types of orbits created by a particle equation dx/Bx = dz/Bz. The parameter d is the scale of the current sheet thickness. The truncation with mass m = 5 and charge q = 1 near a current sheet. (a) Speiser orbits of transient particles, (b) Cucumber orbits of of the field at z = 1 simulates the finite size of the quasitrapped particles and (c) Ring orbits of trapped parti- tail region. cles. Note the different scales of axes. The field vector points to opposite directions on both sides of the equatorial plane z = 0. When a charged particle is released from above, it moves VII. CONCLUDING REMARKS toward the weaker region near |z| = 0 where the adiabaticity condition does not hold. The helix be- Space plasmas provide many case studies which, after comes a “serpentine orbit” that moves in and out proper simplification, can be used in the undergraduate of the equatorial plane. The chaotic dynamics of physics curriculum. We have presented one such case, the these orbits is extensively studied.25,26 basic theory of charged-particle motion under the dipole. Figure 7 shows the three types of orbits that can ex- This paper focuses on visualization and concrete com- ist in such a model.26,27 “Speiser orbits” approach putation, with the hope that students will modify or the equatorial plane and later go beyond z = 1 and rewrite the code to run their own numerical experiments leave the tail region. “Cucumber orbits” alternate on particle motion in magnetic fields. In my opinion, nu- between helical and serpentine orbits. These do not merical simulations provide at least two important ped- form closed orbits because of the breaking of the agogical benefits: First, even if the required analytical first invariant at the equatorial plane. “Ring or- tools are beyond the students’ level, they can use sim- bits” alternate between oppositely-directed fields; ulations to obtain a qualitative understanding. Second, they do not have a helical section. the process of coding the simulation forces students to understand the problem at a basic and operational level. (a) Evaluate ∇B for |z| < 1 and determine the The main body of this article or the exercises can be direction of gradient-curvature drift. incorporated in lectures, or they can be given as ad- vanced assignments to interested students. A natural (b) By trial and error, find initial conditions that place for this subject is a course on electromagnetism create the types of orbits shown in Fig. 7. and/or plasma physics. When the basics are introduced, 10 the instructor can discuss related subjects such as plasma ing charged particle motion. The widely separated time confinement, radiation belts, or space weather. scales of the motion would be a challenge for most of the In advanced mechanics courses, adiabatic invari- schemes. ants are usually presented with an abstract formalism. Charged particle motion provides a natural and concrete case where adiabatic invariants are relevant and indis- pensable. ACKNOWLEDGMENTS The subject can also be incorporated in courses on computational physics. Accuracy and stability of differ- I thank Meral Ozt¨urk¨ for her careful proofreading, and ent numerical integration schemes may be presented us- two anonymous reviewers for their helpful suggestions.
∗ [email protected], [email protected] proton with 10MeV energy has speed v ∼ 0.1c. Then, in 1 L. W. Townsend, “Radiation exposures of aircrew in high SI units, vB ∼ 1000/L3 ≫ E. Therefore the electric drift altitude flight”, Journal of Radiological Protection 21(1), can be neglected near the Earth where L < 6. Alterna- p. 5, doi:10.1088/0952-4746/21/1/003 (2001). tively one can say that low-energy particles that could be 2 Mark Moldwin, An Introduction To Space Weather (Cam- affected by electric drifts have already drifted away, leaving bridge University Press, New York, 2008), pp. 79-94. behind the trapped high-energy particles. 3 Donald A. Gurnett and Amitava Bhattacharjee, Introduc- 16 We are using the Geocentric Solar Ecliptic (GSE) coordi- tion to Plasma Physics: with space and laboratory applica- nate system. The Earth is at the origin, the x-axis points tions (Cambridge University Press, Cambridge, UK, 2005). to the Sun, the z-axis is set so that the dipole moment 4 Peter A. Sturrock, Plasma Physics: An Introduction to the vector is on the xz-plane, and the y-axis is perpendicular theory of astrophysical, geophysical, and laboratory plasmas to both axes. (Cambridge University Press, Cambridge, UK, 1994). 17 See supplementary material at 5 Martin Walt, Introduction to Geomagnetically Trapped Ra-