A third adiabiatic invariant is deduced based on the diagram, showing a charged particle trapped between two “mirror” walls, M1 and M2.

In effect, we are analyzing a system with a trapped charge within a magnetic bottle of length L (or the distance between the mirrors). If the particle is trapped as shown then the particle behavior is periodic.

If we let L change slowly then: v ‖ L = const.

Now, let M1 be stationary and M2 move toward M1 at velocity v_m, then the incident velocity relative to the wall is:

-[(v ‖ + v_m) - v_m]

and:

delta v ‖ = - [-(v ‖ + v_m) - v_m] + v ‖ = 2 v_m

Now, in

#R = v ‖/ 2L

We can then write:

dv ‖/ dt = 2 v_m (v ‖/ 2L) = (v ‖/ L) (-dL/dt) = -(v ‖/ L) (dL/dt)

whence:

L (dv ‖/ dt ) + dv ‖ (dL/ dt) = 0

so:

d/dt( v ‖/ L) = 0 = const.

It should be noted that all the adiabatic invariants are approximations to what are called Poincare invariants. These assume the form:

P = INT_c p*dq

where INT_c denotes an integral for which all points on the closed curve, C, in phase space, conform to the equations of motion.

In respect of the magnetosphere, the time required for a charged particle to move from the equatorial plane to one mirror point (say, M1) and back is given by the bounce period:

t(b) = 4 INT (0 to Θ) [ds/ v ‖ ]

where ds is an element of arc length along the field line (B) and an integration is performed between 0 and Θ, for which we find:

v ‖ = v [1 – B/B_max]^½

Now, the basics presented hitherto ignore the fact that no general solution exists to the equations of motion for a charged particle moving under the influence of the Lorentz force in a dipole B-field. What happens is an approximation is used, called “the guiding center approximation” – assuming magnetic field changes are small over a gyroperiod, gyroradius. Based on this simplification, the electron or ion moves along B-field referenced to a guiding center, such that (Chen, 1977):

(13 a ) x – xo = - i v⊥ exp (iWt)/ W = r _L sin (Wt)

and

(13b) y – yo = ± v⊥ exp (iWt)/ W = r_ L cos (Wt)

The key point is that the guiding center (xo, yo) is fixed. while r_L is the "Larmor radius".

In the GC approximation, particle motion displays three components: 1) gyration about a field line (given by the gyrofrequency, or cyclotron frequency); 2) reflection between two mirror points (embodied by the “bounce period”) and 3) a gradual longitudinal drift, denoted by the (

With “magnetic mirrors” (pinched B-field gradients) present we have the possible loss cone effect. In the auroral context, the loss cone concept has validity in connection with field –aligned potential drops and enhancing parallel current densities (e.g. Birkeland currents). To be specific, in the magnetospheric environment only electrons of small pitch angles contribute to J . Any parallel electric field increases the flux of electrons inside the loss cone and increases J such that (Cravens, p. 430):

(18) J / Jmax = R[ 1 – (1 – 1/R) exp (e φ‖ / k_B T_e ) ]

Lastly, it's useful to be aware of the conditions for which particles are trapped, or quasi-trapped:

If particles-orbits are “trapped” one has the condition:

(14a) (v / v⊥ ) < (B (max) / B(min) - 1)^½

In effect, we are analyzing a system with a trapped charge within a magnetic bottle of length L (or the distance between the mirrors). If the particle is trapped as shown then the particle behavior is periodic.

If we let L change slowly then: v ‖ L = const.

Now, let M1 be stationary and M2 move toward M1 at velocity v_m, then the incident velocity relative to the wall is:

-[(v ‖ + v_m) - v_m]

and:

delta v ‖ = - [-(v ‖ + v_m) - v_m] + v ‖ = 2 v_m

Now, in

*each reflection*the velocity changes by 2 v_m. The number of reflections per second can then be expressed as:#R = v ‖/ 2L

We can then write:

dv ‖/ dt = 2 v_m (v ‖/ 2L) = (v ‖/ L) (-dL/dt) = -(v ‖/ L) (dL/dt)

whence:

L (dv ‖/ dt ) + dv ‖ (dL/ dt) = 0

so:

d/dt( v ‖/ L) = 0 = const.

It should be noted that all the adiabatic invariants are approximations to what are called Poincare invariants. These assume the form:

P = INT_c p*dq

where INT_c denotes an integral for which all points on the closed curve, C, in phase space, conform to the equations of motion.

In respect of the magnetosphere, the time required for a charged particle to move from the equatorial plane to one mirror point (say, M1) and back is given by the bounce period:

t(b) = 4 INT (0 to Θ) [ds/ v ‖ ]

where ds is an element of arc length along the field line (B) and an integration is performed between 0 and Θ, for which we find:

v ‖ = v [1 – B/B_max]^½

Now, the basics presented hitherto ignore the fact that no general solution exists to the equations of motion for a charged particle moving under the influence of the Lorentz force in a dipole B-field. What happens is an approximation is used, called “the guiding center approximation” – assuming magnetic field changes are small over a gyroperiod, gyroradius. Based on this simplification, the electron or ion moves along B-field referenced to a guiding center, such that (Chen, 1977):

(13 a ) x – xo = - i v⊥ exp (iWt)/ W = r _L sin (Wt)

and

(13b) y – yo = ± v⊥ exp (iWt)/ W = r_ L cos (Wt)

The key point is that the guiding center (xo, yo) is fixed. while r_L is the "Larmor radius".

In the GC approximation, particle motion displays three components: 1) gyration about a field line (given by the gyrofrequency, or cyclotron frequency); 2) reflection between two mirror points (embodied by the “bounce period”) and 3) a gradual longitudinal drift, denoted by the (

**) drift referenced in the Introduction.***E X B*With “magnetic mirrors” (pinched B-field gradients) present we have the possible loss cone effect. In the auroral context, the loss cone concept has validity in connection with field –aligned potential drops and enhancing parallel current densities (e.g. Birkeland currents). To be specific, in the magnetospheric environment only electrons of small pitch angles contribute to J . Any parallel electric field increases the flux of electrons inside the loss cone and increases J such that (Cravens, p. 430):

(18) J / Jmax = R[ 1 – (1 – 1/R) exp (e φ‖ / k_B T_e ) ]

Lastly, it's useful to be aware of the conditions for which particles are trapped, or quasi-trapped:

If particles-orbits are “trapped” one has the condition:

(14a) (v / v⊥ ) < (B (max) / B(min) - 1)^½

If “quasi-trapped” then:

(14b) (v / v⊥ ) > (B (max) / B(min) - 1)^½

If transient:

(14c) (v / v⊥ ) < (B (max) / B(min) - 1)^½

If transient:

(14c) (v / v⊥ ) < (B (max) / B(min) - 1)^½

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