1. ASEN 5335 Aerospace Environments -- Radiation Belts1 The Radiation Belts A radiation belt is a population of energetic particles stably-trapped by the magnetic field.

2. ASEN 5335 Aerospace Environments -- Radiation Belts2 THE RADIATION BELTS Particles in Magnetic Fields The motions of electrons and ions in a constant magnetic field (without external forces) are governed by a balance between the so- called Lorenz force (F = qV x B) and the centrifugal force, which results in circular motion: Plasma particles are diamagnetic in that their orbital gyrations produce a magnetic field that opposes the background magnetic field.

3. ASEN 5335 Aerospace Environments -- Radiation Belts3 gyroradius q = charge B = magnetic field strength external force independent of charge external force dependent on charge non-uniform B-field curvature in B-field geometry converging/diverging field lines Now, we will consider the influences of an external force and a non-uniform B-field. Five cases will be considered: gyrofrequency m = mass v = velocity perpendicular to B

4. ASEN 5335 Aerospace Environments -- Radiation Belts4 1. CHARGE-INDEPENDENT FORCE effective B is larger larger gravitational force adds to F here gravitational force subtracts from F here effective B is smallersmaller

5. ASEN 5335 Aerospace Environments -- Radiation Belts5 + + - - This represents current flow to the right

6. ASEN 5335 Aerospace Environments -- Radiation Belts6 2. CHARGE-DEPENDENT FORCE Recall that this assumes only the Lorenz force acting on the particle. In the presence of a neutral gas, this will be true if Therefore both + and - particles move in the same direction and there is no current. which is charge independent

7. ASEN 5335 Aerospace Environments -- Radiation Belts7 r larger here (effective B decreased) r smaller here (effective B increased) r smaller here (effective B increased) r larger here (effective B decreased)

8. ASEN 5335 Aerospace Environments -- Radiation Belts8 Drift of + and - particles due to E-field in presence of B-field + + - -

9. ASEN 5335 Aerospace Environments -- Radiation Belts9 3. NON-UNIFORM MAGNETIC FIELD Now consider a non-uniform magnetic field (i.e., |B| varies spatially). <--------- net current flow + + - - The gyroradius is smaller when |B| is larger: (reason in-part for “ring current”)

10. ASEN 5335 Aerospace Environments -- Radiation Belts10 In this case the force on the particle is proportional to the gradient of B : where is the particle's velocity around the magnetic field. Note: for “cold particles” (i.e., ionosphere/plasmasphere)  is small  gradient drift is small (i.e., they co-rotate) The so-called gradient drift is where  = the particle's "magnetic dipole moment"

11. ASEN 5335 Aerospace Environments -- Radiation Belts11 We will be referring to the magnetic moment of a particle later in these lectures, so it is appropriate to take a minute to explain the origin and meaning of this term. The magnetic moment of a current loop is just defined as (current in X (area of the loop) The equivalent current of a gyrating particle (going around once per ) is Substituting the following, We obtain "magnetic moment" of a particle

12. ASEN 5335 Aerospace Environments -- Radiation Belts12 4. MAGNETIC FIELD CURVATURE As a gyrating particle moves along a B - field that is curved, some additional force must act on the particle and make it turn and follow the field line geometry. R B Since this depends on the sign of q, positive and negative particles “curvature drift” in opposite directions, thus producing electric current. Note that these are “collisionless currents”, and so do not produce ohmic losses. Complements the non-uniform, B-field Gradient Drift R = radius of curvature of field line. v = v || = velocity || to B As a particle follows B, a force is exerted F

13. ASEN 5335 Aerospace Environments -- Radiation Belts13 5. CONVERGING/DIVERGING FIELD LINES Rather, the net force is now in the direction of weaker B- field (diverging field lines). For a proton in a diverging B-field as shown in the following figure, the force acting at right angles to the B-vector does not lie in the plane of circular motion of the charged particle.

14. The same holds true for an electron; that is, the net force is away from the region of stronger (converging) field strength. When the magnitude and duration of the force are sufficient to actually cause the charged particle to reverse direction of motion along the line of magnetic force, the effect is known as mirroring, and the location of the particle's path reversal is known as the mirror point for that particle. Therefore, as a charged particle moves into a region of converging B, a force acts to slow the particle down.

15. ASEN 5335 Aerospace Environments -- Radiation Belts15 The three types of charged particle motion in Earth’s magnetic field

16. Some typical periods of particle motion for 1Mev* particles at 2000 km altitude at the magnetic equator TgTbTd gyroperiod bounce period drift period electrons 7  s (r =.3km).1s50 min. protons4ms (r = 10km) 2s30 min. Note that Td >> Tb >> Tg, so that these processes can essentially be assumed decoupled from each other * 1 eV = energy acquired by a particle of unit charge when accelerated through a potential of 1 volt = 1.6 x 10 -12 ergs.

17. ASEN 5335 Aerospace Environments -- Radiation Belts17 According to Faraday's Law of Magnetic Induction, a time rate of change of magnetic flux will induce an electric field ( and hence a force on the particle ): Therefore, the requirement that no forces act in the direction of motion of the rotating particle demands that Let us examine the mirroring problem more quantitatively. Assuming that the Lorenz force F = qV x B is the only force acting on the particle (i.e., no external forces), the kinetic energy (K.E.) of the particle does not change (Lorenz force is perpendicular to V and therefore does no net work); only the direction of motion changes. Constants of Particle Motion

18. ASEN 5335 Aerospace Environments -- Radiation Belts18 In other words, that the magnetic flux enclosed by the cyclotron path of the charged particle is constant: We previously defined the magnetic moment, implying Assuming B does not vary spatially within the gyropath, where r = gyroradius =

19. ASEN 5335 Aerospace Environments -- Radiation Belts19 Note: same as saying that K.E. does not change if there are no forces parallel to v  First Adiabatic Invariant Or,  = constant. This is called the first adiabatic invariant of particle motion in a magnetic field. We should note that the above has assumed that     constant within at least one orbital period of the particle. This is only approximately true, and the term "invariant" is also an approximation, but one that reflects the first-order constraints on the particle motion.

20. A charged particle in a magnetic ‘bottle’ Conservation of the first adiabatic invariant can cause the spiraling particle to be reflected where the magnetic field is stronger. This causes the particle to be trapped by the magnetic field. Consequences of Conservation of the First Adiabatic Invariant

21. ASEN 5335 Aerospace Environments -- Radiation Belts21 If  increases to 90° before the particle collides vigorously with the neutral atmosphere, the direction of v || will change sign (at the "mirror point") and the particle will follow the direction of decreasing B. Since, (i.e, the K.E. of the particle remains constant since the only forces act  to V), then  must increase as B increases, and correspondingly the distribution of K.E. between and changes:

22. ASEN 5335 Aerospace Environments -- Radiation Belts22 For a given particle the position of the mirror point is determined by the pitch angle as the particle crosses the equator (i.e., where the field is weakest) since Therefore, the smaller  eq the larger B M, and the lower down in altitude is the altitude of B M.

23. ASEN 5335 Aerospace Environments -- Radiation Belts23 Loss Cone and Pitch Angle Distribution Obviously this will happen if  eq is too small, because that requires a relatively large B M (|B| at the mirror point). Particles will be lost if they encounter the atmosphere before the mirror point. B loss cone The equatorial pitch angles that will be lost to the atmosphere at the next bounce define the loss cone, which will be seen as a depletion within the pitch angle distribution.

24. ASEN 5335 Aerospace Environments -- Radiation Belts24 Magnetic Mirroring in a Dipolar magnetic Field Trajectory of particle outside the atmospheric bounce loss cone. This particle will bounce between mirror points Trajectory of particle inside the loss cone. this particle will encounter the denser parts of the atmosphere (i.e., below 100 km) and precipitate from the radiation belts.