PhD thesis - Institute for Space Research - University of Calgary

8 2: Background
rection along B 0 at low altitudes in the polar regions as a consequence
of the magnetic mirror force,
where
F m = −(µ · ∇)B 0 , (2.7)
|µ| = mv2 ⊥
2|B 0 |
(2.8)
is an integral of motion called the “first adiabatic invariant” (see Jursa
[1985]). The adiabatic invariant is approximately constant over the
course of the particle’s motion as long as energy and momentum transfer
to the particle is small [Jursa, 1985]. It is clear from Equation 2.8
that as the particle moves into a region of stronger magnetic field, its
perpendicular velocity increases as well. Since energy is conserved, there
is a corresponding decrease in the particle’s parallel velocity, and the
main effect of the mirror force is to cause particles to reflect from the
low-altitude polar regions, giving a “bounce-like” quality to their motion.
An approximate expression for the altitude at which the particles
reflect (the “mirror point”) can be obtained from Equation 2.8 by approximating
the magnetic field with a dipole, which varies with radius
r as 1/r 3 . Equating µ at the equator with µ at the mirror point,
( ) 3
B eq Rm
= = sin2 α eq
B m R eq sin 2 π 2
(2.9)
where B eq and B m are the magnetic fields at the equator and mirror
point, respectively, and R eq and R m are the geocentric heights at the
equator and mirror point, respectively. The perpendicular velocity is
related to total velocity by the sine of the particle’s pitch angle α peq at
the equator. The mirror altitude h m is then
h m = R m − R E = 3 √sin 2 α peq R eq − R E , (2.10)
where R E is the radius of the Earth. Particles whose pitch angles are
small at equatorial latitudes can mirror low enough into the ionosphere
that they collide with neutral atoms and molecules. Since these particles
do not return to the magnetosphere they are said to be within the “loss
cone” of pitch angles. In the steady state at a given altitude, no particles
have pitch angles within the loss cone, and it is said to be “empty”. An
observation of a “full” loss cone indicates that some physical process
is decreasing the pitch angles of some particles, thereby putting them
inside the loss cone. These particles are said to be “precipitating” into
the upper atmosphere.