Van Allen Belt "mirror points"-- How do these work?

[Chris Frisella]

I was studying the Van Allen Belt, and I get it except for the mirror points (see picture). What would cause the particle to be deflected or turned around? Seems to me it would just continue towards the Earth getting tighter and tighter around the field line.

http://www.tethers.com/WebImages/RadiationTrapping.jpg

 

[Chris Frisella]

I found an answer:

 

[Ken G]

One way to think of what is happening there is to notice that a static magnetic field cannot change the energy, or the speed, of the charge, because the Lorentz force is always perpendicular to the velocity. But when the field pinches, the strength of the Lorentz force increases. The velocity can be decomposed into a component along the field, and perpendicular to the field, but the total magnitude stays the same. When the Lorentz force increases, the radius of curvature of the circle around the field drops. But that can't happen and keep the perpendicular velocity the same, because the product of the perpendicular velocity times the radius of curvature is an "adiabatic invariant", meaning it is basically the angular momentum that cannot change without a torque. The angular momentum doesn't change because the Lorentz force is along the radius, so the torque is zero. So for the radius of curvature to drop, the perpendicular velocity has to rise, and for the total speed to stay the same, that rise must come at the expense of the velocity along the field. Thus, that latter velocity eventually goes to zero, and that's when it goes no further along the field-- the mirror point.