Magnetosphere particle motion

The ions and electrons of a plasma interacting with the Earth's magnetic field generally follow its magnetic field lines. These represent the force that a north magnetic pole would experience at any given point. (Denser lines indicate a stronger force.) Plasmas exhibit more complex second-order behaviors, studied as part of magnetohydrodynamics.Rg = mv/(qB)For protons         Rg = (144/B) √W⊥ The ions and electrons of a plasma interacting with the Earth's magnetic field generally follow its magnetic field lines. These represent the force that a north magnetic pole would experience at any given point. (Denser lines indicate a stronger force.) Plasmas exhibit more complex second-order behaviors, studied as part of magnetohydrodynamics. Thus in the 'closed' model of the magnetosphere, the magnetopause boundary between the magnetosphere and the solar wind is outlined by field lines. Not much plasma can cross such a stiff boundary. Its only 'weak points' are the two polar cusps, the points where field lines closing at noon (-z axis GSM) get separated from those closing at midnight (+z axis GSM); at such points the field intensity on the boundary is zero, posing no barrier to the entry of plasma. (This simple definition assumes a noon-midnight plane of symmetry, but closed fields lacking such symmetry also must have cusps, by the fixed point theorem.) The amount of solar wind energy and plasma entering the actual magnetosphere depends on how far it departs from such a 'closed' configuration, i.e. the extent to which Interplanetary Magnetic Field field lines manage to cross the boundary. As discussed further below, that extent depends very much on the direction of the Interplanetary Magnetic Field, in particular on its southward or northward slant. Trapping of plasma, e.g. of the ring current, also follows the structure of field lines. A particle interacting with this B field experiences a Lorentz Force which is responsible for many of the particle motion in the magnetosphere. Furthermore, Birkeland currents and heat flow are also channeled by such lines — easy along them, blocked in perpendicular directions. Indeed, field lines in the magnetosphere have been likened to the grain in a log of wood, which defines an 'easy' direction along which it easily gives way. The simplest magnetic field B is a constant one– straight parallel field lines and constant field intensity. In such a field, if an ion or electron enters perpendicular to the field lines, it can be shown to move in a circle (the field only needs to be constant in the region covering the circle). If q is the charge of the particle, m its mass, v its velocity and Rg the radius of the circle ('gyration radius'), all one needs do is notice that the centripetal force mv2/Rg must equal the magnetic force qvB. One gets If the initial velocity of the particle has a different direction, one only needs resolve it into a component v⊥perpendicular to B and a component v// parallel to B, and replace v in the above formula with v⊥. If W⊥=m v⊥2/2 is the energy associated with the perpendicular motion in electron-volts (all calculations here are non-relativistic), in a field of B nT (nanotesla), then Rg in kilometers is The velocity parallel to the field v// is not affected by the field, because no magnetic force exists in that direction. That velocity just stays constant (as long as the field does), and adding the two motions together gives a spiral around a central guiding field line. If the field curves or changes, the motion is modified, but the general character of spiraling around a central field line persists: hence the name 'guiding center motion.' Because the magnetic force is perpendicular to the velocity, it performs no work and requires no energy—nor does it provide any. Thus magnetic fields (like the Earth's) can profoundly affect particle motion in them, but need no energy input to maintain their effect. Particles may also get steered around, but their total energy remains the same.