Magnetosphere particle motion

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The region where magnetic forces can be detected is commonly known as a "magnetic field", and similarly the presence of electric forces identifies an "electric field." There also exists a deeper meaning to "field"— a modification of space itself, enabling it to transmit energy and momentum in the form of electromagnetic waves— but in what follows, the simple definitions will suffice.

The magnetic field at some given location in space is usually characterized by the force B which would be sensed by a magnetic pole of unit magnitude and "northward seeking" ("N") polarity, were it placed at that location. "Unit pole" is suitably defined in texts on magnetism, and we ignore now the complicating fact that magnetic poles always exist in N-S pairs (a more fundamental definition can also be made, relating B to electric currents). What matters here is that B is a force and therefore a vector, a quantity with both direction and magnitude. Vectors will be denoted here by bold face letters (other notations also exist, such as underline or arrow on top), while the magnitude of a vector like B will be denoted by plain text letters ("B"). The electric field is similarly defined by a vector E, equal to the force (at the given location) on a unit of positive electric charge.

The most common way of specifying a magnetic field is to provide the vector B (in some way) for every point in space. Mathematically B behaves very much like the distribution of the velocity v in a volume of incompressible fluid (e.g. water is nearly incompressible): both satisfy the same consistency condition. Some special classes of vectors are completely specified by a simpler representation. In the atmosphere, where no electric currents circulate, the distribution in space of a single numerical quantity (a "scalar potential") suffices to define B. A similar representation can give the velocity v of an incompressible fluid, if no swirling motions occur ("irrotational flow").

But there also exists an altogether different way of describing the motion of an incompressible fluid, namely by giving its streamlines, lines marking the motion of particles of the fluid. Every point in space where a velocity v' exists is threaded by a streamline, and at any location, v is always directed along the streamline which threads that point.

Magnetic field lines emanate primarily from the north pole of a magnet and curve around to the south pole

Michael Faraday, one of the founders of electromagnetism, described magnetic fields using a completely analogous description, namely lines which everywhere pointed in the direction of the vector B. He named them "lines of (magnetic) force" but nowadays the term "magnetic field lines" is generally used. The field lines of a bar magnet are a familiar example, since their pattern resemble the one formed by iron filings on a flat sheet of cardboard, when that sheet is placed on top of a horizontal bar magnet. They spread out from one pole, curve around and converge on the other pole, and the patterns near the pole are mirror images of each other.

Field lines provide a quick qualitative description of the structure of magnetic fields in space, and that was their original use. They can also be represented mathematically in terms of "Euler potentials," though that representation is nonlinear and has other problems. In designing electromagnetic devices, different ways of specifying the field are generally preferred.

In the magnetosphere and other space plasmas, however, field lines are extremely useful, since they help visualize not only the field, but also its natural processes. That is because field lines naturally guide the motion of ions and electrons in the plasma. Such charged particles naturally and easily flow along them, but flow perpendicular to them only in certain restricted cases.

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 get separated from those closing at midnight; at such points the field intensity on the boundary is zero, posing no barrier to the entry of plasma. (The above 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 IMF field lines manage to cross the boundary. As discussed further below, that extent depends very much on the direction of the IMF, in particular on its southward or northward slant.

Trapping of plasma, e.g. of the ring current, also follows the structure of field lines (next section below). 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.

[edit] Motion of Charged Particles

A simulated charged particle, its trajectory determined primarily by the Earth's magnetosphere.

A charged particle (an ion or electron) with electric charge q (coulombs), in the presence of an electric field E, will experience an electric force qE.

If a magnetic field B is also present and the particle moves with velocity v, it experiences in addition a magnetic force F = q(v×B). Here "×" ("cross product" or "vector product") identifies F as a vector perpendicular to both v and B , i.e. to the plane these two define, of magnitude Bv_⊥ , where v_⊥ ("v-perp") is the vector component of v perpendicular to B (all calculations here are non-relativistic).

The above formula needs one more clarification, since the plane of v and B has two perpendicular directions, opposed to each other (analogous to "up" and "down"). The one implied here is obtained by the right hand rule: if the thumb of the right hand rule points in the direction of the component v_⊥, and the index finger in the direction of B, then the middle finger points in the direction of F. (You get the opposite direction if you use the left hand.)

The electric field is for now ignored (its turn comes later). One may also ask, "what if the moving particle is studied in some moving frame of reference, where v is different--would the force on it change?" The quick answer here is, the force would remain the same, but its division into electric and magnetic contributions will differ.

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 R_g the radius of the circle ("gyration radius"), all one needs do is notice that the centripetal force mv²/R_g 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 R_g 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."

Note in passing that 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.

[edit] Magnetic Mirroring and Magnetic Drift

The spacing between field lines is an indicator of the relative strength of the magnetic field. Where magnetic field lines converge the field grows stronger, and where they diverge, weaker.

Now, it can be shown that in the motion of gyrating particles, the "magnetic moment" μ = W_⊥/B (or relativistically, p_⊥²/2B) stays very nearly constant. The "very nearly" qualifier sets it apart from true constants of motion, such as energy, reducing it to merely an "adiabatic invariant." For most plasmas in the magnetosphere, the deviation from constancy is negligible.

The conservation of μ is tremendously important (in laboratory plasmas as well as in space). Suppose the field line guiding a particle, the axis of its spiral path, belongs to a converging bundle of lines, so that the particle is led into an increasingly larger B. To keep μ constant, W_⊥ must also grow.

However, as noted before, the total energy of a particle in a "purely magnetic" field remains constant. What therefore happens is that energy is converted, from the part associated with the parallel motion v_// to the perpendicular part. As v_// decreases, the angle between v and B then increases, until it reaches 90°. At that point W_⊥ contains all the available energy, it can grow no more and no further advance into the stronger field can occur.

The result is known as magnetic mirroring. The particle briefly gyrates perpendicular to its guiding field line, and then retreats back to the weaker field, the spiral unwinding again in the process. It may be noted that such motion was first derived by Henri Poincaré in 1895, for a charged particle in the field of a magnetic monopole, whose field lines are all straight and converge to a point. The conservation of μ was only pointed by Alfvén about 50 years later, and the connection to adiabatic invariant was only made afterwards.

Magnetic mirroring makes possible the trapping in the dipole-like field lines near Earth of particles in the radiation belt and in the ring current. On all such lines the field is much stronger at their ends near Earth, compared to its strength when it crosses the equatorial plane. Assuming such particles are somehow placed in the equatorial region of that field, most of them stay trapped, because every time their motion along the field line brings them into the strong field region, they "get mirrored" and bounce back and forth between hemispheres. Only particles whose motion is very close to parallel to the field line, with near-zero μ, avoid mirroring--and these are quickly absorbed by the atmosphere and lost. Their loss leaves a bundle of directions around the field line which is empty of particles--the "loss cone."

In addition to gyrating around their guiding field lines and bouncing back and forth between mirror points, trapped particles also drift slowly around Earth, switching guiding field lines but staying at approximately the same distance (another adiabatic invariant is involved, "the second invariant"). That motion was mentioned earlier in connection with the ring current.

One reason for the drift is that the intensity of B increases as Earth is approached. The gyration around the guiding field line is therefore not a perfect circle, but curves a little more tightly on the side closer to the Earth, where the larger B gives a smaller R_g. This change in curvature makes ions advance sideways, while electrons, which gyrate in the opposite sense, advance sideways in the opposite direction. The net result, as already noted, produces the ring current, though additional effects (like non-uniform distribution of plasma density) also affect the result.

[edit] See also

Magnetosphere
Magnetospheric convection and magnetic storms
Aurora (astronomy)
Magnetosphere history

[edit] External links

• "3D Earth Magnetic Field Charged-Particle Simulator" Tool dedicated to the 3d simulation of charged particles in the magnetosphere.. [VRML Plug-in Required]