Abstract
This chapter develops the adiabatic theory of particle motion from first principles. It is presented as a prime example of “physics as the art of modeling”, in which a complex real system is replaced by a highly simplified virtual, i.e., imagined one. In our case the original system is a charged particle in complex multi-periodic motion in a magnetic field, replaced by a so-called guiding center particle (a model!) of equal mass and charge, moving in a smooth, uncomplicated way (having averaged out the smaller scale turns and loops of the real particle). The instantaneous position of this virtual particle is the guiding center, a point in space whose coordinates depend on the local magnetic field and the properties of the real particle. In addition, we show why the guiding center particle is endowed with a magnetic moment which impersonates average electromagnetic properties of the rapidly cycling motion of the original particle. The notion of drift velocity is introduced, and other fundamental physical magnitudes germane to adiabatic motion are defined; drift velocities are classified into zero order (independent of the particle’s dynamic properties), first order and higher order. The first adiabatic invariant is defined and a simple demonstration of its near-constancy under certain restrictions (the adiabatic conditions) is given. We then discuss in detail what happens to a charged particle in its cyclotron motion when the magnetic field is slowly time-dependent, demonstrating the fundamental role of the often neglected magnetic vector potential. Examples are given for zero and first order drifts in simple field configurations. However simple, these examples (e.g., ion pick-up; adiabatic breakdown; closed vs. open drift orbits and their separatrix; co-rotating vs. convecting regions in an externally imposed electric field) illustrate some important basic properties of particles trapped in the equatorial region of the magnetosphere. In the course of this chapter, we emphasize that magnetic field lines are purely geometric entities, useful for one’s mental representation of magnetic field configurations but, like the notion of a guiding center particle, devoid of any physical reality; nonetheless we give a phenomenological definition of field line velocity again putting to good use the vector potential and its local time variation.
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Notes
- 1.
We shall use rationalized SI (Système International) units throughout this book. q is thus expressed in Coulombs (elementary charge \(= 1.6021 \times 1{0}^{-19}\) Coulombs), B in Tesla ( = 104 Gauss)—see Sect. A.1.
- 2.
For the time being we’ll just consider τ C as a priori known. As we shall see in the next section, in a non-relativistic situation τ C is independent of the particle’s dynamic state, depending only on mass, charge and the local magnetic field.
- 3.
Later we will run into the cyclotron phase average operator, \(\langle \,\,\ldots \rangle _{\varphi } = 1/2\pi \int _{0}^{2\pi }\ldots d\varphi = 1/\tau _{C}\int _{0}^{\tau _{C}}\ldots (d\varphi /dt)dt\). When the cyclotron motion in the GCS is uniform, both operators are identical and will be designated as \(\langle \,\,\rangle _{C} =\langle \,\,\rangle _{\tau } =\langle \,\,\rangle _{\varphi }\). A compilation of all phase averages used in this book is given in a footnote on page 200 of Sect. A.3.
- 4.
In regions where B → 0 (e.g., near a neutral line) the concept of “transforming away E ⊥ ” breaks down. See page 73.
- 5.
Henceforth, whenever the charge q appears in the expression of a scalar quantity as a ⊥ , it will be meant to represent the absolute value of q (unless explicitly stated to the contrary). If on the other hand q appears in the expression of a vector quantity, it is assumed to carry its actual sign; otherwise it will be explicitly written as | q | .
- 6.
Yet another model is to imagine the charge q smeared evenly over the cyclotron circle in the GCS, rotating uniformly with period τ C . See page 92.
- 7.
In Hamiltonian mechanics (e.g., [1]) of point charges in a magnetic field, it is demonstrated that for cyclic variables like the arc \({\boldsymbol l}\) (see Fig. 1.5) the so-called canonical path or action integral \(J =\oint ({\boldsymbol p} + q{\boldsymbol A}) \cdot {\boldsymbol dl}\) is a constant of motion for a single particle (provided that the fields and the forces change very little during one cycle). Taking \({\boldsymbol dl}\) in the direction of a positive particle (Fig. 1.5) and carefully considering that, therefore, the magnetic flux through the cyclotron loop \(\oint {\boldsymbol A} \cdot {\boldsymbol dl}\) is negative, we have \(J_{c} =\oint ({\boldsymbol p} + q{\boldsymbol A}) \cdot {\boldsymbol dl} = 2\pi \rho _{C}mv_{\perp }^{{\ast}}-\pi {\rho _{C}}^{2}B = (2\pi m/q)M\), therefore \(M = \mathrm{const.}\)
- 8.
If we place a charged particle in a B& E field with zero initial velocity (for instance, by ionizing a neutral atom at rest), it will start moving in an open cycloid (Fig. 1.11), with a drift velocity U given by (1.10) and a Larmor radius \(\rho _{C} = mE/q{B}^{2}\). The maximum kinetic energy of the particle (at point P) will be \(T = 2m{(E/B)}^{2} = 2m{U}^{2}\), and the average energy, according to (1.35), will be mU 2. This so-called “ion pick-up” process plays an important role in space physics.
- 9.
Without the first condition we would run into the undesirable situation of having “moving field lines” in a static magnetic field crossed by a static electric field, as for instance in the case of two oppositely charged plates placed parallel to \({\boldsymbol B}\) in the gap of a magnet. It can be shown that the above definition is not only independent of the particular position of the guiding center along the field line but that it is magnetic flux-preserving (i.e., flux tubes preserve their identity). There is nothing artificial with a definition using probe particles placed along a field line: after all, as we know from elementary textbooks, the electric and magnetic field vectors themselves are formally defined by forces on probe particles! Neither is the condition that all ρ be turned off artificial: in elementary electromagnetism books, the self-field of probe charges are also “turned off” (ignored). While it is tempting to exaggerate the physical significance of a purely mathematical-geometric concept such as a field line, we must never lose sight of the fact that the only “physical reality” in electromagnetism is mutually interacting electrically charged matter. Here are some perhaps too strong words from Richard Feynman et al. [6]: “ … not only is it not possible to say whether field lines move or do not move with charges—they may disappear completely in certain coordinate frames …”
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Roederer, J., Zhang, H. (2014). Particle Drifts and the First Adiabatic Invariant. In: Dynamics of Magnetically Trapped Particles. Astrophysics and Space Science Library, vol 403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41530-2_1
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