© CAMBRIDGE UNIVERSITY PRESS 2000 |

**13.5. Equations for the guiding centers**

**13.5.1. Larmor oscillations and drifts**

We briefly recall the basic idea about the guiding center description of
particle orbits ^{(9)} (see
Fig. 13.6). Consider a
charged particle (charge *q*, and mass *m*) moving in a constant
magnetic field **B** in the presence of a constant force **F**

(43) |

One can separate the equations in the parallel and perpendicular directions (with respect to the magnetic field)

(13.44) |

(13.45) |

then one separates the motion of the guiding center
**v**_{0} from the Larmor oscillation

(13.46) |

so that

(13.47) |

The resulting guiding center motion is given by the relation

(13.48) |

from which it is evident that a perpendicular force produces a velocity and not an acceleration.

The more general, inhomogeneous and/or time-dependent, case where
**B** and
**F** are not constant can still be worked out with a similar strategy if
the resulting drifts turn out to be slow and the basic gyration frequency is
sufficiently fast. For example, a gradient in **B** can be absorbed
in **F**_{} in
Eq. (45). Some effects, such as those
due to the curvature of the field, may require some detailed
analysis, which we do not need to record here
^{(10)}. When the curvature is
negligible, the parallel motion is basically reduced to solving the
one-dimensional problem

(13.49) |

where *U* is the potential associated with **F**_{||} and
*µ* = *mv*_{}^{2} / 2*B* is the adiabatic invariant
associated with
the Larmor gyration. Essentially the condition at the basis of this
type of description is that the variations in the magnetic field
encountered by the particle on its orbit be slow with respect to the
relevant cyclotron frequency.

**13.5.2. Star drifts and stellar hydrodynamics**

Much like in the case of Larmor gyrations of charged particles in a magnetic
field, the epicyclic theory can be
extended to the case of weakly non-axisymmetric and weakly time-dependent
fields. This tool allows one
to gain a physical perception of the general properties of orbits without
resorting to numerical surveys and well beyond the simple "small
oscillations" that are considered in the standard stability analysis of the
Lagrangian points, as we briefly gave in the previous section. The specific
theory in the context of stellar dynamics
^{(11)},
formally developed for cool disks in the
vicinity of the relevant corotation radius, is based on the
generalization of the definitions of angular and epicyclic frequency as:

(13.50) |

where now is the sum of
an axisymmetric potential and of a weakly non-axisymmetric and weakly
time-dependent perturbation. Thus the *guiding center orbits*
[*r*_{0}(*t*),
_{0}(*t*)] can
be derived from an *effective* Jacobi integral

(13.51) |

where *E*_{0} generalizes the concept of energy associated
with the circular orbits Eq. (12):

(13.52) |

The relevant equations of the motion are:

(13.53) |

(13.54) |

The second term on the right-hand-side of Eq. (53)
corresponds to the well known *polarization drift* of plasma
physics. It
should be stressed that in the presence of rotation the perturbation forces
induce *drift velocities* in the motion of the guiding centers and
*not accelerations*.

The complete properties of the orbits are then obtained by combining the
information on the motion of the guiding centers with the fact that the
adiabatic invariant *µ* is essentially constant:

(13.55) |

where *H* is the star "Jacobi integral", which depends on the
physical coordinates (*r*,
) and on the conjugate momenta.

When the perturbing potential is time-independent, the guiding center
orbits are then simply obtained as contours of the function
*H*_{0}(*r*_{0},
_{0}). The
Lagrangian points of the star orbit analysis described above are then
recovered as stationary points for *H*_{0}; their stability
properties are easily reconstructed by inspection of the relevant
*Hessian*. If one takes as a
perturbation a two-armed spiral potential of the form given in
Eq. (32), one finds that the contours of constant *H*_{0}
identify two
islands at corotation connected by a *separatrix* passing through the
unstable potential minima (see Fig. 13.7).
Outside the islands, moving
away from corotation either inside or outside the corotation circle, the
contours are in
the form of distorted circles. This new shear flow configuration should be
compared with the unperturbed shear flow. The trapped orbits at the stable
Lagrangian points define some kind of "cat's eyes"
^{(12)}.
There is a clear analogy with the structure of
magnetic islands that originate in plasma configurations via *magnetic
reconnection*
^{(13)}.

The concepts introduced in the present short Section find many
applications in the context of magnetically confined plasmas, in
particular in the description of trapped and circulating particles in
toroidal plasma configurations
^{(14)}.
We will not pursue here these
sources of analogies any further. In closing, we may just briefly refer to
another development, which is conceptually very interesting. For a
collisionless
system it is possible to construct fluid equations from the moments of the
collisionless Boltzmann equation (see Chapter 8). In the absence of
collisions, a well known problem is how to *close* the fluid
equations into a finite set, or, in more physical terms, how to define
an appropriate equation of state.
This question has found a solution in plasma physics in terms of the
so-called *double adiabatic theory*
^{(15)}, which makes use
of the conservation of the adiabatic invariant to set a constraint
equivalent to that of an equation of state. This leads to the
justification of MHD-like equations
for a collisionless plasma, with the peculiarity that pressure is to be
considered *anisotropic*. It should be stressed that the closure is
obtained under a set of assumptions that make the double adiabatic theory
applicable only to a rather limited class of perturbations. Still the
procedure is very interesting, especially from the physical point of view. A
similar theory
has been worked out for the context of the stellar dynamics of galaxy
disks ^{(16)}.

^{9} For example, see Schmidt, G.
(1979), *Physics of High Temperature Plasmas*,
2nd edition, Academic Press, New York.
Back.

^{10} See the book by Schmidt
mentioned earlier; see also Krall, N.A., Trivelpiece, A.W.
(1973), *Principles of Plasma Physics*,
McGraw-Hill, New York
Back.

^{11} Berman, R.H. (1975),
*Ph. D. Thesis*, Massachusetts Institute of Technology; Berman,
R.H., Mark, J. W-K.
(1977), *Astrophys. J.*, **216**, 257;
Berman, R.H., Mark, J. W-K.
(1979), *Astrophys. J.*, **231**, 388
Back.

^{12} Kelvin, Lord
(1880), *Nature*, **23**, 45
Back.

^{13} See, e.g., White, R.
(1983), *Handbook of Plasma
Physics*, *I*, edited by A.A. Galeev, R.N. Sudan, North-Holland.
Back.

^{14} Bertin, G., Coppi, B., Taroni, A.
(1977), *Astrophys. J.*, **218**, 92
Back.

^{15} Chew, G.F., Goldberger, M.L., Low, F.E.
(1956), *Proc. R. Soc. London A*, **236**, 112
Back.