F [i irl

where r = \Jp1 + z2 and J2 = 0.01667. The magnetic stream function, including both dipole and quadrupole terms, is with Q2 — 0.01642. Note that the quadrupole part vanishes on the equatorial plane.

Radiation pressure is measured by the dimensionless coefficient ' = <5>

where d is the Sun-planet distance and ¡3 = JoQpr/Pgrg, with solar constant Jo = 5.7 x 10~5 and Qpr ranges from ~ 0.3 for a 100 nm dielectric grain to more than 2.0 for a 1 pm conducting grain [7], We shall take pg = 1 g cm-3 throughout.

The equilibrium (circular) orbits are given by V[/e = 0; stability is determined by the Hessian det D2Ue, which gives the type of each critical point. Each stable critical point is surrounded by a potential well, bounded by one or more saddle points. Thus, one speaks of saddle point confinement [8], in which all orbits with total energy E < Es are confined. Figure la depicts a positively charged conducting grain in a prograde equatorial orbit at po = 2 with $ = 400, corresponding to grain radius a = 158 nm. As shown in [1] these orbits destabilize at a critical value of q/m which is easily calculated analytically for Jv — 92 — k = 0. In each case either a pitchfork or a tangent bifurcation of the effective potential occurs. In [2] we showed that non-equatorial 'halo' orbits exist for both positively and negatively charged grains, but that retrograde orbits were impossible for negatively charged grains. Analytic stability boundaries were derived and corresponding bifurcations identified, again all with J2 = g? = k = 0. Figure lb shows a typical halo orbit for a prograde positively charged conducting grain, with r0 = 5 and $ = 1100 (a = 95 nm).

Figure 1. Trapped orbits for conducting grains near Saturn: (a; left) equatorial po — 2, $ = 400, (b; right) halo, r0 = 5, $ = 1100
Figure 2. Potential profiles (arbitrary units) for the two orbits of Figure 1: (a; left) equatorial, (b; right) halo.

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