The dynamics of small electrically charged dust grains within the rigidly corotating regions of planetary magnetospheres such as those of Jupiter and Saturn is considered. Depending on whether one is inside or outside the synchronous orbit, it is possible to have different populations of both positively and negatively charged particles moving in equilibrium circular orbits either in the prograde or retrograde sense. Not all these are stable, however, to small perturbations, such as would be produced by the gravitational tug of a neighboring satellite. The stable perturbed grains will perform a motion that can be described as an elliptical gyration about a guiding center which is in uniform circular motion. For different values of the specific charge, the ratio of the semiaxes of this ‘epicyclic’ ellipse lies between 1/2 and 1, while the gyration frequency ω of the grain about the guiding center lies between the Kepler frequency ΩK and ω0 (which is the grain gyrofrequency in a nonrotating frame). In the environments of Jupiter and Saturn, where the grains are expected to be negatively charged both in the sunlit side and in the shadow and which move in the prograde sense, their guiding centers must have speeds intermediate to the Kepler speed and the corotation speed. Such particles with a unique specific charge (and therefore a specific size) could have a 1:1 magneto‐gravitational resonance with a neighboring satellite. A dispersion relation between ω and the wavelength λ of the perturbed orbits in the frame of the perturbed satellite has been derived. This result has been used to discuss the appearance and disappearance of the waves in the F ring of Saturn elsewhere. We merely point out here that, while the existence of a single well‐defined wavelength implies a dust size distribution sharply peaked at a diameter of about 1 µ, the present theory also anticipates this situation. The only collisionless (and therefore nonevolving) state of small electrically charged dust grains moving in the same orbit is when they have precisely the same specific charge and therefore the same size (assuming the same density), since the electrical potential is the same for all sizes.
The boundary between the inner and outer parts of Saturn' s B ring is located at the theoretical limit of stability of dust grains with large negative charge to mass ratio. A grain inside of this stability limit will move along (pseudo) magnetic field lines and strike Saturn if given a slight velocity component normal to the ring plane. Outside of this marginal stability radius, a perturbed grain merely oscillates back and forth through the ring plane. The theoretical location of the marginal stability radius is at 1.625 Rs. Observations by Pioneer 11 and Voyager 2 in the infared see the boundary as a prominent change in ring brightness at this radius (within the spatial resolution). The occultation of &Scorpii by the rings in the ultraviolet seen by Voyager 2 shows about a factor of 2 change in optical depth beginning very close to this radius. This close agreement is conceivably a numerical coincidence. Whether this is the case, or whether the instability is actually the physical reason for the boundary's existence, requires further study of the implications.
Dust grains in the ring systems and rapidly rotating magnetospheres of the outer planets such as Jupiter and Saturn may be sufficiently charged that the magnetic and electric forces on them are comparable with the gravitational force. The adiabatic theory of charged particle motion has previously been applied to electrons and atomic size particles. But it is also applicable to these charged dust grains in the micrometer and smaller size range. We derive here the guiding center equation of motion, drift velocity, and parallel equation of motion for these grains in a rotating magnetosphere. The effects of periodic grain charge-discharge have not been treated previously and have been included in this analysis. Grain charge is affected by the surrounding plasma properties and by the grain plasma velocity (among other factors), both of which may vary over the gyrocircle. The resulting chargedischarge process at the gyrofrequency destroys the invariance of the magnetic moment and causes a grain to move radially. The magnetic moment may increase or decrease, depending on the gyrophase of the charge variation. If it decreases, the motion is always toward synchronous radius for an equatorial grain. But the orbit becomes circular before the grain reaches synchronous radius, a conclusion that follows from an exact constant of the motion. This circularization can be viewed as a consequence of the gradual reduction in the magnetic moment. This circularization also suggests that dust grains leaving Io could not reach the region of the Jovian ring, but several effects could change that conclusion. Excellent qualitative and quantitative agreement is obtained between adiabatic theory and detailed numerical orbit integrations.1.
The sharp, 90-km wide transition from an optical depth of 0.2 in the C ring to 1 in the B ring begins at 91,970 km from Saturn's center. This radius is found to be almost exactly at the inward stability limit of charged particles launched in the ring plane at the local Kepler velocity, provided these particles have large charge to mass ratio. The zonal harmonic models of Saturn's magnetic field from the Voyager data and the gravitational field model from Pioneer data are essential to get the very close agreement between theory and observation. The theoretical stability limits are 91,973 + 145 km from Voyager 1 magnetic field data and 91,991 ñ 145 km from Voyager 2 magnetic data. The zonal harmonic magnetic field lines are not perpendicular to the ring plane. Therefore, in addition to the magnetic mirror, gravitational, and centrigual forces, an unknown force must be postulated to produce equilibrium in the ring plane and make the stability calculation meaningful.
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