## 3. The plasma drag## 3.1. Calculation of the plasma drag forceSaturn's magnetosphere co-rotates with the planet with the angular velocity . At the distance of Enceladus from Saturn the linear speed of a dust grain on the circular Keplerian orbit is 3.2 times less than that of the plasma particles picked up by the planetary magnetic field. Thus E ring dust is accelerated by the plasma. To make numerical estimates of the plasma effect on a grain we use
Saturn's plasma model of Richardson (1995). The number density of
charged particles over a range of distances from Saturn is reproduced
in Fig. 4a. The most abundant in the inner magnetosphere is the
fraction of the "cold" (thermal) electrons. They are followed by the
heavy ions, supposedly the combinations of water group ions
(O
Striking contrast of the spatial densities suggests that the interaction of dust grains with some of the fractions can be safely neglected. In order to estimate the drag force due to each fraction, we represent the motion of the grain around Saturn by a circular Keplerian orbit and calculate the direct drag forces (Banaszkiewicz et al. (1994) and the Coulomb drag forces (Northrop & Birmingham (1990) for each fraction Fig. 4d presents estimates of the plasma drag forces due to various fractions. Heavy ions dominate all other fractions. The solid line shows the drag force due to direct collisions of the heavy ions with the grain, i.e. without Coulomb drag. It is easy to see that the Coulomb drag gives negligible contribution to the plasma drag force. Thus when evaluating the plasma drag force it is necessary and sufficient to account for the direct drag due to heavy ions. The E ring dust grains move with the supersonic speed with respect to the heavy ion fraction (see Fig. 4c). In this case (Morfill & Grün (1979) and the sophisticated expression (17) can be replaced with its asymptotic The value of the direct drag evaluated with (19) is given in the Fig. 4d (dashed line). ## 3.2. The main features of the planar motionIn the E ring region the plasma drag force is at least 15
times weaker than the radiation pressure (Fig. 4d). Therefore, if
an orbital element is perturbed by the radiation pressure or a
like-strength force, the plasma drag is unimportant. This is the case
for the eccentricity and the solar angle, which are subject to changes
by the three forces discussed in Sect. 2. In contrast, the semimajor
axis is perturbed neither by the radiation pressure, planetary
oblateness nor the Lorentz force, but it experiences a secular
perturbation due to the plasma drag. The growth of the semimajor axis
is especially interesting since apart from the spreading dust in space
it leads to remarkable effects in the elements The perturbation of the semimajor axis of the grain's circular orbit () due to the plasma drag force (19) is The semimajor axis of a micron-sized grain launched from Enceladus
increases by 30% after one planetary
year. (This value is obtained using (17).) The growth of the semimajor
axis makes the parameters Rewrite Eqs. (14)-(15) using new variables , and linearize them for small eccentricities (around ). We have where . For constant
and However, the growth of the semimajor axis makes
and and inserting these equations into (21)-(22) allows one to obtain the "perturbed" solution in explicit form. This solution is valid for moderate time intervals, until the assumption of small eccentricities is broken. However this analytical solution proves to be quite difficult to interpret and does not make the problem more tractable, hence we choose a different way. Looking at the numerical solution of the problem one can see that in most cases the trajectories conserve the character of harmonic oscillations but the center of the oscillations drifts in the plane. This fact can be reproduced analytically without solving (21)-(22). Indeed, it is easy to find the center of curvature of the solution at the point : where ,
,
,
are derivatives with respect to
to be substituted from Eqs. (21)-(22). Expansion to Taylor series with
respect to the variables As the second term in (25) is small, it is evident that the
solution rotates about the
center . Fig. 5 shows the
evolution of the eccentricity
for the grains of various sizes in the case of a constant potential.
The solid line presents . Until
the eccentricity gets large, traces
the instantaneous center of the oscillations for all grain sizes
except for a narrow interval around
Note that neglecting the planetary oblateness and the Lorentz force
still does not stop the drift of the center of the oscillation.
Indeed, means
whereas ## 3.3. Radial drift of the E ring dustThe growth of the semimajor axis of the grain orbits due to the
plasma drag is responsible for remarkable transportation of dust
launched from Enceladus outward from Saturn. With the plasma
parameters of Richardson's (1995) model, the growth rate of the
semimajor axis of a grain launched from Enceladus is estimated to be
times the initial value per
planetary year, where The growth rate of the semimajor axis decreases with the increasing
grain radius When the grain radius decreases the Lorentz force comes into play. It causes the orbital regression which cancels out the orbital precession due to the planetary oblateness roughly at . The radiation pressure sends the grains to highly eccentric orbits which allow them to avoid recollision with the satellite for a long time. According to Horányi et al. (1992) the orbits of the grains
with quickly regress due to the
Lorentz force which does not permit the radiation pressure to
introduce large eccentricities. This was thought to be the explanation
for the absence of small dust grains
() in the E ring. Inclusion of
the plasma drag force necessitates corrections to this model regarding
the small grains. Indeed, the amplitude of the eccentricity
oscillations decreases proportional
to the radius of the grains, the motion of which is dominated by the
Lorentz force. Decrease of the amplitude narrows the recollision zone
(i.e. the range of Special computations show that almost all submicron particles avoid recollision with Enceladus and spread in space beyond its orbit. They can be eliminated later due to collision with other satellites, sputtering and mutual collisions. The most rapidly drifting grains (), which are still within the applicability limits of the orbit-averaging technique, can cross the ring as fast in planetary year. Finally, the tiniest dust grains of radii are small enough to let the planetary magnetic field pick them up and accelerate to the co-rotational speed. They do not leave the neighbourhood of Enceladus' orbit but move several times faster than the parent moon and should reimpact it very soon. © European Southern Observatory (ESO) 1999 Online publication: June 17, 1999 |