Astron. Astrophys. 346, 1011-1019 (1999)

3. The plasma drag

3.1. Calculation of the plasma drag force

Saturn'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⁺, OH⁺, H₂O⁺, H₃O⁺) and N⁺ ones which have the atomic number about 16. The much less dense fraction of Saturn's magnetosphere is that of the hydrogen ions (protons) and the least dense one is composed of the "hot" (suprathermal) electrons, the number density of which does not exceed and is not presented in the graph. Calculation of the mass density (number density times particle's mass) shows the dominance (up to 2 orders of magnitude, see Fig. 4d) of the heavy ion fraction.

Fig. 4a-d.  Saturn's plasma and its effect on the motion of the E ring grains: (a) number density of the charged particles versus distance from Saturn (hot electrons are not shown), (b) mass density, (c) the grain speed relative to the bulk of plasma (shown are the maximum and minimum values for a grain on a Keplerian orbit) and the thermal speed of ions, (d) plasma drag force due to various fractions in the units of the solar radiation pressure (vertical crosses are heavy ions, diagonal crosses are protons, "bugs" are cold electrons, bars are hot electrons, solid line is the plasma drag force due to direct collisions with the heavy ions, and the dashed line is the same force in supersonic approximation.

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 i, determined by the number density  and thermal speed  of the charged particles of mass . A grain is defined by its radius s, charge q (assumed to be ) and the velocity v relative to the bulk of plasma. The latter allow us to calculate the Mach number . Note that in (18) e temporarily denotes the elementary charge (CGS units). Symbol stands for the minimum Debye length  among all the fractions. For Saturn's inner magnetosphere it is the one of the thermal electrons which ranges from 70 cm at the distance to 12 m at .

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 motion

In 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 e, .

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 C, W, L, V dependent on time.

Rewrite Eqs. (14)-(15) using new variables , and linearize them for small eccentricities (around ). We have

where . For constant and C, the solution of (21)-(22) with the initial data is

However, the growth of the semimajor axis makes and C functions of time, or equivalently, of . Assuming linear dependence

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 h, k up to the terms of the first order gives

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 s such that is close to zero and the assumption of small eccentricities is broken very soon.

Fig. 5. The evolution of eccentricity () for different-sized grains with the constant charge of (dots). Solid line shows . Time is measured in Saturnian years.

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 C remains a function of time due to the growth of the semimajor axis. This is the case for the Hyperion dust discussed by Banaszkiewicz & Krivov (1997).

3.3. Radial drift of the E ring dust

The 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 s is measured in microns. This rate is large enough to contribute considerably to the formation of the E ring structure (Dikarev & Krivov (1998).

The growth rate of the semimajor axis decreases with the increasing grain radius s. Thus the orbital evolution of large grains can be described in terms of the "radiation pressure + planetary oblateness" model. (The Lorentz force decreases and does not play an important role for large grains either.) The planetary oblateness causes quick precession of the orbit which does not permit the radiation pressure to develop large eccentricities. These grains orbit Saturn close to Enceladus and collide with the satellite in short time (Horányi et al. (1992). This mechanism explains the absence of large grains () in the E ring.

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 a when the grain's orbit intersects that of the satellite). The rate of growth of the semimajor axis is , so that a decrease of the grain radius decreases the period of stay in the zone.

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
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