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

2. The charge variations

2.1. Charge and its variability

The E ring dust is immersed deeply in Saturn's magnetosphere which is characterized by a strong magnetic field (Acuña et al. (1983) and dense plasma (see Richardson (1995) and references therein). As long as the grains move in the ambient plasma, they collect charge which raises the Lorentz force in the planetary magnetic field. To account for the Lorentz force properly, one needs to know the grain's charge as well as its variations. In practice the electric potential of the grain  is evaluated. The charge of the grain of radius s is then calculated using the equation  (q and s are in CGS units, in Volts).

Evaluation of a grain's potential is not an easy task. It requires knowledge of the physical, chemical and geometrical properties of the grain, as well as the ambient plasma parameters and the grain's velocity relative to the bulk of plasma. E ring grains' potentials were calculated in a number of papers (Fig. 1). We have selected some of them for comparison of the relevant particle dynamics (Jurac et al. (1995), their Fig. 6-b; Horányi et al. (1992), their Fig. 1, curve for secondary electron emission parameters  eV, ; Morfill et al. (1993), their Fig. 4, curve for  eV, ). Also discussed is the constant charge -5.6 V since it is close to the "best guess" charge of Horányi et al. (1992).

 Fig. 1. The grain potentials versus planetocentric distance.

The grain's potential is usually found by solving the current balance equation

where  are e.g. electron and ion fluxes incident on a grain, photoelectron current, etc. In so doing, circular equatorial orbits of the grains are usually assumed. First, it results in the constancy of ambient plasma parameters along such orbits. Second, it makes the grain's velocity relative to the bulk of plasma independent of time. Thus the potential of such a grain does not depend on time and is determined solely by the radius of its orbit. The fact that the real dust grains can move along eccentric paths is ignored in most papers. However, the eccentric orbits (i) allow grains to penetrate different plasma environments and (ii) introduce periodic variations of the grain's velocity relative to the bulk of plasma.

In what follows, we will represent the variable potential of the grain moving along an eccentric orbit by a function of distance, , where is the solution of the balance Eq. (1) at the distance r from the planet. However, we will assume that does not depend on time explicitly. To justify this assumption, consider the time variation of the potential, determined by the equation

where  are the rates of the grain's potential variations at the distance r from the planet due to a number of processes of charge exchange between the grain and environment (cf. (1)). The weak dependence of the rates on the grain's velocity relative to the bulk of plasma was reported by Horányi et al. (1992) and is ignored in (2). Usually are proportional to the ambient plasma density. (The exception is the photoelectron emission which is relatively small near Saturn.) Thus, when the plasma density increases, the rate of the grain's potential variation becomes high and the equilibrium potential such that is reached soon. Indeed, a dust grain of size with the potential , a value close to that favoured by most authors for a grain in the vicinity of Enceladus, carries extra electrons on its surface. The number density of thermal electrons near the orbit of Enceladus is . Their temperature (about 3 K) corresponds to the thermal speed  km sec-1. Simple calculation gives the rate of collisions of thermal electrons with the grain  events per second. The amount of extra electrons necessary to develop the potential is thus collected during  minutes or times the grain's orbital period. This means that the grain's potential responds to changes of the plasma parameters very quickly.

The case of low plasma density is discussed in (Burns & Schaffer (1989)). When the orbital anomaly phase lag of the grain's potential becomes large, the Lorentz force is able to spread grains in space due to both semimajor axis and eccentricity variations. This is the case for so-called resonant charge variations. However, this is not the case for Saturn, where plasma is dense enough to reduce the phase lag to small values which we can safely neglect.

2.2. A supplement to the Hamilton-Krivov integral of the motion

Hamilton & Krivov (1996) developed the analytical theory for the investigation of the circumplanetary dust dynamics. The authors assumed that the grain's orbital plane, planet's equatorial plane and the planet's orbital plane all coincide. Then the orbit-averaged equations of motion under the radiation pressure, planetary oblateness (the coefficient of the multipole expansion of the gravitational potential) and the Lorentz force in the case of constant charge take the form (the solar tidal term is not considered in this paper)

where is the angle between the directions from Saturn to the Sun and to the pericenter of the orbit (solar angle), e is the orbital eccentricity, is the solar longitude which substitutes time in the equations of the motion. Note that the equations are written in the reference frame synchronously rotating with the Sun around the planet. This is the reason for the `-1' term in the first equation. The perturbation parameters are defined as

for the radiation pressure, planetary oblateness and the Lorentz force in the case of constant charge, respectively. Here n and are the mean motions of the grain and the Sun, a is the semimajor axis of the grain's orbit, s, m and are the radius, mass and the density of the grain assumed to be an ice ball, is the radiation efficiency factor (assumed to equal unity), is the flux of radiation energy at the heliocenteric distance of Saturn, , and are the gravitational parameter, equatorial radius and the magnitude of the planet's spin vector, c is the speed of light. The Lorentz force depends on the grain's charge  and the induction of the planetary magnetic field assumed to match that of a dipole and to equal  G at the Saturnian equator.

Hamilton & Krivov showed that Eqs. (3)-(4) are semicanonical with the "Hamiltonian"

which gives the integral of the motion . They used the Hamiltonian formalism to investigate the existence and stability of fixed points. They found that Eqs. (3)-(4) have up to five stationary points, depending on the values of the parameters C, W and L. Two points and correspond to one and the same point in the - space. These stationary points are not fixed points of (3)-(4). However, they are physically meaningful: radiation pressure causes orbits with very tiny eccentricities to precess or regress rapidly away from toward . The other three points are shown on the upper left panel in Fig. 2. The plot gives an example of how the stationary points determine the appearance of trajectories in phase space. The detailed analysis of topology of phase portraits is given in (Hamilton & Krivov (1996).

 Fig. 2. Phase portraits of the motion of the grains with variable potential  in Saturn's magnetic field with account for the radiation pressure, planetary oblateness and the Lorentz force. Top to bottom: constant charge  (left to right: , 1.1, 1.2); charge of Jurac et al. (1995) (0.6, 0.85, 1); charge of Horányi et al. (1992) (1.4, 1.6, 1.8); charge of Morfill et al. (1993) (3.0, 3.15, 3.3). The numerical solution of Newton's equations of the motion with the initial data is overplotted with dots.

Now let us generalize the foregoing approach to a variable grain's charge . One can easily expand it to the series in

where are constants. The Lorentz force in the magnetic field acting on a grain which moves at the distance with the velocity in the planetocentric reference frame

after substitution of (9) takes the form of the series

On the base of the latter equation, the orbit-averaged equations of the motion can be obtained for the Lorentz force in the case of a distance-dependent charge

In accordance with (Burns & Schaffer (1989) for all integer k (not only positive) one gets . Therefore, acting alone, the Lorentz force due to a constant or distance-dependent charge causes precession or regression of the solar angle but does not alter the eccentricity. As follows from Eqs. (3)-(4), it is only the radiation pressure which binds together the evolution of the eccentricity and the behavior of the solar angle. Note, however, that in a general three-dimensional problem the Lorentz force due to variable charge is able to change the eccentricity and the semimajor axis of the grain's orbit as well.

In practice the linear approximation for the function can be used instead of the infinite series (9) and the functions  can be expanded in powers of eccentricity. Table 1 shows several first terms of such expansion () together with the errors introduced by ignoring the higher-order terms. It is easy to see that for the residuals grow rapidly enough with the eccentricity that makes it impossible to use the truncated -series. Fortunately, is relatively simple and have already been analyzed in (Hamilton & Krivov (1996). Replacement of the  for with the first terms of the relevant expansions leads to reasonably small residuals for eccentricities up to 0.6 and can be used instead of awkward precise expressions.

Table 1. Approximate right-hand sides of the equations of the motion accounting for the charge variations and their residuals for (the ratio of the angular velocity of Saturn's rotation to the mean motion of Enceladus).

To account for the charge variability, Eqs. (3)-(4) are therefore extended to

where is the term due to the variable charge, and . Eqs. (14)-(15) have the integral of the motion

It is easy to see that introducing the variable charge which depends only on the distance from the planet does not complicate the integral of (Hamilton & Krivov (1996). Indeed, adjusting the coefficients , , , or performing multiplication of the Hamiltonian by the - the operation which does not affect the curves - we reduce the new integral (16) to Eq. (8). It is clear that the sign of each primed coefficient matches the sign of the non-primed one. However, the term can now be either positive or negative, depending on the sign of .

The case of a positive sign (, ) is carefully analyzed in (Hamilton & Krivov (1996). In the case of opposite sign there are specific combinations of parameters when the phase portrait can be mirrored against the axis. This case is demonstrated by the second row of plots in Fig. 2. Note that the case of negative sign can be interpreted as the backward motion of the Sun.

2.3. Dynamics for different charge models

With the grain's potentials shown in Fig. 1 and using the integral (16) the phase portraits of the motion (the curves ) were built for several E ring dust charging models (Fig. 2). In order to obtain the phase portraits, the potentials were approximated by linear functions of the distance in the neighbourhood of the orbit of Enceladus. The corresponding values of L and V are given in Table 2, as well as the values of C and W shared by all charging models.

Table 2. Numerical values of the parameters C, W, L, and for various charging models. The grain size s is in microns.

Fig. 2 also presents the numerical solutions of the full Newtonian equations of the motion , taking into account the planetary obliquity and non-zero grain's orbital inclinations. In these numerical integrations, the potentials were calculated by linear interpolation, using the voltage values sampled by . Shown in the figure are the phase trajectories for the grains initially in circular orbits. Despite many simplifications used to derive Eqs. (14)-(15), the numerical and analytical results are in fairly good agreement.

Variation of the charging model parameters (i.e. the function ) leads to variation of the grain's radius , for which the maximum orbital eccentricity, developed by an initially circular orbit, is the largest. The value of this largest eccentricity depends on the charging model as well (Fig. 3).

 Fig. 3. Evolution of eccentricity for an initially circular orbit. The sizes and the charging models are the same as those in Fig. 2. The data were obtained from the numerical solution of Newton's equations of the motion.

The difference between the phase portraits for different charging models gives rise to the question: which charge models are consistent (provided that the dynamical model is correct) with the large radial extent and narrow size distribution of the ring derived from observations? Let us check which charging models are suitable in this sense, i.e. allow the grains with radii (Nicholson et al. (1996) to attain large orbital eccentricities for the problem under consideration (radiation pressure + planetary oblateness + Lorentz force + variable charge).

As seen from the obtained solutions, suitable models are the constant charge and the variable charge of Jurac et al. (1995). They allow grains of radii to develop large eccentricities (). The charge given by Horányi et al. (1992) evaluated for is less compatible with the observational data on the E ring since it gives too low maximum eccentricity for the grains of . Indeed, Horányi et al. (1992) reject this model and choose another one for which gives almost constant charge . The least suitable model is one favoured by Morfill et al. (1993) who used a different dynamical model accounting for the plasma drag only. In our dynamical model their potential allows attaining considerable orbital eccentricities for the grains of , i.e. outside the interval .

© European Southern Observatory (ESO) 1999

Online publication: June 17, 1999