## 2. The charge variations## 2.1. Charge and its variabilityThe 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 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).
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 where are the rates of the
grain's potential variations at the distance 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 motionHamilton & 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), for the radiation pressure, planetary oblateness and the Lorentz
force in the case of constant charge, respectively. Here 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
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
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.
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 modelsWith 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
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).
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 |