The Christiansen Effect in Saturn’s narrow dusty rings and the spectral identification of clumps in the F ring

Stellar occultations by Saturn’s rings observed with the Visual and Infrared Mapping Spectrometer (VIMS) onboard the Cassini spacecraft reveal that dusty features such as the F ring and the ringlets in the Encke and the Laplace Gaps have distinctive infrared transmission spectra. These spectra show a narrow optical depth minimum at wavelengths around 2.87 μm. This minimum is likely due to the Christiansen Effect, a reduction in the extinction of small particles when their (complex) refractive index is close to that of the surrounding medium. Simple Mie-scattering models demonstrate that the strength of this opacity dip is sensitive to the size distribution of particles between 1 and 100 μm across. Furthermore, the spatial resolution of the occultation data is sufficient to reveal variations in the transmission spectra within and among these rings. In both the Encke Gap ringlets and F ring, the opacity dip weakens with increasing local optical depth, which is consistent with the larger particles being concentrated near the cores of these rings. The Encke Gap ringlets also show systematically weaker opacity dips than the F ring and Laplace Gap ringlet, implying that the former has a smaller fraction of grains less than ∼30 μm across. However, the strength of the opacity dip varies most dramatically within the F ring; certain compact regions of enhanced optical depth lack an opacity dip and therefore appear to have a greatly reduced fraction of grains in the few-micron size range. Such spectrally-identifiable structures probably represent a subset of the compact optically-thick clumps observed by other Cassini instruments. These variations in the ring’s particle size distribution can provide new insights into the processes of grain aggregation, disruption and transport within dusty rings. For example, the unusual spectral properties of the F-ring clumps could perhaps be ascribed to small grains adhering onto the surface of larger particles in regions of anomalously low velocity dispersion.     

Physical collisions of moonlets and clumps with the Saturn's F-ring core

Since 2004, observations of Saturn's F-ring have revealed that the ring's core is surrounded by structures with radial scales of hundreds of kilometers, called “spirals” and “jets.” Gravitational scattering by nearby moons was suggested as a potential production mechanism; however, it remained doubtful because a population of Prometheus-mass moons is needed and, obviously, such a population does not exist in the F-ring region. We investigate here another mechanism: dissipative physical collisions of kilometer-size moonlets (or clumps) with the F-ring core. We show that it is a viable and efficient mechanism for producing spirals and jets, provided that massive moonlets are embedded in the F-ring core and that they are impacted by loose clumps orbiting in the F-ring region, which could be consistent with recent data from ISS, VIMS and UVIS. We show also that coefficients of restitution as low as ∼0.1 are needed to reproduce the radial extent of spirals and jets, suggesting that collisions are very dissipative in the F-ring region. In conclusion, spirals and jets would be the direct manifestation the ongoing collisional activity of the F-ring region.     

The vertical structure of the F ring of Saturn from ring-plane crossings

We present a photometric model of the rings of Saturn which includes the main rings and an F ring, inclined to the main rings, with a Gaussian vertical profile of optical depth. This model reproduces the asymmetry in brightness between the east and west ansae of the rings of Saturn that was observed by the Hubble Space Telescope (HST) within a few hours after the Earth ring-plane crossing (RPX) of 10 August 1995. The model shows that during this observation the inclined F ring unevenly blocked the east and west ansae of the main rings. The brightness asymmetry produced by the model is highly sensitive to the vertical thickness and radial optical depth of the F ring. The F-ring model that best matches the observations has a vertical full width at half maximum of 13 ± 7 km and an equivalent depth of 10 ± 4 km. The model also reproduces the shape of the HST profiles of ring brightness vs. distance from Saturn, both before and after the time of ring-plane crossing. Smaller asymmetries observed before the RPX, when the Earth was on the dark side of the rings, cannot be explained by blocking of the main rings by the F ring or vice versa and are probably instead due to the intrinsic longitudinal variation exhibited by the F ring.     

Nongravitational forces and meteoroid streams

The action of the solar electromagnetic radiation on the moving interplanetary dust particles in its more complete form than the special case known as the Poynting-Robertson effect is theoretically discussed in application to meteoroid stream of comet Encke.Normal and transversal components of the perturbing nongravitational force are used due to the action of the solar electromagnetic radiation. It is shown that the normal component of the force is negligible. However, transversal component is very important: it can probably completely explain all the observed meteoroid streams situated along the orbit of comet Encke (and, possibly, some asteroids) as the product of the comet Encke alone. Much shorter time is required for producing such a meteoroid stream than is a general conception.If the idea about the significance of the transversal component of the nongravitational force (may be, not produced by electromagnetic radiation) is correct, it may have important consequences for our understanding of ageing of comets, global evolution of the cometary (and, partially, asteroidal) system, and, of course, for a long-term evolution of small interplanetary particles.     

The volatile fraction of the cometary nucleus

In order to prepare a flyby mission to Comet Encke, six different sources of information on the possible chemical composition of the cometary nucleus are compared. These are: the neutral and charged radicals and molecules observed in cometary spectra; the chemical composition of type I carbonaceous chondrites; the meteor spectra; the metallic ions collected in the upper atmosphere and correlated with the meteor shower associated with Comet Encke; and finally the volatile molecules observed in a volatile-rich sample of lunar soil, that were interpreted as a possible cometary impact. Possible molecular abundances for the volatile fraction of Comet Encke are tentatively proposed.     

Rapid changes of brightness in P/Halley

Scanner observations of the coma of periodic comet Encke (P/Encke) are presented for four nights in March 1984 during its post-perihelion period. The strong emission features of CN and C₂ molecules have been identified and the abundances of CN and C₂ are estimated. The production rates of these molecules have also been derived from their band luminosities. No trace of sodium emission has been found in this comet.     

Steady state populations of Apollo-Amor objects

The steady-state distribution of orbits of Apollo-Amor objects is calculated for a variety of possible sources. These include asteroids near the inner edge of the belt, cometary orbits similar to Encke, and hypothetical extinct cometary orbits with perihelia larger than that of Encke. In all but one case, the steady-state distributions are similar for all these sources, and predict Amor/Apollo ratios of 1.5 to 3. These ratios are lower than those predicted by work in which the effects of the ν₆ secular resonance were not considered. These results are in general agreement with observation, although the higher (～3) Amor/Apollo ratios found for many of the sources may turn out to be unacceptably high. The absolute number of Apollo-Amors observed is found to require an injection rate of ～15 objects/(10⁶ years). This rate is easily achieved if the present existence of Encke is assumed to be a reasonably probable event, and if Encke becomes a ～1-km-diameter Apollo object following exhaustion of its volatile material; best estimates of the injection rate from the asteroid belt [～1.5/(10⁶ years)] are too low. Hence a dominant cometary component is suggested. The predicted number of Apollo objects in small (q < 1.0 AU, a < 1.4 AU orbits is in agreement with observation. Predicted lunar and terrestrial cratering rates agree approximately with observation. An unexplained difference between the lunar and terrestrial results is probably caused by uncertainties in the scaling laws or crater counts used. This discrepancy precludes an exact test of these calculations using cratering data.     

On the origin of the unusual orbit of Comet 2P/Encke

The orbit of Comet 2P/Encke is difficult to understand because it is decoupled from Jupiter—its aphelion distance is only 4.1 AU. We present a series of orbital integrations designed to determine whether the orbit of Comet 2P/Encke can simply be the result of gravitational interactions between Jupiter-family comets and the terrestrial planets. To accomplish this, we integrated the orbits of a large number of objects from the trans-neptunian region, through the realm of the giant planets, and into the inner Solar System. We find that at any one time, our model predicts that there should be roughly 12 objects in Encke-like orbits. However, it takes roughly 200 times longer to evolve onto an orbit like this than the typical cometary physical lifetime. Thus, we suggest that (i) 2P/Encke became dormant soon after it was kicked inward by Jupiter, (ii) it spent a significant amount of time inactive while rattling around the inner Solar System, and (iii) it only became active again as the ν₆ secular resonance drove down its perihelion distance.     

The ultraviolet spectrum of periodic comet encke (1980 XI)

A comparison of the ultraviolet spectrum of periodic Comet Encke (1980 XI), recorded by the IUE between 24 October and 5 November 1980 with similar spectra of short- and long-period comets shows the gaseous composition of P/Encke to be nearly identical to that of the other comets observed by the IUE. If P/Encke is indeed the remains of a once giant comet, this similarity implies a homogeneous radial structure for the cometary ice nucleus. The OH brightness distribution shows a spatial variation similar to the visible fan-shaped image of the comet, suggestive of a nonuniform distribution of volatile ices on the surface of the nucleus. The total derived water production rate appears to be a factor of 5 higher than that derived from HI Lyman-α observations made during the 1970 apparition and shows a variation with heliocentric distance (r) as r^?3.3 over the range 0.81 to 1.02 AU.     

A tale of two very different comets: ISO and MSX measurements of dust emission from 126P/IRAS (1996) and 2P/Encke (1997)

We present the characteristics of the dust comae of two comets, 126P/IRAS, a member of the Halley family (a near-isotropic comet), and 2P/Encke, an ecliptic comet. We have primarily used mid- and far-infrared data obtained by the ISOPHOT instrument aboard the Infrared Space Observatory (ISO) in 1996 and 1997, and mid-infrared data obtained by the SPIRIT III instrument aboard the Midcourse Space Experiment (MSX) in 1996. We find that the dust grains emitted by the two comets have markedly different thermal and physical properties. P/IRAS's dust grain size distribution appears to be similar to that of fellow family member 1P/Halley, with grains smaller than 5 microns dominating by surface area, whereas P/Encke emits a much higher fraction of big (20 μm and higher) grains, with the grain mass distribution being similar to that which is inferred for the interplanetary dust population. P/Encke's dearth of micron-scale grains accounts for its visible-wavelength classification as a “gassy” comet. These conclusions are based on analyses of both imaging and spectrophotometry of the two comets; this combination provides a powerful way to constrain cometary dust properties. Specifically, P/IRAS was observed preperihelion while 1.71 AU from the Sun, and seen to have a 15-arcmin long mid-infrared dust tail pointing in the antisolar direction. No sunward spike was seen despite the vantage point being nearly in the comet's orbital plane. The tail's total mass at the time was about 8×10⁹ kg. The spectral energy distribution (SED) is best fit by a modified greybody with temperature T=265±15 K and emissivity ε proportional to a steep power law in wavelength λ: ε∝λ^−α, where α=0.50±0.20(2σ). This temperature is elevated with respect to the expected equilibrium temperature for this heliocentric distance. The dust mass loss rate was between 150-600 kg/s (95% confidence), the dust-to-gas mass loss ratio was about 3.3, and the albedo of the dust was 0.15±0.03. Carbonaceous material is depleted in the comet's dust by a factor of 2-3, paralleling the C₂ depletion in P/IRAS's gas coma. P/Encke, on the other hand, observed while 1.17 AU from the Sun, had an SED that is best fit by a Planck function with T=270±15 K and no emissivity falloff. The dust mass loss rate was 70-280 kg/s (95% confidence), the dust-to-gas mass loss ratio was about 2.3, and the albedo of the dust was about 0.06±0.02. These conclusions are consistent with the strongly curved dust tail and bright dust trail seen by Reach et al. (2000; Icarus 148, 80) in their ISO 12-μm imaging of P/Encke. The observed differences in the P/IRAS and P/Encke dust are most likely due to the less evolved and insolated state of the P/IRAS nuclear surface. If the dust emission behavior of P/Encke is typical of other ecliptic comets, then comets are the major supplier of the interplanetary dust cloud.     

R- and J-band photometry of Comets 2P/Encke and 9P/Tempel 1

Near-simultaneous R- and J-band photometric measurements of the short-period Comets 2P/Encke and the Deep Impact mission target 9P/Tempel 1 were obtained. The resulting R-J colors are +0.82±0.08 mag and +1.46±0.13 mag for Encke and Tempel 1, respectively. Tempel 1's color is redder than the solar R-J color index of +0.76. The Tempel 1 observations directly detected the nucleus while the Encke observations likely suffered from coma contamination.     

Error propagation in the numerical solutions of the differential equations of orbital mechanics

The relationship between the eigen values of the linearized differential equations of orbital mechanics and the stability characteristics of numerical methods is presented. It is shown that the Cowell, Encke, and Encke formulation with an independent variable related to the eccentric anomaly all have a real positive eigen value when linearized about the initial conditions. The real positive eigen value causes an amplification of the error of the solution when used in conjunction with a numerical integration method. In contrast an element formulation has zero eigen values and is numerically stable.     

Infrared photometry of periodic comets Encke,Chernykh, Kearns-Kwee,Stephan-Oterma,and Tuttle

Nearly simultaneous photometry of the reflected and thermal infrared spectra of periodic comets Encke, Chernykh, Kearns-Kwee, Stephan-Oterma, and Tuttle are presented. The 10-μm, silicate emission feature has been detected for the first time in periodic comets and was observed in three of these objects. The albedo of the dust particles in the comae of these comets is calculted and compared to that of Comet Kohoutek. The peculiar behavior of the dust in Comet Encke is discussed.     

Studies in the application of recurrence relations to special perturbation methods

Using the rectangular equations of motion for the restricted three-body problem a comparison is made of the integration of these equations by the Encke method and by a set of perturbational equations. Each set of differential equations is integrated using Taylor series expansions where the coefficients of the powers of time are determined by recurrence relations. It is shown that for very small perturbations the use of the perturbational equations is more efficient than the use of the Encke method. A discussion is also given of when Cowell's method is more efficient than either of these techniques.     

Computation of Partial Derivatives of Perturbed Motion Using the Arcs of Osculating and Superosculating Orbits

The methods for analytical determination of partial derivatives of the current parameters of motion with respect to their initial values are described. The methods take into account principal perturbations and are based on the use of the osculating and superosculating intermediate orbits constructed earlier by the author. These orbits ensure the first-, second-, and third-order contact to the real trajectory at the initial time. The solution for parameters of the intermediate motion and partial derivatives of these parameters is given in a universal closed form. The partial derivatives on long time intervals are computed using a step-by-step procedure combined with the Encke method of special perturbations, in which the intermediate orbits are used as the reference. The numerical results show that the new approach can be efficiently used for solving the problem of differential correction of orbits of asteroids and comets on the basis of observational data.     

Expectations for Cassini observations of ring material with nearby moons

This paper describes N-body simulations of two regions of the saturnian ring system and examines what we might expect the Cassini orbiter to see in those areas. The first region is the edge of the Encke gap in the A ring that is perturbed by the satellite, Pan. Our previous simulations of this region neglected particle self-gravity [Lewis and Stewart, 2000a, Bull. Am. Astron. Soc. 34, 883]. Here we examine the interactions of the wakes caused by Pan with the wakes that form from local gravitational instabilities. We find that the two phenomena do not normally coexist and predict that measurements of particle sizes between the moon wakes should reflect the true particle size distribution of the region and not what is caused by gravitational aggregation. The region between the Encke gap edge and the first wake peak is an exception to this rule because our simulations exhibit the formation of exceptionally large gravity-induced wakes in this region. We also describe simulations of the F ring and explain the nature of braid-like structures that form naturally when the ring is perturbed by a single moon on an eccentric orbit. Finally, we discuss the very dynamic nature of the F ring system and how this should be taken into account when interpreting observations and even when planning future observations of this system.     

Cassini imaging of Saturn's rings: II. A wavelet technique for analysis of density waves and other radial structure in the rings

We describe a powerful signal processing method, the continuous wavelet transform, and use it to analyze radial structure in Cassini ISS images of Saturn's rings. Wavelet analysis locally separates signal components in frequency space, causing many structures to become evident that are difficult to observe with the naked eye. Density waves, generated at resonances with saturnian satellites orbiting outside (or within) the rings, are particularly amenable to such analysis. We identify a number of previously unobserved weak waves, and demonstrate the wavelet transform's ability to isolate multiple waves superimposed on top of one another. We also present two wave-like structures that we are unable to conclusively identify. In a multi-step semi-automated process, we recover four parameters from clearly observed weak spiral density waves: the local ring surface density, the local ring viscosity, the precise resonance location (useful for pointing images, and potentially for refining saturnian astrometry), and the wave amplitude (potentially providing new constraints upon the masses of the perturbing moons). Our derived surface densities have less scatter than previous measurements that were derived from stronger non-linear waves, and suggest a gentle linear increase in surface density from the inner to the mid-A Ring. We show that ring viscosity consistently increases from the Cassini Division outward to the Encke Gap. Meaningful upper limits on ring thickness can be placed on the Cassini Division (3.0 m at r∼118,800 km, 4.5 m at r∼120,700 km) and the inner A Ring (10-15 m for r<127,000 km).     

The structure of cometary atmospheres

The temperature distributions in cometary atmospheres at various heliocentric distances for comets of Bennett and Encke types have been calculated by taking into account heating due to the absorption of solar ultraviolet radiation, cooling by H₂O far infrared emission, and various dynamical processes (expansion, advection, and thermal conduction). The agreement of the results with the observations is in general satisfactory. The conversion of CH₄ and NH₃ to CO and N₂ through thermochemical reaction with H₂O is concluded to be impossible, since the temperature is too low at a heliocentric distance 1.5 AU where CO⁺ ions begin to be observable.     

Implicit single-sequence methods for integrating orbits

The solutions of \(\ddot x = F(x,t)\) , and also \(\dot x = F(x,t)\) , are developed in truncated series in timet whose coefficients are found empirically. The series ending in thet ⁶ term yields a position at a final prechosen time that is accurate through 9th order in the sequence size. This is achieved by using Gauss-Radau and Gauss-Lobatto spacings for the several substeps within each sequence. This timeseries method is the same in principle as implicit Runge-Kutta forms, including some not described previously. In some orders these methods are unconditionally stable (A-stable). In the time-series formulation the implicit system converges rapidly. For integrating a test orbit the method is found to be about twice as fast as high-order explicit Runge-Kutta-Nyström-Fehlberg methods at the same accuracies. Both the Cowell and the Encke equations are solved for the test orbit, the latter being 35% faster. It is shown that the Encke equations are particularly well-adapted to treating close encounters when used with a single-sequence integrator (such as this one) provided that the reference orbit is re-initialized at the start of each sequence. This use of Encke equations is compared with the use of regularized Cowell equations.     

Multiband photometry of Comets Kohoutek,Bennett, Bradfield,and Encke

Observations of Comets Kohoutek (1973f), Bradfield (1974b), and P/Encke have been made at a number of wavelengths between 0.55 and 18 μm. The silicate feature first observed in Comet Bennett (1969i) seems to be a common characteristic of cometary material. The comas of these comets radiate infrared with an effective temperature higher than the black-body temperature at the given distance from the Sun. The albedo of the dust particles is between 0.10 and 0.20. The particles in the coma and tail are small (diameter less than 2 μm), but the particles in the anti-tail of Comet Kohoutek must be larger than about 10 μm diameter. The observations give an absolute upper limit to the diameter of Comet Kohoutek of 30 km. A consistent interpretation would indicate that Comets Kohoutek and Bradfield have nuclear diameters of 5 to 10km, that Bennett was several times larger, and that P/Encke is 10 times smaller. The peculiar behavior of Bradfield showed that the coma of a single comet can abruptly change its dust composition.