Saturn's rings: Particle composition and size distribution as constrained by microwave observations: I. Radar observations

We have calculated the radar backscattering characteristics of a variety of compositional and structural models of Saturn's rings and compared them with observations of the absolute value, wavelength dependence, and degree of depolarization of the rings' radar cross section (reflectivity). In the treatment of particles of size comparable to the wavelength of observation, allowance is made for the nonspherical shape of the particles by use of a new semiempirical theory based on laboratory experiments and simple physical principles to describe the particles' single scattering behavior. The doubling method is used to calculate reflectivities for systems that are many particles thick using optical depths derived from observations at visible wavelengths. If the rings are many particles thick, irregular centimeter- to meter-sized particles composed primarily of water ice attain sufficiently high albedos and scattering efficiencies to explain the radar observations. In that case, the wavelength independence of radar reflectivity implies the existence of a broad particle size distribution that is well characterized over the range 1 cm ? r ? m by n(r)dr = n₀r^?3dr. A narrower size distribution with $\text{a}\text{～ 6}\text{cm}$ is also a possibility. Particles of primarily silicate composition are ruled out by the radar observations. Purely metallic particles, either in the above size range and distributed within a many-particle-thick layer or very much larger in size and restricted to a monolayer, may not be ruled out on the basis of existing radar observations. A monolayer of very large ice “particle” that exhibit multiple internal scattering may not yet be ruled out. Observations of the variation of radar reflectivity with the opening angle of the rings will permit further discrimination between ring models that are many particles thick and ring models that are one “particle” thick.     

Theoretical twin meteor showers

Attention is drawn to, and theoretical radiants calculated for, 27 hypothetical twin showers to Cook's (1973, in Evolutionary and Physical Properties of Meteoroids, NASA SP-319, U.S. Govt. Printing Office, Washington, D.C.) working list of meteor streams. Of these twin showers, 7 are previously known, 1 is a night-time twin to the ? Geminids, 9 are found to be contained among Sekanina's (1976, Icarus27, 265–321) radar streams, and the remaining 10 are undetected daytime showers. Minimum radii are computed for all of Cook's streams and are used to assess the possibility of detecting the twin showers.     

A study of geomagnetic storms at Beijing in terms of solar wind

With the aid of the Akasofu's energy coupling function between the solar wind and the magnetosphere, we have made in this paper an analysis of about 20 geomagnetic storms recorded at Beijing during the period of years 1966 to 1972. There is a close correlation between the energy coupling function ? and the geomagnetic indices a_p and K_p. All in all an empirical formula as ? ～ 1?2 × 10¹⁷a_p has been found for the geomagnetic storms occurred in a low latitude station, i.e. Beijing of China. Comparisons of the horizontal component H_max (in γ) and ?(10¹⁸ erg s^?1) in Table 1 indicate that the development of storm main phase at Beijing depends very much on the ? values thus involved. Also, these are well illustrated for several individual storms as mentioned in the second section of the paper. In concluding this paper some brief discussions are made and included. It is hoped that geomagnetic observations in the middle and low latitudes from our vast country should make further contributions to the study of solar wind-magnetosphere coupling, including the Akasofu's energy coupling function.     

Convex profiles from asteroid lightcurves

Integral geometry is used to solve a two-dimensional simplification of the three-dimensional lightcurve inversion problem, and a method is introduced for obtaining a convex profile, $\text{P}$, from asteroid lightcurve data. Whenever four ideal conditions are satisfied, $\text{P}$ is an estimator for the asteroid's “mean cross section,” $\text{C}$, a convex set defined as the average of all cross sections $\text{C}$ cut by planes a distance z above the asteroid's equatorial plane. $\text{C}$ is therefore a two-dimensional average of the asteroid's three-dimensional shape. The ideal conditions are that (A) each curve $\text{C}\text{(z)}$ is convex. (B) the asteroid's scattering law is uniform and geometric, (C) the astrocentric declinations of the Sun and Earth are zero, and (D) the solar phase angle θ ≠ 0. If Condition C is known to hold, the extend to which the lightcurve can be accounted for by a geometrically scattering convex object can be quantified in terms of an appropriate “goodness-of-fit” static. If the solar phase angle is zero, as for radar “lightcurve,” then (i) method yields a profile $\text{P}{}_{\mathrm{<mtext>s</mtext>}}$ the symmetrization $\text{C}{}_{\mathrm{<mtext>s</mtext>}}$ of $\text{C}$; (ii) Condition A need not hold and if it does not, then the inversion yields the symmetrization of the asteroid's mean convex hull; and (iii) Fourier analysis of the lightcurve can reveal violation of Condition B. Doppler-frequency resolution of radar echoes at several rotational phases adds information by constraining the convex hull $\text{H}{}_{\mathrm{<mtext>p</mtext>}}$ of the asteroid's (not necessarily convex) polar silhouette. Estimation of a convex profile from a photoelectric or radar lightcurve is a problem in weighted-least-squares optimization subject to inequality constraints. The solution uses a recursive quadratic programming algorithm to derive a Fourier parameterization for $\text{P}$ from the coefficients in the lightcurve's Fourier expansion. The method has been tested by inverting analytically generated lightcurves for geometrically scattering ellipsoids with semiaxes a ? b ? c, and the inversion yields $\text{P}\text{=}\text{P}{}_{\mathrm{<mtext>s</mtext>}}\text{?}\text{C}\text{=}\text{C}{}_{\mathrm{<mtext>s</mtext>}}\text{=}\text{H}{}_{\mathrm{<mtext>p</mtext>}}$ when the viewing geometry (Condition C) is close to ideal. For situations when the asteroid's pole direction is unknown, a test is offered of the hyphothesis that a given lightcurve can be due to a geometrically scattering ellipsoid with $\text{a}\text{c}\text{? ?}$, where ? is an priori upper bound on the maximum axis ratio. Convex profiles are presented for 15 Eunomia, 118 Peitho, 246 Asporina, 281 Lucretia. 790 Pretoria, 1685 Toro, and 1978 CA.     

Saturn's rings: Particle size distributions for thin layer models

The small physical thickness of Saturn's rings requires that radio occultation observations be interpreted using scattering models with limited amounts of multiple scatter. A new model in which the possible order of near-forward scatter is strictly limited allows for the small physical thickness, and can be used to relate Voyager 1 observations of 3.6-and 13-cm wavelength microwave scatter from Saturn's rings to the ring particle size distribution function n(a), for particles with radius 0.001 ≤ a ≤ 20 m. This limited-scatter model yields solutions for particle size distribution functions for eight regions in Saturn's rings, which exhibit approximately inverse-cubic power-law behavior, with large-size cutoffs in particle radius ranging from about 5 m in ring C to about 10 m in parts of ring A. The power-law index is about 3.1 in ring C, about 2.8 in the Cassini division, and increases systematically with radial location in ring A from 2.7 at 2.10R_s to slightly more than 3.0 at 2.24R_s. Corresponding mass densities are 32–43 kg/m² in ring C, 188 kg/m² in the Cassini division, and 244–344 kg/m² in ring A, under the assumption that the material density of the particles is 0.9 g/cm³. These values are a factor of 1 to 2 lower than first-order mass loading estimates derived from resonance phenomena. In view of the uncertainties in the measurements and in the linear density wave model, and the strong arguments for icy particles with specific gravity not greater than about 1, we interpret this discrepancy as being indicative of possible differences in the regions studied, or systematic errors in the interpretation of the scattering results, the density wave phenomena, or some combination of the above.     

A general cratering-history model and its implications for the lunar highlands

Through analysis of a large number of Monte Carlo and Markov Chain simulations, a model for determining crater accumulation and crater obliteration histories has been derived. The model generally applies to populations of large craters. It predicts that the following relationships hold for subequilibrium-density crater populations: (1) the more negative the production function's exponent, α, (N～D^α) the lower the crater density at which the population size-frequency distribution will significantly depart from its production function; (2) the more negative the production function's exponent, the less obliteration a crater population will sustain after a set number of impacts. Application of the model to the lunar highlands implies (1) the production function for the large craters is highly structured, resembling the observed size-frequency distribution and not the function N～D^?2; (2) even the densely cratered highlands have not attained crater saturation or equilibrium. Direct simulations of the highlands' crater population supports the model's implications.     

Radar and photoelectric observations of asteroid 2100 Ra-Shalom

Results of 13-cm-wavelength radar observations and V-filter photoelectric observations of Ra- Shalom during its 1981 Aug–Sep apparition are reported. The radar data yid detections of echoes in the same sense of circular polarization as transmitted (i.e., the SC sense) as well as in the opposite (OC) sense. The estimate of the ratio of SC to OC echo power, μ_c = 0.14 ± 0.02, indicates that most, but certainly not all, of the backscattering is due to single reflections from surface elements that are fairly smooth at decimeter scales. The value obtained for the OC radar cross section on Aug 26 (1.2 ± 0.3 km²) is about three times larger than those obtained on Aug 23, 24, and 25. The echo bandwidth appears to be within about 1.5 Hz of 5.0 Hz on each date. The photoelectric data suggest a value, P_syn = 19.79 hr, for the synodic rotation period, and yield a composite lightcurve with two pairs of extrema. Combining this value for P_syn with a firm lower bound (4 Hz) on the maximum echo bandwidth yields a lower bound of 1.4 km on the maximum distance between Ra-Shalom's spin axis and any point on its surface.     

On the origin and initial temperature of Jupiter and Saturn

A two-stage growth of the giant planets, Jupiter and Saturn, is considered, which is different from the model of contraction of large gaseous protoplanets. In the first stage, within a time of ～3 × 10⁷ years in Jupiter's zone and ～2 × 10⁸ years in Saturn's zone, a nucleus forms from condensed (solid) material having the mass, ～10²⁸ g, necessary for the beginning of acceleration. The second stage may gravitating body, and a relatively slow accretion begins until the mass of the planet reaches ～10 m_⊕. Then a rapid accretion begins with the critical radius less than the radius of the Hill lobe, so that the classical formulae for the rate of accretion may be applied. At a mass m > m₁ ≈ 50 m_⊕ accretion proceeds slower than it would according to these formulae. When the planet sweeps out all the gas from its nearest zone of feeding (m = m₂ ≈ 130 m_⊕), the width of the exhausted zone being $\text{～}\text{built1}\text{3}$ of the whole zone of the planet) growth is provided the slow diffusion of gas from the rest of the zone (time scale increases to 10⁵?10⁶ years and more). The process is terminated by the dissipation of the remnants of gas. In Saturn's zone m₁ > m₂ ≈ 30 m_⊕. The initial mass of the gas in Jupiter's zone is estimated. Before the beginning of the rapid accretion about 90% of the gas should have been lost from the solar system, and in the planet's zone less than two Jupiter masses remain. The highest temperature of Jupiter's surface, ≈5000°K, is reached at the stage of rapid accretion, m < 100 m_⊕, when the luminosity of the planet reaches 3 × 10^?3 L_⊙. This favors an effective heating of the inner parts of the accretionary disk and the dissipation of gas from the disk. The accretion of Saturn produced a temperature rise up to 2000?2400° K (at m ≈ 20?25 m_⊕) and a luminosity up to 10^?4 L_⊙.     

Tidal rigidity of phobos

The Martian satellites Phobos and Deimos move along nearly circular coplanar, stable orbits and have created surfaces older than ～ 10⁹ years. The accretion hypothesis suggests that their primordial orbits were also very regular. However, tides raised on Mars and Phobos can substantially alter the semimajor axis a of Phobos' orbit over time. The effect of the Martian tidal torque alone on Phobos' orbit implies that the primordial e was ～0.1 to 0.2 about 4.6 × 10⁹ years ago if the present observed e = 0.015 is naively interpreted as a tidally damped remnant. Significant tidal friction in Phobos reduces the time scale for Phobos to achieve a crossing orbit with Deimos to less than 10⁹ years and permits the primodial e to approach unity. The consequences of orbital intersections cannot easily be resolved by assuming either a catastrophic origin for both satellites (namely, that both are fragments of a common parent body fractured by an impact) or that they were captured sequentially by Mars. Either hypothesis is difficult to accept, given that Deimos' orbit, which is only slightly affected by tides, is now so regular. An alternative scenario is proposed in this paper in which the observed e of Phobos results from several gravitational resonance excitations within the last 10⁹ years, assuming tidal friction in Phobos has had only a small effect on its orbit. In facr, both the primordial e and the inclination i may have been much smaller than presently observed. The constraints imposed on tidal friction in Phobos by both the apparent age of Phobos' surface (> 10⁹yrs) and the above scenario can be satisfied only of μQ > 10¹²dynes/cm². Since the Q factor is ～10², the rigidity μ > 10¹⁰dynes/cm². Thus Phobos should have substantial internal strength.     

Delay-Doppler radar observations of Saturn's rings

The relative radar reflectiveness of Saturn's classical ring sections were estimated from delay-Doppler observations made at 12.6-cm wavelength. The A and B rings are responsible for most, if not all, of the radar echo. The average radar reflectivity per unit projected area of the A ring is nearly (～90%) as large as that of the B ring. The outer half of the B ring contains the most reflective part of the ring system. There is no firm evidence for detection of radar backscattering from particles interior to the B ring, exterior to the A ring, or from the planet itself. The radar reflectivity of the C ring is certainly no more than one-half that of the B ring, and probably is much less. Unexpectedly large amounts of power at Doppler shifts near the center of the echo spectrum, previously reported at both λ3.5 and λ12.6 cm for ring-plane tilt angles δ ≥ 24.4°, are not apparent in λ3.5- or 12.6-cm spectra obtained at δ ≤ 21.4°.     

Spectrophotometry of Saturn and its rings from 60 to 180 microns

We observed Saturn at far-infrared and submillimeter wavelengths during the Earth's March 1980 passage through the plane of Saturn's rings. Comparison with earlier spectroscopic observations by D. B. Ward [Icarus32, 437–442 (1977)], obtained at a time when the tilt angle of the rings was 21.8°, permits separation of the disk and ring contributions to the flux observed in this wavelength range. We present two main results: (1) The observed emission of the disk between 60 and 180 μm corresponds to a brightness temperature of 104 ± 2°K; (2) the brightness temperature of the rings drops approximately 20°K between 60 and 80 μm. Our data, in conjunction with the data obtained by other observers between 1 μm and 1 mm, permit us to derive an improved estimate for the total Saturnian surface brightness of (4.84 ± 0.32) × 10^?4W cm^?2 corresponding to an effective temperature of 96.1 ± 1.6°K. The ratio of radiated to incident power, P_R/P_I, is (1.46 ± 0.08)/(1 - A), where A is the Bond albedo. For A = 0.337 ± 0.029, P_R/P_I = 2.20 ± 0.15 and Saturn's intrinsic luminosity is L_S = (2.9 ± 0.5) × 10^?10L_⊙.     

Germane in the atmosphere of Jupiter

High-altitude spectra of Jupiter obtained from the Kuiper Airborne Observatory are analyzed for the presence of germane (GeH₄) in Jupiter's atmosphere. Comparison with laboratory spectra shows that the strong Q branch of the ν₃ band of germane at 2111 cm^?1 is prominent in the Jovian spectra. The abundance of germane in Jupiter's atmosphere is 0.006 (±0.003) cm-am corresponding to a mixing ratio of 0.6 ppb. This trace amount of germane is consistent with chemical equilibrium calculations if the germane present at ～1000°K is carried up by convection to the spectroscopically observable region at ～300°K.     

A theoretical model of the bow shock and the magnetosphere of Jupiter

Using the data obtained from the Pioneer 10 and 11 observations, a theoretical model is proposed for the bow shock and the magentosphere of Jupiter. This indicates that the distance of the magnetopause from Jupiter on the sunlit side is (50–55) × r_J (r_J: Jupiter radius, = 7 × 10⁹ cm) and that the ratio of the stand-off distance to this distance is about equal to or slightly larger than unity. Hence the Mach number of the solar wind seems to be less than 1.5 at Jupiter's orbit. This result necessarily leads to a blunt body model of the Jovian magnetosphere, the tail region of which is not as extended as observed in the Earth's case.     

Axel dust on Saturn and Titan

The scattering and absorption properties of Axel dust were investigated by means of Mie theory. We find that a flat distribution of particle radii between 0 and 0.1 μm, and an imaginary part of the index of refraction which varies as λ^?2.5 produce a good fit to the variation of Titan's geometric albedo with wavelength (λ) provided that τ_ext, the extinction optical depth of Titan's atmosphere at 5000 Å, is about 10. The real part of the complex index is taken to be 2.0. The model assumes that the mixing ratio of Axel dust to gas is uniform above the surface of Titan. The same set of physical properties for Axel dust also produces a good fit to Saturn's albedo if τ_ext = 0.7 at 5000 Å. To match the increase in albedo shortward of 3500 Å, a clear layer (containing about 7 km-am H₂) is required above the Axel dust. Such a layer is also required to explain the limb brightening in the ultraviolet. These models can be used to analyze the observed equivalent widths of the visible methane bands. The analysis yields an abundance of the order of 1000 m-am CH₄ in Titan's atmosphere. The derived CH₄/H₂ mixing ratio for Saturn is about 3.5 × 10^?3 or an enhancement of about 5 over the solar ratio.     

Ionospheric currents obtained from the Chatanika radar and ground magnetic perturbations at the Auroral latitude

The relationship between substorm ionospheric currents and the corresponding ground magnetic perturbations is examined, by using the height-integrated ionospheric current density deduced from the Chatanika incoherent scatter radar and the simultaneous magnetic variations along the Alaska meridian chain of stations. Although time variations of the H component near the radar site on the Earth's surface are in good agreement with those of the east-west ionospheric current, there is a substantial disagreement between the current deduced from the D perturbations and the observed north-south current in the evening sector. It is shown that the disagreement can be removed by introducing a new finding by Yasuhara et al. (1975) that the upward field-aligned current on the poleward side of the auroral oval in the evening sector is more intense than its counterpart fieldaligned current and that it contributes greatly to the ground D perturbations.     

Models,figures, and gravitational moments of the Galilean satellites of Jupiter and icy satellites of Saturn

A general theory for the figures of satellites, which are synchronously rotating in the gravitational field of a planet, is developed to the first approximation. Love numbers, figure parameters, and gravitational moments for two- and three-layer models of the Galilean satellites, Titan, and Saturn's icy satellites are calculated. With the assumed accuracy for flyby measurements of gravitational moments it should be possible to determine the degree of differentiation of Ganymede. The differences between equatorial a and polar c semiaxes, as derived from the observational data, appear to be exaggerated for Io and Mimas (although better agreement between calculated and observed values of (a?c) could be obtained if this satellite had a larger mass). For Enceladus the observed value of (a?c) is in satisfactory agreement with calculations, based on different types of trial models. However, in order to discriminate between different Enceladus trial models, it is necessary to determine the figure parameters more precisely.     

A physical model of Titan's clouds

We have constructed a model of the physical processes controlling Titan's clouds. Our model produces clouds that qualitatively match the present observational constraints in a wide variety of model atmospheres, including those with low atmospheric pressures (25 mbar) and high atmospheric pressures. We find the following: (1) high atmospheric temperatures (160°K) are important so that there is a large scale height in the first few optical depths of cloud; (2) the aerosol mass production occurs at very low aerosol optical depth so that the cloud particles do not directly affect the photochemistry producing them; (3) the production rate of aerosol mass by chemical processes is probably greater than 3.5 × 10^?14 g cm^?2 sec^?1; (4) and the eddy diffusion coefficient is less than 5 × 10⁶ cm² sec^?1 except perhaps in the top optical depth of the cloud. Our model is not extremely sensitive to particle shape, but it is sensitive to particle density. Higher particle densities require larger aerosol mass production rates to produce satisfactory clouds. Particle densities of unity require a mass production rate on the order of 3.5 × 10^?13 g cm^?2 sec^?1. We also show that an increase in mass input causes a decrease in the mean particle size, as required by J. B. Pollack et al. (1980, Geophys. Res. Lett. 7, 829–832), to explain the observed correlation between the solar cycle and Titan's albedo; that coagulation need not be extremely inefficient in order to obtain realistic clouds as proposed by M. Podolak and E. Podolak (1980, Icarus43, 73–83); that coagulation could be inefficient due to photoelectric charging of the particles; and, that the lifetime of particles near the altitude of unit optical depth is a few months, as required to explain the temporal variability observed by S. T. Suess and G. W. Lockwood and D. P. Cruikshank and J. S. Morgan (1979, Bull. Amer. Astron. Soc.11, 564). Although Titan's aerosols are ottically thick in the vertical direction, the atmosphere is so extended that the horizontal visibility is greater than that found anywhere at Earth's surface.     

Turbulent viscosity and Jupiter's tidal Q

A recent estimate of tidal dissipation by turbulent viscosity in Jupiter's convective interior predicts that the current value of the planet's tidal Q ～ 5 × 10⁶. We point out a fundamental error in this calculation, and show that turbulent dissipation alone implies that at present Q ～ 5 × 10¹³. Our reduced estimat for the rate of tidal dissipation shows conclusively that tidal torques have produced only negligible modifications of the orbits of the Galilean satellites over the age of the solar system.     

Does Io have an ammonia atmosphere?

An atmosphere containing 0.5 cm atm of ammonia is assumed on Io. Such an atmosphere will be frozen at the unilluminated pole during the solstices, but will evaporate at the equinoctial seasons. The ammonia atmosphere will explain: (1) the posteclipse brightenings and their observed times of occurrence and nonocurrence; (2) the observed departure from a two-layer model beating curve upon emergence from eclipse; (3) the discordant temperatures obtained at 10 and 20 μm; and (4) discordant temperatures obtained at 10 and 20 μm during the total phase of an eclipse by Jupiter.In order to explain items 3 and 4 above, a proton flux in Jupiter's magnetosphere of 1.1 × 10⁹ cm^?2s^? at an energy of 0.5MeV at io's distance from Jupiter is assumed. This flux is 40 times the flux in Divine's (1972) “upper-limit” model of the Jovian radiation belts, while the proton energy is eight times less. The proton flux, plus the solar ultraviolet and infrared flux absorbed by the ammonia, will heat the atmosphere to 245 ± 10°K. At this temperature the occultation atmospheric upper limit allows the addition of 4 cmatm of nitrogen.     

Saturn's rings: II. Condensations of light and optical thickness of Cassini's division

“Condensations” of light have been observed when Saturn's rings are seen almost edge on, and the Sun and the Earth are on opposite sides of the ring plane. These condensations are associated with ring C and Cassini's division. If the relative brightness between the two condensations and the optical thickness of ring C are known, we can calculate the optical thickness of Cassini's division, τ_CASS. Using Barnard's and Sekiguchi's measurements, we have obtained 0.01 ? τ_CASS ? 0.05. A brightness profile of the condensations which agrees well with visual observations is also presented.We are able to set an upper limit of about 0.01 for the optical thickness of any hypothetical outer ring. This rules out a ring observed by C. Cragg in 1954, but does not eliminate the D′ ring observed by Feibelman in 1967.It is known that the outer edge of ring B is almost at the position of the 1/2 resonance with Mimas. Franklin, Colombo, and Cook explained this fact in 1971, postulating a total mass of ring B of 10^?6M_SATURN. We have derived a formula for the mass of the rings, which is a linear function of the mean particle size. We find that 10^?6M_SATURN implies large particles (～70m). If the particles are small (～10cm), as currently believed, the total mass of ring B is not enough to shift the outer edge. We conclude that the above explanation and current size estimates are inconsistent.