Results for “Mixture modeling” — Page 32

NASA NTRS1993

The Fe/Mn constraint on precursors of basaltic achondrites

Most achondritic meteorites have Fe/Mn ratios that are lower than those of carbonaceous chondrites and of course are lower than the solar system abundance ratio of these elements. Models of the origin of achondritic assemblages must, therefore, account for these ratios. Fe/Mn ratios are suggested to be distinctive for samples from each achondrite parent body and for the Earth and Moon, but the correspondence between the Fe/Mn systematics of achondrites and chondritic precursors is unclear. Most models of achondrite genesis involve magmatic differentiation of chondritic precursors. The Fe/Mn difference between achondrites and chondrites is particularly significant since Fe and Mn are geochemically similar elements with similar partitioning behavior in familiar magmatic systems and are generally coupled during crystal-liquid fractionation. In contrast, however, Mn is more volatile than Fe in a nebular setting. Variation of Fe/Mn ratios based on the relative volatility of these elements in the early nebula provides a constraint for models by which the basaltic achondrites (with Fe/Mn ratios approximately = 25-50) are derived from mixtures of nebular components that were enriched in volatile components such as Mn. However, such volatile enriched components have not been identified in chondrites. When the abundance in achondrites of elements of similar volatility is examined, anomalies appear. For example, Na is massively depleted in basaltic achondrites when compared to Mn. These anomalies might be explained using current models but the alternative hypothesis, that Fe/Mn ratio is controlled not by nebular volatility constraints, but by planetary differentiation should be explored.

Delaney, Jeremy S.↗

NASA NTRS1993

Computer modeling of pulsed CO2 lasers for lidar applications

The object of this effort is to develop code to enable the accurate prediction of the performance of pulsed transversely excited (TE) CO2 lasers prior to their construction. This is of particular benefit to the NASA Laser Atmospheric Wind Sounder (LAWS) project. A benefit of the completed code is that although developed specifically for the pulsed CO2 laser much of the code can be modified to model other laser systems of interest to the lidar community. A Boltzmann equation solver has been developed which enables the electron excitation rates for the vibrational levels of CO2 and N2, together with the electron ionization and attachment coefficients to be determined for any CO2 laser gas mixture consisting of a combination of CO2, N2, CO, He and CO. The validity of the model has been verified by comparison with published material. The results from the Boltzmann equation solver have been used as input to the laser kinetics code which is currently under development. A numerical code to model the laser induced medium perturbation (LIMP) arising from the relaxation of the lower laser level has been developed and used to determine the effect of LIMP on the frequency spectrum of the LAWS laser output pulse. The enclosed figures show representative results for a laser operating at 0.5 atm. with a discharge cross-section of 4.5 cm to produce a 20 J pulse with aFWHM of 3.1 microns. The first four plots show the temporal evolution of the laser pulse power, energy evolution, LIMP frequency chirp and electric field magnitude. The electric field magnitude is taken by beating the calculated complex electric field and beating it with a local oscillator signal. The remaining two figures show the power spectrum and energy distribution in the pulse as a function of the varying pulse frequency. The LIMP theory has been compared with experimental data from the NOAA Windvan Lidar and has been found to be in good agreement.

Spiers, Gary D.↗

NASA NTRS1966

Establishment of concentration profile for the flow of a binary fluid mixture.

Concentration profile of heavy species for binary fluid mixture under body force from uniformly mixed upstream condition, considering inviscid hydrodynamical model

FLOW DISTRIBUTION↗

NASA NTRS1984

Flow of nitrogen-pressurized Halon 1301 in fire extinguishing systems

Halon 1301 which is a halocarbon fire extinguishing agent (CBrF3) used by the U.S. Army for vehicle fire suppression is discussed. Halon 1301 is discharged under nitrogen pressure, and the Halon-nitrogen mixture is a two phase, two component mixture that obeys compressible fluid laws and exhibits choking effects. A computer model was developed to analyze the discharge of Halon and nitrogen from a storage bottle through pipes and nozzles. The model agrees well with data from Halon 1301 discharge tests. The discharge time depends mainly on nozzle area and pipe volume, for given initial conditions. Graphs were developed for estimating discharge times. A nozzle employing multiple concentric converging/diverging nozzles was developed which gave hemispherical coverage.

Elliott, D. G.↗

NASA NTRS1985

A comparison of measured and calculated thermal stresses in a hybrid metal matrix composite spar cap element

A hybrid spar of titanium with an integrally brazed composite, consisting of an aluminum matrix reinforced with boron-carbide-coated fibers, was heated in an oven and the resulting thermal stresses were measured. Uniform heating of the spar in an oven resulted in thermal stresses arising from the effects of dissimilar materials and anisotropy of the metal matrix composite. Thermal stresses were calculated from a finite element structural model using anisotropic material properties deduced from constituent properties and rules of mixtures. Comparisons of calculated thermal stresses with measured thermal stresses on the spar are presented. It was shown that failure to account for anisotropy in the metal matrix composite elements would result in large errors in correlating measured and calculated thermal stresses. It was concluded that very strong material characterization efforts are required to predict accurate thermal stresses in anisotropic composite structures.

Jenkins, J. M.↗

NASA NTRS1985

Chemistry of the outer planets: Investigations of the chemical nature of the atmosphere of Titan

It is clear from the experiments that a variety of complex organic models can be produced by lightning in a Titan-like gas mixture. The dominant products were found to be acetylene and hydrogen cyanide, with smaller amounts of many other species. Any aerosol produced by lightning inititated process will consist of a complex mixture of organic compounds, many of which should easily be identified by pyrolytic gas chromatography. Work will continue to expand the data base of molecules produced by lightning and other processes in order to assist in the design of appropriate analytical instruments for the upcoming Saturn/Titan mission and any other planetary probes.

Scattergood, Thomas W.↗

NASA NTRS1990

Shock temperature measurements of planetary ices - NH3, CH4, and 'synthetic Uranus'

Shock temperature measurements have been performed on several materials which have relevance to the modeling of the outer planets. These materials are methane, ammonia and a mixture of water, ammonia, and isopropanol known as synthetic Uranus. Temperatures have been measured in these materials over the pressure range 33-76 GPa for which there also exists measurements of equation of state and electrical conductivity. The temperatures are found to agree well with available calculations, with small discrepancies between data and theory ascribed to energy absorbing processes such as dissociation and molecular ionization.

Radousky, H. B.↗

NASA NTRS1992

Reactant conversion in homogeneous turbulence: Mathematical modeling, computational validations and practical applications

Closed form analytical expressions are obtained for predicting the limited rate of reactant conversion in a binary reaction of the type F + rO yields (1 + r) Product in unpremixed homogeneous turbulence. These relations are obtained by means of a single point Probability Density Function (PDF) method based on the Amplitude Mapping Closure. It is demonstrated that with this model, the maximum rate of the reactants' decay can be conveniently expressed in terms of definite integrals of the Parabolic Cylinder Functions. For the cases with complete initial segregation, it is shown that the results agree very closely with those predicted by employing a Beta density of the first kind for an appropriately defined Shvab-Zeldovich scalar variable. With this assumption, the final results can also be expressed in terms of closed form analytical expressions which are based on the Incomplete Beta Functions. With both models, the dependence of the results on the stoichiometric coefficient and the equivalence ratio can be expressed in an explicit manner. For a stoichiometric mixture, the analytical results simplify significantly. In the mapping closure, these results are expressed in terms of simple trigonometric functions. For the Beta density model, they are in the form of Gamma Functions. In all the cases considered, the results are shown to agree well with data generated by Direct Numerical Simulations (DNS). Due to the simplicity of these expressions and because of nice mathematical features of the Parabolic Cylinder and the Incomplete Beta Functions, these models are recommended for estimating the limiting rate of reactant conversion in homogeneous reacting flows. These results also provide useful insights in assessing the extent of validity of turbulence closures in the modeling of unpremixed reacting flows. Some discussions are provided on the extension of the model for treating more complicated reacting systems including realistic kinetics schemes and multi-scalar mixing with finite rate chemical reactions in more complex configurations.

Madnia, C. K.↗

NASA NTRS1996

Thermocapillary Motion in an Emulsion

The phenomenological model for the motion of an emulsion or a gas-liquid mixture exposed to thermocapillary forces and micro-acceleration is formulated. The analytical and numerical investigation of one-dimensional flows for these media is fulfilled, the structure of discontinuous motion is studied. The stability conditions of a space-uniform state and of the interface between an emulsion and a pure liquid are obtained.

Pukhnachov, Vladislav V.↗

NASA NTRS2005

Interstellar PAH Analogs in the Laboratory: Comparison with Astronomical Data

Polycyclic Aromatic Hydrocarbons (PAHs) are an important and ubiquitous component of carbon-bearing materials in space. PAHs are the best-known candidates to account for the IR emission bands (UIR bands) and PAH spectral features are now being used as new probes of the ISM. PAHs are also thought to be among the carriers of the diffuse interstellar absorption bands (DIBs). In the model dealing with the interstellar spectral features, PAHs are present as a mixture of radicals, ions and neutral species. PAH ionization states reflect the ionization balance of the medium while PAH size, composition, and structure reflect the energetic and chemical history of the medium. A major challenge for laboratory astrophysics is to reproduce (in a realistic way) the physical conditions that exist in the emission and/or absorption interstellar zones. An extensive laboratory program has been developed at NASA Ames to assess the physical and chemical properties of PAHs in such environments and to describe how they influence the radiation and energy balance in space and the interstellar chemistry. In particular, laboratory experiments provide measurements of the spectral characteristics of interstellar PAH analogs from the ultraviolet and visible range to the infrared range for comparison with astronomical data. This paper will focus on the recent progress made in the laboratory to measure the direct absorption spectra of neutral and ionized PAHs in the near-UV and visible range. Intrinsic band profiles and band positions of cold gas-phase PAHs can now be measured with high-sensitivity spectroscopy and directly compared to the astronomical data. Preliminary conclusions from the comparison of the laboratory data with astronomical observations will also be presented.

Salama, Farid↗

NASA NTRS2005

Interstellar PAH in the Laboratory and in Space. What have we Learned from the New Generation of Laboratory and Observational Studies?

Polycyclic Aromatic Hydrocarbons (PAHs) are an important and ubiquitous component of carbon-bearing materials in space. PAHs are the best-known candidates to account for the IR emission bands (UIR bands) and PAH spectral features are now being used as new probes of the ISM. PAHs are also thought to be among the carriers of the diffuse interstellar absorption bands (DIBs). In the model dealing with the interstellar spectral features, PAHs are present as a mixture of radicals, ions and neutral species. PAH ionization states reflect the ionization balance of the medium while PAH size, composition, and structure reflect the energetic and chemical history of the medium. A major challenge for laboratory astrophysics is to reproduce (in a realistic way) the physical conditions that exist in the emission and/or absorption interstellar zones. An extensive laboratory program has been developed at NASA Ames to assess the physical and chemical properties of PAHs in such environments and to describe how they influence the radiation and energy balance in space and the interstellar chemistry. In particular, laboratory experiments provide measurements of the spectral characteristics of interstellar PAH analogs from the ultraviolet and visible range to the infrared range for comparison with astronomical data. This paper will focus on the recent progress made in the laboratory to measure the direct absorption spectra of neutral and ionized PAHs in the gas phase in the near-UV and visible range in astrophysically relevant environments. These measurements provide data on PAHs and nanometer-sized particles that can now be directly compared to astronomical observations. The harsh physical conditions of the IS medium - characterized by a low temperature, an absence of collisions and strong VUV radiation fields - are simulated in the laboratory by associating a molecular beam with an ionizing discharge to generate a cold plasma expansion. PAH ions are formed from the neutral precursors in an isolated environment at low temperature (of the order of 100 K). The spectra of neutral and ionized PAHs are measured using the high sensitivity methods of cavity ring down spectroscopy (CRDS). These experiments provide unique information on the spectra of free, cold large carbon molecules and ions in the gas phase. Intrinsic band profiles and band positions of cold gas-phase PAHs can now be measured with high-sensitivity spectroscopy and directly compared to the astronomical data. The electronic bands measured for ionized PAH are found to be intrinsically broad (about 20 cm(sup -1)) while the bands associated with the neutral precursors are narrower (of the order of 2 - 10 cm(sup -1)). The laboratory data are discussed and compared with recent astronomical spectra of large and narrow DIBs and with the spectra of circumstellar environments of selected carbon stars and the implications for the interstellar PAH population are derived. Preliminary results also show that carbon nanoparticles are formed during the short residence time of the precursors in the plasma.

Salama, Farid↗

NASA NTRS2005

PAHs and the Diffuse Interstellar Bands. What have we Learned from the New Generation of Laboratory and Observational Studies?

Polycyclic Aromatic Hydrocarbons (PAHs) are an important and ubiquitous component of carbon-bearing materials in space. PAHs are the best-known candidates to account for the IR emission bands (UIR bands) and PAH spectral features are now being used as new probes of the ISM. PAHs are also thought to be among the carriers of the diffuse interstellar absorption bands (DIBs). In the model dealing with the interstellar spectral features, PAHs are present as a mixture of radicals, ions and neutral species. PAH ionization states reflect the ionization balance of the medium while PAH size, composition, and structure reflect the energetic and chemical history of the medium. A major challenge for laboratory astrophysics is to reproduce (in a realistic way) the physical conditions that exist in the emission and/or absorption interstellar zones, An extensive laboratory program has been developed at NASA Ames to characterize the physical and chemical properties of PAHs in astrophysical environments and to describe how they influence the radiation and energy balance in space and the interstellar chemistry. In particular, laboratory experiments provide measurements of the spectral characteristics of interstellar PAH analogs from the ultraviolet and visible range to the infrared range for comparison with astronomical data. This paper will focus on the recent progress made in the laboratory to measure the direct absorption spectra of neutral and ionized PAHs in the gas phase in the near-W and visible range in astrophysically relevant environments. These measurements provide data on PAHs and nanometer-sized particles that can now be directly compared to astronomical observations. The harsh physical conditions of the IS medium - characterized by a low temperature, an absence of collisions and strong V W radiation fields - are simulated in the laboratory by associating a molecular beam with an ionizing discharge to generate a cold plasma expansion. PAH ions are formed from the neutral precursors in an isolated environment at low temperature (of the order of 100 K). The spectra of neutral and ionized PAHs are measured using the high sensitivity methods of cavity ring down spectroscopy (CRDS). These experiments provide unique information on the spectra of free, cold large carbon molecules and ions in the gas phase. Intrinsic band profiles and band positions of cold gas-phase PAHs can now be measured with high- sensitivity spectroscopy and directly compared to the astronomical data. The electronic bands measured for ionized PAH are found to be intrinsically broad (about 20/cm) while the bands associated with the neutral precursors are narrower (of the order of 2 - 10/cm).

Salama, Farid↗

NASA NTRS2005

Spectroscopy of neutral and ionized PAHs. From laboratory studies to astronomical observations

Polycyclic Aromatic Hydrocarbons (PAHs) are an important and ubiquitous component of carbon-bearing materials in space. PAHs are the best-known candidates to account for the IR emission bands (UIR bands) and PAH spectral features are now being used as new probes of the ISM. PAHs are also thought to be among the carriers of the diffuse interstellar absorption bands (DIBs). In the model dealing with the interstellar spectral features, PAHs are present as a mixture of radicals, ions and neutral species. PAH ionization states reflect the ionization balance of the medium while PAH size, composition, and structure reflect the energetic and chemical history of the medium. A major challenge for laboratory astrochemistry is to reproduce (in a realistic way) the physical conditions that are associated with the emission and absorption interstellar zones. An extensive laboratory program has been developed at NASA Ames to assess the physical and chemical properties of PAHs in such environments and to describe how they influence the radiation and energy balance in space and the interstellar chemistry. PAHs, neutrals and ions, are expanded through a pulsed discharge nozzle (PDN) and probed with high-sensitivity cavity ringdown spectroscopy (CRDS). These laboratory experiments provide unique information on the spectra of free, cold large carbon molecules and ions in the gas phase from the ultraviolet and visible range to the near-infrared range. Intrinsic band profiles and band positions of cold gas-phase PAHs can now be measured with high-sensitivity spectroscopy and directly compared to the astronomical data. Preliminary conclusions from the comparison of the laboratory data with astronomical observations of interstellar and circumstellar environments will also be discussed.

Salama, Farid↗

NASA NTRS2006

Astrochemistry: Recent Advances in the Study of Carbon Molecules in Space

Carbon molecules and ions play an important role in space. Polycyclic Aromatic Hydrocarbons (PAHs) are the best-known candidates to account for the infrared emission bands (UIR bands) and PAH spectral features are now being used as probes of the interstellar medium in Galactic and extra-galactic environments. PAHs are also thought to be among the carriers of the diffuse interstellar absorption bands (DIBs). In the model dealing with the interstellar spectral features, PAHs are present as a mixture of radicals, ions and neutral species. PAH ionization states reflect the ionization balance of the medium while PAH size, composition, and structure reflect the energetic and chemical history of the medium. A major challenge for laboratory Astrochemistry is to reproduce (in a realistic way) the physical conditions that exist in the emission and absorption interstellar zones. An extensive laboratory program has been developed in various laboratories to characterize the physical and chemical properties of PAHs in astrophysical environments and to describe how they influence the radiation and energy balance in space and the interstellar chemistry. In particular, laboratory experiments provide measurements of the spectral characteristics of interstellar PAH analogs from the ultraviolet and visible range to the infrared range for comparison with astronomical data. The harsh physical conditions of the interstellar medium - characterized by a low temperature, an absence of collisions and strong ultraviolet radiation fields - are simulated in the laboratory by associating a molecular beam with an ionizing discharge to generate a cold plasma expansion. PAH ions are formed from the neutral precursors in an isolated environment at low temperature (of the order of 100 K). The spectra of neutral and ionized PAHs are measured using the high sensitivity methods of cavity ring down spectroscopy (CRDS). These experiments provide unique information on the spectra of free, cold large carbon molecules and ions in the gas phase.

Salama, Farid↗

NASA NTRS2011

Interstage Flammability Analysis Approach

The Interstage of the Ares I launch platform houses several key components which are on standby during First Stage operation: the Reaction Control System (ReCS), the Upper Stage (US) Thrust Vector Control (TVC) and the J-2X with the Main Propulsion System (MPS) propellant feed system. Therefore potentially dangerous leaks of propellants could develop. The Interstage leaks analysis addresses the concerns of localized mixing of hydrogen and oxygen gases to produce deflagration zones in the Interstage of the Ares I launch vehicle during First Stage operation. This report details the approach taken to accomplish the analysis. Specified leakage profiles and actual flammability results are not presented due to proprietary and security restrictions. The interior volume formed by the Interstage walls, bounding interfaces with the Upper and First Stages, and surrounding the J2-X engine was modeled using Loci-CHEM to assess the potential for flammable gas mixtures to develop during First Stage operations. The transient analysis included a derived flammability indicator based on mixture ratios to maintain achievable simulation times. Validation of results was based on a comparison to Interstage pressure profiles outlined in prior NASA studies. The approach proved useful in the bounding of flammability risk in supporting program hazard reviews.

Little, Jeffrey K.↗

NASA NTRS2014

Pre-Test CFD for the Design and Execution of the Enhanced Injection and Mixing Project at NASA Langley Research Center

With the increasing costs of physics experiments and simultaneous increase in availability and maturity of computational tools it is not surprising that computational fluid dynamics (CFD) is playing an increasingly important role, not only in post-test investigations, but also in the early stages of experimental planning. This paper describes a CFD-based effort executed in close collaboration between computational fluid dynamicists and experimentalists to develop a virtual experiment during the early planning stages of the Enhanced Injection and Mixing project at NASA Langley Research Center. This projects aims to investigate supersonic combustion ramjet (scramjet) fuel injection and mixing physics, improve the understanding of underlying physical processes, and develop enhancement strategies and functional relationships relevant to flight Mach numbers greater than 8. The purpose of the virtual experiment was to provide flow field data to aid in the design of the experimental apparatus and the in-stream rake probes, to verify the nonintrusive measurements based on NO-PLIF, and to perform pre-test analysis of quantities obtainable from the experiment and CFD. The approach also allowed for the joint team to develop common data processing and analysis tools, and to test research ideas. The virtual experiment consisted of a series of Reynolds-averaged simulations (RAS). These simulations included the facility nozzle, the experimental apparatus with a baseline strut injector, and the test cabin. Pure helium and helium-air mixtures were used to determine the efficacy of different inert gases to model hydrogen injection. The results of the simulations were analyzed by computing mixing efficiency, total pressure recovery, and stream thrust potential. As the experimental effort progresses, the simulation results will be compared with the experimental data to calibrate the modeling constants present in the CFD and validate simulation fidelity. CFD will also be used to investigate different injector concepts, improve understanding of the flow structure and flow physics, and develop functional relationships. Both RAS and large eddy simulations (LES) are planned for post-test analysis of the experimental data.

Drozda, Tomasz G.↗

NASA NTRS2016

Calibration of the Fluorine, Chlorine and Hydrogen Content of Apatites With the ChemCam LIBS Instrument

Determining the composition of apatites is important to understand the behavior of volatiles during planetary differentiation. Apatite is an ubiquitous magmatic mineral in the SNC meteorites. It is a significant reservoir of halogens in these meteorites and has been used to estimate the halogen budget of Mars. Apatites have been identified in sandstones and pebbles at Gale crater by ChemCam, a Laser-Induced Breakdown Spectroscometer (LIBS) instrument onboard the Curiosity rover. Their presence was inferred from correlations between calcium, fluorine (using the CaF molecular band centered near 603 nm, whose detection limit is much lower that atomic or ionic lines and, in some cases, phosphorus (whose detection limit is much larger). An initial quantification of fluorine, based on fluorite (CaF2)/basalt mixtures and obtained at the LANL laboratory, indicated that the excess of F/Ca (compared to the stoichiometry of pure fluorapatites) found on Mars in some cases could be explained by the presence of fluorite. Chlorine was not detected in these targets, at least above a detection limit of 0.6 wt% estimated from. Fluorapatite was later also detected by X-ray diffraction (with CheMin) at a level of approx.1wt% in the Windjana drill sample (Kimberley area), and several points analyzed by ChemCam in this area also revealed a correlation between Ca and F. The in situ detection of F-rich, Cl-poor apatites contrasts with the Cl-rich, F-poor compositions of apatites found in basaltic shergottites and in gabbroic clasts from the martian meteorite NWA 7034, which were also found to be more Cl-rich than apatites from basalts on Earth, the Moon, or Vesta. The in situ observations could call into question one of the few possible explanations brought forward to explain the SNC results, namely that Mars may be highly depleted in fluorine. The purpose of the present study is to refine the calibration of the F, Cl, OH and P signals measured by the ChemCam LIBS instrument, initiated for F, for Cl in soils, for P, and estimate their limit of detection. For this purpose, different types of apatites and mixtures of basalt powder and apatites were analyzed using ChemCam Engineering Qualification Model (EQM) at IRAP, Toulouse. The present abstract presents the initial results from the laboratory analyses. Differences between the response function of the EQM and the Flight Model of ChemCam are still to be refined to apply these new results to the Martian dataset.

Meslin, P.-Y.↗

NASA NTRS2016

HST-STIS Spectra of Saturn's Rings and Implications for Their Reddening Agent

We obtained HST-STIS spectra of Saturn's main rings in May 2011, using the G230L (and G430L) gratings, with final averaged radial resolution of 160 (and 330) km/pixel. The dataset filled a previous 200-330nm "spectral gap" between Cassini and ground-based spectra. The data provide radial profiles as a function of wavelength, but our most basic product at this point is a set of very low-noise spectra, radially averaged over broad regions of the rings (A, B, C, and Cassini Division). The raw spectra required special processing to remove artifacts due to extended-source grating scatter. We have modeled the spectra using a new particle surface model, which corrects for on-surface shadowing due to the likely very rough ring particle surfaces, and avoids overestimation of intra-mixed "neutral absorber". We correct for non-classical layer effects and finite ring optical depth, and relate our observed reflectivities to the spherical albedos of individual smooth particles. We model these smooth particle albedos using standard Hapke theory for regolith grain mixtures that are either homogeneous and "intramixed" (nonicy absorbers dispersed in water ice regolith grains) or heterogeneous "intimate" mixtures. As candidates for the nonicy contaminants we have considered amorphous carbon, aromatic-rich and aliphatic-rich organic tholins, silicates, hematite and iron metal. For the A and B rings, we find that iron metal (including a new theoretical estimate of the refractive indices of nanometer-sized grains of iron) is not spectrally steep enough in the 200-300nm range, and that aliphatic-rich tholins are either too steep at short wavelengths or too flat at long wavelengths. However, less than 1% by mass of aromatic-rich tholins provides a very good fit across the entire spectral range with no gratuitous "neutral absorber" needed, and a minimum of additional free parameters. The best fits require forward-scattering regolith grains. For the C Ring and Cassini Division, additional absorbers are needed (updated results will be given).

Saturn's Rings↗