Richard J Greenberg

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• Greenberg, R., & Sak, P. B. (2014). The ridges of Europa: Extensions of adjacent topography onto their flanks. Earth and Planetary Science Letters, 389, 43-51.
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  Abstract: The surface of Europa displays numerous generations of intersecting arrays of linear ridges. At some locations along these ridges, older ridges on adjacent terrain appear to extend up the flank of a more recent ridge. It has thus been suggested that the ridges may have formed by upturning of that adjacent terrain. However, the newer ridges generally appear to be material deposited over the older terrain. Here we consider how the morphology of the overprinted topography may have been inherited by the more recent ridges. An analogous process occurs along some sediment-starved convergent plate boundaries on Earth, where the poorly consolidated material of a frontal prism of an overriding plate is pushed over preexisting ridges and seamounts on the downgoing plate. The overriding plate inherits the morphology of the downgoing plate even though the actual extension of that topography has been underthrust and buried. A well-studied example lies offshore of Costa Rica where the Caribbean plate overrides the Cocos plate. Experiments show other mechanisms as well: mass-wasting down a flank can result in extensions of adjacent ridges thanks to the geometry imposed by a constant angle of repose; in addition, more pronounced extensions of the older ridges result if the new ridge grows as it is bulldozed from behind (i.e., from the central groove of a double ridge on Europa). The shapes of the ridge extensions are distinctly different in these two cases. If tidal pumping extrudes material to the surface at the center of a double ridge, it might drive the latter mechanism. The ridge extensions observed on the flanks of more recent ridges may provide a crucial diagnostic of dominant ridge-building mechanisms when and if additional images are obtained at high resolution from future exploration. In additional to their morphology, the distribution of ridge extensions at only isolated locales may also provide constraints on ridge formation processes and their diversity. © 2013 Elsevier B.V.
• Van, L. C., Barnes, R., & Greenberg, R. (2014). Tides, planetary companions, and habitability: habitability in the habitable zone of low-mass stars. Monthly Notices of the Royal Astronomical Society, 441(3), 1888-1898.
• Greenberg, R., Laerhoven, C. V., & Barnes, R. (2013). Spin-driven tidal pumping: Tidally driven changes in planetary spin coupled with secular interactions between planets. Celestial Mechanics and Dynamical Astronomy, 117(4), 331-348.
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  Abstract: In a multiplanet system, tides acting on the inner planet can significantly affect the orbital evolution of the entire system. While tides usually damp eccentricities, a novel mechanism identified by Correia et al. (Astrophys J Lett 744, article id. L23, 2012) tends to raise eccentricities as a result of the tides' effect on the inner planet's rotation. Our analytical description of this spin-driven tidal (SDT) effect shows that, while the inner planet's eccentricity undergoes pumping, the process is more completely described by an exchange of strength between the two eigenmodes of the dynamical system. Our analysis allows derivation of criteria for two-planet coplanar systems where the SDT effect can reverse tidal damping, and may preclude the effect's being significant for realistic systems. For the specific case quantified by Correia et al., the effect is strong because of the large adopted tidal time lag, which may not be appropriate for the assumed Saturn-like inner planet. On the other hand, the effective Q for any given planet in exotic circumstances is very uncertain, so the SDT effect could play a role in planetary evolution. © 2013 Springer Science+Business Media Dordrecht.
• Laerhoven, C. V., & Greenberg, R. (2013). Small inner companions of warm jupiters: Lifetimes and legacies. Astrophysical Journal, 778(2).
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  Abstract: Although warm Jupiters are generally too far from their stars for tides to be important, the presence of an inner planetary companion to a warm Jupiter can result in tidal evolution of the system. Insight into the process and its effects comes form classical secular theory of planetary perturbations. The lifetime of the inner planet may be shorter than the age of the system, because the warm Jupiter maintains its eccentricity and hence promotes tidal migration into the star. Thus a warm Jupiter observed to be alone in its system might have previously cleared away any interior planets. Before its demise, even if an inner planet is of terrestrial scale, it may promote damping of the warm Jupiter's eccentricity. Thus any inferences of the initial orbit of an observed warm Jupiter must include the possibility of a greater initial eccentricity than would be estimated by assuming it had always been alone. Tidal evolution involving multiple planets also enhances the internal heating of the planets, which readily exceeds that of stellar radiation for the inner planet, and may be great enough to affect the internal structure of warm Jupiters. Secular theory gives insight into the tidal processes, providing, among other things, a way to constrain eccentricities of transiting planets based on estimates of the tidal parameter Q. © 2013. The American Astronomical Society. All rights reserved.
• Timpe, M., Barnes, R., Kopparapu, R., Raymond, S. N., Greenberg, R., & Gorelick, N. (2013). Secular behavior of exoplanets: Self-consistency and comparisons with the planet-planet scattering hypothesis. Astronomical Journal, 146(3).
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  Abstract: If mutual gravitational scattering among exoplanets occurs, then it may produce unique orbital properties. For example, two-planet systems that lie near the boundary between circulation and libration of their periapses could result if planet-planet scattering ejected a former third planet quickly, leaving one planet on an eccentric orbit and the other on a circular orbit. We first improve upon previous work that examined the apsidal behavior of known multiplanet systems by doubling the sample size and including observational uncertainties. This analysis recovers previous results that demonstrated that many systems lay on the apsidal boundary between libration and circulation. We then performed over 12,000 three-dimensional N-body simulations of hypothetical three-body systems that are unstable, but stabilize to two-body systems after an ejection. Using these synthetic two-planet systems, we test the planet-planet scattering hypothesis by comparing their apsidal behavior, over a range of viewing angles, to that of the observed systems and find that they are statistically consistent regardless of the multiplicity of the observed systems. Finally, we combine our results with previous studies to show that, from the sampled cases, the most likely planetary mass function prior to planet-planet scattering follows a power law with index -1.1. We find that this pre-scattering mass function predicts a mutual inclination frequency distribution that follows an exponential function with an index between -0.06 and -0.1. © 2013. The American Astronomical Society. All rights reserved.
• Greenberg, R., & Laerhoven, C. V. (2012). Aligned major axes in a planetary system without tidal evolution: The 61 Virginis example. Monthly Notices of the Royal Astronomical Society, 419(1), 429-435.
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  Abstract: Tidal damping of one of the orbits in a planetary system can lead to aligned major axes (the so-called 'fixed-point' condition), but currently aligned major axes do not necessarily imply such a history. An example is the nominal orbital solution for the 61Virginis system where two orbits librate about alignment, but evaluation of the eigenmodes of the secular theory shows it could not be the result of tidal damping but rather of initial conditions. The amplitudes of the eigenmodes suggest that, at most, this system may have undergone some modest degree of tidal damping. © 2011 The Authors Monthly Notices of the Royal Astronomical Society © 2011 RAS.
• Laerhoven, C. v., & Greenberg, R. (2012). Characterizing multi-planet systems with classical secular theory. Celestial Mechanics and Dynamical Astronomy, 113(2), 215-234.
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  Abstract: Classical secular theory can be a powerful tool to describe the qualitative character of multi-planet systems and offer insight into their histories. The eigenmodes of the secular behavior, rather than current orbital elements, can help identify tidal effects, early planet-planet scattering, and dynamical coupling among the planets, for systems in which mean-motion resonances do not play a role. Although tidal damping can result in aligned major axes after all but one eigenmode have damped away, such alignment may simply be fortuitous. An example of this is 55 Cancri (orbital solution of Fischer et al. in Astophys J 675:790-801, 2008) where multiple eigenmodes remain undamped. Various solutions for 55 Cancri are compared, showing differing dynamical groupings, with implications for the coupling of eccentricities and for the partitioning of damping among the planets. Solutions for orbits that include expectations of past tidal evolution with observational data, must take into account which eigenmodes should be damped, rather than expecting particular eccentricities to be near zero. Classical secular theory is only accurate for low eccentricity values, but comparison with other results suggests that it can yield useful qualitative descriptions of behavior even for moderately large eccentricity values, and may have advantages for revealing underlying physical processes and, as large numbers of new systems are discovered, for triage to identify where more comprehensive dynamical studies should have priority. © 2012 Springer Science+Business Media B.V.
• Pasek, M. A., & Greenberg, R. (2012). Acidification of Europa's subsurface ocean as a consequence of oxidant delivery. Astrobiology, 12(2), 151-159.
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  PMID: 22283235;Abstract: Oxidants are formed at the surface of Europa and may be delivered to the subsurface ocean, possibly in great quantities. Whether these substances would be available for biological metabolism is uncertain, because they may react with sulfides and other compounds to generate sulfuric and other acids. If this process has been active on Europa for much of its age, then not only would it rob the ocean of life-supporting oxidants but the subsurface ocean could have a pH of ∼2.6, which is so acidic as to present an environmental challenge for life, unless organisms consume or sequester the oxidants fast enough to ameliorate the acidification. © 2012, Mary Ann Liebert, Inc.
• Pasek, M., & Greenberg, R. (2012). Acidification of Europa’s subsurface ocean as a consequence of oxidant delivery. Astrobiology, 12(2), 151-159.
• Van, L. C., & Greenberg, R. (2012). Characterizing multiplanet systems with classical secular theory. Celestial Mechanics , 113, 215-234.
• Barnes, R., Greenberg, R., Quinn, T. R., McArthur, B. E., & Benedict, G. F. (2011). Origin and dynamics of the mutually inclined orbits of v andromedae c and d. Astrophysical Journal, 726(2).
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  Abstract: We evaluate the orbital evolution and several plausible origin scenarios for the mutually inclined orbits of v And c and d. These two planets have orbital elements that oscillate with large amplitudes and lie close to the stability boundary. This configuration, and in particular the observed mutual inclination, demands an explanation. The planetary system may be influenced by a nearby low-mass star, v And B, which could perturb the planetary orbits, but we find it cannot modify two coplanar orbits into the observed mutual inclination of 30°. However, it could incite ejections or collisions between planetary companions that subsequently raise the mutual inclination to >30°. Our simulated systems with large mutual inclinations tend to be further from the stability boundary than v And, but we are able to produce similar systems. We conclude that scattering is a plausible mechanism to explain the observed orbits of v And c and d, but we cannot determine whether the scattering was caused by instabilities among the planets themselves or by perturbations from v And B. We also develop a procedure to quantitatively compare numerous properties of the observed system to our numerical models. Although we only implement this procedure to v And, it may be applied to any exoplanetary system. © 2011 The American Astronomical Society. All rights reserved.
• Greenberg, R. (2011). Exploration and protection of Europa's biosphere: Implications of permeable ice. Astrobiology, 11(2), 183-191.
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  PMID: 21417946;Abstract: Europa has become a high-priority objective for exploration because it may harbor life. Strategic planning for its exploration has been predicated on an extreme model in which the expected oceanic biosphere lies under a thick ice crust, buried too deep to be reached in the foreseeable future, which would beg the question of whether other active satellites might be more realistic objectives. However, Europa's ice may in fact be permeable, with very different implications for the possibilities for life and for mission planning. A biosphere may extend up to near the surface, making life far more readily accessible to exploration while at the same time making it vulnerable to contamination. The chances of finding life on Europa are substantially improved while the need for planetary protection becomes essential. The new National Research Council planetary protection study will need to go beyond its current mandate if meaningful standards are to be put in place. © Mary Ann Liebert, Inc.
• Greenberg, R., & Laerhoven, C. V. (2011). Tidal evolution of a secularly interacting planetary system. Astrophysical Journal, 733(1).
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  Abstract: In a multi-planet system, a gradual change in one planet's semimajor axis will affect the eccentricities of all the planets, as angular momentum is distributed via secular interactions. If tidal dissipation in the planet is the cause of the change in semimajor axis, it also damps that planet's eccentricity, which in turn also contributes to the evolution of all the eccentricities. Formulae quantifying the combined effects on the whole system due to semimajor axis changes, as well as eccentricity damping, are derived here for a two-planet system. The CoRoT 7 system is considered as an example. © 2011. The American Astronomical Society. All rights reserved..
• Greenberg, R., Morris, M., & Foley, M. A. (2011). The effect of sub-pixel slope variations on photoclinometry. Icarus, 214(1), 348-350.
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  Abstract: Sub-pixel slope variations can have significant effects on the intensity of light scattered from a planetary surface. As a result, determination of the surface slope from the apparent brightness of a given pixel can be confounded by uncertainly in such variations. Under a wide range of conditions, the average slope across the pixel may be different from what is inferred by photoclinometry. Because topography is derived from photoclinometry by integrating the slope across an image composed of many pixels, topographic elevation could in principle be distorted considerably by this effect. As actually applied, photoclinometry generally includes strategies designed to mitigate these effects substantially. Nevertheless, the potential always exists for unknown variations in the character of sub-pixel topography (among other uncertainties such as albedo variations) to introduce errors. The results shown here show the importance of the mitigating strategies and of considering the magnitude of uncertainties in determination of topography. © 2011 Elsevier Inc.
• Barnes, R., Raymond, S. N., Greenberg, R., Jackson, B., & Kaib, N. A. (2010). CoRoT-7b: Super-earth or super-Io?. Astrophysical Journal Letters, 709(2 PART 2), L95-L98.
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  Abstract: CoRoT-7b, a planet about 70% larger than the Earth orbiting a Sun-like star, is the first-discovered rocky exoplanet, and hence has been dubbed a "super-Earth." Some initial studies suggested that since the planet is so close to its host star, it receives enough insolation to partially melt its surface. However, these past studies failed to take into consideration the role that tides may play in this system. Even if the planet's eccentricity has always been zero, we show that tidal decay of the semimajor axis could have been large enough that the planet formed on a wider orbit which received less insolation. Moreover, CoRoT-7b could be tidally heated at a rate that dominates its geophysics and drives extreme volcanism. In this case, CoRoT-7b is a "super-Io" that, like Jupiter's volcanic moon, is dominated by volcanism and rapid resurfacing. Such heating could occur with an eccentricity of just 10-5. This small value could be driven by CoRoT-7c if its own eccentricity is larger than ∼ 10-4. CoRoT-7b may be the first of a class of planetary super-Ios likely to be revealed by the CoRoT and Kepler spacecraft. © 2010. The American Astronomical Society. All rights reserved.
• Greenberg, R. (2010). The icy Jovian satellites after the Galileo mission. Reports on Progress in Physics, 73(3).
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  Abstract: The icy satellites of Jupiter, Callisto, Ganymede, Europa and Amalthea have diverse and remarkable characteristics. Their initial compositions were determined by conditions in the circum-Jovian nebula, just as the planets' initial properties were governed by their formation within the circumsolar nebula. The satellites subsequently evolved under the complex interplay of orbital and geophysical processes, especially the effects of orbital resonances, tides, internal differentiation and heat. The history and character of the satellites can be inferred from consideration of the formation of planets and the satellites, from studies of their plausible orbital evolution, from measurements of geophysical properties, especially gravitational and magnetic fields, from observations of the compositions and geological structure of their surfaces and from theoretical modeling of the processes that connect these lines of evidence. The three large icy satellites probably contain significant liquid water: Europa has a deep liquid water ocean under a thin surface layer of ice; Ganymede and Callisto likely have relatively thin liquid water layers deep below their surfaces. Models of formation are challenged by the surprising properties of the outermost and innermost of the group: Callisto is partially differentiated, with rock and ice mixed through much of its interior; and tiny Amalthea also appears to be largely composed of ice. Each of the four moons is fascinating in its own right, and the ensemble provides a powerful set of constraints on the processes that led to their formation and evolution. © 2010 IOP Publishing Ltd.
• Greenberg, R. (2010). Transport rates of radiolytic substances into europa's ocean: implications for the potential origin and maintenance of life. Astrobiology, 10(3), 275-283.
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  PMID: 20446868;Abstract: Bombardment of the surface of Europa produces oxidants and other biologically useful substances, but they can only contribute to the habitability of the ocean if they are delivered down through the icy crust. Previous estimates of the thickness of the oxygenated layer of ice assumed that impact gardening is the dominant factor and concluded that the ocean may be habitable if the oxidant delivery time, via undefined mechanisms, is sufficiently short. Consideration of the types of processes that continually resurface Europa suggests that the oxygenated layer is thicker than ̃300m, far greater than the few meters indicated by impact gardening alone, and possibly includes the entire ice crust. The estimated delivery rate to the ocean is such that the oxygen levels could now be high enough to support macrofauna; and, at ̃31011mol/yr of oxygen, it could maintain 3 million tons of macrofauna, assuming respiration rates similar to terrestrial marine organisms. These values are independent of any additional contributions due to possible photosynthesis. Initial formation of life would be difficult with so much oxygen, but the start of oxidant delivery into the ocean would have been delayed by 1-2 billion years while the crust became loaded with oxidants. In the ocean, this delay would have allowed time for prebiotic assemblages and anaerobic biological development prior to the increasing oxidant concentration to otherwise toxic levels. Copyright 2010, Mary Ann Liebert, Inc.
• Jackson, B., Barnes, R., & Greenberg, R. (2010). Tides and Exoplanets. Formation and Evolution of Exoplanets, 243-266.
• Jackson, B., Miller, N., Barnes, R., Raymond, S. N., Fortney, J. J., & Greenberg, R. (2010). The roles of tidal evolution and evaporative mass loss in the origin of CoRoT-7 b. Monthly Notices of the Royal Astronomical Society, 407(2), 910-922.
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  Abstract: CoRoT-7 b is the first confirmed rocky exoplanet, but, with an orbital semimajor axis of 0.0172 au, its origins may be unlike any rocky planet in our Solar system. In this study, we consider the roles of tidal evolution and evaporative mass loss in CoRoT-7 b's history, which together have modified the planet's mass and orbit. If CoRoT-7 b has always been a rocky body, evaporation may have driven off almost half its original mass, but the mass loss may depend sensitively on the extent of tidal decay of its orbit. As tides caused CoRoT-7 b's orbit to decay, they brought the planet closer to its host star, thereby enhancing the mass loss rate. Such a large mass loss also suggests the possibility that CoRoT-7 b began as a gas giant planet and had its original atmosphere completely evaporated. In this case, we find that CoRoT-7 b's original mass probably did not exceed 200 Earth masses (about two-third of a Jupiter mass). Tides raised on the host star by the planet may have significantly reduced the orbital semimajor axis, perhaps causing the planet to migrate through mean-motion resonances with the other planet in the system, CoRoT-7 c. The coupling between tidal evolution and mass loss may be important not only for CoRoT-7 b but also for other close-in exoplanets, and future studies of mass loss and orbital evolution may provide insight into the origin and fate of close-in planets, both rocky and gaseous. © 2010 The Authors. Journal compilation © 2010 RAS.
• Barnes, R., Jackson, B., Greenberg, R., & Raymond, S. N. (2009). Tidal limits to planetary habitability. Astrophysical Journal Letters, 700(1 PART 2), L30-L33.
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  Abstract: The habitable zones (HZs) of main-sequence stars have traditionally been defined as the range of orbits that intercept the appropriate amount of stellar flux to permit surface water on a planet. Terrestrial exoplanets discovered to orbit M stars in these zones, which are close-in due to decreased stellar luminosity, may also undergo significant tidal heating. Tidal heating may span a wide range for terrestrial exoplanets and may significantly affect conditions near the surface. For example, if heating rates on an exoplanet are near or greater than that on Io (where tides drive volcanism that resurfaces the planet at least every 1 Myr) and produce similar surface conditions, then the development of life seems unlikely. On the other hand, if the tidal heating rate is less than the minimum to initiate plate tectonics, then CO2 may not be recycled through subduction, leading to a runaway greenhouse that sterilizes the planet. These two cases represent potential boundaries to habitability and are presented along with the range of the traditional HZ for main-sequence, low-mass stars. We propose a revised HZ that incorporates both stellar insolation and tidal heating. We apply these criteria to GJ 581 d and find that it is in the traditional HZ, but its tidal heating alone may be insufficient for plate tectonics. © 2009. The American Astronomical Society.
• Barnes, R., Jackson, B., Raymond, S. N., West, A. A., & Greenberg, R. (2009). The HD 40307 planetary system: Super-Earths or mini-Neptunes?. Astrophysical Journal, 695(2), 1006-1011.
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  Abstract: Three planets with minimum masses less than 10 M⊕ orbit the star HD 40307, suggesting these planets may be rocky. However, with only radial velocity data, it is impossible to determine if these planets are rocky or gaseous. Here we exploit various dynamical features of the system in order to assess the physical properties of the planets. Observations allow for circular orbits, but a numerical integration shows that the eccentricities must be at least 10-4. Also, planets b and c are so close to the star that tidal effects are significant. If planet b has tidal parameters similar to the terrestrial planets in the solar system and a remnant eccentricity larger than 10-3, then, going back in time, the system would have been unstable within the lifetime of the star (which we estimate to be 6.1 ± 1.6 Gyr). Moreover, if the eccentricities are that large and the inner planet is rocky, then its tidal heating may be an order of magnitude greater than extremely volcanic Io, on a per unit surface area basis. If planet b is not terrestrial, e.g., Neptune-like, these physical constraints would not apply. This analysis suggests the planets are not terrestrial-like, and are more like our giant planets. In either case, we find that the planets probably formed at larger radii and migrated early-on (via disk interactions) into their current orbits. This study demonstrates how the orbital and dynamical properties of exoplanet systems may be used to constrain the planets' physical properties. © 2009. The American Astronomical Society. All rights reserved.
• Greenberg, R. (2009). Frequency dependence of tidal q. Astrophysical Journal Letters, 698(1 PART 2), L42-L45.
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  Abstract: For studies of tidal evolution, values of the key parameter Q, and its frequency dependence, are often derived from estimates of internal energy dissipation when a satellite, planet, or star is physically distorted. Such estimates come from geophysical or astrophysical modeling, from seismic data, from ad hoc assumptions, or from constraints based on current spins and orbits. In a standard procedure, Q values are used to determine the lag in the response to each Fourier component of the tidal potential. The separate components are then co-added. The basis for this procedure is the analogy of the damped, driven, harmonic oscillator. However, this lag-and-add procedure would not be justifiable even for such a simple system, except for a very specific dependence of Q on frequency. There is no reason to expect the lag-and-add procedure to be relevant for a complex system, because the relationship between dissipation rates and tidal lags is unknown. This widely applied type of model is a reasonable approximation only if the decomposed tidal potential involves a narrow range of frequencies, and thus may only be appropriate for analyses to low order in orbital eccentricity and inclination. © 2009. The American Astronomical Society.
• Hurford, T. A., Bills, B. G., Helfenstein, P., Greenberg, R., Hoppa, G. V., & Hamilton, D. P. (2009). Geological implications of a physical libration on Enceladus. Icarus, 203(2), 541-552.
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  Abstract: Given the non-spherical shape of Enceladus [Thomas et al., 2007], the satellite will experience gravitational torques that will cause it to physically librate as it orbits Saturn. Physical libration would produce a diurnal oscillation in the longitude of Enceladus' tidal bulge, which could have a profound effect on the diurnal tidal stresses experienced by the surface of the satellite. Although Cassini ISS has placed an observational upper limit on the amplitude of Enceladus' libration, smaller amplitudes can still have geologically significant consequences. Here we present the first detailed description of how physical libration affects tidal stresses and how those stresses might then affect geological processes including crack formation and propagation, south polar eruption activity, and tidal heating. Our goal is to provide a framework for testing the hypothesis that geologic features on Enceladus are produced by tidal stresses from diurnal librations of the satellite.
• Hurford, T. A., Sarid, A. R., Greenberg, R., & Bills, B. G. (2009). The influence of obliquity on europan cycloid formation. Icarus, 202(1), 197-215.
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  Abstract: Tectonic patterns on Europa are influenced by tidal stress. An important well-recognized component is associated with the orbital eccentricity, which produces a diurnally varying stress as Jupiter's apparent position in Europa's sky oscillates in longitude. Cycloidal lineaments seem to have formed as cracks propagated in this diurnally varying stress field. Maps of theoretical cycloid patterns capture many of the characteristics of the observed distribution on Europa. However, a few details of the observed cycloid distribution have not been reproduced by previous models. Recently, it has been shown that Europa has a finite forced obliquity, so Jupiter's apparent position in Europa's sky will also oscillate in latitude. We explore this new type of diurnal effect on cycloid formation. We find that stress from obliquity may be the key to explaining several characteristics of observed cycloids such as the shape of equator-crossing cycloids and the shift in the crack patterns in the Argadnel Regio region. All of these improvements of the fit between observation and theory seem to require Jupiter crossing Europa's equatorial plane 45° to 180° after perijove passage, suggestive of complex orbital dynamics that locks the direction of Europa's pericenter with the direction of the ascending node at the time these cycloids were formed.
• Raymond, S. N., Barnes, R., Veras, D., Armitage, P. J., Gorelick, N., & Greenberg, R. (2009). Planet-planet scattering leads to tightly packed planetary systems. Astrophysical Journal Letters, 696(1 PART 2), L98-L101.
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  Abstract: The known extrasolar multiple-planet systems share a surprising dynamical attribute: they cluster just beyond the Hill stability boundary. Here we show that the planet-planet scattering model, which naturally explains the observed exoplanet eccentricity distribution, can reproduce the observed distribution of dynamical configurations. We calculated how each of our scattered systems would appear over an appropriate range of viewing geometries; as Hill stability is weakly dependent on the masses, the mass-inclination degeneracy does not significantly affect our results. We consider a wide range of initial planetary mass distributions and find that some are poor fits to the observed systems. In fact, many of our scattering experiments overproduce systems very close to the stability boundary. The distribution of dynamical configurations of two-planet systems may provide better discrimination between scattering models than the distribution of eccentricity. Our results imply that, at least in their inner regions which are weakly affected by gas or planetesimal disks, planetary systems should be "packed," with no large gaps between planets. © 2009. The American Astronomical Society. All rights reserved.
• Barnes, R., Raymond, S. N., Jackson, B., & Greenberg, R. (2008). Tides and the evolution of planetary habitability. Astrobiology, 8(3), 557-568.
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  PMID: 18598142;Abstract: Tides raised on a planet by the gravity of its host star can reduce the planet's orbital semi-major axis and eccentricity. This effect is only relevant for planets orbiting very close to their host stars. The habitable zones of low-mass stars are also close in, and tides can alter the orbits of planets in these locations. We calculate the tidal evolution of hypothetical terrestrial planets around low-mass stars and show that tides can evolve planets past the inner edge of the habitable zone, sometimes in less than 1 billion years. This migration requires large eccentricities (>0.5) and low-mass stars (≲0.35 M⊙). Such migration may have important implications for the evolution of the atmosphere, internal heating, and the Gaia hypothesis. Similarly, a planet that is detected interior to the habitable zone could have been habitable in the past. We consider the past habitability of the recently discovered, ∼5 M⊕ planet, Gliese 581 c. We find that it could have been habitable for reasonable choices of orbital and physical properties as recently as 2 Gyr ago. However, when constraints derived from the additional companions are included, most parameter choices that indicate past habitability require the two inner planets of the system to have crossed their mutual 3:1 mean motion resonance. As this crossing would likely have resulted in resonance capture, which is not observed, we conclude that Gl 581 c was probably never habitable. © 2008 Mary Ann Liebert, Inc.
• Greenberg, R. (2008). Ero's Rahe Dorsum: Implications for internal structure. Meteoritics and Planetary Science, 43(3), 435-449.
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  Abstract: An intriguing discovery of the NEAR imaging and laser-ranging experiments was the ridge system known as Rahe Dorsum and its possible relation with global-scale internal structure. The curved path of the ridge over the surface roughly defines a plane cutting through Eros. Another lineament on the other side of Eros, Calisto Fossae, seems to lie nearly on the same plane. The NEAR teams interpret Rahe as the expression of a compressive fault (a plane of weakness), because portions are a scarp, which on Earth would be indicative of horizontal compression, where shear displacement along a dipping fault has thrust the portion of the lithosphere on one side of the fault up relative to the other side. However, given the different geometry of Eros, a scarp may not have the same relationship to underlying structure as it does on Earth. The plane through Eros runs nearly parallel to, and just below, the surface facet adjacent to Rahe Dorsum. The plane then continues lengthwise through the elongated body, a surprising geometry for a plane of weakness on a battered body. Moreover, an assessment of the topography of Rahe Dorsurn indicates that it is not consistent with displacement on the Rahe plane. Rather, the topography suggests that Rahe Dorsum results from resistance of the Rahe plane to impact erosion. Such a plane of strength might have formed in Eros' parent body by a fluid intrusion (e.g., a dike of partial melt) through undifferentiated material, creating a vein of stronger rock. Albedo, color and near-infrared spectra could be consistent with a distinct material composition and such a history, although the instruments' resolution was not adequate for a definitive detection of such a spatially limited component. However the plane of strength formed, such structural reinforcing might have enabled and controlled the elongated irregular shape of Eros, as well as Rahe Dorsum. © The Meteoritical Society, 2008.
• Greenberg, R., & Barnes, R. (2008). Tidal evolution of Dysnomia, satellite of the dwarf planet Eris. Icarus, 194(2), 847-849.
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  Abstract: The past tidal evolution of the satellite Dysnomia of the dwarf planet Eris can be inferred from the current physical and orbital properties of the system. Preliminary considerations, which assumed a circular orbit for the satellite, suggested that the satellite formed close to the planet, perhaps as a result of a giant impact, and that it is thus unlikely that smaller satellites lie further out. However, if the satellite's orbit is eccentric, even if the eccentricity is very small, a qualitatively different past tidal evolution may be indicated. Early in the Solar System's history, the satellite may have been on a highly eccentric orbit much farther from the planet than it is now, suggestive of a capture origin. Additional satellites farther out cannot be ruled out. © 2008 Elsevier Inc. All rights reserved.
• Jackson, B., Barne, R., & Greenberg, R. (2008). Planetary transits and tidal evolution. Proceedings of the International Astronomical Union, 4(S253), 217-229.
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  Abstract: Transiting planets are generally close enough to their host stars that tides may govern their orbital and thermal evolution. We present calculations of the tidal evolution of recently discovered transiting planets and discuss their implications. The tidal heating that accompanies this orbital evolution can be so great that it controls the planet's physical properties and may explain the large radii observed in several cases, including, for example, TrES-4. Also, since a planet's transit probability depends on its orbit, it evolves due to tides. Current values depend sensitively on the physical properties of the star and planet, as well as on the system's age. As a result, tidal effects may introduce observational biases in transit surveys, which may already be evident in current observations. Transiting planets tend to be younger than non-transiting planets, an indication that tidal evolution may have destroyed many close-in planets. Also the distribution of the masses of transiting planets may constrain the orbital inclinations of non-transiting planets. © 2009 International Astronomical Union.
• Jackson, B., Barnes, R., & Greenberg, R. (2008). Tidal heating of terrestrial extrasolar planets and implications for their habitability. Monthly Notices of the Royal Astronomical Society, 391(1), 237-245.
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  Abstract: The tidal heating of hypothetical rocky (or terrestrial) extrasolar planets spans a wide range of values depending on stellar masses and initial orbits. Tidal heating may be sufficiently large (in many cases, in excess of radiogenic heating) and long-lived to drive plate tectonics, similar to the Earth's, which may enhance the planet's habitability. In other cases, excessive tidal heating may result in Io-like planets with violent volcanism, probably rendering them unsuitable for life. On water-rich planets, tidal heating may generate subsurface oceans analogous to Europa's with similar prospects for habitability. Tidal heating may enhance the outgassing of volatiles, contributing to the formation and replenishment of a planet's atmosphere. To address these issues, we model the tidal heating and evolution of hypothetical extrasolar terrestrial planets. The results presented here constrain the orbital and physical properties required for planets to be habitable. © 2008 RAS.
• Jackson, B., Greenberg, R., & Barnes, R. (2008). Tidal evolution of close-in extrasolar planets. Astrophysical Journal, 678(2), 1396-1406.
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  Abstract: The distribution of eccentricities e of extrasolar planets with semimajor axes a > 0.2 AU is very uniform, and values for e are relatively large, averaging 0.3 and broadly distributed up to near 1. For a < 0.2 AU, eccentricities are much smaller (most e < 0.2), a characteristic widely attributed to damping by tides after the planets formed and the protoplanetary gas disk dissipated. Most previous estimates of the tidal damping considered the tides raised on the planets, but ignored the tides raised on the stars. Most also assumed specific values for the planets' poorly constrained tidal dissipation parameter Qp. Perhaps most important, in many studies the strongly coupled evolution between e and a was ignored. We have now integrated the coupled tidal evolution equations for e and a over the estimated age of each planet, and confirmed that the distribution of initial e values of close-in planets matches that of the general population for reasonable Q values, with the best fits for stellar and planetary Q being ∼105.5 and ∼106.5, respectively. The accompanying evolution of a values shows most close-in planets had significantly larger a at the start of tidal migration. The earlier gas disk migration did not bring all planets to their current orbits. The current small values of a were only reached gradually due to tides over the lifetimes of the planets. These results may have important implications for planet formation models, atmospheric models of "hot Jupiters," and the success of transit surveys. © 2008. The American Astronomical Society. All rights reserved.
• Jackson, B., Greenberg, R., & Barnes, R. (2008). Tidal heating of extrasolar planets. Astrophysical Journal, 681(2), 1631-1638.
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  Abstract: Extrasolar planets close to their host stars have likely undergone significant tidal evolution since the time of their formation. Tides probably dominated their orbital evolution once the dust and gas cleared away, and as the orbits evolved there was substantial tidal heating within the planets. The tidal heating history of each planet may have contributed significantly to the thermal budget governing the planet's physical properties, including its radius, which in many cases may be measured by observing transit events. Typically, tidal heating increases as a planet moves inward toward its star and then decreases as its orbit circularizes. Here we compute the plausible heating histories for several planets with measured radii, using the same tidal parameters for the star and planet that have been shown to reconcile the eccentricity distribution of close-in planets with other extrasolar planets. Several planets are discussed, including, for example, HD 209458b, which may have undergone substantial tidal heating during the past billion years, perhaps enough to explain its large measured radius. Our models also show that GJ 876d may have experienced tremendous heating and is probably not a solid, rocky planet. Theoretical models should include the role of tidal heating, which is large but time-varying. © 2008. The American Astronomical Society. All rights reserved.
• Barnes, R., & Greenberg, R. (2007). Apsidal behavior among planetary orbits: Testing the planet-planet scattering model. Astrophysical Journal, 659(1 II), L53-L56.
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  Abstract: Planets in extrasolar systems tend to interact such that their orbits lie near a boundary between apsidal libration and circulation, a "separatrix," with one eccentricity periodically reaching near zero. One explanation, applied to the v And system, assumed three original planets on circular orbits. One is ejected, leaving the other two with near-separatrix behavior. We test that model by integrating hundreds of hypothetical, unstable planetary systems that eject a planet. We find that the probability that the remaining planets exhibit near-separatrix motion is small (
• Barnes, R., & Greenberg, R. (2007). Extrasolar planet interactions. Proceedings of the International Astronomical Union, 3(S249), 469-478.
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  Abstract: The dynamical interactions of planetary systems may be a clue to their formation histories. Therefore, the distribution of these interactions provides important constraints on models of planet formation. We focus on each system's apsidal motion and proximity to dynamical instability. Although only 25 multiple planet systems have been discovered to date, our analyses in these terms have revealed several important features of planetary interactions. 1) Many systems interact such that they are near the boundary between stability and instability. 2) Planets tend to form such that at least one planet's eccentricity periodically drops to near zero. 3) Mean-motion resonant pairs would be unstable if not for the resonance. 4) Scattering of approximately equal mass planets is unlikely to produce the observed distribution of apsidal behavior. 5) Resonant interactions may be identified through calculating a system's proximity to instability, regardless of knowledge of angles such as mean longitude and longitude of periastron (e.g. GJ 317 b and c are probably in a 4:1 resonance). These properties of planetary systems have been identified through calculation of two parameters that describe the interaction. The apsidal interaction can be quantified by determining how close a planet is to an apsidal separatrix (a boundary between qualitatively different types of apsidal oscillations, e.g. libration or circulation of the major axes). This value can be calculated through short numerical integrations. The proximity to instability can be measured by comparing the observed orbital elements to an analytic boundary that describes a type of stability known as Hill stability. We have set up a website dedicated to presenting the most up-to-date information on dynamical interactions: http://www.lpl.arizona.edu/~rory/research/xsp/dynamics. © 2008 International Astronomical Union.
• Barnes, R., & Greenberg, R. (2007). Stability limits in resonant planetary systems. Astrophysical Journal, 665(1 PART 2), L67-L70.
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  Abstract: The relationship between the boundaries for Hill and Lagrange stability in orbital element space is modified in the case of resonantly interacting planets. Hill stability requires the ordering of the planets to remain constant, while Lagrange stability also requires all planets to remain bound to the central star. The Hill stability boundary is defined analytically, but no equations exist to define the Lagrange boundary, so we perform numerical experiments to estimate the location of this boundary. To explore the effect of resonances, we consider orbital element space near the conditions in the HD 82943 and 55 Cnc systems. Previous studies have shown that, for nonresonant systems, the two stability boundaries are nearly coincident. However, the Hill stability formulae are not applicable to resonant systems, and our investigation shows how the two boundaries diverge in the presence of a mean-motion resonance, while confirming that the Hill and Lagrange boundaries are similar otherwise. In resonance the region of stability is larger than the domain defined by the analytic formula for Hill stability. We find that nearly all known resonant interactions currently lie in this unexpectedly stable region, i.e., where the orbits would be unstable according to the nonresonant Hill stability formula. This result bears on the dynamical packing of planetary systems, showing how quantifying planetary systems' dynamical interactions (such as proximity to the Hill stability boundary) provides new constraints on planet formation models. © 2007. The American Astronomical Society.
• Hurford, T. A., Helfenstein, P., Hoppa, G. V., Greenberg, R., & Bills, B. G. (2007). Eruptions arising from tidally controlled periodic openings of rifts on Enceladus. Nature, 447(7142), 292-294.
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  PMID: 17507977;Abstract: In 2005, plumes were detected near the south polar region of Enceladus, a small icy satellite of Saturn. Observations of the south pole revealed large rifts in the crust, informally called 'tiger stripes', which exhibit higher temperatures than the surrounding terrain and are probably sources of the observed eruptions. Models of the ultimate interior source for the eruptions are under consideration. Other models of an expanding plume require eruptions from discrete sources, as well as less voluminous eruptions from a more extended source, to match the observations. No physical mechanism that matches the observations has been identified to control these eruptions. Here we report a mechanism in which temporal variations in tidal stress open and close the tiger-stripe rifts, governing the timing of eruptions. During each orbit, every portion of each tiger stripe rift spends about half the time in tension, which allows the rift to open, exposing volatiles, and allowing eruptions. In a complementary process, periodic shear stress along the rifts also generates heat along their lengths, which has the capacity to enhance eruptions. Plume activity is expected to vary periodically, affecting the injection of material into Saturn's E ring and its formation, evolution and structure. Moreover, the stresses controlling eruptions imply that Enceladus' icy shell behaves as a thin elastic layer, perhaps only a few tens of kilometres thick. ©2007 Nature Publishing Group.
• Hurford, T. A., Sarid, A. R., & Greenberg, R. (2007). Cycloidal cracks on Europa: Improved modeling and non-synchronous rotation implications. Icarus, 186(1), 218-233.
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  Abstract: Cycloidal crack patterns on Europa are influenced by tides induced by orbital eccentricity, which in turn is driven by the Laplace orbital resonance. Their shapes potentially record the location of their formation (relative to the direction of Jupiter), as well as the parameters of crack formation. Hoppa et al. [Hoppa, G., Tufts, B.R., Greenberg, R., Geissler, P., 1999a. Icarus 141, 287-298] modeled several cycloid chains using a fixed set of material parameters, but some details did not fit. We now allow material parameters to vary for each arc of an observed cycloid. In general, with minimal variation of model parameters between the arcs, fits are greatly improved. Furthermore, accounting for tidal stress accumulated during non-synchronous rotation, in addition to diurnal stress, allows even better fits. Even with the added freedom in the model our fits allow us to better constrain the location where each cycloid may have formed. Our results support Hoppa et al.'s finding that only a few cracks form ridges per cycle of non-synchronous rotation in the region examined, probably because cracking relieves built up stress until further substantial rotation occurs. © 2006 Elsevier Inc. All rights reserved.
• Jackson, B., Greenberg, R., & Barnes, R. (2007). Tidal evolution of close-in extra-solar planets. Proceedings of the International Astronomical Union, 3(S249), 187-196.
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  Abstract: The distribution of eccentricities e of extra-solar planets with semi-major axes a > 0.2 AU is very uniform, and values for e are generally large. For a < 0.2 AU, eccentricities are much smaller (most e < 0.2), a characteristic widely attributed to damping by tides after the planets formed and the protoplanetary gas disk dissipated. We have integrated the classical coupled tidal evolution equations for e and a backward in time over the estimated age of each planet, and confirmed that the distribution of initial e values of close-in planets matches that of the general population for reasonable tidal dissipation values Q, with the best fits for stellar and planetary Q being 105.5 and 106.5, respectively. The current small values of a were only reached gradually due to tides over the lifetimes of the planets, i.e., the earlier gas disk migration did not bring all planets to their current orbits. As the orbits tidally evolved, there was substantial tidal heating within the planets. The past tidal heating of each planet may have contributed significantly to the thermal budget that governed the planet's physical properties, including its radius, which in many cases may be measured by observing transit events. Here we also compute the plausible heating histories for a few planets with anomalously large measured radii, including HD 209458 b. We show that they may have undergone substantial tidal heating during the past billion years, perhaps enough to explain their large radii. Theoretical models of exoplanet interiors and the corresponding radii should include the role of large and time-variable tidal heating. Our results may have important implications for planet formation models, physical models of hot Jupiters, and the success of transit surveys. © 2008 International Astronomical Union.
• Preblich, B., Greenberg, R., Riley, J., & O'Brien, D. (2007). Tidally driven strike-slip displacement on Europa: Viscoelastic modeling. Planetary and Space Science, 55(10), 1225-1245.
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  Abstract: The process of tide-driven walking, proposed as a major mechanism for strike-slip displacement on Europa, is modeled using a finite-element numerical simulation of the behavior of viscoelastic material. For material parameters that are plausible for the water ice composing Europa's crust, the simulation confirms earlier analytic results for strike-slip displacement along a crack that penetrates down to the liquid water substrate. The finite element code permits testing other cases as well. Of considerable interest is whether tidal walking can operate if a crack penetrates not to liquid but only as far as warm, relatively viscous ice. In such a case, significant displacement can be driven, but only if the threshold value of the compressive force needed to lock the fault is near the value of the overburden stress at the bottom of the crack. Such special conditions are not needed for displacement if the crack penetrates to the underlying ocean. © 2007 Elsevier Ltd. All rights reserved.
• Barnes, R., & Greenberg, R. (2006). Behavior of apsidal orientations in planetary systems. Astrophysical Journal Letters, 652(1 II), L53-L56.
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  Abstract: A widely considered characteristic of extrasolar planetary systems has been a seeming tendency for major axes of adjacent orbits to librate in stable configurations. Based on a new catalog of extrasolar planets (Butler et al.) and our numerical integrations, we find that such small-amplitude oscillations are actually not common but in fact quite rare; most pairs of planets' major axes are consistent with circulating relative to one another. However, the new results are consistent with studies that find that two-planet systems tend to lie near a separatrix between libration and circulation. Similarly, in systems of more than two planets, many adjacent orbits lie near a separatrix that divides modes of circulation. © 2006. The American Astronomical Society. All rights reserved.
• Barnes, R., & Greenberg, R. (2006). Extrasolar planetary systems near a secular separatrix. Astrophysical Journal Letters, 638(1 I), 478-487.
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  Abstract: Extrasolar planetary systems display a range of behavior that can be understood in terms of the secular theory of classical celestial mechanics, including the motions of the major axes. Four planet pairs in the seventeen known extrasolar planetary systems with multiple planets (v And, 47 UMa, 55 Cnc, and HD 128311), have trajectories in orbital element space that lie close to the separatrix between libration and circulation. Here we examine the dynamics of the first two, which are not in mean motion resonance. The basics of secular theory are reviewed in order to develop insight into this behavior. The definition of a secular resonance is discussed, correcting misconceptions in the literature; it is not synonymous with libration and is not a commensurability of eigenfrequencies. The behavior of these two near-separatrix systems is evaluated with updated orbital elements by comparing both analytical and numerical results. We find that the apsidal motion from secular theory does not match the predictions from N-body simulations and conclude that first-order secular theory should be used with caution on extrasolar planetary systems. While the existence of one near-separatrix system could be explained simply by chance initial conditions, the fact that there are several is improbable unless some physical process tends to set up systems near the separatrix. Explanations based on an impulsive increase in the eccentricity of one planet are promising, but key issues remain open. © 2006. The American Astronomical Society. All rights reserved.
• Barnes, R., & Greenberg, R. (2006). Stability limits in extrasolar planetary systems. Astrophysical Journal Letters, 647(2 II), L163-L166.
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  Abstract: Two types of stability boundaries exist for any planetary system consisting of one star and two planets. Lagrange stability requires that the planets remain bound to the star, conserves the ordering of the distance from the star, and limits the variations of orbital elements like semimajor axis and eccentricity. Hill stability only requires that the ordering of the planets remain constant; the outer planet may escape to infinity. A simple formula defines a region in orbital element space that is guaranteed to be Hill-stable, although Hill-stable orbits may lie outside the region as well. No analytic criteria describe Lagrange stability. We compare the results of 1000 numerical simulations of planetary systems similar to 47 UMa and HD 12661 with these two types of boundaries. All cases are consistent with the analytic criterion for Hill stability. Moreover, the numerically determined Lagrange boundary lies close to the analytic boundary for Hill stability. This result suggests an analytic formulation that may describe the criterion for Lagrange stability. © 2006. The American Astronomical Society. All rights reserved.
• O'Brien, D. P., Greenberg, R., & Richardson, J. E. (2006). Craters on asteroids: Reconciling diverse impact records with a common impacting population. Icarus, 183(1), 79-92.
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  Abstract: O'Brien and Greenberg [O'Brien, D.P., Greenberg, R., 2005. Icarus 178, 179-212] developed a self-consistent numerical model of the collisional and dynamical evolution of the main-belt and NEA populations that was tested against a diverse range of observational and theoretical constraints. In this paper, we use those results to update the asteroid cratering model of Greenberg et al. [Greenberg, R., Nolan, M.C., Bottke, W.F., Kolvoord, R.A., Veverka, J., 1994. Icarus 107, 84-97; Greenberg, R., Bottke, W.F., Nolan, M., Geissler, P., Petit, J., Durda, D.D., Asphaug, E., Head, J., 1996. Icarus 120, 106-118], and show that the main-belt asteroid population from the O'Brien and Greenberg collisional/dynamical evolution modeling is consistent with the crater records on Gaspra, Ida, Mathilde, and Eros, the four asteroids that have been observed by spacecraft. © 2006 Elsevier Inc. All rights reserved.
• Riley, J., Greenberg, R., & Sarid, A. (2006). Europa's South Pole Region: A sequential reconstruction of surface modification processes. Earth and Planetary Science Letters, 248(3-4), 808-821.
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  Abstract: High-resolution images of a region near the south pole of Europa allow reconstruction of a series of events, tectonic and thermal, that displaced and modified the surface. This approach is complementary to regional geological mapping, which is based on less detailed images, but covers a broader sample of the surface. The events reconstructed in this locale resurfaced about half the area or more, so they probably ranged over a substantial fraction of the age of Europa's surface, which in itself is quite young (< 50 Myr) due to the cumulative effects of such local (or regional) reprocessing events acting over the entire globe. Chaotic terrain was formed (probably by thermal processes) both early and late in the sequence. Trends in changing geological processes that had been inferred from lower-resolution images in regional geological mapping efforts, especially an increase in formation of chaotic terrain with time, which had been widely interpreted as implying a thickening of the ice crust, are not evident in this area. The impression that chaotic terrain is a relatively recent phenomenon may come from the fact that older chaotic terrain can be more difficult to identify, especially in the lower-resolution images used for geological mapping. This study reinforces the conclusion that both chaotic terrain and tectonic features have been formed by competing processes that have resurfaced Europa through much of its geologic history. © 2006 Elsevier B.V. All rights reserved.
• Sarid, A. R., Greenberg, R., & Hurford, T. A. (2006). Crack azimuths on Europa: Sequencing of the northern leading hemisphere. Journal of Geophysical Research E: Planets, 111(8).
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  Abstract: Nonsynchronous rotation of Europa should result in a periodic change in stress azimuths at any given location, which may result in a corresponding change in crack azimuths. An early study seemed to provide evidence for such variation within a crack sequence in the northern trailing hemisphere. However, more recent work with higher-resolution imagery resulted in a more complete sequence of cracks in the region and showed that the azimuth variation with time was not systematic. Similar results were found in the southern leading region. Here we investigate whether the northern leading region contains a crack record, on the basis of crosscutting relationships, that is adequate to indicate nonsynchronous rotation. As in the previous studies of other regions, we find no clear evidence of nonsynchronous rotation within crack azimuth patterns. However, there are other lines of evidence in support of nonsynchronous rotation. Therefore the lack of a continuous pattern of azimuth change with time is likely due to only a few cracks forming per cycle, thus hiding any discernable signal from nonsynchronous rotation in those regions. In that case, the cracks would have formed over numerous cycles of nonsynchronous rotation. The study also reveals a distinctive distribution of crack azimuths, with gaps at certain values. This distribution may reflect a combination of observation selection due to illumination and less-favorable cracking orientations. Copyright 2006 by the American Geophysical Union.
• Greenberg, R., Stone, W. C., Kantor, G., Wettergreen, D., Durda, D. D., & Franke, E. A. (2005). The depthx project: Pioneering technologies for exploration of extraterrestrial aqueous channels. International Astronautical Federation - 56th International Astronautical Congress 2005, 1, 190-200.
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  Abstract: Liquid water is not unique to planet Earth, and the strategy of following the path to water may lead us to extraterrestrial life. On Jupiter's oceanic moon Europa, the most interesting aqueous environments may be within confined channels of liquid through solid strata, such as where Europa's ocean may penetrate hot springs on its floor or tidal cracks in its surface ice. Those settings may provide habitable conditions. Exploration will require development of vehicles with in situ analysis tools for probing such channels. The issues are quite different from those involved in drilling through solids or moving through wide-open submarine spaces. The Deep Phreatic Thermal Explorer (DEPTHX) project is developing such a fully autonomous underwater vehicle, with architecture for intelligent sample collection and in situ discrimination of micro-organisms. Prototypes are being tested in an unusual terrestrial analog to Europa, a site in Mexico with irregular vertical surfaces, an open water column, and potential hydrothermal vents: the 330-meter-deep (or more) Zacatón Cenoté, which contains diverse microbial mats, but remains uncharted. With advanced navigation, detector, sampling, and analysis systems, DEPTHX will be pushing the limits of autonomous robotics. The adaptability and autonomy at the heart of this project will be potentially applicable in a wide range of settings on Earth and in various other sites in the solar system.
• Hurford, T. A., Beyer, R. A., Schmidt, B., Preblich, B., Sarid, A. R., & Greenberg, R. (2005). Flexure of Europa's lithosphere due to ridge-loading. Icarus, 177(2), 380-396.
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  Abstract: Using photoclinometry, topographic profiles across europan ridges have been produced. These profiles allow the identification of bulges in the terrain adjacent to the ridges. The bulges are assumed to have been produced by flexure of the elastic lithosphere due to the load of the ridges, which lie along cracks in the crust. The distance from the crack to these "fore-bulges" depends on the thickness of the elastic plate being flexed. Based on a survey of ridges in Galileo images with resolution
• O'Brien, D. P., & Greenberg, R. (2005). The collisional and dynamical evolution of the main-belt and NEA size distributions. Icarus, 178(1), 179-212.
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  Abstract: The size distribution of main belt of asteroids is determined primarily by collisional processes. Large asteroids break up and form smaller asteroids in a collisional cascade, with the outcome controlled by the strength-size relationship for asteroids. In addition to collisional processes, the non-collisional removal of asteroids from the main belt (and their insertion into the near-Earth asteroid (NEA) population) is critical, and involves several effects: strong resonances increase the orbital eccentricity of asteroids and cause them to enter the inner planet region; chaotic diffusion by numerous weak resonances causes a slow leak of asteroids into the Mars- and Earth-crossing populations; and the Yarkovsky effect, a radiation force on asteroids, is the primary process that drives asteroids into these resonant escape routes. Yarkovsky drift is size-dependent and can modify the main-belt size distribution. The NEA size distribution is primarily determined by its source, the main-belt population, and by the size-dependent processes that deliver bodies from the main belt. All of these effects are simulated in a numerical collisional evolution model that incorporates removal by non-collisional processes. We test our model against a wide range of observational constraints, such as the observed main-belt and NEA size distributions, the number of asteroid families, the preserved basaltic crust of Vesta and its large south-pole impact basin, the cosmic ray exposure ages of meteorites, and the cratering records on asteroids. We find a strength-size relationship for main-belt asteroids and non-collisional removal rates from the main belt such that our model fits these constraints as best as possible within the parameter space we explore. Our results are consistent with other independent estimates of strength and removal rates. © 2005 Elsevier Inc. All rights reserved.
• Richardson Jr., J. E., Melosh, H. J., Greenberg, R. J., & O'Brien, D. P. (2005). The global effects of impact-induced seismic activity on fractured asteroid surface morphology. Icarus, 179(2), 325-349.
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  Abstract: Impact-induced seismic vibrations have long been suspected of being an important surface modification process on small satellites and asteroids. In this study, we use a series of linked seismic and geomorphic models to investigate the process in detail. We begin by developing a basic theory for the propagation of seismic energy in a highly fractured asteroid, and we use this theory to model the global vibrations experienced on the surface of an asteroid following an impact. These synthetic seismograms are then applied to a model of regolith resting on a slope, and the resulting downslope motion is computed for a full range of impactor sizes. Next, this computed downslope regolith flow is used in a morphological model of impact crater degradation and erasure, showing how topographic erosion accumulates as a function of time and the number of impacts. Finally, these results are applied in a stochastic cratering model for the surface of an Eros-like body (same volume and surface area as the asteroid), with craters formed by impacts and then erased by the effects of superposing craters, ejecta coverage, and seismic shakedown. This simulation shows good agreement with the observed 433 Eros cratering record at a Main Belt exposure age of 400 ± 200 Myr, including the observed paucity of small craters. The lowered equilibrium numbers (loss rate = production rate) for craters less than ∼100 m in diameter is a direct result of seismic erasure, which requires less than a meter of mobilized regolith to reproduce the NEAR observations. This study also points to an upper limit on asteroid size for experiencing global, surface-modifying, seismic effects from individual impacts of about 70-100 km (depending upon asteroid seismic properties). Larger asteroids will experience only localized (regional) seismic effects from individual impacts. © 2005 Elsevier Inc. All rights reserved.
• Sarid, A. R., Greenberg, R., Hoppa, G. V., Brown Jr., D. M., & Geissler, P. (2005). Crack azimuths on Europa: The G1 lineament sequence revisited. Icarus, 173(2), 469-479.
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  Abstract: The tectonic sequence in the anti-jovian area covered by regional mapping images from Galileo's orbit E15 is determined from a study of cross-cutting relationships among lineament features. The sequence is used to test earlier results from orbit G1, based on lower resolution images, which appeared to display a progressive change in azimuthal orientation over about 90° in a clockwise sense. Such a progression is consistent with expected stress variations that would accompany plausible non-synchronous rotation. The more recent data provide a more complete record than the G1 data did. We find that to fit the sequence into a continual clockwise change of orientation would require at least 1000° (> 5 cycles) of azimuthal rotation. If due to non-synchronous rotation of Europa, this result implies that we are seeing back further into the tectonic record than the G1 results had suggested. The three sets of orientations found by Geissler et al. now appear to have been spaced out over several cycles, not during a fraction of one cycle. While our more complete sequence of lineament formation is consistent with non-synchronous rotation, a statistical test shows that it cannot be construed as independent evidence. Other lines of evidence do support non-synchronous rotation, but azimuths of crack sequences do not show it, probably because only a couple of cracks form in a given region in any given non-synchronous rotation period. © 2004 Elsevier Inc. All rights reserved.
• Bart, G. D., Turtle, E. P., Jaeger, W. L., Keszthelyi, L. P., & Greenberg, R. (2004). Ridges and tidal stress on Io. Icarus, 169(1), 111-126.
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  Abstract: Sets of ridges of uncertain origin are seen in twenty-nine high-resolution Galileo images, which sample seven locales on Io. These ridges are on the order of a few kilometers in length with a spacing of about a kilometer. Within each locale, the ridges have a consistent orientation, but the orientations vary from place to place. We investigate whether these ridges could be a result of tidal flexing of Io by comparing their orientations with the peak tidal stress orientations at the same locations. We find that ridges grouped near the equator are aligned either north-south or east-west, as are the predicted principal stress orientations there. It is not clear why particular groups run north-south and others east-west. The one set of ridges observed far from the equator (52° S) has an oblique azimuth, as do the tidal stresses at those latitudes. Therefore, all observed ridges have similar orientations to the tidal stress in their region. This correlation is consistent with the hypothesis that tidal flexing of Io plays an important role in ridge formation. © 2004 Elsevier Inc. All rights reserved.
• Greenberg, R. (2004). The evil twin of Agenor: Tectonic convergence on Europa. Icarus, 167(2), 313-319.
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  Abstract: The dilemma of the surface-area budget on Europa is resolved by identification of sites of crustal convergence, which have balanced the continual and common creation of new surface along dilational bands and pull-aparts. Convergence bands are characterized by a distinctive, albeit subdued, morphology. The prominent, unusual lineament Agenor is one of several examples. We also find diametrically opposite Agenor a similar bright linear feature surrounded by markings that allow reconstruction, which shows it to be a convergence feature. Until recently, identification of convergence sites was difficult because these features are subtle and do not exhibit structures (like the Himalayas or plate subduction) familiar from convergence of thick solid crusts on terrestrial planets. © 2003 Elsevier Inc. All rights reserved.
• Richardson, J. E., Melosh, H. J., & Greenberg, R. (2004). Impact-induced seismic activity on asteroid 433 Eros: A surface modification process. Science, 306(5701), 1526-1529.
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  PMID: 15567856;Abstract: High-resolution images of the surface of asteroid 433 Eros revealed evidence of downslope movement of a loose regolith layer, as well as the degradation and erasure of small impact craters (less than ∼100 meters in diameter). One hypothesis to explain these observations is seismic reverberation after impact events. We used a combination of seismic and geomorphic modeling to analyze the response of regolith-covered topography, particularly craters, to impact-induced seismic shaking. Applying these results to a stochastic cratering model for the surface of Eros produced good agreement with the observed size-frequency distribution of craters, including the paucity of small craters.
• Sarid, A. R., Greenberg, R., Hoppa, G. V., Geissler, P., & Preblich, B. (2004). Crack azimuths on Europa: Time sequence in the southern leading face. Icarus, 168(1), 144-157.
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  Abstract: The formation sequence of prominent ridges and other tectonic lineaments on the southern portion of the leading hemisphere of Europa is determined from cross-cutting relationships. These selected features formed fairly recently relative to most of the surface; older lineaments no longer retain clear evidence of cross-cutting. If we assume that this sequence represents the order of formation of cracks that underlie each lineament, and that the orientation of each crack was determined by tidal stress whose azimuth varies monotonically counter-clockwise with time, the azimuth must have rotated more than 740°, which would correspond to the change in tidal stress over two periods of nonsynchronous rotation (relative to the direction of Jupiter). However, that interpretation is not necessarily compelling, because the observed orientations of cross-cutting lineaments are not densely spaced over these cycles; in fact, the sequence would fit nearly as well into an arbitrary model with rotation in the opposite sense from that predicted by theory. This tectonic record may have formed over many more rotational cycles, such that typically only a few cracks form per cycle, which would be consistent with evidence from considerations of cycloidal crack patterns. Sets of cracks that cluster near certain azimuth orientations appear to be parts of globe-encircling lineament systems and may result from other effects, perhaps polar wander that occurred rapidly relative to nonsynchronous rotation. © 2003 Elsevier Inc. All rights reserved.
• Greenberg, R., Hoppa, G. V., Bart, G., & Hurford, T. (2003). Tidal stress patterns on Europa's crust. Celestial Mechanics and Dynamical Astronomy, 87(1-2), 171-188.
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  Abstract: The ice crust of Europa probably floats over a deep liquid-water ocean, and has been continually resurfaced by tectonic and thermal processes driven by tides. Tidal working causes rotational torque, surface stress, internal heating, and orbital evolution. The stress patterns expected on such a crust due to reorientation of the tidal bulge by non-synchronous rotation and due to orbital eccentricity, which introduces periodic ('diurnal') variations in the tide, are shown as global maps. By taking into account the finite rate of crack propagation, global maps are generated of cycloidal features and other distinctive patterns, including the crack shapes characteristic of the wedges region and its antipode on the sub-Jovian hemisphere. Theoretical maps of tidal stress and cracking can be compared with observed tectonics, with the possibility of reconstructing the rotational history of the satellite.
• Greenberg, R., Leake, M. A., Hoppa, G. V., & Tufts, B. R. (2003). Pits and uplifts on Europa. Icarus, 161(1), 102-126.
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  Abstract: A survey of depression and uplift features on Europa, based on Galileo regional mapping images, shows that these features come in a wide range of sizes, with numbers increasing greatly with decreasing size, down to the limits of resolution. Size distributions are similar in the northern leading and southern trailing hemispheres, where they are distinctly different from the southern leading and northern trailing hemispheres, suggesting an oblique, antipodal symmetry pattern, similar to that of chaotic and tectonic terrain. This pattern is suggestive of polar wander. Uplifts are usually polygonal or irregular in shape and rarely are cracked. Patches of chaotic terrain, which we had surveyed earlier, are not included in the current study because their topography is generally unclear, and because there is no a priori known genetic linkage with the pits and uplifts. These results contradict generalizations based on the earlier "pits, spots, and domes" (PSD) taxonomy. Most of the type examples for PSDs were simply patches of chaotic terrain selected from a limited portion of their full size range. The use of the term lenticula to collectively describe PSDs is inconsistent with the IAU definition of lenticula: a small dark spot seen at low resolution. Pits and uplifts do not correlate with lenticulae, although chaos often does. Properties of PSDs that have been widely cited as primary evidence for convective upwelling in thick ice (e.g., that uplifts are generally dome-shaped and often cracked; that pits and domes are regularly spaced; that there is a typical diameter of ∼10 km) were premature and not supported by subsequent data. Most pits and uplifts are less than 10 km across so, if they formed by diapirism or convective upwelling, the sources must have been very shallow, less than 5 km deep. How they actually formed remains unknown. © 2003 Elsevier Science (USA). All rights reserved.
• O'Brien, D. P., & Greenberg, R. (2003). Steady-state size distributions for collisional populations: Analytical solution with size-dependent strength. Icarus, 164(2), 334-345.
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  Abstract: The steady-state population of bodies resulting from a collisional cascade depends on how material strength varies with size. We find a simple expression for the power-law index of the population, given a power law that describes how material strength varies with size. This result is extended to the case relevant for the asteroid belt and Kuiper belt, in which the material strength is described by 2 separate power laws - one for small bodies and one for larger bodies. We find that the power-law index of the small body population is unaffected by the strength law for the large bodies, and vice versa. Simple analytical expressions describe a wave that is superimposed on the large body population because of the transition between the two power laws describing the strength. These analytical results yield excellent agreement with a numerical simulation of collisional evolution. These results will help to interpret observations of the asteroids and KBOs, and constrain the strength properties of those objects. © 2003 Elsevier Inc. All rights reserved.
• Greenberg, R. (2002). Tides and the biosphere of Europa. American Scientist, 90(1), 48-55.
• Greenberg, R., & Geissler, P. (2002). Europa's dynamic icy crust. Meteoritics and Planetary Science, 37(12), 1685-1710.
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  Abstract: Europa's icy crust records active resurfacing by tectonic and thermal processes over tens of millions of years, as rapidity demonstrated by a paucity of craters. Tidal working causes rotational torque, surface stress, internal heating, and orbital evolution, which can explain the formation of observed tectonic crack patterns, ridges, crustal displacement, and chaotic terrain by processes involving connections between the surface and the underlying ocean through cracks, melt sites, and occasional impacts. These processes were recent, and thus most likely continue today. The permeability of the crust allows exchange of materials, including oxidants and exogenic organics from the surface and endogenic substances from the ocean, such that a habitable biosphere might extend to within a few centimeters of the surface. Continual changes in environmental conditions in the ice crust, such as deactivation of individual cracks after thousands of years (due to non-synchronous rotation) and crustal thawing (releasing any trapped organisms), could provide drivers for biological adaptation, as well as opportunity for evolution.
• Greenberg, R., Geissler, P., Hoppa, G., & Tufts, B. R. (2002). Tidal-tectonic processes and their implications for the character of Europa's icy crust. Reviews of Geophysics, 40(2), 1-1 - 1-33.
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  Abstract: Jupiter's satellite Europa is covered by an icy crust, likely overlying a deep, global liquid water ocean. The mutually dependent relationship between the orbit and tidal processes controls Europa's rotation, heating, and stress. Observations of the surface and investigation of tidal-tectonic processes indicate that the dominant types of surface terrain have been created by frequent repeated exposure of an underlying ocean to the surface. Surface lineaments, including ubiquitous cycloidal (chains of arcs) features, are correlated with tidal stress patterns, demonstrating that they form by crustal cracking but only if a substantial ocean is present to give adequate tidal amplitude. Stratigraphy of tectonic features, combined with models of how and where they formed, demonstrates a moderate rate of nonsynchronous rotation. Tidal driving of strike-slip displacement (in which surface areas shear passed one another) by daily (85 hours on Europa) tidal stresses suggests that cracks penetrate to a liquid layer. The characteristic ridge sets that cover tectonic terrain may be built by tidal pumping of fluid and slush to the surface on a daily basis. Widespread tectonic dilation creates new surface as material rises from below. In addition to those tectonic terrains, nearly half of the surface is chaotic terrain, with morphology and other characteristics indicative of melt-through from below, consistent with the tectonic indications of thin ice and plausible tidal heating rates. Formation of both chaotic and tectonic terrains has continually resurfaced the satellite, while connecting the ocean to the surface. Surface colorants correlate with locations where ocean water reached the surface, such as along large-scale ridge systems and around chaotic terrain. As a result of tides, liquid water may have bathed crustal cracks and surfaces with heat, transporting and mixing substances vertically. Such daily transported materials would include substances from the oceanic reservoir, oxidants, and fuels created at the surface, as well as any organisms and their chemical products. Thus a variety of habitable environments likely exist in the crust. Deactivation of individual cracks after thousands of years (due to the tidally driven nonsynchronous rotation) constrains processes like ridge building and strike-slip displacement and would require any living organisms to adapt to such change.
• Greenberg, R., Hoppa, G. V., Geissler, P., Sarid, A., & Tufts, B. R. (2002). The rotation of Europa. Celestial Mechanics and Dynamical Astronomy, 83(1-4), 35-47.
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  Abstract: Theoretical predictions of non-synchronous rotation and of polar wander on Europa have been tested by comparing tectonic features observed in Voyager and Galileo spacecraft images with tidal stresses. Evidence for non-synchronous rotation comes from studying changes in global scale lineaments formed over time, from the character of strike-slip faults, and from comparison of distinctively shaped cycloidal cracks with the longitudes at which such shapes should have formed, in theory. The study of cycloids constrains the rotation period (relative to the direction of Jupiter) to less than 250 000 years, while direct comparison of the orientation of Europa in Voyager and Galileo images shows the rotation is slow, with a period of > 12 000 years. Comparison of strike-slip faults with their theoretical locations of formation provides evidence for substantial polar wander, supported by the distribution of various thermally produced features.
• O'Brien, D. P., Geissler, P., & Greenberg, R. (2002). A melt-through model for chaos formation on Europa. Icarus, 156(1), 152-161.
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  Abstract: The character of chaotic terrain on Europa is consistent with its formation by the melting of a thin conducting ice shell from below. Tidal dissipation can provide adequate energy for such a process. For example, only a few percent of Europa's predicted tidal heat, spread over a region 200 km in diameter, can lead to large melt regions within a few tens of thousands of years. Stronger, more localized concentrations result in melt-through in significantly shorter times (i.e., a few hundred years). The time scale for melt-through is shorter than the time scale for the solid-state viscous inflow of ice by several orders of magnitude. In general, modest concentrations of tidal heat can melt ice away faster than viscous inflow, leading to melt-through. A mechanism to transmit these heat concentrations through the ocean is required for this model. Such heat transport could be the result of convective plumes in the ocean driven by seafloor volcanism or by the destabilization of a stratified ocean. © 2002 Elsevier Science (USA).
• Sarid, A. R., Greenberg, R., Hoppa, G. V., Hurford, T. A., Tufts, B. R., & Geissler, P. (2002). Polar wander and surface convergence of Europa's ice shell: Evidence from a survey of strike-slip displacement. Icarus, 158(1), 24-41.
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  Abstract: Two global issues regarding Europa are addressed by a survey of strike-slip faults. First, a common type of terrain that appears to represent convergent sites of surface removal, which may help compensate for substantial widespread dilation along tectonic bands elsewhere, thus helping resolve the problem of conserving global surface area, is identified. Second, evidence for polar wander may provide the first confirmation of that theoretically predicted phenomenon. These results, among others, come from an extensive survey of strike-slip faults over the portion of the surface where Galileo images at 200-m/pixel resolution were obtained for regional mapping purposes. The images cover two broad swaths that run from the far north to the far south, one in the leading hemisphere and the other in the trailing hemisphere. Among the faults that have been mapped are a fault 170 km long with a strike-slip offset of 83 km, the greatest yet identified on Europa, and a quasi-circular strike-slip fault that surrounds a 500-km-wide plate, which has undergone rotation as a rigid unit. Reconstruction of specific examples of strike slip reveals sites of lateral convergence. Because Europa is unique in many ways, these sites are not similar to compression features on other bodies, which may explain why they had previously been difficult to identify. The distribution of strike slip in both hemispheres, when compared with predictions of the theory of tidal walking, provides evidence for polar wander: The crust of Europa appears to have slid as a single unit relative to the spin axis, such that the site on the crust that was previously at the north rotational pole has wandered, probably during the last few million years, to a location currently in the leading hemisphere, about 30° away from the spin axis. Such polar wander probably also explains symmetry patterns in the distribution of chaotic terrain, pits, and uplift features. © 2002 Elseviier Science (USA).
• Spitale, J., & Greenberg, R. (2002). Numerical evaluation of the general Yarkovsky effect: Effects on eccentricity and longitude of periapse. Icarus, 156(1), 211-222.
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  Abstract: J.N. Spitale and R. Greenberg (2001, Icarus 149, 222-234) developed a nonlinearized, finite-difference solution to the heat equation that yields orbital rates of change due to the Yarkovsky effect for small, spherical, bare-rock asteroids and used it to investigate changes in semimajor axis caused by the Yarkovsky effect. Here, we present results for changes in eccentricity and longitude of periapse. These results may be useful as benchmarks for simplified analytical solutions. Moreover, we explore a range of parameters, some of which are inaccessible to most other approaches. Instantaneous rates can be quite fast: For a 1-m scale body rotating with a 5-h period, de/dt can be as fast as 0.1 per million years (da/dt rates for similar test bodies were reported in J.N. Spitale and R. Greenberg (2001, Icarus 149, 222-234)). For more typical rotation periods, these rates would be considerably slower. Output from our calculation method could be used in simulations of asteroid population evolution such as that by W.F. Bottke, D.P. Rubincam, and J.A. Burns (2000, Icarus 145, 301-331). On long time scales, impacts would randomize the spin axis before significant orbital evolution could occur. Nevertheless, occasional favorable rotation states might persist long enough for substantial eccentricity changes to accumulate (1) if the body is decoupled from the main belt (e.g., many near-Earth asteroids), (2) if the population of very small (mm-scale) main-belt impactors is less than expected, or (3) if our numerical results are scaled up to km-size bodies. © 2002 Elsevier Science (USA).
• Greenberg, R., & Tufts, B. R. (2001). Infecting other worlds. American Scientist, 89(4), 296-299.
• Hoppa, G. V., Greenberg, R., Riley, J., & Tufts, B. R. (2001). Observational Selection Effects in Europa Image Data: Identification of Chaotic Terrain. Icarus, 151(2), 181-189.
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  Abstract: The identification of terrains on images of Europa is highly dependent on the image resolution and illumination angle, both of which are quite variable due to the observational selection constraints of the Galileo mission. Higher resolution images allow for the identification of smaller patches of chaotic terrain, which are indistinguishable from surrounding tectonic terrain in most other Galileo imagery. The incidence angle of light also introduces an observational bias in the identification of chaotic terrain: Images acquired closer to Europa's terminator generally reveal more chaos features than images taken with higher Sun. Proper accounting for selection effects has implications for the size distribution and chronology of chaotic terrain, and the geologic history of the satellite: At ∼200 m/pixel 30% of Europa's surface appears as chaotic terrain, but with the effects of lighting and resolution taken into account, the portion is 40% or more; bias correction eliminates the 10-km-size peak that had been the basis of an earlier thick-ice model; "mottled terrain" and "lenticulae" appear to be the same as chaotic terrain, even though they have been mapped separately in the past; and accounting for how observational bias affects the freshness of appearance of chaotic terrain and the identification of tectonic modification of chaotic terrain demonstrates that the formation of tectonic and chaotic terrain have been concurrent processes throughout Europa's surface history. © 2001 Academic Press.
• Hoppa, G. V., Tufts, B. R., Greenberg, R., Hurford, T. A., O'Brien, D., & Geissler, P. E. (2001). Europa's rate of rotation derived from the tectonic sequence in the Astypalaea region. Icarus, 153(1), 208-213.
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  Abstract: The most recent visible tectonic features in the Astypalaea region in the southern hemisphere of Europa are a set of cycloidal ridges, three of which have cross-cutting relationships that define a time sequence for their formation. The longitudes at which each of these features formed, which may be different from the current location due to rotation of Europa, are constrained by models of their formation. Reconciling the time sequence with the inferred longitudes of formation appears to require that
• Nolan, M. C., Asphaug, E., Greenberg, R., & Melosh, H. (2001). Impacts on asteroids: Fragmentation, regolith transport, and disruption. Icarus, 153(1), 1-15.
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  Abstract: We use a numerical hydrocode model to examine the outcomes of various size impacts into targets the sizes of Asteroids 951 Gaspra and 243 Ida, which were imaged by the Galileo spacecraft. A shock wave fractures the asteroid in advance of crater excavation flow; thus, for impactors larger than 100 m impacting at 5.3 km s-1, tensile strength is unimportant in these bodies, whether they are initially intact or are "rubble piles." Because of the shock-induced fracture, impact results are controlled by gravity. Therefore these asteroids are much more resistant to catastrophic disruption than predicted by previous estimates, which had assumed that strength was controlling these processes for rock targets. The rubble and regolith produced by this fracture can be "jolted" by the impact, redistributing surface material and globally erasing craters. The crater ejecta can produce tens of meters of regolith per large event, likely consisting of 100-m-size boulders mixed with smaller particles. The response of kilometer-size asteroids to impacts is qualitatively different from that of few-centimeter targets in terrestrial experiments, making prediction based on such experiments difficult. The compositional distribution of delivered meteorites depends on the outcomes of such asteroid impacts. © 2001 Academic Press.
• Spitale, J., & Greenberg, R. (2001). Numerical Evaluation of the General Yarkovsky Effect: Effects on Semimajor Axis. Icarus, 149(1), 222-234.
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  Abstract: The Yarkovsky effect may play a key role in the orbital evolution of asteroids and near-Earth objects. To evaluate the acceleration under a wide range of conditions, a three-dimensional finite-difference solution to the heat equation is applied to homogeneous, spherical stony bodies with 1-, 10-, and 100-m diameters. This approach employs neither the linearized boundary conditions, the plane-parallel heat flow approximation, nor the assumption of fast rotation used in earlier work. Thus we can explore a wide range of orbital elements and physical properties. Our work agrees well with earlier results in the regimes where their approximations are valid. We investigate a wide range of spin states, including both the "seasonal" (very fast rotation) and "diurnal" (zero obliquity) extremes of the Yarkovsky effect. We find that, for orbits with high eccentricity, the semimajor axis can change much faster than for circular orbits. For such orbits, the orientation of the rotation axis with respect to the direction of pericenter is critical in determining the evolution. A stony main-belt asteroid of diameter 1 m on a high-eccentricity orbit could change its semimajor axis by more than 1 AU in 1.5 Myr. © 2001 Academic Press.
• Belton, M. J., Anger, C. D., Carr, M. H., Chapman, C. R., Davies, M. E., Greeley, R., Greenberg, R. J., III, J. H., Klaasen, K. P., Neukum, G., Pilcher, C. B., Thomas, P. C., Veverka, J., Gierasch, P. J., Ingersoll, A. P., Fanale, F. P., McEwen, A. S., Morrison, D., Schubert, G., , Beebe, R. F., et al. (2000). Results of the Galileo solid state imaging (SSI) experiment. Advances in Space Research, 26(10), 1641-1647.
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  Abstract: We present a brief synopsis of the nature of SSI data that was taken during the nominal Galileo mission. Significant results are briefly described. These include evidence for geologic activity on Europa that supports the hypothesis of liquid water under a thin ice shell; a demonstration of the ubiquitous presence of a thick crust and high temperature silicate volcanism on Io; a demonstration that, together with results from other experiments, the molecular composition in Jupiter's atmosphere depends on local meteorology; the resolution of lightning flashes and their association with cloud type; the identification of cloud structures at or below the 4-bar level that are presumably composed of water; the location and morphology of visible aurorae in Jupiter's Northern hemisphere; improved geodetic control nets for the Galilean satellites; measurement of the sizes and shapes of Metis, Adrastea and Amalthea; new features of the main and gossamer rings; discovery of visible Io-glow and surface hotspots especially at the sub- and anti-jove regions on Io; evidence for solid state convection in Europa's ice shell and its probable non-synchronous rotation; the discovery of visible Europa-glow; evidence that tectonism has dominated the resurfacing of bright grooved terrain on Ganymede; the discovery of a thick blanket of dark material on Callisto's surface and that mass-wasting and sublimation are significant degradation processes on this satellite. © 2000 COSPAR. Published by Elsevier Science Ltd. All rights reserved.
• Greenberg, R., Geissler, P., Tufts, B. R., & Hoppa, G. V. (2000). Habitability of Europa's crust: The role of tidal-tectonic processes. Journal of Geophysical Research E: Planets, 105(E7), 17551-17562.
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  Abstract: Jupiter's satellite Europa has been identified as one of the most likely sites for life in the solar system. The tidal-tectonic processes that appear to have governed Europa's geology seem to require interaction with an ocean under only a very thin crust, providing a variety of evolving environmental niches. The mutually dependent relationship between orbital evolution and tidal processes in turn controls Europa's rotation, heating, and stress. Surface lineaments are correlated with global stress patterns, demonstrating that they form by crustal cracking, but only if a substantial ocean is present to give adequate tidal amplitude. Tidal driving of strike-slip faulting indicates that cracks penetrate to a fluid layer, which is possible only with a very thin ice crust. The characteristic ridge sets that cover tectonic terrain are likely built by tidal pumping of fluid and slush to the surface on a daily basis. Widespread tectonic dilation creates new surface as material rises from below. Chaotic terrain has morphology and other characteristics indicative of melt-through from below. Surface colorants correlate with locations, such as along large-scale ridge systems and around chaotic terrain, where ocean water reached the surface. This model implies that as a result of tides, liquid water regularly bathed crustal cracks and surfaces with heat and whatever nutrients are included in the oceanic chemistry, creating a variety of habitable environments. The processes were recent and thus most likely continue today. Longer-term evolution of environmental conditions provided the need for adaptation and opportunity for evolution. Copyright 2000 by the American Geophysical Union.
• Hoppa, G., Greenberg, R., Tufts, B. R., Geissler, P., Phillips, C., & Milazzo, M. (2000). Distribution of strike-slip faults on Europa. Journal of Geophysical Research E: Planets, 105(E9), 22617-22627.
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  Abstract: Study of four different regions on Europa imaged by the Galileo spacecraft during its first 15 orbits has revealed 117 strike-slip faults. Europa appears to form preferentially right-lateral faults in the southern hemisphere and left-lateral faults in the northern hemisphere. This observation is consistent with a model where diurnal tides due to orbital eccentricity drive strike-slip motion through a process of "walking," in which faults open and close out of phase with alternating right-and left-lateral shear. Lineaments that record both left-and right-lateral motion (e.g., Agave Linea) may record the accommodation of compression in nearby chaotic zones. Nearly all identified strike-slip faults were associated with double ridges or bands, and few were detected along ridgeless cracks. Thus the depth of cracks without ridges does not appear to have penetrated to the low-viscosity decoupling layer, required for diurnal displacement, but cracks that have developed ridges do extend down to such a level. This result supports a model for ridge formation that requires cracks to penetrate to a decoupling layer, such as a liquid water ocean. Copyright 2000 by the American Geophysical Union.
• Hurford, T. A., & Greenberg, R. (2000). Tidal evolution by elongated primaries: Implications for the ida/dactyl system. Geophysical Research Letters, 27(11), 1595-1598.
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  Abstract: The age of asteroid 243 Ida's satellite Dactyl is < 100 myr according to the conventional formula for the rate of tidal evolution outward from Ida, contrary to estimates based on the likely age of the asteroid itself (~ 1 byr). We investigate whether this discrepancy may be due to the conventional formula for tidal evolution being based on a spherical primary, whereas Ida is actually highly elongated. A model for Ida consisting of three spheres connected by damped springs is used to estimate what effects the elongation may have on the tidal dissipation. In fact, our model gives torques and energy dissipation similar to that in an equivalent sphere, indicating that the spherical model gives reasonable results even when applied to an elongated or irregular body.
• Riley, J., Hoppa, G. V., Greenberg, R., Tufts, B. R., & Geissler, P. (2000). Distribution of chaotic terrain on Europa. Journal of Geophysical Research E: Planets, 105(E9), 22599-22615.
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  Abstract: The locations of chaotic terrain are mapped over all regions of Europa for which we have adequate resolution (∼200 m or better) and appropriate lighting geometry (Sun angle
• Tufts, B. R., Greenberg, R., Hoppa, G., & Geissler, P. (2000). Lithospheric Dilation on Europa. Icarus, 146(1), 75-97.
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  Abstract: Lithospheric dilation on Europa has occurred at ridges, bands, and various hybrid lineaments on a global scale over a large part of the geological age of the surface. Dilational ridges (Class 2 in the R. Greenberg et al. (1998, Icarus135, 64-78) taxonomy) are elevated, are usually a few kilometers across, and may have a lineated or hummocky interior and a pronounced medial groove. Bands are lower and usually wider than Class 2 ridges, and may have a lineated interior with no prominent medial groove. Some lineaments have characteristics of both ridges and bands. The character of Class 2 ridges, bands, and hybrid forms suggests that they are dilational gaps in the lithosphere, filled from below, and that they constitute a morphological continuum with Class 2 ridges and bands as end-members. These relationships may be explained by a model in which external forcing superimposes a secular dilation on the tidal cycle that opens and closes cracks each Europan day, resulting in incomplete closure with accumulation and possible extrusion of new ice fill. Where the lineament ultimately falls on the morphological continuum - especially how much it is elevated above ambient terrain - depends upon the ratio of daily secular dilation to the amplitude of the cyclic tidal separation. We call this ratio the "dilation quotient". Changes in the dilation quotient during the active life of the lineament will create variable lineament forms. One driver for dilation is tidal "walking" of strike-slip faults, which dilates linked nonparallel cracks. That process is prominent in the 800-km-long strike-slip fault Astypalaea Linea. A subsurface liquid water ocean allows the decoupling needed for horizontal displacements and is the source for the ice that fills the dilated lineaments. © 2000 Academic Press.
• Greenberg, R., Hoppa, G. V., Tufts, B. R., Geissler, P., Riley, J., & Kadel, S. (1999). Chaos on Europa. Icarus, 141(2), 263-286.
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  Abstract: The characteristics of chaos regions on Europa suggest they may be sites of melt-through from below. They are wide ranging in size, location, and age. The largest are hundreds of kilometers across. Most are similar to Conamara with a matrix reminiscent of frozen slush and often rafts of preexisting crust. Edges are of two types: ramps, perhaps the tapering of crustal thickness to zero, or cliffs, where rafts appear to have broken clear from the shore. The small features called lenticulae generally appear to be small chaoses with textured matrix and occasional rafts, and many domes may be small chaoses raised by isostatic compensation following refreezing of the crust. The extent of chaoses often appears to be limited by ridge systems with the coastline parallel and set back by a distance comparable to the width of the ridge system. Preexisting ridges often survive as causeways or chains of rafts. Boundaries of chaoses are apparently not controlled by preexisting cracks, consistent with formation by a thermal, rather than mechanical, process. Ridges may thicken the crust such that melt-through is more likely (but not always) between ridge systems. Subsequent cracks and ridges form across preexisting chaoses, ranging from fresh cases with few cracks or ridges across them (with paths somewhat jagged as they meander among rafts) to heavily dissected examples. Isolated tilted raft-like blocks surrounded by densely ridged terrain may be relics of former chaotic terrain. Thus two fundamental resurfacing processes have alternated over Europa's geological history: melt-through (at various places and times) forming chaos terrain, and tectonic cracking and dilation building densely ridged and banded terrain. Mapping of chaos features based on morphology at 200 m shows that they correlate, albeit imperfectly, with dark regions in global (2-km resolution) mosaics (except dark regions due to ridge margins or craters). Extrapolating from our mapping of the 5% of Europa covered by appropriate images, at least 18% of the surface of Europa is fresh appearing chaos, an additional 4% is slightly modified chaos, and much more older chaotic terrain has been overprinted by tectonic structures. Considerable area has been available globally to accommodate the expansion of crust that occurs along extensional ridges and bands. Chaos ubiquity suggests that europan geology has been dominated by the effects of having liquid water under a very thin ice shell, with chaos regions being widespread indicators of occasional zero shell thickness. © 1999 Academic Press.
• Hoppa, G. V., Tufts, B. R., Greenberg, R., & Geissler, P. E. (1999). Formation of cycloidal features on Europa. Science, 285(5435), 1899-1902.
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  PMID: 10489365;Abstract: Cycloidal patterns are widely distributed on the surface of Jupiter's moon Europa. Tensile cracks may have developed such a pattern in response to diurnal variations in tidal stress in Europa's outer ice shell. When the tensile strength of the ice is reached, a crack may occur. Propagating cracks would move across an ever-changing stress field, following a curving path to a place and time where the tensile stress was insufficient to continue the propagation. A few hours later, when the stress at the end of the crack again exceeded the strength, propagation would continue in a new direction. Thus, one arcuate segment of the cycloidal chain would be produced during each day on Europa. For this model to work, the tensile strength of Europa's ice crust must be less than 40 kilopascals, and there must be a thick fluid layer below the ice to allow sufficient tidal amplitude.
• Hoppa, G., Greenberg, R., Geissler, P., Tufts, B. R., Plassmann, J., & Durda, D. D. (1999). Rotation of Europa: Constraints from Terminator and Limb Positions. Icarus, 137(2), 341-347.
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  Abstract: Tidal torques on Europa due to Jupiter would tend to drive Europa's rotation to a rate slightly faster than synchronous, unless synchroneity is maintained by a permanent asymmetry in Europa's mass distribution. Moreover, global fracture patterns on Europa's surface, as well as the complex fine-scale tectonics revealed by Galileo high-resolution imagery, may be evidence of stress due to nonsynchronous rotation. A direct measurement of Europa's rotation rate is made from the positions of surface features relative to the terminator in a Galileo image and comparing the results with similar measurements of the positions of the same features relative to the terminator in a Voyager 2 image taken 17 years earlier. Any nonsynchronous rotation must be slower than a few tenths of a degree in 17 years (a complete rotation in >104years) relative to the direction of Jupiter. © 1999 Academic Press.
• Hoppa, G., Tufts, B. R., Greenberg, R., & Geissler, P. (1999). Strike-slip faults on Europa: Global shear patterns driven by tidal stress. Icarus, 141(2), 287-298.
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  Abstract: Diurnal tides due to orbital eccentricity may drive strike-slip motion on Europa through a process of "walking" in which faults open and close out of phase with alternate right- and left-lateral shear. Mapping of five different regions on Europa has revealed 121 strike-slip faults, including Astypalaea Linea, a 800-km-long fault with 42 km of right-lateral offset. At high southern latitudes near Astypalaea Linea all of the strike slip faults identified were right-lateral. Europa appears to preferentially form right-lateral faults in the southern hemisphere and left-lateral faults in the northern hemisphere, consistent with tidal walking. At the five locations, nonsynchronous rotation explains the azimuthal orientations and distribution of sense of shear, which fit formation ∼60° to 90° west of their current positions. Alternatively, stress due to differential rotation might also explain the observed shear patterns. Nearly all identified strike-slip faults were associated with double ridges or bands, but few were detected along ridgeless cracks (even older ones). Thus, cracks without ridges may not have penetrated to a decoupling layer, consistent with the models for ridge formation that require cracks to penetrate to a liquid water ocean. © 1999 Academic Press.
• Pappalardo, R. T., Belton, M. J., Breneman, H. H., Carr, M. H., Chapman, C. R., Collins, G. C., Denk, T., Fagents, S., Geissler, P. E., Giese, B., Greeley, R., Greenberg, R., Head, J. W., Helfenstein, P., Hoppa, G., Kadel, S. D., Klaasen, K. P., Klemaszewski, J. E., Magee, K., , McEwen, A. S., et al. (1999). Does Europa have a subsurface ocean? Evaluation of the geological evidence. Journal of Geophysical Research E: Planets, 104(E10), 24015-24055.
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  Abstract: It has been proposed that Jupiter's satellite Europa currently possesses a global subsurface ocean of liquid water. Galileo gravity data verify that the satellite is differentiated into an outer H2O layer about 100 km thick but cannot determine the current physical state of this layer (liquid or solid). Here we summarize the geological evidence regarding an extant subsurface ocean, concentrating on Galileo imaging data. We describe and assess nine pertinent lines of geological evidence: impact morphologies, lenticulae, cryovolcanic features, pull-apart bands, chaos, ridges, surface frosts, topography, and global tectonics. An internal ocean would be a simple and comprehensive explanation for a broad range of observations; however, we cannot rule out the possibility that all of the surface morphologies could be due to processes in warm, soft ice with only localized or partial melting. Two different models of impact flux imply very different surface ages for Europa; the model favored here indicates an average age of ∼50 Myr. Searches for evidence of current geological activity on Europa, such as plumes or surface changes, have yielded negative results to date. The current existence of a global subsurface ocean, while attractive in explaining the observations, remains inconclusive. Future geophysical measurements are essential to determine conclusively whether or not there is a liquid water ocean within Europa today. Copyright 1999 by the American Geophysical Union.
• Raphael, J., Tobias, S., & Greenberg, R. (1999). Research experience as a component of science and mathematics teacher preparation. Journal of Science Teacher Education, 10(2), 147-158.
• Tufts, B. R., Greenberg, R., Hoppa, G., & Geissler, P. (1999). Astypalaea Linea: A Large-Scale Strike-Slip Fault on Europa. Icarus, 141(1), 000-.
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  Abstract: Astypalaea Linea is an 810-km strike-slip fault, located near the south pole of Europa. In length, it rivals the San Andreas Fault in California, and it is the largest strike-slip fault yet known on Europa. The fault was discovered using Voyager 2 images, based upon the presence of familiar strike-slip features including linearity, pull-aparts, and possible braids, and upon the offset of multiple piercing points. Fault displacement is 42 km, right-lateral, in the southern and central parts and probably throughout. Pull-aparts present along the fault trace probably are gaps in the lithosphere bounded by vertical cracks, and which opened due to fault motion and filled with material from below. Crosscutting relationships suggest the fault to be of intermediate relative age. The fault may have initiated as a crack due to tension from combined diurnal tides and nonsynchronous rotation, according to the tectonic model of R. Greenberg et al. (1998a, Icarus135, 64-78). Under the influence of varying diurnal tides, strike-slip offset may have occurred through a process called "walking," which depends upon an inelastic lithospheric response to displacement. Alternatively, fault displacement may have been driven by currents in the theorized Europan ocean, which may have created simple shear structures such as braids. The discovery of Astypalaea Linea extends the geographical range of lateral motion on Europa. Such motion requires the presence of a decoupling zone of ductile ice or liquid water, a sufficiently rigid lithosphere, and a mechanism to consume surface area. © 1999 Academic Press.
• Carr, M. H., J., M., Chapman, C. R., Davies, M. E., Geissler, P., Greenberg, R., McEwen, A. S., Tufts, B. R., Greeley, R., Sullivan, R., Head, J. W., Pappalardo, R. T., Klaasen, K. P., Johnson, T. V., Kaufman, J., Senske, D., Moore, J., Neukum, G., Schubert, G., , Burns, J. A., et al. (1998). Evidence for a subsurface ocean on Europa. Nature, 391(6665), 363-365.
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  PMID: 9450749;Abstract: Ground-based spectroscopy of Jupiter's moon Europa, combined with gravity data, suggests that the satellite has an icy crust roughly 150 km thick and a rocky interior. In addition, images obtained by the Voyager spacecraft revealed that Europa's surface is crossed by numerous intersecting ridges and dark bands (called lineae) and is sparsely cratered, indicating that the terrain is probably significantly younger than that of Ganymede and Callisto. It has been suggested that Europa's thin outer ice shell might be separated from the moon's silicate interior by a liquid water layer, delayed or prevented from freezing by tidal heating; in this model, the lineae could be explained by repetitive tidal deformation of the outer ice shell. However, observational confirmation of a subsurface ocean was largely frustrated by the low resolution (>2 km per pixel) of the Voyager images. Here we present high-resolution (54 m per pixel) Galileo spacecraft images of Europa, in which we find evidence for mobile 'icebergs'. The detailed morphology of the terrain strongly supports the presence of liquid water at shallow depths below the surface, either today or at some time in the past. Moreover, lower- resolution observations of much larger regions suggest that the phenomena reported here are widespread.
• Durda, D. D., Greenberg, R., & Jedicke, R. (1998). Collisional Models and Scaling Laws: A New Interpretation of the Shape of the Main-Belt Asteroid Size Distribution. Icarus, 135(2), 431-440.
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  Abstract: Numerical models of the collisional evolution of the main-belt asteroids lead to a new interpretation of the shape of the observed size distribution. Using recent hydrocode predictions of shattering and disruption criteria as starting points, we adjust the size-strength scaling relation for asteroidal strengths within our collisional model until a best least-squares fit to the observed size distribution is obtained. We show for the first time general agreement between the predictions of hydrocodes, results of numerical collisional models, and the observed size distribution of the main-belt population. © 1998 Academic Press.
• Geissler, P. E., Greenberg, R., Hoppa, G., Helfenstein, P., McEwen, A., Pappalardo, R., Tufts, R., Ockert-Bell, M., Sullivan, R., Greeley, R., Belton, M. J., Denk, T., Clark, B., Burns, J., & Veverka, J. (1998). Evidence for non-synchronous rotation of Europa. Nature, 391(6665), 368-370.
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  PMID: 9450751;Abstract: Non-synchronous rotation of Europa was predicted on theoretical grounds, by considering the orbitally averaged torque exerted by Jupiter on the satellite's tidal bulges. If Europa's orbit were circular, or the satellite were comprised of a frictionless fluid without tidal dissipation, this torque would average to zero. However, Europa has a small forced eccentricity e ≃ 0.01 (ref. 2), generated by its dynamical interaction with Io and Ganymede, which should cause the equilibrium spin rate of the satellite to be slightly faster than synchronous. Recent gravity data suggest that there may be a permanent asymmetry in Europa's interior mass distribution which is large enough to offset the tidal torque; hence, if non-synchronous rotation is observed, the surface is probably decoupled from the interior by a subsurface layer of liquid or ductile ice. Non-synchronous rotation was invoked to explain Europa's global system of lineaments and an equatorial region of rifting seen in Voyager images. Here we report an analysis of the orientation and distribution of these surface features, based on initial observations made by the Galileo spacecraft. We find evidence that Europa spins faster than the synchronous rate (or did so in the past), consistent with the possibility of a global subsurface ocean.
• Geissler, P. E., Greenberg, R., Hoppa, G., McEwen, A., Tufts, R., Phillips, C., Clark, B., Ockert-Bell, M., Helfenstein, P., Burns, J., Veverka, J., Sullivan, R., Greeley, R., Pappalardo, R. T., III, J. H., Belton, M. J., & Denk, T. (1998). Evolution of Lineaments on Europa: Clues from Galileo Multispectral Imaging Observations. Icarus, 135(1), 107-126.
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  Abstract: Four distinct classes of lineaments can be described on the basis of Galileo's improved spectral and spatial coverage of Europa: (1) incipient cracks are narrow (
• Greeley, R., Sullivan, R., Klemaszewski, J., Homan, K., W., J., Pappalardo, R. T., Veverka, J., Clark, B. E., Johnson, T. V., Klaasen, K. P., Belton, M., Moore, J., Asphaug, E., Carr, M. H., Neukum, G., Denk, T., Chapman, C. R., Pilcher, C. B., Geissler, P. E., , Greenberg, R., et al. (1998). Europa: Initial Galileo Geological Observations. Icarus, 135(1), 4-24.
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  Abstract: Images of Europa from the Galileo spacecraft show a surface with a complex history involving tectonic deformation, impact cratering, and possible emplacement of ice-rich materials and perhaps liquids on the surface. Differences in impact crater distributions suggest that some areas have been resurfaced more recently than others; Europa could experience current cryovolcanic and tectonic activity. Global-scale patterns of tectonic features suggest deformation resulting from non-synchronous rotation of Europa around Jupiter. Some regions of the lithosphere have been fractured, with icy plates separated and rotated into new positions. The dimensions of these plates suggest that the depth to liquid or mobile ice was only a few kilometers at the time of disruption. Some surfaces have also been upwarped, possibly by diapirs, cryomagmatic intrusions, or convective upwelling. In some places, this deformation has led to the development of chaotic terrain in which surface material has collapsed and/or been eroded. © 1998 Academic Press.
• Greenberg, R. (1998). How asteroids come to earth. Science, 281(5385), 1971-1973.
• Greenberg, R., Geissler, P., Hoppa, G., Tufts, B. R., Durda, D. D., Pappalardo, R., Head, J. W., Greeley, R., Sullivan, R., & Carr, M. H. (1998). Tectonic Processes on Europa: Tidal Stresses, Mechanical Response, and Visible Features. Icarus, 135(1), 64-78.
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  Abstract: Europa's orbital eccentricity, driven by the resonance with Io and Ganymede, results in "diurnal" tides (3.5-day period) and possibly in nonsynchronous rotation. Both diurnal variation and nonsynchronous rotation can create significant stress fields on Europa's surface, and both effects may produce cracking. Patterns and time sequences of apparent tectonic features on Europa include lineaments that correlate with both sources of stress, if we take into account nonsynchronous rotation, after initial crack formation, by amounts ranging up to several tens of degrees. For example, the crosscutting time sequence of features in the Cadmus and Minos Linea region is consistent with a combined diurnal and nonsynchronous tensile-stress field, as it evolves during tens of degrees of nonsynchronous rotation. Constraints on the rotation rate from comparing Voyager and Galileo images show that significant rotation requires 104yr, but could be fast enough to have allowed significant rotation since the last global resurfacing, even if such resurfacing was as recent as a few million years ago. Once cracking is initiated, diurnal tides work cracks so that they open and close daily. Although the daily effect is small, over 105yr double ridges could plausibly be built along the cracks with sizes and morphologies consistent with observed structures, according to a model in which underlying liquid water fills the open cracks, partially freezes, and is extruded during the daily closing of the cracks. Thus, several lines of observational and theoretical evidence can be integrated if we assume nonsynchronous rotation and the existence of a liquid water layer. © 1998 Academic Press.
• Greenberg, R., Raphael, J., Keller, J. L., & Tobias, S. (1998). Teaching High School Science Using Image Processing: A Case Study of Implementation of Computer Technology. Journal of Research in Science Teaching, 35(3), 297-327.
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  Abstract: An in-depth case study of teachers' use of image processing (a state-of-the-art computer technology used by research scientists) in biology, earth science, and physics classes within one high school science department explored issues surrounding technology implementation. The study, conducted within a districtwide, schoolwide, and classroom context, explored four areas related to the teacher's adoption of image processing: (a) teachers' background with computers outside of instructional use, (b) teachers' attitudes toward educational technology and insights gained from their experience using computers within the science curriculum, (c) training and perceived influence of district and school administrators, and (d) teachers' classroom and computer lab practices. The following factors were deemed critical in teachers' decision to use or not use image processing with their students: (a) time to thoroughly explore and master the technology so they could use it with students to explore science concepts; (b) classroom management skills specific to technology use; (c) perception of the teaching value of the technology; (d) perception of the reasonableness of administrators' expectations for technology use; and (e) understanding of how to implement inquiry-based science teaching, independent of technology issues. These factors have implications for how to help teachers use computer technology to teach high school science.
• Helfenstein, P., Currier, N., Clark, B. E., Veverka, J., Bell, M., Sullivan, R., Klemaszewski, J., Greeley, R., Pappalardo, R. T., III, J. H., Jones, T., Klaasen, K., Magee, K., Geissler, P., Greenberg, R., McEwen, A., Phillips, C., Colvin, T., Davies, M., , Denk, T., et al. (1998). Galileo Observations of Europa's Opposition Effect. Icarus, 135(1), 41-63.
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  Abstract: During Galileo's G7 orbit, the Solid State Imaging (SSI) camera acquired pictures of the spacecraft shadow point on Europa's surface as well as a comparison set of images showing the same geographic region at phase angle α = 5°. Coverage, obtained at three spectral bandpasses (VLT, 0.41 μm, GRN, 0.56 μm; and 1MC, 0.99 μm) at a spatial resolution of 404 m/pixel, shows a 162 × 220-km region of Europa's surface located at 30°N, 162°W. We have used these images to measure the near-opposition spectrophotometric behavior of four primary europan terrain materials: IR-bright icy material, IR-dark icy material, dark lineament material, and dark spot material. The high spatial resolution of the G7 images reveal low-albedo materials in dark spots that are among the darkest features (17% albedo at 0.56 μm and 5° phase) yet found on icy Galilean satellites. While material of comparable albedo is found on Ganymede and Callisto, low-albedo europan materials are much redder. All europan surface materials exhibit an opposition effect; however, the strength of the effect, as measured by the total increase in reflectance as phase angle decreases from α = 5° to α = 0°, varies among terrains. The opposition effects of IR-bright icy and IR-dark icy materials which dominate Europa's surface are about 1.5 times larger than predicted from pre-Galileo studies. Low-albedo materials in dark spots exhibit unusually intense opposition effects (up to four times larger than bright icy europan terrains), consistent with the presence of a strong shadow-hiding opposition surge. The strengths of the opposition surges among average europan terrains systematically vary with terrain albedo and can be explained in terms of the simultaneous contributions of shadow-hiding and coherent-backscatter to the total opposition effect. Coherent backscatter introduces a narrow angular contribution (
• Bottke Jr., W. F., Greenberg, R., Carusi, A., & Valsecchi, G. B. (1997). Mapping the effects of distant perturbations on particle-planet interactions. Icarus, 125(2), 288-301.
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  Abstract: Monte-Carlo codes generally treat planestesimal-planet encounters using the two-body scattering approximation, which can be inaccurate when relative velocities are low; however, Monte-Carlo codes using the two-body approximation frequently produce results consistent with more accurate codes using numerical integration. To better understand why this breakdown occurs at low velocities, and to test a hypothesis from Greenberg et al. (1988, Icarus 75, 1-29) that may explain the unexpected accuracy of Monte-Carlo codes, we numerically integrate test body trajectories using a unique set of orbital elements defined by the geometry of the two-body approximation. This new coordinate system is ideal for examining the effects of distant planetary perturbations on particle trajectories all the way to encounter with the planet. Our results show that the failure of the two-body approximation is caused by distant planetary perturbations modifying the approach geometry of the test bodies; behavior at encounter follows two-body scattering even at very low relative velocities. By testing particle swarms encountering a planet, we found that some test bodies, whose approach orbits were shifted by distant planetary perturbations, were then replaced by similarly shifted nearby test bodies. The "particle replacement" mechanism explains why Monte-Carlo codes frequently yield outcome results comparable to numerical integration results. Moreover, we found that the relative velocity of a test body at encounter is not the critical parameter in determining the "breakdown" of two-body scattering outcome statistics; instead, we found that the semimajor axis of the test body relative to the size of the planet's Hill sphere (or the synodic period of the test body when mass is included) is much more diagnostic. Thus, our results verify that Monte-Carlo models can yield statistically accurate results, even if individual particles do not behave as assumed in those codes. © 1997 Academic Press.
• Greenberg, R., Fischer, M., Valsecchi, G. B., & Carusi, A. (1997). Sources of planetary rotation: Mapping planetesimals' contributions to angular momentum. Icarus, 129(2), 384-400.
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  Abstract: A systematic study of the motion backward and forward in time from a uniform distribution of bodies entering a planet's Hill sphere yields a mapping of the contribution to the planet's rotation from planetesimals as a function of their source in heliocentric orbital element space. The mapping allows determination of the contribution to rotation from any assumed distribution of heliocentric planetesimal orbits. For example, it can systematically reproduce earlier results from Monte-Carlo studies of contributions to rotational angular momentum. With our method of calculation, contributions to planetary rotation can be interpreted in terms of the arrival geometries at the planet's Hill sphere and the subsequent two-body motion inside the Hill sphere leading to impact. Prograde rotation is strongly favored if a planet grows in a relatively quiescent population of planetesimals with accretion nibbling at the edges of its feeding zone. However, if the impacting population were dominated by large bodies with high relative velocity, the direction and magnitude of rotation would be random. © 1997 Academic Press.
• Petit, J. -., Durda, D. D., Greenberg, R., Hurford, T. A., & Geissler, P. E. (1997). The long-term dynamics of dactyl's orbit. Icarus, 130(1), 177-197.
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  Abstract: Asteroid Ida's satellite Dactyl was observed over 512hr by the Galileo spacecraft imaging system. The observed motion fits a family of orbits parameterized by the mass of Ida. We have tested the stability of these orbits by numerically integrating motion about a realistically shaped model for Ida. Those with pericenter distanceq≲ 65 km (corresponding to Ida's density ≳ 3.1 g cm-3) are unstable over time scales of a few days to a few months, placing a strong upper limit on Ida's density. Moreover, at the opposite extreme of density, orbits corresponding to densities less than 2.3 g cm-3are chaotic and become unstable after about 1000 years. For density between 2.3 and 2.5 g cm-3, Galileo family orbits are chaotic but there is no indication of instability over thousands of years. Dactyl likely formed at the same time as Ida, so its orbit must be stable over time scales much longer than we have been able to explore numerically. As a start toward understanding long-term stability, we have investigated the character of orbits commensurate with the rotation of Ida within the Galileo family. We found that the overlap of high-order resonances for low densities of Ida explains the chaotic behavior of orbits. The low-orderp:1 andp:2 resonances, corresponding to a high density for Ida, are distinct and stable and are all consistent with the longitudinal position of Dactyl at the epoch of the Galileo encounter. However, there is no evidence of preferential stability of resonant orbits against collison with Ida or escape over 6000 years. If a resonant orbit is actually occupied, it may have been selected by a longer-term stability or by dissipative processes. © 1997 Academic Press.
• Belton, M. J., III, J. H., Ingersoll, A. P., Greeley, R., McEwen, A. S., Klaasen, K. P., Senske, D., Pappalardo, R., Collins, G., Vasavada, A. R., Sullivan, R., Simonelli, D., Geissler, P., Carr, M. H., Davies, M. E., Veverka, J., Gierasch, P. J., Banfield, D., Bell, M., , Chapman, C. R., et al. (1996). Galileo's first images of Jupiter and the Galilean satellites. Science, 274(5286), 377-385.
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  Abstract: The first images of Jupiter, Io, Europa, and Ganymede from the Galileo spacecraft reveal new information about Jupiter's Great Red Spot (GRS) and the surfaces of the Galilean satellites. Features similar to clusters of thunderstorms were found in the GRS. Nearby wave structures suggest that the GRS may be a shallow atmospheric feature. Changes in surface color and plume distribution indicate differences in resurfacing processes near hot spots on lo. Patchy emissions were seen while Io was in eclipse by Jupiter. The outer margins of prominent linear markings (triple bands) on Europa are diffuse, suggesting that material has been vented from fractures. Numerous small circular craters indicate localized areas of relatively old surface. Pervasive brittle deformation of an ice layer appears to have formed grooves on Ganymede. Dark terrain unexpectedly shows distinctive albedo variations to the limit of resolution.
• Bottke Jr., W. F., Nolan, M. C., Melosh, H. J., Vickery, A. M., & Greenberg, R. (1996). Origin of the spacewatch small earth-approaching asteroids. Icarus, 122(2), 406-427.
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  Abstract: Recent discoveries of small Earth-approaching asteroids by the 0.9 m Spacewatch telescope (referred to here as S-SEAs) reveal 16 objects which have diameters ∼50 m or smaller. Approximately half of these objects lie in a region where few large near-Earth asteroids are found, with perihelia (q) and aphelia (Q) near 1 AU, e < 0.35, and i from 0° to ∼30°. Possible origins for these objects are examined by tracking the orbital evolution of test bodies from several possible source regions using an Öpik-type Monte Carlo dynamical evolution code, modified to include (a) impact disruption, based on a map in orbital (a, e, i) space of collision probabilities and mean impact velocities determined using actual main-belt and near-Earth asteroid orbits, (b) fragmentation, and (c) observational selection effects. Amor asteroid fragments evolving from low eccentricity Mars-crossing orbits beyond the q = 1 AU line provide a reasonable fit to S-SEA orbital data. Planetary ejecta from Mars is only consistent with low and moderately inclined S-SEA orbits. Asteroidal fragments from the main-belt via the 3:1 or v6 chaotic resonance zones rarely achieve low-e orbits before planetary impacts, comminution, or ejection remove them from the system. This source could produce the observed moderate-to-high eccentricity S-SEAs. Plantary ejecta from the Earth-Moon system and Venus are only consistent with low-inclination S-SEA orbits. Moreover, constraints set by the planetary cratering record and the meteorite record suggest that the Earth, Moon, and Venus are unlikely to provide many S-SEAs. All of these results are predicated on the observational bias computations (Rabinowitz, D.L. 1994. Icarus 111, 364-377) that provide the current definition of the S-SEA population. © 1996 Academic Press, Inc.
• Geissler, P., Petit, J., Durda, D. D., Greenberg, R., Bottke, W., Nolan, M., & Moore, J. (1996). Erosion and ejecta reaccretion on 243 Ida and its moon. Icarus, 120(1), 140-157.
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  Abstract: Galileo images of Asteroid 243 Ida and its satellite Dactyl show surfaces which are dominantly shaped by impact cratering. A number of observations suggest that ejecta from hypervelocity impacts on Ida can be distributed far and wide across the Ida system, following trajectories substantially affected by the low gravity, nonspherical shape, and rapid rotation of the asteroid. We explore the processes of reaccretion and escape of ejecta on Ida and Dactyl using three-dimensional numerical simulations which allow us to compare the theoretical effects of orbital dynamics with observations of surface morphology. The effects of rotation, launch location, and initial launch speed are first examined for the case of an ideal triaxial ellipsoid with Ida's approximate shape and density. Ejecta launched at low speeds (V ≪ Vesc) reimpact near the source craters, forming well-defined ejecta blankets which are asymmetric in morphology between leading and trailing rotational surfaces. The net effect of cratering at low ejecta launch velocities is to produce a thick regolith which is evenly distributed across the surface of the asteroid. In contrast, no clearly defined ejecta blankets are formed when ejecta is launched at higher initial velocities (V ∼ Vesc). Most of the ejecta escapes, while that which is retained is preferentially derived from the rotational trailing surfaces. These particles spend a significant time in temporary orbit around the asteroid, in comparison to the asteroid's rotation period, and tend to be swept up onto rotational leading surfaces upon reimpact. The net effect of impact cratering with high ejecta launch velocities is to produce a thinner and less uniform soil cover, with concentrations on the asteroids' rotational leading surfaces. Using a realistic model for the shape of Ida (P. Thomas, J. Veverka, B. Carcich, M. J. S. Belton, R. Sullivan, and M. Davies 1996, Icarus 120, 000-000), we find that an extensive color/albedo unit which dominates the northern and western hemispheres of the asteroid can be explained as the result of reaccretion of impact ejecta from the large and evidently recent crater "Azzurra." Initial ejection speeds required to match the color observations are on the order of a few meters per second, consistent with models (e.g., M. C. Nolan, E. Asphaug, H. J. Melosh, and R. Greenberg 1996, Icarus, submitted; E. Asphaug, J. Moore, D. Morrison, W. Benz, and R. Sullivan 1996, Icarus 120, 158-184) that multikilometer craters on Ida form in the gravity-dominated regime and are net producers of locally retained regolith. Azzurra ejecta launched in the direction of rotation at speeds near 10 m/sec are lofted over the asteroid and swept up onto the rotational leading surface on the opposite side. The landing locations of these particles closely match the distribution of large ejecta blocks observed in high resolution images of Ida (P. Lee, J. Veverka, P. Thomas, P. Helfstein, M. J. S. Belton, C. Chapman, R. Greeley, R. Pappalardo, R. Sullivan, and J. W. Head 1996, Icarus 120, 87-105). Ida's shape and rotation allow escape of ejecta launched at speeds far below the escape velocity of a nonrotating sphere of Ida's volume and presumed density. While little ejecta from Ida is captured by Dactyl, about half of the mass ejected from Dactyl at speeds of up to 20 m/sec eventually falls on Ida. Particles launched at speeds just barely exceeding Dactyl's escape velocity can enter relatively long-term orbit around Ida, but few are ultimately reaccreted by the satellite. Because of its low gravity, erosion of Dactyl would take place on exceedingly short time scales if unconsolidated materials compose the satellite and crater formation is in the gravity regime. If Dactyl is a solid rock, then its shape has evolved from a presumably irregular initial fragment to its present remarkably rounded figure by collision with a population of impactors too small to be detected by counting visible craters. As the smallest solar system object yet imaged by a spacecraft, the morphology of Dactyl is an important clue to the asteroid population at the smallest sizes. © 1996 Academic Press, Inc.
• Greenberg, R., Bottke, W. F., Nolan, M., Geissler, P., Petit, J., Durda, D. D., Asphaug, E., & Head, J. (1996). Collisional and dynamical history of Ida. Icarus, 120(1), 106-118.
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  Abstract: The history of Ida is constrained by its membership in the Koronis family, its satellite Dactyl, the record of impacts left on its surface, and other dynamical, morphological, and spectral properties. Models of crater production and comparably effective erasure processes, combined with the current size - frequency distribution of craters, suggest that the age of the surface is either about 50 myr or > 1 byr. The younger age may be inconsistent with the degraded appearance of many craters, while the older age conflicts with the collisional life expectancy of Dactyl. Consideration of Dactyl's evolution may resolve this issue as well as shed light on the formation of Dactyl, the density of Ida, and possible source regions for Ida and Dactyl within the Koronis parent body. © 1996 Academic Press, Inc.
• J., M., E., B., D'Amario, L. A., Byrnes, D. V., Klaasen, K. P., Synnott, S., Breneman, H., Johnson, T. V., Thomas, P. C., Veverka, J., Harch, A. P., Davies, M. E., Merline, W. J., Chapman, C. R., Davis, D., Denk, T., Neukum, G., Petit, J., Greenberg, R., , Storrs, A., et al. (1996). The discovery and orbit of 1993 (243)1 Dactyl. Icarus, 120(1), 185-199.
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  Abstract: Dactyl was discovered in solid state imaging (SSI) data on February 17, 1994, during the long playback of approach images from the Galileo spacecraft's encounter with the asteroid 243 Ida. Forty-seven images of the Ida-Dactyl pair were obtained. A detailed search for other satellites was made. No confirmed detections were made, all other candidate features being consistent with radiation hits. We deduce a manifold of osculating two-body orbits that approximate Dactyl's motion over the observed orbital arc depending on the assumed mass of Ida. At the time of Galileo's encounter, Dactyl was found to be 85 km from the center of Ida, moving at ∼6 m · sec-1 in the same direction as Ida's retrograde spin. The inclination of its orbit is ∼172° in Ida's equatorial system (IAU definition). It was not possible to obtain a definitive orbit or measure of Ida's mass from the observed motion even though supplemental techniques (search for Dactyl's shadow on Ida, changes in angular diameter and brightness, and attempts to determine the spin of Dactyl) were explored. The influence of Ida's irregular gravitational field and solar perturbations on two-body motion are evaluated and found to be undetectable in the observed orbital arc. These effects may, however, strongly influence the motion over orbital time scales. Limits to the value of Ida's gravitation parameter, GM, are derived. A robust lower limit, GM > 0.0023 km3 · sec-2, is obtained by requiring Dactyl's orbit to be bound. Hubble Space Telescope observations, which show no evidence of Dactyl on a hyperbolic orbit, excludes values of GM in the range 0.00216 < GM < 0.0023 km3 · sec-2. An upper limit, GM < 0.0031 km3 · sec-2, deduced by requiring that the orbital motion has a long lifetime in a realistic approximation to Ida's gravitational field, is illustrated with numerical calculations. Ida's mass is therefore constrained to the range 4.2 ± 0.6 × 1019 g, which, together with a volume of 16,100 ± 1900 km3 (Thomas P. C., M. J. S. Belton, B. Carcich, C. R. Chapman, M. E. Davies, R. Sullivan, and J. Veverka 1996. Icarus 120, 20-32.) yields a bulk density of 2.6 ± 0.5 g · cm-3 (Belton, M. J. S., C. R. Chapman, P. C. Thomas, M. E. Davies, R. Greenberg, K. Klaasen, D. Byrnes, L. D'Amario, S. Synnott, T. V. Johnson, A. McEwen, W. Merline, D. R. Davis, J-M. Petit, A. Storrs, J. Veverka, and B. Zellner 1995. Nature 374, 785-788.). © 1996 Academic Press, Inc.
• Nolan, M. C., Asphaug, E., Melosh, H. J., & Greenberg, R. (1996). Impact craters on asteroids: Does gravity or strength control their size?. Icarus, 124(2), 359-371.
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  Abstract: The formation of kilometer-size craters on asteroids is qualitatively different from the formation of meter-size (laboratory -and weapons-scale) craters on Earth. A numerical hydrocode model is used to examine the outcomes of various-size cratering impacts into spheres and half-spaces. A shock wave fractures the target in advance of the crater excavation flow; thus, for impactors larger than 100 m, impacting at typical asteroid impact velocities, target tensile strength is irrelevant to the impact outcome. This result holds whether the target is initially intact or a "rubble pile," even ignoring the effects of gravity. Because of the shock-induced fracture, crater excavation is controlled by gravity at smaller sizes than would otherwise be predicted. Determining the strength-gravity transition by comparing the physical strength of the material to the force of gravity will not work, because strength is eliminated by the shock wave. © 1996 Academic Press, Inc.
• Petit, J. -., & Greenberg, R. (1996). Viscosity in keplerian disks: Steady-state velocity distribution and non-local collision effects. Icarus, 123(2), 524-535.
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  Abstract: A kinetic model based on a numerical algorithm, rather than Boltzman's equation, yields the viscosity and velocity distribution for colliding, finite-size particles in a planetary ring. Results are similar to those of many-particle simulations, and show that non-local effects due to the finite size are dominant in many cases of interest. Only for small particles does the viscosity decrease with increasing optical thickness sufficiently for the standard ringlet instability model to apply. This numerical kinetic theory will allow study of multi-size particle distributions, as well as scattering due to gravitational interactions or alternative collision models. © 1996 Academic Press, Inc.
• Sullivan, R., Greeley, R., Pappalardo, R., Asphaug, E., Moore, J. M., Morrison, D., Belton, M. J., Carr, M., Chapman, C. R., Geissler, P., Greenberg, R., Granahan, J., III, J. H., Kirk, R., McEwen, A., Lee, P., Thomas, P. C., & Veverka, J. (1996). Geology of 243 Ida. Icarus, 120(1), 119-139.
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  Abstract: The surface of 243 Ida is dominated by the effects of impacts. No complex crater morphologies are observed. A complete range of crater degradation states is present, which also reveals optical maturation of the surface (darkening and reddening of materials with increasing exposure age). Regions of bright material associated with the freshest craters might be ballistically emplaced deposits or the result of seismic disturbance of loosely-bound surface materials. Diameter/depth ratios for fresh craters on Ida are ∼1:6.5, similar to Gaspra results, but greater than the 1:5 ratios common on other rocky bodies. Contributing causes include rim degradation by whole-body "ringing," relatively thin ejecta blankets around crater rims, or an extended strength gradient in near-surface materials due to low gravitational self-packing. Grooves probably represent expressions in surface debris of reactivated fractures in the deeper interior. Isolated positive relief features as large as 150 m are probably ejecta blocks related to large impacts. Evidence for the presence of debris on the surface includes resolved ejecta blocks, mass-wasting scars, contrasts in color and albedo of fresh crater materials, and albedo streaks oriented down local slopes. Color data indicate relatively uniform calcium abundance in pyroxenes and constant pyroxene/olivine ratio. A large, relatively blue unit across the northern polar area is probably related to regolith processes involving ejecta from Azzurra rather than representing internal compositional heterogeneity. A small number of bluer, brighter craters are randomly distributed across the surface, unlike on Gaspra where these features are concentrated along ridges. This implies that debris on Ida is less mobile and/or consistently thicker than on Gaspra. Estimates of the average depth of mobile materials derived from chute depths (20-60 m), grooves (≥30 m), and shallowing of the largest degraded craters (20-50 m minimum, ∼100 m maximum) suggest a thickness of potentially mobile materials of ∼50 m, and a typical thickness for the debris layer of 50-100 m. © 1996 Academic Press, Inc.
• Belton, M. J., Chapman, C. R., Thomas, P. C., Davies, M. E., Greenberg, R., Klaasen, K., Byrnes, D., D'Amario, L., Synnott, S., Johnson, T. V., McEwen, A., Merline, W. J., Davis, D. R., Petit, J. -., Storrs, A., Veverka, J., & Zellner, B. (1995). Bulk density of asteroid 243 Ida from the orbit of its satellite Dactyl. Nature, 374(6525), 785-788.
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  Abstract: DURING its reconnaissance of the asteroid 243 Ida, the Galileo spacecraft returned images of a second object, 1993(243)1 Dactyl1 - the first confirmed satellite of an asteroid. Sufficient data were obtained on the motion of Dactyl to determine its orbit as a function of Ida's mass. Here we apply statistical and dynamical arguments to constrain the range of possible orbits, and hence the mass of Ida. Combined with the volume of Ida2, this yields a bulk density of 2.6 ± 0.5 g cm-3. Allowing for the uncertainty in the porosity of Ida, this density range is consistent with a bulk chon-dritic composition, and argues against some (but not all) classes of meteoritic igneous rock types that have been suggested as compositionally representative of S-type asteroids like Ida. © 2002 Nature Publishing Group.
• Belton, M. J., Chapmant, C. R., Thomas, P. C., Davies, M. E., Greenberg, R., Klaasen, K., Byrnes, D., D'Amario, L., Synnott, S., Johnson, T. V., McEwen, A., Merline, W. J., Davis, D. R., Petit, J. -., Storrs, A., Veverka, J., & Zellner, B. (1995). Bulk density of asteroid 243 Ida from the orbit of its satellite Dactyl. Nature, 374(6525), XI-788.
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  Abstract: DURING its reconnaissance of the asteroid 243 Ida, the Galileo spacecraft returned images of a second object, 1993(243)1 Dactyl1 - the first confirmed satellite of an asteroid. Sufficient data were obtained on the motion of Dactyl to determine its orbit as a function of Ida's mass. Here we apply statistical and dynamical arguments to constrain the range of possible orbits, and hence the mass of Ida. Combined with the volume of Ida2, this yields a bulk density of 2.6±0.5 g cm-3. Allowing for the uncertainty in the porosity of Ida, this density range is consistent with a bulk chondritic composition, and argues against some (but not all) classes of meteoritic igneous rock types that have been suggested as compositionally representative of S-type asteroids like Ida.
• Geissler, P., Thompson, W. R., Greenberg, R., Moersch, J., McEwen, A., & Sagan, C. (1995). Galileo multispectral imaging of Earth.. Journal of geophysical research, 100(E8), 16895-16906.
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  PMID: 11539416;Abstract: Nearly 6000 multispectral images of Earth were acquired by the Galileo spacecraft during its two flybys. The Galileo images offer a unique perspective on our home planet through the spectral capability made possible by four narrowband near-infrared filters, intended for observations of methane in Jupiter's atmosphere, which are not incorporated in any of the currently operating Earth orbital remote sensing systems. Spectral variations due to mineralogy, vegetative cover, and condensed water are effectively mapped by the visible and near-infrared multispectral imagery, showing a wide variety of biological, meteorological, and geological phenomena. Global tectonic and volcanic processes are clearly illustrated by these images, providing a useful basis for comparative planetary geology. Differences between plant species are detected through the narrowband IR filters on Galileo, allowing regional measurements of variation in the "red edge" of chlorophyll and the depth of the 1-micrometer water band, which is diagnostic of leaf moisture content. Although evidence of life is widespread in the Galileo data set, only a single image (at approximately 2 km/pixel) shows geometrization plausibly attributable to our technical civilization. Water vapor can be uniquely imaged in the Galileo 0.73-micrometer band, permitting spectral discrimination of moist and dry clouds with otherwise similar albedo. Surface snow and ice can be readily distinguished from cloud cover by narrowband imaging within the sensitivity range of Galileo's silicon CCD camera. Ice grain size variations can be mapped using the weak H2O absorption at 1 micrometer, a technique which may find important applications in the exploration of the moons of Jupiter. The Galileo images have the potential to make unique contributions to Earth science in the areas of geological, meteorological and biological remote sensing, due to the inclusion of previously untried narrowband IR filters. The vast scale and near global coverage of the Galileo data set complements the higher-resolution data from Earth orbiting systems and may provide a valuable reference point for future studies of global change.
• Belton, M. J., Chapman, C. R., Veverka, J., Klaasen, K. P., Harch, A., Greeley, R., Greenberg, R., III, J. H., McEwen, A., Morrison, D., Thomas, P. C., Davies, M. E., Carr, M. H., Neukum, G., Fanale, F. P., Davis, D. R., Anger, C., Gierasch, P. J., Ingersoll, A. P., & Pilcher, C. B. (1994). First images of asteroid 243 Ida. Science, 265(5178), 1543-1547.
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  PMID: 17801529;Abstract: The first images of the asteroid 243 Ida from Galileo show an irregular object measuring 56 kilometers by 24 kilometers by 21 kilometers. Its surface is rich in geologic features, including systems of grooves, blocks, chutes, albedo features, crater chains, and a full range of crater morphologies. The largest blocks may be distributed nonuniformly across the surface; lineaments and dark-floored craters also have preferential locations. Ida is interpreted to have a substantial regolith. The high crater density and size-frequency distribution (-3 differential power-law index) indicate a surface in equilibrium with saturated cratering. A minimum model crater age for Ida - and therefore for the Koronis family to which Ida belongs - is estimated at 1 billion years, older than expected.
• Belton, M. J., Greeley, R., Greenberg, R., McEwen, A., Klaasen, K. P., III, J. H., Pieters, C., Neukum, G., Chapman, C. R., Geissler, P., Heffernan, C., Breneman, H., Anger, C., Carr, M. H., Davies, M. E., Fanale, F. P., Gierasch, P. J., Ingersoll, A. P., Johnson, T. V., , Pilcher, C. B., et al. (1994). Galileo multispectral imaging of the north polar and eastern limb regions of the moon. Science, 264(5162), 1112-1115.
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  PMID: 17744892;Abstract: Multispectral images obtained during the Galileo probe's second encounter with the moon reveal the compositional nature of the north polar regions and the northeastern limb. Mare deposits in these regions are found to be primarily low to medium titanium lavas and, as on the western limb, show only slight spectral heterogeneity. The northern light plains are found to have the spectral characteristics of highlands materials, show little evidence for the presence of cryptomaria, and were most likely emplaced by impact processes regardless of their age.
• Bottke Jr., W. F., Nolan, M. C., Greenberg, R., & Kolvoord, R. A. (1994). Velocity Distributions among Colliding Asteroids. Icarus, 107(2), 255-268.
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  Abstract: The probability distribution for impact velocities between two given asteroids is wide, non-Gaussian, and often contains spikes according to our new method of analysis in which each possible orbital geometry for collision is weighted according to its probability. An average value would give a good representation only if the distribution were smooth and narrow. Therefore, the complete velocity distribution we obtain for various asteroid populations differs significantly from published histograms of average velocities. For all pairs among the 682 asteroids in the main-belt with D > 50 km, we find that our computed velocity distribution is much wider than previously computed histograms of average velocities. In this case, the most probable impact velocity is ∼4.4 km/sec, compared with the mean impact velocity of 5.3 km/sec. For cases of a single asteroid (e.g., Gaspra or Ida) relative to an impacting population, the distribution we find yields lower velocities than previously reported by others. The width of these velocity distributions implies that mean impact velocities must be used with caution when calculating asteroid collisional lifetimes or crater-size distributions. Since the most probable impact velocities are lower than the mean, disruption events may occur less frequently than previously estimated. However, this disruption rate may be balanced somewhat by an apparent increase in the frequency of high-velocity impacts between asteroids. These results have implications for issues such as asteroidal disruption rates, the amount/type of impact ejecta available for meteoritical delivery to the Earth, and the geology and evolution of specific asteroids like Gaspra. © 1994 Academic Press. All rights reserved.
• Greenberg, R., Nolan, M. C., Bottke Jr., W. F., Kolvoord, R. A., & Veverka, J. (1994). Collisional History of Gaspra. Icarus, 107(1), 84-97.
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  Abstract: Interpretation of the impact record on Asteroid 951 Gaspra requires understanding of the effects of collisions on a target body of Gaspra's size and shape. Recent hydrocode models show that major impacts on Gaspra may leave craters larger than previously thought possible and that they can create substantial regolith and produce global jolting capable of erasing many smaller craters. A Gaspra-size body has a mean lifetime of about 109 years and is likely to have several impact craters of 4 km diameter or larger, which is consistent with the number of observed concavities on Gaspra. The steep size distribution of small craters on Gaspra implies an even steeper distribution for small asteroids, and the relative paucity of kilometer-size craters seems to require a twice-in-a-Gaspra-lifetime impact to have occurred about 50 myr ago. All of these considerations consistently point to a scenario in which several of the very large concavities (>4 km diameter) on Gaspra are probably impact craters. © 1994 Academic Press. All rights reserved.
• Lazzaro, D., Sicardy, B., Roques, F., & Greenberg, R. (1994). Is There a Planet around β Pictoris? Perturbations of a Planet on a Circumstellar Dust Disk. 2. The Analytical Model. Icarus, 108(1), 59-80.
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  Abstract: The evolution of micrometer-sized circumstellar grains orbiting β Pictoris is studied, taking into account the combined effects of first order resonances due to a hypothetical planet and the dissipative effect due to Poynting-Robertson drag, We first derive the averaged equations of motion of the grain near a resonance, and we describe qualitatively and quantitatively the capture into the resonance (mechanism of entrance, time scales for capture, etc.). It appears that the probability of capture cannot be derived analytically, because of the nonadiabaticity of the motion at the entrance into the resonance, at least for micrometer-sized particles and planet masses smaller than about one saturnian mass. We show that the capture of a grain into a resonance critically depends (i) on the orbital eccentricity and (ii) on the value of the critical argument of resonance just at the entrance into the resonance. Maps of capture/noncapture regions vs these two parameters are derived numerically for the 1:2, 2:3, and 3:4 resonances. They show the complexity of the capture regions, and indicate that uranian or larger planets are able to trap most of the grains into the 1:2 resonance, while ∼5 Earth masses are sufficient to trap grains into the 3:4 resonance for any grain with initial eccentricities smaller than a few percent. These results underline the dynamical importance of small planetary objects embedded in circumstellar dust disks. © 1994 Academic Press. All rights reserved.
• Veverka, J., Thomas, P., Simonelli, D., Belton, M. J., Carr, M., Chapman, C., Davies, M. E., Greeley, R., Greenberg, R., Head, J., Klaasen, K., Johnson, T. V., Morrison, D., & Neukum, G. (1994). Discovery of Grooves on Gaspra. Icarus, 107(1), 72-83.
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  Abstract: We report the discovery of grooves in Galileo high-resolution images of Gaspra. These features, previously seen only on Mars' satellite Phobos, are most likely related to severe impacts. Grooves on Gaspra occur as linear and pitted depressions, typically 100-200 m wide, 0.8 to 2.5 km long, and 10-20 m deep. Most occur in two major groups, one of which trends approximately parallel to the asteroid's long axis, but is offset by some 15°; the other is approximately perpendicular to this trend. The first of these directions falls along a family of planes which parallel three extensive flat facets identified by Thomas et al., Icarus 107. The occurrence of grooves on Gaspra is consistent with other indications (irregular shape, cratering record) that this asteroid has evolved through a violent collisional history. The bodywide congruence of major groove directions and other structural elements suggests that present-day Gaspra is a globally coherent body. © 1994 Academic Press. All rights reserved.
• Greenberg, R., Kolvoord, R. A., Magisos, M., Strom, R. G., & Croft, S. (1993). Image processing for teaching. Journal of Science Education and Technology, 2(3), 469-480.
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  Abstract: The Image Processing for Teaching (IPT) project provides a powerful medium to excite students about science and mathematics, especially children from minority groups and others whose needs have not been met by traditional "coded" ways of teaching these subjects. Using professional-quality software on microcomputers, students explore a variety of scientific data sets, including biomedical imaging, Earth remote sensing and meteorology data, and planetary exploration images. They also learn about the many mathematical concepts that underlie image processing, such as coordinate systems, slope and intercept, pixels, binary arithmetic, along with many others. We have developed curriculum materials in all areas of mathematics and science for the upper elementary and secondary levels, allowing this tool to be used across a variety of grade levels and student interests. Preliminary indications show image processing to be an effective and fun way to study the application of science and mathematics to "real world" applications, as represented by digital imagery. The use of image processing is also an effective method with which to engage students in inquiry and discovery learning. © 1993 Plenum Publishing Corporation.
• Sears, W. D., Lunine, J. I., & Greenberg, R. (1993). Equilibrium Nonsynchronous Rotation of Titan. Icarus, 105(1), 259-262.
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  Abstract: A satellite on an eccentric orbit experiences a zero net tidal torque when the rotation is slightly faster than synchronous, if there is no significant permanent asymmetry. Calculations indicate that for Titan this period should be 15.82 to 15.93 days, depending on the tidal dissipation model. This range includes the period recently determined by radar observations. A water-ice highland plateau with a diameter of less than 180 km standing just above the expected hydrocarbon ocean is consistent with both the radar data and the asymmetry constraints. © 1993 Academic Press. All rights reserved.
• Belton, M. J., Gierasch, P., Klaasen, K. P., Anger, C. D., Carr, M. H., Chapman, C. R., Davies, M. E., Greeley, R., Greenberg, R., Head, J. W., Neukum, G., Pilcher, C. B., Veverka, J., Fanale, F. P., Ingersoll, A. P., Pollock, J. B., Morrison, D., Clary, M. C., Cunningham, W., & Breneman, H. (1992). Imaging of Venus from Galileo: Early results and camera performance. Advances in Space Research, 12(9), 91-103.
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  Abstract: Three images of Venus have been returned so far by the Galileo spacecraft following an encounter with the planet on UT February 10, 1990. The images, taken at effective wavelengths of 4200 and 9900 Å, characterize the global motions and distribution of haze near the Venus cloud tops and, at the latter wavelength, deep within the main cloud. Previously undetected markings are clearly seen in the near-infrared image. The global distribution of these features, which have maximum contrasts of 3%, is different from that recorded at short wavelengths. In particular, the "polar collar," which is omnipresent in short wavelength images, is absent at 9900 Å. The maximum contrast in the features at 4200 Å is about 20%. The optical performance of the camera is described and is judged to be nominal. © 1992.
• Belton, M. J., Klaasen, K. P., Clary, M. C., Anderson, J. L., Anger, C. D., Carr, M. H., Chapman, C. R., Davies, M. E., Greeley, R., Anderson, D., Bolef, L. K., Townsend, T. E., Greenberg, R., W., J., Neukum, G., Pilcher, C. B., Veverka, J., Gierasch, P. J., Fanale, F. P., , Ingersoll, A. P., et al. (1992). The Galileo Solid-State Imaging experiment. Space Science Reviews, 60(1-4), 413-455.
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  Abstract: The Solid State Imaging (SSI) experiment on the Galileo Orbiter spacecraft utilizes a high-resolution (1500 mm focal length) television camera with an 800 × 800 pixel virtual-phase, charge-coupled detector. It is designed to return images of Jupiter and its satellites that are characterized by a combination of sensitivity levels, spatial resolution, geometric fiedelity, and spectral range unmatched by imaging data obtained previously. The spectral range extends from approximately 375 to 1100 nm and only in the near ultra-violet region (∼ 350 nm) is the spectral coverage reduced from previous missions. The camera is approximately 100 times more sensitive than those used in the Voyager mission, and, because of the nature of the satellite encounters, will produce images with approximately 100 times the ground resolution (i.e., ∼ 50 m lp-1) on the Galilean satellites. We describe aspects of the detector including its sensitivity to energetic particle radiation and how the requirements for a large full-well capacity and long-term stability in operating voltages led to the choice of the virtual phase chip. The F/8.5 camera system can reach point sources of V(mag) ∼ 11 with S/N ∼ 10 and extended sources with surface brightness as low as 20 kR in its highest gain state and longest exposure mode. We describe the performance of the system as determined by ground calibration and the improvements that have been made to the telescope (same basic catadioptric design that was used in Mariner 10 and the Voyager high-resolution cameras) to reduce the scattered light reaching the detector. The images are linearly digitized 8-bits deep and, after flat-fielding, are cosmetically clean. Information 'preserving' and 'non-preserving' on-board data compression capabilities are outlined. A special "summation" mode, designed for use deep in the Jovian radiation belts, near Io, is also described. The detector is 'preflashed' before each exposure to ensure the photometric linearity. The dynamic range is spread over 3 gain states and an exposure range from 4.17 ms to 51.2 s. A low-level of radial, third-order, geometric distortion has been measured in the raw images that is entirely due to the optical design. The distortion is of the pincushion type and amounts to about 1.2 pixels in the corners of the images. It is expected to be very stable. We discuss the measurement objectives of the SSI experiment in the Jupiter system and emphasize their relationships to those of other experiments in the Galileo project. We outline objectives for Jupiter atmospheric science, noting the relationship of SSI data to that to be returned by experiments on the atmospheric entry Probe. We also outline SSI objectives for satellite surfaces, ring structure, and 'darkside' (e.g., aurorae, lightning, etc.) experiments. Proposed cruise measurement objectives that relate to encounters at Venus, Moon, Earth, Gaspra, and, possibly, Ida are also briefly outlined. The article concludes with a description of a 'fully distributed' data analysis system (HIIPS) that SSI team members intend to use at their home institutions. We also list the nature of systematic data products that will become available to the scientific community. Finally, we append a short 'historical' note outlining the responsibilities and roles of institutions and individuals that have been involved in the 14 year development of the SSI experiment so far. © 1992 Kluwer Academic Publishers.
• Belton, M. J., Veverka, J., Thomas, P., Helfenstein, P., Simonelli, D., Chapman, C., Davies, M. E., Greeley, R., Greenberg, R., Head, J., Murchie, S., Klaasen, K., Johnson, T. V., McEwen, A., Morrison, D., Neukum, G., Fanale, F., Anger, C., Carr, M., & Pilcher, C. (1992). Galileo encounter with 951 Gaspra: First pictures of an asteroid. Science, 257(5077), 1647-1652.
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  Abstract: Galileo images of Gaspra reveal it to be an irregularly shaped object (19 by 12 by 11 kilometers) that appears to have been created by a catastrophic collisional disruption of a precursor parent body. The cratering age of the surface is about 200 million years. Subtle albedo and color variations appear to correlate with morphological features: Brighter materials are associated with craters especially along the crests of ridges, have a stronger 1-micrometer absorption, and may represent freshly excavated mafic materials; darker materials exhibiting a significantly weaker 1-micrometer absorption appear concentrated in interridge areas. One explanation of these patterns is that Gaspra is covered with a thin regolith and that some of this material has migrated downslope in some areas.
• J., M., W., J., Pieters, C. M., Greeley, R., Mcewen, A. S., Neukum, G., Klaasen, K. P., Anger, C. D., Carr, M. H., Chapman, C. R., Davies, M. E., Fanale, F. P., Gierasch, P. J., Greenberg, R., Ingersoll, A. P., Johnson, T., Paczkowski, B., Pilcher, C. B., & Veverka, J. (1992). Lunar impact basins and crustal heterogeneity: New western limb and far side data from galileo. Science, 255(5044), 570-576.
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  PMID: 17792379;Abstract: Multispectral images of the lunar western limb and far side obtained from Galileo reveal the compositional nature of several prominent lunar features and provide new information on lunar evolution. The data reveal that the ejecta from the Orientale impact basin (900 kilometers in diameter) lying outside the Cordillera Mountains was excavated from the crust, not the mantle, and covers pre-Orientale terrain that consisted of both highland materials and relatively large expanses of ancient mare basalts. The inside of the far side South Pole-Aitken basin (>2000 kilometers in diameter) has low albedo, red color, and a relatively high abundance of iron- and magnesium-rich materials. These features suggest that the impact may have penetrated into the deep crust or lunar mantle or that the basin contains ancient mare basalts that were later covered by highlands ejecta.
• Kolvoord, R. A., & Greenberg, R. (1992). A critical reanalysis of planetary accretion models. Icarus, 98(1), 2-19.
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  Abstract: Planet formation models and simulations appear to be converging toward a consensus on how growth may have occured, at least up to the emergence of a few dominant planetary embryos from the vast swarm of planetesimals, which is similar to the scenario described by Greenberg et al. (1978, Icarus 35, 1-26). The various approaches to simulation differ in many details, so comparison is now needed to understand the relation of assumed physical processes and computational approximations to the modeled evolutionary behavior of the system. The algorithms of Greenberg et al. for (a) mutual velocity stirring between planetesimals and (b) mass shifting between adjacent numerical "size-bins" are described in detail. Experiments with the original code, as well as with subsequent revisions, show that the computational methods were accurate to the level of precision appropriate at the time, and the model incorporated the relevant physics including dynamical friction, so its then-novel conclusions remain robust and generally in agreement with various, more recent models. We demonstrate that use of continuum equations is inappropriate for modeling growth with initially all-equal masses; numerical models with discrete building blocks allow incipient runaway growth even with such an initial condition. © 1992.
• Greenberg, R. (1991). Miranda. Uranus, 693-735.
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  Abstract: Drawing on the observed geology, photometry and geophysical data, examines various processes and properties that may have contributed to Miranda's evolution. Global tectonics and surface flow features constrain the possible heating mechanisms and materials. Statistics of impact craters and comparisons with other satellites suggest that the impactor-source population evolved through time and that ejecta mantling has resurfaced significant portions of the surface. -from Authors
• Greenberg, R., Bottke, W. F., Carusi, A., & Valsecchi, G. B. (1991). Planetary accretion rates: Analytical derivation. Icarus, 94(1), 98-111.
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  Abstract: Analytic formulae for estimating planetary accretion rates have long been available for conditions where random motion of planetesimals controls velocities relative to a growing planet. Recent Monte Carlo studies have explored a wider range of conditions, but gave impact rates only for the specific conditions under which those numerical experiments were performed. Now analytical formulae, which are valid even when keplerian shear controls the approach velocity, and which accurately reproduce the Monte Carlo results, give accretion rates over a wide range of conditions relevant to the problem of planet formation. © 1991.
• J., M., Gierasch, P. J., Smith, M. D., Helfenstein, P., Schinder, P. J., Pollack, J. B., Rages, K. A., Ingersoll, A. P., Klaasen, K. P., Veverka, J., Anger, C. D., Carr, M. H., Chapman, C. R., Davies, M. E., Fanale, F. P., Greeley, R., Greenberg, R., W., J., Morrison, D., , Neukum, G., et al. (1991). Images from Galileo of the Venus cloud deck. Science, 253(5027), 1531-1536.
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  PMID: 17784096;Abstract: Images of Venus taken at 418 (violet) and 986 [near-infrared (NIR)] nanometers show that the morphology and motions of large-scale features change with depth in the cloud deck. Poleward meridional velocities, seen in both spectral regions, are much reduced in the NIR. In the south polar region the markings in the two wavelength bands are strongly anticorrelated. The images follow the changing state of the upper cloud layer downwind of the subsolar point, and the zonal flow field shows a longitudinal periodicity that may be coupled to the formation of large-scale planetary waves. No optical lightning was detected.
• Carusi, A., Valsechi, G. B., & Greenberg, R. (1990). Planetary close encounters: geometry of approach and post-encounter orbital parameters. Celestial Mechanics and Dynamical Astronomy, 49(2), 111-131.
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  Abstract: Öpik's assumptions on the geometry of particle trajectories leading to and through planetary close encounters are used to compute the distribution of changes in heliocentric orbital elements that result from such encounters for a range of initial heliocentric orbits. Behaviour at encounter is assumed to follow two-body (particle-planet) gravitational scattering, while before and after encounter particle motion is only governed by the force of the Sun. Derivation of these distributions allows precise analysis of the probability of various outcomes in terms of the physical characteristics of the bodies involved. For example, they allow an explanation and prediction of the asymmetry of the extreme energy perturbations for different initial orbits. The formulae derived here may be applied to problems including the original accumulation of planets and satellites, and the continuing evolution of populations of small bodies, such as asteroids and comets. © 1990 Kluwer Academic Publishers.
• Ojakangas, G. W., & Greenberg, R. (1990). Viscosity and mass transport in nonuniform Keplerian disks. Icarus, 88(1), 146-171.
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  Abstract: A new quantitative formalism describing the dynamics of a Keplerian particulate disk, based on a heuristic description of viscous transport, permits study of rings with a wide range of ensemble and individual particle properties. Here the formalism is developed and applied to the case of a ring with a radial gradient in optical thickness. A steady-state solution for the velocity distribution directly gives the radial mass transport, as well as the viscosity. The formula for viscosity is identical to that derived a decade earlier by Goldreich and Tremaine for a uniform disk, thus validating the assumption by various workers that it could be applied to nonuniform disks, for example in consideration of ringlet instabilities. Our analytical method involves solving a novel form of the Krook equation by separating the distribution of collisional products in phase space into a symmetrical component and a remainder that can be approximated by delta functions. Unlike most past approaches, a Gaussian form for the solution is not assumed. In the case described here, the model is simplified in common with past work (e.g., small, uniform particles and Krook-type treatment of collisions), but the general approach is extendable to less artificially restricted cases. © 1990.
• Weidenschilling, S. J., Chapman, C. R., Davis, D. R., Greenberg, R., Levy, D. H., Binzel, R. P., Vail, S. M., Magee, M., & Spaute, D. (1990). Photometric geodesy of main-belt asteroids. III. Additional lightcurves. Icarus, 86(2), 402-447.
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  Abstract: We present observations of asteroids obtained in the period 1982-1989, comprising 107 complete or partial lightcurves of 59 different objects. These include additional data on "Photometric Geodesy" program asteroids (Weidenschilling et al. 1987, Icarus 70, 191-245), obtained to determine their shapes and pole orientations (Drummond et al. 1988, Icarus 76, 19-77), and targets of opportunity observed during the course of that program. We derive new or refined rotation periods for 16 asteroids, and estimates of, or constraints on, periods for 8 more. Data presented here and in Paper I (Weidenschilling et al. 1987) suggest that unusual lightcurves with unequal maxima and minima at large amplitudes are seen preferentially for M-type asteroids. © 1990.
• Greenberg, R., & Nolan, M. C. (1989). Delivery of asteroids and meteorites to the inner solar system. Asteroids II, 778-804.
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  Abstract: Critical observational constraints on the delivery of asteroids include orbital distributions, exposure ages and mineralogy. Orbital maturity in the inner solar system is indicated by the AM/PM distribution of meteorite falls and fireballs: Ordinary chondrites have short exposure ages, but their AM/PM fall statistics indicate significant orbital maturity. Hence, many may be collisional offspring of slightly larger parents that emigrated from the main belt. The required size distribution, extrapolated up to multi-km-size bodies, would also yield numbers of planet-crossing asteroids comparable to those astronomically observed. How multi-km Apollo-Amor-Atens are launched from the main belt is a major outstanding problem, as is identification of a parent population for the ordinary chondrites now that S-type asteroids once again are considered to be different from this most common kind of meteorite on the basis of reflectance spectroscopy. -from Authors
• Greenberg, R. (1988). Particle properties and the large-scale structure of planetary rings: Rebound characteristics and viscosity. Icarus, 75(3), 527-539.
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  Abstract: In a swarm of particles on Keplerian orbits, like a planetary ring, systematic flow is circular with velocity decreasing with distance from the planet. Viscosity ordinarily transports momentum from a faster to a slower moving region, i.e., outward. But any individual at apocenter of its orbit moves slower, and at pericenter faster, than the mean flow, suggesting a reversed momentum transport or negative viscosity. Resolution of this seeming paradox illuminates the importance of physical properties of particles. A conventional model has uniform, highly elastic, perfectly slippery, spherical particles, with a particular dependence of elasticity on impact velocity. Even slight deviation from that ideal affects viscosity and leads to inconsistencies with observed ring structure. If particles are less elastic they may clump into large temporary agglomerations. The size of the larger bodies in the rings probably determines random velocities, viscosity, and much about ring structure. © 1988.
• Greenberg, R., Carusi, A., & Valsecchi, G. B. (1988). Outcomes of planetary close encounters: A systematic comparison of methodologies. Icarus, 75(1), 1-29.
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  Abstract: Motion during a planetary close encounter can be described reasonably well by a two-body approximation, in which the effect of the Sun is temporarily neglected. The approximation is suspect for very slow encounters, such as occur in the quiescent swarm of planetesimals that encountered early planetary embryos. Öpik's formulation of the two-body approximation fails under those conditions, predicting motion different from the correct three-body behavior. The differences are generally due to small distant perturbations of heliocentric orbits during approach toward encounter, rather than due to failure of the two-body model at close encounter. This conclusion comes from intercomparison of several ways of computing and estimating the outcomes of encounters. Comparison and physical interpretation is facilitated by a strategy of studying suites of trajectories in a revealing graphical display. Reinterpretation of past Monte Carlo studies also shows that the two-body approximation can be valid even where the Öpik method is not. The approximation may apply to the quiescent planetesimal swarm if the physical effects that cause failure of the Öpik method are accounted for. The prevalent effects of distant perturbations often yield horseshoe orbits, which may substantially reduce accretion from a wide band within the feeding zone of a planetary embryo. Use of the two-body approximation in swarms will require that encounters offset by distant perturbations be statistically replaced and compensated by other perturbed trajectories. In some cases the two-body model will need to account for partial rotation of relative velocity vectors, rather than full asymptote-to-asymptote rotation. Further studies are needed to find the real limits of the two-body model. Special attention will need to be focused on close encounters of tangential and not-quite-crossing orbits, which are moderately probable and potentially deviate from two-body behavior. © 1988.
• Lebofsky, L. A., Greenberg, R., Tedesco, E. F., & Veeder, G. J. (1988). Infrared lightcurves of asteroids 532 Herculina and 45 Eugenia: Proof of the absence of significant albedo markings. Icarus, 75(3), 518-526.
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  Abstract: Time-resolved near-infrared (J and K bands) and mid-infrared (N and Q bands) photometric measurements of asteroids 532 Herculina and 45 Eugenia show reflected and thermal lightcurves that are in phase: i.e., thermal maxima and minima occur at nearly the same time as the maxima and minima in reflected light. Model lightcurves for Herculina based on the albedo distribution of Taylor et al. (1987, Icarus 69, 354-369) are predicted to be 90° out of phase. A comparison of the model and observed lightcurves leads us to conclude that the light variations of Herculina and Eugenia are due primarily to shape rather than to surface albedo variations. © 1988.
• Marcialis, R., & Greenberg, R. (1988). Warming of miranda during chaotic rotation. Nature, 328(6127), 227-229.
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  Abstract: Miranda, a satellite of Uranus, appears in Voyager images to have had an active geological history, seemingly characterized by relaxation of very large-scale topography or the sinking of large blocks of material accreted by the satellite, with associated extrusion, eruption, and flow of the icy material on the body1. The low temperature of Miranda (
• Binzel, R. P., Cochran, A. L., Barker, E. S., Tholen, D. J., Barucci, A., Martino, M. D., Greenberg, R., Weidenschilling, S. J., Chapman, C. R., & Davis, D. R. (1987). Coordinated observations of asteroids 1219 Britta and 1972 Yi Xing. Icarus, 71(1), 148-158.
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  Abstract: Identified as possible flyby targets for the Galileo spacecraft, Asteroids 1219 Britta and 1972 Yi Xing became the focus of a coordinated observing program. Although a subsequent change in the launch date removed these asteroids from consideration for the Galileo mission, the ground-based observing program yielded a substantial amount of information on these previously unobserved asteroids. Britta's sideral rotation period is found to be 5.57497 ± 0.00013 hr and its rotation is retrograde. The lightcurve amplitude ranged from 0.60 to 0.70 mag, depending on phase angle. Britta can be classified as an S-type asteroids based on its measured spectra and albedo. The absolute magnitude and slope parameter derived from the lightcurve maxima are H0 = 11.67 ± 0.03 and G0 = 0.03 ± 0.04. A 0.002 mag deg-1 phase reddening in B·V was also measured. 1972 Yi Xing was less well observed but a unique synodic period of 14.183 ± 0.003 hr was determined. The observed lightcurve amplitude was 0.18 mag. Five-color measurements are consistent with an S-type classification. For an assumed slope parameter G = 0.25, Yi Xing's (lightcurve maximum) absolute magnitude H0 = 13.32 ± 0.01. © 1987.
• Greenberg, R. (1987). Galilean satellites: Evolutionary paths in deep resonance. Icarus, 70(2), 334-347.
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  Abstract: The Laplace resonance among the inner three Galilean satellites (mean motions n1 - 3n2 + 2n3 = 0) has stable configurations in "deep resonance," i.e., where mean motions taken by pairs are in ratios very close to 2:1. The present satellite configuration, with the resonance variable φ ≡ λ1 - 3λ2 + 2λ3 stable at 180°, is unstable near this exact commensurability. But there is a continuous path of stable conditions branching from φ = 180° to higher and lower values of φ and toward very deep resonance, according to a theory extended to third order in orbital eccentricity. This path provides a track for tidal evolution of the system. Thus, scenarios involving evolution (probably episodic) from deep resonance are viable, and eliminate the requirement by the alternative equilibrium hypothesis for rapid tidal dissipation in Jupiter. Evolution out from deep resonance is consistent with the free eccentricity of Ganymede, the free libration of φ, and observational constraints on Io's secular acceleration. Also, the relatively large forced eccentricities in deep resonance may have controlled geophysical processes in the satellites by much greater tidal heating and global stress than at present. © 1987.
• Leake, M. A., Chapman, C. R., Weidenschilling, S. J., Davis, D. R., & Greenberg, R. (1987). The chronology of Mercury's geological and geophysical evolution: The vulcanoid hypothesis. Icarus, 71(3), 350-375.
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  Abstract: Previous discussions of Mercury's evolution have assumed that its cratering chronology is tied to that of the Moon, i.e., with Caloris forming about 3.9 Gyr ago as part of a late heavy bombardment that affected all of the terrestrial planets. That assumption requires that Mercury's core formed very early, because associated expansion features are not visible, and must have been erased before the cratering flux declined. Moreover, the modest amount of global shrinkage inferred from visible compressional features on Mercury's surface implies that the core is either largely molten at present, or had largely solidified before the end of the bombardment. The former interpretation requires a significant volatile content or implausibly large internal heat sources, while the latter raises questions about how to generate the planet's magnetic field. We have investigated whether constraints on Mercury's chronology could be relaxed by effects of a Mercury-specific bombarding population of planetesimals interior to its orbit, encountering the planet only occasionally due to secular perturbations. Such "vulcanoids" could have been a significant source of early cratering. However, those in orbits that can cross Mercury's are depleted by mutual collisions in {less-than or approximate}1 Gyr, and can provide at most a modest extension of the period of heavy bombardment. Further inside Mercury's orbit, lower collisional velocities might allow survival of vulcanoids to the present. We report on a search for such bodies and on observational limits to such a population. We also review evidence that Mercury's intercrater plains are of volcanic origin and mainly predate Caloris, and that scarp formation (and global contraction) mainly postdates Caloris and has continued to recent times. If global lineaments are the product of tidal despinning, they constrain core formation to the first half of the planet's lifetime. While some questions and inconsistencies remain, the preponderance of evidence suggests that Mercury differentiated early, and at least half of its core volume is presently molten, probably due to a significant content of some light element such as sulfur. © 1987.
• Spaute, D., & Greenberg, R. (1987). Collision mechanics and the structure of planetary ring edges. Icarus, 70(2), 289-302.
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  Abstract: A numerical simulation of collisional evolution, originally developed to model planetary accretion processes, is applied to a hypothetical ring with parameters modeled after Saturn's rings in order to study changes in radial structure near ring edges. The tendency of rings to spread so as to conserve angular momentum while energy is dissipated in collisions is confirmed if random motion is in equilibrium. Even with no energy loss (coefficient of restitution in velocity ε = 1), spreading occurs becase random motion is increasing. With a moderately side-scattering collisional model, characteristic of collisions of nonrotating spheres (the slippery "billiard-ball" model), random motion increases for ε > 0.63, in agreement with analytical models. For isotropic scattering, which may be more realistic given particle rotation, damping dominates for ε up to 0.83. As long as random motion is damped, ring edges may contract rather than spread, producing concentrations of material just inside the ring edges reminiscent of results of earlier stimulation which did not precisely conserve angular momentum. © 1987.
• Weideenschilling, S. B., Chapman, C. R., Davis, D. R., Greenberg, R., Levy, D. G., & Vail, S. (1987). Photometric Geodesy of main-belt asteroids. I. Lightcurves of 26 large, rapid rotators. Icarus, 70(2), 191-245.
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  Abstract: We present 5 years of lightcurve data on a sample of large, rapidly rotating asteroids selected for a program of "photometric geodysy." A total of 257 complete or partial lightcurves were obtained for 26 asteroids, numbers 9, 15, 16, 22, 22, 29, 39, 41, 43, 45, 55, 65, 87, 88, 107, 125, 129, 130, 201, 216, 337, 349, 354, 511, 584, and 694. The observing protocol was designed to obtain precise absolute photometry at a wide variety of orbital longitudes and phase angles. The purpose of this data set is to allow determination (in future publications) of pole positions and shapes, and to constrain densities, strength, and other geophysical traits of these bodies. © 1987.
• Greenberg, R., Goldstein Jr., S. J., & Jacobs, K. C. (1986). Orbital acceleration and the energy budget in the galilean satellite system. Nature, 323(6091), 789-791.
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  Abstract: Prodigious amounts of heat are generated in Jupiter's satellite Io, as evidenced by its spectacular volcanism1 and infrared heat flux 2-6 (∼7.6×l013 W (ref. 7)). Such heating had been predicted from tidal flexing of Io due to the eccentricity of its orbit8, which is maintained by its resonant gravitational interaction with the satellites Europa and Ganymede. The energy for tidal heating must come from Io's orbit, but due to the resonance the energy loss is shared among the three satellites' orbits9,10. This effect modifies the rate at which Io's semi-major axis would be expected to decrease, and also tends to drive the system out of resonance. Yoder9 suggested that the resonance is maintained by tides raised on Jupiter by Io, which act to transfer planetary rotational energy and angular momentum into Io's orbit, but the jovian torque may be insufficient for this purpose. Greenberg10 proposed that with smaller jovian torque the system may be evolving away from resonance, and that measurement of Io's orbital acceleration could show whether the evolution is in fact dominated by heating in Io. An acceleration value n.1/n 1=4.6×10-l0 yr-1±20%, where n 1 is Io's mean motion, derived11 from historical records of Io's motion, is in accord with the expected value n.1/n 1=3.2×10-10±28% based on the measured thermal radiation and the assumption of negligible jovian torque. A much smaller negative acceleration has been determined by Lieske12,13, which would imply significant jovian torque, but not enough to maintain the equilibrium proposed by Yoder, unless the average tidal disruption in Io is much less than is indicated by the thermal flux. © 1986 Nature Publishing Group.
• Sykes, M. V., & Greenberg, R. (1986). The formation and origin of the IRAS zodiacal dust bands as a consequence of single collisions between asteroids. Icarus, 65(1), 51-69.
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  Abstract: The zodiacal dust bands discovered by IRAS can be explained as products of single collisions between asteroids. Debris from such a collision is distributed about the plane of the ecliptic as particles experience differential precession of their ascending nodes due to dispersion of their semimajor axes. For each collision, two bands, one on each side of the ecliptic, are formed on time scales of 105 to 106 years. The band pairs observed by IRAS are most likely the result of collisions between asteroids ∼15 km in diameter that occured within the last several million years. Further analysis of the IRAS sky survey data and of any future, more sensitive surveys should reveal additional, fainter band pairs. Our model suggests that asteroid collisions are sufficient to account for the bulk of the observed zodiacal thermal emission. © 1986.
• Burns, J. A., Schaffer, L. E., Greenberg, R. J., & Showalter, M. R. (1985). Lorentz resonances and the structure of the Jovian ring. Nature, 316(6024), 115-119.
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  Abstract: Charged dust orbiting through spatially periodic planetary magnetic fields will experience time-variable electromagnetic forces. When the forcing frequencies are nearly commensurate with the particle's orbital frequency, the particle undergoes large out-of-plane and radial excursions. Specific 'Lorentz' resonances, corresponding to particular spatial periodicities in the magnetic field, occur on either side of synchronous orbit. We describe here Lorentz resonance locations and strengths for the jovian and saturnian rings. The boundaries of the halo of the jovian ring, and perhaps other ring structure, are near resonances. © 1985 Nature Publishing Group.
• Davis, D. R., Chapman, C. R., Weidenschilling, S. J., & Greenberg, R. (1985). Collisional history of asteroids: Evidence from Vesta and the Hirayama families. Icarus, 62(1), 30-53.
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  Abstract: Collisional evolution studies of asteroids indicate that the initial asteroid population at the time mean collisional velocities were pumped up to ∼5 km/sec was only modestly larger than it is today; i.e., the asteroid belt was already depleted relative to the mean surface density elsewhere in the planetary region. Numerical simulations of the collisional evolution of hypothetical initial asteroid populations have been run, subject to three constraints: they must (a) evolve to the present observed asteroid size distribution, (b) preserve Vesta's basaltic crust, and (c) produce at least the observed number of major Hirayama families. A "runaway growth" initial asteroid population distribution is found to best satisfy these constraints. A new model is presented for calculating the fragmental size distribution for the disruption of large, gravitationally bound bodies in which the material strength is increased by hydrostatic self-compression. This model predicts that large asteroid behave as intrinsically strong bodies, even if they have had a history of being collisionally fractured. This model, when applied to the breakup of the Themis and Eos family parent bodies, gives size distributions in reasonably good agreement with those observed. © 1985.
• Davis, D. R., Weidenschilling, S. J., Chapman, C. R., & Greenberg, R. (1984). Saturn ring particles as dynamic ephemeral bodies. Science, 224(4650), 744-747.
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  Abstract: Although Saturn's rings are within the Roche zone, the accretion of centimeter-sized particles into large aggregates many meters in diameter occurs readily, on a time scale of weeks. These aggregates are disrupted when tidal stresses exceed their very low strengths; thus most of the mass of the ring system is continually processed through a population of large "dynamic ephemeral bodies," which are continually forming and disintegrating. These large aggregates are not at all like the idealized ice spheres often used in modeling Saturn's ring dynamics. Their coefficient of restitution is low, hence they form a monolayer in the ring plane. The optically observable characteristics of the rings are dominated by the swarm of centimeter-sized particles.
• Greenberg, R. (1984). SATELLITE MASSES IN THE URANUS AND NEPTUNE SYSTEMS.. NASA Conference Publication, 463-480.
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  Abstract: The present knowledge of satellite masses, and how they are derived, is reviewed with emphasis on implications for bulk densities and albedos. In general for the Uranian system, the inner satellites have lower densities and/or higher albedos than the outer ones. However, uncertainties are great enough that all five satellites may have nearly equal densities. In that case, albedo would generally (but not monotonically) decrease with semi-major axis. A more severe constraint than previously published is here placed on Miranda's mass, and hence on its density and albedo. The recent radiometric value for Triton's diameter, combined with now-rather-old mass determinations, yields a density greater than 4 gm/cm**3, but systematic errors are possible in both mass and diameter.
• Greenberg, R., & Chapman, C. R. (1984). Asteroids and meteorites: Origin of stony-iron meteorites at mantle-core boundaries. Icarus, 57(2), 267-279.
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  Abstract: Stony-iron meteorites formed at the core/mantle interfaces of small asteroidal parents. The mesosiderites formed when the thick crust of a largely molten parent body (100-200 km in diameter) foundered and sank through the mantle to the core. Pallasites formed in smaller parent bodies (50-100 km) in which olivine crystals from the partially molten mantle sank to the core/mantle interface and rafted there. Subsequent collisions stripped away the rocky mantles of both kinds of parent bodies, exposing the stony-iron surfaces of their cores to direct impacts, which continue to knock off meteorite fragments. © 1984.
• Greenberg, R., & Weidenschilling, S. J. (1984). How fast do Galilean satellites spin?. Icarus, 58(2), 186-196.
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  Abstract: Each of the Galilean satellites, as well as most other satellites whose initial rotations have been substantially altered by tidal dissipation, has been widely assumed to rotate synchronously with its orbital mean motion. Such rotation would require a small permanent asymmetry in the mass distribution in order to overcome the small mean tidal torque. Since Io and Europa may be substantially fluid, they may not have the strenght to support the required permanent asymmetry. Thus, each may rotate at the unknown but slightly nonsynchronous rate that corresponds to zero mean tidal torque. This behaviour may be observable by Galileo spacecraft imaging. It may help explain the longitudinal variation of volcanism on Io and the cracking of Europa's crust. © 1984.
• Greenberg, R., Weidenschilling, S. J., Chapman, C. R., & Davis, D. R. (1984). From icy planetesimals to outer planets and comets. Icarus, 59(1), 87-113.
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  Abstract: Numerical simulations of planet growth in the outer solar system shows thatgrwoth of Uranus and Neptune occurs in reasonably short time, well below the actual age of the system, without the need for ad hoc assumptions about excess mass or artificially low relative velocities among the icy planetesimals. Low velocities, which speed accretion, are a natural consequence of the non-power-law size distribution of planetesimals, just as in our earlier simulations of terrestial planet growth. Initial planetesimals of size ∼ 100 km, predicted by formal expressions for gravitational instability in a thin disk of solid material, failed to produce sufficient debris in the size range 1 to 10 km to account for population of the Oort cloud with comet-sized bodies. However, our model of nonhomologous settling of grains to the midplane of the solar system shows that gravitational clumping did not wait until all solid material had settled to the midplane, as had been assumed in earlier models. Rather, the clumping occurred in successive portions of the material that reached the midplane, producing "initial" planetesimals probably of comet-like sizes. Models of subsequent collisional evolution show that such an initial size distribution, similar to known comets, would have been required in order to have an adequate comet-like size distribution available to feed the Oort cloud as the other planets reach full size. Comets are probably unaltered remnants of the initial population of planetesimals in the outer solar system, not fragments of larger bodies. © 1984.
• Greenberg, R. (1983). The role of dissipation in shepherding of ring particles. Icarus, 53(2), 207-218.
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  Abstract: In the absence of any damping mechanism, a shepherd satellite would force oscillations in the motion of a ring particle (relative to circular motion) that are symmetrical with respect to the encounter geometry. No net torque would be exerted by the satellite on the rings. Only in the presence of some damping mechanism (such as density wave propagation or a dissipative medium) can a particle's response lag so as to provide the asymmetry that permits a torque. Remarkably, the standard formula for the confining torque exerted by a sheperd satellite seems to be independent of damping. Moreover, many heuristic derivations of the formula tend to obscure the role of damping. In fact, the torque on any given particle does depend on the degree of damping, but that dependence disappears when the torque is averaged over a range of orbits that span resonances if the degree of damping is within a certain range. If damping is too weak or too strong, the torque can be much less than is given by the standard formula. © 1983.
• Greenberg, R., & Chapman, C. R. (1983). Asteroids and meteorites: Parent bodies and delivered samples. Icarus, 55(3), 455-481.
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  Abstract: Meteorites may be pieces of main-belt asteroids, derived by cratering collisions. The physical strength of an asteroid critically affects the quantity of ejecta that can be placed in orbits (probably resonant) that evolve to cross the Earth's. Asteroid strengths very widely due to initial composition and size (e.g., weak carbonaceous material or strong rock), subsequent geophysical evolution (e.g., formation of a strong iron core), and subsequent collisional evolution (e.g., conversion of a strong rocky body into a weak rubble pile). The meteorite yield on Earth further depends on meteorite strength, which affects longevity in space and survival through the atmosphere. We show that meteorites may be derived mainly by cratering rather than by disruptive fragmentation and from large main-belt asteroids rather than from small, Earth-approaching bodies. The model combines a wide variety of evidence from various disciplines to yield results consistent with meteorite statistics. However, no claim is made for uniqueness of this model, and many elements still admit considerable uncertainty. © 1983.
• Torbett, M., Greenberg, R., & Smoluchowski, R. (1982). Orbital resonances and planetary formation sites. Icarus, 49(3), 313-326.
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  Abstract: Orbital resonances may have played an important role in determining the locations where the planetesimal swarm eventually accreted into full-size planets. Several pairs of planets do indeed have commensurable orbital periods at present, but the case for control of planet formation by resonances is weakened by the fact that many pairs are not commensurable and that those which are do not necessarily exist at the strongest resonances. However, the mass loss and redistribution that occurred in the early solar system evolution can substantially alter the positions of planets and planetary embryos within the swarm. A cascaded resonance structure is hypothesized where planetesimal growth was accelerated at 2:1 interior and 1:2 exterior resonances with an early-formed Jupiter producing runaway growth of planetary embryos. These embryos produce their own resonances which, in turn, lead to additional embryos in a process that successively propagates inward and outward to generate a resonant configuration of embryos. In this manner, the early presence of Jupiter imposed a harmonic structure on the accumulating planetesimal swarm. For an accretion disk with surface density obeying a power law of index -1.2 the positions of the planetary embryos can be moved into a reasonably good agreement with most of the present planetary positions that is as good as that given by the Titius-Bode law. © 1982.
• Davis, D. R., Housen, K. R., & Greenberg, R. (1981). The unusual dynamical environment of Phobos and Deimos. Icarus, 47(2), 220-233.
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  Abstract: The nonintuitive dynamical environment of Phobos and Deimos is explored using a three-dimensional numerical model. Surface gravity, escape speeds, and ejecta impact contours are calculated, both for the satellites at their present orbit distances and for orbit distances they may have had in the past. Impact loci for Stickney ejecta are computed and compared with the observed groove locations in order to evaluate a possible secondary impact origin for the grooves on Phobos. Possible effects of the dynamical environment on shaping the satellites' surfaces are discussed. © 1981.
• Greenberg, R. (1981). Tidal evolution of the Galilean satellites: A linearized theory. Icarus, 46(3), 415-423.
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  Abstract: The Laplace resonance among the Galilean satellites Io, Europa, and Ganymede is traditionally reduced to a pendulum-like dynamical problem by neglecting short-period variations of several orbital elements. However, some of these variations that can now be neglected may once have had longer periods, comparable to the "pendulum" period, if the system was formerly in deep resonance (pairs of periods even closer to the ratio 2:1 than they are now). In that case, the dynamical system cannot be reduced to fewer than nine dimensions. The nine-dimensional system is linearized here in order to study small variations about equilibrium. When tidal effects are included, the resulting evolution is substantially the same as was indicated by the pendulum approach, except that evolution out of deep resonance is found to be somewhat slower than suggested by extrapolation of the pendulum results. This slower rate helps support my hypothesis that the system may have evolved from deep resonance, although other factors still need to be considered to determine whether that hypothesis is quantitatively viable. © 1981.
• Greenberg, R. (1980). Collisional growth of planetesimals. The Moon and the Planets, 22(1), 63-66.
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  Abstract: Safronov's (1972) demonstration that relative velocities of planetesimals would be comparable to the dominant size bodies' escape velocities, combined with a plausible size distribution that has most mass in the largest bodies, yielded his evolution model with limited growth of the largest planetesimal with respect to its next largest neighbors. A numerical simulation of planetesimal accretion (Greenberg et al., 1978) suggests that at least over one stage of collisional accretion, velocities were much lower than the escape velocity of the largest bodies, because the bulk of the mass still resided in km-scale bodies. The low velocities at this early stage may conceivably have permitted early runaway growth, which, in turn, would have kept the velocities low and permitted continued runaway growth of the largest bodies. © 1980 D. Reidel Publishing Company.
• Greenberg, R. (1979). Growth of large, late-stage planetesimals. Icarus, 39(2), 141-150.
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  Abstract: The late stage of terrestrial planets' growth determined many of their fundamental properties, including their thermal properties and petrology, their impact records, and possibly the existence of the Moon. A critical result of late-stage models, which bears on observable properties, is the size of the largest planetesimals that grew near, and later impacted,those that became full-size planets. There has been considerable misinterpretation of previous models regarding the relation between the size of planetesimals and their relative velocities. Furthermore, some models neglect the possible decrease in relative velocity as control is transferred from the largest to the second-largest body in an accreation zone. Evidence that Venus helped stir Earth-zone planetismals is not copelling. When models are evaluated, the results are found to depend strongly on uncertain initial conditions. The size of the second-largest planetesimal in the Earth's zone might range from ∼300 to ∼2500 km, with corresponding accretion times of ∼7 × 106 and ∼108 years, respectively. Both extremes are generated from plausible initial conditions and both seem consistent with observed planetary properties. © 1979.
• Housen, K. R., Wilkening, L. L., Chapman, C. R., & Greenberg, R. (1979). Asteroidal regoliths. Icarus, 39(3), 317-351.
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  Abstract: We develop a physical model for the evolution of regoliths on small bodies and apply it to the asteroids and meteorite parent bodies. The model considers global deposition of that fraction of cratering ejecta that is not lost to space. It follows the build up of regolith on a typical region, removed from the larger craters which are the source of most regolith blankets. Later in the evolution, larger craters saturate the surface and are incorporated into the typical region; their net ejection of materials to space causes the elevation of the typical region to decrease and once-buried regolith becomes susceptible to ejection or gardening. The model is applied to cases of both strong, cohesive bodies and to bodies of weak, unconsolidated materials. Evolution of regolith depths and gardening rates are followed until a sufficiently large impact occurs that fractures the entire asteroid. (Larger asteroids are not dispersed, however, and evolve mergaregoliths from multiple generations of surficial regoliths mixed into their interiors.) We find that large, strong asteroids generate surficial regoliths of a few kilometers depth while strong asteroids smaller than 10-km diameter generate negligible regoliths. Our model does not treat large, weak asteroids, because their cratering ejecta fail to surround such bodies; regolith evolution is probably similar to that of the Moon. Small, weak asteroids of 1- to 10-km diameter generate centimeter- to meter-scale regoliths. In all cases studied, blanketing rates exceed excavation rates, so asteroid regoliths are rarely, if ever, gardened and should be very immature measured by lunar standards. They should exhibit many of the characteristics of the brecciated, gas-rich meteorites; intact foreign clasts, relatively low-exposure durations to galactic and solar cosmic rays low solar gas contents, minimal evidence for vitrification and agglutinate formation, etc. Both large, strong asteroids and small, weak ones provide regolith environments compatible with those inferred for the parent bodies of brecciated meteorites. But from volumetric calculations, we conclude that most brecciated meteorites formed on the surfaces of, and were recycled through the interiors of, parent bodies at least several tens of kilometers in diameter. The implications of our regolith model are consistent with properties inferred for asteroid regoliths from a variety of astronomical measurements of asteroids, although such data do not constrain regolith properties nearly as strongly as meteoritical evidence Our picture of substantial asteroidal regoliths produced predominantly by blanketing differs from earlier hypotheses that asteroidal regoliths might be thin or absent and that short surface exposure of asteroidal materials is due chiefly to erosion rather than blanketing. © 1979.
• Greenberg, R. (1978). Orbital resonance in a dissipative medium. Icarus, 33(1), 62-73.
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  Abstract: Orbital resonances tend to force bodies into noncircular orbits. If a body is also under the influence of an eccentricity-reducing medium, it will experience a secular change in semimajor axis which may be positive or negative depending on whether its orbit is exterior or interior to that of the perturbing body. Thus a dissipative medium can promote either a loss or a gain in orbital energy. This process may explain the resonant structure of the asteroid belt and of Saturn's rings. For reasonable early solar system parameters, it would clear a gap near the 2:1 resonance with Jupiter on a time scale of a few thousand years; the gap width would be comparable to the Kirkwood gap presently at the location in the asteroid belt. Similarly, a gap comparable in width to Cassini's division would be cleared in Saturn's rings at the 2:1 resonance with Mimas in ∼106 yr. Most of the material from the gap would be deposited at the outer edge of ring B. The process would also affect the radial distribution of preplanetary material. Moreover, it provides an explanation for the large amplitude of the Titan-Hyperion libration. Consideration of the effects of dissipation on orbits near the stable L4 and L5 points of the restricted three-body problem indicates that energy loss causes particles to move away from these points. This results explains the large amplitude of Trojan asteroids about these points and the possible capture of Trojan into orbit about Jupiter. © 1978.
• Greenberg, R., Wacker, J. F., Hartmann, W. K., & Chapman, C. R. (1978). Planetesimals to planets: Numerical simulation of collisional evolution. Icarus, 35(1), 1-26.
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  Abstract: A simulation of collisional and gravitational interaction in the early solar system generates planets ∼500 km in diameter from an initial swarm of kilometer-sized planetesimals, such as might have resulted from gravitational instabilities in the solar nebula. The model treats collisions according to experimental and theoretical impact results (such as rebound, cratering, and catastrophic fragmentation) for a variety of materials whose parameters span plausible values for early solid objects. Ad hoc sticking mechanisms are avoided. The small planets form in ∼104 yr, during which time most of the mass of the system continues to reside in particles near the original size. The relative random velocities remain of the order of a kilometer-sized body's escape velocity, with random velocities of the largest objects somewhat depressed because of damping by the bulk of the material. The simulation is terminated when the largest objects' random motion is of smaller dimension than their collision cross sections, so that the "particle-in-a-box" statistical methods of the model break down. The few 500-km planets, in a swarm still dominated by kilometer-scale planetesimals, may act as "seeds" for the subsequent, gradual, accretional growth into full-sized planets. © 1978.
• Greenberg, R. (1977). Orbit - orbit resonances in the solar system: Varieties and similarities. Vistas in Astronomy, 21(PART 3), 209-239.
• Greenberg, R., Davis, D. R., Hartmann, W. K., & Chapman, C. R. (1977). Size distribution of particles in planetary rings. Icarus, 30(4), 769-779.
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  Abstract: Harris (Icarus 24, 190-192) has suggested that the maximum size of particles in a planetary ring is controlled by collisional fragmentation rather than by tidal stress. While this conclusion is probably true, estimated radius limits must be revised upward from Harris' values of a few kilometers by at least an order of magnitude. Accretion of particles within Roche's limit is also possible. These considerations affect theories concerning the evolution of Saturn's rings, of the Moon, and of possible former satellites of Mercury and Venus. In the case of Saturn's rings, comparison of various theoretical scenarios with available observational evidence suggests that the rings formed from the breakup of larger particles rather than from original condensation as small particles. This process implies a distribution of particle sizes in Saturn's rings possibly ranging up to ∼100 km but with most cross-section in cm-scale particles. © 1977.
• Greenberg, R. (1976). The Laplace relation and the masses of Uranus' satellites. Icarus, 29(3), 427-433.
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  Abstract: The theory of the effect of Ariel and Umbriel on Miranda's orbit is completed with a generalization of Souillart's theory of the Laplace relation. Comparison of observations of Miranda's motion with the theory yields an upper limit for the product of Ariel and Umbriel's masses of about 10-9, where Uranus' mass is unity. Therefore the albedos of Ariel and Umbriel cannot both be as low as the albedos of the darkest asteroids. © 1976.
• Greenberg, R. (1975). The dynamics of Uranus' satellites. Icarus, 24(3), 325-332.
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  Abstract: Current knowledge of the dynamics of Uranus' satellites is reviewed in support of preliminary planning for a mission to that planet. The determination of past and present orbital and rotational behavior is discussed. Improved understanding in this area is important not only for its own sake, but also for the implications with regard to the structure of the planet and to the general dynamical history of the solar system. A program of Earth-based observations over the next few years would permit most effective use of a Uranus probe. © 1975.
• Hartmann, W. K., Davis, D. R., Chapman, C. R., Soter, S., & Greenberg, R. (1975). Mars: Satellite origin and angular momentum. Icarus, 25(4), 588-594.
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  Abstract: The origin of Phobos and Deimos is considered with a view to accounting for the existence of very small satellites with circular orbits in the Martian equatorial plane, and simultaneously for the suspected angular momentum deficiency of the Mars system. All models considered failed to satisfy at least one requirement, and the problem is considered more puzzling than is at first apparent. The Martian angular momentum deficiency, if physically significant, may be unrelated to the present satellites' origin, but might relate to a large ancient satellite, long ago destroyed. Accretion onto Mars of large amounts of asteroidal dust brought in by Poynting-Robertson drag may have some bearing on the angular momentum problem. © 1975.
• Greenberg, R. (1974). Outcomes of tidal evolution for orbits with arbitrary inclination. Icarus, 23(1), 51-58.
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  Abstract: Tides raised by a satellite on a rotating planet dissipate energy and result in an exchange of angular momentum between the orbit and the spin. A set of diagrams is constructed which shows the evolution of the angular momentum vectors. The results are applied to possible histories of the Uranus system. © 1974.
• Greenberg, R. J., III, C. C., & Shapiro, I. I. (1972). Orbit-orbit resonance capture in the solar system. Science, 178(4062), 747-749.
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  Abstract: A realistic model involving mutual gravitation and tidal dissipation for the first time provides a detailed explanation for satellite orbit-orbit resonance capture. Although applying directly only to Saturn's satellites Titan and Hyperion, the model reveals general principles of resonance capture, evolution, and stability which seem applicable to other orbit-orbit resonances in the solar system.

Poster Presentations

• Hurford, T., Cooper, J., Paranicas, C., Greenberg, R., & Sturner, S. (2014, 5-7 February). Local Topographic Shielding and Radiation Shadows from Electron Irradiation on Europa. Workshop on the Habitability of Icy Worlds. Pasadena, CA.
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  LPI Contribution No. 1774
• Greenberg, R. J., & Dykhuis, M. (2013, Fall). Reflectance Properties and Age of the Baptistina Family. DPS meeting. Denver.
• Greenberg, R. J., & Hurford, T. (2013, Fall). Local Topographic Shielding and Radiation Shadows from Electron Irradiation on Europa. DPS meeting. Denver.
• Greenberg, R. J., & Laerhoven, V. (2013, Fall). The high multiplicity systems Gliese 667C and KOI 3158. DPS meeting. Denver.
• Hurford, T., Cooper, J., Paranicas, C., Greenberg, R., & Sturner, S. (2013, Fall). Local Topographic Shielding and Radiation Shadows from Electron Irradiation on Europa. DPS meeting #45American Astronomical Society.
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  #501.05
• Timpe, M., Barnes, R., Kopparapu, R., Raymond, S., & Greenberg, R. (2013, Fall). The Initial Mass Distribution for Exoplanetary Systems. DDA meeting #44American Astronomical Society.
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  #100.03
• Greenberg, R. (2012, May). Kinematics and dynamics of a high order mean-motion resonance:consideration of the Kepler-16 system. AAS/DPS Meeting. Mt. Hood, OR.
• Greenberg, R. (2012, May). Propagation of coupled changes in orbital e and a via secular perturbations. AAS/DPS Meeting. Mt. Hood, OR.
• Greenberg, R. J., & Laerhoven, V. (2012, May). Secular dynamics of the Kepler-11 system. AAS/DPS Meeting. Mt. Hood, OR.
• Van, L. C., & Greenberg, R. (2012, October). Changes in one planet's mass or semi-major axis affects all planets' eccentricities. AAS/DPS Meeting. Reno, NV.
• Van, L. C., & Greenberg, R. (2012, October). Ridges on ridges: Extensions of adjacent topography onto the flanks of Europan ridges. AAS/DPS Meeting. Reno, NV.