Approximation Theory of Multivariate Spline Functions in Sobolev Spaces

In this paper we study some approximation theory questions which arise from the analysis of the discretization error associated with the use of the Rayleigh-Ritz-Galerkin method for approximating the solutions to various types of boundary value problems, cf. [13, [2], [33, [43, [7], [8], [93, [12], [143, [18], [19], [20] and [22]. In particular, we consider upper and lower bounds for the error in approximation of certain families of functions in Sobolev spaces, cf. [15], by functions in finite-dimensional "polynomial spline types" subspaces, cf. [16]. In doing this, we directly generalize, improve, and extend the corresponding results of[1], [17], [18], [19], [20], and [21]. Throughout this paper, the symbol K will be used repeatedly to denote a positive constant, not necessarily the same at each occurrence and the symbol μ will be used repeatedly to denote a nonnegative, continuous function on [0,∞], not necessarily the same at each occurrence

Atmospheric effects on crater growth on Venus

Laboratory experiments allow examining the consequences of complex processes operating over a wide range of scales (both temporal and spatial) and frequently reveal effects that are obvious only in hindsight. Even though all processes may not scale directly, isolation of the controlling variables allows assessing first-order effects through analytical approximations. This approach can be illustrated by the systematic sequence of ballistic ejection, the response of an atmosphere to a strong energy source, the scaling of ejecta thickness, and the role of secondary cratering. Here it is proposed that the effects of atmospheric pressure and density on crater growth (hence, scaling) observed in laboratory experiments has particular relevance for craters on Venus

The role of impact cratering for Mars sample return

The preserved cratering record of Mars indicates that impacts play an important role in deciphering Martian geologic history, whether as a mechanism to modify the lithosphere and atmosphere or as a tool to sample the planet. The various roles of impact cratering in adding a broader understanding of Mars through returned samples are examined. Five broad roles include impact craters as: (1) a process in response to a different planetary localizer environment; (2) a probe for excavating crustal/mantle materials; (3) a possible localizer of magmatic and hydrothermal processes; (4) a chronicle of changes in the volcanic, sedimentary, atmospheric, and cosmic flux history; and (5) a chronometer for extending the geologic time scale to unsampled regions. The evidence for Earth-like processes and very nonlunar styles of volcanism and tectonism may shift the emphasis of a sampling strategy away from equally fundamental issues including crustal composition, unit ages, and climate history. Impact cratering not only played an important active role in the early Martian geologic history, it also provides an important tool for addressing such issues

Effect of impact angle on central-peak/peak-ring formation and crater collapse on Venus

Although asymmetry in ejecta patterns and craters shape-in-plan are commonly cited as diagnostic features of impact angle, the early-time transfer of energy from impactor to target also creates distinctive asymmetries in crater profile with the greatest depth uprange. In order to simulate gravity-controlled crater-growth, laboratory experiments use loose particulate targets as analogs for low-strength material properties following passage of the shock. As a result, impact crater diameter D in laboratory experiments generally is many times greater than the impactor diameter 2r (factor of 40), and early-time asymmetries in energy transfer from oblique impacts are consumed by subsequent symmetrical crater growth, except at the lowest angles (less than 25 deg). Such asymmetry is evident for oblique (less than 60 deg from horizontal) impacts into aluminum where D/2r is only 2 to 4. Because cratering efficiency decreases with increasing crater size and decreasing impact angle, large scale planetary craters (4080 km) should have transient excavation diameters only 6-10 times larger than the impactor. At basin scales, D/2r is predicted to be only 3-5, i.e., approaching values for impacts into aluminum in laboratory experiments. As a result, evidence for early-time asymmetry in impactor energy transfer should become evident on planetary surfaces, yet craters generally retain a circular outline for all but the lowest impact angles

Styles of ejecta emplacement under atmospheric conditions

Laboratory experiments provide essential first-order constraints on processes affecting ballistic ejecta and styles of ejecta emplacement under different atmospheric environments at planetary scales. The NASA-Ames Vertical Gun allows impacting different fine-grained particulate targets under varying atmospheric pressure and density, thereby helping to isolate controlling variables. Further analysis now permits characterizing distinct modes of emplacement that reflect the degree of ejecta entrainment within a turbidity flow created by ejecta curtain movement through the atmosphere

Impact-generated winds on Venus: Causes and effects

The pressure of the dense atmosphere of Venus significantly changes the appearance of ejecta deposits relative to craters on the Moon and Mercury. Conversely, specific styles and sequences of ejecta emplacement can be inferred to represent different intensities of atmospheric response winds acting over different timescales. Three characteristic timescales can be inferred from the geologic record: surface scouring and impactor-controlled (angle and direction) initiation of the long fluidized run-out flows; nonballistic emplacement of inner, radar-bright ejecta facies and radar-dark outer facies; and very late reworking of surface materials. These three timescales roughly correspond to processes observed in laboratory experiments that can be scaled to conditions on Venus (with appropriate assumptions): coupling between the atmosphere and earlytime vapor/melt (target and impactor) that produces an intense shock that subsequently evolves into blast/response winds; less energetic dynamic response of the atmosphere to the outward-moving ballistic ejecta curtain that generates nonthermal turbulent eddies; and late recovery of the atmosphere to impact-generated thermal and pressure gradients expressed as low-energy but long-lived winds. These different timescales and processes can be viewed as the atmosphere equivalent of shock melting, material motion, and far-field seismic response in the target. The three processes (early Processes, Atmospheric Processes, and Late Recovery Winds) are discussed at length

Atmospheric effects on oblique impacts

Laboratory experiments and theoretical calculations often use vertical impact angles (90 deg) in order to avoid the complicating effect of asymmetry. Nevertheless, oblique impacts represent the most likely starting condition for planetary cratering. Changing both impact angles and atmospheric pressure not only allows testing previous results for vertical impacts but also reveals phenomena whose signatures would otherwise be masked in the planetary cratering record. The laboratory studies were performed for investigating impact cratering processes. Impact angles can be increased from 0 to 90 deg in 15 deg increments while maintaining a flat target surface. Different atmospheres (nitrogen, argon, and helium) characterized the effects of both gas density and Mach number. Targets varied according to purpose. Because of the complexities in atmosphere-impactor-ejecta interactions, no single combination allows direct simulation of a planetary-scale (10-100 km) event. Nevertheless, fundamental processes and observed phenomena allow formulating first-order models at such broad scales

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Crater ejecta morphology and the presence of water on Mars

The possible effects of projectile, target, and environment on the cratering process is reviewed. It is suggested that contradictions in interpreting Martian crater ejecta morphologies reflect over simplifying the process as a singular consequence of buried water. It seem entirely possible that most ejecta facies could be produced without the presence of liquid water. However, the combination of extraordinary ejecta fluidity, absence of secondaries, and high ejection angles all would point to the combined effects of atmosphere and fluid rich substrates. Moreover, recent experiments revealing the broad scour zone associated with rapid vapor expansion may account for numerous craters in the circumpolar regions with subtle radial grooving extending 10 crater radii away with faint distal ramparts. Thus certain crater ejecta morphologies may yet provide fundamental clues for the presence of unbound water

Non-random cratering flux in recent time

Proposed periodic cycles of mass mortality have been linked to periodic changes in the impact flux on Earth. Such changes in the impact flux, however, also should be recorded on the Moon. Previous studies have concluded that the impact flux on the Moon over the last 1 to 2 billion years has been reasonably constant, but sudden changes in the impact flux over time intervals as short as 30 my could not be detected in these studies unless the added crater population greatly exceeded the cumulative cratering record. Consequently this study focuses only on bright-rayed craters larger than 1 km thereby not only limiting the study to recent craters but also largely eliminating contamination by secondary craters. Preservation of ray patterns and other fine-scale surface textures in the ejecta provides first-order culling of craters younger than Tycho, i.e., about 100 my. Although a periodic change in the impact flux in the Earth-Moon system cannot yet be confirmed from the data, a non-random component appears to exist with an increased flux around 7 and 15 my. The concentrations in different quadrants of the lunar hemisphere would be consistent with a shower of debris generally smaller than 0.5 km