A study of lunar impact crater size-distributions

Discrepancies in published crater frequency data prompted this study of lunar crater distributions. Effects modifying production size distributions of impact craters such as surface lava flows, blanketing by ejecta, superposition, infilling, and abrasion of craters, mass wasting, and the contribution of secondary and volcanic craters are discussed. The resulting criteria have been applied in the determination of the size distributions of unmodified impact crater populations in selected lunar regions of different ages. The measured cumulative crater frequencies are used to obtain a general calibration size distribution curve by a normalization procedure. It is found that the lunar impact crater size distribution is largely constant in the size range 0.3 km ?D ? 20 km for regions with formation ages between ≈ 3 × 10⁹ yr and ? 4 × 10⁹ yr. A polynomial of 4th degree, valid in the size range 0.8 km ?D ? 20 km, and a polynomial of 7th degree, valid in the size range 0.3 km ?D ? ? 20 km, have been approximated to the logarithm of the cumulative crater frequencyN as a function of the logarithm of crater diameterD. The resulting relationship can be expressed asN ～D ^α(D) where α is a function depending onD. This relationship allows the comparison of crater frequencies in different size ranges. Exponential relationships with constant α, commonly used in the literature, are shown to inadequately approximate the lunar impact crater size distribution. Deviations of measured size distributions from the calibration distribution are strongly suggestive of the existence of processes having modified the primary impact crater population.     

Estimates of the physical parameters of supernova remnants (SNR)

The various physical parameters of a SNR can easily be worked out from graphs established on the basis of the recalibrated relation between the radio surface brightness , and the linear diameterD of a SNR (Ilovaisky and Lequeux, 1972). These graphs lead to the estimation of the distancer (kpc), linear diameterD (pc), monochromatic power at 1 GHz,P ₁ GHz (W Hz^–1); and total powerP _tot (a) (erg. s^–1) of a SNR, given its mean angular diameter <> (arc min), flux density at 1 GHz, S₁ GHz (f.u.) and spectral indexa. Three SNR (W28A₁, Monoceros SNR, W49B) are used to illustrate the case. The radio spectrum of one of these (W49B), curved at low frequencies, is explained in terms of absorption by the diffuse interstellar medium. Various cases are discussed and some physical parameters of the absorbing matter are established.     

Moon: ‘ghost’ craters formed during mare filling

This paper discusses formation of pathological cases of crater morphology due to interaction of craters with molten lavas. Terrestrial observations of such a process are discussed. In lunar maria, a number of small impact craters (D < 10 km) may have been covered by thin layers of fluid lavas, or formed in molten lava. Some specific lunar examples are discussed, including unusual shallow rings resembling experimental craters deformed by isostatic filling.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.     

The luminosity-diameter relation in normal ellipticals: A tool for environmental effects

We have examined a sample of 150 elliptical galaxies taken from a recent survey (Bursteinet al., 1987) which includes also new CCD photometric data. The fits of total luminosityL versus three different diameter indicators (i.e., the effective diameterD _e, theD _n diameter proposed by Dressleret al. (1987), and the isophotal diameterD ₀ at 25 mag arc sec^–2) had given no sign of environmental effect on such quantities. A comparison with previous results based on lower quality data was carried out in order to check the reliability of different photometries and statical tools used.Our main conclusion is that the luminosity-diameter relation seems to be universal in the range of luminosities spanned by our sample; so environmental effects, if present, must be smaller than observational errors.     

Transitional impact craters on the Moon: Insight into the effect of target lithology on the impact cratering process

We studied a data set of 28 well‐preserved lunar craters in the transitional (simple‐to‐complex) regime with the aim of investigating the underlying cause(s) for morphological differences of these craters in mare versus highland terrains. These transitional craters range from 15 to 42 km in diameter, demonstrating that the transition from simple to complex craters is not abrupt and occurs over a broad diameter range. We examined and measured the following crater attributes: depth (d), diameter (D), floor diameter (D_f), rim height (h), and wall width (w), as well as the number and onset of terraces and rock slides. The number of terraces increases with increasing crater size and, in general, mare craters possess more terraces than highland craters of the same diameter. There are also clear differences in the d/D ratio of mare versus highland craters, with transitional craters in mare targets being noticeably shallower than similarly sized highland craters. We propose that layering in mare targets is a major driver for these differences. Layering provides pre‐existing planes of weakness that facilitate crater collapse, thus explaining the overall shallower depths of mare craters and the onset of crater collapse (i.e., the transition from simple to complex crater morphology) at smaller diameters as compared to highland craters. This suggests that layering and its interplay with target strength and porosity may play a more significant role than previously considered.     

A description of the crater Tsiolkovsky on the lunar far side

Although researchers in the last decade have been primarily concerned with the exotic findings of the more distant planets and moons in our solar system, as given by the Voyager series, there is still much work to be done on our nearer neighbours, including the Moon. This paper summarizes some basic age dating of a portion of the lunar surface, namely the mare in the crater Tsiolkovsky on the lunar far side.Using the Apollo 15 panoramic camera photographs, the cumulative crater frequency (N km^-2) relative to crater diameter (D) distribution has been obtained for the mare in the crater Tsiolkovsky. The diameter size range sampled was 0.07 km < D < 1 km. A total of 12 604 craters were counted and their average apparent diameters measured. There were 85 sample areas on the mare surface which were chosen at random, after exclusion of blanketed, volcanic or secondary cratered areas. It was found that a large proportion of the crater floor contains endogenic features, especially volcanic vents at approximately D = 0.3 km. An additional 7 areas of interest were also examined in detail for comparison with areas of purely primary impact craters. Evidence for up to 8 lava floodings can be detected from the size-frequency distributions although no visual data, e.g., flow lobes, can be seen on the mare surface.The total size-frequency distribution for all the areas is coincident with Neukum et al. (1975a and b) Calibration Distribution in the size range 0.25 km < D < 1 km which is at the smallest crater diameters that they obtained. Neukum et al. (1975a and b) give their distribution as a polynomial of 7th degree. However, in this present study a variation is indicated in the steepening of the curve for D < 0.1 km.The results also approximate (but only for D < 0.6 km) the distribution obtained by Shoemaker et al. (1970) in the range 100 m < D < 3 km where N ~ D ^-2.9. The best fit line reached for the data given here is N ~ D ^-2.682.Comparison of the distribution with plots for the maria at Apollo 11, 12, and 15 landing sites show that Tsiolkovsky mare is 3.51 ± 0.1 × 10⁹ yr old. This agrees with other workers (see Gornitz, 1973) who place it between Mare Tranquillitatis (Apollo 11 radiometric dating: 3.5 to 3.9 aeons) and Oceanus Procellarum (Apollo 12: 3.5 to 3.4 aeons). There are no rock samples from Tsiolkovsky to given an absolute age.This places Tsiolkovsky mare within the weighted mean of the age range (1.0 to 4.3 × 10⁹ yr old) of the maria on the Moon. From this it can be concluded that the processes producing the vast basalt outpourings seen on the Moon's face apply for the far side also and that there is a linking factor for the whole Moon.     

Morphological characteristics of lunar craters with small depth/diameter ratio. II

The morphological characteristics of craters with relatively small ratio depth/diameter are discussed. It is observed that many morphological similarities exist among craters possessing ratiosd/D which do not differ considerably. The distribution of 1933 craters with respect to diameter, depth and the depth/diameter ratio is presented.     

Critical crater diameter and asteroid impact seismology

Abstract— If impact stress reverberation is the primary gradational process on an asteroid at global scales, then the largest undegraded crater records an asteroid's seismological response. The critical crater diameter D_crit is defined as the smallest crater whose formation disrupts all previous craters globally up to its size; it is solved for by combining relationships for crater growth and for stress attenuation. The computation for D_crit gives a simple explanation for the curious observation that small asteroids have only modest undegraded craters, in comparison to their size, whereas large asteroids have giant undegraded craters. D_crit can even exceed the asteroid diameter, in which case all craters are “local” and the asteroid becomes crowded with giant craters. D_crit is the most recent crater to have formed on a blank slate; when it is equated to the measured diameter of the largest undegraded crater on known asteroids, peak particle velocities are found to attenuate with the 1.2–1.3 power of distance—less attenuative than strong shocks, and more characteristic of powerful seismic disturbances. This is to be expected, since global degradation can result from seismic (cm s^?1) particle velocities on small asteroids. Attenuation, as modeled, appears to be higher on asteroids known to be porous, although these are also bodies for which different crater scaling rules might apply.     

On critical observations that constrain models of terrestrial hypervelocity impact craters

Impact structures in the crystalline rocks of the Canadian Shield range over two orders of magnitude in size and display morphologies recognized elsewhere in the solar system. This contribution draws upon new examinations of drill core from Canadian craters to reaffirm some relationships, modify others, and refine the transitions from simple to complex with central peak to peak‐ring structures. These include recognizing the hyperbolic form of transient craters, sharpening the allochthon–parautochthon distinction, and proposing new formulae for key relationships. It emphasizes the role of dynamic tensile strength and the attenuation of tensile rarefaction waves in determining the size of both transient and final crater dimensions. On Earth, depth (d) to diameter (D) ratios are not invariant at about 1:10 but change smoothly with size from 1:3 at Brent through 1:5 to 1:10 in the largest; that is, d = 0.4 D^0.75. In craters in crystalline rocks, the central peak grows at about uplift = 0.175 D until, at D about 28 km, the uplift rises above the original surface then collapses to form a peak‐ring structure. These relationships demonstrate the dominant role of gravity in attenuating tensile rarefaction waves and controlling transient crater depth and overall size relative to the volume shocked.     

Radial variation of lunar crater rim topography

The variation of rim topography as a function of range from the crater rim has been determined for a group of morphologically fresh lunar craters (D = 10–140 km) using the recent series of Lunar Topographic Orthophotomaps. The rate at which exterior crater topography converges with the surrounding surface is highly variable along different radial directions at individual craters as well as between different craters. At several craters, oblique impact appears to have contributed to azimuthal elevation/range variations. The topographic expression of a crater above the surrounding surface typically decreases to one-tenth of the estimated rim height at a range of 1.3R–1.7R, well within the rough-textured ejecta deposit surrounding the crater. Comparisons with terrestrial craters suggest that the topographic crater rim is predominantly a structural feature. In most craters large portions of the hummocky facies and virtually all of the radial facies, in spite of their rough appearance and local topographic variations, provide no significant net topographic addition to the preexisting surface. The extreme variability of crater rim topography strongly suggests that ejecta thicknesses are highly variable and that a unique power-law expression cannot truly represent the radial variation of ejecta deposit thickness.     

Comparison of central pit craters on Mars,Mercury, Ganymede,and the Saturnian satellites

We report on the first results of a large‐scale comparison study of central pit craters throughout the solar system, focused on Mars, Mercury, Ganymede, Rhea, Dione, and Tethys. We have identified 10 more central pit craters on Rhea, Dione, and Tethys than have previously been reported. We see a general trend that the median ratio of the pit to crater diameter (D_p/D_c) decreases with increasing gravity and decreasing volatile content of the crust. Floor pits are more common on volatile‐rich bodies while summit pits become more common as crustal volatile content decreases. Uplifted bedrock from below the crater floor occurs in the central peak upon which summit pits are found and in rims around floor pits, which may or may not break the surface. Peaks on which summit pits are found on Mars and Mercury share similar characteristics to those of nonpitted central peaks, indicating that some normal central peaks undergo an additional process to create summit pits. Martian floor pits do not appear to be the result of a central peak collapse as the median ratio of the peak to crater diameter (D_pk/D_c) is about twice as high for central peaks/summit pits than D_p/D_c values for floor pits. Median D_pk/D_c is twice as high for Mars as for Mercury, reflecting differing crustal strength between the two bodies. Results indicate that a complicated interplay of crustal volatiles, target strength, surface gravity, and impactor energy along with both uplift and collapse are involved in central pit formation. Multiple formation models may be required to explain the range of central pits seen throughout the solar system.     

Extreme sensitivity of the YORP effect to small-scale topography

Radiation recoil (YORP) torques are shown to be extremely sensitive to small-scale surface topography, using numerical simulations. Starting from a set of “base objects” representative of the near-Earth object population, random realizations of three types of small-scale topography are added: Gaussian surface fluctuations, craters, and boulders. For each, the expected relative errors in the spin and obliquity components of the YORP torque caused by the observationally unresolved small-scale topography are computed. Gaussian power, at angular scales below an observational limit, produces expected errors of order 100% if observations constrain the surface to a spherical harmonic order l?10. For errors under 10%, the surface must be constrained to at least l=20. A single crater with diameter roughly half the object's mean radius, placed at random locations, results in expected errors of several tens of percent. The errors scale with crater diameter D as D² for D>0.3 and as D³ for D<0.3 mean radii. Objects that are identical except for the location of a single large crater can differ by factors of several in YORP torque, while being photometrically indistinguishable at the level of hundredths of a magnitude. Boulders placed randomly on identical base objects create torque errors roughly 3 times larger than do craters of the same diameter, with errors scaling as the square of the boulder diameter. A single boulder comparable to Yoshinodai on 25143 Itokawa, moved by as little as twice its own diameter, can alter the magnitude of the torque by factors of several, and change the sign of its spin component at all obliquities. Most of the total torque error produced by multiple unresolved craters is contributed by the handful of largest craters; but both large and small boulders contribute comparably to the total boulder-induced error. A YORP torque prediction derived from groundbased data can be expected to be in error by of order 100% due to unresolved topography. Small surface changes caused by slow spin-up or spin-down may have significant stochastic effects on the spin evolution of small bodies. For rotation periods between roughly 2 and 10 h, these unpredictable changes may reverse the sign of the YORP torque. Objects in this spin regime may random-walk up and down in spin rate before the rubble-pile limit is exceeded and fissioning or loss of surface objects occurs. Similar behavior may be expected at rotation rates approaching the limiting values for tensile-strength dominated objects.     

Intra-crater glacial processes in central Utopia Planitia, Mars

We describe and interpret a series of previously unidentified glacial-like lobes (34-43°N; 107-125°E) that were discovered as part of a survey of large (D > 5 km) impact craters in Utopia Planitia, one of the great northern plains of Mars. The lobes have characteristics that are consistent with a glacial origin. Evidence includes curvilinearity of form, lineations and ridges, and surface textures that are thought to form by the sublimation of near-surface volatiles. The lobes display morphologies that range from wedge-shaped to near-circular to elongate. The flow directions are towards the northern walls in the case of craters with large single lobes, and in all directions in the case of the largest (D > 30 km) craters. Concentric crater fill is also interspersed within craters of our study region, with such craters having much higher filling rates than those with flow lobes. We suggest that the impact crater population in south-west Utopia Planitia demonstrates a spectrum of glacial modifications, from low levels of filling in the case of craters with elongate lobes at one extreme, to concentric crater fill in highly-filled craters at the other.     

A photometric investigation of the lunar crater rays

This investigation deals with accurate photometric data concerning a number of rays of Tycho, Copernicus, Kepler, and Aristarchus. They have been derived from plates taken at the Yerkes Observatory in a night of a total lunar eclipse near phase angle 0°. By comparing the normal albedo with that of the surroundings of the rays we found that they can be interpreted as samples of telescopically unresolved bright patches. The fractional areak covered by these patches varies along the ray and shows that they are composed of a number of separate ray elements. The observed value ofk is in accordance with counts on a Ranger photograph.The distribution of the brightness along the rays has also been compared with the mass distribution of the ejecta in the rays around terrestrial explosion craters. The mean length of the lunar rays is in full accordance with its extrapolated terrestrial value. We cannot assume, however, that the rays are regions covered with a homogeneous layer of white powder, because the comparison with the terrestrial explosion craters gives an unprobable value for the height of the layer of the ejecta. The same results follow now from the photometric properties of the rays.From a comparison with the difference in albedo at the Surveyor's footprints follows the suggestion that the lunar rays are composed of bright patches, where the surface material came into a state of lower porosity, while it has a higher porosity in the dark halos around the craters. A suspected dark halo around Tycho has photometrically been measured and the results prove that it really exists. Kepler also shows a very weak halo.     

Crater size-shape profiles for the moon and mercury: Terrain effects and interplanetary comparisons

New crater size-shape data were compiled for 221 fresh lunar craters and 152 youthful mercurian craters. Terraces and central peaks develop initially in fresh craters on the Moon in the 0–10 km diameter interval. Above a diameter of 65 km all craters are terraced and have central peaks. Swirl floor texture is most common in craters in the size range 20–30 km, but it occurs less frequently as terraces become a dominant feature of crater interiors. For the Moon there is a correlation between crater shape and geomorphic terrain type. For example, craters on the maria are more complex in terms of central peak and terrace detail at any given crater diameter than are craters in the highlands. These crater data suggest that there are significant differences in substrate and/or target properties between maria and highlands. Size-shape profiles for Mercury show that central peak and terrace onset is in the 10–20 km diameter interval; all craters are terraced at 65 km, and all have central peaks at 45 km. The crater data for Mercury show no clear cut terrain correlation. Comparison of lunar and mercurian data indicates that both central peaks and terraces are more abundant in craters in the diameter range 5–75 km on Mercury. Differences in crater shape between Mercury and the Moon may be due to differences in planetary gravitational acceleration (gMercury=2.3gMoon). Also differences between Mercury and the Moon in target and substrate and in modal impact velocity may contribute to affect crater shape.     

Lunar craters evolution and meteoroidal flux in pre-mare and post-mare times

In order to study the geomorphic evolution and lifetimes of lunar craters, data were collected from (i) 32mare andterra provinces of the nearside of the Moon using the L.P.L. catalog; (ii) amare area in Sinus Medii, using direct observations of Lunar Orbiter photos, and (iii) aterra area on the farside using direct observations of Zond-8 photos. The theory presented in a previous publication is expanded and applied to the data.The following conclusions are obtained. (1) Steady-state conditions occur on the studiedmare surfaces for craters of diameter up to approximately 220 m, and on the studiedterra surfaces for craters of diameter up to at least 50 km. (2) The average lifetime of a crater, in addition of being a function of the meteoroidal flux, is a steep function of the diameter of the crater. (3) The correlation is good between a geomorphic classification of craters based on visual comparison with standard craters and a classification of craters based on their depth-diameter ratio, resulting in a coefficient of rank correlation of 0.64. (4) When craters are classified as young, mature, and old, the length of time spent as young is less than a few percent of the total lifetime of the crater; the time spent as mature is 10 to 30%; and as much as 80% is spent as an old crater. Within the error of the calculations, these values are independent of crater diameter and apply to both pre-mare and post-mare craters, indicating that they are also independent of the intensity of the meteoroidal flux. (5) The average lifetime of a 50 km crater in pre-mare times is estimated to be less than 0.3×10⁹ years. (6) The average lifetime of a 50 km crater in post-mare times is estimated to be between 3×10¹¹ and 10¹⁴ years. (7) The average meteoroidal flux in pre-mare times is estimated to be three to six orders of magnitude more intense than in post-mare times.     

Periodicity and decay of craters over the past 600 myr

The data set of Grieve which provides diameters and ages of craters is analyzed to obtain periodicity of the formation rate and decay constant of craters. It is confirmed that large craters (D 10 km) do not exhibit any periodicity while small ones appear to satisfy the Broadbent criterion for quantum (periodicity) hypothesis at P 29.5 myr. The result is consistent with a recent study of Bailey who showed that large craters are largely due to asteroids. Allowing for the decay of craters, an excess of 4 small craters within the nearest past is detected. In this sense, one may argue that the solar system is now in a moderate comet shower.     

Fractal properties of outflows from Venusian impact craters

This article presents fractal analyses of 28 outflow margins from 18 Venusian impact craters. The fractal dimensions of the second parts of R-plots of the outflow outlines were measured by a three-step method. The fractality values for the same outflow measured from images which have only a small difference in resolution are very similar, while large differences in image resolution may result in differences in fractality, possibly due to the fact that we are actually studying geological processes on different scales. The outflows were classified into three general categories: single outflows, multiple outflows and outflow fields. Three conclusions were drawn on the relations between fractality and crater diameter, which may be related to the greater effects caused by the immediate local environment on the outflows from small craters than on those from larger craters. Investigation of the relations between the regional topography and fractality indicates that there are substantially less effects on outflows originating from large craters than on those from small craters. The smooth bending in the R-plot and the higher D-value for the multiple outflows could result from the mixing of various fractal or non-fractal units. When comparing our results with the fractality of terrestrial lava flows, outflows from craters of diameter greater than 50 km seem to resemble aa-type lava flows in their fractal dimensions and outflows from craters of diameter below 50 km tend to be more pahoehoe-like. This preliminary result is based on 28 outflows, however, and the pattern should be investigated more carefully by further more extensive work.     

On the rate of ancient meteoroidal flux: Studies of the pre-mare crater population on the Moon

According to radiometric dating of lunar rocks, meteoroidal bombardment and accompanying cratering on the Moon were intensive in the first 0.7×10⁹ y, the so-calledterra stage. Recently the hypothesis of a terminal cataclysm has been gaining acceptance, meaning that a sharp increase in the bombardment followed by a steep decay occurred at the end of theterra stage.The purpose of this paper is to investigate possible variations in the intensity of the bombardment during theterra stage by analyzing the population of large (3–1000 km)terra craters and comparing it with results obtained by theoretical models. The proportion of fresh craters is specifically used.Observational data were obtained by studying the craters on an oldterra surface photographed by Zond 8 and/or covered by LTO and LM maps and by conducting a statistical study of theterra listed in the Lunar and Planetary Laboratory Catalog. Mathematical models were developed in such a way as to make them dimensionless, and as such applicable even without the knowledge of the physical constants and variables involved. Particularly powerful is the method of measuring time as multiples of the average lifetime of a crater of that size.The following conclusions and/or interpretations are reached. (1) A crater remains fresh for less than 20% of the average life of a crater of that size. (2) A condition of equilibrium (i.e., on the average, a new impact does not cause a net increase in the total number of craters) is reached or almost reached on lunarterrae. (3) The age of theterra surface is more than three average lifetimes of the 2 km to at least 20 km size craters (4) The observedterra crater population isnot the result of two cataclysms, one at the beginning of the stage and one at the end. (5) This population, however, could be the result of a constant or slowly decaying flux continuing until the end of the stage, when the terminal cataclysms occurred. This sequence of events cannot be proven or disproven. (6) If no terminal cataclysm occurred, then the meteoroidal flux during theterra stage was slowly decaying or constant. (7) The formation of all the large multi-ringed basins essentially contemporaneously is doubtful.     

Viscosity of the lithosphere of Enceladus

High resolution Voyager II images of Enceladus reveal that some regions on its surface are highly cratered; the most heavily cratered surfaces probably date back to a period of heavy bombardment. The forms of many of the craters on Enceladus are similar to those of fresh lunar craters, but many of the craters are much shallower in depth, and the floors of some craters are bowed up. The flattering of craters and bowing up of the floors are indicative of viscous relaxation of the topography. Analysis of the forms of the flattened craters suggests that the viscosity at the top of the lithosphere, in the most heavily cratered regions, is between 10²⁴ and 10²⁵ P. The exact time scale for the collapse of the craters is not known, but probably was between 100 my and 4 gy. The flattened craters are located in distinct zones that are adjacent to zones, of similar age, where craters have not flattened. The zones where flattened craters occur possibly are regions in which the heat flow was (or is) higher than that in the adjacent terrains. Because the temperature at the top of the lithosphere of Enceladus would be less than or equal to that of Ganymede and Callisto, if it is covered by a thick regolith, and because the required viscosity on Enceladus is one to two orders of magnitude less than that for Ganymede and Callisto, it can be concluded that the lithospheric material on Enceladus is different from that of Ganymede and Callisto. Enceladus probably has a mixture of ammonia ice and water ice in the lithosphere, whereas the lithospheres of Ganymede and Callisto are composed primarily of water ice.