Distribution and covariance function of elevations on a cratered planetary surface

We derive the distribution and covariance function of elevations on a cratered planetary surface from a representation of the surface as the moving average of a random point process. It is assumed that an initially plane surface is excavated by primary impact craters with an inverse-power law size distribution. Crater rim height and rim-to-floor depth are assumed to be power functions of crater diameter. Crater shapes studied include rimless cylinders and paraboloidal bowls, and paraboloidal bowls with power-law external rims and ejecta blanket. The inverse-power law diameter distribution induces a positively skewed stable law elevation distribution, with heavy inverse-power law tails whose exponent (for small craters) is two smaller than the crater diameter distribution exponent. The covariance function (equivalently, power spectral density) is shown to be a power-law at moderate distances, whose exponent also depends on the parameters of the cratering process. Observations of lunar elevations and elevation spectral densities on a meter scale agree well with theory.This work summarizes and extends Bellcomm Technical Reports TR-68-340-3, 4, 5, which were supported by the National Aeronautics and Space Administration, Contract NASw-417.Now at Johns Hopkins University, Baltimore, Md.     

The importance of being cratered: The new role of meteorite impact as a normal geological process

Abstract— This paper is a personal (and, in many ways, incomplete) view of the past development of impact geology and of the newly recognized importance of impact events in terrestrial geological history. It also identifies some exciting scientific challenges for future investigators: to determine the full range of impact effects preserved on the Earth, to apply the knowledge obtained from impact phenomena to more general geological problems, and to continue the merger of the once exotic field of impact geology with mainstream geosciences. Since the recognition of an impact event at the Cretaceous‐Tertiary (K‐T) boundary, much current activity in impact geology has been promoted by traditionally trained geoscientists who have unexpectedly encountered impact effects in the course of their work. Their studies have involved: 1) the recognition of additional major impact effects in the geological record (the Chesapeake Bay crater, the Alamo breccia, and multiple layers of impact spherules in Precambrian rocks); and 2) the use of impact structures as laboratories to study general geological processes (e.g., igneous petrogenesis at Sudbury, Canada and Archean crustal evolution at Vredefort, South Africa). Other research areas, in which impact studies could contribute to major geoscience problems in the future, include: 1) comparative studies between low‐level (≤7 GPa) shock deformation of quartz, and the production of quartz cleavage, in both impact and tectonic environments; and 2) the nature, origin, and significance of bulk organic carbon (“kerogen”) and other carbon species in some impact structures (Gardnos, Norway, and Sudbury, Canada).     

‘High entropy’ accretion onto the surface of a white dwarf

The accretion of hydrogen rich matter onto the surface of a white dwarf, assuming that the accreted matter keeps a small part of its gravitational falling energy, is investigated. The influence of this energy excess on the hydrogen ignition and the thermal instability that follows is described and discussed.     

The Shannon entropy as a measure of diffusion in multidimensional dynamical systems

In the present work, we introduce two new estimators of chaotic diffusion based on the Shannon entropy. Using theoretical, heuristic and numerical arguments, we show that the entropy, S, provides a measure of the diffusion extent of a given small initial ensemble of orbits, while an indicator related with the time derivative of the entropy, \(S'\), estimates the diffusion rate. We show that in the limiting case of near ergodicity, after an appropriate normalization, \(S'\) coincides with the standard homogeneous diffusion coefficient. The very first application of this formulation to a 4D symplectic map and to the Arnold Hamiltonian reveals very successful and encouraging results.     

A comparison of the cratered terrain escarpments on Miranda with the thrust faults on Mercury

Large escarpments on the Uranian satellite Miranda are similar in structure to thrust faults on Mercury result of crustal contraction. The tectonic lineations may be considered evidence of a crustal contraction on Miranda result of differentiation of the satellite.     

Information entropy

In this paper we present an indicator of chaos that takes advantage of the Information Entropy concept. We develop the mathematical formulation and test the results of its application with those obtained by other methods for the 2D Hénon-Heiles system and the 3D Contopoulos, Galgani and Giorgilli potential.     

Planck absolute entropy of a rotating BTZ black hole

In this paper, the Planck absolute entropy and the Bekenstein–Smarr formula of the rotating Banados–Teitelboim–Zanelli (BTZ) black hole are presented via a complex thermodynamical system contributed by its inner and outer horizons. The redefined entropy approaches zero as the temperature of the rotating BTZ black hole tends to absolute zero, satisfying the Nernst formulation of a black hole. Hence, it can be regarded as the Planck absolute entropy of the rotating BTZ black hole.     

Entanglement entropy of a black hole and isolated horizon

Using Unruh-Verlinde temperature obtained by entropic force, we directly calculate partition functions of quantum field in Schwarzschild spacetime via quantum statistical method and derive the expression of the black hole statistical entropy. In our calculation the lower limit of integral is the location of isolated horizon introduced in loop quantum gravity and the upper limit of integral is infinity. So the obtained entropy is the statistical entropy from isolated horizon to the infinite. In our calculation there are not the cutoff and approximation. The results showed that, as long as proper Immirzi parameters are selected, the entropy obtained by loop quantum gravity is consistent with the quantum statistical entropy outside the black hole horizon. Therefore the black hole entropy is a quantum entanglement entropy outside the isolated horizon.     

Crater modification and geologic activity in Enceladus' heavily cratered plains: Evidence from the impact crater distribution

Although we can observe current activity on Saturn's satellite Enceladus with Cassini, insight into past activity is best achieved (for now) through studying the impact crater distributions. Furthermore, approximation of terrain ages can only be attained through calculations using crater densities and estimations of impact rates in the saturnian system. Here we focus on what the impact crater distribution in Enceladus' heavily cratered plains can tell us about Enceladus' geologic history. We use Cassini ISS images to count craters in the heavily cratered plains on Enceladus, along with Rhea, Dione, Tethys and Mimas as references, to develop and compare their size-frequency distributions. Comparisons of our counts show that Enceladus' cratered plains distribution is unique in that it appears to have a relative deficiency of craters for diameters ?2 km and ?6 km compared to the other satellites' heavily cratered plains. Our data also indicates that the impact crater density within the cratered plains changes with latitude. Specifically, both the north and south mid-latitude regions have approximately three times higher density than the equatorial region. We hypothesize that the “missing” small and large craters in Enceladus' cratered plains is due to a combination of viscous relaxation of the larger craters, and burial of the relaxed large craters and small craters by south polar plume and possibly E-ring material. We also conclude that the spatial density distribution is not consistent with recent polar wander.     

The role of expansion for the entropy of the Universe

Under assumption of the closed FRW-universe, the idea is presented that the cosmological expansion/contraction on its own, has an entropy balancing effectively the changing entropy of the cosmic fluid in such a way that at every epoch the total entropy of the Universe remains constant.     

The effect of surface roughness on the transmission of microwave radiation through a planetary surface

To account for surface roughness, the transmission of microwave radiation through a planetary surface to an observer is treated by a Monte Carlo technique. Sizable effects are found near the limb of the planet, and they should be included in analyses of high-resolution observations and high-precision integrated disk observations.     

The entropy and energy of intergalactic gas in galaxy clusters

Studies of the X-ray surface brightness profiles of clusters, coupled with theoretical considerations, suggest that the breaking of self-similarity in the hot gas results from an 'entropy floor', established by some heating process, which affects the structure of the intracluster gas strongly in lower-mass systems. By fitting analytical models for the radial variation in gas density and temperature to X-ray spectral images from the ROSAT PSPC and ASCA GIS, we have derived gas entropy profiles for 20 galaxy clusters and groups. We show that, when these profiles are scaled such that they should lie on top of one another in the case of self-similarity, the lowest-mass systems have higher-scaled entropy profiles than more massive systems. This appears to be due to a baseline entropy of depending on the extent to which shocks have been suppressed in low-mass systems. The extra entropy may be present in all systems, but is detectable only in poor clusters, where it is significant compared with the entropy generated by gravitational collapse. This excess entropy appears to be distributed uniformly with radius outside the central cooling regions.
We determine the energy associated with this entropy floor, by studying the net reduction in binding energy of the gas in low-mass systems, and find that it corresponds to a pre-heating temperature of 0.3 keV. Since the relationship between entropy and energy injection depends upon gas density, we are able to combine the excesses of 70140 keV cm² and 0.3 keV to derive the typical electron density of the gas into which the energy was injected. The resulting value of implies that the heating must have happened prior to cluster collapse but after a redshift z 710. The energy requirement is well matched to the energy from supernova explosions responsible for the metals which now pollute the intracluster gas.     

The dependence of the kolmogorov entropy of mappings on coordinate systems

It is now known that the Lyapunov characteristic numbers (LCNs) are a good indicator of stochasticity of mappings or dynamical systems. But the LCNs and the Kolmogorov entropy depend on the norm. In this paper we investigate under what coordinate transformations the LCNs and the Kolmogorov entropy are kept constant and give some numerical examples.     

The volume and surface area of a uniformly rotating polytrope

This paper develops algebraic expressions for the volume and surface area of a uniformly rotating polytrope. The expressions depend on an analytic theory for boundary shape developed in previous papers. A comparison with the calculations of James indicates the present theory improves on the original Chandrasekhar theory by a factor 10 or more.     

The effects of a disc field on bulge surface brightness

Collisionless N -body simulations are used in an effort to reproduce the observed tendency of the surface brightness profile of bulges to change progressively from an R ^1/4 law to an exponential, going from early- to late-type spirals. A possible cause for this is the formation of the disc, later in the history of the galaxy, and this is simulated by applying on the N -body bulge the force field of an exponential disc the surface density of which increases with time. It is shown that n , the index of the Sersic law Σ _n  ( r ) ∝ exp [−( r / r ₀)^{1/ n} ] that best describes the surface brightness profile, does indeed decrease from 4 (de Vaucouleurs law) to smaller values; this decrease is larger for more massive and more compact discs. A large part of the observed trend of n with B/D ratio is explained, and many of the actual profiles can be matched exactly by the simulations. The correlation between the disc scalelength and bulge effective radius, used recently to support the 'secular evolution' origin for bulges, is also shown to arise naturally in a scenario like this. This mechanism, however, saturates at around n  = 2 and exponential bulges cannot be produced; as n gets closer to 1, the profile becomes increasingly robust against a disc field. These results provide strong support to the old-bulge hypothesis for the early-type bulges. The exponential bulges, however, remain essentially unexplained; the results here suggest that they did not begin their lives as R ^1/4 spheroids, and hence were probably formed, at least in part, by different processes from those of early-type spirals.     

Phase correlations in chaotic dynamics: a Shannon entropy measure

In the present work, we investigate phase correlations by recourse to the Shannon entropy. Using theoretical arguments, we show that the entropy provides an accurate measure of phase correlations in any dynamical system, in particular when dealing with a chaotic diffusion process. We apply this approach to different low-dimensional maps in order to show that indeed the entropy is very sensitive to the presence of correlations among the successive values of angular variables, even when it is weak. Later on, we apply this approach to unveil strong correlations in the time evolution of the phases involved in the Arnold’s Hamiltonian that lead to anomalous diffusion, particularly when the perturbation parameters are comparatively large. The obtained results allow us to discuss the validity of several approximations and assumptions usually introduced to derive a local diffusion coefficient in multidimensional near-integrable Hamiltonian systems, in particular the so-called reduced stochasticity approximation.     

The emergence of a spherical magnetogasdynamic shock wave at the surface of a star

The propagation of a magnetogasdynamic shock wave originating in a stellar interior, is ocnsidered when it approaches the surfaces of the star. The flow behind the shock wave is assumed isothermal rather than adiabatic to stimulate the conditions of large radiative transfer near the stellar surface. The product solution of McVittie has been used to obtain exact solution of the problem. It has been obtained that velocity, density, pressure and magnetic field increases as we move from shock surface towards the nucleus of the star.     

Kolmogorov entropy as a measure of disorder in some non-integrable Hamiltonian systems

This paper is devoted to study the stochastic behaviour of some Hamiltonian systems with closed velocity curves. We investigate Hamiltonians already studied by Ali and Somorjai (1). These authors, by discussing Poincaré's surfaces of section for several energy values, gave a qualitative evaluation of the stochasticity of the systems.Here we present a quantitative study of this stochastic behaviour. For each energy we compute the Lyapunov characteristic exponents of fifty orbits chosen at random, in order to calculate the Kolmogorov entropy by Pesin's formula. Our results are in agreement with those of Ali and Somorjai: the disorder does not increase monotonically with increasing energy. However, we find that the largest entropy does not necessarily correspond to the maximum of the stochastic volume. The Kolmogorov entropy thus appears to be a good measure of the degree of disorder of dynamical systems.