Maximum entropy image reconstruction-a practical non-information-theoretic approach

The maximum entropy method (MEM) of image reconstructtion is discussed in the context of incomplete Fourier information (as in aperture synthesis). Several current viewpoints on the conceptual foundation of the method are analysed and found to be unsatisfactory. It is concluded that the MEM is a form of model-fitting, the model being a non-linear transform of a band-limited function. A whole family of ’entropies’ can be constructed to give reconstructions which (a) are individually unique, (b) have sharpened peaks and (c) have flattened baselines. The widely discussed 1nB and - B1nB forms of the entropy are particular cases and lead to Lorentzian and Gaussian shaped peaks respectively. However, they hardly exhaust the possibilities-for example, B^1/2 is equally good. The two essential features of peak sharpening and baseline flattening are shown to depend on a parameter which can be controlled by adding a suitable constant to the zero spacing correlation ρ₀₀. This process, called FLOATing, effectively tames much of the unphysical behaviour noted in earlier studies of the MEM. A numerical scheme for obtaining the MEM reconstruction is described. This incorporates the FLOAT feature and uses the fast Fourier transform (FFT), requiring about a hundred FFTs for convergence. Using a model brightness distribution, the MEM reconstructions obtained for different entropies and different values of the resolution parameter are compared. The results substantiate the theoretically deduced properties of the MEM. To allow for noise in the data, the least-squares approach has been widely used. It is shown that this method is biased since it leads to deterministic residuals which do not have a Gaussian distribution. It is suggested that fitting the noisy data exactly has the advantage of being unbiased even though the noise appears in the final map. A comparison of the strengths and weaknesses of the MEM and CLEAN suggests that the MEM already has a useful role to play in image reconstruction.     

Maximum entropy restoration of noisy star images

We employ Friden's algorithm based on the maximum likelihood principle to restore one-dimensional star image data in the presence of noise. The algorithm is made iterative and its performance discussed.     

Energy albedo for multiregion slabs

An asymptotic solution for the equation of radiative transfer in an inhomogeneous medium was obtained on the basis of the corresponding solutions for homogeneous sub-layers in the slowing down region. Function relations between the reflection and transmission coefficients for the whole slab and those of the sublayers are given. The invariant embedding concepts are used to get the reflection and transmission coefficients for the sub-layers. We assumed different models for the slowing-down kernels. Laplace transform was used to transform the Boltzmann equation to one velocity approximation with re-scaled mean-free path and single-scattering albedo. Numerical results are given for energy albedo as a function of the mass number of the host medium.     

Improved luminosity model and albedo for Saturn

The globe of Saturn is brighter than previously reported as indicated by its absolute opposition magnitudes in the UBVRI band passes which are ?7.080, ?7.842, ?8.914, ?9.587 and ?9.606, respectively. The geometric albedo in V is 0.501 and it peaks in the R filter at 0.567. The brightest apparent magnitude of the system in the V-band is ?0.61 and the faintest is +1.31. The B–V color index can be as blue as +0.93 and as red at +1.12.     

Maximum energy of cosmic rays accelerated by supernova shocks

The expression for the cutoff momentum of CR, accelerated by the supernova blast wave is derived. Geometrical factors (finite increase with time shock size, slowing shock speed and CR adiabatic effect in the downstream region) are shown to determine the value of the cutoff momentum. These factors are stronger than the time restriction and have a significant dynamical effect: the supernova blast wave cannot be completely smoothed by the CR backreaction even at very high Mach numbers. The shock transition always includes a pure gas subshock which strongly influences CR acceleration and shock evolution. It is shown that maximum particle energy achievable during CR acceleration by supernova shock can be as large as _max ≈ Z × 10¹⁵ eV, if the diffusion coefficient is as small as the Bohm limit. Due to nonlinear effect and adiabatic heating in the downstream region in the free expansion phase the actual value of _max is an order of magnitude higher than that from previous estimates based on the plane-wave consideration and is high enough to consider CR diffusive acceleration in SNRs as a main source of galactic CR at least up to the knee energy 3 × 10¹⁵ eV.     

Holography, dark energy and entropy of large cosmic structures

As is well known, black hole entropy is proportional to the area of the horizon suggesting a holographic principle wherein all degrees of freedom contributing to the entropy reside on the surface. In this note, we point out that large scale dark energy (such as a cosmological constant) constraining cosmic structures can imply a similar situation for the entropy of a hierarchy of such objects.     

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 albedo of Titan

The irradiance of Titan has been measured from 0.50 to 1.08μ in 30 Å band-passes spaced 0.01–0.02μ apart. Geometric albedos have been computed at the wavelenghts of measurement using a standard solar flux distribution after Labs and Neckel. The maximum value of p_λ(0) is 0.37 at 0.68, 0.75, and 0.834μ, the minimum value, in the centers of the strongest methane absorption bands, is 0.10 at 0.887 and 1.012μ.The brightness of Titan at the time of the present measurements has been compared with that of previous modern photoelectric measurements. Within the apparent consistency of the different photoelectric systems, the brightness of Titan appears to undergo changes with time.A provisional curve of the geometric albedo from 0.30 to 4.0μ has been made by combining the present results with those of other authors, i.e., relative measurements of Titan from 0.30 to 0.50μ, and measurements of Jupiter and Saturn from 1.08 to 4.00μ. The latter are used to estimate the strengths of the methane absorption bands of Titan in that spectral range. The bolometric geometric albedo, $\text{p}{}^1\text{(0)}$, is computed to be 0.21. A variety of current measurements of Titan indicate a substantial atmosphere, suggesting a value of the phase integral $\text{q}\text{= 1.30 ± 0.20}$. The bolometric Bond albedo, $\text{A}{}^1$, is then 0.27 ± 0.04, giving an effective radiative temperature T_e= 84 ± 2°K.The absorption band contours of Titan have been compared with those of Jupiter and Saturn at the same resolution. The bands of the planets are known to be due primarily to methane, and they show a very regular relationship, with those of Saturn being consistently deeper and wider. For Titan, the strengths of the bands are equal or less than those of Jupiter in the band centers, while the wings are stronger than those of Saturn.Previous photoelectric and photographic spectra have been examined for evidence of temporal variation of the methane path length in the atmosphere of Titan. Differences in measurement techniques prohibit detection of small differences. The only potential differences beyond experimental uncertainties are those of Kuiper (1944) and Harris (mid-fifties). Taking Kuiper's results at face value, Titan appears to have a shorter methane path length in 1972. Harris's results can be reconciled only by the doubtful hypothesis of an almost complete absence of methane at that time.     

The magnitude and albedo of Mars

A comprehensive set of magnitudes obtained between 1954 and 2006 are analyzed. The martian brightness and its variations are characterized empirically at UBVRI wavelengths. Geometrical factors including phase angle, orbital longitude and rotation angle are distinguished from geophysical factors including dust storms and changing albedo features. The phase function indicates a brightness surge near opposition at all wavelengths except possibly in the U band. The color indices reveal increased reddening with phase angle. No significant brightness difference between morning and evening hemisphere observations is indicated with the possible exception of the I band. There is no conclusive evidence for inter-annual brightness variation during the years from 1991 to 2006 when abundant photometry is available. Major dust storms caused brightness excesses that were strongest in the R band at an average of 0.15 mag more luminous than the empirical model for dust-free conditions. The storm of 2001 produced a rapid increase at the onset followed by a slower decline, while the 2003–2004 event show a more gradual increase. The return to normal brightness was linear in magnitude for both storms. Brightness excesses at longer wavelengths were about 0.20 to 0.25 mags at the peak of the 2001 storm. The observed geometric albedo of Mars is 0.059±0.001 in U, 0.089±0.001 in B, 0.170±0.002 in V, 0.289±0.003 in R, and 0.330±0.003 in I. The corresponding albedo values for all five colors exceed those recorded in the literature, with larger percentage increases at shorter wavelengths.     

The radius and albedo of hyperion

The radius and surface geometric albedo of Hyperion are calculated using the photometric/ radiometric method and a new measurement of the 20-μm thermal flux of the satellite. The results are R = 112 ± 15 km and p_v = 0.47 ± 0.11.     

Time variability of Martian bolometric albedo

A time-sequential set of bolometric albedo maps for Mars has been constructed from Viking Infrared Thermal Mapper data. The maps provide global coverage in longitude for latitudes -60° to +60° at 1° by 1° spatial resolution. Individual maps are constructed under strict geometric constraints for a narrow range of L_s. The set of albedo maps spans a Martian year and includes maps before, during, and after the global dust storms of 1977. Transient brightenings associated with local dust storms or condensate clouds are apparent in some of the maps. During dust-free periods, bolometric albedo maps are generally similar to classical, visual albedo map of Mars. The distribution of bolometric surface albedos is bimodal with typical, clear-sky, Lambert albedos of 0.27 and 0.16 for bright and dark areas, respectively. Atmospheric effects strongly influence apparent surface albedos, especially for dark areas. Neither bright nor dark regions show measurable, long-term variations of bolometric albedos during clear periods.     

An explication of the radiometric method for size and albedo determination

A new radiometric model for disk-integrated photometry of asteroids is presented. With empirical support from photometry of Mercury and the Moon, the model assumes that observed sunward beaming of the infrared emission is due to craters. In contrast to earlier theoretical studies of the lunar emission, the observable flux ratio between a cratered sphere and a smooth sphere is calculated for large ranges in wavelength, temperature, and phase angle. Revised diameters and albedos based on the crater model are given for 84 asteroids. The revised values are in good agreement with Morrison's (1977) radiometric results. It is shown that the systematic discrepancy between radiometric and polarimetric albedos (Zellner and Gradie, 1976) is probably a double-valued function of albedo. Some typical geometric albedos from this paper, Morrison (1977), and Zellner and Gradie (1976), respectively, are: Ceres (0.050 ± 0.005, 0.053 ± 0.004, 0.068), Vesta (0.235 ± 0.032, 0235 ± 0.11, 0.271), mean C type (0.031 ± 0.009, 0.035 ± 0.009, 0.061 ± 0.005), mean S type (0.117 ± 0.030, 0.136 ± 0.032, 0.181 ± 0.23), and mean M type (0.105 ± 0.037, 0.115 ± 0.033, 0.157 ± 0.079). Areas of disagreement between radiometry and polarimetry are underscored, and research to resolve them is suggested.     

Study of Non-spinning black holes with reference to the change in energy and entropy

In this research paper, we have derived the formula for both the changes in energy (δE) and entropy (δS) and thereafter calculated the change in entropy (δS) with corresponding change in energy (δE) taking account the first law of the black hole mechanics relating the change in mass M, angular momentum J, horizon area A and charge Q, of a stationary black hole, when it is perturbed, given by formula satisfying in the vacuum as dM = \frack8p dA + WdJ - udQ\delta M = \frac{k}{8\pi} \delta A + \Omega\delta J - \upsilon\delta Q, specially for Non-spinning black holes.     

The albedo and diameter of 1862 Apollo

We report infrared thermal emission measurements of 1862 Apollo, which is the type example of an Earth-crossing asteroid. We derive a geometric albedo of 0.21 ± 0.02 which is within the albedo range of the S class of asteroids. The effective diameter was observed to vary with rotation from 1.2 ± 0.1 to 1.5 ± 0.1 km.     

The diameter and albedo of 1943 Anteros

We report the results of broadband visual and infrared photometry of the Apollo-Amor asteroid 1943 Anteros during its 1980 apparition. By means of a radiometric model, we calculate a diameter of 2.3 ± 0.2 km and a visual geometric albedo of 0.13 ± 0.03. The albedo and reflectance spectrum of Anteros imply that it is a type S asteroid. Thus, Anteros may have a silicate surface similar to other Apollo-Amor asteroids as well as some stony-iron meteorites.     

Ultraviolet reflectivity and geometrical albedo of Titan

Intermediate resolution (6Å) photoelectric spectral scans of Titan, Saturn, Saturn's Rings and the Moon appear in two forms: ratio spectra of Titan vs the Rings and of Saturn vs the Rings, and relative reflectivities, which are compared to previously published results. Titan's geometrical albedo of 0.094 ± 0.012 was measured at 4255Å with a 50Å bandpass. From this and the spectral measurements, we derived the geometrical albedo as a function of wavelength. We find that the wavelength dependences of Titan's uv spectrum and the spectrum of Saturn's Rings are remarkably similar. No trace of any absorption bands is apparent. These results imply that uv gaseous absorption and Rayleigh scattering play a strongly subdued role in Titan's atmosphere. Any homogeneous atmospheric model implies that the absorber responsible for Titan's uv spectral albedo varies strongly with wavelength. On the other hand, we find that the uv observations can be satisfied by an absorber having a relatively weak dependence upon wavelength if an inhomogeneous atmospheric model is employed. In particular, a fine dust, which absorbs as 1/λ, can explain the uv observations provided that it is preferentially distributed high up in Titan's atmosphere where the optical depth from Rayleigh scattering is low. The likely presence of such a dust in Jupiter's atmosphere and the difficulty in explaining the nature of a continuous uv absorber which varies rapidly with wavelength suggest that the gas and aerosol in Titan's atmosphere are inhomogeneously distributed.     

The solar albedo of hard X-ray flares

The calculations of Compton backscattering from the solar surface of flare X-rays performed by Tomblin (1972) are extended to higher energies. It is shown that the effect is even more pronounced in the 40 keV region and that it can lead to substantial corrections to the observed X-ray spectra.