The projected areas of ellipsoids and cylinders

Vickers, G.T.

doi:10.1016/0032-5910(95)03049-2

Cited by 41 publications

(18 citation statements)

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“…At conjunction -either transit or eclipse-we are seeing the planet's smallest projected area (πR 2 p , in the case of a perfectly edge-on orbit). The projected area of an ellipsoid on an edge-on orbit varies as (e.g., Vickers 1996):…”

Section: Interpreting the Ellipsoidal Variations At 45 µMmentioning

confidence: 99%

THERMAL PHASE VARIATIONS OF WASP-12b: DEFYING PREDICTIONS

Cowan

Machálek

Croll

et al. 2012

ApJ

206

292

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We report Warm Spitzer full-orbit phase observations of WASP-12b at 3.6 and 4.5 µm. This extremely inflated hot Jupiter is thought to be overflowing its Roche lobe, undergoing mass loss, accretion onto its host star, and has been claimed to have a C/O ratio in excess of unity. We are able to measure the transit depths, eclipse depths, thermal and ellipsoidal phase variations at both wavelengths. The large amplitude phase variations, combined with the planet's previously-measured day-side spectral energy distribution, is indicative of non-zero Bond albedo and very poor day-night heat redistribution. The transit depths in the mid-infrared -(R p /R * ) 2 = 0.0123(3) and 0.0111(3) at 3.6 and 4.5 µm, respectively-indicate that the atmospheric opacity is greater at 3.6 than at 4.5 µm, in disagreement with model predictions, irrespective of C/O ratio. The secondary eclipse depths are consistent with previous studies: F day /F * = 0.0038(4) and 0.0039(3) at 3.6 and 4.5 µm, respectively. We do not detect ellipsoidal variations at 3.6 µm, but our parameter uncertainties -estimated via prayer-bead Monte Carlo-keep this non-detection consistent with model predictions. At 4.5 µm, on the other hand, we detect ellipsoidal variations that are much stronger than predicted. If interpreted as a geometric effect due to the planet's elongated shape, these variations imply a 3:2 ratio for the planet's longest:shortest axes and a relatively bright day-night terminator. If we instead presume that the 4.5 µm ellipsoidal variations are due to uncorrected systematic noise and we fix the amplitude of the variations to zero, the best fit 4.5 µm transit depth becomes commensurate with the 3.6 µm depth, within the uncertainties. The relative transit depths are then consistent with a Solar composition and short scale height at the terminator. Assuming zero ellipsoidal variations also yields a much deeper 4.5 µm eclipse depth, consistent with a Solar composition and modest temperature inversion. We suggest future observations that could distinguish between these two scenarios. 11 We follow Agol et al. (2010), who compared many centroiding algorithms and found this one to be optimal. Using fluxweighted centroiding instead of PSF-fitting results in slightly worse χ 2 , commensurate correlated noise as measured using β (see first Section 4.1), and consistent astrophysical parameters.

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Section: Interpreting the Ellipsoidal Variations At 45 µMmentioning

confidence: 99%

THERMAL PHASE VARIATIONS OF WASP-12b: DEFYING PREDICTIONS

Cowan

Machálek

Croll

et al. 2012

ApJ

206

292

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“…If on the other hand the particles are oblate ( a < b = c ), then the projected image will be of the b ‐ c plane and further independent evaluation of the height ( a ) of the particle is required in order to characterize its shape. Ongoing studies involving the determination of the 3‐D shape from a 2‐D projection [ Vickers , 1996; Podczeck , 1997; Brown and Vickers , 1998; Vickers and Brown , 2001] may improve particle shape characterization by photographing methods, but at present the information retrieved from a single 2‐D projection is quite limited when concerning a particle of even idealized ellipsoidal shape. Other disadvantages of photographic methods are the expensive equipment required and the time consuming nature of the procedure, requiring 25 or more measurements on single grains to provide a statistically valid shape value for each granular material.…”

Section: Theoretical Considerationsmentioning

confidence: 99%

Particle shape characterization using angle of repose measurements for predicting the effective permittivity and electrical conductivity of saturated granular media

Friedman

Robinson

2002

Water Resources Research

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[1] The particle shape of sediments, soils, and rocks affects the packing of the material and the subsequent pore geometry; it therefore influences important transport properties, such as electrical conductivity, dielectric permittivity, diffusion coefficient, thermal conductivity, and hydraulic conductivity. It is difficult to quantify the ''average'' shape characteristics of a granular material so a method, which would capture the threedimensional shape characteristics of a granular material in regards to its packing geometry, would be of great benefit. In this article we examine the use of the angle of repose and the maximum angle of stability of a slope of granular material poured into water, as a simple and rapid method of quantifying its ''effective'' particle shape. A two-stage concept is discussed relating the measured slope angles to the aspect ratio of an equivalent oblate ellipsoidal particle and using this aspect ratio for predicting the effective permittivity and bulk electrical conductivity of isotropic packings of granular materials. The slope angleaspect ratio relationships were established using measurements of bulk electrical conductivity of monosized spherical glass beads and nonspherical sand and tuff grains packed in water to different porosities. This empirical relationship, the choice of an oblate geometry to represent particle shape, and a heuristic mixing model accounting for the effect of neighboring particles on the internal electrical field were tested using measurements of the effective permittivity for those granular media. Good agreement was found between the model predictions and measurements, indicating the physical significance of the above relationships and quantifiers. We have also tested and found high correlations between slope angles and a shape factor determined by image analysis of twodimensional particle micrographs. We do not, however, propose to use it for routine particle shape characterization because of the insufficiently informative viewing of oblates from above and because of the expensive equipment and extensive work involve with it. Instead, we propose to use the easily measurable slope angles for routine particle shape characterization because of its simplicity and also because of its beneficial sensitivity to surface roughness and particle angularity effects beyond the aspect ratio of a smooth equivalent ellipsoid.INDEX TERMS: 1866 Hydrology: Soil moisture; 5109 Physical Properties of Rocks: Magnetic and electrical properties; 5139 Physical Properties of Rocks: Transport properties; KEYWORDS: particle shape, geometry, angle of repose, granular media, dielectric permittivity, electrical conductivity Citation: Friedman, S. P., and D. A. Robinson, Particle shape characterization using angle of repose measurements for predicting the effective permittivity and electrical conductivity of saturated granular media, Water Resour.

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“…and the projected area in spherical coordinates as given by Vickers (1996) for a triaxial ellipsoid is…”

Section: Introductionmentioning

confidence: 99%

A generalized geometric approach to anisotropic peak broadening due to domain morphology

2015

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This article reports the derivation of a physically based geometric description of the mean diameter of orthogonal shapes and provides an efficient formalism to relate these to reciprocal lattices and corresponding apparent crystallite sizes. The following descriptions provide a reasonable approximation for the simulation and refinement of anisotropic domain morphology in powder diffraction techniques.

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The projected areas of ellipsoids and cylinders

Cited by 41 publications

References 1 publication

THERMAL PHASE VARIATIONS OF WASP-12b: DEFYING PREDICTIONS

THERMAL PHASE VARIATIONS OF WASP-12b: DEFYING PREDICTIONS

Particle shape characterization using angle of repose measurements for predicting the effective permittivity and electrical conductivity of saturated granular media

A generalized geometric approach to anisotropic peak broadening due to domain morphology

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