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H. H. Kieffer

Publications and source records attributed to H. H. Kieffer.

At least 19 recordsLinked to original sources
USGS2012

H2O grain size and the amount of dust in Mars' residual North polar cap

In Mars' north polar cap the probable composition of material residual from the annual condensation cycle is a mixture of fine dust and H 2 O grains of comparable size and abundance. However, metamorphism of such material will gradually lower its albedo by increasing the size of the H 2 O grains only. If the cap is undergoing net annual sublimation (as inferred from water vapor observations), late summer observations should be of old ice with H 2 O grain sizes of 100 μm or more. Ice of this granularity containing 30% fine dust has a reflectivity similar to that of dust alone; the observed albedo and computed ice grain size imply dust concentrations of 1 part per 1000 or less. The brightness of the icy areas conflicts with what would be expected for a residual cap deposited by an annual cycle similar to that observed by Viking and aged for thousands of years. The residual cap surface cannot be “old dirty” ice. It could be old, coarse, and clean; or it could be young, fine, and dirty. This brings into question both the source of the late summer water vapor and the formation rate of laminated terrain.

Journal of Geophysical Research↗
USGS2010

A new look at photometry of the Moon

We use ROLO photometry (Kieffer, H.H., Stone, T.C. [2005]. Astron. J. 129, 2887-2901) to characterize the before and after full Moon radiance variation for a typical highlands site and a typical mare site. Focusing on the phase angle range 45??. <. ??<. 50??, we test two different physical models, macroscopic roughness and multiple scattering between regolith particles, for their ability to quantitatively reproduce the measured radiance difference. Our method for estimating the rms slope angle is unique and model-independent in the sense that the measured radiance factor I/. F at small incidence angles (high Sun) is used as an estimate of I/. F for zero roughness regolith. The roughness is determined from the change in I/. F at larger incidence angles. We determine the roughness for 23 wavelengths from 350 to 939. nm. There is no significant wavelength dependence. The average rms slope angle is 22.2?? ?? 1.3?? for the mare site and 34.1?? ?? 2.6?? for the highland site. These large slopes, which are similar to previous "photometric roughness" estimates, require that sub-mm scale "micro-topography" dominates roughness measurements based on photometry, consistent with the conclusions of Helfenstein and Shepard (Helfenstein, P., Shepard, M.K. [1999]. Icarus 141, 107-131). We then tested an alternative and very different model for the before and after full Moon I/. F variation: multiple scattering within a flat layer of realistic regolith particles. This model consists of a log normal size distribution of spheres that match the measured distribution of particles in a typical mature lunar soil 72141,1 (McKay, D.S., Fruland, R.M., Heiken, G.H. [1974]. Proc. Lunar Sci. Conf. 5, Geochim. Cosmochim. Acta 1 (5), 887-906). The model particles have a complex index of refraction 1.65-0.003. i, where 1.65 is typical of impact-generated lunar glasses. Of the four model parameters, three were fixed at values determined from Apollo lunar soils: the mean radius and width of the log normal size distribution and the real part of the refraction index. We used FORTRAN programs from Mishchenko et al. (Mishchenko, M.I., Dlugach, J.M., Yanovitskij, E.G., Zakharova, N.T. [1999]. J. Quant. Spectrosc. Radiat. Trans. 63, 409-432; Mishchenko, M.I., Travis, L.D., Lacis, A.A. [2002]. Scattering, Absorption and Emission of Light by Small Particles. Cambridge Univ. Press, New York. <. http://www.giss.nasa.gov/staff/mmishchenko/books.html>) to calculate the scattering matrix and solve the radiative transfer equation for I/. F. The mean single scattering albedo is ??=0.808, the asymmetry parameter is ???cos. ?????=0.77 and the phase function is very strongly peaked in both the forward and backward scattering directions. The fit to the observations for the highland site is excellent and multiply scattered photons contribute 80% of I/. F. We conclude that either model, roughness or multiple scattering, can match the observations, but that the strongly anisotropic phase functions of realistic particles require rigorous calculation of many orders of scattering or spurious photometric roughness estimates are guaranteed. Our multiple scattering calculation is the first to combine: (1) a regolith model matched to the measured particle size distribution and index of refraction of the lunar soil, (2) a rigorous calculation of the particle phase function and solution of the radiative transfer equation, and (3) application to lunar photometry with absolute radiance calibration. ?? 2010 Elsevier Inc.

Icarus↗
USGS2008

Validation of the ASTER instrument level 1A scene geometry

An independent assessment of the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) instrument geometry was undertaken by the U.S. ASTER Team, to confirm the geometric correction parameters developed and applied to Level 1A (radiometrically and geometrically raw with correction parameters appended) ASTER data. The goal was to evaluate the geometric quality of the ASTER system and the stability of the Terra spacecraft. ASTER is a 15-band system containing optical instruments with resolutions from 15- to 90-meters; all geometrically registered products are ultimately tied to the 15-meter Visible and Near Infrared (VNIR) sub-system. Our evaluation process first involved establishing a large database of Ground Control Points (GCP) in the mid-western United States; an area with features of an appropriate size for spacecraft instrument resolutions. We used standard U.S. Geological Survey (USGS) Digital Orthophoto Quads (DOQS) of areas in the mid-west to locate accurate GCPs by systematically identifying road intersections and recording their coordinates. Elevations for these points were derived from USGS Digital Elevation Models (DEMS). Road intersections in a swath of nine contiguous ASTER scenes were then matched to the GCPs, including terrain correction. We found no significant distortion in the images; after a simple image offset to absolute position, the RMS residual of about 200 points per scene was less than one-half a VNIR pixel. Absolute locations were within 80 meters, with a slow drift of about 10 meters over the entire 530-kilometer swath. Using strictly simultaneous observations of scenes 370 kilometers apart, we determined a stereo angle correction of 0.00134 degree with an accuracy of one microradian. The mid-west GCP field and the techniques used here should be widely applicable in assessing other spacecraft instruments having resolutions from 5 to 50-meters. ?? 2008 American Society for Photogrammetry and Remote Sensing.

Photogrammetric Engineering and Remote Sensing↗
USGS2006

Physical properties of the Mars Exploration Rover landing sites as inferred from Mini-TES-derived thermal inertia

The Miniature Thermal Emission Spectrometer (Mini‐TES) on board the two Mars Exploration Rovers provides the first opportunity to observe thermal properties from the Martian surface, relate these properties to orbital data, and perform soil conductivity experiments under Martian conditions. The thermal inertias of soils, bedforms, and rock at each landing site were derived to quantify the physical properties of these features and understand geologic processes occurring at these localities. The thermal inertia for the Gusev plains rock target Bonneville Beacon (∼1200 J m −2 K −1 s −1/2 ) is consistent with a dense, basaltic rock, but the rocks at the Columbia Hills have a lower thermal inertia (∼620 J m −2 K −1 s −1/2 ), suggesting that they have a volcaniclasic origin. Bedforms on the floors of craters at both landing sites have thermal inertias of 200 J m −2 K −1 s −1/2 , consistent with a particle diameter of ∼160 μm. This diameter is comparable to the most easily moved grain size in the current atmosphere on Mars, suggesting that these bedforms may have formed under current atmospheric conditions. Along the Meridiani plains, the thermal inertia is lower than that derived from TES and Thermal Emission Imaging System (THEMIS) orbital data. This discrepancy is not well understood. Mini‐TES–derived thermal inertias at Gusev along a ∼2.5 km traverse follow trends in thermal inertia measured from orbit with TES and THEMIS. However, along the traverse, there are variability and mixing of particle sizes that are not resolved in the orbital thermal inertia data due to meter‐scale processes that are not identifiable at larger scales.

Journal of Geophysical Research E: Planets↗
USGS2005

The spectral irradiance of the moon

Images of the Moon at 32 wavelengths from 350 to 2450 nm have been obtained from a dedicated observatory during the bright half of each month over a period of several years. The ultimate goal is to develop a spectral radiance model of the Moon with an angular resolution and radiometric accuracy appropriate for calibration of Earth-orbiting spacecraft. An empirical model of irradiance has been developed that treats phase and libration explicitly, with absolute scale founded on the spectra of the star Vega and returned Apollo samples. A selected set of 190 standard stars are observed regularly to provide nightly extinction correction and long-term calibration of the observations. The extinction model is wavelength-coupled and based on the absorption coefficients of a number of gases and aerosols. The empirical irradiance model has the same form at each wavelength, with 18 coefficients, eight of which are constant across wavelength, for a total of 328 coefficients. Over 1000 lunar observations are fitted at each wavelength; the average residual is less than 1%. The irradiance model is actively being used in lunar calibration of several spacecraft instruments and can track sensor response changes at the 0.1% level. ?? 2005. The American Astronomical Society. All rights reserved.

Astronomical Journal↗
USGS2004

Assessment of uncertainty in ROLO lunar irradiance for on-orbit calibration

A system to provide radiometric calibration of remote sensing imaging instruments on-orbit using the Moon has been developed by the US Geological Survey RObotic Lunar Observatory (ROLO) project. ROLO has developed a model for lunar irradiance which treats the primary geometric variables of phase and libration explicitly. The model fits hundreds of data points in each of 23 VNIR and 9 SWIR bands; input data are derived from lunar radiance images acquired by the project's on-site telescopes, calibrated to exoatmospheric radiance and converted to disk-equivalent reflectance. Experimental uncertainties are tracked through all stages of the data processing and modeling. Model fit residuals are ???1% in each band over the full range of observed phase and libration angles. Application of ROLO lunar calibration to SeaWiFS has demonstrated the capability for long-term instrument response trending with precision approaching 0.1% per year. Current work involves assessing the error in absolute responsivity and relative spectral response of the ROLO imaging systems, and propagation of error through the data reduction and modeling software systems with the goal of reducing the uncertainty in the absolute scale, now estimated at 5-10%. This level is similar to the scatter seen in ROLO lunar irradiance comparisons of multiple spacecraft instruments that have viewed the Moon. A field calibration campaign involving NASA and NIST has been initiated that ties the ROLO lunar measurements to the NIST (SI) radiometric scale.

Conference Paper↗
USGS2003

Morphology and composition of the surface of Mars: Mars Odyssey THEMIS results

The Thermal Emission Imaging System (THEMIS) on Mars Odyssey has produced infrared to visible wavelength images of the martian surface that show lithologically distinct layers with variable thickness, implying temporal changes in the processes or environments during or after their formation. Kilometer-scale exposures of bedrock are observed; elsewhere airfall dust completely mantles the surface over thousands of square kilometers. Mars has compositional variations at 100-meter scales, for example, an exposure of olivine-rich basalt in the walls of Ganges Chasma. Thermally distinct ejecta facies occur around some craters with variations associated with crater age. Polar observations have identified temporal patches of water frost in the north polar cap. No thermal signatures associated with endogenic heat sources have been identified.

Science↗
USGS2002

Absolute irradiance of the Moon for on-orbit calibration

The recognized need for on-orbit calibration of remote sensing imaging instruments drives the ROLO project effort to characterize the Moon for use as an absolute radiance source. For over 5 years the ground-based ROLO telescopes have acquired spatially-resolved lunar images in 23 VNIR (Moon diameter ???500 pixels) and 9 SWIR (???250 pixels) passbands at phase angles within ??90 degrees. A numerical model for lunar irradiance has been developed which fits hundreds of ROLO images in each band, corrected for atmospheric extinction and calibrated to absolute radiance, then integrated to irradiance. The band-coupled extinction algorithm uses absorption spectra of several gases and aerosols derived from MODTRAN to fit time-dependent component abundances to nightly observations of standard stars. The absolute radiance scale is based upon independent telescopic measurements of the star Vega. The fitting process yields uncertainties in lunar relative irradiance over small ranges of phase angle and the full range of lunar libration well under 0.5%. A larger source of uncertainty enters in the absolute solar spectral irradiance, especially in the SWIR, where solar models disagree by up to 6%. Results of ROLO model direct comparisons to spacecraft observations demonstrate the ability of the technique to track sensor responsivity drifts to sub-percent precision. Intercomparisons among instruments provide key insights into both calibration issues and the absolute scale for lunar irradiance.

Conference Paper↗
USGS2002

Initial lunar calibration observations by the EO-1 Hyperion imaging spectrometer

The Moon provides an exo-atmospheric radiance source that can be used to determine trends in instrument radio-metric responsivity with high precision. Lunar observations can also be used for absolute radiometric calibration; knowledge of the radiometric scale will steadily improve through independent study of lunar spectral photometry and with sharing of the Moon as a calibration target by increasing numbers of spacecraft, each with its own calibration history. EO-1 calibration includes periodic observation of the Moon by all three of its instruments. Observations are normally made with a phase angle of about 7 degrees (or about 12 hours from the time of Full Moon). Also, SeaWiFS has been making observations at such phase angles for several years, and observations of the Moon by instrument pairs, even if at different times, can be used to transfer absolute calibration. A challenge for EO-1 is pointing to include the entire full Moon in the narrow Hyperion scan. Three Hyperion observations in early 2001 covering an order-of-magnitude difference in lunar irradiance show good agreement for responsivity; the SWIR detector has undergone some changes in responsivity. Small discrepancies of calibration with wavelength could be smoothed using the Moon as a source. Off-axis scattered light response and cross-track response variations can be assessed using the lunar image.

Conference Paper↗
USGS2002

Traceable Radiometry Underpinning Terrestrial- and Helio- Studies (TRUTHS)

The Traceable Radiometry Underpinning Terrestrial- and Helio- Studies (TRUTHS) mission offers a novel approach to the provision of key scientific data with unprecedented radiometric accuracy for Earth Observation (EO) and solar studies, which will also establish well-calibrated reference targets/standards to support other EO missions. This paper will present the TRUTHS mission and its objectives. TRUTHS will be the first satellite mission to calibrate its instrumentation directly to SI in orbit, overcoming the usual uncertainties associated with drifts of sensor gain and spectral shape by using an electrical rather than an optical standard as the basis of its calibration. The range of instruments flown as part of the payload will also provide accurate input data to improve atmospheric radiative transfer codes by anchoring boundary conditions, through simultaneous measurements of aerosols, particulates and radiances at various heights. Therefore, TRUTHS will significantly improve the performance and accuracy of Earth observation missions with broad global or operational aims, as well as more dedicated missions. The provision of reference standards will also improve synergy between missions by reducing errors due to different calibration biases and offer cost reductions for future missions by reducing the demands for on-board calibration systems. Such improvements are important for the future success of strategies such as Global Monitoring for Environment and Security (GMES) and the implementation and monitoring of international treaties such as the Kyoto Protocol. TRUTHS will achieve these aims by measuring the geophysical variables of solar and lunar irradiance, together with both polarised and un-polarised spectral radiance of the Moon, and the Earth and its atmosphere.

Conference Paper↗
USGS2002

On-orbit radiometric calibration over time and between spacecraft using the moon

The Robotic Lunar Observatory (ROLO) project has developed a spectral irradiance model of the Moon that accounts for variations with lunar phase through the bright half of a month, lunar librations, and the location of an Earth-orbiting spacecraft. The methodology of comparing spacecraft observations of the Moon with this model has been developed to a set of standardized procedures so that comparisons can be readily made. In the cases where observations extend over several years (e.g., SeaWiFS), instrument response degradation has been determined with precision of about 0.1% per year. Because of the strong dependence of lunar irradiance on geometric angles, observations by two spacecraft cannot be directly compared unless acquired at the same time and location. Rather, the lunar irradiance based on each spacecraft instrument calibration can be compared with the lunar irradiance model. Even single observations by an instrument allow inter-comparison of its radiometric scale with other instruments participating in the lunar calibration program. Observations by SeaWiFS, ALI, Hyperion and MTI are compared here.

Conference Paper↗
USGS2001

Mars' "White Rock" feature lacks evidence of an aqueous origin: Results from Mars Global Surveyor

The “White Rock” feature on Mars has long been viewed as a type example for a Martian playa largely because of its apparent high albedo along with its location in a topographic basin (a crater). Data from the Mars Global Surveyor Thermal Emission Spectrometer (TES) demonstrate that White Rock is not anomalously bright relative to other Martian bright regions, reducing the significance of its albedo and weakening the analogy to terrestrial playas. Its thermal inertia value indicates that it is not mantled by a layer of loose dust, nor is it bedrock. The thermal infrared spectrum of White Rock shows no obvious features of carbonates or sulfates and is, in fact, spectrally flat. Images from the Mars Orbiter Camera show that the White Rock massifs are consolidated enough to retain slopes and allow the passage of saltating grains over their surfaces. Material appears to be shed from the massifs and is concentrated at the crests of nearby bedforms. One explanation for these observations is that White Rock is an eroded accumulation of compacted or weakly cemented aeolian sediment.

Journal of Geophysical Research E: Planets↗
USGS2001

Mars Global Surveyor Thermal Emission Spectrometer experiment: Investigation description and surface science results

The Thermal Emission Spectrometer (TES) investigation on Mars Global Surveyor (MGS) is aimed at determining (1) the composition of surface minerals, rocks, and ices; (2) the temperature and dynamics of the atmosphere; (3) the properties of the atmospheric aerosols and clouds; (4) the nature of the polar regions; and (5) the thermophysical properties of the surface materials. These objectives are met using an infrared (5.8- to 50-μm) interferometric spectrometer, along with broadband thermal (5.1- to 150-μm) and visible/near-IR (0.3- to 2.9-μm) radiometers. The MGS TES instrument weighs 14.47 kg, consumes 10.6 W when operating, and is 23.6×35.5×40.0 cm in size. The TES data are calibrated to a 1-σ precision of 2.5 −6 ×10 −8 W cm −2 sr −1 /cm −1 , 1.6×10 −6 W cm −2 sr −1 , and ∼0.5 K in the spectrometer, visible/near-IR bolometer, and IR bolometer, respectively. These instrument subsections are calibrated to an absolute accuracy of ∼4×10 −8 W cm −2 sr −1 /cm −1 (0.5 K at 280 K), 1–2%, and ∼1–2 K, respectively. Global mapping of surface mineralogy at a spatial resolution of 3 km has shown the following: (1) The mineralogic composition of dark regions varies from basaltic, primarily plagioclase feldspar and clinopyroxene, in the ancient, southern highlands to andesitic, dominated by plagioclase feldspar and volcanic glass, in the younger northern plains. (2) Aqueous mineralization has produced gray, crystalline hematite in limited regions under ambient or hydrothermal conditions; these deposits are interpreted to be in-place sedimentary rock formations and indicate that liquid water was stable near the surface for a long period of time. (3) There is no evidence for large-scale (tens of kilometers) occurrences of moderate-grained (>50-μm) carbonates exposed at the surface at a detection limit of ∼10%. (4) Unweathered volcanic minerals dominate the spectral properties of dark regions, and weathering products, such as clays, have not been observed anywhere above a detection limit of ∼10%; this lack of evidence for chemical weathering indicates a geologic history dominated by a cold, dry climate in which mechanical, rather than chemical, weathering was the significant form of erosion and sediment production. (5) There is no conclusive evidence for sulfate minerals at a detection limit of ∼15%. The polar region has been studied with the following major conclusions: (1) Condensed CO 2 has three distinct end-members, from fine-grained crystals to slab ice. (2) The growth and retreat of the polar caps observed by MGS is virtually the same as observed by Viking 12 Martian years ago. (3) Unique regions have been identified that appear to differ primarily in the grain size of CO 2 ; one south polar region appears to remain as black slab CO 2 ice throughout its sublimation. (4) Regional atmospheric dust is common in localized and regional dust storms around the margin and interior of the southern cap. Analysis of the thermophysical properties of the surface shows that (1) the spatial pattern of albedo has changed since Viking observations, (2) a unique cluster of surface materials with intermediate inertia and albedo occurs that is distinct from the previously identified low-inertia/bright and high-inertia/dark surfaces, and (3) localized patches of high-inertia material have been found in topographic lows and may have been formed by a unique set of aeolian, fluvial, or erosional processes or may be exposed bedrock.

Journal of Geophysical Research E: Planets↗
USGS2000

Detection of crystalline hematite mineralization on Mars by the Thermal Emission Spectrometer: evidence for near-surface water

The Thermal Emission Spectrometer (TES) instrument on the Mars Global Surveyor (MGS) mission has discovered a remarkable accumulation of crystalline hematite (α-Fe 2 O 3 ) that covers an area with very sharp boundaries approximately 350 by 350–750 km in size centered near 2°S latitude between 0° and 5°W longitude (Sinus Meridiani). Crystalline hematite is uniquely identified by the presence of fundamental vibrational absorption features centered near 300, 450, and >525 cm −1 and by the absence of silicate fundamentals in the 1000 cm −1 region. Spectral features resulting from atmospheric CO 2 , dust, and water ice were removed using a radiative transfer model. The spectral properties unique to Sinus Meridiani were emphasized by removing the average spectrum of the surrounding region. The depth and shape of the hematite fundamental bands show that the hematite is crystalline and relatively coarse grained (>5–10 μm). Diameters up to and greater than hundreds of micrometers are permitted within the instrumental noise and natural variability of hematite spectra. Hematite particles <5–10 μm in diameter (as either unpacked or hard-packed powders) fail to match the TES spectra. The spectrally derived areal abundance of hematite varies with particle size from ∼10% (>30 μm diameter) to 40–60% (10 μm diameter). The hematite in Sinus Meridiani is thus distinct from the fine-grained (diameter <5–10 μm), red, crystalline hematite considered, on the basis of visible, near-IR data, to be a minor spectral component in Martian bright regions like Olympus-Amazonis. Sinus Meridiani hematite is closely associated with a smooth, layered, friable surface that is interpreted to be sedimentary in origin. This material may be the uppermost surface in the region, indicating that it might be a late stage sedimentary unit or a layered portion of the heavily cratered plains units. We consider five possible mechanisms for the formation of coarse-grained, crystalline hematite. These processes fall into two classes depending on whether they require a significant amount of near-surface water: the first is chemical precipitation that includes origin by (1) precipitation from standing, oxygenated, Fe-rich water (oxide iron formations), (2) precipitation from Fe-rich hydrothermal fluids, (3) low-temperature dissolution and precipitation through mobile ground water leaching, and (4) formation of surface coatings, and the second is thermal oxidation of magnetite-rich lavas. Weathering and alteration processes, which produce nanophase and red hematite, are not consistent with the coarse, crystalline hematite observed in Sinus Meridiani. We prefer chemical precipitation models and favor precipitation from Fe-rich water on the basis of the probable association with sedimentary materials, large geographic size, distance from a regional heat source, and lack of evidence for extensive groundwater processes elsewhere on Mars. The TES results thus provide mineralogic evidence for probable large-scale water interactions. The Sinus Meridiani region may be an ideal candidate for future landed missions searching for biotic and prebiotic environments, and the physical characteristics of this site satisfy all of the engineering requirements for the missions currently planned.

Journal of Geophysical Research E: Planets↗
USGS2000

Generation of data acquisition requests for the ASTER satellite instrument for monitoring a globally distributed target: glaciers

The advanced spaceborne thermal emission and reflection radiometer (ASTER) instrument is scheduled to be launched on the EOS Terra platform in 1999. The Global Land Ice Measurements from Space project has planned to acquire ASTER images of most of the world's land ice annually during the six-year ASTER mission. This article describes the process of creating the data acquisition requests needed to cover approximately 170,000 glacier targets.

IEEE Transactions on Geoscience and Remote Sensing↗
USGS2000

TES observations of the martian surface and atmosphere

The TES instrument is a Fourier transform Michelson interferometer operating with 10 or 5 cm -1 sampling int he thermal infared spectral region from 1700 to 200 cm -1 (~6 to 50 &mu;m) where virtually all minerals have characteristic fundamental vibrational absorption bands (1, 2, 3, 4, 5, 6, 7, 8). The TES data used in this paper are among the 6x10 7 spectra collected during the early mapping phase of the MGS mission from southern hemisphere winter to early summer (aerocentric longitude, L s , 107&deg; to 297&deg;. The methodology for separating the surface and atmospheric components of the radiance from Mars, which allows detailed analysis and interpretation of surface mineralogy (9, 10), is described in previous papers (10, 11).

Book↗
USGS1999

Real-time control of the robotic lunar observatory telescope

The US Geological Survey operates an automated observatory dedicated to the radiometry of the Moon with the objective of developing a multispectral, spatially resolved photometric model of the Moon to be used in the calibration of Earth-orbiting spacecraft. Interference filters are used with two imaging instruments to observe the Moon in 32 passbands from 350-2500 nm. Three computers control the telescope mount and instruments with a fourth computer acting as a master system to control all observation activities. Real-time control software has been written to operate the instrumentation and to automate the observing process. The observing software algorithms use information including the positions of objects in the sky, the phase of the Moon, and the times of evening and morning twilight to decide how to observe program objects. The observatory has been operating in a routine mode since late 1995 and is expected to continue through at least 2002 without significant modifications.

Publications of the Astronomical Society of the Pa↗