Limb-darkening Effect Research Articles

Gravitational Microlensing near Caustics. II. Cusps

We present a rigorous, detailed study of the generic, quantitative properties of gravitational lensing near cusp catastrophes. Concentrating on the case in which the individual images are unresolved, we derive explicit formulas for the total magnification and centroid of the images created for sources outside, on, and inside the cusped caustic. We obtain new results on how the image magnifications scale with respect to separation from the cusped caustic for arbitrary source positions. Along the axis of symmetry of the cusp, the total magnification μ scales as μ ∝ u-1, where u is the distance of the source from the cusp, whereas perpendicular to this axis, μ ∝ u-2/3. When the source passes through a point u0 on a fold arc abutting the cusp, the image centroid has a jump discontinuity; we present a formula for the size of the jump in terms of the local derivatives of the lens potential and show that the magnitude of the jump scales as |u|1/2 for |u| 1, where |u| is the horizontal distance between u0 and the cusp. The total magnifications for a small extended source located both on and perpendicular to the axis of symmetry are also derived, for both uniform and limb-darkened surface brightness profiles. We find that the difference in magnification between a finite and point source is 5% for separations of 2.5 source radii from the cusp point, while the effect of limb darkening is 1% in the same range. Our predictions for the astrometric and photometric behavior of both pointlike and finite sources passing near a cusp are illustrated and verified using numerical simulations of the cusp-crossing Galactic binary lens event MACHO-1997-BUL-28. Our results can be applied to any microlensing system with cusp caustics, including Galactic binary lenses and quasar microlensing; we discuss several possible applications of our results to these topics.

The microlensing signatures of photospheric starspots

Point lens microlensing events with impact parameter close to the source stellar radius allow the observer to study the surface brightness profile of the lensed source. We have examined the effects of photospheric starspots on multicolour microlensing light curves and investigated the detectability of such spots in different wavebands as a function of spot temperature, position, radius and lens trajectories. We include the effects of limb darkening and spot projection as a function of position on the stellar disc. In particular we apply the updated, state-of-the-art ‘NextGen’ stellar atmosphere models of Hauschildt et al. which predict very strong limb darkening, and which are likely to be applicable to the source stars considered here. Our results indicate that starspots generally give a clear signature only for transit events. Moreover, this signature is strongly suppressed by limb darkening for spots close to the limb, although the spots may still be clearly detected for favourable lens trajectories. It is also clear that intensive temporal sampling throughout the duration of the transit is necessary in order for such events to be effective as a tool for imaging stellar photospheres. None the less, with sufficiently well-sampled light curves of good photometric precision, microlensing can indeed place useful constraints on the presence or otherwise of photospheric starspots.

Direct multi-wavelength limb-darkening measurements of three late-type giants with the Navy Prototype Optical Interferometer

We present direct measurements of the limb-darkened intensity profiles of the late-type giant stars HR5299, HR7635, and HR8621 obtained with the Navy Prototype Optical Interferometer (NPOI) at the Lowell Observatory. A triangle of baselines with lengths of 18.9 m, 22.2 m, and 37.5 m was used. We utilized squared visibility amplitudes beyond the first minimum, as well as triple amplitudes and phases in up to 10 spectral channels covering a wavelength range of ~650 nm to ~850 nm. We find that our data can best be described by featureless symmetric limb-darkened disk models while uniform disk and fully darkened disk models can be rejected. We derive high-precision angular limb-darkened diameters for the three stars of 7.44 mas +/- 0.11 mas, 6.18 mas +/- 0.07 mas, and 6.94 mas +/- 0.12 mas, respectively. Using the HIPPARCOS parallaxes, we determine linear limb-darkened radii of 114 R$_\odot \pm $13 R$_\odot$, 56 R$_\odot \pm $4 R$_\odot$, and 98 R$_\odot \pm $9 R$_\odot$, respectively. We compare our data to a grid of Kurucz stellar model atmospheres, with them derive the effective temperatures and surface gravities without additional information, and find agreement with independent estimates derived from empirical calibrations and bolometric fluxes. This confirms the consistency of model predictions and direct observations of the limb-darkening effect.

Development of meteorological parameters and total ozone during the total solar eclipse of August 11, 1999

During the total eclipse of August 11, 1999 frequent showers occurred due to a unstable stratification of the air mass. At different observation sites, meteorological effects from the eclipse (99.4% coverage at Hohenpeissenberg) and from showers were superimposed making it partly difficult to unambiguously interpret the observations. The weather radar at Hohenpeissenberg observatory provided a general overview of the distribution of clouds and precipitation in this area (200 km diameter). From the Garching site in the zone of totality (100%) temperature and wind data taken on a 50 m mast were evaluated. By selecting periods with relatively low cloud cover it was possible to approximately follow the development of the vertical temperature and wind profiles during the eclipse. The minimum temperature at Hohenpeissenberg (about 450 m above the altitude of Garching) during the eclipse was comparable to that during the previous night, the corresponding value measured at Garching remained about 2 K above the minimum observed during clear sky conditions in the previous night. Showers before, during or after the eclipse may have induced vertical exchange of air parcels. Temperatures during a shower change towards the same direction at all altitudes, thus no inversion forms. Additionally, air parcels with relatively lower concentrations of trace constituents were transported down from aloft for time periods of 10-15 minutes. These mixing processes significantly determined the temporal variations of various trace substances measured during the eclipse. Total ozone measurements at Hohenpeissenberg were performed with both DOBSON and BREWER spectrophotometers and at another site within the zone of totality by using a portable Microtops II filter instrument. Different results were obtained for both sites. These differences can be to a large extend, but not exclusively, attributed to eclipse induced shifts (limb darkening and straylight effects in the atmosphere) in the wavelength pairs used to calculate total ozone. However, there also appeared to be real fluctuations in total ozone during the reemerging phase of the sun which may be related to eclipse induced gravity waves.

Precision Measurements of the Diameters of α Orionis and ο Ceti at 11 Microns

The angular diameters of α Orionis and o Ceti were measured at a wavelength of 11.15 μm using the two-telescope Infrared Spatial Interferometer (ISI). Based on fitting the visibility data to uniform disk models, the diameter of α Orionis is 54.7 ± 0.3 mas and that of o Ceti at phase 0.90 is 47.8 ± 0.5 mas. These diameters are the most precise ever measured in the mid-infrared, due in part to the addition of a 56 m baseline to the ISI, which provided sufficient resolution to observe the first zero in the visibility function of the stellar disks. Moreover, the effects of limb darkening and stellar hot spots are small at these wavelengths. Theoretically estimated limb-darkening effects indicate the actual diameters are approximately 1% ± 0.5% larger than the uniform disk approximation, or 55.2 ± 0.5 and 48.2 ± 0.6 mas for α Orionis and o Ceti, respectively.

Changes in surface solar UV irradiances and total ozone during the solar eclipse of August 11, 1999

During the solar eclipse of August 11, 1999, intensive measurements of UV solar irradiance and total ozone were performed at a number of observatories located near the path of the Moon's shadow. At the Laboratory of Atmospheric Physics (LAP) of the Aristotle University of Thessaloniki, Greece, global and direct spectra of UV solar irradiances (285–365 nm) were recorded with a double monochromator, and erythemal irradiances were measured with broadband pyranometers. In addition, higher‐frequency measurements of global and direct irradiances at six UV wavelengths were performed with a single Brewer spectrophotometer. Total ozone measurements were also performed with Dobson and Brewer spectrophotometers at Hradec Kralove (Czech Republic), Ispra (Italy), Sestola (Italy), Hohenpeissenberg (Germany), Bucharest (Romania), Arosa (Switzerland), and Thessaloniki (Greece). From the spectral UV measurements the limb darkening effect of the solar disk was tentatively quantified from differences of measured solar spectral irradiances at the peak of the eclipse (near to limb conditions) and before the eclipse. Two blackbody curves were fit to the preeclipse and peak eclipse spectra, which have shown a difference in effective temperatures of about 165°K between the limb and the whole of the solar disk. The limb darkening effect is larger at the shorter UV wavelengths. The ratio of the diffuse to direct solar irradiances during the eclipse shows that the diffuse component is reduced much less compared to the decline of the direct solar irradiance at the shorter wavelengths. Moreover, a 20‐min oscillation of erythemal UV‐B solar irradiance was observed before and after the time of the eclipse maximum under clear skies, indicating a possible 20‐min fluctuation in total ozone, presumably caused by the eclipse‐induced gravity waves. This work also shows that routine total ozone measurements with a Brewer or a Dobson spectrophotometer should be used with caution during a solar eclipse. This is because the diffuse light increases by more than 30% with respect to the direct solar radiation, increasing more at the shorter wavelength side of the UV spectrum. This plausible mechanism introduces an artificial decrease in total ozone during solar eclipse of more than 30 Dobson units (DU), which is confirmed by all Brewer and Dobson measurements. Changes in total ozone cited earlier in the refereed literature have not been confirmed in the present study.

Limb-darkening corrections for interferometric uniform disc stellar angular diameters

Stellar angular diameters determined interferometrically are generally established by fitting the observed visibility data with a curve appropriate for a uniformly illuminated disc. The resulting uniform-disc diameters must be corrected for the effects of limb darkening in order to determine the true angular diameters of the stars. An extensive grid of limb-darkening corrections, based directly on the centre-to-limb intensity variations for Kurucz model stellar atmospheres, has been computed without the intermediate step of a parametrized representation of the centre-to-limb variation. The limitations of this method of correction are discussed.

Chromaticity of gravitational microlensing events

In this paper, we investigate the color changes of gravitational microlensing events caused by the two different mechanisms of differential amplification for a limb-darkened extended source and blending. From this investigation, we find that the color changes of limb-darkened extended source events (color curves) have dramatically different characteristics depending on whether the lens transits the source star or not. We show that for a source transit event, the lens proper motion can be determined by simply measuring the turning time of the color curve instead of fitting the overall color or light curves. We also find that even for a very small fraction of blended light, the color changes induced by the blending effect is equivalent to those caused by the limb-darkening effect, causing serious distortion in the observed color curve. Therefore, to obtain useful information about the lens and source star from the color curve of a limb-darkened extended source event, it will be essential to eliminate or correct for the blending effect. We discuss about the methods for the efficient correction of the blending effect.

Wavelength dependence of angular diameters of M giants: an observational perspective

We discuss the wavelength dependence of angular diameters of M giants from an observational perspective. Observers cannot directly measure an optical-depth radius for a star, despite this being a common theoretical definition. Instead, they can use an interferometer to measure the square of the fringe visibility. We present new plots of the wavelength-dependent centre-to-limb variation (CLV) of intensity of the stellar disk as well as visibility for Mira and non-Mira M giant models. We use the terms ``CLV spectra'' and ``visibility spectra'' for these plots. We discuss a model-predicted extreme limb-darkening effect (also called the narrow-bright-core effect) in very strong TiO bands which can lead to a misinterpretation of the size of a star in these bands. We find no evidence as yet that this effect occurs in real stars. Our CLV spectra can explain the similarity in visibilities of R Dor (M8IIIe) that have been observed recently despite the use of two different passbands. We compare several observations with models and find the models generally under-estimate the observed variation in visibility with wavelength. We present CLV and visibility spectra for a model that is applicable to the M supergiant alpha Ori.

Spectrum Synthesis of Accretion Disks in Partial Eclipse

We have modified a spectrum synthesis code for accretion disks in binary star systems, to deal with the case of a disk that is partially eclipsed by the mass-losing star. The known Roche geometry of the binary system permits the correspondence to be made between the spectrum observed over some phase interval and the regions of the disk that contribute to the spectrum. The code can generate simulated observations corresponding to arbitrary phase intervals, with arbitrary spectral sampling and resolution. This capability clearly includes generating broadband light curves as a limiting case. The use of angle-dependent specific intensities means that limb-darkening effects are automatically taken into account. We present trailed-spectrogram and other representations of the partially eclipsed disk spectra of representative models of cataclysmic variables. The spectra are shown versus orbital phase, for portions of the ultraviolet spectral region accessible to the Hubble Space Telescope. We illustrate the behavior of blended features through eclipse and comment on the sensitivity of the synthesized spectra to the parameters of the input model, such as the stellar masses, mass transfer rate, and orbital inclination. Compared with out-of-eclipse spectra of the entire disk, observations during eclipse should have the advantage for diagnostic purposes of showing (1) deeper individual lines, (2) reduced blending of lines, and (3) an effectively smaller range of effective temperatures contributing to the observed spectrum. These advantages are especially important in analysis of the ultraviolet spectra of disks.

Application of Realistic Model Atmospheres to Eclipse Maps of Accretion Disks: The Effective Temperature and Flare of the Disk in the Dwarf Nova Z Chamaeleontis

In a previous paper we presented ultraviolet (λeff=1550 A) high-speed photometry of the eclipsing dwarf nova Z Cha obtained near the peak of a normal eruption, and we derived a maximum-entropy disk map from the eclipse light curve. In the present paper we report a new calculation of the disk map and temperature profile. The calculation differs from the previous one—and all other calculations of disk maps—in two ways: first, we have used realistic model atmospheres to convert the surface brightness of the disk to effective temperature, and second, we have included the effects of limb darkening, which is large at ultraviolet wavelengths and strongly affects the inferred temperature and geometry of the disk. Inclusion of limb darkening in the models made the distribution of surface brightness with radius much steeper than the distribution derived without limb darkening, but the use of realistic model spectra instead of blackbody spectra to convert the surface brightness to temperature flattened the distribution again so that the net effect was to leave the temperature distribution nearly unchanged. The distribution remains much flatter than the r−3/4 law predicted for optically thick, steady-state disks. In agreement with our earlier results, we find that the disk in Z Cha becomes flared during eruptions. The derived flare angle is large, 6°, but not as large as the flare needed for a disk with zero limb darkening. For comparison, model atmospheres for disks similar to the disk in Z Cha give photosphere heights corresponding to flare angles between 3° and 4°.

Mutual irradiation and absorption and surface temperature variation in detached binary systems

In the paper, we calculate the absorbed flux of the mutual irradiation of the components in detached binary systems, and obtain the surface temperature distribution on the components. The maximum temperature difference over the surface is a measure of the deviation between isothermal and equipotential surfaces. For 33 late-type binary systems with known elements, we evaluate this maximum difference. The result shows that the isothermal-equipotential deviation is always larger in the cooler than in the hotter components. This result supports Zhou and Leung's idea that the large difference between “star-spots” and sunspots is due to cyclones and anticyclones. The effect of limb darkening (amounting to over 18% and always larger in the hotter component) is included in the consideration.

Average spectral curves of extraterrestrial solar irradiance in the (180–400) nm wavelength range

Twenty-three spectral curves of extraterrestrial solar irradiance in the (180–400) nm wavelength range, derived from measurements taken on rockets and airborne platforms by various research groups, were examined in order to determine the average spectral curves of extraterrestrial solar irradiance. Such curves can be more reliably used in computational programmes prepared for simulating the radiative-transfer processes occurring in the terrestrial atmosphere. For each spectral curve, a careful examination was performed of the errors due to instrumental characteristics, calibration procedures, atmospheric extinction, solar limb-darkening effects, as well as to the use of normalization procedures to zero-air-mass and the mean Earth-Sun distance. These error estimates were used to obtain realistic evaluations of the standard deviations at the various wavelengths. The resulting data sets were employed to calculate the weighted mean values of the full-disk solar irradiance outside the atmosphere, for spectral resolution values of 0.4 and 1 nm. The values are presented in two tables. A further analysis was performed in order to evaluate the variability of the extraterrestrial solar irradiance as a function of solar activity, but indistinct and contradictory results were found, presumably because the experimental errors were greater than the periodical changes in the density flux of extraterrestrial solar radiation.

A Quantitative Study of Limb Darkening in Accretion Disks

AbstractThe UV emission of accretion disks was synthesized for a grid of optically thick models that cover the high-Ṁ nova-like regime, aiming to quantify the effect of limb darkening in the integrated and radially resolved spectrum of the disk.

The effect of oblateness and gravity darkening on the radiation driving in winds from rapidly rotating B stars

We calculate the radiative driving force for winds around rapidly rotating oblate B stars, and we estimate the impact these forces should have on the production of a wind compressed disk. The effects of limb darkening, gravity darkening, oblateness, and an arbitrary wind velocity field are included in the computation of vector 'oblate finite disk' (OFD) factors, which depend on both radius and colatitude in the wind. The impact of limb darkening alone, with or without rotation, can increase the mass loss by as much as 10% over values computed using the standard uniformly bright spherical finite disk factor. For rapidly rotating stars, limb darkening makes 'sub-stellar' gravity darkening the dominant effect in the radial and latitudinal OFD factors, and lessens the impact of gravity darkening at other visible latitudes (nearer to the oblate limb). Thus, the radial radiative driving is generally stronger over the poles and weaker over the equator, following the gravity darkening at these latitudes. The nonradial radiative driving is considerably smaller in magnitude than the radial component, but is directed both away from the equatorial plane and in a retrograde azimuthal direction, acting to decrease the effective stellar rotation velocity. These forces thus weaken the equatorward wind compression compared to wind models computed with nonrotating finite disk factors.

The standard systems of stellar rotation measurements (II)

The limb darkening effect on the measurement of stellar rotation is discussed in this paper. It is shown that this effect plays an important role in the measurement of V e sin i. In the extreme case with the limb darkening coefficient of 1.0, it may cause a difference up to 17%. As a sequel to paper [1], this work presents the following new explanation for the systematic differences between Slettebak's new and old systems. The main causes of the systematic differences are: (1) The old system made an inadequate twice repeated correction for the limb darkening. (2) Owing to the historical reason, the old system used too large limb darkening coefficients.

Limb darkening effect included in a simple formula for solar energy flux

AbstractIt is proposed a formula for the solar energy flux including the limb darkening effect and beeing valid at any distance from the Sun.

Photon escape probabilities from resolved sources - Application to solar C III limb emission

Monodirectional effective escape probabilities, involving line-of-sight-integrated escape factors, are calculated for the highly anisotropic source represented by C III emission at the solar limb as a concrete example of the application of such probabilities. Predicted intensity variations of C III lines with limb distance are obtained using a spherical shell model, showing several interesting features which may be observable with appropriate instruments. Among these are expected enhancements of extreme ultraviolet lines at 884.5 and 1308.7 A, and local limb-brightening and limb-darkening effects in optically thick lines, which may provide sensitive constraints on the solar C III emission distribution. Comparisons are made with available observations.

A simple method for preliminary analysis of the light curves of partial eclipsing binaries

view Abstract Citations (4) References (4) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Simple Method for Preliminary Analysis of the Light Curves of Partial Eclipsing Binaries Riazi, Nematullah Abstract A simple method is proposed for solving light curves of partial eclipsing binaries for approximate values of the light and orbital elements, which only requires that the two components have different color indices. Although the method neglects the limb darkening and proximity effects, results are suitable as the starting points in more advanced algorithms and rigorous numerical methods. Publication: The Astronomical Journal Pub Date: July 1992 DOI: 10.1086/116234 Bibcode: 1992AJ....104..228R Keywords: Circular Orbits; Eclipsing Binary Stars; Light Curve; Orbital Elements; Stellar Luminosity; Algorithms; Iterative Solution; Stellar Orbits; Astrophysics; BINARIES: ECLIPSING; METHODS: OBSERVATIONAL; TECHNIQUES: PHOTOMETRIC full text sources ADS | data products SIMBAD (1)

Nucleus of Comet IRAS-Araki-Alcock (1983 VII)

Optical, radar, infrared, UV, and microwave-continuum observations of Comet IRAS-Araki-Alcok were obtained in May 1983, the week of the comet's close approach to earth. The comet has a nucleus dimension and a rotation period which are similar to those of Comet Halley, but a different morphological signature (a persisting sunward fan-shaped coma). Time variations are noted in the projected nucleus cross section. Results suggest significant limb-darkening effects in the relevant domains of radio waves, and that the comet's interior must be extremely cold. It is found that the thermal-infrared fluxes from the inner coma of the comet are dominated by the nucleus.