Finite Angular Size Research Articles

Hyper Illumination of Exoplanets: Analytical and Numerical Approaches

This work describes the illumination of exoplanets whose orbits are close enough to their host star that the finite angular size of their host star causes hyper illumination, in which more than 50% of the planet receives light. Such exoplanets include the hot Jupiters KELT-9 b (64.5% illuminated) and Kepler-91 b (69.6% illuminated). We describe the geometry of three primary illumination zones: the fully illuminated zone, penumbral zone, and unilluminated zone. The integrals required to determine the incident radiation as a function of position from the substellar point on the exoplanet are explained and derived, and the analytical solution is presented within the fully illuminated zone. We find that the illumination predicted by our model is greater at the substellar point than the typical plane-parallel ray model used would suggest. In addition, it is greater within the region of the penumbral zone extending into the antistellar side of the exoplanet. Finally, we compare our model to that used in starry, an open-source software package used to create albedo maps. It appears that starry may be overestimating the illumination of closely orbiting exoplanets because the foreshortening of the area element of the host star is not included in its calculation.

Pattern Recognition Using SVM for the Classification of the Size and Distance of Trans-Neptunian Objects Detected by Serendipitous Stellar Occultations

We present a new pipeline based on the Support Vector Machine algorithm to confirm the detection and perform classification of small solar system objects by serendipitous stellar occultations. This pipeline is designed to analyze light curves and to identify the occultation events and the classification of the occulting bodies according to their size, typically from a fraction to a few kilometers, and their distance from the Sun, typically a few tens of astronomical units. The input light curves for this pipeline were obtained from the event simulator for the Trans-Neptunian Automated Occultation Survey (TAOS II). We explore parameters affecting occultation light curves such as spectral type, apparent magnitude and finite angular size of the occulted star, angle from opposition, and readout cadence for the observations; also we assumed a Poisson noise distribution as expected from the TAOS II project. We find that occultation events, especially by trans-Neptunian objects with diameters ≥2 km are detected with 99.99%, 99.53%, and 86% efficiency for stars with a visual apparent magnitude of 12, 14, and 16, respectively at 0.05 s of exposure time. In terms of size and distance classification the overall accuracy is 94%. However, for smaller objects the confirmation and classification depends mostly upon the signal-to-noise ratio.

Analytic Light Curves in Reflected Light: Phase Curves, Occultations, and Non-Lambertian Scattering for Spherical Planets and Moons

We derive efficient, closed-form, differentiable, and numerically stable solutions for the flux measured from a spherical planet or moon seen in reflected light, either in or out of occultation. Our expressions apply to the computation of scattered light phase curves of exoplanets, secondary eclipse light) curves in the optical, or future measurements of planet–moon and planet–planet occultations, as well as to photometry of solar system bodies. We derive our solutions for Lambertian bodies illuminated by a point source, but extend them to model illumination sources of finite angular size and rough surfaces with phase-dependent scattering. Our algorithm is implemented in Python within the open-source starry mapping framework and is designed with efficient gradient-based inference in mind. The algorithm is ∼4–5 orders of magnitude faster than direct numerical evaluation methods and ∼10 orders of magnitude more precise. We show how the techniques developed here may one day lead to the construction of two-dimensional maps of terrestrial planet surfaces, potentially enabling the detection of continents and oceans on exoplanets in the habitable zone. 6 6 https://github.com/rodluger/starrynight

Occultations by Small Non-spherical Trans-Neptunian Objects. I. A New Event Simulator for TAOS II

We present a new occultation event simulator for the Trans-Neptunian Automated Occultation Survey (TAOS II). We have developed a method to compute occultation shadows by small objects with non-circular apparent shapes (as may result from an intrinsic morphology or from the projection of a contact binary). The new simulator calculates diffraction features in the occultation shadows, as well as resulting light curves as would be measured by the TAOS II survey system. We include effects such as the spectral type and finite angular size of the occulted star. We find that occultation events, especially by Trans-Neptunian Objects with diameters ∼3 km may be misidentified or mischaracterized when not taking non-spherical shapes into account.

Source Structure and Measurement Noise Are as Important as All Other Residual Sources in Geodetic VLBI Combined

AbstractThe celestial sources observed by geodetic very‐long‐baseline interferometry (VLBI), most of which are quasars, are resolved—they have a finite angular size as measured by the interferometer—yet modern geodetic VLBI analysis techniques still treat these sources as if they are point‐like. The errors introduced by source structure, the angular distribution of emission, have traditionally been believed to be a small fraction of the total measurement error budget. We analyzed the VLBI data of the CONT14 continuous observing campaign using standard geodetic and imaging techniques, along with an analysis of closures, combinations of interferometric measurements where the error contributions from station‐based effects such as tropospheric and ionospheric errors, clock and cable length errors, station position errors, and so on, cancel exactly, leaving only nonclosing errors, principally source structure and measurement noise. Our statistical analysis of the closure information, based on actual VLBI observations revealed that nearly half (40%) of the total geodetic VLBI error budget (in terms of variance) in CONT14 came from source‐structure effects, nearly the same variance as all station‐based error sources (43%). The remaining error due to other nonsource‐structure, nonclosing errors is also significant (17%). Previous studies, which were primarily simulation‐based, did not account for the full range of effects of source structure on real measurements. Source structure is a major contributor to errors in geodetic VLBI, and source structure must be taken into account in the entire VLBI operational chain, from scheduling to analysis, in order to mitigate its effects on astrometric and geodetic measurements.

Exoplanet phase curves at large phase angles. Diagnostics for extended hazy atmospheres

At optical wavelengths, Titan's brightness for large Sun-Titan-observer phase angles significantly exceeds its dayside brightness. The brightening that occurs near back-illumination is due to moderately large haze particles in the moon's extended atmosphere that forward-scatter the incident sunlight. Motivated by this phenomenon, here we investigate the forward scattering from currently known exoplanets, its diagnostics possibilities, the observational requirements to resolve it, and potential implications. An analytical expression is derived for the amount of starlight forward-scattered by an exponential atmosphere that takes into account the finite angular size of the star. We use this expression to tentatively estimate how prevalent this phenomenon may be. Based on numerical calculations that consider exoplanet visibility, we identify numerous planets with predicted out-of-transit forward scattering signals of up to tens of parts-per-million provided that aerosols of $>$1 micron size form over an extended vertical region near the optical radius level. We propose that the interpretation of available optical phase curves should be revised to constrain the strength of this phenomenon that might provide insight into aerosol scale heights and particle sizes. For the relatively general atmospheres considered here, forward scattering reduces the transmission-only transit depth by typically less than the equivalent to a scale height. For short-period exoplanets the finite angular size of the star severely affects the amount of radiation scattered towards the observer at mid-transit.

Achromatic interfero-coronagraph with variable rotational shear: reducing of star leakage effect, white light nulling with lab prototype

Stellar coronagraphs with a small inner working angle have limited possibilities to detect exoplanets due to the star leakage effect caused by incomplete suppression of the star of finite angular size. We consider an improved instrument for direct imaging of exoplanets: common-path achromatic interfero-coronagraph with variable rotational shear (CP-ARC). CP-ARC reduces the star leakage effect by several orders of magnitude (with small angles of rotational shear relative to a fixed angle of 180 deg) with telescope size more than 1 m. Operation capacity of CP-ARC was experimentally verified by laser light nulling and white light nulling with lab prototype treated.

The opposition effect in the outer Solar system: A comparative study of the phase function morphology

In this paper, we characterize the morphology of the disk-integrated phase functions of satellites and rings around the giant planets of our solar system. We find that the shape of the phase function is accurately represented by a logarithmic model [Bobrov, M.S., 1970. Physical properties of Saturn's rings. In: Dollfus, A. (Ed.), Surfaces and Interiors of Planets and Satellites. Academic, New York, pp. 376–461]. For practical purposes, we also parametrize the phase curves by a linear-exponential model [Kaasalainen, S., Muinonen, K., Piironen, J., 2001. Comparative study on opposition effect of icy solar system objects. Journal of Quantitative Spectroscopy and Radiative Transfer 70, 529–543] and a simple linear-by-parts model [Lumme, K., Irvine, W.M., 1976. Photometry of Saturn's rings. Astronomical Journal 81, 865–893], which provides three morphological parameters: the amplitude A and the half-width at half-maximum (HWHM) of the opposition surge, and the slope S of the linear part of the phase function at larger phase angles. Our analysis demonstrates that all of these morphological parameters are correlated with the single-scattering albedos of the surfaces. By taking more accurately into consideration the finite angular size of the Sun, we find that the Galilean, Saturnian, Uranian and Neptunian satellites have similar HWHMs ( ≲ 0 . 5 ∘ ), whereas they have a wide range of amplitudes A. The Moon has the largest HWHM ( ∼ 2 ∘ ). We interpret that as a consequence of the “solar size bias”, via the finite angular size of the Sun which varies dramatically from the Earth to Neptune. By applying a new method that attempts to morphologically deconvolve the phase function to the solar angular size, we find that icy and young surfaces, with active resurfacing, have the smallest values of A and HWHM, whereas dark objects (and perhaps older surfaces) such as the Moon, Nereid and Saturn's C ring have the largest A and HWHM. Comparison between multiple objects also shows that solar system objects belonging to the same planet have comparable opposition surges. This can be interpreted as a “planetary environmental effect” that acts to locally modify the regolith and the surface properties of objects which are in the same environment.

Observable effects during gravitational microlensing of stars by NUT objects

We consider observable effects during the gravitational microlensing of stars of finite angular size with a given brightness distribution across their disks by exotic objects whose space-time environment is described with the NUT metric. We demonstrate that, under certain conditions, the microlensing light curves and chromatic effects due to the wavelength dependence of the disk brightness distributions can differ considerably from those observed for a Schwarzschild gravitational lens, so that their analysis can facilitate the identification of such objects.

Fitting Formula for Flux Scintillation of Compact Radio Sources

We present a fitting function to describe the statistics of flux modulations caused by interstellar scintillation. The function models a very general quantity: the cross-correlation of the flux observed from a compact radio source of finite angular size observed at two frequencies and at two positions or times. The formula will be useful for fitting data from sources such as intra-day variables and gamma-ray burst afterglows. These sources are often observed at relatively high frequencies (several gigahertz) where interstellar scattering is neither very strong nor very weak, so that asymptotic formulae are inapplicable.

Light Propagation in Inhomogeneous Universes. III. Distributions of Image Separations

Using an analytical model, we compute the distribution of image separations resulting from gravitational lensing of distant sources, for seven COBE-normalized cold dark matter models with various combinations of Ω0 and λ0. Our model assumes that multiple imaging results from strong lensing by individual galaxies. We model galaxies as nonsingular isothermal spheres, whose parameters are functions of the luminosity and morphological type, and take into account the finite angular size of the sources. Our model neglects the contribution of the background matter distribution to lensing and assumes that lensing is entirely caused by galaxies. To test the validity of this assumption, we performed a series of ray-tracing experiments to study the effect of the background matter on the distribution of image separations. Our results are the following: (1) The presence of the background matter tends to increase the image separations produced by lensing galaxies, making the distributions of image separations wider. However, this effect is rather small and independent of the cosmological model. (2) Simulations with galaxies and background matter often produce a secondary peak in the distribution of image separations at large separations. This peak does not appear when the background matter is excluded from the simulations. (3) The effect of the background matter on the magnification distribution is negligible in low-density universes (Ω0 = 0.2) with small density contrast (σ8 = 0.4), but becomes very important as Ω0 and σ8 increase, resulting in a significant widening of the distribution. (4) Multiple imaging is caused primarily by early-type galaxies (elliptical and S0), with a negligible contribution from spiral galaxies. (5) Our analytical model, which has only two free parameters, is in good agreement with the results of ray-tracing experiments, successfully reproducing the distributions of image separations, and also the multiple-imaging probability, for all cosmological models considered. (6) The analytical model predicts that the distributions of image separations are virtually indistinguishable for flat, cosmological-constant models with different values of Ω0. (7) For models with no cosmological constant, the distributions of image separations do depend upon Ω0, but this dependence is weak. We conclude that while the number of multiple-imaged sources can put strong constraints on the cosmological parameters, the distribution of image separations does not constrain the cosmological models in any significant way and mostly provides constraints on the structure of the galaxies responsible for lensing.

Jet quenching versus jet enhancement: a quantitative study of the BDMPS-Z gluon radiation spectrum

We study the gluon radiation spectrum off a hard in-medium produced quark in the multiple soft-rescattering formalism of Baier–Dokshitzer–Mueller–Peigné–Schiff and of Zakharov (BDMPS-Z). Its dependence on the quark and gluon energy, on the gluon transverse momentum, on the in-medium pathlength and on the rescattering properties of the nuclear medium is analyzed quantitatively. The two components of gluon radiation, the hard vacuum radiation associated to the quark production vertex, and the medium-induced rescattering contribution interfere destructively. For small spatial extensions of the medium, this destructive interference overcompensates the hard vacuum radiation, and the total medium-induced radiative energy loss decreases as ΔE∝− L 3 . Medium-induced gluon production dominates only above a finite critical length L>L crit which varies between 3 and more than 6 fm depending on the rescattering properties of the medium. Deviations from the BDMPS- L 2 -behaviour persist above L crit . The medium-dependence of the angular gluon distribution is dominated by transverse brownian k ⊥ -broadening. This results in a depletion of the low transverse momentum part of both the hard and the medium-induced contribution. As a consequence, the medium-induced energy loss outside a finite angular cone size Θ can be more than a factor two larger than the total medium-induced radiative energy loss. We discuss implications of these results for the jet quenching signal in relativistic heavy-ion collisions at RHIC and LHC.

Spectral Properties of Accretion Disks around Galactic and Extragalactic Black Holes: Erratum

We derive a self-consistent set of equations that describe the propagation of nonlinear shear Alfvén waves in the solar wind. The equations include the interaction between the waves and wind plasmas and has an exact energy conservation law. Using a truncated Fourier series, we further reduce the system to a simplified nonlinear model with multiple waves. We solve the model numerically in a radially expanding box with a finite angular size. We find that for linear waves in a time-varying plasma background, the for the WKB are violated in practically the entire radial domain (even at high frequencies), although the scaling of wave amplitudes (δB) with radial distance and the equipartition between the magnetic and kinetic energy are similar to the WKB results. For nonlinear waves, equipartition still holds, but nonlinear wave reflection enhances the backward propagating waves (in the rest frame of the wind), which leads to a faster drop in δB than the linear waves. As a result, there is a "soft" saturation in which δB approaches the background magnetic field gradually at large distance. The simplified nonlinear model produces reasonably high wind speed (between 600 and 700 km s<SUP>-1</SUP>).

The dynamics of solar sails with a non-point source of radiation pressure

The form of the solar radiation pressure on a heliocentric orbiting solar sail is obtained for a finite angular sized and limb darkened solar disk by the use of the radiation pressure tensor. It is found that the usual inverse square variation of the solar radiation pressure is modified by the finite angular size, and to a lesser extent by the solar limb darkening. The actual magnitude of the modification is in itself small, except at close heliocentric distances. However, its existence has implications for the dynamical stability of solar sails both in parked and circular orbital configurations and for the accuracy of trajectory calculations, particularly for sails in the inner solar system.

Solar sail dynamics with an extended source of radiation pressure

The exact form of the solar radiation pressure on a heliocentric orbiting solar sail is obtained for a finite angular sized solar disk by the use of a specific intensity calculation. It is found that the usual inverse square variation of the solar radiation pressure is modified by this finite angular size. The actual magnitude of the modification is in itself small, except at close heliocentric distances. However, its existence has implications for the dynamical stability of solar sails both in parked and circular orbital configurations and for the accuracy of trajectory calculations, particularly for sails in the inner solar system.

The shape of a scatter-broadened image - I. Numerical simulations and physical principles

We investigate the appearance of the scatter-broadened image of a point source in the presence of strong scattering. We present numerical simulations wiih a one-dimensional scattering screen and interpret the results using a physical model of the optics of scattering. For strong scattering and bandwidths less than the scintillation dccorrelation bandwidth, it is known that there are two characteristic time-scales for flux scintillation, namely the slow refractive time-scale, tref and the fast diffractive time-scale, tdiff. Associated with these arc length-scales, rref and rdiff. We show that there are related time-scales (and length-scales) in the imaging problem as well and we demonstrate that the shape of the image depends critically on the relative magnitude of the integration time, tint, compared to these time-scales. We consider first the case when the telescope has an infinitesimally narrow bandwidth and filled UV coverage of characteristic radius σt. We assume |$\sigma _\text t \gt r _\text{diff}$|⁠, since the scatter-broadened image is not resolved otherwise. In the regime tref ≪ tint, such a telescope will measure the ‘ensemble-average’ image, provided we assume ergodicity. There is a well-known result relating the shape of the ensemble-average image to the phase structure function in the scattering medium. The next important regime is |$(\sigma_\text t/r_\text {diff}) \enspace t_\text {diff}$| ≪ tint ≪ |$(r_\text {diff}/\sigma_\text t) \enspace t_\text {ref}$|⁠. In this regime, the telescope measures what we call the ‘average image’. The appearance of the average image depends on the spectrum of density fluctuations in the scattering medium. For ‘shallow’ power-law spectra the image is relatively smooth, but with characteristic low-level substructure, whereas for ‘steep’ spectra the image is strongly fragmented with a fractal morphology. The regime corresponding to the average image does not appear to have been explored before. Finally, the regime, |$t_\text {int} \le t_\text {diff}$|⁠, leads to the ‘snapshot image’. This image displays deeply modulated structure down to the resolution limit of the telescope. When the telescope is displaced, the structure decorrelates for shifts greater than σt. In addition to the filled UV coverage case, we also consider the measurement of visibilities using a two-element interferometer. This mode of observation differs in important ways from the previous case; in particular, some of the time-scales that demarcate the different regimes are not the same. We also consider the effect of a finite bandwidth and a finite source angular size. Both the bandwidth and the source size need to be less than very stringent limits if snapshot images are to be observed; much more relaxed criteria apply for average images. In the case of interstellar scintillation, tref ∼ days to years and tdiff ∼ minutes. Since normal VLBI observations extend over a fraction of a day, images obtained by this technique correspond to the average image regime. Some of the ideas explored in this paper are therefore relevant for the interpretation of such observations.

Angular cross-relations of Abell clusters in different distance classes

The angular autocorrelation and cross-correlation functions of the D = 1 ... 4, D = 5, and D = 6 distance class Abell clusters are estimated. There is a strong anticorrelation between the most distant D = 6 and the closest D = 1 ... 4 subsamples. It is suggested that an artifact of the cluster identification process presumably due to the finite angular size of the cluster. This anticorrelation seems to contradict some recent estimations of projection contaminations in the Abell catalog. The angular proximity of a foreground cluster may have caused a background cluster not to be counted as it was thought to be a subcluster or it was erroneously assigned to a nearer distance class.

Interstellar scattering of the radio source 2013+370

We report the results of a six-station, 1663 MHz VLBI observation of the radio source 2013+370. The image of this source is dominated by interstellar scattering and is observed to be elliptical with a major axis (FWHM) of 17 mas and an axial ratio of 0.70. A fit of the observations to a model visibility function yields a value for α, the spectral index of the power-law density irregularity spectrum, of α = 3.79 +/- 0.05. Corrections for the finite angular size of the source would probably produce a downward revision of α. Other important characteristics of the image are its smooth character, with no evidence of multiple images, and stationarity over several hours as well as over a two-year interval since a previous observation. Our observations are consistent with the interstellar density spectrum being close to Kolmogoroff for those scales which determine the angular broadening.

A comment on the insolation at Mercury

E. Van Hemelrijck and J. Vercheval [ Icarus 48, 167–179 (1981)] presented calculations of the insolation at Mercury and Venus which neglect the finite angular size of the Sun. To determine the temperature structure in the subsurface a more accurate calculation is needed, especially at longitudes ±90° on Mercury, where the Sun takes 18 days to rise or set. These calculations are presented here.

Finite sun effect on the interpretation of solar aureole

Although it is usually assumed that solar radiation falls on the earth's atmosphere in the form of plane waves, the finite angular size of the solar disk contradicts this assumption. For most purposes, this finite sun effect on computed or measured radiation quantities is negligible. However, in the region of the solar aureole, which is dominated by aerosol diffraction scattering, measurable effects may be obtained. In this paper, we show that the finite sun effect is related to derivatives of the scattering phase function and that a 1% effect may be obtained close to the sun if enough large particles are present in the atmosphere.