A Spherical Earth

The appearance of a surface depends dramatically on how it is illuminated. In order to interpret properly satellite and aerial imagery, it is necessary to know the position of the sun in the sky. This is particularly important if this interpretation is to be done in an automated fashion. Techniques using relatively straightforward methods are presented here for calculating the position of the sun with more than enough accuracy.

2010

Abstract By means of a simple mathematical model developed by the authors, the apparent movement of the Sun can be studied for arbitrary latitudes. Using this model, it is easy to gain insight into various phenomena, such as the passage of the seasons, dependences of position and time of sunrise or sunset on a specific day of year, day duration for different latitudes and seasons, and time dependence of the solar altitude reckoned from the horizontal plane. We present simulations of the Sun path alongside animated data.

The kinematics of the Solar zenith passage on the Earth was shown as a simple harmonic motion. This analytical model used the uniform circular motion to describe the Earth's rotation relative to its axis and relative to the Sun, also this model considered the sunlight beams parallel and constant the obliquity of the ecliptic. We made a contrast between some data taken from of the software Stellarium 0.12.1 and the model, we obtained an uncertainty $\Delta t=1d$; this result allow to project the model as an astronomical context conducive to develop the kinematics of simple harmonic motion.

The figure of the Sun reflects its inner structure and dynamics, influencing also the perihelion precession of close orbiting bodies, like Mercury. To study the solar figure from ground, the deformation on the solar image induced by the atmosphere has to be known up to one part over a million, and this is done through differential refraction models: a historical review of them is drafted. The solar oblateness has been investigated in order to validate alternative theories to General Relativity and to understand the internal dynamics of our star. The solar figure should possess only micro departures from sphericity according to the standard stellar structure theory and helioseismology data, though variations along the cycle has been observed. Ground-based and satellite data show contrasting observational results. The oblateness measured onboard RHESSI satellite, the one of SDS onboard a stratospheric ballon and that one of the Astrolabe of Rio de Janeiro are presented with their implications in classical and relativistic gravitation. The perspectives offered by the reflecting heliometer in the future measurement of the oblateness are depicted.

Philologia Classica, 2020

This article discusses an intriguing text in Stobaeus' Ecl. 1.25.3i about the sun's movement as a spiral on a cylinder. The author offers an interpretation of this text and argue that it is about Empedocles' conception of the solar trajectory during the year. After a preliminary attempt, an interlude is inserted on some strange theories, ascribed to Empedocles, about the two hemispheres of the heaven and two suns. Two of the more reliable theories attributed to Empedocles that are relevant in the context of this paper, namely the tilting and the eggshape of the heaven, as well as the problems of the size of the sun and the shape of the earth, are discussed in successive sections. This allows the author to illustrate some of his ideas on Presocratic flat earth cosmology. Prior to offering a visualization of the cosmos according to Empedocles, Bollack's earlier attempt is subjected to a critical examination. In two additional sections of the article, the author claims that, according to Empedocles, the moon must move on a cylinder as well and that the image of the cylinder for movements of the sun and moon dates back to Anaximander.

Astronomy and Astrophysics, 2006

Context. It has recently been claimed that analysis of Greenwich sunspot data over 120 years reveals that sunspot activity clusters around two longitudes separated by 180 • ("active longitudes") with clearly defined differential rotation during activity cycles. In previous work we demonstrated that such effects can be observed in synthetic data without such features, as an artefact of the method of analysis. Aims. In the present work we extend this critical examination of methodology to the actual Greenwich sunspot data and also consider newly proposed methods of analysis claiming to confirm the original identification of active longitudes. Methods. We performed fits of different kinematic frames onto the actual sunspot data. Firstly, a cell-counting statistic was used to analyse a comoving system of frames and show that such frames extract useful information from the data. Secondly, to check the claim of century-scale persistent active longitudes in a contramoving frame system, we made a comprehensive exploration of parameter space following the original methodology as closely as possible. Results. Our analysis revealed that values obtained for the parameters of differential rotation are not stable across different methods of analysis proposed to track persistent active longitudes. Also, despite a very thorough search in parameter space, we were unable to reproduce results claiming to reveal the century-persistent active longitudes. Previous parameter space exploration has been restricted to frames whose latitudinal profile is opposite to solar surface differential rotation. Relaxing this restriction we found that the highest values of nonaxisymmetry occur for frames comoving with the solar surface flow. Further analysis indicates that even these solutions are the result of purely statistical fluctuations. Conclusions. We can therefore say that strong and well substantiated evidence for an essential and century-scale persistent nonaxisymmetry in the sunspot distribution does not exist.

Archive for History of Exact Sciences, 2018

From Antiquity through the early modern period, the apparent motion of the Sun in longitude was simulated by the eccentric model set forth in Ptolemy's Almagest III, with the fundamental parameters including the two orbital elements, the eccentricity e and the longitude of the apogee λ A , the mean motion ω, and the radix of the mean longitudeλ 0. In this article we investigate the accuracy of 11 solar theories established across the Middle East from 800 to 1600 as well as Ptolemy's and Tycho Brahe's, with respect to the precision of the parameter values and of the solar longitudes λ that they produce. The theoretical deviation due to the mismatch between the eccentric model with uniform motion and the elliptical model with Keplerian motion is taken into account in order to determine the precision of e and λ A in the theories whose observational basis is available. The smallest errors in the eccentricity are found in these theories: the Mumtah. an (830):

2001

In this paper we first emphasize why it is important to know the successive zonal harmonics of the Sun's figure with high accuracy: mainly fundamental astrometry, helioseismology, planetary motions and relativistic effects. Then we briefly comment why the Sun appears oblate, going back to primitive definitions in order to underline some discrepancies in theories and to emphasize again the relevant hypotheses. We propose a new theoretical approach entirely based on an expansion in terms of Legendre's functions, including the differential rotation of the Sun at the surface. This permits linking the two first spherical harmonic coefficients (J2 and J4) with the geometric parameters that can be measured on the Sun (equatorial and polar radii). We emphasize the difficulties in inferring gravitational oblateness from visual measurements of the geometric oblateness, and more generally a dynamical flattening. Results are given for different observed rotational laws. It is shown that...

Zenodo, 2020

The tropics are the regions of the Earth that lie between the latitude lines of the Tropic of Cancer and the Tropic of Capricorn. In this zone of the Earth, we have the opportunity to see the zenith passage of the sun, that is, we can see the sun passing at noon directly overhead. The zenith passage happens on two days in the year. These days depend upon the latitude of the place of observation. Let us note that, at the Tropic of Cancer, the zenith passage happens on the day of the June solstice and at the Tropic of Capricorn on that of the December solstice; at the equator, the zenithal sun is observed on the two equinoxes. When the sun passes overhead, the shadows disappear and the days when this happens assume a sacred significance for the people that live and lived within the tropics. Being the zenith passage so important, it is not surprising that we can find it evidenced by the local architectures too. We have, for instance, that some monuments possess a "zenith tube" at their apex, such as at Angkor Wat, Cambodia. It is a vertical sighting tube inserted in the vault of the structure, which produces in a dark chamber a perfectly perpendicular beam of light when the sun is at the local zenith. As we will see, alignments of monuments are also possible: we observed them in Sanchi, India, at the Lion Rock in Sri Lanka, in Angkor Wat and at the Sewu, Prambanan and Borobudur Temples in Java. We can observe alignments at the Mesoamerican Sites of Tula and Chichen Itza. Here we also show alignments to the sunrise of the zenithal sun at the Shwedagon Pagoda and in the plan of Brasilia.