True Longitude Research Articles

Captain Cook's longitude determinations and the transit of Mercury — common assumptions questioned

Captain Cook's methods of determining longitude were more error-prone and uncertain than is commonly understood. He relied on astronomical measurements that had inherently large error margins. Contrary to popular perception, there is no longitude value in the surviving records that is attributable to the transit of Mercury observation in 1769. The Mercury transit timings obtained were not accurate enough to derive the true longitude.

Inclusion of Higher Order Harmonics in the Modeling of Optimal Low-Thrust Orbit Transfer

The higher fidelity modeling of minimum-time transfers using continuous constant acceleration low-thrust is depicted by including the higher zonal harmonics J 3 and J 4for the Earth gravity model. The inclusion of these higher order harmonics is of great benefit in carrying out accurate transfer simulations, especially for long duration flights dwelling in low altitudes where the effects of these zonals are greatest. The analysis presented here can also be coded in the flight guidance computer of spacecraft for autonomous operations and on ground computers for solution uploads and resetting during low-thrust transfers. Equinoctial elements are used to avoid singularities when orbits are circular or equatorial and the applicability of the theory is of a general nature regardless of the size, shape and spatial orientation of the orbits provided they are not of the parabolic or hyperbolic types. To this end, two sets of dynamic and adjoint differential equations in terms of nonsingular orbital elements are derived by further considering a more accurate perturbation model in the form of the higher order Earth zonal harmonics J 3and J 4. Previous analyses involved only the first-order J 2 term in order to model optimal lowthrust transfers between any two given circular or elliptic orbits. The first formulation uses the eccentric longitude as the sixth element of an equinoctial set of elements while describing the thrust as well as the zonal accelerations in the so called direct equinoctial frame. The second formulation makes use of the true longitude as the sixth element instead while resolving the thrust and the zonal accelerations in the rotating Euler-Hill frame simplifying considerably the algebraic derivations leading to the generation of the nonsingular differential equations that are also free of any singularity for the important zero eccentricity and zero inclination cases often encountered in Earth orbit transfer problems. The derivations of both nonsingular formulations are mutually validated by generating an optimal transfer example that achieves the same target conditions regardless of which formulation is used.

The streamlined and complete set of the nonsingular J2-perturbed dynamic and adjoint equations for trajectory optimization in terms of eccentric longitude

The complete set of the dynamic and multipliers differential equations for a nonsingular set of equinoctial elements where the eccentric longitude stands for the sixth or fast orbital element are derived in a more straightforward and streamlined manner as compared to a previous derivation also presented by this author. The mathematics are validated through comparison with the original formulation as well as with two other formulations using the mean longitude and the true longitude elements as the respective sixth element of the equinoctial set. Furthermore, this new formulation is used to derive the costate or adjoint differential equations by fully accounting for the secular first-order perturbative effect of the second zonal harmonic J2, and the complete set of the perturbed dynamic and adjoint system of equations are also validated by direct comparison with the two previously derived formulations using the mean and true longitudes respectively. The present formulation as well as the one using the true longitude as the sixth orbital element remove the need to solve Kepler’s transcendental equation at each integration step, a need that is inevitable when the mean longitude formulation is used, because in the latter case the right-hand sides of the various differential equations cannot be written directly in terms of the mean longitude. The inclusion of the J3 and J4 terms can be similarly accounted for with both the eccentric and true longitude sets and mutually validated also. This particular formulation has been adopted by several aerospace contractors in the United States to build specialized flight guidance software to steer payloads released in high energy orbits to their final destinations using highly efficient low-thrust propulsion.

The Standard Scheme of the Moon and Its Mean Quantities

The Papyrus Rylands 27, of the John Rylands Library, Manchester, is a portion of a Greek treatise or tract of the third century A.D., providing rules for the longitude and node of the Moon. Apart from the illuminating analysis provided by Hunt, it has been the subject of a number of profound studies by Neugebauer, van der Waerden and Jones.1 These have culminated in the presentation by Jones of what he calls the Standard Scheme of lunar motion, illustrated by its application in various parts of the Oxyrhynchus Papyri, and elsewhere.2 It is clear from all this that we have here a scheme for the computation of the true longitude and node of the Moon based on a Babylonian type of 'zigzag' model. Jones has provided all that one needs for the practical derivation of the lunar coordinates, including lengthy tables to facilitate the computation of the summation of values from the zigzag functions that determine the coordinates. That much completes the subject as a self-contained model within the context of 'arithmetical' schemes, but leaves untouched the question of the relation between the Standard Scheme and the trigonometrical models, that is, those based on the use of one or more epicycles, etc. The purpose of this article is to develop this relation, one that situates the model in the interface between Babylonian and Greek astronomy.

Spatial distribution of sodium on Mercury

We observed the Mercury sodium exosphere during the period 1997–2003, collecting images of planetary sodium emission covering the full range of true anomaly angles with only a few small gaps. The distribution of sodium emission over the surface was generally non-uniform and changeable. When the dawn terminator was in view, the terminator was generally brighter than the limb, as expected for evaporation of condensed sodium at the dawn terminator. Also, the excess emission reached its largest values when radiation acceleration reached one or the other of its two maxima, as expected for the effect of radiation acceleration on sodium distribution. When the dusk terminator was in view, the limb was generally brighter than the terminator. The difference was larger than would be expected for a uniform sodium exosphere, suggesting that there is a deficit of sodium near the dusk terminator. There was no apparent effect of radiation acceleration on the ratio, which might be the result of a very large deficit of sodium near the dusk terminator. For the northern and southern hemispheres, excess sodium was observed about a third of the time in one or the other hemisphere, appearing at random intervals of true anomaly and longitude. The random nature of these occurrences suggests an external cause, one not correlated with any characteristic of the planetary orbit or planetary geochemistry. We suggest that the northern or southern excess sodium events are the result of solar weather, whereby solar particles are precipitated to the surface at high latitudes, and produce localized sources of sodium. IMF configurations for which solar particles can precipitate to high latitudes on the surface occur about 30% of the time, in general agreement with the observed frequency of north or south excess emission. Near periods of maximum radiation acceleration, some images displayed two peaks of sodium emission, one peak at high northern latitudes, the other at high southern latitudes. One possible cause could be the accumulation of sodium near the terminator, pushed there by radiation acceleration.

The Equant in India: The Mathematical Basis of Ancient Indian Planetary Models

The planetary models of ancient Indian mathematical astronomy are described in several texts.1 These texts invariably give algorithms for computing mean and true longitudes of the planets, but are completely devoid of any material that would inform us of the origin of the models. One way to approach the problem is to compare the predictions of the Indian models with the predictions from other models that do have, at least in part, a known historical background. Since the Indian models compute true longitudes by adding corrections to mean longitudes, the obvious choices for these latter models are those from the Greco-Roman world. In order to investigate if there is any connection between Greek and Indian models, we should therefore focus on the oldest Indian texts that contain fully described, and therefore securely computable, models. We shall see that the mathematical basis of the Indian models is the equant model found in the Almagest, and furthermore, that analysis of the level of development of Indian astronomy contemporary to their planetary schemes strongly suggests, but does not rigorously prove, that the planetary bisected equant model is pre-Ptolemaic.

Further Simplification of the Manifold Correction Method for Orbit Integration

By introducing some quantities defined on the moving orbital plane, we have developed three ways to further reduce the number of variables in our two simplified methods of manifold correction, from nine to seven or six per celestial body. Among these, the simplest option uses a set of six variables consisting of the three-dimensional orbital angular momentum vector, the two independent components of the Laplace integral vector on the moving orbital plane, and a true orbital longitude measured from an origin solely determined from the orbital angular momentum vector. This scheme no longer requires any manifold correction, as does the standard method to integrate the Cartesian coordinates and velocity. However, the new method is much more precise than the standard method. For example, the longitude error of Mercury in a simultaneous integration of the Sun and nine major planets with a step size of 1.4 days exceeds 360° after a few thousand years when using the standard method but remains at the milliarcsecond level if the new method is used. In addition, the new method achieves better performance than any of the manifold correction methods we have developed, as long as round-off errors are negligible.

Minimum-Time Constant Acceleration Orbit Transfer with First-Order Oblateness Effect

The analysis of the minimum-time thrust and J2-perturbed orbit transfer between elliptic orbits of arbitrary size, shape, and orientation, and using continuous constant acceleration whose direction is optimized, is presented for two different formulations using nonsingular equinoctial orbit elements. A previous formulation that used the equinoctial orbital rotating frame for the component resolution of the perturbation vector is extended to a recently developed formulation that uses the true longitude as the sixth state variable and the polar frame for the component resolution of the thrust and J2-induced accelerations. This new analysis is much simpler because it provides the simplest form of the differential equations for the adjoints and removes the need for solving Kepler’s transcendental equationduring the numerical integration of the dynamicand adjoint system of equations. Because of these simpler features, the corresponding software is also more robust resulting in improved convergence to the optimal solution of interest. The constant power and constant Isp case that results in constant thrust is a trivial extension of the constant acceleration case studied here because the acceleration is easily updated by updating the mass of the vehicle during the numerical integration.

Trajectory Optimization Using Nonsingular Orbital Elements and True Longitude

The system and adjoint differential equations needed to solve low-thrust transfer and rendezvous trajectories are derived in terms of nonsingular orbit elements, with the true longitude selected as the sixth variable that also dee nes the radial distance. The perturbation accelerations are resolved in the rotating polar frame such that the equations of motion are written in polarcoordinates. This formulation is particularly convenient forthe treatment of the J2 perturbation acceleration whose components are easily expressed in terms of the true longitude. The iterations needed for the solution of Kepler’ s equation are also eliminated, and the analytic form of the adjoint equations is provided in their simplest aspect thus far.

Mechanics of Trajectory Optimization Using Nonsingular Variational Equations in Polar Coordinates

Thecompletesetofthestateand adjoint differentialequationsforthenonsingularequinoctialorbitelementsexpressed in polar coordinatesand written in terms of the truelongitudeispresented. Previous formulations adopted the equinoctial frame as the orbital frame for component resolution of the various perturbation accelerations. The consideration of the position dependency of the J2 acceleration in the context of precision integrated orbital transfer trajectories led us to adopt the more convenient polar frame coupled with the use of the true longitude as the accessory variable needed in the description of the variational equations, inasmuch as it is more dife cult to generate the contribution of the J2 perturbation to the adjoint equations if the variational equations are left in terms of the eccentric longitude. I. Introduction T HEconsideration of thenonsingularequinoctialorbitelements has been of great benee t in trajectory propagation and optimization.References1 ‐7 madeuseofthecorresponding variational equations resolved in the equinoctial orbital coordinate system. In Refs. 8‐12, modern numerical methods based on the techniques of collocation and nonlinear programming are applied to the solutionoflow-thrustEarth-boundandinterplanetarytrajectories.These direct methods have introduced robustness characteristics to the optimization process and were e rst used with great success by launch vehicle trajectory optimization and performance analysts. In particular, Betts 9,10 uses a variety of equinoctial element sets within the framework of the direct method. Following the more traditional approach based on the calculus of variations and the indirect shooting method,thispaper developsthe differentialequationsforthe adjoint or multiplier variables to be numerically integrated simultaneously withthestatedifferentialequationstosolvethetwo-point-boundaryvalue orbit transfer problem. As noted by Betts, the elements of the matrix of partial derivatives of the Hamiltonian with respect to the equinoctial elements derived here in analytic form can be used to developacertainanalyticJacobianmatrixinconnectionwithnonlinear programming (NLP) solvers used with direct methods, thereby saving considerable computation time with respect to the adoption ofa numerically determined Jacobian. The variational equations are usually expressed in terms of the eccentric longitude, which can either be left as an accessory variable or be adopted as the fastor sixth orbital element. In Ref. 3, the averaged J2 perturbation effect was accounted for in the design of the optimal transfer trajectory. However, when the exact or rather position-dependent J2 effect must be considered in the context of precision integrated transfer solutions, the J2 perturbation acceleration components become dependent on the eccentric longitude. Because these components are given in the rotating Euler‐Hill or polar frame, it is much simpler to use this frame as the orbital frame instead of the equinoctial framebecause otherwise the transformed expressions for these components become more complicated. These expressions are still quite complicated if expressed in terms of the eccentric longitude even if the polar frame is used. This is important because we must take the partial derivatives of these expressions with respect to the elements to generate the adjoint differential equations. This task is made easier by expressing the variational equations for the J2 perturbation in terms of the true

Insolation and Earth's orbital periods

Solar irradiance received on a horizontal surface depends on the solar output, the semimajor axis of the elliptical orbit of the Earth around the sun (a), the distance from the Earth to the sun (r), and the zenith distance (z). The spectrum of the distance, r, for a given value of the true longitude, λ, displays mainly the precessional periods and, with much less power, half precession periods, eccentricity periods, and some combination tones. The zenith distance or its equivalent, the elevation angle (E), is only a function of obliquity (ϵ) for a given latitude, ϕ, true longitude, and hour angle, H. Therefore the insolation at a given constant value of z is only a function of precession and eccentricity. On the other hand, the value of the hour angle, H, corresponding to this fixed value of z varies with ε, except for the equinoxes, where H corresponding to a constant z also remains constant through time. Three kinds of insolation have been computed both analytically and numerically: the instantaneous insolation (irradiance) at noon, the daily irradiation, and the irradiations received during particular time intervals of the day defined by two constant values of the zenith distance (diurnal irradiations). Mean irradiances (irradiations divided by the length of the time interval over which they are calculated) are also computed for different time intervals, like the interval between sunrise and sunset, in particular. Examples of these insolations are given in this paper for the equinoxes and the solstices. At the equinoxes, for each latitude, all insolations are only a function of precession (this invalidates the results obtained by Cerveny [1991]). At the solstices, both precession and obliquity are present, although precession dominates for most of the latitudes. Because the lengths of the astronomical seasons are secularly variable (in terms of precession only), a particular calendar day does not always correspond to the same position relative to the sun through geological time. Similarly, a given longitude of the Sun on its orbit does not correspond to the same calendar day. For example, 103 kyr ago, assuming arbitrarily that the spring equinox is always on March 21, autumn began on September 13, and 114 kyr ago, it began on September 27, the length of the summer season being 85 and 98 calendar days, respectively, at these remote times in the past.

On the unmodeled perturbations in the motion of uranus

In this paper we apply a numerical method to determine unmodeled perturbations in an attempt to explain the observed discrepancies in the motion of Uranus. We find that the estimated perturbation shows some significant periods that could be attributed to insufficient knowledge of the perturbations from some of the known planets. On the assumption that the gravitational attraction of an unknown planet is the origin of the deviations, the best planar solution of the inverse problem is a planet of 0.6 Earth masses, with true longitude of 133° (1990.5), semi major axis a = 44 AU and eccentricity e = 0.007.

Effect of the New Equinox Definition on the Zero-Point of Longitude of the Indian Calendar

Indian calendars follow a sidereal system of astronomy taking a fixed initial point on the ecliptic as the origin from which the longitudes are measured. Its position for the official Indian Calendar has been defined by the Calendar Reform Committee (1955) as the point on the ecliptic whose true tropical longitude was 23°15′00″ as on 21 March 1956, 0h UT. Its position was determined upto the year 1984 in accordance with Newcomb's value for general precession using the relation where T is in centuries of 36525 ephemeris days from 1900 January 0.5 ET. Recent changes in the location and the motion of the equinox with reference to the epoch J2000.0 have necessitated corresponding changes to be included in the determination of the mean and true positions of the above initial point. The new algorithm worked out is where T is in Julian centuries of 36525 days from J2000.0.

Optimum low thrust transfer to geosynchronous orbit

The low thrust transfer for geosynchronous mission has been studied by many investigators from the viewpoint of optimization in case of continuous thrust. This paper discusses the possibility of fuel saving to attain a geosynchronous orbit by introducing coast phases during each revolution. In advance of optimizing the whole transfer mission, optimization during a single revolution is treated, and it is shown that the entrance and the exit of optimal coasting arcs are expressed by a sixth order equation, which, in case of coplanar transfer, degenerates into a cubic equation, with respect to the cosine of true longitude. Then an optimum transfer to a geosynchronous orbit, including coast phases in each revolution, is simulated. Computational results for typical initial conditions are shown to be compared with those for all-propulsion cases.

Lunar perturbations of artificial satellites of the earth

Lunisolar perturbations of an artificial satellite for general terms of the disturbing function were derived by Kaula (1962). However, his formulas use equatorial elements for the Moon and do not give a definite algorithm for computational procedures. As Kozai (1966, 1973) noted, both inclination and node of the Moon's orbit with respect to the equator of the Earth are not simple functions of time, while the same elements with respect to the ecliptic are well approximated by a constant and a linear function of time, respectively. In the present work, we obtain the disturbing function for the Lunar perturbations using ecliptic elements for the Moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The Moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the Earth. We can then apply the results to numerical integration by using coordinates of the Moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the Moon's motion represented by a precessing and rotating elliptical orbit.

On the Greek Origin of the Indian Planetary Model Employing a Double Epicycle

The development of geometric models to explain the inequalities of planetary motion in terms of combinations of circular motions is characteristic of Greek astronomy, and derives its initial bias from statements of the Pythagoreans, Plato, and Aristotle regarding the structure of the universe and, in the case of Aristotle, the difference between the mechanics of the sublunar centre of the cosmos and that of the celestial spheres. It is hardly necessary to point out here that among the devices invented to account for these inequalities are the eccentric circle and the epicycle, whose properties were investigated by Apollonius of Perge in about 200 b.c.1 The occurrence of these devices in Sanskrit astronomical texts of the fifth century a.d. immediately suggests some Greek influence. And the supposition of such an influence is greatly strengthened by the fact that Greek adaptations of Babylonian linear astronomy2 and Greek treatises on genethlialogy3 were translated into Sanskrit between the second and fourth centuries a.d.; it is virtually confirmed by the fact that the earliest Sanskrit siddhânias to employ epicyclic models, the older Romaka (for the luminaries only4) and Paulisa (for the Sun only5), are largely of Greek or GrecoBabylonian origin. It is my intention here to investigate the Greek background of the common Indian model for the star-planets which involves two concentric epicycles.In the Paitâmahasiddhânta of the Visnudharmottarapurâria,6 which is our earliest extant exponent of the Indian double-epicycle planetary model (it was probably composed in the first half of the fifth century a.d.), the pattern was set for all later texts except for those belonging to or aware of the auduyaka System of the Âryapakca. The orbits of the planets are concentric with the centre of the Earth. The single inequalities recognized in the cases of the two luminaries are explained by manda-cpicycles (corresponding functionally to the Ptolemaic eccentricity of the Sun and lunar epicycle respectively), the two inequalities recognized in the case of each of the five star-planets by a mandaepicycle (corresponding to the Ptolemaic eccentricity) and a .%Arn-epicycle (corresponding to the Ptolemaic epicycle). The further refinements of the Ptolemaic models are unknown to the Indian astronomers.If one tries to imagine the geometric model utilized by the Paitâmahasiddhânta, one immediately realizes that it cannot be cinematic ; the planet cannot ride simultaneously on the circumferences of two epicycles. These two epicycles must be regarded simply as devices for calculating the amounts of the equations by which the mean planet on its concentric orbit is displaced to its true position. This interpretation is confirmed by the explanation offered in early texts of the mechanics of the unequal motions of the planets : demons stationed at the manda and ftghra points on their respective epicycles pull at the planets with chords of wind.7 The computation of the total effect of these two independent forces upon the mean planet varies somewhat from one school (pak$a) of astronomers to another, or even from astronomer to astronomer within a paksa. But the fundamental concept remains clear: the planet is always situated on the ciicumference of a deferent circle concentric with' the centre of the Earth, while two epicycles (one each for the Sun and Moon) revolve about it. As the planet progresses with its mean velocity about this deferent circle, at each instant it is pulled by the two epicycles away from its mean to its true longitude. These instantaneous true longitudes are subject to computation, but a true course of the planet over a period of time can only be conceived of as a series of such instantaneous true longitudes.This is not to deny that the equivalence of an eccentric to an epicyclic model had also been transmitted from Greece to India, though it would seem that the transmission was effected by a different text from that from which the doubleepicycje theory is derived. …

Late Medieval Two-Argument Tables for Planetary Longitudes

Previous articleNext article No AccessLate Medieval Two-Argument Tables for Planetary LongitudesMark J. TichenorMark J. Tichenor Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Journal of Near Eastern Studies Volume 26, Number 2Apr., 1967 Article DOIhttps://doi.org/10.1086/371899 Views: 3Total views on this site Citations: 10Citations are reported from Crossref Copyright 1967 The University of ChicagoPDF download Crossref reports the following articles citing this article:Petra G. Schmidl Kāshī: Ghiyāth (al-Milla wa-) al-Dīn Jamshīd ibn Masҁūd ibn Maḥmūd al-Kāshī [al-Kāshānī], (Sep 2014): 1161–1164.https://doi.org/10.1007/978-1-4419-9917-7_743J. Lennart Berggren History of Mathematics in the Islamic World: The Present State of the Art [1985], (Sep 2013): 51–71.https://doi.org/10.1007/978-3-642-36736-6_4Petra Heijden, Ian T. Durham, İhsan Fazlıoğlu, Narahari Achar, Jürgen Hamel, Adriaan Blaauw, Petra G. Schmidl, Ian T. Durham, Derek Jensen, Glenn A. Walsh, Mary Woods Scott, Jordan D. Marché, Adam Jared Apt, Woodruff T. Sullivan, Gillian Knapp, Setsuro Ikeyama, Glen M. Cooper, Alexander A. Gurshtein, Meltem Akbas, David A. King, Y. Tzvi Langermann, Behnaz Hashemipour, Emilia Calvo, Mohammed Abattouy, Glen Van Brummelen, Sonja Brentjes, Nicholas Campion, Marvin Bolt, Naoshi Fukushima, Glen M. Cooper, Richard A. Jarrell, Roland Wielen, Roland Wielen, Roland Wielen, Roland Wielen, Joseph F. MacDonnell, Paul Charbonneau, Frank K. Edmondson, Thomas A. Dobbins, Virginia Trimble, Christof A. Plicht, Richard A. Jarrell, Katherine Bracher, Christof A. Plicht, Virginia Trimble, Thomas Hockey, Jordan D. Marché, Victor K. Abalakin, László Szabados, Jiȓí Grygar, Horst Kant, Thomas A. Dobbins, Virginia Trimble, Mihkel Joeveer, Thomas A. Dobbins, William Sheehan, Daniel Kolak, Fathi Habashi, Hartmut Frommert, Thomas A. Dobbins, Steven J. Dick, Thomas R. Williams, Len Berggren, Daniel W. E. Green, Ursula B. Marvin, Brian Luzum Kāshī: Ghiyāth (al‐Milla wa‐) al‐Dīn Jamshīd ibn Masҁūd ibn Maḥmūd al‐Kāshī [al‐Kāshānī], (Jan 2007): 613–615.https://doi.org/10.1007/978-0-387-30400-7_743Glen Van Brummelen Taking Latitude with Ptolemy: al- Novel Geometric Model of the Motions of the Inferior Planets, Archive for History of Exact Sciences 60, no.44 (Apr 2006): 353–377.https://doi.org/10.1007/s00407-006-0110-zGlen Van Brummelen Mathematical Methods in the Tables of Planetary Motion in Kūshyār ibn Labbān'sJāmicZı̄j, Historia Mathematica 25, no.33 (Aug 1998): 265–280.https://doi.org/10.1006/hmat.1997.2184JULIO SAMSÓ Al-Zarq?l, Alfonso X and Peter of Aragon on the Solar Equation, Annals of the New York Academy of Sciences 500, no.1 From Deferent1 From Deferent (Jun 1987): 467–476.https://doi.org/10.1111/j.1749-6632.1987.tb37218.xJ. L. Berggren History of Mathematics in the Islamic World: The Present State of the Art, Middle East Studies Association Bulletin 19, no.11 (Mar 2016): 9–33.https://doi.org/10.1017/S0026318400014796George A. Saliba The Double-Argument Lunar Tables of Cyriacus, Journal for the History of Astronomy 7, no.11 (Jul 2016): 41–46.https://doi.org/10.1177/002182867600700104DAVID A. KING A Double-Argument Table for the Lunar Equation Attributed to Ibn Y?nus, Centaurus 18, no.22 (Jun 1974): 129–146.https://doi.org/10.1111/j.1600-0498.1974.tb00561.xClaus Jensen The lunar theories of al-Baghd?d?, Archive for History of Exact Sciences 8, no.44 (Jan 1972): 321–328.https://doi.org/10.1007/BF00328436

Tidal Components of Hurricane Development

A SEEMINGLY minor but crucial discrepancy exists between Rudloff's and my distributions of the lunisolar angle on the 269 dates. I used true apparent longitudes of the Moon and Sun at Greenwich noon given by standard ephemerides, rather than cyclical mean values of shortcut-table derivation.

Tidal Components in Hurricane Development

A SEEMINGLY minor but crucial discrepancy exists between Rudloff's and my distributions of the lunisolar angle on the 269 dates. I used true apparent longitudes of the Moon and Sun at Greenwich noon given by standard ephemerides, rather than cyclical mean values of shortcut-table derivation.

The Planetary Theory of Ibn al-Shatir: Reduction of the Geometric Models to Numerical Tables

Previous articleNext article No AccessThe Planetary Theory of Ibn al-Shatir: Reduction of the Geometric Models to Numerical TablesFuad AbbudFuad Abbud Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Isis Volume 53, Number 4Dec., 1962 Publication of the History of Science Society Article DOIhttps://doi.org/10.1086/349635 Views: 4Total views on this site Citations: 9Citations are reported from Crossref Copyright 1962 History of Science Society, Inc.PDF download Crossref reports the following articles citing this article:Ole Ravn, Ole Skovsmose Mathematics in Nature, (Feb 2019): 21–35.https://doi.org/10.1007/978-3-030-01337-0_2S. Mohammad Mozaffari Holding or Breaking with Ptolemy's Generalization: Considerations about the Motion of the Planetary Apsidal Lines in Medieval Islamic Astronomy, Science in Context 30, no.11 (Apr 2017): 1–32.https://doi.org/10.1017/S0269889717000011S. Mohammad Mozaffari A forgotten solar model, Archive for History of Exact Sciences 70, no.33 (Oct 2015): 267–291.https://doi.org/10.1007/s00407-015-0167-7Humberto Antonio de Barros-Pereira Astronomia islâmica entre Ptolomeu e Copérnico: tradição Maraghah, Revista Brasileira de Ensino de Física 33, no.44 (Dec 2011).https://doi.org/10.1590/S1806-11172011000400021Sonja Brentjes Between doubts and certainties: on the place of history of science in Islamic societies within the field of history of science, NTM International Journal of History and Ethics of Natural Sciences, Technology and Medicine 11, no.22 (Jun 2003): 65–79.https://doi.org/10.1007/BF02908588Ahmad Dallal A Non-Ptolemaic Lunar Model From Fourteenth-Century Central Asia, Arabic Sciences and Philosophy 2, no.22 (Oct 2008): 237–297.https://doi.org/10.1017/S095742390000165XO. Neugebauer On the Planetary Theory of Copernicus, (Jan 1983): 491–505.https://doi.org/10.1007/978-1-4612-5559-8_40O. Neugebauer On the planetary theory of Copernicus, Vistas in Astronomy 10 (Jan 1968): 89–104.https://doi.org/10.1016/0083-6656(68)90040-8Harry Woolf, and Aleida Cattell Eighty-Eighth Critical Bibliography of the History of Science and Its Cultural Influences (To 1 January 1963), Isis 54, no.44 (Oct 2015): 515–667.https://doi.org/10.1086/349789