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Optimization of low-thrust trajectories is necessary in the design of space missions using electric propulsion systems. We consider the problem of limited power trajectory optimization, which is a well-known case of the low-thrust optimization problem. In the article, we present an indirect approach to trajectory optimization based on the use of the maximum principle and the continuation method. We introduce the concept of auxiliary longitude and use it as a new independent variable instead of time. The use of equations of motion in the equinoctial elements and a new independent variable allowed us to simplify the optimization of limited power trajectories with a fixed angular distance and free transfer duration. The article presents a new form of necessary optimality conditions for this problem and describes an efficient new numerical method to solve the limited power trajectory optimization problem. We show the existence of several trajectories with a fixed transfer duration and free angular distance that satisfy the necessary optimality conditions. Using numerical examples, we confirm the existence of the limiting values of the characteristic velocity and the product of the cost function value and the transfer duration as the angular distance increases. The high computational performance of the developed technique makes it possible to carry out and present an analysis of the angular flight range and initial true longitude impact on the cost function, transfer duration, and characteristic velocity.

We introduce six quantities that generalize the equinoctial orbital elements when some or all the perturbing forces that act on the propagated body are derived from a disturbing potential. Three of the elements define a non-osculating ellipse on the orbital plane, other two fix the orientation of the equinoctial reference frame, and the last allows one to determine the true longitude of the body. The Jacobian matrices of the transformations between the new elements and the position and velocity are explicitly given. As a possible application we investigate their use in the propagation of Earth's artificial satellites showing a remarkable improvement compared to the equinoctial orbital elements.

Early Hindu astronomers have applied four corrections to mean longitude for mean sunrise at Laṅkā to get true longitude for true sunrise at the local place.

(ProQuest: ... denotes formulae omitted.)(ProQuest: ... denotes non-USASCII text omitted.)The present paper deals with the methods proposed and the values achieved for the eccentricity and the longitude of apogee of the (apparent) orbit of the Sun in the Ptolemaic context in the Middle East during the medieval period. The main goals of this research are as follows: first, to determine the accuracy of the historical values in relation to the theoretical accuracy and/or the intrinsic limitations of the methods used; second, to investigate whether medieval astronomers were aware of the limita- tions, and if so, which alternative methods (assumed to have a higher accuracy) were then proposed; and finally, to see what was the fruit of the substitution in the sense of improving the accuracy of the values achieved.In Section 1, the Ptolemaic eccentric orbit of the Sun and its parameters are introduced. Then, its relation to the Keplerian elliptical orbit of the Earth, which will be used as a criterion for comparing the historical values, is briefly explained. In Section 2, three standard methods of measurement of the solar orbital elements in the medieval period found in the primary sources are reviewed. In Section 3, more than twenty values for the solar eccentricity and longitude of apogee from the medieval period will be classified, provided with historical comments. Discus- sion and conclusions will appear in Section 4 (in Part 2), followed there by two discussions of the medieval astronomers' considerations of the motion of the solar apogee and their diverse interpretations of the variation in the values achieved for the solar eccentricity.1. IntroductionPtolemy, in Almagest ΠΙ, presented a solar model consisting of an eccentric circle whose centre is displaced from the centre of the Earth by an amount of eccentricity e = TO expressed in terms of an arbitrary length R-60 for the radius of the eccentre (Figure 1). The Sun revolves on the eccentre around the Earth but its motion is uniform with respect to the centre of eccentre, and completes one revolution in a tropical year TY (counted in days), i.e., with a mean motion of ? = 360/7T. Due to the eccentricity of its orbit, the true longitude of the Sun (i.e., its longitude as seen from the Earth T) does not match its mean longitude (its longitude as it appears from the centre O of uniform motion) except at apogee (A) and perigee (Π). The difference between the two, called the 'equation of centre' q, is defined as a one-variable function of the solar mean anomaly c, which is the difference between its mean longitude and the longitude Aap of the apogee:Therefore, the solar eccentric model has three parameters (TY or ?, e, and Aap) but only one anomaly due to its eccentricity.Ptolemy assumes that all three parameters are constant.' Since, at least, the latter part of the ninth century, the Middle Eastern astronomers found (cf. Tables 1 and 2) that the longitude of the solar apogee increases with the passage of time and they (seemingly, first of all, the Banu Musa or Thabit b. Qurra and Habash) assumed its rate of change to be the same as the rate of precession, as is the case with the motion of the apogee of the planets in the Ptolemaic context (the relevant discussion will be presented in Section 4). Also, different values for e and TY were observed during the medieval period. The variety in the values obtained for the two was so large that, for example, Copernicus in the prologue of his De revolutionibus complains that "[the astronomers] are uncertain of the motion of the Sun and Moon to such an extent that they cannot demonstrate or observe a constant magnitude for the tropical year",2 while BirunT was forced to resolve his readers' worries about the huge differences in the values measured for the solar eccentricity over a short period (see Section 4).In fact both e and TY are changing with the passage of time, e decreases slowly by the average amount of 4. …

The growing level of autonomy of unmanned space missions has attracted a significant amount of research in the aerospace field towards feedback orbit control. Existing Lyapunov-based controllers can be used to to transfer a spacecraft between two elliptic orbits of given size and orientation, but do not consider the stabilization of the spacecraft phase angle along the orbit, which is a key requirement for application to formation flying missions. This paper presents a control law based on the orbital element parametrization, which is able to track a given true longitude (i.e. a reference phase angle), in addition to the parameters describing the reference orbit shape and orientation. A numerical simulation of an orbital rendezvous demonstrates the effectiveness of the proposed approach.

The calculation of the total solar energy received during a given interval of time over the year requires more attention than the calculation of the daily incoming solar radiation (daily insolation). It depends indeed upon whether the time interval is defined in terms of the true longitude or of the calendar date. The details of such a calculation based on elliptic integrals and on the sum of the daily irradiations are given in this paper. Numerical examples show the very high accuracy of both methods. The analytical expression and the spectral analysis of the long-term variations of such irradiation received during any time interval over the year show that they depend almost exclusively upon obliquity with a very small contribution of eccentricity, not at all upon precession. The eccentricity signal comes from the variation of the so-called solar constant through the variation of the mean distance from the Earth to the Sun. The precession signal is eliminated through the second law by Kepler. The correlation between total irradiation and obliquity reverses its sign at a specific latitude of the summer hemisphere which is a function of the selected time interval. At this latitude the obliquity signal disappears and only a pure eccentricity signal is left in the total irradiation, but with a very weak amplitude. Amplitude of the continuous wavelet transforrm shows that the amplitude variation of both the daily irradiance (mostly precession) and of the total irradiation (obliquity) vanishes around the Mid-Pleistocene Transition. (C) 2010 Elsevier Ltd. All rights reserved.

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.

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 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 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.

Two-line elements (TLE) of an artificial space object can be generated from optical observations with modest instruments. In this preliminary study, ObsReduce was utilized for astrometric analysis to produce positional observations in IOD (Interactive Orbit Determination) format and ELFIND to generate the expected TLE. It turned out that analysis of a single image captured with a point-and-shoot camera in 16 seconds exposure time in a cloudy sky was sufficient to produce a TLE of the International Space Station (ISS) with comparable results in semi-major axis, eccentricity, inclination, RAA Node, and true longitude.

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.

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.

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

The latitude of the Moon, multiplied by the equinoctial shadow and divided by 12, has to be subtracted from the true longitude of the Moon if the latitude is north at sunrise. At sunset it has to be added.

The procedure for obtaining the madhyama-graha i.e. the mean longitude of a planet from the Ahargana, was explained in the previous chapter. Two corrections, namely manda-samskāra and śīghra-samskāra, have to be applied to the madhyama-graha to obtain the sphuta-graha or the true longitude of the planet. In these two samskāras, to be described later in this chapter, two angles, namely the manda-kendra (manda anomaly or mean anomaly) and the śīghra-kendra (śīghra-anomaly or anomaly of conjunction or solar anomaly) play important roles. In the above verse, the kendras and their sines and cosines (known as bāhus and kotis) pertaining to both the samskāras are dealt with. For this, two quantities, namely the ucca and the kendra, are introduced.

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., Facultad de Ciencias Astronómicas y Geofísicas

In Chap. 5 we have studied Mayer’s lunar theory, and we have seen how Mayer had derived, from differential equations of motion and Newton’s law of gravitation, a mathematical formula expressing the true longitude of the moon in terms of its mean motion. His solution of the differential equations amounts to what we have termed a single-stepped one. In contrast, we have seen in Chap. 4 that his lunar tables implemented a multistepped procedure, using not only mean motion arguments: several tables had to be entered with arguments that had been modified in the course of calculation. Chapter 6 demonstrated that the multistep procedure is rooted in Newton’s Theory of the Moon’s Motion (NTM) and that Mayer is unlikely to have had a theoretical justification of it. Thus, a gap shows up between theory and tables.

We numerically integrated orbits around the L4 and L5 Lagrangian points of Jupiter (Trojans), and Saturn (‘Bruins’): 40 Trojans and 350 Bruins, all of inclination less than 12°. Four Bruins remained stable until the integration was stopped at 412 Myrs. Properties of these stable orbits were: (1) proper eccentricities less than 0.028; (2) librations in true longitude about L4 and L5 of more than 80°; and (3) librations of the longitude of perihelion (ϖ) relative to Saturn’s perihelion, such that they were never close when the forced eccentricity was near 0.08. Orbits with librations in true longitude less than 80° were unstable, the time to instability being correlated with libration angle. No unstable region was found around Jupiter’s Lagrangian points, and stable Trojans may have longitudes of perihelion that either circulate or librate with respect to jupiter’s.

The original publication is available at http://iopscience.iop.org/1538-3881/; International audience; Pluto and its main satellite, Charon, occulted the same star on 2008 June 22. This event was observed from Australia and La Réunion Island, providing the east and north Charon Plutocentric offset in the sky plane (J2000): X= + 12,070.5 ± 4 km (+ 546.2 ± 0.2 mas), Y= + 4,576.3 ± 24 km (+ 207.1 ± 1.1 mas) at 19:20:33.82 UT on Earth, corresponding to JD 2454640.129964 at Pluto. This yields Charon's true longitude L= 153.483 ± 0fdg071 in the satellite orbital plane (counted from the ascending node on J2000 mean equator) and orbital radius r= 19,564 ± 14 km at that time. We compare this position to that predicted by (1) the orbital solution of Tholen & Buie (the "TB97" solution), (2) the PLU017 Charon ephemeris, and (3) the solution of Tholen et al. (the "T08" solution). We conclude that (1) our result rules out solution TB97, (2) our position agrees with PLU017, with differences of ΔL= + 0.073 ± 0fdg071 in longitude, and Δr= + 0.6 ± 14 km in radius, and (3) while the difference with the T08 ephemeris amounts to only ΔL= 0.033 ± 0fdg071 in longitude, it exhibits a significant radial discrepancy of Δr= 61.3 ± 14 km. We discuss this difference in terms of a possible image scale relative error of 3.35 × 10-3in the 2002-2003 Hubble Space Telescope images upon which the T08 solution is mostly based. Rescaling the T08 Charon semi-major axis, a = 19, 570.45 km, to the TB97 value, a = 19636 km, all other orbital elements remaining the same ("T08/TB97" solution), we reconcile our position with the re-scaled solution by better than 12 km (or 0.55 mas) for Charon's position in its orbital plane, thus making T08/TB97 our preferred solution.

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.

A theoretical plasma is supposed to be contained between two perfectly conducting cylinders. A magnetic field from a hypothetical line pole is postulated because the metric coefficients have similar properties to those of a dipole field. It is found, as expected, that the asymmetric hydromagnetic eigenmodes involve coupling of torsion and compression. Nevertheless the behaviour is found to be still roughly like that of a uniform field model. The modes are found to be either roughly torsional or roughly compressional, and the roughly torsional eigenfrequencies are found to be insensitive to the eigenmode structure across the field lines. It is therefore suggested that estimates of micropulsation spectra from the axisymmetric toroidal mode in the dipole field may be fairly valid whatever the true longitude distribution of the micropulsations.

TWO RECENT PAPERS have described the models for motion worked out by Ibn al-Shatir (fl. 1350) of Damascus. These studies have also shown the close similarity between his models and those of Copernicus (fl. 1500). Both systems are made up of linkages of constantlength vectors rotating at constant angular velocity; both astronomers abandoned the Ptolemaic equant, and the lengths of corresponding vectors in the two systems are nearly equal, in many cases identical. Under the circumstances, it is desirable to compare further the work of these two scientists, and a logical next step is the examination of numerical tables in their respective writings. The true longitude of a planet can be regarded as a function of two independent variables, the mean longitude and the anomaly. An individual confronted with the problem of preparing tables for the determination of true longitudes could solve his problem by computing a value each for every combination of values in the chosen domain of both variables. But the

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. …

The paper describes a recursive formulation for the expansion of the gravitational potential valid for both the tesseral and zonal harmonics. The expansion is primarily in rectangular coordinates, but the classical orbit elements or equinoctial orbit elements can be easily substituted. The equations of motion for the zonal harmonics in both classical and equinoctial orbital elements are described in a form which will result in closed-form expressions for the first-order perturbations. In order to achieve this result, the true longitude or true anomaly have to be used as independent variables.

The author's previous theory of the artificial satellite is derived in terms of the. disturbed eccentric anomaly. The present development, in terms of the orbital true longitude, is a substantial improvement over the earlier work in that it leads to the faster convergence for large eccentricities and to a smaller number of terms in the series representing the perturbations. Moreover, each approximation of the radius vector and of the parameters determining the position of the orbit plane is obtained not in the form of a truncated infinite series but in the form of trigonometric polynomials in two arguments. These arguments are the mean true anomaly and the mean argument of the latitude. The present theory, like the previous one, permits the computation of perturbations of any desired order. Thus, any future information about earth's gravitational field can easily be included.