The literal solution of the main problem of the lunar theory

Eckert, W. J.; Eckert, D.

doi:10.1086/110409

Cited by 8 publications

(5 citation statements)

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“…The idea is simple: the best available solution is substituted into the relevant equations of motion, and the necessary corrections are determined by varying the coefficients in the expansion of the solution. Eckert and Smith (1966) started from Brown's solution and achieved residuals in the 13th to 15th decimals by solving 10 000 equations of variation. Although the best computers of the time were run for several hundred hours, the report of the results still creates the impression of a tremendous effort in manual labor.…”

Section: F New Analytical Solutions For the Main Problem Of Lunar Thmentioning

confidence: 99%

“…Various modern methods for solving the main problem of lunar theory were described in the preceding section. Whereas SALE and ELE are at least partially analytic, ELP aims directly at finding the Fourier expansion of the lunar trajectory with purely numerical coefficients, not unlike Airy's method (see Eckert and Smith, 1966). All three calculations eventually yield the expansions for the polar coordinates of the Moon: longitude, latitude, and sine parallax.…”

Section: G Extent and Accuracy Of The Analytical Solutionsmentioning

confidence: 99%

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Moon-Earth-Sun: The oldest three-body problem

1998

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The daily motion of the Moon through the sky has many unusual features that a careful observer can discover without the help of instruments. The three different frequencies for the three degrees of freedom have been known very accurately for 3000 years, and the geometric explanation of the Greek astronomers was basically correct. Whereas Kepler's laws are sufficient for describing the motion of the planets around the Sun, even the most obvious facts about the lunar motion cannot be understood without the gravitational attraction of both the Earth and the Sun. Newton discussed this problem at great length, and with mixed success; it was the only testing ground for his Universal Gravitation. This background for today's many-body theory is discussed in some detail because all the guiding principles for our understanding can be traced to the earliest developments of astronomy. They are the oldest results of scientific inquiry, and they were the first ones to be confirmed by the great physicist-mathematicians of the 18th century. By a variety of methods, Laplace was able to claim complete agreement of celestial mechanics with the astronomical observations. Lagrange initiated a new trend wherein the mathematical problems of mechanics could all be solved by the same uniform process; canonical transformations eventually won the field. They were used for the first time on a large scale by Delaunay to find the ultimate solution of the lunar problem by perturbing the solution of the two-body Earth-Moon problem. Hill then treated the lunar trajectory as a displacement from a periodic orbit that is an exact solution of a restricted three-body problem. Newton's difficultly in explaining the motion of the lunar perigee was finally resolved, and the Moon's orbit was computed by a new method that became the universal standard until after WW II. Poincaré opened the 20th century with his analysis of trajectories in phase space, his insistence on investigating periodic orbits even in ergodic systems, and his critique of perturbation theory, particularly in the case of the Moon's motion. Space exploration, astrophysics, and the landing of the astronauts on the Moon led to a new flowering of celestial mechanics. Lunar theory now has to confront many new data beyond the simple three-body problem in order to improve its accuracy below the precision of 1 arcsecond; the computer dominates all the theoretical advances. This review is intended as a case study of the many stages that characterize the slow development of a problem in physics from simple observations through many forms of explanation to a high-precision fit with the data. [S0034-6861(98)00802-2] CONTENTS I. Introduction 590 A. The Moon as the first object of pure science 590 B. Plan of this review 591 II. Coordinates in the Sky 591 A. The geometry of the solar system 591 B. Azimuth and altitude-declination and hour angle 592 C. Right ascension-longitude and latitude 592 D. The vernal equinox 593 E. All kinds of corrections 593 F. The measurement of time 593 G. The Earth's rota...

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Section: F New Analytical Solutions For the Main Problem Of Lunar Thmentioning

confidence: 99%

Section: G Extent and Accuracy Of The Analytical Solutionsmentioning

confidence: 99%

Moon-Earth-Sun: The oldest three-body problem

1998

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“…We have defined r =r 3 . Now we take the limit m 3 → 0 in the expression (7); this means that the primary and the secondary are sent at an infinite distance, and their total mass becomes infinite. After some computations the limiting Hamiltonian becomes…”

Section: The Limit Case and The Equations Of Motionmentioning

confidence: 99%

“…First of all, is closely connected to the classical lunar Hill problem, introduced in [14], and subsequently studied in many papers, e.g. [7,18,11,29,13]. The spatial version of the problem has been studied in, e.g., [12,2,20,10].…”

Section: Introductionmentioning

confidence: 99%

Hill’s approximation in a restricted four-body problem

Burgos-Garcia

Gidea²

2015

Celest Mech Dyn Astr

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We consider a restricted four-body problem on the dynamics of a massless particle under the gravitational force produced by three mass points forming an equilateral triangle configuration. We assume that the mass m 3 of one primary is very small compared with the other two, m 1 and m 2 , and we study the Hamiltonian system describing the motion of the massless particle in a neighborhood of m 3 . In a similar way to Hill's approximation of the lunar problem, we perform a symplectic scaling, sending the two massive bodies to infinity, expanding the potential as a power series in m 1/3 3 , and taking the limit case when m 3 → 0. We show that the limiting Hamiltonian inherits dynamical features from both the restricted three-body problem and the restricted four-body problem. In particular, it extends the classical lunar Hill problem. We investigate the geometry of the Poincaré sections, direct and retrograde periodic orbits about m 3 , libration points, periodic orbits near libration points, their stable and unstable manifolds, and the corresponding homoclinic intersections. The motivation for this model is the study of the motion of a satellite near a jovian Trojan asteroid.

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“…For the moon, residuals in the radar tracking data [Smith etal, 1968;Mulholland and Devine, 1968] and radio tracking of the lunar orbiters [Cary and Sjogren, 1968] revealed errors in the existing lunar ephemeris of 200 to 500 meters in geocentric distance and position along the orbit. Barton [1967] had already experi mented with improving the ephemeris through a com puter-aided literal development of the lunar theory, and Eckert and Eckert [1967] had extended the Hill-Brown theory to higher order. However, no purely theoretical development has yet approached the accuracy available from radar and radio tracking, and a special perturbation theory employing numerical techniques seems appro priate for improving the lunar ephemeris [Mulholland snd Devine, 1968;Holdridge et al, 1969].…”

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confidence: 99%

The mass distributions of meteoroids and asteroids

Jones¹

1968

Can. J. Phys.

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A theory of the mass distribution of interplanetary particles has been developed and it has been shown that, provided the solid debris resulting from the erosion due to mutual collisions is not important, then the asymptotic mass distribution is a power law. The predictions of this theory are compared with observations, and very good agreement is found in the case of meteor streams and asteroids > 2km in diameter. For sporadic meteoroids it was found that a second process was needed to explain the observed mass distribution, and the condensation of some components of the interplanetary gas is shown to be consistent with observation.

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The literal solution of the main problem of the lunar theory

Cited by 8 publications

References 0 publications

Moon-Earth-Sun: The oldest three-body problem

Moon-Earth-Sun: The oldest three-body problem

Hill’s approximation in a restricted four-body problem

The mass distributions of meteoroids and asteroids

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