scholarly journals On Construction of a Long-Term Moon’s Theory

1999 ◽  
Vol 172 ◽  
pp. 415-416
Author(s):  
T.V. Ivanova
Keyword(s):  
Lunar Theory ◽  
Planetary Theory ◽  
The Moon ◽  
Planetary Orbits ◽  
Small Parameters ◽  
The One ◽  
General Planetary Theory ◽  
Laplace Type

An analytical long-term theory of the motion of the Moon is constructed within the framework of the general planetary theory (Brumberg, 1995). A method, different from the one of (Ivanova, 1997) designated below as (*), for the determination of the perturbations depending on the eccentricities and inclinations of lunar and planetary orbits is used which allows to obtain the solution of the problem in the purely trigonometric form up to any order with respect to the small parameters.The aim of this paper is to construct the long-term Lunar theory in the form consistent with the general planetary theory (Brumberg, 1995). For this purpose the Moon is considered as an additional planet in the field of eight major planets (Pluto being excluded). In the result the coordinates of the Moon may be represented by means of the power series in the evolutionary eccentric and oblique variables with trigonometric coefficients in mean longitudes of the Moon and the planets. The long-period perturbations are determined by solving a secular system in Laplace-type variables describing the secular motions of the lunar perigee and node and taking into account the secular planetary inequalities.

XV. — Researches in physical astronomy

1831 ◽  
Vol 121 ◽  
pp. 231-282
Keyword(s):  
Differential Equations ◽  
The Other ◽  
Lunar Theory ◽  
System Of Equations ◽  
Planetary Theory ◽  
The Moon ◽  
Celestial Bodies ◽  
Independent Variable ◽  
The Mean ◽  
The One

The method pursued by Clairaut in the solution of this important problem of Physical Astronomy, consists in the integration of the differential equations furnished by the principles of dynamics, upon the hypothesis that in the gravitation of the celestial bodies the force varies inversely as the square of the distance, and in which the true longitude of the moon is the independent variable ; the time is thus obtained in terms of the true longitude, and by the reversion of series the longitude is afterwards obtained in terms of the time, which is necessary for the purpose of forming astronomical tables. But while on the one hand this method possesses the advantage, that the disturbing func­tion can be developed with somewhat greater facility in terms of the true lon­gitude of the moon than in terms of the mean longitude, yet on the other hand, the differential equations in which the true longitude is the independent variable are far more complicated than those in which the time is the inde­pendent variable. The latter equations are used in the planetary theory ; so that the method of Clairaut has the additional inconvenience, that while the lunar theory is a particular case of the problem of the three bodies, one system of equations is used in this case, and another in the case of the planets. The method of Clairaut has been adopted, however, by Mayer, by Laplace, and by M. Damoiseau. The last-mentioned author has arranged his results with remarkable clearness, so that any part of his processes may be easily verified by any one who does not shrink from this gigantic undertaking; and the immense labour which this method requires, when all sensible quantities are retained, may be seen in his invaluable memoir. Mr. Brice Bronwin has recently communicated to the Society a lunar theory, in which the same method is adopted.

Stochastic Models for Actuarial Use: The Equilibrium Modelling of Local Markets

2009 ◽  
Vol 39 (1) ◽  
pp. 339-370 ◽  
Author(s):  
Robert J. Thomson ◽  
Dmitri V. Gott
Keyword(s):  
Local Market ◽  
Market Participants ◽  
Market Portfolio ◽  
One Year ◽  
The One ◽  
Mean Variance ◽  
Annual Time ◽  
Time Subject

AbstractIn this paper, a long-term equilibrium model of a local market is developed. Subject to minor qualifications, the model is arbitrage-free. The variables modelled are the prices of risk-free zero-coupon bonds – both index-linked and conventional – and of equities, as well as the inflation rate. The model is developed in discrete (nominally annual) time, but allowance is made for processes in continuous time subject to continuous rebalancing. It is based on a model of the market portfolio comprising all the above-mentioned asset categories. The risk-free asset is taken to be the one-year index-linked bond. It is assumed that, conditionally upon information at the beginning of a year, market participants have homogeneous expectations with regard to the forthcoming year and make their decisions in mean-variance space. For the purposes of illustration, a descriptive version of the model is developed with reference to UK data. The parameters produced by that process may be used to inform the determination of those required for the use of the model as a predictive model. Illustrative results of simulations of the model are given.

Robert E. Horton Memorial Lecture

1978 ◽  
Vol 59 (3) ◽  
pp. 258-266 ◽  
Author(s):  
J. P. Bruce
Keyword(s):  
Water Quality Management ◽  
Water System ◽  
Memorial Lecture ◽  
Project Design ◽  
Drought Frequency ◽  
Upper Great Lakes ◽  
Water Project ◽  
The One

The field of hydrometeorology has always been a difficult one to define. For the present purposes, the term will be used to include those fields in which meteorologists and water specialists must interact closely to solve problems. Traditionally, these interactions have been on topics such as flood forecasting and determination of storm, flood, and drought frequency for water project design. Some of the most important needs of water managers for meteorological information and advice are emphasized here. Internationally, a growing recognition of the importance of hydrometeorology is attested to by the work programs of the World Meteorological Organization and of the U.N. Water Conference of March 1977. Although flood and drought topics remain important in hydrometeorology, there are two subjects whose significance has gained recognition in recent years. One is the need for much greater collaboration on long-term climatic change between meteorologists and climatologists, on the one hand, and water scientists working in hydrology, glaciology, and lake sediments, on the other. Water planners are increasingly anxious to incorporate better estimates and predictions of longer-term climatic probabilities in project design. The second subject of increasing concern is the transport of contaminants and nutrients to water systems through the atmosphere. In some basins, such as the Upper Great Lakes, atmospheric sources of some contaminants and nutrients dominate the chemical budgets of the water system. Meteorological knowledge of cycling and transport of such substances will be an essential key to future water quality management programs.

scholarly journals High Order Resonances in the Evolution of the Lunar Orbit

1983 ◽  
Vol 74 ◽  
pp. 3-17
Author(s):  
J. Kovalevsky
Keyword(s):  
Major Axis ◽  
Lunar Theory ◽  
The Moon ◽  
Tidal Effects ◽  
Semi Major Axis ◽  
Natural Satellite ◽  
Variation Of Constants ◽  
The Mean ◽  
Mean Elements

AbstractThis paper deals with the long term evolution of the motion of the Moon or any other natural satellite under the combined influence of gravitational forces (lunar theory) and the tidal effects. We study the equations that are left when all the periodic non-resonant terms are eliminated. They describe the evolution of the-mean elements of the Moon. Only the equations involving the variation of the semi-major axis are considered here. Simplified equations, preserving the Hamiltonian form of the lunar theory are first considered and solved. It is shown that librations exist only for those terms which have a coefficient in the lunar theory larger than a quantity A which is function of the magnitude of the tidal effects. The solution of the general case can be derived from a Hamiltonian solution by a method of variation of constants. The crossing of a libration region causes a retardation in the increase of the semi-major axis. These results are confirmed by numerical integration and orders of magnitude of this retardation are given.

scholarly journals XVI. Researches in physical astronomy

1831 ◽  
Vol 121 ◽  
pp. 283-298
Keyword(s):  
Analytical Form ◽  
Numerical Results ◽  
Lunar Theory ◽  
Great Care ◽  
Planetary Theory ◽  
Disturbing Force

I Propose in this paper to extend the equations I have already given for determining the planetary inequalities, as far as the terms depending on the squares and products of the eccentricities, to the terms depending on the cubes of the eccentricities and quantities of that order, which is done very easily by a Table similar to Table II. in my Lunar Theory; and particularly to the determination of the great inequality of Jupiter, or at least such part of it as depends on the first power of the disturbing force. That part which depends on the square of the disturbing force may I think be most easily calculated by the methods given in my Lunar Theory; but not without great care and attention can accurate numerical results be expected. I have how­-ever given the analytical form of the coefficients of the arguments in the development of R, upon which that inequality principally depends. It is I think particularly convenient to designate the arguments of the planetary disturbances by indices. The system of indices adopted in this paper is given as appearing better adapted for the purpose than that used in my former paper on the Planetary Theory; but it is not advisable to make use of the same indices in this as in the Lunar Theory.

scholarly journals Researches in physical astronomy

1837 ◽  
Vol 3 ◽  
pp. 128-129
Keyword(s):  
Differential Equations ◽  
The Other ◽  
Disturbing Function ◽  
Direct Integration ◽  
Planetary Theory ◽  
The Moon ◽  
Disturbing Force ◽  
Other Hand ◽  
The Stability ◽  
The One

The present paper contains some further developments of the theory of the moon, which are given at length, in order to save the trouble of the calculator, and to avoid the danger of mistake. The author remarks, that while it seems desirable, on the one hand, to introduce into the science of physical astronomy a greater degree of uniformity, by bringing to perfection a theory of the moon founded on the integration of the equations employed in the planetary theory, it is also no less important, on the other hand, to complete, in the latter, the method hitherto applied solely to the periodic inequalities. Hi­therto those terms in the disturbing function which give rise to the secular inequalities, have been detached, and the stability of the system has been inferred by means of the integration of certain equations, which are linear when the higher powers of the eccentri­cities are neglected and from considerations founded on the varia­tion of the elliptic constants. But the author thinks that the stability of the system may be inferred also from the expressions which result at once from the direct integration of the differential equations. The theory, he states, may be extended, without any analytical difficulty, to any power of the disturbing force, or of the eccentricities, ad­mitting the convergence of the series; nor does it seem to be limited by the circumstance of the planet’s moving in the same direction.

scholarly journals Researches in physical astronomy

1837 ◽  
Vol 3 ◽  
pp. 51-52
Keyword(s):  
Differential Equations ◽  
Great Distance ◽  
Original Position ◽  
Lunar Theory ◽  
Great Length ◽  
The Moon ◽  
The Earth ◽  
Method Of Solution ◽  
Independent Variable ◽  
The One

The first part of this paper relates to the theory of the moon. The method of solution pursued by Clairaut consisted in the inte­gration of differential equations, in which the true longitude of the moon is the independent variable: the time is then obtained in terms of the true longitude; and by the reversion of series, the lon­gitude afterwards obtained in terms of the time. This method is the one adopted by Mayer, Laplace, and Damoiseau. The au­thor has been led, by reflecting on the difficulties of this problem, to believe that the integration of the differential equations in which the time is the independent variable would be at least as easy as the former process; and it would possess the advantage of employing the same system of equations for the moon as for the planets. The lunar theory proposed by the author, and developed in this paper, is an extension of the equations given in his former Researches in Physical Astronomy, already published in the Philosophical Trans­actions; by including those terms, which, in consequence of the great eccentricity of the moon’s orbit, are sensible; and by sup­pressing those which are insensible from the great distance of the sun, the disturbing body. He has not yet attempted to obtain numerical results, but proposes at some future time to engage in their computation. In the second part of the paper, he investigates the precession of the equinoxes, on the supposition that the earth revolves in a re­sisting medium; an investigation which may also be considered as a sequel to the author’s last paper on Physical Astronomy. The effects of the resistance of such a medium is to increase the latitude of the axis of rotation (reckoned from the equator of the figure) till it reaches 90°. Such is now the condition of the axis of the earth: but as the chances are infinitely great against this having been its original position, may not its attainment of this position be ascribed to the resistance of a medium of small density acting for a great length of time, —a supposition which may account for many geological indications of changes having taken place in the climates of the earth ? The operation of such a cause would be also sen­sible in the case of comets: and the accuracy with which the ec­centricity of the Halleian comet of 1759 is known, would appear to afford a favourable opportunity of verifying this hypothesis.

Researches in physical astronomy

1837 ◽  
Vol 3 ◽  
pp. 59-60
Keyword(s):  
Lunar Theory ◽  
Disturbing Force ◽  
The One

The author extends, in the present paper, the equations he has already given for determining the planetary inequalities, as far as the terms depending on the squares and products of the eccentricities, to the terms depending on the cubes of the eccentricities and quantities of that order, which he does by means of a table, similar to the one given in his lunar theory; and applies them particularly to the determination of the great inequality of Jupiter, or at least such part of it as depends on the first power of the disturbing force. That part which depends on the square of the disturbing force may, he thinks, be most easily calculated by the methods given in his lunar theory.

scholarly journals New Approach to the Earth’s Rotation Problem Consistent with the General Planetary Theory

1997 ◽  
Vol 165 ◽  
pp. 301-306 ◽  
Author(s):  
V.A. Brumberg ◽  
T. V. Ivanova
Keyword(s):  
Earth’S Rotation ◽  
Planetary Theory ◽  
The Moon ◽  
Euler Parameters ◽  
Earth's Rotation ◽  
New Approach ◽  
Poisson Equations ◽  
Translatory Motion ◽  
Lunar Evolution ◽  
General Planetary Theory

AbstractThe equations of the translatory motion of the major planets and the Moon and the Poisson equations of the Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and lunar orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and lunar evolution).

scholarly journals XVII. Researches in physical astronomy

1832 ◽  
Vol 122 ◽  
pp. 361-381
Keyword(s):  
Disturbing Function ◽  
Planetary Theory ◽  
The Moon ◽  
The Stability ◽  
The One

I subjoin some further developments in the Theory of the Moon, which I have thought it advisable to give at length, in order to save the trouble of the calculator and to avoid the danger of mistake, although they may be obtained with great readiness and facility by means of the Table which I have given for the purpose. While on the one hand it seems desirable to introduce into the science of Physical Astronomy a greater degree of uniformity, by bringing to perfection a Theory of the Moon, founded on the integration of the equations which are used in the planetary theory, it seems also no less important to complete in the latter the method hitherto applied solely to the periodic inequalities. Hitherto those terms in the disturbing function which give rise to the secular inequalities have been detached, and the stability of the system has been inferred by means of the integration of certain equations, which are linear when the higher powers of the eccentricities are neglected, and from considerations founded on the variation of the elliptic constants.

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