I. Dynamical considerations - Dynamics of the Moon

We know the mass of the Moon very well from the amount it pulls the Earth about in the course of a month; this is measured by the resulting apparent displacements of an asteroid when it is near us. Combining this with the radius shows that the mean density is close to 3.33 g/cm 3 . The velocities of earthquake waves at depths of 30 km or so are too high for common surface rocks but agree with dunite, a rock composed mainly of olivine (Mg, Fe II ) 2 SiO 4 . This has a density of about 3.27 at ordinary pressures. The velocities increase with depth, the rate of increase being apparently a maximum at depth about 0.055 R in Europe and 0.075 R in Japan. It appeared at one time that there was a discontinuity in the velocities at that depth, corresponding to a transition of olivine from a rhombic to a cubic form under pressure. It now seems that the transition, though rapid, is continuous, presumably owing to impurities, but the main point is that the facts are explained by a change of state, and that the pressure at the relevant depth is reached nowhere in the Moon, on account of its smaller size. There will, however, be some compression, and we can work out how much it would be if the Moon is made of a single material. It turns out that the actual mean density of the Moon would be matched if the density at atmospheric pressure is 3.27—just agreeing with the specimen of dunite originally used for comparison. The density at the centre would be 3.41. Thus for most purposes the Moon can be treated as of uniform density. With a few small corrections the ratio 3 C /2 Ma 2 would be 0.5956 ± 0.0010, as against 0.6 for a homogeneous body. To make it appreciably less would require a much greater thickness of lighter surface rocks than in the Earth.

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The motion of a lunar orbiter

Symposium - International Astronomical Union ◽

10.1017/s0074180900105686 ◽

1966 ◽

Vol 25 ◽

pp. 373

Author(s):

Y. Kozai

Keyword(s):

Degrees Of Freedom ◽

Dimensional Space ◽

Artificial Satellite ◽

Equations Of Motion ◽

Triaxial Ellipsoid ◽

The Moon ◽

Secular Motion ◽

The Earth ◽

Close Satellite ◽

The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

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The mean distance to the Moon as determined by radar

Symposium - International Astronomical Union ◽

10.1017/s0074180900104826 ◽

1965 ◽

Vol 21 ◽

pp. 81-93 ◽

Cited By ~ 6

Author(s):

B. S. Yaplee ◽

S. H. Knowles ◽

A. Shapiro ◽

K. J. Craig ◽

D. Brouwer

Keyword(s):

The Moon ◽

The Earth ◽

Lunar Topography ◽

Radar Measurements ◽

The Mean

The results of 1959-1960 radar measurements of the distance of the Moon are given. The method of reduction of the data is described The possible effects of lunar topography and errors of other origins are discussed, as well as the effects of different constants such as the radii of the Earth and of the Moon.

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XXXII. On the precession of the equinoxes produced by the sun's attraction

Philosophical Transactions of the Royal Society of London ◽

10.1098/rstl.1779.0033 ◽

1779 ◽

Vol 69 ◽

pp. 505-526

Keyword(s):

Uniform Density ◽

The Sun ◽

The Moon ◽

The Earth ◽

Different Parts

If the actions of the Sun and the Moon upon the different parts of the earth were equal; of if the earth itself were perfectly spherical, and of an uniform density from the center to the surface; in either case the attractions of those remote bodies would have no effect on the position of the terrestrial equator, and the equinoctial points would constantly be the same in the heavens.

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III.—On the Rate of Increase of Underground Temperature

Geological Magazine ◽

10.1017/s0016756800179907 ◽

1901 ◽

Vol 8 (11) ◽

pp. 502-504

Author(s):

W. J. Sollas

Keyword(s):

British Association ◽

British Isles ◽

The Earth ◽

Rate Of Increase ◽

Rate Of Escape ◽

The Mean

In the 22nd Report of the Committee appointed to investigate the rate of increase of underground temperature, read this year before the British Association in Glasgow, some remarks previously made by me are animadverted upon; and as the Secretary, Professor Everett, has invited me to discuss the matter with him, I take the opportunity of entering somewhat more fully into the question of conductivity than has hitherto seemed necessary. We read in the Report “… . in view of the fact that the President of Section C last year characterised the variation in the British Isles ‘from 1° in 34 feet to 1° in 92 feet’ as ‘a surprising divergence of extremes from the mean,’ it is well to emphasise the connection between gradient and conductivity. If there is anything like uniformity in the annual escape of heat from the earth at different places, there must necessarily be large differences in geothermic gradients, since the rate of escape is jointly proportional to the gradient and the conductivity.”

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On The lunar Secular Acceleration: A Possible Approach

Symposium - International Astronomical Union ◽

10.1017/s0074180900086836 ◽

1990 ◽

Vol 141 ◽

pp. 201-202

Author(s):

V. Protitch-Benishek

Keyword(s):

Celestial Mechanics ◽

Numerical Evaluation ◽

Quadratic Term ◽

The Moon ◽

Moon System ◽

Secular Acceleration ◽

Mean Motion ◽

The Earth ◽

Exact Agreement ◽

The Mean

The secular quadratic term in the expression of the Moon's longitude has been introduced empirically after the conclusion that its mean motion is not constant (Halley, 1695).But, the explanation of this term and also of its numerical evaluation presented and still presents in our time great difficulties. All efforts, namely, to obtain an exact agreement between observed and theoretical value of Moon's secular acceleration were unsuccessful: the first of these two values exceeds always the second one by a very large amount. This discordance and unexplained residuals (O – C) in the mean longitude of the Moon gave rise finally to the statement that these are due to a retardation and irregularity in the Earth's rotation. But, after hardly a fifty years, this hypothesis revealed even more new difficulties and questions concerning also the problem of stability of the Earth-Moon system. It seems that there is a true reason for which this problem occurs as one of the unsolved problems of Celestial Mechanics (Brumberg and Kovalevsky, 1986; Seidelmann, 1986).

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A first analysis of the mean motion of CHAMP

Advances in Geosciences ◽

10.5194/adgeo-1-95-2003 ◽

2003 ◽

Vol 1 ◽

pp. 95-101

Author(s):

F. Deleflie ◽

P. Exertier ◽

P. Berio ◽

G. Metris ◽

O. Laurain ◽

...

Keyword(s):

Orbital Motion ◽

Surface Forces ◽

The Moon ◽

Orbital Elements ◽

Champ Satellite ◽

Mean Motion ◽

The Earth ◽

The Mean ◽

Low Altitude

Abstract. The present study consists in studying the mean orbital motion of the CHAMP satellite, through a single long arc on a period of time of 200 days in 2001. We actually investigate the sensibility of its mean motion to its accelerometric data, as measures of the surface forces, over that period. In order to accurately determine the mean motion of CHAMP, we use “observed" mean orbital elements computed, by filtering, from 1-day GPS orbits. On the other hand, we use a semi-analytical model to compute the arc. It consists in numerically integrating the effects of the mean potentials (due to the Earth and the Moon and Sun), and the effects of mean surfaces forces acting on the satellite. These later are, in case of CHAMP, provided by an averaging of the Gauss system of equations. Results of the fit of the long arc give a relative sensibility of about 10-3, although our gravitational mean model is not well suited to describe very low altitude orbits. This technique, which is purely dynamical, enables us to control the decreasing of the trajectory altitude, as a possibility to validate accelerometric data on a long term basis.Key words. Mean orbital motion, accelerometric data

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1. The Mean Pressure of the Atmosphere over the Globe for the Months and for the Year. Part I.—January, July, and the Year

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600045971 ◽

1869 ◽

Vol 6 ◽

pp. 303-307 ◽

Cited By ~ 2

Author(s):

Alexander Buchan

Keyword(s):

North America ◽

The Netherlands ◽

South America ◽

Atmospheric Pressure ◽

Russian Empire ◽

Antarctic Ocean ◽

The Earth ◽

Mean Pressure ◽

The Mean ◽

The Russian Empire

The three charts which were exhibited, showing, by isobarometric lines, the mean atmospheric pressure over the globe, during January, July, and the year, were constructed from observations made at 358 places thus distributed over the earth,—167 in Europe; 51 in Asia; 22 in Africa and adjoining islands; 35 in South America, West Indian Islands, and Atlantic; 63 in North America; and 20 in Australasia and Antarctic Ocean. Of the European stations, 12 are in Scotland, 14 in England, 27 in Austria, 12 in Italy, 10 in France, 10 in the Netherlands, 9 in Norway, and 57 in the Russian empire, &c. The list might have been largely increased; thus a larger number might have been given from the 80 Scottish stations; but the 12 given were judged sufficient to represent the mean atmospheric pressure of this country.

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On the Absolute Orientation of the Selenodetic Reference Frame

International Astronomical Union Colloquium ◽

10.1017/s0252921100002700 ◽

1981 ◽

Vol 63 ◽

pp. 281-286

Author(s):

V. S. Kislyuk

Keyword(s):

Coordinate System ◽

Center Of Mass ◽

The Moon ◽

Coordinate Systems ◽

Inertial Coordinate System ◽

Reference Coordinate System ◽

The Earth ◽

Principal Axes ◽

The Mean ◽

Selection Of

The selection of selenodetic reference coordinate system is an important problem in astronomy and selenodesy. For the purposes of reduction of observations, planning and executing space missions to the Moon, it is necessary, in any case, to know the orientation of the adopted selenodetic reference system in respect to the inertial coordinate system.Let us introduce the following coordinate systems: C(ξc, ηc, ζc), the Cassini system which is defined by the Cassini laws of the Moon rotation;D(ξd, ηd, ζd), the dynamical coordinate system, whose axes coincide with the principal axes of inertia of the Moon;Q(ξq, ηq, ζq), the quasi-dynamical coordinate system connected with the mean direction to the Earth, which is shifted by 254" West and 75" North from the longest axis of the dynamical system (Williams et al., 1973);S(ξs, ηs, ζs), the selenodetic coordinate system, which is practically realized by the positions of the points on the Moon surface given in Catalogues;I(X,Y,Z), the space-fixed (inertial) coordinate system. All the systems are selenocentric with the exception of S(ξs, ηs, ζs On the whole, the origin of this system does not coincide with the center of mass of the Moon.

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Time Scales for Theory and Practice

Highlights of Astronomy ◽

10.1017/s1539299600008868 ◽

1992 ◽

Vol 9 ◽

pp. 141-149

Author(s):

Gernot M. R. Winkler

Keyword(s):

Theory And Practice ◽

The Sun ◽

The Moon ◽

Vernal Equinox ◽

Perceived Needs ◽

The Earth ◽

Cyclical Process ◽

The Mean ◽

Complete Revolution ◽

Early Human

Very early human experience has suggested a practical definition for the measurement of time: We define a unit of time by defining a standard (cyclical) process. Whenever this process completes its cycle identically, a unit of time has elapsed. This is the origin for the various measures of time in classical astronomy. Nature suggests strongly that we use as such standard processes the year (defined as a complete revolution of the earth around the Sun), the month (the completion of a revolution of the moon around the earth), and the day which again can be measured in several different ways. While the sidereal day is measured by a rotation in respect to the vernal equinox, the mean solar day is measured in respect to the mean. Sun. More recently, we have distinguished many more different ways of defining measures of time, partly in response to perceived needs of the applications, but in part also from purely aesthetic principles.

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Influence of some resonances on the inclination of a satellite

10.5194/epsc2020-440 ◽

2020 ◽

Author(s):

Timothée Vaillant ◽

Alexandre C. M. Correia

Keyword(s):

Major Axis ◽

The Moon ◽

Semi Major Axis ◽

Tidal Dissipation ◽

The Earth ◽

Hamiltonian Model ◽

The Future ◽

The Mean ◽

Over Time ◽

Probability Of Capture

Knowing if the inclination of a satellite with respect to the equator of its planet is primordial can give hints on its origin and its formation. However, several mechanisms are able to modify its inclination during its evolution. The orbit of a satellite evolves over time and because of the tidal dissipation its semi-major axis can notably decrease or increase. Therefore the satellite can encounter several resonances in which it can potentially be captured. Some resonances are able to modify the equatorial inclination of a satellite. Touma and Wisdom (1998) noted that a resonance called &#8216;eviction&#8217; between the mean motion of the Earth and the ascending node frequency of the Moon could increase by several degrees the equatorial inclination of the early Moon and could explain the present orientation of its orbit. Yokoyama (2002) studied these resonances for Phobos and Triton and observed that several resonances of this type can increase the equatorial inclination of Phobos in the future. &#160; In this work, we study the different existing &#8216;eviction&#8217; resonances to determine their possible influence on the equatorial inclination of a satellite. When a satellite goes through such a resonance, the capture is not certain and as noted by Yokoyama (2002), the probability of capture depends on several parameters as the obliquity of the planet and the interaction between other resonances. We consider the case of Phobos where we search to estimate the probability of a capture in an &#8216;eviction&#8217; resonance by using an analytical Hamiltonian model and numerical simulations. This work will then notably estimate the probability that Phobos will be captured in the future in an &#8216;eviction&#8217; resonance able to modify significantly its inclination and will measure the influence of the different parameters over the probability of capture. &#160; Acknowledgments: The authors acknowledge support from project POCI-01-0145-FEDER-029932 (PTDC/FIS-AST/29932/2017), funded by FEDER through COMPETE 2020 (POCI) and FCT. &#160; References: &#160; Touma J. and Wisdom J., Resonances in the Early Evolution of the Earth-Moon System. The Astronomical Journal, 115:1653&#8211;1663, 1998. Yokoyama T., Possible effects of secular resonances in Phobos and Triton. Planetary and Space Science, 50:63&#8211;77, 2002.

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