The motion of a lunar orbiter

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.

Spinning strings of the Neveu-Schwarz-Ramond type in 3, 4, 6, and 10 space-time dimensions

1999 ◽  
Vol 77 (6) ◽  
pp. 427-446
Author(s):  
S B Phillips
Keyword(s):  
Degrees Of Freedom ◽  
Lagrangian Density ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Physical Space ◽  
Space Time ◽  
Coordinate Space ◽  
Time Dimension ◽  
Light Cone Gauge ◽  
Fermionic Sector

A model of a spinning string with an internal coordinate index is proposed and studied. When the action for this model is taken to be diagonal in this internal coordinate space and quantized in the light-cone gauge it is found to be Lorentz covariant in four-dimensional space-time provided that the internal coordinate space is four dimensional.This combination of space-time dimension, D, and internal coordinate space dimension, N, is just one of four possible sets, the other three corresponding to D = 3, 6, and 10, precisely the same values for which it is possible to formulate Yang-Mills theories with simple supersymmetry. By comparing the number of propagating degrees of freedom at the zero-mass level in the open string bosonic and fermionic sectors it is found that a supersymmetric interpretation of this model is possible provided that all physical states in the bosonic sector have even G-parity and the ground-state spin or in the fermionic sector have positive chirality. A possible interpretation of the connection betweenthe N components of each of the D space-time coordinates is presentedon the basis that the space-time coordinates are scalars in the internal coordinate space. This interpretation would appear to be reasonable given the fact that the field variables in the Lagrangian density do not necessarily have to represent physically measurable quantities but can, instead, only represent physically measurable quantities when combined in some manner, the simplest of which being a linear combination. The Lagrangian density simply produces the equations of motion and the constraint equations for the independent variables, only linear combinations of which represent the four dimensions of physical space-time.PACS Nos.: 11.17.+y, 11.10.Qr, 1.30.Cp, 11.30.Pb

scholarly journals The mean distance to the Moon as determined by radar

1965 ◽  
Vol 21 ◽  
pp. 81-93 ◽  
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.

I. Dynamical considerations - Dynamics of the Moon

1967 ◽  
Vol 296 (1446) ◽  
pp. 245-247 ◽  
Keyword(s):  
Atmospheric Pressure ◽  
Uniform Density ◽  
The Moon ◽  
Homogeneous Body ◽  
Cubic Form ◽  
The Earth ◽  
Change Of State ◽  
Rate Of Increase ◽  
The Mean ◽  
Single Material

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.

Detection of Cracks in Mistuned Bladed Disks Using Reduced-Order Models and Vibration Data

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Chulwoo Jung ◽  
Akira Saito ◽  
Bogdan I. Epureanu
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Three Dimensional ◽  
Cost Savings ◽  
Forced Response ◽  
Response Analysis ◽  
Reduced Order Models ◽  
Reduced Order

A novel methodology to detect the presence of a crack and to predict the nonlinear forced response of mistuned turbine engine rotors with a cracked blade and mistuning is developed. The combined effects of the crack and mistuning are modeled. First, a hybrid-interface method based on component mode synthesis is employed to develop reduced-order models (ROMs) of the tuned system with a cracked blade. Constraint modes are added to model the displacements due to the intermittent contact between the crack surfaces. The degrees of freedom (DOFs) on the crack surfaces are retained as active DOFs so that the physical forces due to the contact/interaction (in the three-dimensional space) can be accurately modeled. Next, the presence of mistuning in the tuned system with a cracked blade is modeled. Component mode mistuning is used to account for mistuning present in the uncracked blades while the cracked blade is considered as a reference (with no mistuning). Next, the resulting (reduced-order) nonlinear equations of motion are solved by applying an alternating frequency/time-domain method. Using these efficient ROMs in a forced response analysis, it is found that the new modeling approach provides significant computational cost savings, while ensuring good accuracy relative to full-order finite element analyses. Furthermore, the effects of the cracked blade on the mistuned system are investigated and used to detect statistically the presence of a crack and to identify which blade of a full bladed disk is cracked. In particular, it is shown that cracks can be distinguished from mistuning.

scholarly journals On The lunar Secular Acceleration: A Possible Approach

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).

scholarly journals Changing inclination of earth satellites using the gravity of the moon

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
Karla de Souza Torres ◽  
A. F. B. A. Prado
Keyword(s):  
Dimensional Space ◽  
Semimajor Axis ◽  
Three Dimensional ◽  
Main Idea ◽  
Close Approach ◽  
The Moon ◽  
The Earth ◽  
Plane Change ◽  
Patched Conics ◽  
Time Required

We analyze the problem of the orbital control of an Earth's satellite using the gravity of the Moon. The main objective is to study a technique to decrease the fuel consumption of a plane change maneuver to be performed in a satellite that is in orbit around the Earth. The main idea of this approach is to send the satellite to the Moon using a single-impulsive maneuver, use the gravity field of the Moon to make the desired plane change of the trajectory, and then return the satellite to its nominal semimajor axis and eccentricity using a bi-impulsive Hohmann-type maneuver. The satellite is assumed to start in a Keplerian orbit in the plane of the lunar orbit around the Earth and the goal is to put it in a similar orbit that differs from the initial orbit only by the inclination. A description of the close-approach maneuver is made in the three-dimensional space. Analytical equations based on the patched conics approach are used to calculate the variation in velocity, angular momentum, energy, and inclination of the satellite. Then, several simulations are made to evaluate the savings involved. The time required by those transfers is also calculated and shown.

scholarly journals Everything Is A Circle: A New Universal Orbital Model

2021 ◽  
Author(s):  
AslıPınar Tan
Keyword(s):  
Solar System ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Ecliptic Plane ◽  
The Sun ◽  
The Moon ◽  
Circular Orbits ◽  
Orbital Parameters ◽  
The Earth ◽  
Angular Velocities

Based on measured astronomical position data of heavenly objects in the Solar System and other planetary systems, all bodies in space seem to move in some kind of elliptical motion with respect to each other. According to Kepler’s 1st Law, “orbit of a planet with respect to the Sun is an ellipse, with the Sun at one of the two foci.” Orbit of the Moon with respect to Earth is also distinctly elliptical, but this ellipse has a varying eccentricity as the Moon comes closer to and goes farther away from the Earth in a harmonic style along a full cycle of this ellipse. In this paper, our research results are summarized, where it is first mathematically shown that the “distance between points around any two different circles in three dimensional space” is equivalent to the “distance of points around a vector ellipse to another fixed or moving point, as in two dimensional space”. What is done is equivalent to showing that bodies moving on two different circular orbits in space vector wise behave as if moving on an elliptical path with respect to each other, and virtually seeing each other as positioned at an instantaneously stationary point in space on their relative ecliptic plane, whether they are moving with the same angular velocity, or different but fixed angular velocities, or even with different and changing angular velocities with respect to their own centers of revolution. This mathematical revelation has the potential to lead to far reaching discoveries in physics, enabling more insight into forces of nature, with a formulation of a new fundamental model regarding the motions of bodies in the Universe, including the Sun, Planets, and Satellites in the Solar System and elsewhere, as well as at particle and subatomic level. Based on the demonstrated mathematical analysis, as they exhibit almost fixed elliptic orbits relative to one another over time, the assertion is made that the Sun, the Earth, and the Moon must each be revolving in their individual circular orbits of revolution in space. With this expectation, individual orbital parameters of the Sun, the Earth, and the Moon are calculated based on observed Earth to Sun and Earth to Moon distance data, also using analytical methods developed as part of this research to an approximation. This calculation and analysis process have revealed additional results aligned with observation, and this also supports our assertion that the Sun, the Earth, and the Moon must actually be revolving in individual circular orbits.

scholarly journals A first analysis of the mean motion of CHAMP

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

scholarly journals A supplementary note on constructing the general Earth's rotation theory

2014 ◽  
Vol 9 (S310) ◽  
pp. 13-16
Author(s):  
Victor A. Brumberg ◽  
Tamara V. Ivanova
Keyword(s):  
Equations Of Motion ◽  
Combined Treatment ◽  
Earth’S Rotation ◽  
Combined System ◽  
Euler Parameters ◽  
Earth's Rotation ◽  
The Earth ◽  
Practical Development ◽  
The Mean ◽  
General Planetary Theory

AbstractRepresenting a post-scriptum supplementary to a previous paper of the authors Brumberg & Ivanova (2011) this note aims to simplify the practical development of the Earth's rotation theory, in the framework of the general planetary theory, avoiding the non–physical secular terms and involving the separation of the fast and slow angular variables, both for planetary–lunar motion and Earth's rotation. In this combined treatment of motion and rotation, the fast angular terms are related to the mean orbital longitudes of the bodies, the diurnal and Euler rotations of the Earth. The slow angular terms are due to the motions of pericenters and nodes, as well as the precession of the Earth. The combined system of the equations of motion for the principal planets and the Moon and the equations of the Earth's rotation is reduced to the autonomous secular system with theoretically possible solution in a trigonometric form. In the above–mentioned paper, the Earth's rotation has been treated in Euler parameters. The trivial change of the Euler parameters to their small declinations from some nominal values may improve the practical efficiency of the normalization of the Earth's rotation equations. This technique may be applied to any three-axial rigid planet. The initial terms of the corresponding expansions are given in the Appendix.

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