The shape of the Moon, its internal structure and moments of inertia

1967 ◽  
Vol 296 (1446) ◽  
pp. 254-265 ◽  
Keyword(s):  
Convective Motion ◽  
The Moon ◽  
Moments Of Inertia ◽  
Lunar Interior ◽  
A Value ◽  
Principal Axes ◽  
Thermoelastic Effects ◽  
The Mean ◽  
The Stability ◽  
Sufficient Strength

Harmonic analysis of the Moon’s shape based on all available sets of hypsometric data disclose that the surface of the Moon, far from being a mere spheroid or ellipsoid, contains many significant harmonic terms, the single largest of which are of fourth order (being about three times as large as the second harmonics). Their sum makes the Moon to deviate from a mean sphere by ± 2 km over extensive regions; and local differences attaining 8 to 9 km in eleva­tion have been noted on the limb. These facts reveal that the lunar globe must possess sufficient strength to sustain stress differences of the order of 10 9 dyn/cm 2 ; and this could scarcely be the case if the large part of the Moon’s interior were molten. As melting should be expected if the Moon contained the same proportion of radioactive elements as chondritic meteroites, it is concluded that the mean radioactive content of the lunar interior must be less than that found in stony meteorites, or the terrestrial crust. The moments of inertia about the principal axes of inertia of the lunar globe, as determined from the Moon’s physical librations, are seriously at variance with a state of hydrostatic equilibrium—for any distance between the Earth and the Moon—of a homogeneous body, and can be accounted for only by assuming an asymmetric nonhomogeneity of the lunar globe, or the existence of internal processes which could support nonequilibrium from hydrodynamically. However, an application of Chandrasekhar’s theory of viscous convection in fluid globes reveals that, if such a globe is to possess the same difference, C – A , of momenta as the Moon, the velocity of convective motion should be of the order of 10 –8 cm/s (i. e. too small for the establishment of steady flow in 10 9 y); and the 'observed' value of the Rayleigh number characteristic of the Moon is several hundred times as large as that required theoretically for the stability of the respective flow. Thermoelastic effects due to secular insolation of the lunar globe, considered recently by Levin, are shown incapable to account for a value of the ratio (C – A)/B exceeding 0∙00005; while its empirical value deduced from librations is close to 0∙00063.

Thermal effects on the figure of the Moon

1967 ◽  
Vol 296 (1446) ◽  
pp. 266-269 ◽  
Keyword(s):  
Thermal Effects ◽  
Hydrostatic Equilibrium ◽  
Moment Of Inertia ◽  
The Moon ◽  
Moments Of Inertia ◽  
The Earth ◽  
Principal Moments Of Inertia ◽  
Long Time ◽  
The Stability ◽  
Lunar Rotation

Before discussing its cause, one must be clear in exactly what respect the lunar figure deviates from the equilibrium one. This is necessary because there has been confusion over the question for a long time. It was known early that the Moon’s ellipsoid of inertia is triaxial and that the differences of the principal moments of inertia determined from observations are several times larger than the theoretical values corresponding to hydrostatic equilibrium. The stability of lunar rotation requires that the axis of least moment of inertia point approximately towards the Earth and the laws of Cassini show that it is really so.

THE MICROWAVE SPECTRUM AND MOLECULAR STRUCTURE OF CHF2Cl

1962 ◽  
Vol 40 (1) ◽  
pp. 61-73 ◽  
Author(s):  
David B. McLay ◽  
C. R. Mann
Keyword(s):  
Bond Length ◽  
Electric Quadrupole ◽  
Microwave Spectrum ◽  
Molecular Structures ◽  
Moments Of Inertia ◽  
A Value ◽  
Principal Axes ◽  
Quadrupole Coupling ◽  
Out Of Plane ◽  
Tetrahedral Angle

Sufficient Q- and R-branch rotational transitions have been measured in the microwave spectrum of the asymmetric top molecule CHF2Cl to determine the moments of inertia Ia, Ib, and Ic for each of the three isotopic species C12HF2Cl35, C12HF2Cl37, and C13HF2Cl35. The value of Ia+Ic−Ib, which should depend only upon the out-of-plane co-ordinate of the fluorine atoms, is sufficiently constant to give a very accurate value for the co-ordinate of 1.08524 ± 0.00006 Å but involves some variations which limit the accuracy of determining molecular structures by isotopic substitutions. The components of the nuclear electric quadrupole coupling dyadics in the principal axes systems for the moments of inertia ellipsoids have been calculated accurately from the hyperfine structure of the C12HF2Cl35,37 spectra. In both cases, eQVaa+eQVbb+eQVcc = 0 and the ratio of any component for Cl35 to the same component for Cl37 is equal to 1.269 within the experimental error. The low-field Stark splitting of the 00,0 → 11,0 transition indicates a dipole moment component μc = 1.5 debyes while the absence of a-type transitions indicates a value for μa, which is less than 0.1 μc. The C—Cl bond length and the angle between this bond and the a-axis have been calculated by Kraitchman's formulae for isotopic substitution to be 1.747 ± 0.002 Å and 14.3 ± 0.2°. This angle and the values of eQVaa, eQVbb, and eQVcc lead to eQVzz = eQq = −71.5 ± 0.3 Mc/s and quadrupole coupling asymmetry parameter η = −0.012 ± 0.006, which indicates very little π-electron character. If the C—H bond length is assumed to be 1.095 ±.015 Å and the angle H—C—Cl is assumed to be 108 ± 1°, then the calculated values for the C—F bond length and for the angle F—C—F are 1.350 ±.003 Å and 107.0 ± 0.4°, a value considerably less than the tetrahedral angle. The values for bond lengths, bond angles, and nuclear electric quadrupole coupling components are compared with those in other halogenated methanes.

Dynamical arguments which concern melting of the Moon

1977 ◽  
Vol 285 (1327) ◽  
pp. 561-568 ◽  
Keyword(s):  
Global Ocean ◽  
The Moon ◽  
Molten Material ◽  
Moments Of Inertia ◽  
Crustal Rocks ◽  
Chemical Differentiation ◽  
Principal Axes ◽  
Lunar Maria

This paper points out that the observed differences of the moments of inertia of the lunar globe about its principal axes - determined astronomically and verified more recently by laser rangings - are inconsistent with the assumption that the whole Moon was ever covered by a global layer of molten material, extending to a depth of a few hundred kilometres. Moreover, laser determinations of the shape of the Moon (along the tracks overflown by Apollo 15 -17 missions) make it quite clear that the Moon’s surface did not solidify from a global ocean of lava even 10-20 km deep. Therefore, any melting which occurred on the Moon (and produced the observed chemical differentiation of the crustal rocks) could have taken place only locally — over areas of the size of the lunar maria, but not over the Moon as a whole at the same time.

Analysis of Stability and Bifurcation of an Asymmetrical Rotor

2015 ◽  
Author(s):  
Majid Shahgholi ◽  
S. E. Khadem ◽  
Mahsa Asgarisabet
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Moments Of Inertia ◽  
Rotor Systems ◽  
Unequal Mass ◽  
Principal Axes ◽  
Analysis Of Stability ◽  
Stability And Bifurcations ◽  
The Stability

The effect of shaft and disk asymmetry on the harmonic resonances of a rotor system with the in-extensional nonlinearity and large amplitude are investigated. Two rotor systems, one of which has been comprised of a symmetrical shaft and an asymmetrical disk (SA), and the other one has been comprised of an asymmetrical shaft and an asymmetrical disk (AA) are investigated. The shaft in the AA rotor has unequal mass moments of inertia and flexural rigidities in the direction of principal axes. Also, in the AA system the rigid disk is asymmetric with unequal mass moments of inertia. The equations of motion are derived by the Hamiltonian principle. The stability and bifurcations are obtained using the multiple scales method. The influences of asymmetry of shaft, asymmetry of disk, inequality between two eccentricities corresponding to the principal axes, disk position and external damping on the stability and bifurcations of SA and AA rotors are investigated. The results achieved from multiple scales method show a good agreement with those of numerical simulations.

scholarly journals On the Absolute Orientation of the Selenodetic Reference Frame

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.

scholarly journals Determination of Some Physical Parameters of the Moon with Lunar Laser Ranging Data

1997 ◽  
Vol 165 ◽  
pp. 221-226
Author(s):  
Jin Wenjing ◽  
Li Jinling
Keyword(s):  
Gravitational Field ◽  
Laser Ranging ◽  
Physical Parameters ◽  
The Moon ◽  
Lunar Laser Ranging ◽  
Moments Of Inertia ◽  
Lunar Interior ◽  
Lunar Laser

AbstractInformation about the structure of lunar interior and evolution could be obtained from measurements of lunar free librations, gravitational field, dissipation etc. In this paper the precision of determining free librations from Lunar Laser Ranging (LLR) data are estimated. Using the observing data from four telescopes for eighteen years, the amplitudes and phases of free librations, and ratios of the moments of inertia of the Moon were determined.

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.

Injuries to the Coronoid Process of the Ulna with Involvement of the Lesser Sigmoid Notch

2020 ◽  
Author(s):  
Valentin Rausch ◽  
Sina Neugebauer ◽  
Tim Leschinger ◽  
Lars Müller ◽  
Kilian Wegmann ◽  
...  
Keyword(s):  
Fragment Size ◽  
Coronoid Process ◽  
Annular Ligament ◽  
Concomitant Injuries ◽  
Proximal Ulna ◽  
Sigmoid Notch ◽  
Coronoid Fractures ◽  
The Mean ◽  
The Stability ◽  
Arthritic Changes

Abstract Introduction This study aimed to describe the involvement of the lesser sigmoid notch in fractures to the coronoid process. We hypothesized that injuries to the lateral aspect of the coronoid process regularly involve the annular ligament insertion at the anterior lesser sigmoid notch. Material and Methods Patients treated for a coronoid process fracture at our institution between 06/2011 and 07/2018 were included. We excluded patients < 18 years, patients with arthritic changes or previous operative treatment to the elbow, and patients with concomitant injuries to the proximal ulna. In patients with involvement of the lesser sigmoid notch, the coronoid height and fragment size (anteroposterior, mediolateral, and craniocaudal) were measured. Results Seventy-two patients (mean age: 47 years ± 17.6) could be included in the study. Twenty-one patients (29.2%) had a fracture involving the lateral sigmoid notch. The mean anteroposterior fragment length was 7 ± 1.6 mm. The fragment affected a mean of 43 ± 10.8% of the coronoid height. The mean mediolateral size of the fragment was 10 ± 5.0 mm, and the mean cranio-caudal size was 7 ± 2.7 mm. Conclusion Coronoid fractures regularly include the lesser sigmoid notch. These injuries possibly affect the anterior annular ligament insertion which is important for the stability of the proximal radioulnar joint and varus stability of the elbow.

Instant 99mTc-Labelled Glucoheptonate Kit for Kidney Imaging

1983 ◽  
Vol 22 (05) ◽  
pp. 246-250 ◽  
Author(s):  
M. Al-Hilli ◽  
H. M. A. Karim ◽  
M. H. S. Al-Hissoni ◽  
M. N. Jassim ◽  
N. H. Agha
Keyword(s):  
Sodium Hydroxide Solution ◽  
Normal Subjects ◽  
Organ Distribution ◽  
Stable Complex ◽  
Scintillation Camera ◽  
Ph Adjustment ◽  
The Mean ◽  
The Stability ◽  
Kidney Imaging ◽  
Mean Concentration

Gelchromatography column scanning has been used to study the fractions of reduced hydrolyzed 99mTc, 99mTc-pertechnetate and 99mTc-chelate in a 99mTc-glucoheptonate (GH) preparation. A stable high labelling yield of 99mTc-GH complex in the radiopharmaceutical has been obtained with a concentration of 40-50 mg of glucoheptonic acid-calcium salt and not less than 0.45 mg of SnCl2 2 H2O at an optimal pH between 6.5 and 7.0. The stability of the complex has been found significantly affected when sodium hydroxide solution was used for the pH adjustment. However, an alternative procedure for final pH adjustment of the preparation has been investigated providing a stable complex for the usual period of time prior to the injection. The organ distribution and the blood clearance data of 99mTc-GH in rabbits were relatively similar to those reported earlier. The mean concentration of the radiopharmaceutical in both kidneys has been studied in normal subjects for one hour with a scintillation camera and the results were satisfactory.

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