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.

Low selenocentric orbits stability analysis

2020 ◽  
Author(s):  
Chongrui Du ◽  
O.L. Starinova
Keyword(s):  
Rate Of Change ◽  
The Sun ◽  
Space Systems ◽  
The Moon ◽  
Orbital Structure ◽  
Motion Modeling ◽  
The Earth ◽  
Long Time ◽  
The Stability

The tasks of studying the Moon require long-term functioning space systems. Most of the low selenocentric orbits are known to be unstable, which requires a propellant to maintain the orbital structure. For these orbits, the main disturbing factors are the off-center gravitational field of the Moon and the gravity of the Earth and the Sun. This paper analyzes the stability of low selenocentric orbits according to passive motion modeling and takes into account these main disturbing factors. We put forward a criterion for determining the stability of the orbit and used it to analyze the circular orbit of the Moon at an altitude of 100 kilometers. According to different initial data and different dates, we obtained ranges of the Moon’s orbits with good stability. At the same time, we analyzed the rate of change in the longitude of the ascending node, and found a stable low lunar orbit which can operate for a long time.

scholarly journals Commission 19: Rotation of the Earth

1985 ◽  
Vol 19 (1) ◽  
pp. 193-205 ◽  
Author(s):  
Ya. S. Yatskiv ◽  
W. J. Klepczynski ◽  
F. Barlier ◽  
H. Enslin ◽  
C. Kakuta ◽  
...  
Keyword(s):  
Time Frame ◽  
Laser Ranging ◽  
Earth’S Rotation ◽  
The Moon ◽  
Earth's Rotation ◽  
The Earth ◽  
Long Time ◽  
Terrestrial Reference System ◽  
Rotation Of The Earth

During the period, work on the problem of the Earth’s rotation has continued to expand and increase its scope. The total number of institutions engaged in the determination of the Earth’s rotation parameters (ERP) by different techniques has been increased significantly. The rotation of the Earth is currently measured by classical astrometry, Doppler and laser satellite tracking, laser ranging of the Moon, and radio interferometry. Several long time series of the ERP are available from most of these techniques, in particular, those made during the Main Campaign of the MERIT project. The various series have been intercompared and their stability, in the time frame of years to days, has been estimated for the purposes of establishing a new conventional terrestrial reference system (COTES). On the other hand, the difficulties of maintaining a regular operation for laser ranging to the Moon (LLR) have been recognized. It resulted in the proposal to organize an one-month campaign of observations in 1985 in order to complement the COTES collocation program and to allow additional intercomparisons with other techniques.

scholarly journals Boundary conditions for the formation of the Moon

2015 ◽  
Vol 95 (2) ◽  
pp. 131-139 ◽  
Author(s):  
M. Reuver ◽  
R.J. de Meijer ◽  
I.L. ten Kate ◽  
W. van Westrenen
Keyword(s):  
Angular Momentum ◽  
Boundary Conditions ◽  
Total Angular Momentum ◽  
Nuclear Explosion ◽  
Moment Of Inertia ◽  
The Moon ◽  
Moon System ◽  
Giant Impact ◽  
The Earth ◽  
The Impact

AbstractRecent measurements of the chemical and isotopic composition of lunar samples indicate that the Moon's bulk composition shows great similarities with the composition of the silicate Earth. Moon formation models that attempt to explain these similarities make a wide variety of assumptions about the properties of the Earth prior to the formation of the Moon (the proto-Earth), and about the necessity and properties of an impactor colliding with the proto-Earth. This paper investigates the effects of the proto-Earth's mass, oblateness and internal core-mantle differentiation on its moment of inertia. The ratio of angular momentum and moment of inertia determines the stability of the proto-Earth and the binding energy, i.e. the energy needed to make the transition from an initial state in which the system is a rotating single body with a certain angular momentum to a final state with two bodies (Earth and Moon) with the same total angular momentum, redistributed between Earth and Moon. For the initial state two scenarios are being investigated: a homogeneous (undifferentiated) proto-Earth and a proto-Earth differentiated in a central metallic and an outer silicate shell; for both scenarios a range of oblateness values is investigated. Calculations indicate that a differentiated proto-Earth would become unstable at an angular momentum L that exceeds the total angular momentum of the present-day Earth–Moon system (L0) by factors of 2.5–2.9, with the precise maximum dependent on the proto-Earth's oblateness. Further limitations are imposed by the Roche limit and the logical condition that the separated Earth–Moon system should be formed outside the proto-Earth. This further limits the L values of the Earth–Moon system to a maximum of about L/L0 = 1.5, at a minimum oblateness (a/c ratio) of 1.2. These calculations provide boundary conditions for the main classes of Moon-forming models. Our results show that at the high values of L used in recent giant impact models (1.8 < L/L0 < 3.1), the proposed proto-Earths are unstable before (Cuk & Stewart, 2012) or immediately after (Canup, 2012) the impact, even at a high oblateness (the most favourable condition for stability). We conclude that the recent attempts to improve the classic giant impact hypothesis by studying systems with very high values of L are not supported by the boundary condition calculations in this work. In contrast, this work indicates that the nuclear explosion model for Moon formation (De Meijer et al., 2013) fulfills the boundary conditions and requires approximately one order of magnitude less energy than originally estimated. Hence in our view the nuclear explosion model is presently the model that best explains the formation of the Moon from predominantly terrestrial silicate material.

scholarly journals Synchronization of Two Non-Identical Coupled Exciters in a Non-Resonant Vibrating System of Linear Motion. Part I: Theoretical Analysis

2009 ◽  
Vol 16 (5) ◽  
pp. 505-515 ◽  
Author(s):  
Chunyu Zhao ◽  
Hongtao Zhu ◽  
Ruizi Wang ◽  
Bangchun Wen
Keyword(s):  
Zero Solution ◽  
Moment Of Inertia ◽  
Moments Of Inertia ◽  
Vibrating System ◽  
Frequency Capture ◽  
Synchronization Problem ◽  
Small Parameters ◽  
Existence And Stability ◽  
Synchronous Operation ◽  
The Stability

In this paper an analytical approach is proposed to study the feature of frequency capture of two non-identical coupled exciters in a non-resonant vibrating system. The electromagnetic torque of an induction motor in the quasi-steady-state operation is derived. With the introduction of two perturbation small parameters to average angular velocity of two exciters and their phase difference, we deduce the Equation of Frequency Capture by averaging two motion equations of two exciters over their average period. It converts the synchronization problem of two exciters into that of existence and stability of zero solution for the Equation of Frequency Capture. The conditions of implementing frequency capture and that of stabilizing synchronous operation of two motors have been derived. The concept of torque of frequency capture is proposed to physically explain the peculiarity of self-synchronization of the two exciters. An interesting conclusion is reached that the moments of inertia of the two exciters in the Equation of Frequency Capture reduce and there is a coupling moment of inertia between the two exciters. The reduction of moments of inertia and the coupling moment of inertia have an effect on the stability of synchronous operation.

Orbital and Attitude Stability of Space Shuttles in Parking Orbits

1975 ◽  
Vol 26 (1) ◽  
pp. 20-24
Author(s):  
R Arho
Keyword(s):  
Gravitational Field ◽  
Stationary State ◽  
Circular Orbit ◽  
Principal Axis ◽  
Orbital Plane ◽  
Moments Of Inertia ◽  
Attitude Stability ◽  
Principal Moments Of Inertia ◽  
Bounded Perturbation ◽  
The Stability

SummaryA unified treatment is given of the orbital and attitude stability of space shuttles in parking orbits (in vacuo) in the earth’s gravitational field. A shuttle in a circular orbit with a principal axis aligned with the horizontal in the orbital plane is found to be in stationary geostatic equilibrium. The demand for stability leads to a condition which must be satisfied by the principal moments of inertia. The stability which is achieved is not asymptotic without control. The stationary state is a stable centre about which a bounded perturbation oscillation without damping may exist.

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.

Equatorial flattening and principal moments of inertia of the earth

1984 ◽  
Vol 28 (1) ◽  
pp. 9-10 ◽  
Author(s):  
Milan Burša ◽  
Zdislav Šíma ◽  
M. Pick
Keyword(s):  
Moments Of Inertia ◽  
The Earth ◽  
Principal Moments Of Inertia

Thermodynamicsof giant planetary impacts from ab initiosimulations

2021 ◽  
Author(s):  
Razvan Caracas ◽  
Sarah T. Stewart
Keyword(s):  
Stability Field ◽  
The Moon ◽  
Supercritical Regime ◽  
Horizon 2020 ◽  
Research And Innovation ◽  
The Earth ◽  
Magma Oceans ◽  
The Stability ◽  
Dynamics Simulations ◽  
Planetary Impacts

&lt;h3&gt;Impacts are highly energetic phenomena. They abound in the early stages of formation of the solar system, when they actively participated to the formation of large bodies in the protoplanetary disk. Later on, when planetesimals and embryo planets formed, impacts merged smaller bodies into the large planets that we know today. Giant impacts dominated the last phase of the planetary accretion, with some of these impacts leaving traces observable even today (planets tilts, moon, missing mantle, etc). The Earth was not spared, and its most cataclysmic event also contributed to the formation of the Moon.&lt;/h3&gt;&lt;h3&gt;Here we present the theoretical tools used to explore the thermodynamics of the formation of the protolunar disk and the subsequent condensation of this disk. We show how ab initio-based molecular dynamics simulations contribute to the determination of the stability field of melts, to the equilibrium between melts and vapor and the positioning of the critical points. Together all this information helps building the liquid-vapor stability dome. Next we investigate the supercritical regime, typical of the post-impact state. We take a focused look to the transport properties, the formation of the first atmosphere, and compare the properties of the liquid state typical of magma oceans, to the super-critical state, typical of protolunar disks.&lt;/h3&gt;&lt;h3&gt;We apply this theoretical approach on pyrolite melts, as best approximants for the bulk silicate Earth. These simulations help us retrace the thermodynamic state of the protolunar disk and infer possible condensation paths for both the Earth and the moon.&lt;/h3&gt;&lt;h3&gt;&amp;#160;&lt;/h3&gt;&lt;p&gt;RC acknowledges support from the European Research Council under EU Horizon 2020 research and innovation program (grant agreement 681818 &amp;#8211; IMPACT) and access to supercomputing facilities via the eDARI gen6368 grants, the PRACE RA4947 grant, and the Uninet2 NN9697K grant. STS was supported by NASA grants NNX15AH54G and 80NSSC18K0828; DOE-NNSA grants DE-NA0003842 and DE-NA0003904.&lt;/p&gt;

THE STABILITY OF THE UNSYMMETRICAL HEAVY GYRO WITH DAMPING TORQUE

1992 ◽  
Vol 16 (1) ◽  
pp. 83-89
Author(s):  
Zheng-Ming E ◽  
Shih-Ming Chiu
Keyword(s):  
Fixed Point ◽  
Rigid Body ◽  
Direct Method ◽  
Nonlinear Damping ◽  
Center Of Gravity ◽  
Moments Of Inertia ◽  
Principal Moments Of Inertia ◽  
The Stability ◽  
Damping Torque

The conditions of the unsymmetrical heavy gyro are that A ≠ B ≠ C where, A, B and C are the principal moments of inertia of the rigid body and the center of gravity G of the rigid body does not coincide with the fixed point, i.e. zc ≠ 0, xc = yc = 0 in this paper. The conditions of the stability and of the instability of the unsymmetrical heavy gyro with and without linear or nonlinear damping torque are obtained by the direct method of Lyapunov.

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