The lunar mean moment of inertia and the size of the Moon's core

In this paper the relation between the uncertainty of the Moon's mean moment of inertia (I/Ma ²) and that of the core density _c is discussed with a two-layer model of the Moon - a mantle obeying Roche's law of the density distribution and a homogeneous core (Fe-core or Fe-FeS-core). When the uncertainty of I/Ma ² is 0.0023 (that is the accuracy in present observation), a core with radius of 450 km will be appropriate to the limitation of _c about 1 g cm^–3. Considering the accuracy obtained in space explorations, and the compressibility and the quasi-homogeneity of the Moon, we suggest that the parameters C ₂₀, , , a, and GM of the Moon should define as primary constants, but C ₂₂ and C/Ma ² as derived constants. Therefore, the ratio of mass of Moon to that of Earth in the IAU (1976) system of astronomical constants will become a deducible constant.     

Jupiter’s moment of inertia: A possible determination by Juno

The moment of inertia of a giant planet reveals important information about the planet’s internal density structure and this information is not identical to that contained in the gravitational moments. The forthcoming Juno mission to Jupiter might determine Jupiter’s normalized moment of inertia NMoI = C/MR² by measuring Jupiter’s pole precession and the Lense–Thirring acceleration of the spacecraft (C is the axial moment of inertia, and M and R are Jupiter’s mass and mean radius, respectively). We investigate the possible range of NMoI values for Jupiter based on its measured gravitational field using a simple core/envelope model of the planet assuming that J₂ and J₄ are perfectly known and are equal to their measured values. The model suggests that for fixed values of J₂ and J₄ a range of NMoI values between 0.2629 and 0.2645 can be found. The Radau–Darwin relation gives a NMoI value that is larger than the model values by less than 1%. A low NMoI of ∼0.236, inferred from a dynamical model (Ward, W.R., Canup, R.M. [2006]. Astrophys. J. 640, L91–L94) is inconsistent with this range, but the range is model dependent. Although we conclude that the NMoI is tightly constrained by the gravity coefficients, a measurement of Jupiter’s NMoI to a few tenths of percent by Juno could provide an important constraint on Jupiter’s internal structure. We carry out a simplified assessment of the error involved in Juno’s possible determination of Jupiter’s NMoI.     

Nonhydrostatic effects and the determination of icy satellites’ moment of inertia

We compare the moment of inertia (MOI) of a simple hydrostatic, two layer body as determined by the Radau–Darwin Approximation (RDA) to its exact hydrostatic MOI calculated to first order in the parameter q = Ω²R³/GM, where Ω, R, and M are the spin angular velocity, radius, and mass of the body, and G is the gravitational constant. We show that the RDA is in error by less than 1% for many configurations of core sizes and layer densities congruent with those of solid bodies in the Solar System. We then determine the error in the MOI of icy satellites calculated with the RDA due to nonhydrostatic effects by using a simple model in which the core and outer shell have slight degree 2 distortions away from their expected hydrostatic shapes. Since the hydrostatic shape has an associated stress of order ρΩ²R² (where ρ is density) it follows that the importance of nonhydrostatic effects scales with the dimensionless number σ/ρΩ²R², where σ is the nonhydrostatic stress. This highlights the likely importance of this error for slowly rotating bodies (e.g., Titan and Callisto) and small bodies (e.g., Saturn moons other than Titan). We apply this model to Titan, Callisto, and Enceladus and find that the RDA-derived MOI can be 10% greater than the actual MOI for nonhydrostatic stresses as small as ∼0.1 bars at the surface or ∼1 bar at the core–mantle boundary, but the actual nonhydrostatic stresses for a given shape change depends on the specifics of the interior model. When we apply this model to Ganymede we find that the stresses necessary to produce the same MOI errors as on Titan, Callisto, and Enceladus are an order of magnitude greater due to its faster rotation, so Ganymede may be the only instance where RDA is reliable. We argue that if satellites can reorient to the lowest energy state then RDA will always give an overestimate of the true MOI. Observations have shown that small nonhydrostatic gravity anomalies exist on Ganymede and Titan, at least at degree 3 and presumably higher. If these anomalies are indicative of the nonhydrostatic anomalies at degree 2 then these imply only a small correction to the MOI, even for Titan, but it is possible that the physical origin of nonhydrostatic degree 2 effects is different from the higher order terms. We conclude that nonhydrostatic effects could be present to an extent that allows Callisto and Titan to be fully differentiated.     

On the polar moment of inertia of a compressible body

The rotational dynamics of a body are governed by the values of its principle moments of inertia. These quantities are not directly observable, but they are related to the harmonic coefficients of the external gravity field and to the density distribution within the body, both of which can be inferred from appropriate observations. It is shown that, for the particular case of a spherical planet whose density varies as a power of the radial distance, the principal moment of inertia has an elegantly simple form. Application of this simplified case to the Jovian planets suggests that the density profiles outside the central core are approximately linear, with the apparent exception of Neptune.Presented at the Symposium Star Catalogues, Positional Astronomy and Celestial Mechanics, held in honor of Paul Herget at the U.S. Naval Observatory, Washington, November 30, 1978.     

The moment of inertia of low mass stars

The radius of gyration of Main-Sequence stars is an important parameter in studies of circularization and synchronization of close binaries. We present new values calculated, on the basis of updated models of stellar structure with the latest opacity tables. The importance of evolution in calculating moments of inertia is emphasized. Our results are presented in such a way that they can be directly used to compare theoretical predictions with actual observations.Paper presented at the 11the European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

Moment of inertia of a neutron star. I. Relativistic equation for the accumulated moment of inertia

A relativistic, first-order differential equation is derived for the accumulated moment of inertia of a spherically symmetric celestial body. An approximate equation is proposed to describe the contribution of relativistic effects to the moment of inertia of a superdense star. For configurations of an incompressible fluid, this approximation describes the results of the numerical calculations of Chandrasekhar and Miller to within 5% in the entire range of central pressures from 0 to ∞. Translated from Astrofizika, Vol. 40, No. 1, pp. 87–96, January–March, 1997.     

Properties of the moment of inertia in the problem of three bodies

Sundman's and Birkhoff's results are combined with a recently developed inequality and new qualitative results are given for the problem of three bodies.     

Simulation of precise orbit determination of lunar orbiters

Based on the ongoing Chinese lunar exploration mission, i.e. the “Chang'e 1” project, precise orbit determination of lunar orbiters is analyzed for the actual geographical distribution and observational accuracy of the Chinese united S-band (USB) observation and control network as well as the very long baseline interferometry (VLBI) tracking network. The observed data are first simulated, then solutions are found after including the effects of various error sources and finally compared. We use the space data analysis software package, GEODYN, developed at Goddard Space Flight Center, NASA, USA. The primary error source of the flight orbiting the moon is the lunar gravity field. Therefore, the (formal) error of JGL165P1, i.e. the model of the lunar gravity field with the highest accuracy at present, is first discussed. After simulating the data of ranging and velocity measurement as well as the VLBI data of the time delay and time delay rate, precise orbit determination is carried out when the error of the lunar gravity field is added in. When the orbit is determined, the method of reduced dynamics is adopted with the selection of appropriate empirical acceleration parameters to absorb the effect of errors in the lunar gravity field on the orbit determination. The results show that for lunar missions like the “Chang'e 1” project, that do not take the lunar gravity field as their main scientific objective, the method of reduced dynamics is a simple and effective means of improving the accuracy of the orbit determination of the lunar orbiters.     

Dynamical determination of the quadrupole mass moment of a white dwarf

In this paper we dynamically determine the quadrupole mass moment Q of the magnetic white dwarf WD 0137-349 by looking for deviations from the third Kepler law induced by Q in the orbital period of the recently discovered brown dwarf moving around it in a close 2-hr orbit. It turns out that a purely Newtonian model for the orbit of WD 0137-349B, assumed circular and equatorial, is adequate, given the present-day accuracy in knowing the orbital parameters of such a binary system. Our result is Q=(−1.5±0.9)×10⁴⁷ kg m² for i=35 deg. It is able to accommodate the 3-sigma significant discrepancy of (1.0±0.3)×10⁻⁸ s⁻² between the inverse square of the phenomenologically determined orbital period and the inverse square of the calculated Keplerian one. The impact of i, for which an interval Δ i of possible values close to 35 deg is considered, is investigated as well.     

The determination of the derivatives in Brown's lunar theory

The fundamental matrix solutionT for the variational equations of a Hamiltonian system is symplectic. We use this fact to completeT when it is only partially known. We discuss three cases. The last one gives an easy proof for the method invented by Brown in his lunar theory.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.Dedicated to Victor Szebehely.     

Helioseismic determination of the solar gravitational quadrupole moment

One of the most well-known tests of general relativity (GR) results from combining measurements of the anomalous precession of the orbit of Mercury with a determination of the gravitational quadrupole moment of the Sun J ₂. The latter can be done by inference from an integral relation between J ₂ and the solar internal rotation. New observational data of high quality obtained from the Solar Heliospheric Satellite ( SoHO ) and from the Global Oscillations Network Group (GONG) allow the determination of the internal rotation velocity of the Sun as a function of radius and latitude with unprecedented spatial resolution and accuracy. As a consequence, a number of global properties of the Sun can also be determined with much higher accuracy, notably the gravitational quadrupole moment of the Sun. The anomalous precession of the orbit of Mercury is primarily due to GR effects, but there are classical corrections, the largest of which is that due to J ₂. It is shown here that the data are currently consistent with the predictions of GR.     

Evolution of the moment of inertia in the many-body problem in the field of general attraction

For the many-body problem in the field of general attraction, the evolution of the moment of inertia is studied, we obtain the bifurcation curves between the attainable domain and the forbidden domain in the plane (,t), where is the mean quadratic distance (proportional to the square root of the moment of inertia). That is an extension of the conclusion in [1]. Moreover, the dependence of the bifurcation curves on the generalized potential is obtained.This project is supported by the National Science Foundation of China.     

Constraining the mass and moment of inertia of neutron stars from quasi-periodic oscillations in X-ray binaries

Neutron stars are the densest objects known in the Universe. Being the final product of stellar evolution, their internal composition and structure is rather poorly constrained by measurements.     

Moments of inertia of the lunar globe,and their bearing on chemical differentiation of its outer layers

It is pointed out that the observed moments of inertia of the Moon, disclosed by its librations, are influenced mainly by the distribution of mass in the outer zone in which the lithostatic pressure is less than 10 kb (i.e., in the outer shell not more than 200 km deep); and a conspicuous departure of such moments from those expected in hydrostatic equilibrium disclosed that these layers could never have been fluid. In the same way, the actual shape of the lunar surface cannot represent a solidified surface of a fluid, petrified at any distance from the Earth.The shape of the Moon, and differences of its moments of inertia must reflect the way in which the initial process of cold accretion fell short of producing a globe with strictly spherically-symmetrical stratification of material; and has nothing to do with tides - present or fossil. Such melting or lava flows as may have occurred at the Moon's surface from time to time must have remained localized, and without much effect on the dynamical properties of the Moon. A global ocean of molten magma some 200 km in depth (postulated sometimes to provide a reservoir in which the differentiation of elements exhibited by surface rocks could have taken place) at any time in the past is incompatible with the dynamical evidence on the motion of the Moon about its center of gravity.Bellcomm, Inc., 955 L'Enfant Plaza North, S.W. Washington, D.C. 20024, U.S.A.     

A determination of the intensity of the ancient lunar magnetic field

Thermal demagnetization of lunar breccia 15498,36 shows that the natural remanent magnetization is a simple thermoremanence carried by metallic iron. Using the classical Thellier-Thellier method the strength of the magnetizing field at the time of sample formation was found to be 2100 ±80 gammas.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.     

Characterization of lunar dust and a synopsis of available lunar simulants

Understanding the formation and evolution of the soil and dust of the Moon addresses the fundamental question of the interactions of space with the surface of an airless body. The physical and chemical properties of the lunar dust, the <20 μm portion of lunar soil, are key properties necessary for studies of the toxicity and the electrostatic charging of the dust. These properties have been largely overlooked until recent years. Although chemical and physical studies of the <20 μm portion of lunar soil have been the topic of several studies, there is still need for further studies, primarily of the <1 μm particles. This paper presents a review of the studies of lunar dust that have been conducted to date. As many preparations for future exploration or science activities on the Moon require testing using lunar soil/dust simulants, we also include a brief review of past and current simulants.     

Morphology of lunar craters: A test of lunar erosional models

This paper is a test of published theoretical and experimental studies of crater erosion by micrometeorite bombardment which predict systematic variations in the morphology of lunar craters as a function of crater diameter and crater age. Numerical, ranking-type degradation classifications indicate that the craters on Mare Imbrium and Mare Tranquillitatus confirm these predictions by showing a systematic increase in degradation with decreasing diameter for craters smaller than a few kilometers in diameter but larger than the equilibrium diameter, and by showing fixed proportions of fresh, moderately degraded and very degraded craters under equilibrium conditions. Furthermore, the relative ages of the two mare surfaces may be determined using a diameter/mean-degradation-number curve. These determinations of relative age and process of crater erosion are both essentially independent of the traditionally studied crater diameter/frequency relationships. Morphologies of terra craters near Mare Humorum suggest a young, non-equilibrium crater population superposed on a perimordial population with about equilibrium proportions of fresh, moderately degraded and very degraded craters. The primordial population has been modified by pre-Imbrian or early Imbrian deposition of blanketing deposits. A comparative study of several crater degradation classifications indicates that all are essentially interchangeable.