Mean Moment Of Inertia Research Articles

Inversion of Venus internal structure based on geodetic data

Understanding the internal structure of Venus promotes the exploration of the evolutionary history of this planet. However, the existing research concerning the internal structure of Venus has not used any inversion methods. In this work we employed an inversion method to determine the internal structure of Venus using observational or hypothetical geodetic data; these data include mass, mean radius, mean moment of inertia and second degree tidal Love number k2. To determine the core state of Venus, we created two models of Venus, an isotropic 3-layer model with entire liquid core and an isotropic 4-layer model with liquid outer core and a solid inner core, assuming that the interior of Venus is spherically symmetric and in hydrostatic equilibrium. A series of the sensitivity analysis of interior structure parameters to the geodetic data considered in here shows that not all of the parameters can be constrained by the geodetic data from Venus. On this basis, a Markov Chain Monte Carlo algorithm was used to determine the posterior probability distribution and the optimal values of the internal structure parameters of Venus with the geodetic data. We found that the 3-layer model is more credible than the 4-layer model via currently geodetic data. For the assumption of the 3-layer model with the k2 = 0.295 ± 0.066, I/MR2 = 0.33 ± 0.0165, and kgm−3, the liquid iron-rich core of Venus has a radius of km, which suggests a larger core than previous research has indicated. The average density of the mantle and liquid core of Venus are and kgm−3, respectively.

On the principal building blocks of Mars and Earth

The terrestrial planets are believed to have been formed from primitive material sampling a broad region of the inner solar system. Several meteoritic mixing models attempting to reconcile isotopic characteristics of Mars and Earth have been proposed, but, because of the inherent non-uniqueness of these solutions, additional independent observations are required to resolve the question of the primary building blocks of the terrestrial planets. Here, we consider existing isotopic measurements of Δ′17O, ϵ48Ca, ϵ50Ti, ϵ54Cr, ϵ62Ni, and ϵ84Sr for primitive chondrites and differentiated achondrites and mix these stochastically to reproduce the isotopic signatures of Mars and Earth. For both planets we observe ∼ 105 unique mixing solutions out of 108 random meteoritic mixtures, which are categorised into distinct clusters of mixtures using principal component analysis. The large number of solutions implies that isotopic data alone are insufficient to resolve the building blocks of the terrestrial planets. To further discriminate between isotopically valid mixtures, each mixture is converted into a core and mantle component via mass balance for which geophysical properties are computed and compared to observations. For Mars, the geophysical parameters include mean density, mean moment of inertia, and tidal response, whereas for Earth upper mantle Mg/(Mg+Fe) ratio and core size are employed. The results show that Mars requires an oxidised, FeO-rich differentiated object next to chondritic material as main building blocks. In contrast, Earth's origin remains enigmatic. From a redox perspective, it appears inescapable that enstatite chondrite-like matter constitutes a dominant proportion of the building blocks from which Earth is made. The apparent need for compositionally distinct building blocks for Mars and Earth suggests that dissimilar planetesimal reservoirs were maintained in the inner Solar System during accretion.

Viscoelastic Tides of Mercury and the Determination of its Inner Core Size

AbstractWe computed interior structure models of Mercury and analyzed their viscoelastic tidal response. The models are consistent with MErcury Surface, Space Environment, GEochemistry, and Ranging mission inferences of mean density, mean moment of inertia, moment of inertia of mantle and crust, and tidal Love number k2. Based on these constraints we predict the tidal Love number h2 to be in the range from 0.77 to 0.93. Using an Andrade rheology for the mantle the tidal phase‐lag is predicted to be 4° at maximum. The corresponding tidal dissipation in Mercury's silicate mantle induces a surface heat flux smaller than 0.16 mW/m2. We show that, independent of the adopted mantle rheological model, the ratio of the tidal Love numbers h2 and k2 provides a better constraint on the maximum inner core size with respect to other geodetic parameters (e.g., libration amplitude or a single Love number), provided it responds elastically to the solar tide. For inner cores larger than 700 km, and with the expected determination of h2 from the upcoming BepiColombo mission, it may be possible to constrain the size of the inner core. The measurement of the tidal phase‐lag with an accuracy better than ≈0.5° would further allow constraining the temperature at the core‐mantle boundary for a given grain size and therefore improve our understanding of the physical structure of Mercury's core.

Prepubertal Di-n-Butyl Phthalate Exposure Alters Sertoli and Leydig Cell Function and Lowers Bone Density in Adult Male Mice.

Phthalate exposure impairs testis development and function; however, whether phthalates affect nonreproductive functions is not well understood. To investigate this, C57BL/6J mice were fed 1-500 mg di-n-butyl phthalate (DBP) in corn oil, or vehicle only, daily from 4 to 14 days, after which tissues were collected (prepubertal study). Another group was fed 1-500 mg/kg·d DBP from 4 to 21 days and then maintained untreated until 8 weeks for determination of adult consequences of prepubertal exposure. Bones were assessed by microcomputed tomography and dual-energy X-ray absorptiometry and T by RIA. DBP exposure decreased prepubertal femur length, marrow volume, and mean moment of inertia. Adult animals exposed prepubertally to low DBP doses had lower bone mineral content and bone mineral density and less lean tissue mass than vehicle-treated animals. Altered dynamics of the emerging Leydig population were found in 14-day-old animals fed 100-500 mg/kg·d DBP. Adult mice had variable testicular T and serum T and LH concentrations after prepubertal exposure and a dose-dependent reduction in cytochrome p450, family 11, subfamily A, polypeptide 1. Insulin-like 3 was detected in Sertoli cells of adult mice administered the highest dose of 500 mg/kg·d DBP prepubertally, a finding supported by the induction of insulin-like 3 expression in TM4 cells exposed to 50 μM, but not 5 μM, DBP. We propose that low-dose DBP exposure is detrimental to bone but that normal bone mineral density/bone mineral content after high-dose DBP exposure reflects changes in testicular somatic cells that confer protection to bones. These findings will fuel concerns that low-dose DBP exposure impacts health beyond the reproductive axis.

Bifurcations of the relative equilibria of a gyrostat satellite for a special case of the alignment of its gyrostatic moment

The bifurcations of the equilibria of a gyrostat satellite with a centre of mass moving uniformly in a circular Kepler orbit around an attracting centre are investigated. It is assumed that the axis of rotation of a statically and dynamically balanced flywheel rotating at a constant relative angular velocity is fixed in the principal central plane of inertia of the gyrostat containing the axis of its mean moment of inertia and that it is not collinear with any principal central axis of inertia of the system. The problem is solved in a direct formulation, that is, the whole set of equilibria with respect to the orbital system of coordinates of the gyrostat satellite is determined using the given moments of inertia, the value of the gyroscopic moment and the direction cosines of the axis of rotation of the flywheel and the changes in this set are investigated as a function of the bifurcation parameter, that is, the magnitude of the gyrostatic moment of the system. A parametric analysis of the relative equilibria of the three possible classes of equilibria for a system in a circular orbit in a central Newtonian force field is carried out using computer algebra facilities.

Models of density and mechanical Q-factor distributions according to new data on the nutations and free oscillations of the Earth: 1. Ambiguity in the inverse problem

Ambiguity in the inverse problem of retrieval of the mechanical parameters of the Earth’s shell and core from the set of data on the velocities V p and V S , of longitudinal and transverse seismic body waves, the frequencies f i and quality factors Q i , of free oscillations, and the amplitudes and phases of forced nutation is considered. The numerical experiments show that the inverse problem of simultaneous retrieval of the density profile ρ in the mantle-liquid core system and the mechanical quality factor Q μ of the mantle (if the total mass M and the total mean moment of inertia I of the Earth, and V p and V S are constant at all depths) has most unstable solutions. An example of depth distributions of ρ and Q μ which are alternative to the well-known PREM model is given. In these distributions, the values of M and I and the velocities V p and V S at all depths for the period of oscillations T = 1 s exactly coincide with their counterparts yielded by PREM model (T = 1 s); however, the maximum deviations of the ρ and Q μ profiles from those in the PREM model are about 3% and 40%, respectively; the mass and the moment of inertia of the liquid core are smaller than those for the PREM model by 0.75% and 0.63%, respectively. In this model, the root mean square (rms) deviations of all the measured values of f i and Q i from their values predicted by theory are half to third the corresponding values in the PREM model; the values of Δ for natural frequencies of the fundamental tone and overtones of radial oscillations, the fundamental tones of torsional oscillations, and the fundamental tones of spheroidal oscillations, which are measured with the highest relative accuracy, are smaller by a factor of 30, 6.6, and 2 than those in the PREM model, respectively. Such a large ambiguity in the solution of the inverse problem indicates that the current models of the depth distribution of density have relatively low accuracy, and the models of the depth distribution of the mechanical Q in the mantle are extremely unreliable. It is shown that the ambiguity in the models of depth distribution of density considerably decreases after the new data on the amplitudes and phases of the forced nutation of the Earth are taken into account. Using the same data, one may also refine by several times the recent estimates of the creep function for the lower mantle within a wide interval of periods ranging from a second to a day.

Effect of molecular structure nonrigidity on anisotropy of dipole absorption cross section

The influence of the nonrigidity of free complex asymmetric top molecules on the anisotropy of the absorption cross section over various rotational configurations has been studied under conditions where the total angular momentum is conserved. It is shown that the nonrigidity-induced cross section anisotropy is maximal for the optical transition with the dipole moment collinear to the axis of the molecule’s mean moment of inertia and rapidly decreases with the rise of the intramolecular flexibility. For all types of tops, disturbances from internal motions are stronger in the region where the principal axes of inertia are collinear with the total rotational angular momentum.

On the Hydrostatic Theory of the Figure of the Earth

Summary A table is provided of the external and dynamical ellipticities of the Earth for different values of the ratio I/Mb2, I being the mean moment of inertia and b the mean radius, on the first order Radau theory. A simplified model is used to find the corrections for the slight inaccuracy of the Radau approximation and the terms of the second order. The dynamical ellipticity and J2 as given by artificial satellites are used to estimate I/Mb2, and it is shown that on the hydrostatic theory 106J2 would be 1072.1 ± 0.4, in contradiction to the direct determination 1082.78 ± 0.05. The departures from the hydrostatic state indicated by J2 and J3 imply stress differences of 4 × 107 to 8 × 107 dyn/cm2 in the interior.

Models, figures, and gravitational moments of Jupiter’s satellites Io and Europa

Two types of trial three-layer models have been constructed for the satellites Io and Europa. In the models of the first type (Io1 and E1), the cores are assumed to consist of eutectic Fe-FeS melt with the densities ρ1 = 5.15 g cm−3 (Io1) and 5.2 g cm−3 (E1). In the models of the second type (Io3 and E3), the cores consist of FeS with an admixture of nickel and have the density ρ1 = 4.6 g cm−3. The approach used here differs from that used previously both in chosen model chemical composition of these satellites and in boundary conditions imposed on the models. The most important question to be answered by modeling the internal structure of the Galilean satellites is that of the condensate composition at the formation epoch of Jupiter’s system. Jupiter’s core and the Galilean satellites were formed from the condensate. Ganymede and Callisto were formed fairly far from Jupiter in zones with temperatures below the water condensation temperature, water was entirely incorporated into their bodies, and their modeling showed the mass ratio of the icy (I) component to the rock (R) component in them to be I/R ∼ 1. The R composition must be clarified by modeling Io and Europa. The models of the second type (Io3 and E3), in which the satellite cores consist of FeS, yield 25.2 (Io3) and 22.8 (E3) for the core masses (in weight %). In discussing the R composition, we note that, theoretically, the material of which the FeS+Ni core can consist in the R accounts for ∼25.4% of the satellite mass. In this case, such an important parameter as the mantle silicate iron saturation is Fe# = 0.265. The Io3 and E3 models agree well with this theoretical prediction. The models of the first and second types differ markedly in core radius; thus, in principle, the R composition in the formation zone of Jupiter’s system can be clarified by geophysical studies. Another problem studied here is that of the error made in modeling Io and Europa using the Radau-Darvin formula when passing from the Love number k2 to the nondimensional polar moment of inertia \(\bar C\). For Io, the Radau-Darvin formula underestimates the true value of \(\bar C\) by one and a half units in the third decimal digit. For Europa, this effect is approximately a factor of 3 smaller, which roughly corresponds to a ratio of the small parameters for the satellites under consideration αIo/αEuropa ∼ 3.4. In modeling the internal structure of the satellites, the core radius depends strongly on both the mean moment of inertia I* and k2. Therefore, the above discrepancy in \(\bar C\) for Io is appreciable.

Geophysical constraints on the composition and structure of the Martian interior

Most recent Martian interior structure models are based on the planet's polar moment of inertia C, although the mean moment of inertia I is required for constructing spherically symmetric models of planetary interiors. Using the improved value of C/MpRp2 recently obtained from a combined reanalysis of the entire set of radio science data collected during the last three decades and accounting for the rotationally and topographically induced shape of the planet's gravitational field, we find a mean moment‐of‐inertia factor of I/MpRp2 = 0.3635 ± 0.0012. The new lower value suggests a core radius several tens of kilometers larger if other parameters like core density, crust density, and crust thickness are fixed. It further implies that the Martian crust is several tens of kilometers thicker than previously thought if crust and mantle densities and core size are given. Moreover, the Martian mantle may be less dense, about several tens of kg m−3, with a smaller iron content than previously thought if crust thickness and core size are specified. The mantle density of Mars is relatively well determined by the planet's moment of inertia factor if crust thickness and density are specified.

Does the Moon possess a molten core? Probing the deep lunar interior using results from LLR and Lunar Prospector

It is the main purpose of this study to examine the deeper structure of the Moon in the light of four numbers. These are lunar mass M, mean moment of inertia I, second degree tidal Love number k2, and the quality factor Q, accounting for tidal dissipation within the solid body of the Moon. The former two have been measured by Lunar Prospector to high precision, and more than 30 years of lunar laser ranging (LLR) data have led to an estimate of the second degree tidal Love number and quality factor. The inverse problem dealt with here of obtaining information on the lunar density and S wave velocity profile from the four numbers follows our earlier investigations by employing an inverse Monte Carlo sampling method. We present a novel way of analyzing the outcome using the Bayes factor. The advantage lies in the fact that rather than just looking at a subset of sampled models, we investigate all the information sampled in different runs, i.e., take into account all samples, in order to estimate their relative plausibility. The most likely outcome of our study, based on the data, their uncertainties, and prior information, is a central core with a most probable S wave velocity close to 0 km/s, density of ∼7.2 g/cm3 and radius of about 350 km. This is interpreted as implying the presence of a molten or partially molten Fe core, in line with evidence presented earlier using LLR regarding the dissipation within the Moon.

Internal Structure Models and Dynamical Parameters of the Galilean Satellites

Radio Doppler data generated by the Deep Space Network (DSN) from the recent encounters of the Galileo spacecraft with the Galilean satellites have been used to determine the mass (GM) and unnormalized quadruple gravity coefficients in the external gravitational fields of the Galilean satellites. Therefore, a series of internal structure models, which satisfy the given constraints for the mean density and the mean moment of inertia, can be presented by solving Emden equations. And some dynamical parameters can also be calculated based on the density distributions given by internal structure models.

Constraints on density models from radial moments: Applications to Earth, Moon, and Mars

Few objective constraints exist on radial density variations in planetary interiors. Even if the mean density and mean moment of inertia were known with perfect accuracy, they would only provide integral constraints on the density models. However, among the family of monotonic radial density models with a given mean density and mean moment of inertia, the simple two‐layer piecewise constant models have useful extremal properties. The model has only three parameters, an inner density, an outer density, and a transition radius. Once the radial moment constraints are applied, there is only one remaining degree of freedom. If the outer region density is somehow specified, then the inner region density of the two‐layer model provides a firm lower bound on central density of monotonic models. Likewise, if an upper bound on central density can be provided, then the two‐layer model provides a lower bound on outer region density. An envelope of acceptable density models can be generated by scanning the transition depth from the center to the surface. Any monotonic model with specified mean density and mean inertial moment must lie within that envelope. Resulting extremal density envelopes for the Earth, Moon, and Mars are compared to published radial density profiles. For the Moon, the moment constraints are quite restrictive. For the Earth, which is more centrally condensed, the allowed envelope of density profiles is rather broad. Mars is an intermediate case. Present geodetic and astrometric observations only constrain the Martian mean moment of inertia to lie somewhere in the range 0.325 ≤ I/MR2 ≤ 0.365. However, our analysis shows that any value within that range can be accommodated without invoking geochemically implausible density minima or maxima.

Geodetic constraints on the composition of Mars

The mean density and mean moment of inertia are remotely accessible geodetic parameters which provide two integral constraints on the radial density profile of Mars. The mean density is presently well known, but until the axial precession rate is better determined, the moment of inertia will remain uncertain. Values anywhere in the interval from 0.345 MR2 to 0.365 MR2 can be obtained from the observed gravitational field, depending on how that field is partitioned into hydrostatic and nonhydrostatic components. Even if the moment were accurately known, a considerable degree of nonuniqueness would still remain concerning the density and composition. This nonuniqueness is illustrated by construction of a family of orthogonal polynomials in the normalized radius which make no contribution to either the mean density or the mean inertial moment. If the mean moment is near to the lower end of its plausible range, the nonuniqueness in density and composition is considerably diminished. In order to avoid unrealistically high densities in the deep interior, the Martian mantle must have a density appreciably lower than the terrestrial mantle and must consequently be significantly depleted in iron. On the other hand, if the mean moment is near the upper end of its plausible range, a much broader range of density profiles can be accommodated, including (but not restricted to) iron enriched mantles.

On coil–stretch transitions in dilute polymer solutions

In this paper we examine molecular stretching in the inception of uniaxial elongational flow of dilute polymer solutions. The polymer molecules are modeled as bead–spring chains with finitely extensible nonlinear elastic springs, and we use the Peterlin approximation. This work is distinguished from earlier work because we model the macromolecules with chains instead of dumbbells, and we examine the time dependence of three average quantities describing the chain conformation in unsteady flows: root-mean-square end-to-end distance, root-mean-square extensions of the individual links, and mean moment of inertia about the axis of elongation. We observe a gradual transition from the coiled equilibrium state of the chain to the stretched state after the inception of strong uniaxial elongational flow, and we describe the nature of this transition which takes place in roughly four stages: I equilibrium coil; II deformed coil; III spring stretched (‘‘locally unraveled’’); and IV unfolded chain. Inclusion of hydrodynamic interaction changes the macromolecular response quantitatively but not qualitatively.

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 δ(I/Ma 2) and that of the core density δπc 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 δπc 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, β, γ, 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.

Dielectric relaxation of some rigid polar molecules in a polystyrene matrix

Dielectric absorption studies have been carried out on solutions of twelve polar solutes dispersed in an atactic polystyrene matrix over a range of temperatures in the frequency region 50 to 10 5 Hz and in one case also in the range of 10 4 to 10 7 Hz. Rigid molecules of varying size and molecular dipole moments halobenzenes, p-halotoluenes, p-halobiphenyls, p-nitrobiphenyl and p-bromoethylbenzene - have been examined. All these solute molecules have in common the fact that the molecular dipole moment lies along the main principal axis. The relaxation times and energy barrier parameters for molecular dipole relaxations are determined. The Eyring enthalpy of activation, ΔH E , is found to range from 9 for fluorobenzene to 102 kJ mol −1 for p-iodobiphenyl. Relaxation data of these molecules indicate a linear correlation between enthalpy and entropy of activation, and the former is found to depend upon the volume needed for the reorientation of molecules. Present results for halobenzenes and p-halotoluenes reveal a linear dependence of log (relaxation time) on the mean moment of inertia. The enthalpy data for these rigid molecules are useful in the study of flexible molecules when a decision has to be made as to whether the absorption is to be ascribed to molecular or intramolecular or overlap of the two relaxation processes.

On the Routh theorem and the Chetaev method for constructing the liapunov function from the integrals of the equations of motion: PMM vol. 33, n≗5, 1969, pp. 904–912

The bifurcations of the equilibria of a gyrostat satellite with a centre of mass moving uniformly in a circular Kepler orbit around an attracting centre are investigated. It is assumed that the axis of rotation of a statically and dynamically balanced flywheel rotating at a constant relative angular velocity is fixed in the principal central plane of inertia of the gyrostat containing the axis of its mean moment of inertia and that it is not collinear with any principal central axis of inertia of the system. The problem is solved in a direct formulation, that is, the whole set of equilibria with respect to the orbital system of coordinates of the gyrostat satellite is determined using the given moments of inertia, the value of the gyroscopic moment and the direction cosines of the axis of rotation of the flywheel and the changes in this set are investigated as a function of the bifurcation parameter, that is, the magnitude of the gyrostatic moment of the system. A parametric analysis of the relative equilibria of the three possible classes of equilibria for a system in a circular orbit in a central Newtonian force field is carried out using computer algebra facilities.

Water-proofing concrete: Anon. ( Sci. Amer., cx, No. 21, 425.)

The bifurcations of the equilibria of a gyrostat satellite with a centre of mass moving uniformly in a circular Kepler orbit around an attracting centre are investigated. It is assumed that the axis of rotation of a statically and dynamically balanced flywheel rotating at a constant relative angular velocity is fixed in the principal central plane of inertia of the gyrostat containing the axis of its mean moment of inertia and that it is not collinear with any principal central axis of inertia of the system. The problem is solved in a direct formulation, that is, the whole set of equilibria with respect to the orbital system of coordinates of the gyrostat satellite is determined using the given moments of inertia, the value of the gyroscopic moment and the direction cosines of the axis of rotation of the flywheel and the changes in this set are investigated as a function of the bifurcation parameter, that is, the magnitude of the gyrostatic moment of the system. A parametric analysis of the relative equilibria of the three possible classes of equilibria for a system in a circular orbit in a central Newtonian force field is carried out using computer algebra facilities.