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NASA NTRS1973

Thermally driven mass flows in the convection zone of the sun

A formulation of the fluid dynamics of convective regions is developed which leads to an analytical description of the solar rotation, the Evershed flow, and the supergranulation. The starting point of the present formulation is the mixing length picture of convective equilibrium, but the earlier point mass model for convective molecules is replaced here by a model with both inertia and intrinsic moment of inertia. This extension introduces three rotational degrees of freedom into the dynamics of individual convective molecules, which enter into the dynamical equations for a mixing length fluid in the form of a separate vector field which we term the spin field. It is shown that for convective molecules having a spherically symmetric mass distribution, the spin field is proportional to the local vorticity.

Dijkhuis, G. C.↗

NASA NTRS2001

Pluto and Triton: Interactions Between Volatiles and Dynamics

Volatiles moving across the surfaces of Pluto and Triton can give rise to interesting dynamical consequences. Conversely, measurement of dynamical states can help constrain the movement of volatiles and interior structure of both bodies. Polar wander may theoretically occur on both Triton and Pluto. Triton's obliquity is low, so that the equatorial regions receive more insolation than the poles. Hence there is a tendency for nitrogen ice to sublime at the equator and condense at the poles, creating polar caps. If the nitrogen supply is large enough, then these caps could move in approximately 10(exp 5) years the global equivalent of 200 m of ice to the poles. At this point the equatorial moment of inertia becomes larger than the moment of inertia measured about the rotation axis, so that Triton overbalances and becomes dynamically unstable. The satellite then undergoes polar wander, restoring stability when the new equator contains the excess matter. Hence the pole may be continually wandering. Neptune raises a permanent tidal bulge on Triton, so that the satellite's surface is elongated like a football, with the long axis pointing at Neptune. This is expected to be the axis about which the pole wanders. Volatile migration would resurface the satellite to some depth and wandering would disturb leading side/trailing side crater statistics. Additional information is contained in the original extended abstract.

Rubincam, D. P.↗

NASA NTRS1980

A low-order global gravity field of Venus and dynamical implications

An estimate of sixth-degree and sixth-order harmonic coefficients of the global gravity field of Venus is obtained by processing the long periodic variations of the mean orbital elements of the Pioneer Venus orbiter. Approximately 220 days of data are included in this reduction, which provides almost complete longitudinal coverage. Oblateness is estimated to be -5.97 + or - 3.2 x 10 to the -6. It is noted that the amplitudes of other coefficients are similar to the predicted coefficients using Kaula's rule under equal stress assumption. Atmospheric density values as a function of altitude are obtained to help model drag perturbation. A radial acceleration map at 100 km above the Venus surface is generated, and correlation between gravity anomalies and major topographic features is observed. Spectral analysis of the harmonic model suggests that the interior density anomalies are like those on earth. The orientation angles of the principal axes of the moments of inertia are computed, and deviation of the maximum moment of inertia axis from the spin axis is observed to be small (less than 5 deg). It is found that the minimum moment of inertia axis passes through the Aphrodite and Beta regions.

Ananda, M. P.↗

DOE OSTI2026

Quantum Choreography of the Nucleus: Rotations, Vibrations, and Emergent Structure

Abstract Nuclei are complex many-body quantum systems where interactions of the neutrons and protons via the strong, the weak, and the electromagnetic forces lead to the emergence of simple patterns of energy states that have been described by various theoretical approaches. One of the goals of all the theoretical models is the development of a universal theory that can be applied across the entire chart of nuclides. Significant progress has been made by experiments as well as the increasing sophistication of models, but a universal theory has yet to be established. A recent review of nuclei in the Z = 50–82 region of the chart of nuclides has analyzed all the available compiled data from several decades of studies towards a clarification of the low-lying structure of nuclei. Other reviews have reported and explained the emergence of multiple different shapes in nuclei at somewhat higher excitation energies than the ground state. Some have challenged the interpretation of the first excited K π = 0 + band as a vibration of ground state. This work attempts to provide a guide to determining the nature of the first excited K π = 0 + band in nuclei by the combined use of nuclear lifetimes, energy level evolutions, dynamic moments of inertia, and intrinsic quadrupole moments extracted from transition probabilities. The result is that for a subset of the nuclei in this region, the K π = 0 + band is consistent with the traditional 𝛽-vibration description of an oscillation built on the ground state.

CHFB+5DCH↗

NASA NTRS2012

Algorithms for Determining Physical Responses of Structures Under Load

Ultra-efficient real-time structural monitoring algorithms have been developed to provide extensive information about the physical response of structures under load. These algorithms are driven by actual strain data to measure accurately local strains at multiple locations on the surface of a structure. Through a single point load calibration test, these structural strains are then used to calculate key physical properties of the structure at each measurement location. Such properties include the structure s flexural rigidity (the product of the structure's modulus of elasticity, and its moment of inertia) and the section modulus (the moment of inertia divided by the structure s half-depth). The resulting structural properties at each location can be used to determine the structure s bending moment, shear, and structural loads in real time while the structure is in service. The amount of structural information can be maximized through the use of highly multiplexed fiber Bragg grating technology using optical time domain reflectometry and optical frequency domain reflectometry, which can provide a local strain measurement every 10 mm on a single hair-sized optical fiber. Since local strain is used as input to the algorithms, this system serves multiple purposes of measuring strains and displacements, as well as determining structural bending moment, shear, and loads for assessing real-time structural health. The first step is to install a series of strain sensors on the structure s surface in such a way as to measure bending strains at desired locations. The next step is to perform a simple ground test calibration. For a beam of length l (see example), discretized into n sections and subjected to a tip load of P that places the beam in bending, the flexural rigidity of the beam can be experimentally determined at each measurement location x. The bending moment at each station can then be determined for any general set of loads applied during operation.

Richards, W. Lance↗

NASA NTRS1976

Centroid and moments of an area using a digitizer

The centroid and moments of an area program provides the centroid, moments of inertia, product of inertia, radii of gyration, and area of any closed planar geometric figure. The figure must be available in graphic form and is digitized once with chart digitizer (graphic tablet). The digitizer origin may be set anywhere on the digitizer table. After digitizing, fifteen quantities are calculated and displayed: (1) area (2) moment of inertia of area with respect to digitizer x-axis, (3) moment of inertia of area with respect to digitizer y-axis, (4) product of inertia of area with respect to digitizer axes, (5) first moment of x for digitizer axes, (6) first moment of y for digitizer axes, (7) x coordinate of centroid, (8) y coordinate of centroid, (9) moment of area inertia of with respect to x axis through centroid, (10) moment of inertia of area with respect to y axis through centroid, (11) product inertia of area with respect to x and y axes through centroid, (12) polar moment of inertia of area around centroid, (13) radius of gyration about digitizer x axis, (14) radius of gyration about digitizer y-axis; and (15) variance in the x-direction.

Patch, R. W.↗

NASA NTRS1943

The Effect of Mass Distribution on the Lateral Stability and Control Characteristics of an Airplane as Determined by Tests of a Model in the Free-Flight Tunnel

The effects of mass distribution on lateral stability and control characteristics of an airplane have been determined by flight tests of a model in the NACA free-flight tunnel. In the investigation, the rolling and yawing movements of inertia were increased from normal values to values up to five times normal. For each moment-of-inertia condition, combinations of dihedral and vertical-tail area representing a variety of airplane configurations were tested. The results of the flight tests of the model were correlated with calculated stability and control characteristics and, in general, good agreement was obtained. The tests showed the following effects of increased rolling and yawing moments of inertia: no appreciable change in spiral stability; reductions in oscillatory stability that were serious at high values of dihedral; a reduction in the sensitivity of the model to gust disturbances; and a reduction in rolling acceleration provided by the ailerons, which caused a marked increase in time to reach a given angle of bank. The general flight behavior of the model became worse with increasing moments of inertia but, with combinations of small effective dihedral and large vertical-tail area, satisfactory flight characteristics were obtained at all moment-of-inertia conditions.

STABILITY & CONTROL, LATERAL↗

NASA NTRS1981

Tidal variations of earth rotation

The periodic variations of the earths' rotation resulting from the tidal deformation of the earth by the sun and moon were rederived including terms with amplitudes of 0.002 millisec and greater. The series applies to the mantle, crust, and oceans which rotate together for characteristic tidal periods; the scaling parameter is the ratio of the fraction of the Love number producing tidal variations in the moment of inertia of the coupled mantle and oceans (k) to the dimensionless polar moment of inertia of the coupled moments (C). The lunar laser ranging data shows that k/C at monthly and fortnightly frequencies equals 0.99 + or - 0.15 and 0.99 + or - 0.20 as compared to the theoretical value of 0.94 + or - 0.04.

Yoder, C. F.↗

NASA NTRS1976

Flutter of asymmetrically swept wings

Two formulations of the oblique wing flutter problem are presented; one formulation allows only simple wing bending deformations and rigid body roll as degrees of freedom, while the second formulation includes a more complex bending-torsional deformation together with the roll freedom. Flutter is found to occur in two basic modes. The first mode is associated with wing bending-aircraft roll coupling and occurs at low values of reduced frequency. The second instability mode closely resembles a classical bending-torsion wing flutter event. This latter mode occurs at much higher reduced frequencies than the first. The occurrence of the bending-roll coupling mode is shown to lead to lower flutter speeds while the bending-torsion mode is associated with higher flutter speeds. The ratio of the wing mass moment of inertia in roll to the fuselage roll moment of inertia is found to be a major factor in the determination of which of the two instabilities is critical.

Weisshaar, T. A.↗

NASA NTRS1971

Angular momentum desaturation for Skylab using gravity gradient torques

An angular momentum desaturation method for momentum exchange devices of orbiting spacecraft is described. The specific application of the method is to the Skylab which contains three double-gimbaled control moment gyros for precise attitude control and maneuvering. It is assumed that the attitude reference is inertially fixed and that two of the vehicle principal moments of inertia are much larger than the third. Gravity gradient torques and resultant angular momentum accumulation are developed for small deviations from the reference. The assumed moment-of-inertia distribution allows desaturation about all axes with only two attitude angles each for the two axes with large moments of inertia. The necessary desaturation maneuvers can be decoupled for a special set of orbital coordinates. All maneuvers are made during the night portion of the orbit, and the percentage utilized for desaturation is selectable. Expressions for the attitude angle commands are developed assuming infinite vehicle rates. The effect of finite rates introduces an efficiency into the desaturation. Expressions for this efficiency are developed and means for compensation are treated. Arbitrary misalignments between geometric vehicle axes and principal moment-of-inertia axes are permissible. An angle bias about the sun line minimizes the angular momentum accumulation about the sun line projection into the orbital plane. Adaptive desaturation maneuver limiting consistent with the available maneuver momentum is included.

Kennel, H. F.↗

NASA NTRS1974

Experimental determination of airplane mass and inertial characteristics

Current practices are evaluated for experimentally determining airplane center of gravity, moments of inertia, and products of inertia. The techniques discussed are applicable to bodies other than airplanes. In pitching- and rolling-moment-of-inertia investigations with the airplane mounted on and pivoted about knife edges, the nonlinear spring moments that occur at large amplitudes of oscillation can be eliminated by using the proper spring configuration. The single-point suspension double-pendulum technique for obtaining yawing moments of inertia, products of inertia, and the inclination of the principal axis provides accurate results from yaw-mode oscillation data, provided that the sway-mode effects are minimized by proper suspension rig design. Rocking-mode effects in the data can be isolated.

Wolowicz, C. H.↗

DOE OSTI2025

A small core in Vesta inferred from Dawn’s observations

Vesta’s large-scale interior structure had previously been constrained primarily using the gravity and shape data from the Dawn mission. However, these data alone still allow a wide range of possibilities for the differentiation state of the body. The moment of inertia is arguably the most diagnostic parameter related to the radial density distribution of a planetary body, making it crucial for assessing the body’s state of internal differentiation. Determining the moment of inertia requires additional measurements of the amplitudes of small rotational motions, such as precession and nutation. Here we report an updated estimate of the moment of inertia of Vesta inferred from Dawn’s Doppler tracking via the Deep Space Network and onboard imaging data. The recovered value for Vesta’s normalized polar moment of inertia is $\overline{C}$/MR 2 = 0.4208 ± 0.0047 (where M is the mass of Vesta and R is the reference radius), which is only 6.6% lower than the homogeneous value of 0.4505. This value, combined with the gravity field and global shape, suggests that Vesta’s interior has limited density stratification beneath its howardite–eucrite–diogenite-dominated crust. We propose two possible origin scenarios that are consistent with the observed constraints. In the first scenario, Vesta’s interior did not undergo full differentiation due to late accretion. In the second scenario, Vesta originated as an impact remnant of a larger differentiated body re-accreted with non-chondritic bulk composition produced from a catastrophic impact. Vesta did not experience complete differentiation in either scenario, suggesting that its current state reflects a complex interplay between its accretion timing, thermal evolution, redistribution of 26 Al bearing melt and/or impact processes.

CNEOS 2014-01-08 bolide↗

NASA NTRS2015

Fuel Distribution Estimate via Spin Period to Precession Period Ratio for the Advanced Composition Explorer

The spin period to precession period ratio of a non-axisymmetric spin-stabilized spacecraft, the Advanced Composition Explorer (ACE), was used to estimate the remaining mass and distribution of fuel within its propulsion system. This analysis was undertaken once telemetry suggested that two of the four fuel tanks had no propellant remaining, contrary to pre-launch expectations of the propulsion system performance. Numerical integration of possible fuel distributions was used to calculate moments of inertia for the spinning spacecraft. A Fast Fourier Transform (FFT) of output from a dynamics simulation was employed to relate calculated moments of inertia to spin and precession periods. The resulting modeled ratios were compared to the actual spin period to precession period ratio derived from the effect of post-maneuver nutation angle on sun sensor measurements. A Monte Carlo search was performed to tune free parameters using the observed spin period to precession period ratio over the life of the mission. This novel analysis of spin and precession periods indicates that at the time of launch, propellant was distributed unevenly between the two pairs of fuel tanks, with one pair having approximately 20% more propellant than the other pair. Furthermore, it indicates the pair of the tanks with less fuel expelled all of its propellant by 2014 and that approximately 46 kg of propellant remains in the other two tanks, an amount that closely matches the operational fuel accounting estimate. Keywords: Fuel Distribution, Moments of Inertia, Precession, Spin, Nutation

Moment of Inertia↗

NASA NTRS1976

Does Mercury have a molten core

A feasible nonseismic observational experiment is proposed for determining the existence and extent of a conducting molten core within Mercury. This experiment would utilize the effects of a liquid core on the dynamics of Mercury's rotation; two necessary conditions for performing it are that the core must not follow the mantle's forced librations in longitude but must follow the mantle on the timescale of the 250,000-yr precession. A method is developed by assuming these conditions to be satisfied, and bounds are established on the core viscosity for which they are satisfied. It is shown that the value of the ratio of the moment of inertia of the mantle to the largest principal moment of inertia of the entire planet would indicate whether the core is most probably solid, partially fluid, or entirely fluid. Techniques are suggested for determining the unknowns required to compute the necessary ratio.

Peale, S. J.↗

NASA NTRS2012

Precision Mass Property Measurements Using a Five-Wire Torsion Pendulum

A method for measuring the moment of inertia of an object using a five-wire torsion pendulum design is described here. Typical moment of inertia measurement devices are capable of 1 part in 10(exp 3) accuracy and current state of the art techniques have capabilities of about one part in 10(exp 4). The five-wire apparatus design shows the prospect of improving on current state of the art. Current measurements using a laboratory prototype indicate a moment of inertia measurement precision better than a part in 10(exp 4). In addition, the apparatus is shown to be capable of measuring the mass center offset from the geometric center. Typical mass center measurement devices exhibit a measurement precision up to approximately 1 micrometer. Although the five-wire pendulum was not originally designed for mass center measurements, preliminary results indicate an apparatus with a similar design may have the potential of achieving state of the art precision.

Swank, Aaron J.↗

NASA NTRS2013

Lunar Science from Lunar Laser Ranging

Variations in rotation and orientation of the Moon are sensitive to solid-body tidal dissipation, dissipation due to relative motion at the fluid-core/solid-mantle boundary, tidal Love number k2, and moment of inertia differences. There is weaker sensitivity to flattening of the core/mantle boundary (CMB) and fluid core moment of inertia. Accurate Lunar Laser Ranging (LLR) measurements of the distance from observatories on the Earth to four retroreflector arrays on the Moon are sensitive to variations in lunar rotation, orientation and tidal displacements. Past solutions using the LLR data have given results for Love numbers plus dissipation due to solid-body tides and fluid core. Detection of the fluid core polar minus equatorial moment of inertia difference due to CMB flattening is weakly significant. This strengthens the case for a fluid lunar core. Future approaches are considered to detect a solid inner core.

Love numbers↗

NASA NTRS1995

Determination of Eros Physical Parameters for Near Earth Asteroid Rendezvous Orbit Phase Navigation

Navigation of the orbit phase of the Near Earth steroid Rendezvous (NEAR) mission will re,quire determination of certain physical parameters describing the size, shape, gravity field, attitude and inertial properties of Eros. Prior to launch, little was known about Eros except for its orbit which could be determined with high precision from ground based telescope observations. Radar bounce and light curve data provided a rough estimate of Eros shape and a fairly good estimate of the pole, prime meridian and spin rate. However, the determination of the NEAR spacecraft orbit requires a high precision model of Eros's physical parameters and the ground based data provides only marginal a priori information. Eros is the principal source of perturbations of the spacecraft's trajectory and the principal source of data for determining the orbit. The initial orbit determination strategy is therefore concerned with developing a precise model of Eros. The original plan for Eros orbital operations was to execute a series of rendezvous burns beginning on December 20,1998 and insert into a close Eros orbit in January 1999. As a result of an unplanned termination of the rendezvous burn on December 20, 1998, the NEAR spacecraft continued on its high velocity approach trajectory and passed within 3900 km of Eros on December 23, 1998. The planned rendezvous burn was delayed until January 3, 1999 which resulted in the spacecraft being placed on a trajectory that slowly returns to Eros with a subsequent delay of close Eros orbital operations until February 2001. The flyby of Eros provided a brief glimpse and allowed for a crude estimate of the pole, prime meridian and mass of Eros. More importantly for navigation, orbit determination software was executed in the landmark tracking mode to determine the spacecraft orbit and a preliminary shape and landmark data base has been obtained. The flyby also provided an opportunity to test orbit determination operational procedures that will be used in February of 2001. The initial attitude and spin rate of Eros, as well as estimates of reference landmark locations, are obtained from images of the asteroid. These initial estimates are used as a priori values for a more precise refinement of these parameters by the orbit determination software which combines optical measurements with Doppler tracking data to obtain solutions for the required parameters. As the spacecraft is maneuvered; closer to the asteroid, estimates of spacecraft state, asteroid attitude, solar pressure, landmark locations and Eros physical parameters including mass, moments of inertia and gravity harmonics are determined with increasing precision. The determination of the elements of the inertia tensor of the asteroid is critical to spacecraft orbit determination and prediction of the asteroid attitude. The moments of inertia about the principal axes are also of scientific interest since they provide some insight into the internal mass distribution. Determination of the principal axes moments of inertia will depend on observing free precession in the asteroid's attitude dynamics. Gravity harmonics are in themselves of interest to science. When compared with the asteroid shape, some insight may be obtained into Eros' internal structure. The location of the center of mass derived from the first degree harmonic coefficients give a direct indication of overall mass distribution. The second degree harmonic coefficients relate to the radial distribution of mass. Higher degree harmonics may be compared with surface features to gain additional insight into mass distribution. In this paper, estimates of Eros physical parameters obtained from the December 23,1998 flyby will be presented. This new knowledge will be applied to simplification of Eros orbital operations in February of 2001. The resulting revision to the orbit determination strategy will also be discussed.

Miller, J. K.↗

DOE OSTI2023

New insights into backbending in the symmetry-adapted shell-model framework

Here we provide insights into the backbending phenomenon within the symmetry-adapted framework which naturally describes the intrinsic deformation of atomic nuclei. For 20 Ne, the canonical example of backbending in light nuclei, the ab initio symmetry-adapted no-core shell model shows that while the energy spectrum replicates the backbending from experimental energies under the rotor-model assumption, there is no change in the intrinsic deformation or intrinsic spin of the yrast band around the backbend. For the traditional example of 48 Cr, computed in the valence shell with empirical interactions, we confirm a high-spin nucleus that is effectively near-spherical, in agreement with previous models. However, we find that this spherical distribution results, on average, from an almost equal mixing of deformed prolate shapes with deformed oblate shapes. Microscopic calculations confirm the importance of spin alignment and configuration mixing, but surprisingly unveil no anomalous increase in moment of inertia. Furthermore, this finding opens the path toward further understanding the rotational behavior and moment of inertia of medium-mass nuclei.

20 ≤ A ≤ 38↗