Gravity harmonics from a resonant two-hour satellite
Resonant gravity harmonics of TIROS 9 satellite
Cole, I. J.↗
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Use of non-spherical gravity harmonics for relative motion GN/C (category 2)
Detailed analysis of the Automatic Rendezvous and Capture problem indicate a need for three different regions of mathematical description for the GN&C algorithms: (1) multi-vehicle orbital mechanics to the rendezvous interface point, i.e., within 100 nm; (2) relative motion solutions (such as Clohessy-Wiltshire type) from the far-field to the near-field interface, i.e., within 1 nm and; (3) close proximity motion - the near-field motion where the relative differences in the gravitational and orbit inertial accelerations can be neglected from the equations of motion. Limit boundaries to these regions can be precisely defined by further analysis and will be functions of the tracking measurement accuracies and the computer resources available for the solution of the algorithms. This paper analyzes the relative motion in Regions 2 and 3 above and present the derivation and discussion of the general case of non-spherical gravitational perturbed relative motion. Mathematical deviations from the numerically integrated spherical gravity case and solutions from the Clohessy-Wiltshire equations are presented in the analysis. Based upon this preliminary analysis, it is recommended that further efforts be used to assess the relative position and velocity differences in Region 2 due to non-spherical gravity harmonics and that viable GN&C algorithms be developed to include these gravity perturbations (especially the effects of the first gravity harmonic, J2).
Henderson, David M.↗
Determination of low-order resonant gravity harmonics from the drift of two Russian 12- hour satellites.
Low order resonant gravity harmonics determination from Soviet 12-hour satellites drift
Wagner, C. A.↗
Resonant gravity harmonics from three and a half years of tracking data on three 24 hour satellites
Resonant gravity harmonics of synchronous satellites
Wagner, C. A.↗
Venus Spherical Harmonic Gravity Model to Degree and Order 60
The Magellan and Pioneer Venus Orbiter radiometric tracking data sets have been combined to produce a 60th degree and order spherical harmonic gravity field.
x-band↗
Venus spherical harmonic gravity model to degree and order 60
The Magellan and Pioneer Venus Orbiter radiometric tracking data sets have been combined to produce a 60th degree and order spherical harmonic gravity field. The Magellan data include the high-precision X-band gravity tracking from September 1992 to May 1993 and post-aerobraking data up to January 5, 1994. Gravity models are presented from the application of Kaula's power rule for Venus and an alternative a priori method using surface accelerations. Results are given as vertical gravity acceleration at the reference surface, geoid, vertical Bouguer, and vertical isostatic maps with errors for the vertical gravity and geoid maps included. Correlation of the gravity with topography for the different models is also discussed.
Konopliv, Alex S.↗
A new constraint on Saturn's zonal gravity harmonics from Voyager observations of an eccentric ringlet
The results obtained by Porco et al. (1984) from an analysis of the kinematics of an eccentric ringlet in Saturn's inner ring are used to add an additional constraint to the gravity solution of Null et al. (1981) and thus to derive the first experimental estimate of the Saturn's J6 gravitational field. On the assumption that the eccentricity of this ringlet is forced by the satellite Titan, the location of the Titan apsidal resonance is determined to an accuracy of + or - 13 km, which in turn provides a strong constraint on Saturn's zonal gravity harmonics. Combining this constraint with the results of Null et all. and adopting a priori values for J8, J10, and J12 based on interior models, new estimates are obtained for the first three zonal gravity coefficients J2, J4, and J6.
Nicholson, Philip D.↗
Zonal gravity harmonics from long satellite arcs by a seminumeric method.
A zonal geopotential is presented to degree 21 from evaluation of mean elements for 21 satellites, including two of low (less than 20 deg) inclination. Each satellite is represented by an arc of at least one apsidal rotation. The lengths range from 50 to 660 days. Differential correction of the initial elements in all of the arcs, together with radiation pressure and atmospheric drag coefficients, is accomplished simultaneously with the correction for the zonal harmonics. The satellite orbits and their variations are generated by numerical integration of the Lagrange equations for mean elements. Disturbances due to precession and nutation, atmospheric drag, radiation pressure, and lunisolar gravity are added at 1- to 8-day intervals in the integrated orbits.
Wagner, C. A.↗
Combined solution for low degree longitude harmonics of gravity from 12- and 24-hour satellites.
Low degree earth gravity harmonics effect on 12 and 24 hour orbits of high altitude communications satellites
Wagner, C. A.↗
Adjoint Sensitivity Analysis of Orbital Mechanics: Application to Computations of Observables' Partials with Respect to Harmonics of the Planetary Gravity Fields
An approach is presented to the inversion of gravity fields based on evaluation of partials of observables with respect to gravity harmonics using the solution of adjoint problem of orbital dynamics of the spacecraft. Corresponding adjoint operator is derived directly from the linear operator of the linearized forward problem of orbital dynamics. The resulting adjoint problem is similar to the forward problem and can be solved by the same methods. For given highest degree N of gravity harmonics desired, this method involves integration of N adjoint solutions as compared to integration of N2 partials of the forward solution with respect to gravity harmonics in the conventional approach. Thus, for higher resolution gravity models, this approach becomes increasingly more effective in terms of computer resources as compared to the approach based on the solution of the forward problem of orbital dynamics.
gravity fields↗
Lunar gravity - A harmonic analysis
A sixteenth-degree and sixteenth-order spherical harmonic lunar gravity field has been derived from the long-term Keplerian variations in the orbits of the Apollo subsatellites and Lunar Orbiter 5. This model resolves the major mascon gravity anomalies of the lunar near side and is in very good agreement with line-of-sight acceleration results. The far-side map shows the major ringed basins to be strong localized negative anomalies located in broad regions of positive gravity which correspond closely to the highlands. The rms pressure levels calculated from equivalent-surface height variations show that the moon and earth support nearly equal pressures, whereas Mars is appreciably stronger. The moon appears to support larger loads than earth owing to its weaker central gravity field and perhaps a colder upper lithosphere. Significant differences between the low-degree gravity and topography spectra indicate that the longer-wavelength topographic features are isostatically compensated.
Ferrari, A. J.↗
Effects of differential rotation on the gravitational figures of Jupiter and Saturn
It is assumed that observed zonal currents in the atmospheres of Jupiter and Saturn correspond to a state of permanent rotation, and that the angular velocity is constant on cylindrical surfaces parallel to the rotation axis. The equation of hydrostatic equilibrium for a rotating planet is solved under these restrictive assumptions, and the effect of the hypothesized rotation state on the planet's gravity harmonics and external shape is investigated. Spacecraft data on zonal currents are used to derive nearly model-independent corrections to the first four zonal gravity harmonic coefficients, which can be used to correct observed gravity harmonics to values appropriate for solid-body rotation. If the assumed rotation state is applicable, then zonal currents lead to measurable topography of isopycnic surfaces with respect to the reference figure defined by the magnetospheric rotation period and the gravity harmonics. The amplitude of the topography is on the order of 5 km for Jupiter and 60 km for Saturn.
Hubbard, W. B.↗
Combined solution for low degree longitude harmonics of gravity from 12 to 24 hour satellites
Combined solution for low degree longitude harmonics of gravity from twelve and twenty-four hour satellites
Wagner, C. A.↗
Analysis of harmonic spline gravity models for Venus and Mars
Methodology utilizing harmonic splines for determining the true gravity field from Line-Of-Sight (LOS) acceleration data from planetary spacecraft missions was tested. As is well known, the LOS data incorporate errors in the zero reference level that appear to be inherent in the processing procedure used to obtain the LOS vectors. The proposed method offers a solution to this problem. The harmonic spline program was converted from the VAX 11/780 to the Ridge 32C computer. The problem with the matrix inversion routine that improved inversion of the data matrices used in the Optimum Estimate program for global Earth studies was solved. The problem of obtaining a successful matrix inversion for a single rev supplemented by data for the two adjacent revs still remains.
Bowin, Carl↗
Venus gravity - A harmonic analysis
An improved model of Venusian global gravity has been obtained by fitting an eighteenth-degree and eighteenth-order spherical harmonic series to 78 orbital arcs of high altitude (950-1350 km at periapsis) tracking data and 351 orbital arcs of lower-altitude (150-200 km at periapsis) data from the Pioneer Venus Orbiter (PVO). Compared to a recently published tenth-degree model of Mottinger et al. (1985), which is based on the 78 high-altitude arcs only, the current model provides a significant improvement in resolution and fidelity. As a measure of this improvement, it is noted that for the low-altitude arcs alone, the variance of the residuals for the present model is reduced to 19 percent of the data variance, compared to 51 percent for the tenth-degree model. Venus differs significantly from the earth in that it exhibits a significant correlation between long-wavelength topography and gravity. The gravity/topography spectral admittances are inconsistent with either Airy or Pratt isostasy, but are consistent with dynamic support by mantle convection.
Bills, Bruce G.↗
Venus gravity - A harmonic analysis and geophysical implications
An improved theoretical model of Venusian global gravity has been obtained by fitting a tenth degree spherical harmonic series to 78 orbital arcs of Doppler tracking data from the Pioneer Venus Orbiter. Maps of the free-air anomaly and its formal error are presented. Isostatic anomaly and 'Geoid' maps are also presented, and their geophysical implications are discussed in details. Comparison with equivalent resolution topographic models reveals a strong correlation between long wavelength gravity and the topography of Venus. Analysis of the second degree harmonics showed two aspects of the orientation of the inertial axes of Venus: (1) a significant (about three degrees) departure of the axis of greatest inertia from the rotational axis; and (2) a near alignment of the axis of least inertia with the location of the subterrestrial point at the time of the next inferior conjunction with earth (December 16, 2101). A series of contour maps of the Venusian free-air anomalies is provided.
Mottinger, N. A.↗
Finite amplitude gravity waves: Harmonics, advective steepening, breaking and saturation
A simple theory is presented which determines details of the breaking and saturation of a gravity wave as it propagates upward in the atmosphere. Breaking and saturation are here due to nonlinear advection analogous to the breaching of a surface wave and to the breaking of a planetary wave. Much simplification is obtained by the assumption that in a wave packet consisting of a primary wave and its harmonics, the primary wave remains dominant. This assumption, referred to a quasi-monochromatic approximation, is suggested by observations. Determined by this approximate theory are: a detailed picture of the waveform as it steepens and breaks; harmonics of the wave; the turbulence generation; and an underlying relationship between superadiabatic lapse rate and saturation by wave-wave interactions.
Weinstock, J.↗
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.↗