Earth's Gravitational Potential Research Articles

Satellite Motion about an Oblate Earth

A general theory of the method of averaging is used to study the effect of the Earth's oblateness on the motion of an artificial satellite. The first-order harmonic J2 and the second-order harmonics J3 and J4 are included in the analysis. This paper illustrates a thorough and straightforward application of a second-order method of averaging to the orbital motion of a satellite about an oblate Earth with the well-known conventional orbital elements as the time-dependent variables. The first-order periodic variations together with the second-order averaged and secular variations are derived for all the six orbital elements. Essential transformations and initialization procedures are also outlined. T is well known that for a close-Earth satellite, the inclusion of the perturbations due to the Earth's oblateness and the atmospheric drag is of vital importance to predict the lifetime and the trajectory of the satellite accurately. Except for the pioneer work of Brouwer,1 Lane,2 and Cranford,3 relatively little analytical work has been done in studying the joint effects of the Earth's oblateness and atmospheric drag on an orbiting satellite. Although this study will not develop a joint treatment, it does present a second-order theory of an artificial satellite about an oblate Earth. This theory establishes a foundation for the joint treatment of the Earth's oblateness, atmospheric drag, and other perturbations (e.g., extra-terrestrial gravitation) through application of the general theory of the method of averaging. This approach is feasible not only because the transformations involved need not be canonical, but also because it provides a rigorous, systematic and straightforward procedure for studying a dynamical system perturbed by conservative and/or nonconservative forces. Further, this approach is also general in the sense that both the Von Zeipel method and the two-variable asymptotic expansion procedures are particular cases of the method of averaging.4'5 Specifically, the present task is to develop a second-order theory which includes the effects contributed by the second, third, and fourth harmonics of the Earth's gravitational potential by applying the general theory of the method of averaging with the conventional orbital elements as its variables. General perturbation theories applied to a close-Earth satellite under the influence of the oblateness of the Earth have been published by a number of authors to predict satellite motions. To name a few, second-order treatments have been given by Kozai,6 Brouwer,7 Blitzer,8 Kyner,9 Lorell et al.,10 and Arsenault et al. 11 These and a large number of other theories and their results have been compiled and compared in Ref. 11. Kozai6 used an averaging technique to study the second-order solutions by introducing his mean elements. Brouwer7 used the Von Zeipel method to study the same problem. Blitzer8 introduced dimensionless variables and adopted the method of Bogoliuboff and Mitropolsky12 to obtain a second-order solution. Kyner9 used the method of averaging to study the problem with the use of spherical coordinates and angular momentum. In Ref. 10, the authors also applied the method of averaging to obtain the averaged variational equations to the second order. In their analysis, however, the second harmonic of the Earth's gravitational potential was the only perturbation under consideration and the mean anomaly, rather than time t, was treated as the

Fifteenth-order harmonics in the geopotential

THE Earth's gravitational potential is usually expressed as a double infinite series of spherical harmonics. Analysis of thousands of photographic and laser observations of satellites, together with gravity measurements has yielded global models of the geoid1–3 accurate to about 3 m on average. The harmonic coefficients have been evaluated complete up to degree and order 16 in the Goddard Earth Models (GEM) 4 and 6 (refs 1 and 2), and up to degree and order 18 in the Smithsonian Standard Earth III(SSE)3. Models currently being evaluated will extend to degree 25 (D. E. Smith, personal communication). Although these comprehensive global solutions give the overall geoid shape or gravitational field quite accurately, the individual harmonic coefficients of order between 10 and 20 are poorly determined among the matrix of 300 or more coefficients, and accurate methods for evaluating coefficients of a particular order are badly needed.

Basic theory for PROD, a program for computing the development of satellite orbits

This paper presents the basic theory underlying the computer program PROD for predicting the long-term development of drag-free orbits of eccentricity up to about 0.9 under the influence of the Earth's gravitational potential and the gravitational attractions of the Sun and Moon. To prevent excessive use of computer time, the disturbing functions are averaged analytically with respect to the mean anomaly of the satellite and recurrence relations are given wherever possible for the evaluation of the various coefficients.

A geopotential model (APL 5.0-1967) determined from satellite Doppler data at seven inclinations

Results of a determination of the nonzonal harmonic coefficients of the earth's gravitational potential are presented. This model is based on an analysis of Doppler data obtained from tracking satellites at seven different inclinations. These coefficients are combined with a set of zonal harmonic coefficients determined by R. J. Anderle and S. J. Smith of the Naval Weapons Laboratory to produce a geoidal-heights contour map.

Equations for 15th-Order Harmonics in the Geopotential

BY analysing the orbit of Ariel 3, Gooding1 obtained two equations between 15th-order coefficients in the Earth's gravitational potential from the changes in orbital inclination as the satellite passed through 15th-order resonance (when the successive tracks over the Earth are exactly 24° apart). Gooding's equations, simplified by omitting terms which should be small, are: illustration where the quantities C and S are normalized coefficients in the infinite series expansion of the geopotential, as defined, for example, by Gaposchkin and Lambeck2. Open image in new window

Earth Satellites and the Gravitational Potential

ANALYSIS of the orbit of the artificial Earth satellite Ariel 3 (1967-42A) at the Royal Aircraft Establishment1 has shown that useful determinations of the 15th-order harmonics of the Earth's gravitational potential can be made if the analysis is carried out for the period when the orbit passes through resonance with these harmonics. Several satellites have orbits suitable for such studies.

Representation of the Earth's gravitational potential

The paper represents the Earth's gravitational potentialV, outside a sphere bounding the Earth, by means of its difference δV from the author's spheroidal potential. The difference δV is in turn represented as arising from a surface density σ on the sphere bounding the Earth. Because of the slow decrease with ordern of the normalized coefficients in the spherical harmonic expansion ofV, the density anomalies from which the higher coefficients arise must occur in regions close to the Earth's surface. The surface density σ is thus an idealization of the product of the density anomaly Δϱ and the crustal thicknessb. Values of σ are computed from potential coefficients obtained from two sources, Rapp and the Smithsonian Astrophysical Observatory. The two sources give qualitative agreement for the values of σ and for its contour map. The numerical values obtained for σ are compatible with the idea that the responsible density anomalies are reasonably small, i.e., less than 0.05 g/cm3, and occur in the crust alone.

Satellite geodesy: Interpretation

This report outlines the activity during the past 4 years in the area of interpreting geodetic data derived through procedures of satellite geodesy. No attempt has been made to synthesize the ideas of the various investigators; the report simply points out where the activity lies and briefly what it is.The procedures of dynamic satellite geodesy generally analyze the variations of the orbital elements of artifical satellites for various quantities of interest. Such analysis may yield estimates of the spherical harmonic coefficients of the earth's gravitational potential, their time variation, if any, observation station coordinates in a geocentric system, and other quantities on which the orbits and observation of satellites depend. Although many such quantities are of interest, in this paper we shall regard the potential coefficients and quantities derived from them as the basis for information that can be interpreted to learn more about the structure and behavior of the earth.

Depth of density variations responsible for features of the satellite geoid

By postulating that the non-hydrostatic features of the Earth's gravitational potential are due to harmonic undulations of a nearly spherical internal surface with a density contrast, and that the stresses arising at this surface are of comparable magnitudes for all harmonic terms, it is found that the density anomalies responsible for the geoid must be in the upper mantle, at depths less than 1000 km.

On the computation of the spherical harmonic terms needed during the numerical integration of the orbital motion of an artificial satellite

A method is presented for the accurate and efficient computation of the forces and their first derivatives arising from any number of zonal and tesseral terms in the Earth's gravitational potential. The basic formulae are recurrence relations between some solid spherical harmonics,V n,m, associated with the standard polynomial ones.

Comments on paper by Milo Wolff, ‘Direct measurements of the Earth's gravitational potential using a satellite pair’

Comments on paper by Milo Wolff, ‘Direct measurements of the Earth's gravitational potential using a satellite pair’

Orbit computation with the Vinti potential and universal variables

When the Vinti potential and spheroidal coordinates are used in the solution of the equations of motion of an Earth satellite, the Hamilton-Jacobi differential equation is separable, permitting computation of the free flight path in closed form. If the parameters are chosen to include the effect of the Jz and Js terms of the Earth's gravitational potential, the Vinti potential also allows for a major part of the effect of the term in J4. The model can serve as an improved reference orbit in perturbation theory, or it can be used to compute moderately accurate trajectories (extremely accurate over short time intervals) with a considerable saving of machine time over numerical integration. The development in universal variables is valid for orbits of eccentricity greater than unity as well as for periodic orbits. The main purpose of this paper is to generalize the results of Vinti and others to include the nonperiodic cases. Included is a description of a tested method for practical calculation of trajectories and a numerical example for illustration.

Direct measurements of the Earth's gravitational potential using a satellite pair

It is possible to easily measure variations of intensity of the earth's gravity field by orbiting two geometrically identical satellites spaced about 200 kilometers apart and equipped to measure their relative velocity. The velocity difference is related to the space gradient of the potential field, which is a totally different concept of measurement, compared with present methods that measure the long-term integrated effects of gravity on orbit elements. It is expected that one month of satellite time would yield one set of measurements of the gravity anomalies of the entire earth with sensitivity at least 1 mgal ten times better than present world surveys. The cost is comparatively small and, in view of cost reduction for other related programs, may even be a substantial saving in total.

Some transformation properties for the coefficients in a spherical harmonics expansion of the earth's external gravitational potential

The earth's external gravitational potential may be mathematically expressed as an infinite series of spherical harmonics. The coefficients in such series depend numerically on the location and orientation of the adopted system of reference with respect to the body of the earth. The present subject is to derive general formulae for the numerical changes of the harmonic coefficients due to both a re-orientation and a re-location of the reference frame. Such formulae could be of value in critical studies of the impact of artificial satellites on dynamical geodesy, especially when combining the results with those from terrestrial geodesy. Use has been made of a transformation property of spherical harmonics which seems to be not a part of standard mathematics. Some simple numerical examples are given. DOI: 10.1111/j.2153-3490.1969.tb00464.x

Determining the coefficients of the Legendre polynomial expansion of the Earth's gravitational potential: PMM vol. 33, n≗4, 1969, pp. 749–752

Determining the coefficients of the Legendre polynomial expansion of the Earth's gravitational potential: PMM vol. 33, n≗4, 1969, pp. 749–752

Some transformation properties for the Coefficients in a spherical harmonics expansion of the earth’s external gravitational potential

The earth's external gravitational potential may be mathematically expressed as an infinite series of spherical harmonics. The coefficients in such series depend numerically on the location and orientation of the adopted system of reference with respect to the body of the earth. The present subject is to derive general formulae for the numerical changes of the harmonic coefficients due to both a re-orientation and a re-location of the reference frame. Such formulae could be of value in critical studies of the impact of artificial satellites on dynamical geodesy, especially when combining the results with those from terrestrial geodesy. Use has been made of a transformation property of spherical harmonics which seems to be not a part of standard mathematics. Some simple numerical examples are given. DOI: 10.1111/j.2153-3490.1969.tb00464.x

New Evaluation of Odd Zonal Harmonics in the Geopotential

IN 1966 we determined the coefficients of the odd zonal harmonics in the Earth's gravitational potential from an analysis of fourteen satellite orbits1, chosen to give the widest and most uniform possible coverage in inclination to the equator and semi major axis. We found quite large values for the fifteenth and twenty-first harmonic coefficients, J15 and J21 in the usual notation1. This indicated the need to carry the evaluation to coefficients of still higher order.

Lifetimes of satellites in large-eccentricity orbits

A method has been developed for predicting the long-term development of orbits having moderate to large eccentricities (0.33 < e < 0.9) and semi major axes up to 10 Earth radii. For many orbits of this type the lifetime depends entirely on the interaction of the lunisolar perturbations with the effects of the Earth's gravitational potential. Lifetime estimates are given for ten satellites.

Odd zonal harmonics in the geopotential, determined from fourteen well-distributed satellite orbits

Coefficients of the odd zonal harmonics in the Earth's gravitational potential are evaluated by analysing the oscillations in orbital eccentricity of seventeen satellites, chosen to give the widest and most uniform possible distribution in inclination and semi major axis. Three of the satellites are excluded for various reasons; the other fourteen yield various sets of values for the odd zonal harmonic coefficients J 3, J 5, J 7… The best representations of the odd harmonics in the potential appear to be in terms of either seven or ten coefficients, assuming that higher-degree coefficients are zero. The two sets of values are: 7-Coefficient 10-Coefficient 10 6 J 3 −2.53 ± 0.02 −2.50 ± 0.01 10 6 J 5 −2.22 ± 0.04 −0.26 ± 0.01 10 6 J 7 −0.41 ± 0.06 −0.40 ± 0.02 10 6 J 9 +0.09 ± 0.06 0 ± 0.06 10 6 J 11 −0.14 ± 0.05 −0.27 ± 0.06 10 6 J 13 +0.29 ± 0.06 +0.36 ± 0.08 10 6 J 15 −0.40 ± 0.06 −0.65 ± 0.10 10 6 J 17 +0.30 ± 0.08 10 6 J 19 0 ± 0.11 10 6 J 21 +0.58 ± 0.11 The errors in the last few coefficients of each set may be slightly greater than the standard deviations suggest, because no allowance is made for the neglected higher harmonics. The large magnitudes of J 15 and J 21 are noteworthy: neither of these coefficients has been determined before. Previously accepted values of J 3 to J 9, are in agreement with those in the more complete sets of coefficients above.

Odd Zonal Harmonics in the Earth's Gravitational Potential, determined from Fourteen Well Distributed Satellite Orbits

THE zonal harmonics in the Earth's gravitational field indicate its variations with latitude, averaged over all longitudes. The odd zonal harmonics, of degree 3, 5, 7, …, specify that part of the gravitational field which is not symmetrical about the equator and gives rise to the “pear-shape” effect.