Development of a new theory for determination of geopotential from the orbital motion of artificial satellites

Abstract

A new theory has been developed to exploit the satellite data particularly the position vector and the relative velocity of a satellite in the problem of obtaining the terrestrial gravity field with special consideration to its localised anomalous features. The new theory makes use of the fact that the dynamical variable Hamiltonian, associated with the satellite motion is time-invariant in the ideal case when all the perturbing forces are neglected. With this as a working premise, it is possible to take into account the effects of perturbing forces such as lunar attraction, air drag, radiation pressure and solar attraction. The ideal case ignoring all the perturbing forces, here called the 'simplified theory' and the more factual case allowing for the effect of the important perturbing forces, here called the 'extended theory' are both discussed in detail. The potential function of the earth appears additively in the Hamiltonian function and can be determined from observations of the position vector and the relative velocity of a satellite at a number of points along a small segment of the orbit. Minimally, there must be as many observations as there are unknown coefficients in the expansion of geopotential but an abundance of measurements is desirable for the application of the least squares method. In case the position vector and the relative velocity of a satellite are not available as directly observed quantities, the equations can be expressed in terms of the orbital elements of the satellite. The theory emphasizes the local features of the gravity field by allowing for the fact that a satellite gives information weighted primarily by conditions in its immediate proximity and thus provides expressions for describing the gravitational potential of regions immediately below its orbit. Theoretically, it appears possible to cover the surface of the earth by overlapping expressions of this type and hence to obtain an adequate description of the gravity field of the earth. The equations of condition obtained when the theory is developed to include the effects of lunar attraction and air drag, are shown to remain valid when all the important perturbations; i.e., lunar attraction, air drag, radiation pressure, solar attraction, etc., are taken into consideration. The method of setting up the equations of condition appears to have the advantage of eliminating the necessity of quantifying the perturbing factors, thus enabling us to avoid some of the poorer approximations involved in the process. The new theory appears to offer the possibility of exploiting the 'short wavelength sensing potentiality' of the low altitude satellites which cannot be used with advantage in the perturbation theory. If the geopotential coefficients can be determined to a fairly high degree of accuracy, the theory theoretically has the potential for determining the time-variant part of the earth's gravity field and may be used to give some idea as to the differential rotation of the core and mantle if the core has a radial asymmetry of mass distribution as one resulting from convection currents within the core. For purposes of comparison, a short review of the existing method to determine the geopotential using perturbation theory, is included as well as the results obtained by some other investigators in the field.