Improved expressions for the ellipsoidal harmonic series representing the Earth gravitational potential and its first and second derivatives on and outside the reference ellipsoid of revolution

The Legendre functions of the second kind, renormalized by Jekeli, are considered in the external space on a set of ellipsoids of revolution which are confocal with respect to the normal ellipsoid. Among these ellipsoids a reference one is chosen which bounds the Earth. New expressions for the first and second order derivatives of the Legendre functions are derived. They depend on two very quickly convergent Gauss hypergeometric series which are obtained by transforming the slowly convergent initial hypergeometric series. The derived expressions are applied for constructing the ellipsoidal harmonic series for the Earth disturbing gravitational potential and its derivatives of the first and second orders. Since outside the chosen reference ellipsoid there are no Earth masses (as compared to the normal ellipsoid) then it is more appropriate for constructing the boundary-value equation and solving it on the basis of surface gravity data reduced to this ellipsoid.     

Compact non-singular expansions in the exterior space of a reference ellipsoid for the gravitational gradients in the local north-oriented ellipsoidal and spherical reference frames

Due to the complicated structure of their expressions, the ellipsoidal harmonic series for the derivatives of the Earth’s gravitational potential are commonly applied only on a reference ellipsoid. They depend on the first- and second-order derivatives of the associated Legendre functions of both kinds and contain a few singular terms. We construct ellipsoidal harmonic expansions in the exterior space for the first and second potential derivatives, which are similar to the series on the reference ellipsoid enveloping the Earth. We take a point P at an arbitrary altitude above the reference ellipsoid and construct the ellipsoid of revolution confocal to it, which passes through this point. The conventional complicated singular expressions for the first and second potential derivatives in the local north-oriented ellipsoidal reference frame, with the origin at the point P, are transformed into non-singular ellipsoidal harmonic series, which do not contain the first- and second-order derivatives of the associated Legendre functions. The resulting series have an accuracy of the squared eccentricity. These series can be applied for constructing a geopotential model, which is based, simultaneously, on the surface gravity data and the data of satellite missions, which provide measurements of the accelerations and/or the gravitational gradients. When the eccentricity of the considered external ellipsoid is equated to zero, the ellipsoid becomes an external sphere passing through the point P and the constructed ellipsoidal harmonic expansions are converted into non-singular spherical harmonic series for the first and second potential derivatives in the local north-oriented spherical reference frame.     

Expansions of the derivatives of the disturbing function in planetary problems

A method is suggested to develop literal expansions of derivatives of the disturbing function especially for the case of large values of the major axis ratio . The series remain convergent as well if =1, unless the eccentricities vanish at the same time. The treatment holds true in the case when usual analytical expansions are not valid, that is if the orbits have points equidistant from the primary. The general case is considered too, the intersecting orbits being included.     

Evolution of the cometary cloud under the action of stellar perturbations

The probability of variation of the integrals of the orbit as a result of an encounter was found for a two dimensional system. A method of solution of the Kolmogorov-Feller's equation is obtained using this probability function as a kernel, and it allows us to obtain the distribution of the integrals of the orbit as a function of time. The method is applied to the investigation of the evolution of orbits in the outer cometary cloud under the action of galactic stars. We consider the variations of orbits as a purely discontinuous random process, so we take into account not only distant but also close interactions.     

Non-Singular Expressions for the Gravity Gradients in the Local North-Oriented and Orbital Reference Frames

The conventional expansions of the gravity gradients in the local north-oriented reference frame have a complicated form, depending on the first- and second-order derivatives of the associated Legendre functions of the colatitude and containing factors which tend to infinity when approaching the poles. In the present paper, the general term of each of these series is transformed to a product of a geopotential coefficient and a sum of several adjacent Legendre functions of the colatitude multiplied by a function of the longitude. These transformations are performed on the basis of relations between the Legendre functions and their derivatives published by Ilk (1983). The second-order geopotential derivatives corresponding to the local orbital reference frame are presented as linear functions of the north-oriented gravity gradients. The new expansions for the latter are substituted into these functions. As a result, the orbital derivatives are also presented as series depending on the geopotential coefficients multiplied by sums of the Legendre functions whose coefficients depend on the longitude and the satellite track azimuth at an observation point. The derived expansions of the observables can be applied for constructing a geopotential model from the GOCE mission data by the time-wise and space-wise approaches. The numerical experiments demonstrate the correctness of the analytical formulas.An erratum to this article can be found at     

Generalization of Laplace’s expansion to the earth’s surface

The external expansion of the Earth's potential V in spherical harmonics is generalized to the Earth's surface. Some additional expansions are also proposed which represent the potential of a finite body practically in the whole space. The series developed can be used for the combined evaluation of the Earth's potential from both satellite and gravimetric measurements.