Four widely used algorithms for the computation of the Earth's gravitational potential and its first-, second- and third-order gradients are examined: the traditional increasing degree recursion in associated Legendre functions and its variant based on the Clenshaw summation, plus the methods of Pines and Cunningham--Metris, which are free from the singularities that distinguish the first two methods at the geographic poles. All four methods are reorganized with the lumped coefficients approach, which in the cases of Pines and Cunningham-Metris requires a complete revision of the algorithms. The characteristics of the four methods are studied and described, and numerical tests are performed to assess and compare their precision, accuracy, and efficiency. In general the performance levels of all four codes exhibit large improvements over previously published versions.
From the point of view of numerical precision, away from the geographic poles Clenshaw and Legendre offer an overall better quality. Furthermore, Pines and Cunningham--Metris are affected by an intrinsic loss of precision at the equator and suffer from additional deterioration when the gravity gradients components are rotated into the East-North-Up topocentric reference system
We present orbit analysis for a sample of 8 inner bulge globular clusters, together with one reference halo object. We used proper motion values derived from long time base CCD data. Orbits are integrated in both an axisymmetric model and a model including the Galactic bar potential. The inclusion of the bar proved to be essential for the description of the dynamical behavior of the clusters. We use the Monte Carlo scheme to construct the initial conditions for each cluster, taking into account the uncertainties in the kinematical data and distances. The sample clusters show typically maximum height to the Galactic plane below 1.5 kpc, and develop rather eccentric orbits. Seven of the bulge sample clusters share the orbital properties of the bar/bulge, having perigalactic and apogalatic distances, and maximum vertical excursion from the Galactic plane inside the bar region. NGC 6540 instead shows a completely different orbital behaviour, having a dynamical signature of the thick-disc. Both prograde and prograde-retrograde orbits with respect to the direction of the Galactic rotation were revealed, which might characterize a chaotic behaviour.
This contribution deals with the derivation of explicit expressions of the gradients
of first, second and third order of the gravitational potential. This is accomplished in the framework of
tensor analysis which naturally allows to apply general formulae to the specific coordinate
systems in use in geodesy. In particular it is recalled here that when the potential field is expressed in
general coordinates on a 3D manifold, the gradient operation leads to the definition of the
covariant derivative and that the covariant derivative of a tensor can
be obtained by application of a simple rule. When applied to the gravitational potential or to any of its gradients,
the rule straightforwardly provides the expressions of the higher-order gradients.
It is also shown that the tensor approach offers a clear distinction
between natural and physical components of the gradients.
Two fundamental reference systems---a global, bodycentric system and a local, topocentric system, both body-fixed---are
introduced and transformation rules are derived to convert quantities between the two systems.
The results include explicit expressions for the gradients of the first three orders in both
reference systems
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