Principal moments of inertia and secular love number of Phobos

A dynamic homogeneous model of Phobos is used: its boundary is an equipotential surface specified by the second zonal and the second sectorial harmonics plus the constant part of the tidal potential due to Mars. The principal moments of inertia, the hydrostatic second zonal harmonic and the secular Love number of Phobos have been estimated. They support the hypothesis that Phobos is formed out of primordial matter by accretion in orbit.     

The differences in structure of the gravity field and the figure of the earth in the northern and southern hemispheres

Summary It was shown in [5] that the flattening of parallel sections of the geoid (ϕ=const.) differs in the northern and southern hemispheres. This leads up to the idea of studying further the size and shape of the Earth and the structure of the gravity field separately for the northern and for the southern hemispheres. In this paper attention is devoted especially to the mean values of the radius-vectors, to the best fitting ellipsoid parameters and to the mean values of gravity for the whole hemispheres, on the one hand, and for their ϕ=const. sections, on the other. The symbols used are the same as in [5,6]. Address: Politickych vězňů 12, Praha 1-Nové Město.     

The four primary geodetic parameters

Summary The four primary geodetic parameters defining the geodetic reference system are discussed from the point of view of their physical meaning and current estimation of their actual accuracy. The geopotential scale factor has been treated as the primary geodetic parameter defining the Earth's dimensions.

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¶rt;am m nu¶rt;uu naama, n¶rt;u¶rt;u um mumu, mu u uu a u mmu. ama amnmuaa ¶rt;am am nu¶rt;u naama, n¶rt; a u.

     

Satellite orbit computations using gravity anomalies

Summary A typical geodetic satellite orbit has been computed by numerical integration for a period of thirty hours. The gravitational potential of a standard orbit was represented by the SAO 1969 Standard Earth potential coefficients taken to degree 18. Other orbits were generated using the generalized Stokes' equations and the coating method applied to gravity anomalies and surface densities, in 5°, 10°, 15° and 30° equal-area blocks, derived from the given potential coefficients. The differences between these orbits yield the position differences to be expected when representing the potential field by using gravity data instead of potential coefficients. Using 10°, 15°, and 30° blocks and the generalized Stokes' equations, the position error at the end of thirty hours was 89 meters, 224 meters, and 2060 meters respectively. This error is primarily due to the integration error in computing the gravitational field by summation over a finite number of areas.     

Secular love and tidal numbers of synchronously orbiting satellites

The secular Love and the secular tidal numbers have been computed for eight synchronously orbiting satellites in the solar system for which the triaxiality parameters and satellite-centric gravitational constant are available. Excepting Deimos the total mass of which should be first refined, the secular Love and tidal numbers are rather close to unity, as a rule. That is why, the centrifugal and tidal distortions can be assumed responsible for the actual figures of the synchronously orbiting satellites resulting from the primordial spheres, as well as, their static equilibrium nearly satisfied. The hypothesis of the origin of synchronously orbiting satellites by accretion in orbits is supported by the results obtained.     

The Phobos gravitational field modeled on the basis of its topography

The parameters of the best-fitting ellipsoid have been derived using the latest spherical harmonics of the Phobos topography (Duxbury, 1989) by solution of non-linear overdetermined inverse problem. The lengths of the equatorial axes of the ellipsoid have been determined (a = 12.9 km, b = 11.4 km). They are nearly the same as established by Duxbury (ibid.) on the basis of the linearized relationship between the squared lengths of ellipsoidal axes and the topography coefficients C ₂₀ and C _22. The length of the polar axis (c = 9.1 km) differs of about 20% from Duxbury's value. Supposing mass homogeneity of Phobos, the Stokes parameters of the external gravitational field have been derived up to those of the sixth degree and order. The large irregularities in the Phobos figure cause the values of the Duxbury's potential coefficients be fairly inaccurate except the harmonics C ₂₀, C ₃₂, S ₄₃ and S ₅₁, i.e. linearized relationship between gravity and topography cannot be applied for Phobos. Finally, positions of the centre of figure and the directions of the principal axes of inertia have been established.     

Tidal potential due to general bodies

The exact definition of the tidal potential is given on the basis of the theory of the restricted three-body problem. There are no limitations as regards the perturbing terms in the gravitational fields of the tide forming bodies; andno a priori conditions are needed in the definition.