Mass function

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The mass functions are important parameters of the earth's gravitational field , which can be determined by a harmonic analysis of the gravitational potential ( series expansion according to spherical functions ).

The term was coined by geodesist Karl Ledersteger around 1965, who examined for the first time how far the essential results of dynamic satellite geodesy can be reconciled with theoretical equilibrium figures of the earth's interior .

Physically, the mass functions represent those parameters of the gravitational potential with which one can describe the essential deviations of the earth's gravitational field from that of a mean globe or a theoretical earth ellipsoid . Mathematically, they are the harmonic coefficients of a series expansion of the gravitational potential, which is carried out on the earth's surface by length and width dependent spherical surface functions up to a certain degree  n . This two-dimensional solution is transformed into the range of the orbital levels of the geodetic earth satellites by means of suitable methods of field continuation .

Individual mass functions or coefficients

However, only the following coefficients J3, J4 etc. are designated as actual mass functions, which are significantly smaller (effect on the earth's figure only a few millionths or less than 20 meters). They describe the mean vertical deviation of the geoid from the global earth ellipsoid in certain latitude zones . This rotationally symmetrical potential model (averaging of the geopotential along the parallel circles ) is obtained through harmonic development according to zonal spherical functions that are built up from width-dependent Legendre polynomials .

  • J3 means a north-south asymmetry of the earth figure, which corresponds to a different flattening of the two hemispheres and amounts to about 15 meters. It was discovered by minor orbital disruptions of the first US satellite Explorer 1 and received the unfortunate nickname " pear shape of the earth" in the media , although it is less than a thousandth of the Earth's flattening itself and cannot be observed from space.
  • The 4th order (J4) mass function can be used as average deviation of the earth meridians be interpreted by the elliptical shape, and makes 45 °  width also about 15 meters from (how J3 about 2 millionths of the earth's radius).
  • The coefficient J5 is almost zero.
  • J6 is only a few meters and the following terms only a few decimeters .

Further development

Since around 1970 it has been possible to expand the simplified zonal series development (J2, J3… J14) of the geoid into a comprehensive, two-dimensional analysis method in which the geographical longitude is also taken into account. In such an earth model with tesseral (checkerboard) spherical surface functions, J2 corresponds to the harmonic coefficient C (2.0), J3 corresponds to C (3.0), etc. The general terms of the harmonic coefficients are then given as C (n, m) and S ( n, m). C stand for a cosine term and S for a sine term, each of the geographical longitude, which occurs in the order  m (m = 0, 1, 2 ...), while n represents the degree of latitude dependence (n = 2, 3, 4 ...).

In such a generalized series development, width- and length-dependent effects on the geoid can therefore easily overlap , so that they can be determined without any specifications ( i.e. only from the available measurements). Today this is possible from the orbits of earth satellites up to around degree and order 70.

Around 1975 people began to determine higher-grade coefficients of the gravitational field by combining the terms  C (n, m) derived from satellite orbits with terrestrially determined gravitational field parameters. For some years this has been possible up to around degree and order 720. This means that the geosciences have a mathematical method at their disposal with which the earth's gravity field can be described by up to 100,000 parameters, as required. This allows the geoid to be represented worldwide using recursive (i.e. easy-to- program ) formulas with a resolution of about 50 km, and the effect on satellite orbits to at least 1 meter.

Finer details can be added by means of local terrestrial measurements, while the series development is terminated at the appropriate point if less accuracy is required. For example, GPS receivers have programmed the earth's gravitational field up to degrees n = 8 or 10, which corresponds to around 100 mass functions and an altitude accuracy of around 5 meters.

literature

See also