Astronomical School’s Report, 2017, Volume 13, Issue 1, Pages 1–4
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UDC 521.82

Research of structure of coefficients' representation of the potential by everywhere convergent series

Fys М.М., Brydun A.M., Sohor A.R.

Lviv Polytechnic National University, Karpinskyi str. 6, 79013 Lviv, Ukraine


This paper offers the representation of the potential of planets by a sequence of non-harmonic continuous functions, which are determined by direct resolve of the inversion radius in the binomial series. The coefficients of such interpretation are, in essence, the quantities determined by the body's shape and its filling, generating the potential by the masses and, therefore, taking into account their features. In connection with this, there is a need to study the nature of these quantities (for example, the possible connection with the parameters of the external gravitational field) and to develop methods and means for their determination. There are introduced algorithms for finding these elements of such representation, investigated the character and structure, connected with the harmony in the middle of a bulk body, what allows to find the coefficients of the resolve of series by means of linear combinations of parameters of the external gravitational field of the celestial body (Stokes constant), which is confirmed by arithmetical experiment on a concrete example. On the contrary, according to the known values for the elements in this representation, it is possible to determine Stokes permanent gravitational fields of planetary bodies. This, in turn, allows us to determine the value of the potential of the force of attraction (gravity, if we take into account the rotating component) throughout the space, including at points that are close to the surface or are on it. Obviously, the advantage of this approach consists in the computational aspect, since such images are not harmonic functions of the external generating body, and therefore the proposed approximation should be considered as an approximation of the continuous function. But, given the slow convergence (or even difference) of approaching ball functions in areas close to the surface, such an approach can be considered as an additional tool (or even alternative) when studying the representation of a gravitational field by classical methods (for example, by means of ball functions). Also the recording of the attraction potential using binomial series can be considered as its analytic continuation in the middle of the generating body with the corresponding distribution of masses and used in the interpretation of planetary geodynamic processes, which consist precisely in the proposed areas.

Keywords: potential; continuous function; everywhere convergent series; Earth's gravitational field


  1. Petrovskaya M.S. (1982). Predstavlenie gravitatsionnogo potentsiala Zemli v vide ryadov, skhodyaschikhsya vo vsem prostranstve. Tr. I Orlovskoy konferentsii “Izuchenie Zemli kak planety metodami astronomii, geodezii i geofiziki”. – K.: Naukova dumka, 1982. – 115 s. .
  2. Fys M.M., Sohor A.R., Fotsa R.S. (2008). Aproksymatsiya potentsi’alu planety neperervnymy funktsiyamy. Heodynamika, 1(6), 11–15.
  3. Fys M.M., Sohor A.R., Fotsa R.S. (2012). Pro mozhlyvist’ vyznachennia koefitsiyentiv potentsi’alu vsiudy zbizhnymy riadamy ta yikh heofizychne tlumachennia. Materi’aly XVII mizhnarodnoho naukovo-tekhnichnoho sympozi’umu “Heoinformatsiynyy Mizhnarodnyy yuvileynyy naukovo-tekhnichnyy monitorynh navkolyshn’oho seredovyshcha GPS i GIS-tekhnolohiyi”, 10–15 veresnia, 2012 r. Alushta (Krym)., 115–119.
  4. Lerch F.I. Putney B., Wagner C.A., Klosko S.M. (1981). Goddard earth models for oceanographic applications (GEM-10B and IOC). Marine Geodesy, 5(2), 145–187.

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