Lviv Polytechnic National University, 79013, Lviv, Karpinskyi street 6

**Abstract**

Formulas and relations for the realization of the method of construction of three-dimensional mass distribution of hydrostatically balanced ellipsoidal planet are derived. First, it is possible to interpret the anomalies of the external gravitational field by deviating the three-dimensional density of mass distribution from the hydrostatically balanced state. Second, conditions that provide a minimum of gravitational energy allow us to draw conclusions about the state of rest or dynamic changes in the middle of the body. Determining the conditions under which the planet is in a state of hydrostatic equilibrium, or deviation from it of the real body makes it possible to establish the causes of the mechanism of mass redistribution. This in turn provides the basis for the study and interpretation of dynamic processes in the mid-celestial body. The state of hydrostatic equilibrium allows to solve a number of problems of the astronomy and physics of the planets. Estimates of potential and energy values in the tectonosphere reveal the mechanisms that cause the movement of continental plates. It is known that the hydrostatic equilibrium condition is ensured by a minimum of gravitational energy E. In this case, the piecewise continuous mass distribution function is described by the expression through Legendre polynomials. The potential of the piecewise-continuous mass distribution function is uniformly convergent for the given class of functions. The expression for gravitational energy E depends on a given function δ^{0}(ρ) and a fixed shape in the form of an ellipsoid. Therefore, the redistribution of the density produced by the change in the coefficients b_{mnk} gives variations in the energy E. Also, the values b_{mnk} are determined by the Stokes constants C_{NK}, S_{NK}. The minimum of gravitational energy E is defined as the problem for the conditional extremum of the Lagrange function. Its intersection with respect to magnitudes λ_{nk}, γ_{nk}, b_{pqs} determines the mass distribution that gives the minimum energy E for a particular gravitational field. A detailed analysis of this system of equations leads to the conclusion that it is divided into eight groups formed by the corresponding Stokes constants. The solution of this system is obtained by using a sequence of matrix transformations to implement the above method. Thus, all the necessary formulas and ratios of the described method of construction of three-dimensional mass distribution of hydrostatically balanced ellipsoidal planet are presented, which allows to perform the necessary studies.

**Keywords:** three-dimensional mass distribution; hydrostatically equilibrated ellipsoidal planet; anomalies of the external gravitational field; condition of hydrostatic equilibrium; function of lump continuous mass distribution; Legendre polynomial; Stokes constant

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