Уравнения движения искусственного спутника в поле тяготения Земли с использованием комплексных координат

A. Fominov

Transactions of IAA RAS, issue 4, 290–299 (1999)

Keywords: AES, the light pressure, the Moon's penumbra, the Lageos, the Etalon-type satellite

Abstract

Возмущающий гравитационный потенциал Земли представлен через комплексные сферические функции. Получены дифференциальные уравнения орбитального движения спутника для комплексных координат. Уравнения движения учитывают возмущения, обусловленные несферичностью геопотенциала. Приведен метод вычисления частных производных любого порядка от возмущающего потенциала по координатам. Устанавливается связь между производными потенциала по прямоугольным координатам и производными потенциала по комплексным координатам. Даны основные рекуррентные соотношения для вычисления комплексных сферических функций произвольных степеней и порядков. The mutually conjugate complex coordinates $ w=x+i\,y\, $, $ w^*=x-i\,y $ and $ w_g=x_g+i\,y_g\, $, $ w^*_g=x_g-i\,y_g $ are added to the real coordinates $ z $ and $ z_g\, $. Here $ x, y, z $ stand for the rectangular equatorial coordinates of an Earth's artificial satellite referred to the inertial coordinate system $ CRS $ (epoch and equinox $ J2000.0\, $), where as $ x_g, y_g, z_g $ are the coordinates of a satellite referred to the Earth-fixed coordinate system $ TRS\, $. The conjugate complex spherical functions $ V_nm\, $, $ V_nm^* $ are introduced by formulas (5)–(6). The complex constants $ d_nm=c_nm+i\,s_nm\, $, $ d_nm^*=c_nm-i\,s_nm $ are used, $ c_nm\, $, $ s_nm $ being the constants of the expansion (3) of the perturbing gravitational potential of the Earth. Basic recurrence relations are given enabling one to compute the complex spherical functions of any degree and order. The complex coordinates permit to express the formula (3) for the Earth's potential as the sum of two conjugate complex quantities (7). A method is described for the computation of the partial derivatives of the arbitrary order of complex functions $ V_nm $ and $ V_nm^* $ with respect to coordinates $ w_g\, $, $\, w_g^*\, $, $ \,z_g\,, $ i.e. formulas (8)–(11), (14), (16)–(18). The similar algorithm for the computation of the derivatives of the Earth's potential of any order with respect to coordinates $ w_g\, $, $ w_g^*\, $, $ z_g $ involves expressions (19)–(23). The relationships between the differential operator with respect to rectangular coordinates and that with respect to complex coordinates are given by (25)–(26). The orbital equations of a satellite in the gravitational field of the Earth are described by formulas (27)–(28), where $ Q_{ij}(i,j=1,2,3) $ are the elements of transformation matrix from the $ TRS $ to the $ CRS $ for the current date. Instead of two real coordinates $ x $ and $ y $ a single complex coordinate $ w $ is used in the equation (27). The use of the equations (27)–(28) for the development of a numerical theory of the motion of an artificial Earth's satellite demands availability of an integrator to operate with the complex quantities. Hence, the integrators commonly used in celestial mechanics should be modified in the appropriate manner.

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