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The motion of artificial satellites of non-spherical earth
ABSTRACTS
183
G. V. SAMOILO~ICH*:The motion of arti&iaI sateIIite3of non-sphericalEarth (pp. 140-153). A computer study is made of periodic and quasi-secular disturbances of Kepler motion of an artificial earth satellite with numerical integration of the differential equations in osculating elements in 10 revolutions of the satellite. The Earth may be represented as a spheroid in many practical problems. The results of the study are used in the article on pages 154-162 of this issue of the Soviet Journal to determine the initial parameters of the orbital perturbations of motion. G. V. SAMOILOVICH: The inhence of the parametersof an orbit on the perturbedmotionof an artificialSateIIite (pp. 154-162). Using the results of the computer study of periodic and quasi-secular disturbances of Kepler motion of an artificial earth satellite (pp. 140-153 of this issue of the journal) the influence of the initial parameters of the orbit on the perturbations of motion is analysed in the field of a spheroid, with gravitational anomalies and in the field of a three-axis ellipsoid. The particular effects are described in 10 tables and 7 diagrams. AKSENOV*, YE. A. GREBENIKOV and V. G. DEMIN: Concerningthe stability of some eIasses of orbits of arti&iaI earth satelIites(pp. 163-172).
YE. P.
The equations of motion of the dynamic problem in the motion of artificial earth satellites in the normal gravitational field in a geocentric system of cylindrical co-ordinates, yield families of partial solutions corresponding to polar elliptical orbits, circular equatorial orbits etc. A study is made of stability, in the Lyapunov sense, of these partial solutions, and of their stability under continuous perturbations. The proof of stability in circular equatorial, ellipsoidal, polar elliptical, hyperboloidal and hyperbolic orbits is given by methods of Chetayev and Demin. YE. P. AKSENOV,YE. A. GREBENIKOV and V. G. DEMIN: QuaIitativeanalysis of the forms of motion in the problem of satellite motion in the normaI gravitationalfield of the Earth (pp. 173-197). In the theory of satellite motion outside the Earth’s atmosphere, the main perturbing factor is the deviation off-centre of the Earth’s gravitational field. The potential of Earth’s pull is approximated by a force function expansion. Equations are obtained which show that three different solutions correspond to the position of the moving point on an ellipsoid of rotation, on a hyperboloid of rotation of one sheet a, or in the meridional plane, to which three classes of motion correspond depending on a constant of integration h associated with the total mechanical energy of the satellite. 12 cases are analysed. Formulae are given for describing the motion of an actual satellite, but they are not always convenient and until suitable tables of elliptic integrals are compiled, it is recommended to expand solutions as power series. A separate article is to be published on this aspect of the problem. S. S. TOKMALAYEVA:On the analysis of tight in the field of one attracting centre (pp. 198-210). This paper contains computer formulae for calculating the elements of the trajectory of motion of a particle passing through two specified points in space with the time of motion fixed. It is assumed that the motion is due to the attraction of a point mass and that the mass of the particle is very small compared with the mass of the attracting centre. Close accuracy between the exact solution and results with Adams’ numerical integration formulae would justify a look at the two examples in the Appendix. I. V. ALEKSAKHIN, A. A. KRASOVSKII, P. I. LEBEDEV and A. I. YAKOVLEVA: Determiningthe parametersof initiaI orbits of artificial earth satelUtes(pp. 21 l-225). To estimate small deviations of the orbit parameters from their nominal values, it is necessary to evaluate the partial derivatives of the parameters of the initial orbits with respect to the parameters of motion of the mass-centre on entry into orbit. For the solution of practical problem these derivatives are calculated with given formulae in a starting and a starting initial frame of reference with transition between frames. V. I. CHARNYI: Concerningisochronousderivatives(pp. 226-237). For estimating the influence of initial deviations of co-ordinates and components of the velocity vector of a point in orbit, the matrix of isochronous derivatives is obtained by integration of equations in variations, which provides compact formulae. Certain results are given relating to the general properties of fundamental solutions of special sets of ordinary differential equations, a special case of which are equations in variations in Hamiltonian sets. Practical interest attaches to the method of inversion of matrices of isochronous derivatives by a special transposition of elements (without any loss of accuracy). This method is closely connected with Siegel’s general theory of canonical transformations of Hamilton equations. V. N. KALININ: Equationsof motion of an artificial earth satellite (pp. 238-245). When it is inconvenient to tackle problems in the guided motion of a satellite by using the equations of motion in osculating elements (e.g. a manoeuvre in a small neighbourhood of the orbit), it is advisable to investigate the motion in a moving system of co-ordinates which is associated in a definite way with the
183
G. V. SAMOILO~ICH*:The motion of arti&iaI sateIIite3of non-sphericalEarth (pp. 140-153). A computer study is made of periodic and quasi-secular disturbances of Kepler motion of an artificial earth satellite with numerical integration of the differential equations in osculating elements in 10 revolutions of the satellite. The Earth may be represented as a spheroid in many practical problems. The results of the study are used in the article on pages 154-162 of this issue of the Soviet Journal to determine the initial parameters of the orbital perturbations of motion. G. V. SAMOILOVICH: The inhence of the parametersof an orbit on the perturbedmotionof an artificialSateIIite (pp. 154-162). Using the results of the computer study of periodic and quasi-secular disturbances of Kepler motion of an artificial earth satellite (pp. 140-153 of this issue of the journal) the influence of the initial parameters of the orbit on the perturbations of motion is analysed in the field of a spheroid, with gravitational anomalies and in the field of a three-axis ellipsoid. The particular effects are described in 10 tables and 7 diagrams. AKSENOV*, YE. A. GREBENIKOV and V. G. DEMIN: Concerningthe stability of some eIasses of orbits of arti&iaI earth satelIites(pp. 163-172).
YE. P.
The equations of motion of the dynamic problem in the motion of artificial earth satellites in the normal gravitational field in a geocentric system of cylindrical co-ordinates, yield families of partial solutions corresponding to polar elliptical orbits, circular equatorial orbits etc. A study is made of stability, in the Lyapunov sense, of these partial solutions, and of their stability under continuous perturbations. The proof of stability in circular equatorial, ellipsoidal, polar elliptical, hyperboloidal and hyperbolic orbits is given by methods of Chetayev and Demin. YE. P. AKSENOV,YE. A. GREBENIKOV and V. G. DEMIN: QuaIitativeanalysis of the forms of motion in the problem of satellite motion in the normaI gravitationalfield of the Earth (pp. 173-197). In the theory of satellite motion outside the Earth’s atmosphere, the main perturbing factor is the deviation off-centre of the Earth’s gravitational field. The potential of Earth’s pull is approximated by a force function expansion. Equations are obtained which show that three different solutions correspond to the position of the moving point on an ellipsoid of rotation, on a hyperboloid of rotation of one sheet a, or in the meridional plane, to which three classes of motion correspond depending on a constant of integration h associated with the total mechanical energy of the satellite. 12 cases are analysed. Formulae are given for describing the motion of an actual satellite, but they are not always convenient and until suitable tables of elliptic integrals are compiled, it is recommended to expand solutions as power series. A separate article is to be published on this aspect of the problem. S. S. TOKMALAYEVA:On the analysis of tight in the field of one attracting centre (pp. 198-210). This paper contains computer formulae for calculating the elements of the trajectory of motion of a particle passing through two specified points in space with the time of motion fixed. It is assumed that the motion is due to the attraction of a point mass and that the mass of the particle is very small compared with the mass of the attracting centre. Close accuracy between the exact solution and results with Adams’ numerical integration formulae would justify a look at the two examples in the Appendix. I. V. ALEKSAKHIN, A. A. KRASOVSKII, P. I. LEBEDEV and A. I. YAKOVLEVA: Determiningthe parametersof initiaI orbits of artificial earth satelUtes(pp. 21 l-225). To estimate small deviations of the orbit parameters from their nominal values, it is necessary to evaluate the partial derivatives of the parameters of the initial orbits with respect to the parameters of motion of the mass-centre on entry into orbit. For the solution of practical problem these derivatives are calculated with given formulae in a starting and a starting initial frame of reference with transition between frames. V. I. CHARNYI: Concerningisochronousderivatives(pp. 226-237). For estimating the influence of initial deviations of co-ordinates and components of the velocity vector of a point in orbit, the matrix of isochronous derivatives is obtained by integration of equations in variations, which provides compact formulae. Certain results are given relating to the general properties of fundamental solutions of special sets of ordinary differential equations, a special case of which are equations in variations in Hamiltonian sets. Practical interest attaches to the method of inversion of matrices of isochronous derivatives by a special transposition of elements (without any loss of accuracy). This method is closely connected with Siegel’s general theory of canonical transformations of Hamilton equations. V. N. KALININ: Equationsof motion of an artificial earth satellite (pp. 238-245). When it is inconvenient to tackle problems in the guided motion of a satellite by using the equations of motion in osculating elements (e.g. a manoeuvre in a small neighbourhood of the orbit), it is advisable to investigate the motion in a moving system of co-ordinates which is associated in a definite way with the