Development of a new form of equations of disturbed motion of a satellite in nearly circular orbits

Abstract

The use of simple physical reasoning instead of the method of varying constants has made it possible to elaborate a short scheme of deriving equations of disturbed motion of a satellite in nearly circular orbits. The use of a circular Keplerian orbit as a reference orbit has ensured the nondegeneracy of equations and their simple relation with time. All this taken together has made it possible to propose a form of equations convenient for carrying out numerical and analytical studies with its variables having a simple physical meaning. A relationship between the introduced variables and the Keplerian elements of the orbits was described for undisturbed motion. It was shown that the variables describing the deviation of the orbit radius from the radius of the reference orbit are proportional to eccentricity and deviation of the focal parameter is proportional to the square of the eccentricity. Relationships were constructed that describe the connection between the introduced variables and the Cartesian coordinates of position and velocity in the inertial coordinate system as well as arguments for choosing the radius of the reference orbit. From the condition of equality of energies of motion along circular reference orbit and in elliptical Keplerian orbit, it is expedient to take the radius of the reference orbit equal to the semi-major axis of the Keplerian ellipse. Approaches to the possible development of the proposed equations were presented. They make it possible to describe changes in the argument of the orbit perigee. The proposed change of variables makes it possible to avoid degeneracy of equations at very small eccentricities when studying the change in the orbit perigee. The advantages of using the proposed equations for numerical and analytical studies of satellite motion in nearly circular orbits were shown on concrete calculation examples. It was shown that the results of numerical integration in the proposed variables give almost five orders of magnitude less error than the results of the integration of equations in Cartesian coordinates