Method of Determination of Rectilinear Orbit for an Object Moving in the Ecliptic Plane

V. B. Kuznetsov

Transactions of IAA RAS, issue 54, 52–62 (2020)

DOI: 10.32876/ApplAstron.54.52-62

Keywords: two-body problem, rectilinear movement, determination of preliminary orbit, method of Davidon – Fletcher – Powell

Abstract

The paper describes the method of determining a preliminary radial elliptic orbit lying in the ecliptic plane. Three observations are needed for the orbit determination in this case. It is a particular case that the observed body and the observer are moving in the same plane, which is more difficult to analyze for rectilinear movement. The study considers the 2 type of trajectories that cross the aphelion of the orbit. Usually, these trajectories can be divided into two observational subtypes according to the moment of their aphelion passage. The 1 type that does not cross the orbit aphelion is also studied. As a rule, these trajectories are grouped according to the types of movement described by elliptic and hyperbolic systems of equations. The aim of the present work is to develop a method that allows us to determine the orbit for cases with any distribution of the observations for any value of the energy of the body observed. The systems of transcendent equations are derived as meeting both elliptic movement of the observed body (a trajectory of the 2nd type) and all the other types of movement (a trajectory of the 1st type). The goal function is produced on the basis of such system of equations. The minimum of the goal function is the solution sought. The technique to determine the borders of two-dimensional area of possible solutions is described. The minimum of the goal function is found using the Davidon – Fletcher – Powell method. The real solution is chosen from the two possible ones. The method is illustrated by using models of finding rectilinear orbits of both elliptic and hyperbolic types.

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