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

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.

### Citation

`V. B. Kuznetsov. Method of Determination of Rectilinear Orbit for an Object Moving in the Ecliptic Plane // Transactions of IAA RAS. — 2020. — Issue 54. — P. 52–62.`

```
@article{kuznetsov2020,
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.},
author = {V.~B. Kuznetsov},
doi = {10.32876/ApplAstron.54.52-62},
issue = {54},
journal = {Transactions of IAA RAS},
keyword = {two-body problem, rectilinear movement, determination of preliminary orbit, method of Davidon – Fletcher – Powell},
pages = {52--62},
title = {Method of Determination of Rectilinear Orbit for an Object Moving in the Ecliptic Plane},
url = {http://iaaras.ru/en/library/paper/2063/},
year = {2020}
}
```

```
TY - JOUR
TI - Method of Determination of Rectilinear Orbit for an Object Moving in the Ecliptic Plane
AU - Kuznetsov, V. B.
PY - 2020
T2 - Transactions of IAA RAS
IS - 54
SP - 52
AB - 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.
DO - 10.32876/ApplAstron.54.52-62
UR - http://iaaras.ru/en/library/paper/2063/
ER -
```