## Revisiting Geometric Method for Determination of the Parabolic Orbit

Transactions of IAA RAS, issue 55, 16–23 (2020)

**DOI**: 10.32876/ApplAstron.55.16-23

**Keywords**:
two-body problem, parabolic orbit, comets, determination of preliminary orbit, method of Nelder-Mead

### Abstract

The determination of a preliminary parabolic orbit is a point of interest in the field of cometary astronomy. That can be useful for both discovering new comets and identification with known comets. The paper describes a geometric method of the preliminary parabolic orbit determination, which is a particular case of Cauchy-Kuryshev-Perov method. It shows how, within the framework of the two-body problem, to determine the parabolic orbit that does not lie in the plane of an observer movement based on four angular observations and geometric constructions. Using this method, we can make a model as solving an algebraic system of equations for two dimensionless variables, with a finite number of solutions. Moreover, it has no restrictions on the orbital arc length and the time intervals between observations. All possible combinations of the comet orbital position are represented by the four cases of equation systems. The algorithm for finding the solutions without any preliminary information about the orbit is considered. The solutions are sought in a limited square area that has covered by two-level triangulation. Such triangulation let us to use a smaller number of triangles without losing small isolated areas. At the same time, triangles are ranked for compliance with the search conditions in order to exclude most of them at the initial stage. The solutions of the system are found by searching for the goal function minima using the Nelder-Mead simplex method. In such a way, all possible solutions are found. The calculated orbits are compared through the representation of observations and the best one is selected. As an example, the results of determining the orbit of the near-parabolic comet C/2020 F8 (SWAN) are given.

### Citation

`V. B. Kuznetsov. Revisiting Geometric Method for Determination of the Parabolic Orbit // Transactions of IAA RAS. — 2020. — Issue 55. — P. 16–23.`

```
@article{kuznetsov2020,
abstract = {The determination of a preliminary parabolic orbit is a point of interest in the field of cometary astronomy. That can be useful for both discovering new comets and identification with known comets.
The paper describes a geometric method of the preliminary parabolic orbit determination, which is a particular case of Cauchy-Kuryshev-Perov method. It shows how, within the framework of the two-body problem, to determine the parabolic orbit that does not lie in the plane of an observer movement based on four angular observations and geometric constructions. Using this method, we can make a model as solving an algebraic system of equations for two dimensionless variables, with a finite number of solutions. Moreover, it has no restrictions on the orbital arc length and the time intervals between observations. All possible combinations of the comet orbital position are represented by the four cases of equation systems. The algorithm for finding the solutions without any preliminary information about the orbit is considered. The solutions are sought in a limited square area that has covered by two-level triangulation. Such triangulation let us to use a smaller number of triangles without losing small isolated areas. At the same time, triangles are ranked for compliance with the search conditions in order to exclude most of them at the initial stage. The solutions of the system are found by searching for the goal function minima using the Nelder-Mead simplex method. In such a way, all possible solutions are found. The calculated orbits are compared through the representation of observations and the best one is selected.
As an example, the results of determining the orbit of the near-parabolic comet C/2020 F8 (SWAN) are given.},
author = {V.~B. Kuznetsov},
doi = {10.32876/ApplAstron.55.16-23},
issue = {55},
journal = {Transactions of IAA RAS},
keyword = {two-body problem, parabolic orbit, comets, determination of preliminary orbit, method of Nelder-Mead},
pages = {16--23},
title = {Revisiting Geometric Method for Determination of the Parabolic Orbit},
url = {http://iaaras.ru/en/library/paper/2073/},
year = {2020}
}
```

```
TY - JOUR
TI - Revisiting Geometric Method for Determination of the Parabolic Orbit
AU - Kuznetsov, V. B.
PY - 2020
T2 - Transactions of IAA RAS
IS - 55
SP - 16
AB - The determination of a preliminary parabolic orbit is a point of
interest in the field of cometary astronomy. That can be useful for
both discovering new comets and identification with known comets.
The paper describes a geometric method of the preliminary parabolic
orbit determination, which is a particular case of Cauchy-Kuryshev-
Perov method. It shows how, within the framework of the two-body
problem, to determine the parabolic orbit that does not lie in the
plane of an observer movement based on four angular observations and
geometric constructions. Using this method, we can make a model as
solving an algebraic system of equations for two dimensionless
variables, with a finite number of solutions. Moreover, it has no
restrictions on the orbital arc length and the time intervals between
observations. All possible combinations of the comet orbital position
are represented by the four cases of equation systems. The algorithm
for finding the solutions without any preliminary information about
the orbit is considered. The solutions are sought in a limited square
area that has covered by two-level triangulation. Such triangulation
let us to use a smaller number of triangles without losing small
isolated areas. At the same time, triangles are ranked for compliance
with the search conditions in order to exclude most of them at the
initial stage. The solutions of the system are found by searching for
the goal function minima using the Nelder-Mead simplex method. In
such a way, all possible solutions are found. The calculated orbits
are compared through the representation of observations and the best
one is selected. As an example, the results of determining the
orbit of the near-parabolic comet C/2020 F8 (SWAN) are given.
DO - 10.32876/ApplAstron.55.16-23
UR - http://iaaras.ru/en/library/paper/2073/
ER -
```