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Determination of the Parabolic Orbit in Coplanar (Near-Сoplanar) Case

Transactions of IAA RAS, issue 70, 25–33 (2024)

Keywords: parabolic orbit, plane of ecliptic, Nelder – Mead method

Abstract

An algorithm for finding a solution for determining a parabolic orbit in the coplanar case is proposed, which takes place, when the plane of the orbit desired coincides with the motion plane of an observer. The case, when all the movements are occurring in the same plane, only complicates the orbit determination. So, in the equation that connected three topocentric distances it is singularity that appears. In practice, this case corresponds to near-parabolic comets whose orbits are placed with very small inclination to the ecliptic. The important difference of all the cases for the determination of parabolic orbit from the case with the arbitrary value of eccentricity, is that only three observations are needed to find it. The determination of parabolic orbit is a solution of a system of algebraic equations for two dimensionless variables. The algorithm of the solution is the modification of the coplanar case, which was described earlier. It is based on the finding the minima of the objective function using the Nelder ‒ Mead simplex algorithm. The main changes are connected with the replacement of variables. In such a case, the components of the normal vector to the plane of a desired orbit were replaced by normal dimensionless topocentric distances. It allowed to avoid the loss of precision owing to small values of coordinates and save the closed and dimensionless of search area. One more important thing is to substitute the coplanar equation that connected three topocentric distances by the equation without singularity in coplanar case. The results of determining the orbit of C/1984 U1 Shoemaker comet, with 179.21° inclination of the orbit to ecliptic plane may serve an example. It has been demonstrated that the use of base algorithm for non-coplanar case does not allow to receive the orbit desired. The replacement of variables from components of normal vector by non-dimensional topocentric distances allows to obtain a solution, that poorly represents the average observation. Finally, the rejection of the coplanar equation that connected the topocentric distances on the algebraic expression without singularity allows to calculate the orbit with sufficient precision.

Citation

Text

BibTeX

RIS

V. B. Kuznetsov. Determination of the Parabolic Orbit in Coplanar (Near-Сoplanar) Case // Transactions of IAA RAS. — 2024. — Issue 70. — P. 25–33.

@article{kuznetsov2024,
  abstract = {An algorithm for finding a solution for determining a parabolic orbit in the coplanar case is proposed, which takes place, when the plane of the orbit desired coincides with the motion plane of an observer. The case, when all the movements are occurring in the same plane, only complicates the orbit determination. So, in the equation that connected three topocentric distances it is singularity that appears. In practice, this case corresponds to near-parabolic comets whose orbits are placed with very small inclination to the ecliptic. The important difference of all the cases for the determination of parabolic orbit from the case with the arbitrary value of eccentricity, is that only three observations are needed to find it.
The determination of parabolic orbit is a solution of a system of algebraic equations for two dimensionless variables. The algorithm of the solution is the modification of the coplanar case, which was described earlier. It is based on the finding the minima of the objective function using the Nelder ‒ Mead simplex algorithm. The main changes are connected with the replacement of variables. In such a case, the components of the normal vector to the plane of a desired orbit were replaced by normal dimensionless topocentric distances. It allowed to avoid the loss of precision owing to small values of coordinates and save the closed and dimensionless of search area. One more important thing is to substitute the coplanar equation that connected three topocentric distances by the equation without singularity in coplanar case.
The results of determining the orbit of C/1984 U1 Shoemaker comet, with 179.21° inclination of the orbit to ecliptic plane may serve an example. It has been demonstrated that the use of base algorithm for non-coplanar case does not allow to receive the orbit desired. The replacement of variables from components of normal vector by non-dimensional topocentric distances allows to obtain a solution, that poorly represents the average observation. Finally, the rejection of the coplanar equation that connected the topocentric distances on the algebraic expression without singularity allows to calculate the orbit with sufficient precision.},
  author = {V.~B. Kuznetsov},
  doi = {10.32876/ApplAstron.70.25-33},
  issue = {70},
  journal = {Transactions of IAA RAS},
  keyword = {parabolic orbit, plane of ecliptic, Nelder – Mead method},
  note = {russian},
  pages = {25--33},
  title = {Determination of the Parabolic Orbit in Coplanar (Near-Сoplanar) Case},
  url = {http://iaaras.ru/en/library/paper/2193/},
  year = {2024}
}

TY  - JOUR
TI  - Determination of the Parabolic Orbit in Coplanar (Near-Сoplanar) Case
AU  - Kuznetsov, V. B.
PY  - 2024
T2  - Transactions of IAA RAS
IS  - 70
SP  - 25
AB  - An algorithm for finding a solution for determining a parabolic orbit
in the coplanar case is proposed, which takes place, when the plane
of the orbit desired coincides with the motion plane of an observer.
The case, when all the movements are occurring in the same plane,
only complicates the orbit determination. So, in the equation that
connected three topocentric distances it is singularity that appears.
In practice, this case corresponds to near-parabolic comets whose
orbits are placed with very small inclination to the ecliptic. The
important difference of all the cases for the determination of
parabolic orbit from the case with the arbitrary value of
eccentricity, is that only three observations are needed to find it.
The determination of parabolic orbit is a solution of a system of
algebraic equations for two dimensionless variables. The algorithm of
the solution is the modification of the coplanar case, which was
described earlier. It is based on the finding the minima of the
objective function using the Nelder ‒ Mead simplex algorithm. The
main changes are connected with the replacement of variables. In such
a case, the components of the normal vector to the plane of a desired
orbit were replaced by normal dimensionless topocentric distances. It
allowed to avoid the loss of precision owing to small values of
coordinates and save the closed and dimensionless of search area. One
more important thing is to substitute the coplanar equation that
connected three topocentric distances by the equation without
singularity in coplanar case.  The results of determining the orbit
of C/1984 U1 Shoemaker comet, with 179.21° inclination of the orbit
to ecliptic plane may serve an example. It has been demonstrated that
the use of base algorithm for non-coplanar case does not allow to
receive the orbit desired. The replacement of variables from
components of normal vector by non-dimensional topocentric distances
allows to obtain a solution, that poorly represents the average
observation. Finally, the rejection of the coplanar equation that
connected the topocentric distances on the algebraic expression
without singularity allows to calculate the orbit with sufficient
precision.
DO  - 10.32876/ApplAstron.70.25-33
UR  - http://iaaras.ru/en/library/paper/2193/
ER  -