## Determination of a Celestial Body Orbit

Transactions of IAA RAS, issue 54, 63–70 (2020)

**DOI**: 10.32876/ApplAstron.54.63-70

**Keywords**:
determination of preliminary orbit, improvement of orbit

### Abstract

The paper describes the main stages of determining a celestial body orbit from its observations. The main difficulties that arise when improving a celestial body orbit are considered. Attention is drawn to the fact that the orbit determination can be considered as a boundary value problem, where the boundary conditions are observations of a celestial body. We determine the initial orbit using formulas specially developed by Yu. V. Batrakov for satellite orbits. An example of calculating the orbit of an artificial satellite is given. The case when there are three positional observations made from three points with a small time interval between observations is considered. A scheme for deriving the main formulas for improving the orbit parameters is presented. We pay attention to the possibility of a rapid increase in the values in the right-hand sides of the differential equations for the matrix of isochronous derivatives, whose elements are the partial derivatives of the current parameters of motion by their initial values (matrizant). It is proposed to use the property of the matrizant, which is that the product of the matrizant calculated on subintervals is equal to the value of the matrizant in the interval. We reduce the length of the interval subinterval to the possible minimum — to the length of the integration step for the equations of motion if a one-step integration method is used. Numerical experiments have shown that this can significantly reduce the computation time while improving the orbit. We note that the choice of the epoch of the initial parameters enables us to increase the conditionality of the normal system when improving the orbit.

### Citation

`Yu. D. Medvedev, D. A. Bulekbaev. Determination of a Celestial Body Orbit // Transactions of IAA RAS. — 2020. — Issue 54. — P. 63–70.`

```
@article{medvedev2020,
abstract = {The paper describes the main stages of determining a celestial body orbit from its observations. The main difficulties that arise when improving a celestial body orbit are considered. Attention is drawn to the fact that the orbit determination can be considered as a boundary value problem, where the boundary conditions are observations of a celestial body. We determine the initial orbit using formulas specially developed by Yu. V. Batrakov for satellite orbits. An example of calculating the orbit of an artificial satellite is given. The case when there are three positional observations made from three points with a small time interval between observations is considered. A scheme for deriving the main formulas for improving the orbit parameters is presented. We pay attention to the possibility of a rapid increase in the values in the right-hand sides of the differential equations for the matrix of isochronous derivatives, whose elements are the partial derivatives of the current parameters of motion by their initial values (matrizant). It is proposed to use the property of the matrizant, which is that the product of the matrizant calculated on subintervals is equal to the value of the matrizant in the interval. We reduce the length of the interval subinterval to the possible minimum — to the length of the integration step for the equations of motion if a one-step integration method is used. Numerical experiments have shown that this can significantly reduce the computation time while improving the orbit. We note that the choice of the epoch of the initial parameters enables us to increase the conditionality of the normal system when improving the orbit.},
author = {Yu.~D. Medvedev and D.~A. Bulekbaev},
doi = {10.32876/ApplAstron.54.63-70},
issue = {54},
journal = {Transactions of IAA RAS},
keyword = {determination of preliminary orbit, improvement of orbit},
pages = {63--70},
title = {Determination of a Celestial Body Orbit},
url = {http://iaaras.ru/en/library/paper/2064/},
year = {2020}
}
```

```
TY - JOUR
TI - Determination of a Celestial Body Orbit
AU - Medvedev, Yu. D.
AU - Bulekbaev, D. A.
PY - 2020
T2 - Transactions of IAA RAS
IS - 54
SP - 63
AB - The paper describes the main stages of determining a celestial body
orbit from its observations. The main difficulties that arise when
improving a celestial body orbit are considered. Attention is drawn
to the fact that the orbit determination can be considered as a
boundary value problem, where the boundary conditions are
observations of a celestial body. We determine the initial orbit
using formulas specially developed by Yu. V. Batrakov for satellite
orbits. An example of calculating the orbit of an artificial
satellite is given. The case when there are three positional
observations made from three points with a small time interval
between observations is considered. A scheme for deriving the main
formulas for improving the orbit parameters is presented. We pay
attention to the possibility of a rapid increase in the values in the
right-hand sides of the differential equations for the matrix of
isochronous derivatives, whose elements are the partial derivatives
of the current parameters of motion by their initial values
(matrizant). It is proposed to use the property of the matrizant,
which is that the product of the matrizant calculated on subintervals
is equal to the value of the matrizant in the interval. We reduce the
length of the interval subinterval to the possible minimum — to the
length of the integration step for the equations of motion if a one-
step integration method is used. Numerical experiments have shown
that this can significantly reduce the computation time while
improving the orbit. We note that the choice of the epoch of the
initial parameters enables us to increase the conditionality of the
normal system when improving the orbit.
DO - 10.32876/ApplAstron.54.63-70
UR - http://iaaras.ru/en/library/paper/2064/
ER -
```