# Statistical inversion method for binary asteroids orbit determination

Irina Kovalenko,, Daniel Hestroffer, A. Doressoundiram, Nicolai Emelyanov, Radu S. Stoica

► To cite this version:

Irina Kovalenko,, Daniel Hestroffer, A. Doressoundiram, Nicolai Emelyanov, Radu S. Stoica. Statistical inversion method for binary asteroids orbit determination. Journées systèmes de référence spatio-temporels, Sep 2014, St Petersburg, Russia. hal-01103904

HAL Id: hal-01103904

https://hal.science/hal-01103904

Submitted on 15 Jan 2015

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
## Introduction

We focus on the study of binary asteroids, which are common in the Solar system from its inner to its outer regions. These objects provide fundamental physical parameters such as mass and density, and hence clues on the early Solar System, or other processes that are affecting asteroid over time. The present method of orbit computation for resolved binaries is based on Markov Chain Monte-Carlo statistical inversion technique. Particularly, we use the Metropolis-Hasting algorithm with Thiele-Innes equation for sampling the orbital elements and system mass through the sampling of observations. The method requires a minimum of four observations, made at the same tangent plane; it is of particular interest for orbit determination over short arcs or with sparse data. The observations are sampled within their observational errors with an assumed distribution. The sampling yields the whole region of possible orbits, including the one that is most probable.

## Orbit determination

The astrometric observations are related to
the theoretical positions through the
observational equation:

<figure><img src="image_2.png" /></figure>

* Observations: $\varphi = (\rho_1, \theta_1; \dots; \rho_N, \theta_N)$

* Sky-plane position: $\psi(\mathbf{X})$

* Orbital elements + system's mass: $\mathbf{X} = (a, e, i, \Omega, \omega, M, m_{\text{sys}})$

* Observational errors: $\varepsilon = (\varepsilon_{\alpha 1}, \varepsilon_{\delta 1}; \dots; \varepsilon_{\alpha N}, \varepsilon_{\delta N})$

$$p(\mathbf{X}|\varphi) \propto p(\varphi|\mathbf{X})p(\mathbf{X})$$

Where

$$\bullet p(\mathbf{X}) \propto \sqrt{\det\Lambda^{-1}}$$

$$\bullet p(\varphi|\mathbf{X}) = p_{\varepsilon}(\text{observational error p. d. f.}) = \exp\left[-\frac{1}{2}(\varphi - \psi(\mathbf{X}))^T \Lambda^{-1} (\varphi - \psi(\mathbf{X}))\right]$$

## Markov Chain Monte-Carlo method

The Metropolis-Hastings algorithm will be used for sampling the parameters $\mathbf{X}$.

<figure><img src="image_3.png" /></figure>

For each $t$ iteration:

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

The algorithm is run for a large number of iterations until the entire possible orbital-element space is mapped

<figure><img src="image_6.png" /></figure>

## References

1) R. Palacios 1958 AJ 63, 395 2) D. Oszkiewicz et al. 2013 ,SF2A 237

## Acknowledgements

This work is supported by the Labex ESEP (ANR N° 2011-LABX-030).