# Orbit determination with the two-body integrals: III

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## Abstract

We present the results of our investigation on the use of the two-body integrals to compute preliminary orbits by linking too short arcs of observations of celestial bodies. This work introduces a significant improvement with respect to the previous papers on the same subject: Gronchi et al. (2010, 2011). Here we find a univariate polynomial equation of degree 9 in the radial distance \(\rho \) of the orbit at the mean epoch of one of the two arcs. This is obtained by a combination of the algebraic integrals of the two-body problem. Moreover, the elimination step, which in Gronchi et al. (2010, 2011) was done by resultant theory coupled with the discrete Fourier transform, is here obtained by elementary calculations. We also show some numerical tests to illustrate the performance of the new algorithm.

## Keywords

Orbit determination Algebraic methods Linkage of too short arcs \(J_{2}\) effect## Notes

### Acknowledgments

We wish to thank the referees, Zoran Knežević and Jean-Marc Petit, whose useful comments allowed us to improve the final version of the manuscript. This work is partially supported by the Marie Curie Initial Training Network Stardust, FP7-PEOPLE-2012-ITN, Grant Agreement 317185.

## References

- Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, New York (2005)Google Scholar
- Farnocchia, D., Tommei, G., Milani, A., Rossi, A.: Innovative methods of correlation and orbit determination for space debris. Cel. Mech. Dyn. Astron.
**107**(1–2), 169–185 (2010)Google Scholar - Gauss, C.F.: Theoria motus corporum coelestium in sectionibus conicis solem ambientium (Theory of the motion of the heavenly bodies moving about the sun in conic sections). Dover publications (1809/1963)Google Scholar
- Gronchi, G.F., Dimare, L., Milani, A.: Orbit determination with the two-body integrals. Cel. Mech. Dyn. Astron.
**107**(3), 299–318 (2010)Google Scholar - Gronchi, G.F., Farnocchia, D., Dimare, L.: Orbit determination with the two-body integrals. II. Cel. Mech. Dyn. Astron.
**110**(3), 257–270 (2011)Google Scholar - Milani, A., Sansaturio, M.E., Chesley, S.R.: The asteroid identification problem IV: attributions. Icarus
**151**, 150–159 (2001)CrossRefADSGoogle Scholar - Milani, A., Gronchi, G.F.: The Theory of Orbit Determination. Cambridge University Press, Cambridge (2010)Google Scholar
- Taff, L.G., Hall, D.L.: The use of angles and angular rates. I—initial orbit determination. Cel. Mech.
**16**, 481–488 (1977)Google Scholar