Celestial Mechanics and Dynamical Astronomy

, Volume 107, Issue 3, pp 299–318 | Cite as

Orbit determination with the two-body integrals

Original Article

Abstract

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.

Keywords

Orbit determination Algebraic methods Attributables Linkage Covariance matrix 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bate R.R., Mueller D.D., White J.E.: Fundamentals of Astrodynamics. Dover publications, New York (1971)Google Scholar
  2. Bini D.A: Numerical computation of polynomial zeros by means of Aberth method, Numer. Algorithms 13(3–4), 179–200 (1997)MathSciNetGoogle Scholar
  3. Cox D.A., Little J.B., O’Shea D.: Ideals, Varieties and Algorithms. Springer, New York (1996)MATHGoogle Scholar
  4. Gauss C.F.: Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium (1809), reprinted by Dover publications, New York (1963)Google Scholar
  5. Gronchi G.F.: On the stationary points of the squared distance between two ellipses with a common focus. SIAM J. Sci. Comp. 24(1), 61–80 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. Gronchi G.F.: Multiple solutions in preliminary orbit determination from three observations. Celest. Mech. Dyn. Astron. 103(4), 301–326 (2009)CrossRefMathSciNetADSGoogle Scholar
  7. Kubica J., Denneau L., Grav T., Heasley J., Jedicke R., Masiero J., Milani A., Moore A., Tholen D., Wainscoat R.J.: Efficient intra- and inter-night linking of asteroid detections using kd-trees. Icarus 189, 151–168 (2007)CrossRefADSGoogle Scholar
  8. Laplace P.S.: Mém. Acad. R. Sci. Paris, in Laplace’s collected works 10, 93–146 (1780)Google Scholar
  9. Leuschner A.O. On the Laplacean orbit methods. Proceedings of the ICM, 209–217 (1912)Google Scholar
  10. Merton G.: A modification of Gauss’s method for the determination of orbits. MNRAS 85, 693–731 (1925)ADSGoogle Scholar
  11. Milani A., Sansaturio M.E., Chesley S.R.: The asteroid identification problem IV: Attributions. Icarus 151, 150–159 (2001)CrossRefADSGoogle Scholar
  12. Milani A., Gronchi G.F., de’Michieli Vitturi M., Knežević Z.: Orbit determination with very short arcs I. Admissible regions. Celest. Mech. Dyn. Astron. 90, 59–87 (2004)MATHCrossRefADSGoogle Scholar
  13. Milani A., Gronchi G.F., Knežević Z., Sansaturio M.E., Arratia O.: Orbit determination with very short arcs II. Identifications. Icarus 79, 350–374 (2005)CrossRefADSGoogle Scholar
  14. Milani A., Gronchi G.F., Knežević Z.: New definition of discovery for solar system objects. Earth Moon Planets 100, 83–116 (2007)MATHCrossRefADSGoogle Scholar
  15. Milani A., Gronchi G.F., Farnocchia D., Knežević Z., Jedicke R., Denneau L., Pierfederici F.: Topocentric orbit determination: algorithms for the next generation surveys. Icarus 195, 474–492 (2008)CrossRefADSGoogle Scholar
  16. Milani A., Gronchi G.F.: Theory of Orbit Determination. Cambridge University Press, Cambridge (2009)Google Scholar
  17. Plummer, H.C.: An Introductory Treatise on Dynamical Astronomy. Cambridge University press (1918), reprinted by Dover publications, New York (1960)Google Scholar
  18. Poincaré H.: Sur la détermination des orbites par la méthode de Laplace. Bull. Astrono. 23, 161–187 (1906)Google Scholar
  19. Taff L.G., Hall D.L.: The use of angles and angular rates. I—Initial orbit determination. Celest. Mech. 16, 481–488 (1977)CrossRefADSGoogle Scholar
  20. Taff L.G., Hall D.L.: The use of angles and angular rates. II—Multiple observation initial orbit determination. Celest. Mech. 21, 281–290 (1980)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Pisa5 PisaItaly
  2. 2.Dipartimento di MatematicaUniversità di Roma ‘La Sapienza’2 RomaItaly

Personalised recommendations