Skip to main content
SpringerLink
Log in

Rigorous estimates for the relegation algorithm

  • Original Article
  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

We revisit the relegation algorithm by Deprit et al. (Celest. Mech. Dyn. Astron. 79:157–182, 2001) in the light of the rigorous Nekhoroshev’s like theory. This relatively recent algorithm is nowadays widely used for implementing closed form analytic perturbation theories, as it generalises the classical Birkhoff normalisation algorithm. The algorithm, here briefly explained by means of Lie transformations, has been so far introduced and used in a formal way, i.e. without providing any rigorous convergence or asymptotic estimates. The overall aim of this paper is to find such quantitative estimates and to show how the results about stability over exponentially long times can be recovered in a simple and effective way, at least in the non-resonant case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Precisely, \(\mathcal {G}_{\varrho }=\big \{z\in \mathbb {C}^{n_1}:\max _{1\le j\le n_1}|z_j|<\varrho \big \}\), \(\mathbb {T}^{n_1}_{\sigma }=\big \{q\in \mathbb {C}^{n_1}:\mathrm{Re}\, q_j\in \mathbb {T},\ \max _{1\le j\le n_1}|\mathrm{Im}\, q_j|<\sigma \big \}\,\), \(\mathcal {B}_R=\{z\in \mathbb {C}^{2n_2}: \max _{1\le j\le 2n_2} |z_j|<R\,\}\).

  2. As reported in Giorgilli (2003) (pag. 86, footnote 2), choosing the parameter K by asking \(e^{-K \sigma }\sim \varepsilon \) is not really convenient. As it will be evident from the optimisation of the parameters in the proof of Theorem 1, the best choice is \(K\sim 1/\sigma \).

References

  • Benettin, G., Galgani, L., Giorgilli, A.: Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory, part II. Commun. Math. Phys 121, 557–601 (1989)

    Article  ADS  MATH  Google Scholar 

  • Ceccaroni, M., Biggs, J.D.: Analytical perturbative method for frozen orbits around the asteroid 433 Eros, IAC2012 International Astronautical Congress - Naples, IAC-12,C1,7,6,x14267 (2012)

  • Ceccaroni, M., Biggs, J.D.: Analytic perturbative theories in highly inhomogeneous gravitational fields. Icarus 224, 74–85 (2013)

    Article  ADS  Google Scholar 

  • Ceccaroni, M., Biscani, F., Biggs, J.D.: Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields: a first approach. Sol. Syst. Res. 48, 33–47 (2014)

    Article  ADS  Google Scholar 

  • Deprit, A., Palacián, J., Deprit, E.: The relegation algorithm. Celest. Mech. Dyn. Astron. 79, 157–182 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Feng, J., Noomen, R., Visser, P.N., Yuan, J.: Modelling and analysis of periodic orbits around a contact binary asteroid. Astrophys. Space Sci. 357, 1–18 (2015)

    Article  Google Scholar 

  • Giorgilli, A.: Notes on exponential stability of Hamiltonian systems. In: Dynamical Systems. Part I: Hamiltonian Systems and Celestial Mechanics, Pubblicazioni della Classe di Scienze, Scuola Normale Superiore, Pisa. Centro di Ricerca Matematica “Ennio De Giorgi” (2003)

  • Giorgilli, A., Locatelli, U., Sansottera, M.: Kolmogorov and Nekhoroshev theory for the problem of three bodies. Celest. Mech. Dyn. Astron. 104, 159–173 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Giorgilli, A., Locatelli, U., Sansottera, M.: On the convergence of an algorithm constructing the normal form for elliptic lower dimensional tori in planetary systems. Celest. Mech. Dyn. Astron. 119, 397–424 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Giorgilli, A., Sansottera, M.: Methods of algebraic manipulation in perturbation theory. Workshop Ser. Asoc. Argent. Astrono. 3, 147–183 (2011)

    ADS  Google Scholar 

  • Gröbner, W.: Die Lie-Reihen und Ihre Anwendungen, SpringerVerlag, Berlin (1960); Italian transl.: Le serie di Lie e leloro applicazioni, Cremonese, Roma (1973)

  • Lara, M., San-Juan, J.F., López-Ochoa, L.M.: Averaging tesseral effects: closed form relegation versus expansions of elliptic motion. Math. Probl. Eng. 2013, 570127 (2013)

    MathSciNet  Google Scholar 

  • Lara, M., San-Juan, J.F., López-Ochoa, L.M.: Delaunay variables approach to the elimination of the perigee in artificial satellite theory. Celest. Mech. Dyn. Astron. 120, 39–56 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Nekhoroshev, N.N.: Exponential estimates of the stability time of near-integrable Hamiltonian systems. English translation: Russ. Math. Surv. 32, 1 (1977)

  • Nekhoroshev, N.N.: Exponential estimates of the stability time of near-integrable Hamiltonian systems, 2. Trudy Sem. Im. G. Petrovskogo, 5, 5 (1979). English translation: Topics in modern Mathematics, Petrovskij Semin., 5, 1–58 (1985)

  • Noullez, A., Tsiganis, K., Tzirti, S.: Satellite orbits design using frequency analysis. Adv. Space Res. 56, 163–175 (2015)

  • Palacián, J.: Teoría del Satélite Artificial: Armónicos Teserales y su Relegación Mediante Simplificaciones Algebraicas, Ph.D. thesis, Universidad de Zaragoza (1992)

  • Palacián, J.: Normal forms for perturbed Keplerian systems. J. Differ. Equ. 180, 471–519 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Pardal, P.C.P.M., de Moraes, R.V., Kuga, H.K.: Effects of geopotential and atmospheric drag effects on frozen orbits using nonsingular variables. Math. Probl. Eng. (2014). doi:10.1155/2014/678015

  • Sansottera, M., Lhotka, C., Lemaître, A.: Effective stability around the Cassini state in the spin-orbit problem. Celest. Mech. Dyn. Astron. 119, 75–89 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Sansottera, M., Lhotka, C., Lemaître, A.: Effective resonant stability of Mercury. MNRAS 452, 4145–4152 (2015)

  • Sansottera, M., Locatelli, U., Giorgilli, A.: A semi-analytic algorithm for constructing lower dimensional elliptic tori in planetary systems. Celest. Mech. Dyn. Astron. 111, 337–361 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Sansottera, M., Locatelli, U., Giorgilli, A.: On the stability of the secular evolution of the planar Sun-Jupiter-Saturn-Uranus system. Math. Comput. Simul. 88, 1–14 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We warmly thank A. Giorgilli for helpful discussions and useful comments. The work of M. S. have been partially supported by the research program “Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite”, PRIN 2010JJ4KPA 009, financed by MIUR.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, 20133, Milano, Italy

    Marco Sansottera

  2. Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133, Roma, Italy

    Marta Ceccaroni

Authors
  1. Marco Sansottera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marta Ceccaroni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Sansottera.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sansottera, M., Ceccaroni, M. Rigorous estimates for the relegation algorithm. Celest Mech Dyn Astr 127, 1–18 (2017). https://doi.org/10.1007/s10569-016-9711-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10569-016-9711-2

Keywords

Search

Navigation