Skip to main content
SpringerLink
Log in

Singularity-Free Orbit Theory

  • Chapter
  • First Online:
GPS

  • 3253 Accesses

Abstract

The previous chapter (Chap. 11) of this book covered the most important content regarding numerical satellite orbit determination theory and algorithms. In this chapter, the emphasis will be on singularity-free orbit theory.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Aarseth SJ (2003) Gravitational N-Body simulations Tools and Algorithms, Cambridge University Press

    Google Scholar 

  • Aarseth SJ, Tout CA, Mardling RA (2008) The Cambridge N-Body Lectures, Springer Heidelsberg

    Google Scholar 

  • Barrow-Green J (1996) Poincare and the Three Body Problem, Amer. Math. Soc.

    Google Scholar 

  • Battin R.H. (1999) An Introduction to the Mathematics and Methods of Astrodynamics, revised version, AIAA Education Series

    Google Scholar 

  • Bernoulli JI (1710) Extrait de la réponse de M. Bernoulli à M. Hermann, datée de Basle le 7 Octobre 1710 Joh. B. Op LXXXVI. Both letters were published in the Mémoires de 1’Ac Royale des Sciences, Boudot, Paris

    Google Scholar 

  • Boccaletti D, Pucacco G (1998) Theory of Orbits (two volumes). Springer-Verlag

    Google Scholar 

  • Boccaletti D, Pucacco G (2001) Theory of Oribts, Vol. 1: Integrable systems and non-perturbative methods; Vol. 2: Perturbative and geometrical methods, Springer Berlin

    Google Scholar 

  • Bronstein IN, Semendjajew KA (1987) Taschenbuch der Mathematik.B. G. Teubner Verlagsgesellschaft, Leipzig, ISBN 3-322-00259-4

    Google Scholar 

  • Brouwer D, Clemence GM (1961) Methods of celestial mechanics. Academic Press, New York

    Google Scholar 

  • Chobotov VA (ed) (1991) Orbital mechanics. Published by AIAA, Washington

    Google Scholar 

  • Cui C (1990) Die Bewegung künstlicher Satelliten im anisotropen Gravitationsfeld einer gleichmässig rotierenden starren Modellerde. Deutsche Geodätische Kommission, Reihe C: Dissertationen, Heft Nr. 357

    Google Scholar 

  • Cui C (1997) Satellite orbit integration based on canonical transformations with special regard to the resonance and coupling effects. Dtsch Geod Komm bayer Akad Wiss, Reihe A, Nr. 112, 128 pp

    Google Scholar 

  • Delaunay (1860) Mem. De l’des Sci. 28

    Google Scholar 

  • Delaunay (1867) Mem. De l’des Sci. 29

    Google Scholar 

  • Diacu, F (1992) Singularities of the N-Body Problem, Les Publications CRM, Montreal

    Google Scholar 

  • Diacu F (1996) The solution of the n-body problem, The Mathematical Intelligencer, 1996, 18, p. 66–70

    Google Scholar 

  • Diacu F, Holmes P (1996) Celestial Encounters: The origins or chaos and stability, Princeton University Press, Princeton, NJ

    Google Scholar 

  • Dvorak R, Lhotka C (2013) Celestial dynamics – chaoticity and dynamics of celestial systems, Wiley, Weinheim

    Google Scholar 

  • Eberle J, Cuntz M, Musielak ZE (2008) The instability trasition for the restricted 3-body problem - I. Theoretical approach, Astronomy & Astrophysics, Vol. 489 No.3

    Google Scholar 

  • Euler L (1767) Nov. Comm. Acad. Imp. Petropolitanae, 10, pp207-242, 11, pp152-184; Memories de IAcad. De Berlin, 11, 228-249

    Google Scholar 

  • Goldstein H (1980) Classical Mechanics (2nd Ed), New York, Addison-Wesley

    Google Scholar 

  • Hagihara, Y: Celestial Mechanics. (Vol I and Vol II pt 1 and Vol II pt 2.) MIT Press, 1970

    Google Scholar 

  • Havel K (2008) N-Body Gravitational Problem: Unrestricted Solution, Brampton: Grevyt Press, 2008

    Google Scholar 

  • Herrick S (1972) Astrodynamics, Vol. II. Van Nostrand Reinhold, London

    Google Scholar 

  • Kaula WM (2000) Theory of satellite geodesy: applications of satellites to geodesy. Courier Corporation

    Google Scholar 

  • Kaula WM (1966, 2001) Theory of satellite geodesy. Blaisdell Publishing Company, Dover Publications, New York

    Google Scholar 

  • Kaula WM (2001) Theory of satellite geodesy. Blaisdell Publishing Company, Dover Publications, New York

    Google Scholar 

  • Lagrange JL (1772) Miscellanea Taurinensia, 4, 118-243; Oeuvres, 2, pp67-121; Mechanique Analytique, 1st Ed, pp262-286; 2nd Ed, 2, pp108-121; Oeuvres, 12, pp101-114

    Google Scholar 

  • Landau LD, Lifshitz EM (1976) Mechanics (3rd Ed), New York, Pergamon Press

    Google Scholar 

  • Liu L, Zhao D (1979) Orbit theory of the Earth satellite. Nanjing University Press, (in Chinese)

    Google Scholar 

  • Mittag-Leffler G. The n-body problem (Price Announcement), Acta Matematica, 1885/1886, 7

    Google Scholar 

  • Newton I(1687): Philosophiae Naturalis Principia Mathematica, London, 1687: also English translation of 3rd (1726) edition by I. Bernard Cohen and Anne Whitman (Berkeley, CA, 1999)

    Google Scholar 

  • Poincare (1892) Les Methodes Nouevelles de la Mechanique ce’leste Guthier villars, Paris, Chap. V, p. 250, (published in English in three volumes)

    Google Scholar 

  • Poincare H (1992) New Methods of Celestial Mechanics, AIP

    Google Scholar 

  • Schneider M, Cui CF (2005) Theoreme über Bewegungsintegrale und ihre Anwendung in Bahntheorien, Bayerischen Akad Wiss, Reihe A, Heft Nr. 121, 132pp, München

    Google Scholar 

  • Shapiro II (1962) The prediction of satellite orbits, in IUTAM Symposium Paris 1962, M Roy (ed) Dynamics of Satellites

    Google Scholar 

  • Sundman K (1906) Recherches sur le problème des trois corps, Acta Soc. Sei. Fenn. 34

    Google Scholar 

  • Sundman K (1909) Nouvelles recherches sur le problème des trois corps, Acta Soc. Sei. Fenn. 35

    Google Scholar 

  • Sundman KE (1912) Memoire sur le probleme de trois corps, Acta Mathematica 36 (1912): 105–179.

    Google Scholar 

  • Tisserand F-F (1894) Mecanique Celeste, tome III (Paris, 1894), ch.III, at p. 27

    Google Scholar 

  • Vallado David A (2007) Fundamentals of Astrodynamics and Applications (3rd Ed), Microcosm Press & Springer

    Google Scholar 

  • Van Kamp PD (1967) Principles of Astronomy. W.H. Freemann and Company, San Francisco, CA/London

    Google Scholar 

  • Wang QD (1991) The global solution of the n-body problem (Celestial Mechanics and Dynamical Astronomy (ISSN 0923-2958), vol. 50, no. 1, 1991, p. 73–88., URI retrieved on 2007-05-05)

    Google Scholar 

  • Wang LX, Fang ZD, Zhang MY, Lin GB, Gu LK, Zhong TD, Yang XA, She DP, Luo ZH, Xiao BQ, Chai H, Lin DX (1979) Mathematic handbook. Educational Press, Peking, ISBN 13012-0165

    Google Scholar 

  • Wnuk E (1990) Tesseral harmonic perturbations in the Keplerian orbital elements, Acta. Astronomica Vol.40, No.1-2, p.191-198

    Google Scholar 

  • Xia ZH (1992) The existence of noncollision singularities in Newtonian system, Annals Math. 135(3): 411–468

    Google Scholar 

  • Xu (2013b) Private communication with an editor of MNRAS concerning two submitted papers and the decision letters of MNRAS editorial board (for review available upon request gcxu@sdu.edu.cn)

    Google Scholar 

  • Xu G (2008) Orbits, Springer Heidelberg, ISBN 978-3-540-78521-7, 230 pages, in English

    Google Scholar 

  • Xu G (2010) Analytic Orbit Theory, chapter 4 in G Xu (Ed) Sciences of Geodesy - I, Advances and Future Directions, Springer, pp 105-154

    Google Scholar 

  • Xu G (2010) (Ed.): Sciences of Geodesy - I, Advances and Future Directions, Springer Heidelberg, chapter topics (authors): Aerogravimetry (R Forsberg), Superconducting Gravimetry (J Neumeyer), Absolute and Relative Gravimetry (L Timmen), Deformation and Tectonics (L Bastos et al.), Analytic Orbit Theory (G Xu), InSAR (Y Xia), Marine Geodesy (J Reinking), Kalman Filtering (Y Yang), Equivalence of GPS Algorithms (G Xu et al.), Earth Rotation (F Seitz, H Schuh), Satellite Laser Ranging (L Combrinck), in English, 507 pages

    Google Scholar 

  • Xu Y (2012) Studies on Antarctic GNSS Precise Positioning. Chang’an University. Xi’an, China

    Google Scholar 

  • Xu G (2012) (Ed.): Sciences of Geodesy - II, Advances and Future Directions, Springer Heidelberg, chapter topics (authors): General Relativity and Space Geodesy (L Comblinck), Global Terrestrial Reference Systems and their Realizations (D Angermann et al), Ocean Tide Loading (M Bos, HG Scherneck), Photogrammetry (P Redweik), Regularization and Adjustment (Y Shen, G Xu), Regional Gravity Field Modelling (H Denker), VLBI (H Schuh, J Boehm), in English, 400 pages

    Google Scholar 

  • Xu G, Xu J (2012) On the Singularity Problem of Orbital Mechanics, MNRAS, 2013, Vol.429, pp1139-1148

    Google Scholar 

  • Xu G, Xu J (2013a) On Orbital Disturbance of Solar Radiation, MNRAS, 432 (1): 584-588 doi:10.1093/mnras/stt483

    Google Scholar 

  • Xu G, Xu J (2013b) Orbits – 2nd Order Singularity-free Solutions, second edition, Springer Heidelberg, ISBN 978-3-642-32792-6, 426 pages, in English

    Google Scholar 

  • Xu G, Xu TH, Yeh TK, Chen W (2010a) Analytic Solution of Satellite Orbit Perturbed by Lunar and Solar Gravitation, MNRAS, Vol. 410, Issue 1, pp 645-653

    Google Scholar 

  • Xu G, Xu TH, Chen W, Yeh TK (2010b) Analytic Solution of Satellite Orbit Perturbed by Atmospheric Drag, MNRAS, Vol. 410, Issue 1, pp 654-662 87.

    Google Scholar 

  • Xu Y, Yang Y, Zhang Q, Xu G (2011) Solar Oblateness and Mercury’s Perihelion Precession, MNRAS, Vol. 415, 3335-3343

    Google Scholar 

  • Xu G, Lv ZP, Shen YZ, Yeh TK (2014) A mathematical derivation of singularity-free Lagrange equations of planetary motion, Special issue for celebration 80th birthday of academician Houze Xu, Journal of Surveying and Mapping, 2014 89.

    Google Scholar 

  • Xu Y, Yang Y, Xu G (2014) Analysis on Tropospheric Delay in Antarctic GPS Positioning. Journal of Geodesy and Geodynamics, 34(1): pp104-107

    Google Scholar 

  • Xu G, Chen W, Shen YZ, Jiang N, Jiang CH (2015) A mathematical derivation of singularity-free Gauss equations of planetary motion, Special issue for celebration 70th birthday of Prof Jikun Ou, Journal of Surveying and Mapping, 2015

    Google Scholar 

  • Xu Y, Jiang N, Xu G, Yang Y, Schuh H (2015) Influence of meteorological data and horizontal gradient of tropospheric model on precise point positioning. Adv. Space Res. 56(11), pp2374-2383

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shandong University at Weihai, Institute of Space Sciences, Weihai, China

    Guochang Xu

  2. GFZ German Research Centre for Geosciences, Geodesy, Potsdam, Germany

    Yan Xu

Authors
  1. Guochang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guochang Xu .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Xu, G., Xu, Y. (2016). Singularity-Free Orbit Theory. In: GPS. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-50367-6_12

Download citation

Publish with us

Policies and ethics

Search

Navigation