Skip to main content
SpringerLink
Log in

A discussion of Hill's method of secular perturbations and its application to the determination of the zero-rank effects in non-singular vectorial elements of a planetary motion

  • Published:
Celestial mechanics Aims and scope Submit manuscript

Abstract

Knowledge of the perturbations of zero-rank is essential for the understanding of the behavior of a planetary or cometary orbit over a long interval of time. Recent investigations show that these zero-rank perturbations can cause large oscillations in both the shape and position of the orbit. At present we lack a complete analytical theory of these perturbations that can be applied to cases where either the eccentricity or inclination is large or has large oscillations. For this reason we here develop formulas for the numerical integration of the zero-rank effects, using a modified Hill's theory and suitable vectorial elements. The scalar elements of our theory are the two components of Hamilton's vector in a moving ideal reference frame and the three components of Gibb's rotation vector in an inertial system. The integration step can be taken to be several hundred years in the planetary or cometary case, and a few days in the case of a near-Earth space probe. We re-discuss Hill's method in modern symbolism and by applying the vectorial analysis in a pseudo-euclidean spaceM 3, we obtain a symmetrical computational scheme in terms of traces of dyadics inM 3. The method is inapplicable for two orbits too close together. In Hill's method the numerical difficulty caused by such proximity appears in the form of a small divisor, whereas in Halphen's method it appears as a slow convergence of a hypergeometric series. Thus, in Hill's method the difficulty can be watched more directly than in Halphen's method. The methods of numerical averaging have, at the present time, certain advantages over purely analytical methods. They can treat a large range of eccentricities and orbital inclinations. They can also treat the free ‘secular’ oscillations as well as the forced ones, and together with their mutual cross-effects. At the present time, no analytical theory can do this to the full extent.

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

Abbreviations

m :

the mass of the disturbed body

M :

the mass of the Sun

f :

the gravitational constant

μ:

f(M+m)

r :

the heliocentric position vector of the disturbed body

r :

|r|

r 0 :

the unit vector alongr

n 0 :

the unit vector normal tor and lying in the orbital plane of the disturbed body

a :

the semi-major axis of the orbit of the disturbed body

e :

the eccentricity of the orbit of the disturbed body

g :

the mean anomaly of the disturbed body

ε:

the eccentric anomaly of the disturbed body

p :

a(1−e 2)

P 1 :

the unit vector directed from the Sun toward the perihelion of the disturbed body

P 2 :

the unit vector normal toP 1 and lying in the orbital plane of the disturbed body

s :

\(\frac{e}{{\sqrt {1 - e^2 } }}P_1 \)

λ:

the true orbital longitude of the disturbed body, reckoned from the departure point of the ideal system of coordinates

X :

the true orbital longitude of the perihelion of the disturbed body in the ideal system of coordinates reckoned from the departure point

σ:

the angular distance of the ascending node from the departure point

R 1,R 2,R 3 :

the unit vectors along the axes of the ideal system of coordinates,R 1 andR 2 are in the osculating orbital plane of the disturbed body,R 3 is normal to this plane. The intersection ofR 1 with the celestial sphere is the departure point

R 3 :

P 1×P 2

S 1,S 2,S 3 :

the initial values ofR 1,R 2,R 3, respectively

q :

the Gibb's vector. This vector defines the rotation of the orbital plane of the disturbed body from its initial position to the position at the given timet

m′ :

the mass of the disturbing body

r′ :

the heliocentric position vector of the disturbing body

a′ :

the semi-major axis of the orbit of the disturbing body

e′ :

the eccentricity of the orbit of the disturbing body

g′ :

the mean anomaly of the disturbing body

ω′:

the eccentric anomaly of the disturbing body

P′1 :

the unit vector directed from the Sun toward the perihelion of the disturbing body

P′2 :

the unit vector normal toP′1 and lying in the orbital plane of the disturbing body

A′1 :

a′ P′1

A′2 :

\(a' \sqrt {1 - e'^2 } P'_2 \)

Δ:

|r′−r|

References

  • Baker, H. F.: 1916, ‘On Certain Differential Equations of Astronomical Interest’,Phil. Trans. A 216, 129.

    Google Scholar 

  • Bohl, P.: 1909, ‘Über ein in der Theorie der säkularen Störungen vorkommendes Problem’,J. reine angew. Math. 135, 189–283.

    Google Scholar 

  • Bohl, P.: 1912, ‘Bemerkungen zur Theorie der säkularen Störungen’,Math. Ann. 72, 295–296.

    Google Scholar 

  • Callandreau, O.: 1885, ‘Calcul des variations séculaires des éléments des orbites’,Ann. Observ. Paris,18, A1-A46.

    Google Scholar 

  • Cody, W. J.: 1965, ‘Chebyshev Approximations for the Complete Elliptic IntegralsK andE’,Math. Comput. 19, 105–112.

    Google Scholar 

  • Gauss, K. F.: 1818, ‘Determinatio attractionis quam in punctum quadris positionis datae exerceret planeta, si ejus massa etc.’,Collected Works, Vol.III, p. 331.

    Google Scholar 

  • Gibbs, J. W.: 1901,Vector Analysis, Yale University Press, pp. 343–345.

  • Goriachev, N. N.: 1937,On the Method of Halphen of the Computation of Secular perturbations (in Russian), University of Tomsk, pp. 1–115.

  • Hagihara, Y.: 1928, ‘On the Theory of the Secular Perturbation of Higher Degrees’,Proc. Phys.-Math. Soc. Japan [3],10, 1–14, 34–59, 87–113, 127–156.

    Google Scholar 

  • Halphen, G. H.: 1888,Traité des fonctions elliptiques, Vol. 2, Paris.

  • Hansen, P. A.: 1857, ‘Auseinandersetzung etc. I’,Abh. Sach. Ges. Wiss. 5.

  • Hill, G. W.: 1882, ‘On Gauss's Method of Computing Secular Perturbations’,Astron. Papers of Am. Ephemeris 1, 315–361.

    Google Scholar 

  • Hirayama, K.: 1923, ‘Families of Asteroids’,Jap. J. Astron. Geoph. 1, 55–93.

    Google Scholar 

  • Jessen, B.: 1935, ‘Über die Säkularkonstanten einer fast periodischen Funktion’,Math. Ann. 111, 355–363.

    Google Scholar 

  • Kozai, Y.: 1954, ‘On the Secular Perturbation of Asteroids’,Publ. Astron. Soc. Japan 6, No. 1, 41–66.

    Google Scholar 

  • Kozai, Y.: 1962, ‘Secular Perturbations of Asteroids with High Inclination and Eccentricity’,Astron. J. 67, No. 9, 591–598.

    Google Scholar 

  • Milankovitch, M.: 1948,Astronomische Theorie der Klimaschwankungen und ihre Anwendung in der Geophysik, Belgrade.

  • Musen, P.: 1961, ‘On Strömgren's Method of Special Perturbations’,J. Astronaut. Sci. 8 (2), 50.

    Google Scholar 

  • Musen, P.: 1963, ‘Discussion of Halphen's Method of Secular Perturbations and Its Application to the Determination of Long-Range Effects in the Motion of Celestial Bodies’,Rev. Geophys. 1, No. 1, 85–122.

    Google Scholar 

  • Poincaré, H.: 1905,Leçons de mécanique céleste, Paris.

  • Shute, B. E.: 1964, ‘Launch Windows for Highly Eccentric Orbits’,Progress in Astronautics and Aeronautics 14, Academic Press, New York, pp. 377–387.

    Google Scholar 

  • Shute, B. E.: 1964, ‘Prelaunch Analysis of High Eccentricity Orbits’, NASA TN-D-2530.

  • Skripnichenko, W. I.: 1968, ‘Determination of Mean Motions of the Elements of Planetary Orbits in the Trigonometrical Theory of Secular Perturbations’ (in Russian),Bull. ITA No. 7 (130), 441–452.

  • Smith, A.: 1964, ‘A Discussion of Halphen's Method for Secular Perturbations and its Application to the Determination of Long-Range Effects in the Motion of Celestial Bodies, part 2’. NASA TR R-194.

  • Tornehave, H. and Jessen, O.: 1945, ‘Mean Motions and Zeros of Almost Periodic Functions’,Acta Math. 77, 138–279.

    Google Scholar 

  • Weyl, H.: 1938, ‘Mean Motion, I’,Amer. J. Math. 60, 889–896.

    Google Scholar 

  • Weyl, H.: 1939, ‘Mean Motion, II’,Amer. J. Math. 61, 143–148.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory for Theoretical Studies, NASA-Goddard Space Flight Center, Greenbelt, Md., USA

    Peter Musen

Authors
  1. Peter Musen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Musen, P. A discussion of Hill's method of secular perturbations and its application to the determination of the zero-rank effects in non-singular vectorial elements of a planetary motion. Celestial Mechanics 2, 41–59 (1970). https://doi.org/10.1007/BF01230449

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01230449

Keywords

Search

Navigation