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The motion of a satellite in an axi-symmetric gravitational field

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Abstract

The equations for the variation of the osculating elements of a satellite moving in an axi-symmetric gravitational field are integrated to yield the complete first-order perturbations for the elements of the orbit. The expressions obtained include the effects produced by the second to eighth spherical harmonics. The orbital elements are presented in the most general form of summations by means of Hansen coefficients. Due to their general forms it is a simple matter to estimate the perturbations of any higher harmonic by simply increasing the index of summation. Finally, this paper gives the respective general expressions for the secular perturbations of the orbital elements. The formulae presented should be useful for the reductions of Earth-satellite observations and geopotential studies based on them.

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Abbreviations

α:

semi-major axis

C njk (ν, ω):

cosine functions of ν and ω

e :

eccentricity of the orbit

f :

acceleration vector of perturbing force

f :

sin2t

i:

inclination of the orbit

J n :

coefficients in the potential expansion

M :

mean anomaly

n :

mean motion

p :

semi-latus rectum of the orbit

R, S, andW :

components of the perturbing acceleration

r :

radius-vector of satellite

r :

magnitude ofr

S njk (ν, ω):

sine functions of ν and ω

T :

time of perigee passage

u :

argument of latitude

U :

gravitational potential

ν:

true anomaly

V :

perturbing potential

μ:

G(M++m) (gravitational constant times the sum of the masses of Earth and satellite)

ρn,k :

coefficients ofR component of disturbing acceleration (funtions off)

σn,k :

coefficients ofS andW components of disturbing acceleration (functions off)

χ:

mean anomaly at timet=0

X n,m0 :

zero-order Hansen coefficients

Ω:

argument of perigee

Ω:

right ascension of the ascending node

References

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Authors and Affiliations

  1. Department of Physics, University of Patras, Greece

    Basil Zafiropoulos

Authors
  1. Basil Zafiropoulos
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Zafiropoulos, B. The motion of a satellite in an axi-symmetric gravitational field. Astrophys Space Sci 139, 111–128 (1987). https://doi.org/10.1007/BF00643817

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  • DOI: https://doi.org/10.1007/BF00643817

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