Abstract
In the present article we develop the theory of the long period tidal effects in the motion of artificial satellites assuming the variability of elastic parameters of the Earth (Love numbers) across the parallels. The dependence of Love numbers on the longitude produces perturbations of the period of one day or less and hence is neglected in the present theory. In this respect we follow in the footsteps of Kaula (1969). If the deviations ofk 2 andk 3 from pure constants are not taken into consideration, then the perturbations caused by the variability ofk 2 andk 3 across the parallels will be misinterpreted as the perturbations caused byk 4...-terms, and the spurious values ofk 4... will be deduced.
It is extremely doubtful, however, that the real effects caused byk 4,k 5,..., are significant enough to be detected. The short period effects with the period of the revolution of the satellite, or less, were removed from the differential equations for the variation of elements of the satellite by the averaging over the orbit of the satellite. These differential equations are in the form convenient for numerical integration over a long interval of time and also suitable for developing the tidal effects into trigonometric series with the arguments ω, Ω of the satellite andl, l′, F, D, Γ of the Moon.
The numerical integration can be performed using some simple quadrature formula, without resorting to a predictor-corrector system.
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Abbreviations
- A :
-
the right ascension of the satellite
- r :
-
the distance of the satellite from the center of the Earth
- r 0 :
-
the unit vector directed from the center of the Earth toward the satellite
- λ, μ, ν:
-
the equatorial components ofr 0
- g :
-
the mean anomaly of the satellite
- f :
-
the true anomaly of the satellite
- ω:
-
the argument of perigee of the satellite
- Ω:
-
the right ascension of the ascending node of the satellite
- π=ω+Ω:
-
the longitude of the perigee of the satellite
- L=g+π:
-
the mean longitude of the satellite
- i :
-
the inclination of the orbital plane of the satellite relative to the equatorial plane of a given epoch
- e :
-
the eccentricity of the satellite's orbit
- a :
-
the semi-major axis of the orbit of the satellite
- n :
-
the mean motion of the satellite
- R :
-
the unit vector normal to the orbital plane of the satellite
- δL, δπ, δΩ, δi :
-
the tidal perturbations inL, π, Ω andi, respectively
- m′:
-
the mass of the Moon (Sun)
- A′:
-
the right ascension of the Moon (Sun)
- x′,y′,z′:
-
the equatorial rectangular coordinates of the Moon (Sun) referred to the mean equator and equinox of a given date
- r′d:
-
the distance of the Moon (Sun) from the center of the Earth
- λ′, μ′, ν′:
-
the equatorial components of the unit vectorr′0 directed from the center of the Earth toward the Moon (Sun)
- a′:
-
the semi-major axis of the lunar (solar) orbit defined in such a manner that the constant part in the expansion ofa′/r′ be equal to one
- G :
-
the gravitational constant
- M :
-
the mass of the Earth
- R :
-
the equatorial radius of the Earth
- ε:
-
the eccentricity of the meridian of the Earth
- k 2,k 3,...:
-
Love numbers
- α:
-
R/a
- α′:
-
R/a′ the lunar (solar) parallactic factor
- Ω:
-
the tidal disturbing function
- \(\frac{{\partial \Omega }}{{\partial Z}}\) :
-
the component of the disturbing force normal to orbital plane of the satellite
- P nm :
-
associated Legendre functions
- I :
-
the idemfactor
References
Kaula, W. M.: 1969, ‘Tidal Friction with Latitude Dependent Amplitude’,Astron. J. 74, 1108–1114.
Kozai, Y.: 1965, ‘Effects of the Tidal Deformation of the Earth on the Motion of Close Earth Satellites’,Publ. Astron. Soc. Japan 17, 395.
Musen, P. and Estes, R.: 1972, ‘On the Tidal Effects in the Motion of Artificial Satellites’, NASA-X-550-71-342, August 1971.Celest. Mech.,6, 4.
Newton, R. R.: 1968, ‘A Satellite Determination of Tidal Parameters and Earth Deceleration’,Geophys. J. Roy. Astron. Soc. 14, 505–539.
Smith, D. E., Kolenkiwwicz, R., and Dunn, P. J.: 1971, ‘Geodetic Studies by Laser Ranging to Satellites’, NASA-X-553-71-360, April 1971.
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Musen, P., Felsentreger, T. On the determination of the long period tidal perturbations in the elements of artificial earth satellites. Celestial Mechanics 7, 256–279 (1973). https://doi.org/10.1007/BF01229951
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DOI: https://doi.org/10.1007/BF01229951