Abstract
The general perturbations in the elliptic and vectorial elements of a satellite as caused by the tidal deformations of the non-spherical Earth are developed into trigonometric series in the standard ecliptical arguments of Hill-Brown lunar theory and in the equatorinal elements ω and Ω of the satellite.
The integration of the differential equations for variation of elements of the satellite in this theory is easy because all arguments are linear or nearly linear in time.
The trigonometrical expansion permits a judgment about the relative significance of the amplitudes and periods of different tidal ‘waves’ over a long period of time.
Graphs are presented of the tidal perturbations in the elliptic elements of the BE-C satellite which illustrate long term periodic behavior. The tidal effects are clearly noticeable in the observations and their comparison with the theory permits improvement of the ‘global’ Love numbers for the Earth.
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Abbreviations
- l, l′, F, D and Γ:
-
the arguments of the lunar theory
- G :
-
the gravitational constant
- M :
-
the mass of the Earth
- m′ :
-
the mass of the Moon
- m″ :
-
the mass of the Sun
- \(\bar r\) :
-
the geocentric position vector of the satellite
- r :
-
\(|\bar r|\)
- \(\bar r^0 \) :
-
the unit vector in the direction of\(\bar r\)
- \(\bar r'\) :
-
the geocentric position vector of the Moon
- r′ :
-
\(|\bar r'|\)
- \(\bar r'^0 \) :
-
the unit vector in the direction of\(\bar r'\)
- \(\bar r''\) :
-
the geocentric position vector of the Sun
- r″ :
-
\(|\bar r''|\)
- \(\bar r''^0 \) :
-
the unit vector in the direction of\(\bar r''\)
- a′ :
-
the mean distance of the Moon from the Earth. Its numerical value is defined in such a manner that the constant part in the expansion of the parallaxa′/r′ is equal to one.
- a″ :
-
the mean geocentric distance of the Sun
- S′ :
-
the angle between\(\bar r\) and\(\bar r'\)
- S″ :
-
the angle between\(\bar r\) and\(\bar r''\)
- λ, μ, ν, λ′, μ′, ν′, λ″, μ″, ν″:
-
the components of\(\bar r^0 \),\(\bar r'^0 \) and\(\bar r''^0 \)respectively in the equatorial system
- e :
-
the eccentricity of the orbit of the satellite
- i :
-
the orbital inclination of the satellite
- Ω:
-
the longitude of the ascending node of the satellite
- ω:
-
the argument of perigee of the satellite
- π:
-
the longitude of the perigee of the satellite, ω+Ω
- c :
-
cosi/2
- s :
-
sini/2
- δM :
-
the perturbations in the mean anomaly of the satellite caused by tides
- δΩ:
-
the perturbations in Ω caused by tides
- δπ:
-
the perturbations in π
- δi :
-
the perturbations ini caused by tides
- n :
-
the mean motion of the satellite
- a :
-
the semi-major axis of the satellite's orbit
- \(\bar R\) :
-
unit vector normal to the orbital plane of the satellite
- u 1,u 2,u 3 :
-
components of\(\bar R\) in the equatorial system
- δu 1, δu 2, δu 3 :
-
perturbations inu 1,u 2,u 3 respectively caused by tides
- ε:
-
the obliquity of the ecliptic
- ɛ:
-
the eccentricity of the Earth's meridian
- k 2,k 3,...:
-
Love numbers
- R :
-
the equatorial radius of the Earth
- α:
-
R/a
- α′:
-
R/a′
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Musen, P., Estes, R. On the tidal effects in the motion of artificial satellites. Celestial Mechanics 6, 4–21 (1972). https://doi.org/10.1007/BF01237442
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DOI: https://doi.org/10.1007/BF01237442