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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
An analytical solution of Lagrange planetary
equations valid also for very low eccentricities
F. Deleflie, G. M'etris, P. Exertier
Observatoire de la C“ote d'Azur, Dpt. CERGA, Grasse, France
We give an analytical solution of the Lagrange planetary equations expressed
in a set of non singular elements for eccentricity: the semi­major axis a, the
inclination i, the ascending node \Omega\Gamma – = !+M where ! denotes the argument of
perigee and M the mean anomaly, and the elements C = e cos ! and S = e sin !
where e denotes the eccentricity.
The perturbations which can be computed are due to the gravity field of a
planet, or to the influence of a third body, through an internal or an external
potential. These potentials are expressed with respect to the former set of ele­
ments, and in particular use a formulation of the Kaula functions of eccentricity
G (Wytrzyszczak, 1986) and H with respect to C and S.
The main study deals with the integration of the differential system verified
by C and S, since the metric element e and the angular element ! are mixed. The
differential system is divided into two parts: a first one which can be analytically
integrated in an exact way, the second part being seen as a perturbation of the
first one. More precisely, the first part of the differential system verified by C and
S corresponds to a harmonic oscillator: our method takes into account, in a first
step, long period variations of 4 elements: C, \Omega\Gamma S, and –.
It is therefore a generalization of the Kaula method(Kaula, 1966) expressed in
`classical' orbital elements, in which only the secular variations of the
angles\Omega\Gamma !, M are computed during a first step.
Moreover, our method enlightens the behavior of the argument of perigee !
with time. Like (Cook, 1966), we show that its variations are not really linear
with time, in particular for low eccentricities. We therefore use another angle,
called `, which is well defined even for eccentricities equal to zero. We use this
angle ` to integrate the long as well as the short period variations of C and S,
and as a consequence of all the other elements.
Here, we show why and how we have divided the differential system verified
by a, C, i, \Omega\Gamma S and – in two components. The first part gives a long period
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solution a 1 , C 1 , i 1
,\Omega 1 , S 1 and – 1 ; the second part gives corrections \Deltaa, \DeltaC, \Deltai,
\Delta\Omega\Gamma \DeltaS and \Delta–. There results: a(t) = a 1
(t) + \Deltaa(t), C(t) = C 1
(t) + \DeltaC(t), ... .
We also show the way used to express \Deltaa, \DeltaC, \Deltai, \DeltaS , ... in a synthetic
formulation.
Like the method built by Kaula, our method is based on a development in
powers of the eccentricity. But, there is no supplementary hypothesis induced by
the expression of our theory in non singular elements for eccentricity: it can be
used for bodies -- artificial satellites or planets -- with an eccentricity greater than
or equal to zero.
Finally, we compare three methods in different dynamic configurations: this
method, expressed in non singular elements for eccentricity (with an integration
hypothesis based on `), the method of Kaula, expressed in classical orbital ele­
ments (with an integration hypothesis based on !), and the numerical integration.
Tests show that, compared to the method of Kaula, our method is more accurate
as soon as the eccentricity of the studied body is smaller than a few 10 \Gamma2 .
References
1. Cook G. E. Perturbations of near­circular orbits by the Earth's gravitational
potential, Planet. Space Sci., 1966, 14, 433--444.
2. Wytrzyszczak I. Non singular elements in description of the motion of small
eccentricity and inclination satellites, Celest. Mech., 1986, 38, 101--109.
3. Nacozy P. E., Dallas S. S. The geopotential in non singular orbital elements,
Celest. Mech., 1977, 15, 453--466.
4. Wnuk E., Recent Progress in analytical Orbit Theories, Adv. Space Res.,
1999, 23, 677--687.
5. Kaula W. M. Theory of satellite geodesy, Blaisdell Publishing Company,
Waltham, Masachusetts, 1966.
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