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\centerline{\bf On the convergence of analytical theories for low  satellites} 

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\centerline {Zoran Kne\v zevi\'c}
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\centerline {\it Astronomical Observatory, Belgrade}
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\centerline{and}
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\centerline {Andrea Milani}
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\centerline {\it Department of Mathematics, Pisa}
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We are developing fully analytical perturbation theories to compute
proper elements for low-altitude satellites. We used a mixed
Lagrangian-Hamiltonian formalism, and separately accounted for the
very short periodic perturbations (associated with the revolution of
the satellite around the central body), the medium periodic
perturbations (associated with the rotation of the central body), and
the long periodic ones. The theory takes a much simpler form if it is
possible to neglect the resonances between short and medium periodic
terms, as it is the case for a slowly rotating central body, such as
the Moon. For a low satellite, truncation with respect to the degree
in eccentricity is legitimate, while the inclination functions have to
be handled in full generality.

Even within these assumptions, and having as goal a moderate accuracy
(e.g.  $0.001$ in eccentricity), a simple first order theory is not
enough. The two main problems are the slow convergence with respect to
the degree and order of the spherical harmonics expansion of the
gravity field of the central body, and the presence of second order
effects (containing the square of the gravitational potential
coefficients). An additional problem arises when the initial
conditions correspond to zero eccentricity; in this case --which
frequently occurs in mission analysis studies-- the truncation with
respect to the degree in eccentricity and with respect to the second
order in the perturbations are mutually inconsistent. We have used a
fixed point technique to avoid this inconsistency and to obtain an
accurate result.  We have also investigated the second order effects
by comparing both the analytical and the numerical solutions with a
semianalytical solution; in the case of the Moon, the theory needs to
be pushed to second order if the orbits have to be analytically
predicted over time spans longer than 1 year.

For the cases where the theory is applicable, the results are accurate
enough to analytically predict the orbit with a given initial
condition; the computation being fully analytic and explicit, it is of
course much more efficient than any conceivable numerical technique.
The proper elements can also be used to study problems such as orbit
maintenance and optimum manoeuver cycles, with significant practical
implications.



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