The Earth

oblateness perturbation is caused by the facts that the Earth is not an ideal sphere and it has uneven internal density distribution.

It is found that the

oblateness has a noticeable effect and this is identified to be related to the resonance case and the associated curves in the mass parameters [mu] verses

oblateness coefficient [A.sub.1] configuration space and represented this measure in the parameter space.

Our objective is a series of the optimal

oblateness parameters {[[zeta].sub.i]} = (i = 1, N) for each ith layer contained in an N-layered cylinder; it is what minimizes the total energy U of the system under a given bending curvature [GAMMA].

Modern studies placed the

oblateness of the Sun at 8.77 x [10.sup.-6] [287].

where C and W are the constant parameters measuring the solar radiation pressure and the

oblateness. The reduced system exists three equilibria, that is, ([e.sub.1], 0), ([e.sub.2], 0) and ([e.sub.3], [pi]) ([e.sub.1] < [e.sub.2] < [e.sub.3]), where ([e.sub.2], 0) is hyperbolic and the others are elliptic, as shown in Figure 4.

Since, owing to Saturn's

oblateness, m > n > k, the positions r [sub.L] and r [sub.v] do not coincide: r [sub.v] < r [sub.L].

But they subsequently discovered that around four years ago the Earth's ``dynamic

oblateness'' - its girth - began to increase.

As [Z.sub.DR] provides information about the

oblateness of particles and [K.sub.DP] depends on liquid water content (Bringi and Chandrasekar 2001), the positive [Z.sub.DR] and [K.sub.DP] column suggest raindrops extending up to 5.5-km altitude, about 0.9 km above the 0[degrees]C level (4.6 km according to sounding data).

Since the Lagrangian satellites in the Earth-Moon system is far away from the Earth, the perturbation caused by atmospheric resistance, temporal change in gravity,

oblateness, and mass distribution is not considered.

To find the roundest object in the observable universe, scientists study the

oblateness of objects in space, which in simple words, is the difference between an object's equatorial length and its polar axis.

Invoking a nifty bit of celestial mechanics, Lowell showed that the discrepancy could be explained if Saturn consisted, beneath the outer rind of its clouds, of a series of nested shells of different

oblateness that spun at different speeds.

Additionally, some sensible assumptions are made for the same purpose: (1) the center of mass of the system moves on a Keplerian orbit, (2) the perturbations such as atmospheric drag and Earth

oblateness, are negligible, and (3) the eccentricity of the orbit is small and would not induce the system into chaos motion [19].