# Dynamics and Mission Design Near Libration Points, Vol. IV: by Gerard Gomez, Angel Jorba, Josep J Masdemont

By Gerard Gomez, Angel Jorba, Josep J Masdemont

The purpose of this paintings is to give an explanation for, research and compute the types of motions that seem in a longer neighborhood of the geometrically outlined equilateral issues of the Earth-Moon approach, as a resource of attainable nominal orbits for destiny house missions. The method constructed this is no longer particular to astrodynamics difficulties. The strategies are built in one of these means that they are often used to check difficulties that may be modelled via dynamical structures.

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**Example text**

In principle the points are on 4-dimensional tori. To simplify the visualization we have used "time the synodic period of the Sun"-map. This is a 6-dimensional canonical map and the invariant tori of maximal dimension, if any, must be 3-dimensional. Starting with the above mentioned initial conditions and #0 = 0, we computed 105, 104, 104 and 5 x 104 iterates, respectively, of the canonical transformation. The patterns seem to correspond to 2-dimensional projections of points sitting on 3-dimensional tori (this becomes more clear if one watches at the apparition of the points on a screen when they are displayed slowly).

Let e be the ecliptic coordinates of a body with respect to the barycenter of the solar system. Let a be some adimensional coordinates to be defined. Put e = k(t)C(t)a + b(t), where k, C and b are a scaling factor, an orthogonal matrix and a vector, respectively, all of them depending on time. We select a couple of bodies (primary and secondary) that in our case are taken to be the Earth and the Moon. Let fi be the corresponding mass ratio. We define k, C and b (and, hence, a) by requiring that, for all time, the primary is located at (/x, 0,0), the secondary at Qu — 1,0,0) and the relative velocity of the secondary with respect to the primary sits in the (x, y)-plane in the a coordinates.

170564. 3: Relevant parameters of orbits in the short, long and vertical families of periodic orbits around L5 (see detailed description in the text). 3 Fig. 3. 38 Fig. 3. 1 The Equations of the Bicircular Problem The bicircular problem is a simplified version of the four-body problem. In this model we suppose that the Earth and the Moon are revolving in circular orbits around their center of mass, and the Earth-Moon barycenter moves in a circular orbit around the center of masses of the Sun-Earth-Moon system.