PO around Unstable EP

* Türkçe’si için …

In the very vicinity around equilibrium points (EP), the linear approximation to the equations of motion is considered sufficient enough. But in wide circulation around EP, the nonlinear terms become more important; the linear phase space is broken and various variety periodic orbits (PO) appear in a new periodic family. The nonlinear terms and computation of POs are especially important for missions around unstable EP, because motion around unstable EP is naturally unstable; and for providing cheap station keeping to spacecraft, the exact routes of these unstable PO must be computed. There are some methods to find and calculate these complext and highly non-linear POs; such as ‘differantial correctors’, ‘centre manifold reduction’ or ‘multiply Poincare section’ method. Using and combaning these method, the Poincaré Map in below, which shows all type of PO and Quasi PO orbits around unstable L2, can be plotted.

So, using upper figures, I want to introduce these interesting orbits, one by one;

Horizontal Lyapunov Orbit; Maybe this is the orbit, which can be found the easiest way. Because this is a 2D periodic orbit, using a simple try-error scheme, we can practically find this orbits very easy. This orbits may also called as Horizontal or 2D Halo orbits. It may be used some special station keeping mission design, but it is mostly used for computation of 2D invariant manifold tubes.

Halo Orbit; A 3D orbits, which look like really a “Halo”. It can be used for science orbits; It is the perfect orbit to sun observation misssion; such as SOHO and Genesis. it is also used for computation of 3D invariant manifold tubes. But finding of this orbit is not as easy as in 2-D Halo orbits. Because it is in 3-D position space, simple try-error scheme is not so practically. So soome other numerical method, such as Differantial corrector mehod is used for finding it.

Quasi-Halo Orbit; This orbit circulate the same path as in Halo Orbit, but its shape become like torus. It is such an invariant tori about the corresponding (Halo) periodic orbit. It is very perfect to various scinece mission with formation fly, such as Terestrial Planet Finder (TPF) and Darwin (spacecraft).

Lissajous Orbit; This is another quasi-periodic orbit. But its differance, it makes a Lissajous curve flow. Several missions have used Lissajous trajectories: ACE at Sun–Earth L1 and WMAP at Sun–Earth L2, and also the European Space Agency (ESA) launched into space the Herschel and Planck observatories, both of which use Lissajous orbits at Sun–Earth L2. Furthermore ESA’s future Gaia mission will also use a Lissajous orbit at Sun–Earth L2. [wiki]

Vertical Lyapunov Orbit; Another periodic orbit, which has a good potential for being a science orbits as others. But it has not used for practical space mission yet, as far as I know. Also it can be used for plotting invariant manifold tubes, too.

Also note that, there one more interesting station keeping way using these unstable EPs; to place the spacecarft exactly on the unstable EPs (L1, L2 or L3) and keep it balance using zero velocity-curves … ^_^

* Figures are taken from; E. Kolemen, N. J. Kasdin & P. Gurfil, Quasi-Periodic Orbits of the Restricted Three Body Problem Made Easy, AIP Conference Proceedings, Vol. 886, 2006, pp. 68-77.


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