#### MontBlanc2012

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So far I've posted only maps using a form of perturbation theory (known as Lindstedt-Poincaré theory). Perturbation theory only takes one so far, however in mapping some of these orbit families. In particular, the work so far posted only maps 'small' Halo orbits and does not map the whole of the Halo orbit families.

Over the weekend, I took a step further in this mapping exercise by using numerical integration (bootstrapped with the existing perturbation theory) to map the whole of the L1 and L2 northern and southern halo orbit families. The result is a set of Excel-based interpolation tables that allow one to calculate (for Orbiter 2016) the state vectors (i.e., position and velocity) of any point on any halo orbit for any time. This seems to me to be quite a useful thing so I'm going to share these results - together with some Python code that allow one to decode the Excel spreadsheets to calculate the orbit state vectors. - But I'll actually post this in a second instalment below this one in this thread.

**The L2 southern halo orbit family**

To whet the appetite, when for example all of the L2 southern halo orbits are pasted together, one developed the following picture of what the southern halo orbit family looks like:

The orange surface is the halo orbit manifold (i.e., just the ensemble of all of the halo orbits in the Circular Restricted Three Body Problem - CR3BP). The white sphere depicts the position of the Moon and is drawn to scale. The black lines on the surface of the orange manifold show some specific examples of halo orbits that lie on this manifold.

The halo orbit marked in green is of particular interest. This is an example of something called a Near Rectilinear Halo Orbit (NRHO). Although it looks like a Keplerian elliptical orbit, it isn't - it's actually a halo orbit. But it is on a NRHO very much like this one that NASA propose to park LOP-G (the Lunar Orbital Platform Gateway station). Consequently, the spreadsheet interpolation tables provide a representation of NRHO orbits for use in Orbiter 2016 - and I'll show how to do this in a subsequent post in this thread. As you can see, though these NRHOs have a perilune that pass close to the Moon and there is an interesting problem to be investigated about how to launch from a point on the Moon and rendezvous with a LOP-G-like station parked in such an NRHO; and to complete the reverse transfer back to a base on the surface of the Moon.

**The halo orbits bifurcate from the planar Lyapunov family**

If we can say that the natural end point of the halo orbit family terminates in the NRHOs, what about their starting point. If we look at the same halo orbit manifold families from the other side, we can identify the originating planar Lyapunov orbit.

The highlighted red 'halo orbit' is, in fact, the planar Lyapunov orbit (lying in the orbital plane of the Earth and the Moon) from which the whole of the L2 halo orbit family originates. In this image, the Moon is 'hidden' behind the halo orbit manifold, but the blue sphere depicts the location of the Earth (also drawn to scale in this image).

In a subsequent post in this thread, I'll return to posting the interpolation tables, assorted decoding Python scripts, and demonstrate how the information can be used to calculate the state vectors of these orbits for use in Orbiter 2016.

(N.B. the images in this post were prepared using

*Mathematica*)