# Low Energy Transfer within Earth-Moon system

Practical aspects of a total low energy transfer to the Moon have been seen in missions like GENESIS, which uses Weak stability Boundary legs of Earth and Sun to reach ESL-2. This four body model transfers justifies that:

It is possible to use the unstable manifolds of the planar Lyapunov periodic orbits about the Sun-Earth L2 point to provide a low energy transfer from the Earth to the stable manifolds of planar Lyapunov periodic orbits around the Earth-Moon L2 point.

Thus transferring the spacecraft around EML2 which

act as separatrices in the energy manifold of the flow through the equilibrium point, provide the dynamical channels in phase space that enable ballistic captures of the spacecraft by the Moon.

Such Transfers have theoretically been proved to have 25-40% delta-v savings.

Now, instead of going 1.5 million miles away, can we use just the Earth-Moon system? Dynamical nature of CRTBP in Earth-Moon system suggests that certain states in phase space if attained, can lead spacecraft to asymptotically approach L-points into periodic/quasiperiodic orbits (In case of Earth-Moon system, lets just have Jacobi Energy constant, just enough to open up zero-velocity surface at both EML-1 and EML-2)

Also, Homo/Heteroclinic connection between Lyapunov orbits between two L-points, like utilized in ARTEMIS allows us to traverse space in a way shown here:

Can we in intermediate stage of such trajectory execute a decay manoeuvrer to get captured around the Moon in some way (because I'm guessing, we cannot get ballistically captured by the Moon from EML-1) ? What margin of delta-v would be needed in such case?

Alternatively, is there a possibility of ballistic capture from a periodic orbit around EML-2, spacecraft being transferred to EML-2 from EML-1 as in the figure ?

Quoted Text reference: Low Energy Transfer to the Moon, W.S.Koon

• From LEO it takes ~3.1 km/s to achieve a high apogee in the lunar neighborhood. I believe this leg is often ignored when 25% to 40% delta V savings are claimed. – HopDavid Aug 25 '15 at 23:47
• If departure and destination orbits differ by more than a factor or 11.94 or more, bi-elliptic costs less than Hohmann. This certainly applies to LEO and EML2. If bi-elliptic is timed just right, sun's tidal forces can raise perigee to EML2 height. If falling from the edge of earth's hill sphere, a payload can do a ballistic slide into EML2. See hopsblog-hop.blogspot.com/2015/05/eml2.html – HopDavid Aug 25 '15 at 23:54
• Still looking for an answer? If you've found one you could post it and accept it. If not, can you share where the two block-quoted text items are from? e.g. "It is possible to use the unstable manifolds of the..." If that's a quote from a book or paper, can you add a citation? If it's your own personal assertion, can you say why you're sure that it's true? A second figure showing the manifolds in question would be helpful as well. Thanks! – uhoh Apr 3 '17 at 7:00
• Updated the reference for block-quotes. The picture is most probably from J. S. Parker and Rodney Anderson's 'Low-Energy Lunar Trajectory Design'. I'll try to get the exact page number to the picture in the reference. – Kuldeep Barad Apr 16 '17 at 12:52
• I thought a Japanese mission had been delivered into an unusable (too low) orbit but raised to a usable orbit by exploiting earth-moon orbital harmonics and judicious small delta-v maneuvers. But I'll be darned if I can find a reference based on my limited recall. I also remember a slide from a presentation showing a proposed swarm of spacecraft raising apogee and maneuvering into formation using the same method, a sort of smart orbital energy harvesting based on small, smart maneuvers. – Kengineer Apr 17 '17 at 21:56

Wow, three years and still no answers!. I will give it a try.

What I am going to answer applies just for transfers between periodic orbits at the Lagrangian points (the OP asked so many things but I think that is the fundamental one)

Let's put that we want to transfer from Earth-Moon L1 to L2 using the CR3BP dynamics

1. Define the periodic orbits around L1 and L2 (recently @uhoh posted a nice answer on how to do that, so I am not entering into details about this). Typically these periodic orbits are not really free to choose. As an example, think that L1 periodic orbit is given by the Earth departure and L2 periodic orbit is a target that has been chosen by some reasons (see NRHOs).

2. Choose a Poincaré section, this can be one of the trickiest parts of the procedure. However, almost everyone I have seen follows a quite simple logic. If you see the figure below, we want to pass from L1 to L2, so a nice and easier Poincaré section would be a YZ plane place at $[1-\rho, 0, 0]^T$, in adimensional units, being $\rho=$$M_2/(M_1+M_2)$, that is this vertical plane contains the Moon and is fairly equidistant from both L1 and L2.

1. Compute the unstable manifolds from L1 and the stable manifolds from L2 (we want to leave L1 and approach L2) and its intersections in the phase space with the Poincaré section. What is the issue here?. We need to have at least a match on the positions, so [$x^U_{L1}$, $y^U_{L1}$, $z^U_{L1}$]$^T$=[$x^S_{L2}$, $y^S_{L2}$, $z^S_{L2}$]$^T$, note that we have chosen the Poincaré section, so $x^U_{L1}$=$x^S_{L2}$=$1-\rho$ and we only have to search for matches on $y$ and $z$ (if any).

For example these illustrative figure (simplified for illustrative purposes for a planar case) shows multiple coincidences of the manifolds in the $y$ coordinate.

1. Suppose these position matches between manifolds exists (typically they appear). What about velocities?, they have to be matched at the Poincaré section also, [$\dot{x}^U_{L1}$, $\dot{y}^U_{L1}$, $\dot{z}^U_{L1}$]$^T$=[$\dot{x}^S_{L2}$, $\dot{y}^S_{L2}$, $\dot{z}^S_{L2}$]$^T$, unfortunately this does not typically happens, so you have to pay the difference and do an impulse $\Delta V$=[$\dot{x}^S_{L2}$, $\dot{y}^S_{L2}$, $\dot{z}^S_{L2}$]$^T$-[$\dot{x}^U_{L1}$, $\dot{y}^U_{L1}$, $\dot{z}^U_{L1}$]$^T$.

2. Explore more intersections!. If after the first intersection with the Poincaré section you continue computing the manifold you will likely find another intersection with the Poincaré section and maybe this will be a more favourable one (in terms of impulsive amount than the first one). The figure (b) of Step 3 shows the first manifold intersection with no match on $\dot{y}$ (remember that is a planar case) but it has also computed the second intersection for both manifolds (at the right) and now two matches at $\dot{y}$ appear!. Multiple combinations of intersections order could be possible.

Figures reference: "Heteroclinic Connections Between Periodic Orbits and Resonance Transitions in Celestial Mechanics", Koon, W. S. et al (2000), Chaos, 10 (2), 427-469 (available here and here)

Recommended reference: Chapter 4 of "KoLoMaRo" (Koon, Lo, Marsden & Ross) Dynamical Systems, The Three-Body Problem, and Space Mission Design: http://www.cds.caltech.edu/~marsden/volume/missiondesign/KoLoMaRo_DMissionBk.pdf