MontBlanc2012
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Over the last couple of months I've been busy working out the parametric representing the closed 'centre manifold' orbits of the L1 and L2 Lagrange points of the Earth-Moon and Sun-Earth systems. And I've just about finished.
Essentially, this is a mapping exercise. Just as Keplerian orbits (circular, elliptical, parabolic and hyperbolic) effectively 'map' trajectories in the restricted two-body problem (e.g., ISS in orbit around the Earth), the centre manifold orbits provide a similar landscape structure for the restricted three-body problem. The problem has always been that unlike the well-trawled Keplerian orbital mechanics, there isn't a ready-made, 'off-the-shelf-package' that one can use to determine:
1. where the centre manifold orbits are;
2. how far from the orbits you are;
3. how to get to them;
4. how to set up transfers (optimal or otherwise) between them; and
5. station-keeping on the centre-manifold orbits.
Without a map of the centre-manifold orbits its impossible to even begin to answer these questions (although tools like IMFD and Lagrange MFD) can provide some qualitative insight in some cases.
Over the next couple of months, I'm going to begin to unfold the results of this mapping exercise. There is a fair amount of tedious maths, analysis and calculation sitting behind all of this - everything in the restricted three-body problem is more complicated than the Keplerian case - but the end result should be tools that allow one to answer the questions listed above.
To illustrate the mapping exercise, and somewhat arbitrarily, I've put together a short video to illustrate the mapping of two particular examples of the centre manifold orbits around the L1 Lagrange point of the Earth-Moon system - a planar Lyapunov orbit; and a vertical Lyapunov orbit:
When working with these orbits, it is often hard to appreciate the scale of some of these orbits in relation to the Earth-Moon system. They are big. The end-to-end distance of the two orbits in the video is around 60,000 km. The red orbit is an example of a planar Lyapunov orbit; and the blue orbit is an example of a vertical orbit. In the video, the yellow line is the orbit of the Moon around the Earth; the white sphere is (obviously) the Moon; and the smaller (but more distant) blue sphere is the Earth.
Now, these two orbits have been used as an example because the present a fairly clean demonstration of the mapping of centre manifold orbits. Using different parameter values in the mapping exercise, one could easily calculate the state vectors of a spacecraft on all of the other Lyapunov orbits - just as one calculate the state vectors of a vessel in a Keplerian orbit. Moreover, the full map of the centre-manifold orbits describes general Lissajous orbits and, of course, the much mentioned Halo orbits.
This mapping of Keplerian orbits, together with the Lambert solver for Lagrange points describe, will allow one to plan transfer between centre-manifold orbits of the same Lagrange points; and help design transfers between the centre manifold of different Lagrange points (e.g. L1-L2 transfers).
Well, that's the plan. But setting all of this out in a useful fashion is going to take some time - but for me at least the effort will have been worth it.
---------- Post added 08-30-18 at 06:47 AM ---------- Previous post was 08-29-18 at 01:04 PM ----------
Just as a bit of an addendum, here is another short YouTube video showing a mix of vertical and planar Lyapunov orbits around both L1 and L2. In all, eight centre manifold orbits are shown.
Aside from the fact that is fun to make these videoettes, the main purpose is to help visualise the shape, location and scale of some of these centre manifold orbits.
So far, I haven't focused on Lissajous (and Halo) orbits, because the vertical and planar orbits illustrate the two basic kinds of motion that these orbits can have - an 'in plane' oscillation and a 'vertical' oscillation. Lissajous orbits are just a mix of these two kinds of motion; and a Halo orbit is just a special kind of Lissajous orbit.
Essentially, this is a mapping exercise. Just as Keplerian orbits (circular, elliptical, parabolic and hyperbolic) effectively 'map' trajectories in the restricted two-body problem (e.g., ISS in orbit around the Earth), the centre manifold orbits provide a similar landscape structure for the restricted three-body problem. The problem has always been that unlike the well-trawled Keplerian orbital mechanics, there isn't a ready-made, 'off-the-shelf-package' that one can use to determine:
1. where the centre manifold orbits are;
2. how far from the orbits you are;
3. how to get to them;
4. how to set up transfers (optimal or otherwise) between them; and
5. station-keeping on the centre-manifold orbits.
Without a map of the centre-manifold orbits its impossible to even begin to answer these questions (although tools like IMFD and Lagrange MFD) can provide some qualitative insight in some cases.
Over the next couple of months, I'm going to begin to unfold the results of this mapping exercise. There is a fair amount of tedious maths, analysis and calculation sitting behind all of this - everything in the restricted three-body problem is more complicated than the Keplerian case - but the end result should be tools that allow one to answer the questions listed above.
To illustrate the mapping exercise, and somewhat arbitrarily, I've put together a short video to illustrate the mapping of two particular examples of the centre manifold orbits around the L1 Lagrange point of the Earth-Moon system - a planar Lyapunov orbit; and a vertical Lyapunov orbit:
When working with these orbits, it is often hard to appreciate the scale of some of these orbits in relation to the Earth-Moon system. They are big. The end-to-end distance of the two orbits in the video is around 60,000 km. The red orbit is an example of a planar Lyapunov orbit; and the blue orbit is an example of a vertical orbit. In the video, the yellow line is the orbit of the Moon around the Earth; the white sphere is (obviously) the Moon; and the smaller (but more distant) blue sphere is the Earth.
Now, these two orbits have been used as an example because the present a fairly clean demonstration of the mapping of centre manifold orbits. Using different parameter values in the mapping exercise, one could easily calculate the state vectors of a spacecraft on all of the other Lyapunov orbits - just as one calculate the state vectors of a vessel in a Keplerian orbit. Moreover, the full map of the centre-manifold orbits describes general Lissajous orbits and, of course, the much mentioned Halo orbits.
This mapping of Keplerian orbits, together with the Lambert solver for Lagrange points describe, will allow one to plan transfer between centre-manifold orbits of the same Lagrange points; and help design transfers between the centre manifold of different Lagrange points (e.g. L1-L2 transfers).
Well, that's the plan. But setting all of this out in a useful fashion is going to take some time - but for me at least the effort will have been worth it.
---------- Post added 08-30-18 at 06:47 AM ---------- Previous post was 08-29-18 at 01:04 PM ----------
Just as a bit of an addendum, here is another short YouTube video showing a mix of vertical and planar Lyapunov orbits around both L1 and L2. In all, eight centre manifold orbits are shown.
Aside from the fact that is fun to make these videoettes, the main purpose is to help visualise the shape, location and scale of some of these centre manifold orbits.
So far, I haven't focused on Lissajous (and Halo) orbits, because the vertical and planar orbits illustrate the two basic kinds of motion that these orbits can have - an 'in plane' oscillation and a 'vertical' oscillation. Lissajous orbits are just a mix of these two kinds of motion; and a Halo orbit is just a special kind of Lissajous orbit.
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