EML2

[Hop_David]

Here's a couple screen capture of starting out in olive orbits and eventually leaving earth's Hill Sphere. Same orbits, different zooms.

[sdsds]

Here's a couple screen capture of starting out in olive orbits and eventually leaving earth's Hill Sphere. Same orbits, different zooms.

Is the place where the paths of the pellets diverge so greatly near the SEL-1 point? Or SE-L2?

[Hop_David]

Here's a couple screen capture of starting out in olive orbits and eventually leaving earth's Hill Sphere. Same orbits, different zooms.

Is the place where the paths of the pellets diverge so greatly near the SEL-1 point? Or SE-L2?

I don't remember where I set the barycentric longitude of the sun for this sim.

It's a little hard to tell which part of the Hill Sphere the pellet apogees are at since they take about 45 or 50 days to get there over which time the Hill Sphere rotates 45 or 50 degrees. Also these pellets spent about 10 days in the earth and lunar realms before exiting to the exterior realm.

It seems like aiming at different parts of the Hill Sphere will subject the pellets to different winds.

Aiming at SEL1 or SEL2 seems to boost apogee, often resulting in escape.

Aiming at 45ş angles seems to subject pellets to a sideways wind that rotates line of apsides. If the sideways push is prograde, apogee is boosted. If the push is retrograde, apogee is lowered.

Aiming at 90ş and 270ş (leading and trailing points of Hill Sphere) seems like I'm shooting against the wind.

Borrowing a Wikipedia illustration for tidal force, I believe these are the winds:

[Hop_David]

I've put a lot of this stuff in a blog post: EML2

[Burninate]

I think that mass for energy storage is likely to be prohibitive, the vehicle will simply shut down propulsion when in the dark, this could significantly increase the time needed to spiral out, but the spiral out is an unmanned flight, almost no amount of time is too much, the only thing were worried about is Van-Allen radiation induced degradation of the vehicle, primarily the solar system.

I assumed this too, but batteries sized for the LEO day-night cycle are actually not huge relative to the other parts of the system, especially modern lithium ions.  At 1C, a conservative 1kg LiPo charges at around 150 watts, and holds 150 watt hours.  5C-10C charging is state of the art, but only a ~1.33C charge rate (a 45 minute charge) will switch the bottleneck over to energy density.  To run a ~6.9kw NEXT thruster (58kg) alongside its battery charging system, you need maybe 15kw peak power draw, which at 150w/kg is 100kg of solar panels.  A ~46kg battery(with unknown mass charger) can power a ~6.9kw NEXT thruster weighing 58kg for one half of an orbit at full depth of discharge.  System mass 202kg.

The alternative is a 58kg thruster and 6.9kw of solar panels at ~46kg, system mass 102kg, providing half of the impulse for the early part of the spiral, but stung with unclear penalties afterwards, as apoapsis is in shadow a larger portion of the time.

Depth of discharge and cycle life (allegedly ~1000 cycles for Tesla now) becomes the limiting factor in a battery system based on this, I think.  Generally you should assume half depth of discharge (by virtue of twice as much battery mass as the minimum) doubles cycle life.  There should be metrics available to compare other battery chemistries on the basis of total lifetime kwh per kilogram.  A battery system doesn't need to last forever, either, the benefit fades as the percentage of time spent in shadow drops.  At Earth surface, you get 50%.  At 0.1 Earth Radius (0.1Rⴲ = 637km) altitude, you're down to 36.3%.  At 0.2 Rⴲ, 31.4%.  At 0.5 Rⴲ, 23.2%.  At 1 Rⴲ, 16.7%.  At 5.62 Rⴲ (geosynchronous orbit), you're down to 4.8%.

And that's presuming things line up.  The farther out the inclined orbit, the greater percentage of the year it will not suffer Earth eclipses at all.

[sdsds]

The images below recreate the first section of the Farquhar three-impulse transfer from LEO to EML2. The first is an overview, shown in the rotating frame of reference used here:
http://clowder.net/hop/TMI/FarquharRoute.jpg

The second zooms in on the Earth-departure burn. No real surprises there. The third shows the same trajectory as the first, but in a non-rotating (inertial) reference frame. The spacecraft is going quite slowly (128.44 m/s) near the apogee of its highly elliptical orbit; the Moon in its circular orbit is rushing towards it at 1011.56 m/s.

[sdsds]

The next plot shows what happens over the next 9.1 hours. The spacecraft does get out of the way of the on-rushing Moon, but its trajectory is altered by the Moon's gravity. The second plot shows what would happen after the lunar encounter were there no propulsive maneuver there. The direction of travel of the spacecraft is substantially altered! The final plot shows the same thing, but it is not as evident in the rotating reference frame.

[sdsds]

The plots below show a "quick path" home from EML-2. The total elapsed time is 10.8 days. The total delta-v is 583 m/s. The trajectory departs EML-2 with a 55 m/s burn at an angle of 148.5 degrees from the line between the Earth and Moon. It travels in the general direction of the Moon, looping around a bit until it reaches an apolune at 6.6 days. There a 158 m/s retrograde burn leads to a deep perilune at 8.3 days. A 370 m/s burn at perilune leads to a trajectory that intersects the Earth at a relative speed of 11,134 m/s.

Tiny navigation differences could lead instead to a "near miss" of the Earth, with a similar relative speed at an arbitrarily chosen perigee.

[gbaikie]

The plots below show a "quick path" home from EML-2. The total elapsed time is 10.8 days. The total delta-v is 583 m/s. The trajectory departs EML-2 with a 55 m/s burn at an angle of 148.5 degrees from the line between the Earth and Moon. It travels in the general direction of the Moon, looping around a bit until it reaches an apolune at 6.6 days. There a 158 m/s retrograde burn leads to a deep perilune at 8.3 days. A 370 m/s burn at perilune leads to a trajectory that intersects the Earth at a relative speed of 11,134 m/s.

I suppose that the 370 m/s burn could not be done with low thrust Ion rocket?
But other burns would have enough time to use a Ion rocket?

[sdsds]

Yes, I think that's correct on both. One interesting aspect is that the large perilune burn sets a lower bound on the thrust required for the entire round trip, once the Earth-departure burn is accomplished (presumably with a dedicated stage). For the other two burns, it sure seems like they could be replaced with some sort of continuous, low-thrust propulsion. Figuring out the "control law" that would make that thrust always be in the optimal direction seems like quite a challenge, though!

[sdsds]

Figuring out the "control law" that would make that thrust always be in the optimal direction seems like quite a challenge, though!

Bounded momentum model on the combined two body (Earth Moon) projective gradient.

That's easy for you to say!

I think by optimal you might mean, "has greatest effect" on energy, looking at something like the Jacobi integral. Decreasing the energy would effectively put the particle down into one of the wells of the effective potential. This would be going "always down hill" on the energy manifold. That gets someplace with optimal efficiency. I don't see how to make certain that someplace is the place we want to go.

For planar motion there is a four-dimensional phase space, (x, y, vx, vy). Consider starting at (x₁, y₁, vx₁, vy₁) with the desire to get to (x₂, y₂, vx₂, vy₂). I don't see an easy solution to that!

[sdsds]

On the topic of effective potential, I really like the plot attached below from http://www.cds.caltech.edu/~marsden/volume/missiondesign/KoLoMaRo_DMissionBook_2011-04-25.pdf showing the location of the Lagrange points. But it uses µ = 0.3 to make the image easy to see. For Earth/Moon,  µ = 0.01215. Attached second is Mathematica's version of the plot, using the realistic value for µ.

[Space Ghost 1962]

Figuring out the "control law" that would make that thrust always be in the optimal direction seems like quite a challenge, though!

Bounded momentum model on the combined two body (Earth Moon) projective gradient.

That's easy for you to say!

I think by optimal you might mean, "has greatest effect" on energy, looking at something like the Jacobi integral.

Yes.

Decreasing the energy would effectively put the particle down into one of the wells of the effective potential. This would be going "always down hill" on the energy manifold. That gets someplace with optimal efficiency. I don't see how to make certain that someplace is the place we want to go.

Due to symmetry of the created manifold (being a 2 or N-body problem),  going "always down hill" on the gradient means that before the next "fold" (or point of inflection where you might do a maneuver), you're either adding to or subtracting from the kinetic energy of such (e.g. you get the clue to the direction and magnitude, but you lose the forward/backward direction (!).

For planar motion there is a four-dimensional phase space, (x, y, vx, vy). Consider starting at (x₁, y₁, vx₁, vy₁) with the desire to get to (x₂, y₂, vx₂, vy₂). I don't see an easy solution to that!

When one collapses/"folds" a higher dimensional space to a fewer (in this case a plane), the "folding" (or projection) can be chosen to occur in only certain ways, many/most of which have infinite solutions (or "poles"). Yes, no obvious solutions.

However, inspection of the way it collapses yields (in this case) two which are orthonormal with a resultant in the plane. Again, differing in forward/backward direction. Not reducible in analytic but selectable.

Hope I'm not being too cryptic.

[sdsds]

Plodding forward with plots of the effective potential, here's a zoomed view of the vicinity around the Moon. Although it's pretty, it's hard to glean much from this other than seeing that the saddle points at EML-1 and EML-2 do exist. The second attached plot shows a contour map of the region. This makes the saddles and their locations more evident. Finally a cross section along the y=0 line, which shows exactly where the saddle points are located, and their relative heights.

[sdsds]

As for orbits being reversible?

I'm still wondering whether the "reverse" of the Farquhar route exists. Does the "mirror image theorem" suggest that the corresponding path from EML-2 to Earth would be on the opposite side of the Earth-Moon line?

EDIT: Apparently it's the "Theorem of Image Trajectories." See attached.

EDIT 2: Attached is a Mathematica "notebook" in .pdf form showing a route much like Farquhar's but in reverse. Total delta-v to a 230 km perigee 9 days after leaving EML-2 is less than 350 m/s.

[sdsds]

Necroposting to link this orbit animation from Shane Ross.
https://twitter.com/i/status/1889503469360263344

[sdsds]

Does anyone happen to know a name that could be used as a google search term for an orbital trajectory like the one shown in the first attached image? Or have a link to someplace where it's described?

The trajectory is balanced between corresponding halo orbits around L1 and L2 with low energy transfers available to and from each (example in second image.)

[sdsds]

And here's the low energy transfer to the corresponding L1 halo orbit.