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Space Travel, Part 4 — L1 Rendezvous
3rd-Nov-2013 06:15 pm

(This is Part 4. The previous installments are Part 1: Another Anniversary, Part 2: Climbing the Wall, Part 3: Rockets Are Stupid )

### Improving on Lunar Orbit Rendezvous

Sometimes all it takes is asking the right stupid question. LOR was the result of one such, i.e., "Why do we need to take all of this crap down to the lunar surface?"

Here back in 2013, with the benefit of 20-20 hindsight and 40+ years worth of bored grad students in physics, control theory, and aero-astro engineering picking away at the various issues, yours truly has another one:

Why are we bothering to go into lunar orbit at all?
Why not just leave Michael Collins and the heat shield at L1?

See, in the LOR plan which was ultimately adopted, when the Command/Service Module/LEM combination gets into the vicinity of the moon, the Service Module has to do this burn that puts both it and the LEM into a low lunar orbit about 110km up from the surface. This may not be going all the way to the "bottom" of the 450km well, but in energy terms it's just like going "down" a bit less than half-way to the bottom — just like low Earth orbit is like being half-way down Earth's gravity well — to −290km (recall that L1 is at −170km). We have to kill velocity in order to do that and so there'll be a cost.

On the other hand, leaving the Command Module behind at L1 means the LEM has to travel all the way from L1 down to the lunar surface and back by itself, which is an extra 60,000 kilometers in each direction, probably another day or two of travel time each way. Which, is a hell of a lot more than the few hours it takes to get down from a lunar orbit that's only 110km up. And, for every day you need a few kg of oxygen per person, and likewise for food and water. Clearly, since every last kilogram matters, this is obviously insane, right? Never mind the challenge of getting Armstrong and Aldrin to survive cramped in the LEM for a few days without the mediating influence of Collins; I'm sure they would have killed each other.

But then you notice that they're going to be spending at least that amount of time on the lunar surface anyway (and later missions were significantly longer), the extra food and life-support, in fact, turn out to be a trivial addition to a LEM ascent stage that's already 2½ tons. And, as is the typical pattern, everything pales in comparison to what the fuel cost is going to be.

Running the numbers, we find that the extra ΔV to get us that 60,000km from L1 to low lunar orbit turns out to be roughly a third of what we need to get us the rest of the way down to the surface. It seems that getting down that last 110km is, by far, the hardest part of the trip; recall that we spend 50% of the spacecraft doing it. When we add in the trip from L1 down to 110km, this cost increases to 60%. And, as noted before, the trip back is the same flight path time-reversed, thus with the same ΔVs needed, so it's another 60% getting tossed in order to get us back to L1. Putting that together with the 2 minute hover time at the bottom, and we find we need an extra 7½ tons of fuel for the LEM.

However, since the 30-ton Command/Service-Module is neither having to do a burn to drop into lunar orbit nor having to get back out again; that turns out to save 12 tons of fuel.

… which, doing the subtraction gives us a net of 4½ tons of fuel saved. Which means the overall LEM+CSM combination that we have to launch from earth to L1, originally 45 metric tons, is now reduced in size by 10%. Even if the LEM part of that needs to be quite a bit bigger than before, the Service Module is reduced even more so.

This shouldn't be that surprising since what we're doing is taking the LOR plan to its logical extreme:  Everything we need for the trip back to Earth stays perched in the saddle at L1. We expend zero effort/fuel taking any of it down into the lunar gravity well and back.

But the real bonus appears when we translate this savings back to the launch pad on Earth, where we find ourselves looking at (…drumroll…)

A Saturn V that's ten percent smaller.

This has got to be a win. The accumulated savings over 8 missions are just enough to fly an Apollo 18. Or maybe we could have saved Skylab. Who knows?

What's more, while Armstrong and Aldrin are puttering around on the surface, Collins remains at L1,… stationary between the moon and the Earth. Or we could put him in a halo orbit around L1, that's doable, too.

Which means he stays in contact with both Houston and Tranquility Base at all times. In fact, the only time anybody gets out of contact in this scenario is when the LEM zips behind the moon for its descent and ascent trajectories. This also has to be a win.

### It's Unstable. We Are All Going to Die.

Now if Lunar Orbit Rendezvous was difficult to sell to NASA management in 1962, I'm sure my Collins-at-L1 Plan would have been that much harder. "Halo orbits? WTF? How the hell can he just be sitting there?"

It's a fair bet that referring them to Robert Farquhar's 1968 Ph.D thesis would have gotten me a quick trip to a padded cell. But even if I'd managed to avoid that, there'd probably still have been someone in the room who'd actually had the physics course:

"Um,…, isn't L1 unstable?"

Ruh roh.
(also an anachronism; Scooby Doo premiere wasn't until 1969)

Now that sounds like a real objection.

"Unstable". It's a scary word, no question. Evidently, any plan involving L1 means things are going to explode and people will die; that's what you get for using proto-matter (but at least we get Spock back — god, that movie was stupid).

Contrast with L5, which, being "stable", must therefore be a nice, safe place to raise your kids; perfect for a space colony. (Hey, it was good enough for O'Neill.)

And I'm sure this psychology has something to do with why what I'm about to tell you remained overlooked for so long, why L1-L3 were originally dismissed as useless curiosities, and why it took us another two to three decades after 1962 to figure out that this instability is a feature, not a bug.

So what do these words actually mean? Here's the deal:

At L1, all of the various forces cancel out. What you're left with are tides. Tides are weird.

To get a better sense of how tides work, let's consider another situation where gravity gets cancelled out: You're in an elevator and somebody cuts the cable. Elevator is falling freely, you and everything else in the elevator are falling freely, all at the same rate, which you can't actually see because you're inside the elevator. As far as you're concerned, it's as if somebody flipped a magic switch that turned the gravity off, and now you and everyone else in the elevator are just floating there. At some point you'll all go splat but let's not worry about that yet.

Now, as it happens the various hats and hairpieces floating at the top of the elevator are all slightly farther away from the center of the earth, thus aren't getting pulled quite as strongly, and thus, from your point of view will be accelerating (very slightly) upwards, away from you. Likewise, any random shoes at the bottom of the elevator will be closer to the center of the earth, getting pulled on more strongly and thus (again) will be accelerating away from you (downwards).

Similarly, the people to your sides are going to get pulled towards you, the problem this time being that, for them, the center of the earth is in a very slightly different angular direction from where it is for you.

If you need another example, consider the Actual Tides. Here, it's the Earth itself, which you now need to imagine being inside of a Very, Very, Very Extremely Large elevator falling around the sun. Nothing on Earth actually feels the sun's gravity, because we're all in the same orbit, falling together. And yet, the oceans at noon and midnight are getting pulled upwards (outwards, away from the center of the earth), while the oceans at 6am and 6pm getting pushed down (inwards, towards the center of the earth) — that these times tend not to corresponding with high and low tide is only because oceans are big and heavy and take A While to react, but it does explain why high and low tide are six hours apart rather than twelve as you might have expected.

Anyway, at L1, it's the same story, except that you don't even need the elevator anymore, because the gravity is cancelled out for real (sort of).

If you move in any of the "sideways" directions off of the earth-moon axis, the "low tide" force pushes you back towards L1 and then you end up oscillating back and forth through L1. And you can also combine oscillations in the different directions away from the axis any way you want. One such combination gives you a (vaguely) circular orbit in the plane perpendicular to the earth-moon axis, which, viewed from earth, will look like you're following a halo around the moon, hence "halo orbit", even though it's something of an optical illusion, i.e., you're circling L1, not the moon.

If, however, you move "up/down", i.e., towards the earth or the moon, then you get hit by the "high tide" force that not only pulls you farther away from L1, but gets stronger the farther away from L1 you are. Hence, "unstable". That is, if you don't start at exactly the right place, or even if you do, but then get bumped by a perturbation as will inevitably happen, you start moving further away and then pick up speed at an exponential rate.

And if you're off diagonally, then you're affected by both forces at the same time, and thus you will be headed away on this horribly weird spiral trajectory as the high-tide force pulls you farther away while the low-tide force keeps you circling the Earth-moon axis. Remember this, I'll get back to it.

Meaning, that the bad Star Trek dialogue ("Oh no, Riley's shut down the engines! Our orbit is going to decay!") actually applies to orbits around L1. If you care about staying there, you have to do active station keeping, firing your maneuvering thrusters every so often.

But so what? That "exponentially" may sound scary, but the flip side of it is when you're really close to L1 radially, it's exponentially small. Meaning, if you're sufficiently close to L1, it's a matter of remembering to sneeze in the right direction once every few days. The amount of fuel involved is utterly trivial.

To be sure, rockets can fail, just like any other piece of equipment. And I suppose it would have been slightly scary to the folks in 1962 that the orbits in the vicinity of L1 are, shall we say, a bit chaotic. Meaning when it comes time to leave, a slight change in how you leave can make a big difference in where you end up. One wild burn and now you're on a spiral trajectory headed basically anywhere.

Like Jupiter.

Or the Sun.

No, really.

When I say that aforementioned weird spiral can go anywhere, I really mean anywhere.

At this point, your bullshit detector is probably going off. "Um, what happened to conservation of energy? When did we ever get to (earth) escape velocity? In fact, you said that at L1, we're still 170 km down from 'the top', i.e., 170km from being out of the Earth's gravity well. So how the hell are we getting out to Jupiter?"

Fair questions, those. Something is indeed rotten in Denmark and I now have to reveal what I have been lying about glossing over.

### The Three-Body Problem and other Danish Zombies a.k.a. The Magic of L1

(to be continued in Part 5)