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(Still waffling on whether this should be Space Travel part 8 or Relativity part 1; we'll see...
If you want to go back to part 1, that's here though none of the prior material really matters for this one.)

It's weird to me how everybody knows there's a problem.

How, if you are, say, Issac Asimov or George Lucas or Gene Roddenberry trying to write your Galactic Empire/Federation/Whatever story, you've got to do some kind of handwave about Relativity. We know this because we were taught at some early age that trying to go faster than lightspeed is Somehow Bad, even if nobody ever explains the details.

It's particularly annoying to me because many of the details actually could be explained without going beyond 6th grade math. If you get how right triangles work, that's pretty much all you really need for Special Relativity, except nobody seems to bother trying.

I suppose part of the problem here is the mystique of General Relativity. True story:

Arthur Eddington is at a meeting of Royal Society in 1919 where Sir J.J. Thomson (President) concludes a talk saying nobody's ever really stated in clear language what Relativity is about. The meeting disperses, Ludwig Silberstein (the author of one of the early books on relativity) comes up to Eddington and essentially says, "So you must one of three people in the world who actually get this stuff." Eddington demurs, but Silberstein pushes further, "C'mon, don't be modest," at which point Eddington replies, "Actually, I'm trying to think who #3 is..."

Then again, given the sheer number of people I've encountered who have Misner, Thorne, and Wheeler's Gravitation sitting on their bookshelves, that's clearly not the case anymore.

But if even Albert Fucking Einstein had to take five years off to get up to speed on Differential Geometry — which to some extent is just the Partial Differentiation Chain Rule on Steroids and working through all of the various consequences, but if you didn't get past high school calculus, even that much is going to be a bit hard to take — what hope is there for the rest of us ordinary mortals?

But it so happens that, for Special Relativity, you don't need differential geometry. So let's get started:

Why is the speed of light constant?

How did they even get this idea that the speed of light is always the same? That's the part They never explain. They figure there's no way anybody's going to get that without having the full-ass physics course, so they don't bother trying.

The short answer is that they've done the experiments and that's the way it actually is. Since that's unsatisfying, I'm going to try giving you the half-assed physics course instead:

Read more...Collapse )
26th-Jun-2014 01:57 am - On Caucus Math
party politics
(... The curse of being a math geek living in a state where they have caucuses instead of primaries (not to mention having spent some time observing party rules committees) is I end up thinking about this stuff...)

So here's a fun issue with caucuses:

First, a quick review of the basic rule for awarding delegates at caucuses.

Your precinct is allocated some number of delegates, D, to elect. Some number of people attend your caucus and each declares a preference for a particular candidate. Twenty minutes later, once the blood has been mopped off of the floor, the battle lines have hardened, and everyone who might have been inclined to change his/her mind has been talked to death, you then compute for each candidate the following number
(# of votes for that candidate)
—————————————  × D
(# attendees)
I'll note first of all that every attendee will in fact be included here, because if you don't ever actually declare a preference, that's treated as equivalent to declaring a preference for "Uncommitted," this extra fake candidate that's always added to the mix. So it's guaranteed that all of these "delegate-share" numbers will indeed add up to D.

The next step is to split each delegate-share into a whole number plus a fractional part. For each candidate, the whole number gets awarded directly, and, if those numbers by themselves don't sum to all of the available delegates, you then rank the fractional parts and distribute any remaining delegates, one each, to the highest candidates in that fractional ranking.

(...And yes, for those of you who know about this, I'm skipping the 15% threshold rule, which some states apply at the precinct level. Thankfully, in Washington, we got rid of that 8 years ago, since it's a complete waste of time at the precinct level [also has any number of bad effects, but that's a whole 'nother discussion].)

So first, I'll present Survival Trick Number One, so that you can survive in a chaotic caucus environment without having to do long division in your head. It goes like this: we rewrite the formula above as follows:
(# of votes for that candidate)
((# attendees)/D)
i.e., just divide numerator and denominator by D, which works because multiplication is commutative (...except that the DNC stupidly ruins the commutativity by including a 3-decimal rounding rule in the process, but, as it happens, this doesn't affect things very often at the precinct level, and in any case this still works fine as a rule of thumb so that you can wrap your head around what's going on...)

Bottom line is there's a certain magic number of votes ((# attendees)/D) that you need to get a "whole delgate".

Meaning you can take your caucus, divide it up into blocs of that many people who are all voting for the same candidate. For each such bloc, that candidate gets a "whole delegate", and then whatever votes you have left over, you rank those, and the candidates that are highest on that ranking get the remaining delegates. The advantage of doing things this way is that you're just counting votes without having to do any long division in your head.

Concrete example:

You're in a precinct that's been allocated two delegates.
Twenty people show up.
14 are Kerry supporters.
6 are Dean supporters.
If you follow the worksheet, then it's (14/20)*2 = 1.4 vs. (6/20)*2 = 0.6.
Kerry gets 1 whole delegate and then, because 0.6 beats 0.4, Dean gets the other one.

Or you can do it my way, seeing that (20 attendees)/(2 delegates) = 10 votes needed to get a whole delegate. Thus, Kerry's 14 votes produce one whole delegate (10 votes) with 4 left over that then lose to Dean's 6 leftovers, and so Dean gets the other delegate out of the "fractional ranking", never mind that we're not having to rank fractions any more.

Now for the problem.
It turns out that the number of votes that you need to get a whole delegate is NOT the same as the number of votes you need to win a delegate out of the fractional ranking.
In fact, if your candidate is getting awarded any whole delegates at all, there's a fair argument that some of your votes are being wasted.

What do I mean by this?

Back to our example: Thus far, it's 14 to 6 with each candidate ending up with one delegate. But then the Kerry folks wonder if they can do better. And it turns out, they can!

After a brief strategy session, 7 of the Kerry voters change their preference to Uncommitted. Which now means the totals are 7 Kerry, 7 Uncommitted, and 6 Dean. Since you need 10 to get a whole delegate, there are now zero whole delegates, the fractional ranking then has to award two and they go to Kerry and Uncommitted.

Except,... since the "Uncommitted" folks are really all Kerry supporters, it's a good bet that "Uncommitted" delegate will be signing in for Kerry at the next caucus level.

Which means Kerry has just effectively cleaned up and claimed both delegates.

WTFF? How did that happen?

On the other hand, he did have more than 2/3 of the vote in that precinct, so there's some argument that this isn't actually a totally unfair outcome and perhaps the real question is why should the Kerry supporters have to jump through this extra hoop to get the delegates that are rightfully theirs?

The problem is that, while the number of votes you need to get a whole delegate is
(# attendees)/D,
the number you need to guarantee one out of the fractional ranking is actually
(# attendees)/(D+1),
which, in our example is 20/3 = 6+2/3.

... or, more precisely, if your candidate has at least (# attendees)/(D+1) and there is at least one candidate with strictly more than (# attendees)/(D+1) (even if it's only the slightest ε more), then you are guaranteed a delegate out of the fractional ranking (since in that case there cannot be more than D groups with (# attendees)/(D+1) votes, and you're one of them, so you win)

Dean with 6 votes isn't quite there, and that makes all the difference in the world.

Meanwhile back in the first scenario, where the Kerry supporters are spending 10 votes to get a "whole delegate", this can now be seen as a ripoff, spending 10 when they only needed to spend 6+2/3, thus wasting 3+1/3 of their votes, which costs them a delegate.

More generally, (# attendees)/(D+1) will always be less than (# attendees)/D and, if you're in a close race, chances are you're going to care about that difference.

And apparently, they've even thought about this in Iowa, or, rather, it's the only way I can account for Iowa's version of the threshold rule which not only makes things way more complicated, but also introduces the nastier features of thresholds into the lower-delegate caucuses where they weren't originally a problem.

My fix, which will most likely never be adopted, is much simpler:

Instead of multiplying by D, multiply by (D+1).

That is, for each candidate you instead compute
(# of votes for that candidate)
—————————————  × (D+1)
(# total number of voters)
and then proceed as before, awarding the whole numbers, and again if the total number of delegates awarded in this way is different from D, use the ranking of the fractions to fix it. The only difference now is the (remote) possibility that there will be too many (i.e., D+1) delegates awarded via the whole numbers, in which case, instead of giving out delegates to whoever is at the top of the ranking, we're taking a delegate away from whoever is at the bottom (except that if all of the whole numbers are indeed adding up to (D+1), that means all of the fractional parts will necessarily be zero, so we don't even have to look at any ranking; you just pick somebody at random to lose one).

So in our example, for Kerry the magic number is 14*3/20 = 42/20 = 2.1 and for Dean it's 6*3/20 = 18/20 = 0.9, Kerry gets 2 and we are done; we don't even need to consult the fractional ranking at all.

Or, calculating things my preferred way, you need 20/3 = 6+2/3 votes to get a whole delegate, Kerry has 2 such blocs, Dean doesn't have any, and again we're done. At which point it should be blindingly obvious that there's absolutely no advantage to be had by splitting your voters up over multiple fake candidates; you'll get the same result either way.

Which is the way caucus rules should be (i.e., they just give you an answer and no amount of gameplaying changes it).

But, even though they're never going to adopt this rule, you can still use it, the point of it being that if you're ever in a situation where the Multiply By (D+1) rule is giving you a different answer than the actual Multiply By D rule, that's where you have to watch out.

This effect is most pronounced in the 2 delegate caucuses but it can show up in higher-delegate caucuses as well.

Example 2

4 delegate caucus
20 people show up
17 for Kerry
3 for LaRouche

So now it takes 20/4 = 5 votes to get a whole delegate. Thus, Kerry supporters can spend 15 to get 3 of the 4 delegates. Unfortunately, that means they only have 2 votes left over, which lose to LaRouche's 3 in the fractional ranking and so we get one LaRouche delegate.

What does the Multiply By (D+1) rule say? In this case (D+1)=5, meaning you only need 20/5 = 4 votes to get a delegate out of the fractional ranking, and, with 17 votes, Kerry supporters can produce four such blocs which will all beat LaRouche's 3 votes.

... the only problem being that those blocs need to be for four different candidates. But this is actually doable:

5 for Kerry
4 for Uncommitted (actually Kerry)
4 for Sharpton (actually Kerry)
4 for Kucinich (actually Kerry)
3 for LaRouche

Now Kerry only gets 1 whole delegate while the fractional ranking awards the rest to Uncommitted, Sharpton, and Kucinich, who all change their votes to Kerry at the next caucus, and thus Kerry cleans up all four delegates.
8th-Jan-2014 12:21 am - Today's game theory intro
William has learned about Rock, Paper, Scissors. In honor of that, a puzzle:

Same game: Rock breaks Scissors, Paper covers Rock, Scissors cut Paper. So far so good. Now we add a couple twists:
  • I do research and find myself a Better Rock, a chunk of pure New England granite that rules, absolutely crushes all lesser Rocks. Even though it still loses to Paper I'm happy with it.
  • Meanwhile you've been doing your own research; being of a technological bent you know there are better ways to do Scissors; carbon steel with a diamond edge; not only cuts Paper but completely destroys other Scissors as well. Granted, even a diamond edge is no match for an any actual Rock, let alone mine, but you still win against everything else, so you're happy.
So... my Rock beats your Rock, your Scissors beat my Scissors, and Paper vs. Paper is the only draw possibility left.

What's my strategy? What's your strategy?

And how does this change if I go out and get really good, battle-ready Paper as well, e.g., some of that Tyvek stuff that they use to insulate houses; something that will entirely shred your Paper, even if your high-tech Scissors will still make short work of it. Meaning that not only is there no longer any possibility of a draw, but out of the 9 possible scenarios, I'm winning in five of them.

Unfair, right?


(This is Part 7. There are previous installments; this one is a digression from something I said in Part 6, though you can also start from the beginning at Part 1)

In Part 6, I said:

The theoretical absolute best we can do with rockets is if we can get the exhaust velocity up to the speed of light. This means our exhaust will be pure radiation, that we are somehow powering a huge-ass laser with 100% efficiency, since that's the only way we get all of the exhaust going the same direction. And, boy howdy, do you not want to be following along directly behind,…

Which leads rather directly to this:

The best possible rocket engine and the best possible directed energy weapon are exactly the same thing.

Remember this the next time you're watching Star Wars, Babylon 5, Battlestar Galactica, etc. All of those little fighter ships where the engine is distinct from the guns? Those scenes where they're accelerating forward, closing with the enemy, firing forward with everything they've got?

Wrong. Wrong. Wrong. No military contractor worth its salt is going to waste resources mounting a second gun on a ship when there's already this totally effective and somewhat expensive first gun.

Conversely, if you've got a phaser/laser/gamma-ray-laser that can do real damage from a distance, then most likely that is your engine. If you're in a universe where rockets are your only form of propulsion, you are definitely not going to be wasting resources on a 2nd engine. It's going to be correspondingly expensive to fire, too. Nor will you get off that many shots before you're hurtling away.

What you need to do is arrange to be headed towards your target with as much velocity as you can manage. Then, when you're really close, you flip around and shove the throttle to maximum. It'll look exactly like you're landing on your target (modulo the small matter that you'll want to not be too predictable about it, see below). Best if, once you've killed all of your relative velocity, you can whip out some (really strong) tethers to attach yourself with before continuing to fire, so that you can be expending as much energy as possible on your target vs. propelling yourself away.

Of course, if you actually can get that close you're probably still better off with a burrowing torpedo that can blow up your target from the inside.

In fact, I'm rather having trouble shaking the conclusion that directed energy weapons aren't at least as stupid as rockets. On the other hand, if you're stuck in a universe where rockets are all you have, then so be it. Swords, planted bombs, and bioweapons are all very nice if you can get close enough to use them, but sometimes you just can't.

Also, to be sure, planet and asteroid-based gamma-ray-laser cannons will be a different story. They'll have room for arbitrarily huge reserves of antimatter compared with what you'll have available in your fighter ship, and the momentum consequences of firing off huge blasts will be negligible for them.

Suffice it to say, you'll want to stay well out of range of those. Except for the small problems that,

  1. being lasers, they'll have lots and lots of range, and
  2. the moment you stop firing your own engine for any length of time, your trajectory becomes immediately predictable; figure by the time we have practical antimatter distilleries, we'll have the software for this worked out just fine, too
  3. given the stupidity of rockets, you won't be able to be constantly firing your engine for any length of time before running out of fuel
So, good luck with that.

Granted, if I were going up against an entire planet, I'd probably want to arrange for a dinosaur-killing asteroid to do the dirty work for me. Hide an armada behind it to take out anyone who tries to come near to divert it.

Which then means that any sensible planet is going to have an entire inventory of asteroids of various sizes lined up at its Lagrange points to be able to deal with any such threat, at which point I'd then concentrate my efforts on subverting the folks in charge of the asteroid inventory.

Or maybe just taking a trip down to the planet itself, sneaking in, and detonating the huge antimatter reserve where the phaser cannon is located.

Of course, if everybody has sufficient resources to be distilling out the insane quantities of antimatter needed to be fighting these battles, I'd have to wonder what the hell they're fighting over. Not that this would be the first time in human history where a war got started for completely stupid reasons (cf. WWI)

And round and round we go.

(and there's a Part 8 now, where we move on to Something Completely Different)


(This is Part 6. There are previous installments, though if you only made it as far as Part 3: Rockets Are Stupid, that's good enough for this one.)

Rockets are Even More Stupid Than You Thought

Meanwhile, back on the launch pad, staring at the 2500 tons of Saturn V that I've just told you how to make smaller, we can ponder what's going to be possible once our technology gets Really Good.

The theoretical absolute best we can do with rockets is if we can get the exhaust velocity up to the speed of light. This means our exhaust will be pure radiation, that we are somehow powering a huge-ass laser with 100% efficiency, since that's the only way we get all of the exhaust going the same direction.

And, boy howdy, do you not want to be following along directly behind,…
… which inspires an observation about fighting space battles, which I'm going to defer to Part 7.

Anyway, this Best Possible Rocket brings the fuel cost for getting your Winnebago-sized Command Module to the moon and back again down to a mere 327 grams.

The catch is that half of that 327 grams will need to be antimatter. This also assumes you've solved the problem of storing it in a reasonable way — and if so, the Fusion Power People would really like to hear from you; and no, they won't necessarily be obsolete, because antimatter is merely a storage medium; you still have to extract the energy from somewhere. It should be noted that the amount of energy needed to make that amount of antimatter — and what you get back when you let it recombine — is roughly that of an 8 megaton bomb.

So, if you're imagining this to be the family car, where you can just hop in and fly to the moon for a week when the kids are off school, guess again. Unless you like the idea of each of your neighbors having an 8 megaton bomb in the garage and DUI being about much more than just the occasional lamp post or pedestrian. Hell, let's just have every auto repair garage, bus station, and airport be a terrorist candy shop, where the stakes in question are not just single office buildings but whole continents and planetary habitability.

Suffice it to say, there's a whole range of social and political problems we're going to need to have completely solved before we get to any kind of ubiquitous space travel regime.

Never mind that said problems will have bitten us in the ass long before we have practical antimatter distilleries. No matter how many countries we can get to sign the nuclear non-proliferation treaty, if you're Joe Sixpack sitting at L1 with a 6 ton rock and the right software, that's an 8 megaton bomb you can drop anywhere on earth, no nuclear tech needed.

To The Stars?

And all of that insane energy expense is just to get to the fucking moon. You want to go to the stars?

Using this theoretically perfect rocket which will never actually exist, accelerating at one g (earth gravity) for a day costs you 0.3% of your ship. That may not sound like a lot but you'll need to keep that up continuously for a year to get to 3/4 the speed of light. At which point 2/3 of your ship is now gone (which means at least 1/3 of it was antimatter to start with).

That may be good enough to reach Alpha Centauri, but to get anywhere real, you need to keep this up. Four more years (proper time) of accelerating at 1g — meaning you'll have to spend 242/243 of your original ship; picture launching one third of a Saturn V half-filled with antimatter — puts you 75 light-years out from earth, with finally enough time dilation (100 to 1) so that you can coast across some reasonable fraction of the galaxy (thousands of light-years) in a single lifetime.

Confined to a Winnebago.

And you thought space exploration was going to be fun and exciting.

Also, you better hope you picked a good destination, because, unless you happen to have a perfectly placed neutron star or black hole in your path — at which point you will then also be needing enough shielding to cope with all of the crap likely to be in the vicinity of any such object, because going at any significant fraction of light-speed means you'll need to get really, really close to make any kind of tight turn (also, good luck with the tides) — course changes will be essentially impossible once you get going fast enough.

Nor will you be able to stop anywhere along the way or even slow down at the end of the trip, unless you've arranged to be able to spend another 242/243 of your ship and allowed for another 5 years (proper time) to do it. Meaning we're now launching something 73 times the size of the Saturn V, half-filled with antimatter, and stuck in the Winnebago for a minimum of ten years, in order to be able to do any kind of interstellar travel beyond 150 light-years.

Rockets just suck.

Why I Shouldn't be Allowed to Write for SF Television

This, by the way, is something else that Star Trek and similar shows get wrong.

((Update: It seems I've done Roddenbury an injustice; he apparently did make a pronouncement about impulse engines early on. So you'll have to read this as, "What Star Trek would have been like if impulse engines were rockets." In any case, it doesn't let any of the other shows off the hook: BSG, I'm lookin' at you.))

It's not that I'm going to fault them for postulating the existence of something like Warp Drive, which you just need if you're going to do Galactic Empire stories. Nor am I going to fault them for not dealing with Relativity properly, because the sad fact is that most SF authors only understand Newtonian Universes anyway, and I'd just as soon they stay in their comfort zone and tell stories that make sense on their own terms, rather than attempt to include Relativity and make an utter, complete hash of it. The other fun thing is that if you try to have Warp Drive and Relativity in the same story, then that generally means you have a time travel story even if you don't realize it, at which point JWZ's Law probably applies.

No, where Star Trek — along with everybody elseactually screws up is with the impulse engines, whether they're called that or "thrusters" or "reaction engines" as on some other shows, they are clearly intended to be rockets of some sort. And then they get used in every episode as a completely routine means of puttering around a planetary system.

To which I say, "Wrong." Firing a rocket is cannibalizing your spacecraft; it uses exponential amounts of fuel; you never want to do it if you have any alternative available. Rockets are the propulsion method of last resort.

Meaning if you actually had anything at all like Warp Drive, you would contrive a way to use it for everything you possibly could. You'd use it in-system, you'd install it on the shuttlecraft, you'd use it in the space dock, you'd use it for going to the grocery store. You would use it everywhere that you didn't have some other reasonable alternative (like space elevators, solar sails, tethers, whatever).


The only proper scene in which the impulse engines would even be brought up would be something like the Battle Aftermath Scene. In which the ship has been wrecked by Commodore Decker's planet killer or some such. They've barely eked out a victory, but half the crew is dead. Bodies are scattered everywhere. Shit is on fire. You can hardly see through the smoke. Sulu and Chekov have big nasty burns. Kirk has his shirt ripped off and is bleeding in a dozen places. McCoy and Chapel are buzzing around doing triage. Warp drive is trashed. Dilithium crystals are hopelessly fused, etc.

And now it transpires that they're spinning out of control into a planet or something. At which point we have a dramatic pause and musical cue as Kirk calls down to Engineering.

"Scotty," he says, "Ready the impulse engines."

Some of the younger bridge crew startle at this. It comes up in the training sessions but you never imagine that you'll hear it for real. Because if you do, it usually means you're about to die.

And then we have the long anguished close-up on Scotty. He looks around at the debris in the engine room and realizes the captain is probably making the right call. And finally...

"Aye, captain."

His children are about to be murdered and there's nothing he can do about it. He motions two of his surviving lieutenants over and together they remove the cover and break the seal on the impulse controls. There's a huge lever there that takes three men to move.

Back on the bridge, Kirk flicks a switch on his armrest, "All hands! Jettison Stations! Level Three Emergency! Unnecessary mass into the tubes! Repeat! Level Three Jettison Emergency! Unnecessary mass into the tubes!"

Cut to scenes around the ship of surviving crew members, rummaging through every room, grabbing everything not nailed down — wreckage, equipment, random belongings — and stuffing it all into chutes specifically designed for this purpose.

"Sulu, what's our time factor?"

"We need 5000 ΔV in the next 30 minutes or we're dead."

Flick. "Engineering?"

"Impulse engines ready, Captain. You'll have 195 seconds of burn time."

"Thank you, Scotty. We'll make it count. Spock, how's our mass situation?"

"Down 23% We need to lose another 5. Another 10 minutes, 42 seconds if we can keep the jettison rate up."

"That's cutting it too close." Flick. "All hands! Level TWO Jettison Emergency! Repeat! Level TWO!" Cut to more scenes around the ship. Now they're gathering the dead bodies and stuffing them into the tubes. Cut to exterior view of the ship with expanding cloud of debris and bodies and crap.

… and so on.

Really. Rockets just suck.

Up next, Part 7: Space Battles


(This is Part 5. The previous installments are Part 1: Another Anniversary, Part 2: Climbing the Wall, Part 3: Rockets Are Stupid, Part 4: L1 Rendezvous )

Sisyphus Revisited

So, before I get on with confessing my sins and explain what I've been lying about, I'm realizing I need to say a bit more about instabilities and why that horribly weird spiral trajectory — which I'm going to guess would have been especially horrifying for those Mercury and Gemini astronauts who all got their start as experimental aircraft test pilots; just mention "spiral dive" to any pilots you know and see what they have to say about it (hint: it's not something one usually lives to tell about) — and about why said trajectory is something to be embraced rather than feared.

Imagine a ball bearing rolling back and forth in a parabolic valley.

This scenario, the Simple Harmonic Oscillator, shows up all over the place, pendulum clock, weight on spring, child on a swing. It's like 90% of physics is about making as many situations as possible look like simple harmonic oscillators. Not suprising since this one of the easiest things to solve; we have this hammer; if we can make everything look like a nail, so much the better.

Qualitatively, there's only one solution:

x ∝ cos(ωt)

The ball just goes back and forth and back and forth and back and forth. Forever and ever; the epitome of stability. Once you know the frequency ω, you've got the whole story.

(I use the wonky "is proportional to" (∝) symbol so as not to have to be writing out lots of pointless constants. If, in what follows, you also want to imagine (t-t0) wherever you see t, feel free. So really what we're saying is x=x0cos(ω(t-t0)), the most general solution. Or you can just ignore the math altogether.)

And now we sneak in and make a little, teentsy sign change. What harm could it do?

The next morning, the security guards wake up and find that the parabolic valley has been turned upside-down/inside-out/whatever and we now have a parabolic hilltop (... along with Spock getting a goatee, Federation → Empire, Cat → Dog, etc...).

Whereas before, we had one simple solution, there are now nine (9) and they're all different qualitatively.

Instead of running away screaming, I'm going to make a table listing them all, so that you can get a sense of what we're up against. You may find it easiest to read them in a clockwise or counter-clockwise order.

x ∝ −e−ωt
The ball has always been rolling in from the left; someday it may reach the top but we won't live to see it.
x ∝ +sinh(ωt)
The ball approaches from the left,
makes it over the top,
and rolls down to the right
x ∝ +e+ωt
The ball has always been rolling away to the right; if it was ever at the top, long ago, nobody remembers that far back.
x ∝ −cosh(ωt)
The ball approaches from the left,
fails to get to the top,
rolls back to the left
x = 0
The ball was, is, and ever shall be perfectly balanced at the top of the hill. Amen.
x ∝ +cosh(ωt)
The ball approaches from the right,
fails to get to the top,
rolls back to the right
x ∝ −e+ωt
The ball has always been rolling away to the left; if it was ever at the top, long ago, nobody remembers that far back.
x ∝ −sinh(ωt)
The ball approaches from the right,
makes it over the top,
and rolls down to the left
x ∝ +e−ωt
The ball has always been rolling in from the right; someday it may reach the top but we won't live to see it.

Tell me where you are and how fast you're going, and I'll tell you which box you're in.

The colorings are energy levels. All of the gray boxes have the same energy. Meaning if, when you're at some particular place and you have just the right velocity, you'll be in one of the gray boxes. If you're going faster, then you're in one of the blue boxes; if you're going slower, you're in one of the red boxes.

The gray boxes are essentially boundaries, drawing the fine line between success (blue boxes) and failure to cross over (red boxes). And you can skirt as close to them as you dare, so if, say, you're in one of the blue boxes, by reducing your energy you can make your trajectory be arbitrarily close to the trajectory in the gray box on either side. The closer you get, the more time it takes, so if you're going over the hill, you can arrange to take exactly as much time as you want by picking the right velocity/energy level.

In the center is chaos. An infinitesimal change to an x=0 scenario has eight possible outcomes. Rounding errors will ruin your day if you're not sufficiently clever.

Now, as I mentioned earlier, L1 is actually a saddle point. That is, assuming we orient our axes the right way, it's a parabolic hilltop in the x direction but in both of the y and z directions it's a parabolic valley. Parabolic valleys are places where we can park arbitrary amounts of energy while we're passing through (well okay, there are limits). In other words, if we're going through L1 from the Earth to the Moon or vice versa, we can make the transit take however long we want by changing the size of the spiral.

And since the various frequencies/periods stay roughly the same if we don't go too far out, that means we can spiral around as many times as we want while going through. But also, since we can mess with the y and z directions independently, that gives us even more choices re what direction we're going once we're out of the neighborhood of L1.

What we need is to build a map. Essentially, you can think of there being a (4-dimensional but never mind that) sphere of possible ways to park energy in the y and z directions. Imagine that sphere as being the center box in the table above (i.e., what the orbits would be if we weren't moving at all in the x direction).

Then you have the upper-right and lower left "Rolling Away" boxes which are now (5-dimensional) tubes leading from the sphere to elsewhere, and then the pair of "Rolling in" boxes, which are tubes from elsewhere back to the sphere. These bound the set of possible useful transit trajectories (the blue stuff) that take us from the elsewheres on the left (in the big hole where Earth is) to the elsewheres on the right (the moon and everything outside), which are what we want out of this.

At which point our agenda is simple (hahahahaha): Solve for where the sphere is. Figure out where the tubes go. Once we know where the tubes go, we know what our choices are.

The Actual Lay of the Land

Something is indeed rotten in Denmark and the core of it is that we're not actually doing the happy two-body problem that Newton solved, where angular momentum is conserved, everything has to move on conic-section-shaped trajectories, and ellipses are forever. Counting on our fingers, we see that Earth is one, Moon is two, and spacecraft makes three (3) bodies there, at which point there are no closed-form solutions, Newton gave up, Lagrange figured out a few things and then gave up.

There are lots of weird nooks and crannies, and we're in the process of stumbling onto one of them. Indeed the very fact that we can even have saddle points where chaotic things are happening is all part of why there can't ever be a closed form solution.

But now we need the Big Picture.

L1 is at minus 170km but "the top" is not at 0km. Why? Because, this whole time, I've actually been using a rotating reference frame where the earth and moon are fixed. Which means, among other things, there's centrifugal force to contend with, which gets stronger the farther out you go.

Meaning that 6000 km deep hole where the Earth is is not in the middle of a plane, but rather at the center of this parabola-shaped hill (well okay, parabola-of-revolution-shaped hill), which turns out to be a volcano with a 6000-km deep crater at the top of it. The circular rim of the crater is where the moon's orbit is; everything slopes downwards in all directions outside of it.

Things are further messed up because the moon is sufficiently big to put the earth off-center. That is, since earth and moon actually revolve around their common center of mass, the earth is displaced somewhat (4600km) in the direction opposite to the moon. Which then tilts that aforementioned circular crater rim; rather than being a constant −160½km altitude, the point opposite the moon on the rim (the L3 Lagrange point) is a bit lower (−161km) because it's closer to the earth.

Now if L3 is the low point on the rim, you might be thinking the place opposite it, where the moon is, should be the high point, but

  1. the moon is there, and
  2. the moon has its own gravity, which we have to add back (450km deep hole, remember?)
So, no.

Where is the high point? Follow the rim 120° from L3 in the direction of the moon's orbit and you get to L5 and O'Neill's space colony. If you'd gone 120° the other way you'd have gotten to L4 instead. L5 and L4 are both at −160km and are the real (twin) hilltops. They are as high as you can go and there is no place that's 0km after all.

Things are actually further messed up because of the Coriolis force, which I haven't told you about, which happens to be crucial for understanding why L4 and L5 are stable even though they are hilltops which should otherwise be completely disastrous from a stability point of view. Fortunately, for L1 and L2, the Coriolis force only messes with the frequencies and tilts the various axes a bit; it doesn't change the overall qualitative picture, so I can skip that part.

(Nor was I never clear on why O'Neill preferred L5 to L4. Everything you can do with L5, trajectoriwise, you can do with L4; it's all symmetric, see. I'm also now wondering if the hilltop genuinely is the best place to be; it's actually the hardest place to get to in the Earth-moon system. There are so many tasks you need to do to maintain a space colony and keep everybody alive; station-keeping was never even remotely the biggest problem. Stability also means it's harder to leave, which will suck if you ever want to move the colony somewhere else. Though I suppose being at hilltop may reduce the probability that random rocks will arrive from infinity and ruin your day. That, to me, would be a much better selling point than stability — if it's actually true; haven't done the math on that one yet...)

So to get out from L1, instead of having to climb 170km as you might have originally thought, it's looking like, depending which direction we go, we only have to climb 10km at the most.

But it gets better.

The presence of the moon actually cuts a huge notch in the crater rim. If we continue our hike along the rim from L3 past the peak at L5 we'll find ourselves headed decisively downwards. Then the rim wall splits, going around either side of the big hole where the moon is. Directly across the moon from where they split, the walls rejoin on the far side and the rim continues around up to L4. L1 is the saddle point on the inner wall; L2 (you knew there had to be an L2) is the saddle point on the outer wall. And that is the last of the flat spots; Lagrange proved that there could only be five and this is where he left things 200 years ago.

L2, at −169km, is a measly one (1) kilometer higher than L1. As long as you have at least 140 m/s (313 mph) of velocity when you get to L1, you'll have enough energy to get to L2. And everything I've said about L1 (i.e., that it's a saddle point, that there are tubes, etc...) is true of L2 as well.

So if you're stationary at L1, you just need to put on 140 m/s of ΔV. But it's actually easier than that. The moon is right there. L1 has two outgoing tubes, one headed back towards Earth, the other outward. L2 likewise has two incoming tubes. See where the outward bound L1 tube intersects L2's from-inwards tube. Find the pair of intersecting orbits that comes closest to the moon. That is where you want to do your burn and chances are it'll be a lot smaller than 140 m/s (because the deeper you are, the faster you're moving and the faster you're moving, the less ΔV you need to achieve a particular energy change, i.e., to gain that last kilometer).

Once you are at L2, you are definitively outside the crater.

At which point we switch to the Earth-Sun rotating frame, where there is an entirely different set of Lagrange points. As it happens, the Earth-Sun L1 and L2 points are each about 1.5 million kilometers from Earth, your being at the Earth-Moon L2 point means you're now moving in a 444,000 km radius circle around the earth — exactly where depending on the time of the month — at something like 1200 m/s which is 300 m/s faster than what you'd ordinarily need to stay in circular orbit around Earth at that distance.

Which means, once you get sufficiently beyond L2 and away from the moon's influence, you're being flung away. Depending on how you timed things — and you can hang out in the halo orbits as long as you need to in order to time things just right — you can arrange to get flung away in any direction you want; and you'll be left with enough energy to both get away from the moon and get up another 63km worth of wall (this "wall" now being the wall around the solar crater whose rim is where the Earth's orbit is). Which is good because the Earth-Sun L1 and L2 points are both only about 50km higher from where you are now.

Or you can view everything from a completely non-rotating frame and see that the Moon just gave you a big gravitational assist. And when you get to Earth-Sun L2, the Earth is going to give you one, too, if you've played your cards right.

Except that, once you know where the tubes are, it's no longer a matter of chance. That is, you know where the outgoing tube from Earth-moon L2 is and where the incoming tube for Earth-sun L2 is and thus where they intersect. You then have a bunch of trajectories you can use.

And from Earth-sun L1 or L2, we can similarly go all sorts of other places, Sun-Mars L1, Sun-Mars L2, Sun-Jupiter L1, Jupiter-Ganymede L2, and on, and on. All pairs of co-orbiting bodies in the solar system, sun-planet, planet-moon, etc. each have their own L1 and L2 points guarding the entrances to their respective craters. Since everything is time-reversible in classical mechanics, you have trajectories going both ways, i.e., for every weird spiral trajectory that sends you away from L1 or L2 off to wherever, there's a corresponding one that brings you back in. You can string these trajectories together playing mix and match with them, giving you a way to visit any planet or moon that you like — admittedly, these low-fuel trajectories tend to be really slow, but if you're an unmanned satellite, you don't care.

This is the essence of the Interplanetary Transportation Network.

(which might seem to be a conclusion, but I actually have more to say about rockets in Part 6)

Random update:
I, of course, forgot to mention the really cool part of Farquhar's thesis, which was the proposal that Collins be put in a halo orbit at L2 behind the Moon — which, as noted above, is a measly 1km worth of additional energy/effort beyond my have-him-orbit-L1 plan.

It then so happens you can make the radius big enough so as to remain visible from Earth at all times.

And then we do Far Side landings with no gaps in communication.


(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)


(This is Part 3. The previous installments are Part 1: Another Anniversary and Part 2: Climbing the Wall)

The Stupid Thing About Rockets

There are so many tropes about rockets that the SF authors take for granted. You'd think a spaceship is just like your car: you put fuel in, you get so many miles to the gallon, multiply to figure out how far you can go, or equivalently, divide to see how many gallons you'll need.

Rockets don't work that way. At all.

Imagine being stuck in the middle of a perfectly smooth, flat ice pond, so slippery you can't even stand up. With no way to get traction, nothing to push off of, the only way you can actually get yourself moving somewhere is to throw something away in the opposite direction.

And, as luck would have it, the bastard who put you there also left you with a large suitcase filled with baseballs. And fortunately, while you might not be a major league pitcher, you still have a pretty good throwing arm. So you throw a baseball, and now you're moving; throw another one and you're moving faster.

This is what a rocket is. Firing your rocket always means throwing away part of your spacecraft. Never forget this. Physics doesn't care how big or bright the flame out the back is. What matters is how much junk you're throwing and how fast you're throwing it.

If this sounds like a completely absurd and stupid way to move around, that's because it is. Constantly cannibalizing your own ship is the absolute worst way to travel. We use rockets because, much of the time, there is no alternative, and, given the fundamental principles involved, there's reason to believe that, outside of various special situations, there never will be.

But if you're going to travel this way, there are things you need to know:

  • You want to be throwing each baseball as fast as you possibly can,
    because once it's thrown it's gone forever. You can't retrieve it, since turning around and going back to retrieve it defeats the very purpose of throwing it in the first place. So you have to make each ball count for everything it's worth. Once you run out of baseballs, you're screwed, unless you want to start throwing clothes, limbs, or vital organs, but I can pretty much guarantee that's not going to end well.

  • You can rest as much as you want between throws.
    The ice is perfectly smooth so once you're moving, you stay moving. You keep the velocity you've earned while you're resting up to throw the next baseball. In this case, Conservation of Momentum is your friend. And since space is really, really, really big, you'll generally have all the time in the world to perform your maneuvers. To be sure, timing is everything, but that just means you have to plan ahead, i.e., start throwing earlier.

  • In space, your destinations are not places but rather trajectories (orbits).
    Once you're on a particular trajectory, you stay there forever and it costs you nothing. It's only when you want to change trajectories that you have to do anything, at which point there will be a specific velocity change ΔV you need to achieve, and it really doesn't matter how you do it.

Putting all of this together, we find that, once you've tuned your rocket engine so that it's always throwing stuff as fast as it possibly can, a velocity change ΔV of any particular magnitude always costs you the same percentage of your ship — thank you, Konstantin Tsiolkovsky — no matter how many engines you have running, no matter what your throttle settings are, i.e., no matter how much you're resting between throws.

To put some numbers on it, imagine that your best fastball happens to be 30 meters per second — not quite Randy Johnson Territory but close enough — and you want to gain (or lose), say, 20 meters per second. Then you need to keep throwing until roughly half of your original mass remains (well, okay, e-20/30≈0.51 of it, if you must know). Or, equivalently, that suitcase of baseballs needed to be at least as big/massive as you are. If, after that, you want to pick up another 20 m/s, you need to be able to toss half of your mass again. And if you can manage it a third time, then you'll be going 60 m/s faster than when you started, but you'll only have 1/8 of your original mass left.

Which meant you had to have started with a suitcase of baseballs that's seven (7) times as massive as you are. And now it's gone. Let's hope you're headed in the right direction.

Notice the exponential growth in reverse.

Moreover, with each maneuver tossing all but some percentage of your ship, if you have multiple maneuvers, you have to multiply those percentages to figure out your total cost. It's not like you can just add the gasoline costs for each leg of the trip.

This is where Conservation of Momentum stops being your friend and how rocket travel is fundamentally different and will never be like driving your car.

Bottom line is, once you know your engine technology (fastball velocity) and your flight plan (sum of all of the ΔV's you need to do), you'll know your fuel cost and it'll be a multiplier that is exponential in the total ΔV you need to do. That is, you take your payload, multiply by this number, and that's how big a ship you need to start with, assuming you can manage it so that everything minus your payload is fuel.

And if that multiplier is very large, then a small change in your payload at the end of the trip can make a big difference in what you need at the start.

The only way to reduce the fraction of your ship you have to toss for a given ΔV is to get a better engine that has a higher exhaust velocity. But whatever engine you install, chances are that's what you'll be stuck with for the rest of your trip.

Nor does refueling work the way you think it would. If the new fuel isn't already travelling at the same velocity you are, then collecting it is going to change your course. To avoid that and get it matching your velocity, if it was launched from the same place you were —the only choice in 1969 and also for the forseeable future until we can, say, start harvesting comets or put a hydrazine refinery on Titan—it's going to have performed a set of maneuvers similar to what you've already done, which means it most likely expended the same percentage of itself catching up with you.

Which means you've saved nothing at all by not bringing that fuel with you in the first place.

Which means that Earth Orbit Rendezvous is completely pointless, at least as far as saving fuel is concerned; it just doesn't. In fact, chances are, it uses more because you have all of this duplicated engine+fuel-tank stuff getting boosted into low Earth orbit, whereas a single humongous rocket could have significant economies of scale once you figure out how to build it.

If you really care about saving fuel, what you need to do is either reduce your payload or come up with a better flight plan, or both.

How Lunar Orbit Rendezvous (LOR) Wins

What the LOR advocates noticed is that there's a heat shield, fuel, air, and other consumables needed for the trip back to Earth, that are not being used for the trip down to the lunar surface. If there were a way to just leave all of that crap behind in lunar orbit, i.e., take down a separate Lunar Excursion Module (LEM) that holds only what you're going to need on the surface, then bring back only what you need to bring back, and finally rejoin the stuff you left behind in orbit -- hence the name for this plan: Lunar Orbit Rendezvous (duh) -- you would save tons of fuel, literally.

How much? Well consider that in the actual Apollo 11 mission, the LEM was 15 metric tons (N.B., all tons are metric from now on).

  • Descent stage was 10 tons, 8 of it of fuel.
  • The ascent stage was 5 tons, half of it fuel.

They also wanted enough fuel to be able to hover for 2 minutes before landing; that was the safety margin. And for Apollo 11, they ended up using every last bit of it to get to a new landing site when the original site turned out, upon closer examination, to be hosting its Annual Large Irregular Boulder Convention that week and was thus slightly less than ideal.

So,... 7 tons hovering for 2 minutes in lunar gravity works out to 500kg of fuel, so the rest of the descent stage fuel, 7½ tons, was for getting down from orbit. Meaning whatever the total ΔV was for getting down from lunar orbit, it cost 50% of the ship (started out as 15 tons, remember). And getting back up evidently cost 50%, too. Actually this is what you'd expect, since one trajectory is a time-reversal of the other, so the ΔV's are all the same and therefore so is the total fuel cost, percentagewise.

How does this change if we try to bring everything down with us?

The Command/Service Module that stayed behind in lunar orbit starts out as 30 tons, but 18 of it is fuel, 13 of which gets spent getting us into lunar orbit. Which means we have 17 tons left, 5 of it fuel for getting back to Earth. And then we work backwards from there:

Lunar Orbit RendezvousDirect Flight
2½ ton empty LEM ascent stage
reaches lunar orbit;
astronauts with moonrocks
crawl back into 17 ton CSM
17 tons of CSM
returns to lunar orbit
÷ (1/2) mass reduction getting to lunar orbit from the surface
5 tons of LEM ascent stage lifts off34 tons of CSM lifts off
leaves behind 2 ton empty LEM descent stage
7 tons of LEM lands36 tons of CSM lands
÷ (14/15) mass reduction hovering for 2 minutes
7½ tons of LEM after descent38½ tons of CSM after descent
÷ (1/2) mass reduction descending from lunar orbit
15 tons of LEM separates from77 tons of CSM enters lunar orbit
17 tons of CSM
32 tons of LEM+CSM enters lunar orbit
÷ (32/45) mass reduction getting into lunar orbit
45 tons of LEM+CSM fully loaded108½ tons of CSM fully loaded

And everything from here on back to the launch pad on Earth is correspondingly bigger.

Which means that for Direct Flight we need a rocket more than twice the size of the Saturn V. And this was all being generous in assuming that, e.g., there was nothing in the LEM ascent stage that wasn't already duplicated in the Command Module and that the 2 ton empty LEM descent stage would not need to be correspondingly bigger.

And even if we split the Supersize-Saturn into two or three rockets as per the Earth-Orbit Rendezvous plan, it's still the same multiplier for each rocket to get into orbit and to get all of that material to the moon. That is, even if it's rockets we can actually build, we're still not saving any fuel. Ultimately, if the overall budget stays the same, instead of 8 moon missions (Apollos 10-17), we only get 3 at best. Meaning Apollo 12 is the last one and we wouldn't have been able to spare Apollo 10 for a dress rehearsal (with possibly disastrous results).

I suppose it's a measure of how much a bureaucracy NASA was even back then that it still took at least a year of lobbying by the LOR proponents to get NASA management to actually accept the math on this.

Never mind that there are better flight plans out there. Which brings us to…

How I Would Have Done It.

(to be continued in Part 4)


(continued from Part 1, introducing the 1961-62 NASA debate on how to get to the moon, which you might want to read first)

Climbing the Wall

I don't know who first got the idea to picture the Earth's gravity well as this huge funnel, with the Earth at the bottom and the moon and various other satellites as ball bearings rolling or sliding around the top. I'd draw it but you've seen it already in every science museum on the planet. For all I know, Newton may well have had it in his Principia even if they hadn't quite figured out how to make ball bearings at that point.

What's annoying is just how deep the hole in the middle is, and this number I first heard from either Arthur C. Clarke or Gerard O'Neill:

Getting out of the earth's gravity well takes the same amount of energy as climbing a wall 6,000 kilometers high (i.e., if you had to climb the whole way against earth's surface gravity).

O'Neill's point was pretty simple: Why would you ever want to live at the bottom of a hole? Let's build space colonies!

Unfortunately, my point is a little more subtle, so we need some more numbers.

The energy you need just to get to low earth orbit is like climbing the first 3000km, half way out. Even though you've only gained a few hundred kilometers in real altitude, it's a huge accomplishment to go from standing still on the surface to going fast enough to stay in orbit. And it's useful enough to get to a place outside most of the atmosphere, where you have time to think about what you want to do next. But you still have at least another 2000km of wall to climb before you can get anywhere useful…

Like, say, geosynchronous orbit, where most of the communication satellites live — which is already much farther away than people give it credit for. At this point you're a bit more than a tenth of the way to the moon and about 500km from "the top" of the wall, though by that point the funnel has flared out pretty far so that you're actually going 50km outward for every one that you're going "up" (yay inverse square law...). Meaning instead of climbing El Capitan, we're now doing the leisurely stroll from the house to the supermarket — in my case this happens to be a fifty foot elevation gain over half a mile that I'll hopefully still be able to do when I'm 70.

Now as it happens, since our goal is just to get to the moon, we don't need to get all of the way out of the hole. The moon itself isn't all the way out; it's still in orbit around the Earth, see. But it's most of the way out.

Continuing outward from geosynchronous orbit towards the moon, the "wall" continues to flatten out until finally, when you get about 5/6 of the way there, it flattens out completely. You're now just 170km from "the top", in this saddle point where, to either side of you, the wall continues to rise, but in front of you it drops off and you're staring at another big hole with the Moon at the bottom.

Welcome to L1, the first of the Lagrange Points, the five magical places in the Earth-moon neighborhood where moon gravity, earth gravity, and centrifugal force all cancel each other. Put a ball bearing here in exactly the right place and, if it could be undisturbed by perturbations from the sun and the other planets, it would just stay there forever, i.e., orbiting the earth once a month exactly in synch with the moon.

And I really do mean magical, here. Forget Stonehenge, the Bermuda Triangle, or even Disneyland; they've got nothing on L1, as we'll see.

So now we're staring down into this second hole. To be sure, it's not anywhere near as deep, only 450km down to get to the lunar surface, child's play after having come this far. But it's still deep enough to inspire the Lunar Orbit Rendezvous (LOR) people to ask this stupid question:

"Why the hell are we taking everything with us down this 450km hole and back out again?"

Keep in mind that not only do you have to reach the bottom, but you want to be standing still when you get there (so as not to make a fresh crater), which means burning off all of the velocity you accumulate as you fall into the hole.

Which you have to do with rockets because there's no atmosphere.

And then, of course, you have to add all of this velocity back in order to get yourself out of there.

Which gets us to

The Stupid Thing About Rockets

(to be continued in Part 3)

Another Anniversary

(... something that's been brewing over the past month; this is going to be Part 1 of n...)

Among other things, I've been reading up on the Interplanetary Transportation Network,

but I was also reminded that last month was the 45th anniversary of the Apollo 7 launch, which in itself wasn't much — fly a Saturn 1B and Command/Service Module into low earth orbit and splash down again just to make sure the basic system works. It was mainly notable for being the first manned flight after the Apollo 1 disaster killed Grissom, White, and Chaffee, which resulted in everything at NASA being grounded for a year while they figured out what they did wrong...

Which I knew nothing about at the time because I was seven years old, as old as William is now. For me, Apollo 7 was an introduction to whole idea that we even had a space program. It started as a stupid little "Hey look, we launched something!" article in the Weekly Reader that was maybe a paragraph long and had pictures of astronauts. Sometime later that year they showed the Disney "Man in Space" series in class, by which point I was vacuuming up all kinds of colorful books on space exploration.

By then we were maybe a year or two away from an actual moon landing. It was a big deal for everyone.

What amazes me now is how much we didn't know.

Take orbital mechanics, the study of how things move in space (or everywhere, really). As it's covered in the physics curriculum, you'd think it was a dead subject. Isaac Newton, Joseph-Louis Lagrange, and William Rowan Hamilton had everything we needed to know worked out by 150 years ago, and then Physics moved on to more exciting topics (electricity! magnetism! relativity! quantum mechanics! quantum electrodynamics! quantum chromodynamics! string theoryloop quantum gravity? hell if I know what's next …)?

Classical Mechanics? Been there, done that. Boring?

Guess again.

Granted, I'd already had some inkling of this, what with one of my friends in grad school being a Control Theory student who burst out laughing at the thought that there wasn't anything more to learn about mechanics. And to be fair, they didn't have computers 150 years ago, nor did they know what we know now about numerical analysis or non-linear differential equations. Never mind the general Playing Around With Numbers that you just couldn't do even 40 years ago. And now that people are actually shooting crap into space and needing it to go to particular places, there's a bit more than just abstract interest, now.

So, to take an example,…

Remember how in the late 1970s there was this serendipitous lineup of the outer planets -- Jupiter, Saturn, Uranus, Neptune, all in a nice row so that if we launched something Just Right, even with very little fuel it could go bing-bing-bing-bing, each planet doing a gravitational assist to boost the probe to the next, and so we get this really cheap odyssey that visits them all? Obviously this was a golden opportunity since it would be hundreds of years before we'd get that kind of lineup again. And so we launched Pioneer 10+11 and then Voyagers 1+2 and got back all of those nice pictures.

Turns out,… that "golden opportunity"? Total lie.

Not only can we can do this any time we want, but if we'd known about this back in the 1970s, we could not only have saved a bit of fuel but also done it in such a way that, instead of being stuck with single flyby for each planet, the probe could instead have been arranged to loop around each one as many times as we wanted before going on to the next.

But before I get into how the Interplanetary Transportation Network works, let's start with something much simpler that I'm pretty sure we'd have done differently had we known what we know now:

The Apollo Debate

In the early 1960s, shortly after John F. Kennedy issued his famous challenge, there was a huge debate about the best/cheapest way to actually get somebody to the moon.

Calculating what it takes to launch from Earth something big enough to hold a few astronauts, land it on the moon, and bring it back—this being the "Direct Flight" scenario—you find out that you need this insane, huge-ass rocket, something like two or three times the size of the Saturn V that was eventually built, something that maybe we'd be able to do by 1975 or 1980 (cue maniacal laughter from The Future), but if the goal is to get to the moon by 1970, we'd have to come up with Some Other Plan.

So they focused on what they could do, which is build smaller rockets. You put the astronauts on one of them, and have the rest carry spare fuel tanks, have them meet in low earth orbit, bolt everything together, and then send that to the moon (and back). Obvious, really. This, the "Earth-Orbit Rendezvous" (EOR) scenario, became NASA's game plan. Wernher von Braun gave his blessing and we were off to the races.

But then there was this annoying group that had this other, completely bizarre idea: "Lunar Orbit Rendezvous" (LOR), they called it.

Keep in mind that at this point, we hadn't rendezvoused anything yet in space, so nobody had any idea how hard EOR might be. Think about how you might go about catching up with a meteor. It's going thousands of meters per second and your job is to match it's velocity. And now we're supposed to be doing this in lunar orbit, instead? WTFF?

And they just would not shut up and get with the program.

It's weird to me now, remembering all of those colorful books with illustrations of all the possible ways to get to the moon. They actually mentioned this debate, even if they didn't do a very good job explaining well what the actual pros and cons were.

Or maybe they did, and it just whooshed over my head because I was seven years old and hadn't had any actual physics, yet, and anyway, hey, look, rockets!

But now that I have, it's bloody obvious. In fact, it's so obvious it's a bit appalling to me that NASA had to spend an entire year figuring this out:

Climbing the Wall

(to be continued in Part 2)

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