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.
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 Rendezvous||Direct 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 off||34 tons of CSM lifts off|
|leaves behind 2 ton empty LEM descent stage|
|7 tons of LEM lands||36 tons of CSM lands|
|÷ (14/15) mass reduction hovering for 2 minutes|
|7½ tons of LEM after descent||38½ tons of CSM after descent|
|÷ (1/2) mass reduction descending from lunar orbit|
|15 tons of LEM separates from||77 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 loaded||108½ 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…