wrog

It is time to expand our universe a bit.

A New Dimension

Let's try adding a dimension. We move from the John Hancock Tower to the Flatiron Building (which, if you haven't seen it, is this mostly flat thing standing on end, hence the name). Each floor is a single south-to-north corridor of offices.

Meet the Moving People

Actually, we're going to be particular about who we associate with. We want Moving People who will likewise be able to say that they don't think of themselves as moving, meaning they can't be accelerating or spinning, either. Which leaves having them coast at some constant velocity that is non-zero (because otherwise they'd be stationary and One of Us) and upwards (just to pick a direction).

We also want them moving slower than lightspeed (STL), for Reasons.

(this is Part 5; Part 1 through Part 4 contain possibly-necessary background, but maybe not)

It is time to start putting the pieces together.

I'll start with my first cardinal sin, which is that I've only seen the TV series, not read the books, so if I missed huge reams of infodump where the authors spelled out their actual numbers, so be it. Then again, authors who are not Andy Weir tend not to do this anyway, the cardinal rule of storytelling being that no matter how much work you put into world-building your iceberg, you need to leave most of it underwater, in order to not inspire your readers to beat you to death. At least not until after the series is hugely successful, to the point where publishers can be assured of there being enough rabid nutjobs out there that copies of a Chris-Tolkien-style dump of all research, drafts, cocktail napkins with half-begun stories on them, etc… accumulated during the life of the author will actually sell.

Perhaps what I'll be complaining about is where the TV writers indeed screwed things up. This seems unlikely to me, but if so, this wouldn't be the first time this sort of thing has happened (and, while you might feel sorry for the book authors, nobody was putting a gun to their heads and saying they had to take the TV deal).

Onward…

What is the Epstein Drive?

a.k.a. Our Propulsion Methods Suck, Our Power Plants Suck, Fission Sucks, Fusion Sucks, and Antimatter May Also Suck

(This is Part 4; continued from Part 3, or you can start at Part 1

Recall how rockets work: Every $\text{d}t$ seconds we toss out some bit of mass $\text{d}m$, with velocity $v_{\text{exhaust}}$ in the exact wrong direction, giving the rest of the ship, whose total mass is $m$, a small kick $\text{d}v$ in the direction we want to go. The momentum accountants then tell us:

(This is "Deconstructing The Expanse, Part 2", continued from Part 1, here.)

As luck would have it, I decided to go back and watch part of an episode to check something, mainly to try to pinpoint where exactly the show goes off the rails, and aside from saving myself from a really embarrasing blunder, it also reminded me of a topic I meant to cover but forgot to.

It's Season 3, Episode 7, the one where Maneo, Speed Demon Belter Guy in his tiny racing pinnace decides he's going to be the first one through The Ring and arranges this whole elaborate sequence of "slingshots" to get there first (what happens when he gets there is a fun scene that I liked because I'm actually 12, but I won't spoil it).

And then I realized…

People have multiple misconceptions about gravitational maneuvers. The "slingshot" is badly named. I blame Star Trek.

(…a curious observation about relativistic light-sails, … followed by a teentsy bit of obligatory, if undeserved, "Stargate: Universe" bashing.)

A couple of questions that come up in trying to imagine how the interstellar transit economy/infrastructure is all going to work:

1. Is there some ideal size for a solar power satellite?
   This to some extent lies at the heart of my questions of what the solar power extraction regime is going to look like, what orbits we're going to prefer, how many satellites we'll be needing to build, how close to the sun they need to fly, etc.
   
   
2. How big a mirror/solar-sail are we going to need to propel our interstellar payloads?
   This will be a central issue in transit tube design since it affects everything else, e.g., how far apart the laser ships need to be, how much antimatter they will need to be supplied with, etc.

Reviewing what I said earlier on the subject of mirrors/sails:

[We'll need them to be] extremely lightweight, totally reflective, micrometeoroid-tolerant, blah-blah-blah. There will be some engineering tradeoff to determine the size (i.e., the smaller we make it, the more accurate the rocket/laser thing has to be, and the more energy density it has to withstand, but the less vulnerable it'll be), which I'll leave the engineers to figure out.

My original thought was that our two applications for mirror-building don't really have much to do with each other beyond their reflecting (pardon the pun) the supply and demand sides of our energy economy, that once we got into the details, the differing goals (energy collection vs. propulsion) would assert themselves and ultimately there'd be very different requirements/designs.

For one thing, for solar satellites, why care about making them lightweight? We're just putting them in orbit, and, beyond occasional questions of servicing and refurbishment, they're not going anywhere. And, if anything, we want them sturdy enough to stand up to the stress of being in however close to the sun we need them to be.

Lightweight only matters in that the less material we use, the more we can make, and we will ultimately have to be making millions of them.

It's not that any of this is wrong; we're just missing an important detail is all.

goddammit.

… we can have a station at L2 behind Mercury and still have a wide variety of orbits available. Which is good because I don't know how else we can do a fixed collection point.

I really should know better.

If we're really going to depend on power satellite orbits following the Kepler rules, then we need to keep them far away from any actual planet. Mercury can't be there; end of story. If Mercury is there, then there's no way around having to solve a 3 body problem, and while it might be possible (like I've been assuming) to perturb the Kepler orbit into an actual orbit that works and stays in synch with Mercury, my gut feeling at the moment is that'll be a huge mess to get right.

Also Mercury's orbit is really eccentric which severely screws up the only 3-body solutions I know about.

And if Mercury is not there, then there's no way to hang a station at aphelion. By default, anything stationary at aphelion falls into the sun, unless we do weird space elevator shit and tether it to something (no idea); but, as with the Earth space elevator, that probably entails unobtanium cables with insane tensile strength and we're in Not Gonna Happen Land.

And I'm realizing now there's a much, much simpler solution available:

Put rockets on all of the antimatter cells.

That's it. Once a cell finishes charging, the satellite releases it and it flies off to wherever it needs to go.

(Yes, Rockets are Still Stupid, but if we put a mass driver on the satellite, then the satellite will nearly always have to do an orbit correction every time it launches a cell and we won't actually have saved anything. So rockets it is…)

Chances are, for the sake of administrative sanity if nothing else, we'll still want to have a collection point somewhere. And we'll probably want it to be a Lagrange point so as to have access to all of the best IPTS orbits. But this can be any of the ones available in the inner solar system.

Hell, we could even use Earth-Moon L2. That might even make sense if it's still fairly early in the agenda and lunar orbit is where we need to have antimatter arriving.

In case you were wondering, the energy cost of moving a kg from, say, Mercury's orbit to Earth's is about 2.5 billionths of a kg, which somebody has to be paying anyway. Unless we want to be really stingy and do some stupid funky Mercury/Venus flyby to bleed off momentum, which I suppose we could do, but bleah.

What to do with the empties is potentially more complicated. Maybe our software will be good enough that calculating ballistic trajectories to get them all back to some power satellite somewhere will be workable.

Or we can just make a point to never discharge a cell below the 2.5 billionth mark (or just keep an extra cell around with power in it to do infinitesimal recharges as needed), so that it'll always have enough energy to go find itself a power satellite. KISS.

After my previous forays into the question of How to do Solar Power Satellites Right, here and here, there seem to yet more ways to simplify the math and get a better sense of the design space. Very annoying.

In particular, that WTF/4A energy flux formula is not, in fact, the final word on simplifying and, it seems, we can have a station at L2 behind Mercury and still have a wide variety of orbits available. Which is good because I don't know how else we can do a fixed collection point.(No no no no no)

So, rewind. Let's try this again:

Doing Solar Power Right

The plan: Have a ring of satellites in a single, not-necessarily-circular orbit around the sun, all soaking up sunlight and using it to charge antimatter cells. How do we get this to produce some number of kg per day? What are our actual choices?

One would think the number and sizes of the individual satellites shouldn't matter a whole lot. To first order, we'll just have some total acreage of solar panels being crammed into the orbit. Multiply by solar power flux and we're done, right?

However, if we make the satellites too small, then we're making more of them to cover the same area, and if they're all vaguely square/circular — the sensible way to build them — and thus correspondingly less wide, they'll use up more linear space in the orbit and eventually start bumping into each other. Which we very much don't want.

It also turns out that there will be nothing gained by creating a concentration of satellites at any point along the orbit, since that concentration will simply circulate around the orbit, just as any individual satellite will. In particular, there will be no way to, say, create a concentration close to the sun that stays there.

In fact, it's fairly easy to show (by taking the sum over a single orbital period in which every satellite visits every point of the orbit exactly once) that, assuming all satellites are the same size, for any (nonconstant) satellite density function you might try, there exists a constant density / "evenly spaced" rearrangement of the satellites that produces the same amount energy per unit time on average.

… we just need to nail down what we mean by "evenly spaced" in an eccentric orbit where, at any given time, individual bodies will necessarily have all different velocities and constantly changing mutual distances.

The key observation is that if they're all in the same orbit, then they're all moving in lock step. Assuming that none of the satellites are big enough to be gravitationally messing with any of the others, we're back at the two-body problem that Newton solved 300+ years ago. They're gonna do what they're gonna do and Ellipses are Forever. If it takes one of them, say, 5 days to go from point A to point B, the same will be true of the next satellite following along. So if we make measurements of the km²/day passing point A, we must get the same numbers at point B 5 days later.

Which means that when we're first injecting satellites into this orbit, we want to be sure to inject them at a constant rate — call this our solar panel current, i.e., make it so that the number of km²/day passing the insertion point remains constant.

Once we do this, the current will then be constant and also be the same constant everywhere else along the orbit. Individual velocities can still vary, but anywhere that we have satellites moving faster, they'll have to be more spread out to keep the current the same.

Thus, the place where we have to worry most about stuff colliding will be at aphelion, that point farthest from the sun where everything is moving slowest and hence most bunched together. A panel density can be obtained from dividing the panel current (km²/day) by the aphelion velocity (km/day). This number (km²/km) is how many km² of panel you'll see in any snapshot of 1 km's worth of orbit — or, rather, in every km of an imaginary circular orbit in which everyone is going at (constant) aphelion velocity and spaced out exactly as how they arrived at aphelion.

… which will necessarily be a lower bound on the actual spacings you see in the real orbit. Since I prefer to think in terms of spacings rather than densities, I'll be working in terms of this inverse number instead. That is, we'll define the spacing to be:

h = (panel current) / (aphelion velocity) = 1/ (panel density)

which gives us a number (km/km²), i.e., the number of km of (imaginary circular) orbit you need to snapshot/grab in order to see/accumlate 1 km² of solar panel.

If each individual satellite has area A, then hA is how far apart the satellites will be at aphelion. And if we're comfortable with that being the distance of closest approach, we'll be good to go.

Getting into the ball park

We can now give the formula for this orbit's total power production:

power = W / (2 R h)

where R (km) is aphelion distance, h (km⁻¹) is the spacing, and W is total output of the sun as before (use whatever energy-per-time units you want).

And, yeah, that's it. Just like the WTF formula, this is an exact solution (*) and somehow has none of the other factors you'd expect to see. Inverse-square-law fields have all of these wacky hidden tricks and stuff that cancels unexpectedly. Gauss is probably laughing at me.

(*) or, at least, as exact as we get in what is actually an n-body problem (Yes! This!) and, given that we're in close to the sun, maybe also some General Relativity bullshit lurking (what causes Mercury's orbit to precess) as well. It's also possible I'll lose because I'm ignoring Mercury's gravity (Yes! This!). we'll probably want (small) engines on the satellites to correct perturbations.

So,… if, say,

1. we want aphelion to be at the L2 point behind Mercury (58,130,000 km) (We don't, but this is just as good an example number as any), that convenient place I keep wanting to put a station — which is annoyingly just outside the total solar eclipse zone by about 18,000 km, but maybe having a mere 84% of the sun blocked might be good enough, and also maybe that distance is short enough that we can do some space-elevator bullshit so that the inhabited part of the station(No no no no no) — assuming there even needs to be one — can be hanging down inside the total eclipse zone anyway — and
   
   
2. we want the spacing factor to be, e.g., 36.635 km⁻¹ (to pick a completely random number),
   
   

… which is roughly the number we were getting before, except we're not counting satellites, computing periods, or velocities or anything.

And, also, you can now easily see that if you want 10 times this much power, one way to get there is to reduce the spacing by a factor of 10, to a mere 3.6 km/km². Which will work reasonably well if the individual satellites are 1 km² (and thus 3.6 km apart), but not so well if they're 1/5 that size (200m square ⇒ area is 0.04 km² ⇒ spacing is 3.6km⁻¹×0.04km² = 144m, which is going to lose badly),
… and even with 1km² satellites we're arguably close to the size limit for this orbit.

… which in general will be a diameter of 4/𝛑h for circular satellites, with square ones being able to get away with being 1/h on a side provided you can keep them from turning.

Getting the Actual Orbit

Note that this does not yet nail down the orbit. If we want to find out, say, how much stuff we have to actually build or how close it'll all be getting to the sun, that's more work. There's one more parameter to specify, which we choose depending on what we most care about:

• if we want to keep our satellites out of the solar corona, we probably care a lot about the perihelion (closest sun approach) distance, which we'll call r, for which the formula that matters is
  
  ε = (R − r) / (R + r)
  
  where ε is
  
  
• eccentricity, which we could just specify directly, if we have a number we like.
  
  This is the measure of how much the orbit is squunched, ε = 0 being zero squunch (circular orbit), and values approach 1 as the orbit gets narrower with satellites diving in closer and closer to the sun, the limit being ε = 1 which would normally be a parabolic escape-velocity orbit except those have infinite aphelion, so if we also specify a finite aphelion, that means we're in this degenerate case where the satellites are just being dropped from aphelion directly into the center of the sun and never heard from again.
  
  Suffice it to say, we really want ε < 1.
  
  
• semi-major axis, usually denoted a, this being the distance from the geometric center of the orbit (not where the sun is) to either perihelion or aphelion, since ellipses are symmetric that way.
  
  In case you were wondering:  R = a(1+ε) and r = a(1−ε).
  
  As it happens, there are lots of other numbers you can use as proxies for the semi-major axis (i.e., they're all conveying the same information), including
  • orbital period, T = 2𝛑a√a/GM, but only if you know the magic constant GM/4𝛑² = 2.509462183311675×10¹⁹ km³/day² (M being mass of the sun and G being the gravitation constant, but for this you don't need those numbers separately)
    
    
  • orbital energy, E = −GM/2a
    
    
  and so on.
  
  
• total satellite area = panel area A × number of satellites N.
  
  If h is how far you have to travel (along that imaginary circular orbit) to see 1 km² worth of panel, and hA is how much linear space one satellite is taking up at aphelion, then hAN is all of the space, the circumference of that imaginary circular orbit, how far you go to see all of the satellites. You can also get this number from the aphelion velocity and the orbital period.
  
  So calculating AN given ε is somewhat straightforward:
  
  hAN = vT = 2𝛑a√(1-ε)/(1+ε) = 2𝛑R√(1-ε)/(1+ε)³
  
  Calculating ε given AN is unavoidably mysterious. If I were sane, I'd just use Newton's Method, but since there actually is a formula for solving cubic polynomials (like the quadratic formula, but people tend not to know this one because it's hideous) we can do this:

  ε = s − k/3s − 1	where s = ∛k(1 + √1 + k/27)
  	and k = (2𝛑R/hAN)²

  As for where that comes from, well,… you can ask. Maybe one of these days I'll do a page on Galois theory.
  
  and AN has its own proxies, notably
  • the aphelion velocity v = hAN/T = √GM(1−ε)/R
    
    
  • the aforementioned panel current = AN/T = v/h
    
    
  and so on.

And then we do the wall of numbers to show the range of possibilities you get where production is 86,400 kg/day, satellite spacing is 36.635 km⁻¹, and aphelion is at the Mercury L2 point our chosen distance:

Radius of the sun is around 700,000 km so it's not really advisable to try for closer approach than a million km. And while I'd like to think the corona doesn't go past 3 million km, it probably will whenever the sun gets in a bad mood.

I still get amused at shows like "The Expanse" where it's imagined that people are going to be out there in space suits doing these jobs with their bare hands. Not that there can't be a role for humans. Person-In-Charge sitting in the Gods-Eye-View office in the total eclipse zone, running some Really High Level Software to monitor things. Maybe.

Personally, I think I'd feel better if actual people were kept well away from these sorts of operations.

All of the above is, of course, for just one orbit. At some point, the power needs will get beyond what one orbit can provide, and then …

… we, of course, start a second one.

which shouldn't be that big a deal at that point. Give it a slightly different orbital plane and the only places the new satellites will have any chance of running into any of the old ones in the first orbit will be at aphelion or perihelion. If we shift both of those numbers by a kilometer or two, that will suffice. We can even have them managed from the same L2 station, (unless we want multiple stations for redundancy, which we will at some point,… probably a lot sooner than we'll need that second orbit).

And then we build a third. You can probably see where this is going.

(Next: Dyson Spheres)

(as I was saying in my tech wish list,)

To put some actual numbers on this, if we have the satellites grazing the sun (1 million km out)

So, this was all based on this theory that I had that there'd be a point to putting solar power satellites in close to the sun, that having them dive through the corona for a few hours is what you want. (I've apparently been watching too much Stargate: Universe.)

Strangely enough, this turns out not to actually matter in the way that I thought it would.

More generally, you would think that the energy flux (energy per m² of solar panel) received by a satellite over the course of a single orbit or portion thereof would be really complicated. What with distance from the sun and velocity of the satellite constantly changing, there'd be this nasty integral to do, there being any number of questions about ellipses, e.g., how to calculate the perimeter, that don't have easy answers in terms of familiar functions.

But this isn't one of them.

Herewith I present the formula for total energy flux received for a satellite traversing some angular fraction F of its full orbit — doesn't matter which F, i.e., it's the same energy for any wedge having a given angle at the center, no matter whether it's pointed towards perihelion (closest approach), aphelion (farthest away) or any other direction — where we let

• W = total wattage of the sun and
• T = the orbital period

(so that WT is the total energy emitted by the sun over a full orbit). Get ready for it...

(Yeah, okay, I'm 12 today. But this is kind of how I feel about it.)

In case you were wondering A is area of the orbit for which there are any number of formulae depending on which information we have, e.g., πab if we have semimajor and semiminor axes. Or πa²√(1-ε²) or πb²/√(1-ε²), if they give us eccentricity instead. And so it goes.

Just to sanity check:  For a circular orbit radius r, area A = πr² and doing a full orbit we get (WT)/4πr², as one might expect from projecting the sun onto a sphere of radius r with the satellite always being on that sphere. But the funky thing is how this all works no matter what the shape of the orbit is.

Short Reason Why:  Angular momentum L = r²dθ/dt is conserved, so when we integrate received power over time to get energy, a whole lot of constants move outside: ∫(W/4πr²)dt = (W/4πL)∫dθ. Also L is twice the area swept out per unit time i.e., L = 2A/T. The rest is setting θ = 2πF and shuffling letters around.

To be sure, we want to be cranking the flux up as high as we can, which means making the area A as small as possible. But you can do that just as easily with a circular orbit as with that weird, highly eccentric one I was using, and then the satellites wouldn't have to deal with corona storms and other nastiness.

The only disadvantage is losing the convenient dark place behind Mercury where service station can hide from getting blasted by the sun. But perhaps if this is 10,000 years from now, our tech may be to the point where building something like that at, say, 1/3 of a Mercury orbit radius without there being a planet to shield it won't be that big a deal (after all, the satellites themselves will have to stand up to all kinds of abuse).

Also, it really won't make much difference having one big-ass satellite vs. lots of little satellites — my original reason for lots of little satellites was so that there could always be one in the corona getting charged up at any given time (on the assumption that the corona was the only place worth bothering having things charge up because that's the only place we get the huge energy flux, which is what turns out to be wrong, i.e., we actually do pretty well using the whole orbit), — …

… other than the usual Economies of Scale vs. the Putting All of Our Eggs in One Basket issues that need way more info about what the technology is actually going to be than we have at present.

(galactic empire, continued from here)

The Colonization Arena

So to recap, we've let loose these self-replicating explorer-cockroaches to visit everything that can possibly be visited, and there will be this sphere expanding at 1/100 lightspeed with us at least vaguely in the middle of it. Everywhere in the interior they're going to be building infrastructure and terraforming whatever they can.

Thus, somewhere within the sphere of Explored Stuff, we'll have the sphere of Terraformed Stuff whose boundary will lag by some distance, be it 20 light-years (i.e., if it's 2000 years before the first terraformings are ready for settlement) or 100 light-years (10,000 years) or more. For the purposes of this discussion it doesn't matter a whole lot which it is.

What matters is that, eventually, we will have new planets coming on line and at a constantly increasing rate. In the 300 years it'll take the radius of the Terraformed sphere to grow from 13 to 16 light years, the number of available planets doubles, and it doubles again in the 400 years after that. (Yes, the doubling rate will be decreasing because this is not exponential growth. It's merely cubic. I don't think anyone will be complaining. Except for the aliens. Meaning in the extremely unlikely event that we encounter them this early in the game, I will, admittedly, be very surprised if they don't have at least a few issues with this plan).

Even if the actual numbers of planets turn out to be depressingly low, say, if, instead of going from 64 to 128 to 256, what I figure is the upper end of plausible, i.e., one planet per system everywhere, we instead go from 2 to 4 to 8, that will still not be a bad outcome. Recall that the main point of this exercise is to get beyond 1. (And, yes, if instead we're going from 0 to 0 to 0, that will indeed suck.)

So let's suppose there will be worlds to settle. Now for the fun part: How do we get there? It being agreed that we need to avoid rockets, what now?

Interlude on Mass Drivers

A mass driver is a method for propelling stuff around, invented by Gerard O'Neill back in the 1970s (i.e., if we're being sufficiently specific about the definition; the basic concept for the railgun, which is really quite similar, goes all the way back to World War I, and catapults go back way further than that, but O'Neill admittedly was most likely the first to consider these things in the context of space colonization).

TL;DR:  What matters most for our purposes is that it's a gun, even if it's using electromagnets to propel the payload. Guns are nice in that they allow the payload to be arbitrarily stupid; it won't need engines, fuel or anything else. You just put it in a shell/bucket and pull the trigger.

In O'Neill's version, there's this really long track for the bucket to accelerate along. Then it reaches the end and lets go of the payload, which sails off into infinity. But also you're doing this in a vacuum using magnets that both handle the propulsion and levitate the bucket above the track so that there's no friction. The end result is that virtually all of the energy you're putting in goes towards moving the payload. This is as about as efficient as it gets.

One annoying disadvantage worth mentioning is that if, for whatever reason, you want to do more acceleration (or deceleration) of the payload later on, you will be out of luck, because the payload will not be there anymore.

I propose to solve that problem by having the track extend all the way to the destination.

Yes, you read that right. Suffice it to say, there will be issues:

• O'Neill's version is set up on the moon (or an asteroid, or Mars) because he's trying to solve a different problem: How to get crap off of the moon (or said asteroid, or Mars), which one can reasonably expect is slightly easier than gettng crap to another star system light-years away.
  
  
• O'Neill's version is on the order of a few miles long. Well, okay, the length was never really specified. It all depends on how fast you need the payload going, and, in theory, at least, you can make the track as long as you want,… until you run out of moon.
  
  My version will definitely be running out of moon.

Just to be clear about why the moon matters, I'll mention two useful moon attributes that will not be working for us in the stellar scenario:

1. Having craploads of mass. When the accelerator pushes on the bucket, conservation of momentum (Newton's 3rd law) requires the accelerator to move in the other direction. An accelerator that has the moon attached to it, is essentially not going to move, so the energy you're applying has no place to go except the bucket.
   
   
2. Being a rigid body. You may not have realized this, but rigid bodies are actually miraculous things:  If, say, I poke something with a 10 foot pole, the pole somehow transmits all of the force I apply without any losses, which really shouldn't be possible, when you think about it (and, to be sure, we lose if we rely on this too much; the pole bends/breaks/whatever).

In case you were wondering, Relativity really hates rigid bodies. (The next time anyone pulls a Relativity problem out of a textbook to try to mystify you and it assumes a rigid body, just remember, "There is no such thing as a rigid body," click your heels together, and you'll have a solution in fairly short order). Meaning even if I were to try to build some 10-light-year long monstrosity out of steel bars bolted together, it will, no matter how tight the bolts are, flap around like the 1940 Tacoma Narrows Bridge. Whack one end of it and the other end won't feel it for another decade. Actual rigidity is quite impossible in this world.

Which is why my accelerator will be lots (billions) of pieces moving independently and we'll just have to cope with that. Any of those pieces that don't have moons attached to them (which will necessarily be nearly all of them) are going to move around, and probably a lot, if we don't do something to keep that from happening, which we can deal with, but it costs us.

Note that I might possibly not care about this. If the act of sending a payload shreds the accelerator and scatters it to the four winds, that won't matter so much if I was only ever intending to use the accelerator just that once. However, (1) this does seem kind of wasteful, and (2) an honest accounting would then have the cost of sending include the cost of (re)building the accelerator, and therefore sending won't be as cheap as you might have originally thought it was.

Once we've gotten away from having it be this rigid thing, you probably won't be all that suprised to find out that I'll be using lasers and light-sails rather than electromagnets. (as the QED folks would say, it's all photons anyway…)

In which we one-up the Roman Army Corps of Engineers

So imagine a conceptual tube, however many light-years long. Let's give it a diameter of, say, 100 km. Mainly, we'll want it to be narrow enough so as to be easy to keep clean, i.e., free of large rocks that will ruin the day of any payload trying to be passing along it at near lightspeed, but wide enough to accommodate at least two lanes of traffic, punting for now on the question of how wide "lanes" actually need to be — which I suspect is not going to have much to do with the actual ship/reflector widths, which will be way smaller than the corridor.

The "walls" of the tube will be streams of laser ships all travelling at some low velocity like the 0.018c we were using for the explorer ships — probably six streams in all, 3 going each direction angularly spaced 120° apart for the sake of having the best control over the payload — individual ships in a stream spaced close enough, let's say 600,000 km, that a payload travelling through the tube will always be in range of one of them. Each ship carries enough energy/antimatter to service its share of tube traffic over the course of its own voyage — 1 to 20 kg dribbled out over the course of 600 years in the case of the Tau Ceti tube.

Among other things, this means any changes to the traffic capacity/configuration of tube will need to be arranged no less than 300 years in advance.

The payload ships are really simple: a corner reflector out in front, pulls a bungee cord attached to the rest of the ship, which will just be the payload surrounded with big-ass sphere of (lightweight!) shielding.

Aaaand…, that's all. No engine. No reaction-mass. No windows. No fuel beyond what's needed to keep the lights on and the occupants alive. Well, okay, I suppose we could put in a small engine for those odd, unexpected emergency maneuvers, but every last bit of non-payload extra mass is going to cost us.

(I suppose the bungee is a bit of a splurge, since we could have just mounted the reflector right on the payload, but it should be possible to make the bungee really light and also the passengers will thank us (1) for converting the probably jerky blasts that hit the reflector into a smooth ride — hmm, I'm guessing there's an interesting Control Theory problem there (i.e., we may need a "smart" bungee) — and (2) for not having the lasers aimed directly at them personally — yes, there'll be shielding but the less we stress it, the better, since there's already a whole lot of other crap in interstellar space they'll need protection from)

(Hmm. Let's hope the bungee doesn't break.)

For that matter the laser ships shouldn't be all that complicated either:  laser + mirrors + antimatter cell + camera/radar + software. Done.

As soon as the payload ship passes a laser ship, the latter begins firing at the reflector and keeps firing until the payload passes the next ship, with the magnitudes of all of the various bursts carefully calculated so that payload does what it's supposed to.

Every time the laser ship fires at a payload, it also sends out an equal burst in the opposite direction so that its own course and speed don't change. So this will be at least double the energy cost of a moon-based accelerator. One might suppose that 2×(best we can do) is still pretty good, but there's still one more issue:

• O'Neill's version is not attempting to boost anything anywhere near the speed of light…

… the problem being that once the the payload ship gets going fast enough, when various laser blasts catch up, they will have been significantly red-shifted, meaning they will need to have been sent with correspondingly more energy to provide the kick needed. That plus the aforementioned doubling makes the overall cost equivalent to what it would be if the payload ship were self propelled, i.e., we're back to rocket economics. So far, so bad.

But then we get to the midway point, the payload ship flips around. From then on it gets fired at by the laser ships it's approaching rather than the ones behind it.

Which means all of the shots from then on are getting blue-shifted, i.e., amplified by the same ridiculous factor that we were losing in the acceleration phase. Deceleration thus turns out to be incredibly cheap. Which is how we win.

Comparatively cheap, anyway. I suspect there will not be very many people living on AlphaC doing a regular commute to a job on Earth. The vast majority of interstellar commerce will take the form of information flows transmitted relatively cheaply at lightspeed. But now, we at least have a story for what happens when there are actual people and perishable goods that need to get places and not be taking centuries to do it.

(next: counting the beans)