Space 16: Building the Tubes / In Your Faces, Romans

(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)

Space 17: Running the Numbers

(galactic empire continued from here)

Some Vaguely Actual Costs

Here, have a wall of numbers:

Shipboard Time (aY)		Distance* for acc+dec (aLY)	System Time (aY)		Mass Cost Ratio
acc+dec	per aLY* of coasting		acc+dec	per aLY* of coasting	acc+dec	acc	dec	rocket
2.5	5/8	1.78	3.20	7/6	3.20	2.49	0.71	11
3.0	8/17	2.70	4.26	11/10	4.26	3.48	0.78	19
3.5	5/14	3.93	5.58	17/16	5.58	4.75	0.83	32
4.0	3/11	5.52	7.25	28/27	7.25	6.39	0.86	54
4.5	3/14	7.59	9.38	46/45	9.38	8.49	0.89	89
5.0	1/6	10.26	12.10	75/74	12.10	11.18	0.92	147
5.5	1/8	13.71	15.58	123/122	15.58	14.64	0.94	244
6.0	1/10	18.14	20.04	202/201	20.04	19.09	0.95	402
2φ	1  sinh φ	2(cosh φ − 1)	2sinh φ	1  tanh φ	2sinh φ	eφ − 1	1 − e−φ	e2φ − 1

*System distances, that is, as measured in the (star) system frame of reference, meaning this is how the planetary/Earth/Tau Ceti/etc folks see them (i.e., and not how the Shipboard/payload folks or any of the laser ships will be measuring).

Assuming we have a place to get to, e.g., Tau Ceti, that's a particular distance away:  11 light-years = 11.34 aLY (see big footnote below about distance and time units). Then we can:

• Pick a row, but for Tau Ceti, we have to ignore the 5.5 and 6.0 rows, because those require too much distance (e.g., the 5.5 row requires 13.71 aLY for acceleration and deceleration which is more than the 11.34  aLY that we have).
• The way to read a row: If we use the 5.0 row, that means we're under power for 5 years (subjective/shiptime) during which time we're going 10.26 aLY, so for the remaining 1.08 aLY we do that much coasting in the middle; ×1/6 means 2 more months of shiptime, coasting at 74/75 lightspeed. And for this we're burning 12.1 kg for every kg of payload we want to send.
• The other rows are cheaper but take longer. The top row takes the longest (8 years, 6 months of shiptime) but for roughly a quarter the price.
• The times in the star system frame (we can calculate these from the 4th and 5th columns) will all be pretty much the same, as it happens.

If each star system is devoting, say, half of its energy supply to interstellar transit (and not all of it because whatever else we're doing in the system that isn't transit that needs power, we'll need to be keeping that going), then, once we get to the point where Tau Ceti has a new planet ready for settlement, and let's suppose this is the First New Planet, so that both ends of that tube will have their entire transit budget available for shipping people from Earth to Tau Ceti to get that new colony off the ground, meaning we'll have a full 86,400 kg/day to spend.

If we like that third row's travel time (3.5 years shiptime under power, plus another 2 years, 8 months of coasting), then the cost of sending a kg is 5.65 kg (yes, I know the table says 5.58, but see other big footnote below about "unbalanced" tubes), which multiplies out to around 15,300 kg we can send every day, or roughly 69,000 people/year.

Keep that up for a century and you've planted a colony of nearly 7 million people — plus whatever we get from a century's worth of sex.

Later colonies will have somewhat less throughput because Earth will now have existing transit tubes to the other places that it will need to maintain and so will have fewer kg to spend on the new ones. But we'll always get at least half that number out the door because the uninhabited destinations will be able to spend their entire transit budget on shipping people from Earth.

Note, by the way, that this is not a way to relieve population pressure on Earth, since we're not going to get here until thousands of years after the point where we've stabilized our population (however we manage to do it), which will most likely stay in the billions.

Eventually, we settle into Commerce Mode, with numerous colonies to talk to. Imagining our various settled systems to be arranged in a vaguely face-centered-cubic lattice so that Earth and everybody else will have around 12 outgoing tubes to the nearest neighbors, then Earth's 43,200 kg/day transit budget allows 3600 kg/day for each tube, which lets us send 645 kg (8 people) per day to Tau Ceti (and now that really is 3600/5.58 because we have balanced traffic).

Granted, economies of scale will probably dictate that these be grouped into monthly shuttles of a few hundred people, or maybe a yearly cruise with a couple thousand. In any event, this will essentially be the First Class cabin on the Concorde and will be priced accordingly.

The rest of this is various big footnotes:

---

On Tubes vs. Rockets

The last column is there to show just how much we're winning over rockets; … unless someone actually wants to visit a system not yet connected by a tube, in which case, that's what they'll need. And, even then, they'll probably want to do some kind of hybrid approach where they're using a rocket but with this trail of Tube-Building Stuff in their wake so that they'll have an easier time returning home.

Except that since most tube construction will effectively be "financed" by the destination systems, the cost of this tube will be borne entirely by the origin system and thus cut into their resources something fierce. So I still can't see anyone wanting to do this without a really compelling reason (i.e., why they can't just send a cockroach and wait the however many centuries for the tube constuction to be completed from the other end.)

One may, however, reasonably ask, since the costs of acceleration to the midpoint are essentially the same as rocketing there, why we don't dispense with the first half of the tube and only build the decelerator half. Various answers:

• If there's to be traffic in two directions, you're going to need both halves anyway.
• If traffic is unidirectional, and we don't have the acceleration part of the tube, then we lose the opportunity for the destination to contribute energy (since laser ships from there are the only reasonable way this happens), which matters for the colonization scenario.
• We win from having a "road", i.e., a string of ships posted in front of us making sure there's nothing substantive in our way, or, in the unlikely event that Something Big shows up that's hard to move, to route us around it.
• It could also be — read: I'm doubtful about this but it's worth mentioning — that regular traversal of the tube by near-lightspeed ships will generate a "mini-solar-wind" of sorts (via their wakes in the interstellar medium) that will help keep the vicinity clearer of junk than you might otherwise expect. Or we could have the laser ships, during downtimes when there's nothing passing through, periodically firing low-power unfocused bursts outward to generate such a wind (no, I haven't yet done any math on this one).

About time and distance units: aYs and aLYs

So the table actually uses wacky time and distance units.

• 1 aY is an "acceleration year" which is roughly 31/32 of a solar year, about 11 days shorter.
• The unit of distance, the "acceleration light-year" (aLY), is correspondingly about 66 Neptune orbits short of an actual light-year.

Since we've changed both the distance and time units, the speed of light remains at 1 (aLY per aY).

The reason to do this is to have g = 1 (in aLY/aY², aLY⁻¹, aY⁻¹, whatever), which simplifies the math all over the place (last row), the same reason as why navy folks prefer nautical miles to actual miles or kilometers. Or why astronomers prefer parsecs to light-years.

In particular, it's not a coincidence that the system time numbers are exactly the same as the mass cost ratios. Likewise, maximum time dilation (coshφ), in this world, is exactly the half the aLY distance (3rd column) plus 1.

And really, once we get out there, it's hard not to imagine all of the colony worlds using the aY as their common "year", since all of the schedules for everything they care about that's coming from outside their respective systems will be based on it, and their own various orbital years will all be different/useless anyway. Also, it'll be baked into the various tube designs much the way our current railroad gauges derive from decisions the Romans made 2000 years ago (yes, I know, partially myth), because the cockroaches will just be out there continuing to build stuff — changing that software will be very hard — so the star charts of human-explored space will just be scaled in aLY, with parsecs and solar-based (light-)years becoming these weird historical artifacts that'll appear on science contest exams and nowhere else. (Naturally, Earth itself will resist changing over for a very long time, perhaps for even longer than the USA will hold onto miles, feet, and inches.)

Also, if we ever develop transhumans — I might want to slap a Not Happening tag on this, too, but that's a longer discussion — who are happy accelerating at a different rate, say 2g instead of 1g, then we can just halve our distance and time units and keep using the same chart. Or, if we're instead accelerating at 31/32g because we decided to be nicer to the old people, then we can read the chart as actual years and light-years.

More math, if you care:  For any given rate of acceleration (like 1g), the corresponding acceleration year is the amount of proper time it takes to increase (decrease) one's velocity angle by 1 by accelerating (decelerating) at that rate.

(Review: "velocity angles" are to velocities as angles are to slopes; you lose if you try to add or subtract slopes, but adding and subtracting angles works just fine. So, e.g., if you're driving down the road with velocity-angle = 2 (96% lightspeed) and you throw a baseball out in front of you with velocity-angle = 3 (99.5% lightspeed), it'll have velocity-angle = 5 (99.991% lightspeed) with respect to the road, which never causes problems because velocity angles can go up to infinity (∞ = actual lightspeed), unlike the velocities themselves which are capped at lightspeed = 1 (or c if you use stupid distance and time units). The conversion is:  (c ×)tanh(velocity angle) = velocity).

In the formulas on the bottom row of the table above, φ is the maximum velocity angle achieved during the trip (= the velocity we're coasting at once we get to the middle, i.e., if we're doing any of that before turning the ship around and decelerating).

About "unbalanced" tubes

All of the costs in the chart above assume that all propulsion is being done by outbound laser ships, i.e., acceleration is being done by ships coming from the source, and decelerating by ships from the destination, and that all ships are firing forward. We schedule payloads, fuel the lasers, and plan the firings so that all laser ships are depleted by the time they reach the midpoint.

In this case, the "tube velocity" (velocity of the laser ships) has no effect, because any effort you put into accelerating the laser ships reduces the redshift (or increases the blueshift) of the beams being fired, and it's a wash.

This changes if any ships are retaining antimatter/fuel past the midpoint, because anything they do from then on means they're firing backwards and there'll be this extra redshift factor due to the tube velocity that gets applied twice. This can be reduced by reducing the tube velocity, but then laser ships will need higher capacity, need to last longer, scheduling gets more rigid, and life gets more annoying.

This is why the colonization scenario costs a bit more than expected (the 5.65 vs 5.58 question). With the tube being used unidirectionally, and because the energy needed for decelerating is so much less, there will be all of this leftover antimatter from the destination side that you'd want to use, but the only place to use it will be accelerating and decelerating more stuff from Earth, and it's the acceleration part of that getting hit with this extra redshift.

(next: leaving the galaxy)