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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)
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Each stream -- recall there are 6 streams, 3 in each direction -- has laser-ships spaced 600,000 km apart and they're going 0.018c (5400 km/s).
So, roughly every 2 minutes, each side has to add 3 new fueled-up laser-ships (which then rocket themselves up to speed and assume their positions in the tube -- this cost is included in the overall mass cost for moving payloads)
and during that same window 3 depleted ships arrive from the other side having coasted all the way from the midpoint (I suppose they could retain a little bit of energy and then do things like scanning for incoming rocks or whatever; they'll all have negligible mass at this point and so when they get to the end they can be easily collected or maybe they use their last bit of fuel to slow down and drop themselves at the charging station).
So yes, we're constantly adding fuel to the tube.