wrog

(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?

We'll use whatever onscreen cues we can. Somewhere in season 4 we get a shot of the inside of a ship reactor, where they are clearly attempting to depict Livermore-style inertial-confinement fusion (or, rather, depicting how it fails when you have aliens who can rewrite physics). Also at no point does anyone mention antimatter, so…

Working fusion reactor. Check.

The engine nozzles glow and rumble. They could be ejecting just about anything, really; we don't know. Just the mysterious Epstein Drive doing its thing. If they really had mastered Big Ass Laser, then the emanations would be almost completely invisible, i.e., unless it were pointed directly at you, in which case you'd be dead, and this would also be their primary weapon, which we never see them using this way. Ion drive is basically a huge particle accelerator; that shit would also be mostly invisible, too,… which narrows down the actual propulsion method to,… no idea, really.

So we'll be generous and give them something close to Best Possible Fusion (recall $𝓅=\mathrm{<b>266</b>}$), but to round the numbers up, we'll throw some ~85% efficient unobtainium-catalyzed whatever propulsion into the drive-train so as to have an overall $𝓅=\mathrm{<b>299.79</b>}$ ($=c/\text{1000 km/s}$), and see what happens. They're also pretty clear that everybody uses the same drive tech (which makes sense, given the huge wall that we see trying to do better than fusion and not having antimatter).

The next thing we notice is that none of the Rocinante, Razorback, or any of the Earth/MCRN battleships are depicted as giant fuel tanks. Not only is the fuel they're carrying not taking up most of the ship, it's pretty well hidden. Which puts a significant limit on their fuel/payload ratio. Note that military ships that need to get places quickly, do exotic maneuvers, and will most often be headed to destinations that are hostile and that can't be expected to refuel them, will need this ratio as high as possible, never mind that higher your fuel/payload ratio is, the higher the probability that someone shooting at you will hit a fuel tank … which won't necessarily be Immediate Death like it would be venting into an oxygen atmosphere, but is still likely to be Bad News. Meaning being 90% fuel is probably a bad idea unless you're small and expendable. But there will also be a floor since you need to be able to keep up with the rest of your fleet, which means the minimum ratio will have to be the same no matter how big the ships are. It'll be an inherent feature of ship design.

The economics get interesting because the most economic way to use fuel is to always be arriving at your destination empty. Commercial ships that run a regular route, whose owners can be part of fuel cooperatives that maintain accounts and fuel caches at each end, will be at an advantage in always being able to run as empty as possible, i.e., not wasting fuel on accelerating other fuel that ends up not getting used. The Belt is depicted as an anarchic sort of place, but these kinds of cooperatives/banks will be their life blood. Contrast this with explorers, investigators, and other kinds of "lone wolf" ships that will always need to be reserving half of their Δv for the trip home. And this gets extra special fun for ships that don't have a "home". Meaning the Roci is probably an extreme case w.r.t. how much fuel they need to carry at any given time.

Fuel Capacity  (% of ship)	𝓅ΔV	ΔV for $𝓅=\mathrm{<b>299.79</b>}$
25.9%	0.3	300 km/s
33.0%	0.4	400 km/s
39.3%	0.5	500 km/s
45.1%	0.6	600 km/s
52.8%	0.75	750 km/s
63.2%	1.0	1000 km/s
71.3%	1.25	1250 km/s

We need to pick a number, so, mainly for convenience, I'm just going to go with 39.3% ($=1-\sqrt{1/e}$) — meaning 39.3% of a fully fueled ship mass is fuel. If you don't like that number, over to the right I have provided a table with some other vaguely plausible numbers you can try; go wild.

So if the Rocinante is the size of a car ferry, 1000t (metric tons), that means we're allowing it up to 393t of fuel. Unfortunately, typical fusion reactor fuel like DT is annoyingly non-dense, Liquid hydrogen is around 0.07g/cm³=t/m³, but pure DT will have 2½ times the molecular weight, so we bump that to 0.175t/m³, so 393t of that fits into a cube slightly larger than 13m on a side. If the ship is 40-50m long, this can work, and, clearly, if the fuel tank gets too much bigger it will definitely be getting in their way.

(hmmmm; …really not sure how they survive all of those railgun hits, but never mind that for now).

The Razorback is more like a Winnebago, 10t, with 3.9t of fuel fitting into a 3m cube, similar in size to a crew compartment that's barely enough for two people, so that sort of works, too. Except for one issue that I will get back to.

So, $e^{-𝓅\mathrm{\Delta v}}=1-0.393$ or $𝓅\mathrm{\Delta v}=0.5$ and using up all of the fuel produces $\mathrm{\Delta v}=0.5c/𝓅=\text{500 km/s}$. It doesn't even matter what the size of the ship is; as long as everybody is using the same tech (which for warfleets will be the best available) and is willing to devote the same percentage to fuel, this is what we get. Not too shabby.

Which is definitely not enough for any of the 1g-all-the-way flight plans (except LEO-Moon), which means all flights will be of the form: Accelerate to use up half of the ΔV you have available (getting to 250 km/s at 1g will take about 7 hours and cover 10 light-seconds of distance), coast, then use up the rest decelerating, with some minor adjustments due to the various planets/asteroids having non-zero relative velocities (which will be under 30 km/s for any two objects outside of Mars' orbit, so maybe one of the legs of the trip gets 10% longer; this is noise).

So, if Mars happens to be 1020s away, you can get there in 2 weeks (this breaks down into 20s (distance) of acceleration and deceleration that takes 14 hours, and 1000s of coasting that takes 1000s×(4𝓅 ≈ 1200))…

… unless they won't have fuel for you (which they won't if you're the Rocinante, i.e., a ship that they think you stole from them), in which case you'll only dare get up to 125 km/s, meaning the coasting part of the trip, which is nearly all of it, will take twice as long, and now you have to allow a month.

But even 125 km/s is still way more than solar escape velocity so no worries about solar gravity and no real limits on where you can go, aside from small matters like having enough time, food, and air to get there.

You can also see that, in this world, accelerating more than 1g for the purposes of getting places faster is completely and utterly pointless. If, say, you accelerate at 2g, that just means you are using up your fuel twice as fast to cut that 14 hour accel+decel period down to 7 hours,… out of a 2 week trip.

I can still see them using short high-g bursts to do the Fighter Pilot Thing where they're dodging missiles in close-quarter combat, if that actually works (I have my doubts), but that's another matter entirely.

No matter what you do, you only get that 500 km/s. The only way the Rocinante gets there faster is by strapping on an external fuel tank — e.g., in order to have 1000 km/s available, we need another 649t of DT, a cube at least 15.7m wide or maybe 1/3 the length of the ship — and we never see them doing this.

Because fuel costs are always proportional to the size of the ship, everybody will be in the same boat. If we take the Donnager to be the size of an aircraft-carrier (100,000t, maybe 300m in length, your basic football stadium; it actually looks larger in comparison with the Roci but we'll be generous), that's an extra 64,900t or a 72m cube filled with fuel (fits just barely between the two 20-yard lines of the football field).

… and this is the problem with the Razorback: If it really is breaking speed records all over the place, it should be much larger, and mostly fuel tank, not mostly engine. Or maybe it has multiple stages — that's what I'd be doing if I wanted to totally optimize for speed — but in that case, those extra stages would be part of the package and we'd be seeing them.

for that matter I'm also not clear what the challenge is in breaking speed records out in space. The winner will be whoever can buy the most fuel and pay for the support teams to recover the multiple stages (this would be ridiculously expensive). Hm, I guess that does work for a Rich People Sport.

This brings up some logistics questions…

Space War is going to be Impressively Stupid

… when all of the Earth/MCRN battleships are rushing to get out to Ganymede. I mean maybe they have this huge fleet of tanker ships that travels with them, but then those folks — or at least the portion of them carrying the fuel they need to get home again — all have to go hide somewhere for the battle in order for the ones doing the battle to be able to do all of the high-g shit. And then you get fun surprise situations where they find and blow up each others' fuel caches and tanker ships, because of course you'd do that if you could. But now I'm thinking about situations where a fleet has won the battle but has lost its getting-home fuel and will run out of food+oxygen before anyone can arrive to resupply them. Or maybe we get the Really Fun situation where there's only enough oxygen left for two-thirds of the crew and so they have to arbitrarily kill every 3rd enlisted person in order for anyone to survive.

Also there need to be a lot more ships getting trashed and then joining Voyagers 1 & 2 on random solar escape trajectories never to be found again because they're going so damn fast.

In general, I expect Space War to be even more stupid than Earth War. At least, when you're retreating from Moscow in the dead of winter, you still have oxygen to breathe, you're not getting whacked by radiation, and you still have some vague hope of finding food and water while you're busy freezing to death.

In contrast, space has so many immediate Game Over scenarios it's not even funny. Realistically, I would expect a much higher percentage of situations where they fight a battle and Space Just Kills Everybody.

That all said, the travel times sort of work for the first two seasons where they're mainly tooling around the inner solar system, never trying to go past Jupiter and you never have more than a couple months travel if you can count on getting fuel at the far end. Saturn's moon Phoebe is mentioned and a ship comes from there, but, cleverly, that all happens before the start of the series.

It gets sketchier for the later seasons, i.e., once The Ring is created, and then everybody has to go out there to fight over it. It is made clear that ProtoMolecule can Arbitrarily Violate Physics; that's not the problem — they hang a lampshade and move on; I can, too.

The problem is what the Earth ships are now able to do. The Ring is way the fuck out there and there's no refueling when you arrive — at least not at first (that supply line is going to take years to set up) — so we're definitely on the Exploration side of the chart and Serious Extra Tank territory for those scout ships that they've sent out in advance.

Now when I wrote the first draft of this, I was sure they had completely blown it seeing as getting the fleet out past Neptune is multiple years no matter how generous we try to be. Doing it in a few months then entails somebody working out the antimatter drive, which would have been Really Big News, or the writers finally deciding they were in a straightjacket, punting, and going ahead on shrinking the solar system.

But when I went back to re-watch that first post-Ring epside, the caption is "Beyond Uranus", not "Beyond Neptune" as I had remembered. The timestamp is roughly six months after the Ring was created and the various ships are all still in transit, some substantial but non-specific portion of the way there. And my chart does allow it taking 250 days to get out to Uranus. Which says to me, somebody did actually give this some thought. I am suitably impressed.

I can even imagine how the TV writers' meeting must have gone: This episode comes up, and some nerd at the back of the room, who is basically me, and who happens to be awake at the right moment, raises their hand, "Uh, guys? You need to have it be 'Beyond Uranus', not 'Beyond Neptune'." Head scriptwriter then rolls his eyes, "Oh c'mon. Seriously?" "Yeah, I did the numbers. Trust me on this."

However,…

The Protomolecule Aliens are Complete Dicks

In particular, what the hell were they thinking in giving The Ring a Speed Limit? Given the economics of rocket travel, this is pretty much the worst possible thing you could do.

Because everybody who wants to travel via The Ring is probably already doing serious Extra Tanks to not have to spend upwards of a year getting there and then, presumably, the same amount of time on the other side to get to Joe Colony World, notably without having to expend extra fuel if they can just coast straight through the Ring with only the most minor of course changes — something the Aliens could easily have arranged given that they are apparently able to bend spacetime like a balloon animal. Requiring everyone to slow down and then speed up after going through essentially squares the amount of fuel you need to get anywhere.

It's especially fun for the explorer ships that are getting there before any fuel production infrastructure is in place at the colony end. Not being able to refuel means they get hit with this cost again on the way back, so their fuel allocation needs to be a fourth power. E.g., even if one is happy with it taking 250 days to get to the ring and therefore, say, 500 days to get to a colony, the Ring speed limit means that instead of arriving at the colony with enough fuel to get home, we're instead arriving with an empty tank and, e.g., the Roci would have needed to load up at the start with an additional 649 tons of fuel to both get home and pay the Ring tax on the way back. Also, without the speed limit that additional 649 tons would have allowed them to coast twice as fast and thus get to the colony in 250 days, whereas to achieve that with the speed limit requires an additional 3875 tons of extra fuel, meaning the Roci is now hauling around an external fuel tank roughly the same size as the Roci.

Either that or we assume someone is able to get a supply line set up to support a refueling station at the Ring. That would, in fact, be a rather obvious use for the Behemoth:  just fill that space-colony-sized interior completely with DT and drive it out there. Now you have something real for people to fight over.

Although, honestly, if you think about the economics of this, the only way Earth, Mars, and the various Belt factions can be fielding viable navies and fighting wars is if each of them controls their own fuel supply. Meaning there is evidently DT production happening in sufficiently many places that no one can monopolize it (*) and each faction then has enough to get by. Anybody who doesn't will, once the shooting starts, run out quickly and all of their ships will be dead in the water.

(*) I have my doubts about this, but, since "DT" is my stand-in for whatever formula the Epstein Drive uses and they never actually establish for certain what that is, there's wiggle room. Thus, all I'm requiring is that, whatever fuel they're using, it, unlike the Alien stuff, obeys conservation of momentum/energy. If they really are intended to be using deuterium and tritium, there are other issues that come up (e.g., tritium has to be made in reactors and it has a relatively short half-life (12 years) so caches won't last that long; deuterium is stable but rare and rather a lot of work to sift out from regular hydrogen…)

Which then means that the very first question anybody will be asking once the Ring appears — and they'll be asking this even before they know about the Speed Limit — will be, "How do we get our fuel out there?" Ideally everybody, being all distrustful and paranoid, would just arrange their own supply lines. This will, of course, be very expensive, so I can see at least the possibility of alliances forming to combine resources and share fuel. But if so, this is the kind of agreement that's going to be worked out long and hard in advance with the possibilities for double-cross being thoroughly gamed out and planned for so that everyone involved knows exactly what the consequences will be. It's not going to be some random detail where someone's going, "Oh, shit, where are we getting our fuel?" after the ships are already out there (hint:  As a matter of general principle, getting people to sign up for suicide missions is going to be Hard, and even amongst hardened military types there will come a point where officers get mysteriously shot in the back).

And once the Speed Limit is known about, then whoever controls the fuel caches is king. Nor will it be in any way difficult to implement a blockade of the colony worlds. If you're not Approved To Get Fuel you won't get fuel. Enjoy your multi-decade voyage; our compliments! The only people "sneaking through" will be the very rich who can afford to buy all of their fuel in advance and bypass all of the bullshit; the impoverished Belters will not be in that category.

Also note:  I'm not saying the Blue Special Effect Protomolecule Aliens can't or shouldn't have a speed limit. They can do whatever the hell they want for whatever Inscrutable Alien Reasons. All I'm saying is they can't not know how stupidly inconvenient this is.

Which is why it's more likely to me that they are indeed complete dicks and it's thus not actually that hard to imagine how the Red&Black Special Effect Aliens got so motivated to wipe them out.

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:

$$v_{\text{exhaust}}\text{d}m=-m\text{d}v$$

However, we're going to be weird and (1) express velocities as fractions of the speed of light, and then (2) replace $v_{\text{exhaust}}$ with a specific mass consumption, $1/𝓅$, which you can think of as a kind of "propulsion efficiency factor" (note to Actual Rocketry People: this is basically thust-specific fuel consumption but with different units because I'm weird), giving us

$$\text{d}m/𝓅=-m\text{d}v$$

the idea being that we're instead taking some small portion of our ship, somehow converting $1/𝓅$^th of that entirely into momentum and then throwing the rest overboard as dead weight that we can no longer use for anything, doing it this way so that we can cover the general case where there's some more complicated reaction going on and the momentum we're getting out of it is less than optimal for whatever reason.

But if you want to think of $1/𝓅$ as an "effective exhaust velocity" (where 1, the speed of light, is the best we can do), go right ahead.

This solves pretty easily (thanks, Tsiolkovsky) to get $$m_{\text{after}}=m_{\text{before}}{e}^{-𝓅\mathrm{\Delta v}}$$ or a mass cost per unit payload of $e^{𝓅\mathrm{\Delta v}}-1$ where Δv is the total velocity change we want to achieve. (If we want to be totally correct, Δv is actually the change to the ship's velocity angle rather than its velocity, but in Non-Relativistic Land this makes pretty much no difference).

Quick aside on how $e^x$ behaves, in case you've forgotten: If $x$ is really small ($x\ll 1$), then $e^x\approx 1+x$, in which case the mass cost ($e^{𝓅\mathrm{\Delta v}}-1$) is just $𝓅\mathrm{\Delta v}$. So, if you have the small propane tank off in the corner supplying your maneuvering thruster, it's enough for one of your maneuvers, and you now need to do ten (10) such manuevers, then you need (ever-so-slightly more than) 10 such tanks and you're good to go. No big deal; it's all linear, right?

This changes when you get to the realm where $𝓅\mathrm{\Delta v}$ is nearing 1 and the compounding starts kicking in. And once it's greater than 1, look out! ($e^2-1\approx 6$, $e^3-1\approx 19$, $e^4-1\approx 54$, …) This is where you start seeing spacecraft that are gigantic fuel/propellant tanks with the little teeny payload capsule in front.

… at which point we are obliged to ask: how big does $𝓅$ actually get?

Part 1: Our power supplies suck

• $𝓅=\mathrm{<b>1</b>}$ is Not Happening, but you knew this already.
  
  
• For actual antimatter fuel cells and/or antimatter rockets, there is an argument (*) that we will probably never be able to recover more than 16% of the energy from a proton-antiproton collision, that the rest disappears far too quickly as useless gamma rays and neutrinos. Which means we are looking at a lower bound of $𝓅=\mathrm{<b>7</b>}$ for this.
  
  And even that is still assuming we've solved the storage problem and have either worked out the 100% efficient Big Ass Laser or otherwise some kind of matter-antimatter combustion chamber that can produce equivalent thrust (details!). But still, not bad, right?
  
  
• Next up is the Best Possible nuclear fusion reactor, which we put in place of the antimatter cell. Looking at all of the likely processes (barring discovery of new ones), deuterium-tritium (DT) seems to be everyone's favorite, given that it has the most energy released per unit mass: $$\text{²H}+\text{³H}⟶\text{⁴He}+\text{¹n}+\text{17.6MeV}$$ meaning 5 nucleons (5×938 MeV) come together and 17.6MeV of that, roughly ¹/₂₆₆ of the total, comes out as energy. Meaning if we want to get 1kg of energy out, we need to start with 266 kg of DT in the correct proportions. Doing that and feeding it all to the Big Ass Laser with zero loss, gets us (surprise!) $𝓅=\mathrm{<b>266</b>}$.
  
  
• Moving on to what we can vaguely actually do, we presently have all manner of working nuclear fission reactors. Plugging in the corresponding energy conversion rate for uranium (²³⁵U), we get something like 1/1000. Hooking this up to the Big-Ass Laser then gets us (minimum) $𝓅=\mathrm{<b>1000</b>}$.
  … with some kind of horror show of radioactive strontium and barium streaming behind the ship.

And now we start seeing another problem:

Part 2: Our propulsion methods suck

… i.e., we are never going to have Big Ass Laser. Which is not to say that lasers won't get better, but currently, at least, we're not even close to what we'd need for the laser to be doing the propulsion itself, and even when/if we do, there is going to be another factor to multiply in here. Now what?

• Probably the best ideas we currently have for rocket propulsion that look to be near-future realizable would be the various ion drive proposals. The most ambitious one I'm finding is DS4G which projects an exhaust velocity of around 200 km/s, which gives us $𝓅=\mathrm{<b>1500</b>}$, mostly.
  
  Yes, external power is required but this turns out to be noise compared to the reaction mass cost. (Essentially, attaining 1 kg of momentum at this velocity requires 1500 kg of reaction mass getting spewed at ¹/₁₅₀₀c, due to the reactor supplying ¹/₃₀₀₀ kg in kinetic energy, which would actually be, ¹/₁₂ kg of DT for a fusion reactor, or ⅓ kg of ²³⁵U for a fission reactor, meaning we're really looking at $𝓅=1500\text{¹/₁₂}$ or $𝓅=1500\text{¹/₃}$, respectively, the point here being that once we are throwing enough actual matter (i.e., stuff with non-zero rest mass), that we could be converting to (insane) energy but aren't, the extra kinetic energy we're having to add is down in the noise compared to that, so the inefficiency of the latter process doesn't matter so much.
  
  The big problem with ion drive at present is the large fixed cost (needs a reactor and each engine has a limited maximum thrust) which then gets accounted for as part of the payload. When then means you have ships that are mostly reactor and engine. This probably isn't what Epstein was able to fix, since we're already looking at the After picture here and we're still down an order of magnitude from what fusion can do.
  
  
• Another class of approaches are the nuclear thermal designs, where you use a reactor to heat up hydrogen gas to the point where it dissociates into atomic hydrogen (around 3000K) and beyond as high as you can stand before everything starts melting — or maybe you do let everything melt/vaporize and go with a liquid-core or gas-core reactor instead of solid-core — and then spew extra-fast hydrogen (and whatever else) out the back.
  
  See, the lighter the molecular weight of whatever you're spewing, the more velocity you get for a given temperature, which turns out to be a big win over chemical rockets that are limited to putting out reaction products like water or carbon-dioxide, which, respectively are 18 or 44 times as massive as hydrogen atoms.
  • Probably the most insane design considered to be in this category even though it doesn't quite fit the profile is the nuclear saltwater rocket, where we just mix the uranium in with the reaction mass, in sufficient concentration and with precise timing so that it's going critical right after it's pushed out the door, but doesn't quite chain-react because it's already out in space and dispersing — so that you're being propelled by this continuously exploding semi-bomb that you're somehow managing to keep control of. Um. Yeah. Supposedly, that gets us 60 km/s ($𝓅=\mathrm{<b>5000</b>}$).
    
    
  • Or there's the open gas-core reactor, where you're maintaining this weird bubble of uranium gas that you somehow managing to keep from getting blown out the back of your ship while it heats up the hydrogen around it to n0000K, apparently allows exhaust velocities up to 50 km/s, or $𝓅=\mathrm{<b>6000</b>}$.
    
    
  • For more mundane — and perhaps safer — solid-core approaches, we have the NERVA program's NRX A6, in which we finally get to the first thing on our list that has actually been built and tested and managed (i.e., actual result) a not unreasonable 8.5 km/s ($𝓅=\mathrm{<b>35000</b>}$), which was pretty good for 1967, and still massively outdoes everything in the next category.
    
    
• Finally, we have the chemical rockets,
  • the best known in theory being this odd mixture of lithium, fluorine, and hydrogen that gets 5.32 km/s ($𝓅=\mathrm{<b>56000</b>}$) but is, unfortunately, rather insane to manage from a chemical point of view, fluorine being ridiculously corrosive.
    
    
  • Which leaves us at the actual state of the art, good ol' liquid hydrogen + oxygen, e.g., the RS-25 developed for the Space Shuttle orbiter and still being used for the SLS, where the exhaust velocity is around 4.5 km/s ($𝓅=\mathrm{<b>67000</b>}$).
    
    
  • … the two problems being that, first of all, hydrogen is annoyingly non-dense, which is why lots of folks are interested in liquid methane + oxygen, which sacrifices some efficiency ($𝓅=\mathrm{<b>87000</b>}$), but at least gets the tank sizes down
    
    
  • … the second problem being that the cryogenic storage is generally a huge PITA — fun, complicated, refrigeration machinery — which is why solid rockets that only get up to 3 km/s but are much cheaper and simpler to operate — there's just an ON switch — are still preferred in some contexts (e.g., Earth launches) and why, for a lot of interplanetary probes and other longer-term applications, the propellants of choice are actually the hypergolic ones that stay liquid at room temperature, e.g., the variant of hydrazine that was used in the Apollo Service and Lunar module engines, which likewise gets 3 km/s or ($𝓅=\mathrm{<b>98000</b>}$).

You'll notice the 𝓅 numbers for what we can actually do are Rather Large. Let's look at what this does to the mass costs:

All of the "Nope." boxes are where the numbers are going over ten million. I leave the others in because there might indeed be cases where you're willing to devote an aircraft carrier's worth of fuel (100,000t) to a few kg of payload.

E.g., if you're trying to get your 10 ton Winnebago to Mars, it's the right time of year where you can do it in 2 days on the 1-g acceleration plan, and you have Perfect Fusion Big Ass Laser, the mass cost being 3.74 means you'll need 37 tons of DT. Note also that this has you arriving at Mars with an empty tank. And even if that much is deemed doable, things get quickly hopeless for going too much farther (Winnebago to Ceres in under 4 days is 164 tons of DT; Winnebago to Neptune in 15 days is somewhere between 10 and 16 aircraft carriers filled with DT). And for all of the stupider forms of propulsion, just forget about it.

Short version:  If you see anyone getting to Uranus in under a month without monster fuel tanks, you know they have working antimatter drives or something equivalent (and hence equally dangerous).

Really, most of the 1g-all-the-way plans are dead in the water if we don't have antimatter.

You can also see just how hosed we are only having chemical rockets. Just getting to the moon is a multiplier between 1.44 and 2.68 (and if, having observed the Saturn V, this number seems low to you, keep in mind that this is for starting from low-earth-orbit, not the ground; the Saturn V had already dumped its first two stages at that point; the getting-to-the-moon part was done by just the 3rd stage (LH+LOX) and part of the Service Module (Aerozine), which between them had to boost a payload of 45 tons to the Moon and used over 100 tons of fuel).

Getting to Mars on LH+LOX using Hohmann is already a mass cost of roughly 2½. But there won't be any refueling when we get there so it's really (3½)²−1 to do the round trip, meaning we need 11¼ tons of fuel for every ton of whatever we're sending (astronauts/etc) that we expect to get back.

If we're going to do Mars at all in the next few decades (we shouldn't), NERVA is probably our best bet (mass cost goes down to 0.921, so the round trip mass cost drops to ${1.921}^2-1=2.7$ tons of ordinary hydrogen per ton of payload),

…in which case Clarke and Kubrick got it right in 2001, i.e., you put the engine at the end of a Very Long Pole and stay the fuck away from it while it's operating. (Also try not to go near anything inhabited.)

---

(*) stealing from Actual Physicist Angela Collier, my new favorite you-tuber, who, in her antimatter video, goes into a bit more detail about what happens when protons and antiprotons collide — evidently these are complicated affairs.

None of what she says should be taken as any kind of endorsement of the practicality of the kinds of antimatter cells/drives that my far-future galactic empire scenario depends on. I'm mostly saying that I don't see how we get out there without them — either they're doable or they're not — not that we necessarily will.

I also feel obliged to point out that our sun's power output is sufficiently ridiculous that there's plenty of space to multiply the powersat network by 7, likewise make the laser-ship fuel tanks 7 times larger, and thus have everything still doable, i.e., if the 16% thing is our only problem (hahahaha).

(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.

In "Tommorrow is Yesterday", the original series episode where they accidentally go back to 1967 and grab an Air Force pilot, there's this bit towards the end where they have to do this odd "breakaway" maneuver in close to the Sun, which then propels them to Ludicrous Speed, which then allows them to Do The Time Warp and Fix Everything. Iconic episode; you already knew about it; fine.

I'm assuming D.C. Fontana is responsible for that mess since she wrote it, and even allowing for the possibility that she got the idea from somewhere else, I'm still going to blame her for popularizing it.

Apparently, gravity is supposed to be this kind of rubber band that if you pull hard enough on it, it snaps, and then you have all the velocity you'll ever want. It's a beautiful, clear and simple explanation/analogy, so wonderful that it stuck with people.…

… desite being utterly and completely wrong.

Let's go back to basics:

What is a Slingshot?

Take a (usually) long strip of cloth with a wide patch in the middle, otherwise known as the "sling". Take a rock, otherwise known as the "shot", fold the patch around it, and then take up the other two ends of the strip together in one hand, and pick the whole thing up. You are now holding this pendulum thing with the rock dangling down at the bottom.

You now start swinging the rock in a circle, usually above your head but that part doesn't matter, faster and faster. The rock builds up some velocity.

At this point the Physics 101 teacher will be pointing out that in order for the rock to be moving in a circle, there has to be this force on it aimed in towards the center of the circle — centripetal, they call it,— i.e., towards your shoulder or hips or whatever it is you're swinging the rock around. The force has magnitude $m{v}^2/r$, so if the velocity $v$ is insane, the force is even more insane, and it's your arm and the strip of cloth that are doing that.

At some point, you let go, preferably of just one end of the cloth, but if you don't care that much about a slightly reduced range/release-velocity or getting your slingshot back, letting go of the whole thing is just fine. The rock then flies off at its now insane velocity,…

… hitting the giant in the forehead and killing him instantly. Yay, congratulations, you have just saved your primitive village. Maybe they will tell stories about you and you'll then be able to leverage that into a bid to become King. Good luck with that.

Ok, so, um… Gravity does not do this. Gravity never does this. This is not what the NASA folks are doing.

Yes, if you take an object and drop/throw it in the general direction of a planet/sun/moon/whatever (but not directly at since we don't want it to hit) it will speed up as it falls. Depending on how close it gets to the surface, it can speed up rather a lot. The ideal case is where you have something really massive that you can get really close to, like a dead star or Jupiter, which is sort of almost a dead star (the "dead" part matters because you don't want to be getting burned up when you get in close, though in Jupiter's case there's still a lot of radiation there that can fry you in other ways).

And then we invoke bullshit. Presumably, once your object gets going fast enough, there'd be some way to immediately turn the gravity off, so that it can then go flying away at ludicrous speed, and,… well,… no. Figure, if you could do that, the planet itself would then be immediately flying apart — i.e., if it's rotating, and still not a whole lot of fun for anyone living there even if it's not rotating — which then means you have Death-Star-like capabilities, in which case why are you dicking around with penny-ante "slingshot" maneuvers?

It's Not a Slingshot

What actually happens is that your object does its Jupiter flyby and then, assuming it hasn't grazed the atmosphere, hit one of the ringlets, or done anything else similarly stupid, it comes back out. We can solve for the trajectory: in general, it will be (one branch of) a hyperbola, like this:

We do have choices about what velocity we give it coming in and how close a flyby $r$ we do, which then determines the angle $\theta$ it comes out. But what gravity giveth, gravity taketh away; energy is conserved. Thus, no matter what we do, once our spacecraft gets back out to the same distance it was dropped/thrown from, the velocity will be the same, $\left|{\mathrm{<b>v</b>}}_2\right|=\left|{\mathrm{<b>v</b>}}_1\right|$.

The sharpness of the turn we're doing can be calculated from $$\cos \theta =\frac{1}{1+\frac{r{v}^2}{\mathrm{GM}}}$$ where, for this formula, $v$ is the "from infinity" velocity, i.e., how fast the spacecraft is going when it's essentially out of range of the planet's gravity (strictly speaking, it never is, but there will be a distance from the planet at which its gravity gets smaller than what we can measure).

And,… that's all. If we're just doing a two-body problem, i.e., where there's just Jupiter and the spacecraft, and nobody's firing the engines anywhere, there is nothing to be gained from this.

So, It Doesn't Do Anything?

Actually, it does, and I just dropped two hints in the previous paragraph as to how/why.

The first is that this is not just a two-body problem. What we need to do is look at what's happening from the Sun's point of view, in which Jupiter is now moving in its orbit:

The essential idea here is that having the spacecraft approach Jupiter from the front means that means switching to the Jupiter frame adds Jupiter's velocity to its horizontal component, this new velocity then determines the angle $\theta$ and hence which direction the spacecraft is going to emerge, and if we can keep that angle small enough, it'll emerge in front of Jupiter going forward, so that when switching back to the Sun frame, we're adding in Jupiter's velocity again.

Jupiter in its orbit goes 13.06 km/s. That means we are never going to be adding more than double the Jupiter velocity or a Δv of 26.1 km/s.

Also, the case $\theta =0$ only happens when something falls towards Jupiter at whatever the escape velocity is for that distance (quite small when far away). From the Sun point of view, this will be Jupiter-velocity minus something small. From Jupiter point of view, it'll be a nearly parabolic orbit where the object will be getting flipped around to go forward in front of Jupiter at escape velocity, or from the Sun point of view Jupiter-velocity plus something small. Or a total velocity gain of twice something small.

So for any kind of substantive velocity gain, we're always going to have θ>0 which will then introduce factors of cosθ that reduce our velocity gains accordingly below that promised 26.1 km/s.

If you're NASA trying to do Pioneer 11 and you can't even afford enough fuel for the 2nd half of the Hohmann transfer, you're stuck with doing just the first half (8.8 km/s Δv) using the Big Rocket to get away from Earth, end up approaching Jupiter at a relative velocity of 5.6 km/s, and hoping for the best. The closest flyby you dare is around 110,000 km from the center of the planet (wiki sez Jupiter has a "radius" of around 72,000 km, but when you're talking about a planet where the notion of "surface" is a gentlemen's agreement that gets you quickly down a philosophical rabbit hole if you poke at it too much — maybe it's where the hydrogen gets compressed to something vaguely liquid-like; there's evidently some point lower where it's a fucking metal whatever that means; maybe 72,000 is the cloud tops but you really don't want to be in those clouds, and there's probably also a whole lot of invisible exosphere above that you don't want to be encountering either. Anyway it looks like 110,000 km is what the Pioneer 11 planners went with, the radiation turned out to be nastier than expected, and so that's evidently the closest flyby that anyone has attempted thus far).

And then you crank the formula above to get θ≈13°, which is nice and small, which means you have no problem arranging things to emerge in front of Jupiter with something like 19 km/s, all without spending any additional fuel.

Which NASA was completely happy with in 1973 and it's also good for rather a lot of other solar system maneuvers they couldn't otherwise do.

But it's not Ludicrous Speed.

What's more, once you have your first velocity boost, yes, there's nothing that says you can't go bouncing off the other planets (Pioneer 11 indeed went on to Saturn to get another [small] boost), but that $v^2$ in the formula above will be making your θ angle quite large (cosine going to 0 means θ goes to 90°, i.e., spacecraft not getting deflected at all), and if your angle is too large, then you have to be approaching Jupiter from behind in order to be getting spit out the front, the relative approach velocity will then be a subtraction rather than an addition, and once what Jupiter is subtracting on the way in gets commensurate with what it's adding on the way out, you're really not gaining much of anything at all.

And the only way to make the angle smaller at higher velocities is to do a closer fly-by, but then you start hitting the atmosphere, and, um…, good luck with that.

Is that really all?

Well, all right, there is actually one more general trick — the second hint dropped above — you can use to squeeze extra velocity out of this, and that's to fire your engine during the flyby.

Quick aside about the flyby being this Exciting Time where everybody's all mushed into their seats and Feeling the Speed:  um, NO. Unless you're hitting the atmosphere or firing your engines, you will be in free fall the whole time, and it be just like all of that other boring space travel. At best, you will have nice stuff to look at if you have windows, but that's about it. Also, you probably don't want to have windows because they can break.

The point to firing your engine during flyby goes like this: Adding $\mathrm{\Delta v}$ to your velocity when your velocity is already $v$ increases your kinetic energy per unit mass by $$(v+\mathrm{½}\mathrm{\Delta v})\mathrm{\Delta v}$$ but if you instead dive in close to a planet to bring your velocity up to $V\gg v$ and then do the same burn, your kinetic energy gain will be $$(V+\mathrm{½}\mathrm{\Delta v})\mathrm{\Delta v}$$or, roughly $V/v$ times as much energy, and, once you've popped out of the gravity well back to the distance away you were before and gravity has taken back everything it gave you, you still get to keep that extra energy. It's not quite so straightforward a gain because velocity is the square root of kinetic energy, but it's still something.

In the case of being near Jupiter, diving down to 110,000 km from infinity gets you going at around 48 km/s, so if you're out in deep space going 6 km/s, you want to add another 2 km/s, you could instead dive in close to Jupiter and boost yourself by a mere ¼ km/s to get the same effect. So essentially you're getting an 8-fold savings in fuel expenditure

…except this can only happen at certain points on your trip (i.e. when you're near a planet), and, again, there are diminishing returns: as your $v$ gets large, the multiplier this maneuver applies to your Δv goes down.

Scorecard

In short, these maneuvers are actually really bad at building up speed beyond a certain point (and we'll soon see what kinds of velocities ships are most likely using in The Expanse universe because of mumble-Epstein-Drive-mumble).

The real reason to do this shit is to save fuel.

Which means it's not going to be the racing pinnaces employing these maneuvers. It's going to be the long term Big Cargo haulers that are pushing comets/whatever, where they need to be using the least fuel possible and thus needing to keep their Δv in the toilet, and also where the cargo is stuff like water or minerals or manufactured goods that don't actually care how long it takes to get there. Most likely they'll use ITN and every trick in the book to engineer an entire automated stream of deliveries where the pipeline is years or decades long, which will make sense once you have a colony in a known place that needs to survive and isn't going anywhere.

(... of course, now what I want to see are the scenarios where the colony died decades ago but the supplies keep arriving because it was too much trouble/expense to go hunt down all of the pipeline ships and divert them, and so the places turn into these treasure troves...)

In the next installment, we clarify what we mean by Ludicrous Speed

(…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.

( Read more... )

ok, so I'm on Mastodon (as @wrog@mastodon.murkworks.net, for now).

I will readily admit I never actually used Twitter (creating an account once upon a time was as far as I got) and still don't understand how it was ever supposed to work. I've got Local and Federated feeds spewing endless streams of spam; it's like sitting in the LambdaMOO living room but with 10,000 people there.

I'll be trying out the Lists feature (something similar worked well enough dealing with Livejournal at its height, but even that was far less traffic...).

I really miss Usenet and trn

link

promoting this out of my FailBook comments just because

So, there's this diagram which has been making the rounds now that the James Webb Telescope has been launched. It's good at showing where the Lagrange points are and, if you're used to reading topographic maps, what the gravity well generally looks like, that volcano with the 6000 mile deep crater that I talk about here, if you were having trouble picturing that (modulo the small matter that that was all about the Earth-Moon system, and this picture shows Sun-Earth L1-L5; but they both work the same way).

Granted, with L1, most people seem to be clear about the idea that, with Sun and Earth pulling in opposite directions, there's got to be some point in the middle where the forces are cancelling out, even if that's not quite what L1 is. But for L2-L5, people seem to be completely mystified, and I've been seeing far too many wacky attempts at explanations in the last few days.

Time to sort things out:

First thing that people seem to forget is that, to get this diagram, we are rotating the reference frame.

Meaning we're putting ourselves on this utterly gigantic amusement park carousel, that's spinning at just the right rate, so that, from our point of view, the Earth, normally considered to be whipping around the Sun at around 30 km/sec (in what we'll assume to be a circular orbit, for now), is actually fixed in place. Meaning that white circle you see in the diagram going through L3-L4-L5 that seems to be showing the Earth's orbit — it's really quite misleading — is not actually Earth's orbit … well okay, it is in the sense that everything that would be sharing Earth's orbit around the Sun in the usual view will be somewhere on that circle, but, in this view, the Earth itself is not actually traversing that path.

Because, again, the Earth is not actually moving.

But we still have gravity.

And, for some weird reason, the Earth isn't actually falling into the Sun, as you would expect. Because, as you'll recall if you've ever been on that amusement park ride, there's this other force that was keeping you plastered against the outer wall of the centrifuge while they opened up the bottom. People like to call centrifugal force a "fake" force, but it's plenty real if you're rotating; you have to account for it if you want to get the same answers that the non-rotating people get (Earth doesn't fall into the Sun).

… and it's pretty easy to account for. Centrifugal force is just like gravity in that the acceleration you experience is going to be the same no matter how massive you are, and it's linear in your distance from The Center. (If you want the math, it's ω²r where ω is the rotation rate in radians per unit time. And if you pick the right unit of time, we can even have ω=1 and then it's just r).

"Linear" means, amongst other things, that centrifugal force gets stronger the farther out you go. The white circle is all of the places where it cancels the Sun's gravity. Thus Earth (and anything else placed on that circle) has no net force on it and doesn't move. Inside the circle, the Sun wins and stuff falls in closer; outside the circle centrifugal force wins and stuff flies away. Thus,

• L1 is where Earth gravity + centrifugal force (both pointed outward) cancel Sun gravity (inward)
  
  
• L2 is far enough out so that centrifugal force is now strong enough to cancel both the Sun and Earth pulling together inward.
  
  
• L3 is where centrifugal force cancels both Sun and Earth pulling together the other way, however since, in this case, the Earth is pulling all the way from the far side of the Sun, its effect is going to be really small. Still, L3, like L2, will be outside the white circle, even if it's only about 600 km outside, a distance way too small to see in this diagram.

The second thing that people fail to mention, and this matters if you want to understand L4 and L5, is that the center of rotation is not the center of the Sun but rather the Sun-Earth center-of-mass.

In other words, if you imagine putting the Earth and Sun on a giant see-saw, then there's the spot where you'd put the fulcrum to make them balance, and that is the point everything revolves around. Admittedly, since the Sun is 1/3 of a million times the mass of the Earth, the Sun will be a whole 450 km left of center while the Earth is hanging out 150 million km to the right, which may not seem like much of a difference from the Sun being in the exact center, but you'll thank me when you see my L4 diagram.

Drumroll… Here, have a diagram:

Note that I have exaggerated the mass of the Earth to be half the mass of the Sun, which is why the center is now 1/3 of the way from the Sun to the Earth rather than being inside the Sun. (and yes, the grey circular dots are probably also the wrong sizes, but who cares?) Now you can see everything that matters.

Yes, all of the triangles that look like they're equilateral are, in fact, equilateral, and that really is a parallelogram in the middle, which is why those vectors at L4 all add to zero. The two blue arrows (acceleration of gravity caused by the Sun on the Earth and whatever's at L4) have to be the same length, and likewise for the two red arrows (acceleration of gravity caused by the Earth on the Sun and whatever is at L4), while the green arrows are proportional to the centrifugal forces (equal, if we do that trick with the time units).

The fun part about this diagram is that all gravitational forces are operating at the same distance (Sun⟷Earth = Sun⟷L4 = Earth⟷L4) so we don't even need to care about whether gravity is inverse-square or anything else; the distance-force relationship is whatever it is. Gravitational acceleration is otherwise proportional to the mass of the thing doing the pulling and completely indifferent to the mass of the thing being pulled. The rest is geometry.

The real magic of L4 (L5) is the way it is stable when the mass of the second body (Earth) is sufficiently small, even though, if you look at the diagram at the top of this post, L4 is clearly a hilltop and, normally, stable means you're at the bottom of a valley. It really does look like if you move even slightly away from L4, the forces accounted for by the diagram (gravity and centrifugal force) will be pushing you outwards from L4, and they will get stronger the farther away you get,…

… which is pretty much the definition of instability.

So, two things:

1. If the mass of the Earth is small compared to the Sun (has to be *much* smaller for this to work, but a ratio of 1/333,000 is plenty small; even in the Earth-Moon case where the ratio is only 1/100, that's still good enough),
   then that hilltop at L4/L5 is extremely large and flat (which you can sort of see in the diagram). Meaning the residual gravity + centrifugal forces that would normally be making L4/L5 unstable are likewise small and can be overwhelmed by Other Stuff, namely:
   
   
2. The force we haven't mentioned yet, … the Coriolis force, which is the other force that comes into play in rotating frames, and which cannot be represented in the diagram because it turns out to be velocity dependent.

Note that if centrifugal force were the whole story, then the moment something gets outside the white circle, it picks up speed, hurtles off to infinity and you never see it again. Which we know isn't what happens: In the non-rotating frame if you launch a probe from Earth that doesn't have solar escape velocity, it goes into an elliptical orbit of its own. Meaning, in the rotating frame, once something is moving away sufficiently fast, there has to be this other force that overwhelms centrifugal force so that it can come back.

You can also get a sense of what Coriolis force does by considering the case where we remove the Earth and the Sun (so that we're back to empty space), and then place a small rock where the Earth used to be, having it be stationary in the non-rotating frame. Once we hop back on board the carousel, we'll see the rock doing a circular orbit around the center (following the white circle) with velocity ωr. Which means something must be double-cancelling the centrifugal acceleration (ω²r outward), to give us ω²r acceleration inwards — what the rock needs to stay on that circular path at that speed — so the magnitude of the Coriolis force needs to be 2ω²r = 2ω(ωr) = 2ωv in this case.

So it shouldn't surprise you too much if I tell you that's what it is generally

a_coriolis = 2 v × ω

where ω is now a vector coming out of the page and we're taking a vector cross product, and, for the rock circularly orbiting in the correct direction, the right hand rule will indeed point us in towards the center. (You can also see this in action by getting on the centrifuge ride and pulling your fist towards your face really quickly).

The rock-in-empty-space scenario is not a bad way to remember how it goes, though, to be sure, if you want to actually derive what the Coriolis force is, there's a bit more work to do.

Anyway, the Coriolus rule along with the centrifugal rule

a_centrifugal = ω²r

turns out to be everything you need to know about the weird forces present in rotating frames.

As for what's happening at L4/L5, the capsule summary is that, with the residual combination of gravity and centrifugal force being so weak over such a wide area, you won't have to be going very fast at all for Coriolis to be the force that matters most.

And if Coriolis is the force that matters most, then you'll be going in circles. Also, velocity dependent means the faster you try to go, the stronger the force will be.

Which then makes L4/L5 rather hard to get away from.

To be sure, the end result is still quite weird+unexpected, and there's no substitute for Actually Doing the Math, which turns out to be kind of like solving the double-jointed pendulum problem. Orbits at L4/L5 have two different modes with different periods (1.05 "months" and 3.35 "months" in the case of Earth-moon, where "month" is the lunar orbital period, if I'm doing this right...) and an orbiting colony there can be in any linear combination of them, so in general there will be this weird Lissajous-figure crap happening …

… meaning if we ever get large numbers of people trying to settle L4/L5 -- not bloody likely, methinks -- there's going to have to be strict regulation of who's in which mode (probably will have to pick one of them and require everybody to be in it), otherwise all of the colonies will be crashing into each other...

(you'll want to read about solar power satellites first, where we cover all of the sensible stuff)

On Dyson Spheres and the other ridiculous things truly advanced civilizations will be doing with all of their free time

I should note that the scenario that I fleshed out in the previous post, i.e., having a cloud of independently orbiting power satellites that would start blocking out the sun if we really were able to populate huge numbers of orbits, is a lot more what Freeman Dyson was talking about in his original paper. The fixed sphere that people tend to imagine when they hear about this stuff really makes no sense at all if you think about it (and Dyson knew this at the time and said so). I mean sure, let's waste a whole lot of effort and materiel fighting solar gravity directly; that should be oodles of fun.

Making only slightly less sense is the flotilla of stationary habitats supported by solar sails. We're doing this why? Because we have this pathological hatred of orbital mechanics and don't want to make use of it? So much better to be depending on a sail that can get a hole in it and then we go plummeting straight into the sun. That sounds so much like a place I would want to live; really. Sign me up, please.

At best, I could see it as a kind of Bite Me, Universe gesture by a civilization that really has Done Everything, is now completely bored, and just wants to build some kind of insane, completely pointless artifact, just for the hell of it, because they can. Sure. Why not?

But I can't see it as something we're going to do while we're still on our way to the stars.

And I still wonder:

Can we really build all that?

Forget the full Dyson sphere, let's consider just one of the solar power satellite orbits. The chart in my previous post is calling for anywhere between 600,000 and 10 million square kilometers of satellite area.

Now, at this point, I don't even know what we're going to be making them from or what the preferred tech is going to be for solar power generation. Photovoltaic? Big-ass mirror driving some kind of heat engine? Or maybe we'll be supplying light to a mini-farm that's going to grow megatons of gerbil food, and then, in the next module over, we have billions of gerbils running their little wheels at top speed (at which point our movie instantly loses its "No Animals Were Harmed..." designation)?

Yeah. No idea.

But we can always make some kind of half-assed estimate. Let's just build aluminum sheeting; whatever the solar collector is, we'll need someplace to mount it. How much sheeting can we make out of the asteroid belt? We can vaguely do this:

• total mass of the asteroid belt (kg) is 2.3910×10²¹ kg.
  
  
• people estimating relative abundances in the universe say aluminum is 58 ppm of everything. Fine, so asteroid belt might have 1.38678×10¹⁷ kg of aluminum.
  
  
• let's be pessimistic and imagine that only 1% of it is mineable, or we lose 99% in the smelting process for whatever reason. Now we're at 1.38678×10¹⁵ kg.
  
  
• the thinnest sheeting you can buy online is 1/32"=0.79375mm thick. Meaning we need 793.75 m³ of aluminum to make a 1 km² sheet.
  
  
• Density of aluminum is 2.7g/cm³=2700kg/m³, so that's 2.143 kg to make a 1 m² sheet or 2.143 million kg to make a km² sheet.
  
  
• So we get 650 million 1 km² aluminum sheets if we harvest the entire asteroid belt. Which is somewhere between 60 and 1000 times the number of km² we need to populate one of our orbits, i.e., if it were the case that collecting solar energy only needs the km² aluminum mirror and everything else on the satellite is really cheap and easily available.
  
  
• So by this completely stupid measure, depending on which orbit we choose, anywhere between 60 and 1000 orbits are doable using the h= 36.635km⁻¹ separation and the Mercury L2 point.
  
  
• Note that these orbits were designed to suck off 86400 kg/day which is 1/4.26×10⁹ of the sun's output. Meaning blotting out the sun will entail filling 4.26 billion orbits, assuming we get the geometry exactly right.
  
  

Which seems to suggest that we'll get our 86400 kg/day power production and maybe even be able to go up to 1000 times that, but as far as blocking out the sun goes, just forget about it.

But then we have this interesting fact that I only learned about recently:

The Asteroid Belt is way smaller than you think

On the off-chance that anybody's still trying to convince you that the asteroid belt is the remains of a planet that got destroyed, here's something that really makes that not work:

The total mass of the asteroid belt is about 3% of the mass of the moon.

To be sure, I always knew the asteroid scenes you see in The Empire Strikes Back were bogus (space is big), but I'm still surprised that there's not even remotely enough there for any kind of respectable planet. (Sorry, James Hogan and whoever else wrote SF stories that had a planet breaking up a million years ago)

Which calls into question some of the premises of asteroid mining.

I will grant that there's stuff that won't be available on the moon. But for what is available, which probably includes all sorts of building materials, why not just mine the moon?

We can obtain 3% of the moon by strip-mining the top 12 km of its surface, and we might even vaguely be able to do that with present-day tech. And it's right here; no needing to travel hundreds of millions of km to get to Ceres or wherever else. Probably get huge economies of scale, too.

Granted, this won't really help with the Dyson Sphere. Even consuming the entire moon is only giving us a factor of 30, which is a long way from the 4.26 billion we need. Even eating all of Jupiter only gets us a factor of 800,000.

I think I can safely say that there is not enough aluminum in the solar system to cover the sun. Switching to a more abundant metal like iron gains us another order of magnitude or two, but I suspect we're still hosed.

Though, again, I'm obliged to point out how dubious this particular estimate is. It's not an impossibility proof by any means, and if the materials science folks do manage to come up with some carbon-nanotube/ceramic bullshit unobtainium that's insanely lightweight and can be built out of anything — much like I'm expecting them to do for the laser ships and light-sails we need for the transit tube — then all bets are off.

But I still think I'm pretty safe in putting the Dyson Sphere on the Not Gonna Happen list. If not because it's impossible, then because I'm not convinced we're going to need it.

Granted it is a bit weird finding myself in the position of wanting to say that 86,400 kg of matter+antimatter per day really ought to be enough for anyone — sounds a little too like that apocryphal Bill Gates quote — and having to stop myself.

(continued from here)

So, now what?

Intergalactic Travel

This is where we get full-on, batshit crazy. (You knew this was coming, right?)

Quick reminder in case you forgot or didn't know: M31, i.e., item #31 on comet enthusiast Charles Messier's list of things that annoyed the shit out of him because they weren't actual comets, otherwise known as the Andromeda galaxy, is one of our nearer galactic neighbors, perhaps not the nearest, but it's the one everyone thinks of, so let's just go there.

It is 2.54 million light-years (2.62 million aLY) away. No problem, right? Let's have a look at the numbers:

accel / decel		total shiptime (aY)	mass Cost
shiptime (aY)	distance (aLY)
14.78	1,310,986.97	29.56	2,621,975.93
14.52	1,015,049.37	29.63	2,030,100.74
14.27	785,915.64	29.87	1,571,833.28
14.01	608,505.71	30.33	1,217,013.42
13.76	471,143.65	31.08	942,289.29
13.50	364,789.19	32.19	729,580.38
13.24	282,442.80	33.77	564,887.59
12.99	218,684.97	35.97	437,371.95
12.73	169,319.61	38.95	338,641.21
12.48	131,097.80	42.95	262,197.59

Yes, these mass costs do look a teentsy bit prohibitive.

And yes, you're reading that right:  Even in the cheap case where we're opting to have the journey take 43 years — presumably, by the time we're contemplating this, we'll have figured out how to make an interstellar cruise ship tolerable for a few decades of travel rather than merely a few years (unless we're doing the Australia Thing, it's all prisoners, and we don't care) — we're still burning an entire Enterprise-class aircraft carrier (100,000 metric tons) for every five people we send.

On the other hand, it's a fair bet we won't even be considering this until we've developed, say, a substantial fraction of the Milky Way, say, 100 billion systems. Note that for this particular purpose "developed" means having gotten as far as building energy collection infrastructure and transit tubes; we don't need for them all to be inhabited. If they've all got the same 43,200 kg/day transit budget and we just declare a one day Transit Holiday for everybody, that's … rather a lot of aircraft carriers filled with antimatter, enough to send 200,000 people.

Do that a few days a year for a century and we're basically sending a Europe or 1/3 of an India. Which is not entirely bad for starting off a new galaxy.

But it's possible I'm being too conservative here. Because of something I haven't mentioned yet (it really only occurred to me today).

The Absolute Best Power Plant Ever

By the time our exploration sphere has expanded 25,000 light-years to include the center of our galaxy, we will then be encountering the Sagittarius black hole…

… which I originally thought we'd be wanting to stay the hell away from, but…

It spins, see, which means if you throw shit into it at just the right angle and arrange for it to break apart inside the ergosphere (weird-ass 2nd-event-horizon thing that spinning black holes have that you can actually get out of) at just the right time, then half of it will be spit out way, way, way, faster than anyone should reasonably expect, and then you collect all of that extra energy at some safe distance (hahahaha) and use it to charge up antimatter cells.

And then we're building the Really Big version of the solar energy conveyor thing to get all of that antimatter shipped out of there.

These guys say the rotational parameter for the SBH is around 0.44, which means that about 2.6% of its million-some-odd solar masses is rotational energy that can be completely extracted in this way, meaning there's about 5×10³⁴ kg theoretically available to be harvested.

That's half a billion billion billion aircraft carriers. Possibly more as long as there's crap continuing to fall in to speed up the hole. If we don't get greedy and, say, only take a trillion aircraft carriers per year, that's 3 trillion star systems worth of power generation right there.

Without having to explore and settle 3 trillion star systems.

… modulo certain engineering issues that I will not be attempting to address at this time. I suspect we will need to be using both sides of the paper for this one.

I'll grant that my original timeline has the explorers taking a few million years to reach the center of the galaxy and the SBH. It may be that once we have sufficient infrastructure in our immediate neighborhood, we may want to splurge and launch a few Really Fast explorers in the general direction of Sagittarius so that they're getting there a lot sooner, say 30,000 years.

This will be expensive: We're talking 5/6 lightspeed, which means mass cost ratio of around 11 and recall the explorer ships are big, ten aircraft carriers each, so that's 110 aircraft carriers to launch each one or 7000 times our annual single-system transit budget. Then again, recall that Earth is going to have thousands of years of downtime waiting for those first terraformed exoplanets to come on line. Or we could wait the 2500 years to have energy infrastructure set up in 100 systems so that launching the SBH-explorers won't be as big of a bite.

There's also be risk in launching explorers that far that fast in that we currently have no idea what the solar systems are like in closer to the center of the galaxy, whether the mining/reproduction stuff is even going to work out there; the advantage of the slower plan would be in knowing what's there long before we arrive.

But it may well be worth it if we can have this fucking huge energy pipeline up and running in a mere 60,000 years.

If we have the FHEP, we're definitely getting to Andromeda.

Or, if we want to be more ambitious some year and can extract just 0.00000000000002% of the SBH energy available, that's enough to send a billion people each to every single galaxy in the Local Group using the completely wasteful Andromeda-in-29.56-years schedule (first line of the chart above).

Doing the Long Range Exploring

The economics get a little weird, because there's no getting around spending a few million years on exploration and tube construction, and if we're willing to wait another million years, then the entire galaxy will be already terraformed and harvesting energy by the time we get there.

Also, Andromeda almost certainly has it's own big-ass black hole, and unlike with the Milky Way, our explorers can be heading straight there first thing to get that pipeline started. Which then gets us started on the next galaxy quite a bit sooner than you might think.

Never mind that we're now operating on a timescale where there'll be time to build some real Earths, not just the fake ones you can throw together in a few thousand years. Assuming we even care about that anymore (we might!).

We're clearly going to need to run the explorers differently, since proceeding at 1/100 lightspeed and waiting 250 million years for the explorers to get to Andromeda is really not optimal anymore, nor is the randomly expanding-in-all-directions thing going to work for us. We need to be a lot more goal-oriented now.

Also, intergalactic space is really, really thin. The most we're going to be able to expect is maybe 1 star every 1000 light-years and the mineralogical pickings may be slim once we're away from places where Type III supernovas have happened (though maybe building everything out of aluminum will work just fine). But there are indeed going to be stars. Assuming we can find 2500 systems vaguely lined up 1000 light years apart, we can then have the explorer taking, say, 500 years to build up some energy infrastructure in a system — again, all we need is a stupid little red dwarf — to accumulate enough energy to do the next 1000 light-year hop at 2/3 lightspeed and then begin construction of that portion of the transit tube backwards towards the Milky Way, so as to generally progress at 50% lightspeed and get there in 5 million years (starting up the Replication Thing and going back to the old explorer schedule once we get close to the end) with the transit tube mostly completed.

And we'll probably want to wait an additional 2.5 million years, because, among other things it might be nice to hear back and find out that our explorers actually made it there and were able to do stuff, and also confirm that the tube's been completed, before sending the first actual payload ships with Actual People, (though I suppose it'll be possible to have the transit tube start turning payloads around if the laser ships entirely lose contact with the far end, or receive an "oops", or attain some other reason to believe there isn't actually a far end).

Because once we get into the intergalactic realm, we're now operating on timescales where evolution can happen.

In other words, this is where we're more likely to be running into Opposition.

(next up: Aliens… oh wait, I'm lying; there are a few other things to cover first)

(galactic empire continued from here)

Some Vaguely Actual Costs

Here, have a wall of numbers:

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)

(continued from here)

How exploration will work

First: the explorer probes will need to be slow. This is a side-effect of how stupid rockets are. Because in the case where you're arriving at a system for the first time, rockets will be the only alternative for slowing down, you'll need exponential amounts of fuel for that, and since the explorer has to carry all of its deceleration fuel with it, you then have to accelerate all of that fuel at the start of the trip,... so there'll actually be this doubled exponential factor in there.

But if we don't try to get anywhere near lightspeed it won't be too bad. Since all of the terraforming is taking thousands of years anyway, nobody's going to care about the explorers being slow.

Numbers for this:
( some numbers, etc )