The Other End of the Scale

(this is Part 3, more background for critiquing The Expanse; here are Part 2 and Part 1)

Let's suppose we have Best Possible Rocket, as much free fuel/reaction-mass/whatever as we want, … and we just want to get there as fast as possible? Interestingly, there will still be limits.

In this case it's about how much we can accelerate. There will be some maximum, whether it's 1g for human passengers, or there's no one on board but the more we accelerate, the more shit breaks, and there's only so much breakage we can tolerate before it gets pointless trying to accelerate that much. Imagine trying to build a skyscraper in 2g or 3g; even if no one's going to live there, you still have to put in extra reinforcements so that it doesn't collapse, which will just make it heavier. I figure only the smallest ships (we call them "missiles") will be built for high-acceleration.

Once we have a limit, the best time is achieved by pointing our ship directly at the destination, immediately putting the foot to the floor, doing that for half the trip, then flipping and spending the rest of the trip slowing down at the same rate.

In this world, for all but the shortest trips, the effects of solar gravity will be down in the noise. Yes, the sun will be bending your path a bit and you still have to correct for that, but for a ballpark answer we are back in the first week of Physics 101 doing $x=\mathrm{½}g{t}^2$.

Except there are two legs of the trip so it's really $$x=2\left(\mathrm{½}g{(t/2)}^2\right)=\mathrm{¼}g{t}^2$$ assuming we never get going fast enough that relativity matters (in which case it will be $x=\left(\cosh \right(\mathrm{½}gt)-1)/g$, but it won't).

Inverting this gives us $t=\sqrt{4x/g}$, taking careful note of that square root because it will have consequences.

Meanwhile, the total ΔV will be twice our maximum velocity $=gt=\sqrt{4xg}$, which is enough to make a new chart, once we set $g$ to the usual Earth gravity (9.80665 m/s²):

1g acceleration flight-plans

Trip	Distance	Time	Total ΔV	Mass Cost
LEO ⟶ Moon	1.3s	3h 30m	124 km/s	0.41 g/kg
Earth ⟶ (nearest)Venus	138s	1d 12h	1274 km/s	4.26 g/kg
Earth ⟶ (nearest)Mars	261s	2d  2h	1753 km/s	5.86 g/kg
Earth ⟶ (nearest)Mercury	306s	2d  6h	1897 km/s	6.35 g/kg
Earth ⟶ Mercury	692s	3d  9h	2853 km/s	9.56 g/kg
Earth ⟶ Venus	860s	3d 18h	3180 km/s	10.66 g/kg
Earth ⟶ (nearest)Ceres	881s	3d 19h	3219 km/s	10.80 g/kg
Earth ⟶ Mars	1259s	4d 13h	3848 km/s	12.92 g/kg
Earth ⟶ Ceres	1879s	5d 13h	4701 km/s	15.80 g/kg
Earth ⟶ (nearest)Jupiter	2096s	5d 21h	4965 km/s	16.70 g/kg
Earth ⟶ Jupiter	3094s	7d  3h	6032 km/s	20.33 g/kg
Earth ⟶ (nearest)Saturn	4259s	8d  8h	7078 km/s	23.89 g/kg
Earth ⟶ Saturn	5258s	9d  7h	7863 km/s	26.58 g/kg
Earth ⟶ Uranus	10071s	12d 20h	10883 km/s	36.97 g/kg
Earth ⟶ Neptune	15499s	15d 22h	13500 km/s	46.06 g/kg

A few more column observations/explanations:

• Trip / Distance:  The Hohmann Transfer trajectories are portions of ellipses, but in this world, we are just going in a mostly straight line to our destination, so, unlike on the Hohmann chart, (1) these distances will be much closer to the actual distance traveled, so (2) for the interplanetary trips at least, there are now different distances depending on the time of year.
  
  For maximum distance, we put the destination planet on the far side of the sun w.r.t. Earth (what astronomers call "Superior Conjunction" for Mercury and Venus and "Conjunction" for the outer planets) and this will be the same as the distance on the Hohmann chart. Minimum distance is the planet being right next to Earth on the same side of the sun ("Inferior Conjunction for Mercury and Venus; "Opposition" for the outer planets), and, since the astronomical terminology is annoying/confusing (because what they care about, conjunction and opposition w.r.t. the Sun, which matters for observing, is different from what we care about), we'll just label this one "(nearest)".
  
  
• Time:  Travel times are much shorter — Mars is now two to four days rather than nine months, which is nice, — but, as we can see, still very much more than the worst-case near-light-speed transit time of 1259 seconds or 21 minutes, and hence we're still nowhere near relativistic.
  
  Meaning if you want to get there faster, the next problem you have to magically solve is not how to do FTL (Not Happening), but rather how to do "inertial dampening", i.e., how to keep from feeling acceleration (Not Happening).
  
  
• Total ΔV:  For comparison, solar escape velocity from Earth's orbit is about 42 km/s, so even on the shorter trip to Mars where we're up to 20 times that at the midpoint, we're barely going to notice (hence why I said we can mostly ignore solar gravity).
  
  Unless, of course, the ship's engines fail right then, which will just suck. (Hi, one-way trip to interstellar nowhere. Rescue will be Very Expensive.
  
  Also, if your ships are getting pirated anywhere other than near the beginning or the end of the trip, the wreckage/evidence will be headed out of the solar system pretty quickly).
  
  
• Mass Cost:  For this chart we are using units (g/kg = 10⁻³) that are 1000 times larger than for the Hohmann Transfer chart.
  
  Getting the 10-ton Winnebago — which looks to be about how big the Razorback is— to get to the Moon in 3½ hours, means burning 30 times as much fuel as before, or 4.1 kg, which is enough to blow up most of New England …
  
  (…and some rich asshole on Lake Winnipesaukee in New Hampshire just leaves this in his garage, unattended, for the whole winter??)
  
  Getting it to Mars in the best case (2 days) costs almost 60 kg, meaning we are starting out with a 10.06-ton Winnebago and 30 kg of that extra mass is antimatter, enough to shred the entire continental US. Think very carefully about this.
  
  

One could imagine accelerating at more than 1g, but there are huge costs to doing so and you don't get as much for it as one would think. The problem with distance covered being proportional to the acceleration and (time)² is that increasing the acceleration by a factor of n only shortens the trip by a factor of √n, e.g., 2g gets you 30% off and you need to go to 4g to get to 50% off, except at that point, everybody will be needing a water tank (200+ kg/person?), which is way more payload, and then you multiply that mass cost by at least 2 and possibly a lot more. It's never going to be economical as compared with 1g where, if the thrust is steady enough, the passengers can walk around the ship without even thinking about it.

Yeah, I know, they've got the Miracle Drugs that keep oxygen flowing to the brain and Repair All Damage. Not buying it. Keep in mind that this is not simply about momentary bursts to dodge missiles in the heat of battle; this is about keeping it up for the entire trip. The worst roller coaster you've ever been on gets up to maybe 3g, but only momentarily, and not days or weeks on end. Fighter pilots only have to last about 30 seconds — if the high-g goes on for too much longer than that, they just die.

Never mind that we are mostly never going to get to use these flight plans. We will see why once we finally address the elephant in the room:

Our Rockets are Not Ideal

Everything Sucks

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:

Trip		ΔV (km/s)	Ideal	Anti matter?	Fusion	Fission	DS4G	NSWR	NERVA	LH+LOX	Aerozine
			𝓅 = 1	7	266	1000	1500	5000	35000	67000	98000
Hohmann	Moon	4.0	13g/t	93g/t	3.5g/kg	13g/kg	20g/kg	68g/kg	592g/kg	1.44	2.68
	Venus	5.2	17g/t	121g/t	4.6g/kg	17g/kg	26g/kg	90g/kg	835g/kg	2.20	4.48
	Mars	5.6	18g/t	130g/t	5.0g/kg	18g/kg	28g/kg	97g/kg	921g/kg	2.49	5.23
	Ceres	11.2	37g/t	260g/t	10.0g/kg	37g/kg	57g/kg	204g/kg	2.69	11.1	37.6
	Jupiter	14.4	48g/t	337g/t	12g/kg	49g/kg	74g/kg	272g/kg	4.39	24.2	111
	Uranus	15.9	53g/t	372g/t	14g/kg	54g/kg	83g/kg	304g/kg	5.43	34.2	182
	Mercury	17.1	57g/t	400g/t	15g/kg	58g/kg	89g/kg	331g/kg	6.40	45.1	270
										Nope.
1g Acceleration	Moon	123.8	413g/t	2.9g/kg	116g/kg	511g/kg	858g/kg	6.89	1900718
	Venus(n)	1274.2	4.3g/kg	30g/kg	2.10	69.1	586	Nope.
	Mars(n)	1753.1	5.9g/kg	41g/kg	3.74	345	6446
	Mercury(n)	1896.5	6.3g/kg	45g/kg	4.38	557	13212
	Mercury	2853.0	9.6g/kg	68g/kg	11.6	13583	1583323
	Venus	3180.1	10g/kg	77g/kg	15.8	40439	8132603
	Ceres(n)	3219.1	10g/kg	78g/kg	16.4	46065	9887315
	Mars	3848.3	12g/kg	94g/kg	29.4	375742	Nope.
	Ceres	4701.0	15g/kg	116g/kg	63.8	6457635
	Jupiter(n)	4965.2	16g/kg	122g/kg	80.9	Nope.
	Jupiter	6032.4	20g/kg	151g/kg	210
	Saturn(n)	7077.5	23g/kg	179g/kg	532
	Saturn	7863.0	26g/kg	201g/kg	1070
	Uranus(n)	10329.7	35g/kg	272g/kg	9558
	Uranus	10883.0	36g/kg	289g/kg	15617
	Neptune(n)	13058.5	44g/kg	356g/kg	107640
	Neptune	13500.4	46g/kg	370g/kg	159316

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).