The Other End of the Scale

[wrog]

(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