Travel in “The Expanse”

At least up to what I’ve seen in season 2, the expanse at least tries to acknowledge Newtonian physics. There are odd bits where they mix up where they should have centrifugal gravity and not, and in which direction, but largely, they try. Thankfully there is no FTL nonsense (yet), but the civilization seems to have access to enormous amounts of fuel and amazing rocket engines that they can accelerate (presumably at 1G or above) around the solar system. I wanted to explore, with back of the envelope calculations, what kind of transit times they would encounter.

Let us take one representative long journey in the solar system: a trip from one end of the asteroid belt to the other. Let us assume that the distance we will travel is 5 AUs which is a reasonable distance, though on the short side for such a trip.

Let us assume we are the bees knees in the latest fusion technology, or antimatter technology or whatever, and we can sustain a acceleration of 1G through out the trip. We will ignore relativistic effects, which are tiny at this level of detail. We do what Captain Calculus did in Explorers on the Moon, and accelerate for half the journey, flip over and then decelerate for the rest, which seems to be the model The Expanse has adopted.

Area of a triangle strikes again.

The time taken to cross a distance d, under acceleration a is given by the formula above, as is Vmax, the peak speed mid-way through the flight.

For a 5 AU (7.48e+11 m) journey under taken at 1G (9.8 m/s2) the maximum, mid-point, velocity is 2,707,471 m/s (.009c) and the time taken to complete the journey is 552,545s or 6.4 days, which is insanely good. Like, you can travel from one end to the other of the asteroid belt in 6 and a half days (or thereabouts)? Sign me up.

One second, one second. Now I am a bit curious about how much propellant we’ll have to lug around with us. I’ll assume we have some advanced engine that can take the propellant and fling it backward at whatever speed we want. I’ll assume that our engine can throttle down smoothly so we can keep the acceleration constant as the mass of the ship decreases as we consume propellant.

ms is the mass of the ship, m(t) is the propellant mass at time t. dv/dt is the acceleration of the ship gained by chucking mass dm overboard at speed ve. We vary dm and Ve to keep dv/dt constant. The initial equation we derive based on the law of conservation of momentum – the momentum gained by the ship is equal to the momentum of the mass we threw out back.

Gah. I’ve forgotten how to solve differential equations. All those years in undergrad, all the aggravation I caused my teachers (including mom) all for nothing. Wait! This is the 21st century. We have computers, websearch and Wolfram Alpha

Wait, I have a boundary condition in my back pocket somewhere. Ah here it is: At time T we have reached our destination and exhausted our fuel, so m(T) = 0. We can use that to solve:

Ion thrusters in current use have a maximum propellant speed of 50km/s. So, plugging in

  • T = 552,545s
  • ms = 10,000 kg (which is really light, probably a 1 person ship with radiation shielding)
  • a = 9.8 m/s2 (as agreed)
  • ve = 50,000 m/s

We get M = 1.08E51 Kg

Well folks, the good news is that the entire universe has enough mass for about 100 such trips, so important business only please.

Kaushik, you say, this is the FUTURE! Our ion drives are much more impressive. They fling out propellant at 1000 km/s!

Ok. Let me pull up my little abacus here and run your futuristic numbers.

M = 2,247,392 kg. That’s about the weight of a Saturn V rocket on the pad.

I do have one more question. How much energy will this engine ejecting propellant at 1000 km/s require (at the minimum)?

The kinetic energy of the propellant is 1/2 m v^2. At the very start of our journey, when we are fully laden, the equation comes out to aVe(ms + m(0)) which is 1.1E13 Watts or 11 TW.

Ummm. I … I … McFly, that’s 10,000x more power than the 1.21 gigawatt machine we built, McFly!

1.21 Gigawatts back to the future |  11; TERAWATTS | image tagged in 121 gigawatts back to the future | made w/ Imgflip meme maker

The largest fission reactors on Earth have outputs in the 7-8 TW range so I suppose …

I SUPPOSE it’s possible. I mean if we got fusion reactors. I don’t know enough physics and materials science to judge how plausible it is, but I guess we can dream.

So, in summary, in order to zoom around the solar system at 1G we need a motor with an insanely high exhaust velocity and a pretty meaty power plant and we’ll be riding on a Saturn V sized ship, with a pretty modest payload.

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