Tag Archives: math

Designing the Marathon: A Case Study

One idea I had for a video to promote Slower Than Light was to build the UESC Marathon from Bungie’s eponymous game.  In that game, Deimos has been converted into an interstellar sleeper ship that flies to Tau Ceti to start a colony there.  Marathon departs Mars in 2472 and arrives at Tau Ceti in 2773 (A 301-year voyage at around 4% the speed of light, or 12,000 km/s).  Since Slower Than Light supports converting small asteroids to starships, I wanted to actually build the Marathon in-game.

This was a great exercise in interface design, because it forced me to go through all the math that a player would have to go through to undertake a task like this, and what I need the software to be able to help them with.

To start with, I’ll presume the Marathon is using conventional rockets, not solar sails or anything like that.  Let’s assume that in launch configuration, Marathon has about the same mass as Deimos did originally.  With different propulsion types, how much of the asteroid has to be hollowed out just for fuel?

Tsiolkovsky’s rocket equation tells us that the thing that really matters is the exhaust velocity of our engines: how fast the reaction mass comes roaring out the back.  For H2/LOX engines in STL (like that Saturn V and other 20th-century rockets used), the exhaust velocity is about  4.17 km/s.  We multiply that times the natural log of the ratio of our total rocket weight to our empty rocket weight to get our delta-v.  Since we’re looking to accelerate up to 12,000 km/s and then back down, we need about 24,000 km/s of delta-v.

We divide 24,000km/s by our exhaust velocity to get 5,755.  We raise e to the 5,755th power to get 2.3 times 10 to 2499th as our fuel/mass ratio.  Using hydrogen and oxygen as fuel, we wouldn’t even be able to send an atom’s mass to Tau Ceti using a tank the size of Deimos.  We’ll need something with some more oomph to it.

Nuclear Pulse Propulsion, sometimes known as the Orion Drive, is frequently cited in literature as the most practical means of moving enormous masses around the solar system and galaxy at our current technology level.  Essentially the engine throws a small nuclear weapon out the back of the ship, detonates it, and lets the shockwave hit a pusher plate that drives the spacecraft forward.  Using this technology, it is theoretically possible to get an equivalent exhaust velocity of 1,000 km/s, 250 times more efficient than H2/LOX engines.

Run the numbers on those (e ^ (24,000 km/s / 1,000 km/s)) and you get 26,489,122,129 as your mass ratio.  Deimos has a mass of 1.4762 * 10^15 kilograms, or 1,476,200,000,000 tons.  To get Deimos to Tau Ceti using Nuclear Pulse Propulsion, we need to remove all but about 55 tons of that moon, and replace virtually its entire mass with nuclear bombs.

So what are our other options?  Antimatter is generally the cure-all for high-science woes.  If we built an antimatter rocket to take Deimos to Tau Ceti, how far and how fast would we need to go?  Well, the highest number I could find for a pure antimatter rocket was 100,000 km/s of exhaust velocity — now we’re talking!  I have no idea if that’s even kind of reasonable, but it is going to make our numbers much nicer, so we won’t ask too many questions (this blog post is already passing 500 words.)  At 100,000 km/s of exhaust velocity, we only need to raise e to 0.24, for a mass ratio of 1.271.  That’s bloody nothing!  We just need to replace 21.3% of Deimos’ mass with matter/antimatter fuel, and we’re off to the races with a tiny planetoid in tow.

Finding a pile of antimatter that weighs as much as a thousand aircraft carriers is left an exercise to the reader.