You Can't Take the Sky from Me

[Yes, I recently rewatched Firefly.]

With social distancing on my mind this week, I started thinking about taking it to an extreme: interstellar travel. Specifically, I wanted to try some calculations with the Tsiolkovsky rocket equation, which applies to any space ship that moves by using conventional rockets.

The idea is this: To get moving, you need to push off of something. On the ground, you can push off the ground or the air, but in space your options are more limited. Rockets accelerate fuel out the back, which pushes the payload forward:

via Wikipedia

The problem is that the more mass you carry, the more fuel you need, but more fuel means more mass. If you work out the particulars, the equation you (or rather, Tsiolkovsky) get is:

This says that the more change in velocity you want, the more fuel you need, but there are diminishing returns.

When my parents were shopping for a camper van, they described their ideal as a "Russian space capsule", so let's take them at their word and imagine attaching rockets to their 2000 Chinook Premier. The manual gives the loaded weight as 10,700 lbs, or 4,853 kg. This is m in the equation above. For the exhaust velocity, ve, we can use the numbers for the (now retired) space shuttle: 4,447 m/s. If our target is the edge of the Solar System, we can relate the travel time to the fuel required:

The dashed line shows the mass of the camper, so to get there in under ~2 months, we need more fuel than cargo.

Of course, there's not much to do at the edge of the Solar System, so how long does it take to get to our nearest neighbor, Proxima Centauri, 4.2 light years away? If we want to get there within a reasonable time, we need to go pretty fast, so we'll use the Special Relativity version of the rocket equation:

Relativity says that as you get closer to light speed, it takes more energy to go any faster. Using the same exhaust velocity from before, the curve looks like this:

For reference, the mass of the Earth is around 6 x 10^24 kg, so burning the entire planet could get our camper there in under 10,000 years!

This is the reason other propulsion techniques have been proposed, like the LightSail I mentioned a couple weeks ago, or Project Orion, which involved accelerating the spacecraft by exploding nuclear bombs behind it. I think I prefer to associate my name with the new shuttle replacement, rather than something that leaves a trail of fallout...