When to Launch a Spaceship to Another Star

I came across a charming little calculation recently, with a counter-intuitive result, that's too good not to pass along -- it's called the "wait calculation." It's the kind of thing you might kick yourself for not thinking of yourself. Anthony Kennedy published it in 2006. (Maybe this is a well known paper, but I just discovered it.) I wanted to do the calculation for myself before I looked at his solution, and almost got it right. Here's what I came up with.

Q: When should we launch a spaceship to another star system?

For example, we could launch a generation or hibernation ship today that would take tens of thousands of year just to reach Proxima Centauri with current technology, the nearest star to us at a distance of 4.2 light-years. (That star might even have a orbiting planet in its habitable zone.) Here's a graph from a section of the interesting site Atomic Rockets:

Or should we wait, and, if so, how long? Technology has been getting better for centuries, with an exponential increase in the maximum velocity we can attain. Let's assume this exponential trend continues.

Kennedy used some assumed values for velocity, but I'll try to make it bit more real.

A straight line on a log-linear plot represents exponential growth. The trendline goes from 8e-8*c in 1800 to 8e-3*c in 2010. (Beyond that is speculation.) These speeds are 27 m/s in 1800 to 24,000 m/s in 2010, which are reasonable -- from the speed of a good horse to the speed of the Juno space probe. 

With these two points you can do an exponential fit for speed, v = v₀*exp(at) and get a=0.0069/yr and v₀ = 0.021 m/s. (Strictly speaking v₀ would be the speed when t=0, viz. 2010 years ago. Not quite right, but it won't matter much.)

Kennedy assumed the perhaps simpler formula v = v₀*2^t/h, which makes the velocity doubling time obvious. But I'm a student of physics, so e-exponentials it will be. In my calculation the doubling time is ln(2)/a = 100 yrs.

Let's set up the timeline:

t₀ is when v=v₀. In the above figure, the Δ's are time intervals. On them, subscript 0 is for the past interval, subscript L is when the last ship leaves the Earth, and subscript D is when the last ship arrives at its destination. W is for wait.

 Δ_W is how long we should wait, starting today (well, from 2010) to launch the last ship. Δ_L is how long the last ship takes to reach the destination.

Let Δ_T be the time interval that a ship launched today would take to get to Proxima Centauri. At today's maximum velocity of 24,000 m/s it equals 53,000 yrs.

Here's where I went astray. I thought we wanted to minimize Δ_L, but actually, as Kennedy says, we want to minimize Δ_W + Δ_L.

By taking the ratio of Δ_T to Δ_L, we get

Δ_L = Δ_T*exp(-aΔ_W)

where a and Δ_T are given via the initial data. So

Δ_W + Δ_L = Δ_W + Δ_T*exp(-aΔW)

We minimize the right-hand side by differentiating with respect to Δ_W and setting the derivative equal to 0. Doing so gives

aΔ_W = ln(aΔ_T) = 5.91
so Δ_W= 852 years

Surprisingly (to me at least, initially), we can wait 852 years before launching a spaceship to the nearest star system, assuming our maximum attainable velocity keeps increasing exponentially.

Then we substitute to get

Δ_L = 144 years

at which point the velocity of the last ship will be v_L = 8.8 M m/s = 0.029 c. That doesn't seem unreasonable. It would arrive in the year 3007.

Surprisingly, waiting even longer to depart after the last launch means an arrival at the destination at a later time than does the last launch. An even faster ship doesn't compensate for a longer wait time. And waiting longer increases the chance that humans will go extinct, either by a natural disaster, a manmade disaster, disease, or warfare. But still, travelers might wait if they want to reach their destination in their own lifetime.

So if we wait we could be at the nearest star in 852+144 = 996 yrs, just under a millenium. The generation ship launched today takes 53,000 yrs, so the ship launched in 852 yrs would pass the generation ship almost immediately after it launches. In fact, it would pass it in (using the above numbers) just 2.3 years after it launched! How dispirating for the generation ship, which would have already been sloging on for 852 years, with a couple of scores of generations having come and gone. This calculation shows there isn't much incentive to launch anytime soon, if a later launch seems socioeconomically feasible.

You might wonder about acceleration and deceleration of the ships, but these are small. Even at the speed of the last launch, it would only take 10 days to start or stop if you desire a rate of ±g. You can add special relativity if you want, but it only affects the time intervals on board the ships, not the amount of time you have to wait on Earth.

However...the sticky point is the assumption that maximum attainable velocity will continue to increase exponentially, forever, or according to whatever function you want to pick. Kennedy has a lot of interesting thoughts on the durability of economic growth (vs available resources) and incentives, and his paper is worth reading just for that. He writes:

Using simple equations of growth, it can be shown that there is a time where the negative incentive to travel turns positive and where departures will beat departures made at all other times. Waiting for fear future technology will make a journey redundant is irrational since it can be shown that if growth rates alter then leaving earlier may be a better option. It considers that while growth is resilient and may follow surprising avenues, a future discovery producing a quantum leap in travel technology that justifies waiting is unlikely.

Kennedy also writes about the the social implications of waiting vs not waiting, and what travelers will find when they land. I'm not going to go into that, but read his paper.

Anyway the math isn't difficult; thinking up the problem was the hard part, and realizing it could be easily solved. I think it's a fun little problem. And, you get to think about space travel.

Note: slight corrections made from the original version of this post.