Friday, January 24, 2020

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*2t/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.

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

Let ΔT 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 ΔL, but actually, as Kennedy says, we want to minimize Δ+ ΔL.

By taking the ratio of ΔT to ΔL, we get

ΔL = ΔT*exp(-aΔW)

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

Δ+ ΔL = Δ+ ΔT*exp(-aΔW)

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

W = ln(aΔT) = 5.91
so ΔW= 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
ΔL = 144 years

at which point the velocity of the last ship will be vL = 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.


Thomas said...

The concept certainly isn't new in science fiction. I remember an old novel (Van Vogt, I think)where the crew of the first interstellar trip got a heroes welcome at their destination by the by now thriwing colony built by later ships travelling much faster.

Victor Venema said...

Interesting idea. One should also have a decreasing function for the probability that humans still have the capacity to build a rocket. As the age of Trump and Bosonaro make clear.

The graph would look different for Germany. German trains are much faster than German cars, even without speed limit on the autobahn for people in a mid-life crisis.

My main problem with the function and its extrapolation is that up to now the main aim of transport was not to achieve maximum speed and that the maximum speed depends much on (air) resistance. So I am not sure you can use historical data for trains and cars to make conclusions about interstellar rockets, where high speed is at least one of the main considerations.

I guess smaller robot probes for terraforming would also be a lot faster than ships with humans.

David Appell said...

Thomas: Kennedy writes about how slower travelers might be received at their destination, to which faster ships have already arrived, perhaps millennia or tens of millennia earlier. He speculates they could be completely out-of-date technology-wise, barely able to function or support themselves in the new, advanced society, and may be seen as worthless, cavemen-like parasites who can contribute nothing but need to be housed, clothed and fed.

Perhaps though if they had been receiving communications from Earth, and had a decent ability to fashion the new technologies they learned of -- they're need a good workshop anyway, to replace what breaks -- it might not be so bad. But are Earthlings 10k or 50k years from now going to care enough to keep sending detailed information on the latest developments? Maybe a good job for AI.

David Appell said...

Victor: Good points. But if you just graph the maximum speed attainable, regardless of its technological type, you'd still get an exponential. I think that's what the graph I included is plotting.

Thomas said...

David, do you think a couple of citizens from the Roman empire somehow showing up today would have trouble supporting themselves? They would be instant celebrities, historians would love to hear their first hand accounts of life in ancient Rome and so on. They could work as consultants for the movie industry, teachers of ancient crafts.

Unless future humanity has lost all interest in history, being being able to find something they could work with seems to be the least of the travellers problems. Adjusting to a very different society, on the other hand, may be hard. There is also the risk that humanity has changed so much or been replaced by machines that they would end up in a zoo.

David Appell said...

Hi Thomas. You make some good points, definitely true for Romans. But what if a group of Neanderthals showed up today, from 50,000 yrs ago, the time it currently takes to get to the nearest star? They couldn't take care of themselves in our society, but of course there'd be tremendous interest in examining them scientifically, at least for awhile. Travelers who took 50,000 yrs to get to their destinations, while the fastest ship took only 144 yrs, would be way out of sorts, unable to even speak the language probably, and perhaps having developed a strong dialect. And how much history would travelers on a generation ship have to share? At first glance it sounds rather boring....

But even a generation ship could be in close contact with Earth and any passing ships and the destination planet, though the delay time might be up to 4 years. But would you want a daily or monthly update on the travelings of a group of Neanderthals? Even if they could speak the latest language it might only be "Today we cross river Urk kill deer Arro still leak slow yellow water." :-)

If you think of the title of the Van Vogt book you mentioned, let me know.

Thomas said...

David, you are right that if it is a generational ship the passengers might be seen as less interesting, although antrolpogists might love to find out how an isolated community had developed over the years. In the story I menationed it was a smaller ship where they used suspended animation so they had first hand experience of the dawn of interstellar travel, something that was of obvious interest to a colony founded by later interstellar ships. The time was also a lot shorter than the 50,000 years you talk about.

I think it's impossible to predict how society will change over 50,000 years and thus impossible to guess how people would be received. Maybe there would be a mission to pick up the passengers on the first ship by a much later, faster ship?

If it wasn't too large I think a group of Neanderthals might do fine today. People would be curious and some nation would be willing to set aside a piece of land for them. If a problem is small and not too hard to solve people like to be generous. 100 refugees is fine, 100 000 and we close the borders.

I'm not even sure the story was by van Vogt, it must have been over 30 years since I read it. This was only a small part of it as the protagonist had a hidden motive to go forward in time to be able to get to a point where he knew there would be a time machine waiting so that he could go further back in time do do something I have forgotten.

On a much smaller scale you also have "The Mouse on the Moon" where Grand Fenwick launches a ship to the moon, the Americans, knowing their ship is ready and much faster, is in no hurry as they know they can overtake the first, and the Russians knowing their ship is faster than either of the other two decide to launch last just to prove it.

Entropic man said...

There was an SF short story by Van Gogh ca!led Far Centaurus in which a crew arrived to find their destination colonised, but could not integrate because of their awful smell.

It later evolved into a novel called Rogue Ship.

David Appell said...

Entropic Man: that's funny. Thanks.

David Appell said...

EM: Except the Wikipedia article says

"In the time it took for their ship to reach Alpha Centauri, human science had progressed to the point where the flight from Earth now takes only three hours."

and that's over 1 million times the speed of light!

David Appell said...

Bill: I agree, hibernated humans could be very interesting, relics of a time probably long lost to those who just traveled to the destination.
Picking up the slower travelers is a good idea, if there's room, and enough fuel to slow down and speed back up.

I never went through a juvenile phase reading science fiction -- mostly then I read books titled like "Great Linebackers of the NFL." So I never read any pulp sci-fi or classics from Asimov, Bradley, etc. I did read a book by Heinlein in my 20s -- don't remember its title, only that it had a strong sexy woman as a leading character. I'm sure she had big breasts. (She certainly did on the cover.) I've been reading a lot of science fiction in recent years, but some of the old stuff seems almost silly now, like The Foundation Trilogy and all its hopping around via fission engines. I need to try harder and just ordered "Stranger in a Strange Land."

I saw Asimov speak at a Sci-Fi conference at Stony Brook about 30 years ago. I don't remember anything he said.

Thomas said...

David, that Heinlein book may be "Friday", not one of his better. Heinlein had a rather odd view of women. Science fiction has a tendency to age if you take the prediction of the future at face value, rather than read it as commentary on the time it was written.

Some suggestions would be Iain M Banks 'Culture' series about an extremely high technology civilization mostly run by AI:s, which in many ways is an utopia, but where the books show how even those run into problems when interacting with other civilizations. It's one of the optimistic views on how humans can coexist with much smarter machines.

Or you could read 'Cyteen' by C J Cherryh about a colony where they were low on population and instead of relying on machines raised clones, specially adapted for their tasks, with the obvious ehtical problems. (It's part of a longer series, but the books are fairly independent).

Maybe you'll be interested by Asimov's speech on the greenhouse effect:

Ned said...

I think most SF stories involving interstellar travel tend to be wildly over-optimistic about the chances of (a) successfully holding things together (physically, socially, and ecologically) through such a journey and (b) ending up on a habitable world.

One exception is Stan Robinson's "Aurora" (2015). It's not a light-hearted, fun read, but probably as realistic as any book you'll ever read about interstellar travel. Almost everything you can think of goes wrong.

Ned said...

Also ... on another point it's a bit understated to refer to [24] m/s as "the speed of a good horse". That's the all-time record speed over a short distance for a horse in a sprint.

Kind of like saying 1228 km/hr is "the speed of a good car"!

David Appell said...

Thomas, you're right, the Heinlein book I was thinking of was "Friday." What do you think is his best book?

I've read two books by Iain Banks, "The Hydrogen Sonata" and "Use of Weapons." I loved the first, was lukewarm about the second. I just bought "The Player of Games" for $1 at a thrift shop, and it' on my reading pile.

I'll try "Cyteen," and thanks much for the link to Asimov's talk on the greenhouse effect.

David Appell said...

Ned, fair enough about the speed of a good horse. How about the speed of a good clipper ship, which would be more sustainable?

I will definitely read "Aurora." Thanks for the suggestion.