Incentive Trap 2: Calculating Minimum Time to Arrival

When to launch a starship, given that improvements in technology could lead to a much faster ship passing yours enroute? As we saw yesterday, the problem has been attacked anew by René Heller (Max Planck Institute for Solar System Research), who re-examined a 2006 paper from Andrew Kennedy on the matter. Heller defines what he calls ‘the incentive trap’ this way:

The time to reach interstellar targets is potentially larger than a human lifetime, and so the question arises of whether it is currently reasonable to develop the required technology and to launch the probe. Alternatively, one could effectively save time and wait for technological improvements that enable gains in the interstellar travel speed, which could ultimately result in a later launch with an earlier arrival.

All this reminds me of a conversation I had with Greg Matloff, author of the indispensable The Starflight Handbook (Wiley, 1989) about this matter. We were at Marshall Space Flight Center in 2003 and I was compiling notes for my Centauri Dreams book. I had mentioned A. E. van Vogt’s story “Far Centaurus,” originally published in 1944, in which a crew arrives at Alpha Centauri only to find its system inhabited by humans who launched from Earth centuries later. I alluded to this story yesterday.

Calling it a ‘terrific story,’ Matloff discussed it in terms of Robert Forward’s thinking:

“Bob had a couple of concepts of technological advancement. He had a famous plot of the velocity of human beings versus time. And he said if this is true, and you launch a thousand-year ship today, in a century somebody could fly the same mission in a hundred years. Theyre going to be passed and will probably have to go through customs when they get to Alpha Centauri A-2.”

Customs! Clearly, we’d rather not be on the slow starship that is superseded by new technologies. What Heller and Kennedy before him want to do is to figure out a rational way to decide when to launch. If we make assumptions about the exponential growth in speed over time, we can address the question by adding the time we spend waiting for better technology to the time of the actual journey. We can then calculate a minimum value for this figure based on the growth rates we find in our historical data.

This is how Kennedy came up with a minimum figure of 712 years (from 2006) to reach Barnard’s Star, which is about 6 light years away. The figure would include a long period of waiting for technological improvement as well as the time of the journey itself. Kennedy used a 1.4 percent annual growth in speed in arriving at this figure but, examining 211 years of data on historical speed records, Heller finds a higher annual growth, some 4.72 percent.

From the Penydarren steam locomotive of 1804 to Voyager 1, we see a speed growth of about four orders of magnitude. Growth like this maintained for another 112 years leads to 1 percent of lightspeed.

Image: A Bussard ramjet in flight, as imagined for ESA’s Innovative Technologies from Science Fiction project. Credit: ESA/Manchu.

But how consistent should we expect the growth in speed over time to be? Heller points out that the introduction of new technologies invariably leads to jumps in speed. We are now in the early stages of conceptualizing the Breakthrough Starshot project, which could create exactly this kind of disruption in the trend. Starshot aims at reaching 20 percent of lightspeed.

Working with the exponential speed doubling law we began with, we would expect that a speed of 20 percent of c would not be achieved until the year 2191. But if Starshot achieves its goal in the anticipated time frame of several decades, its success would see us reaching interstellar speeds much faster than the trends indicate. Starshot, or a project like it, would if successful exert a transformative effect as a driver for interstellar exploration.

We know that speed doubling laws cannot go on forever as we push toward relativistic speeds (we can’t double values higher than 0.5 c). But as we move toward substantial percentages of the speed of light, we see powerful gains in speed as we increase the kinetic energy beamed to a small lightsail like Starshot’s. Thus Heller also presents a model based on the growth of kinetic energy, noting that today the Three Gorges Dam in China can reach power outputs of 22.5 GW. 100 seconds exposure to a beam this powerful would take a small sail probe to speeds of 7.1 percent of c. Further kinetic energy increases could allow relativistic speeds for at least gram-to-kilogram sized probes within a matter of decades.

Usefully, Heller’s calculations also show when we can stop worrying about wait times altogether. The minimum value for the wait plus travel time disappears for targets that we can reach earlier than a critical travel time which he calls the ‘incentive travel time.’ Considered in both relativistic and non-relativistic models, this figure (assuming a doubling of speed every 15 years) works out to be 21.6 years. In Heller’s words, “…targets that we can reach within about 22 yr of travel are not worth waiting for further speed improvements if speed doubles every 15 yr.”

Thus already short travel times mean there is little point in waiting for future speed improvements. And in terms of current thinking about Alpha Centauri missions, Heller notes that there is a critical interstellar speed above which gains in kinetic energy beamed to the probe would not result in smaller wait plus travel times. His equations result in a value of 19.6 percent of c, an interesting number given that Breakthrough Starshot’s baseline is a probe moving at 20 percent of c, for a 20-year travel time. Thus:

In terms of the optimal interstellar velocity for launch, the most nearby interstellar target α Cen will be worthy of sending a space probe as soon as about 20 % c can be achieved because future technological developments will not reduce the travel time by as much as the waiting time increases. This value is in agreement with the 20 % c proposed by Starshot for a journey to α Cen.

We can push this result into an analysis of stars beyond Alpha Centauri. Heller looks at speeds beyond which further speed improvements would not result in reduced wait times for ten of the nearest bright stars. The assumption here would be that Starshot or alternative technologies would be continuously upgraded according to historical trends. Plugging in that assumption, we wind up with speeds as high as 57 percent of lightspeed for 70 Ophiuchi at 16.6 light years.

Thus the conclusion: If something like Breakthrough Starshot’s beaming capabilities become available within 45 years — and assuming that the kinetic energy transferred to the probes it pushes could be increased at the historical rates traced here — then we can reach all ten of the nearest star systems with an interstellar probe within 100 years from today.

Just for fun let me conclude with a snippet from “Far Centaurus.” Here a ship is approaching the ‘slowboat’ that has just discovered that Alpha Centauri has been reached by humans long before. The crew has just puzzled out what happened:

I was sitting in the control chair an hour later when I saw the glint in the darkness. There was a flash of bright silver, that exploded into size. The next instant, an enormous spaceship had matched our velocity less than a mile away. Blake and I looked at each other. “Did they say,” I said shakily, “that that ship left its hangar ten minutes ago?” Blake nodded. ‘They can make the trip from Earth to Centauri in three hours,” he said. I hadn’t heard that before. Something happened inside my brain. “What!” I shouted. “Why, it’s taken us five hund… ” I stopped. I sat there. “Three hours!” I whispered. “How could we have forgotten human progress?”

The René Heller paper discussed in the last two posts is “Relativistic Generalization of the Incentive Trap of Interstellar Travel with Application to Breakthrough Starshot” (preprint).