Richard Trevithick’s name may not be widely known today, but he was an important figure in the history of transportation. A mining engineer from Cornwall, Trevithick (1771-1833) built the first high pressure steam engine, and was able to put it to work on a railway known as the Penydarren because it moved along the tramway of the Penydarren Ironworks, in Merthyr Tydfil, Wales, running 14 kilometers until reaching the canal wharf at Abercynon. The inaugural trip marked the first railway journey hauled by a locomotive, and it proceeded at a blistering 4 kilometers per hour. The year was 1804.


Image: The replica Trevithick locomotive and attendant bar iron bogies at the Welsh Industrial & Maritime Museum in 1983. Credit: National Museum of Wales.

Consider, as René Heller (Max Planck Institute for Solar System Research) does in a new paper, how Trevithick’s accomplishment serves as a kind of bookend for 211 years of historical data on the growth in speed in human-made vehicles from the Penydarren to Voyager 1. The world’s first production car was the Benz Velocipede (1894), whose top speed of 19 kilometers per hour far surpassed the Trevithick railway, but was put to shame by a Stanley Steamer racing car that reached a then incredible 204 kilometers per hour in 1903.

I mused about the nature of speed in a 2013 post called The Velocity of Thought, and Heller’s new paper has me doing it again, though in entirely different directions. A few more waypoints and I’ll explain what I mean. When the Wright Brothers took to the air in 1903, their Wright Flyer first flew at about 11 kilometers per hour, and we began to see how quickly aviation records could be superseded. A Sopwith Camel of World War I vintage could reach 181 kilometers per hour. By 1944, German test pilot Heini Dittmar was able to take a ME-163 rocket plane to 1130 km/h, a number that wouldn’t be reached again for almost ten years.


Image: Typical appearance of a Me-163 Komet after landing, waiting for the airfield’s Scheuch-Schlepper tractor and lifting trailer to tow it back for reattachment of its “dolly” maingear. Credit: Wikimedia Commons.

When we get into space, we can note Voyager 1’s 17 kilometers per second as it leaves the Solar System. The Helios solar probes launched in 1974 and 1976 set the current record at 70.22 km/s. And looking forward, the Solar Probe Plus mission is to perform a close flyby of the Sun, reaching a top heliocentric speed of 195 kilometers per second, which works out to 6.5 × 10 ?4 c. If Breakthrough Starshot realizes its goal, an interstellar lightsail may one day head for Proxima Centauri at fully 20 percent of the speed of light.

Part of what occupies René Heller in his new paper is the exponential growth law we can construct between the 1804 Penydarren locomotive and the 17 kilometers per second of Voyager 1 in 2015. From wind- to steam-driven ships and into the realm of automobiles, then aircraft and, finally, rockets, we can extrapolate speeds that may take us into interstellar probe territory some time in this century or the next. Given that an interstellar mission may take longer than the average human lifetime, we thus need to ask a key question. When do we launch?

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Image: Figure 1 from the Heller paper, showing historical speed records. From the paper: “All these values are symbolized with black-rimmed circles in Figure 1, with additional top speed measurements of trains, cars, planes, and rockets shown with different symbols (see legend). The dashed black line illustrates an exponential growth law connecting the 1 m s ?1 speed of the “Penydarren” steam locomotive in 1804 with the 5.7 × 10 ?5 c Solar System escape speed of Voyager 1 in 2015. Credit: René Heller.

For the problem, a classic in science fiction, is to work out the most efficient timing. If we launch a starship at a particular level in our technology, will it not be caught by a faster ship launched at a much later date? Given sufficient technological improvements, a later launch (incorporating the necessary ‘wait time’) could result in an earlier arrival.

Those who have read A. E. van Vogt’s story “Far Centaurus” will recall precisely that scenario, when an Alpha Centauri mission reaches destination only to find it populated by humans who arrived by faster means. It’s a theme that shows up in Heinlein’s Time for the Stars and many other places.

Heller calls this problem ‘the incentive trap.’ And he refers back to Andrew Kennedy’s 2006 paper, which looked at the problem with the assumption of an exponential growth of the interstellar travel speed. Kennedy was assuming a 1.4 % average growth rate, under which a minimum time to reach Barnard’s Star could be calculated: some 712 years from 2006.

What that means is this: There is a total time that includes the waiting time (waiting for improved technology) and the actual travel time, and we can calculate a minimum value for this total time by using our assumption about the exponential growth of the interstellar travel speed. Calculating the minimum value shows us when we can launch without fear of being overtaken by a faster future probe, in hopes of avoiding that “Far Centaurus” outcome.

But was Kennedy right? Heller’s own take on the incentive trap takes into account the possibility that Breakthrough Starshot may achieve a velocity of 20 percent of lightspeed within several decades, an outcome that would, in Heller’s words, “…fundamentally change both the assumptions and the implications of the incentive trap because the speed doubling and the compounded annual speed growth laws would collapse as v approaches c.” And whatever happens with Breakthrough Starshot, the speed growth of human-made vehicles turns out to be much faster than previously believed.

Intriguing results flow out of Heller’s re-examination of what Kennedy had called the ‘wait equation,’ and tomorrow I want to go deeper into the paper to explain how the scientist uses exponential growth law models to show us a velocity which, once we have attained it, will no longer be subject to the incentive trap of faster, later technologies. The results are surprising, particularly if Breakthrough Starshot achieves its goal in the planned 30 years. The implications for our reaching well beyond Alpha Centauri, as we’ll see, are striking.

The Heller paper is “Relativistic Generalization of the Incentive Trap of Interstellar Travel with Application to Breakthrough Starshot” (preprint). The Kennedy paper is “Interstellar Travel: The Wait Calculation and the Incentive Trap of Progress,” Journal of the British Interplanetary Society Vol. 59, No. 7 (July, 2006), pp. 239-247.