“The very closest stars would require many years to visit, even traveling at the speed of light, which is impossible according to Einstein’s theory of relativity. Today’s fastest spaceships would require 200,000 years to travel to Alpha Centauri, our closest bright star. The energy required to send a hundred colonists to another star, as Frank Drake has pointed out, would be enough to meet the energy needs of the entire United States over a human lifetime. And these estimates are regarding nearby stars. When we consider the distances across the entire galaxy, and between galaxies, interstellar travel seems absolutely untenable.” -David E. Fisher
It’s always easy to point out the difficulties with a dream. The stars, after all, are so incredibly far away, that the distances are, well, astronomical.
Alpha Centauri, the closest bright star (the bright, yellow one, above) to us, is about 270,000 times farther away than the Sun is. And the farthest a human has ever traveled, of course, is to the far side of the Moon, just 0.4% of the distance to the Sun.
That, of course, was something that we did back during the Apollo missions. And there was a very impressive thing that happened. Back during the 1960s and 1970s, we launched human beings into space, for the first time, with sustained accelerations for record-breaking periods of time.
The Saturn V rocket — shown here during the 1972 launch of Apollo 17 — was capable of accelerating humans from rest to a speed of 25,000 miles-per-hour (about 11 km/s), over the timespan of only about 10-15 minutes.
We’ve done it hundreds of times in the United States alone, as the space shuttle now does almost exactly the same thing.
The key, of course, isn’t to get as large of an acceleration as possible. Here on Earth’s surface, we are accelerated towards the Earth’s center at a constant rate of 9.8 m/s2. That’s the force of gravity that pulls us (and everything else) down.
But if we built a device — theoretically — that could accelerate us at that rate for an arbitrarily long amount of time, could we ever reach the speed of light?
Look, Your Worshipfulness, let’s get one thing straight. I take orders from just one person: me Einstein.
You might think that all you have to do is take that acceleration and sustain it, and you’ll eventually reach the speed of light. Let’s take a look at what happens if you could do it!
At first, you’d notice that you’d just be going faster and faster, just like you wanted. You’d need to apply a constant force to your ship, and as long as you did, you’d keep on increasing your speed. The graph above looks like a nice straight line, and as the seconds turned into minutes, and the minutes turned into hours, you’d find yourself speeding up, faster and faster, gaining an extra 9.8 m/s of speed for every second you continued to accelerate. The human speed record, at 11 km/s, would be passed in under 20 minutes.
And as the hours turned into days and eventually weeks, you’d think you were headed on your way to the speed of light. At exactly 299,792.458 km/s, you might look back sometime after a little under three-and-a-half days, and mark the milestone that you were 1% of the way there. And perhaps you’d get giddy, after almost a full week, when you realized you were 2% of the way to the speed of light. After all, at 4.37 light years away, Alpha Centauri doesn’t seem so distant if you can make the trip at the speed of light!
But as the weeks passed by and turned into months, you would notice something very interesting, if not troubling.
Even with the same force and the same thrust, you wouldn’t be accelerating quite as quickly as you were at the beginning. This is very bizarre and counterintuitive, and is a consequence of special relativity. It gets harder and harder to change your speed as you start to get close to the speed of light. But if the curve in the above graph isn’t enough to tip you off to the trouble you’ll encounter trying to get up to the speed of light, perhaps you’ll really be able to see what’s going on if we start looking at this sped-up spacecraft after a few years of traveling.
There’s a limit to how fast you can go! And while you can’t quite tell from the graph, you don’t reach the speed of light, you just approach it.
Even in the graph above, you continue to accelerate the whole time, and approach the speed of light, incrementally. After about 15 months, you’ll have attained 90% the speed of light, which is truly amazing, but not what you would have naïvely expected if you had simply continued your original “straight line” graph of velocity vs. time.
And after about two-and-a-half years, you would reach 99% the speed of light. But will you ever get there? Let’s extend the journey for a whole decade and see.
From this graph, it’s impossible to tell. But if I look at the difference of the speed of light (c) and your speed (v), what I’ll call c – v, perhaps we can tell.
After all, it took about a year-and-a-quarter to reach 90% of c, and about 2.5 years to reach 99% of c. Will this pattern continue? A graph of c – v won’t show that, but a graph of the log of c – v will!
(For those of you aspiring to become scientists, teaching computers to do your calculations/graphs/dirty work is a must!)
A difference of 1 in the log means a factor of 10. So after about 4.5 years, we’re up to 99.99% the speed of light. By time about 6 years and 4 months have passed, you’re moving at 299,791 km/s, just one km/s below the speed of light. But — while the first km/s took only 102 seconds to attain — this last one never comes. After another year, you’ve made it another 0.9 km/s, and after another 3 years, another 0.999 km/s.
So you can continue to approach the speed of light, spending an incredible amount on fuel and thrust, but you’ll never get there.
And, of course, there’s the one thing we haven’t talked about, which is the true cost of this.
All of this was calculated assuming a constant thrust, or force-per-unit-time. Which is a fine assumption. After all, that’s about what the three-stage rocket that launched the Apollo astronauts achieved.
But the big cost comes in terms of energy. The first ten minutes of acceleration takes a certain amount of energy, and by the end of it, you’re moving at about 6 km/s. The second ten minutes, however, will get you up to double the speed at 12 km/s, but takes three times as much energy. The next ten minutes will get you up to 18 km/s, but uses up five times as much energy as the first ten minutes.
And this pattern continues. By time a year has gone by, you’re using over 100,000 times the amount of energy you started out using, and you’re still using it every ten minutes! Not only that, but you’re not even increasing your velocity by the same amount; your attempts to change your speed get progressively less and less effective.
Which isn’t to say that we should give up on interstellar travel! It simply helps us be aware of what the obstacles are that we’re trying to overcome, so that we can figure out how we might do it.
And remember, this is space, where there’s no air, so once you reach your desired speed, you can simply coast at that high velocity until you need to slow yourself down. (And that, of course, takes the same amount of energy it took to speed yourself up!)
The most fun part about this, for me, is that if you work out how time dilation and length contraction work, you can literally — given as much energy as you want — reach any star or galaxy within about 13 billion light years of us in just one human lifetime. That’s the cost and reward of reaching for the stars, and it’s the limit of the speed of light that allows us to understand exactly what it takes to make it happen!