It's you, you're the Rocket Man

July 23, 2009
  • General
  • Physics
  • Technology

This week marked the 40th anniversary of man landing on the moon. Buzz Aldrin, among others, met with President Obama to discuss plans to create a permanent base on the moon and to one day fly a man to Mars (Aldrin lobbied for bypassing the moon part and going directly to the red planet). One day we will put a man on Mars. And we will probably put a man on its moon, Phobos and Deimos. We may one day put a man on Mercury, but thats a much more treacherous trip (its highly unlikely that well ever put a man on Venus because its way too hot; the surface temperature of 480 degrees Celsius can melt lead).

But what then? Maybe we put men on some outer moons, like Europa or Titan. But, after Mars, the excitement of a manned intrasolar trip will be limited. I was asked this week if wed ever be able to send a man to another solar system. I think answering that question is a good opportunity to learn some interesting physics.

Our nearest neighbor star is Alpha Centauri, which his 4.37 light years away. This means, of course, that it takes like 4.37 years to get there. Therefore, if we can send a space ship as fast as fast as possible and get close to the speed of light, it will take the ship 4.37 years to get there, or about 8.8 years for the round trip. Thats not terrible and its certainly conceivable that wed find an astronaut willing to spend about a decade of his life away from our Solar System.

Now, you may say that we dont have the technology to get a space ship that can move near the speed of light, to which I reply, rubbish. Theres no friction in space, so theres nothing to stop a ship from accelerating forever. A ship with a regular engine and enough fuel to burn can simply turn its engines on and wait until it gets close enough to the speed of light.

There is, however, an interesting complication. As an object moves closer and closer to the speed of light, it becomes harder and harder to accelerate. Meaning, if our ship turns on its engines at a constant burn and leaves them on, it will accelerate at a constant speed AT FIRST, but eventually relativistic effects will come into play, and it will accelerate slower and slower as it gradually approaches the speed of light.

Lets say that we want to accelerate our ship at 10 m/s/s. I make this choice, of course, because 10 m/s/s is how fast things accelerate in Earths gravitational field, so a person in a ship accelerating at that speed would weigh as much as he does on Earth (he would fell gravity due to the accelerating ship, just like you feel a pull forward and backward when your car breaks or accelerates quickly). If you ignore relativity, it will take about HALF a year to reach about HALF speed of light. If you include relativity, it will take approximately the same amount of time. The following graph shows that the change of speed is nearly linear for constant acceleration until you get above about .6 the speed of light:

So, theres not problem getting fast enough. You have enough fuel to burn for half a year to get up to speed, and then another half a year to slow down. And then you of course need to double that amount if you plan to get back to Earth. So, you need two years worth of fuel, and the rest of the time you simply coast. It would be a lot, but certainly not completely out of the question. (Im ignoring the change of mass to your ship due to the burning of your fuel. Including that is interesting, but not relevant to this discussion. Just assume that the ship is heavy compared to the mass of the fuel).

But, again, relativity adds an interesting twist. Because the person in the ship is flying at a speed approaching light relative to the Earth, he will experience time differently than his family sitting at home. His will pass through less time than people on Earth and therefore will be younger when he returns to his home planet than people who were his same age when he left. If he moves at half the speed of light, time will pass at the rate for everybody else as it does for him. So, if someone on Earth ages 10 years, he will only 7.5 years and will be 2.5 years younger than someone who was his same age when he left. So, an astronaut will only have to volunteer to be away from Earth for less than 8 years to complete the journey.

If one wants to do some concrete calculations, one can consider the following scenario. The astronaut blasts off in his ship, accelerates at 10m/s/s half way to the planet (he accelerates the entire time) and then starts to decelerate half way through his journey (so hell stop just as he reaches the star). His ship burns fuel with 100% efficiency (meaning you have some sort of nuclear reactor that can convert a mass M of fuel into Mc^2 in energy), and we consider the fact that burning fuel reduces the mass of the ship. This site shows how one can do the relativistic calculations involved and gives some neat results:

Distance Location Time for astronaut Fuel 4.3 ly nearest star 3.6 years 38 kg 27 ly Vega 6.6 years 886 kg 30,000 ly Center of galaxy 20 years 9.55 * 10^8 kg 2,000,000 ly Andromeda 28 years 4.2 *10^12 kg

Finally, this I think is the most interesting part. Consider making a flight to the center of the galaxy. It will take 20 years if you accelerate in the above way, but will cost A LOT of fuel. So, why not just accelerate for about half a year and get very close to the speed of light? Why accelerate the whole way? Since you cant go faster than light, as you approach the speed of light, accelerating increases your speed by asymptotically smaller and smaller amounts. You end up burning tons and tons of fuel just to increase your speed by, say .2 the speed of light.

So, the actual time for someone on early is nearly unaffected by all this extra effort. However, interestingly, it make a HUGE difference for the person on the ship. If he just accelerated to about half the speed of light and then coasted the rest of the way, it would take the man nearly 60,000 years! However, if he accelerates constantly the whole way, it only takes him 20 years!!! The speeds are very close to each other, but the difference in time is enormous. So, to the astronaut, all that extra fuel spend accelerating the whole way is certainly justified (he can actually live through the trip!!).

Of course, if youre sending robots who dont care about the passage of time, you would simply save the money on fuel and only accelerate at the very beginning and the very end. They dont mind twiddling their thumbs for 60,000 years.