Alphacentauri

Peter Fisher

11/26/08

8.033

When trying to learn how to think about special relativity, it is useful to think about an actual problem. Over the weekend, I was thinking about what it would take to get to Alpha Centauri, the nearest star to the Sun, 4.8 light years away. One light year is about 9 ×1015 m . Alpha Centuri 7.00E+00 6.00E+00 5.00E+00 Time Position beta Gamma

4.00E+00 3.00E+00 2.00E+00 1.00E+00

Step

I used excel, divided the trip up into 100 time steps. Each step corresponds to a time dt (22 days in my case). At time step i, the ship’s velocity is vi and in that time step, the ship acquires dv’i=a’dt’ where a’=10 g. Then the velocity in the next time step is v + dvi′ vi+1 = i ′ 1+ vi dvi 2 c using the velocity addition rule. The figure at the left shows position, β and γ as functions of time in the Earth frame. The ship gets to 90% of the speed of light in just over a two years and reaches a top speed of 0.95c or γ=3.4 at the turn around point. The whole journey takes just over six years, impressive when compared to a light ray’s time of 4.8 years. The Minkowski diagram is shown at the left. The clock time for the crew on the ship may be found by integrating interval 5.8ly

s=

∫

2

dt 1 − ( dx / dt ) = 0.39ly

0

so only about five months pass for the crew. This is because they are fairly relativistic and, for a photon’s trajectory (shown as the diagonal dashed line in the figure above), the interval is zero. From this, we can see that a mission could go to Alpha Centauri, spend a year there and return in about 13 years. The returning crew would have aged less than two years. Our calculation has met the minimum requirement of not killing the crew with a tremendous acceleration, but the other obstacles are huge: the amount of fuel needed would be vast and, even that could be solved, the radiation from cosmic rays would be fatal after a year or so. I did a study once for NASA about shielding astronauts from cosmic rays and found the system would weigh 300 tons and that was just for a Mars mission.

96

91

81 86

71 76

61 66

51 56

41 46

26

31 36

21

11 16

1

0.00E+00 6

From a practical standpoint, the greatest acceleration we could use would be about 10g or 100 m/s2, the limit of what a human can withstand. The fastest way of getting there (what NASA calls the “mission profile”) would be to accelerate at 10g until the ship is halfway to Alpha Centauri, turn the rocket around and decelerate until the ship reaches Alpha Centauri at low speed. Remember, once the ship gets there, it need to go into orbit and that means it has to be going at low speeds.