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starship-design: Constant 1 g Acceleration
Okay, I chickened out and didn't write a program, but here is the next best thing...
Original by Philip Gibbs 21-September-1996
The Relativistic Rocket
The theory of relativity sets a severe limit to our ability to explore the galaxy in space-ships. As an object approaches the speed of light more and more energy is needed to accelerate it further. To reach the speed of light an infinite amount of energy would be required. It seems that the speed of light is an absolute barrier which cannot be reached or surpassed by massive objects. Given that the galaxy is about 100,000 light years across there seems little hope for us to get very far in galactic terms unless we can overcome our own mortality.
Science fiction writers can make use of wormholes, or warp drives to overcome this restriction but it is not clear that such things can ever be made to work in reality. Another way to get round the problem may be to use the relativistic effects of time dilation and length contraction to cover large distances within a reasonable time span for those aboard a space-ship. If a rocket accelerates at 1g (9.81 m/s2) the crew will experience the equivalence of a gravitational field the same as that on Earth. If this could be maintained for long enough they would eventually receive the benefits of the relativistic effects which improve the effective rate of travel.
What then, are the appropriate equations for the relativistic rocket?
First of all we need to be clear what we mean by continuous acceleration at 1g. The acceleration of the rocket must be measured at any given instant in a non-accelerating frame of reference travelling at the same instantaneous speed as the rocket. This acceleration will be denoted by a. The proper time as measured by the crew of the rocket will be denoted by T and the time as measured in a the non-accelerating frame of reference in which they started will be denoted by t. We assume that the stars are essentially at rest in this frame. The distance covered as measured in this frame of reference will be denoted by d and the speed v. The time dilation or length contraction factor at any instant is gamma
The relativistic equations for a rocket with constant acceleration a are,
t = - sinh - T
c c 2
d = - ( cosh(aT/c) - 1 ) = - ( sqrt[ 1 + (at/c) ] - 1 )
v = c tanh - T = at / sqrt[ 1 + (at/c) ]
gamma = cosh - T = sqrt[ 1 + (at/c) ]
To do some example calculations it is easy to use units of years and light years. Then c = 1 and g = 1.03. Here are some typical answers for a = 1g.
T t d v gamma
1 year 1.19 yrs 0.56 lyrs 0.77c 1.58
2 3.75 2.90 0.97 3.99
5 83.7 82.7 0.99993 86.2
8 1,840 1,840 0.9999998 1,890
12 113,000 113,000 0.99999999996 117,000
So in theory you can travel across the galaxy in just 12 years of your own time. If you want to arrive at your destination and stop then you will have to turn your rocket round half way and decelerate at 1g. In that case it will take nearly twice as long for the longer journeys. Here are some of the apparent times required to get to a few well-known spacemarks to arrive at low speed:
4.3 ly nearest star: 3.6 years
30,000 ly Center of our galaxy: 21 years
2,000,000 ly Andromeda galaxy: 29 years
For distances bigger than about a billion light years the formulas given here are inadequate because the universe is expanding. General Relativity would have to be used to work out those cases.
Sadly there are a few technical difficulties you will have to overcome before you can head off into space. One is to create your propulsion system and generate the fuel. The most efficient self contained rocket system which is possible is a matter/anti-matter photon drive. Matter and anti-matter fuel are allowed to annihilate and the gamma rays are directed out the back (somehow) to propel the rocket forward. Even then the ratio of the mass of fuel used M to the mass of the payload of the rocket m is given by M/m = gamma v/c. Conservation of momentum forbids you to do any better than this with a self contained drive carrying all its own fuel.
The next problem you have to solve is shielding. As you approach the speed of light you will be heading into an increasingly energetic and intense bombardment of cosmic rays and other particles. After only a few years of 1g acceleration even the cosmic background radiation is Doppler shifted into a lethal heat bath hot enough to melt all known materials.
So as we can easily see, the near stars are nearer than they appear and the far stars are even closer than that! All we need is a drive that can accelerate our ship at a constant 1 g for 12 to 24 years....