The Constant Acceleration Mission

While this may not be the preferred option, it's useful to see how our reference mission could be accomplished at constant acceleration. We accelerate for half the mission time to a maximum velocity then decelerate for the second half to end up stationary when we reach our target. While there will be slight corrections due to relativistic effects we can get a good handle of what this entails using the simple non-relativistic equation for distance travelled under constant acceleration

    d = 1/2 a t2

For d and t we can substitute in half our mission values and we get

    5 x 1016 m = 0.5 a (1.5 x 109)2

This gives us

    a = 5x1016 m / 1.25x1018m2 = 4x10-2m/s2

The gravitational acceleration on Earth is just under 10 m/s so this corresponds to about 4x10^-3g. The acceleration required to reach our goal is very small. If we again ignore relativistic effects we can calculate the maximum velocity as

     vmax = at = 4x10-2 x 1.5x109 m/s = 6x107 m/s

This is 20% of the velocity of light, so given the rough nature of calculation ignoring relativity is probably fine. A consequence of using constant accelerations is that our maximum velocity is exactly twice the average velocity. If we want to go 10 light years in 100 years, our average velocity is manifestly 0.1 c.