Mission design, before we get into the nitty gritty of hardware, requires us to consider the very basics. Energy, power, speed, distance and time. How quickly do we want to get to Proxima Centauri b, which is 267,000 AU (4.224 light-years or 1.295 parsecs) away? If we say “20 years” like most extreme Deep Space missions closer to home, then we have an average speed of 13,350 AU/year or 63,279 km/s. A bit more than 1/5 the speed of light. How quickly we get to that speed will determine more hardware related questions like just how much the engine masses for the amount of power it produces – the specific Power or Power-to-Mass ratio. The Power-to-Mass ratio is the chief performance metric for rockets. For example, a torch is a simple photon rocket, but its battery stored energy is insufficient to accelerate it in a measurable way. Likewise any other reaction engine – the acceleration depends on the average power applied to every newton of thrust and the mass of the engine/generator/battery supplying that power.
Also, acceleration takes time – a bit over 30,000,000 seconds is needed to reach light-speed at 1 gee. Relativity makes that relationship a bit more complicated, though not from the point-of-view of an observer on the rocket. An integrating accelerometer, which adds up every moment of acceleration to give a speed, will tell you that your ship has exceeded light-speed in a bit under a year at 1 gee. This ‘speed’ is otherwise known as the ‘rapidity’ and can be plugged straight into Tsiolkovskii’s Rocket Equation.
The basic equations are available here: The Relativistic Rocket …though Wikipedia’s discussion is pretty decent these days.
Rapidity is related to the speed measured by an observer at the local standard of rest (LSoR) by:
r = gT/c and v = c.TANH(r)
Where T is the ship-measured elapsed time under thrust. The acceleration, g, is constant for this example. So 1 c (measured on-ship) is 0.761 c measured by LSoR observers. Applied to the Tsiolkovskii Rocket Equation:
r = u.LN(Mo/Mf)
where u is the exhaust velocity.
If we want to know the mass-ratio (Mo/Mf) for a rocket that accelerates, then brakes to a stop under thrust, the equation in terms of LSoR measured velocity is quite simple:
(Mo/Mf) = ((v+1)/(v-1))(c/u)
…where v is in c units. The mass-energy in the reaction mass as the exhaust velocity gets close to c is factored into the equation already. With a pure photon exhaust, all pointed in the direction of thrust, the equation is just:
(Mo/Mf) = (v+1)/(v-1)
If v is 0.99 c, then (Mo/Mf) = 1.99/0.01 = 199
That’s some mass-ratio! If you can slam on the space-brakes instead and not need rockets to stop, it’s…
(Mo/Mf) = SQRT((v+1)/(v-1)) = 14.1 for v = 0.99c
To boost all the way to Proxima b at 1 gee will take 5.85 years Earth-time and about 3.535 years via ship’s clocks. Top r is 1.824 c while v is 0.9493 c, and the pure photon-rocket mass-ratio is 38.425 (6.2 with space-brakes.)
For the case consider above, 20 years to Proxima b, the acceleration required can be computed from the trip-time equation:
t = SQRT[ (S/c)2 + 4.S/g ]
…which doesn’t factor in any period of cruising at constant velocity. Big ‘S’ is the ‘displacement’ or distance traveled. Solving for g, we get:
g = 4.S / [t2 – (S/c)2]
It’s interesting to compare with the non-relativistic version:
g = 4.S / t2
…notice how the ‘light-travel time’ (S/c) becomes an additional component of time in the relativistic case.
Thus g is just 0.42 m/s2 and the top-speed at the half-way point is g.t/SQRT(1 + (a.t/(2c))2) = 0.4044 c