Getting to the stars won’t be easy. But nor will it be utterly impossible either, as “Project Icarus” is showing with its research. One option for our hypothetical flight to the stars is using pure fusion propulsion – for acceleration and braking. To that end, the most efficient rocket is a staged rocket, when the exhaust velocity is much smaller than the required mission velocity.
Say we want to fly to Alpha Centauri in 100 years. If we could accelerate and decelerate in an instant, then the total mission velocity is twice the average velocity (I.e. 4.36 ly/100 years = 0.0436c times two, or 0.0872 c, some 26,000 km/s.) Of course we can’t do that, so our vehicle must spend some time accelerating and braking. But what is optimal? If we spent the whole trip under constant acceleration the delta-v (total velocity change) would be twice the instant acceleration figure. Rockets have to carry their propellant so maintaining constant acceleration requires an engine that can be throttled over a wide power range – not easily done and very wasteful as less thrust is needed during the later part of the journey. No point carrying all that extra engine mass uselessly. Thus the Stage Rocket – drop big engines and empty tanks as you go.
Carrying propellant in tanks and figuring out the mass of engine needed to push itself plus tanks, plus propellant plus payload, means the size required is not easy to figure out. We need to be resigned to relatively small mass-ratios or else the mass required goes to infinity – i.e. more propellant means more engine which means more tankage which means more propellant… creating an asymptote. For example if tankage masses 3% of the propellant then the absolute highest, for zero engine and payload mass, is a mass ratio of ~33. But there’s a way to do it – we add stages. Stage mass-ratios multiply which means enough stages can allow us to achieve arbitrarily high mass-ratios.
We also want to minimise the power required and we do that by clever thrust programming. For a given voyage time, exhaust velocity and voyage distance we can find a minima – but it’s not easy. Minima are typically found by setting the derivative of a function with the relevant variables to zero i.e. we find P (power) in terms of thrust time, then find dP/dt and set it to zero, solving that equation for thrust time. And the derivative is horrible! There’s no unique solution – unlike the case for constant acceleration (not constant thrust) in which case minimum power is achieved by thrusting for 1/3 the voyage time.
By graphing and hunting for the minima visually a trip to Alpha Centauri lasting 100 years requires a Power/Mass ratio of 144.6 kW/kg, a mass-ratio of 6.65 and a cruise speed of 0.0632c for an exhaust velocity of 10 million m/s, if the thrust time is 62% of the voyage time. Assuming a 450 tonne “Daedalus” class payload, 3% tankage/propellant mass and an engine specific power of 500 kW/kg means a second stage mass of 5,535 tonnes and a total launch mass of 68,090 tonnes.
However, as in many things, the functions are rather surprising – minimum power doesn’t equate to minimum mass. That’s achieved with a thrust/voyage ratio of 0.34 and a mass of 37,806 tonnes. The P/M ratio is 178.7 kW/kg and the mass-ratio is 4.835, giving a cruise speed of 0.0525c. Assuming a specific engine power of 500 kW/kg remember. What happens if the specific power is 300 kW/kg? Thrust time is 43%, stage 2 masses 6,922 tonnes and launch mass is 106,500 tonnes. A dramatic difference!
Next post in this series I will explore what happens when we change the exhaust velocity and voyage times.
