# Proxima Centauri b: Mission Design Notes – Part II

One problem with high-speed flight to Proxima Centauri is the energy required for extremely high exhaust velocities is a non-trivial fraction of the rest mass of the propellant. Einstein’s famous relation E = mc2 tells us that there’s a lot of energy tied up in mass. But the flip side is that pushing propellant to a significant fraction of c requires a LOT of energy. Exactly how much?

The basic equation is:

β = (2ε-ε2)

Where β is the ratio of the exhaust velocity V to c, and ε is the mass-fraction of the propellant’s kinetic energy.

Solving for ε gives:

ε = [1 – √(1 – β²)]

Converting to a joule value means multiplying by c² i.e. 9 x 1016 for every kilogram of reaction mass expelled at the required exhaust velocity.

Nuclear fission and nuclear fusion reactions convert very tiny fractions of mass into energy – about 0.0009 and 0.004 for Uranium and deuterium/helium-3 reactions, for example. Enough for powering a planetary and interplanetary civilization, but not quite up to the task of powering fast starships, unless very large mass-ratios and very high power levels are deployed. An early paper on interstellar flight computed the performance for starships with mass-ratios up to 1,000,000 (!!) Of course that would require multiple staging and rather exotic storage tanks. If the mass-ratio of each stage was 10, and the dry-mass of the stage was 0.5 (the rest being the next stages) then the mass-fraction of propellant to payload is (20)6 = 64,000,000 for the case of a 1,000,000 mass-ratio. The scale boggles the mind.

For a perfect D/3He rocket the above case means a top speed of 0.55 c, if we’re using rockets to slow down. The authors of the study concluded that speeds up to 0.8 c were physically possible. However when they ran the actual numbers for a realistic fusion rocket, the nominal 5 light-year mission required 50 years travel time, largely due to the incredible heating load from the x-rays produced by the fusion reaction. The acceleration was low while the waste-heat radiators were very large and very hot.

Thus realistic Mission Design requires trade-offs.

References:

D.F. Spencer and L.D. Jaffe. “Feasibility of Interstellar Travel.” Astronautica Acta. Vol. IX, 1963, pp. 49–58. [online]

D. F. Spencer, “Fusion Propulsion for Interstellar Missions,” Annals of the New York Academy of Science, 140, December 1966: 407–18.