Some time ago some of us here were calculating what speed a VASIMR craft could get to if it had a 1 kg/kW power source. It was concluded that the mass of propellant made the trip to alpha centauri prohibitively long (something like 1500 years).

What if we used a Bussard ramjet scoop and got our propellant from the hydrogen sitting in space (1 molecule per cubic centimeter)? If you pair this with an ideal fusion power source (I know, this may be a long time coming), then it appears you can get to 0.10c in 48 years.

From the VASIMR Wikipedia page

http://en.wikipedia.org/wiki/VASIMR For the engine requiring 200 kW, the exhaust velocity of the propellant is 50 km/s and the force produced is 5 N. Assuming propellant is hydrogen (the only thing available in interstellar space):

p = mv = (1.67E-27 kg)(5E4 m/s) = 8.4E-23 kgm/s

Force produced by 1 hydrogen atom:

F = dp/dt = (8.4E-23 kgm/s)/ 1 s = 8.4E-23 N

Assuming the spacecraft is 2E5 kg and the acceleration we want to accelerate at 0.02 m/s^2,

F = ma = (2E5 kg)(0.02 m/s^2) = 4000 N

Getting thrust from these hydrogen atoms,

4000 N/8.4E-23 N/atom = 4.8E25 atoms required per second.

In space

1 H2 atom per cm^3 means 2E6 atoms/m^3

(4.8E25 atoms/s)/(2E6 atoms/m^3) = 2.4E19 m^3/s of space must be taken in

Moving at an average of 0.05c, the area of space that needs to be scooped by the magnetic field is

(2.4E19 m^3/s)/(0.05(3E8 m/s)) = 1.6E12 m^2 = 1.6E6 on a side if area is a cube.

Magnetic field required to scoop this much area

Atoms are moving towards the ship (relatively) at 0.05c, so an acceleration per atom is needed:

0.05c = 1.5E7 m/s

Length atoms move is estimated to be 1/cos (pi/4) of distance needed to be scooped:

v^2 = 2ax

a = v^2/2x = (1.5E7 m/s)^2/2(2.1E7) = 5.4E6 m/s^2

That may seem like a huge acceleration but remember we are accelerating atoms:

F = ma = (5.4E6 m/s^2)(1.67E-27 kg) = 8.9E-21 N

F = qVxB

B = F/qV

B = 8.9E-21 N/(1.6E-19 C)(1.5E7 m/s) = 3.8E-9 T which seems too small...

Assuming 100% efficiency of a fusion reaction reacting 3He and 3He:

3He + 3He 4He + 2p + 12.9 MeV

200 kW needed per VASIMR engine

4000 N/5 N/engine = 800 engines

200 kW x 800 engines = 1.6E5 kW = 1.6E8 W

1 eV = 1.6E-19 J

1.6E8 J/s x 1 eV/1.6E-19 J = 1E27 eV/s needed to be generated

(1E27 eV/s)/(12.9E6eV/reaction) = 7.8E19 reactions/s

3He mass per reaction = 2x3x1.67E-27 kg = 1E-26 kg/reaction

7.8E19 reactions/s x 1E-26 kg/reaction = 7.8E-7 kg/s of 3He needed to react

t = v/a = 3E7 m/s / 0.02 m/s^2 = 1.5E9 s = 48 years of accelerating

1.5E9 s x 1.9E-6 kg/s = 2850 kg of 3He needed for fusion power.

At 1 kg/kW, 800 power sources for the VASIMR engines would have a mass of

1.6E5 kW x 1kg/kW = 1.6E5 kg

Thus 80% of the mass of the craft is just the power generators. Depending on how much power it would take to power the magnetic scoop at the front of the ship, the payload may have to be even smaller. However, what we see here is a way to possibly reach alpha centauri in 140 years (need to decelerate as well, possibly with magnetic scoop) or say epsilon eridani in 200 years or Gliese 581 in 300 years.

Edit: travel times

Edit: formula error