Would ejecting propellant close to light speed result a '"dream engine"?

Question

"Dream engine" I assume as pictured in the old science fiction: a single stage rocket that can take off from the Earth and then freely travel at least over Solar system, landing multiple times wherever it needs without refueling.

The rationale is, the mass increases with the speed. Hence by ejecting with velocity close to speed of light we can convert hydrogen atoms into something more like cannon balls by mass. Hence the rocket would require very little mass of propellant.

The energy problem remains obviously unsolved but at least we do not longer need a lot of mass.

Answer 1

Yes, your conclusion is right, and no, your argument is wrong. If the problem to generate enough electric power is solved, then (near-)lightspeed exhaust velocity is the way to go.

Arguing with relativistic increase of mass is the wrong way to look at it for two reasons: Your exhaust doesn't get heavier for free - it all comes from the energy you put into accelerating it. Relativistic effects just change what happens when you "push harder": Velocity stops increasing as much as at low speeds, instead your energy goes into mass of the object (if you want to stay with this slightly outdated view on relativity). There's nothing to gain here for free.

On the other hand, using faster and faster exhaust velocities pays off even in the total absence of relativistic effects. Taking the basic equation for momentum $p = m v$ we can see that you can always trade propellant mass for exhaust velocity to gain the same amount of momentum. Unfortunately, $E = \frac{1}{2}m v^2$ says that we need to increase our power by a factor of 4 to double the velocity and to halve the mass. This quadratic behavior limits us in reaching extremely high exhaust speeds (among other things as the size of the engines).

So, all we have to do is find the optimal point between mass of propellant and available energy. As all conventional energy sources get heavier the more energy they provide, there necessarily is an optimum between increasing energy source mass and decreasing propellant mass.

With current technology the sweet spot seems to be to use quite heavy ions (i.e. Xenon) at comparably low speed. If we manage to build more lightweight energy sources, this will shift to lighter atoms and higher speeds, e.g. Argon or even Hydrogen. In the extreme case of very cheap energy we would go for propellant with 0 mass: Photons! Now we need to spend exactly no propellant at all, at the cost of an extreme amount of energy, i.e. roughly 10,000 times more than for current ion drives.

This "photon rocket" is closer to reality than it seems: building the engine is trivial (just a high power laser), only the power source is a problem. If we decouple the power source from the rocket we end up with a well-investigated technology that we could build even today: A laser driven solar sail!