This is for you writers out there considering writing hard SF.

The more propellant mass a rocket carries (all else being equal), the faster it can go. But there is a “diminishing returns” aspect to this, since propellant is not only expended to push the rocket mass, it must also push the mass of unconsumed propellant still aboard. In other words, by carrying x times more propellant, you don’t go x times faster. This is summariized by the “rocket equation”.

Vf = Ve ln [ (M1+M2) / M2 ]

Where Vf is the payload’s final velocity when the propellant is all used up
M1 is the mass of the propellant
M2 is the mass of the rocket (payload)
Ve is the exhaust velocity

and ‘ln’ is the natural logarithm function. There’s a key for it on your calculator.

Ve will vary depending on the type of propulsion system you have. The higher the Ve, the faster you go. Ve is the only thing the engineer can improve with better technology. Everything else is physics which he can do nothing about.

As it turns out, the most efficient use of your propellant, that is, when as much as possible of propellant energy is converted to payload kinetic energy, occurs when the propellant mass equals 3.92 times the payload mass. This is when as much propellant energy as possible goes into pushing the final payload, and as little as possible is used to push unburned propellant. This is for a single stage rocket, in zero gravity and with no atmosphere, where empty propellant tanks and their supporting structures cannot be jettisoned as you accelerate.

In this ideal case, where approximately 80% of your rocket mass is propellant, the final payload velocity will be approximately 1.6 times the exhaust velocity, and about 65% of the available propellant energy will be converted to payload kinetic energy. You can always carrry more propellant, but you will get progressively less velocity return for the extra mass cost.

Again, it doesn’t matter what your exhaust velocity is, these relations hold. Once your engineering optimizes your propulsion system, the only way you can go faster (besides packing more fuel) is to use a propellant and engine combination with a higher Ve. For those of you willing to pursue this further, exhaust velocities are often expressed in terms of Specific Impulse, which is approximately Ve/9.8. BTW, my source is “The Starflight Handbook”, (Mallove and Matloff). I did not figure this out all myself.

Ve for chemical rockets is on the order of several kilometers per second, fine for sending automated probes around the solar system, but inadequate for anything else. Solid core fission reactors are somewhat better, and advanced liquid and gas core fission concepts could conceivably give us final payload velocities of several hundred km/sec–great for zipping around the planets, but the stars are still beyond our grasp. Ion drives might be a little better, but not much.

Advanced fusion propulsion systems are more promising, but we must still deal with the fact that only less than a percent of the fuel rest mass is convertible to energy, and only a fraction of that may be accessible to us for propulsion (most will be lost as radiation or out the exhaust). Fusion might get us up to a few thousand km/sec. Or at most, a few percent of the speed of light.

The ultimate propulsion system our physics can conceive of would be direct matter/antimatter to energy conversion, which yields an exhaust velocity of the speed of light! This immediately gives us superluminal payload velocities, until we correct for relativistic effects. Complete mutual annihilation of matter-antimatter yields high energy gamma rays, which I imagine would be very difficult to tap into for propulsion purposes, being as they cannot be manipulated by electromagnetic fields and are capable of penetrating meters of lead shielding. Unless we could persuade them to obediently travel in one direction, they would give us no net thrust, not to mention they would vaporize our spacecraft.

Physicist R.L Forward speculates that it might be possible to direct as much as half of the energy into a propellant fluid, but the propellant would gain its Ve from the annihilation of only tiny amounts of matter/antimatter, the fluid itself would contribute no energy of its own. Our rocket might be 80% propellant by mass, but that is not the same thing as saying it was 80% fuel by mass. The propellant is just a working fluid, something to push against, it contributes no energy, but still has mass that has to be pushed along. It would be faster than fusion, but not spectacularly so. It would not push us up to relativistic speeds.

Unless we can find some way of converting a substantial fraction of antimatter annilation to kinetic energy without bulky, inert propellants, our top speed appears to stay well under 10% that of light.

Let us consider that our propellant, matter/antimatter combination as our fuel source, and that whatever mechanical tricks we have to perform with working fluid is equivalent to some proportion of the extractable energy. We’ll call that the K-factor. So for example,if we can extract all the energy from the matter/antimatter for propulsion, if we need no working fluid for propellant mass, and there are no other losses, our K is 1.0 That is, if we carry a ton of normal matter for every ton of antimatter, our K would be equal to 1.0, since both tons are completely converted to useful energy.

But if we use up 101 tons of normal matter and react it with 1 ton of antimatter, two tons worth of energy are created, but we have to expel 100 tons of inert propellant as a working fluid to tap into the energy. Our K would then be 0.02 (2 annihilated/100 exhaust) = 1/50.

K is the multiplier that allows you to calculate the effective exhaust velocity of your fuel and propellant mix. A K of 1.0 yields a Ve of 300,000 km/sec, or c.. A K of 0.1 yields Ve = 30,000 km /sec. K is the factor to calculate effective exhaust velocity. Generalizing for all antimatter drives, our original velocity equation becomes at optimum mass ratio of 80% propellant:

Vf = Kc ln [ (propellant + payload)/payload ] where c is the speed of light.

So for a K of 0.1, our final velocity is 48000 km/sec.

Now its up to the engineers to start working on maximizing K.

It appears with the physics we now have, speeds higher than a few percent that of light are probably not attainable. This isn’t solvable with clever engineering, we need new physics.

However, this is using the conventional rocket concept, which carries its fuel with it. If we could beam energy to the spacecraft (say with a laser, or harvest the energy already in space (solar sails or scoopships, Bussard-type ramjets that suck up the hydrogen in the interstellar medium as fusion fuel) the tyranny of the natural logarithm might be avoided.

Then there’s always warp drive and wormholes.