Thread by @DonaldM38768041

Let's talk about the Relativistic Rocket Equation.

This can be used to study the problem of reaching very high speeds for interstellar travel...

As a rocket consumes its fuel, it becomes lighter and accelerates faster. It's a simple calculus problem to find the final speed V, given the final mass M and initial fully-fueled mass M1, and the exhaust velocity Vex.

This is Tsiolkovsky's classical rocket equation:

The equations tells us two important things about rocket design: 1) the structure should be made as light as possible so M1/M is big, 2) the exhaust velocity should be as high as possible.

In theory, the highest exhaust speed from chemical rockets would be atomic hydrogen: 2H -> H2.
This is believed to yield an exhaust velocity as high as 15 km/sec, but atomic hydrogen is too unstable to actually be used as a fuel.

To consider much higher exhaust speeds, even light or gamma rays as exhaust, the rocket equation must be derived in keeping with the theory of relativity.
This is done very neatly in Taylor & Wheeler's excellent book.

Wheeler presents a naturally simple formulation of relativity in which velocity (beta = V/c) is also expressed as "rapidity", a hyperbolic angle theta.

Velocity and rapidity are related as beta = tanh(theta).

We can compare the classical and relativistic rocket equations with an example of M1 = 1.0 and M = 0.1.
The classical equation starts to seriously diverge with an exhaust velocity above 10% of of the speed of light.

(Note, atomic hydrogen yields Vex of only 0.005 % c)

Let's consider the problem of a rocket powered by known, technology: atomic fission or atomic fusion.

No anti-matter, no giant space lasers that ignore the laws of diffraction, no warp drives using spinning negative matter. Instead, what might really be possible?

A simple engine is the "flashlight drive", an atomic reactor running incandescent, generating thrust by radiation pressure. Vex is the speed of light (beta=1).

M1 is 10% structure , and 90% plutonium. Final mass M is reduced 0.09%, converted by fission into energy.

The final velocity of the simple flashlight drive would be 0.084 percent light speed. If we use fusion of deuterium and tritium, we get 0.39% mass conversion to energy, and the flashlight drive reaches 0.34 percent light speed.

We can improve by ejecting the nuclear waste as the atomic fuel is consumed. An easy way to do this is carrying many small reactors and ejecting them when they are used up.

This gets us more than a 10x increase of speed, reaching a little more than 1.2 percent of light speed. But still too slow for interstellar travel.

Let's try using the reactor energy to accelerate a propellant, such as with an ion engine. Assume all the reactor energy is converted into kinetic energy of the propellant.

As we increase the propellant mass, we see there is an optimum ratio of propellant to reactor mass. This gets us to about 3.3 percent light speed.

What if we also use the atomic waste as propellant in addition? Now we see the optimal case is to use only the reactor waste as propellant. This gets us to 12.6 percent light speed, which is starting to look useful.

Using fusion, the scheme gets us to 24.4 percent light speed.

12.6 percent light speed assumes perfect efficiency. A real implementation of a fission drive might involve a traveling-wave breeder reactor and an ion accelerator.

A fusion drive would accelerate the helium ions generated by the reactor.