The Tyranny of the Rocket Equation

Podcast Transcript

In 1897, the visionary Russian rocket scientist Konstantin Tsiolkovsky discovered an equation that governed how rockets worked.

His equation, which was independently discovered by several other rocket scientists, immutably governs how we can send rockets into space.

The variables in his equation have determined everything surrounding spaceflight and rocketry since its inception and will for the foreseeable future.

Learn more about the Tyranny of the Rocket Equation on this episode of Everything Everywhere Daily.

Rockets have been around in some form for centuries. If you remember, back to my episode on gunpowder, the Chinese were using rockets in warfare and for fireworks for about 1000 years.

While rockets were a known thing, they left scientists baffled as to how they worked. They weren’t being pushed like a sailboat, and they weren’t using friction with the ground like walking or a horse pulling a wagon.

The late 19th and early 20th centuries saw several people around the world independently tackle the problem.

The first person recognized to have solved the physics behind rockets and, therefore, would get credit as the first rocket scientist is Konstantin Tsiolkovsky.

Tsiolkovsky realized that rockets worked because Newton’s Third Law stipulated that for every action, there is an equal and opposite reaction.

Rockets moved forward because they were expelling something else from behind.

The principle by which rockets moved was demonstrated by Tsiolkovsky with a simple thought experiment.

Suppose you were in a boat in the middle of a pond. The boat had no sail and no oars to paddle. You were basically stuck in the middle of the pond with no way to move.

However, on the boat, you did have a pile of heavy rocks. You could get yourself to shore by throwing the rocks off the back of the boat. Not dropping them into the water but throwing them horizontally. This would give the boat momentum, which, in theory, could get your boat to the shore.

This is the same principle behind a rocket. Instead of rocks, however, you are expelling molecules of gas at very high speeds.

Tsiolkovsky created an equation that explained everything, which is today known as the Tsiolkovsky Equation, or more generally as the Rocket Equation.

Without getting into the math too much, the equation is as follows:

Delta-v, or the change in the rocket’s speed, equals the exhaust velocity of the rocket engine times the natural logarithm of the final mass of the rocket over the initial mass of the rocket.

Tsiolkovsky recorded the date of his discovery of the equation as May 10, 1897.

Other people, such as the English mathematician William Moore, American rocket pioneer Robert Goddard, and German physicist Hermann Oberth independently discovered the Rocket Equation.

Just as an aside, Konstantin Tsiolkovsky, the man who helped usher in the modern world of space flight, lived for most of his life in a log cabin.

You might look at the equation and not think much of it, but it has several profound implications. Implications that govern everything to do with rockets and spaceflight.

One has to do with mass. Set suppose you want to put a 100kg object into orbit.

You would need a certain amount of fuel to do this. However, the fuel you would use also has mass, and that mass would need fuel to launch it, and that fuel would need fuel to launch it…and so on and so on.

The amount of fuel isn’t infinite, but it does mean that you need an enormous amount of fuel to launch something even quite small. That is why rockets are so big and why there are no small rockets. There are no shoulder-mounted or backyard rockets that could fly to space.

Moreover, the more mass you try to fly into space increases, the size of the rocket exponentially. The Apollo program launched something smaller than the size of a truck trailer into space but required a rocket vastly larger than one just carrying a small space capsule.

So long as we are using chemical reactions, the exponential growth in the amount of fuel needed means there is a limit to how large of something we can launch into orbit using conventional rockets. You couldn’t, for all practical purposes, launch something as large as a cruise ship into orbit all at once.

Another implication is the amount of fuel that is part of any rocket. Rockets are almost all fuel. It is common for the mass of a rocket sitting on a launch pad to consist of 85 to 95 percent fuel. The actual amount of payload for a rocket is usually around 2 percent.

To put this into perspective, the mass of a large ship might only be three percent fuel, and your car is around four to five percent. A freight locomotive might be 11 percent, and a commercial airliner might be as high as 35 percent.

A rocket is not dissimilar to a can of soda. An aluminum can of soda is about 94 percent soda and 6 percent can. The external fuel tank on the space shuttle was 96 percent fuel, making it even more efficient than a soda can.

So, using the soda can model, all of the soda is the rocket fuel, and the can is the apparatus of the rocket and all that is necessary to put the pull tab into orbit.

This is why engineers have to worry so much about mass and efficiency when designing anything which is going into space. Every kilogram has to be accounted for because it results in significantly more rocket fuel.

It is also why rockets are usually built in stages. You want to shed as much mass as you can in order to increase your velocity. Getting rid of the lower stages of a rocket when they aren’t needed anymore will help increase velocity.

This is just dealing with the mass part of the equation.

The other parts of the equation have limits as well.

In terms of the exhaust velocity of the rocket, there are several different rocket fuels that can be used. Solid rocket fuel can have an exhaust velocity of 3 kilometers per second, and a methane oxygen rocket can have an exhaust velocity of 4.5 kilometers per second.

All other rocket fuels are somewhere between these values.

So long as we are using rockets that are providing thrust from a chemical reaction, there is a limit to how much exhaust velocity there can be. You can’t just arbitrarily make it larger.

The final variable is delta-v. Delta-v is one of the biggest factors in any orbital flight.

In the case of launching something into space, it is indicative of how much velocity you need to get anywhere.

Just to use approximate numbers, something has to go about 8 kilometers per second to get into low earth orbit, which is only about 250 miles or 400 kilometers away. If you remember back to my episode on how satellites work, getting into orbit is more about speed than it is about altitude. We have to go that high to avoid atmospheric drag. Without drag, you could orbit as high as one meter off the surface.

Once you get into Earth’s orbit, the hard part is done. From there, the extra velocity needed to get to the moon is only an additional 6 kilometers per second, and to Mars is an additional 8.

This is why there has been talk of building a base on the moon. The velocity necessary to escape the gravity of the moon is much less than that to escape Earth. If you want to explore the rest of the solar system, it is much easier to do from the Moon than it is from the Earth.

From an energy standpoint, the biggest leap in spaceflight wasn’t landing on the moon, it was just getting into orbit in the first place.

All of these constraints I’ve mentioned have led engineers and physicists to coin the term the Tyranny of the Rocket Equation.

The rocket equation rules everything when it comes to spaceflight. The design of rockets, spacecraft, satellites, and everything else also has to take the rocket equation into consideration.

Even something novel like Starship by Space X, on which I previously did an episode and which just had its first test launch, is subject to the rocket equation. Starship is more focused on reducing costs but doing so within the confines of the rocket equation.

If the rocket equation has a tyrannical hold on travel by rockets, is there any way to break this tyranny?

The answer is….sort of. You can’t break the laws of physics, but you can break the constrains around how rockets function today.

The first way would be to get rid of chemical reactions to provide thrust. Chemical reactions have a maximum exhaust velocity of 4.5 kilometers per second, which is close to what some rockets are already doing.

A far more efficient rocket would be a nuclear thermal rocket.

Uranium has an energy density a million times greater than hydrogen. A nuclear rocket would work by exposing gas, most probably hydrogen, to an extremely hot nuclear reactor.

If the gas were heated to 3,000 degrees Kelvin, it would have an exhaust velocity of 10 kilometers per second, more than twice that of a chemical reaction.

However, that is just the beginning. The theoretical maximum exhaust velocity for a nuclear thermal reactor is 5,000 kilometers per second.

The meer doubling of the exhaust velocity from a nuclear rocket would change everything, turning the fuel to mass ratio from something like a soda can, to something more like an airliner.

NASA is already investigating nuclear rockets for use outside of the Earth’s orbit.

The other way around the rocket equation is to cheat on the delta-v. Earlier I said that going into orbit was an issue of speed, not altitude. This is true, mostly.

The higher the orbit, the longer it takes to go around the Earth. In low Earth orbit, it takes about 90 minutes. However, if you keep going up you reach a point where it takes one day to go around the Earth, exactly the same as the Earth’s rotation.

This is known as geostationary orbit, a subject on which I’ve done a previous episode.

If you can get 22,236 miles or 35,786 kilometers above the surface of Earth, then you are traveling at orbital speeds, even if you went straight up.

It wouldn’t even matter how long it took you to get to that altitude.

The idea proposed to do this is a space elevator. A space elevator would be a long cable which extended from geostationary orbit to the surface of the Earth. A vehicle would then just need to climb the cable, which could be done with significantly less energy than with a rocket.

A final way around the rocket equation would be the discovery of some form of physics that we currently don’t know. If we learned of some way to nullify the effects of gravity, then we could make the rocket equation moot, but as of now, that is only in the realm of science fiction.

Understanding the rocket equation isn’t rocket science.

…OK, I guess it technically is rocket science, but it isn’t difficult to grasp intuitively.

The next time you watch a rocket blast off into space, keep in mind the rules which govern that rocket. It was engineered to perform within the laws of physics which were outlined in the late 19th century by a Russian who lived in a log cabin.

The Executive Producer of Everything Everywhere Daily is Charles Daniel.

The associate producers are Thor Thomsen and Peter Bennett.

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