Explained: SpaceX and the Rocket Equation (for beginners)

With SpaceX’s launch of the Crew Dragon spaceship fresh on our minds, you’re probably thinking at least 2 of these 3 things.

1. That’s a really REALLY cool name
2. How does a rocket engine actually work?
3. What did those eggheads in California do differently that left the world stumped?

The explanation lies in The Rocket Equation. No, not X Æ A-Xii, the other one.

Don’t be alarmed. I want to break it down so that you can attack it with ABSOLUTELY NO PRIOR KNOWLEDGE at all.

First, definitions:

• ∆v is how fast you want the rocket to go. More specifically, it is the change in speed the rocket can achieve (the triangle before the v is the symbol for delta and just means change). Let’s say the rocket starts at rest with a speed of 0. You want it to be able to achieve 8 metres per second — the speed needed for orbit. The change in speed, also known as the ‘delta v’ is 8m/s.
• Ve is the exhaust velocity and represents the total force that the rocket can create. The intuition behind this is crucial. We normally only think of the rocket’s speed but the fuel it’s burning has a mass and speed too. The mass will be in the form of the “exhaust gas” and the speed will be in the direction opposite to that of the rocket.
• The mass fraction, shown as mo/mf, gives the proportion of the ship’s mass that must be dedicated to its fuel. m0 is initial mass of the rocket and mf is final mass after part of its flight.

The ln is the natural logarithm and is there to make the calculus work, but we don’t need to worry about that.

The first thing to do is ask: what’s the point of it?

It has a few but let’s frame it with the following problem: say I have a rocket and want it to achieve a target velocity. What sort of rocket engine should I use to achieve this change in velocity?

An important calculation in answering this question is: what % of the total mass of the rocket will be fuel? This is what the mass fraction tells us. Now naturally you want an engine which has the smallest percentage possible. If only 80% of your spaceship needs to be fuel, that leaves 20% free to carry other stuff. So you’re more likely to pick the engine with the smallest fraction.

So that explains the use of the mass fraction, but what’s the use of the Ve? Well there are actually lots of different types of rocket engines. We’re probably most familiar with chemical rockets which use a solid or liquid fuel but you also have nuclear and ion rockets. We need a way to compare the amount of force that they produce. This is done by looking at their Ve (exhaust velocity).

How did they come up with the equation?

The short answer is that it’s based on the conservation of momentum (speed x mass). This is the idea that the total momentum before and after a collision in a system is the same. The diagram below shows a simple model of the system with a rocket. Suppose a thought experiment where you burn fuel and it whizzes out the back of the rocket at a velocity of Ve but the rocket doesn’t go anywhere. This breaks the conservation of momentum because there is “negative momentum” of the exhaust but no “positive momentum” of the rocket moving forwards to balance it out. In the real world, the “positive momentum” of the rocket is shown by the velocity of dv in the diagram. Manipulating the momentum equation with calculus will get you to the rocket equation.

The big problem with the rocket equation

Its answers are depressing. The following results tell us typically what proportion of the mass of a rocket needs to be fuel.

• Solid fuel rocket = 93%
• Liquid fuel rocket = 77%
• Ion engine rocket = 20%

While the ion engine seems like a no-brainer on paper, we haven’t been able to make one which produces enough force to be effective. The lesson from all this is that ubiquitous theme throughout all rocket science: mass is a massive deal.

How have SpaceX got around the Rocket Equation?

So the equation tells us that if you want to use a liquid fuel engine, a huge amount of your rocket is just going to be the mass of the fuel. So how have they got around this problem with their liquid oxygen engines?

Perhaps they’re using their fuel more efficiently than others? Actually no, in the same way you would use MPG to measure the efficiency of a car’s engine, rockets have ‘specific impulse’. The specific impulse of the Merlin 1D FT is a respectable 348 seconds (the unit of seconds makes no actual sense but is used for an arbitrary reason steeped in history and politics). But this doesn’t compare with the engine which has the crown: Russia’s RD-0146D with a specific impulse of 470 seconds. In fact, Merlin’s is lower than that of the engines used on most of the Delta Space Shuttles.

Where SpaceX are topping the tables are in the thrust to weight ratio of their engines. The latest Merlin engine has the highest ratio of any engine at a whopping 194.5. The name of the game here is weight. While the Space Shuttle had 3 different types of engines, the Falcon 9 only has 1. The killer edge they have is they’re light and simple.

If the Shuttle is a Bugatti, the Falcon 9 is a Kit car beating it around the track. Famously, SpaceX don’t even file patents on their tech so we may never know how they did it, let’s just be glad it happened.