Rocket science at its most basic is surprisingly simple. If you throw enough stuff fast enough out the back of your rocket, it will make the rocket go forward. The more stuff you throw out and the faster you throw it, the faster your rocket will go.
If you’ve ever blown up a balloon and then let it go, you’ve seen this in action.
But this isn’t very efficient, is it? The balloon just flies around randomly. This is why rocket engines have nozzles at the rear to direct the flow of hot gases created in the engine.
https://www.spacevoyaging.com/wp-content/uploads/2023/06/3382447074_fe0645d5e4_k.jpg
So here is a puzzle for you. The SpaceX Starship uses the same Raptor engine for its first and second stages. But as this image shows, even though the engine hardware for the engines used on the first stage (on the left) is the same as for the engines used on the second stage (on the right), the nozzle is much larger for the second stage engines. Why is that? Fair warning: we are going from Rocket Science 101 to 201 in the next two paragraphs.
https://cdn.wccftech.com/wp-content/uploads/2020/09/SPACEX-RAPTOR-ENGINE-STARSHIP-NASA.jpeg
Take a close look at the diagram below, and focus on the dotted lines representing exhaust gases emerging from the nozzle. If those lines are not parallel to the desired direction of motion, they are not contributing 100% to that motion. Think of all the lines above the black arrow that will push the rocket down, and all the lines below that will push it up. That is wasted effort, and means the rocket is not maximally efficient. That maximum efficiency is reached only when ALL the exhaust gas is ejected in the same direction.
https://lh5.googleusercontent.com/ra83SQfdr-LtTbDOBsiUD6QqdmgfvC65wuNPsv6nEmSt178AlEDmSX29m2LJFYjGDJSQGLcttsTojoYThrb4hSYLCHAI3hDmZf1DzGgpEttvjOFMDw6MSGBQg52AHlsCQkZKU1w1
The velocity with which the gases exit the nozzle is a key measure of how much thrust they can impart. But the direction in which they exit is a key measure of how efficiently they move the rocket forward.
Now to Rocket Science 301.
This is the purpose of the nozzle—it allows the gases to expand before they exit. Ideally, the gases exit the nozzle at the same pressure as the surrounding atmosphere. At sea level, where the atmospheric pressure is greatest, the gases don’t need to expand much to reach that pressure, and the nozzle can be smaller for the first stage. For the second stage, which only ignites in the upper atmosphere where the atmospheric pressure is much lower, the nozzle must be larger to allow the gas to expand more to reach a lower exit pressure.
Of course, the engine will only reach its maximum efficiency at a specific pressure, and therefore at a specific altitude. The size of the nozzle is a compromise meant to operate well enough through a range of altitudes from sea level to about 40 miles high, where the first stage of Starship falls away and the second stage engines ignite. The atmospheric pressure this high up is close to zero, and the second stage engine nozzles are much larger to allow for more expansion and a lower exhaust gas exit pressure.
Screenshot from https://www.youtube.com/watch?v=yMrJl-lJrRI
For a rocket engine meant to operate only in the vacuum of space, the nozzle will be as large as possible given the constraints of weight and practicality. It can’t be infinitely large! Look at the nozzle on the Apollo spacecraft. This engine operated only in a vacuum, and the nozzle is not small relative to the size of the craft it propelled.
https://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Apollo_CSM_lunar_orbit.jpg/1920px-Apollo_CSM_lunar_orbit.jpg
Maybe we really should think rocket scientists are smart!