What Makes a Rocket

Throw something backwards

Forget engines for a moment. You are standing on a cart floating in empty space, with a pile of bricks. There is no ground to push on and no air. The only thing you can do is throw bricks.

Every brick you throw backwards carries momentum away with it, and since the total momentum of you-plus-bricks must stay zero, you move forwards. That is the entire principle of the rocket. The "something" a rocket pushes against is its own propellant.

Play with it for a minute, because three surprises are hiding in the plot.

Throw speed matters most. Double your throw speed $\cvel{u}$ and your final speed doubles exactly. Nothing else you can change is that powerful.

Many small throws beat one big one. Throwing all 60 kg at once gets you 60% of your throw speed. Throwing it as a continuous stream of sand gets you 92%. The reason: each later throw pushes a lighter cart, so the same brick buys more velocity. A rocket engine is the limit of infinitely many infinitely small throws.

More bricks help less and less. The dashed pink curve bends over. The final speed of the stream is

$\cdv{\Delta v} = \cvel{u}\,\ln\frac{m_0}{m_f}$

— the rocket equation, which we will spend all of Lesson 3 on. For now, notice the logarithm: carrying twice as much propellant does not give twice the speed.

Thrust is pressure on metal

A real engine doesn't throw bricks; it throws hot gas, about a tonne of it per second for a large booster, at 3 km/s. But how does throwing gas actually push the rocket? The honest answer is surprisingly concrete: the gas pushes on the inside of the engine, and the engine is bolted to the rocket.

The combustion chamber is a pressure vessel with a hole in one end. Gas presses on every inside surface. On the side walls the forces cancel. But the injector face at the front has no hole opposite it, so there is a large net push forwards. The converging walls near the throat lean the wrong way and push back a little; the diverging bell leans the right way and pushes forward again. Add them all up and you get the thrustThrustF = ṁvₑ + (pₑ − pₐ)Aₑ: the momentum of the exhaust plus the pressure imbalance across the nozzle exit..

Summing wall pressures is hard. Fortunately a control volume drawn around the whole engine gives the same answer in one line:

$F = \cfuel{\dot m}\,\cvel{v_e} + (\cpress{p_e} - \cair{p_a})\,A_e$

The first term is the momentum thrust: mass per second times the velocity it leaves with. The second is the pressure thrust: if the gas leaves at a pressure $\cpress{p_e}$ different from the air pressure $\cair{p_a}$ outside, the exit plane is pushed on by the mismatch. At sea level with a large nozzle it is negative — the atmosphere pushes the engine backwards. Climb, and the air falls away, so the same engine makes more thrust.

Two numbers that describe any rocket

Because thrust has two parts, engineers bundle it into one velocity: the effective exhaust velocity $c = F/\dot m$. It equals the real exhaust velocity when the nozzle is perfectly matched to the air around it.

Divide it by standard gravity $g_0 = 9.807$ m/s² and you get the propellant's miles-per-gallon, the specific impulseSpecific impulse IspThrust per unit weight flow of propellant, in seconds: Isp = F/(ṁg₀) = vₑ/g₀. The propellant’s “miles per gallon”.:

$I_{sp} = \frac{F}{\cfuel{\dot m}\,g_0} = \frac{\cvel{c}}{g_0}$

It is measured in seconds, and it is the most-quoted number in rocketry. A kerosene engine at sea level manages about 300 s; hydrogen in vacuum about 450 s; a Hall thruster 1600 s. Multiply by ten (strictly, 9.81) to get the exhaust velocity in m/s.

The other key number is the total impulseTotal impulseThrust integrated over burn time, I = ∫F dt (N·s). The same impulse can be a short kick or a long gentle push., $I = \int F\,dt$: the whole push the engine delivers. The same impulse can be a violent two-minute shove from a solid booster or a gentle year-long whisper from an ion engine; how it is delivered is the question of what the mission needs.

Every way to throw mass

Now step back. Anything that throws mass backwards fast is a rocket, and people have found many ways to do it. They sort themselves by where the energy comes from:

• Chemical rockets store the energy in the propellant itself: a liquid bipropellant engine, a solid motor, a hybrid. The propellant is both the energy source and the reaction mass, which makes them compact and enormously powerful — but caps the exhaust velocity at around 4.5 km/s, because that is all the energy the chemistry holds.
• Electric thrusters get their energy from outside — solar panels or a reactor — and use it to accelerate a propellant electrically: by heating it (resistojets, arcjets), with electric fields (ion engines), or with crossed electric and magnetic fields (Hall thrusters). The exhaust can be ten times faster, but the thrust is limited by the power supply.
• Nuclear thermal rockets heat hydrogen with a fission reactor. No combustion means the exhaust can be pure, light hydrogen: roughly double the Isp of chemical engines.

The atlas has a striking shape: it is roughly a diagonal band from top-left to bottom-right. The dashed lines explain it. The kinetic power carried off in the jet is

$P_{jet} = \tfrac12\,\dot m\,v_e^2 = \tfrac12\,F\,v_e,$

so at fixed power, thrust and exhaust velocity trade one for the other. Chemical engines sit on the gigawatt lines because their energy source is the propellant, and a tonne per second of propellant can release gigawatts. An electric thruster sits on the kilowatt lines because that is what its solar panels can deliver.

The map of this course

The rest of the course follows one launch vehicle and its engines from end to end. Each lesson lives in one part of the machine:

Next: the part of the engine that turns 3500 K gas into a 3 km/s jet. The nozzle, and the strange fact that to make supersonic gas go faster, you have to give it more room.