Flight Performance

The rocket equation

In Lesson 1 you threw bricks off a cart. Make the throws continuous, and the momentum balance for a rocket of mass $m$ ejecting mass at relative speed $v_e$ reads $m\,dv = -v_e\,dm$. Integrate from the starting mass $m_0$ to the final mass $m_f$ and you get the most famous equation in astronautics, first written down by Konstantin Tsiolkovsky in 1903:

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

It says two things. First, the velocity a rocket can gain is proportional to its exhaust velocity. Second, it grows only with the logarithm of the mass ratioMass ratioInitial mass over final (burnout) mass, m₀/m_f.. To add another $v_e$ worth of speed you must multiply the mass ratio by $e \approx 2.7$. Again and again.

Reaching low Earth orbit takes about 9.4 km/s of Δv once losses are counted. With kerosene at an average Isp of about 330 s ($v_e \approx 3.2$ km/s), that is a mass ratio of $e^{9.4/3.2} \approx 19$: the rocket must be 95% propellant before it carries anything at all. Hydrogen, at 450 s, brings this down to 88% — but hydrogen is so light that its tanks are huge, and tanks weigh something. That tension runs through the whole of rocket design.

Staging: stop carrying empty tanks

The problem with a single stage is that it has to carry its entire structure — every empty tank and every engine — all the way to orbit. Once a tank is empty it is dead mass, and each kilogram of dead mass must still be accelerated to 7.8 km/s.

StagingStagingDiscarding empty tanks and engines during flight so the remaining vehicle does not have to keep accelerating dead mass. throws the dead mass away. Each stage fires, empties, and falls off, and the next stage starts over with a much smaller rocket to push. The rocket equation applies stage by stage, and the Δvs simply add:

$\cdv{\Delta v_{total}} = \sum_i \cvel{v_{e,i}}\,\ln\frac{m_{0,i}}{m_{f,i}}$

The first extra stage makes an enormous difference: it can turn "impossible" into "3% payload". The third helps less, and the fourth less still, while every stage adds engines, separation systems, ignition events and failure modes. That is why nearly all orbital launchers have two or three stages. Single-stage-to-orbit remains possible on paper — the lab shows exactly what it demands of structure and Isp.

Flying through air and gravity

The rocket equation assumes a rocket floating in empty space. A real launch fights two extra enemies:

• Gravity. While the rocket climbs, part of its thrust only holds it up. The gravity lossGravity loss∫ g sin γ dt: velocity spent holding the rocket up against gravity while it climbs. Short, powerful burns lose less. is $\int g \sin\gamma\,dt$, where $\gamma$ is the flight-path angle above horizontal. It shrinks if you burn faster (more thrust) and if you turn horizontal sooner.
• Air. The drag lossDrag loss∫ D/m dt: velocity spent pushing through the atmosphere. Typically 100–200 m/s for a large launcher. $\int D/m\,dt$ grows if you fly fast while the air is still thick. For big rockets it is surprisingly small, 50–150 m/s, but the aerodynamic load peaks sharply at max-QMax-QThe moment of maximum dynamic pressure q = ½ρv², when aerodynamic loads peak — usually around 10–14 km altitude., where dynamic pressure $q = \tfrac12\rho v^2$ is largest.

The two pull in opposite directions. A rocket that climbs slowly wastes Δv holding itself up; a rocket that accelerates hard low down meets thick air at high speed. Real ascents are a compromise, and the classic shape is the gravity turnGravity turnAn ascent in which, after a small initial pitch-over, gravity itself bends the trajectory while the thrust stays aligned with the velocity.: rise vertically, tip over by a few degrees, and then let gravity itself bend the trajectory while the thrust stays aligned with the velocity, so the vehicle never flies sideways through the air.

Play with the two controls and you will find a narrow valley. Too little thrust-to-weight and gravity eats you alive. Too small a kick and the rocket goes nearly straight up, then has to spend Δv turning horizontal later. Too big a kick and a slow rocket falls over. Too much thrust and you slam into max-Q and pull uncomfortable g. The best trajectories lose about 1.5 km/s in total. That is why "9.4 km/s to orbit" is quoted, when orbital speed is only 7.8 km/s.

Orbits and the Δv map

Once in space, distances stop mattering and Δv is everything. A circular orbit at radius $r$ needs speed $v = \sqrt{\mu/r}$ — 7.8 km/s just above the atmosphere. Escape needs $\sqrt{2}$ times that. Changing orbit costs Δv at each engine burn, and burns are most efficient where the vehicle is already moving fastest (the Oberth effect).

Mission planners therefore draw the solar system as a subway map, with Δv on each line:

The map makes some counter-intuitive facts obvious. Geostationary orbit, 36 000 km up, costs almost as much as landing on the Moon. Landing on Mars is cheap only because its atmosphere does most of the braking. And every kilometre per second compounds: the fraction of your mass that arrives is $e^{-\Delta v/v_e}$.

Next: where exhaust velocity comes from. Why hydrogen, with a cooler flame than kerosene, still leaves the nozzle 30% faster.