The equation behind every launch
In 1903 the Russian schoolteacher Konstantin Tsiolkovsky worked out how fast a rocket can go by throwing mass out of its back. The answer, now called the rocket equation, is the starting point of every space mission design:
Δv = Isp × g₀ × ln(m₀ ÷ m_f)
- Δv (delta-v) is the change in velocity the rocket can achieve, ignoring gravity and drag.
- Isp is the engine’s specific impulse in seconds, and g₀ = 9.80665 m/s². Their product is the effective exhaust velocity.
- m₀ is the initial mass with propellant (wet mass) and m_f the final mass once the propellant is burned (dry mass).
Worked example
A stage with engines rated at 350 s of specific impulse, weighing 500 t fully fueled and 50 t empty:
| Result | Value |
|---|---|
| Delta-v | 7,903 m/s |
| In km/h | 28,452 km/h |
| Exhaust velocity | 3,432 m/s |
| Mass ratio | 10.00 |
| Propellant fraction | 90.0% |
Even with 90% of its mass as propellant, this single stage falls short of the roughly 9.4 km/s needed to reach low Earth orbit from the ground.
The tyranny of the logarithm
Because delta-v depends on the logarithm of the mass ratio, each extra km/s costs more and more propellant. With an exhaust velocity of 3.4 km/s:
| Delta-v | Mass ratio needed | Propellant share |
|---|---|---|
| 3.4 km/s | 2.7 | 63% |
| 6.9 km/s | 7.5 | 87% |
| 9.4 km/s | 15.5 | 94% |
A single stage carrying 94% propellant would leave almost no mass for engines, tanks and payload. That is why launch vehicles use staging: dropping empty tanks resets the mass ratio for the next stage, and the delta-v of the stages adds up.
Typical specific impulse
| Engine type | Isp (vacuum) |
|---|---|
| Solid rocket motor | 250–290 s |
| Kerosene and liquid oxygen | 310–350 s |
| Methane and liquid oxygen | 350–380 s |
| Hydrogen and liquid oxygen | 420–465 s |
| Ion thruster | 1,500–4,000 s |
Ion engines are extremely efficient but produce tiny thrust, so they are used for long missions in space, not for launch.
Delta-v budgets
Mission planners add up the delta-v of every maneuver. Rough values from Earth’s surface: about 9.4 km/s to low Earth orbit, about 12 km/s to geostationary transfer orbit, and about 12.5 km/s to escape toward the Moon. Real missions add margins and account for gravity and drag losses, which the ideal equation leaves out.