The rocket equation

One equation governs why launch vehicles are the shape they are, why they have stages, and why payload is such a small fraction of what sits on the pad.

Derivation

A rocket of mass $m$ moving at velocity $v$ expels a mass $\mathrm{d}m_p$ of propellant at exhaust velocity $v_e$ relative to itself. Momentum conservation in the instantaneous rest frame gives

since the vehicle loses exactly the mass it expels. Separating and integrating from initial mass $m_0$ to final mass $m_f$:

The result is brutal in a specific way: delta-v is logarithmic in mass ratio. Doubling the propellant does not double the delta-v, it adds one exhaust velocity worth. Conversely, a linear increase in required delta-v demands an exponential increase in propellant.

Specific impulse

Specific impulse is thrust per unit propellant weight flow, and it comes out in seconds. The seconds are an artefact of dividing by $g_0 = 9.80665$ m/s², a defined constant that has nothing to do with local gravity. The physically meaningful quantity is the effective exhaust velocity $v_e = I_{sp}g_0$; the convention persists because it gives the same number in metric and imperial units.

An engine has two quoted values. Sea-level $I_{sp}$ is degraded by ambient back-pressure on the nozzle exit; vacuum $I_{sp}$ is the higher figure with no ambient pressure. The difference is substantial, typically 10 to 25%, and quoting the wrong one is a routine source of confusion. See Nozzles and thrust.

What the equation costs

Rearranged, the propellant fraction required for a given delta-v is

For the 9.4 km/s that a launch vehicle must actually deliver to reach low Earth orbit, from Delta-v budgets:

At best, 88% of the vehicle is propellant. Everything else, structure, engines, avionics, and the payload, shares the remaining 12%. This is why launch vehicles are thin-walled tanks with engines bolted to the bottom: there is no mass budget for anything else.

Why staging is not optional

Introduce the structural coefficient $\varepsilon$, the ratio of dry structural mass to structure-plus-propellant mass. A well-built stage achieves $\varepsilon$ between 0.06 and 0.12; hydrogen stages sit at the worse end because liquid hydrogen is bulky and needs large insulated tanks.

The payload fraction of a single stage is then

The numerator is the whole story. If $1/\text{MR}$ is smaller than $\varepsilon$, the payload fraction is negative: the stage cannot lift its own structure, let alone anything else.

That table is the single-stage-to-orbit problem stated completely. It is not that SSTO is inefficient; for most propellant and structure combinations it is arithmetically impossible, and for the one combination that closes on paper it closes with no margin at all.

Staging escapes this by discarding structure once it is empty. Dead tankage stops being accelerated, so the mass ratio resets for the next stage:

All figures for $I_{sp} = 350$ s and $\varepsilon = 0.08$, with delta-v split equally.

Two stages turn an impossible mission into a viable one. The third stage adds only 0.7 percentage points, and the fourth 0.3. Since every staging event adds a separation mechanism, an ignition and a new set of failure modes, the returns stop justifying the risk quickly. Nearly every orbital launch vehicle uses two or three stages, and this table is why.

Ideal against delivered

The equation gives ideal delta-v, assuming thrust acts entirely along the velocity vector in a gravity-free vacuum. Reality removes three portions of it.

Gravity loss

The integral of g·sin γ over the flight path angle. Thrust spent holding the vehicle up rather than accelerating it. Typically 1.2 to 1.5 km/s.

Drag loss

Aerodynamic resistance, almost all of it below 50 km. Typically 0.1 to 0.3 km/s.

Steering loss

Thrust vectored away from the velocity direction, so the component along v is reduced by a factor cos α. Typically 0.1 to 0.5 km/s.

Rotation credit

Not a loss. Up to 0.465 km/s of free eastward velocity from Earth rotation, scaling with the cosine of launch site latitude.

Gravity loss dominates and drives the ascent profile. Because it is proportional to the time spent fighting gravity, minimising it means pitching over early and getting the velocity vector horizontal quickly. That is the gravity turn, and it is why launch vehicles begin tipping downrange within the first minute rather than climbing vertically.

There is a genuine tension here. A steeper trajectory reduces drag loss by leaving the atmosphere sooner but increases gravity loss. A shallower trajectory does the reverse. The optimum depends on the vehicle’s thrust-to-weight ratio and its ballistic coefficient, and finding it is a trajectory optimisation problem in its own right.

References

• Tsiolkovsky, K. E. “Исследование мировых пространств реактивными приборами” (Exploration of Outer Space by Means of Rocket Devices), 1903.
• Sutton, G. P. and Biblarz, O. Rocket Propulsion Elements, 9th ed., Wiley, 2017, chapters 2 and 4.
• Humble, R. W., Henry, G. N. and Larson, W. J. Space Propulsion Analysis and Design, McGraw-Hill, 1995.
• Wertz, J. R., Everett, D. F. and Puschell, J. J. (eds.) Space Mission Engineering: The New SMAD, Microcosm Press, 2011, chapter 17.