Illustrated Physics
physics, drawn
Concept

Rocket equation — where it appears

The relation between a rocket's gain in speed and the fraction of its mass burned: the speed gained is the exhaust speed times the logarithm of the mass ratio. It follows from momentum conservation alone, and its logarithm is why high speeds demand so much propellant.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The fuel a starship needs, which is most of the universe. The mass of fuel a rocket must start with, divided by the mass it ends with, against the final speed as a fraction of light's — the vertical axis being the number of decades in that ratio, because the numbers do not fit on any other scale. Each solid curve is one exhaust speed, and each dashed one beside it is what Tsiolkovsky's Newtonian formula would have said. Below about a tenth of light speed the two are indistinguishable; above it they part, and the relativistic curve turns upward without limit as the final speed approaches light's, because what adds linearly is the rapidity rather than the velocity. The photon rocket — exhaust at exactly the speed of light, which is the best any engine can do — needs a mass ratio of 1.7321 to reach half light speed, which is √3 exactly, and 4.4 to reach nine tenths. Those are modest numbers and they are the whole of the good news. A chemical exhaust at 4 km/s needs 10^2968 even to reach a tenth of light speed, which is not a difficult engineering problem but an arithmetic impossibility — there are about 10⁵⁰ atoms in the Earth. What the chart cannot show is the other half of the trip: stopping at the far end squares the ratio, and coming home squares it again.

The fuel a starship needs

Tsiolkovsky's logarithm survives relativity with one substitution: what adds is the rapidity rather than the velocity. The result is that a photon rocket reaches half light speed on a mass ratio of the square root of three, and a chemical one reaches a tenth of it on a mass ratio with three thousand digits.

relativity · Mass-energy The one number a burn depends on. The fraction of a deuterium–tritium fuel ball that burns before it flies apart, against its density times its radius. The fuel is held by nothing: a sound wave crosses it in a time R/c, the burn rate goes as the density, and so the fraction burned goes as the product ρR, flattening as the fuel runs out, f = ρR/(ρR + 7 g/cm²). A disassembly-time estimate at 30 keV gives 7.6 g/cm² for the same constant, close to the fitted value. The radius and the density do not appear separately. A third of the fuel burns at ρR = 3 g/cm², and that value does not depend on how large the ball is; what sets the size of an inertial-fusion capsule is what has to be done to reach it.

The fuel held together only by its inertia

On 5 December 2022 a laser put 2.05 megajoules into a gold can the size of a pencil rubber, and a speck of fuel inside it released 3.15. Nothing held that fuel together for the tenth of a nanosecond it burned. It needed to be crushed to a thousand times the density of solid, and it was pushed there by something light cannot do by itself: light's own pressure is hundreds of times too feeble, so a capsule is a rocket that fires inward.

astrophysics · Radiation pressure

Named alongside it

The objects these essays reach for when they reach for this one.

AblationAreal densityBurn fractionCentral hot spotExhaust velocityFermi pressureHohlraumInertial confinementInterstellar travelE = mc²Mass ratioMomentum

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