Vol. Science · Astronomy · The propellant

Rocket
equation

A change of speed costs propellant, and the cost is exponential. For one ideal stage with constant exhaust speed, Δv = v_e · ln(m₀/m_f). The same line written with specific impulse is Δv = I_sp · g₀ · ln(m₀/m_f), because v_e = I_sp · g₀. Standard gravity in that definition is g₀ = 9.80665 m/s² exactly. It is not Earth’s surface gravity. The Earth Fact Sheet lists mean surface gravity 9.820 m/s², and the Escape desk prints that figure at altitude 0. Turn the equation around and the mass ratio is R = m₀/m_f = exp(Δv/v_e). The fraction that remains, dry mass and payload together, is m_f/m₀ = exp(−Δv/v_e). Propellant is the rest, m_p/m₀ = 1 − 1/R. This page does not choose a new mission Δv. Escape from a classroom circle 400 km up is the 3.178 km/s the Oberth desk already prints, from the fact-sheet GM and volumetric radius. Earth to Mars, as a single impulsive total, is the 5.594 km/s the Hohmann desk already prints. Speed on the coast between those burns is vis-viva. The 300 s and 450 s buttons are a classroom example each. They are not a named engine and not a fact-sheet line.

Classroom schematic. Ideal rocket, constant exhaust speed, one stage. No gravity losses, no drag, no steering losses, and no atmosphere. The burn is already summarized as a Δv. Where that burn happens, and why the same Δv buys more energy deep in a well, is the Oberth desk, not this one. Not a mission planner. The stack is initial mass split into propellant and whatever remains. It is not a vehicle, and the pale band is dry mass plus payload together. It does not say how much of that remainder is payload.

Classroom comparison

Two Δv figures this site already prints, at two specific impulses that are each a classroom example. The 3.178 km/s row is escape from the Oberth desk’s 400 km circle. The 5.594 km/s row is the Hohmann desk’s Earth-to-Mars total. Neither row is a propellant load for a flight. A highlighted row is where the slider sits, when it lands on one of these four.

What the formula is

Words

Ideal rocket

Constant exhaust speed, no outside forces. Momentum the exhaust carries one way is momentum the rocket gains the other way. The equation that follows is Δv = v_e · ln(m₀/m_f). One stage. The empty tanks stay in m_f.

Isp and ve

Specific impulse in seconds, times g₀ = 9.80665 m/s², is exhaust speed. The slider and the field stay tied to that product. g₀ is standard gravity. Earth’s fact-sheet mean surface gravity, 9.820 m/s², is a different number.

Mass ratio

R = m₀/m_f = exp(Δv/v_e). How many times heavier the rocket is with propellant than without it. At 3.178 km/s and a classroom 450 s, R = 2.055. At 5.594 km/s and a classroom 300 s, R = 6.695.

Propellant fraction

m_p/m₀ = 1 − 1/R. The share of the ignition mass that leaves as exhaust. The gold band on the stack.

What remains

m_f/m₀ = exp(−Δv/v_e). Dry mass and payload, not separated. The pale band. Calling it a payload fraction would claim a split this page does not make.

What the picture leaves out

Gravity losses, drag, steering, a finite burn, boil-off, residuals, and every extra stage. A second stage throws the empty first stage away and starts the equation again. This page is one stage. It also leaves out where you burn. The same Δv deep in a well changes orbital energy by v·Δv + ½(Δv)², which is the Oberth desk. The Δv itself, for a circle and for escape, is the Escape desk. The 5.594 km/s comparison is the Hohmann total, not a trajectory. Not a mission planner.

Classroom picture of one ideal stage. At the Oberth desk’s 3.178 km/s and a classroom 450 s, m_f/m₀ = 0.4867. At the Hohmann desk’s 5.594 km/s and the same 450 s, m_f/m₀ = 0.2815. g₀ = 9.80665 m/s². Constant exhaust speed. No gravity losses. No drag. The 300 s and 450 s figures are each a classroom example, not a fact-sheet line. Not a mission. Earthrise preview: NASA / Bill Anders (public domain).

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