Oberth effecteffect

The idea was first clearly described by Hermann Oberth in his 1923 book, although hints also appeared in Tsiolkovsky's work. The law works due to the nonlinear dependence of kinetic energy on speed. Imagine you push a cart. If it is barely moving, your push adds almost no energy to it — most of the energy turns into heat in your shoes. If the cart is racing, you push off the ground, transferring momentum while remaining nearly in place, and almost all the work turns into its acceleration. In space, the planet's gravitational field plays the role of the ground, and the rocket exhaust provides the 'push.' The faster the rocket relative to the planet, the greater the percentage of fuel energy converted into motion.

The Oberth effect explains why a rocket accelerates more efficiently when it is already moving fast. Kinetic energy depends on speed squared: to double the speed, you need four times the energy. When a rocket burns fuel and ejects gases backward, it gets an extra push. But some energy goes into accelerating the gases themselves. When the rocket's speed is low, the gases carry away a lot of energy; when high, the gases in the planet's reference frame are almost stationary, and almost all the chemical energy of the fuel becomes useful kinetic energy of the spacecraft. A simple analogy: jumping onto a moving train from a standstill is harder than jumping off a moving train — the former requires more effort.

The Oberth effect is a consequence of the law of conservation of energy for rocket motion in a gravitational field. When the engine fires, the rocket ejects propellant with relative speed u (exhaust velocity). If the current rocket speed in the inertial frame is v, the kinetic energy gained by the rocket over time dt is dK = m v dv + (1/2) dm v² (to first order). For a short impulse (Δv << v), the increase in kinetic energy is ΔK ≈ m v Δv. Burning the same amount of fuel (Δm), the rocket gains more energy the greater its speed v. In the rocket's reference frame, the energy of the exhaust gases is constant, but in the planetocentric frame their kinetic energy depends on v, and at high v it can even be negative relative to the initial state. Thus, an engine firing at the pericenter of an orbit maximally converts chemical energy into orbital mechanical energy.