Last weekend the twin ESCAPADE spacecraft (Blue and Gold) performed powered flybys of Earth, combining a gravitational “slingshot” and theOberth effect to accelerate their journeys to Mars. (The former name is a science popularization that evokes using a sling to whip a stone in a tight curve before releasing it along a fast tangent, and the latter name is that of its discoverer, physicist and rocket pioneer Hermann Oberth.)
“Slingshot”
A spacecraft passes behind a planet’s motion allowing its gravity to bend its trajectory toward its velocity, thereby gaining speed and kinetic energy relative to its star without expending fuel. Assuming a nearly 180^\circ bend, the motion is quasi-one-dimensional. If v_{ab} is the velocity of a relative to b, and indices r, p, s denote “rocket”, “planet”, “star”, then before the flyby
v_{rs}=v_{rp}+v_{ps}and after the flyby
v_{rp}^\prime=-v_{rp}.But m_p \gg m_r implies v_{ps}^\prime\approx v_{ps}, so
\begin{aligned}v_{rs}^\prime&=v_{rp}^\prime+v_{ps}^\prime\\&\approx -v_{rp}+v_{ps}\\&=-(v_{rs}-v_{ps})+v_{ps}\\&=-v_{rs}+2v_{ps},\end{aligned}and the rocket’s return speed is twice the planet’s speed relative to the star!
Oberth Effect
If a spacecraft’s rocket engine can cause a speed change \Delta v, the corresponding kinetic energy change
\begin{aligned}\Delta K &= \frac{1}{2}m(v+\Delta v)^2-\frac{1}{2}mv^2\\&=mv\Delta v+\frac{1}{2}m(\Delta v)^2,\end{aligned}where the first term is proportional to v, so the larger the speed, the larger the kinetic energy change, and hence the larger speed at infinity. Indeed, if the spacecraft exhausts its burned fuel backward near perigee where it moves fastest, it leaves the burned fuel at minimum orbital speed relative to the stars, stripping it of its kinetic energy and transferring that energy to the spacecraft itself.