Spend the same fuel and add the same speed. Change where you add it and the outcome changes.
Compare a burn at apoapsis, where the craft is slowest, with one at periapsis, where it is fastest. The speed added is the same either way, say km per second.
The energy gained is nonetheless greater at periapsis. The same burn puts the craft on an orbit that reaches farther.
This is the Oberth effect, pointed out by Oberth in the nineteen twenties.
The reason is in the shape of the formula. The energy of motion is , going as the square of the speed.
Raise the speed from to and the energy gained is .
There is a in the first term. For the same burn , the faster the craft was already going, the more it gains.
On a graph, the squared curve steepens toward the right. The same step sideways lifts you higher there.
From this comes a move that looks perverse: to go farther out, first drop inward.
Falling makes the craft faster. At the deepest point, the periapsis, it is fastest of all. That is where to burn.
The same burn then reaches farther than it would at apoapsis. The energy spent falling comes back on the way up, so nothing is lost by the detour. What remains is the gain.
Probes bound for the outer solar system dive toward Jupiter not only for the gravity assist but for this effect as well.
Fix the size of the burn and the energy gained rises in proportion to the speed where it is made.
In low orbit around the Earth a craft moves at km per second. Adding there brings it to escape velocity.
At the height of geostationary orbit the speed is only . Reaching escape from there takes only more, but getting up there has already cost .
Added up, leaving in one push from low down is cheaper. That difference is what burning deep buys you.
A probe bound for the outer planets circles the Earth and burns at perigee several times, stretching the ellipse. It is the same idea: burn only where the craft is fastest.
It works in reverse too. A probe heading in toward the Sun slows at perihelion to lower its orbit. The Parker Solar Probe does this repeatedly, alongside gravity assists at Venus.
At the extreme, a calculation says that a craft dropped close to a black hole and burning there could reach a good fraction of the speed of light on very little fuel. It is the deeper-is-better property pushed as far as it will go.
That closes the third chapter. Every calculation so far has held in a world containing exactly two bodies. In the next chapter we add a third.