Why does a rocket get more push firing at its lowest point than its highest?

"the Oberth effect—whereby a small impulse can cause a large change in a rocket’s orbital energy without violating any conservation laws."

"From power considerations, it turns out that when applying delta- v in the direction of the velocity the specific orbital energy gained per unit delta- v is equal to the instantaneous speed. This is called the Oberth effect."

"Oberth effect – the effect for fuel-saving maneuvers in interplanetary space travel is named after him. He was the first to describe it."

"For example, a satellite in an elliptical orbit is boosted more efficiently at high speed (that is, small altitude) than at low speed (that is, high altitude)."

"He discovered the Oberth effect (powered flyby or Oberth maneuver), a fuel-saving strategy for interplanetary space flight that is commonly used"

"If more speed is needed than available from gravity assist alone, a rocket burn near the periapsis (closest planetary approach) uses the least fuel."

"Another example is that when a vehicle is making a pass of a planet, burning the propellant at closest approach rather than further out gives significantly higher final speed, and this is even more so when the planet is a large one with a deep gravity field, such as Jupiter."

"A given rocket burn always provides the same change in velocity ( Δv ), but the change in kinetic energy is proportional to the vehicle's velocity at the time of the burn. Therefore the maximum kinetic energy is obtained when the burn occurs at the vehicle's maximum velocity (periapsis)."

"The Oberth effect describes this technique in more detail."

"The Oberth Effect is a concept in rocketry that describes the increased efficiency of a rocket engine when it is fired at high speeds, particularly during a powered descent or a gravity-assist maneuver."

"By firing the rocket engines at high speeds during descent, the spacecraft can achieve a greater change in velocity with less fuel consumption."

"By timing the rocket burn during the closest approach to the celestial body, the spacecraft can maximize the Oberth Effect and gain a significant boost in speed."

"During its journey to the outer solar system, the spacecraft performed a gravity-assist maneuver around Jupiter to increase its velocity and reach Pluto in a shorter amount of time."

"Now imagine that you burn prograde at periapsis. Now you'll be traveling faster as you leave than you were as you came in. This means that gravity has less time to act on you on the way out than it did on the way in."

"engine at periapsis, that propellant has to escape the gravity well, meaning it will be traveling slower at infinity that it would have been from an engine firing in deep space. That extra kinetic energy has to go somewhere, and it goes into the rocket."

"The Oberth effect depends on the direction of travel relative to the gravity source at the time of the burn."

"Now you'll be traveling faster as you leave than you were as you came in. This means that gravity has less time to act on you on the way out than it did on the way in ."

"The Oberth effect depends on the direction of travel relative to the gravity source at the time of the burn."

"Thus, a low perihelion burn provides maximum benefit from the Oberth Effect, accelerating the spacecraft rapidly towards the receding 3I/ATLAS."

"including the burn at perihelion (at 7SR) known as the Solar Oberth maneuver."

"The reference mission requires a SOM at 3.2 Solar Radii from the Sun’s centre, with an intercept after 35-50 years."

"This principle establishes that applying a ΔV at the point where the spacecraft has a higher velocity – in this case, the periapsis – will maximize the spacecraft’s gain of kinetic energy [21]."