Oberth effect

The Oberth effect is a feature of astronautics where using a rocket engine close to a gravitational body can give a much bigger change in final speed than the same burn executed further from the body. It is named for Hermann Oberth, the Romanian-born, German physicist and a founder of modern rocketry.

Description

Rocket engines produce the same force regardless of their velocity. A rocket acting on a fixed object, as in a static firing, does no useful work at all; the rocket's stored energy is entirely expended on its propellant. But when the rocket and payload move, the force applied to the payload by the rocket during any time interval acts through the distance the rocket and payload move during that time. Force acting through a distance is the definition of mechanical energy or work. So the farther the rocket and payload move during any given interval, i.e., the faster they move, the greater the kinetic energy imparted to the payload by the rocket. (This is why rockets are seldom used on slow-moving vehicles; they're simply too inefficient.)

In particular as a vehicle falls towards periapsis in any orbit (closed or escape orbits) the velocity relative to the central body increases. Burning the engine prograde at periapsis increases the velocity by the same increment as at any other time, determined by the delta-v. However, since the vehicle's kinetic energy is related to the square of its velocity, this increase in velocity has a disproportionate effect on the vehicle's kinetic energy; leaving it with higher energy than if the burn were achieved at any other time.^[1]

It may seem that the rocket is getting energy for free, which would violate conservation of energy. However, any gain to the rocket's energy is balanced by an equal decrease in the energy the propellant is left with. When expended lower in the gravitational field, the propellant is left with less total energy.

Example

If the ship travels at velocity $v$ at the start of a burn that changes the velocity by $\Delta v$ , then the change in specific orbital energy (SOE) is:

v\Delta v+{\frac {(\Delta v)^{2}}{2}}

Once the space craft is far from the planet again, the SOE is entirely kinetic, since gravitational potential energy tends to zero. Therefore, the larger the $v$ at the time of the burn, the greater the final kinetic energy, and the higher the final velocity.

The effect becomes more pronounced the closer to the central body, or more generally, the deeper in the gravitational field potential the burn occurs, since the velocity is higher there.

For example, a Hohmann transfer orbit from Earth to Jupiter brings a spacecraft into a hyperbolic flyby of Jupiter with a periapsis velocity of 60 km/s, and a final velocity (asymptotic residual velocity) of 5.6 km/s, which is 10.7 times slower. That means a burn that adds one joule of kinetic energy when far from Jupiter would add 10.7 joules at periapsis. Every 1 m/s gained at periapsis adds ${\sqrt {10.7}}=3.3$ m/s to the spacecraft's final velocity. Thus, Jupiter's immense gravitational field has tripled the effectiveness of the space craft's propellant.

\Delta \epsilon =\int v\,d(\Delta v)

where $\epsilon$ is the specific energy of the rocket (potential plus kinetic energy) and $\Delta v$ is a separate variable, not just the change in $v$ .

Detailed proof

If an impulsive burn of $\Delta V$ is performed at periapsis in a parabolic orbit where the escape velocity is $V_{esc}$ , then the specific kinetic energy after the burn is: