Oberth effect

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The Oberth effect is a feature of astronautics where using a rocket engine close to a gravitational body gives a higher final speed than the same burn executed further from the body. It is named for Hermann Oberth, the Austro-Hungarian-born, German physicist and a founder of modern rocketry.

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[edit] Description

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 was achieved at any other time.

If the ship travels at velocity v at the start of a burn that changes the velocity by Δv, 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.

See also specific energy change of rockets:

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

where ε is the specific energy of the rocket (potential plus kinetic energy) and Δv is a separate variable, not just the change in v.

A possibly life-saving use of this effect took place during the Apollo 13 mission. While on its way to the Moon the spacecraft's Service Module was disabled and the Lunar Module was used as a lifeboat. Since supplies were limited, it was desirable to return to Earth as quickly as possible. The most efficient way to use the limited rocket power available was to make a burn right after the closest approach to the Moon.

[edit] Detailed proof

If an impulsive burn of ΔV is performed at periapsis in a parabolic orbit where the escape velocity is Vesc, then the specific kinetic energy after the burn is:

\frac{1}{2} (\Delta V + V_{esc})^2
= \frac{1}{2} \Delta V^2 + \Delta V * V_{esc} + \frac{1}{2} V_{esc} ^ 2

When the vehicle leaves the gravity field, the loss of specific kinetic energy is:

\frac{1}{2} V_{esc}^2

so it retains the energy:

\frac{1}{2} \Delta V^2 + \Delta V * V_{esc}

which is larger than the energy from a burn outside the gravitational field by ΔV * Vesc

the impulse is thus multiplied by a factor of:

\sqrt{1 +\frac{2 V_{esc}}{\Delta V}}

Similar effects happen in closed orbits.

[edit] Paradox

Since energy is conserved, and the vehicle's initial chemical energy is the same this may be considered somewhat paradoxical.

The resolution of the paradox is due to leaving propellant behind lower down in the gravitational field; as the vehicle falls into the field, the propellant gains kinetic energy, but during the burn the propellant loses more kinetic energy due to the high speed, and thus from conservation of energy there can be an equal and opposite gain in the energy in the body of the vehicle.

[edit] See also

[edit] References

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