# Oberth effect: Wikis

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# Encyclopedia

In astronautics, the Oberth effect or powered flyby is an effect by which the use of a rocket engine close to a gravitational body (where the gravity potential is low) can give much more kinetic energy and a much bigger change in final speed than the same burn executed further from the body for the same initial orbit. It is named for Hermann Oberth, the Romanian-born, German physicist and a founder of modern rocketry.

For the Oberth effect to be usefully employed, the vehicle must be able to use as much propellant as possible at the lowest possible altitude; thus the Oberth effect is often far less useful for low-thrust reaction engines such as ion drives, which have a low propellant flow rate.

## 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 accelerating its propellant to hypersonic speed. But when the rocket moves, the thrust of 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 the burn, (i.e. the faster they move), the greater the kinetic energy imparted to the rocket and its payload.

$\Delta E = F * s \;$

Where:

E is the energy, F is the force, s is the distance

Differentiating with respect to time:

$\frac {{dE}} {{dt}} = F * \frac {{ds}} {{dt}}$

Or:

$\frac {{dE}} {{dt}} = F * v$

where v is the velocity.

This can be integrated to calculate the increase in energy of the rocket.

However, integrating this is often unnecessary, if the burn is short. For example as a vehicle falls towards periapsis in any orbit (closed or escape orbits) the velocity relative to the central body increases. Briefly burning the engine (an 'impulsive burn') 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 exhaust is left with. When expended lower in the gravitational field, even if the exhaust is left with more kinetic energy, it is left with less total energy.[2]

## Parabolic example

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.

So if a spacecraft on a parabolic flyby of Jupiter with a periapsis velocity of 50 km/s, and it performs a 5 km/s burn, it turns out that the final velocity at great distance is 16.6 km/s; giving a multiplication of the burn by 3.3 times.

## 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:

$\tfrac{1}{2} (\Delta V + V_\text{esc})^2$
$= \tfrac{1}{2} \Delta V^2 + \Delta V \cdot V_\text{esc} + \tfrac{1}{2} V_\text{esc} ^ 2$

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

$\tfrac{1}{2} V_\text{esc}^2$

so it retains the energy:

$\tfrac{1}{2} \Delta V^2 + \Delta V \cdot V_\text{esc}$

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:

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

Plugging in 50 km/s escape velocity and 5 km/s burn we get a multiplier of 3.3.

Similar effects happen in closed and hyperbolic orbits.

## References

1. ^ Atomic Rockets web site: nyrath@projectrho.com
2. ^ The effect would be even stronger if the exhaust speed could be made equal to the speed of the rocket, then the exhaust would be left without kinetic energy, so the total energy of the exhaust would be as low as its potential energy.