Interstellare Raumfahrt: Oberth-Manöver

Slingshot Manöver

Over-simplified example of gravitational slingshot: the spacecraft's velocity changes by up to twice the planet's velocity

Bei einem Slingshot-Manöver in einem Planetensytem ändert sich die Geschwindigkeit des Raumfahrzeuges bezogen auf die Sonne

A gravity assist or slingshot maneuver around a planet changes a spacecraft's velocity relative to the Sun, even though it preserves the spacecraft's speed relative to the planet—as it must according to the law of conservation of energy. To a first approximation, from a large distance, the spacecraft appears to have bounced off the planet. Physicists call this an elastic collision even though no contact actually occurs.

Suppose that you are a "stationary" observer and that you see: a planet moving left at speed U; a spaceship moving right at speed v. If the spaceship is on the right path, it will pass close to the planet, moving at speed U + v relative to the planet's surface because the planet is moving in the opposite direction at speed U. When the spaceship leaves orbit, it is still moving at U + v relative to the planet's surface but in the opposite direction, to the left; and since the planet is moving left at speed U, the spaceship is moving left at speed 2U + v from your point of view – its speed has increased by 2U, twice the speed at which the planet is moving.

It might seem that this is oversimplified since we have not covered the details of the orbit, but it turns out that if the spaceship travels in a path which forms a hyperbola, it can leave the planet in the opposite direction without firing its engine, the speed gain at large distance is indeed 2U once it has left the gravity of the planet far behind.

This explanation might seem to violate the conservation of energy and momentum, but we have neglected the spacecraft's effects on the planet. The linear momentum gained by the spaceship is equal in magnitude to that lost by the planet, though the planet's large mass makes the resulting change in its speed negligibly small. These effects on the planet are so slight (because planets are so much more massive than spacecraft) that they can be ignored in the calculation.[2]

Realistic portrayals of encounters in space require the consideration of two dimensions. In that case the same principles apply, only adding the planet's velocity requires vector addition, as shown below.

2 dimensional schematic of gravitational slingshot. The arrows show the direction in which the spacecraft is traveling before and after the encounter. The arrows' length shows the spacecraft's speed.

Parabolic 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.

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

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

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

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

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

so it retains the energy:

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

which is larger than the energy from a burn outside the gravitational field by $\Delta V\cdot V_{esc}\;$

The impulse is thus multiplied by a factor of:

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

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.