Gravitational slingshot

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In orbital mechanics and aerospace engineering, a gravitational slingshot is the use of the motion of a planet to alter the path and speed of an interplanetary spacecraft. It is a commonly used maneuver for visiting the outer planets, which would otherwise be prohibitively expensive, if not impossible, to reach with current technologies. It is also known as a "gravity assist".

A 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 do, 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.

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Explanation

Image:Gravity assist still Jupiter.svg Consider a spacecraft on a trajectory that will take it close to a planet, for instance Jupiter. As the spacecraft approaches the planet, Jupiter's gravity will pull on the spacecraft, speeding it up. After passing the planet, the gravity will continue pulling on the spacecraft, slowing it down. The net effect on the speed is zero, although the direction may have changed in the process.

Image:Gravity assist moving Jupiter.svg So where is the slingshot? The key is to remember that the planets are not standing still, they are moving in their orbits around the Sun. Thus while the speed of the spacecraft has remained the same as measured with reference to Jupiter, the initial and final speeds may be quite different as measured in the Sun's frame of reference. Depending on the direction of the outbound leg of the trajectory, the spacecraft can gain up to twice the orbital speed of the planet. In the case of Jupiter, this is over 13 km/s. A slingshot may be simulated by rolling a steel ball past a magnet in one hand that is then moved away. Because both masses must not cross paths, the acceleration is oblique to the field and thus is similar to a sail vehicle tacking to work against the force.

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Background

It is important to understand how spacecraft move from planet to planet. Taking Mars as an example destination, the simplest way to solve this problem is to use a Hohmann transfer orbit, an elliptical orbit with the Earth at perihelion and Mars at aphelion. If launched at the proper moment, the spacecraft will arrive at the aphelion just as Mars is passing by. These types of transfers are commonly used, e.g., for moving between orbits over the Earth, Earth-Moon and Earth-Mars transfers.

A Hohmann transfer to the outer planets requires long times and considerable delta V (the velocity adjustments that consume rocket propellant). This is where the slingshot finds its most common applications. Instead of the Hohmann trajectory directly to, say Saturn, the spacecraft is instead sent in a path that is aimed only as far as, say, Jupiter, and the slingshot is then used to accelerate the spacecraft on towards Saturn. This way, the planet lends the spacecraft additional angular momentum, allowing it to reach Saturn using little or no fuel on top of that required to reach Jupiter. Also, during the spacecraft's close approach to Jupiter, the effectiveness of the rocket's propellant is magnified, such that small thrusts near Jupiter produce large changes in the spacecraft's eventual velocity. Such missions require careful timing, making the launch window a crucial part of the mission.

A Hohmann transfer to Saturn would require a total of 15.7 km/s delta V (disregarding Earth's and Saturn's own gravity wells, and disregarding aerobraking) which is not within the capabilities of our current spacecraft boosters. A trip using multiple gravitational assists may take longer, but will use considerably less delta V, allowing a much larger spacecraft to be sent. Such a strategy was used on the Cassini probe, which passed by Venus twice, then Earth, and finally Jupiter on the way to Saturn. The 6.7-year transit is slightly longer than the six years needed for a Hohmann transfer, but cut the total amount of delta V needed to about 2 km/s, so that the large and heavy Cassini was able to reach Saturn even with the small boosters available.

Image:Cassini Interplanet traject.jpg

Image:Cassini's speed related to Sun.png

Another example is Ulysses, the ESA spacecraft which studied the polar regions of the Sun. All the planets orbit roughly in a plane aligned with the equator of the Sun; to move to an orbit passing over the poles of the Sun, the spacecraft would have to change its 30 km/s of the Earth's orbit to another trajectory at right angles to the plane of the Earth's orbit, a task impossible with current spacecraft propulsion systems. Instead the craft was sent towards Jupiter, aimed to arrive at a point in space just "in front" and "below" the planet.

As it passed the planet, the probe 'fell' through Jupiter's gravity field, borrowing a minute amount of momentum from the planet; after it had passed Jupiter, the velocity change had bent the probe's trajectory up out of the plane of the planetary orbits, placing it in an orbit that passed over the poles of the Sun, rendering that region visible to the probe. This manoeuvre required only enough fuel to send Ulysses to a point near Jupiter, which is well within current technologies.

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Powered slingshots

A well-established way to get more energy from a slingshot is to fire a rocket engine near the periapsis to increase the spacecraft's speed. A given rocket burn always provides the same change in velocity (delta-v), but the change in kinetic energy is proportional to the vehicle's velocity at the time of the burn. Therefore, to get the most kinetic energy from the burn, the burn must occur at the vehicle's maximum velocity, at periapsis. Energy is still conserved. The extra energy comes from the propellant being "left behind" in the planet's gravity well.

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

<math>v \Delta v + \frac{(\Delta v)^2}{2}</math>

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 <math>v</math> at the time of the burn, the greater the final kinetic energy, and the higher the final velocity.

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 <math>\sqrt{10.7} = 3.3</math> 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:

<math>\Delta \epsilon = \int v\, d (\Delta v)</math>

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

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Limits to slingshot use

The main practical limit to the use of a slingshot is the size of the available masses in the mission.

Another limit is caused by the atmosphere of the available planet. The closer the craft can get, the more boost it gets, because gravity falls with the square of distance. If a craft gets too far into the atmosphere, the energy lost to friction can exceed that gained from the planet.

Interplanetary slingshots using the Sun itself are impossible because the Sun is at rest with respect to its own frame of reference and is therefore incapable of donating any angular momentum. However, thrusting when near the Sun has the same effect as the powered slingshot described above. This has the potential to magnify a spacecraft's thrusting power enormously, but is limited by the spacecraft's ability to resist the heat.

An interstellar slingshot using the Sun is conceivable, involving for example an object coming from elsewhere in our galaxy and slingshotting around the Sun to boost its galactic travel. The energy and angular momentum would then come from the Sun's orbit around the Milky Way. The time scales involved for such an operation are considerably beyond current human capabilities, however.

There's also another, theoretical limit based on general relativity. If a spacecraft gets close to the Schwarzschild radius of a black hole (the ultimate gravity well), space becomes so curved that slingshot orbits require more energy to escape than the energy that could be added by the black hole's motion.

Also, a spinning mass produces frame-dragging. A spinning black hole actually is surrounded by a region of space, called the ergosphere, within which standing still (with respect to the black hole's spin) is impossible, as space itself slips in the same direction as the black hole's spin at the speed of light. Suffice it to say that there is a subtle relativistic effect (the Lense-Thirring effect) which can transfer angular momentum between any spinning mass and a passing object.

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See also

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External links

de:Swing-by es:Asistencia gravitatoria fr:Appui gravitationnel it:Fionda gravitazionale nl:Zwaartekrachtsslinger ja:スイングバイ ru:Гравитационный манёвр sk:Gravitačný manéver fi:Painovoimalinko

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