# Gravity assist

The trajectories that enabled NASA's twin Voyager spacecraft to tour the four gas giant planets and achieve velocity to escape our solar system

In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement (e.g. orbit around the sun) and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft, typically in order to save propellant, time, and expense. Gravity assistance can be used to accelerate (both positively and negatively) and/or re-direct the path of a spacecraft.

The "assist" is provided by the motion of the gravitating body as it pulls on the spacecraft.[1] The technique was first proposed as a mid-course manoeuvre in 1961, and used by interplanetary probes from Mariner 10 onwards, including the two Voyager probes' notable fly-bys of Jupiter and Saturn.

## Explanation

A gravity assist around a planet changes a spacecraft's velocity (relative to the Sun) by entering and leaving the gravitational field of a planet. The spacecraft accelerates as it approaches the planet and decelerates while escaping its gravitational pull (which is approximately the same). Because the planet orbits the sun, the spacecraft is affected by this motion during the maneuver. To accelerate, the spacecraft flies with the movement of the planet (taking a small amount of the planet's orbital energy); to decelerate, the spacecraft flies against the movement of the planet. The sum of the kinetic energies of both bodies remains constant (see elastic collision). A slingshot maneuver can therefore be used to change the spaceship's trajectory and speed relative to the Sun.

A close terrestrial analogy is provided by a tennis ball bouncing off a moving train. Imagine throwing a ball at 30 mph toward a train approaching at 50 mph. The engineer of the train sees the ball approaching at 80 mph and then departing at 80 mph after the ball bounces elastically off the front of the train. Because of the train's motion, however, that departure is at 130 mph relative to the station.

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

Translating this analogy into space, then, a "stationary" observer sees a planet moving left at speed U and a spaceship moving right at speed v. If the spaceship has the proper trajectory, 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). Since the planet is moving left at speed U, the total velocity of the spaceship relative to the observer will be the velocity of the moving planet plus the velocity of the spaceship with respect to the planet. So the velocity will be U + ( U + v ), that is 2U + v.

This oversimplified example is impossible to refine without additional details regarding the orbit, but if the spaceship travels in a path which forms a parabola, it can leave the planet in the opposite direction without firing its engine, and 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 the spacecraft's effects on the planet have not been considered. The linear momentum gained by the spaceship is equal in magnitude to that lost by the planet, though the planet's enormous mass compared to the spacecraft 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 three dimensions. The same principles apply, only adding the planet's velocity to that of the spacecraft 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.

Due to the reversibility of orbits, gravitational slingshots can also be used to decelerate a spacecraft. Both Mariner 10 and MESSENGER performed this maneuver to reach Mercury.

Another close terrestrial analogy, demonstrating deceleration, is provided by a tennis ball bouncing off a moving train. Imagine throwing a ball at 130 mph toward the back of a train that is moving away at 50 mph. The engineer of the train sees the ball approaching at 80 mph and then departing at 80 mph after the ball bounces elastically off the back of the train. Because of the train's motion, however, that departure is at 30 mph relative to the station.

If even more speed is needed than available from gravity assist alone, the most economical way to utilize a rocket burn is to do it near the periapsis (closest approach). A given rocket burn always provides the same change in velocity (Δv), but the change in kinetic energy is proportional to the vehicle's velocity at the time of the burn. So to get the most kinetic energy from the burn, the burn must occur at the vehicle's maximum velocity, at periapsis. Powered slingshots describes this technique in more detail.

## Historical origins of the method

In his paper “Тем кто будет читать, чтобы строить” (To whoever will read [this paper] in order to build [an interplanetary rocket]),[3] published in 1938 but dated 1918–1919,[4] Yuri Kondratyuk suggested that a spacecraft traveling between two planets could be accelerated at the beginning of its trajectory and decelerated at the end of its trajectory by using the gravity of the two planets' moons. In his 1925 paper "Проблема полета при помощи реактивных аппаратов: межпланетные полеты" [Problems of flight by jet propulsion: interplanetary flights],[5] Friedrich Zander made a similar argument.

However, neither investigator realized that gravitational assists from planets along a spacecraft’s trajectory could propel a spacecraft and that therefore such assists could greatly reduce the amount of propellant required to travel among the planets.[6] That discovery was made by Michael Minovitch in 1961.[7]

The gravity assist maneuver was first used in 1959 when the Soviet probe Luna 3 photographed the far side of Earth's Moon. The maneuver relied on research performed at the Department of Applied Mathematics of Steklov Institute.[8][9]

## Why gravitational slingshots are used

Plot of Voyager 2's heliocentric velocity against its distance from the sun, illustrating the use of gravity assist to accelerate the spacecraft by Jupiter, Saturn and Uranus. To observe Triton, Voyager 2 passed over Neptune's north pole resulting in an acceleration out of the plane of the ecliptic and reduced velocity away from the sun.[10]

A spacecraft traveling from Earth to an inner planet will accelerate because it is falling toward the Sun, and a spacecraft traveling from Earth to an outer planet will decelerate because it is leaving the vicinity of the Sun.

Although the orbital speed of an inner planet is greater than that of the Earth, a spacecraft traveling to an inner planet, even at the minimum speed needed to reach it, is still accelerated by the Sun's gravity to a speed notably greater than the orbital speed of that destination planet. If the spacecraft's purpose is only to fly by the inner planet, then there is typically no need to slow the spacecraft. However, if the spacecraft is to be inserted into orbit about that inner planet, then there must be some way to slow the spacecraft.

Similarly, while the orbital speed of an outer planet is less than that of the Earth, a spacecraft leaving the Earth at the minimum speed needed to travel to some outer planet is decelerated by the Sun's gravity to a speed far less than the orbital speed of that outer planet. Thus, there must be some way to accelerate the spacecraft when it reaches that outer planet if it is to enter orbit about it. However, if the spacecraft is accelerated to more than the minimum required, less total propellant will be needed to enter orbit about the target planet. Also, accelerating the spacecraft early in the flight will, of course, reduce the travel time.

Rocket engines can certainly be used to accelerate and decelerate the spacecraft. However, rocket thrust takes propellant, propellant has mass, and even a small increment Δv (delta-v) in velocity translates to far larger requirement for propellant needed to escape Earth's gravity well. This is because not only must the primary stage engines lift that extra propellant, they must also lift more propellant still, to lift that additional propellant. Thus the liftoff mass requirement increases exponentially with an increase in the required delta-v of the spacecraft.

Since a gravity assist maneuver can change the speed of a spacecraft without expending propellant, if and when possible, combined with aerobraking, it can save significant amounts of propellant.

As an example, the Messenger mission used gravity assist maneuvering to slow the spacecraft on its way to Mercury; however, since Mercury has almost no atmosphere, aerobraking could not be used for insertion into orbit around it.

Journeys to the nearest planets, Mars and Venus, use a Hohmann transfer orbit, an elliptical path which starts as a tangent to one planet's orbit round the Sun and finishes as a tangent to the other. This method uses very nearly the smallest possible amount of fuel, but is very slow — it can take over a year to travel from Earth to Mars (fuzzy orbits use even less fuel, but are even slower).

Similarly it might take decades for a spaceship to travel to the outer planets (Jupiter, Saturn, Uranus, etc.) using a Hohmann transfer orbit, and it would still require far too much propellant, because the spacecraft would have to travel for 800 million km (500 million miles) or more against the force of the Sun's gravity. As gravitational assist maneuvers offer the only way to gain speed without using propellant, all missions to the outer planets have used it.[citation needed]

## Limits to slingshot use

The Planetary Grand Tour trajectory of Voyager 2.

The main practical limit to the use of a gravity assist maneuver is that planets and other large masses are seldom in the right places to enable a voyage to a particular destination. For example the Voyager missions which started in the late 1970s were made possible by the "Grand Tour" alignment of Jupiter, Saturn, Uranus and Neptune. A similar alignment will not occur again until the middle of the 22nd century. That is an extreme case, but even for less ambitious missions there are years when the planets are scattered in unsuitable parts of their orbits.

Another limitation is the atmosphere, if any, of the available planet. The closer the spacecraft can approach, the more boost it gets, because gravity falls off with the square of distance from a planet's center. If a spacecraft gets too far into the atmosphere, the energy lost to drag can exceed that gained from the planet's gravity. On the other hand, the atmosphere can be used to accomplish aerobraking. There have also been theoretical proposals to use aerodynamic lift as the spacecraft flies through the atmosphere. This maneuver, called an aerogravity assist, could bend the trajectory through a larger angle than gravity alone, and hence increase the gain in energy.

Interplanetary slingshots using the Sun itself are not possible because the Sun is at rest relative to the Solar System as a whole. However, thrusting when near the Sun has the same effect as the powered slingshot described below. 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 swinging past the Sun to boost its galactic travel. The energy and angular momentum would then come from the Sun's orbit around the Milky Way. This concept features prominently in Arthur C. Clarke's 1972 award-winning novel Rendezvous With Rama; his story concerns an interstellar spacecraft that uses the Sun to perform this sort of maneuver, and in the process unnecessarily alarms many nervous humans.

Another theoretical limit is based on general relativity. The deepest gravity wells are those found around black holes, but if a spacecraft gets close to the Schwarzschild radius of a black hole, 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.

A rotating black hole might provide additional assistance, if its spin axis is aligned the right way. General relativity predicts that a large spinning mass-produces frame-dragging — close to the object, space itself is dragged around in the direction of the spin. Any ordinary rotating object produces this effect. While attempts to measure frame dragging about the Sun have produced no clear evidence, experiments performed by Gravity Probe B have detected frame-dragging effects caused by the Earth.[11] General relativity predicts that a spinning black hole is surrounded by a region of space, called the ergosphere, within which standing still (with respect to the black hole's spin) is impossible, because space itself is dragged at the speed of light in the same direction as the black hole's spin. The Penrose process may offer a way to gain energy from the ergosphere, although it would require the spaceship to dump some "ballast" into the black hole, and the spaceship would have had to expend energy to carry the "ballast" to the black hole.

## Timeline of notable examples

### Mariner 10 – first use in an interplanetary trajectory

The Mariner 10 probe was the first spacecraft to use the gravitational slingshot effect to reach another planet, passing by Venus on February 5, 1974, on its way to becoming the first spacecraft to explore Mercury.

### Voyager 1 – farthest human-made object

As of Sept 3, 2013, Voyager 1 is over 125.3 AU (18.7 billion km) from the Sun, and is in interstellar space.[12] It gained the energy to escape the Sun's gravity completely by performing slingshot maneuvers around Jupiter and Saturn.[13][14] [17.4 hours for light signals to arrive from sun.]

### Galileo – a change of plan

The Galileo spacecraft was launched by NASA in 1989 aboard Space Shuttle Atlantis. Its original mission was designed to use a direct Hohmann transfer. However, Galileo's intended booster, the cryogenically fueled (Hydrogen/Oxygen) Centaur booster rocket was prohibited as a Shuttle "cargo" for safety considerations following the loss of the Space Shuttle Challenger. With its substituted solid rocket upperstage, the IUS, which could not provide as much delta-v, Galileo did not ascend directly to Jupiter, but flew by Venus once and Earth twice in order to reach Jupiter in December, 1995.

The Galileo engineering review speculated (but was never able to prove conclusively) that this longer flight time coupled with the stronger sunlight near Venus caused lubricant in Galileo's main antenna to fail, forcing the use of a much smaller backup antenna with a consequent lowering of data rate from the spacecraft.

Its subsequent tour of the Jovian moons also used numerous slingshot maneuvers with those moons to conserve fuel and maximize the number of encounters.

### The Ulysses probe changed the plane of its trajectory

In 1990, NASA launched the ESA spacecraft Ulysses to study the polar regions of the Sun. All the planets orbit approximately in a plane aligned with the equator of the Sun. Thus, to enter an orbit passing over the poles of the Sun, the spacecraft would have to eliminate the 30 km/s speed it inherited from the Earth's orbit around the Sun and gain the speed needed to orbit the Sun in the pole-to-pole plane — tasks that are impossible with current spacecraft propulsion systems alone, making gravity assist maneuvers essential.

Accordingly, Ulysses was first sent towards Jupiter, aimed to arrive at a point in space just "in front of" and "below" the planet. As it passed Jupiter, the probe 'fell' through the planet's gravity field, exchanging momentum with the planet; this gravity assist maneuver bent the probe's trajectory up out of the planetary plane into an orbit that passed over the poles of the Sun. By using this maneuver, Ulysses needed only enough propellant to send it to a point near Jupiter, which is well within current capability.

### MESSENGER

The MESSENGER mission (launched in August 2004) made extensive use of gravity assists to slow its speed before orbiting Mercury. The MESSENGER mission included one flyby of Earth, two flybys of Venus, and three flybys of Mercury before finally arriving at Mercury in March 2011 with a velocity low enough to permit orbit insertion with available fuel. Although the flybys are primarily orbital maneuvers, each provided an opportunity for significant scientific observations.

### The Cassini probe – multiple gravity assists

The Cassini probe passed by Venus twice, then Earth, and finally Jupiter on the way to Saturn. The 6.7-year transit was slightly longer than the six years needed for a Hohmann transfer, but cut the extra velocity (delta-v) needed to about 2 km/s, so that the large and heavy Cassini probe was able to reach Saturn, which would not have been possible in a direct transfer even with the Titan IV, the largest launch vehicle available at the time. 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 current launch vehicles and spacecraft propulsion systems.

Cassini's speed related to Sun. The various gravity assists form visible peaks on the left, while the periodic variation on the right is caused by the spacecraft's orbit around Saturn. The data was from JPL Horizons Ephemeris System. The speed above is in kilometers per second. Note also that the minimum speed achieved during Saturnian orbit is more or less equal to Saturn's own orbital velocity, which is the ~5 km/s velocity which Cassini matched to enter orbit.

### Solar Probe+

The NASA Solar Probe+ mission, scheduled for launch in 2018, uses multiple gravity assists at Venus to remove the Earth's angular momentum from the orbit, in order to drop down to a distance of 9.5 solar radii from the sun. This will be the closest approach to the sun of any space mission.

## Powered slingshots

A well-established way to get more energy from a gravity assist is to fire a rocket engine at periapsis where a spacecraft is at its maximum velocity.

Rocket engines produce the same force regardless of their initial 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. However, when the rocket and its payload are free to move, the force applied 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. 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 are simply too inefficient when used in this manner.)

Energy is still conserved, however. The additional energy imparted to the payload is exactly matched by a decrease in energy imparted to the propellant being expelled behind the rocket. This is because the velocity of the rocket is being subtracted from the propellant exhaust velocity. Since the ultimate fate of the propellant is not a concern, the fastest possible burn is usually the optimal procedure.

To impart the most kinetic energy to a spacecraft whose free-fall velocity varies with time, the burn must occur when the spacecraft is moving fastest, which usually occurs at periapsis (the point of closest approach).

## References

1. ^ Basics of Space Flight, Sec. 1 Ch. 4, NASA Jet Propulsion Laboratory
2. ^ The Slingshot Effect, Durham University
3. ^ Kondratyuk’s paper is included in the book: Mel’kumov, T. M., ed., Pionery Raketnoy Tekhniki [Pioneers of Rocketry: Selected Papers] (Moscow, U.S.S.R.: Institute for the History of Natural Science and Technology, Academy of Sciences of the USSR, 1964). An English translation of Kondratyuk’s paper was made by NASA. See: NASA Technical Translation F-9285, pages 15-56 (Nov. 1, 1965).
4. ^ In 1938, when Kondratyuk submitted his manuscript "To whoever will read in order to build" for publication, he dated the manuscript 1918–1919, although it was apparent that the manuscript had been revised at various times. See page 49 of NASA Technical Translation F-9285 (Nov. 1, 1965).
5. ^ Zander's 1925 paper, "Problems of flight by jet propulsion: interplanetary flights", was translated by NASA. See NASA Technical Translation F-147 (1964); specifically, Section 7: Flight Around a Planet’s Satellite for Accelerating or Decelerating Spaceship, pages 290-292.
6. ^ See page 13 of: Dowling, Richard L.; Kosmann, William J.; Minovitch, Michael A.; and Ridenoure, Rex W., "The origin of gravity-propelled interplanetary space travel" (IAA paper no. 90-630), presented at the 41st Congress of the International Astronautical Federation, which was held 6–12 October 1990 in Dresden, G.D.R. Available on-line at: http://www.gravityassist.com/IAF1/IAF1.pdf .
7. ^ Minovitch, Michael, "A method for determining interplanetary free-fall reconnaissance trajectories," Jet Propulsion Laboratory Technical Memo TM-312-130, pages 38-44 (23 August 1961).
8. ^
9. ^ Egorov, Vsevolod Alexandrovich (1957) "Specific problems of a flight to the moon", Physics - Uspekhi, Vol. 63, No. 1a, pages 73–117. Egorov’s work is mentioned in: Boris V. Rauschenbakh, Michael Yu. Ovchinnikov, and Susan M. P. McKenna-Lawlor, Essential Spaceflight Dynamics and Magnetospherics (Dordrecht, Netherlands: Kluwer Academic Publishers, 2002), pages 146–147. (The latter reference is available on-line at: http://books.google.com/books?id=m22bjIWZU9MC&pg=PA146&lpg=PA146&source=web&ots=U-tZaqVFhE&sig=LT_eEZcCegkd1dDS79glEkT28sw&hl=en#PPA146,M1 .)
10. ^ Basics of space flight: Interplanetary Trajectories
11. ^ Everitt et al. (2011). "Gravity Probe B: Final Results of a Space Experiment to Test General Relativity". Physical Review Letters 106 (22): 221101. arXiv:1105.3456. Bibcode:2011PhRvL.106v1101E. doi:10.1103/PhysRevLett.106.221101. PMID 21702590.
12. ^ http://www.jpl.nasa.gov/interstellarvoyager/
13. ^ Cassini-Huygens: Operations - Gravity Assists
14. ^ http://www.heavens-above.com/solar-escape.asp?lat=0&lng=0&loc=Unspecified&alt=0&tz=CET