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David Shortt

Gravity assist

Posted by David Shortt

27-09-2013 13:30 CDT

Topics: Voyager 1 and 2, Neptune, explaining science, Saturn, Uranus, Jupiter

With the recent announcement by NASA that the 36 year-old spacecraft Voyager 1 has officially entered interstellar space at a distance from the Sun about four times further than Neptune's orbit, and with Voyager 2 not far behind, it seems worthwhile to explore how humans managed to fling objects so far into space.


Mariner-Jupiter-Saturn 1977 Spacecraft Artwork, 1975


Mariner-Jupiter-Saturn 1977 Spacecraft Artwork, 1975
NASA and JPL initially referred to what became the Voyagers as the Mariner-Jupiter-Saturn 1977 Project. The two Voyagers were advanced versions of the Mariner-class spacecraft that JPL had flown successfully to Venus, Mars, and Mercury. Shown here is a 1975 JPL artist's rendering of Voyager after encountering Jupiter and, after a gravity assist, approaching Saturn.

Mission planners use gravity assists because they allow the objective to be accomplished with much less fuel (and hence with a much smaller, cheaper rocket) than would otherwise be required. Lifting extra fuel into orbit, just so it can be used later, is exponentially expensive. Furthermore, the extra speed gained by gravity assists dramatically reduces the duration of a mission to the outer planets.

Gravity assists seem a bit mysterious, like one is getting something for nothing. This feeling can persist even if you know some physics. Since energy is conserved, you reason, how can a spacecraft obtain a net velocity boost by passing by a planet? Energy conservation suggests the spacecraft should speed up while approaching the planet, but then lose the same speed while departing. Recently I was talking with a colleague, an excellent plasma physicist who knew the phrase "gravity assist" but thought it must be marketing hyperbole because he didn't believe it could actually work. The mystery begs to be explained.

The key to understanding how a gravity assist works is to consider the problem from two different points of view, or reference frames. It's convenient to think about reference frames for both the planet and for the sun (or the solar system). For economy of language I'll call them the "planet frame" and the "sun frame."

In the planet frame, the planet sits still (by definition!). More importantly, since the planet is so much more massive than the spacecraft, the planet sits almost exactly at the center of mass of the two objects and does not react by any measurable amount as a result of the encounter. For example, Jupiter is about 10 to the 24th power times more massive than the Voyager spacecraft, so Jupiter ignores an encounter to an extremely high degree of precision. This means the spacecraft's total energy, made up of kinetic energy (energy of motion) plus potential energy (energy due to proximity to a massive object), is conserved throughout the encounter in this frame.

In the planet frame, then, the spacecraft indeed speeds up on approach and slows down by the same amount while departing, just like my colleague thought. During the approach, as the spacecraft falls into the gravity well of the planet, it gains kinetic energy (i.e. speed) and loses gravitational potential energy, trading one for the other just like a ball rolling downhill. After the encounter it climbs back out of the gravity well and loses whatever kinetic energy it gained during the approach, ending up with the same final speed it started with. The direction of the spacecraft changes during the encounter, however, so typically it leaves the planet heading in a different direction. The amount of deflection can be controlled by adjusting how close the spacecraft comes to the planet. The closer it gets, the greater the deflection. It's possible to have a very small deflection, near zero degrees, by arranging a wide miss. The maximum deflection is 180 degrees, sending the spacecraft back where it came from, obtained by arranging an extremely close approach. Mathematically the spacecraft's path is a hyperbola, so we say the spacecraft follows a hyperbolic trajectory in the planet frame.

Now let's consider what the encounter looks like in the Sun frame, where the Sun is stationary and the planet is moving. The difference between the planet frame and the Sun frame is just the velocity of the planet with respect to the Sun. To convert from the planet frame to the Sun frame, we simply add the velocity of the planet to both the planet and the spacecraft. This velocity is a vector, which means direction is important, and it can be in any arbitrary direction depending on the planet's position in its orbit at the time of the encounter (It also changes with time because the planet is following a curved orbit around the sun, but during the relatively short encounter with the spacecraft it's a reasonable approximation to consider the planet as moving in a straight line). Because the direction of the spacecraft changes when it encounters the planet and because the original direction of the spacecraft is also arbitrary, it's not immediately obvious how the encounter will look in the Sun frame. The arbitrariness of the directions gives rise to a rich set of possible behavior in the Sun frame, all in accordance with Newton's laws of motion, even though in the planet frame the encounters are simple hyperbolic trajectories. Crucially, because the direction changes, the speed of the spacecraft is different before and after the encounter when viewed in the Sun frame. The outgoing speed is not the same as the incoming speed, and the spacecraft can either speed up or slow down. Let's see by example how this works.

Figure 1: Example encounter

David Shortt

Figure 1: Example encounter

Figure 1 shows a made-up example of an encounter. The top panel shows the encounter in the Sun frame, in which the planet (in black) is moving to the right, and the spacecraft (in blue) experiences a gravity assist. The bottom panel shows the view from the planet frame, in which the spacecraft approaches the planet from below and the planet sits still. I chose the approach parameters so that the trajectory is bent through approximately 90 degrees in the planet frame. In the planet frame the spacecraft leaves the planet with the same speed with which it approached, but in the Sun frame it's clear the spacecraft gains quite a bit of speed. You can see how the spacecraft approaches the planet from behind, accelerates as it gets closer, and "slingshots" around the planet. In this example the spacecraft gains about 60% of the planet's own velocity. We'll see later on that this example is fairly close to what happened to Voyager 2 at Jupiter, Saturn and Uranus.

How does this happen? Consider that in the bottom panel the spacecraft initially moves vertically with some velocity, call it v. After the encounter it leaves the planet with the same velocity v, but in the horizontal direction. To convert to the sun frame, we add the planet's velocity (which I chose arbitrarily to be v in the horizontal direction) to both the planet and the spacecraft. Using the Pythagorean Theorem, in the Sun frame the spacecraft initially has a total velocity equal to the square root of the sum of the squares of the vertical and horizontal velocities, that is v times the square root of 2, or about 1.4v. It leaves the planet with v + v = 2v in the horizontal direction, having gained about 0.6v, or about 60% of the planet's velocity. This shows clearly why the velocity of the spacecraft in the Sun frame increases during the encounter - it's because the spacecraft's direction of motion changes to point along the planet's direction.

This is a general rule of thumb for gravity assists: if, after the encounter, the spacecraft is pointing more along the planet's direction than it was before the encounter, its speed will increase. But where does the energy come from to accelerate the spacecraft? In fact it comes from the planet's own energy of motion. In the Sun frame, there is a transfer of momentum and kinetic energy from the planet to the spacecraft. The planet slows down very slightly in its orbit, and the spacecraft speeds up. Newton's third law states, "To every action there is an equal and opposite reaction," and that's true in this case. Because the planet is so much more massive than the spacecraft, the transfer doesn't affect the planet to any measurable extent, but to the spacecraft it's a big deal. For example, we can calculate that during the Voyager encounters with Jupiter in 1979, Jupiter slowed down by roughly 10 to the -24th power kilometers per second -- a change much too small to measure. But each Voyager gained about 10 km/s, a pretty big number and enough to put them on a fast path to Saturn (and in the case of Voyager 2, to Uranus and Neptune as well) and eventual escape from the solar system.

Figure 2: Possible outcomes of gravity assist maneuvers

David Shortt

Figure 2: Possible outcomes of gravity assist maneuvers

Depending on the relative direction of motion of the planet and the spacecraft, a gravity assist can either speed up, slow down, or merely change the direction of the spacecraft. Figure 2 shows a gallery of possibilities. The center panel (e) shows the view in the planet frame, and the other panels show the sun frame with 8 different directions for the planet's motion. The trajectories in panels (a), (b) and (d) slow down the spacecraft, those in panels (f), (h) and (i) speed it up, and those in panels (c) and (g) change the direction but not the speed. Panel (f) is the same example we considered in Figure 1. It's worth emphasizing that every panel depicts a correct solution of Newton's laws, so any of these could be arranged by a mission designer if needed.

Before looking at a real mission, let's recap what we know so far. In the planet frame, the trajectory is hyperbolic with the same velocity before and after the encounter but with the path deflected through some angle. In the Sun frame this results in trajectories that can speed up or slow down the spacecraft in addition to changing its direction, depending on the geometry of the encounter. Total energy is conserved, and the planet loses (or gains) an insignificant but real amount of velocity, while the spacecraft's velocity and direction may change by a large amount.

Next let's consider a practical example. Voyager 2 is a good choice because it used gravity assists to visit all four of the outer planets: Jupiter, Saturn, Uranus and Neptune. (Voyager 1 followed a similar trajectory up to Saturn, but then had to leave the plane of the solar system and forgo any more planets because mission planners arranged the encounter to include a close approach of Saturn's large and fascinating moon Titan. Voyager 2 did not have a Titan encounter and went on to visit Uranus and Neptune.)

Figure 3: The path of Voyager 2 from its launch from Earth in 1977 through its encounter with Neptune 12 years later

David Shortt


Figure 3 shows a plot of the path of Voyager 2 from its launch from Earth in 1977 through its encounter with Neptune 12 years later. For simplicity the plot omits the orbits of Mercury, Venus and Mars. The axes are labeled in astronomical units, or AU, with the sun at the center (1 AU is the average distance between the Earth and the sun). Notice the particularly sharp "left turns" Voyager 2 makes at Jupiter and Saturn. Viewed as a whole, though, the path of Voyager 2 is a reasonably smooth spiral from Earth to Neptune. This is no accident. The outer planets line up in such a fortuitous way about every 175 years, and it encourages the idea of using gravity assists repeatedly to direct the spacecraft to the next target.

Figure 4: Voyager 2 encounter with Jupiter

David Shortt

Figure 4: Voyager 2 encounter with Jupiter
Figure 5: Voyager 2 encounter with Saturn

David Shortt

Figure 5: Voyager 2 encounter with Saturn
Figure 6: Voyager 2 encounter with Uranus

David Shortt

Figure 6: Voyager 2 encounter with Uranus
Figure 7: Voyager 2 encounter with Neptune

David Shortt

Figure 7: Voyager 2 encounter with Neptune

Figures 4-7 show close-up animations of the encounters at the four outer planets in both the Sun frame and the planet frame. For all the figures, the frame rate is 1 frame per day, the trajectories are shown for 20 days before and after the moment of closest approach, and all the figures are shown at the same spatial scale for comparison, with a width of about 0.6 AU. The Jupiter encounter in Figure 4 looks a lot like panel (i) in Figure 2. You can see how unusual the Jupiter encounter looks in the Sun frame, with the spacecraft experiencing a "bump" in its trajectory as it first accelerated toward massive Jupiter, then whipped around behind it. At Saturn, the encounter in Figure 5 looks a lot like panel (f) in Figure 2. Notice the high speed of the encounter - Voyager was moving fast due to the speed it gained at Jupiter, and the approach had to be very close in order to execute the screaming left turn needed to reach Uranus. The modest left turn at Uranus, shown in Figure 6, looks tame by comparison. Finally, at Neptune in Figure 7, Voyager 2 actually turned slightly right, thereby losing some speed. The reason is that mission planners wanted to arrange a close flyby of Neptune's large moon Triton, and this necessitated flying mostly over Neptune's north pole and making a slight right turn in addition to plunging down, out of the plane of the solar system (the plunge down is not visible in Figure 7 since it's a view from overhead).

Figure 8: Voyager 2 spacecraft speed as a function distance from the sun

David Shortt


Another insight from the animations is that Voyager 2's speed increased quite a bit during its journey. Figure 8 confirms this by plotting the spacecraft's speed in the sun frame (in blue) vs. its distance from the sun in AU. Also plotted, in red, is the escape velocity from the sun, i.e. the speed necessary at that distance to ensure escape from the solar system. After leaving Earth but before its encounter with Jupiter, Voyager 2 lacked enough speed to escape the sun's gravity (the blue curve lies below the red curve between 1 AU and 5 AU). During the Jupiter encounter, Voyager 2 gained enough speed to enable it to leave the solar system - the blue curve stays above the red curve beyond Jupiter. It gained about 10 km/s at Jupiter, about 5 km/s at Saturn, about 2 km/s at Uranus, and lost about 2 km/s at Neptune. As of September 2013, Voyager 2 is over 102 AU from the sun and still traveling at about 15 km/s. Due to its slightly different trajectory, Voyager 1 is over 125 AU from the sun and traveling about 17 km/s, and NASA recently announced that Voyager 1 has officially entered interstellar space.

Thanks to gravity assists, the Voyagers are headed to the stars.

See other posts from September 2013


Or read more blog entries about: Voyager 1 and 2, Neptune, explaining science, Saturn, Uranus, Jupiter


Gene Van Buren: 09/27/2013 03:17 CDT

The animations, plots, and statistics are superb! Very well done.

Supernaut: 09/27/2013 03:40 CDT

Thank you, well done and very insightful!

V: 09/27/2013 10:05 CDT

I always had the same question. Had some idea, but your explanation has cleared everything in so simple manner. Thanks for the invaluable information.

Bob Ware: 09/29/2013 10:07 CDT

Thank you for the great article! regarding your sentence: ".... The maximum deflection is 180 degrees, sending the spacecraft back where it came from...", Is this what NASA referred to as a "free return trajectory" in the APOLLO lunar mission days, until they needed to depart from it to be able to enter Lunar orbit?

David Shortt: 09/29/2013 11:40 CDT

Yes, Bob, that's right. It's also similar to the orbits of sun-grazing comets, which approach the sun from very far away, pass very close to the sun, and return pretty much the way they came (although in the comet case the orbits are technically ellipses since they are gravitationally bound to the sun).

Ellen W: 09/30/2013 02:38 CDT

I've heard gravity assist explained a few times. This is the best by far. Thank you.

Steve levin: 10/01/2013 03:23 CDT

This is very helpful, and very timely: Juno has an Earth flyby planned for October 9.

Bob Ware: 10/01/2013 08:47 CDT

David S. - Thanks!

Anonymous: 10/02/2013 08:41 CDT

thanks. learned something new.. I have a question. am picking Jupiter becuase of its size and the amount of gravity assist. When voyager encountered Jupiter, is it just using jupiter's gravitation only to deflect or in other words change the direction? How big or helpful was the gravity assist? if we have 100%, could you please tell what would be the power percentages for voyager and jupiter's gravitation? How many miles or fuel did voyager save?

David Shortt: 10/05/2013 06:28 CDT

Anonymous, Voyager used Jupiter not just to change direction, but to increase its speed substantially. Fig. 8 shows Voyager 2 gaining about 10 km/s at Jupiter, which is huge. For example, going directly to Neptune without an assist at Jupiter would have taken a much larger rocket to achieve the higher velocity needed for the Earth-Neptune transfer orbit, quite possibly a larger rocket than was available, and still the journey to Neptune would have taken about 30 years. Instead, Voyager 2 did it in 12 years with an available (though large) rocket.

Bob Edgerton: 10/28/2013 12:05 CDT

Beautiful description of subtle effect.

Mario: 11/09/2013 07:55 CST

Does the planet lose orbital velocity (around the sun) or rotational velocity (around its axis)?

tech55: 11/15/2013 06:48 CST

Planet will lose Angular momentum but not the linear momentum

David Czuba: 01/02/2014 07:49 CST

Bob: Free return trajectory is also what partly saved the Apollo 13 crew, though not the mission, since the Moon slung the spacecraft back to Earth. Since the service module was damaged, however, the engine of the lander was used for a minor correction to enter Earth's atmosphere. That minor correction was necessary in order for the reentry vehicle to not skip off the upper atmosphere like a stone across a pond.

bearinspace: 03/13/2014 09:03 CDT

What would the trajectories look like if the outgoing energy did not equal the incoming energy in the planet's frame? For example, firing your engines during the approach to increase your planet-relative speed, or grazing the atmosphere to reduce your planet-relative speed?

Wolf: 02/02/2015 04:09 CST

Hi David, many thanks for your informative discussion of gravity assist which has mystified many of my colleagues including physics Phds. I have a couple of questions 1. Why do a,b,d slow down the spacecraft as opposed to those that speed it up. Is this due to the proximity, speed and angle of approach or is it merely that the craft crosses in front of the planets direction of travel (or behind) and then crosses the planets trajectory after the encounter? Many thanks Wolf Taylor

ivg: 02/17/2015 06:01 CST

Ep+Ec=constant. indeed . the sum of kinetic and potential energy in a system remain constant according to the laws of physics but I raise an issue: initially the potential energy of the object is 0 (before entering the G. field) so the potential energy is "given" by the planet before being converted to kinetic .

allendoyle: 03/07/2015 05:09 CST

fantastic description and animations. Thanks for satisfying this non-astronomer's curiosity. What remains for me is the incredibly precise trajectory and mathematics needed to come out the other side going in the right direction to the next planetary slingshot. It's kinda like Keanu Reeves whipping around bullets and going on to the next bullet. How close does a satellite need to graze a planet, and how does it precisely adjust from hundreds of thousands of miles away???

Colinsk: 07/25/2015 10:18 CDT

I loved your description. It was very helpful in teaching my children. I have long thought the name for this should be Momentum Assist.

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