Gravity assists explained simply— How the Voyagers escaped the Solar System

If nature offers some free help to reach our destination, we should take it

Traveling the vast distances of space isn’t cheap. It costs spacecraft time, fuel and money. Fortunately, nature offers free help along the way and mission designers always take it.

They’re called gravity assists.

Gravitational slingshots/assists allow spacecrafts to save on all those factors using a simple physical law. NASA’s Voyager 1 & 2 spacecrafts are famous for using the gravity of Jupiter & Saturn to go fast enough to escape the Sun’s gravity and study interstellar space.

Voyager 1 approaching Jupiter. Source: Wikipedia

Why use gravity assists?

Instead of using gravity assists to navigate to a destination, a spacecraft can simply carry more fuel to power itself. But adding more fuel makes it weigh more. This means more fuel needs to be added to the rocket to launch the now-heavier spacecraft. Since including extra fuel also increases the rocket’s mass, more fuel is needed to carry that fuel, and so on. Rocket science.

As a thumb rule, the fuel requirements increase exponentially with more mass added to the spacecraft. Heavier spacecraft may require a more complex rocket to be built to meet the demands. Such increases in cost and technological complexity can be saved by using gravity assists. They also allow us to do things that are beyond our current abilities.

How the Voyagers did it

In the 1970s, some of the most ambitious spacecraft in history were launched: Voyager 1 and 2, both by NASA. They would go on to escape the Sun’s gravity and exit the Solar system. Voyager 1 entered interstellar space in 2013 and Voyager 2 is expected to do the same soon. And it wouldn’t have been possible without gravity assists.

After launch, the twin Voyagers didn’t have enough velocity to escape the Sun’s gravity straightaway. It was and remains impossible for us to build a rocket powerful enough to achieve that. The Titan III rockets that launched the Voyagers (10 days apart) left with them enough energy just to get to Jupiter.

To overcome this problem, the Voyagers were made to swing around the gas-giant to acquire the velocity boost needed to escape the Sun. As each spacecraft approached Jupiter, the planet’s gravity sped it up. Such a close gravitational encounter with a planet is called a flyby.

Voyager 1 & 2 trajectories, showing gravity assist maneuvers at Jupiter & Saturn to escape the Solar system. Credit: NASA

It’s easy to prove how the spacecraft velocity increases in the above case using vectors. Unlike scalar quantities (like speed) which only have magnitude, vectors (like velocity) have both magnitude & direction. A change in direction implies change in velocity, which was quite useful to the Voyagers.

Consider the Voyager spacecraft approaching Jupiter as shown in the diagram below. Let the planet’s velocity around the Sun be v. The spacecraft’s velocity on approaching the planet is v(in) and when leaving the planet is v(out), as shown in cases 1 and 2 respectively.

A spacecraft’s trajectory (dashed lines) when approaching and leaving a planet. Beautifully self-drawn.

v(in) can be calculated by the Pythagorean Theorem (the square root of sum of squares of “horizontal velocity component” of the spacecraft v and “vertical velocity component” u). v(out) can be expressed as simply the sum of v and u, as you can infer from above. Here are the resulting velocity calculations for each cases.

Calculation of v(in) and v(out) assuming a scenario of v=u. u could have been any random number but for simplicity, it is taken to be equal to v.

From this simple calculation, we see that v(out)-v(in) = 2v-1.4v = 0.6v

The spacecraft thus gained 60% of the planet’s velocity after the gravity assist, adding to its own. You can clearly see that a change in direction is causing an increase in velocity here. The spacecraft’s velocity thus increases quite a bit and the goal is achieved using nothing but gravity.

The Voyagers did this twice, once with Jupiter and then with Saturn to achieve enough velocity so that they can escape our Sun’s gravity. And reach for the stars.

Understanding the energy conundrum

The same situation looks quite different if you see from the perspective of the planet though. To the planet, the spacecraft’s velocity gradually increases when coming towards it due to gravity. Similarly the velocity gradually decreases in the same manner when going away from it. So the speed before and after encounter is the same. This is in fact the conservation of energy playing out.

But when seen from the Sun’s perspective, the spacecraft appears to be getting extra velocity out of nowhere, seemingly violating the conservation of energy. So where does the extra velocity come from?

The extra velocity comes from the planet itself. It’s worth remembering that the spacecraft also has some mass, even if insignificant compared to Jupiter. Gravity works both ways: the spacecraft pulls on Jupiter — even as Jupiter pulls on the spacecraft — slowing it ever so slightly in its orbit around the Sun.

Because total momentum, a product of mass and velocity, is always conserved in an interaction, the momentum lost by Jupiter is gained by the spacecraft. The velocity loss for Jupiter in this scheme is so negligible as to be unimportant. But the velocity gained by the spacecraft in just one such interaction is quite significant, as in the case of Voyagers.

Voyager 2 had a velocity of ~10 km/s when it approached Jupiter. After the gravity assist, the velocity increased to ~25 km/s.

Voyager 2 spacecraft velocity as a function distance from the Sun, compared to the Solar system escape velocity. Credit: NASA

Gravity assists can help a spacecraft slow down too

Gravity assists can also be used to slow down a spacecraft. This works when the spacecraft approaches the planet in a direction opposite to the planet’s orbit around the Sun. In this scenario, the spacecraft would lose momentum to the planet.

You can now work out the math and see how the velocity of the spacecraft decreases after an encounter with a planet. NASA’s Messenger spacecraft used gravity assists from Earth and Venus to slow down such that it can be captured in orbit by Mercury.

Earth as viewed by the Messenger spacecraft after a gravity assist to slow itself down.

In the science fiction novel Rendezvous with Rama, Arthur C. Clarke describes an alien race whose interstellar spacecraft uses our Sun to perform a gravity assist (an interstellar gravity assist!), causing humans to worry & contemplate in the process.