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 for a spacecraft. It isn’t cheap in time, isn’t cheap in fuel or cheap in money. And if nature offers us some free help to reach our destination, we should take it.

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 then the spacecraft would weigh more too. The increase in weight means that it requires more fuel in the rocket to launch it. Since the inclusion of more fuel increases the weight of the rocket itself too, we need more fuel to carry that fuel. Rocket science.

As you add more weight to the spacecraft, the fuel requirements thus increase exponentially. This increases both the cost & technological complexity of a mission which can be saved by using gravity assists, in addition to saving time.

How the Voyagers did it

On launch, the Voyagers didn’t have enough velocity to directly escape the Sun’s gravity. It was and still is beyond our technological capability to do so. And so the Voyagers were made to slingshot around Jupiter and Saturn to gain extra velocity to be able to escape the solar system. Its easy to understand how this works in terms of 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. Let’s see how.

Consider a spacecraft approaching a planet with a trajectory 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 i.e the spacecraft 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.

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

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. Its worth remembering that the spacecraft has some mass too, even if its insignificant compared to the planet. Since gravity works both ways, the spacecraft pulls on the planet ever so slightly and slows it down in its orbit around the Sun. Because of conservation of energy, the lost energy of the planet is the energy gained by the spacecraft. The velocity loss for the planet due to this is negligible, but the velocity gained by the spacecraft is quite significant, as we can see in our calculations above.

Gravity assists can help a spacecraft slow down too

Gravity assists can also be used to slow down a spacecraft instead of speeding it up. In the example above, consider that the spacecraft approaches the planet in a direction opposite to it’s orbit around the Sun. The math would thus work out such that the velocity of the spacecraft decreases after the encounter with the 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.

More for the curious:
1:An analogical understanding of how gravitational slingshots work
2:In-depth look of how the Voyager trajectories were planned & executed

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