Gravity Assisted Motion, otherwise known as a “gravity slingshot”, is a concept that has been brought light through many space-exploration movies, such as “Interstellar”, where a spacecraft changes direction and gains speed as it whips past a planet. However, the “gravity slingshot” has not just been limited to epic sci-fi thrillers; it has also been used in order to explore the further parts of our solar system, such as when Voyager 2 used a trajectory boost in 1977 from Jupiter in order to reach Saturn.

Gravity assisted motion, or the fly-by technique can add or subtract momentum in order to increase or decrease the energy of a spacecraft’s orbit. It does this by changing a spacecraft’s velocity by entering and leaving the gravitational field of a planet. As the spacecraft approaches the planet, its velocity increases due to gravity, and as the spacecraft moves away from the planet, its velocity decreases (due to Newton’s 1st law, if additional forces act on the spacecraft, its velocity will change). Keeping this in mind, it seems as though in the end the spacecraft has gained no additional energy or speed, as the energy gained when approaching the planet is expended when moving away from the planet. Note that while the magnitude of the velocity has not changed, the direction has as it follows the curve of the planet.

Credits: Nasa Solar System Exploration

If the net energy has not been changed and the spacecraft’s speed remains the same, how is it that NASA managed to ‘fling’ Voyager 2 past Jupiter to Saturn even though the spacecraft didn’t have enough energy to make it to Saturn without passing by Jupiter? It turns out that in order to understand how the spacecraft gains speed, we have to consider our frame of reference. If we were to take on the planet’s frame of reference, we would see the object increasing in velocity as it approached us and exit with a decreasing velocity in a different direction. The magnitude of the velocity wouldn’t be different as it left us compared to when it approached us. In our frame of reference, we are letting the planet remain stationary as the spacecraft gets “slingshot” away. However, we have to understand that the planet used as the “slingshot” is not stationary; in fact, if we were to take on a helio-centric view of our solar system, we would realize that all of the planets are moving constantly, at great speeds. If we were to take the movement of the planets into account, it can be said that the velocity of the planet is added to the velocity of the spacecraft. After all, a gravity assist involves not just a stationary planet, but one that has enormous angular momentum as it orbits around the Sun. Let us take an example where the spacecraft is flung 90 degrees to its original motion (like in the diagram below) and the planet was stationary. If we were to give the spacecraft a horizontal velocity of v, and thus a vertical velocity of v, the resultant would be \sqrt{v^2 + v^2} \approx 1.41v. Now if we looked at the problem with a moving planet with velocity v, the horizontal velocity of the spacecraft as it travels away from the planet is v+v = 2v, which is approximately 0.59v greater than if the planet was stationary. This proves that the velocity of the spacecraft in the Sun’s frame of reference increases after the encounter.

Motion of both objects in Sun’s frame of reference.
Credits: David Shortt, The Planetary Society

Let us take another example, this time where the spacecraft is heading towards the planet. If the spacecraft’s velocity is v, and the planet is approaching it at velocity u m/s, the velocity at which the spacecraft leaves is 2u+v m/s, due to the spacecraft moving towards the planet at the velocity v+u m/s initially, and then getting launched out with an addition u m/s.

However, we still encounter a few problems; what happens to the momentum of the planet if the spacecraft’s momentum increases? After all, according to GCSE and A level physics, momentum has to be conserved. It turn out that even though the momentum and speed of the spacecraft increases significantly, because the planet’s mass is so much larger than a spacecraft, its velocity barely changes. Since we know this, we can use the conservation of mass and kinetic energy in order to prove that the final velocity of the space craft is indeed 2u+v in the diagram above. The two equations below are the equations for the conservation of kinetic energy and momentum respectively, with the subscript 1 denoting before the interaction with the planet, and 2 denoting after:

Mu12 + mv12 = Mu22 + mv22

Mu1 – mv1 = Mu2 – mv2

If we eliminate u2, we get this equation:

v = \frac{(1-\frac{m}{M})v_1 + 2u_1}{1+\frac{m}{M}}

Since \frac{m}{M} is extremely close to zero (the mass of the spacecraft is almost negligible compared to the mass of the planet), the equation becomes v_2 = v_1 + 2u_1.

However, gravity assist also come with some of its limits. One limitation is that planets that are required in gravity assist maneuvers are rarely at the correct place in time; space explorations such as the Voyager missions in the 1970s were only able to overcome the Sun’s gravity to reach Neptune through using the specific alignment of Jupiter, Saturn, Uranus and Neptune, also known as the “Grand Tour”.  This lineup of planets occurs once every 175 years, so the next time such deep space exploration would be possible would be around 2151-2154 (provided technology doesn’t advance to the point where gravity assisted maneuvers becomes redundant). Another limitation occurs if a space probe is to be sent to distant stars by using the Sun as the massive object. If the Sun were to be used, space probes would have to be designed in order to be able to resist extreme heat as a gravity assist would result in the space probe getting close to the Sun.

Now that I have hopefully uncovered how and why gravity slingshots work, perhaps when the next epic Sci-fi film comes out, or when ‘Interstellar’ undergoes another viewing, we can better understand why it is feasible to use gravity assisted motion.

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