This is a manuscript written last year, published in the February 2013 issue of "Amateur Astronomer." The title of this article was even featured on the cover of that issue, which made me very happy at the time! I am sharing it here with everyone for mutual encouragement.
I believe many astronomy enthusiasts are familiar with the concepts of the first, second, and third cosmic velocities, and many have calculated them by hand. We know that as long as the launch speed reaches 7.9 km/s, a spacecraft can orbit the Earth; exceeding 11.2 km/s allows it to break free from Earth and become an "artificial planet" of the solar system; and if it exceeds 16.7 km/s, it can even escape the Sun’s gravity and head into deep space.
16.7 km/s might not seem very large at first glance, especially since the Earth’s orbital speed around the Sun is already 30 km/s—a speed that is quite ordinary in the universe. However, for our current technology, it is dauntingly high. According to Wikipedia, the Saturn V, the most powerful rocket in history, could only accelerate the Apollo 11 spacecraft to near escape velocity (11.2 km/s) on its way to the Moon. In fact, the third cosmic velocity represents the current speed limit for human-made spacecraft. Without sufficient speed, we cannot send probes to explore deep space, and the "interstellar migration" depicted in science fiction would remain forever in the realm of novels.
Historical Origins of Gravity Assist
Is there really no way? Not necessarily. While we might not be able to achieve it on our own from Earth, we can let other planets lend us a hand! This is known as Gravity Assist, also referred to in physics as the "slingshot effect." The scientist who first proposed the gravity assist method was the Soviet researcher Yuri Kondratyuk. In his paper "To Those Who Will Read in Order to Build," dated between 1918 and 1919, he suggested that a spacecraft flying between two planets could use the gravity of those planets for acceleration at the beginning of the trajectory and deceleration at the end. Friedrich Zander also proposed a similar concept in his 1925 paper "Problems of Flight by Jet Propulsion: Interplanetary Flights."
In 1959, the gravity assist method saw its first application when the Soviet probe Luna 3 used it to orbit to the far side of the Moon and photograph the region. This operational procedure was designed by the Keldysh Institute of Applied Mathematics. However, neither of these early pioneers fully realized that the gravity assist exerted by a planet along the spacecraft’s trajectory could significantly reduce the fuel consumption required for interplanetary flight. The first person to recognize this crucial role was Michael Minovitch, a graduate student at UCLA.
In 1961, the 25-year-old Michael Minovitch was thrilled by the university’s new IBM 7090 computer, the fastest in the world at the time. He decided to use it to study the "three-body problem," one of the greatest challenges in celestial mechanics. Using the computer, Minovitch’s iterative models achieved a breakthrough. He discovered that a spacecraft passing a planet orbiting the Sun could "steal" a bit of the planet’s orbital velocity to accelerate away from the Sun without using any rocket fuel. In the summer of 1965, Minovitch considered whether his discovery could be put into practice. The results were surprising: he learned that in the late 1970s, Jupiter, Saturn, Uranus, and Neptune would all be located on the same side of the Sun. If a spacecraft were launched in 1977, it could pass all four major planets within 12 years—an opportunity that occurs only once every 176 years. Thanks to Minovitch’s lobbying and high-level intervention, NASA began the Voyager program. To date, the two Voyager spacecraft have been flying for 35 years and have reached the edge of the solar system. Voyager 1 is currently 18.4 billion kilometers from Earth and is set to become the first human-made object to leave the solar system.
Physical Principles
Gravity assist is fascinating because it cleverly "skims" speed from a planet to increase its own velocity. This involves the law of conservation of momentum. We can use a simple example to demonstrate this effect. Imagine a train moving left at 100 km/h, and a basketball flying right at 100 km/h (the speed is exaggerated for demonstration). What happens after they collide? The train, being massive, remains unharmed and continues left at 100 km/h. However, the basketball will bounce back at a much higher speed (assuming it is perfectly elastic). What is that speed?
If we view the situation from the train’s frame of reference (treating the train as stationary), the basketball is approaching the train at 200 km/h. It hits the train and bounces back at the same speed relative to the train: 200 km/h. But since this 200 km/h is measured relative to a train moving at 100 km/h, its actual speed relative to the ground is 200 + 100 = 300 km/h!
It is incredible that by simply "bouncing," the speed increased by 200 km/h! In other words, it can "skim" twice the speed of the moving body. Some readers might wonder if this extra 200 km/h comes from nowhere, seemingly violating the law of conservation of energy. In fact, the change of reference frame used above is an approximation. Because the train’s mass is much larger than the basketball’s, the train’s speed actually decreases by a tiny, negligible amount, while the basketball’s speed approaches 300 km/h.
How does this relate to gravity assist in space? We cannot have a spacecraft collide with a planet and bounce back. Instead, scientists use a clever method: the planet’s gravity pulls the spacecraft toward it and then "slings" it out symmetrically. This process is equivalent to the bounce in the previous example. It has been proven that if a spacecraft follows a hyperbolic trajectory, it can leave the planet in a different direction without firing its engines. Once it escapes the planet’s gravitational influence, it can gain up to twice the planet’s orbital velocity in terms of a velocity vector change.
In real space, the encounter between a spacecraft and a planet is two-dimensional. To increase the spacecraft’s speed, a vector gain is required, as shown in the diagram below.
If a spacecraft needs to gain even more kinetic energy, the most economical method is to ignite its rockets when it is at the planet’s periapsis (the closest point). The acceleration provided by the rocket is the same, but the change in kinetic energy is proportional to the spacecraft’s instantaneous speed. This is known as the Oberth effect. For example, applying a force of 1 N to a 1 kg object for one second increases its speed by 1 m/s. If the initial speed is 5 m/s, the kinetic energy increase is (6^2 - 5^2)/2 = 5.5 J. However, if the initial speed is 10 m/s, the increase is (11^2 - 10^2)/2 = 10.5 J—nearly double. Therefore, to maximize the kinetic energy gain from a rocket boost, the engine must be fired when the spacecraft is at its maximum speed—at the periapsis.
Practical Applications
It can be said that almost all current outer planet exploration missions rely on gravity assist technology. Besides the Voyager probes, a classic example is the Cassini mission to Saturn. Based on gravity assist principles, scientists designed a "smart curve" for Cassini. The peculiarity of this trajectory was that it did not fly directly toward Saturn. Instead, it first headed inward toward Venus, then circled the Earth several times before finally heading for Saturn. The total journey covered 3.52 billion kilometers, more than 2.5 times the actual distance between Earth and Saturn. Its flight path was a rotating curve composed of several hyperbolic segments, looking much like the spiral on a snail’s shell.
However, gravity assist is not always used for acceleration; it can also be used for deceleration. Since Earth itself moves at 30 km/s, this speed is too high for missions to the inner planets. Deceleration is required, and gravity assist can achieve this. Mariner 10 in 1974 and the later MESSENGER probe both used gravity assist to decelerate on their way to Mercury. Additionally, under suitable conditions, aerobraking (using a planet’s atmosphere) can also be used to slow down a spacecraft. Combining the two often yields better results.
Other missions that utilized gravity assist include Voyager 1, Galileo, Ulysses, and MESSENGER. In future interstellar exploration, we might even use the Sun as a boosting body, though this is not feasible for exploring the solar system itself since the Sun is nearly stationary relative to the system.
Interestingly, gravity assist is not exclusive to human technology. Some asteroids occasionally pass by large planets and gain speed. For instance, the near-Earth asteroid 3753 Cruithne exchanges energy with Earth through gravity assist, periodically changing its orbit.
Of course, the gravity assist method has its drawbacks. Sometimes, waiting for planets to reach the right positions can take years. To achieve better acceleration, the spacecraft must pass closer to the planet to experience stronger gravity. However, the closer it gets, the more the planet’s atmosphere might cause drag and deceleration—a constant contradiction. Nevertheless, it remains a very economical and feasible solution for now.
Extended Reading: Non-collision Singularity—Flying to Infinity in Finite Time
Having discussed gravity assist, let’s look at the related "non-collision singularity" problem in the N-body problem. We study gravity assist to shorten flight times. What if I told you that by using gravity assist, it is possible to fly to infinity in a finite amount of time?
This is the "non-collision singularity" problem in N-body research. This is discussed under non-relativistic conditions and cannot be achieved in just any situation. The problem was first formally proposed by the French mathematician Paul Painlevé, a contemporary of Poincaré, who conjectured that such a solution existed—this became known as the "Painlevé Conjecture." Surprisingly, such a system does exist. With a clever setup, a celestial body can indeed reach infinity in finite time. The person who first provided a definitive answer to this problem was a Chinese mathematician in his twenties—Zhihong Xia—nearly a century after the problem was first posed.
Xia cleverly constructed a system as shown in the figure. There are four bodies of equal mass m_1, m_2, m_3, m_4. m_1 and m_2 perform high-eccentricity elliptical orbits in a clockwise direction at the bottom, while m_3 and m_4 do the same in a counter-clockwise direction at the top. They have equal and opposite angular velocities. A small mass m_5 is located on the central axis between the two binary systems, oscillating back and forth between them. Every time m_5 passes through the upper plane to enter the space between them, the two bodies of that binary system happen to be at their closest point. Under intense gravitational force, m_5 gains massive acceleration via gravity assist. By the law of conservation of momentum, m_1 and m_2 are forced upward. Then, as m_5 passes through the other binary plane, m_3 and m_4 are forced downward. With each pass, m_5 "steals" enormous acceleration from the binary systems, causing their energy to decrease. This means the distances between m_1 and m_2, and m_3 and m_4, tend toward zero, but the consequence is that the distance between the two binary systems reaches infinity in a finite amount of time.
When Zhihong Xia proposed this solution, the initial reaction from the mathematical and astronomical communities was not excitement, but shock and doubt: how could such an important problem be solved by a student in his twenties? However, after rigorous review, the answer was clear: the conclusion was correct. This great achievement is now known as "Xia’s Theorem." The construction of this result fully demonstrates Xia’s incredible creativity. He integrated nearly a century of achievements in celestial mechanics and proposed many new ideas to reach this answer. In other words, he "stood on the shoulders of giants" to produce his own outstanding results. This also serves as an important lesson for our studies: "A journey of a thousand miles begins with a single step; small streams combine to form the mighty ocean."
Of course, this is primarily a mathematical curiosity with little practical application. Nevertheless, it highlights the immense creativity of young Chinese mathematicians. At the same time, as a beautiful piece of intellectual art, it allows us to admire the wonders of the N-body problem and the incredible power of gravity assist. In a world governed by gravity, truly "nothing is too strange"!
References
Popular Science Book: Celestial Encounters: The Origins of Chaos and Stability
Wikipedia: Gravity Assist
Gravity Assist Special Topic: http://www.gravityassist.com/
Gravity Assist Simulation: MESSENGER Education Animations
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