In the process of studying the Solar System, astronomers sometimes discovered new classes of objects. Many of them appeared at the intersection of already known classes. Such an “intermediate” type is usually considered to be quasi-satellites – small bodies that move around the Sun in independent orbits, but can also orbit one of the larger planets.
Lagrange’s gravitational “traps”
In 1906, the German astronomer Max Wolf spotted a barely visible speck on a photographic plate, which turned out to be an image of a new asteroid. It was later named Achilles (588 Achilles). This object is moving far beyond the Main Asteroid Belt – its “habitat” is the orbit of Jupiter, the largest planet in the Solar System. Their orbital periods around the Sun also coincide, and the two bodies are almost stationary relative to each other.
The discovery was unexpected, but it did not surprise scientists too much. Back in 1772, the French mathematician Joseph-Louis Lagrange, trying to solve the so-called three-body problem analytically, found that it was possible only in a few cases. In a system where a body of lesser mass orbits a much heavier one in a circular orbit, there are five positions in which the third body, much lighter than the first two, will maintain an unchanged position relative to them. These positions are marked with a capital L and called Lagrange points.
Three of these points are located on a line passing through the centers of the two massive bodies. They are unstable: if the orbit of the second component differs from the circular one or if there is a gravitational influence from another body, the object that has fallen into them will eventually be “pushed out”.
But the most interesting points – they are designated L₄ and L₅ – move in the orbit of the second body at the same speed, being 60° ahead and 60° behind it. Together with the first two bodies, they form two equilateral triangles. They are a kind of gravitational “trap”: an object that has fallen into them tends to return to its place, even if you try to push it out. It was at the L₄ point of the Sun-Jupiter system that Achilles was discovered. The first asteroid at the L₅ point, Patroclus (617 Patroclus), was discovered by Wolf’s colleague August Kopff in 1906. Later, it became a tradition to name them after the heroes of the Trojan War. Today, there are several thousand such objects in the orbit of the largest planet.
Of course, Jupiter’s orbit is far from circular and has a significant eccentricity. In addition, bodies in the Lagrange points on it are subject to the gravitational influence of other planets, which invariably throws them out of balance. Therefore, Trojan asteroids do not stay permanently at these points, but fluctuate relative to them, sometimes quite strongly. It happens that an object moves away from them far enough that it begins to be intensively attracted by Jupiter and eventually “runs around” it, but then still returns to one of the Lagrange points. At the same time, such “runaways” describe strange shapes concerning the planet that can hardly be called classical orbits, but since their paths are closed around the massive body, they can be considered its satellites. It is easy to understand that a small external gravitational push is enough for them to turn into a real satellite of a large planet.
Astronomers track the vast majority of such cases on the example of Jupiter, the most massive planet in the Solar System. On the one hand, it has a powerful gravity, and on the other hand, it has a not very short period of orbit around the Sun (less than 12 years). The Trojan asteroids have the same length of time. And those of them that move far from the Lagrange points can “write out loops” for decades. In the case of other giant planets, whose one revolution lasts even longer, such processes stretch for centuries. In addition, due to the distance from our sun, small asteroids in their orbits are much more difficult to observe.
Horseshoes and tadpoles
Bodies of much smaller size (tens and hundreds of meters) can be observed near the Earth. And, more importantly, their orbital evolution time is usually much shorter than the human lifespan, and thus allows us to study it on many examples. Objects that can approach us at a distance of less than 0.3 AU (44.9 million km) are referred to as Near-Earth asteroids. There are currently more than 30 thousand of them known.
Among such a large number, one would expect to find several dozen Trojans in Earth orbit, but in fact, only two are known so far: 2010 TK7 and 2020 XL5. Both of them live in the vicinity of the Lagrange point L₄, and the first of them sometimes “flies” away from it quite far and approaches the point L₃, located on the opposite side of the Sun relative to the Earth. However, in the present era, none of them comes close enough to us to be considered our satellite, at least with the prefix “quasi”.
But this definition fits another category of objects – those whose period of rotation around the Sun is almost equal to the Earth’s year. In the coordinate system that rotates with the Earth, they draw strange closed curves. If our planet is inside such a curve, it looks as if it has a satellite orbiting it at a great distance in the opposite direction to the direction of rotation of the big planets.
Of course, there are no complete coincidences in our Universe – usually, the period of rotation of such a quasi-satellite around the Sun is slightly longer or slightly shorter than the corresponding period of the Earth. In the already mentioned coordinate system, this will look like this: The object’s “quasi-orbit” will gradually change its position, and eventually it will be able to come close to our planet. And then everything depends on the inclination of this orbit to the ecliptic. If it is large enough, the asteroid will pass by the Earth at a safe distance, and this will not have any significant consequences for it. However, if the inclination is less than 5°, or if one of the nodes (the points where the asteroid’s orbit intersects the ecliptic) happens to be near the Earth’s orbit, there is a possibility of a close approach, during which Earth’s gravity will greatly change the trajectory of the celestial body, shortening or lengthening its period. In both of these cases, it will cease to be a quasi-satellite.
The option of gravitational capture into Earth orbit does not work in this case, because the approach to the Earth is too fast. Only two options are possible: braking in the Earth’s atmosphere and interference from the Moon’s gravity. But both of them must work with great precision and are extremely unlikely.
Yet some asteroids manage to enter orbit around the Earth, becoming its real satellites for a while. How do they do it?
As already mentioned, bodies located near the Lagrange points L₄ and L₅ of the Earth-Sun system are quite strongly shaken by the gravity of other planets, especially Jupiter. As a result, they begin to move relative to our planet in complex trajectories resembling a drop with a curved tail or a tadpole. If the disturbing influence was particularly powerful, at a certain part of this trajectory, the object begins to approach the Earth and can be captured by its gravity, since its relative speed is low during such approaches. Often, the Moon helps with this capture.
However, the opposite is more often the case: at the point of greatest convergence, our planet gives the object a gravitational “impulse”, and it begins to move away from us. At the same time, its heliocentric orbit changes insignificantly, but in the Earth-Sun coordinates, the loop it describes is noticeably lengthened. It happens that its tail end extends to the Lagrange point L₃, the asteroid jumps through it… and begins to approach us from the other side. Now its trajectory, from the point of view of ground-based observers, resembles a giant horseshoe with the Earth between its ends. And it can play this gravitational “ping-pong” for centuries, until the gravity of some other planet interferes with these “games”.
Among the asteroids in horseshoe and tadpole orbits, there are not many that could threaten our planet with a collision: it is perfectly protected from them on its own. Scientists are interested in such asteroids for another reason. Sending a spacecraft to them from Earth and then returning requires even less fuel than a trip to the Moon. Unfortunately, the search for and study of “terrestrial Trojans” is complicated by the fact that most of the time they are in the sky no more than 60° from the Sun. Specialized orbiting telescopes can solve this problem, but no space agency is planning to launch such missions.
This article was published in Universe Space Tech magazine #1 (189) 2023. You can buy this issue in the electronic version in our store.
