Image: Thousands of objects are trapped in the gravitational troughs of the Lagrange stability regions. Most of the planets in the Solar system have Trojan asteroids located near the L4 and L5 locations of the Sun-planet system.
Image credit: NASA public domain.
Natural satellites orbit directly around their planet or dwarf planet or asteroid.
The formation of a natural satellite is generally associated with a "capture" (gravitational capture of a small celestial body) or an "accretion" (agglomeration of matter in the original disk).
A natural satellite of natural satellite is called a "secondary satellite".
• In the case of capture, it is very difficult for a celestial object to position itself in the Hill sphere of a natural satellite without being ejected. The Hill sphere of a natural satellite is the region of space around the natural satellite where its gravity is stronger than that of the planet. Inside the Hill sphere, the celestial object will be able to orbit around the natural satellite without being ejected by the gravity of the planet. However, Hill's sphere is not a perfect sphere, it is usually stretched in the direction of rotation.
In addition, the size of the Hill sphere varies depending on the distance between the object and the natural satellite. The further the object is from the natural satellite, the smaller the Hill sphere. This is because the gravitational force decreases with distance.
Finally, the object's orbit can also affect its ability to stay within the natural satellite's Hill sphere. If the object is in a very eccentric orbit, it may pass outside the Hill sphere at certain points in its orbit. In this case, the object will be ejected by the gravity of the planet.
However, it is possible that small asteroids or small objects could be temporarily captured by a natural satellite before continuing their cosmic journey around the Sun. These "mini-moons" can only remain in orbit for a short time before being ejected.
• In the case of accretion, when large amounts of matter come together under the effect of gravity, a celestial body can form and remain in orbit around another body.
Gravitational attraction can "trap" small objects or fragments of past collisions, which can then become temporary natural satellites of natural satellites. But these satellites, much smaller than the primary satellite, tend to have irregular and unstable orbits.
However, some mini-moons can remain in orbit for a very long time provided they are located in the regions of Lagrange gravitational stability.
The diameter of the Hill sphere is roughly equal to the distance between the Lagrange points L1 and L2 of the celestial object.
Lagrange points are associated with positions of equilibrium and it is not uncommon to find natural objects such as small asteroids there in the Solar system.
In the case of the Sun-Jupiter system, around the Lagrange points L4 and L5, around 10,000 asteroids are observed. These asteroids are called Trojan asteroids.
Jupiter's Trojan asteroids are so numerous that some of them have natural satellites in turn. Others present themselves as a binary system in the sense that the main body and its moon are of comparable sizes. This is the case of (617) Patroclus (about 145 km in average diameter) with Menoetius (about 98 km in average diameter) located in the L5 region. (16974) Iphthimé (about 57 km average diameter) with S/2013 (16974) (about 35 km average diameter) located in the L4 region. (29314) Eurydamas (about 40 km average diameter) with S/2005 (29314) (about 24 km average diameter) located in the L5 region.
If these binary systems still exist today, it is because they are relatively stable over considerable time scales. Indeed, the two objects are sufficiently massive relative to each other and their mutual distances are sufficiently close that they can be considered as the same object. The binary system orbits around a common center of mass called the barycenter, far enough away from other celestial objects that may disturb it.
Jupiter's largest Trojan asteroid, (624) Hector (about 250 km average diameter) is located in the L4 region and has an asteroid moon called Scamandrios (about 12 km average diameter). Scamandrios orbits in 71 hours about 957 km from the primary.
In this case, the two objects are not sufficiently massive relative to each other, the mass of the main object is clearly greater than that of the secondary object. The orbit of the smaller object is regular because it is in a sufficiently large region of stability and does not resonate with other objects or external disturbances.
Two natural satellites of Saturn, Tethys and Dione, also have Trojans.
The Trojan satellites of Tethys are Telesto located around the Lagrange point L4 and Calypso located around the Lagrange point L5.
The Trojan satellites of Dione are Hélène (L4) and Pollux (L5).
The Trojan asteroids, located near locations L4 and L5 of the system, are not exactly on the Lagrange point, but orbit around it. This extremely large region on the scale of a satellite allows the presence of many objects. Gravitation and celestial mechanics naturally balance the movements of the Trojans.