Wednesday, April 7, 2010

Lagrange Points

As of 2006, one condition specifying an object as a planet is that the object has "cleared its neighborhood". This refers to bodies large enough to gravitationally fling objects out of their orbits. Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune have all done that. Pluto and its Kuiper belt counterparts, known as Plutinos, often share orbits with one another, ending up in a jumble of asteroids. It is easy to see, by this definition, that Pluto (and other dwarf planets such as Eris and Ceres) is clearly not a planet. However, the gas giants have not completely cleared their orbits. Mysterious "Trojan" asteroids follow them around their orbits. But how can this be?

The answer requires physics. There are actually five points within a system of to bodies in which an object can orbit in stability, despite being within or very near a larger object's orbit. These are known as Lagrange points, named after Joseph Lagrange. The discovery of these points occurred while Lagrange was exploring the famous three-bodies problem in 1772 (calculating a stable system for three bodies with their respective masses, speeds, and positions at any given time). Despite the simplicity of a two body system, an additional object makes the result very complex, and many scientists devoted their time to researching this problem. Contrary to previous belief, there are actually five points at which stability is attained (rather than three, which was the result found by Leonhard Euler 20 years previously). All objects at these points have a 1:1 (read "one to one") orbital resonance with the smaller of the two main bodies, meaning that for every orbit that it makes, an object at any Lagrange point makes one too. They are described in the picture below.

The five Lagrange points are illustrated above, along with a description below. They are denoted by L1, L2, L3, L4, and L5, while the arrows denote the forces acting on the areas around these points.

If a line is drawn between the two massive bodies, L1 will lie on that line, in between the two, at the point where the gravitational pulls of each cancel each other out. In the case of the Earth-Sun system, this point is much closer to Earth, due to the fact that the Sun is much more massive. In most cases, a body orbiting closer to the Sun than Earth, i.e. an object at L1, would orbit faster and have a smaller revolution period. However, the gravitational pull of the Earth keeps an object at L1 from speeding up, and the object is literally held at that point all the way around Earth's orbit. It therefore has the same orbital period as the Earth, at one Earth year.

L2 lies on the same line as L1, but is outside the Earth's orbit rather than inside it. Here, the force from the Sun would usually have such an object orbit slower than the Earth, but the Earth's gravity adds pull to the object, speeding it up, and keeping its year equal to one Earth year. Essentially, both L1 and L2 are points at which the forces of the Earth and Sun add or subtract to from exactly one Earth year.

L3, as with L1 and L2, again lies on the line extended from between the two massive bodies, here the Earth and the Sun. This time, the body is opposite the Earth on its orbit, lying outside the Earth's orbit, but also closer to the Sun than Earth. But how can this be? The above statement appears to be a contradiction, but is actually possible due to the pull that the Earth has on the Sun. Both bodies actually orbit around the barycenter of the Earth and the Sun, which, despite being well inside the Sun, is not at the center. Because of this, the Sun is always slightly farther from the Earth than it would be for a less massive object, such as a body at L3, and a position satisfying both statements is therefore possible. At L3, the forces of the Earth and the Sun again balance, and the body has the exact same orbital period as the Earth.

L4 and L5
L4 and L5 are grouped together because they are very similar and both have the same properties. They are the most notable of the five Lagrange Points. If the Sun and the Earth form two points of two equilateral triangles, then L4 and L5 are located at the third point of each triangle. L4 orbits 60º ahead of Earth, and L5 orbits 60º behind. Due to the fact that the points are equidistant from the Earth and the Sun, they receive as much pull from the Sun as the Earth does, and they are kept stable by the Earth's gravity. However, the points L4 and L5 are stable if and only if the ratio of the larger body to the smaller one exceeds 25. This is true for the system of the Sun and any planet, as well as the Earth-Moon system. Otherwise, an object orbiting at these points will be perturbed and its orbit changed.

The five Lagrange Points are not only in physics, but have been observed. Some spacecraft have used the point as a base for observations. The Earth-Sun L1 point is best suited for observations of the Sun, due to the fact that the view from this point will never be disrupted by the Earth or the Moon. One spacecraft that has used this point to its advantage is SOHO (Solar and Heliospheric Observatory) launched in 1995. The Earth-Sun L2 point is ideal for deep space observations, because its view will never be disrupted by the Earth, the Moon, or the Sun. One spacecraft planned to take advantage of the L2 point is the James Webb Space Telescope, which is planned to be launched in 2014 as the successor of the famous Hubble Space Telescope. Here, no light from the Earth or Sun will damage the equipment, and an unhindered view of the wonders of deep space will be available. L3 does not have any specific use, but some people once believed that another planet similar to Earth, called "Counter-Earth" existed at this point. Since this point was never visible from Earth, the possibility was real until more sophisticated technology disproved this hypothesis. L4 and L5 are possible sights of space colonization, and a base could be set up in a stable orbit in one of these points, if partial or total evacuation of Earth ever became necessary. However, this does not seem imminently likely.

In a perfect system with only three bodies, the points L1, L2, and L3 are technically stable, but in a multiple body system like our own Solar System, an object staying at any of these points over a long period of time is impossible. The gravitational effect from the other planets will eventually disrupt any of these orbits. However, L4 and L5 are a very different story, indeed. They are much broader than the other three points, and perturbations will not always disrupt their orbits. Rather, a perturbation at these two points will change the orbit of the point slightly, increasing the eccentricity, but still keeping the objects in the range of the points. Due to this, there is matter floating around in these areas for both the Sun-Earth system and the Earth-Moon system. However, in both cases, the debris is simply interplanetary dust, and does not form large objects. In the outer planets, however, there are larger objects. Within the orbits of Jupiter and Neptune, there are many asteroids occupying these areas. The are referred to as Trojan asteroids. Also, two moons of Saturn, Tethys and Dione, have two smaller satellites each orbiting in perfect synchronization with them.

The Lagrange points are important for many uses, many of which could be important to humanity's survival in the future.


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