Lagrange points

On 25 December 2021, the James Webb Space Telescope (JWST) was successfully launched. It has now reached its destination at the L2 Lagrange point of the Sun-Earth system. For many years I have considered writing a blog about the five L:agrange points, but I was not sure if I could do that in a relatively simple way.

I am still not sure, but in this blog I will give it a try.

Here is a diagram of the Sun-Earth system (not to scale). The five Lagrange points are marked.

Earth and all other planets orbit the Sun because of the gravitational attraction between a planet and the Sun. Earth orbits the Sun in ~365 days at a distance of 150 million km. The other planets do the same, but at different distances and with different periods. Here is the solar system (not to scale).

It was Kepler who studied the planetary motion. He found a relation between the period and the distance, which is now called Kepler’s Third Law: The square of a planetary period is proportional to the third power of its distance. Let’s take Mars as an example. The distance to the Sun is ~228 million km and a Mars year is ~687 days. The distance is a factor 228/150 = 1.52 larger. The third power of 1.52 is 1.52×1.52×1.52 = 3.512. Kepler’s 3rd law predicts that a Mars year will be  3,512 = 1.88 times longer than an Earth year. = 1,88 x 365 = 686 days.

Now let us consider a spacecraft in the Sun-Earth system. It’s mass is so small compared to the mass of Sun and Earth, that it will not influence their motion. But it will feel the gravitational attraction from the Sun and also from the Earth. Is it possible that the combined attraction of Sun and Earth will result in a period of 1 year?

The answer is yes, there are exactly 5 points where this is the case, the 5 Lagrange points!

Here is the explanation for L2. This point lies farther away from the Sun than Earth, so the attraction from the Sun is weaker and would result in a longer period. But Earth also attracts the spacecraft in the same direction as the Sun and in L2 they give together enough attraction to let this point orbit in 1 year. Calculation gives that L2 is located 1.5 million km from Earth, 151.5 million km from the Sun

For L1 the explanation is similar. Here the attraction of the Sun is stronger resulting in a shorter period. But now Earth “pulls back” and together they give the right amount of attraction. The location of L1 is also 1.5 million km from Earth, 148.5 million km from the Sun (the figure above is not to scale).

L3 lies at the opposite side of the Sun, Here the attraction from Earth is minimal, it contributes only little to the attraction of the Sun, so L3 lies only slightly further away than 150 million km from the Sun.

Before we describe the points L4 and L5, we will first look in a bit more detail at the solar system. When we say that the planets orbit the Sun, it suggests that the Sun doesn’t move itself, while the planets orbit around it. And that is not true. The Sun and a planet both orbit around their common center of mass, often called their barycenter. In this image the barycenter is shown for the Sun and Jupiter. Because the Sun is much more massive than Jupiter, their barycenter lies close to the Sun.

Here are a few animations for different situations, where the barycenter is marked with a red cross. The animations are not to scale. The first image shows the situation of for example two stars of equal mass. The next one shows minor planet Pluto and its large moon Charon. The last image shows Earth and Sun. The mass of Earth is so small that the barycenter lies within the Sun.

Of course the resulting force in the Lagrange points has to be directed to the barycenter and for L1, L2 and L3 this is automatically the case, because these points lie all three on the line connecting Sun and Earth. These points were already found by the famous mathematician Euler in 1720.

In 1772 Lagrange discovered two more “stable” points, where the attraction of Sun and Earth are not in the same direction, but together point to the barycenter. of the Sun-Earth system. .

The mathematics is complicated, I will use some hand waving to make the existence of L4 (and L5) plausible. In the diagram below, the masses of Sun and Earth are S and E , the barycenter is indicated as b, it lies within the Sun because the Sun is much more massive than the Earth. The location of b depends on the ratio of the two masses S and E;.

L4 is the top of a triangle with all sides equal to the distance between Earth and Sun. Because L4 has an equal distance to Earth and Sun, the gravitational forces on L4 are in the same ratio of S and E. Therefore the resulting force is directed to b ! Note that L4 lies outside Earth’s orbit. Similar to L1, the two combined forces give L4 a period of 1 year, same as Earth.

Actually the barycenter of the Sun-Earth system lies extremely close to the Sun’s center of mass, The radius of the Sun is 670.000km and b lies about 450 km from its center! In this diagram this distance has been strongly exaggerated to show the process. In the usual diagrams of the Lagrange points, L4 and L5 are located so close to the Earth orbit, that it is not possible to see their separation.

Until now we have described the 5 Lagrange points as points that orbit the Sun in one year, same as the Earth. Another description is often used, a rotating coordinate system. In such a coordinate system, centered in the barycenter and rotating once a year, Sun, Earth and the 5 Lagrange points are stationary. But it comes at a cost. Because such a coordinate system is not an inertial system, fictitious forces have to be introduced, for example the centrifugal force,

In the diagram below the Lagrange points are indicated, in such a rotating frame. The contour lines give the gravitational field energy. Compare it with the contour lines on a topo map. The blue and red arrows indicate the direction of the force (the direction of the slope in a topo map). In topo map terminology L4 and L5 are located on the top a hill, while the other three are located in so-called saddle points. On first sight it would seem that all Lagrange points are unstable, For the L1-L3 points a small displacement in the x-direction, and for L4 and L5 a small displacement in any direction would be enough to disturb the balance (like a pencil on its tip).

Careful and complicated mathematical analysis (see for example here) leads to a surprising result: the regions around L4 and L5 are actually stable, objects in a large region around these Lagrange points will move in orbits and stay in that region. The regions around the other three Lagrange points are unstable, objects can orbit for a while, but will eventually escape. That is illustrated in the two diagrams below. The left diagram. shows the Sun-Earth system in an inertial frame, the right one in a rotating frame The 5 Lagrange points are marked in red.

Notice the moving tiny points, They are test masses. released near the various Lagrange punts. Look carefully and you will see that the test masses released near L1 and L2 quickly move away. For L3 it takes a bit longer. All these three Lagrange points are unstable. But around L4 and L5 the test masses do not “escape”, these points are stable.

In the introduction of this blog I wrote that the JWST had reached its destination at the L2 Lagrange point of the Sun-Earth system. Actually the space telescope is not positioned in :the Lagrange point itself but orbiting L2. And what an orbit it is! Elliptical, the distance to L2 varies between 250.000 km and 832.000 km. One period takes about 6 months. The orbit is not stable, about every 21 days the thrusters of the JWST must perform minor course corrections.

A more detailed explanation of the WEBB launch and orbit can be found in this brilliant YouTube video: How James Webb Orbits “Nothing”

There also satellites orbiting L1. At the moment for example the SOHO satellite to study the Sun and the DSOVR to study the Earth. Here are two pictures taken by these two spacecraft.

In 1978 the International Sun-Earth Explorer-3 (ISEE-3). was the first spacecraft that went into an orbit around a Lagrange point. It studied the Sun and Earth for 4 years and also here the unstable orbit had to be corrected regularly. Here is a diagram of the launch process.

After its mission was completed, the spacecraft got a new target, to study comets! It was renamed International Cometary Explorer, left its orbit and via amazingly complicated manoeuvres went on its way to a comet., Click on the screenshot to see an animation of the mission. Very informative and fascinating..

What about L3? This Lagrange point is permanently behind the Sun, as seen from the Earth. No scientific use, but it has played a role in science fiction. . Here is an example, a science fiction movie Journey to the Far Side of the Sun, released in 1969 (the same year that humans landed on the Moon). Click on the screenshot to watch the movie.

Synopsis of the movie: In 2069 a planet is discovered in L3 and the director of Eurosec (named Jason Webb !) organises a mission to what turns out to be a mirror-earth. Very interesting to watch.

We now know that L3 is unstable, with a “decay time” of about 150 year. It would be a suitable location for alien enemies to hide, while preparing for an attack 😉

L4 and L5 are stable (under certain conditions) but have no use for science. Possibly in the far future, these regions could be used to build human colonies.

Until here we have concentrated on the Lagrange points of the Sun-Earth system, but the Earth-Moon system has also its Lagrange points and so do for example the Sun and Jupiter.

Jupiter has collected thousands of asteroids around its L4 and L5 points. They are called trojans because they are named after heros of the Trojan war. Here is an animation. The asteroids in front of Jupiter are called the Greeks and the ones trailing Jupiter are called the Trojans.

The name trojan is now generally used for objects in the L4 and L5 points of other planets. In the L4 and L5 points of the Earth until now “only” two Earth trojans. have been observed.

But there may have been one in the early history of Earth!. I will end this blog with a fascinating theory about the origin of the Moon! The theory is called the Giant-Impact Hypothesis. When the Sun and the planets were born, about 4.5 billion year ago, Earth was not alone. It had a Mars-sized sister planet in the L4 (or L5) Lagrange point. About 20-30 million year later, this hypothetical planet, named Theia, possibly disturbed by the other planets, left the L4 region and collided with Earth. It must have been a cataclysmic event From this collision the Moon was born.. Here is the scenario.

And a visualisation

Here is the Wikipedia List of Objects at Lagrange Points

All the images are taken from the Internet, many from Wikipedia.

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