B.V. Bulyubash,
, MSTU im. R.E. Alekseeva, Nizhny Novgorod

Lagrange points

About 400 years ago, astronomers had at their disposal a new instrument for studying the world of planets and stars - the Galileo Galilei telescope. Very little time passed, and the law of universal gravitation and the three laws of mechanics discovered by Isaac Newton were added to it. But only after Newton's death were they developed mathematical methods, which made it possible to effectively use the laws discovered by him and make accurate calculations of the trajectories of celestial bodies. The authors of these methods were French mathematicians. Key figures were Pierre Simon Laplace (1749–1827) and Joseph Louis Lagrange (1736–1813). To a large extent, it was through their efforts that a new science was created - celestial mechanics. This is exactly what Laplace called it, for whom celestial mechanics became the basis for the philosophy of determinism. In particular, the image of a fictional creature described by Laplace, who, knowing the speeds and coordinates of all particles in the Universe, could unambiguously predict its state at any future point in time, became widely known. This creature - “Laplace's demon” - personified the main idea of ​​the philosophy of determinism. And the finest hour new science came on September 23, 1846, with the discovery of the eighth planet of the solar system - Neptune. The German astronomer Johann Halle (1812–1910) discovered Neptune exactly where it should have been according to calculations made by the French mathematician Urbain Le Verrier (1811–1877).

One of outstanding achievements celestial mechanics was the discovery by Lagrange in 1772 of the so-called libration points. According to Lagrange, in a two-body system there are a total of five points (usually called Lagrange points), in which the sum of the forces acting on a third body placed at a point (the mass of which is significantly less than the masses of the other two) is equal to zero. Naturally, we are talking about a rotating frame of reference, in which the body, in addition to the forces of gravity, will also be acted upon by the centrifugal force of inertia. At the Lagrange point, therefore, the body will be in a state of equilibrium. In the Sun–Earth system, the Lagrange points are located as follows. On the straight line connecting the Sun and the Earth, there are three points out of five. Dot L 3 is located on the opposite side of the Earth's orbit relative to the Sun. Dot L 2 is located on the same side of the Sun as the Earth, but in it, unlike L 3, The Sun is covered by the Earth. And period L 1 is on the straight line connecting L 2 and L 3, but between the Earth and the Sun. Points L 2 and L 1 is separated from the Earth by the same distance - 1.5 million km. Due to their characteristics, Lagrange points attract the attention of science fiction writers. So, in the book “Solar Storm” by Arthur C. Clarke and Stephen Baxter, it is at the Lagrange point L 1 space builders are building a huge screen designed to shield the Earth from a super-powerful solar storm.

The remaining two points are L 4 and L 5 are in Earth’s orbit, one is in front of the Earth, the other is behind. These two points are very significantly different from the others, since the balance of the celestial bodies located in them will be stable. That is why the hypothesis is so popular among astronomers that in the vicinity of points L 4 and L 5 may contain the remains of a gas and dust cloud from the era of the formation of the planets of the Solar System, which ended 4.5 billion years ago.

After automatic interplanetary stations began to explore the Solar System, interest in Lagrange points increased sharply. So, in the vicinity of the point L 1 spacecraft are conducting research on the solar wind NASA: SOHO (Solar and Heliospheric Observatory) And Wind(translated from English – wind).

Another device NASA– probe WMAP (Wilkinson Microwave Anisotropy Probe)– located in the vicinity of the point L 2 and studies the cosmic microwave background radiation. Towards L 2 space telescopes “Planck” and “Herschel” are moving; in the near future they will be joined by the Webb telescope, which should replace the famous long-lived space telescope Hubble. As for the points L 4 and L 5, then September 26–27, 2009 twin probes STEREO-A And STEREO-B transmitted to Earth numerous images of active processes on the surface of the Sun. Initial Project Plans STEREO have recently been significantly expanded, and currently the probes are also expected to be used to study the vicinity of Lagrange points for the presence of asteroids there. the main objective Such research involves testing computer models that predict the presence of asteroids at “stable” Lagrange points.

In this regard, it should be said that in the second half of the 20th century, when it became possible to solve numerically on a computer complex equations celestial mechanics, the image of a stable and predictable solar system (and with it the philosophy of determinism) has finally become a thing of the past. Computer modelling showed that from the inevitable inaccuracy in the numerical values ​​of the velocities and coordinates of the planets at a given moment in time, very significant differences in the models of the evolution of the Solar system follow. So, according to one scenario, the solar system may even lose one of its planets in hundreds of millions of years.

At the same time, computer models provide a unique opportunity to reconstruct the events that took place in the distant era of the solar system’s youth. Thus, the model of mathematician E. Belbruno and astrophysicist R. Gotta (Princeton University) became widely known, according to which at one of the Lagrange points ( L 4 or L 5) in the distant past the planet Theia was formed ( Teia). The gravitational influence from the other planets forced Thea at some point to leave the Lagrange point, enter a trajectory towards Earth and eventually collide with it. Gott and Belbruno's model fleshes out a hypothesis that many astronomers share. According to it, the Moon consists of material formed about 4 billion years ago after the collision of a space object the size of Mars with the Earth. This hypothesis, however, has a weak point: the question of where exactly such an object could have formed. If the place of its birth were areas of the solar system remote from the Earth, then its energy would be very large and the result of its collision with the Earth would not be the creation of the Moon, but the destruction of the Earth. Consequently, such an object should have formed not far from the Earth, and the vicinity of one of the Lagrange points is quite suitable for this.

But since events could develop this way in the past, what prevents them from happening again in the future? Will not, in other words, another Theia grow in the vicinity of the Lagrange points? Prof. P. Weigert (University of Western Ontario, Canada) believes that this is impossible, since solar system At present, there are clearly not enough dust particles to form such objects, but 4 billion years ago, when planets were formed from particles of gas and dust clouds, the situation was fundamentally different. According to R. Gott, asteroids may well be discovered in the vicinity of the Lagrange points - the remains of the “building material” of the planet Theia. Such asteroids can become a significant risk factor for the Earth. Indeed, the gravitational influence from other planets (and primarily Venus) may be sufficient for the asteroid to leave the vicinity of the Lagrange point, and in this case it may well enter a collision trajectory with the Earth. Gott's hypothesis has a prehistory: back in 1906, M. Wolf (Germany, 1863–1932) discovered asteroids at the Lagrange points of the Sun–Jupiter system, the first ones outside the asteroid belt between Mars and Jupiter. Subsequently, more than a thousand of them were discovered in the vicinity of the Lagrange points of the Sun–Jupiter system. Attempts to find asteroids near other planets in the solar system were not so successful. Apparently, they are still not near Saturn, and only in the last decade have they been discovered near Neptune. For this reason, it is quite natural that the question of the presence or absence of asteroids at the Lagrange points of the Earth-Sun system is of great concern to modern astronomers.

P. Weigert, using a telescope on Mauna Kea (Hawaii, USA), already tried in the early 90s. XX century find these asteroids. His observations were meticulous, but did not bring success. Relatively recently, automatic search programs for asteroids were launched, in particular, the Lincoln Project to search for asteroids close to the Earth (Lincoln Near Earth Asteroid Research project). However, they have not yet produced any results.

It is assumed that the probes STEREO will bring such searches to a fundamentally different level of accuracy. The probes' flight over the vicinity of the Lagrange points was planned at the very beginning of the project, and after the asteroid search program was included in the project, even the possibility of leaving them forever in the vicinity of these points was discussed.

Calculations, however, showed that stopping the probes would require too much fuel consumption. Considering this circumstance, project managers STEREO We settled on the option of slow flight of these areas of space. This will take months. Heliospheric recorders are placed on board the probes, and it is with their help that asteroids will be searched. Even so, the task remains very difficult, since in future images the asteroids will be just dots moving against a background of thousands of stars. Project managers STEREO count on active assistance in the search from amateur astronomers who will view the resulting images on the Internet.

Experts are very concerned about the safety of the movement of probes in the vicinity of the Lagrange points. Indeed, collisions with “dust particles” (which can be quite large in size) can damage the probes. In their flight the probes STEREO have already repeatedly encountered dust particles - from once to several thousand per day.

The main intrigue of the upcoming observations is the complete uncertainty of the question of how many asteroids the probes should “see” STEREO(if they see it at all). New computer models have not made the situation more predictable: it follows from them that the gravitational influence of Venus can not only “pull” asteroids from Lagrange points, but also contribute to the movement of asteroids to these points. The total number of asteroids in the vicinity of the Lagrange points is not very large (“we are not talking about hundreds”), and their linear sizes are two orders of magnitude smaller than the sizes of asteroids from the belt between Mars and Jupiter. Will his predictions be confirmed? There's only a little time left to wait...

Based on the materials of the article (translated from English)
S. Clark. Living in weightlessness //New Scientist. February 21, 2009

Have any experiments been conducted on placing spacecraft at the Lagrange points of the Earth-Moon system?

Despite the fact that humanity has known about the so-called libration points existing in space and their amazing properties for quite a long time, they began to be used for practical purposes only in the 22nd year of the space age. But first, let's briefly talk about the miracle points themselves.

They were first theoretically discovered by Lagrange (whose name they now bear), as a consequence of solving the so-called three-body problem. The scientist was able to determine where in space there may be points at which the resultant of all external forces becomes zero.

Points are divided into stable and unstable. Stable ones are usually designated L 4 and L 5 . They are located in the same plane with the main two celestial bodies(V in this case- Earth and Moon), forming with them two equilateral triangles, for which they are often called triangular. The spacecraft can remain at triangular points for as long as desired. Even if it deviates to the side, the acting forces will still return it to the equilibrium position. The spacecraft seems to fall into a gravitational funnel, like a billiard ball into a pocket.

However, as we said, there are also unstable libration points. In them, the spacecraft, on the contrary, is located as if on a mountain, being stable only at its very top. Any external influence deflects it to the side. Getting to an unstable Lagrange point is extremely difficult - it requires ultra-precise navigation. Therefore, the device has to move only close to the point itself in the so-called “halo orbit”, from time to time consuming fuel to maintain it, although very little.

There are three unstable points in the Earth-Moon system. Often they are also called rectilinear, since they are located on the same line. One of them (L 1) is located between the Earth and the Moon, 58 thousand km from the latter. The second (L 2) is located so that it is never visible from the Earth - it hides behind the Moon, 65 thousand km from it. The last point (L 3), on the contrary, is never visible from the Moon, since it is blocked by the Earth, from which it is approximately 380 thousand km away.

Although it is more profitable to be in stable points (there is no need to consume fuel), spacecraft have so far become acquainted only with unstable ones, or rather, only with one of them, and even then related to the Sun-Earth system. It is located inside this system, 1.5 million km from our planet and, like the point between the Earth and the Moon, is designated L 1. When viewed from Earth, it is projected directly onto the Sun and can serve as an ideal point for tracking it.

This opportunity was first used by the American ISEE-3, launched on August 12, 1978. From November 1978 to June 1982, it was in a "halo orbit" around the Li point, studying the characteristics of the solar wind. At the end of this period, it was he, but already renamed ICE, who happened to become the first comet researcher in history. To do this, the device left the libration point and, having performed several gravitational maneuvers near the Moon, in 1985 it flew near the comet Giacobini-Zinner. The next year, he also explored Halley's comet, although only at distant approaches.

The next visitor to the L 1 point of the Sun-Earth system was the European solar observatory SOHO, launched on December 2, 1995 and, unfortunately, recently lost due to a control error. During her work, quite a bit of important scientific information was obtained and many interesting discoveries were made.

Finally, the latest apparatus launched to date in the vicinity of L 1 was the American ACE apparatus, designed to study cosmic rays and stellar wind. He launched from Earth on August 25 last year and is currently successfully conducting his research.

What's next? Are there any new projects related to libration points? Of course they do exist. Thus, in the USA, the proposal of Vice President A. Gore was accepted for a new launch in the direction of point L 1 of the Sun-Earth system of the scientific and educational apparatus "Triana", already nicknamed the "Gore Camera".

Unlike his predecessors, he will monitor not the Sun, but the Earth. Our planet from this point is always visible in full phase and is therefore very convenient for observations. It is expected that the images received by the Gora Camera will be uploaded to the Internet almost in real time, and access to them will be open to everyone.

There is also a Russian “libration” project. This is the Relikt-2 device, designed to collect information about the cosmic microwave background radiation. If funding is found for this project, then the L 2 libration point in the Earth-Moon system awaits it, that is, the one that is hidden behind the Moon.

In the rotation system of two cosmic bodies of a certain mass, there are points in space where by placing any object of small mass, you can fix it in a stationary position relative to these two bodies of rotation. These points are called Lagrange points. The article will discuss how they are used by humans.

What are Lagrange points?

To understand this issue, one should turn to the solution to the problem of three rotating bodies, two of which have such a mass that the mass of the third body is negligible in comparison with them. In this case, it is possible to find positions in space in which the gravitational fields of both massive bodies will compensate for the centripetal force of the entire rotating system. These positions will be Lagrange points. By placing a body of low mass in them, you can observe how its distances to each of the two massive bodies do not change for any length of time. Here we can draw an analogy with geostationary orbit, in which the satellite is always located above one point on the earth's surface.

It is necessary to clarify that a body that is located at the Lagrange point (also called a free point or point L), relative to an external observer, moves around each of the two bodies with a large mass, but this movement, together with the movement of the two remaining bodies of the system, has the following character , that relative to each of them the third body is at rest.

How many of these points are there and where are they located?

For a system of rotating two bodies with absolutely any mass, there are only five points L, which are usually designated L1, L2, L3, L4 and L5. All these points are located in the plane of rotation of the bodies in question. The first three points are on the line connecting the centers of mass of the two bodies in such a way that L1 is located between the bodies, and L2 and L3 are behind each of the bodies. Points L4 and L5 are located so that if you connect each of them with the centers of mass of two bodies of the system, you will get two identical triangles in space. The figure below shows all the Earth-Sun Lagrange points.

The blue and red arrows in the figure show the direction of action of the resulting force when approaching the corresponding free point. From the figure it can be seen that the areas of points L4 and L5 are much larger than the areas of points L1, L2 and L3.

Historical reference

The existence of free points in a system of three rotating bodies was first proven by an Italian-French mathematician in 1772. To do this, the scientist had to introduce some hypotheses and develop his own mechanics, different from Newton’s mechanics.

Lagrange calculated the L points, which were named after him, for ideal circular orbits of rotation. In reality, the orbits are elliptical. The latter fact leads to the fact that Lagrange points no longer exist, but there are regions in which a third body of small mass performs a circular motion similar to the motion of each of the two massive bodies.

Free point L1

The existence of the Lagrange point L1 is easy to prove using the following reasoning: take the Sun and the Earth as an example, according to Kepler’s third law, the closer a body is to its star, the shorter its rotation period around this star (the square of the body’s rotation period is directly proportional to the cube of the average distance from body to the star). This means that any body that is located between the Earth and the Sun will orbit the star faster than our planet.

However, it does not take into account the influence of gravity of the second body, that is, the Earth. If we take this fact into account, we can assume that the closer the third low-mass body is to the Earth, the stronger will be the counteraction of the Earth's gravity to the solar one. As a result, there will be a point where Earth’s gravity will slow down the speed of rotation of the third body around the Sun in such a way that the periods of rotation of the planet and the body will be equal. This will be the free point L1. The distance to the Lagrange point L1 from the Earth is equal to 1/100 of the radius of the planet’s orbit around the star and is 1.5 million km.

How is the L1 area used? This is an ideal place to observe solar radiation as there are never any solar eclipses. Currently, there are several satellites located in the L1 region that study the solar wind. One of them is the European artificial satellite SOHO.

As for this Earth-Moon Lagrange point, it is located approximately 60,000 km from the Moon, and is used as a “transshipment” point during spacecraft and satellite missions to the Moon and back.

Free point L2

Reasoning similarly to the previous case, we can conclude that in a system of two bodies of revolution, outside the orbit of a body with a smaller mass, there should be a region where the drop in centrifugal force is compensated by the gravity of this body, which leads to equalization of the periods of rotation of the body with a smaller mass and the third body around the body with more mass. This area is a free point L2.

If we consider the Sun-Earth system, then to this Lagrange point the distance from the planet will be exactly the same as to point L1, that is, 1.5 million km, only L2 is located behind the Earth and further from the Sun. Since in the L2 region there is no influence of solar radiation due to the earth's protection, it is used to observe the Universe, placing various satellites and telescopes here.

In the Earth-Moon system, point L2 is located behind the natural satellite of the Earth at a distance of 60,000 km from it. Lunar L2 contains satellites that are used to observe the far side of the Moon.

Free points L3, L4 and L5

Point L3 in the Sun-Earth system is located behind the star, so it cannot be observed from Earth. The point is not used in any way, since it is unstable due to the influence of gravity of other planets, for example, Venus.

Points L4 and L5 are the most stable Lagrange regions, so there are asteroids or cosmic dust near almost every planet. For example, only cosmic dust exists at these Lagrange points of the Moon, while Trojan asteroids are located at L4 and L5 of Jupiter.

Other uses of free points

In addition to installing satellites and observing space, the Lagrange points of Earth and other planets can also be used for space travel. It follows from the theory that movements through the Lagrange points of different planets are energetically favorable and require little energy expenditure.

One more interesting example Using the L1 point of the Earth became a physics project of one Ukrainian schoolchild. He proposed placing a cloud of asteroid dust in this area, which would protect the Earth from the destructive solar wind. Thus, the point can be used to influence the climate of the entire blue planet.

Whatever goal you set for yourself, whatever mission you plan, one of the biggest obstacles on your way in space will be fuel. Obviously, a certain amount of it is needed in order to leave the Earth. The more cargo needs to be taken out of the atmosphere, the more fuel is needed. But because of this, the rocket becomes even heavier, and it all turns into a vicious circle. This is what prevents us from sending several interplanetary stations to different addresses on one rocket - there simply is not enough space for fuel. However, back in the 80s of the last century, scientists found a loophole - a way to travel around the solar system using almost no fuel. It's called the Interplanetary Transport Network.

Current methods of space flight

Today, moving between objects in the solar system, for example, traveling from Earth to Mars, usually requires a so-called Hohmann ellipse flight. The launch vehicle is launched and then accelerated until it is beyond the orbit of Mars. Near the red planet, the rocket slows down and begins to rotate around its destination. It burns a lot of fuel both for acceleration and braking, but the Hohmann ellipse remains one of the most effective ways moving between two objects in space.

Hohmann Ellipse - Arc I - flight from Earth to Venus. Arc II - flight from Venus to Mars Arc III - return from Mars to Earth.

Gravity maneuvers are also used, which can be even more effective. When performing them, the spacecraft accelerates using the gravitational force of a large celestial body. The increase in speed is very significant almost without the use of fuel. We use these maneuvers every time we send our stations on a long journey from Earth. However, if a ship needs to enter the orbit of a planet after a gravity maneuver, it still has to slow down. You, of course, remember that this requires fuel.

This is exactly why at the end of the last century, some scientists decided to approach the problem from the other side. They treated gravity not as a sling, but as a geographical landscape, and formulated the idea of ​​an interplanetary transport network. The entrance and exit springboards to it were the Lagrange points - five regions near celestial bodies where gravity and rotational forces come into balance. They exist in any system in which one body rotates around another, and without pretense of originality, they are numbered from L1 to L5.

If we place a spaceship at the Lagrange point, it will hang there indefinitely because gravity does not pull it in one direction more than in another. However, not all these points are created equal, figuratively speaking. Some of them are stable - if you move a little to the side while inside, gravity will return you to your place - like a ball at the bottom of a mountain valley. Other Lagrange points are unstable - if you move a little, you will start to be carried away from there. Objects located here are like a ball on top of a hill - it will stay there if it is well placed or if it is held there, but even a slight breeze is enough for it to pick up speed and roll down.

Hills and valleys of the cosmic landscape

Spaceships flying around the solar system take all these “hills” and “valleys” into account during flight and during the route planning stage. However, the interplanetary transport network forces them to work for the benefit of society. As you already know, every stable orbit has five Lagrange points. This is the Earth-Moon system, and the Sun-Earth system, and the systems of all the satellites of Saturn with Saturn itself... You can continue yourself, after all, in the Solar system a lot of things revolve around something.

Lagrange points are everywhere, even though they constantly change their specific location in space. They always follow the orbit of the smaller object in the rotation system, and this creates an ever-changing landscape of gravitational hills and valleys. In other words, the distribution of gravitational forces in the solar system changes over time. Sometimes attraction in certain spatial coordinates is directed towards the Sun, at another point in time - towards some planet, and it also happens that the Lagrange point passes through them, and in this place equilibrium reigns when no one is pulling anyone anywhere .

The hills and valleys metaphor helps us visualize this abstract idea better, so we'll use it a few more times. Sometimes in space it happens that one hill passes next to another hill or another valley. They may even overlap each other. And at this very moment, space travel becomes especially effective. For example, if your gravitational hill overlaps a valley, you can "roll" into it. If your hill overlaps another hill, you can jump from peak to peak.

How to use the Interplanetary Transport Network?

When the Lagrange points of different orbits move closer to each other, it takes almost no effort to move from one to the other. This means that if you are not in a hurry and are ready to wait for their approach, you can jump from orbit to orbit, for example, along the Earth-Mars-Jupiter route and beyond, almost without wasting fuel. It is easy to understand that this is the idea that the Interplanetary Transport Network uses. The constantly changing network of Lagrange points is like a winding road, allowing you to move between orbits with minimal fuel consumption.

In the scientific community, these point-to-point movements are called low-cost transition trajectories, and they have already been used several times in practice. One of the most famous examples is a desperate but successful attempt to save the Japanese lunar station in 1991, when the spacecraft had too little fuel to complete its mission in the traditional way. Unfortunately, we cannot use this technique on a regular basis, since a favorable alignment of Lagrange points can be expected for decades, centuries, and even longer.

But, if time is not in a hurry, we can easily afford to send a probe into space, which will calmly wait for the necessary combinations, and collect information the rest of the time. Having waited, he will jump to another orbit and carry out observations while already in it. This probe will be able to travel throughout the solar system for an unlimited amount of time, recording everything that happens in its vicinity and adding to the scientific knowledge of human civilization. It is clear that this will be fundamentally different from the way we explore space now, but this method looks promising, including for future long-term missions.

> Lagrange points

What they look like and where to look Lagrange points in space: history of discovery, the Earth and Moon system, 5 L-points of a system of two massive bodies, the influence of gravity.

Let's be honest: we are stuck on Earth. We should thank gravity for the fact that we were not thrown into outer space and we can walk on the surface. But to break free, you have to apply a huge amount of energy.

However, there are certain regions in the Universe where a smart system has balanced the gravitational influence. With the right approach, this can be used to develop space more productively and quickly.

These places are called Lagrange points(L-points). They got their name from Joseph Louis Lagrange, who described them in 1772. In fact, he succeeded in extending the mathematics of Leonhard Euler. The scientist was the first to discover three such points, and Lagrange announced the next two.

Lagrange points: What are we talking about?

When you have two massive objects (for example, the Sun and the Earth), their gravitational contact is remarkably balanced in specific 5 areas. In each of them you can place a satellite that will be held in place with minimal effort.

The most notable is the first Lagrange point L1, balanced between the gravitational attraction of two objects. For example, you can install a satellite over the surface of the Moon. The gravity of the earth pushes it into the Moon, but the force of the satellite also resists. So the device will not have to waste a lot of fuel. It is important to understand that this point is between all objects.

L2 is in line with the mass, but on the other side. Why doesn't the combined gravity pull the satellite towards the Earth? It's all about orbital trajectories. The satellite at point L2 will be located in a higher orbit and lags behind the Earth, as it moves around the star more slowly. But the earth's gravity pushes it and helps anchor it in place.

You need to look for L3 on the opposite side of the system. Gravity between objects stabilizes and the device maneuvers with ease. Such a satellite would always be obscured by the Sun. It is worth noting that the three described points are not considered stable, therefore any satellite will sooner or later deviate. So there is nothing to do there without working engines.

There are also L4 and L5 located in front and behind the lower object. An equilateral triangle is created between the masses, one of the sides of which will be L4. If you turn it upside down, you get L5.

The last two points are considered stable. This is confirmed by the asteroids found on large planets like Jupiter. These are Trojans caught in a gravitational trap between the gravity of the Sun and Jupiter.

How to use such places? It is important to understand that there are many types of space exploration. For example, satellites are already located at the Earth-Sun and Earth-Moon points.

Sun-Earth L1 is a great place to host a solar telescope. The device came as close as possible to the star, but did not lose contact with its home planet.

They plan to place the future James Webb telescope at point L2 (1.5 million km from us).

Earth-Moon L1 is an excellent point for a lunar refueling station, which allows you to save on fuel delivery.

The most fantastic idea would be to put the Ostrov III space station in L4 and L5, because it would be absolutely stable there.

Let's still thank gravity and its strange interaction with other objects. After all, this allows you to expand the ways of exploring space.