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

Lagrange points

About 400 years ago, astronomers had new instrument to study the world of planets and stars - a telescope Galileo Galilei... Quite a little time passed, and discovered by Isaac Newton's law of universal gravitation and three laws of mechanics. But it was only after Newton's death that mathematical methods were developed that made it possible to effectively use the laws he discovered and to accurately calculate 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 what Laplace called it, for whom celestial mechanics became the foundation of 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 moment in time, became widely known. This creature - "Laplace's demon" - personified main idea 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. 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 the outstanding achievements of celestial mechanics was the discovery by Lagrange in 1772 of the so-called libration points. According to Lagrange, in a system of two bodies, there are a total of five points (usually called Lagrange points), in which the sum of the forces acting on the third body placed at the point (whose mass 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, in addition to gravitational forces, a centrifugal force of inertia will also act on the body. Thus, at the Lagrange point, the body will be in a state of equilibrium. In the Sun-Earth system, the Lagrange points are located as follows. Three out of five points are located on the straight line connecting the Sun and the Earth. Point L 3 is located on the opposite side of the Earth's orbit relative to the Sun. Point 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 the point L 1 is on the 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 peculiarities, Lagrange points attract the attention of science fiction writers. So, in the book "Solar Storm" by Arthur Clarke and Stephen Baxter, exactly at the Lagrange point L 1 space builders are erecting 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 orbit of the Earth, 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 trapped 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-dust cloud of the epoch 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 conduct research on the solar wind NASA: SOHO (Solar and Heliospheric Observatory) and Wind(translated from English - wind).

Another apparatus NASA- probe WMAP (Wilkinson Microwave Anisotropy Probe)- is in the vicinity of the point L 2 and examines the background radiation. Towards L 2 the space telescopes Planck and Herschel are in motion; in the near future they will be joined by the Webb telescope, which is to replace the famous space long-lived Hubble telescope. As for the points L 4 and L 5, then September 26–27, 2009 twin probes STEREO-A and STEREO-B transmitted to the Earth numerous images of active processes on the surface of the Sun. Initial project plans STEREO were recently significantly expanded, and currently the probes are also expected to be used to study the vicinity of the Lagrange points for the presence of asteroids there. The main goal of this study is to test computer models predicting 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 numerically solve 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 simulations have shown 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 follow in the models of the evolution of the solar system. So, according to one of the scenarios, the solar system in hundreds of millions of years may even lose one of its planets.

At the same time, computer models provide a unique opportunity to reconstruct events that took place in the remote era of the youth of the solar system. Thus, the model of mathematician E. Belbruno and astrophysicist R. Gott (Princeton University), according to which, at one of the Lagrange points ( L 4 or L 5) in the distant past, the planet Thea was formed ( Teia). The gravitational effect from the other planets forced Thea at some point to leave the Lagrange point, enter the trajectory of motion towards the Earth and eventually collide with it. The Gott-Belbruno model fills in the details of a hypothesis shared by many astronomers. According to her, the Moon consists of matter formed about 4 billion years ago after collision with the Earth. space object the size of Mars. This hypothesis, however, has a vulnerability: the question of where exactly such an object could have formed. If the place of his birth were parts of the solar system remote from the Earth, then his energy would be very large and the result of a 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 have developed in such a way in the past, what prevents them from happening again in the future? Will not, in other words, grow another Thea in the vicinity of the Lagrange points? Prof. P. Weigert (University of Western Ontario, Canada) believes that this is impossible, since in Solar system At present, dust particles are clearly not enough for the formation of such objects, and 4 billion years ago, when the planets were formed from particles of gas and dust clouds, the situation was fundamentally different. In the opinion of R. Gott, in the vicinity of the Lagrange points, asteroids, the remnants of the "building material" of the planet Thei, may well be found. Such asteroids can become a significant risk factor for the Earth. Indeed, the gravitational effect 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 the trajectory of collision 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 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 around 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 with the help of a telescope at Mauna Kea (Hawaii, USA) already tried in the early 90s. XX century find these asteroids. His observations were notable for their scrupulousness, but they did not bring success. Relatively recently, programs for automatic search for asteroids were launched, in particular, the Lincoln project for the search for asteroids close to Earth. (Lincoln Near Earth Asteroid Research project)... However, they have not given any result yet.

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

Calculations, however, showed that stopping the probes would require too much fuel. Given this circumstance, the project leaders STEREO settled on the option of a slow flight of these areas of space. It 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 the backdrop of thousands of stars. Project leaders STEREO look forward to active help in the search from amateur astronomers who will view the resulting images on the Internet.

Experts are very concerned about the problem of the safety of movement of probes in the vicinity of the Lagrange points. Indeed, collision with "dust particles" (which can be quite significant in size) can damage the probes. In flight, the probes STEREO have repeatedly encountered dust particles - from times 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 did not make the situation more predictable: it follows from them that the gravitational effect of Venus can not only "pull" asteroids out of the 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 is very little left to wait ...

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

> Lagrange points

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

Let's be honest: we're stuck on Earth. We should thank gravity for not being 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 an intelligent system has balanced the gravitational influence. With the right approach, this can be used for more productive and faster space exploration.

These places are called Lagrange points(L-points). The name was given by Joseph Louis Lagrange, who described them in 1772. In fact, he was able to expand on Leonard Eiler's mathematics. 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), then their gravitational contact is remarkably balanced in specific 5 areas. In each of them, a satellite can be positioned, which will be held in place with minimal effort.

Most notable is the first Lagrange point L1, balanced between the gravitational pull of two objects. For example, you can set up a satellite above the lunar surface. The earth's gravity pushes him into the moon, but the force of the satellite also resists. So the device does 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 ground, but on the other side. Why isn't the combined gravity pulling the satellite toward Earth? It's all about orbital trajectories. The satellite at L2 will be in a higher orbit and lag behind the Earth, as it moves around the star more slowly. But Earth's gravity pushes him and helps him to gain a foothold in place.

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

There are also L4 and L5 located in front and behind the bottom 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 found asteroids on large planets like Jupiter. These are Trojans trapped in a gravitational trap between the gravitations of the Sun and Jupiter.

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

Sun-Earth L1 is a great place to live for a solar telescope. The device has approached the star as much as possible, but does not lose contact with its home planet.

The future James Webb telescope is planned to be placed at L2 point (1.5 million km from us).

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

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

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

From the side of the first two bodies, it can remain motionless relative to these bodies.

More precisely, the Lagrange points are a special case in the solution of the so-called limited three-body problem- when the orbits of all bodies are circular and the mass of one of them is much less than the mass of either of the other two. In this case, we can assume that two massive bodies revolve around their common center of mass with constant angular velocity. There are five points in the space around them, at which a third body with negligible mass can remain stationary in a rotating frame of reference associated with massive bodies. At these points, the gravitational forces acting on the small body are balanced by the centrifugal force.

Lagrange points got their name in honor of the mathematician Joseph Louis Lagrange, who was the first to solve a mathematical problem in 1772, from which the existence of these singular points followed.

All Lagrange points lie in the plane of the orbits of massive bodies and are designated by the capital Latin letter L with a numeric index from 1 to 5. The first three points are located on a line passing through both massive bodies. These Lagrange points are called collinear and are designated L 1, L 2 and L 3. Points L 4 and L 5 are called triangular or Trojan. Points L 1, L 2, L 3 are points of unstable equilibrium, at points L 4 and L 5 the equilibrium is stable.

L 1 is located between two bodies of the system, closer to a less massive body; L 2 - outside, behind a less massive body; and L 3 for the more massive one. In the coordinate system with the origin at the center of mass of the system and with the axis directed from the center of mass to the less massive body, the coordinates of these points in the first approximation in α are calculated using the following formulas:

Point L 1 lies on a straight line connecting two bodies with masses M 1 and M 2 (M 1> M 2), and is located between them, near the second body. Its presence is due to the fact that the gravity of the body M 2 partially compensates for the gravity of the body M 1. Moreover, the more M 2, the further from it this point will be located.

Lunar point L 1(in the Earth-Moon system; removed from the center of the Earth by about 315 thousand km) can be an ideal place for the construction of a manned space station, which, located on the path between the Earth and the Moon, would make it easy to get to the Moon with minimal fuel consumption and become a key node of the cargo flow between the Earth and its satellite.

Point L 2 lies on a straight line connecting two bodies with masses M 1 and M 2 (M 1> M 2), and is located behind a body with a lower mass. Points L 1 and L 2 are located on the same line and in the limit M 1 ≫ M 2 are symmetric with respect to M 2. At the point L 2 gravitational forces acting on a body compensate for the action of centrifugal forces in a rotating frame of reference.

Point L 2 in the Sun - Earth system is an ideal place for the construction of orbiting space observatories and telescopes. Since the object at the point L 2 capable of long time to maintain its orientation relative to the Sun and the Earth, it becomes much easier to screen and calibrate it. However, this point is located a little further than the earth's shadow (in the penumbra region) [approx. 1], so that solar radiation is not completely blocked. At the moment (2020) Gaia and Spektr-RG satellites are in halo orbits around this point. Previously, telescopes such as Planck and Herschel operated there, in the future it is planned to send several more telescopes there, including James Webb (in 2021).

Point L 2 in the Earth-Moon system can be used to provide satellite communication with objects on back side The moon, as well as being a convenient place to place a gas station to ensure the flow of goods between the Earth and the Moon

If M 2 is much less in mass than M 1, then the points L 1 and L 2 are at approximately the same distance r from the body M 2, equal to the radius of the Hill sphere:

Point L 3 lies on a straight line connecting two bodies with masses M 1 and M 2 (M 1> M 2), and is located behind a body with a larger mass. Same as for point L 2, at this point gravitational forces compensate for the action of centrifugal forces.

Before the start of the space age, the idea of ​​existing on the opposite side of the earth's orbit at a point was very popular among science fiction writers L 3 another planet similar to it, called "Counter-Earth", which due to its location was inaccessible for direct observation. However, in fact, due to the gravitational influence of other planets, point L 3 in the Sun-Earth system is extremely unstable. So, during heliocentric conjunctions of the Earth and Venus on opposite sides of the Sun, which occur every 20 months, Venus is only 0.3 a.u. from point L 3 and thus has a very serious effect on its position in relation to the earth's orbit. In addition, due to the imbalance [ clarify] the center of gravity of the Sun - Jupiter system relative to the Earth and the ellipticity of the Earth's orbit, the so-called "Counter-Earth" would still be available for observation from time to time and would definitely be noticed. Another effect that would betray its existence would be its own gravity: the influence of a body already in the order of 150 km or more on the orbits of other planets would be noticeable. With the advent of the possibility of making observations using spacecraft and probes, it was reliably shown that there are no objects larger than 100 m in size at this point.

Orbital spacecraft and satellites located near the point L 3, can constantly monitor various forms of activity on the Sun's surface - in particular, for the appearance of new spots or flares - and quickly transmit information to Earth (for example, as part of the NOAA space weather early warning system). In addition, information from such satellites can be used to ensure the safety of long-range manned flights, for example, to Mars or asteroids. In 2010, several options for launching such a satellite were studied.

If, on the basis of the line connecting both bodies of the system, build two equilateral triangles, two vertices of which correspond to the centers of the bodies M 1 and M 2, then the points L 4 and L 5 will correspond to the position of the third vertices of these triangles located in the plane of the orbit of the second body 60 degrees in front of and behind it.

The presence of these points and their high stability is due to the fact that, since the distances to two bodies at these points are the same, the forces of attraction from the side of two massive bodies correlate in the same proportion as their masses, and thus the resulting force is directed to the center of mass of the system ; in addition, the geometry of the triangle of forces confirms that the resulting acceleration is related to the distance to the center of mass in the same proportion as for two massive bodies. Since the center of mass is at the same time the center of rotation of the system, the resulting force exactly corresponds to that which is needed to keep the body at the Lagrange point in orbital equilibrium with the rest of the system. (In fact, the mass of the third body should not be negligible). This triangular configuration was discovered by Lagrange while working on the three-body problem. Points L 4 and L 5 are called triangular(as opposed to collinear).

Also the points are called Trojan: This name comes from the Trojan asteroids of Jupiter, which are the most striking example of the manifestation of these points. They were named after the heroes of the Trojan War from Homer's Iliad, and the asteroids at L 4 get the names of the Greeks, and at the point L 5- defenders of Troy; therefore they are now called “Greeks” (or “Achaeans”) and “Trojans”.

Distances from the center of mass of the system to these points in the coordinate system with the center of coordinates at the center of mass of the system are calculated using the following formulas:

Bodies placed at collinear Lagrange points are in unstable equilibrium. For example, if an object at point L 1 is slightly displaced along a straight line connecting two massive bodies, the force attracting it to the body it is approaching increases, while the force of attraction from the other body, on the contrary, decreases. As a result, the object will move farther and farther from the equilibrium position.

This feature of the behavior of bodies in the vicinity of the point L 1 plays an important role in close binary stellar systems. The Roche lobes of the components of such systems are in contact at the point L 1, therefore, when one of the companion stars in the process of evolution fills its Roche lobe, matter flows from one star to another precisely through the vicinity of the Lagrange point L 1.

Despite this, there are stable closed orbits (in a rotating coordinate system) around collinear libration points, at least in the case of the three-body problem. If the motion is influenced by other bodies as well (as happens in the solar system), instead of closed orbits, the object will move in quasiperiodic orbits in the form of Lissajous figures. Despite the instability of such an orbit,

Lagrange points are areas in a system of two cosmic bodies with a large mass, in which a third body with a small mass can be motionless for a long period of time relative to these bodies.

In astronomical science, Lagrange points are also called libration points (libration from Latin librātiō - wobble) or L-points. They were first discovered in 1772 by the famous French mathematician Joseph Louis Lagrange.

Lagrange points are most often mentioned when solving the restricted three-body problem. In this problem, three bodies have circular orbits, but the mass of one of them is less than the mass of any of the other two objects. Two large bodies in this system revolve around common center masses, having a constant angular velocity... In the area around these bodies there are five points at which a body whose mass is less than the mass of any of the two large objects can remain motionless. This is due to the fact that the forces of gravity that act on this body are compensated centrifugal forces... These five points are called Lagrange points.

Lagrange points lie in the plane of the orbits of massive bodies. In modern astronomy, they are denoted by the Latin letter "L". Also, depending on its location, each of the five points has its own serial number, which is denoted by a numerical index from 1 to 5. The first three Lagrange points are called collinear, the other two are Trojan or triangular.

Locations of the nearest Lagrange points and examples of points

Regardless of the type of massive celestial bodies, Lagrange points will always have the same location in the space between them. The first Lagrange point is between two massive objects, closer to the one that has less mass. The second Lagrange point is located behind a less massive body. The third Lagrange point is located at a considerable distance behind the body, which has greater mass... The exact location of these three points is calculated using special mathematical formulas individually for each cosmic binary system, taking into account its physical characteristics.

If we talk about the Lagrange points closest to us, then the first Lagrange point in the Sun-Earth system will be at a distance of one and a half million kilometers from our planet. At this point, the Sun's attraction will be two percent stronger than in our planet's orbit, while the decrease in the required centripetal force will be half as much. Both of these effects at a given point will be balanced by the gravitational attraction of the Earth.

The first Lagrange point in the Earth-Sun system is a convenient observation point for the main star of our planetary system - the Sun. It is here that astronomers are seeking to place space observatories to observe this star. So, for example, in 1978, the ISEE-3 spacecraft, designed to observe the Sun, was located near this point. In subsequent years, the spacecraft DSCOVR, WIND and ACE were launched into the area of ​​this point.

Second and third Lagrange points

Gaia, a telescope located at the second Lagrange point

The second Lagrange point is located in a binary system of massive objects behind a body with a lower mass. The use of this point in modern astronomical science is reduced to the placement of space observatories and telescopes in its area. At the moment, such spacecraft as Herschel, Planck, WMAP and others are located at this point. In 2018, another spacecraft, the James Webb, is to go there.

The third Lagrange point is located in the binary system at a considerable distance behind a more massive object. If we talk about the Sun-Earth system, then such a point will be located behind the Sun, at a distance slightly greater than that in which the orbit of our planet is located. This is due to the fact that, despite its small size, the Earth still has an insignificant gravitational effect on the Sun. Satellites located in this area of ​​space can transmit accurate information about the Sun, the appearance of new "spots" on the star, and also transmit data about space weather to the Earth.

Fourth and fifth Lagrange points

The fourth and fifth Lagrange points are called triangular. If in a system consisting of two massive space objects revolving around a common center of mass, on the basis of the line connecting these objects, mentally draw two equilateral triangles, the vertices of which will correspond to the position of two massive bodies, then the fourth and fifth Lagrange points will be in place third vertices of these triangles. That is, they will be in the orbital plane of the second massive object 60 degrees behind and in front of it.

Lagrange triangular points are also called "Trojan" points. The second name for the points comes from the Trojan asteroids of Jupiter, which are the brightest visual manifestations of the fourth and fifth Lagrange points in our solar system.

At the moment, the fourth and fifth Lagrange points in the Sun-Earth binary system are not used in any way. In 2010, at the fourth Lagrange point of this system, scientists discovered a fairly large asteroid. At the fifth point of Lagrange, at this stage, no large space objects are observed, but the latest data tell us that there is a large accumulation of interplanetary dust.

1. In 2009, two STEREO spacecraft flew through the fourth and fifth Lagrange points.
2. Lagrange points are often used in science fiction writing. Often in these areas of space, around binary systems, science fiction writers place their fictional space stations, garbage dumps, asteroids and even other planets.
3. In 2018, scientists plan to place the James Webb Space Telescope at the second Lagrange point in the solar-Earth binary system. This telescope should replace the current space telescope "", which is located at this point. In 2024, scientists plan to place another PLATO telescope at this point.
4. The first Lagrange point in the Moon-Earth system could be an excellent place to place a manned orbital station, which could significantly reduce the cost of resources required to get from Earth to the Moon.
5. The two space telescopes "Planck" and "", which were launched into space in 2009, are currently located at the second Lagrange point in the Sun-Earth system.

Whatever goal you set for yourself, whatever mission you plan, one of the biggest obstacles in your path in space will be fuel. Obviously, a certain amount of it is needed already 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 the same rocket - there simply will not be 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, almost without using fuel. It's called the Interplanetary Transport Network.

Current methods of space travel

Today, moving between objects in the solar system, for example, traveling from Earth to Mars, usually requires a so-called Hohmann ellipse flight. The carrier starts up and then accelerates until it is beyond the orbit of Mars. Near the red planet, the rocket slows down and begins to revolve around its target. It burns a lot of fuel for both acceleration and deceleration, but the Homan's ellipse remains one of the most efficient ways to move between two objects in space.

Homan's ellipse - Arc I - flight from Earth to Venus. Arc II - flight from Venus to Mars Arc III - return from Mars to Earth.

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

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

If we place a spaceship at the Lagrange point, it will hang there indefinitely, since gravity does not pull it in one direction more than in any other. However, not all of these points are, figuratively speaking, created equal. Some of them are stable - if you, while inside, move a little to the side, 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, and you will begin to be carried away from there. The objects here resemble a ball at the top of a hill - it will stick there if it is well installed or if it is held there, but even a slight breeze is enough for it to roll downward, picking up speed.

Hills and valleys of the cosmic landscape

Spaceships flying through the solar system take into account all these "hills" and "valleys" during flight and during the route planning stage. However, the interplanetary transport network makes them work for the good of society. As you already know, each stable orbit has five Lagrange points. This is the Earth-Moon system, and the Sun-Earth system, and the systems of all 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 and everywhere, although they constantly change their specific location in space. They always orbit the smaller object of the rotation system, and this creates an ever-changing landscape of gravity hills and valleys. In other words, the distribution gravitational forces in the solar system changes over time. Sometimes attraction in certain spatial coordinates is directed towards the Sun, at another moment in time - towards a planet, and it also happens that the Lagrange point passes along them, and equilibrium reigns in this place, when no one pulls anyone anywhere ...

The metaphor of hills and valleys helps us better represent this abstract idea, 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 can even overlap. And at this very moment, space travel becomes especially effective. For example, if your gravity hill overlaps with a valley, you can "slide" into it. If another hill overlaps your hill, you can jump from top to top.

How to use the Interplanetary Transport Network?

When the Lagrange points of different orbits approach each other, almost no effort is required to move from one to the other. This means that if you are in no 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 further, almost without wasting fuel. It is easy to understand that this is exactly the idea used by the Interplanetary Transport Network. The ever-changing network of Lagrange points is like a winding road that allows you to move between orbits with a meager 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 the desperate but successful rescue attempt at the Japanese lunar station in 1991, when the spacecraft was low on 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 quite afford to send a probe into space, which will calmly wait for the necessary alignments, and collect information the rest of the time. After waiting, he will jump to another orbit, and carry out observations, being already on it. This probe will be able to travel through the solar system for an unlimited amount of time, registering everything that happens in its vicinity, and replenishing the scientific baggage of human civilization. It is clear that this will be fundamentally different from how we explore space now, but this method looks promising, including for future long-term missions.