libration points. What are "Lagrange points"

In the Earth-Moon system, the first three points of libration are on the rotating line connecting the Earth and the Moon: the point lies between the planets, the second point is behind the Moon, and the third collinear point is located on the far side of the Earth with respect to the Moon. The remaining two libration points and are located on both sides outside the rotating line.

Five points of equilibrium, known as the Lagrange points or libration points are shown in fig. 3. In them, the combined gravitational forces from the first and second bodies are exactly compensated by the centripetal acceleration of the third body. Such points allow the third body to maintain an orbital period equal to the orbital periods of the first and second bodies near their joint center of mass.

Rice. 3. Five points of libration in the Earth-Moon system.

The points, and are unstable. Since, if an object placed at the collinear Lagrange point is slightly displaced along the straight line connecting the Earth and the Moon, then the force that attracts the object to the body it approaches increases, and the force of attraction from the other body, on the contrary, decreases. As a result, the object will be more and more removed from the equilibrium position.

However, there are stable closed quasi-periodic and periodic orbits such as Lissajous and halo orbits that fluctuate around these points. That is, a spacecraft moving along a halo orbit will remain in it for a long time (Fig. 4).

Rice. 4. Halo orbits in the Earth-Moon system.

An object, such as a spacecraft, that is displaced from the libration point will oscillate around the point with a period determined by how far it is displaced in the Y and Z directions (Figure 5). The parameter φ is the angle that determines the position of the spacecraft in a given halo orbit and is similar to the true anomaly when flying in an elliptical orbit. It is measured in the positive direction from the +Z axis around the +X axis from 0 ? up to 360?.

Fig 5.

Among the five points of libration of the Earth-Moon system, the most relevant for human exploration of outer space are the two closest to the Moon - and. They are located near the near and far sides of the Moon, respectively, as seen from Earth. However, it is better to explore the far side of the Moon, which is one of the priority places for the study of cosmogony and the history of the solar system. The moon protects the surface on her reverse side from terrestrial radio noise, which makes it easier to study low-frequency signals (below 100 MHz).

The libration point is an ideal place for the construction of orbiting space observatories and telescopes. Since an object at a point is able to maintain its orientation relative to the Sun and Earth for a long time, it becomes much easier to shield and calibrate it. A point in the Earth-Moon system can be used to provide satellite communications with objects on the far side of the Moon, as well as be a convenient place to place a gas station to ensure the flow of cargo between the Earth and the Moon.

Thus, in this paper, we consider a spacecraft on the lunar surface, located closer to the north (the range in latitude is from 60? to 90?), from which an object is observed moving along a halo orbit at the libration point (Fig. 6 ).

Rice. 6. Spacecraft at the point of libration.

Have experiments been carried out to place spacecraft at the Lagrange points of the Earth-Moon system?

Despite the fact that the so-called libration points that exist in space, and their amazing properties mankind has known for a long time, they began to use them for practical purposes only in the 22nd year 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 managed to determine where in space there can be points at which the resultant of all external forces goes to zero.

Points are divided into stable and unstable. Stable is usually denoted L 4 and L 5 . They are located in the same plane with the main two celestial bodies (in this case, the Earth and the Moon), forming two equilateral triangles with them, for which they are often called triangular. The spacecraft can stay at the triangular points for an arbitrarily long time. 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 aside. It is extremely difficult to get to the unstable Lagrange point - this requires ultra-precise navigation. Therefore, the apparatus has to move only near the very point along the so-called "halo-orbit", from time to time consuming fuel to maintain it, however, 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 about 380 thousand km.

Although it is more profitable to be at stable points (it is not required to consume fuel), spacecraft have so far only become acquainted with unstable ones, or rather, with only one of them, and even then related to the Sun-Earth system. It is located within this system, 1.5 million km from our planet and, like the point between the Earth and the Moon, has the designation L 1 . When viewed from the 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 apparatus ISEE-3, launched on August 12, 1978. From November 1978 to June 1982 he 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 explorer in history. To do this, the device left the libration point and, having made several gravitational maneuvers near the Moon, in 1985 carried out a flyby near the Giacobini-Zinner comet. On next year he also explored Halley's comet, however, only at distant approaches.

The next visitor to point L 1 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 few important scientific information and made many interesting discoveries.

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

And what's next? Are there new projects related to libration points? Certainly they exist. Thus, in the USA the proposal of Vice-President A. Gore on 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 "Chamber of Gore", was accepted.

Unlike his predecessors, he will not follow the Sun, but the Earth. Our planet is always visible from this point in full phase and therefore very convenient for observations. It is expected that the images received by the "Camera Gore" will be sent 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 libration point L 2 in the Earth-Moon system, that is, the one hidden behind the Moon, awaits it.

astronautics

"They are often referred to as 'the place where there is no gravity.' Huge outer space, millions of kilometers long, where gravity does not work, areas that capture and do not release any object that gets there. Astronomers call them the Lagrange points, or in short - L4 and L5. Under the cut - a huge article about them, dear ones.

(Article by Stuart Clark, New Scientist), quite large. Translation abbreviated)

In the 4.5 billion years since the formation of the solar system, everything from dust clouds to asteroids and hidden planets has been able to collect and accumulate in them. Some pseudo-scientific publications even claim about aliens hiding in L4 and L5 and watching the Earth from their saucers.

If we ignore the little green men for a moment, even the very presence of old cosmic rock fragments at the points can make many scientists happy. "I think there's actually a whole 'population' of different objects in L4 and L5," says astrophysicist Richard Gott of Princeton University.

After a century of scientific speculation, we finally came to find out what is hidden in the Lagrange points. This year, a little later, two spacecraft, who until now have been studying the Sun, will reach L4 and L5 spaces.

Astronomers plan to use the instrumentation aboard NASA's STEREO space probes A and B to look for celestial bodies, which, presumably, can be hidden in the regions of the Lagrange points. Their findings could significantly affect our understanding of how the solar system formed, about the colossal interactions that formed the moon, and perhaps warn us against future collisions.

Lagrange points were first discovered in 1772 by the mathematician José Louis Lagrange. He calculated that the Earth's gravitational field should cancel out the Sun's gravitational pull in five regions of space - in fact, the only regions where an object can truly become weightless.
Of the five points, L4 and L5 are the most intriguing. They are the only stable areas, if the satellite enters L1 or L2, after a few months it will “let go” and it will fly further, but any object that falls into the “field of view” of L4 or L5 will remain there for a very long time, if not forever. They are located at a distance of 150 million km from the Earth, on the plane of the Earth's orbit, with L4 orbiting the Sun 60 degrees ahead of the Earth, and L5 at exactly the same angle behind the planet.

There is evidence of similar regions around other planets. In 1906, Max Wolff discovered an asteroid behind the main belt between Mars and Jupiter and realized that it was in L4 Jupiter. Wolf named it Achilles, and thus founded the tradition of naming similar asteroids after the participants. Trojan War. The realization that Achilles could fall into such a trap raised a wave of searches more examples. About 1000 asteroids are now known to have been captured by Jupiter's Lagrange points.

The search for "Trojan" asteroids near other planets has not yet been particularly successful. They could not be found near Saturn, but exactly one was found near Neptune. And, of course, they became interested in the Earth.
The only problem is that points L4 and L5 are difficult to see from Earth. They are located close to the Sun, so that at night the L5 region is above the horizon and drops rapidly, while L4 is eclipsed by dawn rays.

That did not stop Paul Weigert of the University of Eastern Ontario, Canada, from conducting a series of searches in the 1990s using the French-Hawaiian telescope on Mauna Kea, Hawaii. This was quite a difficult task, since L4 and L5 occupy more visible areas in the sky than the full moon. Unfortunately, Weigert's team failed to find any interesting things.

More recently, automatic searches such as the Lincoln Near Earth Asteroid Research project have also begun to pay attention to Lagrange regions, but so far nothing has been found there. “This line of research has languished because everyone sits and waits for someone else to make a discovery,” says Weigert.

STEREO spacecraft could change things - even though the probes weren't specifically designed to search for asteroids. They were launched in 2006, one in front of the Earth, the other behind, so now they can explore the space between the Earth and the Sun, mainly studying solar storms that can disable orbiting satellites or equipment on Earth. Just L4 and L5 are very good "observation points" for solar activity. "We even talked about stopping the probes when they reach these areas, since it still takes several days for accurate records," says Michael Keizer from the Center for Space flying at Goddard in Greenbelt, Maryland, also a member of the STEREO project.
In fact, the STEREO project team believes that stopping their probes in zones L4 and L5 requires too much fuel consumption. Therefore, the probes will be set up for a very slow "flight", though not so slow as to fall into a gravitational trap.

In this regard, Richard Garrison from the Rutford Appleton laboratory in Oxfordshire came up with the idea that the probes could be loaded with another task. He explored all the possibilities and realized that tools designed to obtain heliospheric images can be reconfigured to search for asteroids. Even so, finding a Trojan asteroid would be very difficult, as it would be a dot moving against a backdrop of thousands of stars. Fortunately, a team of volunteers has already been formed who will study the images in detail.

If the asteroid is still found, by changing the reflected by its surface sunlight it will be possible to determine its rotation and predict the location of other asteroids at Lagrange points. And then, perhaps, the answer to the question will appear: why does the Earth have such a massive satellite? Now most scientists are sure that the Moon was formed from space debris, or rather, debris left after an object the size of Mars crashed into the Earth about 4 billion years ago. The problem is to explain where he could come from. Because, as computer models of the situation show, all solar system objects of this size would have to destroy the Earth on impact, instead of breaking into pieces and forming satellites themselves. So such an object had to appear “nearby” in order not to have enough time to accelerate before the collision. Another confirmation of the close location of such a body is the discovery in the lunar substance of the same amount of oxygen isotopes, which is typical for the Earth. Mars, for example, is characterized by a different ratio. But it remains unclear how such a large celestial body could form close to the Earth and not collide with it. If only the formation did not occur at the Lagrange points. And once the object reached a certain size, the attraction of other planets, such as Venus, pulled it out of this area and forced it to crash into the Earth. “The same number of oxygen isotopes as the Earth could be explained by the fact that its formation took place close to the Earth,” says Gott. Also, being in the same orbit, the two planets could not have differed too much in speed when the collision occurred. And, if at the Lagrange points near the Earth it will be possible to detect the remains of the formation of such a planet and prove that the content of oxygen isotopes they have in common with the earth, in fact, the theory will be almost proven.

If asteroids are found, they are unlikely to be more than a kilometer in diameter, Weigert said. Despite the fact that the average size asteroids of the main belt - 100 kilometers.

Finally, you can add a little yellowness to the expectation of discovery: some astronomers suggest that a whole planet can hide in the Lagrange points. “No way,” says Paul Weigrt. - There is no required amount substances to form such a large body",
But 4.5 billion years ago, the situation was different: the planets formed from a mixture of dust and gas, and L4 and L5 were just the right “accumulators” for large celestial bodies to arise there. Nothing of planetary size, but Richard Gott thinks there might still be monstrous asteroids lurking there. "If we could find something big enough, it would be like a wound up bomb," because the gravitational influence of other planets, especially Venus, can " "pull" such an asteroid just far enough to get it out of the Lagrange point. And send it to Earth.

“If we see a big enough asteroid there, we'll just blow it up and take the debris for ourselves,” says Gott.

Lagrange points are areas in a system of two cosmic bodies with 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 - rocking) or L-points. They were first discovered in 1772 by the famous French mathematician Joseph Louis Lagrange.

Lagrange points are most often mentioned in 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 either of the other two objects. Two large body in this system, they revolve around a common center of mass, having a constant angular velocity. In the area around these bodies there are five points where a body whose mass is less than the mass of either of the two large objects can remain motionless. This is due to the fact that the gravitational forces that act on this body are compensated by centrifugal forces. These five points are called the Lagrange points.

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

Regardless of the type of massive celestial bodies, the 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 with less mass. The second Lagrange point is behind the less massive body. The third Lagrange point is located at a considerable distance behind the body with a larger 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 attraction of the Sun will be two percent stronger than in the orbit of our planet, while the decrease in the necessary centripetal force will be half as much. Both of these effects at a given point will be balanced by the gravitational pull 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 seek 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, spacecraft SOHO, DSCOVR, WIND and ACE were launched to the area of this point.

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

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

The fourth and fifth Lagrange points are called triangular. If in a system consisting of two massive space objects rotating around a common center of mass, on the basis of a 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 ahead of it.

Triangular Lagrange points are also called "Trojan" points. The second name of the points comes from the Trojan asteroids of Jupiter, which are the brightest visual manifestation 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 found enough large asteroid. No large space objects are observed at the fifth Lagrange point at this stage, however, recent data tell us that there is a large accumulation of interplanetary dust.

Interesting Facts

In 2009, two STEREO spacecraft flew over the fourth and fifth Lagrange points.

Lagrange points are often used in science fiction. Often in these areas of space, around binary systems, science fiction writers place their fictional space stations, garbage dumps, asteroids and even other planets.

In 2018, scientists plan to place the James Webb Space Telescope at the second Lagrange point in the Sun-Earth binary. This telescope is to replace the current Hubble Space Telescope, which is located at this point. In 2024, scientists plan to place another PLATO telescope at this point.

The first Lagrange point in the Moon-Earth system could become great place to accommodate a manned orbital station, which could significantly reduce the cost of resources needed to get from Earth to the Moon.

The two space telescopes Planck and Herschel, which were launched into space in 2009, are currently at the second Lagrange point in the Sun-Earth system.

"They are often referred to as 'the place where there is no gravity.' Huge expanses of space, millions of kilometers long, where gravity does not work, areas that capture and do not release any object that got there. Astronomers call them the Lagrange points, or in short - L4 and L5. Under the cut - a huge article about them, dear ones.

After a century of scientific speculation, we finally came to find out what is hidden in the Lagrange points. This year, a little later, two spacecraft that have been studying the Sun so far will reach L4 and L5 spaces.

Astronomers plan to use the instrumentation aboard NASA's STEREO A and B space probes to look for celestial bodies that might be lurking in Lagrange point regions. Their findings could significantly affect our understanding of how the solar system formed, about the colossal interactions that formed the moon, and perhaps warn us against future collisions.

There is evidence of similar regions around other planets. In 1906, Max Wolff discovered an asteroid behind the main belt between Mars and Jupiter and realized that it was in L4 Jupiter. Wolf named it Achilles, and thus founded the tradition of naming such asteroids after participants in the Trojan War. The realization that Achilles might have fallen into such a trap set off a wave of searches for more examples. About 1000 asteroids are now known to have been captured by Jupiter's Lagrange points.

If an asteroid is found, it will be possible to determine its rotation by changing the sunlight reflected by its surface and predict the location of other asteroids at Lagrange points. And then, perhaps, the answer to the question will appear: why does the Earth have such a massive satellite? Now most scientists are sure that the Moon was formed from space debris, or rather, debris left after an object the size of Mars crashed into the Earth about 4 billion years ago. The problem is to explain where he could come from. Because, as computer models of the situation show, all objects of this size entering the solar system should have destroyed the Earth in a collision, instead of disintegrating into pieces and forming satellites themselves. So such an object had to appear “nearby” in order not to have enough time to accelerate before the collision. Another confirmation of the close location of such a body is the discovery in the lunar substance of the same amount of oxygen isotopes, which is typical for the Earth. Mars, for example, is characterized by a different ratio. But it remains unclear how such a large celestial body could form close to the Earth and not collide with it. If only the formation did not occur at the Lagrange points. And once the object reached a certain size, the attraction of other planets, such as Venus, pulled it out of this area and forced it to crash into the Earth. “The same number of oxygen isotopes as the Earth could be explained by the fact that its formation took place close to the Earth,” says Gott. Also, being in the same orbit, the two planets could not have differed too much in speed when the collision occurred. And, if at the Lagrange points near the Earth it will be possible to detect the remains of the formation of such a planet and prove that the content of oxygen isotopes they have in common with the earth, in fact, the theory will be almost proven.

If asteroids are found, they are unlikely to be more than a kilometer in diameter, Weigert said. Given that the average size of the main belt asteroids is 100 kilometers.

Finally, you can add a little yellowness to the expectation of discovery: some astronomers suggest that a whole planet can hide in the Lagrange points. “No way,” says Paul Weigrt. “There is not enough material to form such a large body,”
But 4.5 billion years ago, the situation was different: the planets formed from a mixture of dust and gas, and L4 and L5 were just the right “accumulators” for large celestial bodies to arise there. Nothing of planetary size, but Richard Gott thinks there might still be monstrous asteroids lurking there. "If we could find something big enough, it would be like a wound up bomb," because the gravitational influence of other planets, especially Venus, can " "pull" such an asteroid just far enough to get it out of the Lagrange point. And send it to Earth.

“If we see a big enough asteroid there, we'll just blow it up and take the debris for ourselves,” says Gott.