Lagrange point l1. Lagrange points and the distance between them. Using the Lagrange point to influence climate. Lagrange points: What is it about

In the system of rotation of two space bodies of a certain mass, there are points in space, by placing any object of small mass in which, 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 solving the problem of three rotating bodies, two of which have such a mass that the mass of the third body is negligible compared to 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 the Lagrange points. By placing a body of small mass in them, one can observe how its distances to each of the two massive bodies do not change for an arbitrarily long time. Here we can draw an analogy with the geostationary orbit, in which the satellite is always located above one point on the earth's surface.

It should be clarified that the body that is located at the Lagrange point (it is also called the free point or point L), relative to the 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 such a character that with respect to each of them the third body is at rest.

How many of these points 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 denoted L1, L2, L3, L4 and L5. All these points are located in the plane of rotation of the considered bodies. 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 behind each of the bodies. Points L4 and L5 are located in such a way that if we connect each of them with the centers of mass of two bodies of the system, we 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 the resulting force when approaching the corresponding free point. It can be seen from the figure that the areas of points L4 and L5 are much larger than the areas of points L1, L2 and L3.

History reference

The existence of free points in a system of three rotating bodies was first proved 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 revolution. In reality, the orbits are elliptical. The latter fact leads to the fact that there are no longer Lagrange points, but there are areas in which the 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: let's take the Sun and the Earth as an example, according to Kepler's third law, the closer the body is to its star, the shorter its period of rotation around this star (the square of the body's rotation period is directly proportional to the cube of the average distance from bodies to stars). This means that any body that is located between the Earth and the Sun will revolve around 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, then we can assume that the closer to the Earth is the third body of small mass, the stronger will be the opposition to the Earth's solar gravity. As a result, there will be such a point where the 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 become equal. This will be the free point L1. The distance to the Lagrange point L1 from the Earth is 1/100 of the radius of the planet's orbit around the star and is 1.5 million km.

How is the L1 region used? This is an ideal place to observe solar radiation, as there is never solar eclipses. Currently, several satellites are located in the L1 region, which are engaged in the study of 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 "transit" point during missions of spacecraft and satellites to and from the Moon.

Free point L2

Arguing 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 alignment of the periods of rotation of a body with a smaller mass and a third body around the body with more weight. This area is a free point L2.

If we consider the Sun-Earth system, then up to this Lagrange point the distance from the planet will be exactly the same as up to point L1, that is, 1.5 million km, only L2 is located behind the Earth and further from the Sun. Since there is no influence of solar radiation in the L2 region due to the earth's protection, it is used for observing the Universe, having 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 the Earth. The point is not used in any way, because it is unstable due to the influence of the gravity of other planets, such as 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 for free points

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

Another interesting example of the use of the L1 point of the Earth was the physics project of one Ukrainian schoolchild. He proposed to place 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.

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 stationary 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 bodies in this system 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 by the 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.

Location of nearest Lagrange points and examples of points

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, DSCOVR, WIND and ACE were launched to 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 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. At the moment, such spacecraft as Herschel, Plank, WMAP and are located at this point. In 2018, another spacecraft, the James Webb, should go there.

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.

Fourth and fifth Lagrange points

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 planet. 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. 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.

1. In 2009, two STEREO spacecraft flew over the fourth and fifth Lagrange points.
2. 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.
3. 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 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 location for a manned orbital station, which could significantly reduce the cost of resources needed to get from Earth to the Moon.
5. 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.

When Joseph Louis Lagrange worked on the problem of two massive bodies (restricted problem of three bodies), he discovered that in such a system there are 5 points with the following property: if bodies of negligibly small mass are located in them (relative to massive bodies), then these bodies will be immobile relative to those two massive bodies. Important point: massive bodies must rotate around a common center of mass, but if they somehow simply rest, then this whole theory is not applicable here, now you will understand why.

The most successful example, of course, is the Sun and the Earth, and we will consider them. The first three points L1, L2, L3 are on the line connecting the centers of mass of the Earth and the Sun.

Point L1 is between the bodies (closer to the Earth). Why is it there? Imagine that between the Earth and the Sun is some small asteroid that revolves around the Sun. As a rule, bodies inside the Earth's orbit have a higher frequency of revolution than that of the Earth (but not necessarily) So, if our asteroid has a higher frequency of revolution, then from time to time it will fly past our planet, and it will slow it down with its gravity, and eventually the frequency of revolution of the asteroid will be the same as that of the Earth. If the Earth has a higher frequency of revolution, then it, flying past the asteroid from time to time, will pull it along and accelerate it, and the result is the same: the revolution frequencies of the Earth and the asteroid will become equal. But this is only possible if the asteroid's orbit passes through point L1.

Point L2 is behind the Earth. It may seem that our imaginary asteroid at this point should be attracted to the Earth and the Sun, since they were on the same side of it, but no. Do not forget that the system is rotating, and due to this, the centrifugal force acting on the asteroid is balanced by the gravitational forces of the Earth and the Sun. Bodies outside the Earth's orbit, in general, the frequency of revolution is less than that of the Earth (again, not always). So the essence is the same: the asteroid's orbit passes through L2 and the Earth, flying by from time to time, pulls the asteroid along with it, eventually equalizing the frequency of its circulation with its own.

Point L3 is behind the Sun. Remember, earlier science fiction writers had such an idea that on the other side of the Sun there is another planet, such as Counter-Earth? So, the L3 point is almost there, but a little further away from the Sun, and not exactly in the Earth's orbit, since the center of mass of the "Sun-Earth" system does not coincide with the center of mass of the Sun. With the frequency of revolution of the asteroid at point L3, everything is obvious, it should be the same as that of the Earth; if it is less, the asteroid will fall on the Sun, if it is more, it will fly away. By the way, this point is the most unstable, it sways due to the influence of other planets, especially Venus.

L4 and L5 are located in an orbit that is slightly larger than Earth's, and as follows: imagine that from the center of mass of the "Sun-Earth" system we drew a beam to the Earth and another beam, so that the angle between these beams was 60 degrees. And in both directions, that is, counterclockwise and along it. So, on one such beam there is L4, and on the other L5. L4 will be in front of the Earth in the direction of travel, that is, as if running away from the Earth, and L5, respectively, will catch up with the Earth. The distances from any of these points to the Earth and to the Sun are the same. Now, remembering the law of universal gravitation, we notice that the force of attraction is proportional to the mass, which means that our asteroid in L4 or L5 will be attracted to the Earth as many times weaker as the Earth is lighter than the Sun. If the vectors of these forces are constructed purely geometrically, then their resultant will be directed exactly to the barycenter (the center of mass of the "Sun-Earth" system). The Sun and the Earth revolve around the barycenter with the same frequency, and the asteroids in L4 and L5 will also rotate with the same frequency. L4 are called Greeks, and L5 are called Trojans in honor of the Trojan asteroids of Jupiter (more on Wiki).

Whatever goal you set for yourself, whatever mission you plan - one of the biggest obstacles to your path in space will be fuel. It is obvious that a certain amount of it is needed already in order to leave the Earth. The more cargo you need to take out of the atmosphere, the more fuel you need. But because of this, the rocket becomes even heavier, and the whole thing turns into a vicious circle. This is what prevents us from sending several interplanetary stations to different addresses on one rocket - it simply does not have 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 flight

Today, moving between objects in the solar system, such as traveling from Earth to Mars, usually requires a so-called Hohmann ellipse flight. The carrier is launched and then accelerates until it is beyond the orbit of Mars. Near the red planet, the rocket slows down and begins to rotate around the target of its destination. For both acceleration and deceleration, it burns a lot of fuel, but at the same time, the Hohmann ellipse remains one of the most effective ways moving between two objects in space.

Ellipse Goman-Dug 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. Making them, the spacecraft 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 on a long journey from Earth. However, if the ship after the gravitational maneuver needs to enter the orbit of a planet, it still has to slow down. Of course, you remember that this requires fuel.

Exactly for this reason, 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 geographical landscape, and formulated the idea of ​​an interplanetary transport network. The entrance and exit springboards to it were the Lagrange points - five districts next to celestial bodies where gravity and rotational forces come into equilibrium. They exist in any system in which one body revolves around another, and without pretense of originality are numbered from L1 to L5.

If we place a spacecraft at the Lagrange point, it will hang there indefinitely, since gravity doesn't pull it one way more than it does the other. However, not all of these points are, figuratively speaking, created equal. Some of them are stable - if you move a little to the side while inside, gravity will return you to the 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. The objects 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 light breeze is enough for it to pick up speed and roll down.

Hills and valleys of space landscape

Spacecraft flying around the solar system take into account all these "hills" and "valleys" during the flight and at the stage of laying the route. However, the interplanetary transport network forces them to work for the benefit 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 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 and everywhere, even though they constantly change their specific location in space. They always follow the orbit of the smaller object of 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 a planet, and it also happens that a Lagrange point passes through them, and equilibrium reigns in this place, when no one is pulling anywhere .

The hills and valleys metaphor 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 may even overlap. And at this very moment, cosmic movements become especially effective. For example, if your gravity hill overlaps a valley, you can "roll" into it. If another hill overlaps your hill, you can jump from peak to peak.

How to use the Interplanetary Transport Network?

When the Lagrange points of different orbits approach 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 spending fuel. It is easy to understand that this idea is used by the Interplanetary Transport Network. The constantly 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 transfer 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 spacecraft there was too little fuel to complete their mission in the traditional way. Unfortunately, we cannot use this technique on a regular basis, since a favorable combination of Lagrange points can be expected for decades, centuries, and even longer.

But, if time is not in a hurry, we can well 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, being already on it. This probe will be able to travel around 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.

The Lagrange points are named after the famous eighteenth-century mathematician who described the concept of the Three-Body Problem in his 1772 work. These points are also called Lagrangian points, as well as libration points.

But what is the Lagrange point from a scientific, not historical point of view?

A Lagrangian point is a point in space where the combined gravity of two fairly large bodies, such as the Earth and the Sun, the Earth and the Moon, is equal to the centrifugal force felt by a much smaller third body. As a result of the interaction of all these bodies, a point of equilibrium is created where the spacecraft can park and conduct its observations.

We know of five such points. Three of them are located along the line that connects the two large objects. If we take the connection of the Earth with the Sun, then the first point L1 lies just between them. The distance from Earth to it is one million miles. From this point, the view of the Sun is always open. Today it is completely captured by the "eyes" of SOHO - the Observatory of the Sun and the Heliosphere, as well as the Observatory of the Climate of Deep Space.

Then there's L2, which is a million miles from Earth, just like its sister. However, in the opposite direction from the Sun. At this point, with the Earth, the Sun, and the Moon behind it, the spacecraft can get a perfect view of deep space.

Today, scientists measure the cosmic background radiation from the Big Bang in this area. It is planned to move the James Webb Space Telescope to this region in 2018.

Another Lagrange point - L3 - is located in the opposite direction from the Earth. It always lies behind the Sun and is hidden for all eternity. By the way, a large number of science fiction told the world about some secret planet X, just located at this point. There was even a Hollywood movie Man from Planet X.

However, it is worth noting that all three points are unstable. They have an unstable balance. In other words, if the spacecraft drifted away from or away from the Earth, then it would inevitably fall either on the Sun or on our planet. That is, he would be in the role of a cart located on the tip of a very steep hill. So the ships will have to constantly make adjustments so that a tragedy does not happen.

It's good that there are more stable points - L4, L5. Their stability is compared to a ball in a big bowl. These points are located along the earth's orbit sixty degrees behind and in front of our house. Thus, two equilateral triangles are formed, in which large masses protrude as vertices, for example, the Earth or the Sun.

Since these points are stable, cosmic dust and asteroids constantly accumulate in their area. Moreover, the asteroids are called Trojan, as they are called by the following names: Agamemnon, Achilles, Hector. They are located between the Sun and Jupiter. According to NASA, there are thousands of such asteroids, including the famous Trojan 2010 TK7.

It is believed that L4, L5 are great for organizing colonies there. Especially due to the fact that they are quite close to the globe.

Attractiveness of Lagrange points

Away from the heat of the sun, ships at Lagrange points L1 and 2 can be sensitive enough to use infrared rays from asteroids. Moreover, in this case, cooling of the case would not be required. These infrared signals can be used as guiding directions, avoiding the path to the Sun. Also, these points have a fairly high throughput. The communication speed is much higher than when using the Ka-band. After all, if the ship is in a heliocentric orbit (around the Sun), then its too great a distance from the Earth will have a bad effect on the data transfer rate.