Spacecraft at the Lagrangian points of the earth-moon system. Lagrange points and the distance between them. Lagrange point L1. Using the Lagrange point to influence climate Flying at the Lagrange point l1 earth sun
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 solving the so-called restricted 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 any of the other two. In this case, we can assume that two massive bodies revolve around their common center of mass with a constant angular velocity . In the space around them, there are five points where a third body with a negligible mass can remain motionless in the 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 give a solution in 1772 mathematical problem, which implies the existence of these singular points.
All Lagrange points lie in the plane of orbits of massive bodies and are denoted by a capital Latin letter L with a numerical 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 denoted 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. In a coordinate system with the origin at the center of mass of the system and with an axis directed from the center of mass to a less massive body, the coordinates of these points in the first approximation in α are calculated using the following formulas:
Dot L1 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 . In this case, the larger M 2 , the further this point will be located from it.
lunar point L1(in the Earth - Moon system; removed from the center of the Earth by about 315 thousand km) can be an ideal place for building a space manned orbital station, which, located on the path between the Earth and the Moon, would make it easy to get to the Moon with minimal fuel and become a key node in the cargo flow between the Earth and its satellite.
Dot L2 lies on a straight line connecting two bodies with masses M 1 and M 2 (M 1 > M 2), and is located behind the body with a smaller mass. points L1 and L2 are located on the same line and in the limit M 1 ≫ M 2 are symmetrical with respect to M 2 . At the point L2 gravitational forces acting on the body compensate for the action of centrifugal forces in a rotating frame of reference.
Dot L2 in the Sun - Earth system is an ideal place for the construction of orbiting space observatories and telescopes. Because the object at the point L2 is able to maintain its orientation relative to the Sun and the Earth for a long time, it becomes much easier to shield and calibrate it. However, this point is located a little further than the earth's shadow (in the penumbra) [approx. 1], so that solar radiation is not completely blocked. In halo orbits around this point on this moment(2020) Gaia and Spektr-RG devices are located. Previously, such telescopes as Planck and Herschel operated there, in the future several more telescopes are planned to be sent there, including James Webb (in 2021).
Dot L2 in the Earth-Moon system, it can be used to provide satellite communications with objects on the far side of the Moon, and also be a convenient place to place a gas station to ensure cargo flow between the Earth and the Moon
If M 2 is much smaller in mass than M 1 , then the points L1 and L2 are at about the same distance r from the body M 2 equal to the radius of the Hill sphere:
Dot 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 the body with a larger mass. Same as for point L2, at this point the gravitational forces compensate for the centrifugal forces.
Before the beginning space age among science fiction writers, the idea of the existence of opposite side earth orbit at point L 3 another planet similar to it, called " Counter-Earth", which, due to its location, was not available for direct observation. However, in fact, due to the gravitational influence of other planets, the point L 3 in the Sun-Earth system is extremely unstable. So, during the heliocentric conjunctions of the Earth and Venus on opposite sides of the Sun, which happen every 20 months, Venus is only 0.3 a.u. from the point L 3 and thus has a very serious influence on its location relative 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 "Anti-Earth" would still be available for observation from time to time and would certainly be noticed. Another effect that betrays its existence would be its own gravity: the influence of a body with a size of about 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 at this point there are no objects larger than 100 m.
Orbital spacecraft and satellites located near the point L 3, can constantly monitor various forms of activity on the surface of the Sun - in particular, 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 a line connecting both bodies of the system, two equilateral triangles are constructed, 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 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 are related 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 by the same proportion as for two massive bodies. Since the center of mass is also the center of rotation of the system, the resulting force exactly matches that required 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 called triangular(as opposed to collinear).
The dots are also called Trojan: this name comes from Jupiter's Trojan asteroids, which are the most a prime example manifestation of these points. They were named after the heroes of the Trojan War from Homer's Iliad, and the asteroids at the point L 4 get the names of the Greeks, and at the dot L 5- the defenders of Troy; therefore they are now called "Greeks" (or "Achaeans") and "Trojans".
The 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 that attracts it to the body it is approaching increases, and the force of attraction from the other body, on the contrary, decreases. As a result, the object will increasingly move away 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 star systems. The Roche lobes of the components of such systems touch at the point L 1 , therefore, when one of the companion stars fills its Roche lobe in the process of evolution, 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 other bodies also influence the motion (as happens in the Solar System), instead of closed orbits, the object will move in quasi-periodic orbits shaped like Lissajous figures. Despite the instability of such an orbit,
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 object. 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 are measuring the cosmic background radiation in this area, which resulted from big bang. 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, big number 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 were drifting towards 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 the infrared rays coming from asteroids. And in this case case cooling 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.
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. An 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 there is less an asteroid will fall in the Sun, if more - fly away. By the way, given point 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).
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 revolution. 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 must 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 large mass, but this movement, together with the movement of the two remaining bodies of the system, is of such a nature that, relative 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. Last 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 spaceships and satellites to the moon and back.
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 using point L1 of the Earth became a physical project of one Ukrainian student. 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
What they look like and where to look Lagrange points in space: history of discovery, the Earth and 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 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 for more productive and faster development of space.
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 expanding the mathematics of Leonhard Euler. Scientist first discovered three such points, and Lagrange announced the next two.
Lagrange points: What are we talking about?
When you have two massive objects (such as the Sun and the Earth), their gravitational contact is wonderfully balanced in specific 5 areas. In each of them, you can place a satellite that will be held in place with minimal effort.
Most notable is the first Lagrange point L1, balanced between the gravitational attraction of two objects. For example, you can install a satellite above 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 does not have to spend a lot of fuel. It is important to understand that this point exists between all objects.
L2 is in line with ground, but on the other side. Why doesn't the unified 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 lag behind the Earth, as it moves around the star more slowly. But the earth's gravity pushes it and helps it stay in place.
L3 should be looked for on the opposite side of the system. Gravity between objects stabilizes and the craft 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, because any satellite will deviate sooner or later. So without working engines there is nothing to do there.
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 found asteroids on large planets, like Jupiter. These are Trojans caught 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 Earth-Sun and Earth-Moon points.
Sun-Earth L1 is a great place to live for a solar telescope. The device approached the star as close as possible, but does not lose contact with its home planet.
The future James Webb telescope (1.5 million km from us) is planned to be located at point L2.
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 want 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.
