From the side of the first two bodies, it can remain motionless relative to these bodies.
More precisely, Lagrange points represent a special case when 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 either of the other two. In this case, we can assume that two massive bodies are revolving around their common center of mass with a constant angular velocity. In the space around them there are five points at which a third body with 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 provide a solution to 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 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 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 the two bodies of the system, closer to the less massive body; L 2 - outside, behind the less massive body; and L 3 - for the more massive one. 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 to a first approximation in α are calculated using the following formulas:
Dot L 1 lies on the 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 larger M2, the further this point will be located from it.
Lunar point L 1(in the Earth-Moon system; approximately 315 thousand km away from the center of the Earth) could be an ideal place for the construction of a manned space orbital station, which, located on the path between the Earth and the Moon, would allow easy access to the Moon with minimal fuel consumption and to become a key node in the cargo flow between the Earth and its satellite.
Dot 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 the body with a smaller mass. Points L 1 And L 2 are located on the same line and in the limit M 1 ≫ M 2 are symmetrical with respect to M 2. At the point L 2 gravitational forces acting on the body compensate for the action of centrifugal forces in a rotating reference frame.
Dot L 2 in the Sun-Earth system is an ideal place for the construction of orbital space observatories and telescopes. Since the object is at a point L 2 able to maintain its orientation relative to the Sun and Earth for a long time, its shielding and calibration becomes much easier. 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. In halo orbits around this point on this moment(2020) there are devices Gaia and Spektr-RG. Previously, telescopes such as Planck and Herschel operated there; in the future, several more telescopes are planned to be sent there, including James Webb (in 2021).
Dot L 2 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 locate 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 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:
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 L 2, at this point gravitational forces compensate for the action of centrifugal forces.
Before the start of the space age, the idea of existence 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 to direct observations. 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 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 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 “Counter-Earth” would still be available for observation from time to time and would certainly be noticed. Another effect that would reveal its existence would be its own gravity: the influence of a body already on the order of 150 km or more in size on the orbits of other planets would be noticeable. With the advent of the ability to make observations using spacecraft and probes, it was reliably shown that at this point there are no objects larger than 100 m in size.
Orbital spacecraft and satellites located near the point L 3, can constantly monitor various forms activity on the surface of the Sun - in particular, the appearance of new spots or flares - and promptly 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-distance manned flights, for example to Mars or asteroids. In 2010, several options for launching such a satellite were studied.
If, based on a line connecting both bodies of the system, we construct two equilateral triangles, the 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 orbital plane 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 the two bodies at these points are the same, the attractive forces from the two massive bodies are correlated in the same proportion as their masses, and thus the resulting force is directed towards the center of mass of the system ; furthermore, 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 also the center of rotation of the system, the resulting force exactly corresponds to that 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 called triangular(as opposed to collinear).
Also called points 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, with the asteroids at the point L 4 get the names of the Greeks, and at the point L 5- defenders of Troy; that is why they are now called “Greeks” (or “Achaeans”) and “Trojans”.
Distances from the center of mass of the system to these points in coordinate system with the center of coordinates at the center of mass, the systems 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 moves slightly along a straight line connecting two massive bodies, the force attracting 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 move further and further away from its equilibrium position.
This feature of the behavior of bodies in the vicinity of the L 1 point plays an important role in close binary star systems. The Roche lobes of the components of such systems touch at the L1 point, therefore, when one of the companion stars fills its Roche lobe during the evolution process, matter flows from one star to another precisely through the vicinity of the Lagrange point L1.
Despite this, there are stable closed orbits (in a rotating coordinate system) around the collinear libration points, at least in the case of the three-body problem. If the motion is also influenced by other bodies (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,
Whatever goal you set for yourself, whatever mission you plan, one of the biggest obstacles on your way in space will be fuel. Obviously, a certain amount of it is needed in order to leave the Earth. The more cargo needs to be taken out of the atmosphere, the more fuel is needed. But because of this, the rocket becomes even heavier, and it all turns into a vicious circle. This is what prevents us from sending several interplanetary stations to different addresses on one rocket - there simply is not enough space for fuel. However, back in the 80s of the last century, scientists found a loophole - a way to travel around the solar system using almost no fuel. It's called the Interplanetary Transport Network.
Current methods of space flight
Today, moving between objects in the solar system, for example, traveling from Earth to Mars, usually requires a so-called Hohmann ellipse flight. The launch vehicle is launched and then accelerated until it is beyond the orbit of Mars. Near the red planet, the rocket slows down and begins to rotate around its destination. It burns a lot of fuel both for acceleration and braking, but the Hohmann ellipse remains one of the most effective ways moving between two objects in space.
Hohmann Ellipse - Arc I - flight from Earth to Venus. Arc II - flight from Venus to Mars Arc III - return from Mars to Earth.
Gravity maneuvers are also used, which can be even more effective. When performing them, the spacecraft accelerates using the gravitational force of a large celestial body. The increase in speed is very significant almost without the use of fuel. We use these maneuvers every time we send our stations on a long journey from Earth. However, if a ship needs to enter the orbit of a planet after a gravity maneuver, it still has to slow down. You, of course, remember that this requires fuel.
This is exactly why at the end of the last century, some scientists decided to approach the problem from the other side. They treated gravity not as a sling, but as a geographical landscape, and formulated the idea of an interplanetary transport network. The entrance and exit springboards to it were the Lagrange points - five regions near celestial bodies where gravity and rotational forces come into balance. They exist in any system in which one body rotates around another, and without pretense of originality, they are numbered from L1 to L5.
If we place a spaceship at the Lagrange point, it will hang there indefinitely because gravity does not pull it in one direction more than in another. However, not all these points are created equal, figuratively speaking. Some of them are stable - if you move a little to the side while inside, gravity will return you to your place - like a ball at the bottom of a mountain valley. Other Lagrange points are unstable - if you move a little, you will start to be carried away from there. Objects located here are like a ball on top of a hill - it will stay there if it is well placed or if it is held there, but even a slight breeze is enough for it to pick up speed and roll down.
Hills and valleys of the cosmic landscape
Spaceships flying around the solar system take all these “hills” and “valleys” into account during flight and during the route planning stage. However, the interplanetary transport network forces them to work for the benefit of society. As you already know, every stable orbit has five Lagrange points. This is the Earth-Moon system, and the Sun-Earth system, and the systems of all the satellites of Saturn with Saturn itself... You can continue yourself, after all, in the Solar system a lot of things revolve around something.
Lagrange points are everywhere, even though they constantly change their specific location in space. They always follow the orbit of the smaller object in the rotation system, and this creates an ever-changing landscape of gravitational hills and valleys. In other words, the distribution of gravitational forces in the solar system changes over time. Sometimes attraction in certain spatial coordinates is directed towards the Sun, at another point in time - towards some planet, and it also happens that the Lagrange point passes through them, and in this place equilibrium reigns when no one is pulling anyone anywhere .
The hills and valleys metaphor helps us visualize this abstract idea better, so we'll use it a few more times. Sometimes in space it happens that one hill passes next to another hill or another valley. They may even overlap each other. And at this very moment, space travel becomes especially effective. For example, if your gravitational hill overlaps a valley, you can "roll" into it. If your hill overlaps another hill, you can jump from peak to peak.
How to use the Interplanetary Transport Network?
When the Lagrange points of different orbits move closer to each other, it takes almost no effort to move from one to the other. This means that if you are not in a hurry and are ready to wait for their approach, you can jump from orbit to orbit, for example, along the Earth-Mars-Jupiter route and beyond, almost without wasting fuel. It is easy to understand that this is the idea that the Interplanetary Transport Network uses. The constantly changing network of Lagrange points is like a winding road, allowing you to move between orbits with minimal fuel consumption.
IN 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 attempt to save the Japanese lunar station in 1991, when the spacecraft had too little fuel to complete its mission in the traditional way. Unfortunately, we cannot use this technique on a regular basis, since a favorable alignment of Lagrange points can be expected for decades, centuries, and even longer.
But, if time is not in a hurry, we can easily afford to send a probe into space, which will calmly wait for the necessary combinations, and collect information the rest of the time. Having waited, he will jump to another orbit and carry out observations while already in it. This probe will be able to travel throughout the solar system for an unlimited amount of time, recording everything that happens in its vicinity and adding to the scientific knowledge of human civilization. It is clear that this will be fundamentally different from the way we explore space now, but this method looks promising, including for future long-term missions.
Have any experiments been conducted on placing spacecraft at the Lagrange points of the Earth-Moon system?
Despite the fact that humanity has known about the so-called libration points existing in space and their amazing properties for quite a long time, they began to be used for practical purposes only in the 22nd year of the space age. But first, let's briefly talk about the miracle points themselves.
They were first theoretically discovered by Lagrange (whose name they now bear), as a consequence of solving the so-called three-body problem. The scientist was able to determine where in space there may be points at which the resultant of all external forces goes to zero.
Points are divided into stable and unstable. Stable ones are usually designated L 4 and L 5 . They are located in the same plane with the main two celestial bodies (in this case, the Earth and the Moon), forming with them two equilateral triangles, for which they are often called triangular. The spacecraft can remain at triangular points for as long as desired. Even if it deviates to the side, the acting forces will still return it to the equilibrium position. The spacecraft seems to fall into a gravitational funnel, like a billiard ball into a pocket.
However, as we said, there are also unstable libration points. In them, the spacecraft, on the contrary, is located as if on a mountain, being stable only at its very top. Any external influence deflects it to the side. Getting to an unstable Lagrange point is extremely difficult - it requires ultra-precise navigation. Therefore, the device has to move only close to the point itself in the so-called “halo orbit”, from time to time consuming fuel to maintain it, although very little.
There are three unstable points in the Earth-Moon system. Often they are also called rectilinear, since they are located on the same line. One of them (L 1) is located between the Earth and the Moon, 58 thousand km from the latter. The second (L 2) is located so that it is never visible from the Earth - it hides behind the Moon, 65 thousand km from it. The last point (L 3), on the contrary, is never visible from the Moon, since it is blocked by the Earth, from which it is approximately 380 thousand km away.
Although it is more profitable to be in stable points (there is no need to consume fuel), spacecraft have so far become acquainted only with unstable ones, or rather, only with one of them, and even then related to the Sun-Earth system. It is located inside this system, 1.5 million km from our planet and, like the point between the Earth and the Moon, is designated L 1. When viewed from Earth, it is projected directly onto the Sun and can serve as an ideal point for tracking it.
This opportunity was first used by the American ISEE-3, launched on August 12, 1978. From November 1978 to June 1982, it was in a "halo orbit" around the Li point, studying the characteristics of the solar wind. At the end of this period, it was he, but already renamed ICE, who happened to become the first comet researcher in history. To do this, the device left the libration point and, having performed several gravitational maneuvers near the Moon, in 1985 it flew near the comet Giacobini-Zinner. The next year, he also explored Halley's comet, although only at distant approaches.
The next visitor to the L 1 point of the Sun-Earth system was the European solar observatory SOHO, launched on December 2, 1995 and, unfortunately, recently lost due to a control error. During her work, quite a bit of important scientific information was obtained and many interesting discoveries were made.
Finally, the latest apparatus launched to date in the vicinity of L 1 was the American ACE apparatus, designed to study cosmic rays and stellar wind. He launched from Earth on August 25 last year and is currently successfully conducting his research.
What's next? Are there any new projects related to libration points? Of course they do exist. Thus, in the USA, the proposal of Vice President A. Gore was accepted for a new launch in the direction of point L 1 of the Sun-Earth system of the scientific and educational apparatus "Triana", already nicknamed the "Gore Camera".
Unlike his predecessors, he will monitor not the Sun, but the Earth. Our planet from this point is always visible in full phase and is therefore very convenient for observations. It is expected that the images received by the Gora Camera will be uploaded to the Internet almost in real time, and access to them will be open to everyone.
There is also a Russian “libration” project. This is the Relikt-2 device, designed to collect information about the cosmic microwave background radiation. If funding is found for this project, then the L 2 libration point in the Earth-Moon system awaits it, that is, the one that is hidden behind the Moon.
In a system of rotation of two cosmic bodies of a certain mass, there are points in space at which by placing any object of small mass, you can fix it in a stationary position relative to these two bodies of rotation. These points are called Lagrange points. The article will discuss how they are used by humans.
What are Lagrange points?
To understand this issue, one should turn to the solution to the problem of three rotating bodies, two of which have such a mass that the mass of the third body is negligible in comparison with them. In this case, it is possible to find positions in space in which the gravitational fields of both massive bodies will compensate for the centripetal force of the entire rotating system. These positions will be Lagrange points. By placing a body of low mass in them, you can observe how its distances to each of the two massive bodies do not change for any length of time. Here we can draw an analogy with geostationary orbit, in which the satellite is always located above one point on the earth’s surface.
It is necessary to clarify that a body that is located at the Lagrange point (also called a free point or point L), relative to an external observer, moves around each of the two bodies with a large mass, but this movement, together with the movement of the two remaining bodies of the system, has the following character , that relative to each of them the third body is at rest.
How many of these points are there and where are they located?
For a system of rotating two bodies with absolutely any mass, there are only five points L, which are usually designated L1, L2, L3, L4 and L5. All these points are located in the plane of rotation of the bodies in question. The first three points are on the line connecting the centers of mass of the two bodies in such a way that L1 is located between the bodies, and L2 and L3 are behind each of the bodies. Points L4 and L5 are located so that if you connect each of them with the centers of mass of two bodies of the system, you will get two identical triangles in space. The figure below shows all the Earth-Sun Lagrange points.
The blue and red arrows in the figure show the direction of action of the resulting force when approaching the corresponding free point. From the figure it can be seen that the areas of points L4 and L5 are much larger than the areas of points L1, L2 and L3.
Historical reference
The existence of free points in a system of three rotating bodies was first proven by an Italian-French mathematician in 1772. To do this, the scientist had to introduce some hypotheses and develop his own mechanics, different from Newton’s mechanics.
Lagrange calculated the L points, which were named after him, for ideal circular orbits of rotation. In reality, the orbits are elliptical. The latter fact leads to the fact that Lagrange points no longer exist, but there are regions in which a third body of small mass performs a circular motion similar to the motion of each of the two massive bodies.
Free point L1
The existence of the Lagrange point L1 is easy to prove using the following reasoning: take the Sun and the Earth as an example, according to Kepler’s third law, the closer a body is to its star, the shorter its rotation period around this star (the square of the body’s rotation period is directly proportional to the cube of the average distance from body to the star). This means that any body that is located between the Earth and the Sun will orbit the star faster than our planet.
However, it does not take into account the influence of gravity of the second body, that is, the Earth. If we take this fact into account, we can assume that the closer the third low-mass body is to the Earth, the stronger will be the counteraction of the Earth's gravity to the solar one. As a result, there will be a point where Earth’s gravity will slow down the speed of rotation of the third body around the Sun in such a way that the periods of rotation of the planet and the body will be equal. This will be the free point L1. The distance to the Lagrange point L1 from the Earth is equal to 1/100 of the radius of the planet’s orbit around the star and is 1.5 million km.
How is the L1 area used? This is an ideal place to observe solar radiation as there are never any solar eclipses. Currently, there are several satellites located in the L1 region that study the solar wind. One of them is the European artificial satellite SOHO.
As for this Earth-Moon Lagrange point, it is located approximately 60,000 km from the Moon, and is used as a “transshipment” point during missions spaceships and satellites to the Moon and back.
Free point L2
Arguing similarly previous case, we can conclude that in a system of two bodies of rotation, outside the orbit of a body with a smaller mass, there should be a region where the drop in centrifugal force is compensated by the gravity of this body, which leads to equalization of the periods of rotation of the body with a smaller mass and the third body around a body with a larger mass. This area is a free point L2.
If we consider the Sun-Earth system, then to this Lagrange point the distance from the planet will be exactly the same as to point L1, that is, 1.5 million km, only L2 is located behind the Earth and further from the Sun. Since in the L2 region there is no influence of solar radiation due to the earth's protection, it is used to observe the Universe, placing various satellites and telescopes here.
In the Earth-Moon system, point L2 is located behind the natural satellite of the Earth at a distance of 60,000 km from it. Lunar L2 contains satellites that are used to observe the far side of the Moon.
Free points L3, L4 and L5
Point L3 in the Sun-Earth system is located behind the star, so it cannot be observed from Earth. The point is not used in any way, since it is unstable due to the influence of gravity of other planets, for example, Venus.
Points L4 and L5 are the most stable Lagrange regions, so there are asteroids or cosmic dust near almost every planet. For example, only cosmic dust exists at these Lagrange points of the Moon, while Trojan asteroids are located at L4 and L5 of Jupiter.
Other uses of free points
In addition to installing satellites and observing space, the Lagrange points of Earth and other planets can also be used for space travel. It follows from the theory that movements through the Lagrange points of different planets are energetically favorable and require little energy expenditure.
Another interesting example of using the L1 point of the Earth was the physics project of one Ukrainian schoolchild. He proposed placing a cloud of asteroid dust in this area, which would protect the Earth from the destructive solar wind. Thus, the point can be used to influence the climate of the entire blue planet.
What are these “points”, why are they attractive in space projects and is there any practice of using them? The editorial board of the Planet Queen portal addressed these questions to Doctor of Technical Sciences Yuri Petrovich Ulybyshev.
The interview is conducted by Oleg Nikolaevich Volkov, deputy head of the “Great Beginning” project.
Volkov O.N.: Guest of the Internet portal “Planet Korolev” is the Deputy Head of the Scientific and Technical Center of the Energia Rocket and Space Corporation, Head of the Space Ballistics Department, Doctor of Technical Sciences Yuri Petrovich Ulybyshev. Yuri Petrovich, good afternoon!
.: Good afternoon.
V.: The existence of manned systems in low-Earth orbit is not a novelty. This is a common, familiar thing. IN Lately The international space community is showing interest in other space projects in which it is planned to place space complexes, including manned ones, at the so-called Lagrange points. Among them are a project for visited space stations, a project for stations placed to search for dangerous asteroids and monitor the Moon.
What are Lagrange points? What is their essence from the point of view of celestial mechanics? What is the history of theoretical research on this issue? What are the main results of the research?
U.: In our solar system there is a large number of natural effects associated with the movement of the Earth, Moon, and planets. These include the so-called Lagrange points. IN scientific literature they are more often even called libration points. To explain the physical essence of this phenomenon, let us first consider simple system. There is an Earth, and the Moon flies around it in a circular orbit. There is nothing else in nature. This is the so-called limited three-body problem. And in this problem we will consider the spacecraft and its possible movement.
The very first thing that comes to mind is: what will happen if the spacecraft is located on the line connecting the Earth and the Moon. If we move along this line, then we have two gravitational accelerations: the attraction of the Earth, the attraction of the Moon, and plus there is centripetal acceleration due to the fact that this line is constantly rotating. It is obvious that at some point all these three accelerations, due to the fact that they are differently directed and lie on the same line, can become zero, i.e. this will be the balance point. This point is called the Lagrange point, or libration point. In fact, there are five such points: three of them are on the rotating line connecting the Earth and the Moon, they are called collinear libration points. The first one, which we have discussed, is designated L 1, the second is behind the Moon- L 2, and the third collinear point- L 3 is located on the opposite side of the Earth in relation to the Moon. Those. on this line, but in the opposite direction. These are the first three points.
There are two more points that are located on both sides outside this line. They are called triangular libration points. All these points are shown in this figure (Fig. 1). This is such an idealized picture.
Fig.1.
Now, if we place a spacecraft at any of these points, then within the framework of such a simple system it will always remain there. If we deviate a little from these points, then periodic orbits may exist in their vicinity, they are also called halo orbits (see Fig. 2), and the spacecraft will be able to move around this point in such peculiar orbits. If we talk about libration points L 1, L 2 systems Earth - Moon, then the period of movement along these orbits will be about 12 - 14 days, and they can be chosen in completely different ways.
Fig.2.
In fact, if we go back to real life and consider this problem in its exact formulation, then everything will turn out to be much more complicated. Those. a spacecraft cannot remain in such an orbit for a very long time, more than, say, one period, and cannot remain in it, due to the fact that:
Firstly, the Moon’s orbit around the Earth is not circular – it is slightly elliptical;
In addition, the spacecraft will be affected by the gravity of the Sun and the pressure of sunlight.
As a result, the spacecraft will not be able to remain in such an orbit. Therefore, from the point of view of implementing space flight in such orbits, it is necessary to launch the spacecraft into the appropriate halo orbit and then periodically carry out maneuvers to maintain it.
By the standards of interplanetary flights, fuel costs for maintaining such orbits are quite small, no more than 50 - 80 m/sec per year. For comparison, I can say that maintaining the orbit of a geostationary satellite per year is also 50 m/sec. There we keep the geostationary satellite near a fixed point - this task is much simpler. Here we must keep the spacecraft in the vicinity of such a halo orbit. In principle, this task is practically feasible. Moreover, it can be implemented using low-thrust engines, and each maneuver is a fraction of a meter or a unit of m/sec. This suggests the possibility of using orbits in the vicinity of these points for space flights, including manned ones.
Now, from the point of view, why are they beneficial, and why are they interesting, specifically, for practical astronautics?
If you all remember, the American project " APOLLO ", which used a lunar orbit from which the vehicle descended, landed on the surface of the Moon, after some time returned to the lunar orbit and then flew towards the Earth. Circumlunar orbits are of some interest, but they are not always convenient for manned astronautics. We may have various emergency situations, in addition, it is natural to want to study the Moon not only in the vicinity of a certain area, but in general to study the entire Moon. As a result, it turns out that the use of lunar orbits is associated with a number of limitations. Restrictions are imposed on launch dates and on return dates from lunar orbit. The parameters of lunar orbits may depend on the available energy. For example, the polar regions may be inaccessible. But probably the most important argument in favor of space stations in the vicinity of libration points is that:
First, we can launch from Earth at any time;
If the station is at the libration point, and the astronauts must fly to the Moon, they can fly from the libration point, or rather from the halo orbit, to any point on the surface of the Moon;
Now that the crew has arrived: from the point of view of manned astronautics, it is very important to ensure the possibility of a quick return of the crew in the event of any emergency situations, illness of crew members, etc. If we are talking about lunar orbit, we may need to wait, say, 2 weeks for the launch time, but here we can launch at any time - from the Moon to the station at the libration point and then to the Earth, or, in principle, directly to the Earth. Such advantages are quite clearly visible.
Available options: L1 or L2. There are certain differences. As you know, the Moon always faces us with the same side, i.e. The period of its own rotation is equal to the period of its movement around the Earth. Eventually, back side The Moon is never visible from Earth. In this case, you can choose a halo orbit such that it will always be in line of sight with the Earth and have the opportunity to carry out communications, observations and some other experiments related to the far side of the Moon. Thus, space stations located at either the L1 or L2 point may have certain advantages for manned spaceflight. In addition, it is interesting that between the halo orbits of points L1 or L2 it is possible to carry out a so-called low-energy flight, literally 10 m/sec, and we will fly from one halo orbit to another.
V.: Yuri Petrovich, I have a question: point L1 is located on the line between the Moon and the Earth, and, as I understand it, from the point of view of ensuring communication between the space station and the Earth, it is more convenient. You said that L2, the point that is located behind the Moon, is also of interest for practical astronautics. How to ensure communication with the Earth if the station is located at the L2 point?
U.: Any station, being in orbit in the vicinity of the L1 point, has the possibility of continuous communication with the Earth, any halo orbit. For point L2 it is somewhat more complicated. This is due to the fact that space station when moving in a halo orbit, it may appear in relation to the Earth, as it were, in the shadow of the Moon, and communication is then impossible. But it is possible to build a halo orbit that will always be able to communicate with the Earth. This is a specially chosen orbit.
Q: Is it easy to do?
U.: Yes, it can be done, and since nothing can be done for free, it will require slightly higher fuel consumption. Let's say, instead of 50 m/sec it will be 100 m/sec. This is probably not the most critical question.
V.: One more clarifying question. You said that it is energetically easy to fly from point L1 to point L2, and back. Do I understand correctly that it makes no sense to create two stations in the area of the Moon, but it is enough to have one station that energetically easily moves to another point?
U.: Yes, by the way, our partners in the international space station offer one of the options for discussing the development of the ISS project in the form of a space station with the possibility of flying from point L1 to point L2, and back. This is quite feasible and foreseeable in terms of flight time (say, 2 weeks) and can be used for manned astronautics.
I also wanted to say that in practice, flights in halo orbits have currently been implemented by the Americans according to the project ARTEMIS . This is about 2-3 years ago. There, two spacecraft flew in the vicinity of points L1 and L2 maintaining the corresponding orbits. One vehicle flew from point L2 to point L1. All this technology has been implemented in practice. Of course, I wanted us to do it.
V.: Well, we still have everything ahead. Yuri Petrovich, next question. As I understand from your reasoning, any cosmic system consisting of two planets has Lagrange points, or libration points. There are such points for the Sun-Earth system, and what is the attractiveness of these points?
U.: Yes, of course, absolutely correct. There are also libration points in the Earth-Sun system. There are also five of them. In contrast to the cislunar libration points, flight at those points can be attractive for completely different tasks. Specifically speaking, the points L1 and L2 are of greatest interest. Those. point L1 in the direction from the Earth to the Sun, and point L2 in the opposite direction on the line connecting the Earth and the Sun.
So, the first flight to point L1 in the Sun-Earth system was carried out in 1978. Since then, several space missions have been carried out. The main theme of such projects was related to observing the Sun: the solar wind, solar activity, among other things. There are systems that use warnings about some active processes on the Sun that affect the Earth: our climate, people’s well-being, etc. This is what point L1 is about. It is primarily of interest to humanity for the possibility of observing the Sun, its activity and the processes that take place on the Sun.
Now point L2. Point L2 is also interesting, primarily for astrophysics. And this is due to the fact that a spacecraft located in the vicinity of this point can use, for example, a radio telescope, which will be shielded from radiation from the Sun. It will be directed oppositely from the Earth and the Sun and may allow for more purely astrophysical observations. They are not noisy from the Sun or any reflected radiation from the Earth. And it’s also interesting, because... We move around the Sun, making a full revolution in 365 days, then with such a radio telescope we can view any direction of the universe. There are also such projects. Right now at our Physics Institute Russian Academy Science is developing such a project “Millimetron”. At this point, too, a number of missions were implemented, and spacecraft are flying.
Q: Yuri Petrovich, from the point of view of searching for dangerous asteroids that can threaten the Earth, at what point should spacecraft be placed so that they monitor dangerous asteroids?
U.: Actually, it seems to me that there is no such direct, obvious answer to this question. Why? Because moving asteroids in relation to the solar system seem to be grouped into a number of families, they have completely different orbits and, in my opinion, it is possible to place a device for one type of asteroid at the circumlunar point. You can also look at what concerns the libration points of the Sun-Earth system. But it seems to me difficult to give such an obvious, direct answer: “such and such a point in such and such a system.” But, in principle, libration points could be attractive for protecting the Earth.
V.: I understand correctly, the solar system has many more interesting places, not only the Earth - the Moon, the Earth - the Sun. What other interesting places? solar system can it be used in space projects?
U.: The fact is that in the solar system in the form in which it exists, in addition to the effect associated with libration points, there are a number of such effects associated with the mutual motion of bodies in the solar system: the Earth, the planets, etc. d. Here in Russia, unfortunately, I don’t know any work on this topic, but, first of all, Americans and Europeans have discovered that there are so-called low-energy flights in the solar system (moreover, these studies are quite complex in mathematical in terms of operation, and in terms of computation - they require large computing supercomputers).
Here, for example, we return to point L1 of the Earth - Moon system. In relation to this point, it is possible to construct (this is attractive for automatic vehicles) flights throughout the solar system, giving small, by the standards of interplanetary flights, impulses of the order of several hundred m/sec. And then this spacecraft will begin to move slowly. In this case, it is possible to construct a trajectory in such a way that it will bypass a number of planets.
Unlike direct interplanetary flights, this will be a long process. Therefore, it is not very suitable for manned space flight. And for automatic devices it can be very attractive.
Here in the picture (Fig. 3) an illustration of these flights is shown. The trajectories seem to hook into each other. Transition from halo orbit from L1 to L2. He st O a little bit is enough. It's the same there. We seem to be gliding along this tunnel, and at the point of engagement or close to engagement with another tunnel, we give a small maneuver and fly over, go to another planet. In general, a very interesting direction. It's called " Superhighway "(at least that's the term Americans use).
Fig.3.
(drawing from foreign publications)
Practical implementation was partially done by the Americans as part of the project GENESIS . Now they are also working in this direction. It seems to me that this is one of the most promising areas in the development of astronautics. Because after all, with those engines, “propulsors” that we currently have, I mean high-thrust engines and electric jet engines (which still have very little thrust and require a lot of energy), we will make progress in terms of solar development system or further study is very difficult. But such long-term or even ten-year flight problems can be very interesting for research. Just like Voyager. He's been flying since 1978 or 1982, I think ( since 1977 - ed.), has now gone beyond the solar system. This direction is very difficult. Firstly, it is difficult in mathematical terms. In addition, here the analysis and calculations on the mechanics of flights require high computer resources, i.e. It is doubtful to calculate this on a personal computer; you need to use supercomputers.
Q: Yuri Petrovich, can the system of low-energy transitions be used to organize a space solar patrol - a permanent system for monitoring the solar system with the existing fuel restrictions that we have?
U.: Even between the Earth and the Moon, and also, for example, between the Earth and Mars, the Earth and Venus, there are so-called quasi-periodic trajectories. Just as we analyzed the halo orbit, which in an ideal problem exists without disturbance, but when we impose real disturbances, we are forced to adjust the orbit in some way. These quasi-periodic orbits also require small ones, by the standards of interplanetary flights, when the characteristic velocities are hundreds of m/sec. From the point of view of a space patrol for observing asteroids, they could be attractive. The only negative is that they are poorly suited for current manned space flights due to the long flight duration. And from the point of view of energy, and even with the engines that we now have in our century, we can make quite interesting projects.
Q: Do I understand correctly, you assume the libration points of the Earth-Moon system are for manned objects, and the points you spoke about earlier are for automatic machines?
U.: I would also like to add one point, a space station in L1 or L2 can be used to launch small spacecraft (the Americans call this approach “ Gate Way " - "Bridge to the Universe"). The device can, using low-energy flights, somehow periodically move around the Earth at very large distances, or fly to other planets or even fly around several planets.
V.: If you dream a little, then in the future the Moon will be a source of space fuel, and lunar fuel will flow to the libration point of the Earth-Moon system, then you can refuel spacecraft with space fuel and send space patrols throughout the solar system.
Yuri Petrovich, you talked about interesting phenomena. They were examined by the American side ( NASA), and in our country they are working on these projects?
U.: As far as I know, they are probably not involved in projects related to the libration points of the Earth-Moon system. They are working on projects related to the libration points of the Sun-Earth system. We have extensive experience in this direction; the Institute of Applied Mathematics of the Russian Academy of Sciences named after Keldysh, the Institute of Space Research, and some universities in Russia are trying to deal with similar problems. But such a systematic approach, a large program, because the program must begin with the training of personnel, and personnel with very highly qualified, No. In traditional courses on space ballistics and celestial mechanics, the mechanics of the movement of spacecraft in the vicinity of libration points and low-energy flights are practically absent.
I must point out that in times Soviet Union They were more or less actively involved in similar programs, and specialists were, as I already mentioned, at the Institute of Applied Mathematics, IKI, and Lebedev Physical Institute. Now many of them are at this age... And a large number of young people who would deal with these problems are very weakly visible.
I did not mention the Americans in the sense of praising them. The fact is that in the USA very large departments deal with these problems. First of all, in the laboratory JPL NASA a large team is working, and they have probably implemented the majority of American interplanetary space projects. In many American universities, in other centers, in NASA , there are a large number of well-trained specialists with good computer equipment. They are addressing this issue, in this direction, on a very broad front.
In our country, unfortunately, it is somehow crumpled. If such a program were to appear in Russia and would be of great interest overall, then the deployment of this work could take quite a long time, starting with personnel training and ending with research, calculations, and development of appropriate spacecraft.
Q: Yuri Petrovich, what universities train specialists in celestial mechanics in our country?
U.: As far as I know, at Moscow State University, at St. Petersburg University there is a department of celestial mechanics. There are such specialists there. How many there are, I find it difficult to answer.
V.: Because in order to begin to implement the practical side of the issue, you must first become a deep specialist, and for this you need to have the appropriate specialty.
U.: And have a very good mathematical background.
V.: Okay. Can you now provide a list of references that would help those people who currently do not have special mathematical training?
U.: In Russian, as far as I know, there is one monograph by Markeev dedicated to libration points. If my memory serves me correctly, it is called “Libration Points in Celestial Mechanics and Cosmodynamics.” It came out around 1978. There is a reference book edited by Duboshin “Handbook of Celestial Mechanics and Astrodynamics.” It went through 2 editions. As far as I remember, it also contains such questions. The rest can be gleaned, firstly, on the website of the Institute of Applied Mathematics there is an electronic library and its own preprints (separately published articles) in this area. They print freely on the Internet. Using the search engine, you can find relevant preprints and view them. There is a lot of material available on the Internet in English.
V.: Thank you for the fascinating story. I hope this topic will be of interest to our Internet resource users. Thank you very much!
