Also known as the translational pointCelestial mechanicsMedium is restrictiveThree body problemFive special solutions of.A point in space where a small object is under the gravitational force of two large objects. At this point, the small object is basically stationary with respect to the two large objects.The existence of these points was calculated by the Swiss mathematician Euler in 1767LagrangeIn 1772, the derivation proved that there were two left.In 1906, an asteroid moving in the orbit of Jupiter was first discovered (seeTroy group asteroid）OnJupiterandsunlightIs at the Lagrange point.In each system composed of two celestial bodies, there are five Lagrangian points according to inference, but only two are stable, that is, small objects at this point tend to remain at their original positions even if disturbed by external gravity.Each stable point forms an equilateral triangle with the point where the two objects are located.
In 1906, it was first found that small particles moving in the orbit of Jupiterplanet(SeeTroy group asteroid）Under the action of Jupiter and the sun, it is at the Lagrange point.In each system composed of two celestial bodies, there are five Lagrangian points according to inference, but only two are stable, that is, small objects at this point tend to remain at their original positions even if disturbed by external gravity.Each stable point forms an equilateral triangle with the point where the two objects are located.
French mathematician, ergologist and astronomer in the 18th centuryLagrange(Lagrange) In his paper "Three Body Problem" published in 1772, in order to find the general solution of the three body problem, he used a very special example as the result of the problem, that is, if at a certain time, three moving objects are exactly at the three vertices of an equilateral triangle, then given the initial speed, they will always keep the equilateral triangle formation motion.A. D In 1906, astronomers found that asteroid 588 was exactly equidistant from the sun. It moved 60 ° ahead of Jupiter in almost the same orbit. Together, they formed a moving equilateral triangle.The asteroid 617 discovered in the same year also lagged behind Jupiter by about 60 °, forming the second Lagrange equilateral triangle.In the 1980s, astronomers found that Saturn and its large satellites had similar equilateral triangles in their motion system.It is further discovered that there are Lagrange points in various motion systems in nature.
phenomenon
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Lagrange point diagram
L1, L2 and L3 are unstable points on the line between the two celestial bodies.If the test particle is pushed perpendicular to the centerline, a force will push it back to the balance point;But if the test particle drifts towards any star, the gravity of that star will pull it towards itself.However, although they are unstable, appropriate initial disturbances can be selected so that the motion near the corresponding translation point is still periodic or quasi periodic.That is, such initial disturbance is selected to make the original solution of the system degenerate into periodic solution, and the corresponding motion becomes stable. At this time, this stability is called conditional stability.
For L4 and L5, when 0<μ<μ * (where μ * meets μ * (1 - μ *)=1/27), L4 and L5 are linear stable.For each system in the solar system that is treated as a restrictive three body problem, such as the sun wood asteroid, the sun earth moon,..., the corresponding μ meets the condition 0<μ<μ * (μ * meets μ * (1 - μ *)=1/27).For μ *<μ<1/2, it is obviously unstable.
Five Special Solutions
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Calculation formula of five points[1]：
L1
On the line between M1 and M2, and between them.For example, the closer an object revolves around the sun, the shorter its orbital period.But this ignores the influence of the gravitational pull of the earth on its pulling force.If the object is between the earth and the sun, the influence of the earth's gravity will weaken the pull of the sun on the object, thus increasing the orbital period of the object.The closer the object is to the earth, the greater the impact.At L1, the orbital period of the object is exactly equal to the orbital period of the earth.SOHO (NASA website on SOHO project)[2]）That is, it operates around the L1 point of the sun earth system.
L2
It is on the line between two large objects and on the side of the smaller object.
For example, similar effects occur on the other side of the earth.The farther away an object is from the sun, the longer its orbital period is usually.Gravity of the earthThe pulling force reduces the orbital period of the object.At point L2, the orbital period becomes equal to the Earth.
Solar terrestrial Lagrange L2 pointAmong them, point L2 is located on the sun earth line, about 1.5 million kilometers away from the outside of the earth, where satellites can stay for a long time with little fuel consumption. It is an ideal location for detectors and astronomical telescopes to locate and observe the solar system, has important practical application and scientific exploration value in engineering and science, and is a hot spot in international deep space exploration.
L2 is usually used to place space observatories.Because L2 objects can maintain a position away from the sun and the earth, it is easy to protect and calibrate.
Wilkinson Microwave Anisotropy DetectorIt has been running around the L2 point of the sun earth system.James Weber Space TelescopeIt will be placed on L2 point of the sun ground system.In addition, the world's first communication satellite operating at the L2 point of the Earth Moon Lagrange, China's lunar exploration projectChang'e-4Mission Relay Star Magpie Bridge will also run in this orbit.[3]
In addition:Chang'e-2 satelliteAt 16:50:05 on June 9, 2011lunar probeAfter the mission is completed, it will leave the lunar orbit and fly to the second Lagrange point for further exploration. The flight distance is 1.5 million kilometers, which is expected to take 85 days.At 23:27 on August 25, 2011, Beijing time, after 77 days of flight“Chang'e-2”It is the first time in the world to set out from the lunar orbit and enter the sun andGravity of the earthEquilibrium point - the orbit around Lagrange L2 point.
On November 23, 2014, the service module implemented the lunar orbit maneuver control and flew to the Earth Moon L2 point.Since then, on November 27, it entered the Lissajou orbit around the Earth Moon L2 point, which is the first time that a Chinese aircraft has flown to the Earth Moon L2 point (rather than the Earth Sun Lagrange point);Three track maintenance controls were implemented on November 28, December 11 and December 26 respectively[4]。
L3
It is on the line between two large objects and on the side of the larger object.
For example, the third Lagrangian point, L3, is located on the other side of the sun, slightly farther from the sun than the earth.The pull force between the earth and the sun makes the orbital period of the object equal to that of the earth again.
Some science fiction novels and comics often describe an "anti earth" at L3.
L4
Lagrange point
Based on the connection between two celestial bodiesEquilateral triangleOn the third vertex of, and in front of the orbit of the smaller object around the center of mass of the two celestial systems.The reason why this point is stable is that the distance between it and the two objects is equal, and the ratio of their respective gravitations to the two objects is exactly equal to the two objectsMass of objectRatio.Therefore, the resultant force of the two gravitations just points to the center of mass of the system, and the magnitude of the resultant force just provides the centripetal force required for the revolution of the object, so that its rotation period is the same as that of the object with less mass and achieves orbital balance.In this system, two objects and objects on L4 point rotate around the center of mass, and the center of rotation coincides with the center of mass.In fact, the mass of objects at L4 and L5 must be negligible.
L4 and L5 points are sometimes referred to as triangular Lagrange points or Troy points.
L5
On the third vertex of an equilateral triangle with the line between the two celestial bodies as the bottom, and behind the orbit of the smaller celestial body around the larger celestial body.
L4 and L5 are sometimes referred to as "triangular Lagrange points" or "Troy points".
EnceladusThere are two small satellites at L4 and L5 points ofCalypso。Mimas has a moon, Titan, at L4.
Balance
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Strictly speaking, first of all, the Lagrange point is only the stable point in the normal plane of the connection between two stars, but it is unstable in three-dimensional space: consider Lone: If the test particle is moved perpendicular to the centerline, a force will push it back to the balance point (stable balance);However, if the test particle drifts towards any star, the gravity of that star will pull it towards itself (unstable equilibrium).(See balance) L1, L2 and L3 are unstable on this line. If an object is placed on this line, it will leave this point immediately.Therefore, one kind of orbit design is that it is halo orbiting around L2. In this way, our satellite can maintain its orbit with only a few adjustments.
This comparison: if MoneRatio MtwoIf it is greater than 24.96, it is at LfourAnd LfiveThe object is in stable equilibrium: when a test particle deviates from this equilibrium point, the Coriolis force will twist its orbit into a lenticular shape (relative to the rotating coordinates).There are thousands of asteroids in the Sun Jupiter system, commonly known as "Troy asteroids", all of which have such trajectories.Sun Mars, Sun Saturn, Jupiter Ganymede, Saturn Titan and other systems also have similar stars.There is also 2010 TK7 (the first Trojanian asteroid on Earth) in the sun earth system. In the 1950s, it was found that dust mist surrounded LfourAnd Lfive。L of Earth Moon SystemfourAnd LfiveThe point also found that the dust fog was weaker than the sunshine.
The company object of the earth, Clutny, "encircles" the earth in an orbit similar to Troy, but it is not realtrojan moon It basically orbits the sun in an elliptical orbit with a period slightly less than one year. When it approaches the earth, it extracts kinetic energy from the earth's revolution and enters a higher orbit.When Clutny is overtaken by the earth, it will give back this kinetic energy, fall into the low-energy orbit and start the cycle again.
Epimetheus has a similar relationship with Janus, but its mass is similar, so it periodically exchanges orbits.
Another similar configuration is orbital resonance, in which the period of each star is a simple integer ratio due to its interaction.
L of TethysfourAnd LfiveThe point has two small moons, Telesto and Calypso.L of DionefourThe point has a moon Helene.
purpose
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stayCelestial mechanicsLagrange points are five special solutions of the restricted three body problem.For example, when two celestial bodies orbit, there are five positions in space where the third object can be placed (with negligible mass) and kept at the corresponding position of the two celestial bodies.Ideal stateNext, two objects in the same orbit rotate at the same period. The universal gravitation of the two objects provides the centripetal force required at the Lagrange point, making the third object relatively stationary with the first two objects.
Reason shines in space
According to the plan,NASAToHubble Space Telescope(HST) for the 5th repair.After maintenance, it is estimated that it can work for at least another 5 years.HST has not "retired" yet, "successor"James Webb Space Telescope (JWST) had to wait on the ground for several years.
Interestingly, the James Weber Space Telescope will not orbit the Earth as HST does. Its "working place" is set at the "second Lagrange point" of the sun earth system (1.5 million kilometers away from the side of the Earth facing the sun).Lagrange (1736-1813) never expected that his research achievements on the "three body problem" were repeatedly cited in human scientific research and aerospace engineering more than 200 years after their publication.
Some unique stable points on the orbital plane of a binary system, planets and the sun, satellites and planets (or any two celestial bodies orbiting each other due to gravitational traction).For example, there are two Lagrangian points 60 degrees ahead and 60 degrees behind the orbit of Jupiter. If an asteroid is on these two Lagrangian points, it will oscillate near this point, but will not leave these pointsTrojan asteroids (Trojan astroids) is located in these two regions.In fact, any binary system has five Lagrangian points.In addition to the above two points, the other three Lagrangian points are not very stable. Small objects located on other Lagrangian points will leave their positions if slightly disturbed.
The research results of "three body problem" have been used by later generations, and JWST is not the first example.Earlier, the Wilkinson Cosmic Microwave Anisotropy Detection Satellite (WMAP), which was launched in 2001, attracted worldwide attentionCosmic microwave backgroundThe second generation cosmic microwave background detection satellite after the Explorer satellite COBE.What people are curious about is also the positioning of WMAP: it is at the "second Lagrange point" of the sun earth system.
Let's say, what is the "three body problem"?To put it simply, it refers to how the system of "three celestial bodies" such as "Sun Earth Small Mass Object" or "Sun Jupiter Small Mass Object" operates.To be more specific, it is to study such questions: if there are infinitely small mass objects in the "Sun Earth" or "Sun Jupiter" celestial systems, how will these small objects move under the action of universal gravitation?
The simplest type of "three body problem" is "plane circle restricted three body problem".Lagrange solved this problem and got five special solutions: three straight line solutions and two equilateral triangle solutions. Only two equilateral triangle solutions are stable solutions.If a small mass object is located at a certain Lagrangian point, the gravitational force it receives from the sun Jupiter (or sun Earth) is just equal to the centripetal force it needs to rotate with the sun Jupiter (or sun Earth).That is to say, at a certain Lagrangian point, a small mass object can maintain its relative position with the sun Jupiter (or sun Earth) unchanged.
Interestingly, both the "first generation satellite" HST and COBE "revolve" around the earth, and the "second generation satellite" JWST and WMAP set their positions at the "second Lagrange point" of the sun earth system.European Space AgencyThe two satellites "Herschel" and "GAIA" are also optimistic about that "location" and plan to settle there.
In the history of scientific development, there are many funny stories related to the "three body problem".About a hundred years ago, in 1906, German astronomer Max Wolff discovered a strange asteroid.Its orbit is the same as that of Jupiter, but not between the orbit of Mars and that of JupiterAsteroid beltLi.The most wonderful thing is that its windingDaily movementThe period is the same as Jupiter.Seen from the sun, it always rotates 60 ° in front of Jupiter and will not be close to Jupiter.This asteroid is named "Achilles", which is described in Homer's epic "Iliad"Troy WarInGreek Hero。
Astronomer Saliet is sensitive to the fact that the asteroid "Achilles" is probably a special case of French mathematician Lagrange's "three body problem": as long as the small object, the big planet and the sun form an equilateral triangle, the small object and the big planet will always rotate around the sun synchronously, and they will never collide.
Sure enough, astronomers soon found an asteroid 60 degrees behind Jupiter.So far, 700 asteroids have been found at these two Lagrangian points around Jupiter.How wonderful is the prediction of scientific theory!The later discovered asteroids at the Lagrange point were all named after the heroes in the Trojan War.As a result, these hundreds of asteroids have a "collective" title: asteroids in the Trojang Group.This "trojan" is actually the middle and small of ancient Greek mythologyAsiaTroy.
not long ago,French Centre for Space StudiesA new idea was put forward by astronomers of.French scientists proposed to capture some medium-sized "celestial bodies" and "deploy" them to one of the five Lagrangian points of the "Sun Earth" system.After discovering dangerous asteroids to the earth, people can call these "celestial bodies" to intercept dangerous asteroids.
With wonderful theories, beautiful images, and beautiful applications, Lagrange brings us a full range of interests: reason, interest, and ambition.This is our comprehensive aesthetic sense of science.
