Perturbation refers to the movement of one celestial body around anotherTwo body problemDue to other factorscelestial bodiesThe deviation caused by the attraction of or other factors on the orbit is very small compared with the gravity of the centroid, so it is called perturbation.Under perturbation, the celestial bodycoordinate、speedorOrbital elementsThe change component is called perturbation term.
Perturbation theoreticdevelopmentIt has a history of more than 200 years.Euler、Lagrange、Gaussian、PoissonandLaplaceMany famous scholars have made many contributions to its developmentPerturbation methodThere are no fewer than 100 kinds.To sum up, it can be roughly divided into three categories: coordinatesPerturbation method, instantaneous ellipse method andCanonical transformation。Some methods can not be clearly listed in which category, for example, the famous Hansen method has the characteristics of both one and two categories.
Coordinate method
Study the coordinates of celestial bodies in real orbits andIntermediate trackThe difference between the coordinates of is called coordinate perturbation.In the classical method, the coordinate perturbation is often expressed as a small parameter (such as the mass of the perturbed planet)power seriesAnd then calculate item by item.Due to the development of computing technology, pickup in the approximate solution of differential equationsIterative methodIt is gradually replacing the original small parameter power series expansion method.Its main advantage is unifiediterationProcess, so that the calculation process can be highly automated.According to different coordinate systems, coordinate perturbation can be divided into the following methods.
(1) Cartesian coordinates
This is Enke's research in 1858cometIt discusses the coordinate perturbation in theRectangular coordinate systemExpression in, often used for calculationShort period cometandLunar rocketTrack.The advantages of this method are: perturbationequationThe derivation is simple, symmetrical in form, and coordinates can be obtained directly, which is convenient for calculating the ephemeris of celestial bodies.Its disadvantages are: it is difficult to show the geometric characteristics and mechanical meaning of perturbation in rectangular coordinates;As the time span grows, the three perturbations of the direct coordinates tend to increase at the same time, so that the equations they obey cannot be taken asLinearizationOtherwise, the zero point will be replaced many times.
(2) Spherical coordinate
Natural celestial bodies generally move around a main celestial body, such as planetsSolar motion, the satellite is circlingplanetExercise.Therefore, spherical coordinates orpolar coordinatesThe perturbation of has obvious geometric significance.Clello andLaplaceIn studying the movement of comets andMegaplanetsSpherical coordinates were first proposed in motion theoryPerturbation method。Later, Newcomb improved the Laplace method, especially inPerturbation functionThe operator operation is used to make the expansion process not only have simple mathematical expression, but also have regular processing, which is convenient for later calculation on the computer.Newcomb successfully used this method to studyMercury, Venus, Earth and MarsInner planetas well asUranus、NeptuneThe calendar of inner planets based on the motion ofAstronomical almanacThe foundation of.Hill proposed aTrue proximate angleSpherical coordinates of argumentsPerturbation methodIt was successfully used to calculate the perturbation of Ceres, the first asteroid.
(3) Other coordinates
In 1963, Mussen proposed another method to calculate coordinate perturbation, which is used to calculateCelestial coordinatesstayDiametral, speed andangular momentumPerturbation in three directions.Although such decomposition is not orthogonal, it has many advantages, such as obvious mechanical significance, easy derivation, direct integration, operator operation, unified and compact form of perturbation equations of all orders, and easy calculation automation. It is now used to establish newTheory of motion of major planets。
In the study of various coordinate perturbations, ellipses are almost always taken asIntermediate track。Hill is studyingLunar motion theoryThe so-called double average orbit is used as the intermediate orbit, which is a kind ofPeriodic orbit, it avoids the moonPerigeeHourPrecessionThe difficulties brought by fast.Jildang once proposed to use the rotating ellipse as the intermediate orbit in order to eliminate the long-term term in the coordinate perturbation, and expressed the perturbation asTrue proximate angleThe trigonometric series of.His theory once aroused widespread concern, but later research proved that this method is not convergent.
Instantaneous ellipse method
This is based onOrbital elementsAs a basic variablePerturbation method。If the planet is attracted only by the sun, asKepler's lawAs described, it will move along a fixed ellipse, and the six orbital elements that determine the elliptical movement should be constants.If the influence of other factors is taken into account, the planet will deviate from the original ellipse, and the six orbital elements will no longer be constants. They will follow theConstant variation methodThe derived rules change.In this case, a family of ellipses can be obtained, which are tangent to the real orbit one by one. At the tangent point, they not only have the same coordinates, but also have the same speed;But the accelerations are different from each other. One is the real acceleration, and the other is the elliptical acceleration. The difference between the two isPerturbation forcePerturbation acceleration caused by.Due to this perturbation acceleration, the celestial body will leave this ellipse and go to a nearby instantaneous ellipse at the next moment;On the contrary, once the perturbation disappears, the celestial body willVanishing pointThe instantaneous ellipse of is always moving.Celestial bodies insolar radiationThe motion under pressure perturbation is exactly like this: whenRadiation pressureWhen it works, the instantaneous ellipse of the celestial body changes constantly;However, when the celestial body enters a shadow area that cannot be reached by sunlight, the radiation pressure disappears, and the celestial body moves along the instantaneous ellipse of the shadow entry point until it runs out of the shadow.
The real orbit of the celestial body is the instantaneous ellipse familyEnvelope。Compared with coordinate perturbation, ellipseOrbital elementsThe change of is generally much slower, so it is easy to handle.The instantaneous ellipse method was firstEulerIt was proposed in the middle of the 18th century when studying the mutual perturbation between Jupiter and Saturn, and was later improved by Lagrange.He basedConstant variation method, usingLagrange bracketThe perturbation describing the changes of the elements of the elliptical orbit is strictly derivedequation──Lagrange equation。This method is widely used, especially by Le WeierMegaplanetsThe movement of.
Canonical transformation method
This is a method based on analytical mechanics.Its basic idea is to carry out a series of appropriate regular transformations on variables in order to reduceEquation of motionThe order of, so that the new equation has a simpler form, for example, to get a description of constant speed linear motion orSimple harmonic vibrationSo that the problem can be solved.In the 19th century, Deloitte founded the famous Deloitte from this point of viewLunar motion theory。He first put the moon'sPerturbation functionExpand into more than 400 triangular terms, and then carry out a series of regular transformations, so that each transformation can eliminate one of them.He spent almost 20 years making thousands of transformations and found three suitable onesangular velocity, the moon'sOrbital elementsAre expressed in timeTrigonometric polynomial, without any long-term items.Drone's job isCelestial mechanicsInTransformation theoryLaid the foundation.This method is unified by a series of formsCyclic processTherefore, it is very convenient to use an electronic computer for calculation.
The reason why Deloney has to carry out so many transformations is to give strict mathematical treatment to every item in the perturbation function.This is unnecessary in practice, and some higher-order terms can be omitted.Guided by this idea, Zeppel established the Zeppel transformation at the beginning of the twentieth century.He firstPerturbation functionThe angular variables in are queued according to their changing speed, and then the appropriate transformation is found within a certain precision range, so as to eliminate all terms containing fast variables at once, and obtain a group of averagedequationAnd then repeat the similar process for the new equation until all angular variables are eliminated.Compared with the Deloitte method, the workload of this method is much less, so it was successfully used to study the motion of asteroids as soon as it appeared.Artificial satelliteLast day, it was more widely used.However, Chapel transformation also has some shortcomings, the most prominent of which is:Generating functionIt is a mixed type, containing both new and old variables, which is inconvenient to use.In order to overcome this shortcoming, Horihara Ichiro put forward a theory based on the Li transformation in the 1960s - Horihara Li transformation.Its advantages are that not only the transformation between new and old variables hasExplicit functionIn the form ofCanonical transformationIt remains unchanged under, so it is related to which groupRegular variableIt has nothing to do with calculation, but is universal.
theoretical research
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The creation and development of electronic computers have not only greatly improvednumerical calculationIt is widely used today for its precision and speed, and it can replace people to complete a large number of repeated derivation of machineryPerturbation theoryResearch.In recent years, DePritt, Henrad and Rom have compiled an analytical moonCalendar。In terms of calculating the main perturbation term of the sun,Perturbation functionThere are nearly 3000 items, and through Lie transformation, nearly 50000 items of lunar coordinate expressions are obtained.Its scale is far beyond Deloney's theory.
There are various perturbation factors affecting the movement of celestial bodies:Universal gravitationCaused byConservatism, caused by medium dampingDissipative force, there are continuous forces, and there are also forces such asRadiation pressureDiscontinuous force caused, etc.influenceMegaplanetsThe main perturbation factor of motion is the mutual attraction between planets;Earth's atmosphereDamping caused the satellite to fall to the ground;Solar radiation pressureDeterminedCometary tailShape of;Tidal frictionthen isSatellite orbitThe main driving force of evolution.Only by accurately grasping various perturbation factors can we accurately calculate the movement of celestial bodies and interpret various magnificent astronomical phenomena.On the contrary, through precise observation and accurate grasp of the movement law of celestial bodies, we canPerturbation theoryTo understand the mechanical environment around the celestial body, such as measuring the mass of the perturbed celestial body and the mechanics of the main celestial bodyOblatenessandModulus of elasticity、Atmospheric densityAnd variousgravitational fieldParameters and so on, and can even predict the existence andTrack。Therefore, perturbation theory not only has rich theoretical content, but also has high practical value.
application
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Perturbation can be studied mathematically in two different ways: analytical method and numerical method.These two methods are correspondingly formed in the perturbation theoryGeneral perturbationandSpecial perturbationTwo branches.Perturbation theory is not only the main means to study the motion of celestial bodies, but alsoTheoretical physicsIt is also widely used in engineering technology, namely the so-calledPerturbation theory。
Artificial earth satelliteThe state of motion around the earth depends on the various forces it receives.These forces mainly include: the gravity of the earth on satellites, the gravity of the sun and moon on satellites, atmospheric resistance, solar light pressure, earth tidal force, etc.Under the action of multiple forces, the orbit of satellite in space is extremely complex, which is difficult to be expressed by simple and accurate mathematical model.In order to study the basic laws of satellite motion, the forces acting on satellites can be divided into two categories. One is the gravity of the earth's center of mass, that is, the earth is regarded as a sphere with uniform density or composed of an infinite number of concentric spherical layers with uniform density. It can be proved that its gravity on a point outside the sphere is equivalent to the gravity generated by a particle whose mass is concentrated at the center of the sphere. This gravity is called central gravity.However, the earth is actually non spherical symmetry. This non spherical symmetry of the earth's gravitational field produces a non central gravity on satellites. In addition to the sun and moon's gravity, atmospheric resistance, solar light pressure, and earth's tidal force, the second type is calledPerturbation forceNoncentral gravity of[1]。Compared with the central gravity, the perturbation force is only 10-3magnitude.
For satellite precise positioning, the influence of the earth's gravitational field perturbation, the sun and moon perturbation, the atmospheric resistance, the light pressure perturbation, and the tide perturbation on the satellite's motion state must be considered when calculating the satellite's motion state.The satellite motion considering the perturbation force becomes the satellite'sSubject motion。
Research on satellite's motion and researchTwo body problemThe method of is similar to that of. First, the mathematical expression is derived according to the physical characteristics of various forces on the satellite, then the differential equation of the perturbed motion is established, and finally the differential equation is solved to obtain the equation of satellite motion.
