Study and determine the size and change rule of perturbationtheoryAnd methods.One celestial body goes around anotherTwo body problemWhen the orbit of the celestial body is moving, it will deviate from the original orbit due to the attraction of other celestial bodies or the influence of other factors.This deviation is called perturbation.Perturbation can be studied mathematically in two different ways: analytical method and numerical method.These two methods correspondingly form two branches of general perturbation and special perturbation in perturbation theory.Perturbation theory is not only the main means to study the motion of celestial bodies, but also widely used in theoretical physics and engineering technology, namely the so-called perturbation theory.
Chinese name
Perturbation theory
Foreign name
perturbation theory
Application
Celestial motion, theoretical physics and engineering technology
The development of perturbation theory has a history of more than 200 years.Perturbation theory not only[1]It is the main means to study the movement of celestial bodies, and is also widely used in theoretical physics and engineering technology, namely the so-called perturbation theory.Many famous scholars, such as Euler, Lagrange, Gauss, Poisson and Laplace, 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:Coordinate perturbation method, instantaneous ellipse method and canonical transformation method。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 perturbation method
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Coordinate perturbation method
Study the coordinates of celestial bodies in real orbits andIntermediate trackThe difference between the coordinates of is called coordinate perturbation.In classical methods, coordinate perturbation is often expressed as a power series of a small parameter (such as the mass of the perturbed planet), and then calculated item by item.Due to the development of computing technology, the pickup iteration method is gradually replacing the original small parameter power series expansion method in the approximate solution of differential equations.Its main advantage is that it has a unified iterative process, so that the calculation process can be highly automated.
Rectangular coordinate perturbation
Rectangular coordinate perturbation
This is Enke's research in 1858[2]Comet movementIt discusses the expression of coordinate perturbation in rectangular coordinate system and is often used for calculationShort period cometAnd lunar rocket orbits.The advantage of this method is that the derivation of the perturbation equation is simple, the form is symmetrical, and the 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;With the growth of time span, the three perturbations of rectangular coordinates tend to increase at the same time, so that the equations they obey cannot be linearized, otherwise the zero point will be replaced many times.
Spherical coordinate perturbation
Spherical coordinate perturbation
Natural celestial bodies generally move around a main celestial body, for example, planets move around the sun, and satellites move around the planets.Therefore, the perturbation of spherical coordinates or polar coordinates has obvious geometric significance.Clello and Laplace are studying the motion andTheory of motion of major planetsThe spherical coordinate perturbation method was first proposed by Shi.Later,newcon The Laplacian method is improved, especially the operator operation is used to expand the perturbation function, so that the expansion process not only has a simple mathematical expression, but also has a regular processing process, which is convenient for later calculation on the computer.Newcomb successfully used this method to study the motion of Mercury, Venus, Earth and Mars, as well as Uranus and Neptune. The calendar of the inner planets compiled according to this method has been the basis for compiling astronomical calendars since the 20th century.Hill proposed a spherical coordinate perturbation method with the true near point angle as an argument, which was successfully used to calculate the perturbation of Ceres, the first asteroid.
Other coordinate perturbation
In 1963, Mussen proposed another method to calculate the coordinate perturbation, which is used to calculate the perturbations of celestial coordinates in the radial, velocity and angular momentum 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 automation of calculation. It is now used to establish a new theory of planetary motion.
Instantaneous ellipse method
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Instantaneous ellipse method
This is the perturbation method with orbital elements as basic variables.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;Only the accelerations are different from each other. One is the real acceleration, and the other is the elliptical acceleration. The difference between the two is the perturbation acceleration caused by the perturbation force.
Due to the effect of the perturbation acceleration, the celestial body will leave this ellipse at the next moment and go to a nearby instantaneous ellipse;On the contrary, once the perturbation disappears, the celestial body will keep moving along the instantaneous ellipse of the vanishing point.The motion of celestial bodies under the perturbation of solar radiation pressure is exactly like this: when the radiation pressure acts, the instantaneous ellipse of celestial bodies 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 a celestial body is the envelope of a family of instantaneous ellipses.Compared with the coordinate perturbation, the change of the elements of the elliptical orbit is generally much slower, so it is easy to deal with.The instantaneous ellipse method was first proposed by Euler when he studied the mutual perturbation between Jupiter and Saturn in the middle of the 18th century, and was later improved by Lagrange.He usedLagrange bracketThe perturbation equation Lagrange equation, which describes the changes of the elements of the elliptical orbit, is strictly derived.This method has been widely used, especially in the study of the motion of large planets by Le Weier.
Canonical transformation method
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This is[3]A method based on analytical mechanics.Its basic idea is to carry out a series of appropriate canonical transformations on variables in order to reduce the order of the equation of motion, so that the new equation has a simpler form, such as obtaining an equation describing linear motion or simple harmonic vibration at constant speed, so that the problem can be solved.
Drone transformation
In the 19th century, Deloitte founded the famous Deloitte from this point of view[4]Lunar motion theory。He first expanded the perturbation function of the moon into more than 400 trigonometric terms, and then carried 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 angular velocities, which expressed the orbital elements of the moon as trigonometric polynomials of time without any long-term term.Drone's job isCelestial mechanicsThe transformation theory in is the foundation.This method is composed of a series of uniform circular processes, so it is very convenient to use electronic computers 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 first queues up the angular variables in the perturbation function according to their changing speed, and then finds an appropriate transformation within a certain precision range, so as to eliminate all the terms containing the fast variables at one time and obtain a group of averaged equations, and then repeats the similar process for the new equation until all the angular variables are eliminated.
Zeppel transformation
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.The man-made satellite has been used more widely since it was launched.However, Chapel transform also has some shortcomings, the most prominent of which is that the generating function that determines the transformation relationship between new and old variables is mixed, and it contains both new and old variables, which is inconvenient to use.
Horihara Lee transformation
In order to overcome this shortcoming, Horihara Ichiro put forward a theory based on the Li transformation in the 1960s - Horihara Li transformation.Its advantage is that not only the transformation between new and old variables has the form of explicit function, but also its results remain unchanged under the regular transformation, so it has nothing to do with which group of regular variables to calculate, and it is universal.The creation and development of electronic computers not only greatly improve the accuracy and speed of numerical calculation, but also replace people to complete a large number of mechanical repeated derivation. Today, they have been widely used in perturbation theory research.In recent years, DePritt, Henrad and Rom have compiled an analytical lunar calendar using electronic computers.As far as the calculation of the main perturbation terms of the sun is concerned, there are nearly 3000 terms of the perturbation function. Through the Lie transformation, nearly 50000 terms of lunar coordinate expressions are obtained.Its scale is far beyond Deloney's theory.
influence factor
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There are various perturbation factors affecting the movement of celestial bodies: the conservative force caused by gravity, the dissipative force caused by medium damping, the continuous force, and the intermittent force caused by radiation pressure.
The main perturbation factor affecting the motion of the giant planets is the mutual attraction between the planets;The damping of the earth's atmosphere caused the satellite to fall to the ground;The shape of the comet tail is determined by the solar radiation pressure;Tidal friction is the main power of satellite orbit 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 mastery of the motion law of celestial bodies, we can understand the mechanical environment around celestial bodies according to the analysis of perturbation theory, such as measuring the mass of perturbed celestial bodies, the scientific oblateness and elastic modulus of the main celestial bodies, atmospheric density and various gravitational field parameters, etc., and even predict the existence and whereabouts of some unknown celestial bodies.Therefore, perturbation theory not only has rich theoretical content, but also has high practical value.
