The shape of an ellipse and the degree of its similarity to a circle is characterized by the ratio, where c- the distance from the center of the ellipse to its focus (half the interfocal distance), a- semi-major axis. The quantity e called the eccentricity of the ellipse. At c= 0 and e= 0 the ellipse turns into a circle.

Proof of Kepler's first law

Law universal gravitation Newton says that "every object in the universe attracts every other object along a line connecting the centers of mass of objects, in proportion to the mass of each object, and inversely proportional to the square of the distance between objects." This assumes that the acceleration a has the form

In coordinate form, we write

Substituting also into the second equation, we obtain

which is simplified

After integration, we write the expression

for some constant, which is the specific angular momentum (). Let

The equation of motion in the direction becomes

Newton's law of universal gravitation connects force per unit mass with distance as

where G is the universal gravitational constant and M is the mass of the star.

As a result

it differential equation has a general solution:

for arbitrary constants of integration e and θ 0.

Replacing u by 1 / r and putting θ 0 = 0, we get:

We got the equation of a conical section with an eccentricity e and the origin of the coordinate system at one of the focuses. Thus, Kepler's first law follows directly from Newton's law of universal gravitation and Newton's second law.

Kepler's Second Law (Area Law)

Kepler's second law.

Each planet moves in a plane passing through the center of the Sun, and for equal times the radius vector connecting the Sun and the planet sweeps out sectors of equal area.

With regard to our solar system, two concepts are associated with this law: perihelion is the point of the orbit closest to the Sun, and aphelion- the most distant point of the orbit. Thus, it follows from Keppler's second law that the planet moves around the Sun unevenly, having a greater linear velocity at perihelion than at aphelion.

Every year in early January, the Earth, passing through perihelion, moves faster, so the apparent movement of the Sun along the ecliptic to the east also occurs faster than the average for the year. At the beginning of July, the Earth, passing aphelion, moves more slowly, therefore, the movement of the Sun along the ecliptic slows down. The law of areas indicates that the force governing the orbital motion of the planets is directed towards the Sun.

Proof of Kepler's second law

By definition, the angular momentum of a point particle with mass m and the speed is written in the form:

.

where is the radius vector of the particle and is the momentum of the particle.

A-priory

.

As a result, we have

.

Differentiate both sides of the equation in time

insofar as cross product parallel vectors is zero. notice, that F always parallel r since the force is radial, and p always parallel v a-priory. Thus, it can be argued that is a constant.

Kepler's Third Law (Harmonic Law)

The squares of the periods of revolution of the planets around the Sun are referred to as cubes of the semi-major axes of the planets' orbits.

Where T 1 and T 2 - periods of revolution of two planets around the Sun, and a 1 and a 2 - the lengths of the semi-major axes of their orbits.

Newton found that the gravitational attraction of a planet of a certain mass depends only on the distance to it, and not on other properties, such as composition or temperature. He also showed that Kepler's third law is not entirely accurate - in fact, it also includes the mass of the planet: where M Is the mass of the Sun, and m 1 and m 2 - planetary masses.

Since motion and mass turned out to be related, this combination of Kepler's harmonic law and Newton's law of gravitation is used to determine the mass of planets and satellites, if their orbits and orbital periods are known.

Proof of Kepler's third law

Kepler's second law states that the radius vector of a revolving body sweeps out equal areas in equal intervals of time. If now we take very small intervals of time at the moment when the planet is at points A and B(perihelion and aphelion), then we will be able to approximate the area with triangles with heights equal to the distance from the planet to the sun, and a base equal to the product of the planet's velocity and time.

Using the law of conservation of energy for the total energy of the planet in points A and B, we write

Now that we've found V B, we can find the sectorial velocity. Since it is constant, we can choose any point of the ellipse: for example, for a point B get

However, the total area of ​​the ellipse is (which is equal to π ab, insofar as ). The time of a complete revolution is thus equal to

Note that if the mass m is not negligible compared to M, then the planet will revolve around the Sun at the same speed and in the same orbit as the material point revolving around the mass M + m(cm.

“He lived in an era when there was still no certainty of the existence of some general pattern for all natural phenomena ...

What a deep faith he had in such a pattern, if, working alone, not supported and understood by anyone, for many decades he drew strength from it for the difficult and painstaking empirical research the motion of the planets and the mathematical laws of this motion!

Today, when this scientific act has already taken place, no one can fully appreciate how much ingenuity, how much hard work and patience it took to discover these laws and express them so accurately ”(Albert Einstein on Kepler).

Johannes Kepler was the first to discover the law of planetary motion Solar system. But he did this on the basis of an analysis of the astronomical observations of Tycho Brahe. Therefore, let's talk about it first.

Tycho Brahe (1546-1601)

Tycho Brahe - Danish astronomer, astrologer and alchemist of the Renaissance. He was the first in Europe to begin to carry out systematic and high-precision astronomical observations, on the basis of which Kepler derived the laws of planetary motion.

He became interested in astronomy as a child, conducted independent observations, created some astronomical instruments. Once (November 11, 1572), returning home from a chemical laboratory, he noticed in the constellation Cassiopeia an unusually bright star that didn't exist before. He immediately realized that this was not a planet, and rushed to measure its coordinates. The star shone in the sky for another 17 months; at first it was visible even during the day, but gradually its luster faded. This was the first supernova explosion in our Galaxy in 500 years. This event excited the whole of Europe, there were many interpretations of this "heavenly sign" - they predicted catastrophes, wars, epidemics and even the end of the world. Scientific treatises have also appeared containing erroneous statements that it is a comet or atmospheric phenomenon... In 1573 his first book, On a New Star, was published. In it, Brahe reported that no parallax (a change in the apparent position of an object relative to a distant background, depending on the position of the observer) was found for this object, and this convincingly proves that the new star is a star, and it is not near the Earth, but at least at a planetary distance. With the appearance of this book, Tycho Brahe was recognized as the first astronomer in Denmark. In 1576, by decree of the Danish-Norwegian king Frederick II, Tycho Brahe was granted the island of Ven ( Hven), located 20 km from Copenhagen, and significant sums were allocated for the construction of the observatory and its maintenance. It was the first building in Europe specially built for astronomical observations. Tycho Brahe named his observatory "Uraniborg" in honor of the muse of astronomy Urania (this name is sometimes translated as "Sky Castle"). The building was designed by Tycho Brahe himself. In 1584, another castle-observatory was built near Uraniborg: Stjerneborg (translated from Danish as "Star Castle"). Soon, Uraniborg became the best astronomical center in the world, combining observation, student training and the publication of scientific papers. But later, in connection with the change of the king. Tycho Brahe lost his financial support, and then the prohibition to study astronomy and alchemy on the island followed. The astronomer left Denmark and stayed in Prague.

Soon Uraniborg and all the buildings associated with it were completely destroyed (in our time, they are partially restored).

In this tense time, Brahe came to the conclusion that he needed a talented young mathematician assistant to process the data accumulated over 20 years. Having learned about the persecution of Johannes Kepler, whose outstanding mathematical abilities he had already appreciated from their correspondence, Tycho invited him to his place. Scientists had a task: to deduce from observations new system world, which should replace both Ptolemaic and Copernican. He entrusted Kepler with the key planet: Mars, whose motion decisively did not fit not only in Ptolemy's scheme, but also in Brahe's own models (according to his calculations, the orbits of Mars and the Sun crossed).

In 1601, Tycho Brahe and Kepler began work on new, refined astronomical tables, which were named "Rudolphs" in honor of the emperor; they were completed in 1627 and served astronomers and sailors until early XIX century. But Tycho Brahe only managed to name the tables. In October, he suddenly fell ill and died of an unknown illness.

Having carefully studied the data of Tycho Brahe, Kepler discovered the laws of planetary motion.

Kepler's laws of motion

Initially, Kepler planned to become a Protestant priest, but thanks to his outstanding mathematical abilities, he was invited in 1594 to lecture in mathematics at the University of Graz (now Austria). Kepler spent 6 years in Graz. Here in 1596 his first book, The Mystery of the World, was published. In it, Kepler tried to find the secret harmony of the Universe, for which he compared the orbits of the five then known planets (he singled out the sphere of the Earth) various "Platonic bodies" (regular polyhedrons). He presented the orbit of Saturn as a circle (not yet an ellipse) on the surface of a ball, described around a cube. In turn, a ball was inscribed in the cube, which was supposed to represent the orbit of Jupiter. In this ball was inscribed a tetrahedron, described around a ball representing the orbit of Mars, etc. This work, after further discoveries of Kepler, lost its original meaning (if only because the orbits of the planets were not circular); nevertheless, Kepler believed in the presence of a hidden mathematical harmony of the Universe until the end of his life, and in 1621 he republished The Mystery of the World, making numerous changes and additions to it.

Being an excellent observer, Tycho Brahe for many years compiled a voluminous work on the observation of planets and hundreds of stars, and the accuracy of his measurements was significantly higher than that of all his predecessors. To improve the accuracy, Brahe used both technical improvements and a special technique for neutralizing observation errors. Systematic measurements were especially valuable.

For several years, Kepler has been carefully studying Brahe's data and, as a result of careful analysis, comes to the conclusion that the trajectory of Mars is not a circle, but an ellipse, in one of the focuses of which is the Sun - a position known today as Kepler's first law.

Kepler's first law (law of ellipses)

Each planet in the solar system revolves around an ellipse, in one of the focuses of which is the sun.

The shape of an ellipse and the degree of its similarity to a circle is characterized by the ratio, where is the distance from the center of the ellipse to its focus (half the interfocal distance), is the semi-major axis. The value is called the eccentricity of the ellipse. When, and, therefore, the ellipse turns into a circle.

Further analysis leads to the second law. The radius vector connecting the planet and the Sun in equal time describes equal areas. This meant that the farther a planet is from the Sun, the slower it moves.

Kepler's second law (area law)

Each planet moves in a plane passing through the center of the Sun, and for equal intervals of time the radius vector connecting the Sun and the planet describes equal areas.

There are two concepts associated with this law: perihelion is the point of the orbit closest to the Sun, and aphelion- the most distant point of the orbit. Thus, it follows from Kepler's second law that the planet moves around the Sun unevenly, having a greater linear velocity at perihelion than at aphelion.

Every year in early January, the Earth, passing through perihelion, moves faster, so the apparent movement of the Sun along the ecliptic to the east also occurs faster than the average for the year. At the beginning of July, the Earth, passing aphelion, moves more slowly, therefore, the movement of the Sun along the ecliptic slows down. The law of areas indicates that the force governing the orbital motion of the planets is directed towards the Sun.

Kepler's third law (harmonic law)

The squares of the periods of revolution of the planets around the Sun are referred to as cubes of the semi-major axes of the planets' orbits. This is true not only for the planets, but also for their satellites.

Where and are the periods of revolution of the two planets around the Sun, and and are the lengths of the semi-major axes of their orbits.

Newton later found that Kepler's third law is not entirely accurate - it also includes the mass of the planet:, where is the mass of the Sun, and and are the masses of the planets.

Since motion and mass turned out to be related, this combination of Kepler's harmonic law and Newton's law of gravitation is used to determine the mass of planets and satellites, if their orbits and orbital periods are known.

The importance of Kepler's discoveries in astronomy

Discovered by Kepler three laws of planetary motion fully and accurately explained the apparent unevenness of these movements. Instead of numerous contrived epicycles, Kepler's model includes only one curve - an ellipse. The second law established how the speed of the planet changes when moving away or approaching the Sun, and the third allows you to calculate this speed and the period of revolution around the Sun.

Although historically the Keplerian system of the world is based on the Copernican model, in fact they have very little in common (only the daily rotation of the Earth). The circular motions of the spheres carrying the planets disappeared, the concept of a planetary orbit appeared. In the Copernican system, the Earth still occupied a somewhat special position, since only it did not have epicycles. Kepler's Earth is an ordinary planet, the movement of which is subject to three general laws. All orbits celestial bodies- ellipses, the common focus of the orbits is the Sun.

Kepler also derived the "Kepler equation" used in astronomy to determine the position of celestial bodies.

Kepler's laws later served Newton basis for the creation of the theory of gravitation. Newton mathematically proved that all Kepler's laws are consequences of the law of gravitation.

But Kepler did not believe in the infinity of the Universe and offered as an argument photometric paradox(this name arose later): if the number of stars is infinite, then in any direction the gaze would stumble upon a star, and there would be no dark areas in the sky. Kepler, like the Pythagoreans, considered the world to be the realization of a certain numerical harmony, both geometric and musical; revealing the structure of this harmony would give answers to the deepest questions.

Kepler's other achievements

In mathematics he found a way to determine the volumes of various bodies of revolution, proposed the first elements of integral calculus, analyzed in detail the symmetry of snowflakes, Kepler's work in the field of symmetry found later application in crystallography and coding theory. He compiled one of the first tables of logarithms, for the first time introduced the most important concept an infinitely distant point,introduced the concept focus of the conical section and considered projective transformations of conic sections, including those that change their type.

In physicscoined the term inertia as an innate property of bodies to resist the applied external force, came close to the discovery of the law of gravitation, although he did not try to express it mathematically, the first, almost a hundred years earlier than Newton, put forward a hypothesis that the cause of the tides is the influence of the Moon on the upper layers of the oceans.

In optics: optics as a science begins with his works. It describes the refraction of light, refraction and the concept optical image, general theory of lenses and their systems. Kepler found out the role of the lens, correctly described the causes of myopia and hyperopia.

TO astrology Kepler had an ambivalent attitude. There are two of his statements on this matter. First: " Of course, this astrology is a stupid daughter, but, my God, where would her mother, highly wise astronomy, if she did not have a stupid daughter! After all, the world is much more stupid and so stupid that for the good of this wise old mother, the stupid daughter must talk and lie. And the salaries of mathematicians are so negligible that the mother would probably starve if the daughter did not earn anything.". And second: " People are wrong thinking that from heavenly bodies earthly affairs depend". But, nevertheless, Kepler made horoscopes for himself and his loved ones.

The three laws of motion of the planets relative to the Sun were deduced empirically by the German astronomer Johannes Kepler at the beginning of the 17th century. This became possible thanks to many years of observations by the Danish astronomer Tycho Brahe.

Kepler's first law. Each planet moves along an ellipse, in one of the focuses of which is the Sun.

Kepler's second law (law of equal areas). The radius vector of the planet for equal time intervals describes equal areas. Another formulation of this law: the sectorial speed of the planet is constant.

Kepler's third law. The squares of the orbital periods of the planets around the Sun are proportional to the cubes of the semi-major axes of their elliptical orbits.

The modern formulation of the first law is supplemented as follows: in unperturbed motion, the orbit of a moving body is a curve of the second order - an ellipse, parabola or hyperbola. Unlike the first two, Kepler's third law applies only to elliptical orbits. The speed of the planet at perihelion:

where vc is the average or circular speed of the planet at r = a. The speed of movement in aphelion: Kepler discovered his laws empirically. Newton derived Kepler's laws from the law of universal gravitation. To determine the masses of celestial bodies essential has Newton's generalization of Kepler's third law to any system of revolving bodies.

Kepler's third law. The speeds of planets close to the Sun are much higher than those of distant ones. Explanation to the figure on the right - The velocities of planets close to the Sun are much higher than those of distant ones. In generalized form, this law is usually formulated as follows: the squares of the periods T1 and T2 of the revolution of two bodies around the Sun, multiplied by the sum of the masses of each body (M1 and M2, respectively) and the Sun (M), are related as cubes of the semi-major axes a1 and a2 of their orbits: this does not take into account the interaction between the bodies M1 and M2. If we neglect the masses of these bodies in comparison with the mass of the Sun (i.e. M1<< М, M2 << М), то получится формулировка третьего закона, данная самим Кеплером:

Kepler's third law can also be expressed as the relationship between the orbital period T of a body with mass M and the semi-major axis of the orbit a (G is the gravitational constant):

The following remark should be made here. For simplicity, it is often said that one body revolves around another, but this is true only for the case when the mass of the first body is negligible compared to the mass of the second (attracting center). If the masses are comparable, then the influence of a less massive body on a more massive one should also be taken into account. In the coordinate system with the origin at the center of mass, the orbits of both bodies will be conical sections lying in the same plane and with foci at the center of mass, with the same eccentricity. The difference will be only in the linear dimensions of the orbits (if the bodies are of different masses). At any time, the center of mass will lie on a straight line connecting the centers of the bodies, and the distances to the center of mass r1 and r2 of bodies with masses M1 and M2, respectively, are related by the following ratio: r1 / r2 = M2 / M1. The pericenter and apocenters of their orbits (if the movement is finite) of the body will also pass simultaneously. Kepler's third law can be used to determine the mass of binary stars.

An ellipse is defined as a locus of points for which the sum of the distances from two given points (foci F1 and F2) is a constant value equal to the length of the major axis: r1 + r2 = | AA´ | = 2a. The degree of elongation of the ellipse is characterized by its eccentricity e. Eccentricity e = ОF / OA. When the focus coincides with the center e = 0, and the ellipse turns into a circle. The semi-major axis a is the average distance from the focus (of the planet from the Sun): a = (AF1 + F1A ") / 2. Since the total energy is negative when moving along the ellipse, the semi-major axis is greater than zero. The length of the semi-minor axis b depends on the sectorial velocity of the body ( i.e. the rate of change of the area swept out by the radius vector). Circular orbits are a degenerate case of elliptical. Writing Newton's second law, we obtain that the kinetic and potential energy of a body in a circular orbit are related by the ratio: 2K = –U. Applying the law of conservation of energy, it is easy to obtain that K = –E. Thus, in a circular motion, the sum of the total and kinetic energy is always zero. Orbital elements characterize the shape, size and orientation in space of the orbit of a celestial body, as well as the position of the body in this orbit. Osculatory elements are widely used to describe the position of a planet or satellite.

The most important points and lines of the ellipse.

An ellipse is defined as a locus of points for which the sum of the distances from two given points (foci F1 and F2) is a constant value equal to the length of the major axis: r1 + r2 = | AA´ | = 2a. The degree of elongation of the ellipse is characterized by its eccentricity e. Eccentricity e = ОF / OA. When the focus coincides with the center e = 0, and the ellipse turns into a circle. The semi-major axis a is the average distance from the focus (of the planet from the Sun): a = (AF1 + F1A ") / 2. It is related to the mechanical energy of the body as follows:

Since the total energy is negative when moving along an ellipse, the semi-major axis is greater than zero. The length of the minor semiaxis b depends on the sectorial velocity of the body (i.e., the rate of change of the area swept by the radius vector): Circular orbits are a degenerate case of elliptic ones. Writing down Newton's second law, we get that the kinetic and potential energy of a body in a circular orbit are related by the ratio: 2K = –U. Applying the law of conservation of energy, it is easy to find that K = –E. That. in circular motion, the sum of the total and kinetic energy is always zero. Orbital elements characterize the shape, size and orientation in space of the orbit of a celestial body, as well as the position of the body in this orbit. At present, osculating elements are widely used to describe the position of a planet or satellite. The point of the body's orbit that is closest to the attracting center (focus) is generally called the pericenter, and the point farthest from it (only at the ellipse) is called the apocenter. If the Earth is the attracting center, then these points are called perigee and apogee, respectively. The closest point to the Sun is called perihelion, the most distant one is aphelion. For the moon, these points will be perilune (perilune) and apolunium (aposet), for an arbitrary star - periastron and apoastron. The straight line connecting the periapsis with the focus (the major axis of the ellipse, the axis of the parabola, or the real axis of the hyperbola) is called the line of the apses. The distance from the attracting center to the pericenter is AF1 = a (1 - e), to the apocenter - F1A "= a (1 + e). The average distance from the attracting center to the body moving around it in an ellipse is equal to the length of the semi-major axis.

The planets move around the Sun in elongated elliptical orbits, with the Sun at one of the two focal points of the ellipse.

The line segment connecting the Sun and the planet cuts off equal areas at equal intervals of time.

The squares of the periods of revolution of the planets around the Sun are referred to as cubes of the semi-major axes of their orbits.

Johannes Kepler had a sense of beauty. Throughout his adult life, he tried to prove that the solar system is a kind of mystical work of art. At first he tried to pair her device with five regular polyhedra classical ancient Greek geometry. (A regular polyhedron is a three-dimensional figure, all faces of which are equal regular polygons.) At the time of Kepler, six planets were known, which were supposed to be placed on rotating "crystal spheres". Kepler argued that these spheres are located in such a way that regular polyhedrons fit exactly between adjacent spheres. Between the two outer spheres - Saturn and Jupiter - he placed a cube inscribed in the outer sphere, in which, in turn, the inner sphere is inscribed; between the spheres of Jupiter and Mars - a tetrahedron (regular tetrahedron), etc. * Six spheres of the planets, five regular polyhedrons inscribed between them - it would seem, the very perfection?

Alas, comparing his model with the observed orbits of the planets, Kepler was forced to admit that the real behavior of celestial bodies does not fit into the slender framework outlined by him. According to the apt remark of the modern British biologist J. B. S. Haldane, "the idea of ​​the universe as a geometrically perfect work of art turned out to be another beautiful hypothesis, destroyed by ugly facts." The only surviving result of that youthful impulse of Kepler was a model of the solar system, made by the scientist himself and presented as a gift to his patron, Duke Frederick von Württemberg. In this beautifully executed metal artifact, all the orbital spheres of the planets and the regular polyhedrons inscribed in them are hollow containers that do not communicate with each other, which were supposed to be filled with various drinks on holidays to treat the guests of the duke.

Only after moving to Prague and becoming an assistant to the famous Danish astronomer Tycho Brahe (1546–1601), Kepler came across ideas that truly immortalized his name in the annals of science. Tycho Brahe collected astronomical observations all his life and accumulated huge amounts of information about the motion of the planets. After his death, they were taken over by Kepler. These records, by the way, were of great commercial value at that time, since they could be used to compile refined astrological horoscopes (today, scientists prefer to remain silent about this section of early astronomy).

While processing the results of Tycho Brahe's observations, Kepler faced a problem that, even with modern computers, might seem intractable to someone, and Kepler had no choice but to carry out all the calculations manually. Of course, like most astronomers of his time, Kepler was already familiar with Copernicus 'heliocentric system (see Copernicus' principle) and knew that the Earth revolves around the Sun, as evidenced by the above model of the solar system. But how exactly do the Earth and other planets rotate? Let's imagine the problem as follows: you are on a planet that, firstly, revolves around its axis, and secondly, revolves around the Sun in an orbit unknown to you. Looking into the sky, we see other planets, which are also moving in orbits unknown to us. Our task is to determine the geometry of the orbits and the speed of movement of other planets from the data of observations made on our globe rotating around its axis around the Sun. This is exactly what Kepler ultimately managed to do, after which, on the basis of the results obtained, he derived his three laws!

The first law** describes the geometry of the trajectories of planetary orbits. You may remember from the school geometry course that an ellipse is a set of points on a plane, the sum of the distances from which to two fixed points - tricks- is equal to a constant. If this is too difficult for you, there is another definition: imagine a section of the lateral surface of a cone by a plane at an angle to its base, not passing through the base - this is also an ellipse. Kepler's first law just asserts that the orbits of the planets are ellipses, in one of the focuses of which the Sun is located. Eccentricities(degree of elongation) of the orbits and their distance from the Sun in perihelion(point closest to the Sun) and apogelia(the most distant point) all planets are different, but all elliptical orbits have one thing in common - the Sun is located in one of the two foci of the ellipse. After analyzing the observational data of Tycho Brahe, Kepler concluded that planetary orbits are a set of nested ellipses. Before him, it simply did not occur to any astronomer.

The historical significance of Kepler's first law can hardly be overestimated. Before him, astronomers believed that the planets move exclusively in circular orbits, and if this did not fit into the framework of observations, the main circular motion was supplemented by small circles that the planets described around the points of the main circular orbit. This was, I would say, primarily a philosophical position, a kind of immutable fact that is not subject to doubt and verification. Philosophers argued that the heavenly structure, unlike the earthly one, is perfect in its harmony, and since the most perfect of geometric shapes are the circle and the sphere, it means that the planets move in a circle (and this delusion I still have to dispel among my students today). The main thing is that, having gained access to the vast observational data of Tycho Brahe, Johannes Kepler was able to step over this philosophical prejudice, seeing that it does not correspond to the facts - just as Copernicus dared to remove the Earth from the center of the universe, faced with arguments that contradict persistent geocentric notions. also consisted of the "wrong behavior" of the planets in orbits.

Kepler's early geometric model of the Universe: six orbital planetary spheres and five inscribed regular polyhedrons in between

Second law describes the change in the speed of movement of the planets around the sun. In a formal form, I have already cited its formulation, and in order to better understand its physical meaning, remember your childhood. Probably, you had a chance to unwind around a pole on the playground, grabbing it with your hands. In fact, the planets circle the sun in a similar way. The further from the Sun the elliptical orbit leads the planet, the slower the movement, the closer to the Sun the faster the planet moves. Now imagine a pair of line segments connecting two planetary positions in orbit with the focus of the ellipse in which the Sun is located. Together with the segment of the ellipse lying between them, they form a sector, the area of ​​which is precisely the same “area that is cut off by a straight line segment”. It is about her that the second law says. The closer the planet is to the Sun, the shorter the segments. But in this case, in order for the sector to cover an equal area in equal time, the planet must travel a greater distance in its orbit, which means that its speed of movement increases.

The first two laws deal with the specifics of the orbital trajectories of a single planet. The third law Kepler allows you to compare the orbits of the planets with each other. It says that the farther from the Sun the planet is, the longer it takes for its full revolution when moving in orbit, and the longer, accordingly, the "year" lasts on this planet. We know today that this is due to two factors. First, the farther a planet is from the Sun, the longer the perimeter of its orbit. Secondly, as the distance from the Sun increases, the linear velocity of the planet's motion also decreases.

In his laws, Kepler simply stated facts by studying and generalizing the results of observations. If you asked him what caused the ellipticity of the orbits or the equality of the areas of the sectors, he would not have answered you. It simply followed from his analysis. If you asked him about the orbital motion of planets in other stellar systems, he also would not have found an answer to you. He would have to start all over again - to accumulate observational data, then analyze them and try to identify patterns. That is, he simply would not have reason to believe that another planetary system obeys the same laws as the solar system.

One of the greatest triumphs of Newtonian classical mechanics lies precisely in the fact that it provides a fundamental basis for Kepler's laws and asserts their universality. It turns out that Kepler's laws can be derived from Newton's laws of mechanics, Newton's law of universal gravitation and the conservation of angular momentum by rigorous mathematical calculations. If so, we can be sure that Kepler's laws are equally applicable to any planetary system anywhere in the universe. Astronomers looking for new planetary systems in world space (and quite a few of them have already been discovered), over and over again, as a matter of course, use Kepler's equations to calculate the parameters of the orbits of distant planets, although they cannot observe them directly.

Kepler's third law has played and continues to play an important role in modern cosmology. Observing distant galaxies, astrophysicists detect faint signals emitted by hydrogen atoms orbiting very far from the galactic center orbits - much further than stars usually are. Using the Doppler effect in the spectrum of this radiation, scientists determine the rotation speeds of the hydrogen periphery of the galactic disk, and from them - the angular velocities of galaxies as a whole (see also Dark matter). I am glad that the works of the scientist who firmly set us on the path of a correct understanding of the structure of our solar system, and today, centuries after his death, play such an important role in the study of the structure of the immense universe.

* Between the spheres of Mars and the Earth - dodecahedron (dodecahedron); between the spheres of the Earth and Venus - the icosahedron (twenty-sided); between the spheres of Venus and Mercury - an octahedron (octahedron). The resulting construction was presented by Kepler in section on a detailed volumetric drawing (see figure) in his first monograph "Cosmographic Mystery" (Mysteria Cosmographica, 1596). - Translator's note.

** Historically, Kepler's laws (like the principles of thermodynamics) are numbered not according to the chronology of their discovery, but in the order of their understanding in scientific circles. In reality, the first law was discovered in 1605 (published in 1609), the second in 1602 (published in 1609), the third in 1618 (published in 1619). - Translator's note.

Kepler's formulation:

The planet moves along an ellipse, in one of the focuses of which is the Sun.

Newton generalizes it: first, a star-star (double star) system, a planet-satellite can be considered; secondly, a smaller body can move along a parabola or hyperbola (Fig. 33).

Modern wording:

In a gravitationally bound system, the body B moves along an ellipse, in one of the focuses of which is the body A... The ex-centricity of an ellipse is determined by the numerical value of the total energy of the system. In a gravitationally uncoupled system, body B moves along a parabola ( E= 0) or hyperbole ( E> 0), the focus of which is the body A.

Ellipse

An ellipse (Fig. 33) is an elongated circle with the property that there are two points (foci of an ellipse F 1 and F 2, for which the condition is satisfied: the sum of the distances of the foci from any point of the ellipse is constant ( F 1C + F 2C = F 1E + F 2E= const), i.e., does not depend on the point selected on the ellipse).

Section AB called the major axis, respectively, the segment AO = OB- semi-major axis (accepted designation a), segments CD and OC- minor axle and semi-axle b... The size of the ellipse is determined by the semi-major axis, the shape is determined by the eccentricity e = √ (1 - b 2 / a 2). At e= 0 the ellipse degenerates into a circle, for e= 1 - in a parabola, for e> 1 - into a hyperbola, which is better represented as a graph of a function y = 1 / x, rotated 45 °. The ellipse has a semi-major axis a> 0, the parabola has a= ∞, hyperbola a < 0, что, конечно, только математиче-ская абстракция.

The radius vector of the planet for equal time intervals describes equal areas (Fig. 34).

This statement is similar to the fact that the speed of motion decreases with distance from the Sun, or rather, it is the law of conservation of angular momentum.

If we count the number of days from the day of the vernal equinox (March 21) to the autumn day (September 23) and from September 23 to March 21 of the next year, it turns out that the first period is 7 days. longer than the second. In other words, the Earth in winter moves faster than in summer, therefore, it is closer to the Sun in winter. The closest point of its orbit to the Sun - perihelion - the Earth passes on January 6.

The law of conservation of angular momentum

Moment of impulse ( K = mvr) Is a physical quantity convenient for describing the motion of a point along a circle or ellipse, parabola, hyperbola, as well as for describing the rotation of a rigid body. The law of conservation of momentum momentum(as well as the laws of conservation of momentum and energy) is one of the three fundamental laws of nature. According to Noether's theorem, this law is a consequence of the isotropy (equality of all directions) of the universe.

The ratio of the cube of the semi-major axis of the planetary orbit to the cube of the period of the planet's revolution around the Sun is equal to the sum of the masses of the Sun and the planet (in Newton's formulation):

a 3 / T 2 = (G/ 4π 2). ( M + m),Material from the site

where M and m- the masses of the bodies of the system; a and T- semi-major axis and orbital period of a smaller body (planet, satellite); G- gravitational constant.

It is necessary to pay attention to the constant factor on the right-hand side. In the formula, it is given in SI units, but in astronomy, the astronomical unit of length (instead of a meter), year (instead of a second) and the mass of the Sun (instead of a kilogram) are used. Then, as it is easy to be convinced, if we neglect the mass of the planet in relation to the mass of the Sun, the constant factor in this formula is equal to one.

Kepler's third law provides the only way to directly determine the mass of a celestial body (for example,