“He lived in an era when there was still no confidence in the existence of some general pattern for all natural phenomena...
How deep was his faith in such a pattern, if, working alone, not supported or understood by anyone, for many decades he drew strength from it for a difficult and painstaking empirical study of the movement of planets and the mathematical laws of this movement!
Today, when this scientific act has already been accomplished, no one can fully appreciate how much ingenuity, how much hard work and patience was required to discover these laws and express them so precisely” (Albert Einstein on Kepler).
Johannes Kepler was the first to discover the law of motion of the planets of the solar system. But he did this on the basis of an analysis of the astronomical observations of Tycho Brahe. So let's talk about him first.
Tycho Brahe (1546-1601)
Tycho Brahe - Danish astronomer, astrologer and alchemist of the Renaissance. Kepler was the first in Europe to begin conducting 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, and created some astronomical instruments. One day (November 11, 1572), returning home from a chemical laboratory, he noticed an unusually bright star in the constellation Cassiopeia, which had not been there 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 shine dimmed. 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” - disasters, wars, epidemics and even the end of the world were predicted. Scientific treatises also appeared containing erroneous statements that this was a comet or an atmospheric phenomenon. In 1573, his first book, “On the New Star,” was published. In it, Brahe reported that no parallax (changes in the apparent position of an object relative to a distant background depending on the position of the observer) was detected in this object, and this convincingly proves that the new luminary is a star, and it is not located near the Earth, but at least at a planetary distance. With the appearance of this book, Tycho Brahe was recognized as Denmark's first astronomer. In 1576, by decree of the Danish-Norwegian king Frederick II, Tycho Brahe was granted the island of Ven for life use ( 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 specifically built for astronomical observations. Tycho Brahe named his observatory "Uraniborg" in honor of the muse of astronomy Urania (the name is sometimes translated as "Castle in the Sky"). The design of the building was drawn up by Tycho Brahe himself. In 1584, another observatory castle was built next to Uraniborg: Stjerneborg (translated from Danish as “Star Castle”). Uraniborg soon became the world's best astronomical center, combining observations, teaching students and publishing scientific works. But later, in connection with the change of king. Tycho Brahe lost financial support, and then there was a ban on practicing astronomy and alchemy on the island. The astronomer left Denmark and stopped in Prague.
Soon Uraniborg and all the buildings associated with it were completely destroyed (in our time they have been partially restored).
During this tense time, Brahe came to the conclusion that he needed a young, talented mathematician assistant to process the data accumulated over 20 years. Having learned about the persecution of Johannes Kepler, whose extraordinary mathematical abilities he had already appreciated from their correspondence, Tycho invited him to his place. The scientists were faced with a task: to deduce from observations a new system of the world, which should replace both the Ptolemaic and Copernican ones. He entrusted Kepler with the key planet: Mars, whose movement strongly did not fit not only into Ptolemy’s scheme, but also into Brahe’s own models (according to his calculations, the orbits of Mars and the Sun intersected).
In 1601, Tycho Brahe and Kepler began work on new, refined astronomical tables, which were called “Rudolph” in honor of the emperor; they were completed in 1627 and served astronomers and sailors until the beginning of the 19th century. But Tycho Brahe only managed to give the tables a name. In October he unexpectedly fell ill and died of an unknown illness.
After carefully studying the data of Tycho Brahe, Kepler discovered the laws of planetary motion.
Kepler's laws of planetary motion
Initially, Kepler planned to become a Protestant priest, but thanks to his extraordinary mathematical abilities, he was invited in 1594 to lecture on mathematics at the University of Graz (now Austria). Kepler spent 6 years in Graz. Here in 1596 his first book, “The Secret of the World,” was published. In it, Kepler tried to find the secret harmony of the Universe, for which he compared various “Platonic solids” (regular polyhedra) to the orbits of the five then known planets (he especially singled out the sphere of the Earth). He presented the orbit of Saturn as a circle (not yet an ellipse) on the surface of a ball circumscribed around a cube. The cube, in turn, was inscribed with a ball, which was supposed to represent the orbit of Jupiter. A tetrahedron was inscribed in this ball, circumscribed around a ball representing the orbit of Mars, etc. This work, after further discoveries by Kepler, lost its original meaning (if only because the orbits of the planets turned out to be non-circular); Nevertheless, Kepler believed in the existence of a hidden mathematical harmony of the Universe until the end of his life, and in 1621 he republished “The Secret of the World”, making numerous changes and additions to it.
Being an excellent observer, Tycho Brahe compiled a voluminous work over many years 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 increase accuracy, Brahe used both technical improvements and a special technique for neutralizing observation errors. The systematic nature of the measurements was especially valuable.
Over the course of several years, Kepler carefully studied Brahe's data and, as a result of careful analysis, came to the conclusion that Mars' trajectory is not a circle, but an ellipse, with the Sun at one of its foci - a position known today as Kepler's first law.
Kepler's first law (law of ellipses)
Each planet in the solar system revolves in an ellipse, with the Sun at one of the focuses.
The shape of the 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), and is the semimajor axis. The quantity 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 describes equal areas at equal times. This meant that the further a planet is from the Sun, the slower it moves.
Kepler's second law (law of areas)
Each planet moves in a plane passing through the center of the Sun, and in equal periods of time, the radius vector connecting the Sun and the planet describes equal areas.
There are two concepts associated with this law: perihelion- the point of the orbit closest to the Sun, and aphelion- the most distant point of the orbit. Thus, from Kepler’s second law it follows that the planet moves unevenly around the Sun, having a greater linear speed at perihelion than at aphelion.
Every year at the beginning of January, the Earth moves faster when passing through perihelion, 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, and therefore the movement of the Sun along the ecliptic slows down. The law of areas indicates that the force governing the orbital motion of 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 related as the cubes of the semimajor axes of the planets' orbits. This is true not only for planets, but also for their satellites.
Where and are the periods of revolution of two planets around the Sun, and and are the lengths of the semimajor axes of their orbits.
Newton later established 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 are found to be related, this combination of Kepler's harmonic law and Newton's law of gravity is used to determine the mass of planets and satellites if their orbits and orbital periods are known.
The significance 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 as it moves away or approaches the Sun, and the third allows us to calculate this speed and the period of revolution around the Sun.
Although historically the Keplerian world system 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 spheres carrying planets disappeared, and the concept of a planetary orbit appeared. In the Copernican system, the Earth still occupied a somewhat special position, since it was the only one without epicycles. According to Kepler, the Earth is an ordinary planet, the movement of which is subject to three general laws. All orbits of celestial bodies are ellipses; the common focus of the orbits is the Sun.
Kepler also derived the “Kepler equation,” used in astronomy to determine the positions of celestial bodies.
The laws discovered by Kepler later served Newton basis for the creation of the theory of gravity. 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, as an argument, proposed photometric paradox(this name arose later): if the number of stars is infinite, then in any direction the gaze would encounter 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 provide answers to the most profound 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 later found application in crystallography and coding theory. He compiled one of the first tables of logarithms, and for the first time introduced the most important concept infinitely distant pointintroduced the concept focus of the conic section and reviewed projective transformations of conic sections, including those changing their type.
In physicscoined the term inertia as the innate property of bodies to resist an 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 the hypothesis that the cause of tides is the influence of the Moon on the upper layers of the oceans.
In optics: optics as a science begins with his works. He describes the refraction of light, refraction and the concept of optical image, the general theory of lenses and their systems. Kepler figured out the role of the lens and correctly described the causes of myopia and farsightedness.
TO astrology Kepler had an ambivalent attitude. Two of his statements are cited on this matter. First: " Of course, this astrology is a stupid daughter, but, my God, where would her mother, the highly wise astronomy, go if she didn’t have a stupid daughter! The world is even much more stupid and so stupid that for the benefit of this old reasonable mother, the stupid daughter must chat and lie. And the salary of mathematicians is so insignificant that the mother would probably starve if her daughter did not earn anything" And second: “ People are mistaken in thinking that earthly affairs depend on the heavenly bodies" But, nevertheless, Kepler compiled horoscopes for himself and his loved ones.
“He lived in an era when there was still no confidence in the existence of some general pattern for all natural phenomena...
How deep was his faith in such a pattern, if, working alone, not supported or understood by anyone, for many decades he drew strength from it for a difficult and painstaking empirical study of the movement of planets and the mathematical laws of this movement!
Today, when this scientific act has already been accomplished, no one can fully appreciate how much ingenuity, how much hard work and patience was required to discover these laws and express them so precisely” (Albert Einstein on Kepler).
Johannes Kepler was the first to discover the law of motion of the planets of the solar system. But he did this on the basis of an analysis of the astronomical observations of Tycho Brahe. So let's talk about him first.
Tycho Brahe (1546-1601)
Tycho Brahe - Danish astronomer, astrologer and alchemist of the Renaissance. Kepler was the first in Europe to begin conducting 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, and created some astronomical instruments. One day (November 11, 1572), returning home from a chemical laboratory, he noticed an unusually bright star in the constellation Cassiopeia, which had not been there 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 shine dimmed. 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” - disasters, wars, epidemics and even the end of the world were predicted. Scientific treatises also appeared containing erroneous statements that this was a comet or an atmospheric phenomenon. In 1573, his first book, “On the New Star,” was published. In it, Brahe reported that no parallax (changes in the apparent position of an object relative to a distant background depending on the position of the observer) was detected in this object, and this convincingly proves that the new luminary is a star, and it is not located near the Earth, but at least at a planetary distance. With the appearance of this book, Tycho Brahe was recognized as Denmark's first astronomer. In 1576, by decree of the Danish-Norwegian king Frederick II, Tycho Brahe was granted the island of Ven for life use ( 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 specifically built for astronomical observations. Tycho Brahe named his observatory "Uraniborg" in honor of the muse of astronomy Urania (the name is sometimes translated as "Castle in the Sky"). The design of the building was drawn up by Tycho Brahe himself. In 1584, another observatory castle was built next to Uraniborg: Stjerneborg (translated from Danish as “Star Castle”). Uraniborg soon became the world's best astronomical center, combining observations, teaching students and publishing scientific works. But later, in connection with the change of king. Tycho Brahe lost financial support, and then there was a ban on practicing astronomy and alchemy on the island. The astronomer left Denmark and stopped in Prague.
Soon Uraniborg and all the buildings associated with it were completely destroyed (in our time they have been partially restored).
During this tense time, Brahe came to the conclusion that he needed a young, talented mathematician assistant to process the data accumulated over 20 years. Having learned about the persecution of Johannes Kepler, whose extraordinary mathematical abilities he had already appreciated from their correspondence, Tycho invited him to his place. The scientists were faced with a task: to deduce from observations a new system of the world, which should replace both the Ptolemaic and Copernican ones. He entrusted Kepler with the key planet: Mars, whose movement strongly did not fit not only into Ptolemy’s scheme, but also into Brahe’s own models (according to his calculations, the orbits of Mars and the Sun intersected).
In 1601, Tycho Brahe and Kepler began work on new, refined astronomical tables, which were called “Rudolph” in honor of the emperor; they were completed in 1627 and served astronomers and sailors until the beginning of the 19th century. But Tycho Brahe only managed to give the tables a name. In October he unexpectedly fell ill and died of an unknown illness.
After carefully studying the data of Tycho Brahe, Kepler discovered the laws of planetary motion.
Kepler's laws of planetary motion
Initially, Kepler planned to become a Protestant priest, but thanks to his extraordinary mathematical abilities, he was invited in 1594 to lecture on mathematics at the University of Graz (now Austria). Kepler spent 6 years in Graz. Here in 1596 his first book, “The Secret of the World,” was published. In it, Kepler tried to find the secret harmony of the Universe, for which he compared various “Platonic solids” (regular polyhedra) to the orbits of the five then known planets (he especially singled out the sphere of the Earth). He presented the orbit of Saturn as a circle (not yet an ellipse) on the surface of a ball circumscribed around a cube. The cube, in turn, was inscribed with a ball, which was supposed to represent the orbit of Jupiter. A tetrahedron was inscribed in this ball, circumscribed around a ball representing the orbit of Mars, etc. This work, after further discoveries by Kepler, lost its original meaning (if only because the orbits of the planets turned out to be non-circular); Nevertheless, Kepler believed in the existence of a hidden mathematical harmony of the Universe until the end of his life, and in 1621 he republished “The Secret of the World”, making numerous changes and additions to it.
Being an excellent observer, Tycho Brahe compiled a voluminous work over many years 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 increase accuracy, Brahe used both technical improvements and a special technique for neutralizing observation errors. The systematic nature of the measurements was especially valuable.
Over the course of several years, Kepler carefully studied Brahe's data and, as a result of careful analysis, came to the conclusion that Mars' trajectory is not a circle, but an ellipse, with the Sun at one of its foci - a position known today as Kepler's first law.
Kepler's first law (law of ellipses)
Each planet in the solar system revolves in an ellipse, with the Sun at one of the focuses.
The shape of the 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), and is the semimajor axis. The quantity 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 describes equal areas at equal times. This meant that the further a planet is from the Sun, the slower it moves.
Kepler's second law (law of areas)
Each planet moves in a plane passing through the center of the Sun, and in equal periods of time, the radius vector connecting the Sun and the planet describes equal areas.
There are two concepts associated with this law: perihelion- the point of the orbit closest to the Sun, and aphelion- the most distant point of the orbit. Thus, from Kepler’s second law it follows that the planet moves unevenly around the Sun, having a greater linear speed at perihelion than at aphelion.
Every year at the beginning of January, the Earth moves faster when passing through perihelion, 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, and therefore the movement of the Sun along the ecliptic slows down. The law of areas indicates that the force governing the orbital motion of 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 related as the cubes of the semimajor axes of the planets' orbits. This is true not only for planets, but also for their satellites.
Where and are the periods of revolution of two planets around the Sun, and and are the lengths of the semimajor axes of their orbits.
Newton later established 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 are found to be related, this combination of Kepler's harmonic law and Newton's law of gravity is used to determine the mass of planets and satellites if their orbits and orbital periods are known.
The significance 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 as it moves away or approaches the Sun, and the third allows us to calculate this speed and the period of revolution around the Sun.
Although historically the Keplerian world system 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 spheres carrying planets disappeared, and the concept of a planetary orbit appeared. In the Copernican system, the Earth still occupied a somewhat special position, since it was the only one without epicycles. According to Kepler, the Earth is an ordinary planet, the movement of which is subject to three general laws. All orbits of celestial bodies are ellipses; the common focus of the orbits is the Sun.
Kepler also derived the “Kepler equation,” used in astronomy to determine the positions of celestial bodies.
The laws discovered by Kepler later served Newton basis for the creation of the theory of gravity. 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, as an argument, proposed photometric paradox(this name arose later): if the number of stars is infinite, then in any direction the gaze would encounter 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 provide answers to the most profound 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 later found application in crystallography and coding theory. He compiled one of the first tables of logarithms, and for the first time introduced the most important concept infinitely distant pointintroduced the concept focus of the conic section and reviewed projective transformations of conic sections, including those changing their type.
In physicscoined the term inertia as the innate property of bodies to resist an 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 the hypothesis that the cause of tides is the influence of the Moon on the upper layers of the oceans.
In optics: optics as a science begins with his works. He describes the refraction of light, refraction and the concept of optical image, the general theory of lenses and their systems. Kepler figured out the role of the lens and correctly described the causes of myopia and farsightedness.
TO astrology Kepler had an ambivalent attitude. Two of his statements are cited on this matter. First: " Of course, this astrology is a stupid daughter, but, my God, where would her mother, the highly wise astronomy, go if she didn’t have a stupid daughter! The world is even much more stupid and so stupid that for the benefit of this old reasonable mother, the stupid daughter must chat and lie. And the salary of mathematicians is so insignificant that the mother would probably starve if her daughter did not earn anything" And second: “ People are mistaken in thinking that earthly affairs depend on the heavenly bodies" But, nevertheless, Kepler compiled horoscopes for himself and his loved ones.
He had extraordinary mathematical abilities. At the beginning of the 17th century, as a result of many years of observations of the movements of the planets, as well as based on an analysis of the astronomical observations of Tycho Brahe, Kepler discovered three laws that were later named after him.
Kepler's first law(law of ellipses). Each planet moves in an ellipse, with the Sun at one focus.
Kepler's second law(law of equal areas). Each planet moves in a plane passing through the center of the Sun, and over equal periods of time, the radius vector connecting the Sun and the planet sweeps out equal areas.
Kepler's third law(harmonic law). The squares of the orbital periods of planets around the Sun are proportional to the cubes of the semimajor axes of their elliptical orbits.
Let's take a closer look at each of the laws.
Kepler's first law (law of ellipses)
Each planet in the solar system revolves in an ellipse, with the Sun at one of the focuses.
The first law describes the geometry of the trajectories of planetary orbits. Imagine a section of the side surface of a cone by a plane at an angle to its base, not passing through the base. The resulting figure will be an ellipse. The shape of the ellipse and the degree of its similarity to a circle is characterized by the ratio e = c / a, where c is the distance from the center of the ellipse to its focus (focal distance), a is the semimajor axis. The quantity e is called the eccentricity of the ellipse. At c = 0, and therefore e = 0, the ellipse turns into a circle.
The point P of the trajectory closest to the Sun is called perihelion. Point A, farthest from the Sun, is aphelion. The distance between aphelion and perihelion is the major axis of the elliptical orbit. The distance between aphelion A and perihelion P constitutes the major axis of the elliptical orbit. Half the length of the major axis, the a-axis, is the average distance from the planet to the Sun. The average distance from the Earth to the Sun is called an astronomical unit (AU) and is equal to 150 million km.
Kepler's second law (law of areas)
Each planet moves in a plane passing through the center of the Sun, and over equal periods of time, the radius vector connecting the Sun and the planet occupies equal areas.
The second law describes the change in the speed of movement of planets around the Sun. Two concepts are associated with this law: perihelion - the point of the orbit closest to the Sun, and aphelion - the most distant point of the orbit. The planet moves around the Sun unevenly, having a greater linear speed at perihelion than at aphelion. In the figure, the areas of the sectors highlighted in blue are equal and, accordingly, the time it takes the planet to pass through each sector is also equal. The Earth passes perihelion in early January and aphelion in early July. Kepler's second law, the law of areas, indicates that the force governing the orbital motion of planets is directed towards the Sun.
Kepler's third law (harmonic law)
The squares of the orbital periods of planets around the Sun are proportional to the cubes of the semimajor axes of their elliptical orbits. This is true not only for planets, but also for their satellites.
Kepler's third law allows us to compare the orbits of planets with each other. The farther a planet is from the Sun, the longer the perimeter of its orbit and when moving along its orbit, its full revolution takes longer. Also, with increasing distance from the Sun, the linear speed of the planet’s movement decreases.
where T 1, T 2 are the periods of revolution of planet 1 and 2 around the Sun; a 1 > a 2 are the lengths of the semi-major axes of the orbits of planets 1 and 2. The semi-axis is the average distance from the planet to the Sun.
Newton later discovered that Kepler's third law was not entirely accurate; in fact, it included the mass of the planet:
where M is the mass of the Sun, and m 1 and m 2 are the mass of planets 1 and 2.
Since motion and mass are found to be related, this combination of Kepler's harmonic law and Newton's law of gravity is used to determine the mass of planets and satellites if their orbits and orbital periods are known. Also knowing the distance of the planet to the Sun, you can calculate the length of the year (the time of a complete revolution around the Sun). Conversely, knowing the length of the year, you can calculate the distance of the planet to the Sun.
Three laws of planetary motion discovered by Kepler provided an accurate explanation for the uneven motion of the planets. The first law describes the geometry of the trajectories of planetary orbits. The second law describes the change in the speed of movement of planets around the Sun. Kepler's third law allows us to compare the orbits of planets with each other. The laws discovered by Kepler later served as the basis for Newton to create the theory of gravitation. Newton mathematically proved that all Kepler's laws are consequences of the law of gravitation.
Kepler's formulation:
The planet moves along an ellipse, at one of the foci of which the Sun is located.
Newton generalizes it: firstly, the system star - star (double star), 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, at one of the foci of which there is a body A. The excentricity of the ellipse is determined by the numerical value of the total energy of the system. In a gravitationally unbound system, body B moves along a parabola ( E= 0) or by hyperbola ( E> 0), the foci of which are the body A.
Ellipse
Ellipse (Fig. 33) is an elongated circle with the property that there are two points (foci of the 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).
Line segment AB is called the major axis, respectively the segment A.O. = O.B.- semimajor axis (accepted designation a), segments CD And O.C.- minor axis and semi-axis b. The size of the ellipse is determined by the semimajor axis, the shape is determined by the excentricity e = √(1 - b 2 / a 2). At e= 0 the ellipse degenerates into a circle, when e= 1 - into a parabola, with e> 1 - into a hyperbola, which is better represented as a graph of the function y = 1 / x, rotated 45°. The ellipse has a major semi-axis a> 0, near a parabola a= ∞, for the hyperbola a < 0, что, конечно, только математиче-ская абстракция.
The radius vector of the planet describes equal areas in equal periods of time (Fig. 34).
This statement is similar to the fact that the speed of motion decreases with distance from the Sun, or rather, this is the law of conservation of angular momentum.
If you count the number of days from the vernal equinox (March 21) to the autumn equinox (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 one. In other words, the Earth moves faster in winter than in summer, therefore, it is closer to the Sun in winter. The Earth passes the closest point of its orbit to the Sun—perihelion—on January 6.
Law of conservation of angular momentum
Momentum ( K = mvr) is a physical quantity convenient for describing the movement of a point along a circle or ellipse, parabola, hyperbola, as well as for describing the rotation of a rigid body. Law of conservation of angular momentum(like 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 semimajor axis of the planetary orbit to the cube of the period of revolution of the planet 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— masses of bodies of the system; a And T— semimajor axis and period of revolution of the smaller body (planet, satellite); G— gravitational constant.
It is necessary to pay attention to the constant factor on the right side. In the formula it is given in SI units, but in astronomy the astronomical unit of length (instead of the meter), the year (instead of the second) and the mass of the Sun (instead of the kilogram) are used. Then, as is easy to see, 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 opportunity to directly determine the mass of a celestial body (for example,
Each planet moves in an ellipse, with the Sun at one focus. The law was also discovered by Newton in the 17th century (it is clear that on the basis of Kepler’s laws). Kepler's second law is equivalent to the law of conservation of angular momentum. Unlike the first two, Kepler's third law applies only to elliptical orbits. The German astronomer J. Kepler at the beginning of the 17th century, based on the Copernican system, formulated three empirical laws of motion of the planets of the solar system.
Within the framework of classical mechanics, they are derived from the solution of the two-body problem by passing to the limit → 0, where, are the masses of the planet and the Sun, respectively. We have obtained the equation of a conic section with eccentricity and the origin of the coordinate system at one of the foci. Thus, from Kepler’s second law it follows that the planet moves unevenly around the Sun, having a greater linear speed at perihelion than at aphelion.
3.1. Movement in a gravitational field
Newton established that the gravitational attraction of a planet of a certain mass depends only on its distance, and not on other properties such as composition or temperature. Another formulation of this law: the sectorial speed of the planet is constant. The modern formulation of the first law has been supplemented as follows: in unperturbed motion, the orbit of a moving body is a second-order curve - an ellipse, parabola or hyperbola.
Despite the fact that Kepler's laws were a major step in understanding the motion of planets, they still remained only empirical rules derived from astronomical observations.
For circular orbits, Kepler's first and second laws are satisfied automatically, and the third law states that T2 ~ R3, where T is the orbital period, R is the orbital radius. In accordance with the law of conservation of energy, the total energy of a body in a gravitational field remains unchanged. At E = E1 rmax. In this case, the celestial body moves in an elliptical orbit (planets of the Solar System, comets).
Kepler's laws apply not only to the movement of planets and other celestial bodies in the Solar System, but also to the movement of artificial Earth satellites and spacecraft. Established by Johannes Kepler at the beginning of the 17th century as a generalization of Tycho Brahe’s observational data. Moreover, Kepler studied the movement of Mars especially carefully. Let's look at the laws in more detail.
At c=0 and e=0, the ellipse turns into a circle. This law, like the first two, is applicable not only to the movement of planets, but also to the movement of both their natural and artificial satellites. Kepler is not given, since this was not necessary. Kepler was formulated by Newton as follows: the squares of the sidereal periods of the planets, multiplied by the sum of the masses of the Sun and the planet, are related as the cubes of the semi-major axes of the planets’ orbits.
17th century J. Kepler (1571-1630) based on many years of observations by T. Brahe (1546-1601). Law of areas.) 3. The squares of the periods of any two planets are related as the cubes of their average distances from the Sun. Finally, he assumed that the orbit of Mars was elliptical, and saw that this curve described observations well if the Sun was placed at one of the foci of the ellipse. Kepler then proposed (although he could not clearly prove it) that all planets move in ellipses with the Sun at the focal point.
KEPLER'S LAW OF AREA. 1st law: each planet moves in an elliptical direction. When a stone falls to Earth, it obeys the law of gravity. This force is applied to one of the interacting bodies and is directed towards the other. In particular, I. Newton came to this conclusion in his mental throwing of stones from a high mountain. So, the Sun bends the movement of the planets, preventing them from scattering in all directions.
Kepler, based on the results of Tycho Brahe's painstaking and long-term observations of the planet Mars, was able to determine the shape of its orbit. The action of the Earth and the Sun on the Moon makes Kepler's laws completely unsuitable for calculating its orbit.
The shape of the 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), and is the semimajor axis. Thus, it can be argued that, and therefore the speed of sweeping the area proportional to it, is a constant. of the Sun, and and are the lengths of the semimajor axes of their orbits. The statement is also true for satellites.
Let's calculate the area of the ellipse along which the planet moves. In this case, the interaction between bodies M1 and M2 is not taken into account. The difference will only be in the linear dimensions of the orbits (if the bodies are of different masses). In the world of atoms and elementary particles, gravitational forces are negligible compared to other types of force interactions between particles.
Chapter 3. Fundamentals of celestial mechanics
Gravity controls the movement of the planets in the solar system. Without it, the planets that make up the solar system would scatter in different directions and get lost in the vast expanses of world space. From the point of view of an earthly observer, the planets move along very complex trajectories (Fig. 1.24.1). The geocentric system of Ptolemy lasted for more than 14 centuries and was only replaced by the heliocentric system of Copernicus in the middle of the 16th century.
In Fig. Figure 1.24.2 shows the elliptical orbit of a planet whose mass is much less than the mass of the Sun. Almost all the planets of the Solar System (except Pluto) move in orbits that are close to circular. Circular and elliptical orbits.
Newton was the first to express the idea that gravitational forces determine not only the movement of the planets of the solar system; they act between any bodies in the Universe. In particular, it has already been said that the force of gravity acting on bodies near the Earth’s surface is of gravitational nature. The potential energy of a body of mass m located at a distance r from a stationary body of mass M is equal to the work of gravitational forces when moving mass m from a given point to infinity.
In the limit as Δri → 0, this sum goes into an integral. The total energy can be positive or negative, or equal to zero. The sign of the total energy determines the nature of the movement of the celestial body (Fig. 1.24.6). If the speed of the spacecraft is υ1 = 7.9·103 m/s and is directed parallel to the Earth’s surface, then the ship will move in a circular orbit at a low altitude above the Earth.
Thus, Kepler's first law follows directly from Newton's law of universal gravitation and Newton's second law. 3. Finally, Kepler also noted the third law of planetary motions. The sun, and and are the masses of the planets. In relation to our Solar system, two concepts are associated with this law: perihelion - the point of the orbit closest to the Sun, and aphelion - the most distant point of the orbit.
