Formulate 3 statements about the solar system. Generalized Kepler's laws. Movement in a gravitational field

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

A straight line connecting the Sun and a planet cuts off equal areas in equal periods of time.

The squares of the periods of revolution of the planets around the Sun are related to the cubes of the semimajor axes of their orbits.

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

Alas, having compared 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 harmonious framework he outlined. As the contemporary British biologist J. B. S. Haldane aptly noted, “the idea of ​​the Universe as a geometrically perfect work of art turned out to be yet another beautiful hypothesis destroyed by ugly facts.” The only result of Kepler's youthful impulse that survived the centuries was a model of the solar system, made by the scientist himself and presented as a gift to his patron, Duke Frederick von Württemburg. In this beautifully executed metal artifact, all the orbital spheres of the planets and the regular polyhedra inscribed in them are hollow containers that do not communicate with each other, which on holidays were supposed to be filled with various drinks to treat the Duke’s guests.

Only after moving to Prague and becoming an assistant to the famous Danish astronomer Tycho Brahe (1546-1601) did Kepler come across ideas that truly immortalized his name in the annals of science. Tycho Brahe spent his entire life collecting data. astronomical observations and accumulated enormous amounts of information about the movements of the planets. After his death they came into the possession of Kepler. These records, by the way, had 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 encountered a problem that, even with modern computers, might seem intractable to someone, and Kepler had no choice but to carry out all the calculations by hand. Of course, like most astronomers of his time, Kepler was already familiar with heliocentric system Copernicus ( cm. Copernican principle) and knew that the Earth revolves around the Sun, as evidenced by the above-described model of the solar system. But how exactly does the Earth and other planets rotate? Let's imagine the problem as follows: you are on a planet that, firstly, rotates around its axis, and secondly, revolves around the Sun in an orbit unknown to you. Looking into the sky, we see other planets that are also moving in orbits unknown to us. Our task is to determine, based on observations made on our planet rotating around its axis around the Sun globe, the geometry of orbits and the speed of movement of other planets. This is exactly what Kepler ultimately managed to do, after which, based on the results obtained, he derived his three laws!

First law describes the geometry of the trajectories of planetary orbits. You may remember from school course geometry that an ellipse is a set of points on a plane, the sum of the distances from which to two fixed points is tricks— equal to a constant. If this is too complicated for you, there is another definition: imagine a section of the side 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 states that the orbits of the planets are ellipses, with the Sun at one of the foci. Eccentricities(degree of elongation) of orbits and their distance from the Sun in perihelion(the point closest to the Sun) and apohelia(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 Tycho Brahe's observational data, Kepler concluded that planetary orbits are a set of nested ellipses. Before him, this simply had not occurred to any astronomer.

The historical significance of Kepler's first law cannot be overestimated. Before him, astronomers believed that the planets moved 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, first of all a philosophical position, a kind of immutable fact, 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 a circle and a sphere, which means the planets move in a circle (and even today I have to dispel this misconception over and over again among my students). The main thing is that, having gained access to the extensive observational data of Tycho Brahe, Johannes Kepler was able to step over this philosophical prejudice, seeing that it did not correspond to the facts - just as Copernicus dared to remove the Earth from the center of the universe, faced with arguments that contradicted persistent geocentric ideas, which also consisted of the “improper behavior” of planets in orbits.

Second Law describes the change in the speed of the planets around the Sun. I have already given its formulation in a formal form, but in order to better understand it physical meaning, remember your childhood. You've probably had the opportunity to spin around a pole on the playground, grabbing it with your hands. In fact, the planets orbit the sun in a similar way. The farther the elliptical orbit takes a planet from the Sun, the slower its movement; the closer it is to the Sun, the faster the planet moves. Now imagine a pair of line segments connecting two positions of the planet in its orbit with the focus of the ellipse in which the Sun is located. Together with the ellipse segment lying between them, they form a sector, the area of ​​which is precisely the “area that is cut off by a straight line segment.” This is exactly what the second law talks about. How closer planet towards 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 its speed of movement increases.

In the first two laws we're talking about about the specifics of the orbital trajectories of a single planet. Third Law Kepler allows you to compare the orbits of planets with each other. It says that the farther a planet is from the Sun, the longer it takes to complete a full revolution when moving in orbit and the longer, accordingly, the “year” lasts on this planet. Today we know that this is due to two factors. Firstly, the farther a planet is from the Sun, the longer the perimeter of its orbit. Secondly, with increasing distance from the Sun, the linear speed movements of the planet.

In his laws, Kepler simply stated facts, having studied and generalized the results of observations. If you had asked him what caused the ellipticity of the orbits or the equality of the areas of the sectors, he would not have answered you. This simply followed from the analysis he carried out. If you asked him about the orbital motion of planets in other star systems, he also would not have anything to answer you. He would have to start all over again - accumulate observational data, then analyze it and try to identify patterns. That is, he simply would have no reason to believe that another planetary system obeys the same laws as the Solar system.

One of the greatest triumphs classical mechanics Newton lies precisely in the fact that it provides a fundamental justification 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 law of conservation of angular momentum through rigorous mathematical calculations. And if so, we can be sure that Kepler's laws are equally applicable to any planetary system anywhere in the Universe. Astronomers searching for new planetary systems in space (and quite a few of them have already been discovered) time after time, 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 played and continues to play an important role in modern cosmology. By observing distant galaxies, astrophysicists detect faint signals emitted by hydrogen atoms orbiting in very distant orbits from the galactic center - much further than stars usually are. Using the Doppler effect in the spectrum of this radiation, scientists determine the rotation rates of the hydrogen periphery of the galactic disk, and from them - the angular velocities of galaxies as a whole ( cm. also Dark Matter). I am glad that the works of the scientist who firmly set us on the path correct understanding devices of our Solar system, and today, centuries after his death, play such an important role in studying the structure of the vast Universe.

Between the spheres of Mars and Earth there is a dodecahedron (dodecahedron); between the spheres of Earth and Venus - the icosahedron (twenty-hedron); between the spheres of Venus and Mercury there is an octahedron (octahedron). The resulting design was presented by Kepler in cross-section in a detailed three-dimensional drawing (see figure) in his first monograph, “The Cosmographic Mystery” (Mysteria Cosmographica, 1596).— Translator's note.

I. Kepler spent his whole life trying to prove that our solar system is some kind of mystical art. Initially, he tried to prove that the structure of the system is similar to regular polyhedra from ancient Greek geometry. In Kepler's time, six planets were known to exist. They were believed to be placed in crystal spheres. According to the scientist, these spheres were located in such a way that polyhedra fit exactly between adjacent ones correct form. Between Jupiter and Saturn a cube was placed, inscribed in the external environment into which the sphere was inscribed. Between Mars and Jupiter there is a tetrahedron, etc. After many years of observing celestial objects, Kepler's laws appeared, and he refuted his theory of polyhedra.

Laws

The geocentric Ptolemaic system of the world was replaced by a heliocentric type system created by Copernicus. Still later, Kepler identified around the Sun.

After many years of observing the planets, Kepler's three laws emerged. Let's look at them in the article.

First

According to Kepler's first law, all the planets in our system move along a closed curve called an ellipse. Our luminary is located at one of the focuses of the ellipse. There are two of them: these are two points inside the curve, the sum of the distances from which to any point of the ellipse is constant. After long observations, the scientist was able to reveal that the orbits of all the planets of our system are located almost in the same plane. Some celestial bodies move in elliptical orbits close to a circle. And only Pluto and Mars move in more elongated orbits. Based on this, Kepler's first law was called the law of ellipses.

Second Law

Studying the movement of bodies allows the scientist to establish that it is greater during the period when it is closer to the Sun, and less when it is at its maximum distance from the Sun (these are the perihelion and aphelion points).

Kepler's second law states the following: each planet moves in a plane passing through the center of our star. At the same time, the radius vector connecting the Sun and the planet under study describes equal areas.

Thus, it is clear that bodies move unevenly around the yellow dwarf, having a maximum speed at perihelion and a minimum at aphelion. In practice, this can be seen in the movement of the Earth. Every year at the beginning of January, our planet moves faster during its passage through perihelion. Because of this, the movement of the Sun along the ecliptic occurs faster than at other times of the year. In early July, the Earth moves through aphelion, causing the Sun to move more slowly along the ecliptic.

Third Law

According to Kepler's third law, a connection is established between the period of revolution of a planet around a star and its average distance from it. The scientist applied this law to all the planets of our system.

Explanation of laws

Kepler's laws could only be explained after Newton's discovery of the law of gravity. According to it, physical objects take part in gravitational interaction. It has universal universality, to which all objects of material type and physical fields are subject. According to Newton, two motionless bodies act on each other with a force proportional to the product of their weight and inversely proportional to the square of the intervals between them.

Indignant movement

The movement of bodies in our solar system is controlled by the gravitational force of the yellow dwarf. If bodies were attracted only by the force of the Sun, then the planets would move around it exactly according to Kepler's laws of motion. This type the displacements are called unperturbed or Keplerian.

In reality, all objects in our system are attracted not only by our star, but also by each other. Therefore, none of the bodies can move exactly in an ellipse, hyperbola or circle. If a body deviates during motion from Kepler's laws, then this is called perturbation, and the motion itself is called perturbed. This is what is considered real.

The orbits of celestial bodies are not fixed ellipses. During attraction by other bodies, the orbital ellipse changes.

Contribution of I. Newton

Isaac Newton was able to derive the law from Kepler's laws of planetary motion universal gravity. To solve cosmic-mechanical problems, Newton used universal gravity.

After Isaac, progress in the field of celestial mechanics consisted of the development of mathematical science applied to the solution of equations expressing Newton's laws. This scientist was able to establish that the gravity of a planet is determined by its distance and mass, but indicators such as temperature and composition do not have any effect.

In his scientific work Newton showed that Kepler's third law was not entirely accurate. He showed that when making calculations it is important to take into account the mass of the planet, since the movement and weight of the planets are related. This harmonic combination shows the connection between Keplerian laws and the law of gravity identified by Newton.

Astrodynamics

The application of Newton's and Kepler's laws became the basis for the emergence of astrodynamics. This is a section of celestial mechanics that studies the movement of artificially created cosmic bodies, namely: satellites, interplanetary stations, and various ships.

Astrodynamics deals with calculations of spacecraft orbits, and also determines what parameters to launch, what orbit to launch, what maneuvers need to be carried out, and planning the gravitational effect on ships. And these are not all the practical tasks that are posed to astrodynamics. All the results obtained are used to carry out a wide variety of space missions.

Celestial mechanics, which studies the movement of natural cosmic bodies under the influence of gravity, is closely related to astrodynamics.

Orbits

An orbit is understood as the trajectory of a point in a given space. In celestial mechanics, it is generally accepted that the trajectory of a body in the gravitational field of another body has a significantly larger mass. In a rectangular coordinate system, the trajectory can have the shape of a conical section, i.e. be represented by a parabola, ellipse, circle, hyperbola. In this case, the focus will coincide with the center of the system.

For a long time it was believed that orbits should be circular. For quite a long time, scientists tried to choose exactly the circular option of movement, but they did not succeed. And only Kepler was able to explain that the planets do not move in a circular orbit, but in an elongated one. This made it possible to discover three laws that could describe the movement of celestial bodies in orbit. Kepler discovered the following elements of the orbit: the shape of the orbit, its inclination, the position of the plane of the body's orbit in space, the size of the orbit, and the time reference. All these elements determine the orbit, regardless of its shape. When calculating the main coordinate plane may be the plane of the ecliptic, galaxy, planetary equator, etc.

Numerous studies show that according to geometric shape orbits can be elliptical or circular. There is a division into closed and open. According to the angle of inclination of the orbit to the plane of the earth's equator, orbits can be polar, inclined and equatorial.

According to the period of revolution around the body, orbits can be synchronous or sun-synchronous, synchronous-daily, quasi-synchronous.

As Kepler said, all bodies have a certain speed of motion, i.e. orbital speed. It can be constant throughout the entire revolution around the body or change.

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. German astronomer I. Kepler in early XVII century, based on the Copernican system, he 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 artificial satellites Earth and spaceships. 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. Sign 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.

Kepler's laws

In the world of atoms and elementary particles, gravitational forces are negligible compared to other types of force interactions between particles. It is very difficult to observe the gravitational interaction between the various bodies around us, even if their masses are many thousands of kilograms. However, it is gravity that determines the behavior of “large” objects such as planets, comets and stars, and it is gravity that keeps us all on Earth.

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.

The patterns of planetary motion have attracted people's attention for a long time. The study of the movement of planets and the structure of the solar system led to the creation of the theory of gravity - the discovery of the law of universal gravitation.

From the point of view of an earthly observer, the planets move along very complex trajectories (Fig. 1.24.1). The first attempt to create a model of the Universe was made Ptolemy(~140 g). At the center of the universe, Ptolemy placed the Earth, around which planets and stars moved in large and small circles, like in a round dance.

Geocentric system Ptolemy lasted more than 14 centuries and was replaced only in the middle of the 16th century heliocentric the Copernican system. In the Copernican system, the trajectories of the planets turned out to be simpler. German astronomer I. Kepler at the beginning of the 17th century, based on the Copernican system, he formulated three empirical laws of motion of the planets of the solar system. Kepler used the results of observations of the planetary movements of the Danish astronomer T. Brahe.

Kepler's first law (1609):

All planets move in elliptical orbits, with the Sun at one focus.

In Fig. Figure 1.24.2 shows the elliptical orbit of a planet whose mass is much less than the mass of the Sun. The sun is at one of the ellipse's foci. Point closest to the Sun P trajectory is called perihelion, dot A, farthest from the Sun – aphelion. The distance between aphelion and perihelion is the major axis of the ellipse.

Almost all the planets of the Solar System (except Pluto) move in orbits that are close to circular.

Kepler's second law (1609):

The radius vector of the planet describes equal areas in equal periods of time.

Rice. Figure 1.24.3 illustrates Kepler's 2nd law.

Kepler's second law is equivalent law of conservation of angular momentum. In Fig. 1.24.3 shows the momentum vector of the body and its components and the area swept by the radius vector in a short time Δ t, approximately equal to the area of ​​a triangle with base rΔθ and height r:

Here – angular velocity ( see §1.6).

Momentum L in absolute value equal to the product of the moduli of vectors and

Therefore, if, according to Kepler’s second law, then the angular momentum L remains unchanged when moving.

In particular, since the velocities of the planet at perihelion and aphelion are directed perpendicular to the radius vectors and from the law of conservation of angular momentum it follows:

Kepler's third law is true for all planets in the solar system with an accuracy of greater than 1%.

In Fig. 1.24.4 shows two orbits, one of which is circular with a radius R, and the other is elliptical with a semimajor axis a. The third law states that if R = a, then the periods of revolution of the bodies in these orbits are the same.

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. Kepler's laws needed theoretical justification. A decisive step in this direction has been taken Isaac Newton, who opened in 1682 law of universal gravitation:

Where M And m– masses of the Sun and planet, r– the distance between them, G= 6.67·10 –11 N·m 2 /kg 2 – gravitational constant. 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.

For circular orbits, Kepler's first and second laws are satisfied automatically, and the third law states that T 2 ~ R 3, where T is the circulation period, R– radius of the orbit. From this we can obtain the dependence of gravitational force on distance. When a planet moves along a circular path, it is acted upon by a force that arises due to the gravitational interaction of the planet and the Sun:

If T 2 ~ R 3 then

The property of conservatism of gravitational forces ( see §1.10) allows us to introduce the concept potential energy . For the forces of universal gravity, it is convenient to count the potential energy from a point at infinity.

Potential energy of a body of massm located at a distancer from a stationary body of massM , is equal to the work of gravitational forces when moving massm from a given point to infinity.

The mathematical procedure for calculating the potential energy of a body in a gravitational field consists of summing up the work on small displacements (Fig. 1.24.5).

The law of universal gravitation applies not only to chiseled masses, but also to spherically symmetrical bodies. The work done by the gravitational force on a small displacement is:

In the limit at Δ r i→ 0 this sum goes into the integral. As a result of calculations for potential energy, we obtain the expression

In accordance with the law of conservation of energy, the total energy of a body in a gravitational field remains unchanged.

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).

At E = E 1 < 0 тело не может удалиться от центра притяжения на расстояние r > r max. In this case, the celestial body moves along elliptical orbit(planets of the solar system, comets).

At E = E 2 = 0 the body can move away to infinity. The speed of the body at infinity will be zero. The body moves along parabolic trajectory.

At E = E 3 > 0 movement occurs along hyperbolic trajectory. The body moves away to infinity, having a reserve of kinetic energy.

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. In this case, the center of gravity is the Earth.

First cosmic speed is the speed at which a satellite moves in a circular orbit near the Earth's surface.

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Second escape velocity is called the minimum speed that must be reported spaceship near the surface of the Earth, so that it, having overcome gravity, turns into an artificial satellite of the Sun (artificial planet). In this case, the ship will move away from the Earth along a parabolic trajectory.

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Rice. 1.24.7 illustrates escape velocities. If the speed of the spacecraft is equal to υ 1 = 7.9 10 3 m/s and is directed parallel to the surface of the Earth, then the ship will move in a circular orbit at a low altitude above the Earth. At initial velocities exceeding υ 1 but lower than υ 2 = 11.2·10 3 m/s, the ship’s orbit will be elliptical. At an initial speed of υ 2, the ship will move along a parabola, and at an even higher initial speed, along a hyperbola.

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 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.