Kepler's laws of planetary motion. Fundamentals of astronomy. Movement of celestial bodies. Kepler's laws

Kepler's laws

In the world of atoms and elementary particles, gravitational forces are negligible compared to other types of force interaction 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, it is gravity that keeps us all on Earth.

As for the ancient Egyptian origin of the obelisk, we learn from the Roman historian Pliny, a contemporary of Caligula, which was created for "Nuncoreus, son of Sesostris". Heliopolis, of course, was the heart of the ancient Egyptian religion of the sun and had the same power for the ancient Egyptians that St. Peter has for the Catholics today.

The site was then restored to become the center of the Catholic world: the Vatican. St. Peter's Basilica was begun in 334 AD by Costantino the Great, but it was only completed in the 16th century by the architect and sculptor Bramante, then Raffaello and finally Michelangelo.

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 the attention of people for a long time. The study of the motion of the planets and the structure of the solar system led to the creation of the theory of gravity - the discovery of the law gravity.

The basilica is located above the middle of the upper part of the ancient Vatican Circus of Caligula. Meanwhile, Caligula's obelisk ended near the south wall of the Basilica, almost forgotten in a small alley partially covered with rubbish and debris from before the 15th century. The Pope wanted the base of the obelisk to stand on four bronze statues in Evangelism, and its point was surmounted by a huge bronze statue of Jesus with a golden cross in his hand.

But Papa Niccolò died before he could commission the work, and the project was shelved. When Bernini began working on the design of St. Peter's Square and its colonnade, the Basilica had long been completed by Bramante and Michelangelo. In any case, when Bernini later planned the square, he corrected this mistake and made sure that the east and west axis of the square was aligned exactly to the east.

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). In the center of the universe, Ptolemy placed the Earth, around which the planets and stars moved in large and small circles, as 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 the motion of the planets of the solar system. Kepler used the results of observations of the movement of the planets by the Danish astronomer T. Brahe.

This difference between the alignments of the basilica and the square can be clearly seen when looking at the basilica to the front. The two fountains may look the same, but they were actually built 50 years apart. The obelisk was later used as a gnomon. Rolling over his shadow at noon different time year was marked on the floor with white marble slabs engraved with the months of the year. Templars, Vatican mysteries, Symbols, Sacred architecture.

Our indefatigable curiosity, getting to know our little friends like friends, attracted the attention to talk to the busy Newton, and in order to deceive the expectation, wrote this application dedicated to coupons who want to deepen the topics concerning in the story.

Kepler's first law (1609):

All planets move in elliptical orbits with the Sun at one of the foci.

On fig. 1.24.2 shows the elliptical orbit of the planet, the mass of which is much less than the mass of the Sun. The sun is at one of the foci of the ellipse. Point closest to the sun P trajectory is called perihelion, dot A, the furthest from the Sun aphelion. The distance between aphelion and perihelion is the major axis of the ellipse.

The reason why astronomical science made no progress for fourteen centuries after Ptolemy can be found for various reasons. First of all, the collapse of the Roman Empire and the clear division between Western and Eastern Europe that characterized it: this contributed to the fact that the West almost completely forgot the scientific baggage of the Greeks. Moreover, the spread of Christianity led him to interpret the Bible literally. The books of Genesis include naive astronomical ideas borrowed from other peoples.

Such statements were not particularly dangerous for the Jewish people, who had never been interested in astronomy, but caused a serious regression and slowed down the development of this science in the West. It is also time to ridicule the sphericity of the Earth, the first great discovery of the Greeks. We must wait for the sixteenth century to start the real astronomical revolution. Although Nicolaus Copernicus is usually given credit for such epochal changes, it must be remembered that he was in practice only the initiator of the long and painful process of cultural renewal that led to the Newtonian vision of the world.

Almost all the planets of the solar system (except Pluto) move in orbits close to circular.

Kepler's second law (1609):

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

Rice. 1.24.3 illustrates Kepler's 2nd law.

Kepler's second law is equivalent to law of conservation of angular momentum. On 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:

The Copernican model, heliocentric, didn't really make any big simplifications in the calculations to predict the positions of the stars. One of the fundamental reasons for Copernicus' refusal was the desire to eliminate the equal, that is, the presence of a circular motion, the speed of which was uniform not with respect to the center of the circle, but at a different point, clearly contrasting with the Platonic dictation. To explain all phenomena without resorting to equations, he still had to introduce eccentric spheres and epicycles, so the Copernican system today is not the known eliocentric system.

Here is the angular velocity ( see §1.6).

angular momentum L in absolute value is equal to the product of the modules of the vectors and

Therefore, if, according to Kepler's second law, then the angular momentum L stays the same while moving.

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

The elliptical motions of the planets were still interpreted as the composition of circular motions. It was only with Kepler's empirical laws that this ancient "prejudice" was overcome. The real revolutionary aspect of Copernicus' work is that after him, many scientists began to believe in the physical reality of the model.

Let's now take a closer look at what were the fixed points on which Copernicus was based: 1 - There is no single center for all celestial orbits. 2 - The center of the Earth is not the center of the Universe, but only the center of gravity of heavy bodies and the lunar orbit3. All orbits "surround" the Sun, which is "in the middle", as the center of the world is close to the Sun. 4 - The distance of the Sun from the Earth is very small compared to the distance of the fixed stars. 5 - Each motorcycle owned fixed stars, comes from them, but from the Earth.

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

On fig. 1.24.4 shows two orbits, one of which is circular with a radius R, and the other is elliptical with a major semiaxis 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 the most important step in understanding the motion of the planets, they still remained only empirical rules derived from astronomical observations. Kepler's laws needed a theoretical justification. A decisive step in this direction has been taken Isaac Newton who discovered in 1682 law of gravity:

Thus, the Earth makes a complete rotation around its axis, while the firmament or the last sky remains motionless. 6 - The movements that seem to us to belong to the Sun are associated with the Earth and its orbit in which it revolves around the Sun, like any other planet. 7 - The retrograde and direct motion of the planets is not caused by them, but by the Earth. A single movement of the Earth is enough to explain the large number of irregularities in the heavens.

From these preliminary axioms, we immediately see that Copernicus' break with the old cosmology is not as clear as one might expect. The center of the Copernican cosmos is not in the center of the Sun, but near the center of the Sun. It is precisely located in the center of the Earth's orbit, which is eccentric for the Sun.

where M and m are the masses of the sun and the planet, r- the distance between them, G\u003d 6.67 10 -11 N m 2 / kg 2 - gravitational constant. Newton was the first to suggest that gravitational forces determine not only the movement of the planets of the solar system; they act between any bodies of the Universe. In particular, it has already been said that the force of gravity acting on bodies near the Earth's surface is of a gravitational nature.

As you can see, the movements are always circular and uniform. Copernicus always uses epicidides, except for the Earth and the Moon. The Earth is shown revolving around the Sun, but in fact the center of its orbit is close to the star, as will be seen in the following figure. The moon revolves around them, and both have no epic. The positions of the fixed stars are not to scale, in fact they should be placed at a much greater distance. The next figure refers to the movement of the Earth to the movement of Mars. It is well known in it that the Earth circulates in an inner circle centered on a point that does not coincide with the center of the Sun.

For circular orbits, Kepler's first and second laws hold automatically, and the third law states that T 2 ~ R 3 , where T is the circulation period, R is the radius of the orbit. From here it is possible to obtain the dependence of the gravitational force on the distance. When the planet moves along a circular trajectory, a force acts on it, which arises due to the gravitational interaction of the planet and the Sun:

This point is the actual center of the system and takes its name from the Mean Sun. The sun is instead the center of Mars' landing. The figure clearly shows another privilege that Copernicus has mentioned agrees with the Earth: she, unlike other planets, is devoid of an epicyclo. Copernicus does not even lower the Earth to the rank of a common planet.

The Earth and the Moon do not have epicycles. In conclusion, we can say that the theory of Copernicus is a synthesis between a bold innovation and the techno-cultural bag of the past. Putting the Sun "almost" in the center of the system, however, made it possible to greatly facilitate the explanation of the obvious irregularities in planetary weather. But the relationship with Ptolemy shows a clear connection with Aristotelian constructions. It was not at all that Kepler should have said that Copernicus would have been a better interpreter of nature than Ptolemy.

If a T 2 ~ R 3 , then

The property of conservation of gravitational forces ( see §1.10) allows us to introduce the notion potential energy . For the forces of universal gravitation, it is convenient to count the potential energy from an infinitely distant point.

Shortly before the beginning of the telescopic era, the Danish astronomer Tycho Brahe achieved an unsurpassed level of accuracy in astronomical observations with the naked eye. He really made science astronomical observation rule of absolute life. Despite the fact that he abandoned the "almost" heliolating system of Copernicus and again considered a stationary Earth, its contribution was undoubtedly one of the indispensable factors for reaching the truth, which would be fully understood later thanks to the observations of Galileo, Kepler's empirical laws and later. with the ingenious synthesis of Newton, who started classical mechanics.

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

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

The Earth is again in the center of space, and the Moon and the Sun revolve around in circular orbits. Mercury, Venus, Mars, Jupiter and Saturn circulate around the sun, as well as fixed stars. The model can be seen as an attempt to reconcile old and new ideas. Moreover, Tycho never developed his system until he had a complete planetary theory like that of Ptolemy and Copernicus, although he may have had an idea. However, although the Moon and the Sun revolve around them directly, the other planets revolve around the Sun and with it around the Earth.

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

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

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

The Sun's orbit becomes deflection and planetary trajectories of huge epicycles. Despite his ideas, Brae's contribution to the success of the heliocentric model was extremely important. In addition to Kepler's discoveries, he completely destroyed the concept of solid balls on which the planets lay, representing a real orbit. In fact, in its model, the orbit of the Sun intersects the orbits of Mercury, Venus and Mars, which would be impossible if such planets were moved in their motion by ancient and never before doubted crystal balls.

Moreover, even if it is stationary, the Earth is no longer the real center of rotation of the entire universe, since this role is mainly assumed by the Sun. The German Johannes Kepler was a mathematician, optician, astronomer and an acclaimed musician. He entered the history of science to formulate the famous three empirical laws that describe the motion of the planets. However, for important reasons, Kepler's laws only give a description of the motion of the planets around the Sun, they give us no indication of why the planets move in this way.

The total energy can be positive and negative, and also equal to zero. Sign full energy determines the nature of the movement of a 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).

First Law: The orbit that a planet moves around the Sun is an ellipse from which the Sun occupies one of the two lights. Second Law: The vector ray connecting the center of the Sun with the center of the planet rotates equal areas in equal times. This means that the planet moves faster when it is closest to the Sun than when it is further away.

Third Law: Cube Ratio big half the axis of the orbit and the square of the period of revolution is a constant valid for all planets. This law is very important: in fact, the periods of rotation of the planets around the Sun are very easy to measure, so it is enough to know the distance from one planet from the Sun to immediately calculate the distances from the Sun of all other planets.

At E = E 2 = 0 the body can move 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 supply of kinetic energy.

Kepler's laws apply not only to the motion of the 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 of the satellite in a circular orbit near the surface of the Earth.

from here

second cosmic speed called the minimum speed to be reported space ship at the surface of the Earth, so that, having overcome the gravity of the earth, it 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.

from here

Rice. 1.24.7 illustrates cosmic velocities. If the speed of the spacecraft is υ 1 = 7.9·10 3 m/s and is directed parallel to the Earth's surface, then the spacecraft will move in a circular orbit at a low altitude above the Earth. At initial velocities exceeding υ 1 but less 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.

The movement of cosmic bodies has been observed by man for a very long time. Also in Ancient Greece models of the motion of the planets of the solar system around the sun were invented. These models were very complex, since the apparent movement of the planets across the sky is described by very complex lines, they were called epicycles. The first attempt to describe the universe was made in ancient Greece in the second century AD by Ptolemy (Fig. 1).

Rice. 1. Geocentric model of K. Ptolemy ()

He proposed to place the Earth at the center of the Universe, and the movements of the planets were described by large and small circles, which were called Ptolemy's epicycles.

It was not until the 16th century that Copernicus proposed replacing Ptolemy's geocentric model of the world with a heliocentric one. That is, place the Sun at the center of the Universe and assume that all the planets and the Earth together with them move around the Sun (Fig. 2).

Rice. 2. heliocentric model N. Copernicus ()

At the beginning of the 17th century, the German astronomer Johannes Kepler, having processed a huge amount of astronomical information received by the Danish astronomer Tycho Brahe, proposed his own empirical laws, which since then have been called Kepler's laws.

All planets solar system move along some curves, which are called an ellipse. An ellipse is one of the simplest mathematical curves, the so-called second-order curve. In the Middle Ages, they were called conical intersections - if you cross a cone or cylinder with a certain plane, we get the very curve along which the planets of the solar system move.

Rice. 3. Curve of planetary motion ()

This curve (Fig. 3) has two highlighted points, which are called foci. For each point of the ellipse, the sum of the distances from it to the foci is the same. At one of these foci is the center of the Sun (F), the point of the curve closest to the Sun (P) is called perihelion, and the farthest (A) is called aphelion. The distance from the perihelion to the center of the ellipse is called the semi-major axis, and the vertical distance from the center of the ellipse to the ellipse is called the minor semi-axis of the ellipse.

As the planet moves along the ellipse, the radius vector connecting the center of the Sun with this planet describes a certain area. For example, during the time ∆t the planet moved from one point to another, the radius vector described a certain area ∆S.

Rice. 4. Kepler's second law ()

Kepler's second law states: for the same time intervals, the radius vectors of the planets describe the same areas.

Figure 4 shows the angle ∆Θ, this is the angle of rotation of the radius vector for some time ∆t and the momentum of the planet (), directed tangentially to the trajectory, decomposed into two components - the momentum component along the radius vector () and the momentum component, in the direction , perpendicular to the radius vector (⊥).

Let's make calculations connected with Kepler's second law. Kepler's statement that equal areas are covered in equal intervals means that the ratio of these quantities is a constant. The ratio of these quantities is often called the sectoral velocity, this is the rate of change in the position of the radius vector. What is the area ∆S swept by the radius vector in time ∆t? This is the area of a triangle whose height is approximately equal to the radius vector, and whose base is approximately equal to r ∆ω, using this statement, we write the value of ∆S as ½ of the height per base and divide by ∆t, we get the expression:

This is the rate of change of angle, that is, the angular velocity.

Final Result:

The square of the distance to the center of the Sun, multiplied by the angular velocity of motion in this moment time is a constant value.

But if we multiply the expression r 2 ω by the mass of the body m, then we get a value that can be represented as the product of the length of the radius vector and the momentum in the direction transverse to the radius vector:

This value, equal to the product of the radius vector and the perpendicular component of the momentum, is called the moment of momentum.

Kepler's second law is the statement that moment of momentum in a gravitational field is a conserved quantity. This implies a simple but very important statement: at the points of least and greatest distance to the center of the Sun, that is, aphelion and perihelion, the velocity is directed perpendicular to the radius vector, so the product of the radius vector and the velocity at one point is equal to this product at another point.

Kepler's third law states that the ratio of the square of the period of revolution of a planet around the Sun to the cube of the semi-major axis is the same value for all planets in the solar system.

Rice. 5. Arbitrary trajectories of the planets ()

Figure 5 shows two arbitrary planetary trajectories. One has an explicit form of an ellipse with a semi-axis length (a), the second has the form of a circle with a radius (R), the time of revolution along any of these trajectories, that is, the period of revolution, is associated with the semi-axis length or radius. And if the ellipse turns into a circle, then the semi-major axis just becomes the radius of this circle. Kepler's third law states that if the length of the semi-major axis is equal to the radius of the circle, the periods of revolution of the planets around the Sun will be the same.

For the case of a circle, this ratio can be calculated using Newton's second law and the law of motion of a body in a circle, this constant is 4π 2 divided by the constant of universal gravitation (G) and the mass of the Sun (M).

Thus, it can be seen that if we generalize gravitational interactions, as Newton did, and assume that all bodies participate in gravitational interaction, Kepler's laws can be extended to the movement of satellites around the Earth, to the movement of satellites around any other planet, and even to the movement of satellites Moon around the center of the moon. Only on the right side of this formula, the letter M will mean the mass of the body that attracts the satellites. All satellites of a given space object will have the same ratio of the square of the period of revolution (T 2) to the cube of the semi-major axis (a 3). This law can be extended to all bodies in the Universe in general and even to the stars that make up our Galaxy.

In the second half of the 20th century, it was noticed that some stars that are far enough from the center of our Galaxy do not obey this Kepler law. This means that we do not know everything about how gravity works in the size of our Galaxy. One possible explanation for why distant stars move faster than required by Kepler's third law is that we don't see all of the mass of the galaxy. A significant part of it may consist of a substance that is not observed by our devices, does not interact electromagnetically, does not emit or absorb light, but participates only in gravitational interaction. Such matter was called hidden mass or dark matter. The problem of dark matter is one of the main problems of physics of the 21st century.

The topic of the next lesson: systems material points, center of mass, law of motion of the center of mass.

Bibliography