Kepler's laws: first, second and third. Kepler's first law Kepler's first law in Newton's formulation

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 of the correct shape fit exactly between the neighboring ones. 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 of movement is 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 of universal gravitation from Kepler's laws of planetary motion. 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 making calculations, the main coordinate plane can be the plane of the ecliptic, galaxy, planetary equator, etc.

Numerous studies show that the geometric shape of the orbits can be elliptical and round. 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.

In the microcosm, during the interaction of elementary particles - atoms, molecules - nuclear and electromagnetic interactions are dominant. It is almost impossible to observe the gravitational interaction of elementary particles. Scientists have to resort to very big tricks in order to measure the gravitational interaction of bodies whose mass is hundreds, thousands of kilograms. However, on a cosmic scale, all other interactions, except gravitational ones, are practically unnoticeable. The movement of planets, satellites, asteroids, comets, stars in the galaxy is completely described by gravitational interaction.

He proposed placing the Earth at the center of the Universe, and the movements of the planets were described by large and small circles, which were called Ptolemaic epicycles.

Only in the 16th century did Copernicus propose 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 along with them move around the Sun (Fig. 2).

Rice. 2. Heliocentric model of N. Copernicus ()

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

All planets of the Solar System move along some curves 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 intersect a cone or cylinder with a certain plane, you will get the same curve along which the planets of the solar system move.

Rice. 3. Planetary motion curve ()

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. The center of the Sun (F) is located at one of these foci; the point of the curve closest to the Sun (P) is called perihelion, and the farthest point (A) is called aphelion. The distance from the perihelion to the center of the ellipse is called the semimajor axis, and the vertical distance from the center of the ellipse to the ellipse is the semiminor axis of the ellipse.

As a planet moves along an 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: over equal periods of time, the radius vectors of the planets describe equal areas.

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

Let us perform calculations related to Kepler's second law. Kepler's statement that equal areas are traversed in equal intervals means that the ratio of these quantities is a constant. The ratio of these quantities is often called sectoral velocity; this is the rate of change in the position of the radius vector. What is the area ∆S that the radius vector sweeps over time ∆t? This is the area of ​​a triangle, the height of which is approximately equal to the radius vector, and the base is approximately equal to r ∆ω, using this statement, we write the value ∆S in the form of ½ the height per base and divide by ∆t, we get the expression:

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

Final result:

,

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

But if we multiply the expression r 2 ω by the body mass m, we obtain 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 quantity, equal to the product of the radius vector and the perpendicular component of the impulse, is called “angular momentum”.

Kepler's second law is a statement that angular momentum in a gravitational field is a conserved quantity. This leads to a simple but very important statement: at the points of the smallest and greatest distance to the center of the Sun, that is, aphelion and perihelion, the speed is directed perpendicular to the radius vector, therefore the product of the radius vector and the speed 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 semimajor axis is the same for all planets in the Solar System.

Rice. 5. Arbitrary trajectories of planets ()

Figure 5 shows two arbitrary trajectories of the planets. One has the explicit form of an ellipse with the length of the semi-axis (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 length of the semi-axis or with the radius. And if the ellipse turns into a circle, then the semimajor axis becomes the radius of this circle. Kepler's third law states that in the case when the length of the semimajor 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 is clear that if we generalize gravitational interactions, as Newton did, and assume that all bodies participate in gravitational interactions, 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 Moons 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 orbital period (T 2) to the cube of the semimajor axis (a 3). This law can be extended to all bodies in the Universe and even to the stars that make up our Galaxy.

In the second half of the twentieth century, it was noticed that some stars that are quite far from the center of our Galaxy do not obey this Kepler law. This means that we don't know everything about how gravity works across the size of our Galaxy. One possible explanation for why distant stars move faster than required by Kepler's third law is the following: we do not see the entire mass of the Galaxy. A significant part of it may consist of matter that is not observable by our instruments, does not interact electromagnetically, does not emit or absorb light, and participates only in gravitational interaction. This substance was called hidden mass or dark matter. Dark matter problems are one of the main problems of physics of the 21st century.

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

Bibliography

1. Tikhomirova S.A., Yavorsky B.M. Physics (basic level) - M.: Mnemosyne, 2012.
2. Kabardin O.F., Orlov V.A., Evenchik E.E. Physics-10. M.: Education, 2010.
3. Open Physics ()

1. Elementy.ru ().
2. Physics.ru ().
3. Ency.info().

Homework

1. Define Kepler's first law.
2. Define Kepler's second law.
3. Define Kepler's third law.

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.

The two greatest scientists, far ahead of their time, created a science called celestial mechanics, that is, they discovered the laws of motion of celestial bodies under the influence of gravity, and even if their achievements were limited to this, they would still have entered the pantheon of the greats of this world. It so happened that they did not intersect in time. Only thirteen years after Kepler's death Newton was born. Both of them were supporters of the heliocentric Copernican system. Having studied the motion of Mars for many years, Kepler experimentally discovered three laws of planetary motion, more than fifty years before Newton discovered the law of universal gravitation. Not yet understanding why the planets move the way they do. It was hard labor and brilliant foresight. But Newton used Kepler’s laws to test his law of gravitation. All three of Kepler's laws are consequences of the law of gravity. And Newton discovered it at the age of 23. At this time, 1664 - 1667, the plague raged in London. Trinity College, where Newton taught, was dissolved indefinitely so as not to worsen the epidemic. Newton returns to his homeland and in two years makes a revolution in science, making three important discoveries: differential and integral calculus, an explanation of the nature of light and the law of universal gravitation. Isaac Newton was solemnly buried in Westminster Abbey. Above his grave stands a monument with a bust and the epitaph “Here lies Sir Isaac Newton, the nobleman who, with the torch of mathematics in his hand, was the first to prove, with the torch of mathematics in his hand, the movements of the planets, the paths of comets and the tides of the oceans... Let mortals rejoice that such an adornment of the human race exists.”

The merit of discovering the laws of planetary motion belongs to the outstanding German scientist, astronomer and mathematician, Johannes Kepler(1571 – 1630) – a man of great courage and extraordinary love for science.

He showed himself to be an ardent supporter of the Copernican system of the world and set out to clarify the structure of the Solar system. Then this meant: to know the laws of planetary motion, or, as he put it, “to trace God’s plan during the creation of the world.” At the beginning of the 17th century. Kepler, studying the revolution of Mars around the Sun, established three laws of planetary motion.

Kepler's first law:Each planet revolves around the Sun in an ellipse, with the Sun at one focus.

Under the influence of gravity, one celestial body moves in the gravitational field of another celestial body along one of the conic sections - a circle, ellipse, parabola or hyperbola.

An ellipse is a flat closed curve that has the property that the sum of the distances of each point from two points, called foci, remains constant. This sum of distances is equal to the length of the major axis of the ellipse. Point O is the center of the ellipse, F1 and F2 are the foci. The Sun is in this case at focus F1.

The point of the orbit closest to the Sun is called perihelion, the farthest point is called aphelion. The line connecting any point of the ellipse with the focus is called the radius vector. The ratio of the distance between the foci to the major axis (to the largest diameter) is called eccentricity e. The greater the eccentricity, the more elongated the ellipse is. The semimajor axis of the ellipse a is the average distance of the planet from the Sun.

Comets and asteroids also move in elliptical orbits. For a circle e = 0, for an ellipse 0< е < 1, у параболы е = 1, у гиперболы е > 1.

The orbits of the planets are ellipses, differ little from circles; their eccentricities are small. For example, the eccentricity of the Earth's orbit is e = 0.017.

Kepler's second law: The radius vector of the planet describes equal areas in equal periods of time (determines the speed of the planet’s orbit). The closer a planet is to the Sun, the faster it is.

The planet travels from point A to A1 and from B to B1 in the same time. In other words, the planet moves fastest at perihelion, and slowest when it is at its greatest distance (at aphelion). Thus, the speed of Comet Halley at perihelion is 55 km/s, and at aphelion 0.9 km/s.

Mercury, which is closest to the Sun, orbits the Sun in 88 days. Venus moves behind it, and a year on it lasts 225 Earth days. The Earth revolves around the Sun in 365 days, that is, exactly one year. The Martian year is almost twice as long as the Earth's. A Jupiter year is equal to almost 12 Earth years, and distant Saturn circles its orbit in 29.5 years! In short, the farther the planet is from the Sun, the longer the year on the planet. And Kepler tried to find a relationship between the sizes of the orbits of various planets and the time of their revolution around the Sun.

On May 15, 1618, after many unsuccessful attempts, Kepler finally established a very important relation known as

Kepler's third law:The squares of the periods of revolution of the planets around the Sun are proportional to the cubes of their average distances from the Sun.

If the orbital periods of any two planets, for example the Earth and Mars, are denoted by Tz and Tm, and their average distances from the Sun are a z and m, then Kepler’s third law can be written as an equality:

T 2 m / T 2 z = a 3 m / a 3 z.

But the period of revolution of the Earth around the Sun is equal to one year (Тз = 1), and the average distance between the Earth and the Sun is taken as one astronomical unit (аз = 1 AU). Then this equality will take a simpler form:

T 2 m = a 3 m

The orbital period of a planet (in our example, Mars) can be determined from observations. It is 687 Earth days, or 1.881 years. Knowing this, it is not difficult to calculate the average distance of the planet from the Sun in astronomical units:

Those. Mars is on average 1,524 times farther from the Sun than our Earth. Consequently, if the orbital time of a planet is known, then its average distance from the Sun can be found from it. In this way, Kepler was able to determine the distances of all the planets known at that time:

Mercury – 0.39,

Venus – 0.72,

Earth – 1.00

Mars – 1.52,

Jupiter – 5.20,

Saturn - 9.54.

Only these were relative distances - numbers showing how many times a particular planet is further from the Sun or closer to the Sun than the Earth. The true values ​​of these distances, expressed in earthly measures (in km), remained unknown, because the length of the astronomical unit - the average distance of the Earth from the Sun - was not yet known.

Kepler's third law connected the entire solar family into a single harmonious system. The search took nine difficult years. The scientist’s perseverance won!

Conclusion: Kepler's laws theoretically developed the heliocentric doctrine and thereby strengthened the position of new astronomy. Copernican astronomy is the wisest of all works of the human mind.

Subsequent observations showed that Kepler's laws apply not only to the planets of the Solar System and their satellites, but also to stars physically connected to each other and revolving around a common center of mass. They formed the basis of practical astronautics, since all artificial celestial bodies move according to Kepler’s laws, starting with the first Soviet satellite and ending with modern spacecraft. It is no coincidence that in the history of astronomy Johannes Kepler is called the “legislator of the sky.”

At the beginning of the 16th century, the Polish astronomer N. Copernicus (1473–1543) substantiated the heliocentric system, according to which the movements of celestial bodies are explained by the movement of the Earth (as well as other planets) around the Sun and the daily rotation of the Earth. Copernicus' theory of observation was perceived as an entertaining fantasy. In the 16th century this statement was considered by the church to be heresy. It is known that G. Bruno, who openly supported the heliocentric system of Copernicus, was condemned by the Inquisition and burned at the stake.

Kepler's first law. All planets move in ellipses, with the Sun at one of the focuses (Fig. 7.6).

Rice. 7.6

Newton solved the inverse problem of mechanics and from the laws of planetary motion obtained an expression for the gravitational force:

As we already know, gravitational forces are conservative forces. When a body moves in a gravitational field of conservative forces along a closed trajectory, the work is zero.
The property of conservatism of gravitational forces allowed us to introduce the concept of potential energy.

Potential energy body mass m, located at a distance r from a large body of mass M, There is

Thus, 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.

At E < 0 тело не может удалиться от центра притяжения на расстояние r 0 < r max. In this case, the celestial body moves along elliptical orbit(planets of the Solar system, comets) (Fig. 7.8)

Rice. 7.8

The period of revolution of a celestial body in an elliptical orbit is equal to the period of revolution in a circular orbit of radius R, Where R– semimajor axis of the orbit.

At E= 0 the body moves along a parabolic trajectory. The speed of a body at infinity is zero.

At E< 0 движение происходит по гиперболической траектории. Тело удаляется на бесконечность, имея запас кинетической энергии.

First cosmic speed is the speed of movement of a body in a circular orbit near the surface of the Earth. To do this, as follows from Newton’s second law, the centrifugal force must be balanced by the gravitational force:

From here

From here
Third escape velocity– the speed of movement at which a body can leave the solar system, overcoming the gravity of the Sun:

υ 3 = 16.7·10 3 m/s.

Figure 7.8 shows the trajectories of bodies with different cosmic velocities.