Kepler's laws. Cosmic speeds
It can be shown that , where s- sectorial speed, i.e. the area described by the radius vector of a moving body per unit time.
Thus, sectorial speed for a moving body is a constant value- this is the wording Kepler's second generalized law , and relation (3.11) is a mathematical expression of this law.
Let some body of mass m moves around a central body of mass M along the ellipse. Then the sectorial speed is 
, where is the area of the ellipse, T is the period of revolution of the body, a And b are the major and minor semi-axes of the ellipse, respectively. The semi-axes of the ellipse are related to each other by the relation: , where e- eccentricity of the ellipse. Taking this into account, as well as formula (3.8), we obtain: 
, Where 
. Hence, after transformations we have:
It's there second recording form Kepler's third generalized law.
If we consider the movement of two planets around the Sun, i.e. around the same body ( M 1 ==M 2), and neglect the masses of the planets ( T 1 =m 2 = 0) in comparison with the mass of the Sun, we obtain formula (2.7), derived by Kepler from observations. Since the masses of the planets are insignificant compared to the mass of the Sun, Kepler’s formula agrees quite well with observations.
Formulas (3.12) and (3.13) play a big role in astronomy: they make it possible to determine the masses of celestial bodies (see § 3.6).
Differential equation (2) has the following first integrals:
Area integral
Where - constant angular momentum vector. Due to constancy, the orbit of the body will be a flat curve. If we enter polar coordinates in this plane r And υ, then the area integral can be written as:
………………….. (4)
from which follows Kepler's second law (law of areas). If is the area described by the radius vector during the time interval, then the sectorial speed:
.
(5)
(6)
In other words, the area described by the radius vector is proportional to the time intervals of movement.
The force included in the equation of relative motion is potential. The potential of this force is determined by the expression
Energy integral. From the equation of motion (2) follows the law of conservation of energy
(7)
Here is a constant equal to the total mechanical energy divided by the mass of the moving body.
Since then when equation (7) will be satisfied for any r , and movement is not limited in space. At ˂ 0, motion is limited in space.
In general, the orbital equation (solution to equation (2)) has the form:
, (8)
where is the true anomaly and is the eccentricity.
The magnitude of the eccentricity is determined by the value of the total energy and is equal to:
.
(9)
the focal parameter is:
(10)
As can be seen from (9), three types of trajectories are possible:
0 ≤ e ˂ 1 (һ˂0)- ellipse ( e = 0– circle);
e = 1 (һ=0) - parabola;
e > 1 (һ>0) - hyperbole.
Formula (8) defines the analytical expression Kepler's first generalized law.(diagram 8)
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.
In general, during elliptical motion, the point of the orbit closest to the central body is called periapsis, and the most distant – apocenter. When moving around the Sun, these points are called perihelion And aphelion.
Kepler's third generalized law. For elliptical motion it is easy to obtain a connection between the sidereal period of revolution T and semi-major axis A orbits. Considering that the area of the ellipse and the radius - the vector describes it over the period T, we have from (5): . On the other hand, from (10) it follows that
……
(11)
Equating these two expressions, we get:
(12)
This relationship represents Kepler's third generalized law. It is valid for any two attracting material bodies, be they planets, double stars or artificial celestial bodies, because the right side of relation (12) includes universal constants.
Let M 1 – mass of the Sun, m 1 – mass of the planet, a 1 And T 1 – respectively, the semimajor axis and the sidereal period of the planet’s revolution around the Sun. If there is another system, such as a planet M 2 and a satellite of the planet with a mass m 2 , which orbits the planet with a period T 2 at medium distance a 2 , then for these two systems the third generalized Kepler’s law (12) is valid, which takes the form:
=
(13)
When two bodies of low mass move around one central body, for example, when planets move around the Sun, in formula (13) we should put M 1 = M 2 , m 1 « M 1 , m 2 « M 2 , and then
that is, we obtain Kepler's third empirical law.
From the expression for eccentricity (9) and (11) it is easy to find that 
Then the energy integral equation (7) takes the form:
(14)
This formula is valid for any type of movement. For an elliptical orbit a > 0, for parabolic orbit a = , and for hyperbolic a ˂ 0.
Characteristic velocities of Keplerian motion. For every distance r from the central body there are two characteristic velocities: one at r = a – circular speed
(15)
having which, the revolving body moves in a circular orbit; the other is parabolic speed
in which a moving body leaves the central body in a parabola a = . Obviously, always.
When a body rotates in an elliptical orbit, the average orbital speed coincides with the circular speed
(16)
Where a - semimajor axis of the orbit and - sidereal period of revolution. From equalities (14) and (16) we find that at any point of the elliptical orbit at a distance r from the central body the orbiting body has a speed
(17)
The speed at the pericenter is determined at r = q = a (1 - e), and the speed at the apocenter is at r = Q = a (1 + e).
In a limited two-body problem, and is determined only by the mass of the central body. Neglecting the mutual attraction of the planets to a first approximation, we can consider the motion of each of them around the Sun under the conditions of a limited two-body problem. Then any planet has an average speed
Two body problem
Equation of motion
= - (M + m)
Integral
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 of whose faces are equal regular polygons.) At the time of Kepler, 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 guests of the duke.
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 collected astronomical observation data throughout his life 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 was faced with 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 the heliocentric system of 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 observational data made on our globe rotating around its axis around the Sun, the geometry of the orbits and speeds 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 your school geometry course that an ellipse is a set of points on a plane, the sum of the distances from which to two fixed points 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 celestial structure, unlike the earthly one, is perfect in its harmony, and since the most perfect of geometric figures are the circle and the sphere, it means that 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 its formal form, but to better understand its 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 a planet's elliptical orbit takes it 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. The closer the planet is to the Sun, the shorter the segments. But in this case, in order for the sector to cover an equal area in equal time, the planet must travel a greater distance in its orbit, which means its speed of movement increases.
The first two laws deal with 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, as the distance from the Sun increases, the linear speed of the planet’s movement also decreases.
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 his analysis. If you asked him about the orbital motion of planets in other star systems, he also would not have anything to answer. 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 of Newton's classical mechanics 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 apply equally 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 has 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 put us on the path to a correct understanding of the structure of our solar system, and today, centuries after his death, play such an important role in the study of the structure of the 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.
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 of the focuses.
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.
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.
The law of universal gravitation was discovered by Newton based on Kepler's three laws.Kepler's first law. All planets move in ellipses, with the Sun at one of the focuses (Fig. 7.6).
Rice. 7.6
Kepler's second law. The radius vector of the planet describes equal areas in equal times (Fig. 7.7).
Almost all planets (except Pluto) move in orbits that are close to circular. For circular orbits, Kepler's first and second laws are satisfied automatically, and the third law states that T 2 ~ R 3 (T– circulation period; R– orbit radius).
Newton solved the inverse problem of mechanics and from the laws of planetary motion obtained an expression for the gravitational force:
| (7.5.2) |
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
Second escape velocity is called the speed of movement of a body along a parabolic trajectory. It is equal to the minimum speed that must be imparted to a body on the surface of the Earth so that it, having overcome gravity, becomes an artificial satellite of the Sun (artificial planet). To do this, it is necessary that the kinetic energy be no less than the work done to overcome the Earth’s gravity:
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.