Solar system visible movements of celestial bodies. Fundamentals of Celestial Mechanics

The two most significant successes of classical natural science, based on Newtonian mechanics, were the almost exhaustive description of the observed motion of celestial bodies and the explanation of the ideal gas laws known from experiment.

Kepler's laws. Initially, it was believed that the Earth is stationary, and the movement celestial bodies seemed very complicated. Galileo was one of the first to suggest that our planet is no exception and also moves around the Sun. This concept was met with rather hostility. Tycho Brahe decided not to take part in discussions, but to take up direct measurements of the coordinates of bodies on celestial sphere. He devoted his whole life to this, but not only did he not draw any conclusions from his observations, but he did not even publish the results. Later, Tycho's data came to Kepler, who found a simple explanation for the observed complex trajectories by formulating three laws of motion of the planets (and the Earth) around the Sun (Fig. 6_1):

1. The planets move in elliptical orbits, in one of the focuses of which is the Sun.

2. The speed of the planet changes in such a way that the areas swept by its radius vector for equal intervals time are equal.

3. The periods of revolution of the planets of one solar system and the semi-major axes of their orbits are related by:

The complex movement of the planets on the "celestial sphere" observed from the Earth, according to Kepler, arose as a result of the addition of these planets in elliptical orbits with the movement of the observer, who, together with the Earth, orbital movement around the sun and daily rotation around the planet's axis.

direct evidence daily rotation Earth was an experiment set by Foucault, in which the plane of oscillation of a pendulum rotated relative to the surface of the rotating Earth.

Law of gravity. Kepler's laws perfectly described the observed movement of the planets, but did not reveal the reasons leading to such movement (for example, it could well be considered that the reason for the movement of bodies in Keplerian orbits was the will of some creature or the desire of the celestial bodies themselves to harmony). Newton's theory of gravitation indicated the cause that determined the movement of cosmic bodies according to Kepler's laws, correctly predicted and explained the features of their movement in more difficult cases, made it possible to describe many phenomena on a cosmic and terrestrial scale in the same terms (the movement of stars in a galactic cluster and the fall of an apple on the Earth's surface).

Newton found the correct expression for gravitational force arising from the interaction of two point bodies (bodies whose dimensions are small compared to the distance between them):

(2)
,

which, together with the second law, if the mass of the planet m is much less mass stars M, led to the differential equation

(3)
,

admitting an analytical solution. Without involving any additional physical ideas, purely mathematical methods it is fashionable to show that under appropriate initial conditions (sufficiently small initial distance to the star and the speed of the planet) cosmic body will rotate along closed, stable elliptical orbit in in full agreement with Kepler's laws (in particular, Kepler's second law is a direct consequence of the law of conservation of angular momentum, which is fulfilled during gravitational interactions, since the moment of force (2) relative to the massive center is always zero). At a sufficiently high initial velocity (its value depends on the mass of the star and initial position) the cosmic body moves along a hyperbolic trajectory, eventually moving away from the star to an infinite distance.

An important property of the law of gravity (2) is its conservation mathematical form when gravitational interaction non-point bodies in the case of a spherically symmetric distribution of their masses over volume. In this case, the role of R is played by the distance between the centers of these bodies.

Movement of celestial bodies in the presence of perturbations. Strictly speaking, Kepler's laws are fulfilled exactly only in the case of motion of only one body near another, which has significantly larger mass, provided that these bodies are spherical. With minor deviations from the spherical shape (for example, due to the rotation of a star, it can “flatten” somewhat), the orbit of the planet ceases to be closed and is an ellipse precessing around the star.

Another common disturbance is gravitational influence planets of one star system Each other. Keplerian orbits are stable with respect to weak perturbations, i.e., having experienced the impact of a close-flying neighbor, the planet tends to return to its original trajectory. In the presence of strong perturbations (the passage of a massive body at a short distance), the problem of motion becomes much more complicated and cannot be solved analytically. numerical calculations show that in this case the trajectories of the planets cease to be ellipses and represent open curves.

According to Newton's third law, there is a force acting on the star from the side of the planets. In the case of M>>m, the acceleration of the star is negligibly small and it can be considered stationary. In the presence of two bodies of commensurate masses that are attracted to each other, their stable joint motion in elliptical orbits around a common center of gravity . It is obvious that a more massive body moves along an orbit of a smaller radius. In the case of planets moving around a star, this effect is hardly noticeable. however, systems were found in space that make the described movement - double stars . A numerical calculation of the motion of the planets in a binary star system shows that their orbits are essentially non-stationary, the distance from the planet to the stars varies rapidly over a very wide range. The inevitable rapid change climate on the planets makes it very problematic the possibility of biological evolution. Even less likely is the emergence of technical civilizations on the planets of systems double stars, since the complex non-periodic motion of the planets leads to the observable motion of bodies on the “celestial sphere” that is difficult to decipher, significantly complicating the formulation of Kepler’s laws and, as a consequence, the development classical mechanics(Fig. 6_2).

The structure of the solar system. It is well known that the bulk of the solar system (about 99.8%) falls on its only star - Sun. Total mass planets is only 0.13% of the total. On other bodies of the system (comets, planetary satellites, asteroids and meteoritic substance) accounts for only 0.0003% of the mass. From the above figures it follows that Kepler's laws for the motion of the planets in our system must be carried out very well. Significant deviations from elliptical orbits can occur only in the case of a close (compared to the distance to the Sun) flight past one of the planets: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune or Pluto (this is especially true for the most massive of the planets - Jupiter). It was observations of the perturbation of the orbit of Neptune that made it possible to predict and then discover Pluto - the most distant of known planets our system.

where is the radius of the Earth's orbit. In contrast to Kepler's laws, relation (4) does not follow from Newton's laws in any way and has not yet received a theoretical justification, although the orbits of all currently known planets are satisfactorily described by this formula. The only exception is the value n=3, for which there is no planet in the calculated orbit. Instead, it was found asteroid belt - small bodies on a planetary scale irregular shape. Empirical laws that are not supported by existing theory can play a positive role in research, since they also reflect objective reality(perhaps not quite accurate and even somewhat distorted).

The hypothesis of the pre-existing fifth planet, Phaethon, seemed attractive, destroyed to pieces by the giant gravitational attraction of its massive neighbor, Jupiter, however quantitative analysis motion of the planet - the giant showed the failure of this assumption. Apparently, the mentioned problem can be solved only on the basis of a complete theory of the origin and evolution of the planets of the solar system, which does not yet exist. A very attractive theory of the joint origin of the sun and planets from a single gas cloud, compressed under the action of gravitational forces, is in conflict with the observed uneven distribution torque(momentum) between the star and the planets. Models of the origin of planets as a result of the gravitational capture by the Sun of bodies arriving from distant space, the effects caused by the explosion of supernovae are discussed. In most “scenarios” for the development of the solar system, the existence of the asteroid belt is somehow associated with its close proximity to the massive planet systems.

The currently known properties of the planets of the solar system allow us to divide them into two groups. First four planets terrestrial group characterized by relatively small masses and high densities the substances that make them up. They consist of a molten iron core surrounded by a silicate shell - the bark. The planets have gaseous atomospheres. Their temperatures are mainly determined by the distance to the Sun and decrease with its increase. Beginning with Jupiter group of giant planets mainly composed of light elements (hydrogen and helium), the pressure of which in inner layers increases to huge values due to gravitational compression. As a result, as they approach the center, the gases gradually pass into a liquid and, possibly, into a solid state. It is assumed that in central regions pressure is so great that hydrogen exists in metal phase, which has not yet been observed on the Earth even in laboratory conditions. The planets of the second group have a large number of satellites. At Saturn, their number is so great that, with insufficient magnification, the planet seems to be surrounded by a system of continuous rings (Fig. 6_3).

The problem of the existence of life on other planets still arouses increased interest in the near-scientific fields. At present, it can be stated with sufficient certainty that in the protein forms familiar to modern natural science, life on the planets of the solar system (of course, with the exception of the Earth) does not exist. The reason for this is primarily the smallness of the physicochemical range of conditions that allow the possibility of the existence organic molecules and the course of vital chemical reactions with their participation (not too high and low temperatures, narrow pressure range, presence of oxygen, etc.). The only planet besides the Earth, the conditions on which do not clearly contradict the possibility of the existence of protein life, is Mars. However, sufficiently detailed studies of its surface using interplanetary stations"Mars", "Marioner" and "Viking" showed that life does not exist on these planets even in the form of microorganisms (Fig. 6_4).

As for the question of the existence of non-protein forms of extraterrestrial life, its serious discussion should be preceded by a strict formulation of the most generalized concept of life, but this problem has not yet received a generally accepted satisfactory solution. (One gets the impression that the discovery of life forms that are significantly different from our usual imagination may not arouse any noticeable interest in the non-scientific public at all. It is not very difficult to imagine the creation of computer viruses that can replicate in networks and can evolve, it is much more difficult to imagine a reaction to this in society, other than the annoyance of users who lost programs).

On the nature of gravitational forces. Newton's law gravity refers to fundamental laws classical natural science. The methodological weakness of Newton's concept was his refusal to discuss the mechanisms leading to the emergence of gravitational forces (“I do not invent hypotheses”). After Newton, attempts were repeatedly made to create a theory of gravity. The vast majority of approaches are associated with the so-called hydrodynamic gravity models , who are trying to explain the emergence of gravitational forces by mechanical interactions of massive bodies with an intermediate substance, to which one or another name is attributed: “ether”, “graviton flow”, “vacuum”, etc. Attraction between bodies arises as a result of rarefaction of the Medium, which occurs either when it is absorbed by massive bodies, or when its flows are screened by them. All these theories have a common significant drawback: correctly predicting the dependence of force on distance (2), they inevitably lead to one more unobservable effect: deceleration of bodies moving relative to the introduced substance.

A significant new step in the development of the concept of gravitational interaction was made by A. Einstein, who created general relativity .

All cosmogonic hypotheses can be divided into several groups. According to one of them, the Sun and all the bodies of the solar system: planets, satellites, asteroids, comets and meteoroids - were formed from a single gas and dust cloud, or dust cloud. According to the second, the Sun and its family have various origins, so that the Sun was formed from one gas and dust cloud (nebulae, globules), and the rest of the celestial bodies of the Solar System - from another cloud, which was captured in some not entirely clear way by the Sun into its orbit and separated in some, even more in an incomprehensible way to many of the most various bodies(planets, their satellites, asteroids, comets and meteoroids) having the most various characteristics: mass, density, eccentricity, the direction of orbit and the direction of rotation around its axis, the inclination of the orbit to the plane of the Sun's equator (or ecliptic) and the inclination of the plane of the equator to the plane of its orbit.
Nine major planets revolve around the Sun in ellipses (slightly different from circles) in almost the same plane. In order of distance from the Sun, these are Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. In addition to them, there are many small planets (asteroids) in the solar system, most of which move between the orbits of Mars and Jupiter. The space between the planets is filled with extremely rarefied gas and space dust. It is pierced by electromagnetic radiation.
Sun 109 times more earth in diameter and about 333,000 times more massive than Earth. The mass of all the planets is only about 0.1% of the mass of the Sun, so it controls the movement of all members of the solar system by the force of its attraction.

Configuration and conditions of visibility of planets

The configurations of the planets are called some more characteristic mutual arrangements planets, earth and sun.
The conditions for the visibility of planets from the Earth differ sharply for the inner planets (Venus and Mercury), whose orbits lie inside the Earth's orbit, and for the outer planets (all the rest).
The inner planet may be between the Earth and the Sun or behind the Sun. In such positions, the planet is invisible, as it is lost in the rays of the Sun. These positions are called conjunctions of the planet with the Sun. At inferior conjunction, the planet is closest to Earth, and at superior conjunction, it is farthest from us.

Synodic periods of planetary revolution and their relationship with sidereal periods

The period of revolution of the planets around the Sun in relation to the stars is called the stellar or sidereal period.
How closer planet to the Sun, the more its linear and angular velocity and a shorter sidereal period of revolution around the Sun.
However, from direct observations, it is not the sidereal period of the planet's revolution that is determined, but the time interval that flows between its two successive configurations of the same name, for example, between two serial connections(oppositions). This period is called the synodic period. Having determined the synodic periods from observations, the sidereal periods of the planets are found by calculation.
The synodic period of the outer planet is the period of time after which the Earth overtakes the planet by 360 ° as they move around the Sun.

Kepler's laws

The merit of discovering the laws of planetary motion belongs to the outstanding German scientist Johannes Kepler(1571 -1630). AT early XVII in. Kepler, studying the circulation of Mars around the Sun, established three laws of planetary motion.

Kepler's first law . Each planet revolves in an ellipse with the Sun at one of its foci.

Kepler's second law (law of areas). The radius-vector of the planet for the same intervals of time describes equal areas.

Kepler's third law . The squares of the sidereal periods of the planets are related as the cubes of the semi-major axes of their orbits.

The average distance of all planets from the Sun in astronomical units can be calculated using Kepler's third law. Having determined the average distance of the Earth from the Sun (i.e., the value of 1 AU) in kilometers, one can find in these units the distances to all the planets of the solar system. The semi-major axis of the earth's orbit is taken as astronomical unit distances (=1 a.e.)
The classic method for determining distances was and remains the goniometric geometric method. They determine the distances to distant stars, to which the radar method is not applicable. The geometric method is based on the phenomenon parallax shift.

Parallactic displacement is a change in direction to an object when the observer moves..

EXAMPLE OF SOLVING THE PROBLEM

A task. Oppositions of some planet are repeated in 2 years. What is the semi-major axis of its orbit?

Given	SOLUTION The semi-major axis of the orbit can be determined from Kepler's third law: , and the sidereal period - from the ratio between the sidereal and synodic periods: ,
- ?

The size and shape of the earth

In photographs taken from space, the Earth looks like a ball illuminated by the Sun.
The exact answer about the shape and size of the Earth is given degree measurements, i.e., measurements in kilometers of the length of an arc of 1 ° at different places on the surface of the Earth. Degree measurements have shown that the length of 1° arc of the meridian in kilometers in the polar region is the largest (111.7 km), and the smallest at the equator (110.6 km). Therefore, at the equator, the curvature of the Earth's surface is greater than at the poles, and this indicates that the Earth is not a ball. The equatorial radius of the Earth is greater than the polar one by 21.4 km. Therefore, the Earth (like other planets) due to rotation is compressed at the poles.
A ball, equal in size to our planet, has a radius of 6370 km. This value is considered to be the radius of the Earth.
The angle at which the radius of the Earth is seen perpendicular to the line of sight is called horizontal parallax.

Mass and density of the Earth

The law of universal gravitation allows you to determine one of the most important characteristics celestial bodies - the mass, in particular the mass of our planet. Indeed, based on the law of universal gravitation, the acceleration free fall g=(G*M)/r 2 . Therefore, if the values of the acceleration of free fall, the gravitational constant and the radius of the Earth are known, then its mass can be determined.
Substituting in the specified formula value g \u003d 9.8 m / s 2, G \u003d 6.67 * 10 -11 N * m 2 / kg 2,

R \u003d 6370 km, we find that the mass of the Earth is M \u003d 6 x 10 24 kg. Knowing the mass and volume of the Earth, we can calculate its average density.

Since ancient times, people have observed in the sky such phenomena as the apparent rotation of the starry sky, the change in the phases of the moon, sunrise and sunset heavenly bodies, the apparent movement of the Sun across the sky during the day, solar eclipses, change in the height of the Sun above the horizon during the year, lunar eclipses. It was clear that all these phenomena are connected, first of all, with the movement of celestial bodies, the nature of which people tried to describe with the help of simple visual observations, correct understanding and the explanation of which evolved over the centuries.

First written references about celestial bodies arose in ancient egypt and Sumer. The ancients distinguished three types of bodies in the firmament of heaven: stars, planets and "tailed stars". The differences come just from observations: Stars remain motionless relative to other stars for quite a long time. Therefore, it was believed that the stars were "fixed" on the celestial sphere. As we now know, due to the rotation of the Earth, each star "draws" a circle in the sky.

The planets, on the contrary, move across the sky, and their movement is visible naked eye within an hour or two. Even in Sumer, 5 planets were found and identified: Mercury, Venus, Mars, Jupiter, Saturn. To them, the Sun and the Moon were added to the heap. Total: 7 planets. "Tailed" stars of the comet. Appeared infrequently, symbolized troubles.

Kepler's Laws I. Each planet moves in an ellipse with the Sun at one of its foci. II.(law equal areas). The radius vector of the planet describes equal areas in equal time intervals. III. The squares of the periods of revolutions of the planets around the Sun are proportional to the cubes of the major semi-axes of their elliptical orbits. The three laws of planetary motion relative to the sun were empirically derived by the German astronomer Johannes Kepler at the beginning of the 17th century. This became possible thanks to many years of observations by the Danish astronomer Tycho Brahe.

The most simply visible movement of the planets and the Sun is described in the frame of reference associated with the Sun. This approach has been called heliocentric system world and was proposed by the Polish astronomer Nicolaus Copernicus (). AT ancient times and up to Copernicus, it was believed that the Earth is located at the center of the Universe and all celestial bodies revolve along complex trajectories around it. This system of the world is called the geocentric system of the world.

After the recognition of the revolutionary heliocentric system of the world of Copernicus, after Kepler formulated the three laws of motion of celestial bodies and destroyed centuries-old naive ideas about simple roundabout planets around the Earth, proved by calculations and observations that the orbits of the movement of celestial bodies can only be elliptical, it finally became clear that the apparent movement of the planets consists of: the movement of the observer on the surface of the Earth the rotation of the Earth around the Sun own movements celestial bodies

The complex apparent motion of the planets in the celestial sphere is due to the revolution of the planets of the solar system around the sun. The very word "planet" in ancient Greek means "wandering" or "tramp". The trajectory of a celestial body is called its orbit. The velocities of the planets in their orbits decrease with the distance of the planets from the Sun. The nature of the movement of the planet depends on which group it belongs to. Therefore, in relation to the orbit and conditions of visibility from the Earth, the planets are divided into internal (Mercury, Venus) and external (Mars, Saturn, Jupiter, Uranus, Neptune, Pluto), or, respectively, in relation to the Earth's orbit, into lower and upper.

The outer planets are always turned to the Earth by the side illuminated by the Sun. The inner planets change their phases like the moon. The greatest angular distance of a planet from the Sun is called elongation. The greatest elongation at Mercury is 28°, at Venus - 48°. At eastern elongation inner planet visible in the west, in the rays of the evening dawn, shortly after sunset. Evening (eastern) elongation of Mercury During the western elongation, the inner planet is visible in the east, in the rays of dawn, shortly before sunrise. The outer planets can be at any angular distance from the Sun.

The phase angle of the planet is called the angle between the beam of light incident from the Sun on the planet and the beam reflected from it towards the observer. The phase angles of Mercury and Venus vary from 0° to 180°, so Mercury and Venus change phases just like the Moon. Near inferior conjunction, both planets have the largest angular dimensions, but look like narrow crescents. At phase angle ψ = 90°, half of the disk of planets is illuminated, phase φ = 0.5. In superior conjunction, the lower planets are fully illuminated, but are poorly visible from the Earth, as they are behind the Sun.

Since, during observations from the Earth, the movement of the planets around the Sun is also superimposed on the movement of the Earth in its orbit, the planets move across the sky from east to west ( direct movement), then from west to east ( backtracking). Moments of direction change are called stops. If you put this path on the map, you get a loop. The size of the loop is the smaller, the greater the distance between the planet and the Earth. The planets describe loops, and not just move back and forth in a single line, solely due to the fact that the planes of their orbits do not coincide with the plane of the ecliptic. Such a complex loop-like character was first noticed and described using the example of the apparent motion of Venus.

It is a known fact that the movement of certain planets can be observed from the Earth in a strictly certain time year, this is due to their position over time in the starry sky. The characteristic mutual arrangements of the planets relative to the Sun and the Earth are called planetary configurations. Internal and outer planets are different: for the lower planets these are conjunctions and elongations (the largest angular deviation of the planet's orbit from the orbit of the Sun), for the upper planets these are quadratures, conjunctions and oppositions.

If T is the Earth, P 1 is the inner planet, S is the Sun, the celestial conjunction is called an inferior conjunction. In the "ideal" inferior conjunction, Mercury or Venus transits across the disk of the Sun. If T is the Earth, S is the Sun, P 1 is Mercury or Venus, the phenomenon is called an upper conjunction. In the “ideal” case, the planet is covered by the Sun, which, of course, cannot be observed due to the incomparable difference in the brightness of the stars. For the Earth-Moon-Sun system, a new moon occurs in the lower conjunction, and a full moon occurs in the upper one.

In their movement in the celestial sphere, Mercury and Venus never go far from the Sun (Mercury no further than 18° 28°; Venus no further than 45° 48°) and can be either to the east or to the west of it. The moment of greatest angular removal of the planet to the east of the Sun is called eastern or evening elongation; to the west by western or morning elongation.

Let us introduce the concepts of specific physical quantities characterizing the motion of the planets and allowing some calculations: full turn around the sun in relation to the stars. The synodic period of a planet's revolution is the time interval S between two successive configurations of the same name.

Used literature: Used literature: 1) D. Ya. Myakishev, B. V. Bukhovtsev. Physics. Grade 11: textbook. for general education institutions 2) Internet resources: planet/ page1.html

From Antiquity to the 15th century. It was believed that the Earth is motionless and is at the center of the universe. N. Copernicus and G. Galileo were among the first in modern times who expressed the idea that our planet revolves around the sun. This concept was met with rather hostility: Galileo was even forced to publicly abandon it under pressure from the church. Great importance for the future discovery of the laws of motion were the observations of T. Brahe, who devoted his whole life to this.

However, he did not draw any conclusions from his observations. Later, the works of T. Brahe came to I. Kepler, who found a simple explanation for the observed complex trajectories, formulating three laws of planetary motion around the Sun:

The planets move in elliptical orbits around the Sun;
planets move unevenly further planet is from the Sun, the slower it moves, and vice versa: the closer it is to the Sun, the faster it moves;
the periods of revolution of the planets around the sun depend on their distance from it: more remote planets move slower than those closer to the sun.

Kepler's laws described the observed motion of the planets, but did not reveal the causes leading to such motion. I. Newton's theory of gravitation indicated the cause that determined the movement of cosmic bodies according to Kepler's laws, correctly predicted and explained the features of their movement, and also made it possible to describe phenomena on a cosmic and terrestrial scale in the same terms. Newton found the correct expression for the gravitational force arising from the interaction of bodies, formulating the law of universal gravitation: between any two bodies there is an attractive force proportional to the product of their masses and inversely proportional to the square of the distance between them.

Kepler's laws are fulfilled exactly only in the case of the motion of one body near another, which has a much larger mass, and under the condition that these bodies are spherical. Even with minor deviations from the spherical shape, the planet's orbit is an ellipse precessing around the star. The precession speed can be calculated quite accurately on the basis of Newton's laws and turns out to be maximum for the planet closest to the Sun - Mercury.

According to Newton's third law, there is a force acting on the star from the side of the planet. In the case when the mass of the star is much greater than the mass of the planet, the acceleration of the star is negligible and it can be considered stationary. However, in the presence of bodies of commensurate masses that are attracted to each other, their stable joint movement around is possible. common center wt. In the case of the motion of planets around a star, this effect is hardly noticeable, however, systems that perform the described motion, binary stars, have been discovered in space.

The bulk of the solar system - about 99.8% - falls on the sun. The total mass of the planets is only 0.13% of total mass solar system. From these figures it follows that Kepler's laws for the motion of the planets in our system must be observed very well. Significant deviations from elliptical orbits can occur only in the case of a close flyby of one of the planets: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus or Neptune.

Newton's law of gravity and Kepler's laws make it possible to relate the sizes of planetary orbits to rotation periods, but they do not allow us to calculate the orbits themselves. Back in the 18th century a formula was proposed for the radii of the orbits of the planets of the solar system: R n = (0.4 + 0.3 x 2 n) x R o , where n = 0, 1, 2, 3...; R o - radius of the Earth's orbit. Unlike Kepler's laws, this ratio does not follow from Newton's laws in any way and has not yet received any theoretical explanation. It is possible that this ratio is coincidence. However, the orbits of the planets known today are satisfactorily described by this formula. The only exception is the value n = 3, for which there is no planet in the calculated orbit. Instead, a belt of asteroids was discovered - irregularly shaped bodies, small on a planetary scale.

The problem of evolution of the solar system. Currently, there is no proven theory of the evolution of the solar system. A very attractive theory of the joint origin of the Sun and planets from a single gas cloud, compressed under the influence of gravitational forces, is in conflict with the observed uneven distribution of rotational moment between the star and the planets. Models of the origin of planets as a result of the gravitational capture by the Sun of bodies arriving from distant space are discussed.

The currently known properties of the planets of the solar system allow us to divide them into two groups. First four planets terrestrial group are characterized by relatively small masses and high densities of their constituent substances. They consist of a molten iron core surrounded by a silicate shell - the bark. The planets have gaseous atmospheres. Their temperatures are determined mainly by the distance to the Sun and decrease with its increase. Starting from Jupiter, the group of giant planets is mainly composed of light elements - hydrogen and helium. As they approach the center of the planet, hydrogen and helium gradually change from gaseous to liquid and solid states.

It is assumed that in the central regions the pressure is so high that hydrogen exists in a metallic phase, which has not yet been observed on Earth even under laboratory conditions. The planets of the second group have a large number satellites. Saturn has such a large number of them that, under insufficient magnification, the planet seems to be surrounded by a system of continuous rings.

Kepler's laws.

Initially, it was believed that the Earth was motionless, and the movement of celestial bodies seemed very complicated. Galileo was one of the first to suggest that our planet is no exception and also moves around the Sun. This concept was met with rather hostility. Tycho Brahe decided not to take part in discussions, but to take up direct measurements of the coordinates of bodies on the celestial sphere. He devoted his whole life to this, but not only did he not draw any conclusions from his observations, but he did not even publish the results. Later, Tycho's data came to Kepler, who found a simple explanation for the observed complex trajectories by formulating three laws of motion of the planets (and the Earth) around the Sun (Fig. 6_1):

1. The planets move in elliptical orbits, in one of the focuses of which is the Sun.

2. The speed of the planet changes in such a way that the areas swept by its radius vector for equal periods of time turn out to be equal.

3. The periods of revolution of the planets of one solar system and large axle shafts their orbits are related by:

The complex movement of the planets on the “celestial sphere” observed from the Earth, according to Kepler, arose as a result of the addition of these planets in elliptical orbits with the movement of the observer, who, together with the Earth, performs orbital motion around the sun and daily rotation around the axis of the planet.

Direct proof of the daily rotation of the Earth was an experiment set by Foucault, in which the plane of oscillation of the pendulum rotated relative to the surface of the rotating Earth.

The law of universal gravitation.

Kepler's laws perfectly described the observed movement of the planets, but did not reveal the reasons leading to such movement (for example, it could well be considered that the reason for the movement of bodies in Keplerian orbits was the will of some creature or the desire of the celestial bodies themselves to harmony). Newton's theory of gravitation indicated the cause that determined the movement of cosmic bodies according to Kepler's laws, correctly predicted and explained the features of their movement in more complex cases, made it possible to describe many phenomena on a cosmic and terrestrial scale in the same terms (the movement of stars in a galactic cluster and the fall of an apple on the Earth's surface) .

Newton found the correct expression for the gravitational force arising from the interaction of two point bodies (bodies whose dimensions are small compared to the distance between them):

which, together with the second law, if the mass of the planet m is much less than the mass of the star M, led to the differential equation

admitting an analytical solution. Without involving any additional physical ideas, it is fashionable to show by purely mathematical methods that under appropriate initial conditions (sufficiently small initial distance to the star and the speed of the planet), the cosmic body will rotate in a closed, stable elliptical orbit in full accordance with Kepler's laws (in particular, Kepler's second law is a direct consequence of the law conservation of angular momentum, which is fulfilled during gravitational interactions, since the moment of force (2) relative to the massive center is always equal to zero). At a sufficiently high initial speed(its value depends on the mass of the star and the initial position) the cosmic body moves along a hyperbolic trajectory, eventually moving away from the star to an infinite distance.

An important property of the law of gravity (2) is the preservation of its mathematical form in the case of gravitational interaction of non-point bodies in the case of a spherically symmetric distribution of their masses over the volume. In this case, the role of R is played by the distance between the centers of these bodies.

Movement of celestial bodies in the presence of perturbations. Strictly speaking, Kepler's laws are fulfilled exactly only in the case of motion of only one body near another, which has a much larger mass, provided that these bodies are spherical. With minor deviations from the spherical shape (for example, due to the rotation of a star, it can “flatten” somewhat), the orbit of the planet ceases to be closed and is an ellipse precessing around the star.

Another common perturbation is the gravitational influence of the planets of one star system on each other. Keplerian orbits are stable with respect to weak perturbations, i.e., having experienced the impact of a close-flying neighbor, the planet tends to return to its original trajectory. In the presence of strong perturbations (the passage of a massive body at a short distance), the problem of motion becomes much more complicated and cannot be solved analytically. numerical calculations show that in this case the trajectories of the planets cease to be ellipses and represent open curves.

According to Newton's third law, there is a force acting on the star from the side of the planets. In the case of M>>m, the acceleration of the star is negligibly small and it can be considered stationary. In the presence of two bodies of commensurate masses that are attracted to each other, their stable joint motion in elliptical orbits around a common center of mass is possible. It is obvious that a more massive body moves along an orbit of a smaller radius. In the case of planets moving around a star, this effect is hardly noticeable. however, in space, systems were found that make the described movement - double stars. A numerical calculation of the motion of the planets in a binary star system shows that their orbits are essentially non-stationary, the distance from the planet to the stars varies rapidly over a very wide range. At the same time, the inevitable rapid climate change on the planets makes it very problematic the possibility biological evolution. The emergence of technical civilizations on the planets of binary star systems is even less likely, since the complex non-periodic motion of the planets leads to the observable motion of bodies on the “celestial sphere” that is difficult to decipher, significantly complicating the formulation of Kepler’s laws and, as a result, the development of classical mechanics (Fig. 6_2).

The structure of the solar system.

It is well known that the bulk of the solar system (about 99.8%) falls on its only star, the Sun. The total mass of the planets is only 0.13% of the total. The remaining bodies of the system (comets, planetary satellites, asteroids and meteoritic matter) account for only 0.0003% of the mass. From the above figures it follows that Kepler's laws for the motion of the planets in our system must be carried out very well. Significant deviations from elliptical orbits can occur only in the case of a close (compared to the distance to the Sun) flight past one of the planets: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune or Pluto (this is especially true for the most massive of the planets - Jupiter). It was observations of the perturbation of Neptune's orbit that made it possible to predict and then discover Pluto, the most distant known planet in our system.

, is the radius of the Earth's orbit. Unlike Kepler's laws, relation (4) does not follow from Newton's laws in any way and has not yet received theoretical substantiation, although the orbits of all currently known planets are satisfactorily described by this formula. The only exception is the value n=3, for which there is no planet in the calculated orbit. Instead, a belt of asteroids was discovered - irregularly shaped bodies, small on a planetary scale. empirical laws, not supported by the available theory, can play positive role in studies, since they also reflect objective reality (perhaps in a not entirely accurate and even somewhat distorted form).

The hypothesis of a pre-existing fifth planet, Phaethon, was shattered into pieces by a giant gravitational attraction its massive neighbor - Jupiter, however, a quantitative analysis of the movement of the giant planet showed the inconsistency of this assumption. Apparently, the mentioned problem can be solved only on the basis of a complete theory of the origin and evolution of the planets of the solar system, which does not yet exist. A very attractive theory of the joint origin of the sun and planets from a single gas cloud, compressed under the influence of gravitational forces, is in conflict with the observed uneven distribution of rotational moment (momentum) between the star and the planets. Models of the origin of planets as a result of the gravitational capture by the Sun of bodies arriving from distant space, the effects caused by the explosion of supernovae are discussed. In most "scenarios" of the development of the solar system, the existence of the asteroid belt is somehow associated with its close proximity to the most massive planet in the system.

The currently known properties of the planets of the solar system allow us to divide them into two groups. The first four planets of the terrestrial group are characterized by relatively small masses and high densities of their constituent substances. They consist of a molten iron core surrounded by a silicate shell - the bark. The planets have gaseous atomospheres. Their temperatures are mainly determined by the distance to the Sun and decrease with its increase. The group of giant planets starting from Jupiter is mainly composed of light elements (hydrogen and helium), the pressure of which in the inner layers increases to enormous values due to gravitational compression. As a result, as they approach the center, the gases gradually pass into a liquid and, possibly, into a solid state. It is assumed that in the central regions the pressure is so high that hydrogen exists in a metallic phase, which has not yet been observed on Earth even under laboratory conditions. The planets of the second group have a large number of satellites. At Saturn, their number is so great that, with insufficient magnification, the planet seems to be surrounded by a system of continuous rings (Fig. 6_3).

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