Laws of motion of celestial bodies. Size and shape of the Earth. §4. Elements of planetary orbits
All cosmogonic hypotheses can be divided into several groups. According to one of them, the Sun and all bodies solar system: planets, satellites, asteroids, comets and meteoroids - formed from a single gas and dust cloud. According to the second, the Sun and its family have different origins, so that the Sun was formed from one gas and dust cloud (nebula, globule), 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 on its own orbit and was divided in some, even more incomprehensible way into many very different bodies (planets, their satellites, asteroids, comets and meteoroids), having the most various characteristics: mass, density, eccentricity, direction of orbit and direction of rotation around its axis, inclination of the orbit to the plane of the equator of the Sun (or ecliptic) and inclination of the equatorial plane to the plane of its orbit.
Nine major planets revolve around the Sun in ellipses (not much different from circles) almost in 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 cosmic dust. It is penetrated by electromagnetic radiation.
Sun 109 times more than Earth in diameter and approximately 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 gravity.
Configuration and visibility conditions of planets
Some planetary configurations are called more characteristic mutual arrangements planets, Earth and Sun.
The conditions for the visibility of planets from Earth differ sharply for the internal planets (Venus and Mercury), whose orbits lie within the Earth’s orbit, and for the external planets (all others).
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 planet-Sun conjunctions. At inferior conjunction, the planet is closest to Earth, and at superior conjunction, it is furthest from us.
Synodic periods of revolution of planets and their connection with sidereal periods
The period of revolution of the planets around the Sun in relation to the stars is called the sidereal or sidereal period.
The closer a planet is to the Sun, the greater its linear and angular velocities and the shorter the sidereal period of revolution around the Sun.
However, from direct observations, it is not the sidereal period of revolution of the planet that is determined, but the period of time that elapses between its two successive configurations of the same name, for example, between two successive conjunctions (oppositions). This period is called the synodic orbital period. Having determined the synodic periods from observations, the sidereal periods of the planets' revolutions are found through calculations.
The synodic period of an 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). IN early XVII V. Kepler, studying the revolution 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 the focuses.
Kepler's second law (law of areas). The radius vector of the planet describes equal areas in equal periods of time.
Kepler's third law . The squares of the sidereal periods of revolution of the planets are related as the cubes of the semimajor 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, we can find in these units the distances to all planets of the Solar System. The semimajor axis of the Earth's orbit is taken as the astronomical unit of distance (= 1 AU)
The classic way to determine distances was and remains the goniometric geometric method. They also determine distances to distant stars, to which the radar method is not applicable. The geometric method is based on the phenomenon parallactic displacement.
Parallax displacement is the change in direction of an object when the observer moves.
EXAMPLE OF SOLVING A PROBLEM
Task. Oppositions of a certain planet are repeated after 2 years. What is the semimajor axis of its orbit?
| Given |
SOLUTION The semimajor axis of the orbit can be determined from Kepler's third law: |
| - ? |
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 Earth's surface. Degree measurements showed that the length of 1° meridian arc in kilometers in the polar region is greatest (111.7 km), and at the equator it is smallest (110.6 km). Consequently, at the equator the curvature of the Earth's surface is greater than at the poles, which means that the Earth is not a sphere. The equatorial radius of the Earth is 21.4 km greater than the polar radius. Therefore, the Earth (like other planets) is compressed at the poles due to rotation.
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 visible from the luminary, perpendicular to the line of sight, is called horizontal parallax.
Mass and density of the Earth
The law of universal gravitation allows us to determine one of the most important characteristics celestial bodies - mass, in particular the mass of our planet. Indeed, based on the law of universal gravitation, acceleration free fall g=(G*M)/r 2 . Consequently, if the values of the acceleration of gravity, the gravitational constant and the radius of the Earth are known, then its mass can be determined.
Substituting the value g = 9.8 m/s 2 into the indicated formula, G = 6.67 * 10 -11 N * m 2 / kg 2,
R = 6370 km, we find that the mass of the Earth is M = 6 x 10 24 kg. Knowing the mass and volume of the Earth, you can calculate its average density.
Based on observations of the motion of the Moon and analyzing the laws of planetary motion discovered by Kepler, I. Newton (1643-1727) established the law of universal gravitation. According to this law, as you already know from your physics course, all bodies in the Universe are attracted to each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them:
here m 1 and m 2 are the masses of two bodies, r is the distance between them, and G is the coefficient of proportionality, called the gravitational constant. Its numerical value depends on the units in which force, mass and distance are expressed. The law of universal gravitation explains the movement of planets and comets around the Sun, the movement of satellites around planets, double and multiple stars around their common center of mass.
Newton proved that under the influence of mutual gravity, bodies can move relative to each other along ellipse(in particular, according to circle), By parabola and by hyperbole. Newton found that the type of orbit that a body describes depends on its speed at a given point in the orbit(Fig. 34).
At a certain speed the body describes circle near the attracting center. This speed is called the first cosmic or circular speed; it is imparted to bodies launched as artificial Earth satellites in circular orbits. (The derivation of the formula for calculating the first cosmic velocity is known from a physics course.) The first cosmic velocity near the Earth's surface is about 8 km/s (7.9 km/s).
If the body is given a speed twice as large as the circular speed (11.2 km/s), called the second cosmic or parabolic speed, then the body will forever move away from the Earth and can become a satellite of the Sun. In this case, the movement of the body will occur according to parabola relative to the Earth. At an even higher speed relative to the Earth, the body will fly in a hyperbola. Moving along a parabola or hyperbole, the body goes around the Sun only once and moves away from it forever.
The average speed of the Earth's orbit is 30 km/s. The Earth's orbit is close to a circle, therefore, the speed of the Earth's motion in orbit is close to circular at the Earth's distance from the Sun. The parabolic speed at the distance of the Earth from the Sun is km/s≈42 km/s. At such a speed relative to the Sun, a body from Earth's orbit will leave the Solar System.
2. Disturbances in the motion of planets
Kepler's laws are strictly observed only when we consider the motion of two isolated bodies under the influence of their mutual attraction. There are many planets in the Solar System, all of them are not only attracted by the Sun, but also attract each other, so their movements do not exactly obey Kepler’s laws.
Deviations from motion that would occur strictly according to Kepler's laws are called disturbances. In the Solar System, disturbances are small because the attraction of each planet by the Sun is much stronger than the attraction of other planets.
The greatest disturbance in the solar system is caused by the planet Jupiter, which is about 300 times more massive than the Earth. Jupiter has a particularly strong influence on the movement of asteroids and comets when they come close to it. In particular, if the directions of the comet’s acceleration caused by the attraction of Jupiter and the Sun coincide, then the comet can develop such a high speed that, moving along the hyperbola, it will leave the Solar system forever. There were cases when the gravity of Jupiter restrained the comet, the eccentricity of its orbit became smaller and the orbital period sharply decreased.
When calculating the apparent positions of planets, disturbances must be taken into account. Now high-speed electronic computers help make such calculations. When launching artificial celestial bodies and when calculating their trajectories, the theory of motion of celestial bodies, in particular the theory of perturbations, is used.
The ability to send automatic interplanetary stations along desired, pre-calculated trajectories, and bring them to the target taking into account disturbances in motion - all these are vivid examples of the knowability of the laws of nature. The sky, which according to believers is the abode of the gods, has become an arena human activity just like the Earth. Religion has always opposed the Earth and the sky and declared the sky inaccessible. Now, artificial celestial bodies created by man move among the planets, which he can control by radio from great distances.
3. Discovery of Neptune
One of bright examples achievements of science, one of the evidence of the unlimited cognition of nature was the discovery of the planet Neptune through calculations - “at the tip of a pen.”
Uranus, the planet next to Saturn, which for many centuries was considered the most distant of the planets, was discovered by W. Herschel at the end of the 18th century. Uranus is hardly visible to the naked eye. By the 40s of the 19th century. accurate observations have shown that Uranus deviates barely noticeably from the path it should follow, taking into account the disturbances from all the known planets. Thus, the theory of the movement of celestial bodies, so strict and accurate, was put to the test.
Le Verrier (in France) and Adams (in England) suggested that if disturbances from the known planets do not explain the deviation in the movement of Uranus, then it is affected by the attraction of an as yet unknown body. They almost simultaneously calculated where behind Uranus there should be an unknown body producing these deviations with its gravity. They calculated the orbit of the unknown planet, its mass and indicated the place in the sky where the unknown planet should have been located at that time. This planet was found through a telescope at the place they indicated in 1846. It was named Neptune. Neptune is not visible to the naked eye. Thus, the disagreement between theory and practice, which seemed to undermine the authority of materialist science, led to its triumph.
4. Tides
Under the influence of mutual attraction of particles, the body tends to take the shape of a ball. The shape of the Sun, planets, their satellites and stars is therefore close to spherical. Rotation of bodies (as you know from physical experiments) leads to their flattening, to compression along the axis of rotation. Therefore it is slightly compressed at the poles Earth, and the rapidly rotating Jupiter and Saturn are most compressed.
But the shape of the planets can also change due to the forces of their mutual attraction. A spherical body (planet) moves as a whole under the influence of the gravitational attraction of another body as if the entire gravitational force was applied to its center. However, certain parts of the planet are located on at different distances from the attracting body, therefore the gravitational acceleration in them is also different, which leads to the emergence of forces tending to deform the planet. The difference in acceleration caused by the attraction of another body at a given point and at the center of the planet is called tidal acceleration.
Consider, for example, the Earth-Moon system. The same element of mass at the center of the Earth will be attracted by the Moon weaker than on the side facing the Moon, and stronger than on the side facing the Moon. opposite side. As a result, the Earth, and primarily the water shell of the Earth, is slightly stretched in both directions along the line connecting it with the Moon. In Figure 35, for clarity, the ocean is depicted as covering the entire Earth. At points lying on the line Earth - Moon, the water level is highest - there are tides. Along the circle, the plane of which is perpendicular to the direction of the Earth - Moon line and passes through the center of the Earth, the water level is lowest - there is low tide. With the daily rotation of the Earth, the ebb and flow of the tides alternately enter different places Earth. It is easy to understand that there can be two high and two low tides per day.
The Sun also causes ebbs and flows on Earth, but due to the great distance of the Sun, they are smaller than the lunar ones and less noticeable.
Huge amounts of water move with the tides. Currently, they are beginning to use the enormous energy of water participating in the tides on the shores of the oceans and open seas.
The axis of tidal protrusions should always be directed towards the Moon. As the Earth rotates, it tends to turn the water tidal bulge. Since the Earth rotates around its axis much faster than the Moon revolves around the Earth, the Moon pulls the water hump towards itself. Friction occurs between the water and the solid ocean bottom. As a result, the so-called tidal friction. It slows down the rotation of the Earth, and the day becomes longer over time (once they were only 5-6 hours). Strong tides caused by the Sun on Mercury and Venus appear to be the reason for their extremely slow rotation around their axis. Tides caused by the Earth have slowed down the Moon's rotation so much that it always faces the Earth with one side. Thus, tides are an important factor in the evolution of celestial bodies and the Earth.
5. Mass and density of the Earth
The law of universal gravitation also allows us to determine one of the most important characteristics of celestial bodies - mass, in particular the mass of our planet. Indeed, based on the law of universal gravitation, the acceleration of free fall 
Consequently, if the values of the acceleration of gravity, the gravitational constant and the radius of the Earth are known, then its mass can be determined.
Substituting the value g = 9.8 m/s 2 , G = 6.67 * 10 -11 N * m 2 / kg 2 , R = 6370 km into the indicated formula, we find that the mass of the Earth is M = 6 * 10 24 kg.
Knowing the mass and volume of the Earth, you can calculate its average density. It is equal to 5.5 * 10 3 kg/m 3. But the density of the Earth increases with depth, and, according to calculations, near the center, in the Earth’s core, it is equal to 1.1 * 10 4 kg/m 3. An increase in density with depth occurs due to an increase in the content of heavy elements, as well as due to an increase in pressure.
(WITH internal structure The Earth, studied by astronomical and geophysical methods, you were introduced to in the course of physical geography.)
Exercise 12
1. What is the density of the Moon if its mass is 81 times and its radius is 4 times less than that of the Earth?
2. What is the mass of the Earth if the angular speed of the Moon is 13.2° per day and the average distance to it is 380,000 km?
6. Determination of the masses of celestial bodies
Newton proved that a more accurate formula for Kepler's third law is:
where M 1 and M 2 are the masses of any celestial bodies, a m 1 and m 2 are the masses of their satellites, respectively. Thus, the planets are considered satellites of the Sun. We see that the refined formula of this law differs from the approximate one in the presence of a factor containing masses. If by M 1 = M 2 = M we mean the mass of the Sun, and by m 1 and m 2 the masses of two different planets, then the ratio 
will differ little from unity, since m 1 and m 2 are very small compared to the mass of the Sun. In this case, the exact formula will not differ noticeably from the approximate one.
Kepler's refined third law allows us to determine the masses of planets with satellites and the mass of the Sun. To determine the mass of the Sun, we will compare the movement of the Moon around the Earth with the movement of the Earth around the Sun:
The masses of planets that do not have satellites are determined by the disturbances that their attraction produces in the movement of neighboring planets, as well as in the movement of comets, asteroids or spacecraft.
Exercise 13
1. Determine the mass of Jupiter by comparing the Jupiter system with a satellite with the Earth - Moon system, if the first satellite of Jupiter is 422,000 km away from it and has an orbital period of 1.77 days. The data for the Moon should be known to you.
2. Calculate at what distance from the Earth on the Earth-Moon line are those points at which the attractions of the Earth and the Moon are equal, knowing that the distance between the Moon and the Earth is equal to 60 radii of the Earth, and the mass of the Earth is 81 times the mass of the Moon.
Since ancient times, people have observed such phenomena in the sky as the visible rotation of the starry sky, the changing phases of the Moon, the rising and setting of celestial bodies, the visible 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 were 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, the correct understanding and explanation of which took centuries to develop.
The first written mentions of celestial bodies appeared in ancient Egypt and Sumer. The ancients distinguished three types of bodies in the firmament: stars, planets and “tailed stars.” The differences come precisely 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.
Planets, on the contrary, move across the sky, and their movement is visible to the naked eye for an hour or two. Even in Sumer, 5 planets were found and identified: Mercury, Venus, Mars, Jupiter, Saturn. The Sun and Moon were added to them in abundance. Total: 7 planets. "Tailed" stars of a comet. They appeared infrequently and symbolized troubles.
Kepler's Laws I. Each planet moves in an ellipse, at one of the foci of which the Sun is located. II.(law of equal areas). The radius vector of the planet describes equal areas in equal periods of time. III. The squares of the periods of revolution of the planets around the Sun are proportional to the cubes of the semi-major axes of their elliptical orbits. The three laws of planetary motion relative to the Sun were derived empirically 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 apparent motion of the planets and the Sun is most simply described in the reference frame associated with the Sun. This approach was called the heliocentric system of the world and was proposed by the Polish astronomer Nicolaus Copernicus (). In ancient times and right up to Copernicus, it was believed that the Earth was located at the center of the Universe and all celestial bodies revolved along complex trajectories around it. This world system is called the geocentric world system.
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 the simple circular motion of planets around the Earth, proved by calculations and observations that the orbits of motion of celestial bodies can only be elliptical, it finally became clear that that the apparent motion of the planets consists of: the movement of the observer on the surface of the Earth, the rotation of the Earth around the Sun, the own movements of celestial bodies
The complex apparent motion of planets on the celestial sphere is caused by the revolution of the planets of the Solar System around the Sun. The word “planet” itself, translated from ancient Greek, means “wandering” or “vagrant”. The trajectory of a celestial body is called its orbit. The speed of movement of planets in orbits decreases as the planets move away from the Sun. The nature of the planet's movement depends on which group it belongs to. Therefore, in relation to the orbit and visibility conditions 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 always face the Earth with 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 for Mercury is 28°, for Venus - 48°. During eastern elongation, the inner planet is visible in the west, in the rays of the evening dawn, shortly after sunset. Evening (eastern) elongation of Mercury During 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 a planet is the angle between a ray of light falling from the Sun onto the planet and a ray reflected from it towards the observer. The phase angles of Mercury and Venus vary from 0° to 180°, so Mercury and Venus change phases in the same way as the Moon. Near inferior conjunction, both planets are at their greatest angular dimensions, but look like narrow sickles. At a phase angle of ψ = 90°, half of the disk of the planets is illuminated, phase φ = 0.5. At superior conjunction, the inferior planets are fully illuminated, but are poorly visible from Earth, as they are behind the Sun.
Since, when observed 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 either from east to west (direct motion), or from west to east (retrograde motion). Moments of change of direction are called stops. If you plot this path on a map, you get a loop. The larger the distance between the planet and the Earth, the smaller the loop is. The planets describe loops, rather than simply moving back and forth along one line, solely due to the fact that the planes of their orbits do not coincide with the plane of the ecliptic. This complex looping pattern was first observed and described using the apparent motion of Venus.
It is a known fact that the movement of certain planets can be observed from the Earth at strictly defined times of the year, this is due to their position over time in the starry sky. The characteristic relative positions of the planets relative to the Sun and Earth are called planetary configurations. The configurations of the inner 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 Earth, P 1 is inner planet, S is Sun, the celestial conjunction is called inferior conjunction. In an “ideal” inferior conjunction, Mercury or Venus transits the disk of the Sun. If T is Earth, S is Sun, P 1 is Mercury or Venus, the phenomenon is called superior 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 at the inferior conjunction, and a full moon at the superior conjunction.
In their movement across 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 east or west of it. The moment at which the planet is at its greatest angular distance 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 that characterize the movement of planets and allow us to make some calculations: The sidereal (stellar) period of revolution of a planet is the time period T during which the planet makes one complete revolution around the Sun in relation to the stars. The synodic period of revolution of a planet is the time interval S between two successive configurations of the same name.
Literature used: Literature used: 1)G. Ya. Myakishev, B.V. Bukhovtsev. Physics.11th grade: textbook. for general education institutions 2) Internet resources: planet/ page1.html
The two most significant successes of classical natural science, based on Newtonian mechanics, were an almost exhaustive description of the observed motion of celestial bodies and an explanation of the laws of ideal gas known from experiment.
Kepler's laws.
Initially, it was believed that the Earth was motionless, and the movement of celestial bodies seemed very complex. 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 quite hostility. Tycho Brahe decided not to take part in the discussions, but to take direct measurements of the coordinates of bodies on the celestial sphere. He devoted his whole life to this, but not only did not draw any conclusions from his observations, but did not even publish the results. Later, Tycho's data came to Kepler, who found a simple explanation for the observed complex trajectories, formulating three laws of motion of planets (and the Earth) around the Sun (Fig. 6_1):
1. The planets move in elliptical orbits, with the Sun at one of the focuses.
2. The speed of the planet’s movement changes in such a way that the areas swept by its radius vector in equal periods of time turn out to be equal.
3. The orbital periods of the planets of one solar system and the semimajor axes of their orbits are related by the relation:
.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 daily rotation Earth was an experiment carried out by Foucault, in which the plane of oscillation of a pendulum was rotated relative to the surface of the rotating Earth.
Law of Universal Gravitation.
Kepler's laws perfectly described the observed motion of the planets, but did not reveal the reasons leading to such motion (for example, it could be considered that the reason for the movement of bodies in Kepler's orbits was the will of some being or the desire of the celestial bodies themselves for harmony). Newton's theory of gravity indicated the reason 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, and made it possible to describe in the same terms many phenomena of cosmic and terrestrial scales (the movement of stars in a galactic cluster and the fall of an apple to the surface of the Earth) .
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, in the case 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 possible to show using purely mathematical methods that, with 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 of conservation of angular momentum, which is fulfilled during gravitational interactions, since the moment of force (2) relative to the massive center 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 infinitely large 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 disturbances. Strictly speaking, Kepler's laws are fulfilled exactly only in the case of the movement of only one body near another, which has a significantly larger mass, provided that these bodies are spherical. With minor deviations from the spherical shape (for example, due to the rotation of the star, it may be somewhat “flattened”), the planet’s orbit ceases to be closed and becomes an ellipse precessing around the star.
Another common disturbance is the gravitational influence of planets in the same star system on each other. Keplerian orbits are stable relative to weak disturbances, i.e., having experienced the impact of a nearby neighbor, the planet tends to return to its original trajectory. In the presence of strong disturbances (flight of a massive body at a short distance), the problem of motion becomes significantly 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 planets. In the case of M>>m, the acceleration of the star is negligible and it can be considered stationary. If there are two bodies of commensurate masses that attract each other, their stable joint motion in elliptical orbits around a common center of mass is possible. Obviously, a more massive body moves in an orbit of smaller radius. In the case of planetary motion around a star, this effect is hardly noticeable. however, systems have been discovered in space that perform the described motion - double stars. Numerical calculations of the motion of planets in a binary star system show that their orbits are significantly unsteady, and the distance from the planet to the stars varies quickly over a very wide range. The inevitable rapid climate changes on the planets make it very problematic to biological evolution. The emergence of technical civilizations on planets of double star systems is even less likely, since the complex non-periodic motion of 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 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%) lies in 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 meteorite matter) account for only 0.0003% of the mass. From the above figures it follows that Kepler’s laws for the motion of planets in our system must be fulfilled 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 (especially the most massive of the planets - Jupiter). It was observations of the disturbance in Neptune's orbit that made it possible to predict and then discover Pluto, the most distant known planet in our system.
Newton's law of gravity and Kepler's laws make it possible to relate the sizes of planetary orbits to rotation periods, but do not allow us to calculate the orbits themselves. Back in the 18th century, an empirical formula was proposed for the radii of the orbits of the planets of the solar system:
, is the radius of the Earth’s orbit. Unlike Kepler's laws, relation (4) does not follow from Newton's laws and has not yet received 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, a belt of asteroids was discovered - small bodies on a planetary scale irregular shape. Empirical laws that are not confirmed by the existing theory can play a positive role in research, since they also reflect objective reality (perhaps in a not entirely accurate and even somewhat distorted form).The hypothesis about a pre-existing fifth planet, Phaeton, destroyed into pieces by the gigantic gravitational attraction of its massive neighbor Jupiter, seemed attractive, but a quantitative analysis of the movement of the giant planet showed the inconsistency of this assumption. Apparently, the mentioned problem can be resolved 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, turns out to be in contradiction with the observed uneven distribution of angular momentum (angular 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 deep space, effects caused by supernova explosions are discussed. In most “scenarios” for the development of the solar system, the existence of the asteroid belt is one way or another 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 the substances composing them. They consist of a molten iron core surrounded by a silicate shell - bark. Planets have gas atomospheres. Their temperatures are mainly determined by the distance to the Sun and decrease as it increases. 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 transform into liquid and, possibly, 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 Zamla even under laboratory conditions. Planets of the second group have a large number satellites. On Saturn, their number is so large that, with insufficient magnification, the planet appears to be surrounded by a system of continuous rings (Fig. 6_3).
Space exploration has long gone beyond imagination:
– every year astronauts go beyond the Earth;
– people launch satellites, some of which have already crossed the solar system;
– huge telescopes observe the stars from the orbit of our planet.
Who was the first pioneer in the sky? What incredible theories are behind our space achievements? What does the future hold for us? This book will briefly and clearly tell you about the most important discoveries in the field of astronomy and about the people who made them.
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Tycho Brahe's observations and measurements allowed his student, the German scientist Johannes Kepler, to take the next step in the development of astronomy.
Geocentric Ptolemaic world system and Copernican heliocentric system
Calculating the orbit of Mars, Kepler discovered that it is not a circle, as Copernicus and other scientists believed, but an ellipse. At first, he did not extend this conclusion to other planets, but later he realized that not only Mars, but all planets have an ellipsoidal orbit. Thus, Kepler's first law of planetary motion was discovered. In modern formulation it sounds like this: each planet of the solar system revolves in an ellipse, at one of the foci of which the Sun is located.
The second law of planetary motion was a logical consequence of the first. Even before the formulation of the first law, while observing the movement of Mars, Kepler noticed that the planet moves slower the further it is from the Sun. The elliptical shape of the orbit fully explains this feature of motion. Over equal periods of time, a straight line connecting a planet to the Sun describes equal areas - this is Kepler’s second law.
The second law explains the change in the speed of the planet, but does not provide any calculations. The formula for calculating how fast the planets rotate and how long it takes to travel around the Sun is Kepler's third law.
Kepler's research put an end to the dispute between the world systems of Ptolemy and Copernicus. He convincingly proved that the Sun, not the Earth, is at the center of our system. After Kepler's scientific world no further attempts were made to revive the geocentric system.
The accuracy of the three laws of planetary motion discovered by Kepler was confirmed by numerous astronomical observations. Nevertheless, the basis and reasons for these laws remained unclear until at the end of the 17th century. Newton's genius did not manifest itself.
Everyone knows the story of how Newton discovered the law of universal gravitation: an apple fell on his head, and Newton realized that the apple was being pulled towards him by the Earth. In the extended version of this legend, there is also the Moon, which the scientist looked at while sitting under an apple tree.
After the apple fell, Newton realized that the force that caused the apple to fall and the force that kept the Moon in Earth's orbit were of the same nature.
In reality, of course, everything was far from so simple. Before the discovery of the famous law, Newton devoted many years to the study of mechanics, the laws of motion and interaction between bodies. He was not the first to suggest the existence of gravitational forces. Galileo Galilei spoke about this, but he believed that attraction to the Earth acts only on our planet and extends only to the Moon. Kepler, who discovered the laws of planetary motion, was sure that they work exclusively in space and have no relation to terrestrial physics. Newton was able to combine these two approaches - he was the first to realize that physical laws, primarily the law of universal gravitation, are universal and applicable to all material bodies.
The essence of the law of universal gravitation comes down to the fact that there is attraction between absolutely all bodies in the Universe. The force of attraction depends on two main quantities - the mass of bodies and the distance between them. The heavier the body, the more strongly it attracts lighter bodies. The Earth attracts the Moon and holds it in its orbit. The Moon also has a certain effect on our planet (it causes tides), but the gravitational force of the Earth, due to its larger mass, is greater.
In addition to the law of universal gravitation, Newton formulated three laws of motion. The first of them is called the law of inertia. It states: if no force is applied to a body, it will remain in a state of rest or uniform rectilinear movement. The second law introduces the concept of force and acceleration, and these two quantities, as Newton proved, depend on the mass of the body. The greater the mass, the less acceleration will be for a certain applied force. Newton's third law describes the interaction of two material objects. Its simplest formulation says: action is equal to reaction.
The discoveries made by Isaac Newton and the formulas he derived gave astronomy a powerful tool that made it possible to advance this science far forward. Many phenomena that had no explanation before have revealed their nature. It became clear why planets revolve around the Sun, and satellites revolve around planets, without flying into outer space: they are held by the force of gravity. The speed of the planets remains uniform due to the law of inertia. The round shape of celestial bodies also received its explanation: it is acquired due to gravity, attraction to a more massive center.
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