The attraction of the earth and the moon. About Attractions - Earthly and Lunar

Briefly, his story is as follows. Even the ancients, observing the movement of the planets in the sky, guessed that all of them, together with the Earth, "walk" around the Sun. Later, when people forgot what they knew before, this discovery was rediscovered by Copernicus. And then a new question arose: how exactly do the planets go around the Sun, what is their movement? Do they go in a circle and the Sun is in the center, or do they move along some other curve? How fast are they moving? And so on.

It turned out not so soon. After Copernicus, troubled times again came and great disputes flared up about whether the planets go along with the Earth around the Sun or the Earth is at the center of the Universe. Then a man named Tycho Brahe (Tycho Brahe (1546-1601) - Danish astronomer) figured out how to answer this question. He decided that he needed to watch very carefully where the planets appear in the sky, write it down exactly, and then already choose between two hostile theories. This was the beginning of modern science, the key to a correct understanding of nature is to observe the object, write down all the details and hope that the information obtained in this way will serve as the basis for one or another theoretical interpretation. And so Tycho Brahe, a wealthy man who owned an island near Copenhagen, equipped his island with large bronze circles and special observation posts and recorded night after night the positions of the planets. Only at the cost of such hard work does any discovery come to us.

When all this data was collected, it fell into the hands of Kepler. (Johannes Kepler (1571-1630) - German astronomer and mathematician, was Brahe's assistant), which tried to solve how the planets move around the sun. He searched for a solution by trial and error. Once it seemed to him that he had already received the answer: he decided that the planets move in a circle, but the Sun is not in the center. Then Kepler noticed that one of the planets, it seems Mars, deviated from the desired position by 8 arc minutes, and realized that the answer he received was incorrect, since Tycho Brahe could not make such a big mistake. Relying on the accuracy of his observations, he decided to revise his theory and eventually discovered three facts.

Laws of planetary motion around the sun

First, Kepler established that the planets move around the Sun in ellipses and the Sun is in one of the foci. An ellipse is a curve that all artists know about because it is a stretched circle. Children also know about it: they were told that if you thread a string into a ring, fasten its ends and insert a pencil into the ring, it will describe an ellipse.

The two points A and B are foci. The planet's orbit is an ellipse. The sun is in one of the foci. Another question arises: how does the planet move along the ellipse? Does it go faster when it is closer to the Sun? Does it slow down moving away from it? Kepler answered this question as well. He discovered that if you take two positions of the planet separated from each other by a certain period of time, say three weeks, then take another part of the orbit and there are also two positions of the planet separated by three weeks, and draw lines (scientists call them radius vectors) from the Sun to the planet, then the area enclosed between the orbit of the planet and a pair of lines that are separated from each other by three weeks is the same everywhere, in any part of the orbit. And for these areas to be the same, the planet must go faster when it is closer to the Sun, and slower when it is far from it.

A few years later, Kepler formulated the third rule, which concerned not the movement of one planet around the Sun, but connected the movements of various planets with each other. It said that the time of a complete revolution of the planet around the Sun depends on the size of the orbit and is proportional to the square root of the cube of this value. And the size of the orbit is the diameter that intersects the widest point of the ellipse.

So Kepler discovered three laws that can be reduced to one, if we say that the orbit of the planet is an ellipse - for equal periods of time, the radius vector of the planet describes equal areas and time (period) of the planet's revolution around the Sun in proportion to the size of the orbit to the power of three second, i.e., the square root of the cube of the magnitude of the orbit. These three laws of Kepler completely describe the motion of the planets around the Sun.

Meanwhile, Galileo discovered the great principle of inertia. Then it was Newton's turn, who decided that a planet orbiting the Sun didn't need force to move forward; if there were no force, the planet would fly tangentially. But in fact, the planet does not fly in a straight line. She always finds herself not in the place where she would have fallen if she had flown freely, but closer to the Sun. In other words, its speed, its movement is deflected towards the Sun.

It became clear that the source of this force (gravitational force) is located somewhere near the Sun.

People looked at Jupiter through a telescope with satellites revolving around it, and it reminded them of a small solar system. Everything looked as if the satellites were attracted to Jupiter. The moon also revolves around the earth and is attracted to it in exactly the same way. Naturally, the idea arose that attraction acts everywhere. It only remained to generalize these observations and say that all bodies attract each other. This means that the Earth must attract the Moon in the same way as the Sun attracts the planets. But it is known that the Earth also attracts ordinary objects: for example, you sit firmly on a chair, although you might like to fly through the air. The gravitation of objects towards the Earth was a well-known phenomenon. Newton suggested that the Moon in orbit is kept by the same forces that attract objects to the Earth.

Why hot flashes happen

First, the tides. The tides are caused by the Moon itself pulling on the Earth and its oceans. So they thought before, but here's what turned out to be inexplicable: if the Moon attracts water and raises them above the near side of the Earth, then only one tide would occur per day - right under the Moon. In fact, as we know, the tides are repeated after about 12 hours, that is, twice a day. There was another school that held opposing views. Its adherents believed that the Moon attracts the Earth, and the water does not keep up with it. Newton was the first to understand what was really happening: the attraction of the Moon acts equally on the Earth and on water, if they are equally distant. But the water at point y is closer to the moon than the earth, and at point x it is farther away. In y, water is attracted to the Moon more strongly than the Earth, and in x it is weaker. Therefore, a combination of the two previous pictures is obtained, which gives a double tide.

In fact, the Earth does the same thing as the Moon - it moves in a circle. The force with which the Moon acts on the Earth is balanced - but with what? Just as the Moon moves in circles to balance the Earth's gravity, so does the Earth move in circles. Both of them revolve around a common center, and the forces on Earth are balanced in such a way that the water in x is attracted by the Moon weaker, in y - stronger, and in both places the water swells. So the hot flashes were explained and why they occur twice a day.

Discovery of the speed of light

With the development of science, measurements were made more and more accurately and the confirmation of Newton's laws became more and more convincing. The first accurate measurements concerned the satellites of Jupiter. It would seem that if you carefully observe their circulation, you can be sure that everything happens according to Newton. However, it turned out that this was not the case. Jupiter's satellites appeared at the calculated points either 8 minutes earlier or 8 minutes later than would have been expected according to Newton's laws. They were found to be ahead of schedule when Jupiter is approaching the Earth, and behind when Jupiter and Earth are moving apart, a very strange phenomenon.

Römer (Olaf Römer (1644-1710) - Danish astronomer), convinced of the correctness of the law of gravity, came to the interesting conclusion that it takes a certain time for light to travel from the satellites of Jupiter to the Earth, and, looking at the satellites of Jupiter, we see them not where they are now, but where they were several minutes ago - as many minutes as it takes light to reach us. When Jupiter is closer to us, the light comes faster, and when Jupiter is farther away, the light travels longer; therefore Römer had to correct his observations for this difference in time, i.e. take into account that sometimes we make these observations earlier, and sometimes later. From this he was able to determine the speed of light. This was the first time it was established that light does not propagate instantaneously.

Discovery of the planet

Another problem arose: the planets should not move in ellipses, because, according to Newton's laws, they not only attract the Sun, but also attract each other - weakly, but still attract, and this slightly changes their movement. Large planets were already known - Jupiter, Saturn, Uranus - and it was calculated how much they should deviate from their perfect Keplerian orbits-ellipses due to mutual attraction. When these calculations were completed and verified by observations, it was found that Jupiter and Saturn were moving in full accordance with the calculations, and something strange was happening with Uranus. It would seem that there is still reason to doubt Newton's laws; But most importantly, don't lose heart! Two people, John Couch Adams (1819-1892) - English mathematician and astronomer; Urbain Le Verrier (1811-1877) French astronomer, who performed these calculations independently and almost simultaneously, suggested that the motion of Uranus is influenced by an invisible planet. They sent letters to observatories suggesting, "Point your telescope this way and you will see an unknown planet." “What nonsense,” they said in one of the observatories, “some boy got a paper and a pencil in his hands, and he tells us where to look for a new planet.” In another observatory, the directorate was easier to climb - and Neptune was discovered there!

People have dreamed of traveling to the stars since ancient times, starting from the time when the first astronomers examined other planets of our system and their satellites in primitive telescopes. Many centuries have passed since then, but alas, interplanetary and even more so flights to other stars are impossible even now. And the only extraterrestrial object that researchers have visited is the Moon.

We know that Gravity is the force with which the Earth attracts various objects.

Gravity is always directed towards the center of the planet. The force of gravity tells the body an acceleration, which is called the acceleration of free fall and is numerically equal to 9.8 m / s 2. This means that any body, regardless of its mass, in free fall (without air resistance) changes its speed for every second of falling by 9.8 m / s.

Using the formula to find the free fall acceleration

The mass of the planets M and their radius R are known through astronomical observations and complex calculations.

and G is the gravitational constant (6.6742 10 -11 m 3 s -2 kg -1).

If we apply this formula to calculate the gravitational acceleration on the Earth's surface (mass M = 5.9736 1024 kg, radius R = 6.371 106 m), we get g \u003d 6.6742 * 10 * 5.9736 / 6.371 * 6.371 \u003d 9.822 m / s 2

The standard (“normal”) value adopted when constructing systems of units is g = 9.80665 m / s 2, and in technical calculations they usually take g = 9.81 m / s 2.

The standard value of g has been defined as "average" in some sense the acceleration of free fall on Earth, approximately equal to the acceleration of free fall at a latitude of 45.5° at sea level.

Due to attraction to the Earth, water flows in rivers. A person, jumping up, falls to the Earth, because the Earth attracts him. The Earth attracts all bodies to itself: the Moon, the water of the seas and oceans, houses, satellites, etc. Due to gravity, the appearance of our planet is constantly changing. Avalanches come down from the mountains, glaciers move, rockfalls fall, rains fall, rivers flow from the hills to the plains.

All living beings on earth feel its attraction. Plants also "feel" the action and direction of gravity, which is why the main root always grows down to the center of the earth, and the stem up.

The Earth and all other planets moving around the Sun are attracted to it and to each other. Not only the Earth attracts bodies to itself, but these bodies also attract the Earth to themselves. Attract each other and all the bodies on Earth. For example, the attraction from the Moon causes the ebb and flow of water on Earth, huge masses of which rise in the oceans and seas twice a day to a height of several meters. Attract each other and all the bodies on Earth. Therefore, THE MUTUAL ATTRACTION OF ALL BODIES IN THE UNIVERSE IS CALLED UNIVERSAL GRAVITATION.

To determine the force of gravity acting on a body of any mass, it is necessary to multiply the acceleration of free fall by the mass of this body.

F=g*m,

where m is the mass of the body, g is the free fall acceleration.

From the formula it can be seen that the value of gravity increases with increasing body weight. It can also be seen that the force of gravity also depends on the magnitude of the free fall acceleration. So, we conclude: for a body of constant mass, the value of gravity changes with a change in the acceleration of free fall.

Using the formula for finding the free fall acceleration g=GM/R 2

We can calculate g values ​​on the surface of any planet. The mass of the planets M and their radius R are known through astronomical observations and complex calculations. where G is the gravitational constant (6.6742 10 -11 m 3 s -2 kg -1).

The planets have long been divided by scientists into two groups. The first is the terrestrial planets: Mercury, Venus, Earth, Mars, and more recently Pluto. They are characterized by relatively small size, a small number of satellites and a solid state. The rest - Jupiter, Saturn, Uranus, Neptune - are giant planets, consisting of gaseous hydrogen and helium. All of them move around the Sun in elliptical orbits, deviating from a given trajectory if a neighboring planet passes nearby.

Our "first space station" is Mars. How much would a person weigh on Mars? It is not difficult to make such a calculation. To do this, you need to know the mass and radius of Mars.

As is known, the mass of the "red planet" is 9.31 times less than the mass of the Earth, and the radius is 1.88 times smaller than the radius of the globe. Consequently, due to the action of the first factor, the force of gravity on the surface of Mars should be 9.31 times less, and due to the second - 3.53 times greater than ours (1.88 * 1.88 = 3.53 ). Ultimately, it is there a little more than 1/3 of the earth's gravity (3.53: 9.31 = 0.38). It is 0.38 g of the earth, which is about half as much. This means that on the red planet you can jump and jump much higher than on Earth, and all weights will also weigh much less. In the same way, one can determine the stress of gravity on any celestial body.

Now let's define the stress of gravity on the Moon. The mass of the Moon, as we know, is 81 times less than the mass of the Earth. If the Earth had such a small mass, then the gravity force on its surface would be 81 times weaker than it is now. But according to Newton's law, the ball attracts as if all its mass is concentrated in the center. The center of the Earth is at a distance of an earth radius from its surface, the center of the moon is at a distance of a lunar radius. But the lunar radius is 27/100 of the earth, and from a decrease in distance by 100/27 times, the force of attraction increases by (100/27) 2 times. So, in the end, the gravitational stress on the surface of the moon is

100 2 / 27 2 * 81 = 1/6 earth

It is curious that if water existed on the Moon, a swimmer would feel in the lunar reservoir just as on Earth. Its weight would decrease by a factor of six, but the weight of the water it displaces would also decrease by the same amount; the ratio between them would be the same as on Earth, and the swimmer would be immersed in the water of the Moon exactly as much as he is immersed in ours.

free fall acceleration on the surface of some celestial bodies, m/s 2

Sun 273.1

Mercury 3.68-3.74

Venus 8.88

Earth 9.81

Moon 1.62

Ceres 0.27

Mars 3.86

Jupiter 23.95

Saturn 10.44

Uranus 8.86

Neptune 11.09

Pluto 0.61

As can be seen from the table, an almost identical value of the acceleration of free fall is present on Venus and is 0.906 of the earth's.

Now let's agree that on Earth an astronaut-traveler weighs exactly 70 kg. Then for other planets we get the following weight values ​​(the planets are arranged in order of increasing weight):

But on the Sun, gravity (attraction) is 28 times stronger than on Earth. A human body would weigh 20,000 N there and would be instantly crushed by its own weight.

If we have a space trip to the planets of the solar system, then we need to be prepared for the fact that our weight will change. The force of attraction also has various effects on living beings. Simply put, when other habitable worlds are discovered, we will see that their inhabitants differ greatly from each other depending on the mass of their planets. For example, if the Moon were inhabited, then it would be inhabited by very tall and fragile creatures, and vice versa, on a planet with the mass of Jupiter, the inhabitants would be very short, strong and massive. Otherwise, on weak limbs in such conditions, you simply cannot survive with all your desire. The force of gravity will play an important role in the future colonization of the same Mars.

Student . The story is widely known that the discovery of Newton's law of universal gravitation was caused by the fall of an apple from a tree. How reliable this story is, we do not know, but the fact remains that the question that we have gathered today to discuss: "Why does the moon not fall to the Earth?" interested Newton and led him to the discovery of the law of gravity. Newton argued that between the Earth and all material bodies there is a gravitational force, which is inversely proportional to the square of the distance.

Newton calculated the acceleration imparted to the Moon by the Earth. The acceleration of freely falling bodies near the Earth's surface is equal to g=9.8 m/s 2 . The Moon is removed from the Earth at a distance equal to about 60 Earth radii. Therefore, Newton reasoned, the acceleration at this distance will be: . The moon, falling with such an acceleration, should approach the Earth in the first second by 0.0013 m. But the moon, in addition, moves by inertia in the direction of the instantaneous velocity, i.e., along a straight line tangent to its orbit at a given point around the Earth (Fig. 25).

Moving by inertia, the Moon should move away from the Earth, as the calculation shows, in one second by 1.3 mm. Of course, such a motion, in which in the first second the Moon would move along the radius to the center of the Earth, and in the second second - tangentially, does not really exist. Both movements add up continuously. As a result, the Moon moves along a curved line close to a circle.

Let's conduct an experiment, which shows how the force of attraction acting on a body at a right angle to the direction of its movement turns a rectilinear movement into a curvilinear one. A ball, having rolled down from an inclined chute, by inertia continues to move in a straight line. If, however, a magnet is placed on the side, then under the influence of the force of attraction to the magnet, the trajectory of the ball is curved (Fig. 26).

The moon revolves around the earth, held by the force of gravity. A steel rope that could keep the moon in orbit would have to have a diameter of about 600 km. But, despite such a huge force of attraction, the Moon does not fall to the Earth, because, having an initial speed, it moves by inertia.

Knowing the distance from the Earth to the Moon and the number of revolutions of the Moon around the Earth, Newton determined the centripetal acceleration of the Moon. We got a number already known to us: 0.0027 m/s2.
Stop the force of attraction of the Moon to the Earth - and the Moon will rush in a straight line into the abyss of outer space. So in the device shown in Figure 27, the ball will fly away tangentially if the thread holding the ball on the circle breaks. In the device you know on a centrifugal machine (Fig. 28), only the connection (thread) keeps the balls in a circular orbit.

When the thread breaks, the balls scatter along the tangents. It is difficult for the eye to catch their rectilinear movement when they are devoid of connection, but if we make a drawing (Fig. 29), it will be seen that the balls move in a rectilinear manner, tangentially to the circle.

Stop moving by inertia - and the moon would fall to the Earth. The fall would have lasted four days, nineteen hours, fifty-four minutes, fifty-seven seconds, Newton calculated.

The teacher present at the class. The report is over. Who has questions?

Question . With what force does the earth pull the moon?

Student . This can be determined by the formula expressing the law of gravity: , where G is the gravitational constant, M and m are the masses of the Earth and the Moon, r is the distance between them. I expected this question and did the calculation beforehand. The earth pulls the moon with a force of about 2 * 10 20 N.

Question . The law of universal gravitation applies to all bodies, which means that the Sun also attracts the Moon. I wonder with what strength?

Answer . The mass of the Sun is 300,000 times the mass of the Earth, but the distance between the Sun and the Moon is 400 times greater than the distance between the Earth and the Moon. Therefore, in the formula, the numerator will increase by 300,000 times, and the denominator - by 400 2, or 160,000 times. The gravitational force will be almost twice as large.

Question . Why doesn't the moon fall on the sun?

Answer . The moon falls on the sun in the same way as on the earth, that is, only enough to remain at about the same distance, revolving around the sun.

- Around the Earth!

- Wrong, not around the Earth, but around the Sun. The Earth revolves around the Sun together with its satellite - the Moon, which means that the Moon also revolves around the Sun.

Question . The moon does not fall to the Earth, because, having an initial speed, it moves by inertia. But according to Newton's third law, the forces with which two bodies act on each other are equal in absolute value and oppositely directed. Therefore, with what force the Earth attracts the Moon to itself, with the same force the Moon attracts the Earth. Why doesn't the Earth fall on the Moon? Or does it revolve around the moon?

Teacher . The fact is that both the Moon and the Earth revolve around a common center of mass. Recall the experience with the balls and the centrifugal machine. The mass of one of the balls is twice the mass of the other. In order for the balls connected by a thread to remain in equilibrium with respect to the axis of rotation during rotation, their distances from the axis, or center of rotation, must be inversely proportional to the masses. The point around which these balls revolve is called the center of mass of the two balls.

Newton's third law is not violated in the experiment with balls: the forces with which the balls pull each other towards a common center of mass are equal. The common center of mass of the Earth and the Moon revolves around the Sun.

Question . Can the force with which the Earth pulls on the Moon be called the weight of the Moon?

Student . No! We call the weight of the body the force caused by the attraction of the Earth, with which the body presses on some support, for example, a scale pan, or stretches the spring of a dynamometer. If you put a stand under the Moon (from the side facing the Earth), then the Moon will not put pressure on it. The moon would not stretch the spring of the dynamometer, if we could hang it. The entire action of the force of attraction of the Moon by the Earth is expressed only in keeping the Moon in orbit, in imparting centripetal acceleration to it. It can be said about the Moon that in relation to the Earth it is weightless in the same way as objects in a space ship-satellite are weightless when the engine stops working and only the force of attraction to the Earth acts on the ship.

Question . Where is the center of mass of the Earth-Moon system?

Answer . The distance from the Earth to the Moon is 384,000 km. The ratio of the mass of the Moon to the mass of the Earth is 1:81. The distances from the center of mass to the centers of the Moon and the Earth will be inversely proportional to these numbers. Dividing 384,000 km by 82, we get approximately 4,700 km. This means that the center of mass is located at a distance of 4700 km from the center of the Earth.

What is the radius of the earth?

– About 6400 km.

– Consequently, the center of mass of the Earth-Moon system lies inside the globe (Fig. 30, point O). Therefore, if you do not pursue accuracy, you can talk about the revolution of the Moon around the Earth.

Question . Which is easier: to fly from the Earth to the Moon or from the Moon to the Earth?

Answer . In order for a rocket to become an artificial satellite of the Earth, it must be given an initial speed approximately equal to 8 km / s. In order for the rocket to leave the Earth's sphere of gravity, the so-called second cosmic velocity, equal to 11.2 km / s, is needed. To launch rockets from the Moon, you need less speed: after all, gravity on the Moon is six times less than on Earth.

Question . I don't understand why bodies inside a rocket have no weight. Maybe it's only at that point on the way to the Moon, at which the force of attraction to the Moon is balanced by the force of attraction to the Earth?

Teacher . No. The bodies inside the rocket become weightless from the moment when the engines stop working and the rocket begins free flight in orbit around the Earth, while being in the Earth's gravitational field. In free flight around the Earth, both the satellite and all objects in it relative to the center of mass of the Earth move with the same centripetal acceleration and therefore are weightless.

1st question. How did balls not connected by a thread move on a centrifugal machine: along a radius or tangent to a circle?

The answer depends on the choice of the frame of reference, i.e., on the choice of the body with respect to which we are considering the motion of the balls. If we take the surface of the table as the reference system, then the balls move along tangents to the circles they describe. If we take the rotating device itself as the reference system, then the balls move along the radius. Without specifying the reference system, the question of the nature of the motion does not make sense. To move means to move relative to other bodies, and we must necessarily indicate with respect to which ones.

2nd question. What does the moon revolve around?

If we consider the movement relative to the Earth, then the Moon revolves around the Earth. If the Sun is taken as the reference body, then it is around the Sun. Let me explain what was said with a drawing from the book "Entertaining Astronomy" by Perelman (Fig. 31). Say, with respect to which body the movement of celestial bodies is shown here.

- Relative to the Sun.

- Right. But it is easy to see that the Moon is constantly changing its position relative to the Earth.

Teacher . Of course they can't. At the position of the Earth or the Moon (note I say "or", not "and") at the point of intersection of the orbits shown, the distance between the Earth and the Moon is 380,000 km. To better understand this, draw a diagram of this complex movement for the next lesson. Draw the Earth's orbit as an arc of a circle with a radius of 15 cm (the distance from the Earth to the Sun, as you know, is 150,000,000 km). On an arc equal to 1/12 of a circle (the monthly path of the Earth), mark five points at equal distances, counting the extreme ones. These points will be the centers of the lunar orbits relative to the Earth in consecutive quarters of the month. The radius of the lunar orbits cannot be plotted on the same scale as the Earth's orbit, as it would be too small. To draw lunar orbits, it is necessary to increase the selected scale by about ten times, then the radius of the lunar orbit will be about 4 mm. Indicate the position of the Moon on each orbit, starting with the full moon, and connect the marked points with a smooth dotted line.

At the next lesson of the circle, one of the students showed the required diagram (Fig. 32).

The story of a student drawing a diagram: “I learned a lot while drawing this diagram. It was necessary to correctly determine the position of the Moon in its phases, to think about the direction of movement of the Moon and the Earth in their orbits. There are inaccuracies in the drawing. I will tell about them now. At the selected scale, the curvature of the lunar orbit is incorrectly depicted. It must always be concave with respect to the Sun, i.e., the center of curvature must be inside the orbit. In addition, there are not 12 lunar months in a year, but more. But one twelfth of a circle is easy to construct, so I conditionally accepted that there are 12 lunar months in a year. And finally, it is not the Earth itself that revolves around the Sun, but the common center of mass of the Earth-Moon system.

13. Movement of celestial bodies under the influence of gravitational forces

1. Cosmic velocities and shape of orbits

Based on observations of the motion of the moon and analyzing the laws of motion of the planets discovered by Kepler, I. Newton (1643-1727) established the law of universal gravitation. According to this law, as you already know from the physics course, all bodies in the Universe are attracted to each other with a force that is 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, binary and multiple stars around their common center of mass.

Newton proved that under the influence of mutual gravitation, bodies can move relative to each other along ellipse(particularly for circle), on 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 some speed the body describes circle near the center of attraction. This speed is called the first cosmic or circular speed, it is reported to bodies launched as artificial satellites of the Earth in circular orbits. (The derivation of the formula for calculating the first cosmic velocity is known from the course of physics.) The first cosmic velocity near the surface of the Earth is about 8 km / s (7.9 km / s).

If the body is given a speed that is twice 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 along parabola relative to the earth. At an even greater speed relative to the Earth, the body will fly along a hyperbola. Moving along a parabola or hyperbole, the body goes around the Sun only once and forever moves away from it.

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 movement along the orbit is close to circular at the distance of the Earth from the Sun. The parabolic velocity at the Earth's distance 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 the planets

Kepler's laws are exactly 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 perturbations. In the solar system, perturbations are small, because the attraction of each planet by the Sun is much stronger than the attraction of other planets.

The biggest perturbation 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 motion of asteroids and comets when they come close to it. In particular, if the directions of the comet's accelerations caused by the attraction of Jupiter and the Sun coincide, then the comet can develop such a high speed that, moving along a hyperbola, it will leave the solar system forever. There were cases when the attraction of Jupiter held back the comet, the eccentricity of its orbit became smaller and the period of revolution sharply decreased.

When calculating the apparent position of the planets, perturbations must be taken into account. Now high-speed electronic computers help to make such calculations. When launching artificial celestial bodies and when calculating their trajectories, they use the theory of motion of celestial bodies, in particular, the theory of perturbations.

The ability to send automatic interplanetary stations along the desired, pre-calculated trajectories, to bring them to the goal, taking into account disturbances in movement - all these are vivid examples of the cognizability of the laws of nature. The sky, which according to the believers is the abode of the gods, has become the arena of human activity in the same way as the Earth. Religion has always contrasted the Earth and the sky and declared the sky inaccessible. Now, among the planets, artificial celestial bodies are moving, created by man, which he can control by radio from great distances.

3. Discovery of Neptune

One of the clearest examples of the achievements of science, one of the evidence of the unlimited cognizability of nature was the discovery of the planet Neptune by calculations - "at the tip of a pen."

Uranus - the planet following Saturn, which for many centuries was considered the most distant of the planets, was discovered by V. Herschel at the end of the 18th century. Uranus is hardly visible to the naked eye. By the 40s of the XIX century. accurate observations have shown that Uranus deviates just barely from the path it should follow, given the perturbations from all the known planets. Thus the theory of motion of celestial bodies, so rigorous and precise, was put to the test.

Le Verrier (in France) and Adams (in England) suggested that if perturbations from the known planets do not explain the deviation in the motion of Uranus, it means that the attraction of an as yet unknown body acts on it. They almost simultaneously calculated where behind Uranus there should be an unknown body that produces these deviations by its attraction. They calculated the orbit of the unknown planet, its mass and indicated the place in the sky where the unknown planet should have been at the given time. This planet was found in a telescope at the place indicated by them in 1846. It was called Neptune. Neptune is not visible to the naked eye. Thus, the disagreement between theory and practice, which seemed to undermine the authority of materialistic 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. The rotation of bodies (as you know from physical experiments) leads to their flattening, to compression along the axis of rotation. Therefore, the globe is slightly compressed at the poles, and the rapidly rotating Jupiter and Saturn are most compressed.

But the shape of the planets can also change from the action of 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 all the force of attraction were applied to its center. However, individual parts of the planet are at different distances from the attracting body, so the gravitational acceleration in them is also different, which leads to the emergence of forces that tend to deform the planet. The difference in accelerations caused by the attraction of another body at a given point and in the center of the planet is called tidal acceleration.

Consider, for example, the Earth-Moon system. The same element of mass in the center of the Earth will be attracted by the Moon weaker than on the side facing the Moon, and stronger than on the opposite side. As a result, the Earth, and primarily the water shell of the Earth, is slightly extended in both directions along the line connecting it with the Moon. In Figure 35, the ocean is depicted covering the entire Earth for clarity. 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 the lowest - there is a low tide. With the daily rotation of the Earth, different places on the Earth alternately enter the tide band. It is easy to understand that there can be two high tides and two low tides in a day.

The sun also causes ebbs and flows on Earth, but due to the great distance of the Sun, they are smaller than the moon and less noticeable.

The tides move a huge amount of water. At present, they are beginning to use the enormous energy of water, which participates in the tides, on the shores of the oceans and open seas.

The axis of the tidal protrusions must 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. There is friction between the water and the solid bottom of the ocean. As a result, the so-called tidal friction. It slows down the rotation of the Earth, and the days become longer over time (once they were only 5-6 hours). The strong tides caused on Mercury and Venus by the Sun, apparently, were the reason for their extremely slow rotation around their axis. The tides caused by the Earth have slowed down the rotation of the Moon so much that it always faces the Earth on 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

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 the value g = 9.8 m / s 2, G = 6.67 * 10 -11 N * m 2 / kg 2, R \u003d 6370 km into the indicated formula, we find that the mass of the Earth is M \u003d 6 * 10 24 kg.

Knowing the mass and volume of the Earth, we 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 core of the Earth, it is equal to 1.1*10 4 kg/m 3 . The 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 the internal structure of the Earth, studied by astronomical and geophysical methods, you got acquainted in the course of physical geography.)

Exercise 12

1. What is the density of the Moon if its mass is 81 times, and the radius is 4 times less than that of the Earth?

2. What is the mass of the Earth, if the angular velocity 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 precise formula for Kepler's third law is:

where M 1 and M 2 are the masses of any celestial bodies, and 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 by the presence of a factor containing masses. If under M 1 =M 2 = M we mean the mass of the Sun, and under 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 noticeably differ from the approximate one.

To compare the masses of the Earth and another planet, such as Jupiter, in the original formula, the index 1 must be attributed to the movement of the Moon around the Earth with a mass M 1, and 2 - to the movement of any satellite around Jupiter with a mass M 2.

The masses of planets that do not have satellites are determined by the perturbations that they produce by their attraction in the motion of their neighboring planets, as well as in the motion 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 attraction of the Earth and the Moon are the same, knowing that the distance between the Moon and the Earth is 60 Earth radii, and the mass of the Earth is 81 times the mass of the Moon.

As science knows, the Moon is a natural satellite of the Earth, a spherical celestial body, cold, but not cooled down (it is believed that the Moon was originally cold). The Moon is located at a distance of 384,000 kilometers from the Earth, its radius is 1738 kilometers. There is no water on the Moon, no atmosphere, and any weight there is six times lighter than on Earth.

There is no water on the moon. But its connection with water is the most direct.

Most of the Earth's surface is covered by seas and oceans. There is a lot of water on our planet. If it were not so, life would hardly have appeared here. All living things need a large amount of fluid. The human body is more than sixty percent water. This is water, which is contained in the composition of every cell of the body, and blood, and other fluids.

The ebb and flow of the terrestrial seas and oceans are connected with the Moon. The moon with great force attracts the water surface of the part of the Earth over which it is located. Imagine: a huge tidal wave all the time "runs" after the Moon on the earth's surface, when the Moon makes a complete revolution around the Earth.

This happens for a completely natural reason - according to the law of universal gravitation, which operates in the entire universe. All celestial bodies, including the Sun, Moon and Earth, have a force of attraction - some more, others less, depending on their size. It is thanks to this force that we all stand firmly on the ground: the forces of gravity, the forces of gravity attract us. Due to the force of solar attraction, the Earth revolves around the Sun and does not fly away from it. And the Earth's gravity keeps the Moon in Earth orbit.

The moon is much smaller than the Earth, and therefore it is, of course, not able to attract the Earth to itself. But it can attract terrestrial water masses. And not only them: scientists have found that the Moon deforms even the solid shell of the Earth by gravity, stretching it by about 50 centimeters! The Earth seems to be breathing all the time, inhaling and exhaling with its different parts following the attraction of the Moon moving around it.

But the deformation of the solid surface of the Earth is less noticeable to us than the ebbs and flows. This phenomenon was observed by everyone who was by the sea. Arriving at the beach in the morning, you see that the water has receded, exposing the coastal stones, leaving algae and jellyfish on the wet pebbles. And after a few days it turns out that the strip of the beach, on which you were conveniently located yesterday for relaxation, today disappeared under water.

The strongest tides occur on the new moon. Why? Because in the new moon, both the Sun and the Moon are on the same side of the Earth. Therefore, on the new moon, the Moon is not visible in the sky: the Sun at this time illuminates its reverse side. At this moment, the attraction of the Sun is added to the attraction of the Moon and both luminaries pull the Earth in one direction. Ground water masses rush in this direction. The tide begins to rise, while on the opposite side of the Earth there is a low tide.

During a full moon, the Sun and Moon are on opposite sides of the Earth; The Earth is between the Sun and the Moon, and both luminaries are on opposite sides of it. Then the water masses partly rush towards the Sun, and partly towards the Moon, tides are observed both there and there, but less than on the new moon.

In other phases of the Moon - when the Moon and the Sun are not on the same side of the Earth, and not on opposite sides, but occupy intermediate positions - the ebbs and flows are almost imperceptible, since the Sun and Moon neutralize each other's attraction and the water shell is distributed evenly over the entire the surface of the earth.

Since there is a lot of water on Earth, the earth's climate depends on the state of water. Oceans and seas are the kitchen where earthly weather is “cooked”. And of course, any change in the state of the seas and oceans immediately affects the state of the weather. Weather changes are directly related to the tides. The behavior of the atmosphere depends on this, the emergence of cyclones and anticyclones in it, and hence the humidity of the air, the direction and speed of the wind, and other factors. And our well-being and many processes in the body depend on the weather: changes in blood pressure, blood flow velocity, activity of various organs - you can’t list everything. Not to mention the mood and state of the nerves, psyche, soul - all this is directly affected by the weather. Sunny, clear weather excites and tones us, quiet, cloudy - soothes, low clouds depress, and a strong wind with dampness and cold can lead to depression.

We depend on the weather, the weather originates in the oceans, and the state of the oceans is related to the moon. It turns out that our state ultimately depends on the moon.

But this is just one example of the not very strong and very indirect influence of the Moon on us - through the ebbs and flows of the seas and oceans. In addition, the Moon affects us in many other ways - absolutely directly and very diversely.

As we already know, the human body is more than sixty percent water. But if the Moon attracts earthly water, then the water that is part of our body is no exception.

At the new moon, at the strongest tides, the water inside the body, together with the water of the seas and oceans, rushes up to the Moon. At this moment, it seems that we have become lighter, that we are not walking, but as if we are flying above the ground, and we even want to jump, our legs themselves come off the ground. At this time, you need to be more careful - not to lose your balance and fulcrum in the physical and mental sense. It is difficult to be active, to go about your usual earthly affairs - after all, the body, as it were, breaks away from the earth, it is pulled upwards.

After the new moon, the attraction of the moon weakens and we quietly descend from heaven to earth. The attraction of the Earth again affects us with the usual force. We again acquire the usual sense of our own weight. You can gradually return to normal activity and daily activities, now it is easier.

As the lunar crescent grows and approaches the full moon, the Sun and Moon diverge further and further. They begin to attract all terrestrial liquids from different directions. And our body begins to burst, as it were, liquids stretch in different directions, the process of expansion is underway. Imagine: you have just been pulled up, then down, and now suddenly to the sides. This is a serious stress for the body: it just needs to have time to rebuild.

During the full moon, the Sun and the Moon act on us from opposite sides. Therefore, all fluids of the human body are attracted closer to the surface of the body. The body bursts as much as possible from the inside, as if a void is formed inside, but energy splashes out from the outside - it literally whips with a powerful stream.

But now the Moon begins to decrease, and the organism, which was expanding before, passes to contraction. All liquids from the surface rush inward, energy also flows inward. Such a restructuring is stress again. But as the fluids rush inward, a person feels stronger and more active: after all, now the energy is concentrated inside, and he is ready to act, to use this energy to achieve various goals in his life.

After the maximum compression of energy inside the body, new changes occur - the new moon comes again, and the fluids again rush to the head.

As you can see, the body is not frozen in immobility: something in it is constantly changing, transforming, moving from one state to another; moreover, changes occur synchronously with the moon, and hence with the entire universe. If we know and take into account the changes taking place in us, then health, inner harmony, and well-being will come. If we live in unison with the Universe, then the Universe helps us and supports us with all its immense forces.

The waning or waxing Moon is not only the cause of terrestrial tides; the well-being of a person depends on it, which can be taken care of in advance by referring to the lunar calendar.

How exactly to take into account the lunar rhythms will be discussed more than once in this book. In the meantime, we will understand to the end the mechanisms of our relationship with the moon.

All that we have talked about is the physical influence of the Moon. But there is another effect - energy.