The structure of the solar system. Earth orbit around the sun Planetary orbit

People have always been interested in the unexplored expanses of space. Exploration of other planets attracted many scientists, and common man interesting is the question of what is there in space? First of all, scientists pay attention to the planets of the solar system. Since they are closest to Earth and easier to study. The mysterious red planet - Mars is especially actively studied. Let's find out which planet is bigger - Mars or Earth, and try to understand why the red celestial body attracts us so much.

Brief characteristics of the planets of the solar system. Their sizes

From Earth, all the planets of our system seem to us to be small luminous points that are difficult to see with the naked eye. It differs from all Mars - it seems to us larger than the rest, and sometimes even without telescopic equipment you can see its orange light.

Which planet is bigger: Mars or Earth? Do we see Mars so well because it is huge, or is it just closer to us? Let's take a look at this issue. To do this, we will consistently consider the sizes of all planets belonging to the solar system. They were divided into two groups.

Terrestrial group of planets

Mercury is the smallest planet. In addition, it is the closest to the sun. Its diameter is 4878 km.

Venus is the planet next in distance from the Sun and closest to the Earth. Its surface temperature reaches +5000 degrees Celsius. The diameter of Venus is 12103 km.

The earth differs in that it has an atmosphere and water reserves, which made it possible for life to arise. Its size is slightly larger than Venus and is 12,765 km .

Mars is the fourth planet from the Sun. Earth and has a diameter at the equator of 6786 km. Its atmosphere is almost 96% composed of Mars has a more elongated orbit than Earth.

Giant planets

Jupiter is the largest planet in the solar system. Its diameter is 143,000 km. It consists of gas, which is in a vortex motion. Jupiter rotates around its axis very quickly, in about 10 Earth hours it makes a complete revolution. It is surrounded by 16 satellites.

Saturn is a planet that can justifiably be called unique. Its structure has the lowest density. Saturn is also known for its rings, which are 115,000 km wide and 5 km thick. It is the second largest planet in the solar system. Its size is 120,000 km.

Uranus is unusual in that it can be seen in blue-green with a telescope. This planet also consists of gases that move at a speed of 600 km / h. The diameter is just over 51,000 km.

Neptune is composed of a mixture of gases, most of which is methane. It is because of this that the planet acquired a blue color. The surface of Neptune is shrouded in clouds of ammonia and water. The size of the planet is 49,528 km.

The most distant planet from the Sun is Pluto, it does not belong to any of the groups of planets in the solar system. Its diameter is half that of Mercury and is 2320 km.

Characteristics of the planet Mars. Features of the Red Planet and comparison of its size with the size of the Earth

So we examined the sizes of all the planets in the solar system. Now you can answer the question of which planet is larger - Mars or Earth. A simple comparison of the planetary diameters can help in this. The sizes of Mars and Earth differ by half. The red planet is almost half the size of our Earth.

Mars is a very interesting space object to study. The mass of the planet is 11% of the Temperature on its surface varies throughout the day from +270 to -700 degrees C. The sharp drop is due to the fact that the atmosphere of Mars is not so dense and consists mainly of carbon dioxide.

The description of Mars begins with an emphasis on its deep red color. I wonder what caused this? The answer is simple - the composition of the soil is rich in iron oxides, and the increased concentration of carbon dioxide in its atmosphere. For such a specific color, the ancient people called the planet bloody and gave it a name in honor of the Roman god of war - Ares.

The planet's surface is mostly deserted, but there are also dark areas, the nature of which has not yet been studied. Mars is a plain, and the southern one is slightly elevated from the middle level and dotted with craters.

Many do not know, but the most high mountain in the entire solar system - Olympus. Its height from base to top is 21 km. The width of this hill is 500 km.

Is it possible

All the works of astronomers are aimed at finding signs of life in space. In order to study Mars for the presence of living cells and organisms on its surface, Mars rovers have visited this planet many times.

Numerous expeditions have already proved that water was previously present on the Red Planet. It is there even now, only in the form of ice, and it is hidden under a thin layer of stony soil. The presence of water is also confirmed by photographs in which the channels of the Martian rivers are clearly visible.

Many scientists want to prove that humans can adapt to life on Mars. The following facts are cited to prove this theory:

1. Almost the same speed of movement of Mars and Earth.
2. The similarity of gravitational fields.
3. Carbon dioxide can be used to produce vital oxygen.

Perhaps, in the future, the development of technology will allow us to easily make interplanetary travel and even settle on Mars. But first of all, humanity must preserve and protect its home planet - the Earth, so that you never have to think about which planet is bigger - Mars or Earth, and whether the red planet will be able to accept all willing migrants.

Do you know what the orbit of a planet is? Geography (grade 6) gave us an idea about but many probably did not understand what it is, what it is for and what will happen if the planet changes its orbit.

Orbit concept

So what is the orbit of the planet? The simplest definition: an orbit is the path of a body around the sun. Gravity forces you to move one by one
the same paths around the star from year to year, from one million years to the next million. On average, the planets have an ellipsoidal orbit. The closer its shape is closer to a circle,
the more stable the weather conditions on the planet.

The main characteristics of the orbit are the orbital period and radius. The mean radius is the average between the minimum orbital diameter and
maximum. The orbital period is the length of time that a celestial body needs to fully fly around the star.
the distance separating the star and the planet, the longer the orbital period will be, since the effect of the star's gravity at the outskirts of the system is much weaker than at its center.

Since no orbit can be absolutely round, during the planetary year the planet is at different distances from the star. Place, where
the planet is closest to the star, it is usually called the periastron. The point farthest from the star, on the contrary, is called the apoaster. For the solar system it is
perihelion and aphelion, respectively.

Orbital elements

It is clear what the planet's orbit is. What do its elements represent? There are several elements that are commonly identified around the orbit. It is by these parameters that scientists determine the type of orbit, the characteristics of the movement of the planet and some other parameters that are insignificant for the average person.

• Eccentricity. This is an indicator that helps to understand how elongated the planet's orbit is. The lower the eccentricity, the more rounded the orbit has, while a celestial body with high eccentricity moves around the star along a strongly elongated ellipse. The planets of the solar system have extremely low eccentricities, which indicates their almost circular orbits. Comets are characterized by unusually high eccentricities.
• Semi-major axis. It is calculated from the planet to the average point halfway along the orbit. This is not synonymous with apastron, since the star is not located in the center of its orbit, but in one of its focuses.
• Mood. For these calculations, the planet's orbit is a kind of plane. The second parameter is the base plane, that is, the orbit of a particular body in the star system or conventionally accepted. So in the solar system, the base is considered to be called the ecliptic. For the planets of other stars, such is the plane that lies on the line of the observer from the Earth. In our system, almost all orbits are located in the plane of the ecliptic. However, comets and some other bodies move at a high angle to it.

Solar system orbits

So, orbiting a star is what is called the orbit of a planet. In our solar system, the orbits of all planets are directed in the same direction in which
the sun rotates. This movement is explained by the theory of the origin of the Universe: after the Big Bang, pratoplasm moved in one direction, matter with the flow
time became denser, but their movement did not change.

Around their own axis, the planets move similarly to the rotation of the Sun. The only exceptions to this are Venus and Uranus, which rotate around their axis in
your own unique mode. Perhaps they were once exposed to celestial bodies, which changed the direction of their rotation around their axis.

The plane of motion in the solar system

As already mentioned, the orbits of the planets in the solar system are located almost on the same plane, close to the plane of the Earth's orbit. Knowing what the orbit of the planet is,
it can be assumed that the reason why the planets move in practically the same plane is most likely the same: once the substance, from which now
all bodies in the solar system consist, it was a single cloud and rotated around its axis under the influence of external gravity. Over time, the substance
divided into the one from which the Sun was formed, and the one that for a long time was a dust disk revolving around the star. The dust gradually formed
planets, and the direction of rotation remains the same.

Orbits of other planets

It's hard to talk about this topic. The fact is that we know what the orbit of a planet is, but until recently we did not know if other stars even have planets.
Only recently, using the latest equipment and modern observation methods, have scientists been able to calculate the presence of planets in other stars. Such planets are called
exoplanets. Despite the incredible power of modern equipment, only a few exoplanets were photographed or seen, and the observation of them surprised
scientists.

The fact is that these few planets seem to be completely unfamiliar with what the planet's orbit is. Geography claims that all bodies move along the eternal
laws. But it seems that the laws of our system do not apply to other stars. There were such planets close to the star that, it seemed to scientists, could
exist only at the very edge of the system. And these planets behave in a completely different way from how they should behave according to calculations: they rotate in the wrong
side that their star, and their orbits lie in different planes and have too elongated orbits.

The sudden stop of the planet

In fact, a sudden, unrelated stop is simply unrealistic. But let's say it happened.

Despite the stop of the whole body, its individual elements cannot stop abruptly either. This means that the magma and the core will continue to move by inertia. Until full
stopping all the filling of the earth will have time to turn more than once, completely breaking the crust of the Earth. This will cause an instantaneous release of a huge amount of lava, enormous
faults and the emergence of volcanoes in extremely unexpected places. Thus, almost instantly, life on Earth will cease to exist.

In addition, even if it is possible to stop the "filling" immediately, there is still an atmosphere. It will continue its inertial rotation. And this is a speed of about 500 m / s.
Such a "breeze" will sweep away all living and nonliving from the surface, carrying it along with the atmosphere itself into Space.

Stop rotation gradually

If the rotation around its axis stops not suddenly, but for a long time, there is a minimal chance of surviving. As a result of the disappearance
centrifugal force, the oceans will rush to the poles, while the land will be at the equator. In this situation, the day will be equal to the year, and the change of seasons will correspond to the onset of the time of day: morning - spring, day - summer, etc. The temperature regime will be much more extreme, since neither the oceans nor the movement of the atmosphere will soften it.

What will happen if the Earth goes out of orbit?

Another fantasy: what happens if the planet goes out of orbit? The planet cannot simply move to another orbit. It means that this collision with another celestial body helped her to make. In this case, an explosion of enormous force will destroy everything and everyone.

If we assume that the planet simply stopped in space, stopping its movement around the Sun, then the following will happen. Under the influence of the sun's gravity, our planet will head towards it. She will not be able to catch up with him, since the Sun also does not stand in one place. But it will fly close enough to the luminary for the solar wind to destroy the atmosphere, evaporate all moisture and burn all the land. An empty burnt ball will fly further. Having reached the orbits of distant planets, the Earth will affect their movement. Once near the giant planets, the Earth is likely to be torn apart into small pieces.

These are the scenarios of likely events when the Earth stops. However, scientists to the question "can the planet go out of orbit" answer unequivocally: no. She is more or
less successfully existed for more than 4.5 billion years, and in the foreseeable future there is nothing that could prevent it from holding out for as long ...

The most important (and most massive!) Member of the solar system is the sun itself. Therefore, it is no coincidence that the great star occupies a central position in the solar system. It is surrounded by numerous companions. The most significant of these are the large planets.

The planets are spherical "heavenly lands". Like the Earth and the Moon, they do not have their own light - they are illuminated exclusively by the sun's rays. Nine major planets are known that are distant from the central star in the following order: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto... Five planets - Mercury, Venus, Mars, Jupiter and Saturn - have been known to people since time immemorial due to their brilliance. Nicolaus Copernicus included our Earth among the planets. And the most distant planets - Uranus, Neptune and Pluto - were discovered using telescopes.

The solar system, a system of cosmic bodies, including, in addition to the central luminary - Suns - nine major planets, their satellites, many minor planets, comets, small meteoric bodies and cosmic dust moving in the area of \u200b\u200bthe prevailing gravitational action of the Sun. The solar system was formed about 4.6 billion years ago from a cold cloud of gas and dust. At present, with the help of modern telescopes (in particular, the Hubble Space Telescope), astronomers have discovered several stars with similar protoplanetary nebulae, which confirms this cosmogonic hypothesis. The general structure of the solar system was revealed in the middle of the 16th century. N. Copernicus, who substantiated the concept of the motion of planets around the Sun. This model of the solar system is called heliocentric... In the 17th century. I. Kepler discovered the laws of planetary motion, and I. Newton formulated the law of universal gravitation. The study of the physical characteristics of cosmic bodies that make up the solar system became possible only after the invention of the telescope by G. Galileo in 1609. So, observing sunspots, Galileo first discovered the rotation of the Sun around its axis.

Our Earth is in third place from the Sun. Its average distance from it is 149.6 million km. It is taken as one astronomical unit (1 AU) and serves as a standard in measuring interplanetary distances. Light travels 1 a. That is, in 8 minutes and 19 seconds, or 499 seconds.

The average distance of Mercury from the Sun is 0.387 AU. That is, that is, it is 2.5 times closer to the central luminary than our Earth, and the average distance of distant Pluto is almost 40 such units. A radio signal sent from Earth towards Pluto would take almost 5.5 hours to "travel". The further the planet is from the Sun, the less radiant energy it receives. Therefore, the average temperature of the planets drops rapidly with increasing distance from the radiant star.

By physical characteristics the planets are clearly divided into two groups. The four closest to the Sun - Mercury, Venus, Earth and Mars - are called planets terrestrial group ... They are relatively small, but their average density is high: about 5 times that of water. After the Moon, the planets Venus and Mars are our closest space neighbors. Far from the Sun, Jupiter, Saturn, Uranus and Neptune are much more massive than the terrestrial planets and even more exceed them in volume. In the bowels of these planets, matter is strongly compressed, nevertheless, their average density is low, and Saturn has even less density of water. Hence, giant planets consist of lighter (volatile) substances than the terrestrial planets.

At one time, astronomers attributed Pluto to planets like the Earth. However, recent studies have forced scientists to abandon this view. Frozen methane was found on its surface by spectroscopy. This discovery testifies to the similarity of Pluto with the large satellites of the giant planets. Some researchers are inclined to think that Pluto is the "escaped" satellite of Neptune.

Even Galileo, who discovered the four largest satellites of Jupiter (they are called Galilean satellites), the remarkable Jupiter family was represented by the solar system in miniature. Today natural satellites are known in almost all major planets (with the exception of Mercury and Venus), and their total number has increased to 137. There are especially many satellites-moons in giant planets.

If we were given the opportunity to look at the solar system from the side of its north pole, then we could observe a picture of the orderly movement of the planets. They all move around the Sun in almost circular orbits in the same direction - opposite to the clockwise rotation. This direction of movement in astronomy is usually called direct movement... But the revolution of the planets takes place not around the geometric center of the Sun, but around the common center of mass of the entire solar system, in relation to which the Sun itself describes a complex curve. And very often this center of mass is outside the solar ball.

The solar system is far from being exhausted by the central luminary - the Sun and nine large planets with their satellites. Needless to say, large planets are the most important representatives of the Sun family. However, our great luminary also has many other "relatives".

The German scientist Johannes Kepler spent almost his entire life searching for the harmony of planetary movements. He was the first to draw attention to the fact that between the orbits of Mars and Jupiter, there is an empty space. And Kepler was right. Two centuries later, in this interval, a planet was indeed discovered, only not large, but small. In its diameter, it turned out to be 3.4 times, and in volume - 40 times smaller than our Moon. The new planet was named after the ancient Roman goddess Ceres, the patroness of agriculture.

Over time, it turned out that Ceres has thousands of celestial "sisters" and most of them move just between the orbits of Mars and Jupiter. There they form a kind minor planet belt... In the bulk, these are crumb planets with a diameter of about 1 km. Second belt of minor planets recently discovered on the outskirts of our planetary system - beyond the orbit of Uranus. It is quite possible that the total number of these celestial bodies in the solar system reaches several million.

But the family of the Sun is not limited to planets alone (large and small). Sometimes tailed "stars" are visible in the sky - comets... They come to us from afar and usually appear suddenly. Scientists believe that on the outskirts of the solar system there is a "cloud" consisting of 100 billion potential, that is, nothing manifested, cometary nuclei. This is what serves as a constant source of the comets we observe.

Occasionally we are "visited" by giant comets. The bright tails of such comets stretch across almost the entire sky. So, in the September comet of 1882, the tail reached 900 million km in length! When the nucleus of this comet flew near the Sun, its tail went far beyond the orbit of Jupiter ...

As you can see, our Sun has a very large family. In addition to nine major planets with their satellites, under the leadership of the great luminary, there are still at least 1 million minor planets, about 100 billion comets, as well as countless meteoric bodies: from blocks several tens of meters in size to microscopic dust particles.

The planets are located at great distances from each other. Even Venus, next to the Earth, is never closer to us than 39 million km, which is 3000 times the diameter of the globe ...

Involuntarily one wonders: what is our solar system? A cosmic desert with separate worlds lost in it? Emptiness? No, the solar system is not a void. In interplanetary space, there is still an innumerable number of particles of solid matter of the most diverse sizes, but mostly very small, with a mass of thousandths and millionths of a gram. it meteor dust... It is formed by the evaporation and destruction of cometary nuclei. As a result of the crushing of the colliding small planets, fragments of various sizes appear, the so-called meteoric bodies... Under the pressure of the sun's rays, the smallest particles of meteoric dust are swept out to the outskirts of the solar system, and the larger ones in a spiral approach the Sun and, before reaching it, evaporate in the vicinity of the central star. Some meteoric bodies fall to Earth in the form meteorites.

The near-solar space is penetrated by all types of electromagnetic radiation and corpuscular flows.

The sun itself is a very powerful source. But on the outskirts of the solar system, radiations coming from the depths of our Galaxy prevail. By the way: how to set the boundaries of the solar system? Where do they go?

It may seem to some that the boundaries of the solar domain are outlined by the orbit of Pluto. After all, there seems to be no major planets beyond Pluto. Here is just the right time to "dig in" the boundary pillars ... But we must not forget that many comets go far beyond Pluto's orbit. Aphelion - the most distant points - their orbits lie in a cloud of pristine ice cores. This hypothetical (supposed) cometary cloud is located at a distance of 100 thousand AU from the Sun. e., that is, 2.5 thousand times farther than Pluto. So here, too, the power of the great luminary extends. Here is the solar system too!

Obviously, the solar system reaches those places in interstellar space where the gravitational force of the sun is commensurate with the force of gravity of the nearest stars. The closest star to us, Alpha Centauri, is 270 thousand AU away from us. e. and by its mass is approximately equal to the Sun. Therefore, the point at which the gravitational forces of the Sun and Alpha Centauri are balanced is approximately in the middle of the distance separating them. And this means that the boundaries of the solar domain are at least 135 thousand AU away from the great luminary. e., or 20 trillion kilometers!

Based on the definition, a planet is a space body revolving around a star. Orbit, in turn, is the trajectory of this planet in the gravitational field of another body, as a rule, most often these bodies are stars. For example, for the Earth, such a body is the Sun.

All the planets of the solar system move along their trajectory in the direction of the rotation of the sun. At the moment, scientists know only one single planet that moves in the opposite direction - an exoplanet called WASP-17b, located in the constellation Scorpio.

Planetary year

The sidereal period of rotation (planetary year) is the time during which a planet makes one revolution around its star. The speed of the planet's movement varies depending on where it is located, the closer to the star the faster the speed, the further away from the star the correspondingly slower the planet moves. Therefore, the length of a planetary year directly depends on the distance at which the planet is located relative to its "Sun". If the distance is short, then the planetary year is relatively short. Since the farther the planet is from the star, the less gravity affects it, which means that the movement becomes slower and the year is correspondingly longer.

Perihelion, aphelion and eccentricity

The orbits of absolutely all planets have the shape of an elongated circle, and how great this elongation is is determined by the eccentricity, if the eccentricity is very small (almost zero), the shape is closest to a circle. Trajectories of motion with an eccentricity close to unity have the shape of an ellipse. For example, the orbits of numerous satellites and exoplanets of the Kuiper belt are elliptical, and all the orbits of the planets of the solar system are almost absolutely circular.

Due to the fact that none of the cosmic orbits known to us is an exact circle, in the process of moving along it, the distance between the planet and the neighboring luminary changes. The point at which the planet is closest to the star is called the periastron. In the solar system, this point is called perihelion. The point of the planet's trajectory farthest from the star is called the apoastron, and in the solar system, the aphelion.

Factor responsible for the change of seasons

The angle between the reference plane and the plane of the orbit is called the inclination of the orbit. The base plane in the solar system is the plane of the Earth's orbit, which is called the ecliptic. There are eight planets in the solar system and their orbits are very close to the plane of the ecliptic.

All the planets of the solar system are located at an angle to the equatorial plane relative to the star. For example, the tilt angle of the Earth's axis is approximately 23 degrees. This factor affects how much light is received by the northern or southern hemisphere of the planet, and is also responsible for the change of seasons.

Change of day and night filmed by Electro-L satellite

10.1. Planetary configurations

The planets of the solar system revolve around the sun in elliptical orbits (see. kepler's laws) and are divided into two groups. Planets that are closer to the Sun than the Earth are called lower... These are Mercury and Venus. Planets that are located further from the Sun than the Earth are called upper... These are Mars, Jupiter, Saturn, Uranus, Neptune and Pluto.

Planets in the process of revolution around the Sun can be located relative to the Earth and the Sun in an arbitrary way. Such a mutual arrangement of the Earth, the Sun and the planet is called configuration... Some of the configurations are highlighted and have special names (see Fig. 19).

The lower planet can be located in line with the Sun and the Earth: either between the Earth and the Sun - bottom connection, or behind the Sun - top connection... At the moment of the lower conjunction, the passage of the planet along the disk of the Sun can occur (the planet is projected onto the disk of the Sun). But due to the fact that the orbits of the planets do not lie in the same plane, such passages do not occur every lower conjunction, but rather rarely. Configurations in which a planet, when viewed from Earth, is at the maximum angular distance from the Sun (these are the most favorable periods for observing the lower planets) are called the largest elongations, western and eastern.

The upper planet can also be in line with the Earth and the Sun: behind the Sun - compound, and on the other side of the Sun - confrontation... Confrontation is the most favorable time to observe the upper planet. Configurations at which the angle between the directions from the Earth to the planet and to the Sun is 90 o are called squares, western and eastern.

The time interval between two successive configurations of the same name of the planet is called its synodic period of circulation P, in contrast to the true period of its revolution relative to the stars, which is therefore called sidereal S... The difference between these two periods arises due to the fact that the Earth also revolves around the Sun with a period T... The synodic and sidereal periods are related:

for the lower planet, and
for the top.

10.2. Kepler's laws

The laws by which the planets revolve around the Sun were empirically (i.e., from observations) established by Kepler, and then theoretically substantiated on the basis of Newton's law of universal gravitation.

First law. Each planet moves along an ellipse, in one of the focuses of which is the Sun.

Second law. When a planet moves, its radius vector describes equal areas in equal time intervals.

Third law. The squares of the sidereal orbital times of the planets relate to each other as cubes of the semi-major axes of their orbits (as the cubes of their average distances from the Sun):

Kepler's third law is approximate, from the law of universal gravitation was obtained refined Kepler's third law:

Kepler's third law is fulfilled with good accuracy only because the masses of the planets are much less than the mass of the Sun.

An ellipse is a geometric figure (see fig. 20) that has two main points - magic tricks F 1 , F 2, and the sum of the distances from any point of the ellipse to each of the foci is a constant value equal to the major axis of the ellipse. The ellipse has center O, the distance from which to the most distant point of the ellipse is called semi-major axis a, and the distance from the center to the nearest point is called semi-minor axis b... The value that characterizes the flattening of the ellipse is called eccentricity. e:

A circle is a special case of an ellipse ( e=0).

The distance from the planet to the Sun varies from the smallest, equal to

perihelion) to the maximum equal to

(this point of the orbit is called aphelion).

10.3. The movement of artificial celestial bodies

The movement of artificial celestial bodies obeys the same laws as natural ones. However, a number of peculiarities should be noted.

The main thing is that the size of the orbits of artificial satellites, as a rule, is comparable to the size of the planet around which they revolve, so they often talk about the height of the satellite above the planet's surface (Fig. 21). It should be borne in mind that the center of the planet is in the focus of the satellite's orbit.

For artificial satellites, the concept of the first and second space velocity is introduced.

First space speed or circular speed is the speed of circular orbital motion near the surface of the planet at an altitude h:

This is the minimum required speed that must be given to a spacecraft to become an artificial satellite of a given planet. For the Earth at the surface vk \u003d 7.9 km / sec.

Second space speed or parabolic speed is the speed that must be given to a spacecraft so that it can leave the sphere of gravity of a given planet in a parabolic orbit:

For the Earth, the second cosmic speed is 11.2 km / sec.

The velocity of a celestial body at any point in an elliptical orbit at a distance R from the gravitating center can be calculated by the formula:

Here everywhere cm 3 / (r s 2) is the gravitational constant.

Questions

4. Could Mars pass through the solar disk? Passage of Mercury? The passage of Jupiter?

5. Can Mercury be seen in the east in the evening? And Jupiter?

Tasks

Decision: The orbits of all planets lie approximately in the same plane, so the planets move along the celestial sphere approximately along the ecliptic. At the moment of opposition, the right ascensions of Mars and the Sun differ by 180 o : ... Let's calculate on May 19. March 21 it is equal to 0 o ... On a day, the right ascension of the Sun increases by about 1 o ... 59 days passed from March 21 to May 19. So, a. On the celestial map, you can see that the ecliptic with such a right ascension passes through the constellations Libra and Scorpio, which means Mars was in one of these constellations.

47. The best evening visibility of Venus (its greatest distance to the east of the Sun) was on February 5. When the next time Venus became visible under the same conditions, if its sidereal orbital period is 225 d ?

Decision: The best evening visibility of Venus occurs during its eastern elongation. Therefore, the next best evening vision will come during the next eastern elongation. And the time interval between two successive eastern elongations is equal to the synodic period of Venus's revolution and can be easily calculated:

or P=587 d ... This means that the next evening visibility of Venus under the same conditions will come in 587 days, i.e. September 14-15 next year.

48. (663) Determine the mass of Uranus in units of the mass of the Earth, comparing the motion of the Moon around the Earth with the motion of the satellite of Uranus - Titania, revolving around it with a period of 8 d .7 at a distance of 438,000 km. The period of the Moon's revolution around the Earth is 27 d .3, and its average distance from the Earth is 384,000 km.

Decision: To solve the problem, it is necessary to use Kepler's third refined law. Since for any body weighing mrevolving around another body with mass at an average distance a with a period T:

(36)

Then we have the right for any pair of celestial bodies revolving around each other to write down the equality:

Taking Uranus with Titania for the first pair, and the Earth with the Moon for the second, and also neglecting the mass of satellites in comparison with the mass of the planets, we get:

49. Taking the Moon's orbit as a circle and knowing the orbital speed of the Moon v Л \u003d 1.02 km / s, determine the mass of the Earth.

Decision: Recall the formula for the square of the circular velocity () and substitute the average distance of the Moon from the Earth a L (see the previous problem):

50. Calculate the mass of the binary star Centaurus, in which the period of rotation of the components around the common center of mass T \u003d 79 years, and the distance between them is 23.5 astronomical units (AU). An astronomical unit is the distance from the Earth to the Sun, equal to approximately 150 million km.

Decision: The solution to this problem is similar to the solution to the problem of the mass of Uranus. Only when determining the masses of binary stars are they compared with the Sun-Earth pair, and their mass is expressed in the masses of the Sun.

51. (1210) Calculate the linear velocities of a spacecraft at perigee and apogee if it flies over the Earth at perigee at an altitude of 227 km above the ocean surface and the major axis of its orbit is 13,900 km. The radius and mass of the Earth is 6371 km and 6.0 10 27 g.

Decision: Let's calculate the distance from the satellite to the Earth at its apogee (the greatest distance from the Earth). For this, it is necessary, knowing the distance at perigee (the smallest distance from the Earth), to calculate the eccentricity of the satellite orbit by the formula () and then determine the required distance using formula (32). We get h a \u003d 931 km.