The configurations of the planets are called some characteristic mutual arrangements of the planets of the Earth and the Sun.

First of all, we note that the conditions for the visibility of planets from the Earth differ sharply for the inner planets (Venus and Mercury), whose orbits lie inside the Earth's orbit, and for the outer planets (all the rest).

The inner planet may be between the Earth and the Sun or behind the Sun. In such positions, the planet is invisible, as it is lost in the rays of the Sun. These positions are called conjunctions of the planet with the Sun. In the lower conjunction, the planet is closest to the Earth, and in the upper conjunction it is the farthest from us (Fig. 26).

It is easy to see that the angle between the directions from the Earth to the Sun and to the inner planet never exceeds a certain value, remaining sharp. This limiting angle is called the greatest distance of the planet from the Sun. The greatest removal of Mercury reaches 28 °, Venus - up to 48 °. Therefore, the inner planets are always visible near the Sun, either in the morning in the eastern side of the sky, or in the evening in the western side of the sky. Due to the proximity of Mercury to the Sun, it is rarely possible to see Mercury with the naked eye (Fig. 26 and 27).

Venus moves away from the Sun in the sky at a greater angle, and it is brighter than all the stars and planets. After sunset, it remains longer in the sky in the rays of dawn and is clearly visible even against its background. It can also be seen well in the rays of the morning dawn. It is easy to understand that neither Mercury nor Venus can be seen in the southern side of the sky and in the middle of the night.

If, passing between the Earth and the Sun, Mercury or Venus are projected onto the solar disk, then they are then visible on it as small black circles. Such passages across the disk of the Sun during the inferior conjunction of Mercury and especially Venus are relatively rare, not more often than after 7-8 years.

The hemisphere of the inner planet illuminated by the Sun at its different positions relative to the Earth is visible to us in different ways. Therefore, for terrestrial observers, the inner planets change their phases, like the Moon. In lower conjunction with the Sun, the planets are turned towards us with their unlit side and are invisible. A little away from this position, they look like a sickle. With an increase in the angular distance of the planet from the Sun, the angular diameter of the planet decreases, and the width of the crescent becomes larger. When the angle at the planet between the directions to the Sun and to the Earth is 90 °, we see exactly half of the illuminated hemisphere of the planet. Such a planet is fully facing us with its daytime hemisphere in the era of the upper connection. But then it is lost in the sun and invisible.

The outer planets can be in relation to the Earth behind the Sun (in conjunction with it), like Mercury and Venus, and then they

Rice. 26. Configurations of the planets.

are also lost in the sun's rays. But they can also be located on the continuation of the direct line Sun - Earth, so that the Earth is between the planet and the Sun. This configuration is called opposition. It is most convenient for observing the planet, since at this time the planet, firstly, is closest to the Earth, secondly, it is turned to it with its illuminated hemisphere and, thirdly, being in the sky in the opposite place to the Sun, the planet is in the upper climax is around midnight and is therefore long visible both before and after midnight.

The moments of planetary configurations, the conditions of their visibility in a given year are given in the "School Astronomical Calendar".

2. Synodic periods.

The synodic period of a planet's revolution is the period of time that elapses between repetitions of its identical configurations, for example, between two oppositions.

The speed of the planets is greater, the closer they are to the Sun. Therefore, after the opposition of Mars, the Earth will overtake him. Every day she will move away from him further and further. When she overtakes him a full turn, then there will be a confrontation again. The synodic period of the outer planet is the period of time after which the Earth overtakes the planet by 360 ° as they move around the Sun. The angular velocity of the Earth (the angle described by it per day) is the angular velocity of Mars where is the number of days in a year, T is the sidereal period of the planet, expressed in days. If is the synodic period of the planet in days, then in a day the Earth will overtake the planet by 360 °, i.e.

If we substitute the appropriate numbers into this formula (see Table V in the appendix), then we can find, for example, that the synodic period of Mars is 780 days, etc. For inner planets that circulate faster than the Earth, one must write:

For Venus, the synodic period is 584 days.

Rice. 27. Location of the orbits of Mercury and Venus relative to the horizon for the observer when the Sun sets (the phases and apparent diameters of the planets are indicated in different positions relative to the Sun at the same position of the observer).

Astronomers did not initially know the sidereal periods of the planets, while the synodic periods of the planets were determined from direct observations. For example, they noted how much time passes between successive oppositions of the planet, that is, between days when it culminates exactly at midnight. Having determined the synodic periods S from observations, they found by calculation the sidereal periods of revolution of the planets T. When Kepler later discovered the laws of planetary motion, using the third law he was able to establish the relative distances of the planets from the Sun, since the sidereal periods of the planets had already been calculated based on the synodic periods.

1 Jupiter's sidereal period is 12 years. After what period of time are his confrontations repeated?

2. It is noticed that oppositions of some planet are repeated in 2 years. What is the semi-major axis of its orbit?

3. The synodic period of the planet is 500 days. Determine the semi-major axis of its orbit. (Reread this assignment carefully.)

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

Kepler's first law. Each planet orbits in an ellipse with the sun at one of its foci.(Fig. 30).

Ellipse(see Fig. 30) is called a flat closed curve, which has such a property that the sum of the distances of each of its points from two points, called foci, remains constant. This sum of distances is equal to the length of the major axis DA of the ellipse. Point O is the center of the ellipse, K and S are foci. The sun is in this case at the focus S. DO=OA=a - the semi-major axis of the ellipse. The semi-major axis is the average distance of the planet from the Sun:

The closest point in the orbit to the Sun is called A. perihelion, and the farthest point from it is D - aphelion.

The degree of elongation of the ellipse is characterized by its eccentricity e. The eccentricity is equal to the ratio of the focus distance from the center (OK=OS) to the length of the semi-major axis a, i.e. When the foci coincide with the center (e=0), the ellipse turns into a circle.

The orbits of the planets are ellipses, little different from circles; their eccentricities are small. For example, the eccentricity of the Earth's orbit is e=0.017.

Kepler's second law(law of areas). The radius-vector of the planet for the same intervals of time describes equal areas, i.e., the areas SAH and SCD are equal (see Fig. 30) if the arcs and are described by the planet for the same time intervals. But the lengths of these arcs bounding equal areas are different: >. Consequently, the linear velocity of the planet is not the same at different points of its orbit. The speed of the planet when moving it in orbit is the greater, the closer it is to the Sun. At perihelion, the speed of the planet is greatest, at aphelion the smallest. Thus, Kepler's second law quantitatively determines the change in the speed of the planet's movement along the ellipse.

Kepler's third law. The squares of the sidereal periods of the planets are related as the cubes of the semi-major axes of their orbits. If the semi-major axis of the orbit and the sidereal period of revolution of one planet are denoted by a 1, T 1, and the other planet by a 2, T 2, then the formula of the third law will be as follows:

This Kepler's law relates the average distances of the planets from the Sun to their sidereal periods and allows you to establish the relative distances of the planets from the Sun, since the sidereal periods of the planets have already been calculated based on the synodic periods, in other words, it allows you to express the semi-major axes of all planetary orbits in units of the semi-major axis earth orbit.

The semi-major axis of the earth's orbit is taken as the astronomical unit of distance (a = 1 AU).

Its value in kilometers was determined later, only in the 18th century.

Problem solution example

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

Exercise 8

2. Determine the period of revolution of an artificial satellite of the Earth, if the highest point of its orbit above the Earth is 5000 km, and the lowest is 300 km. Consider the earth as a sphere with a radius of 6370 km. Compare the movement of the satellite with the revolution of the moon.

3. The synodic period of the planet is 500 days. Determine the semi-major axis of its orbit and sidereal period.

12. Determination of distances and sizes of bodies in the solar system

1. Definition of distances

The average distance of all planets from the Sun in astronomical units can be calculated using Kepler's third law. Having defined average distance of the earth from the sun(i.e., the value of 1 AU) in kilometers, can be found in these units of distance to all the planets of the solar system.

Since the 40s of our century, radio engineering has made it possible to determine the distances to celestial bodies by means of radar, which you know about from a physics course. Soviet and American scientists determined by radar the distances to Mercury, Venus, Mars and Jupiter.

Recall how the distance to an object can be determined from the time it takes for a radar signal to travel.

The classic method for determining distances was and remains the goniometric geometric method. They determine the distances to distant stars, to which the radar method is not applicable. The geometric method is based on the phenomenon parallax shift.

A parallactic displacement is a change in direction to an object when the observer moves (Fig. 31).

Look at the vertically placed pencil, first with one eye, then with the other. You will see how at the same time he changed position against the background of distant objects, the direction towards him changed. The farther you move the pencil, the smaller the parallax shift will be. But the farther the observation points are from each other, i.e., the more basis, the greater the parallactic shift at the same object distance. In our example, the basis was the distance between the eyes. To measure the distances to the bodies of the solar system, it is convenient to take the radius of the Earth as a basis. The positions of a luminary, such as the Moon, are observed against the background of distant stars simultaneously from two different points. The distance between them should be as large as possible, and the segment connecting them should make an angle with the direction to the luminary, as close as possible to a straight line, so that the parallactic shift is maximum. Having determined from two points A and B (Fig. 32) the directions to the observed object, it is easy to calculate the angle p at which a segment equal to the radius of the Earth would be visible from this object. Therefore, in order to determine the distances to celestial bodies, you need to know the value of the basis - the radius of our planet.

2. The size and shape of the Earth

In photographs taken from space, the Earth looks like a ball illuminated by the Sun, and shows the same phases as the Moon (see Fig. 42 and 43).

The exact answer about the shape and size of the Earth is given degree measurements, i.e., measurements in kilometers of the length of an arc of 1 ° at different places on the surface of the Earth. This method is still in the III century, BC. e. used by a Greek scientist living in Egypt Eratosthenes. This method is now used in geodesy- the science of the shape of the Earth and measurements on the Earth, taking into account its curvature.

On flat terrain, two points are chosen that lie on the same meridian, and the length of the arc between them is determined in degrees and kilometers. Then calculate how many kilometers corresponds to the length of the arc, equal to 1°. It is clear that the length of the meridian arc between the selected points in degrees is equal to the difference in the geographical latitudes of these points: Δφ= = φ 1 - φ 2 . If the length of this arc, measured in kilometers, is equal to l, then with the sphericity of the Earth, one degree (1 °) of the arc will correspond to the length in kilometers: Then the circumference of the earth's meridian L, expressed in kilometers, is equal to L = 360°n. Dividing it by 2π, we get the radius of the Earth.

One of the largest meridian arcs from the Arctic Ocean to the Black Sea was measured in Russia and Scandinavia in the middle of the 19th century. under the direction of V. Ya. Struve(1793-1864), director of the Pulkovo Observatory. Large geodetic measurements in our country were made after the Great October Socialist Revolution.

Degree measurements have shown that the length of 1° arc of the meridian in kilometers in the polar region is the largest (111.7 km), and the smallest at the equator (110.6 km). Therefore, at the equator, the curvature of the Earth's surface is greater than at the poles, and this indicates that the Earth is not a ball. The equatorial radius of the Earth is greater than the polar one by 21.4 km. Therefore, the Earth (like other planets) due to rotation is compressed at the poles.

A ball, equal in size to our planet, has a radius of 6370 km. This value is considered to be the radius of the Earth.

Exercise 9

1. If astronomers can determine geographic latitude with an accuracy of 0.1", then what is the maximum error in kilometers along the meridian does this correspond to?

2. Calculate in kilometers the length of a nautical mile, which is equal to the length of the V arc of the equator.

3. Parallax. The value of the astronomical unit

The angle at which the Earth's radius is seen perpendicular to the line of sight is called horizontal parallax..

The greater the distance to the luminary, the smaller the angle ρ. This angle is equal to the parallactic displacement of the star for observers located at points A and B (see Fig. 32), just like ∠CAB for observers at points C and B (see Fig. 31). ∠CAB is conveniently determined by its equal ∠DCA, and they are equal as angles at parallel lines (DC AB by construction).

Distance (see fig. 32)

where R is the radius of the Earth. Taking R as a unit, we can express the distance to the luminary in earthly radii.

The horizontal parallax of the Moon is 57". All the planets and the Sun are much further away, and their parallaxes are seconds of arc. The parallax of the Sun, for example, ρ = 8.8". The parallax of the Sun corresponds to the average distance of the Earth from the Sun, approximately equal to 150,000,000 km. This distance taken as one astronomical unit (1 AU). In astronomical units, the distances between the bodies of the solar system are often measured.

At small angles sinρ≈ρ, if the angle ρ is expressed in radians. If ρ is expressed in seconds of arc, then a factor is introduced where 206265 is the number of seconds in one radian.

Then

Knowing these relationships simplifies the calculation of the distance from a known parallax:

Problem solution example

Task. How far is Saturn from Earth when its horizontal parallax is 0.9"?

Exercise 10

1. What is the horizontal parallax of Jupiter as seen from Earth at opposition if Jupiter is 5 times farther from the Sun than Earth?

2. The distance of the Moon from the Earth at the point of the orbit closest to the Earth (perigee) is 363,000 km, and at the most distant point (apogee) 405,000 km. Determine the horizontal parallax of the Moon at these positions.

4. Determining the size of the luminaries

In Figure 33, T is the center of the Earth, M is the center of the luminary with a linear radius r. According to the definition of horizontal parallax, the Earth's radius R is visible from the sun at an angle ρ. The radius of the luminary r is visible from the Earth at an angle .

Insofar as

If the angles and ρ are small, then the sines are proportional to the angles, and we can write:

This method of determining the size of the luminaries is applicable only when the disk of the luminary is visible.

Knowing the distance D to the luminary and measuring its angular radius, you can calculate its linear radius r: r=Dsin or r=D if the angle is expressed in radians.

Problem solution example

Task. What is the linear diameter of the Moon if it is visible from a distance of 400,000 km at an angle of about 0.5°?

Exercise 11

1. How many times larger is the Sun than the Moon if their angular diameters are the same and the horizontal parallaxes are respectively 8.8" and 57"?

2. What is the angular diameter of the Sun as seen from Pluto?

3. How many times more energy does each square meter of the surface of Mercury receive from the Sun than Mars? Take the necessary data from the applications.

4. At what points in the sky does an earthly observer see the luminary, being at points B and A (Fig. 32)?

5. In what ratio does the angular diameter of the Sun, visible from Earth and Mars, change numerically from perihelion to aphelion if the eccentricities of their orbits are respectively 0.017 and 0.093?

Task 5

1. Measure with a protractor ∠DCA (Fig. 31) and ∠ASC (Fig. 32), with a ruler - the length of the bases. Calculate the distances CA and SC from them, respectively, and check the result by direct measurement from the figures.

2. Measure the angles p and I in Figure 33 with a protractor and determine the ratio of the diameters of the depicted bodies from the data obtained.

3. Determine the periods of revolution of artificial satellites moving in elliptical orbits, shown in Figure 34, by measuring their major axes with a ruler and assuming the radius of the Earth is 6370 km.

synodic circulation period(S) planet is called the time interval between its two successive configurations of the same name.

sidereal or stellar orbital period(T) of the planet is called the period of time during which the planet makes one complete revolution around the Sun in its orbit.

The sidereal period of the Earth's revolution is called a sidereal year (T ☺). Between these three periods, a simple mathematical relationship can be established from the following reasoning. The angular displacement along the orbit per day is the same for the planet, and for the Earth. The difference between the daily angular displacements of the planet and the Earth (or the Earth and the planet) is the apparent displacement of the planet per day, i.e. From here for the lower planets

for the upper planets

These equalities are called the equations of synodic motion.

Directly from observations, only the synodic periods of revolutions of the planets S and the sidereal period of the Earth's revolution can be determined, i.e. sidereal year T ☺ . The sidereal periods of revolutions of the planets T are calculated according to the corresponding equation of synodic motion.

The duration of a stellar year is 365.26 ... mean solar days.

7.4. Kepler's laws

Kepler was a supporter of the teachings of Copernicus and set himself the task of improving his system based on the observations of Mars, which were made by the Danish astronomer Tycho Brahe (1546-1601) for twenty years and for several years by Kepler himself.

Early on, Kepler shared the traditional belief that celestial bodies could only move in circles, and so he spent a lot of time trying to find a circular orbit for Mars.

After many years and very laborious calculations, abandoning the general misconception about the circularity of motion, Kepler discovered three laws of planetary motion, which are currently formulated as follows:

1. All planets move along ellipses, in one of the focuses of which (common to all planets) is the Sun.

2. The radius vector of the planet describes equal areas in equal time intervals.

3. The squares of the sidereal periods of revolutions of the planets around the Sun are proportional to the cubes of the semi-major axes of their elliptical orbits.

As is known, in an ellipse, the sum of the distances from any of its points to two fixed points f 1 and f 2 lying on its axis AP and called foci is a constant value equal to the major axis AP (Fig. 27). The distance PO (or OA), where O is the center of the ellipse, is called the semi-major axis , and the ratio is the eccentricity of the ellipse. The latter characterizes the deviation of the ellipse from the circle, in which e \u003d 0.

The orbits of the planets differ little from circles, i.e. their eccentricities are small. The smallest eccentricity has the orbit of Venus (e = 0.007), the largest - the orbit of Pluto (e = 0.247). The eccentricity of the earth's orbit e = 0.017.

According to Kepler's first law, the Sun is at one of the foci of the planet's elliptical orbit. Let in Fig. 27, and this will be the focus f 1 (C - the Sun). Then the point of the orbit P closest to the Sun is called perihelion, and the most distant point from the Sun A - aphelion. The major axis of the AP orbit is called apsi line d, and the line f 2 P, connecting the Sun and the planet P in its orbit, - radius vector of the planet.

Planet's distance from the Sun at perihelion

q = a (1 - e), (2.3)

Q = a (l + e). (2.4)

The semi-major axis of the orbit is taken as the average distance of the planet from the Sun

According to Kepler's second law, the area СР 1 Р 2 described by the planet's radius-vector over time t near the perihelion is equal to the area СР 3 Р 4 described by him for the same time t near aphelion (Fig. 27b). Since the arc Р 1 Р 2 is greater than the arc Р 3 Р 4 , then, consequently, the planet near perihelion has a greater speed than near aphelion. In other words, its movement around the Sun is uneven.