Example of problem solution
Synodic period of revolution(S) of a planet is the time interval between its two successive configurations of the same name.
Sidereal or sidereal period of revolution(T) of a planet is 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 the sidereal year (T☺). A simple mathematical relationship can be established between these three periods from the following reasoning. The angular movement in orbit per day is equal 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. Hence for the lower planets
for the upper planets
These equalities are called equations of synodic motion.
Only the synodic periods of revolutions of the planets S and the sidereal period of revolution of the Earth can be determined directly from observations, i.e. sidereal year T ☺. The sidereal rotation periods of the planets T are calculated using the corresponding equation of synodic motion.
The duration of a sidereal year is 365.26... average solar day.
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 observations of Mars, which were carried out by the Danish astronomer Tycho Brahe (1546-1601) for twenty years and by Kepler himself for several years.
At first, 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 of very labor-intensive calculations, abandoning the general misconception about the circularity of motions, Kepler discovered three laws of planetary motions, which are currently formulated as follows:
1. All planets move in ellipses, in one of the focuses (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 semimajor 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 semimajor axis 
, and the ratio is the eccentricity of the ellipse. The latter characterizes the deviation of the ellipse from a circle for which e = 0.
The orbits of the planets differ little from circles, i.e. their eccentricities are small. The orbit of Venus has the smallest eccentricity (e = 0.007), the greatest eccentricity is the orbit of Pluto (e = 0.247). The eccentricity of the earth's orbit is e = 0.017.
According to Kepler's first law, the Sun is located at one of the foci of the planet's elliptical orbit. Let in Fig. 27, and this will be the focus f 1 (C - Sun). Then the point of orbit P closest to the Sun is called perihelion, and the point A most distant from the Sun is aphelion. The major axis of the AP's orbit is called apsi line d, and the line f 2 P connecting the Sun and planet P in its orbit is radius vector of the planet.
Distance of the planet from the Sun at perihelion
q = a (1 - e), (2.3)
Q = a (l + e). (2.4)
The average distance of the planet from the Sun is taken to be the semimajor axis of the orbit.
According to Kepler’s second law, the area CP 1 P 2 described by the radius vector of the planet over time 
t near perihelion, equal to the area of CP 3 P 4 described by him for the same time 
t near aphelion (Fig. 27, b). Since the arc P 1 P 2 is greater than the arc P 3 P 4, then, consequently, the planet near perihelion has a speed greater than near aphelion. In other words, its movement around the Sun is uneven.
Planetary configurations refer to some characteristic mutual positions of the planets of the Earth and the Sun.
First of all, we note that the conditions for the visibility of planets from Earth differ sharply for the internal planets (Venus and Mercury), whose orbits lie within the Earth’s orbit, and for the external planets (all others).
The inner planet may be between the Earth and the Sun or behind the Sun. In such positions the planet is invisible, as it is lost in the rays of the Sun. These positions are called planet-Sun conjunctions. At inferior conjunction the planet is closest to the Earth, and at superior conjunction it is 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 acute. This limiting angle is called the greatest distance of the planet from the Sun. The greatest distance 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 in the sky longer in the dawn rays and is clearly visible even against its background. It is also clearly visible in the morning light. It is easy to understand that in the southern part of the sky and in the middle of the night neither Mercury nor Venus can be seen.
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, no more often than every 7-8 years.
The hemisphere of the inner planet illuminated by the Sun is visible to us differently at different positions relative to the Earth. Therefore, for earthly observers, the inner planets change their phases, like the Moon. In inferior conjunction with the Sun, the planets turn their unlit side towards us and are invisible. A little away from this position they have the shape of a sickle. As the angular distance of the planet from the Sun increases, 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 the Earth is 90°, we see exactly half of the illuminated hemisphere of the planet. Such a planet is completely facing us with its daytime hemisphere during the era of superior conjunction. But then she is lost in the sun's rays and invisible.
The outer planets can be located behind the Sun in relation to the Earth (in conjunction with it), like Mercury and Venus, and then they
Rice. 26. Planetary configurations.
are also lost in the sun's rays. But they can also be located on the continuation of the straight 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, its illuminated hemisphere is turned towards it and, thirdly, being in the sky in a place opposite to the Sun, the planet is in the upper culmination is around midnight and, therefore, is visible for a long time both before and after midnight.
Moments of planetary configurations and conditions of their visibility in a given year are given in the “School Astronomical Calendar”.
2. Synodic periods.
The synodic period of revolution of a planet is the period of time that elapses between repetitions of its identical configurations, for example, between two oppositions.
The closer they are to the Sun, the faster the planets move. Therefore, after the opposition of Mars, the Earth will begin to overtake it. Every day she will move further away from him. When she overtakes him by a full turn, 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 it describes per day) is the angular velocity of Mars where is the number of days in a year, T is the sidereal period of revolution 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 corresponding numbers into this formula (see Table V in the Appendix), we can find, for example, that the synodic period of Mars is 780 days, etc. For the inner planets that orbit faster than the Earth, we must write:
For Venus, the synodic period is 584 days.
Rice. 27. The location of the orbits of Mercury and Venus relative to the horizon for an observer when the Sun sets (the phases and apparent diameters of the planets in different positions relative to the Sun are indicated for the same observer position).
Astronomers initially did not 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 The sidereal period of Jupiter's revolution is 12 years. After what period of time are his confrontations repeated?
2. It is noticed that oppositions of a certain planet are repeated after 2 years. What is the semimajor axis of its orbit?
3. The synodic period of the planet is 500 days. Determine the semimajor axis of its orbit. (Re-read 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 17th century. Kepler, studying the revolution of Mars around the Sun, established three laws of planetary motion.
Kepler's first law. Each planet rotates in an ellipse, with the Sun at one focus(Fig. 30).
Ellipse(see Fig. 30) is a flat closed curve that has the property that the sum of the distances of each point 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 the foci. The sun is in this case at the focus S. DO=OA=a is the semimajor axis of the ellipse. The semimajor axis is the average distance of the planet from the Sun:
The point of orbit A closest to the Sun is called perihelion, and the farthest point D from it is aphelion.
The degree of elongation of an ellipse is characterized by its eccentricity e. Eccentricity is equal to the ratio of the distance of the focus from the center (OK=OS) to the length of the semimajor 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 describes equal areas in equal time intervals, i.e., the areas SAH and SCD are equal (see Fig. 30), if the arcs and are described by the planet in equal time intervals. But the lengths of these arcs, delimiting equal areas, are different: >. Consequently, the linear speed of motion of the planet is not the same at different points of its orbit. The closer a planet is to the Sun, the faster it moves in its orbit. At perihelion the planet's speed is greatest, and at aphelion it is least. Thus, Kepler's second law quantifies the change in the speed of a planet's motion along an ellipse.
Kepler's third law. The squares of the sidereal periods of planets are related as the cubes of the semimajor axes of their orbits. If the semimajor axis of the orbit and the sidereal period of revolution of one planet are denoted by a 1, T 1, and of the other planet by a 2, T 2, then the formula of the third law will be as follows: 
This Kepler law connects the average distances of the planets from the Sun with their sidereal periods and allows us 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 us to express the semimajor axes of all planetary orbits in units of the semimajor axis earth's orbit.
The semimajor 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.
Example of problem solution
Task. Oppositions of a certain planet are repeated after 2 years. What is the semimajor axis of its orbit?
Exercise 8
2. Determine the orbital period of an artificial Earth satellite if the highest point of its orbit above the Earth is 5000 km, and the lowest point is 300 km. Consider the earth to be a sphere with a radius of 6370 km. Compare the motion of the satellite with the revolution of the Moon.
3. The synodic period of the planet is 500 days. Determine the semimajor axis of its orbit and stellar orbital period.
12. Determination of distances and sizes of bodies in the solar system
1. Determination of distances
The average distance of all planets from the Sun in astronomical units can be calculated using Kepler's third law. Having determined average distance of the Earth from the Sun(i.e., the value of 1 AU) in kilometers, the distances to all planets in the Solar System can be found in these units.
Since the 40s of our century, radio technology has made it possible to determine distances to celestial bodies using radar, which you know about from a physics course. Soviet and American scientists used radar to clarify the distances to Mercury, Venus, Mars and Jupiter.
Remember how the distance to an object can be determined by the travel time of a radar signal.
The classic way to determine distances was and remains the goniometric geometric method. They also determine distances to distant stars, to which the radar method is not applicable. The geometric method is based on the phenomenon parallactic displacement.
Parallax displacement is the change in direction of an object when the observer moves (Fig. 31).
Look at the vertical pencil first with one eye, then with the other. You will see how he changed his position against the background of distant objects, the direction toward him changed. The farther you move the pencil, the less parallactic displacement there will be. But the farther the observation points are from each other, i.e., the more basis, the greater the parallactic displacement at the same distance of the object. In our example, the basis was the distance between the eyes. To measure distances to solar system bodies, it is convenient to take the radius of the Earth as a basis. The positions of a star, 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 towards the luminary, as close as possible to a straight line, so that the parallactic displacement is maximum. Having determined the directions to the observed object from two points A and B (Fig. 32), 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. Size and Shape of the Earth
In photographs taken from space, the Earth appears as 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 Earth's surface. This method dates back to the 3rd century BC. e. used by a Greek scientist who lived in Egypt Eratosthenes. This method is now used in geodesy- the science of the shape of the Earth and of measurements on the Earth, taking into account its curvature.
On flat terrain, select two points lying on the same meridian and determine the length of the arc between them in degrees and kilometers. Then calculate how many kilometers an arc length of 1° corresponds. 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 if the Earth is spherical, one degree (1°) of the arc will correspond to a 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 mid-19th century. under the direction of V. Ya. Struve(1793-1864), director of the Pulkovo Observatory. Large geodetic measurements in our country were carried out after the Great October Socialist Revolution.
Degree measurements showed that the length of 1° meridian arc in kilometers in the polar region is greatest (111.7 km), and at the equator it is smallest (110.6 km). Consequently, at the equator the curvature of the Earth's surface is greater than at the poles, which means that the Earth is not a sphere. The equatorial radius of the Earth is 21.4 km greater than the polar radius. Therefore, the Earth (like other planets) is compressed at the poles due to rotation.
A ball equal in size to our planet has a radius of 6370 km. This value is considered to be the radius of the Earth.
Exercise 9
1. If astronomers can determine geographic latitude with an accuracy of 0.1", what maximum error in kilometers along the meridian does this correspond to?
2. Calculate the length of a nautical mile in kilometers, which is equal to the length of the V arc of the equator.
3. Parallax. Astronomical unit value
The angle at which the radius of the Earth is visible from the luminary, perpendicular to the line of sight, is called horizontal parallax.
The greater the distance to the star, the smaller the angle ρ. This angle is equal to the parallactic displacement of the luminary for observers located at points A and B (see Fig. 32), just like ∠CAB for observers at points C and B (see Fig. 31). It is convenient to determine ∠CAB by its equal ∠DCA, and they are equal as angles of parallel lines (DC AB by construction).
Distance (see Fig. 32)
where R is the radius of the Earth. Taking R as one, we can express the distance to the star in Earth radii.
The horizontal parallax of the Moon is 57". All planets and the Sun are much further away, and their parallaxes are arcseconds. The parallax of the Sun, for example, is ρ = 8.8". Corresponds to the parallax of the Sun The average distance of the Earth from the Sun is approximately 150,000,000 km. This is the distance is taken as one astronomical unit (1 AU). Distances between solar system bodies are often measured in astronomical units.
At small angles sinρ≈ρ, if the angle ρ is expressed in radians. If ρ is expressed in arcseconds, then the multiplier is introduced 
where 206265 is the number of seconds in one radian.
Then 
Knowing these relationships simplifies the calculation of distance from a known parallax: 
Example of problem solution
Task. How far is Saturn from Earth when its horizontal parallax is 0.9"?
Exercise 10
1. What is the horizontal parallax of Jupiter observed from the Earth at opposition, if Jupiter is 5 times farther from the Sun than the Earth?
2. The distance of the Moon from the Earth at the point of its 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. Determination of luminary sizes
In Figure 33, T is the center of the Earth, M is the center of the luminary of linear radius r. By definition of horizontal parallax, the Earth's radius R is visible from the luminary at an angle ρ. The radius of the star r is visible from the Earth at an angle.
Because the
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 luminaries is applicable only when the disk of the luminary is visible.
Knowing the distance D to the star and measuring its angular radius, you can calculate its linear radius r: r=Dsin or r=D, if the angle is expressed in radians.
Example of problem solution
Task. What is the linear diameter of the Moon if it is visible from a distance of 400,000 km at an angle of approximately 0.5°?
Exercise 11
1. How many times is the Sun larger than the Moon if their angular diameters are the same and their 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 that of Mars? Take the necessary data from the applications.
4. At what points in the sky does the 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 from Mars, change numerically from perihelion to aphelion if the eccentricities of their orbits are respectively equal to 0.017 and 0.093?
Task 5
1. Measure ∠DCA (Fig. 31) and ∠ASC (Fig. 32) with a protractor and the length of the bases with a ruler. Calculate the distances CA and SC from them, respectively, and check the result by direct measurement using the drawings.
2. Measure the angles p and I in Figure 33 with a protractor and, from the data obtained, determine the ratio of the diameters of the depicted bodies.
3. Determine the orbital periods of artificial satellites moving in elliptical orbits shown in Figure 34 by measuring their major axes with a ruler and taking the radius of the Earth to be 6370 km.
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