Planetary configurations. Problem solving.

Key questions: 1) configurations and visibility conditions of planets; 2) sidereal and synodic periods of planetary revolution; 3) formula for the connection between the synodic and sidereal periods.

The student must be able to: 1) solve problems using a formula connecting the synodic and sidereal periods of revolution of the planets.

Theory

Indicate the basic configurations for the upper (lower) planets. Define the synodic and sidereal periods.

Let's say that at the initial moment of time the minute hand and the hour hand coincide. The period of time after which the hands meet again will not coincide with either the period of rotation of the minute hand (1 hour) or the period of rotation of the hour hand (12 hours). This period of time is called the synodic period - the time after which certain positions of the hands are repeated.

The angular speed of the minute hand, and the hour hand - . During the synodic period S the hour hand of the clock will travel

and minute

Subtracting the paths, we get, or

Write down formulas connecting the synodic and sidereal periods and calculate the repetition of configurations for the upper (lower) planet closest to the Earth. The required table values ​​can be found in the appendices.

2. Consider an example:

Determine the sidereal period of the planet if it is equal to the synodic period. What is the real planet solar system comes closest to these conditions?

According to the conditions of the problem T = S, Where T- sidereal period, the time of revolution of the planet around the Sun, and S- synodic period, the time of repetition of the same configuration with a given planet.

Then in the formula

Let's make a replacement S on T: The planet is infinitely far away. On the other hand, having made a similar replacement

The most suitable planet is Venus, whose period is 224.7 days.

Problem solving

1. What is the synodic period of Mars if its sidereal period is 1.88 Earth years?

Mars is an outer planet and the formula is valid for it

2. Mercury's inferior conjunctions repeat after 116 days. Determine the sidereal period of Mercury.

Mercury is an inner planet and the formula is valid for it

3. Determine the sidereal period of Venus if its inferior conjunctions occur every 584 days.

4. After what period of time do Jupiter’s oppositions repeat if its sidereal period is 11.86 g?

Apparent motion of the Sun and Moon

Key questions: 1) daily movement of the Sun at various latitudes; 2) changes in the apparent movement of the Sun throughout the year; 3) apparent movement and phases of the Moon; 4) Solar and lunar eclipses. Eclipse conditions.

The student must be able to: 1) use astronomical calendars, reference books, and a moving star chart to determine the conditions for the occurrence of phenomena associated with the revolution of the Moon around the Earth and the apparent movement of the Sun.

Independent work 20 min

Option 1	Option 2
1. Describe the position of the inner planets	1. Describe the position of the outer planets
2. The planet is observed through a telescope in the shape of a sickle. What planet could this be? [Internal]	2. Which planets and under what conditions can be visible all night (from sunset to sunrise)? [All outer planets in opposition eras]
3. Through observation, it was established that there are 378 days between two successive identical configurations of the planet. Assuming a circular orbit, find the sidereal (stellar) period of revolution of the planet.	3. Minor planet Ceres orbits the Sun with a period of 4.6 years. After what period of time do the oppositions of this planet repeat?
4. Mercury is observed at the position of maximum elongation, equal to 28o. Find the distance from Mercury to the Sun in astronomical units.	4. Venus is observed at the position of maximum elongation, equal to 48o. Find the distance from Venus to the Sun in astronomical units.

LESSON 7. PLANET CONFIGURATIONS,

DISTANCES TO BODIES AND THEIR SIZES.

1. Basic configurations of the lower and upper planets.

2. Sidereal and synodic periods of planets.

3. Determining the size of the Earth

4. Determining distances to bodies.

5. Determination of body sizes.

1. Basic configurations of the inner and outer planets.

The complex apparent motion of planets on the celestial sphere is caused by the revolution of the planets of the Solar System around the Sun. The word “planet” itself, translated from ancient Greek, means “wandering” or “vagrant”. The trajectory of a celestial body is called its orbit.

In relation to the Earth's orbit, the planets are divided into internal (lower) - Mercury, Venus, their orbits are located inside the Earth's orbit, and external (upper) - Mars, Jupiter, Saturn, Uranus, Neptune, their orbits are located outside the Earth's orbit. The outer planets always face the Earth with the side illuminated by the Sun. The inner planets change their phases like the Moon. The orbital planes of all planets in the Solar System lie near the ecliptic plane, deviating from it by less than 7°. The speeds at which planets move in their orbits are different and decrease as the planets move away from the Sun. The Earth moves slower than Mercury and Venus, but faster than all other planets. Due to the difference in the speed of movement of the planets at certain times, different mutual arrangements Sun and planets.

Special, geometrically correct, mutual positions of the Sun, Earth and planets are called configurations. Identical configurations of planets occur at different points in their orbits, opposite different constellations, in different times year. The configurations that are created by the lower and upper planets are different.

The lower planets have this connectionsV1 AndV3 (top and bottom) and elongationV2 AndV4 (eastern and western). For the upper planets it is - quadraturesM2 and M4(eastern and western), compoundM1 and confrontationM3 .

What is behind these terrible names? Conjunctions are the placement of the Sun, Earth and planet on the same line, while the planet is either between the Sun and the Earth (inferior conjunction), or is hiding from the Earth behind the Sun (upper conjunction). The only configuration in which any planet, both inferior and superior, can be is a superior conjunction, and the planet naturally cannot be observed. Inferior conjunction is inherent only to the lower planets, while, although quite rarely, we can observe the passage of Mercury and Venus (in the form of a black circle) against the background of the solar disk.

The apparent motion of the lower planets resembles oscillatory motion near the Sun. The maximum angular distance of the lower planets from the Sun is called elongation. In the case of elongation, the Earth, planet and Sun form right triangle, while at the vertex of the right angle there is a planet. The greatest elongation of Mercury is 28˚, Venus - 48˚. From the Earth at this time, not the entire hemisphere of the planet illuminated by the Sun is visible, but only its part, called the phase. With eastern elongation, the planet is visible in the west shortly after sunset; with western elongation, the planet is visible in the east shortly before sunrise.

The most convenient moment for observing the upper planets is opposition. All three celestial bodies and, as with a conjunction, they are on the same line, but the Earth in this case is located between the Sun and the planet and the entire hemisphere of the planet is illuminated by the Sun. The outer planet can be at any angular distance from the Sun from 0˚ to 180˚. When the angular distance between the Sun and the upper planet is 90˚, then the planet is said to be in quadrature (quadrature is an angular quarter of a circle), respectively in the east or west, as in elongation. In this case, the Earth, the Sun and the planet also form a right triangle, but the Earth is at the vertex of the right angle.

The Earth - Moon - Sun system is special, it has an inferior conjunction, like the inner planets, with a new moon occurring (the Moon between the Sun and Earth), and opposition, like the outer planets, during the full moon.

2. Sidereal and synodic periods of planets.

The period of time during which the planet makes a complete revolution around the Sun in its orbit is called the sidereal (or sidereal) period of revolution of the planet (T), and the period of time between two identical configurations of the planet is called the synodic period ( S). The planets move around the Sun in one direction, and each of them, after a period of time equal to its sidereal period, makes one complete revolution around the Sun. Let the planets be in a certain configuration. Over a period of time equal to the sidereal period of the Earth, any lower planet will make more than one revolution around the Sun and will overtake the Earth, and any upper planet will make less than a full revolution and will lag behind the Earth. Consequently, after an Earth year, the configuration of the planets will not be repeated, i.e., the synodic period is not equal to the sidereal period. However, there is a relationship between periods that is easy to establish. This dependency is called equation of synodic motion.

Let's create an equation for the lower planet. Over the course of an Earth day, the planet shifts by an angle where T is the sidereal period of the planet, and the Earth by an angle , where is the sidereal period of the Earth. The difference between these angles will give the advance angle α, , by which the lower planet will be ahead of the Earth in a day. When an advance of 360º (α·S=360º) accumulates in S days, the configuration of the planets will repeat. S - in in this case- synodic period. The final equation for the lower planet looks like this:

or or

Since the upper planets move slower than the Earth, the equation for them takes the form: or or

Task.Determine the period of revolution of Mars around the Sun, knowing that oppositions of Mars occur every 780 days?

3. Determining the size of the Earth.

The idea of ​​the Earth as a ball that hangs freely in space without any support is certainly one of the greatest achievements of science ancient world. And the first precise definition earthly size was made by Eratosthenes from Egypt. The experiment he performed is one of the ten most beautiful physical experiments invented by mankind. He decided to measure the length of a small arc of the earth's meridian not in degrees, but in units of length, and then determine what part in degrees of the full circle it constitutes. Knowing the part, find the length of the entire circle. Then, using the circumference, determine the radius, which is the radius of the globe.

Obviously, the length of the meridian arc in degrees is equal to the difference in the geographical latitudes of two points located on the same meridian: Δφ=φв – φА. In order to determine this difference, Eratosthenes compared the height of the Sun at its climax on the same day at points A and B (Alexandria and Aswan). In Aswan on this day the Sun illuminated the bottom of the deepest wells, i.e. it was at the zenith, and in Alexandria it was 7.2˚ from the zenith. From simple geometric constructions it followed that the difference in latitude of these cities was Δφ = 7.2˚. In ancient units of measurement, the distance between Alexandria and Aswan was 5000 Greek stadia, modern - 800 km. Having designated the length of the Earth's meridian by L, we have the following proportion: from where we obtain the length of the meridian equal to 40,000 km. Knowing the circumference, we can easily find the radius of the Earth - 6366 km, which differs from the average radius by only 5 km.

The extent to which the shape of the Earth differs from a sphere became clear only at the end of the 18th century as a result of the work of two expeditions in South America in Peru and in Scandinavia near the Arctic Circle. Measurements have shown that the length of 1˚ arc of the meridian in the north and south is greater than at the equator. This meant that the Earth was flattened at the poles. Its polar radius is 21 km shorter than the equatorial one. This means that the cross section of the Earth along the meridian will not be a circle, but an ellipse, the major axis of which lies in the plane of the equator, and the minor axis coincides with the axis of rotation of the Earth. And already in the twentieth century it became clear that earth's equator also cannot be considered a circle. Its oblateness is 100 times less than the oblateness of the meridian, but it still exists. Most accurately, the shape of our planet is conveyed by a figure called an ellipsoid, in which any section by a plane passing through the center of the Earth is not a circle.

4. Determination of distances to bodies.

Determining the geographic latitude of two points turns out to be much easier than measuring the distance between them, which can be hampered by natural obstacles. Therefore, a method based on the phenomenon of parallactic displacement is used. Parallax displacement is the change in direction of an object when the observer moves. First, accurately calculate the length of a conveniently located segment BC, called the basis, and two angles B and C in triangle ABC. Further on the theorem of sines the values ​​AC and AB are easily found. A similar method is used to determine the distance to celestial bodies. It was possible to measure the distance from the Earth to the Sun for the first time only in the 18th century, when it was determined horizontal parallax Sun. Horizontal parallax (p) is the angle at which the radius of the Earth is visible from a star located on the horizon, perpendicular to the line of sight. In essence, this measures the parallactic displacement of an object located outside the Earth, and the basis is the radius of the Earth. The only difference is that the triangle is constructed as a rectangular one, which simplifies the calculations.

From triangle OAS we can express the distance SO=D: where RÅ is the radius of the Earth. Of course, no one observes the radius of the Earth from the luminary, and the horizontal parallax is determined by measuring the height of the luminary at the moment of the upper culmination from two points of the Earth located on the same meridian and having known latitudes, by analogy with the method of Eratosthenes. Obviously, the further away an object is, the less its parallax. Highest value has the Moon's parallax ( rƒ =57΄02΄΄), solar parallax r=8.79′′. This parallax value corresponds to a distance to the Sun equal to km. This distance is taken as one astronomical unit (1AU) and is used when measuring distances between bodies of the Solar System.

For small angles sinp≈ p, while p expressed in radians. If r expressed in seconds, then the formula will take the form: Å, since there are 206265′′ in one radian.

The horizontal parallax method was used to determine the distance to objects until the second half of the 20th century, when new methods for determining distances in the Solar System appeared - radar and laser ranging. Using these methods, the distances to many bodies were clarified with an accuracy of up to a kilometer, and laser ranging of the Moon makes it possible to determine distances with an accuracy of centimeters.

Task. At what distance from Earth is Saturn when its parallax is equal to 0,9’’ ?

5. Determination of body sizes.

2. What is a connection?

3. Is it possible to observe Venus in the east in the morning and in the west in the evening?

4. The angular distance of the planet from the Sun is 55°. Which planet is it, top or bottom?

5. What is configuration?

6. What planets can pass against the background of the solar disk?

7. During what configurations are the lower planets clearly visible?

8. During what configurations are the upper planets clearly visible?

9. What is the sidereal period of the planet?

10. What is the synodic period?

11. What is horizontal parallax?

12. What is called parallactic displacement?

13. When is the superior planet in square?

14. What is elongation?

15. At what conjunction can the inner planet be observed?

All cosmogonic hypotheses can be divided into several groups. According to one of them, the Sun and all the bodies of the solar system: planets, satellites, asteroids, comets and meteoroids - were formed from a single gas and dust cloud. According to the second, the Sun and its family have different origins, so that the Sun was formed from one gas and dust cloud (nebula, globule), and the rest of the celestial bodies of the Solar system - from another cloud, which was captured in some not entirely clear way by the Sun on its own orbit and was divided in some, even more incomprehensible way into many different bodies (planets, their satellites, asteroids, comets and meteoroids), having the most different characteristics: mass, density, eccentricity, direction of orbital rotation and direction of rotation around its axis, the inclination of the orbit to the plane of the Sun's equator (or ecliptic) and the inclination of the equatorial plane to the plane of its orbit.
The nine major planets revolve around the Sun in ellipses (not much different from circles) in almost the same plane. In order of distance from the Sun, these are Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. In addition to them, there are many small planets (asteroids) in the Solar System, most of which move between the orbits of Mars and Jupiter. The space between the planets is filled with extremely rarefied gas and cosmic dust. It is penetrated by electromagnetic radiation.
Sun 109 times more than Earth in diameter and approximately 333,000 times more massive than Earth. The mass of all the planets is only about 0.1% of the mass of the Sun, so it controls the movement of all members of the Solar System by the force of its gravity.

Configuration and visibility conditions of planets

Planetary configurations are some more characteristic mutual positions of the planets, the Earth and the Sun.
The conditions for the visibility of planets from Earth differ sharply for the internal planets (Venus and Mercury), whose orbits lie within the Earth’s orbit, and for the external planets (all others).
The inner planet may be between the Earth and the Sun or behind the Sun. In such positions the planet is invisible, as it is lost in the rays of the Sun. These positions are called planet-Sun conjunctions. At inferior conjunction, the planet is closest to Earth, and at superior conjunction, it is farthest from us.

Synodic periods of revolution of planets and their connection with sidereal periods

The period of revolution of the planets around the Sun in relation to the stars is called the sidereal or sidereal period.
The closer a planet is to the Sun, the greater its linear and angular velocities and the shorter the sidereal period of revolution around the Sun.
However, from direct observations, it is not the sidereal period of revolution of the planet that is determined, but the period of time that elapses between its two successive configurations of the same name, for example, between two successive conjunctions (oppositions). This period is called the synodic orbital period. Having determined the synodic periods from observations, the sidereal periods of revolution of the planets are found by calculations.
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.

Kepler's laws

The merit of discovering the laws of planetary motion belongs to the outstanding German scientist Johannes Kepler(1571 -1630). IN early XVII V. Kepler, studying the revolution of Mars around the Sun, established three laws of planetary motion.

Kepler's first law . Each planet revolves in an ellipse, with the Sun at one of the focuses.

Kepler's second law (law of areas). The radius vector of the planet describes equal areas in equal periods of time.

Kepler's third law . The squares of the sidereal periods of revolution of the planets are related as the cubes of the semimajor axes of their orbits.

The average distance of all planets from the Sun in astronomical units can be calculated using Kepler's third law. Having determined the average distance of the Earth from the Sun (i.e., the value of 1 AU) in kilometers, we can find in these units the distances to all planets of the Solar System. The semimajor axis of the Earth's orbit is taken as the astronomical unit of distance (= 1 AU)
The classic way to determine distances was and remains the goniometric geometric method. They also determine distances to distant stars, to which the radar method is not applicable. The geometric method is based on the phenomenon parallactic displacement.

Parallax displacement is the change in direction of an object when the observer moves.

EXAMPLE OF SOLVING A PROBLEM

Task. Oppositions of a certain planet are repeated after 2 years. What is the semimajor axis of its orbit?

Given	SOLUTION The semimajor axis of the orbit can be determined from Kepler's third law: , and the sidereal period - from the relationship between the sidereal and synodic periods: ,
- ?

Size and Shape of the Earth

In photographs taken from space, the Earth looks like a ball illuminated by the Sun.
The exact answer about the shape and size of the Earth is given degree measurements, i.e. measurements in kilometers of the length of an arc of 1° at different places on the Earth's surface. Degree measurements showed that the length of 1° meridian arc in kilometers in the polar region is greatest (111.7 km), and at the equator it is smallest (110.6 km). Consequently, at the equator the curvature of the Earth's surface is greater than at the poles, which means that the Earth is not a sphere. The equatorial radius of the Earth is 21.4 km greater than the polar radius. Therefore, the Earth (like other planets) is compressed at the poles due to rotation.
A ball equal in size to our planet has a radius of 6370 km. This value is considered to be the radius of the Earth.
The angle at which the radius of the Earth is visible from the luminary, perpendicular to the line of sight, is called horizontal parallax.

Mass and density of the Earth

The law of universal gravitation makes it possible 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 gravity g=(G*M)/r 2 . Consequently, if the values ​​of the acceleration of gravity, the gravitational constant and the radius of the Earth are known, then its mass can be determined.
Substituting the value g = 9.8 m/s 2 into the indicated formula, G = 6.67 * 10 -11 N * m 2 / kg 2,

R = 6370 km, we find that the mass of the Earth is M = 6 x 10 24 kg. Knowing the mass and volume of the Earth, you can calculate its average density.

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1

LESSON 7. PLANET CONFIGURATIONS,

DISTANCES TO BODIES AND THEIR SIZES.

1. 
   Basic configurations of the lower and upper planets.
2. 
   Sidereal and synodic periods of planets.
3. 
   Determining the size of the Earth
4. 
   Determining distances to bodies.
5. 
   Determination of body sizes.

1. 
   Basic configurations of the inner and outer planets.

The complex apparent motion of planets on the celestial sphere is caused by the revolution of the planets of the Solar System around the Sun. The word “planet” itself, translated from ancient Greek, means “wandering” or “vagrant”. The trajectory of a celestial body is called its orbit.

In relation to the Earth's orbit, the planets are divided into internal (lower) - Mercury, Venus, their orbits are located inside the Earth's orbit, and external (upper) - Mars, Jupiter, Saturn, Uranus, Neptune, their orbits are located outside the Earth's orbit. The outer planets always face the Earth with the side illuminated by the Sun. The inner planets change their phases like the Moon. The orbital planes of all planets in the Solar System lie near the ecliptic plane, deviating from it by less than 7°. The speeds at which planets move in their orbits are different and decrease as the planets move away from the Sun. The Earth moves slower than Mercury and Venus, but faster than all other planets. Due to the difference in the speeds of movement of the planets at certain times, different relative positions of the Sun and planets arise.

Special, geometrically correct, mutual positions of the Sun, Earth and planets are called configurations. The same configurations of planets occur at different points in their orbits, opposite different constellations, at different times of the year. The configurations that are created by the lower and upper planets are different.
The lower planets have this connectionsV 1 And V 3 (top and bottom) and elongationV 2 And V 4 (eastern and western). For the upper planets it is - quadraturesM 2 and M 4 (eastern and western), compoundM 1 and confrontationM 3 .

What is behind these terrible names? Conjunctions are the placement of the Sun, Earth and planet on the same line, while the planet is either between the Sun and the Earth (inferior conjunction), or is hiding from the Earth behind the Sun (upper conjunction). The only configuration in which any planet, both inferior and superior, can be located is superior conjunction, and the planet naturally cannot be observed. Inferior conjunction is inherent only to the lower planets, while, although quite rarely, we can observe the passage of Mercury and Venus (in the form of a black circle) against the background of the solar disk.

The apparent motion of the lower planets resembles the oscillatory motion around the Sun. The maximum angular distance of the lower planets from the Sun is called elongation. In the case of elongation, the Earth, planet and Sun form a right triangle, with the planet at the vertex of the right angle. The greatest elongation of Mercury is 28˚, Venus - 48˚. From the Earth at this time, not the entire hemisphere of the planet illuminated by the Sun is visible, but only its part, called the phase. During eastern elongation, the planet is visible in the west shortly after sunset; during western elongation, the planet is visible in the east shortly before sunrise.

The most convenient moment for observing the upper planets is opposition. All three celestial bodies, as in conjunction, are on the same line, but the Earth in this case is located between the Sun and the planet and the entire hemisphere of the planet is illuminated by the Sun. The outer planet can be at any angular distance from the Sun from 0˚ to 180˚. When the angular distance between the Sun and the upper planet is 90˚, then the planet is said to be in quadrature (quadrature is an angular quarter of a circle), respectively in the east or west, as in elongation. In this case, the Earth, the Sun and the planet also form a right triangle, but the Earth is at the vertex of the right angle.

The Earth-Moon-Sun system is special, it has an inferior conjunction, like the inner planets, during which a new moon occurs (the Moon between the Sun and the Earth), and opposition, like the outer planets, during the full moon.

Task.Determine the period of revolution of Mars around the Sun, knowing that oppositions of Mars occur every 780 days?

1. 
   Determining the size of the Earth.

The idea of ​​the Earth as a ball that hangs freely in space without any support is certainly one of the greatest achievements of science of the ancient world. And the first accurate determination of the earth's dimensions was made by Eratosthenes from Egypt. The experiment he performed is one of the ten most beautiful physical experiments invented by mankind. He decided to measure the length of a small arc of the earth's meridian not in degrees, but in units of length, and then determine what part in degrees of the full circle it constitutes. Knowing the part, find the length of the entire circle. Then, using the circumference, determine the radius, which is the radius of the globe.
Obviously, the length of the meridian arc in degrees is equal to the difference in the geographical latitudes of two points located on the same meridian: Δφ = φ in – φ A. In order to determine this difference, Eratosthenes compared the height of the Sun at its culmination on the same day in points A and B (Alexandria and Aswan). In Aswan on this day the Sun illuminated the bottom of the deepest wells, i.e. was at the zenith, and in Alexandria it was 7.2˚ from the zenith. From simple geometric constructions it followed that the difference in latitude of these cities was Δφ = 7.2˚. In ancient units of measurement, the distance between Alexandria and Aswan was 5000 Greek stadia, modern - 800 km. Denoting the length of the Earth's meridian by L, we have the following proportion:
from where we obtain the length of the meridian equal to 40,000 km. Knowing the circumference, we can easily find the radius of the Earth - 6366 km, which differs from the average radius by only 5 km.

To what extent the shape of the Earth differs from a sphere became clear only at the end of the 18th century as a result of the work of two expeditions in South America in Peru and Scandinavia near the Arctic Circle. Measurements have shown that the length of 1˚ arc of the meridian in the north and south is greater than at the equator. This meant that the Earth was flattened at the poles. Its polar radius is 21 km shorter than the equatorial one. This means that the cross section of the Earth along the meridian will not be a circle, but an ellipse, the major axis of which lies in the plane of the equator, and the minor axis coincides with the axis of rotation of the Earth. And already in the twentieth century it became clear that the earth’s equator also cannot be considered a circle. Its oblateness is 100 times less than the oblateness of the meridian, but it still exists. Most accurately, the shape of our planet is conveyed by a figure called an ellipsoid, in which any section by a plane passing through the center of the Earth is not a circle.
4. Determination of distances to bodies.

ABOUT
determining the geographic latitude of two points turns out to be much easier than measuring the distance between them, which may be hampered by natural obstacles. Therefore, a method based on the phenomenon of parallactic displacement is used. Parallax displacement is the change in direction of an object when the observer moves. First, accurately calculate the length of a conveniently located segment BC, called the basis, and two angles B and C in triangle ABC. Further on the theorem of sines
the values ​​AC and AB are easily found. A similar method is used to determine the distance to celestial bodies. It was possible to measure the distance from the Earth to the Sun for the first time only in the 18th century, when it was determined horizontal parallax Sun. Horizontal parallax(p) is the angle at which the radius of the Earth is visible from a star located on the horizon, perpendicular to the line of sight. In essence, this measures the parallactic displacement of an object located outside the Earth, and the basis is the radius of the Earth. The only difference is that the triangle is constructed as a rectangular one, which simplifies the calculations.

From triangle OAS we can express the distance SO=D: where R  is the radius of the Earth. Of course, no one observes the radius of the Earth from the luminary, and the horizontal parallax is determined by measuring the height of the luminary at the moment of the upper culmination from two points of the Earth located on the same meridian and having known latitudes, by analogy with the method of Eratosthenes. Obviously, the farther an object is located, the less its parallax. The parallax of the Moon is of greatest importance ( r  =57΄02΄΄), solar parallax r =8.79′′. This parallax value corresponds to a distance to the Sun equal to 150,000,000 km. This distance is taken as one astronomical unit (1AU) and is used when measuring distances between bodies of the Solar System.

For small angles sinp≈ p, while p expressed in radians. If r expressed in seconds, then the formula will take the form: .
D.z. §7. clause 2,3. problems 8,9 p.35, § 11. problems 1, 5, 6 p.52.
Express survey questions

1. Is it possible to observe Mercury in the evenings in the east?

2. What is a connection?

3. Is it possible to observe Venus in the east in the morning and in the west in the evening?

4. The angular distance of the planet from the Sun is 55°. Which planet is it, up or down?

5. What is configuration?

6. What planets can pass against the background of the solar disk?

7. During what configurations are the lower planets clearly visible?

8. During what configurations are the upper planets clearly visible?

9. What is the sidereal period of the planet?

10. What is the synodic period?

11. What is horizontal parallax?

12. What is called parallactic displacement?

13. When is the superior planet in square?

14. What is elongation?

15. At what conjunction can the inner planet be observed?