What condition must the star's declination satisfy? A guide for teachers of astronomy. Practical work with a moving map of the starry sky

A- the azimuth of the luminary, is measured from the point of the South along the line of the mathematical horizon clockwise in the direction of west, north, east. It is measured from 0 o to 360 o or from 0 h to 24 h.

h- the height of the luminary, measured from the point of intersection of the circle of height with the line of the mathematical horizon, along the circle of height up to the zenith from 0 o to +90 o, and down to the nadir from 0 o to -90 o.

http://www.college.ru/astronomy/course/shell/images/Fwd_h.gifhttp://www.college.ru/astronomy/course/shell/images/Bwd_h.gif Equatorial coordinates

Geographic coordinates help determine the position of a point on Earth - latitude  and longitude . Equatorial coordinates help determine the position of stars on the celestial sphere - declination  and right ascension .

For equatorial coordinates, the main planes are the plane of the celestial equator and the declination plane.

The right ascension is counted from the vernal equinox  in the direction opposite to the daily rotation of the celestial sphere. Right ascension is usually measured in hours, minutes and seconds of time, but sometimes in degrees.

Declination is expressed in degrees, minutes and seconds. The celestial equator divides the celestial sphere into northern and southern hemispheres. The declinations of stars in the northern hemisphere can be from 0 to 90°, and in the southern hemisphere - from 0 to -90°.

Equatorial coordinates take precedence over horizontal coordinates:

1) Created star charts and catalogs. The coordinates are constant.

2) Drawing up geographical and topological maps earth's surface.

3) Implementation of orientation on land, sea space.

4) Checking the time.
Exercises.

Horizontal coordinates.
1. Determine the coordinates of the main stars of the constellations included in the autumn triangle.

2. Find the coordinates  Virgo,  Lyra,  Big Dog.

3. Determine the coordinates of your zodiac constellation, at what time is it most convenient to observe it?

equatorial coordinates.
1. Find on star map and name the objects having coordinates:

1)  \u003d 15 h 12 m,  \u003d -9 o; 2)  \u003d 3 h 40 m,  \u003d +48 o.

2. Determine the equatorial coordinates of the following stars from the star map:

1)  Ursa Major; 2)  China.

3. Express 9 h 15 m 11 s in degrees.

4. Find on the star map and name the objects that have coordinates

1)  = 19 h 29 m,  = +28 o; 2)  = 4 h 31 m,  = +16 o 30 / .

5. Determine the equatorial coordinates of the following stars from the star map:

1)  Libra; 2)  Orion.

6. Express 13 hours 20 meters in degrees.

7. What constellation is the Moon in if its coordinates are  = 20 h 30 m,  = -20 o.

8. Determine the constellation in which the galaxy is located on the star map M 31, if its coordinates are  0 h 40 m,  = 41 o.

4. The culmination of the luminaries.

Theorem about the height of the celestial pole.
Key questions: 1) astronomical methods for determining geographic latitude; 2) using a moving chart of the starry sky, determine the condition of visibility of the stars at any given date and time of day; 3) solving problems using relationships that connect the geographical latitude of the place of observation with the height of the luminary at the climax.
The culmination of the luminaries. Difference between upper and lower climax. Working with the map determining the time of culminations. Theorem about the height of the celestial pole. Practical ways to determine the latitude of the area.

Using the drawing of the projection of the celestial sphere, write down the height formulas in the upper and lower culmination of the luminaries if:

a) the star culminates between the zenith and the south point;

b) the star culminates between the zenith and the celestial pole.

Using the celestial pole height theorem:

- the height of the pole of the world (Polar Star) above the horizon is equal to the geographical latitude of the place of observation

Corner
- both vertical and
. Knowing that
is the declination of the star, then the height of the upper culmination will be determined by the expression:

For the bottom climax of a star M 1:

Give home the task to get a formula for determining the height of the upper and lower culmination of a star M 2 .

Assignment for independent work.

1. Describe the conditions for the visibility of stars at 54° north latitude.

Star	visibility condition
Sirius ( \u003d -16 about 43 /)
Vega ( = +38 o 47 /)	never setting star
Canopus ( \u003d -52 about 42 /)	rising star
Deneb ( = +45 o 17 /)	never setting star
Altair ( = +8 o 52 /)	Rising and setting star
 Centauri ( \u003d -60 about 50 /)	rising star

2. Install a mobile star map for the day and hour of classes for the city of Bobruisk ( = 53 o).

Answer the following questions:

a) which constellations are above the horizon at the time of observation, which constellations are below the horizon.

b) what constellations ascend in this moment are coming in at the moment.
3. Determine the geographical latitude of the observation site if:

a) the star Vega passes through the zenith point.

b) the star Sirius at its upper culmination at an altitude of 64° 13/ south of the zenith point.

c) the height of the star Deneb at its upper climax is 83 o 47 / north of the zenith.

d) the star Altair passes at the lower culmination through the zenith point.

On one's own:

Find the intervals of declination of stars that are at a given latitude (Bobruisk):

a) never rise b) never enter; c) can ascend and set.

Tasks for independent work.
1. What is the declination of the zenith point at the geographical latitude of Minsk ( = 53 o 54 /)? Accompany your answer with a picture.

2. In what two cases does the height of the star above the horizon not change during the day? [Either the observer is at one of the poles of the Earth, or the luminary is at one of the poles of the world]

3. Using the drawing, prove that in the case of the upper culmination of the luminary north of the zenith, it will have a height h\u003d 90 o +  - .

4. The azimuth of the luminary is 315 o, the height is 30 o. In what part of the sky is this luminary visible? In the southeast

5. In Kyiv, at an altitude of 59 o, the upper culmination of the star Arcturus was observed ( = 19 o 27 /). What is the geographical latitude of Kyiv?

6. What is the declination of the stars culminating in a place with a geographical latitude  at the north point?

7. The polar star is 49/46 from the north celestial pole // . What is its declination?

8. Is it possible to see the star Sirius ( \u003d -16 about 39 /) at meteorological stations located on about. Dikson ( = 73 o 30 /) and in Verkhoyansk ( = 67 o 33 /)? [On about. Dixon is not present, not in Verkhoyansk]

9. A star that describes an arc of 180 o above the horizon from sunrise to sunset, during the upper climax, is 60 o from the zenith. At what angle is the celestial equator inclined to the horizon at this location?

10. Express the right ascension of the star Altair in arc meters.

11. The star is 20 o from the north celestial pole. Is it always above the horizon of Brest ( = 52 o 06 /)? [Always]

12. Find the geographical latitude of the place where the star at the top culmination passes through the zenith, and at the bottom it touches the horizon at the north point. What is the declination of this star?  = 45 o; [ \u003d 45 about]

13. Azimuth of the star 45 o, height 45 o. In which side of the sky should you look for this luminary?

14. When determining the geographical latitude of the place, the desired value was taken equal to the height of the Polar Star (89 o 10 / 14 / /), measured at the time of the lower climax. Is this definition correct? If not, what is the error? What correction (in magnitude and sign) must be made to the measurement result in order to obtain the correct latitude value?

15. What condition must the declination of a luminary satisfy in order for this luminary not to set at a point with latitude ; so that it is not ascending?

16. The right ascension of the star Aldebaran (-Taurus) is equal to 68 about 15 /. Express it in units of time.

17. Does the star Fomalhaut (-Golden Fish) rise in Murmansk ( = 68 o 59 /), the declination of which is -29 o 53 / ? [Does not rise]

18. Prove from the drawing, from the lower culmination of the star, that h\u003d  - (90 o - ).

Homework: § 3. q.v.
5. Measurement of time.

Definition of geographic longitude.
Key issues: 1) differences between the concepts of sidereal, solar, local, zone, seasonal and universal time; 2) the principles of determining time according to astronomical observations; 3) astronomical methods for determining the geographical longitude of the area.

Students should be able to: 1) solve problems for calculating the time and dates of the chronology and transferring time from one counting system to another; 2) determine the geographical coordinates of the place and time of observation.

At the beginning of the lesson, independent work 20 minutes.

1. Using a moving map, determine 2 - 3 constellations visible at a latitude of 53 o in the Northern Hemisphere.

patch of sky	Option 1 15. 09. 21 h	Option 2 25. 09. 23 h
Northern part	B. Bear, Charioteer. Giraffe	B. Bear, Hounds Dogs
South part	Capricorn, Dolphin, Eagle	Aquarius, Pegasus, Y. Pisces
Western part	Bootes, S. Crown, Snake	Ophiuchus, Hercules
East End	Aries, Pisces	Taurus, Charioteer
Constellation at its zenith	Swan	Lizard

2. Determine the azimuth and height of the star at the time of the lesson:

1 option.  B. Ursa,  Leo.

Option 2.  Orion,  Eagle.

3. Using a star map, find the stars by their coordinates.

Main material.

To form concepts about days and other units of measurement of time. The occurrence of any of them (day, week, month, year) is associated with astronomy and is based on the duration of cosmic phenomena (the rotation of the Earth around its axis, the revolution of the Moon around the Earth and the revolution of the Earth around the Sun).

Introduce the concept of sidereal time.

Pay attention to the following; moments:

- the length of the day and year depends on the frame of reference in which the movement of the Earth is considered (whether it is associated with fixed stars, the Sun, etc.). The choice of reference system is reflected in the name of the unit of time.

- the duration of time counting units is associated with the conditions of visibility (culminations) of celestial bodies.

- the introduction of the atomic time standard in science was due to the uneven rotation of the Earth, discovered with increasing clock accuracy.

The introduction of standard time is due to the need to coordinate economic activities in the territory defined by the boundaries of time zones.

Explain the reasons for the change in the length of the solar day throughout the year. To do this, it is necessary to compare the moments of two successive climaxes of the Sun and any star. Mentally choose a star that for the first time culminates simultaneously with the Sun. The next time the culmination of the star and the Sun will not happen at the same time. The sun will culminate at about 4 min later, because against the background of stars it will move about 1 // due to the movement of the Earth around the Sun. However, this movement is not uniform due to the uneven movement of the Earth around the Sun (students will learn about this after studying Kepler's laws). There are other reasons why the time interval between two successive climaxes of the Sun is not constant. There is a need to use the average value of solar time.

Give more precise data: the average solar day is 3 minutes 56 seconds shorter than the sidereal day, and 24 hours 00 minutes 00 from sidereal time is equal to 23 hours 56 minutes 4 from the average solar time.

Universal time is defined as local mean solar time at the zero (Greenwich) meridian.

The entire surface of the Earth is conditionally divided into 24 sections (time zones), limited by meridians. The zero time zone is located symmetrically with respect to the prime meridian. Time zones are numbered from 0 to 23 from west to east. The real boundaries of time zones coincide with the administrative boundaries of districts, regions or states. The central meridians of time zones are 15 o (1 h) apart, so when moving from one time zone to another, time changes by an integer number of hours, and the number of minutes and seconds does not change. A new calendar day (as well as a new calendar year) begins on the date change line, which runs mainly along the 180 o meridian. d. near the northeastern border of the Russian Federation. To the west of the date line, the day of the month is always one more than to the east of it. When crossing this line from west to east, the calendar number decreases by one, and when crossing from east to west, the calendar number increases by one. This eliminates the error in the calculation of time when moving people traveling from the Eastern to the Western hemisphere of the Earth and back.

The calendar. Limit yourself to consideration brief history calendar as part of culture. It is necessary to single out three main types of calendars (lunar, solar and lunisolar), tell what they are based on, and dwell in more detail on the Julian solar calendar of the old style and the Gregorian solar calendar of the new style. After recommending relevant literature, invite students to prepare for the next lesson brief messages about different calendars or organize a special conference on this topic.

After presenting the material on the measurement of time, it is necessary to move on to generalizations related to the determination of geographic longitude, and thereby summarize the questions about determining geographic coordinates using astronomical observations.

Modern society cannot do without knowing the exact time and coordinates of points on the earth's surface, without accurate geographical and topographic maps necessary for navigation, aviation and many other practical issues of life.

Due to the rotation of the Earth, the difference between the moments of noon or the culmination of stars with known equatorial coordinates at two points on the earth surface is equal to the difference between the values of the geographical longitude of these points, which makes it possible to determine the longitude of a particular point from astronomical observations of the Sun and other luminaries and, conversely, local time at any point with a known longitude.

To calculate the geographic longitude of the area, it is necessary to determine the moment of climax of any luminary with known equatorial coordinates. Then, using special tables (or a calculator), the observation time is converted from mean solar to stellar. Having learned from the reference book the time of the culmination of this luminary on the Greenwich meridian, we can determine the longitude of the area. The only difficulty here is the exact conversion of units of time from one system to another.

The moments of the climax of the luminaries are determined with the help of a transit instrument - a telescope, strengthened in a special way. The spotting scope of such a telescope can only be rotated around a horizontal axis, and the axis is fixed in the west-east direction. Thus, the instrument turns from the south point through the zenith and the celestial pole to the north point, i.e. it traces the celestial meridian. The vertical thread in the field of view of the telescope tube serves as a mark of the meridian. At the time of the passage of a star through the celestial meridian (in the upper climax), sidereal time is equal to right ascension. The first passage instrument was made by the Dane O. Roemer in 1690. For more than three hundred years, the principle of the instrument has not changed.

Note the fact that the need to accurately determine the moments and intervals of time stimulated the development of astronomy and physics. Up to the middle of the 20th century. astronomical methods of measuring, keeping time and time standards underlay the activities of the World Time Service. The accuracy of the clock was controlled and corrected by astronomical observations. At present, the development of physics has led to the creation of more accurate methods for determining and standards of time. Modern atomic clocks give an error of 1 s in 10 million years. With the help of these watches and other instruments, many characteristics of the visible and true movement of cosmic bodies were refined, new cosmic phenomena were discovered, including changes in the speed of the Earth's rotation around its axis by approximately 0.01 s during the year.
- average time.

- standard time.

- summer time.

Messages for students:

1. Arabic moon calendar.

2. Turkish lunar calendar.

3. Persian solar calendar.

4. Coptic solar calendar.

5. Projects of ideal perpetual calendars.

6. Counting and keeping time.

6. Heliocentric system of Copernicus.
Key questions: 1) the essence of the heliocentric system of the world and the historical prerequisites for its creation; 2) the causes and nature of the apparent motion of the planets.
Frontal conversation.

1. A true solar day is the time interval between two successive climaxes of the same name of the center of the solar disk.

2. A sidereal day is the time interval between two successive culminations of the same name of the vernal equinox, equal to the period of the Earth's rotation.

3. The mean solar day is the time interval between two culminations of the same name of the mean equatorial Sun.

4. For observers located on the same meridian, the culmination of the Sun (as well as any other luminary) occurs simultaneously.

5. A solar day differs from a stellar day by 3 m 56 s.

6. The difference in the values of local time at two points on the earth's surface at the same physical moment is equal to the difference in the values of their geographical longitudes.

7. When crossing the border of two neighboring belts from west to east, the clock must be moved one hour ahead, and from east to west - one hour ago.

Consider an example solution tasks.

The ship, which left San Francisco on the morning of Wednesday, October 12 and headed west, arrived in Vladivostok exactly 16 days later. What date of the month and on what day of the week did he arrive? What should be taken into account when solving this problem? Who and under what circumstances faced this for the first time in history?

When solving the problem, it must be taken into account that on the way from San Francisco to Vladivostok, the ship will cross a conditional line called the international date line. It passes along the earth's meridian with a geographic longitude of 180 o, or close to it.

When crossing the date change line in the direction from east to west (as in our case), one calendar date is discarded from the account.

For the first time, Magellan and his companions encountered this during their trip around the world.

Let on rps. 11 the semicircle represents the meridian, P is the north celestial pole, OQ is the trace of the equatorial plane. The angle PON, equal to the angle QOZ, is the geographical sprat of the place ip (§ 17). These angles are measured by the arcs NP and QZ, which are therefore also yes; the declination of the luminary Mi, which is in the upper culmination, is measured by the arc QAlr. Denoting its zenith distance as r, we obtain for the luminary, culminating - 1, k, increasing (, * south of the zenith:

For such luminaries, obviously, "

If the luminary passes through the meridian north of the zenith (point M /), then its declination will be QM (\ n we get

I! In this case, taking the complement to 90°, we get the height

stars h at the time of the upper cul-,

minacpp. p M, Z

Finally, if b - e, then the star in the upper culmination passes through the zenith.

It is just as easy to determine the height of the luminary (UM,) at the lower M, the climax, i.e., at the moment of its passage through the meridian between the pole of the world (P) and the north point (N).

From fig. 11 it can be seen that the height h2 of the luminary (M2) is determined by the arc LH2 and is equal to h2 - NP-M2R. Arc arc M2R-r2,

i.e., the distance of the luminary from the pole. Since p2 \u003d 90 - 52> then

h2 = y-"ri2 - 90°. (3)

Formulas (1), (2) and (3) have extensive applications.

Exercises for the chapter /

1. Prove that the equator intersects the horizon at points 90° away from the north and south points (at the east and west points).

2. What are the hour angle and zenith azimuth?

3. What are the declination and hourly angle of the west point? East point?

4. What \thol with the horizon forms the equator with a latitude of - (-55 °? -) -40 °?

5. Is there a difference between the north celestial pole and the north point?

6. Which of the points of the celestial equator is above all above the horizon? Why paRiio the zenith distance of this point for latitude<р?

7. If a star rose at a point in the northeast, then at what point on the horizon will it set? What are the azimuths of the points eb of sunrise and sunset?

8. What is the azimuth of the star at the time of the upper culmination for a place under the latitude cp? Is it the same for all stars?

9. What is the declination of the north celestial pole? south pole?

10. What is the declination of the zenith for a place with latitude o? north point declination? south points?

11. In what direction does the star move in the lower climax?

12. The North Star is 1° away from the celestial pole. What is its declination?

13. What is the height of the North Star at the upper culmination for a place under the latitude cp? Same for the bottom climax?

14. What condition must the declination S of a star satisfy in order for it not to set under latitude 9? to make it non-ascending?

15. What hurts the angular radius of the circle of setting stars in Leningrad (“p = - d9°57”)?” In Tashkent (srg-41b18")? "

16. What is the declination of the stars passing through the zenith in Leningrad and Tashkent? Are they visiting for these cities?

17. At what zenith distance does the star Capella (i - -\-45°5T) pass through the upper culmination in Leningrad? in Tashkent?

18. Up to what declination are the stars of the southern hemisphere visible in these cities?

19. Starting from what latitude can you see Canopus, the brightest star in the sky after Sirius (o - - 53 °) when traveling south? Is it necessary to leave the territory of the USSR for this (check the map)? At what latitude will Kapoius become a non-setting star?

20. What is the height of the Chapel at the lower climax in Moscow = + 5-g<°45")? в Ташкенте?

21. Why is the right ascension counting from west to east, and not in the opposite direction?

22. The two brightest stars in the northern sky are Vega (a = 18ft 35m) and Capella (r -13da). In which side of the sky (western or eastern) and what hour angles are they at the time of the upper climax of the vernal equinox? At the moment of the lower climax of the same point?

23. What interval of sidereal time passes from the lower culmination of the Chapel to the upper climax of Bern?

24. What is the hour angle of the Chapel at the moment of the upper climax of the Run? At the moment of her lower climax?

25. At what hour in sidereal time does the vernal equinox point rise? comes in?

26. Prove that for an observer at the earth's equator, the azimuth of a star at the time of sunrise (AE) and at the time of setting (A^r) is very simply related to the declination of the star (i).

- clarification - ideally, the work is done in the computer training program IISS "Planetarium"

Without this program, you can do the work using a moving map of the starry sky: a map and an overlay circle.

Practical work with a moving map
starry sky.

Theme . Apparent motion of the Sun

Lesson Objectives .

Students should be able to:

1. Determine the equatorial coordinates of the luminaries on the map and, conversely, knowing the coordinates, find the luminary and determine its name from the table;

2. Knowing the equatorial coordinates of the Sun, determine its position on the celestial sphere;

3. Determine the time of sunrise and sunset, as well as the time spent above the horizon of stars and the Sun;

4. Calculate the height of the star above the horizon at the upper culmination, knowing the geographical latitude of the place of observation and determining its equatorial coordinates on the map; solve the reverse problem.

5. Determine the declination of the luminaries that do not rise or set for a given latitude of the observation site.

Basic concepts. Equatorial and horizontal coordinate systems.

Demo Material. Moving map of the starry sky. Planetarium. Illustrations.

Independent activity of students. Performing tasks with the help of an electronic planetarium and a moving map of the starry sky.

Worldview aspect of the lesson. Formation of a scientific approach to the study of the world.

5. What does the declination sign show?

6. What is the declination of points lying on the equator?

Find concentric circles on the map, the center of which coincides with the north celestial pole. These circles are parallels, that is, the locus of points that have the same declination. The first circle from the equator has a declination of 30°, the second - 60°. The declination is measured from the celestial equator, if towards the north pole, then δ > 0; if south of the equator, then δ< 0.

For example, find a Charioteer, Chapel. It is located in the middle between the parallels 30° and 60°, so its declination is approximately 45°.

The radial lines on the map correspond to the declination circles. To determine the right ascension of a star, you need to determine the angle from the vernal equinox to the circle of declination passing through this star. To do this, connect the north pole of the world and the luminary with a straight line and continue it until it intersects with the inner border of the map on which the clock is indicated, this is the direct ascension of the luminary.

For example, we connect the Chapel with the north pole of the world, continue this line to the inner edge of the map - approximately 5 hours 10 minutes.

Assignment for students.

Determine the equatorial coordinates of the luminaries and, conversely, find the luminary from the given coordinates. Test yourself with an electronic planetarium.

1. Determine the coordinates of the stars:

1. alion	AND)a= 5h13m,d= 45°
2. aCharioteer	B)a= 7h37m,d= 5°
3. aSmall Dog	AT)a= 19h50min,d= 8°
4. aEagle	G)a= 10h,d= 12°
	D)a= 5h12min,d= -8°
	E)a= 7h42min,d= 28°

2. Based on the approximate coordinates, determine which stars these are:

1. a= 5h 12min,d= -8°	AND)aCharioteer
2. a= 7h 31min,d=32°	B)bOrion
3. a= 5h 52min,d=7°	AT)aGemini
4. a= 4h 32min,d=16°	G)aSmall Dog
	D)aOrion
	E)aTaurus

3. Determine the equatorial coordinates and in which constellations are:

To complete the following tasks, remember how to determine the position of the Sun. It is clear that the Sun is always on the line of the ecliptic. Let's connect the calendar date with a straight line with the center of the chart, and the point of intersection of this line with the ecliptic is the position of the Sun at noon.

Assignment for students.

Option 1

4. Equatorial coordinates of the Sun a = 15 h, d = –15°. Determine the calendar date and the constellation in which the Sun is located.

AND)a= 21 h,d= 0° B)a= -15°,d= 21 h B)a= 21 h,d= -15°

6. Right Ascension of the Sun a = 10h 4min. What is the brightest star that is closest to the Sun on this day?

AND)aSextant B)aHydra B)alion

To determine which luminaries are above the horizon at a given time, it is necessary to impose a moving circle on the map. Combine the time indicated on the edge of the moving circle with the calendar date indicated on the edge of the map, and the constellations that you see in the "window" you will see above the horizon at this time.

During the day, the celestial sphere makes a complete revolution from east to west, and the horizon does not change its position relative to the observer. If you rotate the overlay circle clockwise, imitating the daily rotation of the celestial sphere, then we will notice that some luminaries rise above the horizon, while others set. Rotating the overlaid circle clockwise, note the position of the circle when Aldebaran first appeared above the horizon. Look at what time, marked on the overlay circle, corresponds to the desired date, this will be the desired sunrise time. Determine which side of the horizon Aldebaran rises. Similarly, determine the time and place of the setting of the star and calculate the length of stay of the luminary above the horizon.

Assignment for students.

7. Which of the constellations that the ecliptic crosses are above the horizon in our latitudes at 22:00 on June 25?

A) Eagle B) Ophiuchus C) Lion

8. Determine the time of sunrise and sunset, the length of the day

9. Determine the time of sunrise and sunset, the length of the day

Remember the ratio by which, knowing the equatorial coordinates of the luminaries, you can calculate the height of the luminary at the upper climax. Let's consider the problem. Let's write down the condition: latitude of Moscow j = 55°; since the date is known - March 21 - the day of the vernal equinox, we can determine the declination of the Sun - d \u003d 0 °.

Questions for students.

1. Does the Sun culminate south or north of the zenith? (Becaused < j, then the Sun culminates to the south).

2. What formula should be used to calculate height?

3. (h = δ + (90˚ - φ)

4. Calculate the height of the Sun. h = 0° + 90° – 55° = 35°

Assignment for students. Using an electronic planetarium, determine the equatorial coordinates of the stars and check the correctness of the solution to the problem.

1. At what height is the Sun at noon on December 22 at the latitude of Moscow 55°?

2. What is the height of Vega at the upper climax for Chisinau (j = 47°2`)?

3. At what latitude does Vega culminate at its zenith?

4. What condition must the declination of the Sun satisfy in order for the Sun to pass through the zenith at noon at a given latitude j?

Let's turn to Figure 12. We see that the height of the celestial pole above the horizon is h p =∠PCN, and the geographical latitude of the place is φ=∠COR. These two angles (∠PCN and ∠COR) are equal as angles with mutually perpendicular sides: ⊥, ⊥. The equality of these angles gives simplest way determining the geographical latitude of the area φ: the angular distance of the celestial pole from the horizon is equal to the geographic latitude of the area. To determine the geographic latitude of the area, it is enough to measure the height of the celestial pole above the horizon, since:

2. Daily movement of luminaries at different latitudes

Now we know that with a change in the geographical latitude of the place of observation, the orientation of the axis of rotation of the celestial sphere relative to the horizon changes. Let us consider what will be the visible motions of celestial bodies in the region of the North Pole, at the equator and at the middle latitudes of the Earth.

At the pole of the earth the pole of the world is at its zenith, and the stars move in circles parallel to the horizon (Fig. 14, a). Here the stars do not set and do not rise, their height above the horizon is unchanged.

At middle geographic latitudes exist as ascending and incoming stars, as well as those that never fall below the horizon (Fig. 14, b). For example, circumpolar constellations (see Fig. 10) never set at the geographical latitudes of the USSR. Constellations farther from the north celestial pole appear briefly above the horizon. And the constellations lying near the south pole of the world are non-ascending.

But the further the observer moves south, the more southern constellations he can see. At the earth's equator, if the Sun did not interfere during the day, the constellations of the entire starry sky could be seen in a day (Fig. 14, c).

To an observer at the equator, all stars rise and set perpendicular to the horizon plane. Each star here passes over the horizon exactly half of its path. The north pole of the world for him coincides with the point of the north, and the south pole of the world coincides with the point of utah. The axis of the world is located in the plane of the horizon (see Fig. 14, c).

Exercise 2

1. How can you determine by the appearance of the starry sky and its rotation that you have arrived at the North Pole of the Earth?

2. How are the daily paths of stars relative to the horizon for an observer located at the Earth's equator? How do they differ from the daily paths of stars visible in the USSR, i.e., in middle geographical latitudes?

Task 2

Measure the geographic latitude of your area using the eclimeter using the height of the North Star and compare it with the latitude reading on the geographic map.

3. The height of the luminaries at the climax

During the apparent rotation of the sky, which reflects the rotation of the Earth around its axis, the pole of the world occupies a constant position above the horizon at a given latitude (see Fig. 12). During the day, the stars describe circles above the horizon around the axis of the world, parallel to the celestial equator. Moreover, each luminary crosses the celestial meridian twice a day (Fig. 15).

The phenomena of the passage of luminaries through the celestial meridian relative to the horizon for are called culminations. In the upper climax, the height of the luminary is maximum, and in the lower climax, it is minimal. The time interval between climaxes is equal to half a day.

At not setting at a given latitude φ of the luminary M (see Fig. 15), both culminations are visible (above the horizon), for the stars that rise and set (M 1, M 2, M 3), the lower culmination occurs under the horizon, below the north point. At the luminary M 4, located far south of the celestial equator, both climaxes can be invisible (the luminary non-ascending).

The moment of the upper climax of the center of the Sun is called true noon, and the moment of the lower climax is called true midnight.

Let us find the relationship between the height h of the star M at the upper culmination, its declination δ and the latitude of the area φ. To do this, we will use Figure 16, which shows the plumb line ZZ", the world axis PP" and the projections of the celestial equator QQ" and the horizon line NS onto the plane of the celestial meridian (PZSP"N).

We know that the height of the world pole above the horizon is equal to the geographical latitude of the place, i.e. h p =φ. Therefore, the angle between the noon line NS and the axis of the world PP "is equal to the latitude of the area φ, i.e. ∠PON=h p = φ. It is obvious that the inclination of the plane of the celestial equator to the horizon, measured by ∠QOS, will be equal to 90 ° -φ, since ∠QOZ= ∠PON as angles with mutually perpendicular sides (see Fig. 16) Then the star M with declination δ, culminating south of the zenith, has an altitude at its upper culmination

From this formula it can be seen that the geographical latitude can be determined by measuring the height of any luminary with a known declination δ at the upper climax. In this case, it should be borne in mind that if the luminary at the moment of climax is located south of the equator, then its declination is negative.

Problem solution example

A task. Sirius (α B. Psa, see appendix IV) was at its upper climax at 10°. What is the latitude of the observation point?

Pay attention to the fact that the drawing exactly matches the condition of the problem.

Exercise 3

When solving problems, the geographical coordinates of cities can be counted on a geographical map.

1. At what height in Leningrad does the upper climax of Antares (α Scorpio, see Appendix IV) occur?

2. What is the declination of the stars that culminate at the zenith in your city? at a point south?

3. Prove that the height of the luminary at the lower culmination is expressed by the formula h=φ+δ-90°.

4. What condition must satisfy the declination of a star so that it is not setting for a place with a geographical latitude φ? non-ascending?

TO HELP THE ASTRONOMY TEACHER

(for physics and mathematics schools)

1. subject of astronomy.

Sources of knowledge in astronomy. Telescopes.

Key questions: 1. What does astronomy study. 2. Connection of astronomy with other sciences. 3. The scale of the universe. 4. The value of astronomy in the life of society. 5. Astronomical observations and their features.

Demonstrations and TCO: 1. Earth globe, transparencies: photographs of the Sun and Moon, planets of the starry sky, galaxies. 2. Instruments used for observation and measurement: telescopes, theodolite.

[Astron- luminary; nomos- law]

Astronomy studies the vast world that surrounds the Earth: the Sun, the Moon, planets, phenomena occurring in the solar system, stars, the evolution of stars ...

Astronomy ® Astrophysics ® Astrometry ® Stellar astronomy ® Extragalactic astronomy ® Ultraviolet astronomy ® g Astronomy ® Cosmogony (origin) ® Cosmology (general laws of the development of the universe)

Astrology is a doctrine that states that according to the relative positions of the Sun, planets, against the background of constellations, it is possible to predict phenomena, destinies, events.

The Universe is the entire material world, boundless in space and evolving in time. Three concepts: microcosm, macrocosm, megaworld.

Earth ® Solar system ® Galaxy ® Metagalaxy ® Universe.

The earth's atmosphere absorbs g, x-ray, ultraviolet, a significant fraction of infrared, radio waves 20 m< l < 1 мм.

Telescopes (optical, radio)

Lens telescopes (refractor), mirror telescopes (reflector). Refractus- refraction (lens - lenses), reflectere- reflect (lens - mirror).

The main purpose of telescopes is to collect as much light energy as possible from the body under study.

Optical telescope features:

1) Lens - up to 70 cm, luminous flux ~ D 2 .

2) F is the focal length of the lens.

3) F/D- relative aperture.

4) Magnification of the telescope, where D in millimeters.

The largest D= 102 cm, F= 1940 cm.

Reflector - to study the physical nature of celestial bodies. Lens - a concave mirror of small curvature, made of thick glass, Al the powder is sprayed on the other side under high pressure. The rays are collected in the focal plane, where the mirror stands. The mirror absorbs almost no energy.

The biggest D= 6 m, F= 24 m. Photographs stars 4 × 10 -9 fainter than visible ones.

Radio telescopes - an antenna and a sensitive receiver with an amplifier. The biggest D= 600 m consists of 900 flat metal mirrors 2 ´ 7.4 m.

Astronomical observations.

1 . Does the appearance of a star change when viewed through a telescope with magnification?

No. Due to the great distance, the stars are visible as dots even at the highest possible magnification.

2 . Why do you think, when viewed from Earth, that during the night the stars move around the celestial sphere?

Because the Earth rotates on its axis inside the celestial sphere.

3 . What advice would you give to astronomers who want to study the universe using gamma rays, x-rays, and ultraviolet light?

Raise instruments above the earth's atmosphere. Modern technology makes it possible to observe in these parts of the spectrum with balloons, artificial satellites of the Earth or from more remote points.

4 . Explain the main difference between a reflecting telescope and a refractor telescope.

In lens type. A refractor telescope uses a lens, while a reflecting telescope uses a mirror.

5 . Name the two main parts of a telescope.

Lens - collects light and builds an image. Eyepiece - magnifies the image built by the lens.

For independent work.

Level 1: 1 - 2 points

1 . Which of the following scientists played a major role in the development of astronomy? Indicate the correct answers.

A. Nicolaus Copernicus.

B. Galileo Galilei.

B. Dmitry Ivanovich Mendeleev.

2 . The worldview of people in all eras has changed under the influence of the achievements of astronomy, as it deals with ... (indicate the correct statement)

A. ... the study of objects and phenomena independent of man;

B. ... the study of matter and energy under conditions that are impossible to reproduce on Earth;

B. ... by studying the most general patterns of the Megaworld, of which man himself is a part.

3 . One of the following chemical elements was first discovered using astronomical observations. Specify which one?

A. Iron.

B. Oxygen.

4 . What are the features of astronomical observations? List all correct statements.

A. Astronomical observations are in most cases passive in relation to the objects under study.

B. Astronomical observations are mainly based on conducting astronomical experiments.

B. Astronomical observations are related to the fact that all the luminaries are so far away from us that neither by eye nor through a telescope can one decide which one is closer, which one is farther.

5 . You were offered to build an astronomical observatory. Where would you build it? List all correct statements.

A. Within big city.

B. Far from a major city, high in the mountains.

B. At the space station.

6 What are telescopes used for in astronomical observations? Specify the correct statement.

A. To get an enlarged image of a celestial body.

B. To gather more light and see fainter stars.

B. To increase the angle of view from which a celestial object is visible.

Level 2: 3 - 4 points

1. What is the role of observations in astronomy and with what tools are they performed?

2. What are the most important types of celestial bodies you know?

3. What is the role of astronautics in the study of the Universe?

4. List the astronomical phenomena that can be observed during life.

5. Give examples of the relationship between astronomy and other sciences.

6. Astronomy is one of the oldest sciences in the history of mankind. For what purpose did ancient man observe the heavenly bodies? Write what problems people in ancient times solved with the help of these observations.

Level 3: 5 - 6 points

1. Why do the luminaries rise and set?

2. The natural sciences use both theoretical and experimental research methods. Why is observation the main research method in astronomy? Is it possible to set up astronomical experiments? Justify the answer.

3. What are telescopes used for when observing stars?

4. Why are telescopes used to observe the Moon and planets?

5. Does the telescope increase the apparent size of the stars? Explain the answer.

6. Remember what information on astronomy you received in the courses of natural history, geography, physics, history.

4th level. 7 - 8 points

1. Why, when observing the Moon and planets through a telescope, magnification is not more than 500 - 600 times?

2. According to its linear diameter, the Sun is larger than the Moon by about 400 times. Why are their apparent angular diameters nearly equal?

3. What is the purpose of the lens and eyepiece in a telescope?

4. What is the difference between the optical systems of a refractor, a reflector and a meniscus telescope?

5. What are the diameters of the Sun and the Moon in angular measure?

6. How can you indicate the location of the luminaries relative to each other and relative to the horizon?

2. Constellations. Star cards. Celestial coordinates.

Key questions: 1. The concept of constellation. 2. The difference between stars in brightness (luminosity), color. 3. Magnitude. 4. Apparent diurnal motion of stars. 5. celestial sphere, its main points, lines, planes. 6. Star map. 7. Equatorial SC.

Demonstrations and TCO: 1. Demonstration moving sky map. 2. Model of the celestial sphere. 3. Star atlas. 4. Transparencies, photographs of constellations. 5. Model of the celestial sphere, geographic and stellar globes.

For the first time, the stars were designated by the letters of the Greek alphabet. In the constellation of the Bayger atlas, drawings of the constellations disappeared in the 18th century. The magnitudes are shown on the map.

Ursa Major - a (Dubhe), b (Merak), g (Fekda), s (Megrets), e (Aliot), x (Mizar), h (Benetash).

a Lyra - Vega, a Lebedeva - Deneb, a Bootes - Arcturus, a Charioteer - Chapel, a B. Dog - Sirius.

The sun, moon and planets are not shown on the maps. The path of the Sun is shown on the ecliptic in Roman numerals. The star charts have a grid of celestial coordinates. The observed daily rotation is an apparent phenomenon - caused by the actual rotation of the Earth from west to east.

Proof of the rotation of the earth:

1) 1851 physicist Foucault - Foucault pendulum - length 67 m.

2) space satellites, photographs.

Celestial sphere- an imaginary sphere of arbitrary radius used in astronomy to describe the relative position of the stars in the sky. The radius is taken as 1 PC.

88 constellations, 12 zodiacal. Conditionally can be divided into:

1) summer - Lyra, Swan, Eagle 2) autumn - Pegasus with Andromeda, Cassiopeia 3) winter - Orion, B. Pes, M. Pes 4) spring - Virgo, Bootes, Leo.

plumb line crosses the surface of the celestial sphere at two points: at the top Z – zenith- and at the bottom Z" – nadir.

math horizon- a large circle on the celestial sphere, the plane of which is perpendicular to the plumb line.

Dot N mathematical horizon is called north point, dot S – south point. Line NS- is called noon line.

celestial equator called a great circle perpendicular to the axis of the world. The celestial equator intersects the mathematical horizon at points of the east E and west W.

heavenly meridian called a great circle of the celestial sphere, passing through the zenith Z, pole of the world R, south pole of the world R", nadir Z".

Homework: § 2.

constellations. Star cards. Celestial coordinates.

1. Describe what daily circles the stars would describe if astronomical observations were carried out: at the North Pole; at the equator.

The apparent movement of all stars occurs in a circle parallel to the horizon. The North Pole of the World, as viewed from the North Pole of the Earth, is at its zenith.

All stars rise at right angles to the horizon in the eastern sky and also set below the horizon in the western sky. The celestial sphere rotates around an axis passing through the poles of the world, at the equator located exactly on the horizon line.

2. Express 10 hours 25 minutes 16 seconds in degrees.

The earth makes one revolution in 24 hours - 360 o. Therefore, 360 o corresponds to 24 hours, then 15 o - 1 hour, 1 o - 4 minutes, 15 / - 1 minute, 15 // - 1 s. In this way,

10×15 o + 25×15 / + 16×15 // = 150 o + 375 / +240 / = 150 o + 6 o +15 / +4 / = 156 o 19 / .

3. Determine the equatorial coordinates of Vega on the star map.

Let's replace the name of the star with a letter designation (a Lyra) and find its position on the star chart. Through an imaginary point we draw a circle of declination to the intersection with the celestial equator. The arc of the celestial equator, which lies between the vernal equinox and the point of intersection of the circle of declination of a star with the celestial equator, is the right ascension of this star, counted along the celestial equator towards the apparent daily circulation of the celestial sphere. The angular distance, counted from the circle of declination from the celestial equator to the star, corresponds to the declination. Thus, a = 18 h 35 m, d = 38 o.

We rotate the overlay circle of the star map so that the stars cross the eastern part of the horizon. On the limb, opposite the mark of December 22, we find the local time of its sunrise. By placing the star in the western part of the horizon, we determine the local time of the setting of the star. We get

5. Determine the date of the upper culmination of the star Regulus at 21:00 local time.

We set the overlay circle so that the star Regulus (a Leo) is on the line of the celestial meridian (0 h – 12h overlay circle scales) south of the north pole. On the limb of the overlay circle we find the mark 21 and opposite it, on the edge of the overlay circle, we determine the date - April 10.

6. Calculate how many times Sirius is brighter than the North Star.

It is generally accepted that with a difference of one magnitude, the apparent brightness of the stars differs by about 2.512 times. Then a difference of 5 magnitudes will make a difference in brightness exactly 100 times. So the stars of the 1st magnitude are 100 times brighter than the stars of the 6th magnitude. Therefore, the difference in apparent stellar magnitudes of two sources is equal to one when one of them is brighter than the other in (this value is approximately equal to 2.512). In the general case, the ratio of the apparent brightness of two stars is related to the difference in their apparent magnitudes by a simple relation:

Luminaries whose brightness exceeds the brightness of stars 1 m, have zero and negative magnitudes.

Magnitudes of Sirius m 1 = -1.6 and Polaris m 2 = 2.1, we find in the table.

We take the logarithm of both parts of the above relation:

In this way, . From here. That is, Sirius is 30 times brighter than the North Star.

Note: using the power function, we will also get the answer to the question of the problem.

7. Do you think it is possible to fly on a rocket to any constellation?

A constellation is a conditionally defined section of the sky, within which the luminaries turned out to be located at different distances from us. Therefore, the expression "fly to the constellation" is meaningless.

Level 1: 1 - 2 points.

1. What is a constellation? Choose the correct statement.

A.. A group of stars that are physically related to each other, such as having the same origin.

B. A group of bright stars located in space close to each other

B. A constellation is understood to be an area of the sky within certain established boundaries.

2. Stars have different brightness and color. What kind of stars does our Sun belong to? Specify the correct answer.

A. To white. B. To yellow.

B. To red.

3. The brightest stars were called the stars of the first magnitude, and the weakest - the stars of the sixth magnitude. How many times brighter are 1st magnitude stars than 6th magnitude stars? Specify the correct answer.

A. 100 times.

B. 50 times.

B. 25 times.

4. What is the celestial sphere? Choose the correct statement.

A. The circle of the earth's surface bounded by the horizon line. B. An imaginary spherical surface of arbitrary radius, with the help of which the positions and movements of celestial bodies are studied.

B. An imaginary line that touches the surface the globe at the point where the observer is located.

5. What is called declension? Choose the correct statement.

A. Angular distance of the star from the celestial equator.

B. The angle between the horizon line and the luminary.

B. Angular distance of the luminary from the zenith point.

6. What is called right ascension? Choose the correct statement.

A. The angle between the plane of the celestial meridian and the horizon line.

B. The angle between the noon line and the axis of apparent rotation of the celestial sphere (the axis of the world)

B. The angle between the planes of the great circles, one passing through the celestial poles and the given luminary, and the other through the celestial poles and the vernal equinox lying on the equator.

Level 2: 3 - 4 points

1. Why does the Polar Star not change its position relative to the horizon during the daily movement of the sky?

2. How is the axis of the world relative to the earth's axis? Relative to the plane of the celestial meridian?

3. At what points does the celestial equator intersect with the horizon line?

4. In what direction relative to the sides of the horizon does the Earth rotate around its axis?

5. At what points does the central meridian intersect with the horizon?

6. How does the horizon plane pass relative to the surface of the globe?

Level 3: 5 - 6 points.

1. Find the coordinates of the star map and name the objects that have coordinates:

1) a = 15 h 12 min, d = –9 o; 2) a = 3 h 40 min, d = +48 o.

1) a Big Dipper; 2) β Kita.

3. Express 9 hours 15 minutes 11 seconds in degrees.

4. Find on the star map and name the objects that have coordinates:

1) a = 19 h 29 min, d = +28 o; 2) a = 4 h 31 min, d = +16 o 30 / .

1) a Libra; 2) g of Orion.

6. Express 13 hours 20 minutes in degrees.

7. What constellation is the Moon in if its coordinates are a = 20 hours 30 minutes, d = -20 o?

8. Determine from the star map the constellation in which the galaxy Μ31 is located, if its coordinates are a = 0 h 40 min, d = +41 o.

4th level. 7 - 8 points

1. The faintest stars that can be photographed by the world's largest telescope are stars of the 24th magnitude. How many times weaker are they than 1st magnitude stars?

2. The brightness of a star varies from minimum to maximum by 3 magnitudes. How many times does its brilliance change?

3. find the brightness ratio of two stars if their apparent magnitudes are equal, respectively m 1 = 1.00 and m 2 = 12,00.

4. How many times the Sun looks brighter than Sirius if the magnitude of the Sun m 1 = -26.5 and m 2 = –1,5?

5. Calculate how many times the star a Canis Major is brighter than the star a Cygnus.

6. Calculate how many times the star Sirius is brighter than Vega.

3. Working with the map.

Determining the coordinates of celestial bodies.

Horizontal coordinates.

A- the azimuth of the luminary, is measured from the point of the South along the line of the mathematical horizon clockwise in the direction of west, north, east. It is measured from 0 o to 360 o or from 0 h to 24 h.

h- the height of the luminary, measured from the point of intersection of the circle of height with the line of the mathematical horizon, along the circle of height up to the zenith from 0 o to +90 o, and down to the nadir from 0 o to -90 o.

#"#">#"#">hours, minutes and seconds of time, but sometimes in degrees.

Equatorial coordinates take precedence over horizontal coordinates:

1) Created star charts and catalogs. The coordinates are constant.

2) Compilation of geographical and topological maps of the earth's surface.

3) Implementation of orientation on land, sea space.

4) Checking the time.

Exercises.

Horizontal coordinates.

1. Determine the coordinates of the main stars of the constellations included in the autumn triangle.

2. Find the coordinates of a Virgo, a Lyra, a Canis Major.

3. Determine the coordinates of your zodiac constellation, at what time is it most convenient to observe it?

equatorial coordinates.

1. Find on the star map and name the objects that have coordinates:

1) a = 15 h 12 m, d = –9 o; 2) a \u003d 3 h 40 m, d \u003d +48 o.

2. Determine the equatorial coordinates of the following stars from the star map:

1) a Big Dipper; 2) b China.

3. Express 9 h 15 m 11 s in degrees.

4. Find on the star map and name the objects that have coordinates

1) a = 19 h 29 m, d = +28 o; 2) a = 4 h 31 m, d = +16 o 30 / .

5. Determine the equatorial coordinates of the following stars from the star map:

1) a Libra; 2) g of Orion.

6. Express 13 hours 20 meters in degrees.

7. What constellation is the Moon in if its coordinates are a = 20 h 30 m, d = -20 o.

8. Determine the constellation in which the galaxy is located on the star map M 31 if its coordinates are a 0 h 40 m, d = 41 o.

4. The culmination of the luminaries.

Theorem about the height of the celestial pole.

Key questions: 1) astronomical methods for determining geographic latitude; 2) using a moving chart of the starry sky, determine the condition of visibility of the stars at any given date and time of day; 3) solving problems using relationships that connect the geographical latitude of the place of observation with the height of the luminary at the climax.

The culmination of the luminaries. Difference between upper and lower climax. Working with the map determining the time of culminations. Theorem about the height of the celestial pole. Practical ways to determine the latitude of the area.

Using the drawing of the projection of the celestial sphere, write down the height formulas in the upper and lower culmination of the luminaries if:

a) the star culminates between the zenith and the south point;

b) the star culminates between the zenith and the celestial pole.

Using the celestial pole height theorem:

- the height of the pole of the world (Polar Star) above the horizon is equal to the geographical latitude of the place of observation

Angle - as vertical, a. Knowing that is the declination of the star, then the height of the upper culmination will be determined by the expression:

For the bottom climax of a star M 1:

Give home the task to get a formula for determining the height of the upper and lower culmination of a star M 2 .

Assignment for independent work.

1. Describe the conditions for the visibility of stars at 54° north latitude.

2. Install a mobile star map for the day and hour of classes for the city of Bobruisk (j = 53 o).

Answer the following questions:

a) which constellations are above the horizon at the time of observation, which constellations are below the horizon.

b) which constellations are rising at the moment, setting at the moment.

3. Determine the geographical latitude of the observation site if:

a) the star Vega passes through the zenith point.

b) the star Sirius at its upper culmination at an altitude of 64° 13/ south of the zenith point.

c) the height of the star Deneb at its upper climax is 83 o 47 / north of the zenith.

d) the star Altair passes at the lower culmination through the zenith point.

On one's own:

Find the intervals of declination of stars that are at a given latitude (Bobruisk):

a) never rise b) never enter; c) can ascend and set.

Tasks for independent work.

1. What is the declination of the zenith point at the geographical latitude of Minsk (j = 53 o 54 /)? Accompany your answer with a picture.

3. Using the drawing, prove that in the case of the upper culmination of the luminary north of the zenith, it will have a height h\u003d 90 o + j - d.

4. The azimuth of the luminary is 315 o, the height is 30 o. In what part of the sky is this luminary visible? In the southeast

5. In Kyiv, at an altitude of 59 o, the upper climax of the star Arcturus was observed (d = 19 o 27 /). What is the geographical latitude of Kyiv?

6. What is the declination of the stars culminating in a place with geographic latitude j at the north point?

7. The polar star is 49/46 from the north celestial pole // . What is its declination?

8. Is it possible to see the star Sirius (d \u003d -16 about 39 /) at meteorological stations located on about. Dikson (j = 73 o 30 /) and in Verkhoyansk (j = 67 o 33 /)? [On about. Dixon is not present, not in Verkhoyansk]

10. Express the right ascension of the star Altair in arc meters.

11. The star is 20 o from the north celestial pole. Is it always above the horizon of Brest (j = 52 o 06 /)? [Always]

13. Azimuth of the star 45 o, height 45 o. In which side of the sky should you look for this luminary?

15. What condition must the declination of a luminary satisfy in order for this luminary not to set at a point with latitude j; so that it is not ascending?

16. The right ascension of the star Aldebaran (a-Taurus) is equal to 68 about 15 /. Express it in units of time.

17. Does the star Fomalhaut (a-Golden Fish) rise in Murmansk (j = 68 o 59 /), the declination of which is -29 o 53 / ? [Does not rise]

18. Prove from the drawing, from the lower culmination of the star, that h= d - (90 o - j).

Homework: § 3. q.v.

5. Measurement of time.

Definition of geographic longitude.

Key issues: 1) differences between the concepts of sidereal, solar, local, zone, seasonal and universal time; 2) the principles of determining time according to astronomical observations; 3) astronomical methods for determining the geographical longitude of the area.

At the beginning of the lesson, independent work is carried out for 20 minutes.

1. Using a moving map, determine 2 - 3 constellations visible at a latitude of 53 o in the Northern Hemisphere.

2. Determine the azimuth and height of the star at the time of the lesson:

1 option. a B. Ursa, a Lion.

Option 2. b Orion, a Eagle.

3. Using a star map, find the stars by their coordinates.

Main material.

Introduce the concept of sidereal time.

Pay attention to the following; moments:

- the duration of time counting units is associated with the conditions of visibility (culminations) of celestial bodies.

- the introduction of the atomic time standard in science was due to the uneven rotation of the Earth, discovered with increasing clock accuracy.

The introduction of standard time is due to the need to coordinate economic activities in the territory defined by the boundaries of time zones.

Universal time is defined as local mean solar time at the zero (Greenwich) meridian.

The calendar. Limit ourselves to considering the brief history of the calendar as part of culture. It is necessary to single out three main types of calendars (lunar, solar and lunisolar), tell what they are based on, and dwell in more detail on the Julian solar calendar of the old style and the Gregorian solar calendar of the new style. After recommending relevant literature, invite the students to prepare short reports about different calendars for the next lesson or organize a special conference on this topic.

When consolidating the studied material with students, the following tasks can be solved.

A task 1.

Determine the geographical longitude of the observation site if:

(a) At local noon, the traveler noted 14:13 GMT.

b) according to the exact time signals, 8:00 am 00 s, the geologist recorded 10:13:42 local time.

Taking into account the fact that

c) the navigator of the liner at 17:52:37 local time received the Greenwich time signal at 12:00:00.

Taking into account the fact that

1 h \u003d 15 o, 1 m \u003d 15 / and 1 s \u003d 15 //, we have.

d) the traveler noted 5:35 p.m. at local noon.

Taking into account the fact that 1 h \u003d 15 o and 1 m \u003d 15 /, we have.

A task 2.

The travelers noticed that according to local time the eclipse of the moon began at 15:15, while according to the astronomical calendar it should have taken place at 3:51 GMT. What is the longitude of their location.

A task 3.

On May 25 in Moscow (2nd time zone) the clock shows 10 h 45 m. What is the average, standard and summer time at this moment in Novosibirsk (6 time zone, l 2 = 5 h 31 m).

Knowing the Moscow summer time, we find universal time T o:

At this moment in Novosibirsk:

- average time.

- standard time.

- summer time.

Messages for students:

1. Arabic lunar calendar.

2. Turkish lunar calendar.

3. Persian solar calendar.

4. Coptic solar calendar.

5. Projects of ideal perpetual calendars.

6. Counting and keeping time.

6. Heliocentric system of Copernicus.

Key questions: 1) the essence of the heliocentric system of the world and the historical prerequisites for its creation; 2) the causes and nature of the apparent motion of the planets.

Frontal conversation.

1. A true solar day is the time interval between two successive climaxes of the same name of the center of the solar disk.

2. A sidereal day is the time interval between two successive culminations of the same name of the vernal equinox, equal to the period of the Earth's rotation.

3. The mean solar day is the time interval between two culminations of the same name of the mean equatorial Sun.

4. For observers located on the same meridian, the culmination of the Sun (as well as any other luminary) occurs simultaneously.

5. A solar day differs from a stellar day by 3 m 56 s.

6. The difference in the values of local time at two points on the earth's surface at the same physical moment is equal to the difference in the values of their geographical longitudes.

7. When crossing the border of two neighboring belts from west to east, the clock must be moved one hour ahead, and from east to west - one hour ago.

Consider an example solution tasks.

When crossing the date change line in the direction from east to west (as in our case), one calendar date is discarded from the account.

For the first time, Magellan and his companions encountered this during their trip around the world.

Main material.

Ptolemy Claudius (c. 90 - c. 160), ancient Greek scientist, the last major astronomer of antiquity. Complemented the star catalog of Hipparchus. He built special astronomical instruments: astrolabe, armillary sphere, triquetra. Described the position of 1022 stars. He developed a mathematical theory of the motion of planets around a stationary Earth (using the representation of the apparent motion of celestial bodies using combinations of circular motions - epicycles), which made it possible to calculate their position in the sky. Together with the theory of the motion of the Sun and Moon, it amounted to the so-called. Ptolemaic system of the world. Having achieved high accuracy for those times, the theory, however, did not explain the change in the brightness of Mars and other paradoxes of ancient astronomy. Ptolemy's system is set forth in his main work "Almagest" ("The Great Mathematical Construction of Astronomy in Books XIII") - an encyclopedia of astronomical knowledge of the ancients. The Almagest also contains information on rectilinear and spherical trigonometry, and for the first time a solution to a number of mathematical problems is given. In the field of optics, he studied the refraction and refraction of light. In the work "Geography" he gave a set of geographical information of the ancient world.

For one and a half thousand years, Ptolemy's theory was the main astronomical doctrine. Very accurate for its era, it eventually became a limiting factor in the development of science and was replaced by the heliocentric theory of Copernicus.

The correct understanding of the observed celestial phenomena and the place of the Earth in the solar system has evolved over the centuries. Nicolaus Copernicus finally broke the idea of the immobility of the Earth. Copernicus (Kopernik, Copernicus) Nicholas (1473 - 1543), the great Polish astronomer.

Creator of the heliocentric system of the world. He made a revolution in natural science, abandoning the doctrine of the central position of the Earth, accepted for many centuries. He explained the visible movements of the heavenly bodies by the rotation of the Earth around its axis and the revolution of the planets (including the Earth) around the Sun. He outlined his teaching in the essay “On the Rotations of the Celestial Spheres” (1543), which was banned by the Catholic Church from 1616 to 1828.

Copernicus showed that it was the rotation of the Earth around the Sun that could explain the apparent loop-like motions of the planets. The center of the planetary system is the Sun.

The axis of rotation of the Earth is deviated from the axis of the orbit by an angle equal to approximately 23.5°. Without this tilt, there would be no change of seasons. The regular change of seasons is a consequence of the movement of the Earth around the Sun and the inclination of the axis of rotation of the Earth to the plane of the orbit.

Since, during observations from the Earth, the movement of the planets around the Sun is also superimposed on the movement of the Earth in its orbit, the planets move across the sky from east to west (direct movement), then from west to east (reverse movement). The moment of change of direction is called standing. If you put this path on the map, you get a loop. The size of the loop is the smaller, the greater the distance between the planet and the Earth. The planets describe loops, and not just move back and forth in a single line, solely due to the fact that the planes of their orbits do not coincide with the plane of the ecliptic.

The planets are divided into two groups: the lower ( internal) - Mercury and Venus - and upper ( external) are the other six planets. The nature of the movement of the planet depends on which group it belongs to.

The greatest angular distance of a planet from the Sun is called elongation. The greatest elongation for Mercury is 28°, for Venus it is 48°. At eastern elongation, the inner planet is visible in the west, in the rays of the evening dawn, shortly after sunset. With western elongation, the inner planet is visible in the east, in the rays of dawn, shortly before sunrise. The outer planets can be at any angular distance from the Sun.

The phase angle of Mercury and Venus varies from 0° to 180°, so Mercury and Venus change phases in the same way as the Moon. Near inferior conjunction, both planets have the largest angular dimensions, but look like narrow crescents. At a phase angle j = 90 o, half of the disk of the planets is illuminated, phase Φ = 0.5. In superior conjunction, the lower planets are fully illuminated, but are poorly visible from the Earth, as they are behind the Sun.

planetary configurations.

Homework: § 3. q.v.

7. Configurations of the planets. Problem solving.

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

The student should be able to: 1) solve problems using a formula that connects the synodic and sidereal periods of the planets.

Theory. Specify the main configurations for the upper (lower) planets. Define synodic and sidereal periods.

Suppose at the initial moment of time the minute hand and the hour hand coincide. The time interval after which the hands meet again will not coincide with either the period of revolution of the minute hand (1 hour) or the period of revolution of the hour hand (12 hours). This period of time is called the synodic period - the time after which certain positions of the arrows are repeated.

The angular velocity of the minute hand, and the hour hand -. For the synodic period S the hour hand of the clock will pass the way

and minute

Subtracting the paths, we get, or

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

2. Consider an example:

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

According to the task T = S, where T is the sidereal period, the time it takes for a planet to revolve 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, making a similar substitution

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

Decision tasks.

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. Inferior conjunctions of Mercury are repeated 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 are repeated after 584 days.

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

8. Apparent motion of the Sun and Moon.

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 form of a sickle. What planet could it be? [Internal]	2. What planets and under what conditions can be visible all night (from sunset to sunrise)? [All Outer Planets in Opposition Ages]
3. By observation it has been established that between two successive identical configurations of the planet is 378 days. Assuming a circular orbit, find the sidereal (stellar) period of the planet's revolution.	3. The minor planet Ceres revolves around the Sun with a period of 4.6 years. After what period of time are the oppositions of this planet repeated?
4. Mercury is observed in the position of maximum elongation, equal to 28 o. Find the distance from Mercury to the Sun in astronomical units.	4. Venus is observed in the position of maximum elongation, equal to 48 o. Find the distance from Venus to the Sun in astronomical units.

Main material.

When forming the ecliptic and the zodiac, it is necessary to stipulate that the ecliptic is a projection of the plane of the earth's orbit onto the celestial sphere. Due to the rotation of the planets around the Sun in almost the same plane, their apparent movement on the celestial sphere will be along and near the ecliptic with a variable angular velocity and a periodic change in the direction of motion. The direction of the Sun's motion along the ecliptic is opposite to the daily motion of the stars, the angular velocity is about 1 o per day.

Days of solstice and equinox.

The movement of the Sun along the ecliptic is a reflection of the rotation of the Earth around the Sun. The ecliptic runs through 13 constellations: Pisces, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, Ophiuchus.

Ophiuchus is not considered a zodiacal constellation, although it lies on the ecliptic. The concept of the signs of the zodiac developed several thousand years ago, when the ecliptic did not pass through the constellation Ophiuchus. In ancient times, there were no exact boundaries and the signs corresponded to the constellations symbolically. Currently, the zodiac signs and constellations do not match. For example, the vernal equinox and the zodiac sign Aries are in the constellation Pisces.

For independent work.

Using a mobile map of the starry sky, establish under which constellation you were born, that is, in which constellation the Sun was at the time of your birth. To do this, connect the north pole of the world and the date of your birth with a line and see in which constellation this line crosses the ecliptic. Explain why the result differs from that indicated in the horoscope.

Explain the precession of the earth's axis. Precession is the slow cone-shaped rotation of the earth's axis with a period of 26 thousand years under the influence of gravitational forces from the Moon and the Sun. Precession changes the position of the celestial poles. About 2700 years ago, the star a Draconis was located near the north pole, called the Royal Star by Chinese astronomers. Calculations show that by the year 10,000 the North Pole of the world will approach the star a Cygnus, and in 13600 there will be a Lyra (Vega) in the place of the Polar Star. Thus, as a result of precession, the points of the spring and autumn equinoxes, the summer and winter solstices slowly move through the zodiac constellations. Astrology offers information that is outdated 2 thousand years ago.

The apparent movement of the Moon against the background of stars is due to the reflection of the actual movement of the Moon around the Earth, which is accompanied by a change in the appearance of our satellite. The visible edge of the moon's disk is called limbus . The line separating the sunlit and unlit parts of the moon's disk is called terminator . The ratio of the area of the illuminated part of the visible disk of the Moon to its entire area is called moon phase .

There are four main phases of the moon: new moon , first quarter , full moon and last quarter . In the new moon Φ = 0, in the first quarter Φ = 0.5, in the full moon the phase is Φ = 1.0, and in the last quarter again Φ = 0.5.

At the new moon, the Moon passes between the Sun and the Earth, the dark side of the Moon, not illuminated by the Sun, faces the Earth. True, sometimes at this time the disk of the Moon glows with a special, ashy light. The faint glow of the night part of the lunar disk is caused by sunlight reflected by the Earth towards the Moon. Two days after the new moon, in the evening sky, in the west, shortly after sunset, a thin crescent of the young moon appears.

Seven days after the new moon, the growing moon is visible in the form of a semicircle in the west or southwest, shortly after sunset. The Moon is 90° east of the Sun and is visible in the evenings and in the first half of the night.

The full moon occurs 14 days after the new moon. At the same time, the Moon is in opposition to the Sun, and the entire illuminated hemisphere of the Moon is facing the Earth. On a full moon, the moon is visible all night, the moon rises at sunset, and sets at sunrise.

A week after the full moon, the aging moon appears before us in the phase of its last quarter, in the form of a semicircle. At this time, half of the illuminated and half of the unlit hemisphere of the Moon is facing the Earth. The moon is visible in the east, before sunrise, in the second half of the night

The full moon repeats the daily path of the sun in the sky, which it passed six months before, so in summer the full moon does not move far from the horizon, and in winter, on the contrary, it rises high.

The Earth revolves around the Sun, so from one new moon to the next, the Moon revolves around the Earth not by 360 °, but somewhat more. Accordingly, the synodic month is 2.2 days longer than the sidereal month.

The time interval between two consecutive identical phases of the moon is called synodic month, its duration is 29.53 days. Sidereal same month, i.e. the time it takes the moon to make one revolution around the earth relative to the stars is 27.3 days.

Solar and lunar eclipses.

In ancient times, solar and lunar eclipses caused superstitious horror in people. It was believed that eclipses portend wars, famine, ruin, mass diseases.

The occultation of the sun by the moon is called solar eclipse . This is a very beautiful and rare occurrence. A solar eclipse occurs when the Moon crosses the plane of the ecliptic at the time of the new moon.

If the disk of the Sun is completely covered by the disk of the Moon, then the eclipse is called complete . At perigee, the Moon is closer to the Earth at 21,000 km from the average distance, at apogee - further at 21,000 km. This changes the angular dimensions of the moon. If the angular diameter of the Moon's disk (about 0.5 o) turns out to be slightly less than the angular diameter of the Sun's disk (about 0.5 o), then at the moment of the maximum phase of the eclipse from the Sun, a bright narrow ring remains visible. Such an eclipse is called annular . And, finally, the Sun may not be completely hidden behind the disk of the Moon due to the mismatch of their centers in the sky. Such an eclipse is called private . Such a beautiful formation as the solar corona can only be observed during total eclipses. Such observations, even in our time, can give a lot to science, so astronomers from many countries come to observe the country where there will be a solar eclipse.

A solar eclipse begins at sunrise in the western regions of the earth's surface and ends in the eastern regions at sunset. Usually a total solar eclipse lasts a few minutes (the longest total solar eclipse of 7 minutes 29 seconds will be on July 16, 2186).

The moon moves from west to east, so the solar eclipse starts from the western edge of the solar disk. The degree of coverage of the Sun by the Moon is called solar eclipse phase .

Solar eclipses can be seen only in those areas of the Earth, which passes the band of the moon's shadow. The diameter of the shadow does not exceed 270 km, so the total eclipse of the Sun is visible only on a small area of the earth's surface.

The plane of the lunar orbit at the intersection with the sky forms a large circle - the lunar path. The plane of the earth's orbit intersects with the celestial sphere along the ecliptic. The plane of the lunar orbit is inclined to the plane of the ecliptic at an angle of 5 o 09 / . Period of revolution of the Moon around the Earth (stellar or sidereal period) R) = 27.32166 Earth days or 27 days 7 hours 43 minutes.

The plane of the ecliptic and the lunar path intersect each other in a straight line called knot line . The points of intersection of the line of nodes with the ecliptic are called ascending and descending nodes of the lunar orbit . The lunar nodes continuously move towards the Moon, that is, to the west, making a complete revolution in 18.6 years. The ascending node's longitude decreases by about 20° each year.

Since the plane of the lunar orbit is inclined to the plane of the ecliptic at an angle of 5 o 09 /, the Moon during a new moon or full moon can be far from the ecliptic plane, and the disk of the Moon will pass above or below the disk of the Sun. In this case, the eclipse does not occur. For a solar or lunar eclipse to occur, it is necessary that the Moon during the new moon or full moon be near the ascending or descending node of its orbit, i.e. near the ecliptic.

In astronomy, many signs introduced in ancient times have been preserved. The symbol of the ascending node means the head of the dragon Rahu, which pounces on the Sun and, according to Indian legends, causes its eclipse.

During the full lunar eclipse The moon completely disappears into the shadow of the Earth. The total phase of a lunar eclipse lasts much longer than the total phase of a solar eclipse. The shape of the edge of the earth's shadow during lunar eclipses served the ancient Greek philosopher and scientist Aristotle as one of the strongest proofs of the sphericity of the Earth. Philosophers of ancient Greece calculated that the Earth is about three times the size of the Moon, simply based on the duration of eclipses (the exact value of this coefficient is 3.66).

The moon at the time of a total lunar eclipse is actually deprived of sunlight, so a total lunar eclipse is visible from anywhere in the hemisphere of the Earth. The eclipse starts and ends simultaneously for all geographic points. However, the local time of this phenomenon will be different. Since the Moon moves from west to east, the left edge of the Moon enters the Earth's shadow first.

An eclipse can be total or partial, depending on whether the Moon enters the Earth's shadow completely or passes near its edge. The closer to the lunar node a lunar eclipse occurs, the more phase . Finally, when the disk of the Moon is covered not by a shadow, but by partial shade, there are penumbral eclipses . They cannot be seen with the naked eye.

During an eclipse, the Moon hides in the shadow of the Earth and, it would seem, should disappear from sight every time, because. The earth is not transparent. However, the Earth's atmosphere scatters the sun's rays that fall on the eclipsing surface of the Moon "bypassing" the Earth. The reddish color of the disk is due to the fact that red and orange rays pass through the atmosphere best.

Each lunar eclipse is different in terms of the distribution of brightness and color in the earth's shadow. The color of an eclipsed moon is often estimated on a special scale proposed by the French astronomer André Danjon:

1. The eclipse is very dark, in the middle of the eclipse the Moon is almost or not visible at all.

2. The eclipse is dark, gray, the details of the Moon's surface are completely invisible.

3. The eclipse is dark red or reddish, a darker part is observed near the center of the shadow.

4. The eclipse is brick red, the shadow is surrounded by a grayish or yellowish border.

5. Copper-red eclipse, very bright, outer zone light, bluish.

If the plane of the moon's orbit coincided with the plane of the ecliptic, then lunar eclipses would repeat every month. But the angle between these planes is 5°, and the Moon only crosses the ecliptic twice a month at two points called nodes of the lunar orbit. Ancient astronomers knew about these nodes, calling them the Head and Tail of the Dragon (Rahu and Ketu). In order for a lunar eclipse to occur, the full moon must be near the node of its orbit.

Lunar eclipses occur several times a year.

The time it takes for the moon to return to its node is called dragon month , which is equal to 27.21 days. After such a time, the Moon crosses the ecliptic at a point shifted in relation to the previous crossing by 1.5 o to the west. The phases of the moon (synodic month) repeat on average every 29.53 days. The time interval of 346.62 days, during which the center of the solar disk passes through the same node of the lunar orbit, is called draconian year .

Eclipse return period - saros - will be equal to the time interval after which the beginnings of these three periods will coincide. Saros means "repetition" in ancient Egyptian. Long before our era, even in antiquity, it was established that saros lasts 18 years 11 days 7 hours. Saros includes: 242 draconian months or 223 synodic months or 19 draconian years. During each saros there are 70 to 85 eclipses; of these, there are usually about 43 solar and 28 lunar. There can be at most seven eclipses in a year - either five solar and two lunar, or four solar and three lunar. Minimum number eclipses in a year - two solar eclipses. Solar eclipses occur more often than lunar ones, but they are rarely observed in the same area, since these eclipses are visible only in a narrow band of the moon's shadow. At some specific point on the surface, a total solar eclipse is observed on average once every 200 - 300 years.

Homework: § 3. q.v.

9. Ecliptic. The apparent movement of the sun and moon.

Problem solving.

Key questions: 1) the daily movement of the Sun on different latitudes; 2) change in the apparent motion of the Sun during the year; 3) apparent movement and phases of the moon; 4) Solar and lunar eclipses. eclipse conditions.

The student should be able to: 1) use astronomical calendars, reference books, a moving map of the starry sky to determine the conditions for the occurrence of phenomena associated with the circulation of the Moon around the Earth and the apparent movement of the Sun.

1. How much does the Sun move along the ecliptic every day?

During the year, the Sun describes a circle of 360 o along the ecliptic, therefore

2. Why is a solar day 4 minutes longer than a sidereal day?

Because, rotating around its own axis, the Earth also moves in orbit around the Sun. The Earth must make a little more than one revolution around its axis, so that for the same point on the Earth the Sun is again observed on the celestial meridian.

A solar day is 3 min 56 s shorter than a stellar day.

3. Explain why the moon rises an average of 50 minutes later each day than the day before.

On a given day, at the time of sunrise, the Moon is in a particular constellation. After 24 hours, when the Earth makes one complete revolution around its axis, this constellation will rise again, but the Moon will move about 13 o east of the stars in this time, and its rise will therefore come 50 minutes later.

4. Why before spacecraft circled the moon and photographed it reverse side, people could see only one half of it?

The period of rotation of the Moon around its axis is equal to the period of its revolution around the Earth, so that it faces the Earth with the same side.

5. Why is the Moon not visible from the Earth on the New Moon?

The Moon at this time is on the same side of the Earth as the Sun, so the dark half of the lunar ball, unlit by the Sun, is facing us. In this position of the Earth, Moon and Sun, a solar eclipse can occur for the inhabitants of the Earth. It does not happen every new moon, since the Moon usually passes on a new moon above or below the disk of the Sun.

6. Describe how the position of the Sun in the celestial sphere has changed from the beginning of the school year to the day on which this lesson is held.

Using the star map, we find the position of the Sun on the ecliptic on September 1 and on the day of the lesson (for example, October 27). On September 1, the Sun was in the constellation Leo and had a declination of d = +10 o. Moving along the ecliptic, the Sun crossed the celestial equator on September 23 and moved into the southern hemisphere, on October 27 it is in the constellation Libra and has a declination d = -13 o. That is, by October 27, the Sun moves across the celestial sphere, rising less and less above the horizon.

7. Why are eclipses not observed every month?

Since the plane of the lunar orbit is inclined to the plane of the earth's orbit, then, for example, in the new moon, the Moon does not appear on the line connecting the centers of the Sun and the Earth, and therefore the lunar shadow will pass by the Earth and there will be no solar eclipse. For a similar reason, the Moon does not pass through the cone of the Earth's shadow every full moon.

8. How many times faster does the Moon move across the sky faster than the Sun?

The sun and moon move across the sky in the opposite direction to the daily rotation of the sky. During the day, the Sun passes approximately 1 o, and the Moon - 13 o. Therefore, the Moon moves through the sky 13 times faster than the Sun.

9. How does the morning crescent of the Moon differ in shape from the evening crescent?

The morning crescent of the Moon has a bulge to the left (resembles the letter C). The Moon is located at a distance of 20 - 50 o to the west (to the right) from the Sun. The evening crescent of the Moon has a bulge to the right. The moon is located at a distance of 20 - 50 about east (to the left) of the sun.

Level 1: 1 - 2 points.

1. What is called the ecliptic? Point out the correct statements.

A. The axis of apparent rotation of the celestial sphere, connecting both poles of the world.

B. Angular distance of the luminary from the celestial equator.

B. An imaginary line along which the Sun makes its apparent annual movement against the background of the constellations.

2. Indicate which of the following constellations are zodiacal.

A. Aquarius. B. Sagittarius. B. Hare.

3. Indicate which of the following constellations are not zodiacal.

A. Taurus. B. Ophiuchus. B. Cancer.

4. What is called a sidereal (or sidereal) month? Specify the correct statement.

A. The period of revolution of the Moon around the Earth relative to the stars.

B. The time interval between two total lunar eclipses.

C. The time interval between the new moon and the full moon.

5. What is called a synodic month? Specify the correct statement.

A. Time span between full moon and new moon. B. The time interval between two successive identical phases of the moon.

B. Time of rotation of the Moon around its axis.

6. Specify the duration of the synodic month of the Moon.

A. 27.3 days. B. 30 days. B. 29.5 days.

Level 2: 3 - 4 points

1. Why is the position of the planets not indicated on the star maps?

2. In what direction is the apparent annual movement of the Sun relative to the stars?

3. In what direction is the apparent movement of the Moon relative to the stars?

4. Which total eclipse (solar or lunar) is longer? Why?

6. As a result of which the position of the points of sunrise and sunset changes during the year?

Level 3: 5 - 6 points.

1. a) What is the ecliptic? What constellations are on it?

b) Draw what the moon looks like in the last quarter. At what time of the day is it visible in this phase?

2. a) What determines the annual apparent motion of the Sun along the ecliptic?

b) Draw what the moon looks like between the new moon and the first quarter.

3. a) Find on the star map the constellation in which the Sun is located today.

b) Why are total lunar eclipses observed in the same place on Earth many times more often than total solar eclipses?

4. a) Is it possible to consider the annual movement of the Sun along the ecliptic as proof of the revolution of the Earth around the Sun?

b) Draw what the moon looks like in the first quarter. At what time of the day is it visible in this phase?

5. (a) What is the cause of the visible light of the moon?

b) Draw what the moon looks like in the second quarter. What time of day does she look in this phase?

6. (a) How does the noon height of the Sun change during the year?

b) Draw what the moon looks like between the full moon and the last quarter.

4th level. 7 - 8 points

1. a) How many times during the year can you see all the phases of the moon?

b) The noon altitude of the Sun is 30° and its declination is 19°. Determine the geographic latitude of the observation site.

2. a) Why do we see only one side of the moon from Earth?

b) At what altitude in Kyiv (j = 50 o) does the upper climax of the star Antares occur (d = -26 o)? Make an appropriate drawing.

3. a) There was a lunar eclipse yesterday. When can we expect the next solar eclipse?

b) The Star of the World with a declination of -3 o 12 / was observed in Vinnitsa at an altitude of 37 o 35 / of the southern sky. Determine the geographical latitude of Vinnitsa.

4. a) Why does the total phase of a lunar eclipse last much longer than the total phase of a solar eclipse?

b) What is the noon height of the Sun on March 21 at a point whose geographical height is 52 o?

5. a) What is the minimum time interval between solar and lunar eclipses?

b) At what geographical latitude will the Sun culminate at noon at a height of 45 o above the horizon, if on that day its declination is -10 o?

6. a) The moon is visible in the last quarter. Could there be a lunar eclipse next week? Explain the answer.

b) What is the geographical latitude of the place of observation, if on June 22 the Sun was observed at noon at an altitude of 61 o?

10. Kepler's laws.

Key questions: 1) subject, tasks, methods and tools of celestial mechanics; 2) formulations of Kepler's laws.

The student should be able to: 1) solve problems using Kepler's Laws.

At the beginning of the lesson, independent work is carried out (20 minutes).

Option 1	Option 2
1. Write down the equatorial coordinates of the Sun at the equinoxes.	1. Write down the values of the equatorial coordinates of the Sun on the days of the solstices
2. On a circle representing the horizon line, mark the points of north, south, sunrise and sunset on the day the work is done. Use the arrows to indicate the direction of the displacement of these points in the coming days.	2. On the celestial sphere, depict the course of the Sun on the day the work is done. Use the arrow to indicate the direction of the Sun's displacement in the coming days.
3. What is the maximum height the Sun rises to on the day of the vernal equinox at the North Pole of the earth? Picture.	3. What is the maximum height the Sun rises to on the day of the vernal equinox at the equator? Picture
4. Is the Moon east or west of the Sun from new moon to full moon? [east]	4. Is the Moon east or west of the Sun from full moon to new moon? [west]

Theory.

Kepler's first law .

Each planet moves in an ellipse with the Sun at one of its foci.

Kepler's second law (law of equal areas ) .

The radius vector of the planet describes equal areas in equal time intervals. Another formulation of this law: the sectoral speed of the planet is constant.

Kepler's third law .

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

The modern formulation of the first law is supplemented as follows: in unperturbed motion, the orbit of a moving body is a curve of the second order - an ellipse, parabola or hyperbola.

Unlike the first two, Kepler's third law only applies to elliptical orbits.

The speed of the planet at perihelion

where v c is the average or circular speed of the planet at r = a. Speed at aphelion

Kepler discovered his laws empirically. Newton derived Kepler's laws from the law of universal gravitation. To determine the masses of celestial bodies, Newton's generalization of Kepler's third law to any system of circulating bodies is of great importance.

In a generalized form, this law is usually formulated as follows: the squares of the periods T1 and T2 of the revolution of two bodies around the Sun, multiplied by the sum of the masses of each body (respectively M 1 and M 2) and the Sun ( M), are related as cubes of semi-major axes a 1 and a 2 of their orbits:

In this case, the interaction between bodies M 1 and M 2 is not taken into account. If we consider the motion of the planets around the Sun, in this case, and, then we get the formulation of the third law given by Kepler himself:

Kepler's third law can also be expressed as a relationship between the period T orbiting a body with mass M and the major semiaxis of the orbit a (G is the gravitational constant):

Here it is necessary to make the following remark. For simplicity, it is often said that one body revolves around another, but this is true only for the case when the mass of the first body is negligible compared to the mass of the second (attracting center). If the masses are comparable, then the influence of a less massive body on a more massive one should also be taken into account. In a coordinate system with the origin at the center of mass, the orbits of both bodies will be conic sections lying in the same plane and with foci at the center of mass, with the same eccentricity. The difference will be only in the linear dimensions of the orbits (if the bodies have different masses). At any moment in time, the center of mass will lie on a straight line connecting the centers of the bodies, and the distances to the center of mass r 1 and r 2 bodies mass M 1 and M 2 respectively are related by the following relationship:

The pericenters and apocenters of their orbits (if the motion is finite) of the body will also pass simultaneously.

Kepler's third law can be used to determine the mass of binary stars.

Example.

- What would be the semi-major axis of the planet's orbit if the synodic period of its revolution was equal to one year?

From the equations of synodic motion we find the sidereal period of the planet's revolution. Two cases are possible:

The second case is not implemented. For determining " a»we use Kepler's 3rd law.

There is no such planet in the solar system.

An ellipse is defined as the locus of points for which the sum of the distances from two given points (foci F 1 and F 2) there is a constant value and equal to the length of the major axis:

r 1 + r 2 = |AA / | = 2a.

The degree of elongation of the ellipse is characterized by its eccentricity e. Eccentricity

e = OF/OA.

When the focus coincides with the center e= 0, and the ellipse turns into circle .

Major axis a is the average distance from the focus (the planet from the Sun):

a = (AF 1 + F 1 A /)/2.

Homework: § 6, 7. c.

Level 1: 1 - 2 points.

1. Indicate which of the planets listed below are internal.

A. Venus. B. Mercury. W. Mars.

2. Indicate which of the planets listed below are outer.

A. Earth. B. Jupiter. V. Uranus.

3. In what orbits do the planets move around the Sun? Specify the correct answer.

A. In circles. B. By ellipses. B. By parabolas.

4. How do the periods of revolution of the planets change with the removal of the planet from the Sun?

B. The period of revolution of a planet does not depend on its distance from the Sun.

5. Indicate which of the planets listed below can be in superior conjunction.

A. Venus. B. Mars. B. Pluto.

6. Indicate which of the planets listed below can be observed at opposition.

A. Mercury. B. Jupiter. B. Saturn.

Level 2: 3 - 4 points

1. Can Mercury be seen in the evenings in the east?

2. The planet is visible at a distance of 120 ° from the Sun. Is this planet outer or inner?

3. Why are conjunctions not considered convenient configurations for observing the inner and outer planets?

4. During what configurations are the outer planets clearly visible?

5. During what configurations are the inner planets clearly visible?

6. In what configuration can both inner and outer planets be?

Level 3: 5 - 6 points.

1. a) Which planets cannot be in superior conjunction?

6) What is the sidereal period of Jupiter's revolution if its synodic period is 400 days?

2. a) What planets can be observed at opposition? Which ones can't?

b) How often do oppositions of Mars, whose synodic period is 1.9 years, repeat?

3. a) In what configuration and why is it most convenient to observe Mars?

b) Determine the sidereal period of Mars, knowing that its synodic period is 780 days.

4. (a) Which planets cannot be in inferior conjunction?

b) After what period of time do the moments of the maximum distance of Venus from the Earth repeat if its sidereal period is 225 days?

5. a) What planets can be seen next to the Moon during a full moon?

b) What is the sidereal period of the revolution of Venus around the Sun, if its upper conjunctions with the Sun are repeated after 1.6 years?

6. a) Is it possible to observe Venus in the morning in the west, and in the evening in the east? Explain the answer.

b) What will be the sidereal period of the outer planet's revolution around the Sun if its oppositions are repeated in 1.5 years?

4th level. 7 - 8 points

1. a) How does the value of the planet's velocity change as it moves from aphelion to perihelion?

b) The semi-major axis of the orbit of Mars is 1.5 AU. e. What is the sidereal period of its revolution around the Sun?

2. a) At what point of the elliptical orbit is the potential energy of an artificial satellite of the Earth minimal and at what point is it maximal?

6) At what average distance from the Sun does the planet Mercury move if its period of revolution around the Sun is 0.241 Earth years?

3. a) At what point of the elliptical orbit is the kinetic energy of an artificial satellite of the Earth minimal and at what point is it maximal?

b) Jupiter's sidereal period around the Sun is 12 years. What is the average distance of Jupiter from the Sun?

4. a) What is the orbit of a planet? What shape are the orbits of the planets? Can planets collide as they move around the sun?

b) Determine the length of the Martian year if Mars is 228 million km away from the Sun on average.

5. a) At what time of the year is the linear velocity of the Earth around the Sun the greatest (smallest) and why?

b) What is the semi-major axis of the orbit of Uranus if the sidereal period of the revolution of this planet around the Sun is

6. a) How do the kinetic, potential and total mechanical energy of the planet change as it moves around the Sun?

b) The period of revolution of Venus around the Sun is 0.615 Earth year. Determine the distance from Venus to the Sun.

Visible movement of the stars .

1. What conclusions of Ptolemy's theory turned out to be correct?

The spatial arrangement of celestial bodies, the recognition of their movement, the circulation of the Moon around the Earth, the possibility of mathematical calculation of the apparent positions of the planets.

2. What disadvantages did the heliocentric system of the world of N. Copernicus have?

The world is limited by the sphere of fixed stars, the uniform motion of the planets is preserved, the epicycles are preserved, the insufficient accuracy of predicting the positions of the planets.

3. The absence of what obvious observational fact was used as proof of the incorrectness of the theory of N. Copernicus?

Not detecting the parallactic motion of stars due to its smallness and observational errors.

4. To determine the position of the body in space, three coordinates are needed. In astronomical catalogs, most often only two coordinates are given: right ascension and declination. Why?

The third coordinate in the spherical coordinate system is the modulus of the radius vector - the distance to the object r. This coordinate is determined from more complex observations than a and d. In the catalogs, its equivalent is the annual parallax, hence (pc). For the problems of spherical astronomy, it is sufficient to know only two coordinates a and d or alternative pairs of coordinates: ecliptic - l, b or galactic - l, b.

5. What important circles of the celestial sphere do not have corresponding circles on the globe?

The ecliptic, the first vertical, the colors of the equinoxes and solstices.

6. Where on Earth can any circle of declinations coincide with the horizon?

At the equator.

7. What circles (small or large) of the celestial sphere correspond to the vertical and horizontal threads of the field of view of the goniometric instrument?

Only the great circles of the celestial sphere are projected as straight lines.

8. Where on Earth is the position of the celestial meridian uncertain?

At the poles of the earth.

9. What are the zenith azimuth, hour angle and right ascension of the celestial poles?

Values A, t, a in these cases are undefined.

10. At what points on the Earth does the North Pole of the world coincide with the zenith? with a north point? with nadir?

At the north pole of the earth, at the equator, at south pole Earth.

11. An artificial satellite crosses the horizontal thread of the goniometer at a distance d o to the right of the center of the field of view, the coordinates of which A= 0 o , z = 0o. Determine the horizontal coordinates of the artificial satellite at this point in time. How will the object coordinates change if the tool azimuth is changed to 180 o ?

1) A= 90o, z = d o ; 2) A= 270o, z = d o

12. At what latitude of the Earth can you see:

a) all the stars of the celestial hemisphere at any moment of the night;

b) stars of only one hemisphere (northern or southern);

c) all the stars of the celestial sphere?

a) At any latitude at any moment half of the celestial sphere is visible;

b) at the poles of the Earth, the northern and southern hemispheres are visible, respectively;

c) at the equator of the Earth for a period of less than a year you can see all the stars of the celestial sphere.

13. At what latitudes does the daily parallel of a star coincide with its almucantarat?

At latitudes.

14. Where on the globe do all the stars rise and set perpendicular to the horizon?

At the equator.

15. Where on the globe do all the stars move parallel to the mathematical horizon during the year?

At the poles of the earth.

16. When do the stars at all latitudes move parallel to the horizon during the daily motion?

At the top and bottom climaxes.

17. Where on Earth is the azimuth of some stars never equal to zero, and the azimuth of other stars is never equal to 180 o?

At the earth's equator for stars c, and for stars c.

18. Can the azimuths of a star be the same at the upper and lower culminations? What is it equal to in this case?

In the northern hemisphere, for all declination stars, the azimuths at the upper and lower culminations are the same and equal to 180 o.

19. In what two cases does the height of a star above the horizon not change during the day?

The observer is at one of the Earth's poles, or the star is at one of the world's poles.

20. In what part of the sky do the azimuths of the luminaries change the fastest and in what part the slowest?

The fastest in the meridian, the slowest in the first vertical.

21. Under what conditions does the azimuth of a star not change from its rising to its upper culmination, or, similarly, from its upper culmination to its setting?

For an observer located at the earth's equator and observing a star with declination d = 0.

22. The star is above the horizon for half a day. What is her inclination?

For all latitudes, this is a star with d = 0; at the equator, any star.

23. Can a luminary pass through the points of east, zenith, west and nadir in a day?

Such a phenomenon occurs at the Earth's equator with stars located at the celestial equator.

24. Two stars have the same right ascension. At what latitude do both stars rise and set at the same time?

At the Earth's equator.

25. When does the daily parallel of the Sun coincide with the celestial equator?

On the days of the equinoxes.

26. At what latitude and when does the daily parallel of the Sun coincide with the first vertical?

On the days of the equinoxes at the equator.

27. In what circles of the celestial sphere, large or small, does the Sun move in daily motion on the days of equinoxes and days of solstices?

On the days of the equinoxes, the daily parallel of the Sun coincides with the celestial equator, which is a great circle of the celestial sphere. On the days of the solstices, the daily parallel of the Sun is a small circle, 23 o .5 from the celestial equator.

28. The sun has set at the point of the west. Where did it rise on this day? What dates of the year does this happen?

If we neglect the change in the declination of the Sun during the day, then its rising was at the point of the east. This happens every year on the equinoxes.

29. When does the boundary between the illuminated and unilluminated hemispheres of the Earth coincide with the earth's meridians?

The terminator coincides with the earth's meridians on the days of the equinoxes.

30. It is known that the height of the Sun above the horizon depends on the movement of the observer along the meridian. What interpretation of this phenomenon was given by the ancient Greek astronomer Anaxagoras, based on the concept of a flat Earth?

The apparent movement of the Sun above the horizon was interpreted as a parallactic displacement, and therefore was used to try to determine the distance to the star.

31. How should two places be located on Earth so that on any day of the year, at any hour, the Sun, at least in one of them, is above the horizon or on the horizon? What are the coordinates (l, j) of such a second point for the city of Ryazan? Ryazan coordinates: l = 2 h 39m j = 54 o 38 / .

The desired place is located on the diametrically opposite point of the globe. For Ryazan, this point is in the South Pacific Ocean and has the coordinates of western longitude and j = –54 o 38 / .

32. Why does the ecliptic turn out to be a great circle of the celestial sphere?

The sun is in the plane of the earth's orbit.

33. How many times and when during the year does the Sun pass through the zenith for observers located at the equator and in the tropics of the Earth?

Twice a year during the equinoxes; once a year on the solstices.

34. At what latitudes is twilight the shortest? the longest?

At the equator, twilight is shortest, as the Sun rises and falls perpendicular to the horizon. In circumpolar regions, twilight is the longest, as the Sun moves almost parallel to the horizon.

35. What time does the sundial show?

True solar time.

36. Is it possible to design a sundial that would show the average solar time, maternity, summer, etc.?

Yes, but only for a specific date. For different types time should have its own dials.

37. Why is solar time used in everyday life and not sidereal time?

The rhythm of human life is connected with the Sun, and the beginning of the sidereal day falls on different hours of the solar day.

38. If the Earth did not rotate, what astronomical units of time would be preserved?

The sidereal year and the synodic month would have been preserved. Using them, it would be possible to introduce smaller units of time, as well as build a calendar.

39. When are the longest and shortest true solar days in a year?

The longest true solar day occurs on the days of the solstices, when the rate of change in the right ascension of the Sun due to its movement along the ecliptic is greatest, and in December the day is longer than in June, since the Earth is at perihelion at this time.

The shortest day is obviously on the equinoxes. In September, the day is shorter than in March, because at this time the Earth is closer to aphelion.

40. Why will the longitude of the day on May 1 in Ryazan be greater than at a point with the same geographical latitude, but located in the Far East?

During this period of the year, the declination of the Sun increases daily, and due to the difference in the moments of the onset of the beginning of the day of the same date for the western and eastern regions of Russia, the longitude of the day in Ryazan on May 1 will be greater than in more eastern regions.

41. Why are there so many types of solar time?

The main reason is communication. public life with daylight. The dissimilarity of the true solar day leads to the appearance of the mean solar time. The dependence of the mean solar time on the longitude of the place led to the invention of standard time. The need to save electricity has led to maternity and summer time.

42. How would the duration of the solar day change if the Earth began to rotate in the opposite direction to the actual one?

A solar day would be shorter than a sidereal day by four minutes.

43. Why is the afternoon longer than the first half of the day in January?

This is due to a noticeable increase in the Sun's declination during the day. The sun in the afternoon describes a greater arc in the sky than before noon.

44. Why is the continuous polar day greater than the continuous polar night?

Due to refraction. The sun rises earlier and sets later. In addition, in the northern hemisphere, the Earth passes aphelion in summer and therefore moves more slowly than in winter.

45. Why is the day always longer than the night by 7 minutes at the earth's equator?

Due to refraction and the presence of a disk near the Sun, the day is longer than the night.

46. Why is the time interval from the spring equinox to the autumn equinox longer than the time interval between the autumn equinox and the spring one?

This phenomenon is a consequence of the ellipticity of the earth's orbit. During the summer, the Earth is at aphelion and its orbital velocity is less than during the winter months, when the Earth is at perihelion.

47. The difference of longitudes of two places is equal to the difference of which times - solar or sidereal?

It doesn't matter. .

48. How many dates can be on Earth at the same time?

Tutoring

Need help learning a topic?

Our experts will advise or provide tutoring services on topics of interest to you.
Submit an application indicating the topic right now to find out about the possibility of obtaining a consultation.

What condition must the star's declination satisfy? A guide for teachers of astronomy. Practical work with a moving map of the starry sky

A- the azimuth of the luminary, is measured from the point of the South along the line of the mathematical horizon clockwise in the direction of west, north, east. It is measured from 0 o to 360 o or from 0 h to 24 h.

h- the height of the luminary, measured from the point of intersection of the circle of height with the line of the mathematical horizon, along the circle of height up to the zenith from 0 o to +90 o, and down to the nadir from 0 o to -90 o.

http://www.college.ru/astronomy/course/shell/images/Fwd_h.gifhttp://www.college.ru/astronomy/course/shell/images/Bwd_h.gif Equatorial coordinates

- clarification - ideally, the work is done in the computer training program IISS "Planetarium"

Practical work with a moving map starry sky.

2. Daily movement of luminaries at different latitudes

Exercise 2

Task 2

3. The height of the luminaries at the climax

Problem solution example

Exercise 3

Horizontal coordinates.

A- the azimuth of the luminary, is measured from the point of the South along the line of the mathematical horizon clockwise in the direction of west, north, east. It is measured from 0 o to 360 o or from 0 h to 24 h.

h- the height of the luminary, measured from the point of intersection of the circle of height with the line of the mathematical horizon, along the circle of height up to the zenith from 0 o to +90 o, and down to the nadir from 0 o to -90 o.

#"#">#"#">hours, minutes and seconds of time, but sometimes in degrees.

Tutoring

Practical work with a moving map
starry sky.