What condition must the declination of a star satisfy in order to. To help the teacher of astronomy (for physical and mathematical schools). Problem solution example
The use of astronomical means is possible only by heavenly bodies above the horizon. Therefore, the navigator must be able to determine which luminaries in a given flight will be non-setting, non-ascending, ascending and setting. For this, there are rules that allow you to determine what a given luminary is at the latitude of the observer's place.
On fig. 1.22 shows the celestial sphere for an observer located at a certain latitude. The straight line SU represents the true horizon, and the straight lines and MJ are the daily parallels of the luminaries. It can be seen from the figure that all the luminaries are divided into non-setting, non-ascending, ascending and setting.
The luminaries whose daily parallels lie above the horizon are non-setting for a given latitude, and the luminaries whose daily parallels are below the horizon are non-ascending.
Non-setting will be such luminaries, the daily parallels of which are located between the parallel of the NC and the North Pole of the World. A luminary moving along the daily parallel of the SC has a declination equal to the arc QC of the celestial meridian. The arc QC is equal to the complement geographical latitude observer position up to 90°.
Rice. 1. 22. Conditions for the rising and setting of the luminaries
Consequently, in the Northern Hemisphere, non-setting luminaries will be those luminaries whose declination is equal to or greater than the addition of the latitude of the observer's place to 90 °, i.e. . For the Southern Hemisphere, these luminaries will be non-rising.
Non-ascending luminaries in the Northern Hemisphere will be those luminaries whose diurnal parallels lie between the MU parallel and the South Pole of the World. Obviously, non-rising luminaries in the Northern Hemisphere will be those luminaries whose declination is equal to or less than the negative difference, i.e. . For the Southern Hemisphere, these luminaries will be non-setting. All other luminaries will be ascending and setting. In order for the luminary to rise and set, its declination must be less than 90° minus the latitude of the observer's place in absolute value, i.e. .
Example 1. Star Alioth: star declination latitude of the observer's place Determine which star is at the specified latitude according to the conditions of sunrise and sunset.
Solution 1. Find the difference
2. Compare the declination of the star with the resulting difference. Since the declination of the star is greater than that, the star Aliot at the indicated latitude is not set.
Example 2. Star Sirius; declination of the star latitude of the place of the observer Determine which star is at the specified latitude according to the conditions of sunrise and sunset.
Solution 1. Find the negative difference since the star
Sirius has a negative declination
2. Compare the declination of the star with the resulting difference. Since the star Sirius at the indicated latitude is not ascending.
Example 3. Star Arcturus: declination of the star latitude of the observer's place Determine which star is at the specified latitude according to the conditions of sunrise and sunset.
Solution 1. Find the difference
2. Compare the declination of the star with the resulting difference. Since the star Arcturus rises and sets at the indicated latitude.
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).
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) 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 Virgo, Lyra, 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) \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) 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 ( = 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 /)? [Is 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 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.
|
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 |
|
southern 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.
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.
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 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.
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.
