What condition must the declination satisfy? A manual for astronomy teachers. Tasks for independent work
The branch of astronomy whose main task is to study the geometric, kinematic and dynamic properties of celestial bodies is called...
A) Astrometry
B) Astrophysics
B) Fundamentals of celestial mechanics
D) Cosmology
D) Cosmogony
Science at the intersection of astronomy and physics, studying physical processes in astronomical objects such as stars and galaxies is called...
A) Astrometry
B) Astrophysics
B) Fundamentals of celestial mechanics
D) Cosmology
D) Cosmogony
A) north point
D) east point
D) there is no correct answer
A) noon line.
B) true horizon
B) right ascension
D) declination
D) there is no correct answer
The angle between the planes of great circles, one of which passes through the poles of the world and a given luminary, and the other through the poles of the world and the point of the vernal equinox, is called ...
A) right ascension.
B) stellar magnitude.
B) declination.
D) ascent
D) there is no correct answer
What is the declination of the Sun on the equinoxes?
The third planet from the Sun is...
A) Saturn.
B) Venus.
D) Jupiter
What orbits do the planets take around the Sun?
A) in circles
B) by ellipses. close to circles.
B) along the branches of parabolas
D) by hyperbole
D) there is no correct answer
The point of a planet's orbit closest to the Sun is called...
A) perihelion.
B) aphelion
B) eccentricity
A telescope is needed to...
A) collect light and create an image of the source
B) collect light from a celestial object and increase the angle of view. under which the object is visible.
C) obtain an enlarged image of a celestial body
D) get sunlight
D) there is no correct answer
All giant planets are characterized by...
A) fast rotation.
B) slow rotation
B) ultra-fast rotation
D) reverse rotation
D) there is no correct answer
Asteroids rotate between orbits...
A) Venus and Earth
B) Mars. and Jupiter.
B) Neptune and Pluto
D) Only Mars
D) Only Jupiter
What substances predominate in the atmospheres of stars?
A) helium and oxygen
B) nitrogen and helium
B) hydrogen. and helium.
D) oxygen and nitrogen
D) only hydrogen
What class of stars does the Sun belong to?
A) supergiant
B) yellow dwarf.
B) white dwarf
D) red giant
D) dwarf
How many constellations is the sky divided into?
D) there is no correct answer
Who discovered the laws of planetary motion around the Sun?
A) Ptolemy
B) Copernicus
B) Kepler.
D) Newton
Which layer of the Sun is the main source of visible radiation?
A) Chromosphere
B) Photosphere.
B) Solar corona
D) Atmosphere
D) Troposphere
Express 9 h 15 m 11 s in degrees.
B) 1380.47.45
D) 90̊ 00ʹ 01ʹʹ
D) there is no correct answer
Altair's parallax is 0.20. What is the distance to this star in light years?
A) 20 St. years
B) 0.652 St. of the year
B) 16.3 light years
D) 1400 St. years
D) there is no correct answer
How many times is a star of magnitude 3.4 fainter than Sirius, which has an apparent magnitude of 1.6?
A) 1.8 times
B) 0.2 times
B) 100. times.
D) 10 times
D) there is no correct answer
The branch of astronomy that studies the properties and evolution of the Universe as a whole is called...
A) Astrometry
B) Astrophysics
B) Fundamentals of celestial mechanics
D) Cosmology
D) Cosmogony
the science that studies the origin and development of cosmic bodies and their systems: stars and star clusters, galaxies, nebulae, the solar system including the Sun, planets with satellites, asteroids, comets, meteorites is called ...
A) Astrometry
B) Astrophysics
B) Fundamentals of celestial mechanics
D) Cosmology
D) Cosmogony
The branch of astronomy that studies stars is called...
A) Astrometry
B) Astrophysics
B) stellar astronomy
D) Cosmology
D) Cosmogony
What are the names of the 12 zodiac constellations through which the annual path of the Sun passes:
a) milky way;
b) ecliptic;
c) right ascension;
d) Universe.
D) zodiac signs
All planets have satellites except...
A) Saturn B) Venus C) Earth D) Mars
D) Jupiter
The diameter of the Sun is greater than the diameter of the Earth in
A) 109 times B) 218 times C) 312 times D) 100 times E) 1000 times
The annual parallax serves to:
A) determining the distance to the nearest stars;
B) determining the distance to the planets;
C) the distances the Earth travels per year;
D) evidence of the finiteness of the speed of light;
D) there is no correct answer
While watching the starry sky for an hour at night, you noticed that the stars were moving across the sky. This happens because:
A) The Earth moves around the Sun
B) The sun moves along the ecliptic
B) The earth rotates around its axis
D) stars move around the Earth
D) there is no correct answer
The cube of the semimajor axis of the orbit of a body, divided by the square of its period of revolution and the sum of the masses of the bodies, is a constant value. What is Kepler's law?
a) Kepler's first law;
b) Kepler's second law;
c) Kepler's third law;
d) Kepler's fourth law.
D) there is no correct answer
The distance from the Earth to the Sun is called:
a) light year
b) parsec
c) astronomical unit
d) annual parallax
d) there is no correct answer
Name the main reasons for the change of seasons:
A) change in the distance to the Sun due to the movement of the Earth in an elliptical orbit;
B) the inclination of the earth's axis to the plane of the earth's orbit;
B) rotation of the Earth around its axis;
D) temperature changes
D) there is no correct answer
The ratio of the cubes of the semimajor axes of the planets is 64. What is the ratio of their periods of revolution around the Sun?
A) 8 B) 4 C) 16 D) 2 E) 10
When is the Earth closest to the Sun due to its annual orbital motion?
A) in summer B) at perihelion C) in winter D) at aphelion e) in spring
Terrestrial planets include:
A) Venus; B) Jupiter; B) Saturn; D) Neptune. D) Uranus
The third refined Law of I. Kepler is used mainly to determine in stars:
A) distance B) period C) mass D) radius E) all of the above
The period of time between two new moons is called:
A) synodic month
B) sidereal month
B) full lunar month
D) calendar month
D) there is no correct answer
It is known that the orbit of any planet is an ellipse, at one of the foci of which the Sun is located. The point of the orbit closest to the Sun is called:
A) apogee B) perigee C) apohelium D) perihelion E) no correct answer
The reference frame associated with the Sun, proposed by Nicolaus Copernicus, is called:
a) geocentric;.
b) heliocentric;
c) centric;
d) copernic.
D) there is no correct answer
The highest point in the celestial sphere is called...
A) north point B) zenith
B) nadir. D) east point D) south point
Age of the Sun:
A) 2 billion years
B) 5 billion. years
B) 500 million years
D) 100 million years
D) 10 billion years
The line of intersection of the plane of the celestial horizon and the meridian is called...
A) noon line.
B) true horizon.
B) right ascension.
D) ascent
D) there is no correct option
Find the arrangement of the giant planets in order of distance from the Sun:
A) Uranus, Saturn, Jupiter, Neptune
B) Neptune, Saturn, Jupiter, Uranus
B) Jupiter, Saturn, Uranus, Neptune
D) Saturn, Uranus, Neptune, Jupiter
D) there is no correct answer
What is the value of the astronomical unit?
A) 160 million km. B) 149.6 million km.
B) 135 million km. D) 143.6 million km. e) 150 million km.
A) circular B) hyperbolic
C) elliptical D) parabolic E) spherical
How can we explain the absence of a magnetic field on the Moon?
A) weak attraction
B) slow axial rotation
B) large temperature changes
D) poor electrical conductivity of the mantle
The ratio of the cubes of the semi-axes of the orbits of the two planets is 16. Consequently, the period of revolution of one planet is greater than the period of revolution of the other:
A) 8 times B) 2 times C) 4 times D) 16 times E) the period will not change
The following are the bodies that make up the Solar System. Select an exception.
A) Sun B) major planets and their satellites C) asteroids D) comets E) meteors
Small bodies of the Solar System include:
A) stars B) large planets and their satellites C) asteroids D) planets E) the Sun
How long does it take light from the Sun to reach the Earth?
A) arrives instantly B) Approximately 8 minutes
C) 1st year D) about a day E) 12 hours
The planets are located relative to the Sun as follows:
a) Venus, Earth, Mars, Mercury, Neptune, Saturn, Uranus, Jupiter.
b) Mercury, Venus, Earth, Mars, Neptune, Saturn, Jupiter, Uranus.
c) Mercury. Venus,. Earth,. Mars, Jupiter, Saturn, Uranus, Neptune,
D) Earth, Mars, Venus, Mercury, Jupiter, Saturn, Uranus, Neptune
D) Neptune, Uranus, Saturn, Jupiter, Mercury, Venus, Mars, Earth
Determine the relative position of the planes of the celestial equator and the Earth's equator?
A). Tangent
IN). Perpendicular
WITH). Parallel
D). At an angle
E). Forms an angle of 23"27"
Indicate the intersection points of the circle of the mathematical horizon with the circle of the celestial equator?
A). North South
IN). Zenith-nadir
WITH). East-west
D). Celestial pole
E). Equinox points
Determine the relative position of the axis of the world and the axis of the Earth?
A). At an angle of 30°
IN). At an angle of 90°
WITH). Parallel
D). Floor angle 23°27"
E). Crosses
Indicate the place of the Sun in the Galaxy?
A). At the center of the galaxy
IN). Located at the galactic core
WITH). The Sun is closer to the main plane, at a distance of 10 kpc from the center of the Galaxy.
D). The Sun is closer to the edge of the Galaxy, at a distance of 30 kpc from the center of the Galaxy
E) The Sun is located on the main plane of the Galaxy, at a distance of 15 kpc from the center of the Galaxy.
In what direction does the Sun move in our Galaxy at a speed of 20 km/sec?
A). towards the constellation Draco
B).in the direction of the constellation Leo
WITH). towards the constellation Hercules
D). towards the constellation Orion
E). towards the constellation Aquila
What condition must the declination of a star satisfy in order for it to be non-ascending under geographic latitude (φ).
A). δ< (90°-φ)
IN). | δ | ≥ (90-φ)
WITH). δ ≥-(90-φ)
D). δ< -(90- φ)
E). there is no right answer
What is a sidereal year?
E). I find it difficult to answer
Provide the correct list of giant planets
A). Mars, Earth, Jupiter, Saturn
IN). Mars, Mercury, Neptune, Pluto
C). Venus, Uranus, Saturn, Neptune
D). Jupiter, Saturn, Uranus, Neptune
E). Uranus, Saturn, Neptune, Pluto
In what direction does the daily rotation of the celestial sphere occur?
A). If you stand facing south, then from east to west according to the “clock hand”
B). If you stand facing south, then from west to east according to the “clock hand”
C). If you stand facing south, then from east to west against the “clock hand”
D). If you stand facing south, then from west to east against the “clock hand”
E). I find it difficult to answer.
Which of the following planets does not have a satellite?
IN). Neptune
C). Venus
D). Jupiter
Which of the following planets have two satellites?
A). Earth, Jupiter
IN). Mars, Neptune
C). Venus, Uranus
D). Jupiter, Saturn
E). Uranus, Saturn
When was the Gregorian calendar introduced?
On what day and for what point on the celestial sphere are both right ascension and declination equal to zero?
If today a certain star culminated at 8 pm, when will it culminate in 15 days?
A). 8:15 p.m.
IN). 7:54 pm
WITH). 7 p.m.
D). 7h. 40m. evenings
E). 9 p.m.
What explains the existence of different seasons on Earth and the formation of different thermal zones?
A). Rotation of the Earth
IN). Annual movement of the Earth
WITH). The tilt of the Earth's axis to the plane of its orbit and the movement of the Earth around the Sun
D). The movement of the Sun along the ecliptic
E). There is no right answer
What is Universal Time?
A). The time is equal to the right ascension of the luminary located at the upper culmination.
E. Difficult to answer
In what constellation is the bright star Arcturus located?
WITH). South Cross
D). Bootes
E). Southern fish
In what constellation is the bright star Regulus located?
WITH). South Cross
E). Southern fish
Please indicate the magnification of the telescope?
What's the difference between an astrograph and a regular telescope?
A) the increase is small
B) high magnification
C) no eyepiece
D) no lens
E) gives a photograph of a celestial object
What is a telescope lens used for?
A) To obtain a magnified image of celestial bodies
B) Collect light emitted by celestial bodies
C) To increase the angle of view
D) Collect light emitted by celestial bodies and obtain magnified images of celestial bodies
E) There is no correct answer
A) At the equator
B) At the middle latitude of the Earth
C) At the South Pole
D) In the southeast
E) In the northwest
A) At the equator
B) At the middle latitude of the Earth
C) At the South Pole
D) In the southeast
E) In the northwest
What is the relative position of the Sun, Earth and Moon that explains the phases?
new moon?
A) When the Sun is between the Moon and the Earth
B) When the Earth is between the Sun and the Moon
C) When the Magnifier is between the Sun and the Earth
D) When the Moon is in opposition to the Sun
E) When the Moon is at opposition to the Earth
In how many days will the last quarter of the moon phase occur? (Counting from
new moon)
A) after 7.5 days
B) after 29.5 days
C) after 15 days
D) after 22.5 days
E) After 27.5 days
What is the period of rotation of the Moon around its axis?
A) 29.5 days
B) 30 days
C) 27.32 days
D) 22.5 days
E) 25.5 days
What phase must the Moon be in for a lunar eclipse to occur?
A) In the full moon phase
B) In the first quarter phase
C) In the new moon phase
])) In the last quarter phase
E) In the phase before the new moon
How long does it take for a ray of light to travel from the Sun to the Earth?
A) 3 min 20 sec
B) 57 minutes
C) 10.5 minutes
D) 8 min 18 sec
E)15 minutes
In what constellation is the bright star Vega located?
WITH). South Cross
E). Southern fish
Do sunspots form?
A) In the crown
B) In the chromosphere
C) In the photosphere
D) In the convective zone
E) In the emission zone
Are solar prominences appearing?
A) In the crown
B) In the chromosphere
C) In the photosphere
D) In the convective zone
E) In the emission zone
The division of stars into supergiants and dwarfs is associated with the great difference between them
A) sizes
B) temperature
C) luminosity
EJ chemical composition
Stars similar to the Sun are classified as?
A) Supergiants
B) giants
C) yellow dwarfs
D) subgiants
E) red dwarfs
What does the letter order of spectral classes look like in order of decreasing temperature from hottest to coldest?
A) B. O.A.F.K.G.M.
B) O.A.B.F.K.M. G.
C) O.V.F.A.M.K. G.
D) O.V.A.F. G.K..M.+
E) A.B.O.S.K.M.G.
Which of the following spectral designations for the Sun is correct?
The sequence of supergiants on the H-R diagram is characterized by
A) They belong to the main sequence stars
B) significantly exceeds the Sun in luminosity
C) these are white stars
The sequence of white dwarfs on the H-R diagram is characterized by
A) Belong to main sequence stars
B) significantly exceeds the Sun in luminosity
C) these are very dense stars
D) Characterized by the highest luminosities
E) characterized by the largest dimensions
How many stars are there in our Galaxy?
A) More than 100 million
B) There are as many of them in the largest globular clusters
C) more than 100 billion
D) more than 1 billion
E) about 3 billion.
Who is the creator of the heliocentric system of the world?
A) N. Copernicus
B) G. Galileo
C) Ptolemy
D) D. Bruno
E) I. Kepler
A) At the North Pole (for residents of the northern hemisphere)
B) At the middle latitude of the Earth
C) At the South Pole
D) In the southeast
E) In the northwest
Is it located near the center of the Sun?
A) zone of nuclear reactions
B) chromosphere
C) photosphere
D) convective zone
E) radiant energy zone
In what constellation is the bright star Fomalhaut located?
WITH). South Cross
E). Southern fish
In what constellation is the bright star Spica located?
A). Bootes
IN). Auriga
E) Aldebaran
What is the name of the bright star in the constellation Boötes?
A) Aldebaran
E) Betelgeuse
What is the name of the bright star in the constellation Auriga?
C) Chapel
E) Caster
Having just risen, the star rises at a right angle to the horizon. Where on Earth can this be observed?
A) At the North Pole
B) Beyond the Arctic Circle
C) At the equator
D) At any latitude of the Earth’s northern hemisphere, except the equator and pole
In what constellation is the bright star Antares located?
A).Perseus
WITH). Scorpion
E). Southern fish
Having just culminated, the star moves downwards. Which side of the sky is it on?
A) In the east
B) In the south
C) In the western
D) In the north
E) On the northwestern
Having just culminated, the star moves upward. Which side of the sky is it on?
A) In the east
B) In the western
C) In the north
D) In the south
E) In the northeast
All stars visible to the observer move parallel to the horizon from left to right. Where on Earth will this happen?
A) At the equator
B) At any latitude of the Earth’s northern hemisphere, except the equator and pole.
C) Beyond the Arctic Circle
D) At the North Pole
E) At latitude 23° 27!
We live in some point of the second time zone N1=2n. We now have standard time T1n=14h23m15S. What is the standard time in Novosibirsk at this moment N2 =6 n
B Mockbe Ni = 2n it is now 10 o'clock maternity time. What time is it in Vladivostok N2=9n also according to maternity leave?
For Aizat
Determine the local time at a point, geographic longitude is 7h46m, if the clock running exactly according to Moscow maternity time shows 18h36m (for Moscow n = 2)
At 18:32 local time, the ship's navigator received signals from Moscow daylight time transmitted at 11:00 (determine the ship's longitude).
It is noon in Kharkov, and at the same time in Kazan the clock shows 12:46. What is the longitude of Kazan? (if the longitude of Kharkov is 2.25).
On the shortest day (for a resident of the northern hemisphere) the Sun rises at what point on the horizon?
A) Exactly in the east
B) In the southeast
C) Exactly south
D) In the northeast
C) In the middle of the horizon
On the longest day (for a resident of the northern hemisphere) the Sun rises at what point on the horizon?
A) In the southeast
B) Exactly in the east
C) Exactly south
D) In the northeast
E) In the east
At what point on Earth is the polar star visible above the observer's head?
A) At the equator
B) At the middle latitude of the Earth
C) On the northern pole (for a resident of the northern hemisphere)
D) Pa south pole
E) At latitude 550.
At what place on Earth is the polar star visible at the foot of the observer?
A) On the northern pole (for a resident of the northern hemisphere)
B) At the middle latitude of the Earth
C) At the equator
D) Pa south pole
E) At latitude 450
At what place on Earth is the polar star visible at an angle to the horizon?
A) At the equator
B) At the middle latitude of the Earth
C) On the northern pole (for a resident of the northern hemisphere)
D) Pa south pole
E) At latitude 900
A few days after the new moon, a crescent-shaped light part of the Moon is observed, convex to the right. On which side of the celestial sphere is this phase of the Moon visible?
A) Before sunrise the Sun is in the east
B) On the southern side of the sky, above the horizon after sunset
C) After sunset, on the western side of the sky, closer to the horizon
D) After sunset, on the northern side of the sky slightly above the horizon
E) After sunset, in the west above the horizon
What phase must the Moon be in for a solar eclipse to occur?
A) In the full moon phase
B) In the first quarter phase
C) In the new moon phase
])) In the last quarter phase
E) In the phase before the new moon
Why does the Moon always face the same side towards the Earth?
A) The Moon revolves around the Earth
B) The period of revolution of the Moon around the Earth is 27.32 days
C) The period of revolution of the Moon around its axis and around the Earth is 27.32 days
D) The period of rotation of the Moon around its axis is 29.5 days.
E) The moon rotates around its axis.
How many Lupa moves do you know?
When does an annular solar eclipse occur?
A) When the Moon is at a smaller distance from the Earth
B) When the Moon is at a great distance from the Earth
C) When the Moon is at some distance from the Sun-Earth line
D) When the Moon is at a great distance from the Sun, and the Earth is closer to the Sun.
E) When the Moon is between the Earth and the Sun
Determine the correct order of planets according to the Ptolemaic world system?
A) Mars. Mercury. Moon. Jupiter. Saturn, Venus, Sun
B) Mercury. Mars. Moon. Jupiter. Saturn. Venus. Sun
C) Moon. Mars. Jupiter. Saturn. Venus. Sun, Mercury
D) Moon. Mercury. Venus. Sun. Mars. Jupiter. Saturn +
E) Jupiter, Saturn, Venus. Sun. Moon. Mars. Mercury
By what law is the mass of planets with satellites determined?
A) According to the law of universal gravitation
B) According to Kepler's first law
C) By the disturbance of a given planet from others
D) According to Kepler’s third law, supplemented by Newton
E) Among the answers A-D there is not a single correct one
On which satellite of Jupiter have volcanic phenomena been discovered?
A) Ganymede
C) Callisto
E) There is no correct answer among answers A-D.
Which planets of the solar system rotate around their axis in the opposite direction?
direction, i.e. from east to west?
A) Jupiter and Saturn
B) Mars and Mercury
C) Venus and Uranus
D) Neptune and Pluto
E) Saturn and Neptune
Which of the 9 major planets of the solar system orbits the Sun "lying on its side"
A) Jupiter
B) Mercury
C) Neptune
E) Pluto
Which of the large satellites of the solar system is the only one surrounded by a dense atmosphere?
B) Ganymede is a satellite of Jupiter
C) Iapetus is a satellite of Saturn
D) Titan is a satellite of Saturn
E) Phobos is a satellite of Mars
What is the year 1543 known in astronomy?
A) By decision of the Catholic Church, Giordano Bruno was burned at the stake
B) Galileo invented the telescope
C) The planet Neptune (Galileo) was discovered
D) The book of N. Copernicus was published, which sets out the heliocentric system of the world
What is the year 1846 known for in astronomy?
A) The book of N. Copernicus was published, which sets out the heliocentric system of the world
B) By decision of the Catholic Church, Giordano Bruno was burned at the stake
C) Galileo invented the telescope
D) The planet Neptune (Galileo) was discovered
E) I. Newton discovered the law of universal gravitation
Which point is called the "middle sun"
A) a fictitious point moving uniformly along the celestial equator
B) a fictitious point moving unevenly along the celestial equator
C) a fictitious point moving uniformly along the ecliptic
D) the sun moving uniformly along the ecliptic
E) the sun moving unevenly along the ecliptic
Which expression determines the aperture ratio of a telescope?
What is a tropical year?
A). The time is equal to the right ascension of the luminary located at the upper culmination.
B). Time during which the Sun completes a circle on the celestial sphere
C). Time between two successive passages of the Sun's center through the vernal equinox
D).Mean time of the Greenwich meridian, counted from midnight.
E). I find it difficult to answer
What is Civil Time?
A). The time is equal to the right ascension of the luminary located at the upper culmination.
B). Time measured by the hour angle of the center of the Sun.
C). Time equal to the hour angle of the “mean sun”
D).Mean time of the Greenwich meridian, counted from midnight.
E. Difficult to answer
How many times is the radius of the Sun greater than the radius of the Earth?
On what days is the equation of time equal to zero?
When was Universal Time introduced?
The ratio of the semi-major axes of the planets is 64, what is the ratio of their periods of rotation of the Sun?
The horizontal parallax of the Moon is 57' if the equatorial radius of the Earth is 6378 km, what is the distance from the Moon to the Earth?
In what year did the planet Neptune emerge?
What is the magnitude scale called?
A) logarithmic scale
B) algorithmic scale
C) density
D) equator
How many times is the volume of the Sun greater than the volume of the Earth?
What is the mass of the Earth?
A) Мᶿ=5.98*1024 kg
B) Мᶿ=1.76*1016 kg
C)Mᶿ=7.76*1023 kg
D) Мᶿ=3.56*1015 kg
E) Мᶿ=90.7*1012 kg
What is the average temperature of the Sun?
What constellation is the winter solstice in?
B) Sagittarius and Capricorn
C) Capricorn
D) Sagittarius
E) Aquarius
In what year was the planet Uranus discovered?
How many meteorites are there on the surface of the Earth?
A) 234 large meteorites
B) 5609 large meteorite
C)115 large meteorite
D) 78 large meteorites
E) 183 large meteorites
Relative to what point is sidereal time measured?
A) vernal equinox points
B) points of the autumnal equinox
C) summer solstice points
D) winter solstice points
E) summer solstice points of the celestial pole
Specify the day of the summer solstice
Specify the day of the winter solstice
Specify the day of the vernal equinox
Specify the day of the autumn equinox
What is the rotation of the Earth around its axis called with a period of 24 hours?
A) climax
B) eclipse
C) Daily rotation of the Earth
D) Annual rotation of the Earth
E) there is no correct answer
Number of zodial stars
On what day does the Sun rise higher above the horizon?
Who discovered the laws of planetary motion?
A) Ptolemy
B) Copernicus
E) Galileo
What is the time between two passages through the vernal equinox called?
A) Sidereal year
B) Sagittarius
C) Twins
E) Capricorn
If the sider period of Mars is 1.9 years, then how long does it take for Mars to recur?
A) 1.9 g.
In what orbits do the planets move?
A) circular
B) by hyperbole
C) along an ellipse
D) along a parabola
E) straight
What is the name of the point closest to the Earth from the Sun?
A) in autumn.
B) at perihelion
TO HELP AN ASTRONOMY TEACHER
(for physics and mathematics schools)
1. Subject of astronomy.
Sources of knowledge in astronomy. Telescopes.
Key issues: 1. What astronomy studies. 2. Connection of astronomy with other sciences. 3. The scale of the universe. 4. The importance of astronomy in the life of society. 5. Astronomical observations and their features.
Demonstrations and TSO: 1. Earth globe, transparencies: photographs of the Sun and Moon, planets in the starry sky, galaxies. 2. Instruments used for observation and measurements: telescopes, theodolite.
[Astron- luminary; nomos- law]
Astronomy studies the vast world surrounding 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 patterns of development of the universe)
Astrology is a teaching that states that by the relative positions of the Sun, planets, against the background of constellations, phenomena, destinies, and events can be predicted.
The Universe is the entire material world, limitless in space and developing in time. Three concepts: microworld, macroworld, megaworld.
Earth ® Solar System ® Galaxy ® Metagalaxy ® Universe.
The Earth's atmosphere absorbs g, x-rays, ultraviolet, a significant proportion of infrared, radio waves 20 m< l < 1 мм.
Telescopes (optical, radio)
Lens telescopes (refractor), mirror telescopes (reflector). Refractus– refraction (objective – lenses), reflecter– reflect (lens – mirror).
The main purpose of telescopes is to collect as much light energy as possible from the body being studied.
Features of the optical telescope:
1) Lens – up to 70 cm, luminous flux ~ D 2 .
2) F– focal length of the lens.
3) F/D– relative hole.
4) Magnification of the telescope, where D in millimeters.
The largest D= 102 cm, F= 1940 cm.
Reflector - for studying the physical nature of celestial bodies. The lens is a concave mirror of slight curvature, made of thick glass, Al the powder is sprayed on the other side under high pressure. The rays are collected at the focal plane, where the mirror is located. 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 depending on magnification?
No. Due to their great distance, stars are visible as dots even at the highest possible magnification.
2 . Why, when observed from Earth, do you think that stars move across the celestial sphere during the night?
Because the Earth rotates around 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 these parts of the spectrum from balloons, artificial Earth satellites or from more distant points.
4 . Explain the main difference between a reflecting telescope and a refracting telescope.
In lens type. A refracting 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 created 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? Please 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, since it deals with... (indicate the correct statement)
A. ... the study of objects and phenomena independent of humans;
B. ... the study of matter and energy under conditions impossible to reproduce on Earth;
B. ... the study of the most general laws of the Megaworld, of which man himself is a part.
3 . One of the chemical elements listed below was first discovered through astronomical observations. Please indicate 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 being studied.
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 from us that it is impossible to decide either by eye or through a telescope which of them is closer and which is further.
5 . You have been asked to build an astronomical observatory. Where would you build it? List all correct statements.
A. Within a large city.
B. Far from a large city, high in the mountains.
B. On a space station.
6 , Why are telescopes used for astronomical observations? Please indicate the correct statement.
A. In order to obtain an enlarged image of a celestial body.
B. To collect more light and see fainter stars.
B. To increase the angle of view from which a celestial object is seen.
Level 2: 3 – 4 points
1. What is the role of observations in astronomy and with what instruments are they performed?
2. What are the most important types of celestial bodies you know?
3. What is the role of astronautics in the exploration of the Universe?
4. List 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 human history. For what purpose did ancient man observe the heavenly bodies? Write what problems people in ancient times solved using these observations.
Level 3: 5 – 6 points
1. Why does the sun rise and set?
2. Natural sciences use both theoretical and experimental research methods. Why is observation the main method of research in astronomy? Is it possible to conduct astronomical experiments? Justify your answer.
3. What are telescopes used for when observing stars?
4. What are telescopes used for when observing the Moon and planets?
5. Does a telescope magnify the apparent size of stars? Explain your answer.
6. Remember what information on astronomy you received in natural history, geography, physics, and history courses.
Level 4. 7 – 8 points
1. Why, when observing the Moon and planets through a telescope, do they use a magnification of no more than 500 - 600 times?
2. In terms of its linear diameter, the Sun is approximately 400 times larger than the Moon. Why are their apparent angular diameters almost 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, reflector and meniscus telescope?
5. What are the diameters of the Sun and 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. Difference between stars in brightness (luminosity), color. 3. Magnitude. 4. Apparent daily motion of stars. 5. celestial sphere, its main points, lines, planes. 6. Star map. 7. Equatorial SC.
Demonstrations and TSO: 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, geographical and star globes.
For the first time, stars were designated by letters of the Greek alphabet. In the constellation atlas of Baiger in the 18th century, the drawings of the constellations disappeared. The magnitudes are indicated 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 Auriga - Capella, a B. Canis - Sirius.
The Sun, Moon and planets are not indicated on the maps. The path of the Sun is shown on the ecliptic in Roman numerals. Star maps display 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 Earth's rotation:
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 positions of luminaries in the sky. The radius is taken as 1 Pc.
88 constellations, 12 zodiac. It can be roughly divided into:
1) summer - Lyra, Swan, Eagle 2) autumn - Pegasus with Andromeda, Cassiopeia 3) winter - Orion, B. Canis, M. Canis 4) spring - Virgo, Bootes, Leo.
Plumb line intersects the surface of the celestial sphere at two points: at the top Z – zenith- and at the bottom Z" – nadir.
Mathematical 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 – point south. Line N.S.- 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 the great circle of the celestial sphere passing through the zenith Z, celestial pole R, south celestial pole 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 motion of all stars occurs in a circle parallel to the horizon. The North Pole of the world when observed from the North Pole of the Earth is at the zenith.
All stars rise at right angles to the horizon in the eastern part of the sky and also set below the horizon in the western part. The celestial sphere rotates around an axis passing through the poles of the world, located exactly on the horizon at the equator.
2. Express 10 hours 25 minutes 16 seconds in degrees.
The Earth makes one revolution in 24 hours - 360 degrees. Therefore, 360 o corresponds to 24 hours, then 15 o - 1 hour, 1 o - 4 minutes, 15 / - 1 minute, 15 // - 1 s. Thus,
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 from the star map.
Let's replace the name of the star with a letter designation (a Lyrae) and find its position on the star map. Through an imaginary point we draw a circle of declination until it intersects with the celestial equator. The arc of the celestial equator, which lies between the point of 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, measured along the celestial equator towards the apparent daily rotation of the celestial sphere. The angular distance measured along the declination circle 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 sunset of the star. We get
5. Determine the date of the upper culmination of the star Regulus at 21:00 local time.
We install the overhead circle so that the star Regulus (a Leo) is on the line of the celestial meridian (0 h – 12h scale of the overhead circle) south of the north pole. On the dial of the applied circle we find the mark 21 and opposite it on the edge of the applied circle we determine the date - April 10.
6. Calculate how many times brighter Sirius is than the North Star.
It is generally accepted that with a difference of one magnitude, the apparent brightness of stars differs by approximately 2.512 times. Then a difference of 5 magnitudes will amount to a difference in brightness of exactly 100 times. So 1st magnitude stars are 100 times brighter than 6th magnitude stars. Consequently, the difference in the apparent magnitudes of two sources is equal to unity when one of them is brighter than the other (this value is approximately equal to 2.512). In general, the ratio of the apparent brightness of two stars is related to the difference in their apparent magnitudes by a simple relationship:
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.
Let us take logarithms of both sides of the above relationship:
Thus, . 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 conventionally defined area of the sky within which there are luminaries located at different distances from us. Therefore, the expression “fly to a 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, for example, having the same origin.
B. A group of bright stars located close to each other in space
B. A constellation refers to an area of the sky within some established boundaries.
2. Stars have different brightness and color. What stars does our Sun belong to? Please indicate the correct answer.
A. To the whites. B. To the yellow ones.
B. To the reds.
3. The brightest stars were called stars of the first magnitude, and the faintest were called stars of the sixth magnitude. How many times brighter are 1st magnitude stars than 6th magnitude stars? Please indicate the correct answer.
A. 100 times.
B. 50 times.
B. 25 times.
4. What is the celestial sphere? Choose the correct statement.
A. A circle of the earth's surface limited 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 of the globe at the point where the observer is located.
5. What is called declination? Choose the correct statement.
A. The angular distance of the star from the celestial equator.
B. The angle between the horizon line and the luminary.
B. The 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 celestial axis)
B. The angle between the planes of great circles, one passing through the poles of the world and a given luminary, and the other through the poles of the world and the point of the vernal equinox lying on the equator.
Level 2: 3 – 4 points
1. Why does the North Star not change its position relative to the horizon during the daily movement of the sky?
2. How is the axis of the world located 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?
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 lie relative to the surface of the globe?
Level 3: 5 – 6 points.
1. Find the coordinates on the star map and name the objects that have the coordinates:
1) a = 15 hours 12 minutes, d = –9 o; 2) a = 3 hours 40 minutes, d = +48 o.
1) a Ursa Major; 2) β China.
3. Express 9 hours 15 minutes 11 seconds in degrees.
4. Find on the star map and name objects that have coordinates:
1) a = 19 hours 29 minutes, d = +28 o; 2) a = 4 hours 31 minutes, d = +16 o 30 / .
1) a Libra; 2) g Orion.
6. Express 13 hours 20 minutes in degrees.
7. In what constellation is the Moon located if its coordinates are a = 20 hours 30 minutes, d = –20 o?
8. Using the star map, determine the constellation in which the galaxy M31 is located if its coordinates are a = 0 h 40 min, d = +41 o.
Level 4. 7 – 8 points
1. The faintest stars that can be photographed by the world's largest telescope are stars of 24th magnitude. How many times are they fainter than 1st magnitude stars?
2. The star’s brightness changes from minimum to maximum by 3 magnitudes. How many times does its shine 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 does the Sun appear 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 Majoris is brighter than the star a Cygnus.
6. Calculate how many times the star Sirius is brighter than Vega.
3. Working with the map.
Determination of coordinates of celestial bodies.
Horizontal coordinates.
A– azimuth of the luminary, measured from the point South along the line of the mathematical horizon clockwise in the direction west, north, east. Measured from 0° to 360° or from 0 h to 24 h.
h– the height of the luminary, measured from the point of intersection of the altitude circle with the line of the mathematical horizon, along the altitude circle 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.
Declination is expressed in degrees, minutes and seconds. The celestial equator divides the celestial sphere into the 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 give advantage over horizontal coordinates:
1) Star maps and catalogs have been created. The coordinates are constant.
2) Drawing up geographical and topological maps of the earth's surface.
3) Orientation on land, sea and space.
4) Time check.
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 =3 h 40 m, d = +48 o.
2. Using the star map, determine the equatorial coordinates of the following stars:
1) a Ursa Major; 2) b China.
3. Express 9 h 15 m 11 s in degrees.
4. Find on the star map and name objects that have coordinates
1) a = 19 h 29 m, d = +28 o; 2) a = 4 h 31 m, d = +16 o 30 / .
5. Using the star map, determine the equatorial coordinates of the following stars:
1) a Libra; 2) g Orion.
6. Express 13 h 20 m in degrees.
7. In what constellation is the Moon located if its coordinates are a = 20 h 30 m, d = –20 o.
8. Using the star map, determine the constellation in which the galaxy is located M 31, if its coordinates are a 0 h 40 m, d = 41 o.
4. Climax of the luminaries.
Theorem about the altitude of the celestial pole.
Key questions: 1) astronomical methods for determining geographic latitude; 2) using a moving star map, determine the visibility conditions of the luminaries on any given date and time of day; 3) solving problems using relationships connecting the geographic latitude of the observation site with the height of the star at its culmination.
The culmination of the luminaries. Difference between upper and lower climax. Working with a map to determine the time of climaxes. Theorem about the altitude of the celestial pole. Practical ways to determine the latitude of an area.
Using the drawing of the projection of the celestial sphere, write down the formulas for the heights at the upper and lower culminations 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 altitude theorem:
– the height of the celestial pole (the North Star) above the horizon is equal to the geographic latitude of the observation site
The angle is like vertical, a. Knowing that is the declination of the star, the height of the upper culmination will be determined by the expression:
For the star's bottom climax M 1:
Home give the task to obtain a formula for determining the height of the upper and lower culmination of a star M 2 .
Assignment for independent work.
1. Describe the visibility conditions for stars at 54°N latitude.
2. Install a moving 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 moment of observation, which constellations are below the horizon.
b) which constellations are rising at the moment, setting at the moment.
3. Determine the geographic latitude of the observation site if:
a) the star Vega passes through the zenith point.
b) the star Sirius at the upper culmination at an altitude of 64 o 13 / south of the zenith point.
c) the height of the star Deneb at the upper culmination is 83 o 47 / north of the zenith.
d) the star Altair passes through the zenith point at its lower culmination.
On one's own:
Find the declination intervals of stars that at a given latitude (Bobruisk):
a) never ascend; b) never come in; c) can rise and set.
Tasks for independent work.
1. What is the declination of the zenith point at the geographic latitude of Minsk (j = 53 o 54 /)? Accompany your answer with a drawing.
2. In what two cases does the height of the star above the horizon do 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 a drawing, prove that in the case of the upper culmination of the luminary north of the zenith, it will have a height h= 90 o + j – d.
4. The azimuth of the star is 315 o, altitude 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 (d = 19 o 27 /). What is the geographic latitude of Kyiv?
6. What is the declination of the stars that culminate at a location with latitude j at north?
7. The polar star is 49 / 46 // away from the north pole of the world. What is its declination?
8. Is it possible to see the star Sirius (d = –16 o 39 /) at meteorological stations located on the island. Dikson (j = 73 o 30 /) and in Verkhoyansk (j = 67 o 33 /)? [On about. Dixon no, no in Verkhoyansk]
9. A star that describes an arc of 180 degrees above the horizon from sunrise to sunset is 60 degrees away from the zenith during the upper culmination. 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 degrees away from the north pole of the world. Is it always above the Brest horizon (j = 52 o 06 /)? [Always]
12. Find the geographic latitude of the place where the star at the upper culmination passes through the zenith, and at the lower culmination touches the horizon at the north point. What is the declination of this star? j = 45 o;
13. The azimuth of the luminary is 45 o, the altitude is 45 o. In which direction of the sky should we look for this luminary?
14. When determining the geographical latitude of a place, the required value was taken to be equal to the height of the Polar Star (89 o 10 / 14 //), measured at the moment of the lower culmination. Is this definition correct? If not, what is the mistake? 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 to be non-setting at a point with latitude j; so that it is not ascending?
16. The right ascension of the star Aldebaran (a-Taurus) is 68 o 15 /. Express it in units of time.
17. Is the star Fomalhaut (a-Doradus) rising in Murmansk (j = 68 o 59 /), whose declination 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. k.v.
5. Time measurement.
Determination of geographic longitude.
Key questions: 1) differences between the concepts of sidereal, solar, local, zone, seasonal and universal time; 2) principles for determining time based on astronomical observations; 3) astronomical methods for determining the geographic longitude of an area.
Students should be able to: 1) solve problems on calculating time and dates and converting 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, identify 2 - 3 constellations visible at latitude 53 o in the Northern Hemisphere.
2. Determine the azimuth and altitude of the star at the time of the lesson:
Option 1. a B. Ursa, a Leo.
Option 2. b Orion, a Eagle.
3. Using a star map, find the stars by their coordinates.
Main material.
Develop concepts about days and other units of time. The occurrence of any of them (a day, a week, a month, a year) is associated with astronomy and is based on the duration of a cosmic phenomenon (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 reference system in which the movement of the Earth is considered (whether it is connected with the fixed stars, the Sun, etc.). The choice of reference system is reflected in the name of the time unit.
– the duration of time units is associated with the visibility conditions (culminations) of celestial bodies.
– the introduction of the atomic time standard in science was due to the uneven rotation of the Earth, discovered when the accuracy of clocks increased.
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 changes in the length of sunny days throughout the year. To do this, you should compare the moments of two successive culminations of the Sun and any star. We mentally choose the star that will culminate for the first time simultaneously with the Sun. Next time the star and the Sun will not culminate at the same time. The sun will culminate at about 4 minutes later, because against the background of stars it will move by 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 become aware of this after studying Kepler’s laws). There are other reasons why the time interval between two successive culminations of the Sun is not constant. There is a need to use the average solar time.
Provide more accurate data: the average solar day is 3 minutes 56 s shorter than the sidereal day, and 24 hours 00 minutes 00 s sidereal time is equal to 23 hours 56 min 4 s average solar time.
Universal time is defined as the local mean solar time at the prime (Greenwich) meridian.
The entire surface of the Earth is conventionally divided into 24 areas (time zones), limited by meridians. The zero time zone is located symmetrically relative to the prime meridian. Time zones are numbered from 0 to 23 from west to east. The actual boundaries of time zones coincide with the administrative boundaries of districts, regions or states. The central meridians of time zones are 15 o (1 hour) apart from each other, therefore, when moving from one time zone to another, the time changes by an integer number of hours, but the number of minutes and seconds does not change. A new calendar day (as well as a new calendar year) begins on the date line, which runs mainly along the meridian of 180 o. near the northeastern border of the Russian Federation. West of the date line, the day of the month is always one more than 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 timing the movements of people traveling from the Eastern to the Western hemispheres of the Earth and back.
Calendar. Limit yourself to considering a brief history of the calendar as part of culture. It is necessary to highlight the three main types of calendars (lunar, solar and lunisolar), tell what is their basis, and dwell in more detail on the Julian solar calendar of the old style and the Gregorian solar calendar of the new style. Having recommended relevant literature, invite students to prepare short reports on 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 issues regarding the determination of geographic coordinates using astronomical observations.
Modern society cannot do without knowledge of 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 Earth surface is equal to the difference in the values of the geographic longitude of these points, which makes it possible to determine the longitude of a specific 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 an area, it is necessary to determine the moment of culmination of a star with known equatorial coordinates. Then, using special tables (or a calculator), the observation time is converted from solar mean to stellar. Having found out 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 time units from one system to another.
The moments of the culmination of the luminaries are determined using a passage instrument - a telescope, strengthened in a special way. The telescope 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 point of the south through the zenith and the celestial pole to the point of the north, i.e. it tracks the celestial meridian. A vertical thread in the field of view of the telescope tube serves as a mark of the meridian. At the moment a star passes through the celestial meridian (at the upper culmination), sidereal time is equal to right ascension. The passage instrument was first 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 moments and periods of time stimulated the development of astronomy and physics. Until the middle of the 20th century. astronomical methods of measuring, storing time and time standards underlay the activities of the world Time Service. The accuracy of the clock was controlled and corrected by astronomical observations. Currently, the development of physics has led to the creation of more accurate methods for determining time and standards. Modern atomic clocks give an error of 1 s per 10 million years. With the help of these watches and other instruments, many characteristics of the apparent and true motion of cosmic bodies were clarified, new cosmic phenomena were discovered, including changes in the speed of rotation of the Earth around its axis by approximately 0.01 s during the year.
When consolidating the studied material with students, the following tasks can be solved.
Task 1.
Determine the geographic longitude of the observation location if:
a) at local noon, the traveler noted 14:13 Greenwich time.
b) using the exact time signals of 8:00 m 00 s, the geologist recorded 10:13 m 42 s local time.
Considering that
c) the navigator of the airliner at 17:52:37 local time received a signal of Greenwich time 12:00:00.
Considering that
1 h = 15 o, 1 m = 15 / and 1 s = 15 //, we have.
d) the traveler noted 17:35 at local noon.
Taking into account the fact that 1 h = 15 o and 1 m = 15 /, we have.
Task 2.
Travelers noticed that according to local time, the lunar eclipse began at 15:15, while according to the astronomical calendar it should have taken place at 3:51 Greenwich time. What is the longitude of their location.
Task 3.
On May 25 in Moscow (2nd time zone) the clock shows 10 hours 45 m. What is the average, zone and summer time at this moment in Novosibirsk (6 time zone, l 2 = 5 hours 31 m).
Knowing Moscow summer time, we will 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 storing time.
6. Heliocentric system of Copernicus.
Key questions: 1) the essence of the heliocentric system of the world and the historical background for its creation; 2) the causes and nature of the apparent motion of the planets.
Frontal conversation.
1. A true solar day is the period of time between two successive culminations of the same name at the center of the solar disk.
2. A sidereal day is the period of time between two successive culminations of the same name at the point of the vernal equinox, equal to the period of rotation of the Earth.
3. The average solar day is the period of time between the two culminations of the same name of the average equatorial Sun.
4. For observers located on the same meridian, the culmination of the Sun (like any other luminary) occurs simultaneously.
5. A solar day differs from a sidereal day by 3 m 56 s.
6. The difference in local time values 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 zones from west to east, the clock must be set forward one hour, and from east to west - one hour back.
View 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 day of the month and what day of the week did he arrive? What needs to be taken into account when solving this problem? Who and under what circumstances encountered this for the first time in history?
When solving the problem, you need to take into account that on the way from San Francisco to Vladivostok the ship will cross a conventional 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 international date line in the east to west direction (as in our case), one calendar date is discarded from the count.
Magellan and his companions first 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. Supplemented the star catalog of Hipparchus. He built special astronomical instruments: an astrolabe, an armillary sphere, and a 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 movement of the Sun and Moon, it constituted 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 out in his main work “Almagest” (“The Great Mathematical Construction of Astronomy in the XIII Books”) - an encyclopedia of the astronomical knowledge of the ancients. The Almagest also contains information on rectilinear and spherical trigonometry, and for the first time the 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 collection 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.
A correct understanding of observed celestial phenomena and the Earth’s place in the solar system has evolved over centuries. Nicolaus Copernicus finally broke the idea of the immobility of the Earth. Copernicus (Kopernik, Copernicus) Nicholas (1473 – 1543), 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 celestial 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 work “On the Rotations of the Celestial Spheres” (1543), which was banned by the Catholic Church from 1616 to 1828.
Copernicus showed that it is the rotation of the Earth around the Sun that can explain the visible loop-like movements of the planets. The center of the planetary system is the Sun.
The Earth's rotation axis is tilted from the orbital axis by an angle of approximately 23.5°. If it were not for this tilt, the seasons would not exist. The regular change of seasons is a consequence of the movement of the Earth around the Sun and the inclination of the Earth's axis of rotation to the orbital plane.
Since, when observed 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 either from east to west (direct motion), or from west to east (retrograde motion). Moments of change of direction are called standing. If you put this path on a map, it will turn out a loop. The larger the distance between the planet and the Earth, the smaller the loop is. The planets describe loops, rather than simply moving back and forth along one 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: lower ( internal) – Mercury and Venus – and upper ( external) – the other six planets. The nature of the planet's movement 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 – 48°. During eastern elongation, the inner planet is visible in the west, in the rays of the evening dawn, shortly after sunset. During 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 the inferior conjunction, both planets have their largest angular dimensions, but look like narrow crescents. At a phase angle j = 90 o, half of the planetary disk is illuminated, phase Φ = 0.5. At superior conjunction, the inferior planets are fully illuminated, but are poorly visible from Earth, as they are behind the Sun.
Planetary configurations.
Homework: § 3. k.v.
7. Planetary configurations. Problem solving.
Key questions: 1) configurations and visibility conditions of planets; 2) sidereal and synodic periods of planetary revolution; 3) formula for the connection between the synodic and sidereal periods.
The student must be able to: 1) solve problems using a formula connecting the synodic and sidereal periods of revolution of the planets.
Theory. Indicate the basic configurations for the upper (lower) planets. Define the synodic and sidereal periods.
Let's say that at the initial moment of time the minute hand and the hour hand coincide. The period of time after which the hands meet again will not coincide with either the period of rotation of the minute hand (1 hour) or the period of rotation of the hour hand (12 hours). This period of time is called the synodic period - the time after which certain positions of the hands are repeated.
The angular velocity of the minute hand, and the hour hand is . During the synodic period S the hour hand of the clock will travel
and minute
Subtracting the paths, we get, or
Write down formulas connecting the synodic and sidereal periods and calculate the repetition of configurations for the upper (lower) planet closest to the Earth. Find the necessary 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 conditions of the problem T = S, Where T– sidereal period, the time of revolution of the planet around the Sun, and S– synodic period, the time of repetition of the same configuration with a given planet.
Then in the formula
Let's make a replacement S on T: The planet is infinitely far away. On the other hand, having made a similar replacement
The most suitable planet is Venus, whose period is 224.7 days.
Solution 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. Mercury's inferior conjunctions repeat after 116 days. Determine the sidereal period of Mercury.
Mercury is an inner planet and the formula is valid for it
3. Determine the sidereal period of Venus if its inferior conjunctions occur every 584 days.
4. After what period of time do Jupiter’s oppositions repeat if its sidereal period is 11.86 g?
8. Apparent movement 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 shape of a sickle. What planet could this be? [Internal] | 2. Which planets and under what conditions can be visible all night (from sunset to sunrise)? [All outer planets in opposition eras] |
| 3. Through observation, it was established that there are 378 days between two successive identical configurations of the planet. Assuming a circular orbit, find the sidereal (stellar) period of revolution of the planet. | 3. The minor planet Ceres revolves around the Sun with a period of 4.6 years. After what period of time do the oppositions of this planet repeat? |
| 4. Mercury is observed at the position of maximum elongation equal to 28 o. Find the distance from Mercury to the Sun in astronomical units. | 4. Venus is observed at the position of maximum elongation equal to 48 degrees. 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 revolution of the planets around the Sun in almost the same plane, their apparent motion on the celestial sphere will occur along and near the ecliptic with variable angular velocity and periodic changes in the direction of motion. The direction of the Sun's movement along the ecliptic is opposite to the daily movement of stars, the angular velocity is about 1 o per day.
Solstice and equinox days.
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 zodiac constellation, although it lies on the ecliptic. The idea of the zodiac signs 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 coincide. For example, the vernal equinox and the zodiac sign of Aries are located in the constellation Pisces.
For independent work.
Using a moving star chart, determine under which constellation you were born, i.e., in which constellation the Sun was at the time of your birth. To do this, connect a line between the north celestial pole and your date of birth and see in which constellation this line intersects the ecliptic. Explain why the result differs from that indicated in the horoscope.
Explain the phenomenon of 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 2,700 years ago, near the north pole there was a star called a Draco, called the Royal Star by Chinese astronomers. Calculations show that by 10000 the North Pole of the world will approach the star a Cygnus, and in 13600 the Polar Star will be replaced by a Lyrae (Vega). Thus, as a result of precession, the points of the spring and autumn equinoxes, summer and winter solstices slowly move along the zodiacal constellations. Astrology offers information that was out of date 2 thousand years ago.
The apparent movement of the Moon against the background of stars occurs 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 limbo . The line dividing the illuminated and unilluminated 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 . At the new moon Φ = 0, at the first quarter Φ = 0.5, at the full moon the phase is Φ = 1.0, and at the last quarter again Φ = 0.5.
During 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 from the Earth to the Moon. Two days after the new moon, a thin crescent of the young moon appears in the evening sky in the west, shortly after sunset.
Seven days after the new moon, the waxing Moon is visible in the shape 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.
14 days after the new moon, the full moon occurs. The Moon is in opposition to the Sun, and the entire illuminated hemisphere of the Moon faces the Earth. During a full moon, the Moon is visible throughout the night, the Moon rises during sunset, and sets during sunrise.
A week after the full moon, the aging Moon appears before us in its last quarter phase, in the form of a semicircle. At this time, half of the illuminated and half of the unilluminated hemisphere of the Moon faces 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 across the sky, which it traversed six months earlier, so in the summer the full Moon does not move far from the horizon, but in winter, on the contrary, it rises high.
The Earth revolves around the Sun, so from one new moon to the next, the Moon rotates around the Earth not 360°, but somewhat more. Accordingly, the synodic month is 2.2 days longer than the sidereal month.
The time interval between two successive 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 among people. It was believed that eclipses foreshadow wars, famine, ruin, and mass diseases.
The covering of the Sun by the Moon is called solar eclipse . This is a very beautiful and rare phenomenon. A solar eclipse occurs when the Moon crosses the ecliptic plane at the time of the new moon.
If the disk of the Sun is completely covered by the disk of the Moon, then an eclipse is called complete . At perigee, the Moon is closer to Earth by 21,000 km from the average distance, at apogee - further by 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 smaller than the angular diameter of the Sun's disk (about 0.5 o), then at the moment of the maximum phase of the eclipse a bright narrow ring remains visible from the Sun. This eclipse is called ring-shaped . 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. This eclipse is called private . You can observe such a beautiful formation as the solar corona only during total eclipses. Such observations, even in our time, can give a lot to science, so astronomers from many countries come to 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. Typically, a total solar eclipse lasts several minutes (the longest duration of a total solar eclipse, 7 minutes 29 seconds, will be on July 16, 2186).
The moon moves from west to east, so a solar eclipse begins 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 only be seen in those areas of the Earth through which the Moon's shadow passes. The diameter of the shadow does not exceed 270 km, so a 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 along a straight line called line of nodes . 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 full revolution in 18.6 years. Every year the longitude of the ascending node decreases by about 20 degrees.
Since the plane of the lunar orbit is inclined to the ecliptic plane at an angle of 5 o 09 /, the Moon during a new moon or full moon may be far from the ecliptic plane, and the lunar disk will pass above or below the solar disk. In this case, no eclipse occurs. For a solar or lunar eclipse to occur, the Moon must be near the ascending or descending node of its orbit during the new or full moon, i.e. close to 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 attacks the Sun and, according to Indian legends, causes its eclipse.
During full lunar eclipse The Moon completely disappears into the Earth's shadow. 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 was about three times larger than the Moon, simply based on the duration of eclipses (the exact value of this coefficient was 3.66).
During a total lunar eclipse, the moon is actually deprived of sunlight, so a total lunar eclipse is visible from anywhere in the Earth's hemisphere. The eclipse begins and ends simultaneously for all geographic locations. 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 greater its phase . Finally, when the disk of the Moon is covered not by a shadow, but by a penumbra, it happens penumbra 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 view every time, because The earth is opaque. However, the earth's atmosphere scatters the sun's rays, which fall on the eclipsed 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 the distribution of brightness and color in the Earth's shadow. The color of the eclipsed Moon is often assessed using 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 surface of the Moon are completely invisible.
3. The eclipse is dark red or reddish, with a darker part observed near the center of the shadow.
4. The eclipse is brick-red in color, the shadow is surrounded by a grayish or yellowish border.
5. The eclipse is copper-red, very bright, the outer zone is light, bluish.
If the plane of the Moon's orbit coincided with the plane of the ecliptic, then lunar eclipses would be repeated 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 Moon must be near the node of its orbit during a full moon.
Lunar eclipses occur several times a year.
The period of time after which the Moon returns to its node is called draconic month , which is equal to 27.21 days. After such a time, the Moon crosses the ecliptic at a point shifted relative to the previous intersection by 1.5 o to the west. The phases of the Moon (synodic month) repeat on average every 29.53 days. The period of time of 346.62 days during which the center of the solar disk passes through the same node of the lunar orbit is called draconic year .
Eclipse recurrence period – Saros - will be equal to the period of time after which the beginnings of these three periods will coincide. Saros means "repetition" in ancient Egyptian. Long before our era, even in ancient times, it was established that saros lasts 18 years 11 days 7 hours. Saros includes: 242 draconic months or 223 synodic months or 19 draconic years. During each Saros there are between 70 and 85 eclipses; Of these, there are usually about 43 solar and 28 lunar. Over the course of a year, a maximum of seven eclipses can occur - either five solar and two lunar, or four solar and three lunar. The minimum number of eclipses in a year is two solar eclipses. Solar eclipses occur more often than lunar eclipses, but they are rarely observed in the same area, since these eclipses are visible only in a narrow strip of the Moon's shadow. At any specific point on the surface, a total solar eclipse is observed on average once every 200–300 years.
Homework: § 3. k.v.
9. Ecliptic. Apparent movement of the Sun and Moon.
Problem solving.
Key questions: 1) daily movement of the Sun at various latitudes; 2) changes in the apparent movement of the Sun throughout the year; 3) apparent movement and phases of the Moon; 4) Solar and lunar eclipses. Eclipse conditions.
The student must be able to: 1) use astronomical calendars, reference books, and a moving star chart to determine the conditions for the occurrence of phenomena associated with the revolution of the Moon around the Earth and the apparent movement of the Sun.
1. How much does the Sun move along the ecliptic every day?
During the year, the Sun describes a circle of 360 degrees along the ecliptic, therefore
2. Why are solar days 4 minutes longer than sidereal days?
Because, while 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 minutes 56 seconds shorter than a sidereal day.
3. Explain why the Moon rises every day on average 50 minutes later than the day before.
On a given day at the moment of sunrise, the Moon is in a certain constellation. After 24 hours, when the Earth makes one full revolution around its axis, this constellation will rise again, but during this time the Moon will move approximately 13 degrees east relative to the stars, and its rise will therefore occur 50 minutes later.
4. Why was it that before spacecraft circled the Moon and photographed its far side, people could only see 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 during the new moon?
At this time, the Moon is on the same side of the Earth as the Sun, so the dark half of the lunar globe, not illuminated by the Sun, is facing us. In this position of the Earth, Moon and Sun, a solar eclipse may occur for the inhabitants of the Earth. It does not happen every new moon, since the Moon usually passes above or below the disk of the Sun during the new moon.
6. Describe how the position of the Sun on the celestial sphere has changed from the beginning of the school year to the day on which this lesson is taught.
Using the star chart, 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 d = +10 o. Moving along the ecliptic, the Sun crossed the celestial equator on September 23 and moved to the southern hemisphere; on October 27 it is in the constellation Libra and has a declination d = –13 degrees. 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, on a new moon the Moon is not 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 Earth's shadow cone on every full moon.
8. How many times faster does the Moon move across the sky 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 travels approximately 1 o, and the Moon - 13 o. Therefore, the Moon moves across the sky 13 times faster than the Sun.
9. How does the morning crescent moon differ in shape from the evening one?
The morning crescent moon bulges 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 Moon bulges to the right. The Moon is located at a distance of 20 - 50 o east (to the left) of the Sun.
Level 1: 1 – 2 points.
1. What is called the ecliptic? Please indicate the correct statements.
A. The axis of apparent rotation of the celestial sphere, connecting both poles of the world.
B. The angular distance of the luminary from the celestial equator.
B. The 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? Please indicate 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.
B. The time interval between the new moon and the full moon.
5. What is called a synodic month? Please indicate the correct statement.
A. The time interval between the full moon and the 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. Indicate 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 are the positions of the planets not indicated on star maps?
2. In what direction does the apparent annual motion of the Sun relative to the stars occur?
3. In what direction is the apparent movement of the Moon relative to the stars?
4. Which total eclipse (solar or lunar) lasts longer? Why?
6. As a result of what does the position of the sunrise and sunset points change throughout the year?
Level 3: 5 – 6 points.
1. a) What is the ecliptic? What constellations are there?
b) Draw what the Moon looks like in the last quarter. At what time of day is it visible in this phase?
2. a) What determines the annual apparent movement 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 evidence of the Earth’s revolution around the Sun?
b) Draw what the Moon looks like in the first quarter. At what time of day is it visible in this phase?
5. a) What is the cause of visible light from the Moon?
b) Draw what the Moon looks like in the second quarter. What time of day does she appear in this phase?
6. a) What causes the midday altitude of the Sun to change throughout the year?
b) Draw what the Moon looks like between the full moon and the last quarter.
Level 4. 7 – 8 points
1. a) How many times during the year can you see all phases of the moon?
b) The midday 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 culmination of the star Antares (d = –26 o) occur? Make a corresponding drawing.
3. a) Yesterday there was a lunar eclipse. 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 / southern sky. Determine the geographic 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 midday altitude of the Sun on March 21 at a point whose geographic altitude is 52 o?
5. a) What is the minimum time interval between solar and lunar eclipses?
b) At what geographic latitude will the Sun culminate at noon at an altitude of 45° above the horizon, if on this day its declination is –10°?
6. a) The moon is visible in the last quarter. Could there be a lunar eclipse in a week? Explain your answer.
b) What is the geographic latitude of the observation site 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 must 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 values of the equatorial coordinates of the Sun on the days of the equinoxes. | 1. Write down the values of the equatorial coordinates of the Sun on the days of the solstices |
| 2. On the circle representing the horizon line, mark the points of north, south, sunrise and sunset on the day the work is performed. Use arrows to indicate the direction in which these points will shift in the coming days. | 2. On the celestial sphere, depict the course of the Sun on the day the work is completed. Use an arrow to indicate the direction of the Sun's displacement in the coming days. |
| 3. To what maximum height does the Sun rise on the day of the vernal equinox at the North Pole of the earth? Drawing. | 3. To what maximum height does the Sun rise on the day of the vernal equinox at the equator? Drawing |
| 4. Is the Moon located east or west of the Sun from new moon to full moon? [east] | 4. Is the Moon located 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 focus.
Kepler's second law (law of equal areas ) .
The radius vector of the planet describes equal areas in equal periods of time. Another formulation of this law: the sectoral speed of the planet is constant.
Kepler's third law .
The squares of the orbital periods of planets around the Sun are proportional to the cubes of the semimajor axes of their elliptical orbits.
The modern formulation of the first law has been supplemented as follows: in unperturbed motion, the orbit of a moving body is a second-order curve - an ellipse, parabola or hyperbola.
Unlike the first two, Kepler's third law applies only to elliptical orbits.
The speed of the planet at perihelion
Where v c – 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 systems of orbiting bodies is important.
In a generalized form, this law is usually formulated as follows: the squares of the periods T1 and T2 of 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 the cubes of the semimajor 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 movement 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 the relationship between the period T orbital movement of a body with mass M and the semimajor axis of the orbit a (G– gravitational constant):
The following remark must be made here. For simplicity, it is often said that one body revolves around another, but this is only true for the case when the mass of the first body is negligible compared to the mass of the second (the attracting center). If the masses are comparable, then the influence of the less massive body on the more massive one should be taken into account. In a coordinate system with the origin at the center of mass, the orbits of both bodies will be conical sections lying in the same plane and with foci at the center of mass, with the same eccentricity. The difference will only be in the linear dimensions of the orbits (if the bodies are of different masses). At any moment of time, the center of mass will lie on the straight line connecting the centers of the bodies, and the distance to the center of mass r 1 and r 2 body mass M 1 and M 2 are respectively related by the following relationship:
The periapsis 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 double stars.
Example.
– What would be the semimajor axis of the planet’s orbit if the synodic period of its revolution were equal to one year?
From the equations of synodic motion we find the sidereal period of revolution of the planet. There are two possible cases:
The second case is not realized. 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 = |A.A. / | = 2a.
The degree of elongation of an ellipse is characterized by its eccentricity e. Eccentricity
e = ОF/O.A..
When the foci coincide with the center e= 0, and the ellipse turns into circle .
Major axle shaft a is the average distance from the focus (the planet from the Sun):
a = (A.F. 1 + F 1 A /)/2.
Homework: § 6, 7. k.v.
Level 1: 1 – 2 points.
1. Indicate which of the following planets are internal.
A. Venus. B. Mercury. V. Mars.
2. Indicate which of the following planets are outer planets.
A. Earth. B. Jupiter. V. Uranus.
3. In what orbits do the planets move around the Sun? Please indicate the correct answer.
A. In circles. B. By ellipses. B. By parabolas.
4. How do the orbital periods of planets change as the planet moves away 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 following planets may 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 visible 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 which 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 in 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 revolution 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 maximum distance of Venus from the Earth repeat if its sidereal period is 225 days?
5. a) What planets can be visible near the Moon during a full moon?
b) What is the sidereal period of Venus’ revolution around the Sun if its superior conjunctions with the Sun are repeated every 1.6 years?
6. a) Is it possible to observe Venus in the west in the morning and in the east in the evening? Explain your answer.
b) What will be the sidereal period of the outer planet’s revolution around the Sun if its oppositions are repeated after 1.5 years?
Level 4. 7 – 8 points
1. a) How does the value of the planet’s speed change as it moves from aphelion to perihelion?
b) The semimajor axis of Mars’ orbit is 1.5 a. 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 Earth satellite minimal and at what point is it maximum?
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 Earth satellite minimal and at what point is it maximum?
b) The sidereal period of Jupiter's revolution around the Sun is 12 years. What is the average distance of Jupiter from the Sun?
4. a) What is the orbit of the planet? What shape do the orbits of the planets have? Can planets collide as they move around the Sun?
b) Determine the length of the Martian year if Mars is removed from the Sun by an average of 228 million km.
5. a) At what time of year is the linear speed of the Earth’s movement around the Sun the greatest (smallest) and why?
b) What is the semimajor axis of the orbit of Uranus, if the sidereal period of 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 years. Determine the distance from Venus to the Sun.
Apparent movement of luminaries .
1. What conclusions of Ptolemy’s theory turned out to be correct?
The spatial arrangement of celestial bodies, the recognition of their movement, the revolution of the Moon around the Earth, the possibility of mathematical calculation of the apparent positions of the planets.
2. What shortcomings did N. Copernicus’s heliocentric system of the world have?
The world is limited to the sphere of fixed stars, the uniform motion of the planets is preserved, epicycles are preserved, and insufficient accuracy in predicting the positions of the planets.
3. The absence of what obvious observational fact was used as proof of the incorrectness of N. Copernicus’ theory?
Failure to detect the parallactic motion of stars due to its smallness and observational errors.
4. To determine the position of a 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 radius vector module - the distance to the object r. This coordinate is determined from more complex observations than a and d. In catalogs, its equivalent is the annual parallax, hence (pc). For 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?
Ecliptic, first vertical, colors of equinoxes and solstices.
6. In what place on Earth can any circle of declination coincide with the horizon?
At the equator.
7. Which circles (small or large) of the celestial sphere correspond to the vertical and horizontal lines of the field of view of the goniometer 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 Earth's poles.
9. What are the zenith azimuth, hour angle and right ascension of the poles of the world?
Values A, t, a in these cases are uncertain.
10. At what points on Earth does the North Pole coincide with the zenith? with the north point? with nadir?
At the north pole of the Earth, at the equator, at the south pole of the Earth.
11. An artificial satellite intersects the horizontal thread of a protractor instrument at a distance d o to the right of the center of the field of view, the coordinates of which A= 0 o , z = 0 o . Determine the horizontal coordinates of the artificial satellite at this point in time. How will the coordinates of the object change if the azimuth of the instrument is changed to 180 o?
1) A= 90 o, z = d o ; 2) A= 270 o, 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, half of the celestial sphere is visible at any moment;
b) at the Earth’s poles, the northern and southern hemispheres are visible, respectively;
c) at the Earth’s equator, in 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 almucantar?
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 Earth's poles.
16. When, during daily motion, do stars at all latitudes move parallel to the horizon?
In the upper and lower climaxes.
17. Where on Earth the azimuth of some stars is never equal to zero, and the azimuth of other stars is never equal to 180 o?
On the earth's equator for stars c, and for stars c.
18. Can the azimuths of a star at the upper and lower culminations be the same? What is it equal to in this case?
In the northern hemisphere, for all stars with declination, 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 located at one of the poles of the Earth or the star is located at one of the poles of the world.
20. In what part of the sky do the azimuths of the luminaries change the fastest and in what part the slowest?
Fastest in the meridian, 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 declination?
For all latitudes it is a star with d = 0, at the equator it is any star.
23. Can a star pass through the points of the east, zenith, west and nadir in a day?
This 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 its daily motion on the days of the equinoxes and days of the 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 located 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? On what dates of the year does this happen?
If we neglect the change in the declination of the Sun during the day, then its sunrise was at the point of the east. This happens every year on the equinoxes.
29. When does the boundary between the illuminated and unlit 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 did the ancient Greek astronomer Anaxagoras give to this phenomenon, based on the idea 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 luminary.
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, would be 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 at a diametrically opposite point on the globe. For Ryazan, this point is in the southern part of the Pacific Ocean and has 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 because the Sun rises and falls perpendicular to the horizon. In the 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 construct a sundial that would show average solar time, maternity time, summer time, etc.?
You can, but only for a specific date. There should be different dials for different types of time.
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 remain?
The sidereal year and synodic month would have been preserved. Using them, it would be possible to introduce smaller units of time, as well as construct a calendar.
39. When in the year are there the longest and shortest true solar days?
The longest true solar days occur on solstices, when the rate of change in the direct 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 days are obviously on the days of 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 length of the day on May 1 in Ryazan be greater than at a point with the same geographic 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 beginning of the day of the same date for the western and eastern regions of Russia, the length 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 the connection between social life and daylight hours. The difference in true solar days leads to the appearance of mean solar time. The dependence of mean solar time on the longitude of a place led to the invention of standard time. The need to save electricity led to maternity and summer time.
42. How would the length of the solar day change if the Earth began to rotate in the opposite direction to the actual one?
Solar days would become shorter than sidereal days by four minutes.
43. Why is the day in January longer in the afternoon than in the first half of the day?
This is due to a noticeable increase in the declination of the Sun during the day. The sun traces a larger arc in the sky after noon 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 at the earth’s equator always longer than the night by 7 minutes?
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 one greater than the time interval between the autumn and spring equinox?
This phenomenon is a consequence of the ellipticity of the earth's orbit. In summer, the Earth is at aphelion and its orbital speed is less than the speed in the winter months, when the Earth is at perihelion.
47. The difference in longitude of two places is equal to the difference in which times - solar or sidereal?
Doesn't matter. .
48. How many dates can there be on Earth at the same time?
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The use of astronomical means is only possible based on celestial bodies located 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.
In Fig. Figure 1.22 shows the celestial sphere for an observer located at a certain latitude. The straight line SY represents the true horizon, and the straight lines and MU represent the daily parallels of the luminaries. From the figure it is clear that all luminaries are divided into non-setting, non-rising, ascending and setting.
Luminaries whose daily parallels lie above the horizon are non-setting for a given latitude, and luminaries whose daily parallels are below the horizon are non-rising.
Non-setting luminaries will be those whose daily parallels are located between the NC parallel and the North Pole of the world. A luminary moving along the daily parallel SC has a declination equal to the arc QC of the celestial meridian. The arc QC is equal to the addition of the geographic latitude of the observer to 90°.
Rice. 1. 22. Conditions for sunrise and sunset
Consequently, in the Northern Hemisphere, non-setting luminaries will be those luminaries whose declination is equal to or greater than the addition of the observer’s latitude to 90°, i.e. For the Southern Hemisphere, these luminaries will be non-rising.
Non-rising luminaries in the Northern Hemisphere will be those luminaries whose daily parallels lie between the MU parallel and the South celestial pole. 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 not set. All other luminaries will be rising and setting. In order for a luminary to rise and set, its declination must be in absolute value less than 90° minus the latitude of the observer, i.e.
Example 1. Star Aliot: declination of the star, latitude of the observer’s position. Determine what kind of rising and setting conditions this star is at the specified latitude.
Solution 1. Find the difference
2. We 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 setting.
Example 2. Star Sirius; declination of the star latitude of the observer’s place Determine what kind of rising and setting conditions a given star is at the specified latitude.
Solution 1. Find the negative difference since the star
Sirius has a negative declination
2. We compare the declination of the star with the resulting difference. Since the star Sirius at the indicated latitude is non-rising.
Example 3. Star Arcturus: declination of the star, latitude of the observer’s position. Determine what kind of rising and setting conditions this star is at the specified latitude.
Solution 1. Find the difference
2. We compare the declination of the star with the resulting difference. Since the star Arcturus rises and sets at the indicated latitude.
Let it be on RPS. The 11th 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 equal to yes; the declination of the luminary Mie, located at the upper culmination, is measured by the arc QAlr. Denoting its zenith distance by r, we obtain for the luminary, culmination 1, k, ul- lest (,* 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 addition to 90°, we get the height
stars h at the moment of the upper cul- ,
Minatspp. p M, Z
Finally, if b - e, then the star at the upper culmination passes through the zenith.
The height of the luminary (UM) is also simply determined at the lower M, culmination, i.e. at the moment of its passage through the meridian between the celestial pole (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 ДШ2 and is equal to h2 - NP-М2Р. Arc arc M2P-p2,
i.e., the distance of the star from the pole. Since p2 = 90 - 52> then
h2 = y-"ri2 - 90°. (3)
Formulas (1), (2) and (3) have extensive applications.
Chapter Exercises /
1. Prove that the equator intersects the horizon at points 90° from the points north and south (at points east and west).
2. What are the hour angle and zenith azimuth?
3. What are the declination and hourly angle of the west point and the east point?
4. What kind of plane does the equator form with the horizon at latitude -(-55°? -)-40°?
5. Is there a difference between the north celestial pole and the north point?
6. Which point on the celestial equator is highest above the horizon? What is the zenith distance of this point for latitude?<р?
7. If a star rose at a point in the northeast, at what point on the horizon will it set? What are the azimuths of the sunrise and sunset points?
8. What is the azimuth of the star at the time of the upper culmination for a place under 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 zenith foliation for a place with latitude o? declination of the north point? points of the south?
11. In what direction does the star move at the lower culmination?
12. The polar star is 1° 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 latitude cf? Same for the lower climax?
14. What condition must the declination S of a star satisfy in order for it to not set at latitude 9? so that it is non-ascending?
15. What caused the angular radius of the circle of setting stars in Leningrad (“p = - 9°57”)?” In Tashkent (срг-41ъ18")? "
16. What is the declination of the stars passing through the zenith in Leningrad and Tashkent? Are they suitable for these cities?
17. At what zenith distance does the Capella star pass through the upper culmination (i - -\-45°5T) in Leningrad? in Tashkent?
18. To what declination are the stars of the southern hemisphere visible in these cities?
19. From what latitude can you see Canopus, the brightest star in the sky after Sirius (about - - 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 never setting star?
20. What is the height of the Chapel at the lower culmination in Moscow = + 5-g<°45")? в Ташкенте?
21. Why are right ascensions counted from west to east, and not in the opposite direction?
22. The two brightest stars in the northern sky are Vega (a=18ft 35t) and Capella (g -13da). In which side of the sky (western or eastern) and at what hour angles are they located at the moment of the upper culmination of the vernal equinox? At the moment of the lower climax of the same point?
23. What interval of stellar time passes from the lower culmination of Capella to the upper culmination of Bern?
24. What hour angle does the Capella have at the moment of the upper culmination of Bega? At the moment of her lowest climax?
25. At what time in sidereal time does the vernal equinox rise? coming in?
26. Prove that for an observer at the earth’s equator, the azimuth of the star at the time of sunrise (AE) and at the time of sunset (A^r) is very simply related to the declination of the star (i).
Let's look at Figure 12. We see that the height of the celestial pole above the horizon is h p =∠PCN, and the geographic 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 provides the simplest way to determine the geographic latitude of an 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 an 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 geographic latitude of the observation site, the orientation of the axis of rotation of the celestial sphere relative to the horizon changes. Let's consider what the visible movements of the celestial bodies will be in the area of the North Pole, at the equator and at the middle latitudes of the Earth.
At the Earth's pole the celestial pole is at the zenith, and the stars move in circles parallel to the horizon (Fig. 14, a). Here the stars do not set or rise, their height above the horizon is constant.
At mid-latitudes exist as ascending And coming in stars, and those that never fall below the horizon (Fig. 14, b). For example, circumpolar constellations (see Fig. 10) never set at the geographic 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, within a day one could see the constellations of the entire starry sky (Fig. 14, c).
For an observer at the equator, all stars rise and set perpendicular to the horizon. Each star here passes exactly half of its path above the horizon. For him, the north pole of the world coincides with the point of north, and the south pole of the world coincides with the point of south. The world axis is located in the horizontal plane (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 located 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 mid-latitudes?
Task 2
Using an eclimeter, measure the geographic latitude of your area based on the height of the North Star and compare it with the latitude reading on a geographic map.
3. The height of the luminaries at the climax
The celestial pole, with the apparent rotation of the sky, reflecting the rotation of the Earth around its axis, occupies a constant position above the horizon at a given latitude (see Fig. 12). Over the course of a 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 per day (Fig. 15).
The phenomena of the passage of luminaries through the celestial meridian relative to the horizon are called culminations. At the upper culmination the height of the luminary is maximum, and at the lower culmination it is minimum. The time interval between climaxes is half a day.
U not coming in at a given latitude φ of the luminary M (see Fig. 15), both culminations are visible (above the horizon); for stars that rise and set (M 1, M 2, M 3), the lower culmination occurs below the horizon, below the north point. For the luminary M4, located far south of the celestial equator, both culminations may be invisible (the luminary non-ascending).
The moment of the upper culmination of the center of the Sun is called true noon, and the moment of the lower culmination is called true midnight.
Let us find the relationship between the height h of the luminary M at the upper culmination, its declination δ and the latitude of the area φ. To do this, we 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 celestial pole above the horizon is equal to the geographic latitude of the place, i.e. h p =φ. Consequently, the angle between the noon line NS and the world axis PP" is equal to the latitude of the area φ, i.e. ∠PON=h p =φ. Obviously, 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 a height at the upper culmination
From this formula it is clear that geographic latitude can be determined by measuring the altitude of any star with a known declination δ at the upper culmination. It should be taken into account that if the star at the moment of culmination is located south of the equator, then its declination is negative.
Example of problem solution
Task. Sirius (α B. Canis, see Appendix IV) was at its highest culmination at an altitude of 10°. What is the latitude of the observation site?
Please ensure that the drawing exactly matches the conditions of the problem.
Exercise 3
When solving problems, the geographic coordinates of cities can be calculated from a geographic map.
1. At what altitude in Leningrad is the upper culmination of Antares (α Scorpio, see Appendix IV)?
2. What is the declination of the stars that culminate at their zenith in your city? at the point south?
3. Prove that the height of the star at the lower culmination is expressed by the formula h=φ+δ-90°.
4. What condition must the declination of a star satisfy in order for it to be non-setting for a place with geographic latitude φ? non-ascending?
