Astronomy assignments. Astronomy Olympiad school tour tasks with solutions. Self-study assignments in astronomy

Keys to Olympiad tasks in astronomy 7-8 CLASS

Objective 1. An astronomer on Earth is observing a total lunar eclipse. What can an astronaut observe on the Moon at this time?

Decision: If there is a total lunar eclipse on Earth, an observer on the Moon will be able to see a total solar eclipse - the Earth will cover the solar disk.

Objective 2. What evidence of the sphericity of the Earth could have been known to ancient scientists?

Decision: Evidence of the sphericity of the Earth, known to ancient scientists:

the rounded shape of the edge of the earth's shadow on the lunar disk during lunar eclipses;

gradual appearance and disappearance of ships as they approach and move away from the coast;

change in the height of the Polar Star when changing the latitude of the place of observation;

removing the horizon as you go up, for example, to the top of a lighthouse or tower.

Objective 3.

On an autumn night, a hunter goes to the forest in the direction of the North Star. Immediately after the sun rises, he comes back. How should a hunter navigate by the position of the sun?

Decision: The hunter walked north into the forest. Returning, he must move south. Since the Sun is near the equinox in the fall, it rises near the point to the east. Therefore, you need to walk so that the Sun is on the left.

Problem 4.

What luminaries are visible during the day and under what conditions?

Decision: The Sun, Moon and Venus are visible to the naked eye, and the stars are up to 4 m - using a telescope.

Task 5. Determine which celestial objects do not change right ascension, declination, azimuth and altitude due to the daily rotation of the Earth? Do such objects exist? Give an example:

Decision: If the star is located at the North or South Pole of the world, all four coordinates for an observer anywhere on Earth will be unchanged due to the rotation of the planet around its axis. Near the North Pole of the world there is such a star - Polaris.

Keys to Olympiad tasks in astronomy CLASS 9

Objective 1. The steamer, leaving Vladivostok on Saturday 6 November, arrived in San Francisco on Wednesday 23 November. How many days did he travel?

Decision: The steamer on its way to San Francisco crossed the date line from west to east, with one day subtracted. The number of days on the way is 23 - (6 - 1) \u003d 18 days.

Objective 2. The height of a star located on the celestial equator at the moment of its upper climax is 30. What is the height of the Pole of the World at the place of observation? (You can draw a picture for clarity).

Decision: If the star is at the highest climax at the celestial equator,h = 90 0 - . Therefore, the latitude of the place  \u003d 90 0 – h = 60 0 ... The height of the Pole of the World is equal to the latitudeh p =  = 60 0

Problem 3 . On March 4, 2007, a total lunar eclipse occurred. What and where was the moon in the sky two weeks after sunset?

Decision . A lunar eclipse is observed during the full moon phase. Since a little less than two weeks pass between the full moon and the new moon, two weeks immediately after sunset, the moon will be visible as a narrow crescent above the horizon in its western side.

Problem 4 . q = 10 7 J / kg, mass of the Sun 2 * 10 30 kg, and luminosity 4 * 10 26

Decision . Q = qM = 2*10 37 t = Q: L = 2 *10 37 /(4* 10 26 )= 5 * 10 10

Task 5. How to prove that the Moon is not made of cast iron if it is known that its mass is 81 times less than the mass of the Earth, and its radius is about four times less than that of the Earth? Read the density of cast iron approximately 7 times the density of water.

Decision . The simplest thing is to determine the average density of the Moon and compare it with the tabular density value for different materials: p \u003dm/ V. Then, substituting the mass and volume of the Moon into this expression in fractions of terrestrial dimensions, we get: 1/81: 1/4 3 \u003d 0.8 The average density of the Moon is only 0.8 of the Earth's density (or 4.4 g / cm 3 -the true value of the average density of the moon 3.3 g / cm 3 ). But even this value is less than the density of cast iron, which is approximately 7g / cm 3 .

Keys to Olympiad tasks in astronomy 10-11 CLASS

Objective 1. The sun rose at the North Pole on the meridian of Yekaterinburg (λ \u003d 6030` E). Where (approximately) will it rise next?

Decision: With the sunrise at the North Pole, the polar day began. The next time the Sun will rise at the beginning of the next polar day, i.e. exactly one year later.

If the Earth made a whole number of revolutions around its axis in a year, then the next sunrise would also be on our meridian. But the Earth makes about a quarter more turnover (leap year is taken from here).

This quarter of a turn corresponds to a 90 0 and since its rotation is from west to east, the sun will rise on the meridian with a longitude of 60.5 0 v.d. - 90 0 = - 29.5 0 , i.e. 29.5 0 h.d. The eastern part of Greenland is located at this longitude.

Objective 2. The travelers noticed that local time, the eclipse of the moon began at 5 hours 13 minutes, while according to the astronomical calendar, this eclipse should begin at 3 hours 51 minutes GMT. What is the geographic longitude of the travelers' observation site?

Decision: The difference in geographic longitudes of two points is equal to the difference in local times of these points. In our problem, we know the local time at the point where the eclipse of the moon was observed at 5 hours 13 minutes and the local Greenwich (Universal) time of the beginning of the same eclipse at 3 hours 51 minutes, i.e. the local time of the prime meridian.

The difference between these times is 1 hour 22 minutes, which means that the longitude of the place of observation of the lunar eclipse is 1 hour 22 minutes east longitude, because the time at this longitude is longer than Greenwich.

Objective 3. At what speed and in what direction should the plane fly at the latitude of Yekaterinburg so that the local solar time for the plane's passengers stops?

Decision: The plane must fly west at the speed of the Earth's rotationV \u003d 2πR/ T

At the latitude of YekaterinburgR = R eq  cos ,  E  57 0

V \u003d 2π  6371 cos 57 0 / 24  3600 \u003d 0.25 km / s

Problem 4. At the end of the XIX century. Some scientists believed that the source of the sun's energy was the chemical reactions of combustion, in particular, the combustion of coal. Assuming that the specific heat of combustion of coalq = 10 7 J / kg, mass of the Sun 2 * 10 30 kg, and luminosity 4 * 10 26 Wet, please provide strong evidence that this hypothesis is wrong.

Decision: Heat reserves without oxygen areQ = qM = 2 *10 37 J. This supply is enough for a whilet = Q: L = 2* 10 37 / 4* 10 26 = 5* 10 10 c \u003d 1700 years. Julius Caesar lived more than 2000 years ago, dinosaurs froze out about 60 million years ago, so that due to chemical reactions the sun cannot shine. (If someone mentions a nuclear power source, that would be great.)

Task 5. Try to find a complete answer to the question: under what conditions does the change of day and night occur nowhere on the planet.

Decision: So that the change of day and night does not occur anywhere on the planet, three conditions must be met simultaneously:

a) the angular velocities of the orbital and axial rotation must coincide (the length of the year and the sidereal day are the same),

b) the axis of rotation of the planet must be perpendicular to the plane of the orbit,

c) the angular velocity of orbital motion must be constant, the planet must have a circular orbit.

Tasks.

I. Introduction.

2. Telescopes.

1. Diameter of the refractor objective D \u003d 30 cm, focal length F \u003d 5.1 m. What is the theoretical resolution of the telescope? What magnification will you get with a 15mm eyepiece?

2. On June 16, 1709, according to the old style, the army led by Peter I defeated the Swedish army of Charles XII near Poltava. What is the date of this historic event on the Gregorian calendar?

5. The composition of the solar system.

1. What celestial bodies or phenomena in ancient times were called "wandering star", "hairy star", "shooting star". What was it based on?

2. What is the nature of the solar wind? What celestial phenomena does it cause?

3. How can one distinguish an asteroid from a star in the starry sky?

4. Why does the number density of craters on the surface of Jupiter's Gallilean satellites monotonically increase from Io to Callisto?

II. Mathematical models. Coordinates.

1. Using a moving map of the starry sky, determine the equatorial coordinates of the following objects:

a) α Dragon;

b) The Orion Nebula;

c) Sirius;

d) the Pleiades star cluster.

2. As a result of the precession of the earth's axis, the North Pole of the world describes a circle along the celestial sphere for 26000 years centered at a point with coordinates α \u003d18h δ \u003d + 67º. Determine which bright star will become polar (near the north pole of the world) in 12,000 years.

3. At what maximum height above the horizon can the Moon be observed in Kerch (φ \u003d 45 º)?

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

a) α \u003d 15 h 12 min δ \u003d - 9˚;

b) α \u003d 3 h 40 min δ \u003d + 48˚.

5. At what altitude does the upper culmination of the star Altair (α Eagle) occur in St. Petersburg (φ \u003d 60˚)?

6. Determine the declination of a star if in Moscow (φ \u003d 56˚) it culminates at an altitude of 57˚.

7. Determine the range of latitudes in which polar day and polar night can be observed.

8. Determine the visibility condition (declination range) for VZ - ascending-setting stars, NS - non-setting, NV - non-ascending at different latitudes, corresponding to the following positions on Earth:

9. How has the position of the Sun changed from the beginning of the school year to the day of the Olympiad, determine its equatorial coordinates and the height of the culmination in your city today.

10. Under what conditions will the seasons not change on the planet?

11. Why is the Sun not assigned to any constellation?

12. Determine the geographic latitude of the place where the star Vega (α Lyrae) may be at its zenith.

13. In what constellation is the Moon, if its equatorial coordinates are 20 h 30 min; -18º? Determine the date of observation, as well as the moments of its rising and setting, if the moon is known to be at a full moon.

14. On what day were the observations carried out if it is known that the midday height of the Sun at the geographic latitude 49º was 17º30´?

15. Where is the Sun higher at noon: in Yalta (φ \u003d 44º) on the vernal equinox or in Chernigov (φ \u003d 51º) on the summer solstice?

16. What astronomical instruments can be found on the star chart in the form of constellations? What other devices and mechanisms are named?

17. A hunter in the fall goes to the forest at night in the direction of the Pole Star. After the sunrise, he comes back. How should a hunter move for this?

18. At what latitude will the Sun climax at noon at 45º on April 2?

III. Elements of mechanics.

1. Yuri Gagarin on April 12, 1961, ascended to an altitude of 327 km above the Earth's surface. By what percentage has the force of gravity of the astronaut to the Earth decreased?

2. At what distance from the center of the Earth should there be a stationary satellite orbiting in the plane of the Earth's equator with a period equal to the period of the Earth's revolution.

3. The stone was thrown to the same height on Earth and on Mars. Will they simultaneously descend on the surface of the planets? A speck of dust?

4. The spacecraft landed on an asteroid with a diameter of 1 km and an average density of 2.5 g / cm3 ... The astronauts decided to go around the asteroid along the equator in an all-terrain vehicle in 2 hours. Will they be able to do it?

5. The explosion of the Tunguska meteorite was observed on the horizon in the city of Kirensk, 350 km from the explosion site. Determine at what height the explosion occurred.

6. At what speed and in what direction should the plane fly in the equatorial region to stop the solar time for the plane passengers?

7. At what point of the comet's orbit is its kinetic energy maximum, and at what point is it minimum? And the potential one?

IV. Planet configurations. Periods.

12. Configurations of planets.

1. Determine for the positions of the planetsa, b, c, d, e, f marked on the diagram, the corresponding descriptions of their configurations. (6 points)

2. Why is Venus called the morning and the evening star?

3. “After sunset, it began to get dark quickly. The first stars had not yet lit up in the dark blue sky, and Venus was already blindingly shining in the east. Is everything correct in this description?

13. Sidereal and synodic periods.

1. The stellar period of Jupiter's orbital is 12 years. After what period of time are his confrontations repeated?

2. It is noticed that oppositions of some planet are repeated after 2 years. What is the semi-major axis of its orbit?

3. The synodic period of the planet is 500 days. Determine the semi-major axis of its orbit.

4. After what time interval are the oppositions of Mars repeated if the stellar period of its revolution around the Sun is 1.9 years?

5. What is the period of revolution of Jupiter, if its synodic period is 400 days?

6. Find the average distance of Venus from the Sun if its synodic period is 1.6 years.

7. The period of revolution around the Sun of the shortest-period comet Encke is 3.3 years. Why are the conditions of its visibility repeated with a characteristic period of 10 years?

V. Moon.

1.October 10 saw a lunar eclipse. What date will the moon be in the first quarter?

2. Today the moon rose at 2000 when to expect her sunrise the day after tomorrow?

3. What planets can be seen near the Moon during the full moon?

4. What are the names of scientists, whose names are present on the map of the moon.

5. In what phase and at what time of day the Moon was observed by Maximilian Voloshin, described by him in a poem:

The earth will not destroy the reality of our dreams:

In the park of rays dawns quietly melt,

The murmur of mornings will merge in the daytime chorus,

the flawed sickle will decay and burn ...

6. When and on which side of the horizon is it better to observe the Moon a week before the lunar eclipse? Until sunny?

7. In the encyclopedia "Geography" it is written: "Only twice a year the Sun and the Moon rise and set exactly in the east and west - on the days of the equinoxes: March 21 and September 23". Is this statement true (perfectly true, more or less true, generally false)? Give an extended explanation.

8. Is the full Earth always visible from the surface of the Moon, or does it, like the Moon, undergo a successive phase change? If there is such a change in the earth's phases, then what is the relationship between the phases of the moon and the earth?

9. When will Mars be brighter in conjunction with the Moon: in the first quarter or in the full moon?

Vi. The laws of planetary motion.

17. Kepler's First Law. Ellipse.

1. The orbit of Mercury is essentially elliptical: the planet's perihelion distance is 0.31 AU, the aphelion distance is 0.47 AU. Calculate the semi-major axis and eccentricity of Mercury's orbit.

2. The perihelion distance of Saturn to the Sun is 9.048 AU, aphelion 10.116 AU. Calculate the semi-major axis and eccentricity of Saturn's orbit.

3. Determine the height of the IZS moving at an average distance from the Earth's surface of 1055 km, at the points of perigee and apogee, if the eccentricity of its orbit is e \u003d 0.11.

4. Find the eccentricity from the known a and b.

18. Kepler's Second and Third Laws.

2. Determine the orbital period of an artificial Earth satellite if highest point its orbit above the Earth is 5000 km, and the lowest is 300 km. Consider the earth as a ball with a radius of 6370 km.

3. Halley's comet makes a complete revolution around the Sun in 76 years. At the point of its orbit closest to the Sun, at a distance of 0.6 AU. from the Sun, it moves at a speed of 54 km / h. At what speed does it move at the point of its orbit farthest from the Sun?

4. At what point of the comet's orbit is its kinetic energy maximum, and at what point is it minimum? And the potential one?

5. The period between two oppositions of the celestial body is 417 days. Determine its distance from the Earth in these positions.

6. The greatest distance from the Sun to a comet is 35.4 AU, and the smallest 0.6 AU. The last passage was observed in 1986. Could the Star of Bethlehem be this comet?

19. Refined Kepler's law.

1. Determine the mass of Jupiter by comparing the Jupiter system with the satellite with the Earth-Moon system, if the first satellite of Jupiter is 422,000 km away from it and has an orbital period of 1.77 days. The data for the Moon should be known to you.

2 Calculate at what distance from the Earth on the Earth - Moon line are those points at which the attraction of the Earth and the Moon is the same, knowing that the distance between the Moon and the Earth is 60 Earth radii, and the Earth and Moon masses are 81: 1.

3. How would the duration of the terrestrial year change if the mass of the Earth were equal to the mass of the Sun, and the distance would remain the same?

4. How will the length of the year on Earth change if the Sun turns into a white dwarf with a mass equal to 0.6 that of the Sun?

Vii. Distances. Parallax.

1. What is the angular radius of Mars in opposition, if its linear radius is 3 400 km, and the horizontal parallax is 18 ′ ′?

2. On the Moon from Earth (distance 3.8 * 105 km) with the naked eye, objects 200 km long can be distinguished. Determine what size objects will be visible on Mars with the naked eye during the opposition.

3. Parallax Altair 0.20 ′ ′. What is the distance to a star in light years?

4. The galaxy located at a distance of 150 Mpc has an angular diameter of 20 ″. Compare its linear dimensions of our Galaxy.

5. How long does it take for a spacecraft flying at a speed of 30 km / h to reach the closest star to the Sun, Proxima Centauri, whose parallax is 0.76 ′ ′?

6. How many times is the Sun larger than the Moon, if their angular diameters are the same, and the horizontal parallaxes are, respectively, 8.8 ″ and 57 ′?

7. What is the angular diameter of the Sun as seen from Pluto?

8. What is the linear diameter of the Moon, if it is visible from a distance of 400,000 km at an angle of approximately 0.5˚?

9. How many times more energy is received from the Sun for each square meter of the surface of Mercury than Mars? Take the necessary data from the applications.

10. At what points of the firmament does the terrestrial observer see the star, being at points B and A (Fig. 37)?

11. In what ratio does the angular diameter of the Sun, visible from Earth and Mars, numerically change from perihelion to aphelion, if the eccentricities of their orbits are respectively equal to 0.017 and 0.093?

12. Are the same constellations visible from the Moon (are they visible in the same way) as from the Earth?

13. At the edge of the moon, a 1 "pronged mountain is visible. Calculate its height in kilometers.

14. Using the formulas (§ 12.2), determine the diameter of the lunar circus Alphonse (in km) by measuring it in Figure 47 and knowing that the angular diameter of the Moon, as seen from Earth, is about 30 ′, and the distance to it is about 380,000 km.

15. Objects 1 km in size are visible from the Earth on the Moon through a telescope. What is the smallest size of detail visible from Earth on Mars with the same telescope during opposition (at a distance of 55 million km)?

VIII. Wave nature of light. Frequency. Doppler effect.

1. The wavelength corresponding to the hydrogen line is longer in the spectrum of the star than in the spectrum obtained in the laboratory. Is a star moving towards us or away from us? Will there be a shift in the spectral lines if the star moves across the line of sight?

2. In the photograph of the spectrum of the star, its line is shifted relative to its normal position by 0.02 mm. How much has the wavelength changed if a distance of 1 mm in the spectrum corresponds to a change in wavelength of 0.004 μm (this value is called the dispersion of the spectrogram)? How fast is the star moving? Normal wavelength 0.5 μm \u003d 5000 Å (angstroms). 1 Å \u003d 10-10 m.

IX. Stars.

22. Characteristics of the stars. Pogson's law.

1. How many times is Arcturus larger than the Sun if the luminosity of Arcturus is 100 and the temperature is 4500 K? The sun's temperature is 5807 K.

2. How many times does the brightness of Mars change if its apparent magnitude ranges from +2.0m to -2.6 m?

3. How many stars of the Sirius type (m \u003d -1.6) would it take for them to shine the same way as the Sun?

4. The best modern ground-based telescopes have objects up to 26m ... How many times weaker objects can they fix in comparison with the naked eye (the limiting magnitude is taken as 6m)?

24. Classes of stars.

1. Draw the evolutionary path of the Sun on the Hertzsprung-Russell diagram. Please explain.

2. The spectral types and parallaxes of the following stars are given. Distribute them

a) in order of decreasing temperature, indicate their colors;

b) in order of distance from the Earth.

25. Evolution of the stars.

1. Under what processes in the Universe are heavy chemical elements formed?

2. What determines the rate of evolution of a star? What are the possible end stages of evolution?

3. Draw a qualitative graph of the brightness variation of a binary star if its components are the same size, but the companion has a lower brightness.

4. At the end of its evolution, the Sun will begin to expand and turn into a red giant. As a result, the temperature of its surface will drop by half, and the luminosity will increase 400 times. Will the sun swallow up any of the planets?

5. In 1987, a supernova explosion was registered in the Large Magellanic Cloud. How many years ago did the explosion occur if the distance to the BMO is 55 kiloparsecs?

H. Galaxies. Nebulae. Hubble's Law.

1. The redshift of the quasar is 0.8. Assuming that the motion of a quasar obeys the same pattern as galaxies, taking the Hubble constant H \u003d 50 km / sec * Mpc, find the distance to this object.

2. Compare the relevant items regarding the type of object.

Examples of solving problems in astronomy

§ 1. The star Vega is located at a distance of 26.4 s. years from Earth. How many years would a rocket fly towards it at a constant speed of 30 km / s?

The rocket's speed is 10 0 0 0 times less than the speed of light, so astronauts will fly to Run 10,000 times longer.

Solutions:

§ 2. At noon your shadow is half your height. Determine the height of the Sun above the horizon.

Solutions:

Sun height h measured by the angle between the plane of the horizon and the direction to the luminary. From a right-angled triangle, where the legs areL (shadow length) and H (your height), we find

§ 3. How much does the local time in Simferopol differ from Kiev time?

Solutions:

In winter

That is, in winter, local time in Simferopol is ahead of Kiev time. In spring, the hands of all clocks in Europe are moved 1 hour forward, so Kiev time is 44 minutes ahead of local time in Simferopol.

§ 4. The asteroid Amur moves along an ellipse with an eccentricity of 0.43. Could this asteroid collide with the Earth if its period of rotation around the Sun is 2.66 years?

Solutions:

An asteroid can collide with Earth if it crosses orbitEarth, that is, if the distance at perihelionrmin \u003d< 1 а. o .

Using Kepler's third law, we determine the semi-major axis of the asteroid's orbit:

where a 2 - 1 a. o .- semi-major axis of the Earth's orbit;T 2 \u003d 1 year-period

rotation of the Earth:

Figure: P. 1.

Answer.

Asteroid Cupid will not cross the Earth's orbit, so it cannot collide with the Earth.

§ 5. At what height above the Earth's surface a geostationary satellite should rotate, hanging over one pointEarth?

Rose LS (X - H LIL

1.Using Kepler's third law determine the semi-major axis of the satellite orbit:

where a2 \u003d 3 80,000 km is the semi-major axis of the Moon's orbit; 7i, \u003d 1 day - the period of rotation of the satellite around the Earth; T "2 \u003d 27.3 days - the period of the Moon's revolution around the Earth.

a1 \u003d 41900 km.

Answer. Geostationary satellites rotate from west to east in the equatorial plane at an altitude of 35,500 km.

§ 6. Can cosmonauts see the Black Sea with the naked eye from the surface of the Moon?

Rosv "yazannya:

Determine the angle at which the Black Sea is visible from the Moon. From a right-angled triangle, in which the legs are the distance to the Moon and the diameter of the Black Sea, we determine the angle:

Answer.

If it is day in Ukraine, then the Black Sea can be seen from the Moon, because its angular diameter is greater than the resolving power of the eye.

§ 8. On the surface of which planet of the terrestrial group will the weight of the astronauts be the least?

Solutions:

P \u003d mg; g \u003d GM / R 2,

where G - gravitational constant; M is the mass of the planet,R is the radius of the planet. The least weight will be on the surface of the planet where the acceleration of the freefalling. From the formulag \u003d GM / R we determine that on Mercury # \u003d 3.78 m / s2, on Venus # \u003d 8.6 m / s2, on Mars # \u003d 3.72 m / s2, on Earth # \u003d 9.78 m / s2.

Answer.

The weight will be the smallest on Mars, 2.6 times less than on Earth.

§ 12. When, in winter or summer, more solar energy gets into your window at noon? Consider the cases: A. The window faces south; B. The window faces east.

Solutions:

A. The amount of solar energy that a unit of surface receives per unit of time can be calculated using the following formula:

E \u003d qcosi

where q - solar constant; i is the angle of incidence of the sun's rays.

The wall is located perpendicular to the horizon, so in winter the angle of incidence of sunlight will be less. So, oddly enough, in winter more energy comes from the sun into your apartment window than in summer.

Would. If the window faces east, then sun rays at noon never light up your room.

§ 13. Determine the radius of the star Vega, which emits 55 times more energy than the Sun. The surface temperature is 1 1000 K. What kind would this star have in our sky if it shone in the place of the Sun?

Solutions:

The radius of the star is determined using the formula (13.11):

where Др, \u003d 6 9 5 202 km is the radius of the Sun;

Sun surface temperature.

Answer.

The star Vega has a radius 2 times greater than that of the Sun, so in our sky it would look like a blue disk with an angular diameter of 1 °. If Vega shone instead of the Sun, then the Earth would receive 55 times more energy than it is now, and the temperature on its surface would be above 1000 ° C. Thus, conditions on our planet would become unsuitable for all forms of life.

". On our site you will find the theoretical part, examples, exercises and answers to them, subdivided into 4 main categories, for the convenience of using the site. These sections cover: the basics of spherical and practical astronomy, the basics of theoretical astronomy and celestial mechanics, the basics of astrophysics and the characteristics of telescopes.

By clicking the mouse on the right side of our site on any of the subsections in 4 categories, you will find in each of them the theoretical part, which we advise you to study before the crime to direct problem solving, then you will find the "Examples" item, which we have added for better understanding theoretical part, directly the exercises themselves to consolidate and expand your knowledge in these areas and also the "Answers" item to test the knowledge gained and correct errors.

Perhaps, at first glance, some tasks will seem outdated, since the geographical names of the countries, regions and cities mentioned on the site have changed over time, while the laws of astronomy have not undergone any changes. Therefore, in our opinion, the collection contains many useful information in theoretical parts that contain timeless information available in the form of tables, graphs, charts and text. Our site gives you the opportunity to start learning astronomy from scratch and continue learning by solving problems. The collection will help you lay the foundations for hobby astronomy and maybe one day you will discover a new star or fly to the nearest planet.

FOUNDATIONS OF SPHERICAL AND PRACTICAL ASTRONOMY

The culmination of the luminaries. View of the starry sky at different geographic parallels

At every place on the earth's surface, the height hp of the pole of the world is always equal to the geographical latitude φ of this place, i.e., hp \u003d φ (1)

and the plane of the celestial equator and the planes of celestial parallels are inclined to the plane of the true horizon at an angle

Azimuth "href \u003d" / text / category / azimut / "rel \u003d" bookmark "\u003e azimuth AB \u003d 0 ° and hour angle tB \u003d 0 ° \u003d 0h.

Figure: 1. The upper culmination of the luminaries

When δ\u003e φ, the luminary (M4) in the upper culmination crosses the celestial meridian north of the zenith (above the north point Ν), between the zenith Z and the north pole of the world P, and then the zenith distance of the star

height hв \u003d (90 ° -δ) + φ (7)

azimuth AB \u003d 180 °, and hour angle tB \u003d 0 ° \u003d 0h.

At the moment of the lower culmination (Fig. 2), the luminary crosses the celestial meridian under the north pole of the world: a non-setting star (M1) -above the north point N, a setting star (M2 and M3) and a non-ascending star (M4) -under the north point. In the lower climax, the height of the luminary

hн \u003d δ- (90 ° -φ) (8)

its zenith distance zn \u003d 180 ° -δ-φ (9)

), at latitude φ \u003d + 45 ° 58 "and at the Arctic Circle (φ \u003d + 66 ° 33"). Capella declination δ \u003d + 45 ° 58 ".

Data: Capella (α Auriga), δ \u003d + 45 ° 58 ";

northern tropic, φ \u003d + 23 ° 27 "; place with φ \u003d + 45 ° 58";

arctic Circle, φ \u003d + 66 ° 33 ".

Decision: The Capella declination δ \u003d + 45 ° 58 "\u003e φ of the northern tropic, and therefore you should use formulas (6) and (3):

zv \u003d δ-φ \u003d + 45 ° 58 "-23 ° 27" \u003d 22 ° 31 "N, hv \u003d 90 ° -zv \u003d 90 ° -22 ° 31" \u003d + 67 ° 29 "N;

therefore, the azimuth is Av \u003d 180 °, and the hour angle tv \u003d 0 ° \u003d 0h.

At the geographic latitude φ \u003d + 45 ° 58 "\u003d δ the zenith distance of the Capella is zв \u003d δ-φ \u003d 0 °, that is, in the upper culmination it is at the zenith, and its height is hв \u003d + 90 °, the hour angle tв \u003d 0 ° \u003d 0h, and AB azimuth is undefined.

The same values \u200b\u200bfor the Arctic Circle are calculated by formulas (4) and (3), since the declination of the star δ<φ=+66°33":

zv \u003d φ-δ \u003d + 66 ° 33 "-45 ° 58" \u003d 20 ° 35 "S, hv \u003d 90 ° -zv \u003d + 90 ° -20 ° 35" \u003d + 69 ° 25 "S, and therefore Av \u003d 0 ° and tv \u003d 0 ° \u003d 0h,

Calculations of the height hн and zenith distance zн Chapels in the lower culmination are carried out according to formulas (8) and (3): in the northern tropics (φ \u003d + 23 ° 27 ")

hн \u003d δ- (90 ° -φ) \u003d + 45 ° 58 "- (90 ° -23 ° 27") \u003d -20 ° 35 "N,

i.e., at the bottom climax, the Capella goes beyond the horizon, and its zenith distance

zн \u003d 90 ° -hн \u003d 90 ° - (- 20 ° 35 ") \u003d 110 ° 35" N, azimuth An \u003d 180 ° and hour angle tн \u003d 180 ° \u003d 12h,

At the geographic latitude φ \u003d + 45 ° 58 "near the star hн \u003d δ- (90 ° -φ) \u003d + 45 ° 58" - (90 ° -45 ° 58 ") \u003d + 1 ° 56" N,

that is, it is already non-setting, and its zn \u003d 90 ° -hn \u003d 90 ° -1 ° 56 "\u003d 88 ° 04" N, An \u003d 180 ° and tn \u003d 180 ° \u003d 12h

In the Arctic Circle (φ \u003d + 66 ° 33 ")

hн \u003d δ- (90 ° -φ) \u003d + 45 ° 58 "- (90 ° -66 ° 33") \u003d + 22 ° 31 "N, and zн \u003d 90 ° -hн \u003d 90 ° -22 ° 31" \u003d 67 ° 29 "N,

that is, the star also does not go beyond the horizon.

Example 2.On what geographical parallels does the star Capella (δ \u003d + 45 ° 58 ") not go beyond the horizon, is it never visible, and at the bottom climax passes in nadir?

Data: Capella, δ \u003d + 45 ° 58 ".

Decision. By condition (10)

φ≥ + (90 ° -δ) \u003d + (90 ° -45 ° 58 "), whence φ≥ + 44 ° 02", that is, on the geographic parallel, with φ \u003d + 44 ° 02 "and north of it, up to the North Pole of the Earth (φ \u003d + 90 °), Capella is a non-setting star.

From the condition of symmetry of the celestial sphere, we find that in the southern hemisphere of the Earth the Capella does not rise in areas with a latitude from φ \u003d -44 ° 02 "to the southern geographic pole (φ \u003d -90 °).

According to formula (9), the lower culmination of the Capella in nadir, that is, at zΗ \u003d 180 ° \u003d 180 ° -φ-δ, occurs in the southern hemisphere of the Earth, at the geographical parallel with latitude φ \u003d -δ \u003d -45 ° 58 " ...

Objective 1.Determine the height of the pole of the world and the inclination of the celestial equator to the true horizon at the earth's equator, in the northern tropic (φ \u003d + 23 ° 27 "), at the north polar circle (φ \u003d + 66 ° 33") and at the north geographic pole.

Objective 2.The declination of the star Mizara (ζ Ursa Major) is + 55 ° 11 ". At what zenith distance and at what altitude it is in the upper climax in Pulkovo (φ \u003d + 59 ° 46") and Dushanbe (φ \u003d + 38 ° 33 ") ?

Objective 3.At what is the smallest zenith distance and highest altitude in Evpatoria (φ \u003d + 45 ° 12 ") and Murmansk (φ \u003d + 68 ° 59") the stars Aliot (ε Ursa Major) and Antares (a Scorpio), the declination of which is respectively + 56 ° 14 "and -26 ° 19"? Indicate the azimuth and hour angle of each star at these times.

Problem 4.At some point of observation, a star with a declination of + 32 ° 19 "rises above the point of the south to a height of 63 ° 42". Find the zenith distance and height of this star at the same place at an azimuth of 180 °.

Task 5.Solve the problem for the same star under the condition of its smallest zenith distance 63 ° 42 "north of the zenith.

Task 6.What declination should the stars have in order to pass at the zenith in the upper climax, and in the nadir, the north point and the south point of the observation site, in the lower climax? What is the geographical latitude of these places?

I will again use the brochure "Didactic Material on Astronomy" written by G.I. Malakhova and EK Straut and published by the publishing house "Prosveshchenie" in 1984. This time, the first tasks of the final test on page 75 are being distributed.

To visualize formulas, I will use the LаTeX2gif service, since the jsMath library is not able to draw formulas in RSS.

Task 1 (Option 1)

Condition: The planetary nebula in the constellation Lyra has an angular diameter of 83 ″ and is located at a distance of 660 pc. What are the linear dimensions of the nebula in astronomical units?

Decision: The parameters specified in the condition are related to each other by a simple relationship:

1 pc \u003d 206265 AU, respectively:

Task 2 (Option 2)

Condition: Parallax of the star Procyon 0.28 ″. Distance to the star Betelgeuse 652 St. of the year. Which of these stars and how many times is farther from us?

Decision: Parallax and distance are related by a simple relationship:

Next, we find the ratio of D 2 to D 1 and we get that Betelgeuse is about 56 times farther than Procyon.

Task 3 (Option 3)

Condition: How many times has the angular diameter of Venus observed from Earth changed as a result of the planet moving from the minimum distance to the maximum? Consider the orbit of Venus as a circle with a radius of 0.7 AU.

Decision: We find the angular diameter of Venus for the minimum and maximum distances in astronomical units and then their simple ratio:

We get the answer: decreased by 5.6 times.

Task 4 (Option 4)

Condition: What is the angular size of our Galaxy (with a diameter of 3 × 10 4 pc) an observer in galaxy M 31 (the Andromeda nebula) at a distance of 6 × 10 5 pc?

Decision: The expression connecting the linear dimensions of the object, its parallax and angular dimensions is already in the solution of the first problem. Let's use it and, slightly modifying it, substitute the necessary values \u200b\u200bfrom the condition:

Problem 5 (Option 5)

Condition: The resolution of the naked eye is 2 ′. What size objects can an astronaut discern on the lunar surface, flying over it at an altitude of 75 km?

Decision: The problem is solved similarly to the first and fourth:

Accordingly, the astronaut will be able to distinguish details of the surface of 45 meters.

Problem 6 (Option 6)

Condition: How many times is the Sun larger than the Moon if their angular diameters are the same and the horizontal parallaxes are 8.8 ″ and 57 ′, respectively?

Decision: This is a classic task of determining the size of the stars from their parallax. The formula for the connection between the parallax of a star and its linear and angular dimensions has been repeatedly found above. As a result of reducing the repeating part, we get:

In response, we find that the Sun is almost 400 times larger than the Moon.