Astronomy assignments. Assignments for the municipal stage of the Astronomy Olympiad

Keys to Olympiad assignments in astronomy 7-8 GRADES

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

Solution: 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.

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

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

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

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

change in the altitude of the North Star when changing the latitude of the observation site;

Removing the horizon as you ascend upward, for example, to the top of a lighthouse or tower.

Task 3.

On an autumn night, a hunter walks into the forest in the direction of the North Star. Immediately after sunrise he returns back. How should a hunter navigate by the position of the sun?

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

Task 4.

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

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

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

Solution: If the star is located in 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 assignments in astronomy GRADE 9

Task 1. The steamer, having left Vladivostok on Saturday, November 6, arrived in San Francisco on Wednesday, November 23. How many days was he on the road?

Solution: On its way to San Francisco, the steamer crossed the international date line from west to east, subtracting one day. The number of days on the way is 23 – (6 – 1) = 18 days.

Task 2. The altitude of a star located on the celestial equator at the time of its upper culmination is 30. What is the height of the Celestial Pole at the observation location? (You can make a drawing for clarity).

Solution: If the star is at its highest culmination on the celestial equator,h = 90 0 - . Therefore, the latitude of the place  = 90 0 – h = 60 0 . The height of the Celestial Pole 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 immediately after sunset?

Solution . A lunar eclipse occurs during the full moon phase. Since a little less than two weeks pass between the phases of the full moon and the new moon, then two weeks immediately after sunset, the Moon will be visible in the form of a narrow crescent above the horizon on its western side.

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

Solution . 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 approximately four times less than that of the Earth? Consider the density of cast iron to be approximately 7 times the density of water.

Solution . The simplest thing is to determine the average density of the Moon and compare it with the table density value for different materials: p =m/V. Then, substituting the mass and volume of the Moon into this expression in fractions of Earth sizes, we get: 1/81:1/4 3 =0.8.The average density of the Moon is only 0.8 of the Earth’s density (or 4.4 g/cm 3 -true value of the average density of the Moon 3.3 g/cm 3 ). But this value is also less than the density of cast iron, which is approximately 7g/cm 3 .

Keys to Olympiad assignments in astronomy 10-11 GRADES

Task 1. The sun at the north pole rose on the meridian of Yekaterinburg (λ= 6030` east). Where (approximately) will it rise next?

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

If in a year the Earth made an integer number of revolutions around its axis, then the next sunrise would also be on our meridian. But the Earth makes about a quarter more revolutions (hence the leap year).

This quarter turn corresponds to the rotation of the Earth by 90 0 and since its rotation occurs from west to east, the sun will rise on the meridian with longitude 60.5 0 e.d. – 90 0 = - 29.5 0 , i.e. 29.5 0 w.d. At this longitude is the eastern part of Greenland.

Task 2. Travelers noticed that according to local time the lunar eclipse began at 5 hours 13 minutes, while according to the astronomical calendar this eclipse should begin at 3 hours 51 minutes Greenwich time. What is the geographic longitude of the place where the travelers are observed?

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

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

Task 3. At what speed and in what direction should a plane fly at the latitude of Yekaterinburg for local solar time to stop for the plane's passengers?

Solution: The plane must fly west at the speed of the Earth's rotationV= 2πR/T

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

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

Task 4. At the end of the 19th century. Some scientists believed that the source of the sun's energy was chemical combustion reactions, in particular the combustion of coal. Assuming that the specific heat of combustion of coalq = 10 7 J/kg, solar mass 2 * 10 30 kg, and the luminosity is 4 * 10 26 W, provide strong evidence that this hypothesis is incorrect.

Solution: Heat reserves excluding oxygen areQ = qM = 2 *10 37 J. This supply will last for a whilet = Q: L = 2* 10 37 / 4* 10 26 = 5* 10 10 c = 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 talks about a nuclear power source, that will be great.)

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

Solution: To ensure that there is no change of day and night anywhere on the planet, three conditions must be met simultaneously:

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

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

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

Tasks.

I. Introduction.

2. Telescopes.

1. Refractor lens diameter D = 30 cm, focal length F = 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 historical event according to the Gregorian calendar?

5. Composition of the Solar System.

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

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

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

4. Why does the numerical density of craters on the surface of Jupiter’s Galilean satellites monotonically increase from Io to Callisto?

II. Mathematical models. Coordinates.

1. Using a moving star chart, determine the equatorial coordinates of the following objects:

a) α Dragon;

b) 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 26,000 years with a center at a point with coordinates α =18h δ = +67º. Determine which bright star will become polar (close to 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 = 45 º)?

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

a) α = 15 hours 12 minutes δ = – 9˚;

b) α = 3 hours 40 minutes δ = + 48˚.

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

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

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

8. Determine the visibility condition (declination range) for EO – rising-setting stars, NS – non-setting stars, NV – non-rising stars at various latitudes corresponding to the following positions on Earth:

9. How 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 there be no change of seasons on the planet?

11. Why is the Sun not classified as one of the constellations?

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

13. In what constellation is the Moon located if its equatorial coordinates are 20 hours 30 minutes; -18º? Determine the date of observation, as well as the moments of its rising and setting, if it is known that the Moon is full.

14. On what day were the observations carried out, if it is known that the midday altitude of the Sun at a geographic latitude of 49º turned out to be equal to 17º30´?

15. Where is the Sun higher at noon: in Yalta (φ = 44º) on the day of the spring equinox or in Chernigov (φ = 51º) on the day of the summer solstice?

16. What astronomical instruments can be found on a star map in the form of constellations? And the names of what other devices and mechanisms?

17. A hunter walks into the forest at night in the fall towards the North Star. After sunrise he returns back. How should the hunter move for this?

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

III. Elements of mechanics.

1. Yuri Gagarin on April 12, 1961 rose to a height of 327 km above the surface of the Earth. By what percentage did the astronaut's gravitational force to the Earth decrease?

2. At what distance from the center of the Earth should a stationary satellite be located, orbiting in the plane of the Earth’s equator with a period equal to the period of rotation of the Earth.

3. A stone was thrown to the same height on Earth and on Mars. Will they descend to the surface of the planets at the same time? What about 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/cm 3 . The astronauts decided to travel 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 altitude the explosion occurred.

6. At what speed and in what direction must a plane fly near the equator for solar time to stop for the plane’s passengers?

7. At what point in the comet’s orbit is its kinetic energy maximum and at what point is it minimum? What about potential?

IV. Planetary configurations. Periods.

12. Planetary configurations.

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

2. Why is Venus called either the morning or evening star?

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

13. Sidereal and synodic periods.

1. The sidereal period of Jupiter’s revolution is 12 years. After what period of time are his confrontations repeated?

2. It is noticed that oppositions of a certain planet are repeated after 2 years. What is the semimajor axis of its orbit?

3. The synodic period of the planet is 500 days. Determine the semimajor axis of its orbit.

4. After what period of time do the oppositions of Mars repeat if the sidereal period of its revolution around the Sun is 1.9 years?

5. What is the orbital period 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 do the conditions of its visibility repeat with a characteristic period of 10 years?

V. Moon.

1. On October 10, a lunar eclipse was observed. What date will the Moon be in the first quarter?

2. Today the moon rose at 20 00 when to expect it to rise the day after tomorrow?

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

4. Name the names of the scientists whose names are on the map of the Moon.

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

The earth will not destroy the reality of our dreams:

In the park of rays the dawns are fading quietly,

The murmur of the morning will merge into the daytime chorus,

the damaged sickle will decay and burn...

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

7. The Geography encyclopedia says: “Only twice a year, the Sun and 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 (completely true, more or less true, not at all true)? 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 change of phases? 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 brightest in conjunction with the Moon: in the first quarter or in the full moon?

VI. Laws of planetary motion.

17. Kepler's First Law. Ellipse.

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

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

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

4. Find the eccentricity using known a and b.

18. Kepler's Second and Third Laws.

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

3. Halley's Comet takes 76 years to complete a revolution around the Sun. 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 in the comet’s orbit is its kinetic energy maximum and at what point is it minimum? What about potential?

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

6. The greatest distance from the Sun to the comet is 35.4 AU, and the smallest is 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 a 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 are equal, knowing that the distance between the Moon and the Earth is equal to 60 radii of the Earth, and the masses of the Earth and the Moon are in the ratio 81: 1.

3. How would the length of the earth’s year change if the mass of the Earth were equal to the mass of the Sun, but the distance remained 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 solar masses?

VII. Distances. Parallax.

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

2. On the Moon from Earth (distance 3.8 * 10 5 km) with the naked eye one can distinguish objects with a length of 200 km. Determine what size objects will be visible on Mars with the naked eye during opposition.

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

4. A galaxy located at a distance of 150 Mpc has an angular diameter of 20′′. Compare it with the linear dimensions of our Galaxy.

5. How much time 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 their 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 does each square meter of the surface of Mercury receive from the Sun than that of Mars? Take the necessary data from the applications.

10. At what points in the sky does an earthly observer see the luminary, being at points B and A (Fig. 37)?

11. In what ratio does the angular diameter of the Sun, visible from Earth and from Mars, change numerically 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. On the edge of the Moon, a tooth-shaped mountain 1′′ high is visible. Calculate its height in kilometers.

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

15. From the Earth, objects 1 km in size are visible on the Moon through a telescope. What is the smallest size of features visible from Earth on Mars through 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 the star moving towards us or away from us? Will a shift in the spectrum lines be observed if the star moves across the line of sight?

2. In the photograph of the star’s spectrum, its line is shifted relative to its normal position by 0.02 mm. How much has the wavelength changed if in the spectrum a distance of 1 mm 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 is 0.5 µm = 5000 Å (angstrom). 1 Å = 10-10 m.

IX. Stars.

22. Characteristics of 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 temperature of the Sun is 5807 K.

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

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

4. The best modern ground-based telescopes can reach objects up to 26 m . How many times fainter objects can they detect compared to the naked eye (take the limiting magnitude to be 6 m)?

24. Classes of stars.

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

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

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

b) in order of distance from the Earth.

25. Evolution of stars.

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

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

3. Draw a qualitative graph of the change in brightness of a binary star if its components are the same size, but the satellite 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, its surface temperature will drop by half and its luminosity will increase 400 times. Will the Sun absorb any of the planets?

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

X. Galaxies. Nebulae. Hubble's law.

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

2. Match the corresponding points regarding the type of object.

Assignments for independent work on astronomy.

Topic 1. Study of the starry sky using a moving map:

1. Set the moving map for the day and hour of observation.

date of observation_________________

observation time ___________________

2. list the constellations that are located in the northern part of the sky from the horizon to the celestial pole.

_______________________________________________________________

5) Determine whether the constellations Ursa Minor, Bootes, and Orion will set.

Ursa Minor___

Bootes___

______________________________________________

7) Find the equatorial coordinates of the star Vega.

Vega (α Lyrae)

Right ascension a = _________

Declension δ = _________

8)Indicate the constellation in which the object with coordinates is located:

a=0 hours 41 minutes, δ = +410

9. Find the position of the Sun on the ecliptic today, determine the length of the day. Sunrise and sunset times

Sunrise____________

Sunset___________

10. Time of stay of the Sun at the moment of the upper culmination.

________________

11. In which zodiacal constellation is the Sun located during the upper culmination?

12. Determine your zodiac sign

Date of birth___________________________

constellation __________________

Topic 2. Structure of the Solar System.

What are the similarities and differences between the terrestrial planets and the giant planets. Fill in table form:

2. Select a planet according to the option in the list:

Compose a report about the planet of the solar system according to the option, focusing on the questions:

How is this planet different from others?

What mass does this planet have?

What is the position of the planet in the solar system?

How long is a planetary year and how long is a sidereal day?

How many sidereal days fit into one planetary year?

The average life expectancy of a person on Earth is 70 Earth years; how many planetary years can a person live on this planet?

What details can be seen on the surface of the planet?

What are the conditions on the planet, is it possible to visit it?

How many satellites does the planet have and what kind?

3.Select the required planet for the corresponding description:

Topic 3. Characteristics of stars.

Select a star according to the option.

Indicate the position of the star on the spectrum-luminosity diagram.

Required formulas:

Average Density:

Luminosity:

Life time:

Distance to star:

Topic 4. Theories of the origin and evolution of the Universe.

Name the galaxy we live in:

Classify our galaxy according to the Hubble system:

Draw a diagram of the structure of our galaxy, label the main elements. Determine the position of the Sun.

What are the names of the satellites of our galaxy?

How long does it take for light to travel through our Galaxy along its diameter?

What objects are components of galaxies?

Classify the objects of our galaxy from photographs:

What objects are the components of the Universe?

Universe

Which galaxies make up the population of the Local Group?

What is the activity of galaxies?

What are quasars and at what distances from Earth are they located?

Describe what you see in the photographs:

Does the cosmological expansion of the Metagalaxy affect the distance from Earth...

To the Moon; □

To the center of the Galaxy; □

To the M31 galaxy in the constellation Andromeda; □

To the center of a local galaxy cluster □

Name three possible options for the development of the Universe according to Friedman's theory.

References

Main:

Klimishin I.A., “Astronomy-11”. - Kyiv, 2003

Gomulina N. “Open Astronomy 2.6” CD - Physikon 2005 r.

Workbook on astronomy / N.O. Gladushina, V.V. Kosenko. - Lugansk: Educational book, 2004. - 82 p.

Additional:

Vorontsov-Velyaminov B. A.
“Astronomy” Textbook for 10th grade of high school. (Ed. 15th). - Moscow "Enlightenment", 1983.

Perelman Ya. I. “Entertaining astronomy” 7th ed. - M, 1954.

Dagaev M. M. “Collection of problems in astronomy.” - Moscow, 1980.

Problem 1

The focal length of the telescope lens is 900 mm, and the focal length of the eyepiece used is 25 mm. Determine the magnification of the telescope.

Solution:

The magnification of the telescope is determined from the relation: , where F– focal length of the lens, f– focal length of the eyepiece. Thus, the magnification of the telescope will be once.

Answer: 36 times.

Problem 2

Convert the longitude of Krasnoyarsk to hourly units (l=92°52¢ E).

Solution:

Based on the relationship between the hourly unit of angle and the degree unit:

24 hours =360°, 1 hour =15°, 1 minute =15¢, 1 s = 15², and 1°=4 minutes, and taking into account that 92°52¢ = 92.87°, we get:

1 hour · 92.87°/15°= 6.19 hours = 6 hours 11 minutes. e.d.

Answer: 6 hours 11 minutes e.d.

Problem 3

What is the declination of the star if it culminates at an altitude of 63° in Krasnoyarsk, whose latitude is 56° N?

Solution:

Using the relationship connecting the height of the luminary at the upper culmination, culminating south of the zenith, h, declination of the luminary δ and latitude of the observation site φ , h = δ + (90° – φ ), we get:

δ = h + φ – 90° = 63° + 56° – 90° = 29°.

Answer: 29°.

Problem 4

When it is 10 hours 17 minutes 14 seconds in Greenwich, the local time at some point is 12 hours 43 minutes 21 seconds. What is the longitude of this point?

Solution:

Local time is mean solar time, and local Greenwich time is universal time. Using the relationship relating the mean solar time T m, universal time T0 and longitude l, expressed in hourly units: T m = T0 +l, we get:

l = T m – T 0 = 12 hours 43 minutes 21 seconds. – 10 hours 17 minutes 14 seconds = 2 hours 26 minutes 07 seconds.

Answer: 2h 26 min 07 s.

Problem 5

After what period of time do the moments of maximum distance of Venus from the Earth repeat if its sidereal period is 224.70 days?

Solution:

Venus is the lower (inner) planet. The planetary configuration at which the inner planet is at its maximum distance from the Earth is called superior conjunction. And the period of time between successive configurations of the same name on the planet is called the synodic period S. Therefore, it is necessary to find the synodic period of the revolution of Venus. Using the equation of synodic motion for the lower (inner) planets, where T– sidereal, or sidereal period of revolution of the planet, TÅ – sidereal period of rotation of the Earth (sidereal year), equal to 365.26 average solar days, we find:

=583.91 days.

Answer: 583.91 days.

Problem 6

The sidereal period of Jupiter's revolution around the Sun is about 12 years. What is the average distance of Jupiter from the Sun?

Solution:

The average distance of a planet from the Sun is equal to the semi-major axis of the elliptical orbit a. From Kepler's third law, comparing the motion of a planet with the Earth, for which taking the sidereal period of revolution T 2 = 1 year, and the semimajor axis of the orbit a 2 = 1 AU, we obtain a simple expression for determining the average distance of the planet from the Sun in astronomical units based on the known sidereal period of revolution, expressed in years. Substituting the numerical values ​​we finally find:

Answer: about 5 AU

Problem 7

Determine the distance from Earth to Mars at the moment of its opposition, when its horizontal parallax is 18².

Solution:

From the formula for determining geocentric distances , Where ρ – horizontal parallax of the luminary, RÅ = 6378 km – the average radius of the Earth, let’s determine the distance to Mars at the moment of opposition:

» 73×10 6 km. Dividing this value by the value of the astronomical unit, we get 73 × 10 6 km / 149.6 × 10 6 km » 0.5 AU.

Answer: 73×10 6 km » 0.5 AU

Problem 8

The horizontal parallax of the Sun is 8.8². At what distance from Earth (in AU) was Jupiter when its horizontal parallax was 1.5²?

Solution:

From the formula it is clear that the geocentric distance of one star D 1 is inversely proportional to its horizontal parallax ρ 1, i.e. . A similar proportionality can be written for another luminary for which the distance D 2 and horizontal parallax are known ρ 2: . Dividing one ratio by the other, we get . Thus, knowing from the conditions of the problem that the horizontal parallax of the Sun is 8.8², while it is located at 1 AU. from Earth, you can easily find the distance to Jupiter from the known horizontal parallax of the planet at this moment:

=5.9 a.u.

Answer: 5.9 a.u.

Problem 9

Determine the linear radius of Mars if it is known that during great opposition its angular radius is 12.5² and its horizontal parallax is 23.4².

Solution:

Linear radius of luminaries R can be determined from the relationship, r is the angular radius of the star, r 0 is its horizontal parallax, R Å is the radius of the Earth, equal to 6378 km. Substituting the values ​​from the problem conditions, we get: = 3407 km.

Answer: 3407 km.

Problem 10

How many times is the mass of Pluto less than the mass of the Earth, if it is known that the distance to its satellite Charon is 19.64 × 10 3 km, and the satellite’s orbital period is 6.4 days. The distance of the Moon from the Earth is 3.84 × 10 5 km, and its orbital period is 27.3 days.

Solution:

To determine the masses of celestial bodies, you need to use Kepler's third generalized law: . Since the masses of the planets M 1 and M 2 significantly less than the masses of their satellites m 1 and m 2, then the masses of the satellites can be neglected. Then this Kepler law can be rewritten as follows: , Where A 1 – semimajor axis of the orbit of the satellite of the first planet with mass M 1, T 1 – period of revolution of the satellite of the first planet, A 2 – semimajor axis of the orbit of the satellite of the second planet with mass M 2, T 2 – period of revolution of the satellite of the second planet.

Substituting the corresponding values ​​from the problem conditions, we get:

= 0,0024.

Answer: 0.0024 times.

Problem 11

The Huygens space probe landed on Saturn's moon Titan on January 14, 2005. During the descent, he transmitted to Earth a photograph of the surface of this celestial body, on which formations similar to rivers and seas are visible. Estimate the average temperature on the surface of Titan. What kind of liquid do you think the rivers and seas on Titan might consist of?

Note: The distance from the Sun to Saturn is 9.54 AU. The reflectivity of the Earth and Titan is assumed to be the same, and the average temperature on the Earth's surface is 16°C.

Solution:

The energies received by Earth and Titan are inversely proportional to the square of their distances from the Sun r. Some of the energy is reflected, some is absorbed and goes to heat the surface. Assuming that the reflectivity of these celestial bodies is the same, then the percentage of energy spent on heating these bodies will be the same. Let us estimate the surface temperature of Titan in the black body approximation, i.e. when the amount of absorbed energy is equal to the amount of energy emitted by the heated body. According to the Stefan-Boltzmann law, the energy emitted by a unit surface per unit time is proportional to the fourth power of the absolute temperature of the body. Thus, for the energy absorbed by the Earth we can write , Where r h – distance from the Sun to the Earth, T h is the average temperature on the Earth’s surface, and Titan – , Where r c – distance from the Sun to Saturn with its satellite Titan, T T is the average temperature on the surface of Titan. Taking the relation, we get: , from here 94°K = (94°K – 273°K) = –179°C. At such low temperatures, the seas on Titan may consist of liquid gas, such as methane or ethane.

Answer: From liquid gas, for example, methane or ethane, since the temperature on Titan is –179°C.

Problem 12

What is the apparent magnitude of the Sun as seen from the nearest star? The distance to it is about 270,000 AU.

Solution:

Let's use Pogson's formula: , Where I 1 and I 2 – brightness of sources, m 1 and m 2 – their magnitudes, respectively. Since brightness is inversely proportional to the square of the distance to the source, we can write . Taking logarithm of this expression, we get . It is known that the apparent magnitude of the Sun from Earth (from a distance r 1 = 1 a.u.) m 1 = –26.8. You need to find the apparent magnitude of the Sun m 2 from a distance r 2 = 270,000 a.u. Substituting these values ​​into the expression, we get:

, hence ≈ 0.4 m.

Answer: 0.4 m.

Problem 13

The annual parallax of Sirius (a Canis Majoris) is 0.377². What is the distance to this star in parsecs and light years?

Solution:

Distances to stars in parsecs are determined from the relation , where π is the annual parallax of the star. Therefore = 2.65 pcs. So 1 pc = 3.26 sv. g., then the distance to Sirius in light years will be 2.65 pc · 3.26 sv. g. = 8.64 sv. G.

Answer: 2.63 pcs or 8.64 sv. G.

Problem 14

The apparent magnitude of the star Sirius is –1.46 m, and the distance is 2.65 pc. Determine the absolute magnitude of this star.

Solution:

Absolute magnitude M related to apparent magnitude m and distance to the star r in parsecs with the following ratio: . This formula can be derived from Pogson's formula , knowing that absolute magnitude is the magnitude that a star would have if it were at a standard distance r 0 = 10 pcs. To do this, we rewrite Pogson’s formula in the form , Where I– the brightness of a star on Earth from a distance r, A I 0 – brightness from a distance r 0 = 10 pcs. Since the apparent brightness of a star will change in inverse proportion to the square of the distance to it, i.e. , That . Taking logarithms, we get: either or .

Substituting the values ​​from the problem conditions into this relation, we obtain:

Answer: M= 1.42 m.

Problem 15

How many times is the star Arcturus (a Boötes) larger than the Sun if the luminosity of Arcturus is 100 times greater than the solar one and the temperature is 4500° K?

Solution:

Star luminosity L– the total energy emitted by a star per unit time can be defined as , where S is the surface area of ​​the star, ε is the energy emitted by the star per unit surface area, which is determined by the Stefan-Boltzmann law, where σ is the Stefan-Boltzmann constant, T– absolute temperature of the star’s surface. Thus, we can write: , where R– radius of the star. For the Sun we can write a similar expression: , Where L c – luminosity of the Sun, R c – radius of the Sun, T c is the temperature of the solar surface. Dividing one expression by the other, we get:

Or you can write this relationship this way: . Taking for the Sun R c =1 and L with =1, we get . Substituting the values ​​from the problem conditions, we find the radius of the star in radii of the Sun (or how many times the star is larger or smaller than the Sun):

≈ 18 times.

Answer: 18 times.

Problem 16

In the spiral galaxy in the constellation Triangulum, Cepheids are observed with a period of 13 days, and their apparent magnitude is 19.6 m. Determine the distance to the galaxy in light years.

Note: The absolute magnitude of a Cepheid with the indicated period is equal to M= – 4.6 m.

Solution:

From the relation , relating the absolute magnitude M with apparent magnitude m and distance to the star r, expressed in parsecs, we get: = . Hence r ≈ 690,000 pc = 690,000 pc · 3.26 light. city ​​≈2,250,000 St. l.

Answer: approximately 2,250,000 St. l.

Problem 17

The quasar has a redshift z= 0.1. Determine the distance to the quasar.

Solution:

Let's write down Hubble's law: , where v– radial velocity of removal of the galaxy (quasar), r- distance to it, H– Hubble constant. On the other hand, according to the Doppler effect, the radial velocity of a moving object is equal to , с is the speed of light, λ 0 is the wavelength of the line in the spectrum for a stationary source, λ is the wavelength of the line in the spectrum for a moving source, is the red shift. And since the red shift in the spectra of galaxies is interpreted as a Doppler shift associated with their removal, Hubble's law is often written in the form: . Expressing the distance to the quasar r and substituting the values ​​from the problem conditions, we get:

≈ 430 Mpc = 430 Mpc · 3.26 light. g. ≈ 1.4 billion St.L.

Answer: 1.4 billion St.L.