Astronomical Olympiads - V.G. Surdin

Astronomical Olympiads Problems with solutions V. G. Surdin translated by M.V.Safonova September 4, 2000

Table of Contents 1) Table of Contents ............................................................................................................................................... 2 2) The Starry Sky ................................................................................................................................................... 3 3) The Earth: - Coordinates, Shape, Motion. ............................................................................................................. 6 4) The Moon. Solar and lunar eclipse. ...................................................................................................................... 8 5) The Apparent Motion of the Celestial Bodies ..................................................................................................... 11 6) Time and Calendar ........................................................................................................................................... 14 7) Coordinates of the Celestial Bodies.................................................................................................................... 16 8) Brightness, Luminosity, Distances. .................................................................................................................... 17 9) Astronomical Instruments and Observations. ...................................................................................................... 21 10) Motion of Celestial Bodies.............................................................................................................................. 25 11) Space Travel and Nature of the Planets ............................................................................................................ 28 12) Astrophysics. ................................................................................................................................................. 30 13) Astronomical Quiz ......................................................................................................................................... 33 14) PROBLEMS OF CAPTAIN FIBBER (with a multiple choice)......................................................................... 33 15) Old Hottabych Problems................................................................................................................................. 35

The Starry Sky 16) Which constellation can be found twice on a star map? Can it be seen in March in the evening? 17) Total solar eclipse is happening in the second fortnight of June around the local noon. Draw the map of the position of the planets and the brightest stars that can be seen during the eclipse, taking into account that Venus is in its maximum western elongation, Mercury is in lowest conjunction, Mars is in opposition. 18) Why is it that the closer you are to the Milky Way, the more you see stars and the less distant galaxies? 19) Which star, Betelgeuse or Deneb, can be seen longer in the Moscow sky? 20) Which means of orientation by the stars you know (apart from orientation by the Polaris)? 21) What do you know about the constellations of Scorpius (The Scorpion) and Ophiuchus (The Serpent Holder)? 22) What do you know about the constellations of Andromeda, Pegasus and Cassiopeia? (YOU Call- vary such questions, choosing more or less known constellations, depending on the location and level of the children's knowledge) 23) Show approximate location of constellations of Scutum (The Shield), Vulpecula (The Fox), Sagitta (The Arrow), Equuleus (The Little Horse), Lepus (The Hare), Lynx (The Lynx), Corvus (The Crow). 24) Describe the star sky during the total solar eclipse of June 30th, 1954 and mark ' the position of the Sun 25) Can you see in the March of this year in the early evening the Ursa Major (The Great Bear), Leo (The Lion), Auriga (The Charioteer) and Orion (The Hunter) constellations? Can you see the Moon and the planets? 26) Which constellations are named after the physical devices? 27) Name 7 brightest objects on the sky. 28) Which non-zodiacal constellation the Sun passes through on its annual path around the sky and when'? 29) What is zodiacal light? What time of the day and which location on the Earth are the best for observing it'? 30) Describe the Mars's night sky. 31) Describe the night sky of lo, which is orbiting the Jupiter at a distance of 6 Jupiter's radii. 32) How many lunar disks are necessary to cover all sky? 33) When on the sky can you see Sun? Moon? Stars? 34) Here is the unfinished verse by A. S. Pushkin. [choose one couplet] … the azure height above me One little star sends tender light-To my dark-red horizon, and a close moon to the right. Just a little star has risen To the azure sky so soonTo my right, dark-red horizon 7o my left, a close moon. Determine: a) which part of tile world tile poet was facing,

b) what time of the day it was, c) in which phase was the Moon, d) why the Moon was called "Close" e) what "little star" could shine at tile poet.' 35) What kinds of relations have tile mythological characters, after whom the planets of the Solar System arc named? 36) A planet is at the angular distance of 100° from the Sun. Which planet is it- inner or outer? 37) Why Leonids and Quadrantids meteor showers have these names? 38) Can you observe Mars from the Eridanus (The River) constellation? 39) The Moon is in Sagittarius (The Archer). In which constellation the observer from the Moon will see the Earth? 40) .Name all the objects of the Solar System in the order of the decreasing size (do not count the comets tails). 41) Explain why there are observed more meteors from midnight to dawn than from evening to midnight. 42) Estimate how often the Moon occults the stars visible to the naked eye. 43) Venus is in its western elongation. Which time of the day is the best for its observations? 44) A hunter on the autumn night goes to the forest in the direction of the Polaris. Immediately after the sunrise he returns back. How should a hunter orient himself . by the position of the Sun? 45) Why many star observers insist that Earth's artificial satellites (EAS) move in zigzags across the sky? 46) F. Pollack's "Course of general astronomy" states that "All stars seen with the naked eye, and many of those seen through a telescope, are long ago counted, catalogued and placed on the astronomical charts." Why then the number of stars seen with the naked eye is always given only approximate? 47) What can be the maximum angle between the Polaris and the pole of the sky (north celestial pole) as a result of the precession of the Earth's axis? When this angle was reached last? Did then the Polaris set behind the horizon on the latitude of Moscow'? 48) If you have decided to observe the Moon a week before the eclipse, where on the sky would you look for It immediately after the sunset? 49) Using-, the skeleton (outline) maps of the constellations oil tile figures 1.1 - 1.3 Perform the following exercises: a) write the latin and the hindi names of the constellation, the name of the brightest star and its stellar magnitude; b) '' connect the stars of the constellation to obtain its contour; c) at what time of the year this constellation is the best to observe at your latitude?; d) what interesting objects in this constellation you know? Mark their position on the map.

Figure 1 Constellation Maps

The Earth: - Coordinates, Shape, Motion. 1) Who and where went the closest to the centre of the Earth? 2) On March 21st on a true noon the shadow of the vertical pole is equal to its height. On what latitude it is happening? 3) Find the width in km of the meteor shower, which was observed from July 16th to August 24th. Assume that the Earth is moving perpendicularly to the stream. 4) In winter the Earth is several million km closer to the Sun than in summer. Why then the winter is colder than the summer? 5) What is the period of the rotation of the Earth around its axis? 6) Two travellers started the round-the-world tour from the same poi:: on the equator and with the same velocity: one went along the equator, another-along the meridian. Will they reach the starting point simultaneously? 7) Orbits of two comets lie it the ecliptic. Comets have perihelion radii of 0.5 AU and 1.5 AU Lengths of their tails in the perihelion exceed 100 million km. Can the Earth pass through the ;ails of these comets? 8) Maximum angular radius of the Sun (θmax = 16'17.53" is observed on January 4th, minimum radius θmax = 15'45.34" is observed on the 5th of July. Where on the orbit is the Earth on these dates? What are the distances from these points to the Sun? 9) If the period of the Earth's rotation were 12 hours, how many lunar tides we would have observed during this period? 10) For the french traveller Jean Atone, who started alone in summer 1986 the journey from Canada to the North Pole, one Swiss firm has manufactured a special watch, whose hand was moving round the face not in 12 hours, but in 24. Why had he needed such a watch? (Unfortunately, in the article ''bout that event the newspaper "Izvestiya" has not mentioned that the fact exactly the same watch was manufactured in Moscow for russian researchers in Antarctica). 11) An observer standing on the equator sees an EAS. always right above himself. At what distance from the surface is the orbit of this satellite and what is its linear velocity? 12) I low many heliostationary satellites arc necessary to maintain round-the-clock communication between the North and the South poles? 13) What should be the height of the TV antenna on the South Pole to receive the signal from the heliostationary satellite? 14) From which point on the Earth you walk 100 km to the south, 100 km to the cast and then 100 km to the north and you return to the original point? 15) In the Jules Verne novel "Air ship" it is mentioned that near midnight above the ship "Albatross" was seen the constellation Southern Cross, "four stars of which were shining brightly in zenith", and approximately an hour later the ship flew over the Southern Pole. Knowing that the four brightest stars of the Southern Cross have declinations -62.6°, -59.2°, -58.2° and -56.6°, correspondingly, find the astronomical inaccuracy in the narration and find the approximate geographical latitude of the location of the "Albatross" at midnight, assuming that the ship was flying by the shortest route to the Pole. 16) If the Earth stopped its rotation, would we have the change of day and night? 17) In the Jules Verne's novel "The Mystery Island" when the survivors found the the chest and having examined the sea surface exclaimed: "There are no traces of the shipwreck around for a 100 miles!". What was the magnification of the spyglass?

18) Why is the period of the change of the seasons (i.e.. tropical year) approximately 20 min shorter than the period of the Earth's revolution around the Sun (i.e.. sidereal year)? 19) I-low can one prove that the Earth is rotating around its axis and around the Sun? 20) Tunguska's meteor to explosion was observed at the horizon in the town Quirks (river Lent) 3510 km from the place of explosion. Determine at what height above the surface was the explosion. 21) From the North Pole to the South Pole was made the vertical shaft. One projectile is dropped into the shaft without the initial velocity, another one is launched on the low polar orbit (see Fig. 2.1). Which one of them will reach the South Pole first? 22) Why the shape of the Earth and other planets is spheroids, but of the asteroids and comets nucleiirregular? 23) In 24 hours the Earth is receiving around 10 thousands tons of space substance 'meteorites', cosmic dust, etc). Estimate how this could have changed the duration of the day since the beginning of the evolution of our planet. 24) Spectral studies show that the radial velocity of Regulus (a Leo) relative to the Earth changes from V1 = -27 km/sec to V2 = 33 km/sec with the period of 1 year (negative velocity means that the object approaches us). Knowing that this star is on the ecliptic and that the distance from the Earth to the Sun is 1.5 x 1011 m, calculate the gravitational constant. Consider our orbit to be circular and neglect the Earth's mass as compared to the Sun's mass (M~ = 2.1030 kg). 25) The end was close: the beast got tired and stood still. 100 m to the south from the bear the hunter appeared, walked 150 m to the east and, turning, fired a shot exactly to the north. Stricken animal fell dead. The question is: what colour was the bear?

Figure 2 What is the fastest way to reach antipodes

The Moon. Solar and lunar eclipse. 1) Are there places on the Moon where the Earth rises and sets? 2) What maximum altitude can the Moon reach on Moscow's latitude (55045'); on which date, at what hour and in which constellation it happens? 3) Why at the middle latitudes the crescent of the "young" Moon in the spring is always high above the horizon and its horns are turned up, but in the autumn this happens only with the "old" Moon? 4) Why solar eclipses are happening more often in summers? 5) Why the longest solar eclipses are observed only in the tropical countries? 6) Can you observe a solar eclipse on the North Pole? 7) Can a plane catch up with a lunar shadow moving across the Earth? 8) Can one observe a solar eclipse from the board of the space station? If not, then why? And if yes, then what is its duration? 9) Goring the solar eclipse a traveller noticed that the eclipse began right from the bottom. Where and when could it have been? 10) An observer noticed that the solar eclipse started from the upper edge of the solar disk. Where and when could it have been? 11) If in one year the solar eclipse was on a 2nd of January, could during this year be another solar eclipse? 12) A lunar eclipse is happening when the Sun's declination is +20050' and Moon's declination is -21" 16'. How the Moon is passing through the Earth's shadow? 13) Can one during the total lunar eclipse observe from the same point on Earth both the Moon and the Sun? 14) When the apparent diameter of the Moon is larger: when it is observed near the horizon or in zenith? 15) Where one can observe annular solar eclipse more often: in zenith or on the horizon? 16) On which side of the Moon the day is brighter-on the nearside or on the farside? And the night? 17) On the geographic pole of the Earth the Sun is half a year above the horizon and half a year below the horizon. What about the Moon? 18) Specialists hold that future inhabitants of lunar bases will prefer 25-hour diurnal cycle, and not 24-hour. Why? 19) It is assumed that exactly half of the lunar globe is illuminated by the Sun, and that the other half is in the shadow. Is it true? 20) In which terrestrial regions one can observe the lunar eclipse today? 21) Usually total solar eclipses arc observed in the band of 200 km width and 10,000 km length. On average on the Earth is observed one total solar eclipse a year. Estimate after how many years a coca solar eclipse will repeat in the same location, for example, in your town. 22) What are the "white nights" and why they happen in St.Petersburg and not in Odes? (Geographic latitude of St. Petersburg is 60°, of Odes is 46.50.) 23) Where on the Moon it is better to build a solar observatory? 24) At what time of the year the Moon rises on a full Moon night higher above the horizon-in winter or in summer?

25) Do solar eclipses happen on Uranus? 26) The total solar eclipse is happening in the beginning of January in Moscow. How the Moon looked half and hour after the first contact with the shadow? How it looked for the observer in Antarctica? 27) How long can the lunar occultation of a star continue? 28) Arthur Clark in his novel "2001: a Space Odyssey" is describing the beginning of a sunrise on the-Moon: "...a thin bow of unbearable incandescence had thrust itself .::love the eastern horizon. Though it would take more than a hour for the Sun to clear the edge of the slowly turning Moon, the stars were already banished." Determine on which lunar latitude it is happening. 29) If the Moon has risen at 23"45"' on Tuesday, when will be the next moonrise? 30) How many times a year the Moon turns around its axis? 31) How long the Moon stays above the horizon on the equator (without taking refraction in the account)? 32) In the novel of F. Panferov "In the Name of the Young" (1960) there are such lines: "Then the rockets to the Moon were launched and its far, always in the darkness, side was photographed and that evoked a lot of enthusiasm in all countries." Find the inaccuracies. 33) Can one observe the solar eclipse on November 15th from the North Pole? 34) What the period of the rotation of the Moon around its axis is equal to: syndic month or sidereal month? 35) Due to the tidal interaction with the Earth, the Moon is receding from the Earth by 3cm every year along the spiral orbit. After how many years we won't be able to observe the solar eclipses? 36) In the system Earth-Moon a paradoxical phenomenon is happening: as a result of the tidal friction the angular velocity of the Earth as well as the Moon is decreasin.. Does it contradict the law of the conservation of the angular momentum? 37) Can one observe from Saturn the eclipse of the Sun by Titan, which is revolving at the distance of 1222000 km from the centre of Saturn and has a diameter of 5150 km? 38) How often the Earth rises on the Moon in the vicinity of the crater Tight? 39) Do solar eclipses happen on Mars and Jupiter? 40) Due to the tidal effects the Moon is slowly receding from the Earth in the present epoch. In the past the radius of its orbit was smaller. Estimate the duration of the total solar eclipses at a time when its radius was half the present ore. 41) Astronomers are observing the total solar eclipse from the board of the plane flying to the east along the eclipse path with the speed of 900 km/h. The eclipse :s happening around noon on the latitude of 600 and its full phase on the surface lasted for 5 mins. For how long it was observed on the board of the plane? 42) At what time of the lay one can observe a waxing moon and at what time a waning moon? 43) Is it possible to launch a stationary lunar satellite? 44) Can one in the course of 24 hours observe both an old and a young moon? 45) How many times a year the Moon is in zenith on the equator? 46) lohann Kepler in his work "Dream, or Posthumous composition about lunar astronomy" wrote "The Moon in Iceland is often not seen during the time when other countries observe a full Moon". What time of the year this statement is referred to? Iceland is situated on the latitudes from 63° to 66°. 47) How one can prove that "ashen" lunar light is caused by the sunlight reflected from the Earth?

48) Describe the phenomena that could be observed on the Moon during the total solar eclipse on the Earth. 49) For radio communication with the Earth from the Moon it was decided to mount a directed antenna. Does this antenna need a self-guided device or is it enough to orient it only once at the time of installation? 50) An observer is on the boarder between the nearside and farside of the Moon. How the positions of stars and the Earth relative to the horizon are changing with time for this observer? Describe briefly. 51) If the lunar albedo in an optical range were exactly 0% (and not 7%), how could we know about the presence of the Moon near the Earth ? 52) On which side of the Moon-nearside or farside-the night is longer? 53) In the story by A.P.Chekhov "The Chemist's Wife" there are such lines: "Suddenly an immense, broadfaced moon crawled out from behind some bushes in the distance. The moon was bright red: indeed, whenever the moon crawls out from behind any shrubbery, for some reason it always looks terrible embarrassed." The question is: why the Moon was red and broad-faced? And is it important that the bushes were in the distance? 54) Estimate from the point of view of an astronomer a couplet from the song by You Kim And on the Moon, on the Moon, from a blue lunar dune, Strange lunar folks observe something exciting: Far at the end of the Moon, like a fantastic balloon, The Earth is beautifully setting and rising. 55) On which planet one can observe the longest total solar eclipse?

The Apparent Motion of the Celestial Bodies 1) What is the apparent path of the Sun on the Uranus sky? 4.2 2) What apparent paths would have due to the parallax the following stars: α Leo, δ Orion, γ Draco and Polaris? 3) Explain the meaning of the folk saying: "The Crescent moves in winter like the Sun in summer". 4) On December 22nd during the total lunar eclipse there was an occultation of the Jupiter by the Moon. In what constellation it happened and what was the position of the Jupiter relative to the Sun and the Earth? 5) American solo-yachtsman Steven Kalahari had suffered a ship-wreck in Atlantic. He found himself on an inflatable raft and was judging (finding?) his course by the stars: "In the night I determine my position by the two referencepoints simultaneously: Polaris and Southern Cross." On approximately what latitudes and at what time of the year it happened? Prompt: coordinates of the Southern Cross are α = -12.5 h and δ = - 60° ± 5°. 6) Why in tropical countries the venetian blinds with the vertical slats arc preferred, whereas on middle latitudes-with horizontal? (Sec Fig.1). 7) Can a day be longer than a year? Figure 3 Which blinds are better?

8) Describe the motion of the 5th jovian satellite (Amalthea)diurnal and relative to the stars-for an observer from the equator of Jupiter, assuming for simplicity that the motion of the satellite is happening in the ecliptic and also in the plane of the Jupiter's orbit, which lies in the ecliptic. Assume the period of the satellite revolution to be 12 hours and the period of rotation of Jupiter-!0 hours. 9) If the Earth's rotation axis were perpendicular to the ecliptic, what would be the duration of the day on the 40° N.L. on the day of summer solstice? 10) On the Earth the solar day is longer than stellar day, but on Venus it is vice versa. Why? 11) What is the interval between the "disappearance" of the saturnian rings (the plane of the rings turning edge on to us)? 12) Does the Sun have annual parallax like other stars? 13) It is considered that Venus is seen either in the mornings or in the evenings. But is it possible to observe Venus in the morning and evening of the same day? 14) In what basic configurations the Earth is seen from Mercury and Mars and what is the largest angular distance between the Earth and the Moon that can be observed from these planets? 15) In the tear-off calendar it was written that the Iength of the day in Moscow on March 21st 1990 is 12 hours 14 minutes. Why then this day is called the day of the vernal equinox? 16) How can the inhabitants of the farside of the Moon guess that the Moon is revolving around the Earth ? 17) A camera, fixed in the direction of the southern part of the sky, takes a short exposure exactly at 12" of civil time on the same photoplate. What trajectory will describe the center of the Sun on that photo? 18) Why the great martian oppositions always happen at the same time of the year? What is that time? 19) How to find out whether the observed celestial object is a comet or a distant nebula?

20) To what maximum height above the horizon the Sun rises on the martian pole? The tilt of planet's axis to the ecliptic is 65°. 21) Explain why sometimes one can see the lunar crescent in the spring a day earlier after the new moon than in the autumn. How does it depend on the latitude of the observation? 22) What is the time and the height of Mars's culmination, when it is observed at opposition in Moscow ((P = 56°} on June 22nd? In which constellation it is seen at that moment? 23) What is the time and the height of Jupiter's culmination, when it is observed at the opposition in Moscow (¢ = 56") on June 22nd? In which constellation it is seen at that moment? 24) A sidereal period of a superior planet is 417 days. What is the average distance of this planet to the Sun? What is this planet? 25) At what phase the Moon rises to the maximum height above the horizon? 26) The distance between a star and the northern celestial pole is 45°. Can the star be always seen on the latitudes of Yalta (¢ = 44.5°), St. Petersburg (¢ = 60°), Archangels (p = 64.6°) and Tashkent (¢ = 64.6°)? 27) On which latitudes the circumpolar stars culminate in zenith? 28) Under what conditions there is no day and night change on a planet? 29) 12 zodiacal constellations are evenly distributed along the ecliptic. In which of them the Sun spends the least amount of times? 30) What is the duration of crossing the horizon line by the solar disk on the equator and on the pole? 31) In middle latitudes the lunar disk rises completely above the horizon in three minutes. How long it takes for the Earth's disk to rise on the Moon` 32) On March 21st at 9"00' local solar time on the equator was observed the total solar eclipse. At what height above: the horizon was the Sun it that moment? 33) In the novel by M. Bulgakov "Master and Margarita" in 3rd chapter a May evening on Patriarh Ponds is described: "The: sky above Moscow was as if bleached, and absolutely distinctly was seen in the height the full moon, not yet golden but white." What inaccuracy the writer committed here? 34) What astronomical event is depicted on the Fig. 4.2? 35) A celestial object has a synodic period of 1.25 years. What is its sidereal period? Between orbits of which two planets it moves? 36) Apparent solar day is a period between two subsequent culminations of the Sun, for example, between two local noons. Is the duration of the apparent solar day constant within a year? 37) How the diurnal and annual motion of the Sun look for the observer on the Earth's South Pole. 38) On the 22nd of June you are going along the smooth road in the northern direction. Suddenly you are blinded by the patch of the sunlight reflected from the windscreen of the coming from the opposite direction car. The glass of the windscreen was smooth and at an angle of 30° to the vertical. At what time of the day it happened? At what height above the horizon was the Sun? And on what approximately latitude it happened? 39) Following cities have latitudes: a) Greenwich (England): S l° n.l. b) St. Luis (USA): 38° n.l. c) Calcutta (India): 23° n.l.

d) Mbdanaka (Zaire): 0° e) Rio-de-Zhaneiro (Brazil): ?3° s.{. In which of them: a) Sun is near the horizon on June 2lst? b) Sun is near zenith on December 21st? c) Polaris is near the horizon? b) Stars with declination +67° never set? e) Polar star is never seen? c) Stars with declination +51° pass through zenith? Give the answers in the table form: letters of questions - a b c d e f numbers of answers 40) What is the hour angle of a star at the moment of its lower culmination?

Time and Calendar 1) When and where the polar day is longer-on the North or on the South Pole? 2) Is it true that on March 21st and on September 23rd everywhere on Earth duration of the day is equal to the duration of the night? 3) What do you need to know in order to verify the city clocks by the sundial? 4) On March 20th at 18.00 Moscow standard time the Moon is occulting the Pleiades. Describe how this event would have been observed in Moscow (2nd time zone), in Ekaterinburg (4th time zone) and in Komsomolsk-on-Amur (9th time zone). 5) Where could an observer be for whom on the day of summer solstice the sunrise is happening at 6.00 Greenwich time? 6) At what hour Moscow time on April 11th the Algol culminates (α = 3h05m)? 7) At what geographical longitude the mean local noon coincides with the time of receiving the sixth point of 10.00 Moscow time radio broadcast? 8) What would be the duration of the lunar month if the distance to the Moon were four times the present one? Is this possible at all? 9) Vega is in its upper culmination at 20h0m0s. At what time after 8 days will be its next upper culmination? 10) It is known that Magellan's companions upon returning from their round-the-earth travel lost one day in their day count. How it can be explained? How this problem is avoided nowadays? If an astronaut starts at 10 am on May 10th and makes one trip around the Earth in 1.5 hours on which date and at which hour he would return back to the Earth? 11) X-ray telescope on the orbital station "Salut-7" was not being used to observe the objects within the angular radius less than 60° from the Sun in order to keep the telescope in working condition (Fig. S.1). What is the minimum time of the expedition on "Salut-7" during which the whole sky x-ray would be explored? 12) Astronomical spring of 1983 has begun on March 21st-at 4h39m Greenwich time. What was the Moscow time at that moment? What is the stellar time on March 21st at noon in Moscow? 13) At 22.00 Moscow time the stellar clocks showed 8h56m02s. What would the stellar clocks show at 22.00 Moscow time next day? 14) "In the middle of the polar night a plane has taken, us to the North Pole. Only narrow lunar crescent was illuminating the endless ice plain". In which year did the author of these lines visit the North Pole?

Figure 4 Do not look at the Sun

15) Is the duration of the day on both lunar hemispheres the same, assuming that the day begins when the upper edge of the Sun rises above the horizon in the center of the hemisphere? 16) The total solar eclipse was observed on March 21st at 9h local true solar time on the equator. At what time on this day the Moon will set? 17) There are 365.24 mean solar days in a year. How will this number change if the rotation of the Earth were happening with the same period, but in the reverse direction?

18) On the New Year's Eve the orbital station is roving in a circular stationary orbit with 1.5 hrs period. What is the probability that the astronauts will fly over certain region of the Earth right at the moment of the beginning of the New Year in this region? How many times in one day can this happen? 19) At what place on a horizon the point of vernal equinox rises and sets on June 22nd and on September 22nd at the latitude φ = 56°? What it the stellar time at the moments of its rise and setting? 20) sub A traveller bent over a well and saw a reflection of Capella on the surface of the water (α = 5h13m, J =+45°57'). At what geographical latitude was the traveller and what was the stellar time at that moment? 21) If January 1st is Monday, on which day of the week would the ordinary year and the leap year end? 22) How many days w ill be in February 2100? 23) When will the 21st century start? 24) The point of vernal equinox has risen one hour ago (by stellar time). What is the stellar time now? 25) A plane flying from San-Francisco to Tokyo is approaching the Date Line. On its board the exact zonal time is I6h22m and the date is December 31st. After 5 min it crossed the Date Line. What is the tonal time and the date immediately after crossing? 26) With what speed and in which direction shall fly the plane in the equatorial region in order to stop the local solar time on the board?

Coordinates of the Celestial Bodies. 1) This chapter is small since most of the problems on determination of the coordinates are published in many places and are of routine character, as a rule. We decide not to overload the book with them and selected only few original ones. But we do not consider the skill in solving these problems to be of a second rate rather otherwise-it is the basis of observational astronomy, the alphabet, without which one should not go deep into this science. In each round of our Olympiad there are by all means such problems. 2) On September 22nd 1985 the Halley's comet had coordinates α = 6h13m and δ = +19°45'. Determine at what hour by Moscow time and at what height on this day did its culmination happen in Moscow (φ = 56°, α = 38°)? 3) What are the names of the points of crossing of almukantarat with the horizon? 4) On location with the latitude +49° the 22 day-old Moon culminated at the height of 46°. Determine the longitude of the rising lunar node. 5) Two weak stars with coordinates α1 = 18h, δ1 = +40° and α2 = 6h, δ2 = +10° were observed simultaneously at one almukantarat; one-in upper culmination and the other-in the low culmination. At what latitude, at what stellar time and in what time of the year did it happen? 6) Is it possible to observe in Moscow (φ = 560) in the course of one day both upper and low culmination's of the following stars: Deneb (α = 20h40m, δ = +45°) Betelgeuse (α = 6h6, δ = +7°) 7) In which of the cities listed below the lunar disk will not disappear behind the horizon during the whole day: Moscow (φ = 56°), St. Petersburg (φ = 59°56'), Vyborg (φ = 60°43'), Yakutsk (φ = 62°02') 8) During the night the culmination of a star was observed twice; in the upper culmination its height was hU = 80°, in the lower it was hL = 60°. On what latitude the observation was made and in which season of the year? 9) Calculate the height, azimuthal and zonal (hourly) angle of the star 13 UMi (α = 20h40m, δ = +45°) at the moments of its lower and upper culminations at the point located on a northern polar circle. Does this star ever sets at this location? What is the stellar time at the moments of its culminations? 10) Measurements of the height of the midday Sun on June 22nd were h1 = 57° and on the December 22nd h2 = 100. Determine the latitude of the location of the observation and the declination of the Sun on these days. 11) How shall the telescope with equatorial mounting be oriented to observe a star with coordinates α = 13h52m and δ = 30°, if the stellar time is ts = 12h19m? In what time the star would cross the field of the 45' telescope if the tracking were switched off. 12) What is the declination of the stars which can be seen at the horizon at all places on Earth?

Brightness, Luminosity, Distances. 1) The distance to Sirius (2.7 Kpc) is decreasing every second by 8 km. In how many years will the brightness of Sinus double? 2) What is the ratio of the radii of the stars in the eclipsing binary of Algol type (Fig. 7.1), if the eclipse is central, the companion is dark and the ratio of brightness in maximum to the brightness in minimum is n? 3) What is the change in the radius of a Cepheid if the amplitude of variation luminosity is 1.5m and the specific luminosity remains constant? 4) Determine the diameter of the star Cet in km; its angular apparent diameter is 0.0065" and parallax is 0.024". 5) How much time passed from the conjunction to the opposition if the brightness of a planet changed by 1m? 6) At maximum of the eclipse an eclipsing binary has brightness of 6m and in minimum of 8m. Assuming the eclipse to be central and the companion to be dark, find the ratio of the volumes of the components of the binary. 7) Parallax of the Sun is 8.80”; of a star is 0.44”. How much farther is the star from the Earth than the Sun is? 8) What is the angular diameter and stellar magnitude of the sun for an observer on Pluto and what is the difference between the illumination. of this planet by the Sun. and the illumination of the Earth by the full moon? Reference data: for an observer on Earth the angular diameter of the Sun is 32"; the distance from Pluto to the Sun is 40 AU, the apparent stellar magnitude of the Sun is m~ = -26.7 and of the Moon is m0 = -12.6. 9) A globular cluster (Fig. 7.2) has a million main sequence stars, each of them having an absolute stellar magnitude of Mms = 6, and also ten thousand red giants with the magnitudes Mrg = 1. Can one see this cluster with the naked eye from the distance of 10 Kpc? 10) When Mars is in opposition one can see in the telescope on its surface details of the size of no less than 100 km. Details of what size could be seen when Mars is in conjunction? 11) The supernova (S And) was observed in 1885 in Andromeda galaxy. Taking into account that the distance to the galaxy is 690 Kpc, estimate when it actually happened.

5) Figure 5 Globular Cluster

12) In 1987 the supernova was registered in the Large Magellanic Cloud. How many years ago happened the explosion if the distance to LMC is 55 Kpc? 13) Which of the two stars is brighter on our sky: a star with the apparent stellar magnitude m = 2 or a star with the absolute magnitude M = -5 and located at the distance of 100 Kpc from the Earth? 14) Halley's cornet in perihelion passes the Sun at the distance of 0.6 AU How much more is the illumination of the comet by the Sun in perihelion than in aphelion? 15) How could astronomers in the pre-spaceflight epoch determine the reflective properties (albedo) of the Earth?

16) Estimate the apparent stellar magnitude of the nucleus of a comet at a distance 40 AU from the Sun, if radius of the nucleus is 5 km and albedo is 0.4. 17) At the moment of first observation in October 1982 the Halley's comet had a magnitude of 24m and was at a distance of 11 AU from the Earth end the Sun. Estimate the radius of the nucleus of the comet, assuming that it reflects 40% of incoming light. 18) What illuminates the Earth better: Sirius (-1.5m) or stars with magnitude from 5m to 6m, the number of which on the night sky hemisphere is about 1600? 19) The number of stars with the magnitudes from 3m to 4m is 400 and with the magnitudes from 4m to 5m is about 1100. Which ones illuminate the Earth more? 20) How much brighter is the full moon night compared with the no moon night? 21) How the Earth and the Moon look like for an observer on Mars? 22) Venus in its maximum elongation has brightness of -4.1m. What maximum brightness will have the Earth for an observer on an artificial satellite around Venus? Take albedo of Venus as 0.8 and of Earth as 0.4. 23) On the photographs of the Earth with the Moon taken from the interplanetary satellites the Earth looks very bright, whereas the Moon looks dark. Why? 24) Why the brightness of a star increases as it rises above the horizon? 25) Why the connection between the luminosity and the period of brightness variation of a Cepheid was discovered from the observations of the stars in the large Magellanic Cloud and not in our Galaxy? 26) Explain why the accuracy of the measuring the distances to the faraway galaxies depends on the accuracy with which we know the distance to the Sun? 27) A supernova in the maximum of its brightness reaches the stellar magnitude M = -21. If the monitoring of the sky is performed with the limiting magnitude of m = 14, how often we would register the supernova? Assume that in the typical galaxy the supernova occurs on average once it 100 years and that the number density of the galaxies is 1 in 10 Mpc3. 28) Imagine that behind your back is the full Moon and in front of you is the polished metal bail in which it reflects (Fig. 7.3). Estimate the stellar magnitude of the lunar reflection if the distance to the ball L = 2 m, its radius R = 0.5 cm, ball's albedo k = 0.7 and the Moon's brightness is m0 = -12.7.

Figure 6 Mirror ball as an astronomical instrument

29) One of the Io's hemispheres is constantly facing Jupiter. Can an astronaut read the book without additional light on that hemisphere' 30) How many weak stars of 6m can replace Venus in brightness? 31) The brightness of Mars in average opposition is about -2m. What would be the brightness of the Earth, observed from Venus in inferior conjunction? (Mars's albedo is 0.16, Earth's albedo is 0.36). 32) Angular diameter of the solar spot of round shape, observed not far from the center of the disk is 17". What is its linear size? 33) In the vicinity of the Sun due to the absorption by the interstellar dust the light from a star by passing a distance of 10 pc decreases by 1 %. Considering the dust panicles as opaque balls of radius r = 2 x 10-5 cm, find the average distance between the particles.

34) The Moon reflects about 7 % of the sunlight. Why then is its brightness hundred thousand times less than solar brightness? (Nontrivial variant of this question: why then it illuminates the Earth hundred thousand times weaker than the Sun?) 35) The globular cluster M92 in Hercules has angular diameter α = 8' and apparent brightness in m ~ 6; distance to the cluster is r = 10 Kpc. Find the absolute stellar magnitude of the cluster and its linear size. Assuming that the cluster consists of the solar type stars find the number of stars in the cluster, their average density and the average distance between them. 36) At the moment of observation a galaxy is at a distance of 330 Mpc and has a velocity V = 30,000 km/sec. What was the distance to it when the light reaching us now left the galaxy? 37) The brightness of Sirius is 22 times greater than of the Sun. At what distance an observer on his way to Sirius would notice that their magnitudes became comparable? (Parallax of Sirius π = 0.373".) 38) An astronomer has noticed a meteor in zenith. The same meteor in the direction to the center of the Earth was observed by an astronaut from the orbit. Estimate the radius of the astronaut's orbit if the magnitude of the meteor for the astronaut was 1.5m weaker than for the astronomer. 39) American artist Author Woods proposed to create on the near earth orbit the "Sculpture of the Earth" Figure 7 A Galaxy inflatable torus made from thin aluminium fail reflecting 70 % of the light. The diameter of the torus is proposed to be 800 in and thickness 50 m. Estimate the brightness of this sculpture assuming that it is an heliosynchronous orbit with the radius 1000 km, 40) G. Galileo in his book "The master of test-tubes affairs" writes: "To ...the question about why the Moon is not smooth and slippery I will answer that the Moon and all other planets are dark themselves (inside) and only shine when illuminated by the sun. Therefore their surface shall be rough, for if it was smooth and slippery as a mirror, the reflected light would not have reached us, and the} would have remained invisible." Is Galileo correct? 41) Can one notice heliostationary satellite of 3 m diameter by the naked eye? 42) Which star will be the brightest on the sky for an astronaut in the vicinity of Proxima Centaurus (apart from Proxima Centaurus itself)? 43) Up to what distance the human eye is still capable to notice stars like Sun? 44) A star is at a distance of 5.6 light years from the Sun and is moving towards the Sun along the straight line, passing through it, with the velocity 111km/sec. In how many years this star will be twice as bright as it is now? 45) Calculate the size of the flat mirror, which if installed on the Moon, will reflect the sunlight as a star of 3m. The apparent stellar magnitude of the Sun is -2.7m, the distance to the Moon is 384,000 km and the albedo of the mirror is 100%. 46) If we assume that albedo of Phobos is the same as of Mars (14%), then the radius of the satellite, calculated from the visual brightness, is 7 km. However, the images of Phobos, obtained from the close distance by the interplanetary probes showed that actual diameter is 10 km. What is the albedo of the satellite in this case? 47) While calculating parallaxes of all stars, astronomers determined that 165 stars have parallax above 0.100". Estimate how many stars have parallaxes greater than 0.025". 48) How far from the Solar System one should move away for the Sun to look like a star of 18m if: (Assume

absolute stellar magnitude of the Sun M = +5m) a) there is no interstellar absorption of light; b) absorption is 3m in 1 Kpc. 49) A telescope can see stars with 19"'. Can one observe through it a globular cluster consisting of a million solar type stars in a neighboring galaxy at a distance 10 Kpc from us? 50) If instead of the Moon on its place there would be a flat mirror of the same size, how would the Earth be illuminated? 51) It is suspected that behind the Pluto's orbit there is a belt of large asteroids and comets (Kuiper belt). Can one discover an asteroid of 350 km diameter with albedo about 7% at a distance of 100 AU from the Sun with the help of the ground-based telescope with limiting magnitude 24m?

Astronomical Instruments and Observations. 1) A fly sat on a telescope's objective. How the view of the solar surface, which was observed through it, changed? 2) How will the photograph of the full Moon change, if one will close right half of the telescope's objective? 3) Why till the end of XIX century in the observatories were installed mostly telescopes - refractors, but in XX century – mostly reflectors? 4) Which systems of telescopes-reflectors you know? Draw the light paths in these telescopes. Why in the large telescopes it is possible to change the optical system? 5) What kind of spectrum has a fast rotating planet if the slit of a spectrograph is directed along its equator? 6) How, in principle, A. A. Belopolskii managed with the help of a spectrograph determine the meteorite structure of saturnian ring? 7) The radial velocity of a planet near the point of summer solstice around twentieth of March was determined from the spectrum to be 70 km/sec. In half a yeas this velocity was determined to be 130 km/sec. Determine on the basis of these data the distance from the Earth to the Sun (this method was first proposed by the academician A. A. Belopolskii). 8) What interesting objects you can see tonight in the 60 mm telescope with the magnification 40? 9) Some solar telescopes has vacuum inside the tube. Why it has been done? 10) Can you see stars in the day from the deep well? 11) What is Wolf number (W)? If observations showed W = 200 and the number of solar spots is 100, what can you say about the distribution of the solar spots on the disk? 12) The resolution of a human eye is about 100" and the quality of the images in the best observatories on Earth is, about 1". It would seem that there is no need for the visual observations to make the telescopes with magnifications of more than 100-I50. Why then the magnification of 300-600 is often used? 13) Why telescopes of the XVIII century had very long tubes? 14) What is the maximum exposure with which one should photograph the surface of the Earth to be able to resolve the devils of the 10 m size, if the camera is fixed on a satellite orbiting the Earth at 300 km above the surface? 15) Observations showed that the angular diameter of the Crab nebula increases by 0.4" every year. At the same time the lines in the spectrum of the nebula are doubled with the distance between the lines Δλ/λ=0.008. Why instead of one line we see double line? Estimate the distance to the nebula. 16) The observations in the x-ray of the lunar occupations of the Crab nebula showed that duration of the coverage of half of the nebula is 1 min, estimate the diameter of the part of the nebula that emits in x-ray, assuming the distance to the nebula to be 1.7 Kpc. 17) A space telescope is capable of registering much weaker stars than a ground-bused telescope of the same size. Why? 18) The distance between the components of a Capella binary is 0.054". Which objectives one should use to observe them separately through the telescope with D = 1 m and focal length F = 10 m and through telescope with D = 5 m and focal length F = 30 m? 19) Why the image of a star on a photoplate has noticeable diameter, moreover, why is that the brighter the star the larger is the diameter? 20) Recently on Hawaii the 10 m telescope Keck started its work. Estimate its limiting stellar magnitude for

the visual observations. Which magnification one should use at that? 21) Which telescope is more suitable for the study of the small details on the Moon: 4 m ground-based or 2 m space telescope? 22) Why the apparent brightness of the artificial earth satellites is varying? Which instrument you will choose for the observation of these satellites: binoculars with 80 mm diameter of the objectives and magnification of 42 or 50 mm telescope with the magnification of 50? 23) Can one with the help of a space telescope working in the near-earth orbit measure the diameter of the nucleus of the Halley's cornet, which will pass the Earth at the distance of 0.5 AU? Estimate the parameters of such telescope. 24) Was Arthur Clark correct when he wrote in his novel "2001: a space odyssey" that ..soon as the day began on the Moon and the first sun rays appeared, "the stars were ... banished"? 25) In 1991 one American engineer published a sensational book where it was stated -.pat American astronauts never reached the Moon in 1969-72 and that all photo and tele-documents were made in a special film studio on Earth. One of the main argument was that on the photographs presented by astronauts on the black sky there were no stars. Can you explain what is the problem here? 26) Why all the domes of the observatories are white? 27) Why one can see the unilluminated surface of the young Moon (the ashen moonlight), but during the solar eclipse we cannot see it? And, by the way, why the pirates had black sails? 28) From the letter of Pliny the Younger, who was the witness of the famous eruption of Vesuvius on August 24th 1979: "When I looked back (at the volcano-V.S.) I saw the thick darkness moving at us-not the kind of darkness as on a new moon or on a cloudy night, but the kind of darkness inside the dosed room when the lights are off." (cited from the book by Zenon Kosidovskii "When the Sun was a God", Moscow, Nauka, 1968). The question is: why and in which way is the darkness of a new moon or of a cloudy night different from the darkness of a closed room? 29) While observing a star in zenith it was found that the absorption in photographic range of the spectrum was 0.45m. What would be the absorption when the star is at the horizon? 30) Why we say: spectral lines and not "spectral squares, circles, arcs" or even, say, some "spectral flourishes"? 31) If our extraterrestrial friends from the neighboring star are measuring the radial velocity of the Sun with the accuracy of 10 m/sec, can they notice the existence of a solar planetary system? 32) Which of the two telescopes diameter D and focal length F one should use to photograph the binary with the distance between the components 0.8", if the size of the photoemulsion is 30 μm: a) D=35 cm, F=4 m; b)D=10cm, F=12 m? 33) If the inhabitants of the planet a Cen systematically measure the position of the Sun with the accuracy of 0.01", can they notice the variations in the solar motion, caused by the movements of the planets of the Solar System? Parallax of α Cen π = 0.751". 34) Describe qualitatively how the visible size and the shape of the solar disk change due to the atmospheric refraction. 35) On the photographs of the globular clusters the stars in the core are merged together. Is it really means that the density of the stars there is so high that they are touching each other? 36) Amateur astronomers in order to increase the magnification of the telescopes sometimes suggest to observe the images through the microscope and not through the eye-piece. Is it advisable to do that? 37) How to distinguish a solar spot from the shadow of the disk of the planet?

38) How the limiting stellar magnitude will change if one is photographing on the Moon, where the brightness of the sky is half the brightness of the Earth sky? 39) Due to the precession the pole of the sky performs one round about the axis of ecliptic in 26,000 years with a radius of 23.5°. How often one shall correct the direction of the polar axis in the equatorial mount so that the angle between the polar axis and the celestial axis wouldn't exceed 0.1°? 40) Can one take a photograph of a solar spot of a diameter 2' with the camera....(obscure)? 41) Can one construct a telescope for a visual observations consisting of only one lens? If yes then what shall be the parameters of this lens in order to obtain the magnification of 50 with the 10' field of view? 42) The solar spot has an angular size of 1' for the observer on the Earth. Can this spot be seen with the naked eye by an observer from Mercury? Venus? Mars`? 43) A theoretical astronomer after coming to his work place in the morning learned that the solar eclipse is just beginning. How, having nothing but the paper and the pencil at hand, can he observe the partial phases of the eclipse? 44) The photon flux from 0m star is about 106 photons/cm2/sec. How many photons will fall on the photoplate from 20m star in one hour if the diameter of the objective is 1m? 45) Using the conditions for the previous problem, estimate the quantum efficiency of the human eye, taking into account that in especially favourable conditions (like at the high altitudes after prolonged adaptation) the eye can see stars up to the 8m, and the time of the light collection by the retina is 0.1 sec. 46) Prove that the brightness of an extended object on the eye retina during observations through the telescope is the same as during the unaided observations. 47) Why the photographing of the planets always fails to register small details on their disks which otherwise are noticeable by visual observations? 48) In a multiple system (Hydra the farthest component D has brightness of 12.5m. lts distance from the brightest component A, which has brightness of 3.7m is 20". A telescope of which diameter one shall use for the observations of these stars? 49) In 1975 a signal to one of the globular clusters was sent from the radio-telescope in Arecibo (Puerto-Rico) (diameter D = 305 m) on a wavelength A = 21 cm. Was it necessary to take into account the displacement of the cluster during the time of the travel of a signal, in other words, to send a signal with an angular forestalling? 50) If quantities A = 112 and B = 102 are measured with an accuracy of 1 %, with what accuracy the quantity (A - B) is calculated? 51) If the place of the spectral line on a photoplate is measured with an accuracy 0.02 mm, with what accuracy one can determine the radial velocity of a star by its spectrum taken with the dispersion: a) 200 A/mm; b) 1 A/mm? 52) How will the number of stars on a photoplate increase if the exposure of a photoplate is increased 2.5 times? 53) In order to increase the angular resolution of the observations astronomers sometimes use lunar occulations of sources. Estimate which part of the sky is, in principle, available for the observations from the Earth using this method? Figure 8 Halley's Comet

54) On a photograph of Halley's comet (Fig. 8.1), obtained in December 1985 in Crimean Astrophysical Observatory, one can see short streaks-the traces of the motion of the images of the stars. A correspondent of one newspaper, which published the photograph,

wrote: "The time of exposure was a whole hour, that is why due to the rotation of the Earth stars turned into dashes". And what do you think? 55) In 1991 one Japanese amatuer astronomer, while photographing the Moon on a video camera, registered a black dot which in a few seconds crossed the surface of the Moon. Newspapers presented this fact as a sensation asserting (insisting) that it was a discover} of a giant UFO in a near-moon orbit: "This UFO has a diameter of 20 km and moves with the speed of 200 km/sec". And what do you think the camera had registered? 56) Orion nebula looks colourless during visual observations in the telescope but on the coloured photographs it has a variety of colours? Why is it so`? 57) In an observatory was working a 2m telescope, which with the 20 min exposure could photograph sars upto 20m. Then a new multi-mirrored telescope was constructed with 6 mirrors 3 m diameter each. What exposure is necessary now to photograph the same stars? 58) Why is it possible to conduct radioastronomical observations during the day but in the optical range only solar observations are possible?

Motion of Celestial Bodies. 1) A spaceship landed on 1 km diameter asteroid, which had the average density of 2.5 g/cm3 . Astronauts decided to go round the equator on a Landrover in 2 hours. Can they do that'? 2) Astronauts landed on an asteroid and the communication with them broke down. They have oxygen reserves for 2 hours. Can their friends on a mothership fly round the asteroid in search for the lost astronauts in this time if its density is 2.5 g/cm3 ? 3) Determine the height of the orbit of a telesatellite which is always in zenith above Moscow. 4) Can a comet be all night on the (near the) meridian of an observer? 5) Halley's comet revolves around the Sun once in 76 years; Neptune does it once in 165 years. Whose aphelion is farther from the Sun? 6) In a Julie Verne's novel "Hector Servadak" a Halley's comet is described, which has a distance from the Sun in aphelion of 820 million km and a period of 2 years. Could such a comet exist? 7) How the orbital velocity of a planet depends on a radius of its orbit? 8) Imagine a scientific base on a surface of an asteroid. Astronauts are playing soccer and one of them, having hit the ball a bit too strongly, made it fly away with a speed equal to the first space velocity for that asteroid. Will the ball become a satellite of this asteroid? 9) What would be a period of diurnal rotation of Mercury if its orbit were absolutely circular? 10) Why Pluto and its satellite Chiron have a synchronized period of rotation whereas the Earth's period is much shorter than the Moon's? 11) Why astronomers couldn't determine the mass of Venus by the observations from the Earth using the same technique as was used for the mass determination of the most of the planets? 12) To illustrate how great is the distance from the Sun to the Earth poet Hebel in his "Treasure-house" used such an example: "A gunner (artilleryman) on the Sun is directing his cannon right at you. You are fleeing in fright. But do not worry: you don't have to hurry up as you have a lot of time to avoid the shell (projectile)." (from V. Litzman "Giants and midgets in the world of numbers", M., Fizmat, 1959, p. 17). Determine the time of travel of the missile launched from the Sun towards the Earth with the initial velocity 5000 km/hr. 13) Determine the mass of the Moon in units of the Earth's mass, if the first artificial lunar satellite Luna-10 had a period of 2h58m in an orbit which lay within the range from 361 to 1007 km above the lunar surface. Take the radius of the Moon Rc = 1737 km. 14) It was decided to launch a spaceship from the Earth to the center of the Galaxy. With what velocity and in what direction it shall be launched? 15) Because of the crash of the American spaceship Challenger, which happened in the beginning of 1986, the planned space launches of the automatic stations Galileo to Jupiter and Ullis to the polar regions of the Sun were postponed for 13 months. Why exactly 13 months? 16) During the supernova explosion a star ejected the shell, mass of which was this star survive? 17) The Moon through its tidal effect is braking the diurnal rotation of the Earth. In a few billions of years their rotation will synchronize and the Earth will be constantly facing the Moon with one side (just like the Moon now is facing the Earth). Then the sidereal lunar month will be equal to the present-day value of twenty-four hours. What would be the duration of the solar day on the Earth then ?

18) Suppose that around each star there is a planet with Jupiter's mass and orbital period. Determine at a star of what mass it is easier to discover the presence of a planet if one measures: a) a) angular displacement of a star, b) b) variation in a radial velocity of a star. 19) Elliptical orbit of a comet was cut into two parts by a straight line passing through the Sun (Fig. 9.1). Prove that the comet is receiving an equal amount of a solar heat in each of the two parts of the trajectory. 20) How the duration of a Year will change on Earth when the Sun will turn into a white dwarf with mass MWD = 0.61M~?

10) Figure 9 Are the orbits parts equivalent

21) Three equal mass stars form an equilateral triangle with the side L and move around the common center of mass in a circular orbits with the period P (Fig. 9.2). Find masses of these stars. 22) Altair (α Aquilae) has an annular parallax π = 0.198", proper motion y = 0.658", radial velocity V = -26 km/sec and brightness m = 0.89m. When Altair will move to the closest distance from the Sun, what is this distance and what would be its brightness? 23) How one can prove that the Sun together with the neighboring stars is rotating around the center of the Galaxy?

Figure 10 Determine the masses of the stars

24) Determine the velocity of the possible collision of a spaceship flying to Jupiter with a meteor in the asteroid hell. Assume that the spaceship is flying along a straight line from the Earth with turned off engines. 25) Let the Sun rotate around the center of the Galaxy in circular orbit with the radius R~ = 10 Kpc and average velocity V~ = 250 km/sec. Assuming all the mass of the Galaxy (MG) to he concentrated in the center, find MG, the duration of the galactic year (i.e. the period of the rotation of the Sun) and the minimum velocity that the spaceship shal1 acquire to leave the Galaxy. 26) From which side of the Earth - day or night - it is energetically (energy) advantageous to launch the spaceship to Mercury? 27) A spaceship was launched in such a manner that after leaving the sphere of the Earths gravitational attraction it started a free fall to the Sun along a practically straight line. How many days this fall would take? 28) Three stars with the same mass M are on a straight line and move in such a way that one of them is in the center of the circle along which two others rotate. Determine the period of rotation. 29) With what velocity and in which direction a spaceship shall be launched from the Earth in order to fall or. the Sun? 30) In the science-fiction novel of A. Rundberg "A Journey to the Earth" (Alma-Ata, "Kazakhstan", 1987) the king of the planet Lunx tells to the defeated rivals from the planet Ricket: "I grant you the peace only on the condition that every ninth year Ricket will send seven noble maidens and seven noble youths to Lunx. I intend to offer these maidens and youths to Taurus... Each full moon one young life of a noble ricketian

will be sacrificed to a monster for a torments." The question is: could in reality the duration of the lunar month be as connected with the duration of a year as on the planet Lunx? 31) What would happen to the Solar System if the Sun suddenly disappears (Fig. 9.3)? 32) Is it possible to send the probe to the Moon by shooting it from the cannon from the surface of the Earth? To Mars? To the Sun? In the orbit of artificial earth satellites? 33) In which case the energy expenditures are less: launching the space probe from the spaceship or from the cannon?

Figure 11 What would happen to the solar system

34) While observing a comet near the Sun, astronomers determined that its orbit is hyperbolic. Will this comet leave the Solar System forever? 35) Recall "Little prince" by A. de St. Exupery. When Prince was visiting different wonderful planets, he visited the planet of a Lamplighter, which "makes one full rotation about its axis in one minute". Estimate the density of that planet. To which class of astronomical objects can it belong? 36) Calculate the distance from the Earth at which a point is located where the gravitational attraction between the Earth and the Moon is the same. Assume the distance between the Earth and the Moon to be equal to 60 Earth radii and the ratio of masses as 1:81. Can a spaceship remain motionless at this point? 37) If all the momentum of the Solar System were concentrated in the Sun, what would be its angular velocity? 38) Would the meteorite showers be observable in the second half of the night, if the meteors would always only following the Earth in its orbital motion? 39) From the board of the EAS, rotating in the circular equatorial orbit, a weightless rope was stretched to the Earth and :he end was fixed on the equator (Fig. 9.4). What would be the fate of the satellite after that? 40) Estimate maximum and minimum velocities of a satellite entering the Earth's atmosphere. 41) An EAS was launched in the polar orbit. Is it possible that the Sun does not illuminate the whole orbit; illuminates hail of it or illuminates the whole orbit? 42) Mizar (ζ Ursae Majoris) has a radial velocity of 9 km/sec, proper motion of 0.13" a year and parallax of 0.038". What is its spatial velocity? In what time will it move away from Alcor (companion of Mizar).

Figure 12 What is the fate of the satellite?

43) You are hanging motionlessly in the center of a spherical orbital station 20 m diameter and holding a rubber ball in your hand, the mass of which is 1000 times less than yours. If you throw the ball with the velocity of 10 m/sec what is the minimum time it will take for to reach the wall? 44) An EAS was first in a geostationary orbit. Then it was moved in an circular equatorial orbit of a double diameter. How often now it is crossing the meridian of a stationary terrestrial observer? 45) In a binary each component has a mass 2M and they arc moving in a circular orbit at a distance of I AU from each other. What is the maximum difference in their radial velocities for an observer on the Earth, if the inclination of the binary orbit (i) is: a) 0°, b) 90°, c) 45°?

Space Travel and Nature of the Planets 1) A spaceship left the Solar System with the velocity of 30 km/sec in the direction of the Galaxy center. In how many year it will reach the center? 2) There are a lot of old satellite in the Earth's orbit which represent a danger to the acting satellites and astronauts working in space. Suggest a method of removing them from the orbit. 3) What height would have taken the pole-vaulter on the Moon? 4) American astronauts working on the orbital station Skylab (1973) were jogging along the inner surface of the station, which was actually a cylinder with the diameter 6 m. With what speed one shall run in these conditions to experience the terrestrial gravity? How shall the station be oriented at that? 5) How much time will take a spaceship, moving with the velocity 30 km/sec, to reach the nearest to the Sun star Proxima Centaurus, which parallax is 0.76"? 6) Having approached an unknown planet, a spaceship went in a low circular orbit. Would astronauts be able to determine the average density of the planets using only a watch (Fig. 10.1)? 7) Why the density of craters on the surface of jovian satellites is increasing monotonically from Io to Callisto? 8) Can one observe meteors on Mercury? Would it be possible to discover meteorites on its surface? 9) On which Solar System bodies was found the volcanic activity?

Figure 13 How to measure the density with a watch?

10) Why for some time after the landing the astronauts sleep without the pillows? 11) American astronauts on the orbital station Skylab during the recreation hours were trying to play darts with the suction cup but failed. Why? 12) On the Earth a small spark is enough to inflame the gas methane. Why on Jupiter, where the powerful electric discharges are occurring constantly, the methane in the atmosphere does not inflame? 13) Astronauts on Mars decided to warm-up near the small fire. They have a bucket of kerosene, thermite packet and wooden boxes. What would you advise them to use as a fuel for the fire? 14) On Venus the cloud cover is so dense that from its surface the Sun and stars are never seen. How then to tell night from day there? 15) Describe briefly the advantages that usage satellites and spaceships give to astronomy. 16) Where it is easier to swim in the water: on the earth or on the Moon? 17) On the Earth a candle burns for 2 hours. How many hours it will burn on the board of a satellite? 18) Two manned satellites are moving in the same circular orbit at some distance from each other. What should do the pilot of the rear satellite in order to catch up with the front one? What is the minimum number of corrections? 19) Station Mars-3 has a period of rotation of 12 days and its minimum distance from the center of Mars is 5000 km. What is its maximum distance from the surface of Mars? 20) A rocket engine can develop a thrust equal exactly to the weight of the rocket. Can this rocket be launched

into space? 21) Due to the modern views the water on Mars (if there is a water) is concentrated in the layer of permafrost on the polar caps. But if there is water on Mars, why then in the tropical regions where the temperature can be above 0°, there are no open reservoirs with the liquid water? 22) On April 12th 1961 Yu. A. Gagarin took off in a spaceship Vostok; from the spacedrome Baikonur. The ship was moving in an orbit of artificial Earth satellite with 1.5 hrs period and had landed after one revolution. In what place relative to Baikonur it landed? 23) Imagine yourself on a spaceship orbiting an unknown planet above the terminator line. How to determine the period of the rotation of the ship if you only have a watch and optical instruments? 24) How in the conditions of complete cloudiness could venusian inhabitants determine the duration of a year? 25) How to orientate yourself on Venus? (How to determine parts of the world on Venus?) 26) An interplanetary spaceship is launched into the interstitial (intermediate) earth orbit. At which point of the orbit it is more advantageous to start the engine to accelerate to the second space velocity? 27) A shaft is dug through the Earth. How to use this shaft to launch the rockets into space? How much fuel it will save? (What fuel economy it will give?) What shape shall have this shaft if it is built between: a) North and South Pole; b) Diametrically opposite points on the equator (Fig. 10.2)? 28) What should have been the length of the cannon in the J. Verne's novel "From the cannon to the Moon" in order for the acceleration for the travellers not to exceed 10g? How much will increase the weight of a human body at the start? 29) It was decided to turn our earth into a giant spaceship. Is it possible to use modern chemical rocket propulsion engines for that purpose? 30) Why rocket fuel shall have the maximum possible combustion heat?....? 31) It is proposed to use water as a rocket fuels first decompose water electrically (break) into oxygen and hydrogen, then mix the resulting gases, and burn the generated "fire-damp" ("detonating mixture") in the rocket engine. Estimate this project. 32) Three identical pendulum clocks were placed: a) on the Earth; b) on the Moon in the air; c) on the Moon in the vacuum. Which one of them will go faster and which-slower? Figure 14 Through the shaft to the

33) Astronomers monitoring the Sun on the Earth see the solar flare stars simultaneously with astronauts observing the Sun. What is then the point in having the terrestrial solar flare patrol, created specially to warn the astronauts about the danger?

34) A rocket is moving vertically away from the Earth with the constant acceleration = 9.8 m/sec2. How the weight of the rocket changes as it moves farther and farther from the Earth? 35) Suppose the properties of the Earth's atmosphere have changed in such a way that it started to reflect the visible light and transmit infrared. How the temperature of the Earth's surface would change? Where and when on the Earth similar conditions do arise?

Astrophysics. 1) Astronomers know about the comets which pass the Sun at the distance of 1-2 solar radii from its surface. Why they do not evaporate in the solar corona, where the temperature exceeds million degrees? 2) It is supposed that the Moon, like the Earth, had an atmosphere at some point. How to explain why the Moon lost its atmosphere and the Earth has not? 3) At the place of supernova explosion in 1987 in LMC the optical pulsar with the period 0.002 sec was discovered. Assuming that superluminal velocity is impossible, estimate the size of the pulsar. To which class of Mars belongs this compact object? {Note: this problem wag suggested in-the 1989 Olympiad, when astrophysicists had no doubt about the discovery. However, later the discovery of the newborn pulsar was not confirmed: the reason for the variability of the optical signal was lying in the high voltage interference in the electronic blocks of the telescope. Still; in the middle of 1994 again the reports appeared confirming the earlier discovery. But majority of specialists are still skeptical and consider that upto now there are found no signs of the compact object at the place on the supernova explosion in LMC. It is possible that in the nearest future this controversy will be resolved by new observations.} Nevertheless, this problem has a meaning and a proper solution. Moreover, such fast pulsars do exist and are of tremendous interest to astrophysicists. 4) It was found from the observations of the radiopulsars that the intervals between their pulses are changing periodically, besides, for all pulsars this period is the same and equal to one stellar (sidereal) year. Explain this phenomenon. Estimate how the period between the pulses of a Crab pulsar (PPSR = 0.033 sec) changes. In which month this period is minimum? 5) A pulsar has an extremely periodic signal and is moving uniformly and rectilinear with respect to the observer. Prove that the observed period between the pulses will either increase or remain constant with time independent on the direction and the velocity of its motion. Do not take into account the motion of the Earth in the Solar System. 6) There is a hypothesis that the periodicity in the solar activity is connected with the tidal interaction of the Sun with the rest of the planets. The maximum interaction the Sun is experiencing from Jupiter and Venus. Determine the period of the maximum tides on the solar surface. Is this period close to the duration of the solar activity cycle? 7) Molecules of hydroxyl (OH) in a dense interstellar gas cloud emit variable maser radiation at the wavelength A =18 cm. Estimate the maximum size of the radiating region if variation have a characteristic time of 5 minutes. 8) Why UV space telescope is necessary to observe the interstellar absorption lines of most of the chemical elements, whereas the absorption lines of the same elements in stellar atmospheres one can study by usual ground optical telescope? 9) How do astronomers distinguish hot stars, reddened by the interstellar light absorption, from the actual cold red stars? 10) In spectra of normal stars we can see absorption lines because relatively cold atmosphere gas of a star screens its much hotter inside regions. What then you can say about the structure of stars, in whose spectra we see emission lines? 11) Suppose that iron and calcium have a spectral line with the same wavelength. How would you determine which element is present in the atmosphere of a star if you found this line in its spectrum? 12) Two galaxies, similar to our Galaxy, collide with a velocity of 1000 km/sec. Estimate how many stars will collide with each other. 13) English proverb says: "The neighbor's yard is always greener". Is this observation justified, and if yes, which reasons, apart from an ordinary envy, could explain it? With what phenomenon on the surface of the Sun can you find analogy?

14) Observations show that light from a star, occulted by saturnian rings, is attenuated by approximately 1m, and the width of the rings, do not exceed 3 km. Using these data, estimate how often and with what velocities particles of the rings collide with each other. 15) How to prove that stars shine by themselves, but planets shine by reflecting the light? 16) The brightness of a nova, blazed up in Vela constellation on August 29th 1975, had increased from 21m to 2m. In the spectrum of this star a hydrogen line with λ = 4861 A was blueshifted by 41 A. Determine the increase of the luminosity of the star at the moment of explosion and the velocity with which the shell was ejected (thrown off). 17) Is there a connection between planetary nebulae and planets? 18) An asteroid is rotating around the Sun and around its axis in the same direction. How the radius of its orbit changes due to the action of the radiation from the Sun (both incident and reflected)? 19) In the center of the Sun the temperature; 15.106 K and thermonuclear reactions are occurring there. Why then Sirius, which is a white dwarf, does not have these reactions, though the temperatures inside it are estimated to be 40.106 K? 20) In our Galaxy a star of a spectral class B is born on average once in 50 years. How many such stars are now in the Galaxy if the lifetime of such star is about 108 years? 21) Solar wind consists of protons moving with a velocity of 300 km/sec and its density in the interplanetary space near the Earth's orbit is 10 particles in 1 cm3. What is the force with which this wind "presses" the Moon? Protons mass is mP = 1.6 x 10-24g. 22) What observational data contradicts the statement that the source of the energy of the stars is the fission of the radioactive elements? 23) Which of the stars of the same spectral class have higher surface temperature: a giant or a dwarf? 24) Angular diameter of an elliptical galaxy is d = 3'. Hydrogen absorption line Hβ in its spectrum has a wavelength λ = 4866A and a width σ = 3A. Estimate mass of the galaxy. A laboratory wavelength of a line Hβ is λ0 = 4861A. 25) How much the luminosity of the Sun will change if half of its surface area would be covered by the solar spots? 26) Two radio sources are ejected from the nucleus of a distant galaxy in the opposite directions along a line, having 60° inclination to the line of sight, with constant identical velocities V = 2c/3. How much closer one source will seem to be to the center of the galaxy than the other? 27) Why do larger mass stars have a shorter lifetime than stars of smaller mass? 28) A radio source was ejected from a quasar nucleus in the direction of an observer at an angle of 30° to the line of sight with a near-luminal velocity. What would be the apparent velocity of the source if the observer can measure only its angular displacement? 29) Effective temperature of the solar photosphere is 4580 K; of a solar spot (on average inside the penumbra} 4500 K. How much less is the brightness of the spot than that of the photosphere? 30) It is known that energy can be transmitted by (heat) conduction (diffusion), convection and radiation. In which way is the energy from the nuclear source in the center of the Sun transmitted to its photosphere? 31) Gravitational attraction of the Earth to the Sun is 6.1010 times greater than the pressure of the solar radiation on it. Determine the radius of the spherical particles, which would move rectilinear near the Sun in any direction, assuming that the density and the absorption coefficient of these particles is the same as of the Earth. 32) What is the average density of the white dwarf which has a mass equal to solar mass, luminosity-thousand

times less than solar and surface temperature-twice of the solar surface temperature? 33) An artificial planet was moved from the Earth orbit to the Mars's orbit and was painted black. Its temperature at that hasn't changed. What fraction of the light the planet was reflecting initially? 34) At the end of its evolution the Sun will start expanding and turn into a red giant. In the result of that the temperature of its surface will fall to half of its value and its luminosity will increase 400 times. Will Sun swallow any of the planet? 35) How would a star sky look for the observer flying on a subluminal spaceship (i.e.. with the velocity close to the velocity of light)? 36) What would happen to a drop of water placed instantly into the open space? 37) Explain why Titan-saturnian satellite-managed to preserve its atmosphere, but Mercury didn't? 38) Why a hydrogen bomb explodes, while the Sun does not, though in both cases the energy is produced in the result of the thermonuclear reaction of the conversion of hydrogen into helium? 39) A spherical galaxy in the constellation Sculptor has a mass of 4.1039 g and its distance from our Galaxy is R = 85 Kpc. Mass of our Galaxy is M = 2.1044 g. Determine what could be the maximum radius r of the galaxy in Sculptor so that it would not be destroyed by a tidal interaction with our Galaxy? 40) List and briefly explain the methods with which the diameters of stars are determined. 41) The semi-major axis of an asteroid orbit is 10 AU, its eccentricity is 0.85 and albedo of its surface is 0.14. Calculate the maximum and the minimum surface temperature of the asteroid during its orbital period using the fact that the Moon's albedo is 0.07 and average temperature is 0° C. 42) A white dwarf with mass 0.9 M~ and radius 6000 km is moving towards the Sun at a speed of 60 km/sec. Are the lines of its spectrum blueshifted or redshifted?

Astronomical Quiz In this section are collected the simple problems, which rather require intuition, experience and sense of humour than detailed calculations. They can be of a use for the evenings of "science for fun" ("science for entertainment") and other such affairs. 1) July, noon. Can you see stars in the sky? 2) An astronomer is observing the full Moon with a telescope. A naughty boy is covering the right half of the objective. Draw, how the view of the Moon changed. 3) An astronaut has pushed off from the orbital station, which is moving at the height of 400 km, and started moving towards the Earth with the velocity 4 km/sec. How long will it take for him to reach the Earth? 4) Can one receive a New Year presents 4 times a year? 5) XXI century. Martian observatory measured the parallax of a star to be 0.1". What is the distance to the star? 6) An astronomer was regularly measuring the radial velocity of a star by the position of its spectral lines. Half a year ago it was 100 km/sec, but today it is only 20 km/sec. What does it mean? 7) The first image of an asteroid from a very close distance was obtained in 1991. How then long before that astronomers knew that asteroids have an irregular shape? 8) Suppose the Earth stopped in its orbit. In what time will it fall on the Sun? GUESS THE NUMBER! 9) As it is known, an ant is stronger than an elephant, since it is lifting the weight many, times greater its own, whereas an elephant cannot do that. An ant even runs faster:: it runs the length equal to the length of its body in a lesser time than an elephant. Let us look at the celestial bodies from this point of view. a) In what time the Earth in its orbital motion travels the distance equal to its diameter? b) In what time the Sun in its orbit around the galactic center travels the distance equal to its diameter? 10) With what acceleration does the Sun moves in the Galaxy? 11) How many neutrino pass through the tip of your nose every second, if all the solar energy is obtained by the conversion of the hydrogen into helium? 12) How much antimatter is inside the Sun? 13) Surface of which planet the Sun illuminates as bright as a table lamp illuminates the pages of your book? 14) Is it possible to do poker-work on a full moon in the focus of a 10-m telescope, like it is done with a lens on a sunny day?

PROBLEMS OF CAPTAIN FIBBER (with a multiple choice) Captain Fibber is a well known and loved cartoon character, who, nevertheless, is quite a boaster and not to be trusted. These problems are based on some of his adventures. 15) Captain Fibber on his Yacht "Trouble", while travelling in a northern seas, entered a fjord and moored in a narrow strafe. In the. night there was a low tide, the water went away and the yacht hung in the air, wedged between the two sides of the strafe. "There is an abyss of 40 feet under the keel!"-writes Captain Fibber. a) Do you believe him? VARIANTS: YES NO b) What is the maximum height the tides reach in the open sea? VARIANTS: 1 cm, 10 cm, 50 cm, 3 m, 10 m, 15 m

c) What is the maximum height of a tide near the shore? VARIANTS: 1 cm, 10 cm, 50 cm, 3 m, 10 m, 15 m 16) While leaving St. Petersburg port, captain Fibber put his chronometer correctly by the local time. But in the sea, as narrates captain Fibber, happened the following: "...going down to my cabin, taking out my chronometer and discovering a strange change in its character: the instrument was, as I said, tame, but after, you know, lying around without tending and care, became quite wild, showing god knows what: at sunrise, it is shows noon, Sun goes to the noon, it shows 6 pro...". Can you determine in which ocean was the yacht "Trouble" then? VARIANTS: Pacific Indian Atlantic 17) Do you remember how captain Fibber was sailing "with cocks"? When he left England, he took with him two young cocks, bought in Greenwich. As soon as the cocks start crowing, captain Fibber starts observations with the sextant. What was the role of the cocks? a) Cocks crow was signifying the same universal time, which captain Fibber must know in order to determine the coordinates (position) of the yacht; b) Cocks were waking the captain at the moment of the sunrise, which is necessary to determine the coordinates (position) of the yacht; c) Cocks are crowing only in the clear weather, when the stars are seen, which the captain was using to determine the coordinates (position) of the yacht.

Old Hottabych Problems In this chapter are collected a few problems which were born during reading the fantasy story "Old Hottabych" by L. I. Lagin. Adventures of old genie and his young friends - Volka and Zhenya - are known to all Russian children. But how many remember that Volka Kostylkov was an amateur astronomer, the acting member of astronomical society at Moscow planetarium and even its senior member? Most probably because of that there are a lot of episodes in the story, which would make ponder over each astronomer amateur. It is possible that the example of "Old Hottabych" story will urge you on to the scientific analysis of other fantasy and science-fiction books. Sometimes their content turn out to be considerably richer than it would seem from the first sight. May be you would like to prepare in your school the evening "Magic and Science" or "Contest of Sorcerers", where children can discuss and give a scientific appraisal to the miracles, performed by the heroes of different books. And may be one of you would decide to perform a miracle yourself? Acquaintance with science will help you in it! 18) The first gift from genie Hottabych to Volka was a wrist watch. First, since genie was unfamiliar with the modem technology (he was very very ancient), the watch was made- of a one piece of gold and without any mechanism inside. It, of course, didn't show any time. "Is there really shall be something, inside?"-the old genie began to feel anxious. Instead of the answer Volka silently took off the watch and handed it back to Hottabych. -Very well,-meekly consented genie.-I will present such a watch to you that would not have anything inside. Golden watch again materialized on Volka's wrist, but this time it was thin and flat. The glass cover has disappeared and instead of second, minute and hour hands appeared in the center of the dial-plate a small vertical golden pin and on the places, where the hour marks should have been, the wonderful, clear water emeralds were sparkling. -Never and no one, even the richest sultans of the Universe, have ever had a wrist solar watch!-was boasting again the old genie.-There were solar clocks in city squares, there were on the markets, in the gardens, in the palaces, and all of them were cut out of stone. But these one I just invented myself. Is it not bad, really? -Indeed, to happen to be the first and the only owner in the whole world of a wrist solar watch was very alluring." So, can one make a wrist solar watch? If yes, then why even the richest sultans never had ones? 19) "-Blessed Volka,-said after the breakfast Hottabych, blissfully basking in the sun,-all the time I am making presents to you, to my understanding-valuable, and each time they turn out to be not to your heart (desire). May be we shall do this: you yourself will tell me what would you ... like to get as a gift from me, and I would be happy ... to immediately realize the desired. -Then present me a large naval binoculars,--replied Volka immediately." Why Volka selected exactly such a present? 20) Volka and Hottabych went on a flying carpet to rescue Zhenya from the slavery: "Evening twilight blanketed the city, but here, in the. sky, was still seen the crimson solar disk, slowly sinking behind the horizon. -Interesting...-thoughtfully uttered Volka, it is interesting, at what height we are row? -Six or seven hundred loktei,-answered Hottabych, while counting something on his fingers". Was Hottabych correct in determining the height of the flight, if for the ground observer the Sun has already set, but from the flying carpet it was seen nearly entirely? Lokot (elbow)-old Russian measure of a length equal to approximately half a meter).