IAO final papers – Probl em Sets Sets Theory Theory Tests Tests Cat
Question
J1
1.
Why is it sometimes better to use a small telescope in orbit around the Earth than it is to use a large telescope on a mountain top?
J1
2.
A thick black fly has dotten onto the object lens of a 5 cm telescope. What will an observer looking to the Moon through the telescope see?
J1
3.
Explain why we see more meteors from midnight to dawn than from evening to midnight.
J1
4.
The 12 Zodiacal signs are equally extended on the ecliptic. In which of them does the Sun lie in for the shortest period?
J1
5.
On 1 cm2 of Pluto's surface fall approximately 10,000 photons per second from a star of the fifth magnitude. How many photons would fall on a detector from a star of 20m during half an hour, if BTA at the Earth is used (the diameter of the main mirror is 6 m)?
J1
6.
The sun has a parallax of ps = 8".8, and a star with the same absolute brightness – p* = 0".022. Is it possible to observe the star at night sky visually? visually?
J1
7.
The moon set in St. Petersburg (60° North, 30° East) yesterday just at midnight. In what region of the Earth will there be an opportunity to observe a total solar eclipse sometime next week?
J1
8.
A spaceship landed on an asteroid 2.2 km in diameter with an average density of 2.2 g/cm . The asteroid is slowly rotating. The cosmonauts decided to travel along the equator of the asteroid in a rover in 2.2 hours. Will it be possible for them to do such a thing? If the answer is negative, why? If the answer is positive, what do they take into account?
S1
9.
Why might some stars appear double in blue light through they could not be resolved in red light?
S1
10. Why can radio astronomers observe during the day, whereas optical astronomers are (for the most part) limited to nighttime observing?
S1
11. Why is it better for some purposes to use a medium size telescope on a mountain instead of a telescope on a spaceship at low orbit near the Earth?
S1
12. What are the reasons why the Hubble Space Telescope is able to observe fainter objects than we can study from the ground?
S1
13. The moon set in St.Petersburg (60° North, 30° East) yesterday just at midnight. In what region of the Earth will there be an opportunity to observe a total solar eclipse sometime next week?
S1
14. Altair (a Aquila) has a parallax of p = 0".198, pr oper motion µ = 0".658/year, radial velocity Vr = 26 km/s and visible brightness m = 0m.89. When and what would be the minimum distance of Altair to the Sun? Also find the brightness of Altair at that point.
S1
15. Recently the Ten-meter Keck telescope began to operate on Mauna Kea (Hawaii), where the diameter of stellar images may be as small as 0".3. Can you evaluate the limiting stellar magnitude for visual observation with this telescope?
J2
16. Two stars have the same absolute magnitude. One is thousand times farther away than the other. What is the difference in apparent magnitudes? Which magnitude larger?
3
J2
17. What would an observer have seen sitting on the Moon and looking at the Earth, Earth, when the total eclipse of the Sun took place on the Solovetz Islands (34°45' East, 65°01' North) at 5 a.m. July 22, 1990? Illustrate your answer with a drawing.
J2
18. The duration of the day on Mars is only approximately 2.5 % longer than on Earth. The orbital period of Mars is 687 days. Find (approximately) the difference between the duration of the sidereal day and the mean solar day on Mars.
J2
19. On the day of the all-the-world holiday (fortieth anniversary of the launch of the first satellite), October 4, 1997, Venus was not far from its Eastern elongation, its coordinates were approximately a = 15h20m, d = -22°. Using the above data, estimate its coordinates and position relative to the Sun on the day of the launch o f the first satellite, October 4, 1957. The orbital period of Venus is 0.61521 of the tropical year.
J2
20. Let us consider that observer is sitting on a planet of Sirius. Which object is brighter one in "his sky": either our Sun or the stars of the Big Dipper?
J2
21. Let us say that the Sun is in Zenith, if it covers the Zenith by its disc. Where is it possible to see such an event more often - in Quito (latitude = 0°) or in San-Paulo (latitude = -23.5°)? Explain.
S2
22. If a star is moving away from the Earth at very high speed, will the star have a continuous spectrum that appears hotter or cooler than it would if the star were at rest? Explain
S2
23. In the course of star war one crazy civilization cut a star in two halfs (without varying substance density and its temperature). How did this lofty deed change the star's magnitude? What is the magnitude of the resulting double star compared to that of the original star
S2
24. What limits the resolving power of the 6-meter telescope BTA in SAO? Calculate it. Explain your calculations
J3
25. What can one see in the Moon's sky more often – the Sun or the Earth?
J3
26. In a new postal service a huge cannon shots a postal shell from England to New Zealand. Can you estimate the duration of the shells flight?
J3
27. It is known that the equatorial coordinates of vernal equinox are 0 hr and 0 deg. Which are the North Ecliptic Pole coordinates? coordinates?
J3
28. Suppose that the Sun collapsed suddenly to a black hole. How would the orbital period of the Earth be affected?
J3
29. Can we distinguish the lunar Mare Crisium, which diameter is 520 km, by a naked eye?
J3
30. There are about of 250 millions of stars in the elliptical galaxy M32 (a satellite of Andromeda galaxy). The visual magnitude of this galaxy is 9m. If luminosities of all stars are equal, what is the visual magnitude of one star in this galaxy?
S3
31. Is it possible to observe solar eclipses, meteors, comets, auroras, rainbows, noctilucent clouds and artificial satellites on the Moon?
S3
32. There are Cepheids variables in our own Galaxy as well as in other galaxies. Why was the "periodluminosity" relation first recognized for Cepheids in the Magellanic Clouds?
S3
33. Because precession, the vernal equinox point moves slowly (50" per year) in the sky. Along what celestial circle does it move – the equator or the ecliptic?
S3
34. Artificial Earth satellite moves with a speed of 6.9 km/sec along the circular equatorial orbit in the direction of the Earth rotation. What is the period of the satellite appearance above any fixed
equatorial point? S3
35. Can we distinguish the lunar Mare Crisium, which diameter is 520 km, by a naked eye?
S3
36. There are about of 250 millions of stars in the elliptical galaxy M32 (a satellite of Andromeda galaxy). The visual magnitude of this galaxy is 9m. If luminosities of all stars are equal, what is the visual magnitude of one star in this galaxy?
J4
37. The apparent diameter of the Moon, as seen from the Earth, is 31'. What is the image diameter in the objective focal plane if its focal length is 254 cm and the objective diameter is 40 cm? Draw a figure (a few figures) to explain your calculations.
J4
38. A photometer is mounted on a 125 cm (focal length) telescope. Can you observe a star with magnitude - 5m, 10m, 15m in a cluster if a count from a star of a similar spectral type with magnitude 8m gives 4000 counts/second? The level of white noise of the photometer (instrumental noise) is 500 counts/second; the upper limiting value for observations is 200000 counts/second. Explain your calculations.
JS4
39. Where (on the Earth) and when is it possible to observe the sunrise with the longest duration? Estimate its duration.
JS4
40. Usually we consider that there are about 6000 stars in the whole sky which are visible by our eyes. Estimate, how many visible stars are circumpolar (which means that they never set): Note: formula for sphere's area calculation: S = 4pR2. a.
if you are placed 1° from the North Pole.
b.
if you are placed 1° from the Equator (to the North).
JS4
41. An airship started from a cosmodrome located near the equator of the earth at the moment of a sunset. A pilot of the airship wants to continue to observe the sun on the horizon. What should the speed of moving of the airship be? Describe in detail the motion of the air ship
JS4
42. Suppose that a total solar eclipse is observed from a place at the equator when the sun is in zenith. Also, suppose that the shadow of the moon moves along the equator. Calculate the speed of the shadow relative to the observer.
S4
43. One star peaks at 2000 A. Another peaks at 10000 A. Which one emits more radiation at 2000 A? Which one emits more radiation at 10000 A? What is the ratio of the total radiation emitted by the two stars? Consider the stars as black bodies.
S4
44. Engineers from the Simferopol University describe a new method to utilize old military ships: to construct very small black holes from their material (patent yzarc-048UA7). Estimate the diameter of a black hole constructed using this patent from a ship with the mass of 5000 tn (1 tn = 1000 kg). What physical object has a size of the same order of magnitude? Describe propagation of visible light near this black hole.
J5
45. As you know, the most widely used calendar in the middle centuries was Julian. Just now most countries use the Gregorian calendar and the difference between Julian and Gregorian calendars is 13 days: for the same days dates in the Julian calendar fall behind the dates in the Gregorian calendar by 13. Last time the dates in these calendars coincided were in the 3rd century. Calculate in what century such a difference will be 1 year and the 22nd of October (for example) in Gregorian calendar will coincide with the 22nd of October in the Julian once again.
J5
46. Two stars have the same apparent magnitude and are of the same spectral type. One is twice as far away as the other. What is the relative size of the two stars?
JS5
47. There are two photos of the Moon taken by the same camera mounted on the same telescope (the
(the telescope is placed on the Earth). The first photo has been made while the Moon was near its perigee and the second one – near the apogee. Find from these data the value of the Moon's orbit eccentricity. Estimate the minimal period between the moments at which these two photos could be taken.
JS5
48. A cosmonaut in a spacecraft is moving over the Moon surface through the Mare Frigoris at an altitude of 100 km. An astronaut is walking on the Moon's surface at Mare Frigoris and it is daytime at that place (the Sun is shining). Can the cosmonaut register the astronaut using binoculars with a magnification of 20x. Take Take into account all the possibilities.
JS5
49. There is a radio source placed on a satellite of some planet named "Olympia". The radio source is working all the time but an observer does not register the signal all the time due to eclipses. The figure shows the level of the receiving signal by the observer vs time. Find from these data the average density of the planet. Take into account that the orbit of the satellite is circular, the observer is in the plane of the satellite's orbit and "Olympia" is far away from the observer. (Picture Missing!!!)
JS5
50. An 1.2-meter Schmidt camera has a 6° ´ 6° field of view. Estimate how many photographs you would have to take to cover the whole sky. (Please, make an estimation of the maximum and minimum number of photos.) Explain your calculations. Where do you have to place your telescope to be able to do this?
S5
51. A quasar is observed and it is found that a line whose rest wavelenght is 3000 Ao is observed at 15000 Ao. Estimate: a.
How fast is the quasar receding?
b.
How far away is it if its distance is given by the Hubble relation (The Hubble constant is H = 75 km/s/Mpc)?
Both answers may be done with an accuracy of 30 %. S5
52. Young scientists from the Komi-Republic territory (in the Russian Federation) registered a few days ago a new object looking like an eclipsing binary star. But the period of this star was not stable: the stellar magnitude of the object is usually equal to 24.32m. Once every 7-11 seconds it is rising to 24.52m for 0.2-0.3 seconds. After investigations it was clear that the shining object is eyes of a group of absolutely black cats cats sitting on a small absolutely black body in our Solar System and looking towards the Sun! And one of the cats is blinking! Calculate the number of cats in the group sitting on the small body and looking to the Sun. Draw a picture explaining your solution. Consider that all the cats are equal in size.
Practi Practi cal Tests Tests Cat
Question
J1
1.
Paper with a drown circle and table of maximal elongations of Mercury and Venus had been done for participants. a.
The figure show a circle, that is the orbit of Earth. By using data from the table, draw orbits of Mercury and Venus.
b.
Estimate radii of the orbits (in a.u.).
Table: Greatest Elongations of Mercury and Venus. Mercury: 1989-1990 Date East 8 Jan 89 19° 18 Feb 89 30 Apr 89 21° 18 Jun 89 28 Aug 89 27° 10 Oct 89 22 Dec 89 20° 1 Feb 90 13 Apr 90 20° 31 May 90 11 Aug 90 27° 24 Sep 90 5 Dec 90 21° S1
2.
West 26° 23° 18° 25° 25°
Venus: 1983-1990 Date East 15 Jun 83 45° 4 Nov 83 21 Jan 85 47° 12 Jun 85 26 Aug 86 46° 15 Jan 87 2 Apr 88 46° 22 Aug 88 8 Nov 89 47° 30 Mar 90
West 47° 46° 47° 46° 46°
18°
Tackle the following: a.
At what time did you see Venus on Saturday, November 2, 1996...
b.
... and at what angular distance from the Sun
JS2
3.
Problem is about Doppler-effect, to find velocities of two stars using their spectra and spectrum of our Sun, to estimate speed of the Earth moving around the Sun.
JS3
4.
The masses of Capella's components: The 6-meter telescope of telescope of SAO is one of a few to carry out speckle-interferometric observations of visual binary stars. The purpose is a direct measurement of stellar masses. You are proposed to estimate the masses of Capella's components, using our observational data. Capella (Alpha Aur) is a very close visual pair. Fig.1 shows a relative orbit of the component B as observed over many years by observations by different o bservatories. The points obtained in SAO are marked in red. red. The position of the component A is marked by a cross and is connected to the periastron point with a straight line. The radial velocity curves of the b oth components are shown in Fig.2. Capella's parallax is p = 0".077, the revolution period of its components is P = 104d.
JS3
5.
a.
Consider a three-dimensional model of the system allowing, in particular, for the orbital eccentricity and the inclination of the orbital plane towards the line of sight.
b.
Estimate the masses of the components, using the Kepler's Third Law.
c.
Consider the possible causes of the errors of your estimates.
A galaxy's mass: Edge-on spiral galaxies are suitable for the determination of th eir masses.Prof. masses.Prof. I.D.Karachentsev and I.D.Karachentsev and his colleagues complied a catalog of such galaxies and obtained their mass spectroscopically. The spectrum of one galaxy, FGC 1908 in Dragon, is given below. It was obtained on the 4-th of March 1997, with the help of a spectrograph installed at the primary focus of the 6-meter telescope. telescope. As the figure shows, the spectrograph slit was aligned with the major axis of
belong to the galaxy. Their laboratory wavelengths wavelengths are indicated. When determining the the galaxy's mass at the SAO the Hubble constant was taken to be H = 74 km/(s×Mpc). You are required to repeat the estimate of the galaxy's mass. Recall that 1 pc = 3.09×1018 cm, the mass of the Sun is MO = 2×1033 g and the gravity constant is G = 6.67×10-8 dyn×cm2/g2.
JS4
6.
a.
Explain why a two-dimensional spectrum of the galaxy looks exactly like that.
b.
Estimate the mass of FGC 1908 and compare it to the mass of our galaxy.
c.
Consider the possible causes of the errors of your estimates.
Write the observational program for this night on a 40cm telescope. What stars would you observe this nigth? Write sequence of the observations. Stars to observe:
V (mag) 1
Aql
m
(1950) h
m
3 .5
19
49 . 9
2. 1
3 04. 9
40 46
0° 53′
2
Per
3
Cyg R Aqr
14. 2
19 48. 48. 6
32 47
12. 4
23 41. 41. 2
- 15 34 34
Ori RW Vir
1. 3
5 52. 4
7 24
6. 7
12 04. 04. 7
- 6 29
4 5 6
Moonrise: 21:17:26 Phase 0.77
Star
Interval of observation
Sidereal time for 0h: s0 = 0:31:04. You may use all observational time (since 20h till 5h 30 min).
Stars that could not be included in the program: Explanations and notes:
Local time LT=UT+3 0
Coordinates of observatory λ = 2h16m ϕ = 44 43’ Put your result into the table (right-below): JS4
7.
A light curve of eclipsing binary is shown in the figure. Estimate the ratio of R/r in assuming that the eclipse is central and the small component is fainter.
J5
8.
A Supernova In The Galaxy Ngc 3184.
Introduction: Special Astrophysical Observatory of Russian Academy of Sciences participates in an international program on investigation of supernovas bursting in distant galaxies. Stellar magnitudes of supernovas are measured with the help of CCD at 1-meter and 60-centimeter telescopes. For the brightest of galaxies the spectra are also obtained. Fig.1 presents a recent recent sample of a CCD image image obtained with the 60-centimeter telescope. It was taken on October 7, 2000. It shows a part of a galaxy cluster, in one of which a supernova burst. Galaxies are marked with the letter G. Their images differ from sharp images of stars by fuzzy edges. The visible stellar magnitudes of two brightest galaxies and the supernova (it is marked with the letters SN) are indicated. And now the task itself: In Fig.2 is shown an image of a spiral galaxy NGC 3184 in which on December 10, 1999 a supernova was noticed. The image was taken before the burst! You can see stars up to 23rd magnitude in it. Fig.3 is a CCD image with a part of the galaxy and the supernova taken on January 28, 2000 with the 1-meter telescope of SAO. Galaxy NGC 3184 is in the constellation of Ursa Major. Its
coordinates (1950) are as follows: α = 10h 15m, δ = 41° 40´. It is similar to the famous galaxy of M33 in the Triangle (M33 is closer to us, the distance to it is 700 kpc, its angular size is about 50´). By the images of 28.01.2000 in SAO the apparent stellar magnitudes of the supernova in blue, green and red filters were determined. Its apparent stellar magnitude was 14m.67.
J5
9.
a.
Find the supernova in Fig.3.
b.
Estimate its absolute magnitude.
c.
Professional astronomers don't doubt that they deal with a supernova. In particular, a spectrum was obtained, by which it was assigned to type II. But nevertheless – for non-professionals – prove that it is not a burst of a nearby (foreground)-star on the background of a galaxy.
A "Drilling" Of The Moon With The Help Of Ratan
Introduction: Dear students, you are to see the first colonies on the Moon. The areas of the Moon surface consisting of oxygen-bearing rock are already being searched for them. First of all these are ilmenite basalts. Beside oxygen (10% from the weight) the ilmenite (FeTiO3) can give a pure iron. With the help of the radio telescope RATAN-600 a "radio drilling" of the Moon was carried out: unlike the visible light, radio waves come from under its surface. The depth of the "drilling" increases with the increase of wavelength. At the wavelength of 1 cm the Moon is transparent down to 0.5 m, at 30 cm – down to 10 m. The regions with increased content of ilmenite are also distinguished by an increased radio emission level. Fig.1 presents radio cuts of the Moon Moon obtained at its passage through the "knife" beams of RATAN. From the 6 used wave lengths only 3 are presented, since the cut at 2.1 cm almost repeats the cut at 1.4 cm, and the cuts at 3.9 and 31 cm differ little from the cut at 8.2 cm. The last could be said also about 13-cm cut, but at the moment of observation the transmitters left at the Moon by American astronauts were operating, exactly at this wavelength. The cuts at different wavelengths can be compared both to each other and to the seas and continents known to you in the image of the Moon in visible light. A band covered in the observations at 1.4 cm is marked in it. In other wavelengths the beams cover the entire disk of the Moon Moon in altitude. The positions of transmitters transmitters operating at the wavelength of 13 cm are also shown. Task: a.
What transmitters were operating at the moment of observation?
b.
What is the difference in the image of the Moon in cm wavelength from its image in visible wavelengths? What is the reason for it?
c. What region would you recommend for the settlement of first colonies? S5
53. Spectra Of A Planetary Nebula "Cat Eye" And Its Central Star Introduction: Planetary nebula NGC 6543 ("Cat eye") played its special part in the history of astrophysics. On August 16, 1865, an English amateur astronomer William Heggins looked at it with a spectroscope and, as he writes, "did not see an expected total spectrum, but only one bright line!" Shortly after it was resolved into two lines with the wavelengths of 4959 å and 5007 å. Heggins assigned it to a new element "nebulium". Later it was found out that these are "forbidden" lines of doubly ionized oxygen that were never observed before - neither in terrestrial laboratory nor in stars. Unlike "permitted" lines, only a very thin gas emits such lines. Recall that if an atom is neutral, then a Roman numeral I follows the symbol of its chemical element, if the atom lost one electron then it is Roman II, etc…. for example, the neutral nitrogen is N I, the ionized nitrogen is N II. In n otation of forbidden line the symbols of atoms and ions are taken in square brackets, for example: the ionized nitrogen is [N II], doubly ionized oxygen is [O III]. The spectrum of the central star generating the nebula NGC 6543 and exciting its glow, as well as the spectrum of the nebula itself was obtained in the course of study of the late stages of stellar evolution. The high-resolution spectrograph of the 6 -meter telescope of SAO was used. The star and a peripheral part of the nebula were alternately projected onto its slit. These positions of the slit marked by a sign ' * ' for the first one and by a sign ' @ ' fo r the second one in Fig 1a, 1b, 1c. Task: The upper parts of Fig. 1a, 1b, 1c show the parts of spectra as intensity vs wavelength given from the mentioned regions. In each of them the interesting lines are selected, and their profiles of are presented in the lower fragments as a relation between intensity and radial velocity. The The figures show also the values of radial velocities measured by the tops of profiles of separate lines. Determine which lines belong to the star and which to the nebula; fill in the appended table:
Lines in the spectra of NGC 6543 and its central star: Figure 2a 2b 2c
Lines belonging to the nebula
Lines belonging to the star
a.
What is the difference between them?
b.
What can be said about the character of motion in the nebula and in th e star's atmosphere from the form of the line profiles?
c.
Estimate the velocity of these motions.
Observational Tests Tests Cat
Question
JS2
10. Every participant has to show all the constellations in the following parts of the sky (sky was not open, some parts of it were covered by trees): North-East, South & North.
JS3
11. The Sun in optical and radio range: With the help of a "school telescope" examine and draw the details visible on the disk of the Sun. Orientate the Sun according to the cardinal points. Identify the details of your picture with the details of one-dimensional radio-sections of the Sun fulfilled in previous days with RATAN-600. RATAN-600. Radio observations were carried out at noon, the knife beam was directed vertically and covered all disk of the Sun
JS3
12. The Star-Splitter : ...That telescope was christened the Star-Splitter, Because it didn't do a thing but split A star in two or three, the way you split A globule of quicksilver in your hand With one stroke of your finger in the middle... Robert Frost “The Star-Splitter” With the help of a "school telescope" find in the sky and divide several (£ 5) binaries into components. Pay attention to the brightness and colors of the components. Explain the observed relation between the brightness and colors of the components (fill the table below).
Explanation of the Color of Color of Object observed main comp. companion facts (color, etc.) 1 2 3 4 5
JS4
13. Name the distance between the Moon and Jupiter in degree. Show how did you find the result.
JS4
14. Find and name the five brightesr stars of autumn sky. Which one is the brightest?
JS4
15. Define the period since new moon for today.
JS5
16. Stars On The Daytime Sky : Introduction: You cannot see stars at daytime with a naked eye. And can they be seen with a telescope? If yes, explain why? Write here your explanation: You have an opportunity to test that with the help of a guide of 1-meter telescope of SAO. Its objectiveglass diameter is 20 cm, focus distance is 3 m, and field of seeing is 21´. Preparation to observation: Using the map of stars, select a star. Point here your object: Its coordinates: Observation: An operator input the coordinates of your object into a computer controlling the 1-m telescope. After the telescope is pointed on it, check if it is seen into the guide. The winner of the observational round competition is one who sees the faintest star at a minimum distance from the Sun.
JS5
17. Observation by Photos:
Photo 1. What nebulae do you recognize? Why most of them are red, but some are blue? Photo 2. What stellar clusters are seen in the photo? What constellations do the photo borders cross? What part of the Galaxy is seen in Photo 1 and 2? Photo 3. What objects are seen in the photo, their names or at least the types? In what region of the Galaxy they are? Photo 4. Australian aborigines call this system of dark nebulae "Emu" (Australian ostrich). In what constellations are its body, neck and head? What object of the photo is the nearest and the most distant?
