The
Paradoxical
Universe
250 Problems in Astronomy
V. V. Ivanov, A. V. Krivov, P. A. Denissenkov
Saint-Petersburg State University
Explanations, councils and the manual
This training management is formally addressed to the students of low-order it is course the astronomical departments of universities, that study its first astronomical course - the course of general astronomy. But the authors hope that on these electronic pages interesting for themselves they will find everything, coma road astronomy - from the qualified amateurs to the astronomers - professionals. As the basis of benefit is assumed the many-year experience of teaching by the authors of the course of general astronomy for the students - the astronomers of the first course of St. Petersburg university. Was included also certain additional, only more complex material.
Tasks are anticipated by the list of values of the thoroughly selected most important astronomical and physical quantities. This list, named by us the "universe in the numbers", is planned as the "numerical portrait" of the universe at all its structural levels - from the atomic nucleus to the metagalaxy. Student, who attentively studied the numbers, their analyzed, comparable with each other, etc., will obtain excellent idea about how the peace surrounding us is arranged. We recommend to always have a copy of the "universe in the numbers" with itself - in the briefcase, on the working table, in the computer. But are still better the data by the "universe in the numbers" to learn by heart. Having a minimum of constants near at hand, and those more holding them in the memory, it is possible with the ease to make different estimations, and estimation - important tool of scientific research. In many instances, when it does not be required precise or special information, the "universe in the numbers" will completely replace thick reference book, for example, the known book of Allen "astrophysical values" [ 1 ] or the encyclopedia "physics of space" [ 6 ] , although, of course, and without them sometimes not to manage. Subsequently we assume the numbers, which entered the "universe in the numbers", known and we do not indicate them in the formulations of tasks.
The tasks (sometimes, faster, questions for the reflections) and their detailed solutions compose major portion of this teaching aid. They are frequently accompanied by extensive commentaries and additions, which are the important component element of benefit and make its not only with the assignment book, but also with the book for reading - by something like the reader. However, the aspect of the tasks - standard (frequently somewhat reformulated), undertaken from the known benefits (for example , [ 3 ]), majorities belongs to the authors.
Special support is made to the tasks of estimated nature, whose solution develops in students the skill to make simple and efficient estimators and thus rapidly and without the
cumbersome calculations to obtain correct idea about the diverse astronomical objects and the phenomena, which then, if this is necessary, they can be investigated more accurately and fully. We tried to follow known motto the "purpose of calculations - not the number, but understanding". Even microcalculator is not necessary during the solution of almost all problems. The authors know according to the experience of teaching that to instruction in accurate calculations with the high accuracy both on the average and in the higher school is given considerably more attention, than to the ordinal estimations or to estimate calculations with one significant place. Many student- freshmen, using the calculator (and all the more computer), diligently extract from the indicator (or the display screen) the long tail of the significant places and propose this as the answer. Let us give a characteristic example. On one of first practical training to students it is proposed to estimate the mass of air in the audience. The question excitedly is posed from the places: and are such the dimensions of audience? Answer: you will estimate "to the view". And even after this frequently is given an answer of the type: 438.75 kg instead of of correct, matched with the accuracy of initial data, answer: near polutonny.
Solution of many problems actually is reduced to combine several numbers of the "universe in the numbers" in one - two simple formulas, brought out at the lectures, and to obtain the required answer. The task can serve as typical illustration: to estimate the mass of the earth's atmosphere. The answer in an obvious manner is obtained, for example, from radius of the earth, the height of homogeneous atmosphere and air density on the earth's surface (all three numbers in the "universe in the numbers" are). It is possible, it goes without saying, to solve problem and differently (as?).
Let us give three useful advice. The first - to produce calculations with the reasonable accuracy - we already discussed. Second council - in the solution of one problem consecutively using one and the same system of units. It is possible to consider cgs system de facto standard in astronomy; use, for example, it. In this case it is possible not to follow the dimensionality of the intermediate results: it is clear, in what units will come out the answer (if mass - that in the grams). The third council - is more ideological, than technical: do not forget to evaluate everythingthat you make and which in you is obtained, from the positions of the common sense. And intermediate results, and final answer compulsorily must be comprehended and estimated, they are plausible. Commonplace council? But students continue to pass the solved tasks with the following answers to instructors: a radius of the Moon = 2.54 parsecs; the mass of the Earth = 5 . 10-5 grams; the number of
stars in our galaxy =732 (all examples real). The authors of assignment book, checking tests, see such answers each yr! Truly, the collection of such results could compose the alternative "universe in the numbers" - the "universe in the number1s from the point of view of student, which does not follow to commonplace councils"...
We relate to the proposed tasks as to the means, which makes it possible to develop the habits of logical thinking, creative relation to the problem, the fantasy. For this purpose we frequently propose tasks with a question "why?", and also namerenno by the "illegible" ("neshkol'nymi") formulations, which contrast with the standard "it is given - to find". As an example let us give the following task: "to estimate the time, for which a change in the outlines of constellations will become noticeable by rule of thumb". It is possible to discuss
as follows. It is clear that a change in the outlines of constellations is caused by the presence in the stars of proper motions. Star drift? ? (into ugl.sek./god) it is connected with its tangential velocity?Vt (in km/s) and with distance to star R in the parsecs by known
formula?: Vt= 4.74 R ?. It is obvious that the stars, which determine the outlines of
constellations, be among bright, and it means, on the average, the closest to us. Therefore it is possible to consider that the distances to the stars comprise order of tens of parsecs. What?Vt, then, after recalling the typical three-dimensional speeds of stars in the galaxy, it
is possible to accept?Vt ~of 10 km/s. Substituting these values into the formula, we do find
?~0.2? ugl.sek./god. Further, completely it is possible to consider that the constellation by
noticeable means does change when the determining its form stars do displace on?1o= 3600". We immediately conclude that the desired time interval is about 20000, and it is
better to say, tens of thousands of years this is - correct result. See another, even simpler approach in the task.
But even when the formulation of task is maximally clear and its solution, it would seem, does not require special creation, in student the possibility to appear initiative remains. Thus, in task about mass mentioned above of the Earth's atmosphere simple calculation does give the answer: ~ 5. 1021 g. but not are more interesting whether, without being
limited to this (correct) answer, to compare the obtained number with some others? Natural to take the mass of Earth itself for the comparison: ~ 6. 1027 g. We can say as the
answer: the "mass of the earth's atmosphere - one millionth of the mass of the Earth". What answer is better?
Some tasks have the increased difficulty, and their solution will be hardly accessible to student- freshmen. They are noted by asterisk (*), and separately difficult - by two asterisks (**). All such tasks are original and nonstandard. Their start, as we hope, will make this benefit interesting to the wider circle of the readers, and the students of low-order it is course, after becoming acquainted with the solutions, they will enlarge their horizon. We already said that partly this benefit is planned not only as assignment book, but simultaneously and as its kind reader.
Contents
Problems
1. First acquaintance 2. Kinematics of the sky 3. Tools
4. Kinematics of the solar system 5. The universal gravitation 6. Earth, the Moon the planet 7. The stellar magnitude 8. Emission
9. Spectra of the sun and stars 10. Sun
11. Fundamental characteristics of the stars 12. Physics of the stars
13. Dual and variable stars 14. The interstellar medium 15. Galaxy
16. Beyond the limits of the galaxy 17. New tasks
Solutions
1. First acquaintance 2. Kinematics of the sky 3. Tools
4. Kinematics of the solar system 5. The universal gravitation 6. Earth, the Moon the planet
7. Solutions to chapter 7 not available in English 8. Emission
9. Spectra of the sun and stars 10. Sun
11. Fundamental characteristics of the stars 12. Physics of the stars
13. Dual and variable stars 14. The interstellar medium 15. Galaxy
1. First acquaintance
1.1 of the substance of the Earth made a wire length from the Earth to
a) of the sun; b) Cen; c) of the Andromeda nebula. To estimate the diameters of these wires ( V.V.Sobolev).
1.2 You will estimate how many stars in our galaxy and how many galaxies in the entire universe are fallen to one person.
1.3 As you do think what more - stars in the galaxy or gnats on the Earth?
1.4 universe are entirely small: stars in the entire universe as much of, atoms as in the drop of water! Prover'te.
1.5 Which is more: the Coulomb force of the attraction of electron to the proton in
hydrogen atom or the Newtonian attracting force of two cosmonauts, who are exchanged handshake in open space?
1.6 Which the depth of that potential pit, in which we do live?
1.7 Which distance modulus Saint Petersburg - Moscow (in the stellar magnitude)?
1.8 is evident whether the sun by naked eye from the Pleiades?
1.9 That is greater - the angular dimension of the disk of the Moon or Andromeda nebula?
1.10 Why it is at night dark?
2. Kinematics of the sky
2.1 After awaking on 21 March, you unexpectedly revealed that they were transferred by some miracle in the uninhabited atoll. Shines the sun, on you - only bathing suit. You will be able in the hour - other to say, in what hemisphere you are located - northern or southern?
2.2 After awaking from the lethargic sleep, you revealed that you are located in the uninhabited atoll, which lies accurately on the equator. Shines the sun, on you - only bathing suit. Will be able you in the hour - to other to say that now you have house in Petersburg - winter or summer?
2.3 After awaking, you unexpectedly revealed yourselves among the primitive people. How you would begin to convince them, that the Earth - this is sphere and that it does revolve around its axis?
2.4 In what direction - from left to right or from right to left - is moved because of the precession the spring equinox point?
2.5 are such minimum and maximum midday altitude of sun in your city?
2.6 In which hour will today visit the sun?
2.7 On 22 March the sun visited to later than the day before. Where you - in Armavire or in Arzamase?
Indication: sin = sin 231
/2o = 0.40 .
2.8 What fraction of the total number of stars on this latitude never does go? Stars to consider evenly distributed in the celestial sphere.
2.9 Is visible the Moon into the June plenilune at the North Pole?
2.10 Why in summer the Moon in the plenilune is always dim and yellowish and is visible low above the horizon (in our latitudes), and in winter, on the contrary, it is bright and does rise highly?
2.11 Which the greatest and smallest height of the upper culmination of the Moon in your city?
2.12 is it possible in Russia to somewhere see the Moon in the zenith?
2.13 Moon is visible in last fourth. To what on the average are equal the radial velocities of stars, find on the sky not far from it?
On 2.14 7 February the Moon was in last fourth. To what did be equal its right ascension?
2.15constellation of Orion has equatorial coordinates near , ~ 0o, and the constellation
of southern crown - , ~ -40o. Are such the visibility conditions of these constellations in
Petersburg now? What they will be in 13 000 years?
On 2.16 22 June on the solar disk from the right side is visible damage. What these are - beginning or the end of the eclipse?
2.17 today occurred the solar eclipse. It means, the angular distance of the sun from the node of moon's orbit is less than 17o. You will explain, why.
Indication: the angle of the slope of lunar orbit to the plane of ecliptic .
2.18 v which to hour (Moscow time) today will begin true noon in that place, where you are located?
2.19 sun is located in the upper culmination in universal time. To find length (by equation of time to disregard).
2.20 Earth - spheroid with compression by 1/300. A) To determine, on how much in the twenty-four hours will leave forward pendulum hours at the terrestrial pole in comparison with the analogous hours at the equator. b) On how much should be lengthened pendulum it is hour at the pole, so that the hours you dispatch so?
Centrifugal force not to consider.
2.21 It is known that the duration of twenty-four hours increases by seconds in 1000 years. Therefore in 1000 years of indication it is hour, synchronized in the direction of the rotation of the Earth taking into account its age-long retarding, they will differ from the indications of atomic hour, synchronized with a constant initial velocity of the rotation of the Earth. To it is how much? (EXPLOSIVES. Vityazev)
3. Tools
3.1 to estimate the safe speed of the motion of the controlled from the Earth Mars rover, equipped with the television camera, which "sees" only on 10 m.
3.2 Where on the Earth it is necessary to place telescope so that its German mounting would be simultaneous and azimuthal?
3.3 will be sufficient one disk in order to write down the entire the information about the celestial sphere, which humanity was located up to the moment of the invention of telescope? (Sun, the Moon, planet and comet - not on account.)
3.4 Which smallest linear dimension of the formations on the Moon, which can be distinguished by the naked eye?
3.5 satellite does fly above the earth's surface at the height of 200 km which minimum size of the details on the Earth, which can be from it photographed?
Indication: the size of the disk of the vibration of star because of the influence of the earth's atmosphere comprises .
3.6 beyond the orbit of Neptune, at heliocentric distances from 30 to 50 AU, is located that opened in 1992. Kuiper's belt, which consists of the small bodies of the type of comets and asteroids. At the beginning of 1996 in the belt are discovered 32 objects; in 1996 it is opened 7 more. It is assumed that there there are tens of thousands of bodies of larger than 100 km they are such the approximately minimal sizes of the bodies of Kuiper's belts, which can be revealed with the aid of the khabblovskogo space telescope?
3.7 To estimate the photon flow (number of fotonov/(sm s)), that is necessary from: A) the suns; b) Cen; c) of star .
Indication: flow from the source of zero stellar magnitude composes approximately fotonov/(sm s).
3.8 how many photons do fall in 1 second on the mirror of the largest in the world telescope of cake (D = of 10 m) from Vegi and from the star ?
4. Kinematics of the solar system
4.1 as in the task , you again among the primitive people. Continuing to sow reasonable, good, eternal, you explained by them, as the solar system was arranged, but they revealed to horror their that distance from the sun and the periods of revolution of Venus and Jupiter you forgot. What observations you should be produced in order to determine their values?
4.2 na how much approximately degrees was displaced Pluto along its orbit from the time of its discovery?
4.3 period of revolution of Neptune around the sun is 165 years. Will be in the sun visible disk, if we look at it from Neptune by the naked eye, or it will be point target?
4.4 to depict in the projection on the plane of ecliptic the orbit of celestial body, which has the elements: AU, e = 0 . For the comparison on the same drawing and on the same scale to depict the orbit of the Earth.
4.5 How angular distance of the Earth from the sun in the elongation with the observation from Jupiter?
4.7 sidereal period of the axial rotation of the sun (at the equator) of --- of 25 days. To what is equal the synodic period of its rotation with the observation with A) mercury; b) the Earth; c) of Pluto?
4.8 You will estimate the characteristic transit time of Venus on the solar disk. How it is moved along the disk with the observation from middle latitudes of the northern hemisphere of --- from right to left or from left to right?
4.9 diameter of Pluto composes 2300 km, distances from the sun (to the end of the current millenium) of --- 30 AU. You will estimate zone width on the earth's surface, in which it is possible to observe coating star with Pluto, and its duration.
4.10 astronomers of future produce the radar of the asteroid, which moves along the circular orbit by radius 5/3 AU. By how many times more powerful pulse they must send in the
connection, than in the opposition in order to register the signal of identical force reflected? To how many stellar magnitude this asteroid brighter in the opposition than in the connection?
5. The universal gravitation
5.1 which would become with the solar system, if the mass of the sun did instantly decrease doubly?
5.2 as would change the orbit of the Earth, if the mass of the sun suddenly was doubled?
5.3 why in all without the exception of the bodies of the solar system, from the sun to the asteroids and the satellites of planets inclusively, the minimum times of their circling do have one and the same order of magnitude?
5.4 speed of the motion of comet in the perihelion three times as great than in the aphelion. To what is equal eccentricity of its orbit?
5.5 as does depend the linear speed of the motion of planet along the circular orbit on a radius of orbit R? No matter how unexpectedly, a question refers straight to the famous problem of the "concealed mass" in the galaxies
5.6 to determine the semimajor axis, the revolution period, eccentricity and perihelion distance of comet, which at a distance of 1 AU from the sun has a speed, directed perpendicularly to radius-vector, also, along the value ten times less than the speed of the Earth.
5.7 satellite "Molniya"; in the perigee it is separated from the center of the earth by 7000 km, in the apogee - on 46000 km you will calculate circling time of satellite around the Earth and eccentricity of its orbit.
5.8 you will estimate the time of the overflight of automatic spacecraft from the Earth to Mars on the semi-ellipse, in the perihelion which is concerned the orbit of the Earth, and in the aphelion - the orbit of Mars (Hohmann ellipse). Semimajor axis of the orbit of Mars to accept equal to 1.5 AU.
5.9 Geliostatsionarnoy is called the circular orbit, which lies at the equatorial plane of the sun, with the revolution period, equal to the sidereal period of the axial rotation of the sun. To find its semimajor axis.
5.10 which the minimum speed of a drop in the meteorite on the Moon?
5.11 what must be the rate of accretion to Jupiter (in the masses of Jupiter per year) so that its bolometric luminosity would grow doubly?
5.12 when people intended to land on the Moon, actively was discussed a question, it was not covered with the thick layer of dust, in which it is possible to sink. Then in students this question arose: how long must on the Earth work vacuum cleaner, so that the isolated energy would be sufficient for its delivery to the Moon?
5.13 which limit of size of the asteroid, from which still it is possible to jump into space?
5.14 it is improbable, but the fact: from the friction against air the satellite, which flies in the rarefied layers of the atmosphere, is not impeded, but it is accelerated! To understand, as so it can be, after proving that energy, which the satellite expends on overcoming of air resistance, exactly is equal to an increase (positive) in its kinetic energy.
5.15 how must be distance to the Moon, so that in its orbit in the solar system would be points of inflection? To consider that the Moon dvizhetsya strictly in the plane of ecliptic, and its orbit relative to the Earth - circular.
5.16 spacecraft investigates neutron star. At what approximately distance from it tidal forces yet will not create the danger in the health of cosmonaut?
5.17 several years ago in the solar system is opened the new class of the objects of --- dual asteroids. You will roughly estimate the greatest possible distance from the 100-kilometer asteroid to its satellite - the asteroid of smaller size.
5.18 surface, on which the attracting forces to the Earth and to the sun are equal, is called the sphere of the attraction of the Earth relative to the sun. Is the sphere of attraction actually sphere? With what mass of the Earth the sphere of attraction would be plane?
5.19 raised stone to height R above the surface of the planet of radius R and mass M and let go there with the zero speed. From dimensional considerations to show that the time of its free fall on the surface of planet exists
where --- dimensionless coefficient (order of one). Without attempting to find accurately, to obtain the following estimation:
5.20 Task- parody to the fantastic subjects about the starry wars.
Reconnaissance reported, that the nuclear-powered submarine of enemy must float at the north pole for 10 minutes exactly at noon on our time. DAN order it to destroy. For this to the pole in the morning arrived our submarine, from onboard of which was vertically upward neglected the combat missile. In the very short time (count - instantly) rocket collected the speed, equal to the first space, after which its engines were opened. Our submarine here was immersed and swam away. After a certain time - exactly at noon as prevented the reconnaissance, on the surface appeared enemy submarine. But it did not pass even five minutes after its emersion, as it was destroyed by rocket fallen to it from space.
How much did show the hours of the commander of our valiant submarine, when it did return command "launching!"?
6. Earth, the Moon the planet
6.1 to what is equal the temperature of the surface of the Moon at sunflower point?
6.2 with the building of lunar observatory muddler- astronaut at hot noon spilled to the lain on the soil thick iron sheet the bucket of remarkable white- white whiting (albedo A=0.99). It was revealed after a certain time that to work, costing on this sheet, is completely impossible. How you do think, why?
6.3 let us assume that around the star with the temperature and the average density
dvizhetsya the planet. Let it not have an atmosphere, it is inverted to the star by one side and has the same albedo A, as on the Moon. To what is equal the period of revolution of this planet along the orbit, if temperature on its surface is the same as on the Moon? How does depend
temperature on the planetary disc on the zenith distance of star? It is considered that also the star, and planet emit as blackbody.
6.4 is there sufficient whether the kinetic kinetic energy of the Earth along the orbit in order to ionize the mass of hydrogen, equal to the mass of the Earth?
6.5 what fraction of the mass of the Earth is fallen to its atmosphere?
6.6 with a good approximation (by what?) it is possible to consider that the sun illuminates the Earth by parallel light beams. As then to explain the picture, which you for sure repeatedly observed: from under the cloud, which closes the sun, do go the clearly divergent rays? 6.7 the earth's atmosphere has optical thickness along the normal (at the wavelength
). Visualize that the mass of the atmosphere grew 5. With what will see the cloudless sky of peterburzhets at noon and in midnight? But that if the mass of the atmosphere does decrease 5?
6.8 with a smooth increase in the mass of the atmosphere from zero skies would be in the daytime first black, then everything would become brighter, then again by less bright and finally would begin the dark. At what optical thickness is reached the maximum of the brightness of sky?
6.9 sky from the morning to the evening is tightened by the overcast, but the disciplined astronomer honestly works --- on its solar telescope it it attempts to be occupied by the determination of the chemical composition of the sun. How you do think, can this something leave?
6.10 sun arises from behind the sea horizon. One person admires rise from deck of ship, and another of --- from the mountain peak with a height of 4 km on the located in ocean volcanic island (let us say, from the observatory of Mauna Kea in Hawaii). In which of the observers the solar disk will be more brightly, also, into how much approximately once?
6.11 everyone knows that on the rise and with the sunset is red. To show that this in a sense illusion, in fact it infrared. More precisely saying, to show that the maximum in the energy distribution in the spectrum of the direct radiation of the sun, recorded at the moments of rise and sunset, lies at the IR range. To consider that 1) in the exoatmospheric spectrum of the sun the energy distribution blackbody with the maximum at the wavelength and 2) extinction of light in the atmosphere is caused by molecular scattering.
Note. The calculator will be required for the solution of this problem. The solution is sufficiently long.
7.1 to largest ground-based telescopes (in particular, to two largest in the world 10- meter telescopes of cake) are accessible stars . By how many times they are weaker than stars, hardly distinguished by the naked eye?
7.2 the variable o cet (peace of whale) in the maximum has visual luster , in the minimum of --- . In how much approximately once do change its luminosity in the visible region? 7.3 difference in the stellar magnitude of two stars of identical luminosity is equal . By how many times one of them is further another?
7.4 where it is brighter than --- in the daytime on Pluto or on the lunar night on the Earth?
7.5 of the substance of the Moon into the plenilune made millions of identical spherical satellites, after leaving them approximately in the same place, but so that they would not
overshade each other. Which the stellar magnitude of the received cluster? Stellar magnitude of full moon is known to you.
7.6 binary star has components and . To find the summary stellar magnitude of dual. 7.7 Zatmenno - binary system has identical components. On how many stellar magnitude do change the luster of system at the moment of the total eclipse of one component of another? 7.8 v star cluster N of the stars of stellar magnitude m each. To find the summary stellar magnitude of accumulation.
7.9 before the sky of approximately 6000 stars is brighter than the 6th stellar magnitude. Considering, that in all stars identical luminosity and that they are distributed in the space evenly, to estimate, it is how much before the sky of stars of brighter than mof -1 stellar magnitude.
7.10 * on what approximately distance it is necessary to place 100 - watt lamp so that it would look like the star of 0-1 stellar magnitude?
7.11 * is it possible from the Moon with the naked eye to see cities on the Earth?
7.12 * one of four Galilean satellites of Jupiter --- Europe --- has a radius of 1600 km and it is completely covered with ice. You will estimate the stellar magnitude of Europe at that moment, when Jupiter nearest of all to the Earth.
7.13 to show that with a small change in the distance to the luminous object ( ) its visible stellar magnitude changes on
7.14 the stellar magnitude of planet in the opposition to 3.43m is less than in the connection.
What this after planet?
7.15 how long did pass from the connection to the opposition of planet, if its stellar magnitude in this time did decrease by 0.85m ?
7.16 you will estimate the maximum distance (in PK), from which the sun by the also evidently naked eye.
7.17 as you do think, there will be the sun in the number of one - two ten brightest stars for the observer, who lives in environments Cen?
7.18 are accessible to the largest telescopes of the stars, similar to the sun, in the Andromeda nebula?
7.19 to estimate the absolute bolometric stellar magnitude of the working vacuum cleaner.
8. Emission
8.1 * the very good estimation of blackbody intensity in the maximum of Planckian curve is obtained, if we use ourselves the approximation of wine. You will explain, why this so, and give estimation of error. How far beyond maximum, i.e., in the region , still it is possible to use the approximation of wine? How in the maximum it does grow with T? 8.2 * you will examine the family of Planckian curves in the scale of the wavelengths, which correspond to the different values of the temperature (see any course of general physics). They do not intersect, those, which correspond to higher temperature, lying above (prove!), and
maximum in them is located to the left (Wein's displacement law). In this case maximum in these Planckian curves the "sharper", the higher the temperature. Why this thus? But as does change with an increase in the temperature the form of Planckian curves in the frequency scale? 8.3 ** to obtain the following exponential approximation of the dependence of blackbody intensity on the temperature in the environment :
It is hence clearly evident that upon transfer into the short-wave part of the spectrum the sensitivity to variations in the temperature becomes stronger of --- the very important property of the Planck function.
8.4 * why the medium energy of one blackbody photon is not equal ? To find it. 8.5 * to show that photon concentration in the equilibrium field of emission with temperature
T is equal .
8.6 upon what transfer of electron in hydrogen atom is formed the lying at the red part of the
spectrum line ) ?
Indication: to consider known the maximum wavelength of the photon, even capable of ionizing hydrogen atom from the ground state: .
8.7 are possible the ground observations of the line of interstellar hydrogen H (passage of electron from the 11th to the 10th level)?
8.8 to estimate the wavelength of the recombination radio link of hydrogen H . 8.9 * to prove that the recombination radio links of hydrogen , , are equidistant in the frequency.
8.10 is capable whether emission of ionizing hydrogen atoms, which are located on the second level?
8.11 * why in the spectrum of the solar chromosphere the evidently much larger number of lines of a Balmer series, than in the spectra of white dwarfs?
9. Spectra of the sun and stars
9.1 to estimate the minimum width of Fraunhofer lines.
9.2 star of class B0V have a period of axial rotation . To find the characteristic widths of lines in the spectrum of this star in the visible region of the spectrum (in the angstroms).
9.3 * lines H and K of ionizovannogo calcium of --- the strongest in the visible part of the spectrum suns. (their equivalent widths of --- 14 and 19 , respectively.) The strongest of the hydrogen lines of --- these are Balmer lines and (their equivalent widths they are close they are about 4 ). Why lines H and K are considerably stronger than hydrogen lines, although hydrogen of --- fundamental component of solar atmosphere, and calcium of --- the minor constituent?
9.4 ** the prevailing detail of the spectra of the stars of class A0 in the optical part of the spectrum of --- large Balmer jump on . Is great whether the darkening of the disks of these stars to the edge na and on ?
but 9.5 ** energy distribution in the spectrum of sun (G2V) is close to the blackbody with k. why in the spectrum Of vegi (A0V) it in no way similarly to the Planckian s k?
9.6 ** the atmosphere of the sun can be considered gray. This means that the emission of all wavelengths (in the visible part of the spectrum) is weakened by it equally. Why then the darkening of solar disk to the edge with the decrease of wavelength does become more? 9.7 ** as is known, the fundamental component the atmosphere suns of --- these are hydrogen and helium. The number of atoms of all other elements, together, --- so-called "metals" of --- is from the number of hydrogen atoms. How will change the mass of solar atmosphere, i.e., those it is layer, from where emission does come to us directly, if the content of "metals"
decreased 10 times?
10. Sun
10.1 ** as does be born daylight? It is more precise: which that concrete elementary atomic process, proceeding in the atmosphere of the sun, with which are emitted the photons, received by us as sunlight?
10.2 visualize improbable: the gigantic spot, which shut entire solar disk, developed in the sun. It will become in the daytime darker than on the lunar night?
10.3 prove that the gravitational pole of the Sun not is capable of holding the electrons of the solar corona.
10.4 ** the brightness of crown from the brightness of the sun. It shines due to the scattering of light of the sun on the free electrons. To estimate the mass of crown.
10.5 approximately days after the chromospheric flare appear different geophysical disturbances. Which kinetic energy of their causing protons (in eV)?
11. Fundamental characteristics of the stars
11.1 stars of the main sequence of the spectral class B0V ( K) have a mass . After using mass-ratio of --- luminosity, to estimate their average density.
11.2 along luminosity and radius of the sun to calculate flow with 1 cm2 of its surface, and on it
--- effective temperature.
11.3 as will change the luminosity of star, if its radius slightly decreased (let us say, to 2%), and the effective temperature of --- on so many increased?
11.4 to find a difference in the absolute stellar magnitude of two stars of identical size, whose effective temperatures differ to 11%.
11.5 luminosity of one of two identical with respect to size stars to 4% is more than another. Considering that emission of both stars blackbody, to find, on are how much characterized by A) temperatures of stars; b) the wavelengths, which correspond to maximum in the energy
distribution in the spectrum; c) of the intensity of emission in the maximum of spectrum; g) the intensity of emission at one wavelength in Rayleigh -djinsovsko1 the region of the spectrum. 11.6 * to find height of the uniform hydrogen atmosphere for A) the suns and b) white dwarf s
k, and .
11.7 white dwarf has to and . To find his absolute bolometric stellar magnitude.
11.8 40 Eri B of --- one of the first open white dwarfs (and one of most studied at the present time). It has effective temperature 17000 k and absolute stellar magnitude . To find its radius.
11.9 * to estimate the thickness of foil from shokoladki, sowing to which in the environment of the star of class O5, gnat could fly away on it as on the photon sail to other stars. At the moment, when gnat sits down itself on the foil, it rests.
12.1 rate of energy release per unit of mass in the human tele- several orders higher than in the sun. Why we are much colder?
12.2 molar mass of the substance of solar depths is close to 0.6. Why it is less than one? You will estimate the total number of particles, which compose the sun.
12.3 density and temperature in the center of the sun are equal, correspondingly, 150 g/sm3 and K. What it is there more than --- of photons or particles?
12.4 * average solar density (1.4 g/sm3) is more than water density. Why it it is possible to
consider gas sphere?
12.5 mass - particle approximately to 1 % is lower than summary mass of four protons. You will ascertain that the thermonuclear synthesis reactions - particles are capable of supporting the present luminosity of the sun at duration of billions years.
12.6 to estimate the relation of the numbers of photons and neutrino, second-by-second radiated by the sun. With synthesis of one - particle is separated the energy 26.7 MEV, the neutrino taking away only the small part of this energy (of about 2%).
12.7 * you will determine the rate of accretion (in the masses of the sun per year), which could ensure the observed luminosity of the sun. How this accretion it would affect the duration of year?
12.8 temperatures in the centers of two stars are equal TO K and K. To estimate the ratio of the corresponding velocities of energy release.
12.9 which smallest possible period of the axial rotation of the red giant s , which has a
radius km?
12.10 in the quite short-period pulsar PSR 1937---214 period is one-and-a-half milliseconds. To estimate its density.
12.11 * visualize impossible: in interior of the suns gas pressure ceased to exist. In what time it will collapse into the point? But how long does depart to the compression of interstellar cloud with an initial density of g/smof 3?
12.12 ** when began interstellar overflights, about one of the stars was opened the planet, which consists of the uncommon substance. A study of the models of this substance showed that it is compressed with the pressing; moreover pressure P and density of substance are connected
with the relationship , where K of --- certain constant, which was measured in the laboratory.
Many other properties of this substance made with its truly priceless. Therefore, in spite of the distance of planet and the high cost of transport, was begun the output of this substance on enormous, truly space scales.
But here was revealed something improbable: how much not took away substances from that surprising planet, its size from this did not change!
Prove that this "miracle" is possible, and find a radius of this planet. (just in case is immediately it prevented, that relativistic effects here not with what; gravity - Newtonian).
Indication. To use the method of dimensional analysis. For obtaining the precise value of a radius use an equation of hydrostatic equilibrium.
13. Dual and variable stars
13.1 * let --- the radial velocity of component spectral- dual with period P. Understand the physical sense of the values
and
In the second case we count the orbit of circular even we assume that the center of the masses of dual rests.
13.2 * dependence the period of --- absolute stellar magnitude in strip V for the cepheids takes the form
< Mv> = -1.3m -3.0m lg P,
where P of --- a day and
< Mv> of --- the average value of absolute stellar magnitude. To estimate distance to Cep (period 5.3 d).
13.3 * in cepheid with the average period are observed its sine flutters with an amplitude of 43 seconds, proceeding strictly with the annual period. The period of cepheid is
maximum on 20 December. Why it is possible to assert that this star is located near the spring equinox point?
13.4 * the effective temperature of cepheid changes between 5500 and 7000 k, and its
bolometric stellar magnitude of --- on . By how many times does change a radius of star in the course of pulsations? Phase lag in the curve of radial velocities to disregard.
13.5 Malomassivnyy component of binary star s , and by period
is X-ray pulsar with the true period of 0.1 s. find, within what limits the observed period of pulsations because of the Doppler effect changes. Orbit circular, slope angle , i.e., line of sight lies in the orbital plane of orbit.
13.6 to make, which prove, that, live we in system Cen, to astronomers would be difficult to see planets near the sun. See, however, task .
13.7 which must be the accuracy of the determination of A) radial velocities and b) proper motions, so that the astronomer with Cen could suspect existence of planets in the sun?
14. The interstellar medium
14.1 * in the star of class A0 interstellar absorption in the visible range is equal . Which observed color index of this star?
14.2 there is a planetary nebula, whose nucleus --- very hot star (T = 80000 K). The large part of the emission of star is absorbed in the fogginess. Why through the fogginess are well visible distant galaxies?
14.3 * how many interstellar atoms of hydrogen does ionize second-by-second star with WITH T = 100000 k and ?
14.4 * star with T = 16000 K are submerged in the interstellar medium. You will estimate, what portion of the energy emitted by it proceeds with the ionization of interstellar hydrogen. To consider that the star shines as blackbody.
14.5 * to show that a radius of zone HII depends on the concentration of hydrogen n as . 14.6 to estimate the mass of the molecular cloud, which consists in essence of H2, if its size
14.7 * you will estimate the mass of neutral hydrogen on the line of sight (into g/sm2), which
generates noticeable ( %) absorption in the center of line L . Absorption coefficient in the center of line taking into account one atom (at a temperature of gas of
approximately 100 k).
15. Galaxy
15.1 za how many years the Earth in its annual motion around the sun does pass the way, equal to distance to Cen? This is - the important characteristic time: the form of the celestial sphere on the Earth substantially changes in this time interval. Why?
15.2 to estimate eccentricity of the parallactic ellipses of Spica ( Vir) and by polar.
15.3 "flying Barnard's star" have a record large proper motion 10" V and a parallax 0.5". To estimate its tangential velocity.
15.4 star, which is been located at a distance of 10 PK, dvizhetsya with a speed of about 50 km/s perpendicular to line of sight. To what is approximately equal to its proper motion?
15.5 Vegas - this is the star of the spectral class A0V. Its visible stellar magnitude . To estimate parallax.
15.6 the Crab Nebula - this a remainder of the supernova, which flared up in 1054 it it has, speaking in general terms, the form of the luminous ellipsis with the size of ''.kh 6'. The measurement of the radial velocities of gas of fogginess showed that it is enlarged with a speed of about 1200 km/s. To estimate distance to the fogginess.
15.7 two cepheids exactly of one and the same period are located in the Andromeda nebula of --- one on the nearest, another on outermost from us the edges of its disk. You will estimate a difference in the stellar magnitude of these cepheids.
15.8 * five of seven stars of the ladle of Ursa Major (besides "extreme") have practically identical proper motions and nearly equal radial velocities. After recalling, as appears ladle before the sky in the comparison with other stars, to estimate distance to the Ursa Major (more precise, to these five stars of ladle).
15.9 astronomers for the first time carried out observations in the ultra-unknown (UN) and infranevedomom (IT WAS OTHER) ranges. In UN they found discrete sources, moreover it turned out that they were distributed throughout entire celestial sphere evenly. In they were also other discovered previously unknown objects, moreover it turned out that these sources are
concentrated to the galactic plane (but not to it to center). What it is possible to speak about object distances, opened in UN range? But about the distances to the Other- sources? 15.10 on the distance to the center of galaxy and velocity of the motion of the sun along its galactic orbit to estimate the mass of galaxy.
16. Beyond the limits of the galaxy
16.1 you, of course, remember, as appear the usual photographs of the Andromeda nebula (Mey). How you it does seem, which approximately angle of the slope of the galactic plane Of mey to the line of sight? After you, anywhere without glancing, made an estimation, prover'te yourselves, after taking the necessary measurements of the photograph of this fogginess in the answer.
16.2 as would appear the Andromeda nebula for the naked eye, if we did see it from the edge?
16.3 you will estimate the absolute stellar magnitude of the supernova, which flared up in 1987 in the large Magellan cloud. In the maximum of luster it had the visible stellar magnitude of approximately 3m.
16.4 width of line in the spectrum of the nucleus of Seyfert galaxy are about 30 . Which spread of characteristic velocities of the motion of the clouds of the radiating gas in the nucleus of this galaxy?
16.5 radio source in the nucleus of active galaxy has angular dimension of 0.001", the amount of red shift z = 0.5. You will estimate the linear dimensions of source in PK.
16.6 galaxy are moved away from us at a rate of 5000 km/s. It is visible as object with size of 1'. To estimate its linear dimension.
16.7 * what more in the universe - protons or relict photons? Average density of substance in the universe to accept equal to 10-30 g/sm3. The temperature of relict emission is equal to 2.7 K.
New tasks
1.2. Surprising fact: The Earth obtains from the sun more heat, than Venus despite the fact that is located 1.4 times further
from it! And the matter here completely not in the fact that the Earth only is more than Venus - the discussion deals with the heat, which is fallen per the unit surface area, and not to the planet as a whole. So in what the matter, as so it can be? And why then on Venus nevertheless much to zharche, than on the Earth?
3. If the sun was wooden and oxygen it would be in the abundance, then in it not nuclear, but usual fire would blaze. Through how many days the firewood will burn down and the sun will go out? To consider that the solar firewood bonfire blazes with such force, which ensures the same energy release as in the present sun. The mass of firewood is equal to the mass of the sun.
4. After arriving in the machine of time to the ancient Greeks, you met in Rhodes with Hipparchus and told to it, as the solar system was arranged in reality. Some of its parameters, in particular, the orbit eccentricity of the Earth (e= 0.0167), you,
unfortunately, forgot. However, to take with yourselves into the past at least reference book Allen [1], to say nothing of the computer connected to the Internet, you did not find time. To find in the antiquity it was for long cannot, without risking to remain there forever, your desheven'kaya machine of time it was entirely primitive. Therefore to describe to Hipparchus about the telescopes did not be sufficient time. It was necessary to urgently return to present, hardly after having time in the last minute to present to shaken Hipparchus its wrist quartz hours.
Hipparchus, after believing in Kepler's laws, decided to determine itself from the observations of the sun, to what it is equal to e. What it could dostich', if soon it did not die?
5. 17 June midday altitude of sun to less than on 22 June. On how much approximately it does grow from 22 December until 1 January?
6. We will indicate that the sun in the zenith, if it shuts zenith with its disk. Where such it is possible to see more frequently - in Quito or into San -Paulu?
7. For several days before the new year the fashionable hotel has on famous oceanic Kapakabana on the bank, that in the center of Rio de Janeiro, appeared this advertising panel: "nowhere in the world you will be able never to tan after sea bathing more rapidly than on Kapakabane on the following week!"
To prove that this is truth. Latitude of Rio de Janeiro .
8. Russian astronomer arrived to work on the contract to Rio de Janeiro. Manifesting enviable diligence, whereas during the first workday it observed on its astrometric tool the sun from the rise to the sunset. In this case with the consternation (but on the truth speaking, even with a certain fear - "already I do not descend 4 from the mind?") it revealed that from the morning the sun was moved on the sky from left to right, and in the after-dinner time - from right to left.
What was days 10 prior to far - Easter or Christmas? But will not be able you to say, what precisely number our suffering from the incredible heat Russian did approach the work in
Brazil? The latitude of the university observatory, on which it observed, was equal .
9. In 1997 the declination of the Moon varied each lunar month approximately from
to . In what year the declination of the Moon will vary from to ? But here is a question potrudneye: in what constellation find in 1997 the ascending node of moon's orbit?
10. Are such the smallest and greatest duration of sunrise, i.e., the time interval, during which the solar disk "does come out" from behind the horizon? Refraction not to consider.
11. To estimate the duration of sunrise on the Moon on the average lunar latitudes. 12. How many radians one hour?
13. Twenty-four hours on Mars are nearly equal terrestrial - in all to 2.5% is more prolonged than our, and the period of its revolution around the sun - about two years. To how much approximately on Mars the sideral days are shorter than average solar?
14. That that yr - this is the period of revolution of the earth around the sun, any child knows. This assertion is inaccurate in reality. Why? To estimate a relative error in this well-known truth.
15. The so-called equation of time is the difference between the "present", i.e., evenly current time t and by apparent solar time , i.e., by the hour angle of the sun at moment t:
With an accuracy to it is given by the following expression: where
How the duration of apparent solar days it does change during the year? When it greatest and when - smallest? To the how much longest solar days of more prolonged than the short?
16. Student- astronomers decided to finish celebrating the astronomical beginning of the third millenium. What number and into how much Moscow time they had to raise glasses with champagne, if it is known that astronomically 1997 it began on 31 December, 1996, v on Greenwich.
17. In one and the same date, on 27 February, 1900 , at noon local time one steamship left from Vladivostok to San Francisco and, etc. - from San Francisco to Vladivostok. Both find in the way exactly on 10 days. Immediately on the arrival into the ports of
designation the captains of each of the steamships sent on the telegram, which notified their shipowners about the satisfactory completion of voyage. What date was written on the telegram, which left from Vladivostok? But on that that it was sent from San
18. Live Hipparchus on Mars, that from that opened by them from Rhodes it, observing sky from Mars, would not reveal?
19. In Durlandii from 1 January is produced the denomination of the speed of light. Its value is reduced 1000 , with 300000 km/s to 300 km/s. How from this did change the
arrangement of stars before the sky?
20. Let us visualize that in the northern pole of the ecliptic is opened the belonging to the solar system object, which they observed exactly during the year. In this case it was explained that, in contrast to all adjacent stars, the object does not show annual
aberrational displacement. Why it is possible to assert that the period of revolution of this object around the sun is not less??? years?
21. In 2098 the astronomers Of futurlandii, using old ground-based dvadtsatimetrovym telescope, opened in the solar system object remarkable in many respects. It turned out that the visible way, passed by object before the sky in five years of observations, takes the form of the piece of the stretched chain with five by elongated components, as if assumed on top on the rope.
Version 1 (complex): To estimate the amplitude of changes in the luster of the object opened by futurlandtsami.
Version 2 (simpler): To estimate angular dimension of "major axis" is sectional this chain and period of revolution around the sun of opened by our futurlandskimi associates asteroid. To estimate also its sizes, if it is known that the visible stellar magnitude of object of approximately 31m. To find the amplitude of its changes. To mark on the "chain"
of the points, in which the luster of object reaches extrema.
22. Futurlandskiye astronomers conduct astrometric observations on the small expeditionary telescope, which they temporarily installed at one of the bodies of Kuiper's belt. Body has a period of revolution around the sun of approximately 730 years and it dvizhetsya along the circular orbit. To what is there equal aberration constant?
23. We will consider that the Earth dvizhetsya along the circular orbit with a velocity of 30 km/s, or , where c - speed of light. Let us assume that there is a star, which is been located at a distance of parsecs, which is visible in the direction of the spring equinox point. More precisely saying, it accomplishes small harmonic oscillations along the ecliptic relative to the mean vernal equinox, i.e., by such, whose position is corrected for the influence of precession and nutation. The oscillatory period - it is exact yr. Twice in the year star is visible accurately in the direction (of average) point of spring.
It is understandable what to observe this star conveniently in autumn (why? When it culminates in autumn?). After how many days to (or after) the day of the autumnal equinox the star does penetrate (average) spring equinox point? To what is equal the amplitude of the annual fluctuations of the right ascension of this star?
24. The equation of synodic motion for the outer space planet takes the form
where S - sidereal period, P - synodic (in the years). It is possible to rewrite it thus:
To give visual, so to say the "physical" interpretation of this expansion S in the series (type "Achilles never it will overtake tortoise").
For the inner planets analogous decomposition takes the form
Try to give its visual interpretation.
25. The period of revolution of Venus around the sun is 0.61521 tropical years. It
accomplishes thirteen revolutions around the sun almost accurately in 8 terrestrial years: . This generates specific feature in the apparent motion of Venus. Of what it does consist?
During November 2005 there will be the opposition of Mars. The visible way of Mars among the stars near this opposition will take the form, schematically depicted in figure. In what constellation is located the ascending node of the orbit of Mars?
26.
In which of two planets - Neptune or Pluto - the fraction of year, during which they dvizhutsya backwards, is more? To estimate it for Pluto.
The visible way of comet among the stars takes the form of the rapidly being untwisted spiral. One turn is accomplished exactly in yr, motion occurs clockwise. To find the coordinates of the center of spiral. What fate is prepared to this uncommon comet?
28. Flying through the solar system, galactic newcomer for some purpose carried off with himself Venus and Mars, and in the Earth eccentricity of its orbit increased twenty, after leaving all the remaining orbital parameters constant. How as a result did change climate on the Earth?
29. What such VLA, VLBI and VLBA?
30. The oppositions of Mars, proceeding during August or during September, are considered great, the same as they occur during February or during March, it is possible to name anti-great. If we calculate all oppositions of Mars from the times, let us say, Hipparchus, then it will be explained that, in to the disappointment of astronomers, then loving the closest approaches, them it was substantially less than anti-great. Why?
To estimate the ratio of the numbers occurred in the last 3000 years of the "greatest" oppositions (occurred into one of ten nights - from 25 August through 3 September) to the number of unfavorable, that were being observed into one of ten nights half a year later. (calendar we consider ideal, so that the "slidings down of dates" in 10 000 years does not occur.)
31. Comet dvizhetsya on the ellipsis, which has eccentricity e = 0.5. By how many times its speed in the perihelion is greater than in the aphelion?
32. One planet dvizhetsya along the circular orbit of radius a, another - along the elliptic orbit with perigel'nym distance of a and eccentricity e. To show that in the perihelion the second planet dvizhetsya once of more rapid than the first.
33. Asteroid dvizhetsya on the ellipsis, in by right focus of which is located the sun (see figure). Times and passages of semi-ellipses BPB' and B'AB relate as by 1:3. To find eccentricity.
Answer: .
34. To what the equally time average value of distance during the motion along the Keplerian ellipsis?
35. The Earth - blue planet, and Mars red. To what are approximately equal their color indices B- b-v? After recalling, as appear Jupiter and Saturn, to give the estimation also of their color indices.
36. Where it is more than carbon dioxide - in the atmosphere of Mars, which almost wholly does consist of , or in the atmosphere of the Earth, where its content do compose about 0.03%?
37. Observer is located on the surface of Venus. If in its atmosphere not there would be clouds, then A) would be visible there at night stars (by naked eye)? b) what color there would be there sky in the daytime? c) there would be evidently the sun before the sky? 38.
Before you the usual photograph of Saturn with its rings. You will estimate the mass of rings, if it is known that they consist of small bodies with the significant dimension of R. Radius of Saturn of approximately 60 000 km, its albedos high (why?).
39. To estimate the temperature of star, which half of all photons emits in the Lyman
continuum. To find also the temperature of such star, in which in the Lyman continuum is emitted half of the entire emitted by it energy. In both cases it is considered that the emission is blackbody.
40. Lyman and Balmer series do not overlap: the wavelength of the boundary of a Balmer series is more than the wavelength of line (1216 A). Do not overlap also Balmer and Paschen series. Many naively assume that does not happen to have
Paschen series its first line (passage between levels n= 3 and n= 4) has a wavelength, greater than at the boundary of the following (brekketovskoy) series (passages to level n= 4).
To estimate, how many lines n- oy series, n"1, are assigned on (na +y)-uyu series. [ Otvet: if 4 it did not be mistaken in the calculation, then . ]
41. After using the Saha formula, to obtain the exponential approximation of the dependence of value from temperature T in the environment .
42. The fact that they usually call energy distribution in the spectrum (let us say, the suns), in reality is a flow distribution along the wavelengths, but not on the energies. At what wavelength is located the maximum in the distribution of solar flow on the energies? To consider that the sun emits as blackbody with K.
43. To show that for the star, which is been in mechanical equilibrium, the following useful relationship occurs:
where P=P(R) - pressure, M and R - mass and a radius of star, respectively. In the course (of hydrostatic) evolution of the star of a constant mass both its radius Rand form of dependence P=P(R) change. However, to the left the integral confronting remains constant.
44. How you do think, there is much hydrogen in the center ? But in the center of Betelgeuse?
45. You, probably, repeatedly read about the fact that the neutrino is capable of flying, without being absorbed, through the inconceivable thick layers of the substance. To show that the protons, which are located in the center of the sun, far exceed the neutrino: from the generation of the sun to that moment, when proton perishes with the proton-proton reaction, pass about 1010 years. In this time it manages to fly in the substance with density
on the order 150 g/sm3 - it is such density in the center of the sun - much larger way. You
will estimate this way and will comprehend the obtained result.
46. In one book, dedicated to starry evolution, there is this phrase: "the sun, deprived of the energy sources, will be approximately one years compressed... to the degenerate state". To verify this assertion. To consider that the luminosity of the sun in the
compression stroke does not change.
47. What it is more in the center of the sun - carbon or nitrogen?
48. If suddenly convection envelopped entire today's sun from the center to the surface, the chemical composition of its atmosphere noticeably would change: hydrogen it would become less, and helium it is more. In reality in the solar atmosphere the part by weights of hydrogen and helium are equal respectively X= 0.70, Y= 0.27. What they would become they were equal after mixing? What still perceptible changes would occur in the chemical composition of solar atmosphere? We recall that the age of the sun - about 5 billion years, of the same order and the age of life on the Earth.
49. According to calculations, in the depths of today's sun contents C, N and O change from the center to the surface, as shown in figure [ to give figure ]. How do appear the
corresponding curves, which relate to the moment, when the sun was twice younger than now? But which there will be their form for the moment, when the sun does complete its life on the main sequence?
50. How would appear solar spectrum, if the temperature in the photosphere did diminish in depth?
51. Is it possible according to the form of the region of the spectrum of the center of solar disk to estimate the darkening of tsentr/kray in this spectral region?
52. Why in the long-wave part of solar spectrum there is not one very deep spectral line? 53. In the atmosphere of the sun the radiation absorption in the optical continuum is caused
in essence by negative hydrogen ion . Corresponding coefficient of absorption (taking into account one ion ) of order cm 2. Using this, to estimate in order of
magnitude pressure in the solar photosphere. The acceleration of gravity on the "surface" of the sun to you is known.
54. Which must be the tension of turbulent magnetic field in the photosphere, so that it would create a pressure, equal to gas?
55. What in order of magnitude fraction of the mass of the sun is fallen to its atmosphere? 56. They indicate that the star possesses the gray atmosphere, if the coefficient of absorption
of its gases in the continuum does not depend on wavelength. If the substance of the atmosphere is located in the local thermodynamic equilibrium, then in the so-called approximation of Eddington- Barbier energy distribution in the spectrum of star is given by the formula of Planck s , where - effective temperature. The dependence of the temperature in this atmosphere on the optical depth takes the form
.
Let us assume that the star with the gray atmosphere has with tens of the spectral lines, scattered along entire spectrum. The absorption coefficient in the center of all these lines is one and the same; moreover it is 1000 times more than the absorption coefficient in the continuum. Then spectrum will take approximately this form, as shown in figure [ to give figure ]. Dotted line showed envelope the line of centerss of absorption, existing in the spectrum. By how many times area under this curve is less than under the curve, describing energy distribution in the continuum?
Answer: 3 times.
By what formula is described broken curve?
57. To estimate, what maximum depth they can have lines of absorption in the spectrum of the center of solar disk in A region, assuming that the atmosphere of the sun gray and in it occurs FOR LTR.
58. Why in the spectrum of the sun in A region there is not one line with the residual flow of less than 0.3, whereas in region A of such lines ochen'-ochen' there are many?
59. At a distance of 10 PK from the sun flared up II type supernova. Prehistoric boy began to observe it. What he could describe to his grandsons about that seen by them before the sky?
60. What is higher - air density at the apex of everest or the average density ?
Relationship period- density for the classical cepheids takes the form , where P - period a day and - average density into g/sm3.
61. Star emits as blackbody with temperature T. What portion of all emitted by it photons is fallen to the Lyman continuum?
62. To estimate the sizes of the specks, which the light pressure "vymetayet" from the environments of the star of the main sequence of mass M.
63. How the mass of dust on the line of sight, which does cause a certain fixed interstellar absorption, it does depend on the size of specks? To consider that is applicable geometric optics, i.e., the sizes of specks are great in comparison with the light wavelength.
64. To estimate the mass of interstellar dust on the line of sight (g/sm ), if it is known that the absorption in the visible part of the spectrum (let us say, at the wavelength ) comprises . The size of the specks of order see because of the diffraction their effective area it approximately twice exceeds geometric cross section.
65. To estimate the total mass of that interstellar dust, which causes attenuation of the emission of the star of a radius on . It is considered that dust is distributed in the space between us and star evenly.
66. In one assignment book there is this interesting task.
Speck with a density of g/smof 3 and size of a= 1 m was formed at a distance
from the star, which possesses luminosity erg/s and mass . To determine the speed of speck at infinity, if it moved in the vacuum.
Its following solution further is there given.
The equation of motion of speck in the absence of resistance of medium takes the form
Integrating, we obtain
In this solution to eat an error. Where it? To give the correct solution.
67. Kh.Gyuygens in the middle THE XVII century for the first time measured the angular dimensions of planetary discs, in particular, Saturn . Was known also the period of its revolution around the sun of years) and, of course, then that the luster of Saturn much the same as in bright stars. Hence Newton knew how to obtain the
estimation of distances to the nearest stars (in the astronomical units) correct in order of magnitude. By the way, to what is equal astronomical unit itself, at that time it was known still very bad.
This - historically the first method, which reliably evaluated distances to the nearest stars. Measurements of annual parallaxes in those times gave only the very rough lower
estimates these distances, quantitative photometry yet not did not exist, and about which the visible stellar magnitude of the sun, not there was idea whatever.
Try to reproduce Newton's reasonings even you will ascertain that they actually give a good estimation of interstellar distances.
68. After solving the previous task , glance again at the Ursa Major. Could not you now say, at what approximately distance from each other are located those five stars of ladle, which do have the general proper motion?
69. If you solved task , then easily you will answer this question: to what are approximately equal the masses of five "internal" stars of ladle?
70. What does shine more brightly in the sky of Sirius - sun or the star of the ladle of Ursa Major?
71. < to give the diagram GR of shzs 47 tuc (contemporary, with long GP). > To estimate distance to the accumulation: A) on the position of horizontal branch; b) on the position of turning point.
72. The star drifts of the open cluster Of giady are directed to one point of celestial sphere - apex, and therefore angular distances between the stars of accumulation are gradually decreased. To what extent will increase the stellar magnitude of the brightest stars Of giad, when the angular dimension of accumulation before the sky does decrease 10? 73. With what astronomical object is associated in you number 273 ? Describe it on the
