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Please work all of the following problems. Show your work and work individually. Considerable partial credit will be given for a correct approach even if you miss the actual numerical answer. If your answer sounds preposterous to you say so and suggest what might be wrong. Express your answers, where appropriate, to the accuracy implied by the statement of the problem (usually 2 figures) and use scientific notation. You will probably not be able to work some of the problems until we cover the appropriate material in class. You would be well advised, however, not to wait to the last minute. Work what you can early on. Each problem is individually simple, but the total time required can add up. 1) a) What are the approximate lifetimes of stars of 0.8, 1.0, and 2.0 solar masses (calculate it; don't try to look it up). b) An open cluster of stars, all born at the same time, originally, with stars of all masses from 0.1 to 100 solar masses, is now found to have as its heaviest still living member, a star of 6 solar masses. How old is the cluster? 2) In class we discussed Population I and II stars. Which population is older? To which population would the following belong: a) a star in a globular cluster; b) the sun; c) a 0.8 solar mass star just born in the Orion nebula; and d) the progenitor star of a supernova in the disk of our galaxy 3) The Lick Observatory is located on Mt. Hamilton at a latitude of approximately 37 degrees north. If one can only observe stars that are more than 10 degrees above the horizon, which of the following stars would be visible some night of the year from Mt. Hamilton: a) Polaris, declination +90 degrees; b) alpha-Centauri, declination - -61 degrees; c) Vega, declination +39 degrees; d) Canopus, declination -52 degrees; and e) Formalhaut. declination -30 degrees? 4)Lets recreate the event that may have killed the dinosaurs 65 million years ago. Consider a small rocky asteroid of radius 7 kilometers and density 4 g/ cm**3. You may assume that it is spherical in shape. a) Compute its total mass in grams. Now presume this asteroid, starting at rest relative to the Earth, falls from a great distance (say infinity) and strikes the Earth's surface. Neglect, for the moment, the motion of the asteroid and the Earth around the sun. Just think of the Earth and asteroid both initially at rest and then accelerating toward one another due to their mutual gravitational attraction (actually the asteroid does all the moving). b) With what velocity in km/s would the asteroid hit the earth? c) What energy will be dissipated in the impact (as heat and kinetic energy of the ejecta)? Express your answer in ergs and in Megatons of high explosives (1 Megaton = 4.2 x 10**22 erg). a) If the latitude of Santa Cruz is 37 degree N what will be the 5) Long period comets orbit the sun following paths that are elongated ellipses. Halley's comet comes well inside the Earth's orbit about once every 77 years and then goes very far (many AU) back out again - completing an orbit around the sun. What is the approximate distance from Halley's comet to the sun when it is farthest away? (hint: just calculate the semi-major axis of its elliptical orbit and multiply by two. This neglects the short distance between the comet and the sun when it passes near the sun). You may express your answer in AUs. 6) Consider a mass m in a stable circular orbit with radius r about a very large mass M (M >> m) to which it is bound by gravity. Equating the gravitational force to the ``centrifugal force'' gives an equation that can be solved for the orbital speed, v. Do so. In terms of the variables m and M and r (and the constant G), what is the total energy, kinetic plus gravitational of the mass m? Is it negative or positive or zero? Now consider the mass m moving radially away from M at radius r at the escape speed. Solve again for the total energy. Is it negative, positive, or zero? What is the ratio of the escape speed to the orbital speed? 7) The four fundamental forces were discussed in class. Which of the four forces is chiefly responsible for: a) Holding the neutrons and protons together in a helium nucleus? b) Holding together a small bar of steel? c) The chemistry of life? d) The energy obtained from the combustion of gasoline? e) Binding your fingers to your hand? f) Holding your feet to the ground? g) Allowing a free, unbound neutron to decay into a proton. 8) The small moon Phobos orbits Mars with a semimajor axis of 9.38 x 10**8 cm and a period of only 7.66 hours. Derive the mass of Mars in grams. Compare this with the earth which has a mass of 5.98 x 10**27 grams. 9) What is the fastest the earth could rotate without flying apart? What would be the rotation period of a marginally stable earth? One way to work this would be to consider the period of a satellite with an orbit having the radius equal to that of the earth (you will, of course, need to assume - look up on your constant sheet - the mass and radius of the earth). This is also the period of most near-Earth spacecraft. The calculation ignores the fact that rotation would deform the earth and change the answer, but for now, that's OK. Express your answer as the rotation period (in minutes, hours, days, or whatever is most appropriate) the earth would have. Another way of asking this question would be to say ``What is the shortest period a satellite can have without crashing into the Earth?''. Since many astronomical instruments are in near earth orbit, this is their typical period. 10) Supernovae are very bright, but do you think that they pose a biological hazard? Consider a typical supernova with Lpeak = 10**8 solar luminosities. At what distance, in parsecs, would that supernova have a brightness equal to that of the sun? At what distance would it be 10 times fainter? Compare that to the distance to the nearest star, Alpha Centauri. (nb. 1 AU = 1/206265 pc). |