Physics Final Exam with Useful Constants, Exams of Nuclear Physics

Download Physics Final Exam with Useful Constants and more Exams Nuclear Physics in PDF only on Docsity! MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department and Department of Earth, Atmospheric, & Planetary Sciences Physics 8.282J – EAPS 12.402J May 22, 2003 Final Exam Name (please print) Last First 1. Work all problems. 2. Closed book exam; you may use two pages of notes. 3. Wherever possible, try to solve the problems using general analytic expressions. Plug in numbers only as a last step. Problem Grade Grader 1 2 3 4 5 6 7 8 Total USEFUL CONSTANTS Constant cgs units mks units c (speed of light) G (gravitation constant) k (Boltzmann’s constant) h (Planck’s constant) mproton eV (electron Volt) M� (solar mass) L� (solar luminosity) R� (solar radius) σ (Stefan­Boltzmann cons) Å(Angstrom) 3 × 1010 7 × 10−8 1.4 × 10−16 6.6 × 10−27 1.6 × 10−24 1.6 × 10−12 2 × 1033 4 × 1033 7 × 1010 6 × 10−5 10−8 cm/sec dyne­cm2/g2 erg/K erg­sec g erg g erg/sec cm erg/cm2­sec­K4 cm 3 × 108 7 × 10−11 1.4 × 10−23 6.6 × 10−34 1.6 × 10−27 1.6 × 10−19 2 × 1030 4 × 1026 7 × 108 6 × 10−8 10−10 m/sec N­m2/kg2 J/K J­sec kg J kg J/sec m J/m2­sec­K4 m km (kilometer) 105 cm 103 m pc (parsec) kpc (kiloparsec) Mpc (megaparsec) year day 3 × 1018 3 × 1021 3 × 1024 3 × 107 86400 cm cm cm sec sec 3 × 1016 3 × 1019 3 × 1022 3 × 107 86400 m m m sec sec AU 1� (arc minute) 1�� (arc second) 1.5 × 1013 2.9 × 10−4 4.9 × 10−6 cm rad rad 1.5 × 1011 2.9 × 10−4 4.9 × 10−6 m rad rad Problem 3 (Short Answer Questions – 12 points) a. An extrasolar Jupiter­like planet is in a circular orbit of radius 5 AU about a solar­type star, which lies at a distance of 5 pc from an observer. When the planet is at its maximum angular separation from the star as seen by the observer, what diameter optical telescope (operating above the Earth’s atmosphere) would be needed so the image of the planet would lie in the first dark ring of the diffraction pattern of the stellar image? b. Use a dimensional analysis of the equation of radiative transport dT 3κρL = dr − 64πσr2T 3 to find a proportionality between a star’s luminosity and its mass. In this expression σ is a constant, and you may also take κ to be a constant. From other dimensional analyses that we did in lecture, we found that the mean temperature of a star is given by T ∝ M/R. Problem 4 (Short Answer Questions – 12 points) a. A cluster of galaxies contains 103 individual galaxies. Assume they all have the same mass, Mgal, and are distributed uniformly over a spherical volume of radius Rclus. Use the virial theorem to estimate the characteristic velocity, Vgal, of galaxies within the cluster. First, find a general expression for Vgal in terms of Mgal and Rclus, then obtain a numerical estimate for Vgal if Mgal = 1012 M� and Rclus = 1 Mpc. b. Describe what the Eddington luminosity limit is. Specific what the physics and assump­ tions are that underlie the derivation of this limit. Problem 5 (Short Answer Questions – 13 points) a. A white dwarf of radius RWD = 3000 km and mass MWD = 1.4M� collapses to become a neutron star of radius RNS = 10 km. Compute, to order of magnitude, the gravitational energy released in the collapse. In what form is most of this energy emitted? b. A quasar is observed to have a luminosity of 1046 ergs s−1 (1039 Watts). The “central object” in the quasar is a black hole of mass 108 M�. Find the rate at which mass must be accreted, ṁ, in order to produce this much luminosity. [Hint: Assume that the accretion disk, from which the radiation is emitted, has an inner radius of 3RS , where RS = 2GMBH/c2.]