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### solution_4

Course: AST 294, Fall 2009
School: National Taiwan University
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Word Count: 802

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294Z: ASTRONOMY The History of the Universe Professor Barbara Ryden SOLUTIONS TO PROBLEM SET # 4 1) [20 points] Today, the average density of matter in the universe is = 3 10-27 kg/m3 . If the matter consisted entirely of hydrogen atoms, how many hydrogen atoms (on average) would be contained in a cubic meter of the universe? If the matter consisted entirely of regulation baseballs (of mass M = 0.145 kg...

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294Z: ASTRONOMY The History of the Universe Professor Barbara Ryden SOLUTIONS TO PROBLEM SET # 4 1) [20 points] Today, the average density of matter in the universe is = 3 10-27 kg/m3 . If the matter consisted entirely of hydrogen atoms, how many hydrogen atoms (on average) would be contained in a cubic meter of the universe? If the matter consisted entirely of regulation baseballs (of mass M = 0.145 kg apiece), how many baseballs (on average) would be contained in a cubic astronomical unit of the universe? The mass of a single hydrogen atom is m = 1.7 10-27 kg. In order for the mass density to be = 3 10-27 kg/m3 , the number of hydrogen atoms per cubic meter would have to be n= 3 10-27 kg/m3 = = 1.76/m3 . m 1.7 10-27 kg (1) Fewer than two atoms, on average, per cubic meter of space that's a very low density. The mass of a single baseball is M = 0.145 kg. In order for the mass density to be = 3 10-27 kg/m3 , the number of baseballs per cubic AU would have to be n= = M 3 10-27 kg/m3 0.145 kg 1.5 1011 m 1 AU 3 = 69,800,000/AU3 . (2) 2) [20 points] The temperature of the cosmic background light today is T 3 K. At the time the universe became transparent, the temperature of the cosmic background light was T 3000 K. This means that the universe has expanded by a factor of 1000 since it became transparent. If the density of matter today is = 3 10-27 kg/m3 , what was the density of matter when the universe became transparent? Consider a cube that is expanding along with the general expansion of the universe. The edges of the cube currently have a length now = 1 m. The volume of the cube is 3 = 1 m3 , and thus the mass it contains is now M = now 3 = 3 10-27 kg. Since matter is not created or destroyed, now the mass M inside the cube remains constant as the universe expands. At the time the universe became transparent, the expanding cube had sides of length 1m now = = 0.001 m . (3) trans = 1000 1000 The density within the box when the universe became transparent was trans = M 3 trans = 3 10-27 kg 3 10-27 kg = = 3 10-18 kg/m3 . (0.001 m)3 10-9 m3 (4) If lengths in the universe increase by a factor of one thousand, then the density of matter decreases by a factor of one billion. 3) [20 points] At the time of primordial nucleosynthesis, the temeprature of the cosmic background light was T 4.8 108 K. By what factor has the universe expanded since the time of primordial nucleosynthesis? If the density of matter today is = 3 10-27 , kg/m3 what was the density of matter at the time of primordial nucleosynthesis? The temperature at the time of primordial nucleosynthesis was Tnuc = 4.8 108 K. The temperature now is Tnow = 3 K. The universe has expanded by a factor of Tnuc 4.8 108 K = = 1.6 108 (5) Tnow 3K since the time of primordial nucleosynthesis. That is, a cube that currently has sides of length now = 1 m had sides of length nuc = now 1.6 108 = 1m = 6.25 10-9 m . 1.6 108 (6) Using same argument as in the previous problem, we deduce that the matter density at the time of primordial nucleosynthesis was nuc = M 3 nuc = 3 10-27 kg = 0.012 kg/m3 . (6.25 10-9 m)3 (7) 4) [20 points] Look up the density of the Earth's air at sea level. Is the density of matter at the time of primordial nucleosynthesis greater than or less than the density of the Earth's air at sea level? According to the Wikipedia article "Density of Air", dry air has a density air 1.2 kg/ m3 at sea level at a temperature T = 20 C = 68 F . (It becomes denser when cooler, and less dense when warmer.) Thus, the density of matter at the time of primordial nucleosynthesis is less than the density of the Earth's air at sea level, by a factor of a hundred. (Although the universe started out in an extraordinarily dense state, by the time several minutes have passed since the Big Bang, the density of matter has dropped to something that's not outlandishly high, by terrestrial standards.) 5) [20 points] Suppose that y...

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National Taiwan University - AST - 294
ASTRONOMY 294Z The History of the Universe Professor Barbara RydenProblem Set # 7: The Last Problem Set Due Tuesday, March 4 at class timeNAME (please print clearly):SCORE (instructor use only):Reminder: The Final Exam for this class will be
National Taiwan University - AST - 294
ASTRONOMY 294Z: The History of the Universe Professor Barbara RydenSOLUTIONS TO PROBLEM SET # 61) [20 points] Potassium-40 is an unstable atomic nucleus; it decays to argon40 with a half-life of 1.3 billion years. Suppose that a rock contains 1,00
National Taiwan University - AST - 294
ASTRONOMY 294Z: The History of the Universe Professor Barbara RydenSOLUTIONS TO PROBLEM SET # 11) [20 points] Two Martian astronomers, Marvin and Marla, are located due north and south of each other on the planet Mars. Marvin sees the Sun directly
National Taiwan University - AST - 294
ASTRONOMY 294Z The History of the UniverseWinter Quarter 2008 Tuesday &amp; Thursday, 3:00 4:48 pm Hitchcock 131Where Do We Come From? What Are We? Where Are We Going? Paul Gauguin, 1897 Instructor: Professor Barbara Ryden Oce: 4035 McPherson Lab (4
National Taiwan University - AST - 294
ASTRONOMY 294Z: The History of the Universe Professor Barbara RydenSOLUTIONS TO PROBLEM SET # 31) [20 points] You are given a ticket for running a red traffic light. For an observer halted at the light, the light emitted has a wavelength 0 = 700 n
National Taiwan University - AST - 294
ASTRONOMY 294Z: The History of the Universe Professor Barbara RydenMIDTERM EXAM: SOLUTIONSShort-Answer Problems 1) Arrange the following scientists in order of when they lived, earliest to latest: Isaac Newton, Aristarchus, Nicolaus Copernicus. Ar
National Taiwan University - AST - 294
ASTRONOMY 294Z The History of the Universe Professor Barbara RydenProblem Set # 6 Due Tuesday, February 26 at class timeNAME (please print clearly):SCORE (instructor use only):Only four questions on this problem set! However, the final questi
National Taiwan University - AST - 294
ASTRONOMY 294Z: The History of the Universe Professor Barbara RydenPRACTICE MINI-EXAMShort Answer Problems 1) Who lived first: Kepler or Copernicus? Copernicus. (Kepler modified the Copernican model by adding elliptical orbits.) 2) If the density
National Taiwan University - AST - 294
ASTRONOMY 294Z The History of the Universe Professor Barbara RydenProblem Set # 4 Due Tuesday, February 5 at class timeNAME (please print clearly):SCORE (instructor use only):Reminder: The midterm exam will be on Thursday, February 7, at clas
National Taiwan University - AST - 294
ASTRONOMY 294Z The History of the Universe Professor Barbara RydenProblem Set # 3 Due Tuesday, January 29 at class timeNAME (please print clearly):SCORE (instructor use only):1) [20 points] You are given a ticket for running a red traffic lig
National Taiwan University - AST - 294
ASTRONOMY 294Z: The History of the Universe Professor Barbara RydenSOLUTIONS TO PROBLEM SET # 71) [20 points] The age of the universe (that is, the time since the Big Bang) is 14 billion years. The age of the Solar System is 4.56 billion years. Th
National Taiwan University - AST - 294
ASTRONOMY 294Z The History of the Universe Professor Barbara RydenProblem Set # 5 Due Tuesday, February 19 at class timeNAME (please print clearly):SCORE (instructor use only):1) [20 points] Einstein showed that mass (M ) and energy (E) are i
National Taiwan University - AST - 822
1Wednesday, September 21: IntroductionThis course deals with continuum radiation processes of astrophysical interest. (Processes that produce spectral lines, rather than continuum radiation, are dealt with in Astronomy 823: Theoretical Spectrosco
National Taiwan University - AST - 822
1Monday, November 21: Inverse Compton ScatteringWhen I did the calculations for the scattering of photons from electrons, I chose (for the sake of simplicity) the inertial frame of reference in which the electron was initially at rest. However, a
National Taiwan University - AST - 822
1Wednesday, November 16: Return of SynchrotronBoth bremsstrahlung and synchrotron radiation can be emitted by hot ionized gas. How can you tell whether the light you observe from a distant galaxy is bremsstrahlung or synchrotron? One way of disti
National Taiwan University - AST - 822
AST 822: Electromagnetic RadiationProblem Set 6: due Wednesday, November 23 1) A highly relativistic electron initially has Lorentz factor 0 1. It is moving in a magnetic field of constant flux density B with a pitch angle . What is (t) as the elect
National Taiwan University - AST - 822
11.1ASTRONOMY 822: Electromagnetic RadiationProblem Set 4: due Wednesday, October 261) Commercial radio stations often broadcast their programs using 2 to 12 dipole antennas arranged in long rows:a) Why do they use this arrangement, rather th
National Taiwan University - AST - 822
ASTRONOMY 822 Electromagnetic RadiationProblem Set 2 due Wednesday, October 12 at class time 1) Consider a plane electromagnetic wave of the form E = ey E0 cos(kx - t) ^ and B = ez E0 cos(kx - t) . ^ The wavelength of the light is measured to be =
National Taiwan University - AST - 822
ASTRONOMY 822 Electromagnetic RadiationProblem Set 3 due Wednesday, October 19 at class time 1) A particle of charge q is moving at a constant speed v q along a straight line. You are located so that the particle's distance at closest approach is b
National Taiwan University - AST - 822
1Monday, October 3: Stellar AtmospheresThere exist entire books written about stellar atmospheres; I will only give a brief sketch of the simplest approximations used in studying stellar atmospheres. In particular, I want to discuss that most use
National Taiwan University - AST - 822
1Monday, November 28: ComptonizationWhen photons and electrons coexist in the same volume of space, their scattering interactions can transfer energy from photons to electrons (Compton scattering) or from electrons to photons (inverse Compton sca
National Taiwan University - AST - 822
1Monday, October 17: Multi-particle SystemsFor non-relativistic charged particles, we have derived a useful formula for the power radiated per unit solid angle in the form of electromagnetic radiation: dP q2 2 2 [a sin ] , = d 4c3 q (1)where q
National Taiwan University - AST - 822
1Monday, October 24: Special Relativity ReviewPersonally, I'm very fond of classical physics. As we've seen recently, you can derive some very useful electromagnetic formulae without taking into account quantum mechanics or special relativity. Ho
National Taiwan University - AST - 822
1Monday, September 26: Radiative TransferAs light travels through the universe, things happen to it. By interacting with charged particles, photons can gain energy; they can lose energy; they can change their direction of motion. Photons can also
National Taiwan University - AST - 822
1Monday, November 7: Synchrotron Radiation for BeginnersAn accelerated electron emits electromagnetic radiation. The most effective way to accelerate an electron is to use electromagnetic forces. Since electrons have mass, they can also be accele
National Taiwan University - AST - 822
1Monday, October 31: Relativistic Charged ParticlesAs I was saying, before the midterm exam intervened, in an inertial frame of reference K there exists an electric field E and a magnetic field B at a spacetime location (r, t). Another inertial f
National Taiwan University - AST - 822
1Monday, October 10: PotentialsI've already remarked on the similarity between the electrostatic force between two point charges, q1 q2 (1) F = 2 , r and the gravitational force between two point masses, F = -G m1 m2 . r2 (2)The similarities be
National Taiwan University - AST - 822
AST 822: Electromagnetic RadiationProblem Set 5: due Wednesday, November 9 1) An observer in the inertial frame of reference K sees a particle moving in the xy plane, with velocity ux = u cos and uy = u sin . The inertial frame of reference K is mo
National Taiwan University - AST - 822
ASTRONOMY 822 Electromagnetic RadiationProblem Set 1 due Wednesday, October 5 at class time1) If you could construct a box opaque to neutrinos, then at a temperature T , the thermal neutrinos inside the box would have the specific intensity I (T )
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
ASTRONOMY 143: The History of the Universe Professor Barbara RydenPRACTICE MINI-EXAMWITH SOLUTIONSShort-Answer Problems 1) Which has the greater energy: a photon of infrared light or a photon of ultraviolet light? ULTRAVIOLET light has the great
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 825
Radiative Gas Dynamics Problem Set 3 Due Thursday, February 191) The atmospheres of Venus and Mars are made of carbon dioxide, which has a kinematic viscosity = 0.070 cm2 s1 (T /100 K)1/2 . Solar energy is added to the Venusian atmosphere at a rat
National Taiwan University - AST - 825
Ast 825: Radiative Gas Dynamics Problem Set 1 Due Thursday, January 221) Lets start simply, with a hydrostatics problem, rather than a hydrodynamics problem. The Earths atmosphere may be approximated as a static, plane-parallel atmosphere, with a u
National Taiwan University - AST - 825
Chapter 13 Astrophysical JetsIn an astrophysical context, jets may be dened as well-collimated outows from a compact object. Jets are commonly bipolar, with two back-to-back jets streaming away from the central compact object. There are two types of
National Taiwan University - AST - 825
Chapter 7 Basic TurbulenceThe universe is a highly turbulent place, and we must understand turbulence if we want to understand a lot of whats going on. Interstellar turbulence causes the twinkling of radio sources, just as turbulence in the earths a
National Taiwan University - AST - 825
Chapter 12 Winds from Hot &amp; Cool StarsHot luminous stars, such as O and B supergiants, are known to have stellar winds. In the UV, they display P Cygni line proles.1 P Cygni proles show absorption at short wavelengths and emission at longer waveleng
National Taiwan University - AST - 825
ASTRONOMY 825 Radiative Gas DynamicsWinter Quarter 2009 Tu Th 11:10 - 12:30 4054 McPherson Lab Professor: Barbara Ryden Oce: 4035 McPherson Lab Phone number: 292-4562 E-mail: ryden@astronomy.ohio-state.edu Textbook: Radiative Gas Dynamics, by Barbar
National Taiwan University - AST - 825
Chapter 8 Spherical AccretionAccretion may be dened as the gravitational attraction of material onto a compact object. The compact object may be a black hole with a Schwarzschild radius R = 2GM /c2 3 km(M / M ). Another possible compact object is a
National Taiwan University - AST - 825
Chapter 4 Radiative &amp; Magnetohydrodynamic ShocksI have been dealing, so far, with non-radiative shocks. Since, as we have seen, a shock raises the density and temperature of the gas, it is quite likely, under astrophysical conditions, that the post-
National Taiwan University - AST - 825
Chapter 10 Advanced Accretion DisksThe total luminosity of a disk with the viscous dissipation rate D(R) isLdisk = 2RD(R)RdR = 1 GM M . 2 R(10.1) The disk luminosity is half of the total accretion luminosity Lacc = GM M /R ; the other h
National Taiwan University - AST - 825
Ast 825: Radiative Gas Dynamics Problem Set 2 Due Thursday, February 51) Its what youve been waiting for: a radiative MHD shock! Suppose that a radiative, steady-state, plane parallel shock has a bulk velocity u perpendicular to the shock front and
National Taiwan University - AST - 825
Chapter 9 Accretion Disks for BeginnersWhen the gas being accreted has high angular momentum, it generally forms an accretion disk. If the gas conserves angular momentum but is free to radiate energy, it will lose energy until it is on a circular or
National Taiwan University - AST - 825
Chapter 11 The Solar WindA stellar wind consists of particles emitted from the stellar atmosphere with a suciently large velocity to escape the stars gravitational attraction. The escape velocity at the surface of a star with mass M and radius R is
National Taiwan University - AST - 825
Astronomy 825: Radiative Gas Dynamics Winter 2009Barbara Ryden January 5, 2009iiContentsNotes on the Notes 1 Fundamentals of Gas Dynamics 2 Viscosity &amp; Heat Conduction 3 Sound &amp; Shocks 4 Radiative &amp; MHD Shocks 5 Spherical Blastwaves &amp; SNRs 6 Io
National Taiwan University - AST - 825
Radiative Gas Dynamics Problem Set 4 Due Thursday, March 51) Near the inner edge of an accretion disk around a neutron star, the pressure is provided by the radiation pressure: Prad = 4SB 4 T 3c (1)and the opacity is provided by Thomson scatterin
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143
National Taiwan University - AST - 143