CC0937 THE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 2913 ASTROPHYSICS AND RELATIVITY (ADVANCED) SEMESTER 2, 2014 TIME ALLOWED: 2 HOURS ALL QUESTIONS HAVE THE VALUE SHOWN INSTRUCTIONS: This paper consists of 2 sections. Section A Special Relativity 45 marks Section B Cosmology 50 marks Candidates should attempt all questions. USE A SEPARATE ANSWER BOOK FOR EACH SECTION. In answering the questions in this paper, it is particularly important to give reasons for your answer. Only partial marks will be awarded for correct answers with inadequate reasons. No written material of any kind may be taken into the examination room. Nonprogrammable calculators are permitted.
CC0937 SEMESTER 2, 2014 Page 2 of 8 Table of constants Avogadro s number N A = 6.022 10 23 mole 1 speed of light c = 2.998 10 8 m.s 1 electronic charge e = 1.602 10 19 C electron rest mass m e = 9.110 10 31 kg electron rest energy E = 511 kev electron volt 1 ev = 1.602 10 19 J protron rest mass m p = 1.673 10 27 kg neutron rest mass m n = 1.675 10 27 kg Atomic mass unit amu = 1.661 10 27 kg Planck s constant h = 6.626 10 34 J.s Boltzmann s constant k B = 1.380 10 23 J.K 1 Universal gas constant R = 8.315 J.mol 1 K 1 Stefan-Boltzmann constant σ = 5.670 10 8 W.m 2.K 4 Radiation Constant α = 7.565 10 16 J m 3 K 4 gravitational constant G = 6.673 10 11 N.m 2.kg 2 Solar mass M = 1.99 10 30 kg Solar radius R = 6.96 10 8 m Solar luminosity L = 3.826 10 26 W Solar absolute visual magnitude M V = 4.83 Solar absolute bolometric magnitude L bol = 4.75 astronomical unit AU = 1.496 10 11 m parsec pc = 3.086 10 16 m light year ly = 9.461 10 15 m arc seconds/radian = 206265
CC0937 SEMESTER 2, 2014 Page 3 of 8 SECTION A: SPECIAL RELATIVITY γ = FORMULAS 1 1 u 2 /c 2 x y z x y = γ(x ut) = y = z t = γ ( t ux/c 2) = γ( x u t) = y z = z t = γ ( t u x/c 2) I = (c t) 2 + ( x) 2 + ( y) 2 + ( z) 2 v x = v x u 1 uv x /c 2 c = fλ c ± u f = c u f 0 p = γm v F = d p dt F = maγ 3 ( v) F = maγ ( v) E = γmc 2 = mc 2 + K E 2 = (mc 2 ) 2 + (pc) 2 a µ = (a 0,a 1,a 2,a 3 ) length squared(a µ ) = (a 0 ) 2 + (a 1 ) 2 + (a 2 ) 2 + (a 3 ) 2 x µ p µ = (ct, x) = (E/c, p)
CC0937 SEMESTER 2, 2014 Page 4 of 8 Please use a separate book for this section. Answer ALL QUESTIONS in this section. 1. (a) In a few sentences, briefly explain the two Principles of Einstein s Theory of Special Relativity. (b) In one or two lines, briefly explain what is meant by an inertial frame of reference. (c) In one or two lines, briefly explain what is meant by the proper time interval between two events and describe how it might be measured. (d) An inertial frame of reference S moves with a constant velocity of v = 3 2 c relative to another inertial frame of reference, S, parallel to the x axis of S. The origins of the two frames coincide at t = t = 0. (i) What is the Lorentz factor of frame S relative to S? (ii) A light bulb fixed to the origin of frame S sends out a flash of light at time t = 1 sec. What are the coordinates of this event as measured by an observer at rest in frame S? (iii) An observer is located at the origin of frame S and is at rest in that frame. At what time will they receive the flash? (iv) The observer in S measures the speed of the flash as it goes past. What value do they find? (15 marks) 2. (a) Alexis, Berat and Cynthia are in three different spacecraft. Alexis is in the middle and she observes Berat and Cynthia to be moving away from her in opposite directions, both at speed 0.7c relative to her. Thus, according to the measurements in her frame, the distance between Berat and Cynthia is increasing at the rate 1.4c. (i) What is the speed of Alexis as measured by Berat? (ii) What is the speed of Cynthia as measured by Berat? (b) Using a diagram, briefly explain how the Equivalence Principle in General Relativity can be used to show that signals sent upwards in a gravitational field are received at a slower rate than they were emitted (this effect is called gravitational redshift ). (c) Write down the total energy and total momentum of a photon that has frequency f. (15 marks)
CC0937 SEMESTER 2, 2014 Page 5 of 8 3. A particle of rest mass M is at rest in a laboratory when it spontaneously decays into three identical particles, each of rest mass m, in the directions shown in the diagram. Two of the particles (labelled #1 and #2) have speeds v 1 = 0.8c and v 2 = 0.6c. #3 #1 θ (a) Write down the energy-momentum 4-vectors for particles #1 and #2. #2 (b) Calculate the length squared of each of the two 4-vectors in part (a). Briefly comment on your answers. (c) Calculate the speed and direction of particle #3. Be sure to note any conservation laws used in the calculation. (15 marks)
CC0937 SEMESTER 2, 2014 Page 6 of 8 SECTION B: COSMOLOGY FORMULAS [ ] dr ds 2 = c 2 dt 2 a 2 2 (t) 1 kr 2 + r2 (dθ 2 + sin 2 θdφ 2 ) 1 + z = λ obs = a(t obs) λ emit a(t emit ) (ȧ ) 2 = 8πGρ kc2 a 3 a 2 + Λ 3 ( ä a = 4πG ρ + 3p ) 3 c 2 + Λ 3 ρ + 3ȧ (ρ + p ) a c 2 = 0 d a = d c (1 + z) = d l (1 + z) 2 N = ǫ(ν)dν = 8πh c 3 1 exp(hν/k B T) 1 ν 3 dν exp(hν/k B T) 1 ǫ = αt 4
CC0937 SEMESTER 2, 2014 Page 7 of 8 Please use a separate book for this section. Answer ALL QUESTIONS in this section. 4. (a) Write down the first Friedmann equation for an empty universe, with no radiation, matter or cosmological constant, and then solve it to show that the relationship between scale factor, a, and time, t, is a(t) = a 0 ( t t 0 ). (1) (b) Using equation (1), in an empty universe, find the relationship between the Hubble parameter, H = ȧ/a, and the age of the universe. (c) The currently observed Hubble parameter is approximately H 0 = 70 km s 1 Mpc 1. Using this, calculate the age of the Universe in the case of an empty Universe and compare it to the current best estimate of the age of the Universe, t 0 = 13.8 billion years. Briefly discuss why the two values are (or are not) close in value. (12 marks) 5. (a) The Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is [ ] dr ds 2 = c 2 dt 2 a 2 2 (t) 1 kr + 2 r2 (dθ 2 + sin 2 θdφ 2 ). (2) In a sentence or two for each, briefly explain the main terms in the metric. (b) Considering the path of a photon in the FLRW metric, derive the relationship between expansion factor and redshift. (c) Gamma ray bursts (GRBs) are intense and short lived bursts of high energy photons that are seen at high redshift and are thought to come from the end states of stars (e.g. supernovae or merging neutron stars). A galaxy at z = 6 is discovered to host a GRB, whose burst of radiation is observed to last 3.5 ± 0.6 seconds (RMS error) by a satellite orbiting the Earth. A second, more nearby galaxy, with z = 1.5, is later found to also host a GRB. The same satellite observes the second GRB to be of length 1.6 ± 0.25 seconds. By calculating the duration of each burst in the rest frame of its galaxy, state whether it is plausible that both GRBs have the same physical mechanism. (13 marks)
CC0937 SEMESTER 2, 2014 Page 8 of 8 6. (a) Describe two modern cosmological observations which have led to our current Λ Cold Dark Matter (ΛCDM) model. Make sure to discuss both the observations and the reasons they lead to our current model. (b) From the first Friedmann equation derive the critical density, ρ c and go on to show that ( ) 3H 2 3 0 a ρ = Ω m,0 (3) 8πG a 0 where Ω m = ρ/ρ c. (c) Use the second Friedmann equation and the equation of state for Λ of ρ Λ = p Λ /c 2 to show that Λ = 8πGρ Λ (4) (d) Finally, use the solutions to parts (b) and (c) to find the redshift of transition between acceleration and deceleration for a flat universe with Ω m,0 = 0.3 and Ω Λ,0 = 0.7 and no relativistic matter. (12 marks) 7. (a) Briefly describe three problems with the standard cosmological model, and discuss how inflation is able to solve them. (b) Write the FLRW metric for the radial path of a photon in a flat universe, and recall that in an Einstein-de Sitter (EdS) universe the scale factor evolves as ( ) 2/3 t a(t) = a 0. (5) t 0 From these, show that the particle horizon, d h = ra(t), is d h = 2c/H, where r is the comoving radial distance in the metric and H is the Hubble parameter (it will help to recall that H = ȧ/a). (c) In a simple flat inflationary universe, dominated by a cosmological constant with no other significant density component, the scale factor grows exponentially. During this exponential growth, how does the Hubble parameter, H, change? (d) Given that the particle horizon is typically order of H 1 in any cosmology, and that inflation causes exponential growth in a, use your answer to the previous question to explain why inflation fixes the horizon problem. (13 marks) THERE ARE NO MORE QUESTIONS.