Posted: February 8th, 2019

Physics 342 Problem Set #5 – Astrophysics

Physics 342: Problem Set #5due April 9You are encouraged to work in groups on these problems, but each student must write upthe solutions individually. You must also list your collaborators on your solutions and citeany external sources you used (other than the course notes or textbook).You will receive partial credit for partial answers, but only if you show your work and explainyour reasoning. Be careful with units.1. Each part of this question covers a key concept. Each requires at most a few sentencesto answer; some are much shorter. Please be concise.(a) Why does helium fusion require a higher temperature than hydrogen fusion?(b) What is the heaviest element that can be fused in the cores of massive stars?What prevents heavier elements from forming? What happens to these stars oncethis heaviest element is created through fusion?(c) Why can’t a white dwarf have a mass greater than 1.4 M?(d) Why is degenerate matter so difficult to compress?2. The lower limit to the main sequence occurs when the core temperature of a star is nolonger sufficient to fuse hydrogen into helium. We have previously estimated the coretemperature for a star with mass M and radius R and the temperature required forp ? p fusion via quantum tunneling (see Lecture from March 24) asTcore ?GMmpkR and Tfusion ?2e4mp3kh2Use these and the mass-radius relationship for main sequence stars to estimate thelowest mass star capable of fusion. Give your answer in units of M.3. (a) The neutrinos produced in a supernova explosion have been observed only forSN 1987A, which occurred in the Large Magellanic Cloud about 50 kiloparsecsfrom Earth. About 1.3×1010 neutrinos per square centimeter passed through theEarth, and the average energy of each neutrino was about 10 MeV. Estimate thetotal number of neutrinos that were emitted into space by SN 1987A, and theirtotal energy (in ergs). Explain your reasoning and assumptions.(b) Models of SN 1987A show that about 1.5 × 1051 erg of this energy went into thekinetic energy of the explosion, with an ejected mass of about 20 M. Estimatethe typical velocity of the ejecta. How long would it take for the ejecta to expandto subtend a radius of 0.1 arcsec on the sky such that we would resolve the debrisas a supernova remnant? (1 arcsec = 1/3600 of a degree).14. We have assumed that the gas in a white dwarf is cold and degenerate, while thegas inside the Sun is not degenerate. Let’s check the numbers. Recall that the keyto determining whether the gas is degenerate is to compare the thermal and Fermienergies. For each of the following cases, calculate ET and EF to determine whetherthe gas is degenerate.Tc (K) ?c (kg/m3) core composition(a) Sun today 1.6×107 1.5×105 ?50/50 H/He mix(b) Sun on giant branch 2.7×107 5.1×107 He(c) 5 M star on giant branch 1.1×108 7.7×106 He(d) 0.6 M white dwarf 1.1×107 1.1×109 C/O5. (a) Use the equation of hydrostatic equilibrium (one of the equations of stellar structure),and a known boundary condition, to show that the central pressure of astar is given by P(r) = 23?G?2R2. Assume a constant density throughout thestar.(b) In a white dwarf, the central pressure is given by the non-relativistic electrondegeneracy pressure. Again assuming constant density, estimate the radius of awhite dwarf in terms of its mass. What happens to the size of the white dwarf asthe mass increases?(c) Your result in (b) indicates a maximum mass where the volume of the whitedwarf goes to zero. In reality, the electrons become ultra-relativistic at that point.Derive an expression of this maximum mass by assuming the central pressure isinstead given by the relativistic electron degeneracy pressure. Solve the equationfor the maximum mass in M. (Your book does a much more precise derivation,and finds this limiting mass to be 1.44 M, the Chandresekar mass limit.)6. Type Ia supernovae are the thermonuclear explosions of accreting white dwarfs thathave reached the Chandrasekhar limit. In this problem, consider an exploding whitedwarf of mass 1.38 M and radius 104 km.(a) The explosive fusion occurs in several steps, but the ultimate result is that carbonis fused into nickel: 14 12C ? 356Ni. The optical light we see from a type Iasupernova is produced mainly by the radioactive decay of 56Ni, first to 56Co, andthen to 56Fe. Let’s examine the first step, from nickel to cobalt. This reactionreleases energy, because the atomic mass of cobalt-56 is 55.939839 amu. Determinethe total amount of energy, Edecay, released from the radioactive decay of all thenickel to cobalt.2(b) The radioactive energy is not released all at once, but over a few days. The rateof radioactive energy release is the optical luminosity, with L(t) = L0 e?t/? , and? = 8.764 days. Use your results from (a), and the fact that Edecay =R ?0L(t) dtto calculate the initial optical luminosity L0 in units of L.3