Physics 342 Problem Set #5 – Astrophysics
Physics 342: Drawback Set #5due April 9You are inspired to work in teams on these issues, however every scholar should write upthe options individually. You need to additionally listing your collaborators in your options and citeany exterior sources you used (aside from the course notes or textbook).You'll obtain partial credit score for partial solutions, however provided that you present your work and explainyour reasoning. Watch out with items.1. Every a part of this query covers a key idea. Every requires at most a number of sentencesto reply; some are a lot shorter. Please be concise.(a) Why does helium fusion require a better temperature than hydrogen fusion?(b) What's the heaviest aspect that may be fused within the cores of large stars?What prevents heavier parts from forming? What occurs to those stars oncethis heaviest aspect is created via fusion?(c) Why can’t a white dwarf have a mass larger than 1.four M?(d) Why is degenerate matter so troublesome to compress?2. The decrease restrict to the primary sequence happens when the core temperature of a star is nolonger enough to fuse hydrogen into helium. We have now beforehand estimated the coretemperature for a star with mass M and radius R and the temperature required forp ? p fusion by way of quantum tunneling (see Lecture from March 24) asTcore ?GMmpkR and Tfusion ?2e4mp3kh2Use these and the mass-radius relationship for important sequence stars to estimate thelowest mass star able to fusion. Give your reply in items of M.three. (a) The neutrinos produced in a supernova explosion have been noticed solely forSN 1987A, which occurred within the Massive Magellanic Cloud about 50 kiloparsecsfrom Earth. About 1.3×1010 neutrinos per sq. centimeter handed via theEarth, and the common vitality of every neutrino was about 10 MeV. Estimate thetotal variety of neutrinos that had been emitted into house by SN 1987A, and theirtotal vitality (in ergs). Clarify your reasoning and assumptions.(b) Fashions of SN 1987A present that about 1.5 × 1051 erg of this vitality went into thekinetic vitality of the explosion, with an ejected mass of about 20 M. Estimatethe typical velocity of the ejecta. How lengthy wouldn't it take for the ejecta to expandto subtend a radius of zero.1 arcsec on the sky such that we'd resolve the debrisas a supernova remnant? (1 arcsec = 1/3600 of a level).14. We have now assumed that the fuel in a white dwarf is chilly and degenerate, whereas thegas contained in the Solar shouldn't be degenerate. Let’s verify the numbers. Recall that the keyto figuring out whether or not the fuel is degenerate is to match the thermal and Fermienergies. For every of the next instances, calculate ET and EF to find out whetherthe fuel is degenerate.Tc (Ok) ?c (kg/m3) core composition(a) Solar at the moment 1.6×107 1.5×105 ?50/50 H/He combine(b) Solar on big department 2.7×107 5.1×107 He(c) 5 M star on big department 1.1×108 7.7×106 He(d) zero.6 M white dwarf 1.1×107 1.1×109 C/O5. (a) Use the equation of hydrostatic equilibrium (one of many equations of stellar construction),and a recognized boundary situation, to point out that the central stress of astar is given by P(r) = 23?G?2R2. Assume a continuing density all through thestar.(b) In a white dwarf, the central stress is given by the non-relativistic electrondegeneracy stress. Once more assuming fixed density, estimate the radius of awhite dwarf when it comes to its mass. What occurs to the scale of the white dwarf asthe mass will increase?(c) Your end in (b) signifies a most mass the place the quantity of the whitedwarf goes to zero. In actuality, the electrons turn into ultra-relativistic at that time.Derive an expression of this most mass by assuming the central stress isinstead given by the relativistic electron degeneracy stress. Clear up the equationfor the utmost mass in M. (Your e-book does a way more exact derivation,and finds this limiting mass to be 1.44 M, the Chandresekar mass restrict.)6. Sort Ia supernovae are the thermonuclear explosions of accreting white dwarfs thathave reached the Chandrasekhar restrict. On this downside, think about an exploding whitedwarf of mass 1.38 M and radius 104 km.(a) The explosive fusion happens in a number of steps, however the final result's that carbonis fused into nickel: 14 12C ? 356Ni. The optical mild we see from a kind Iasupernova is produced primarily by the radioactive decay of 56Ni, first to 56Co, andthen to 56Fe. Let’s study step one, from nickel to cobalt. This reactionreleases vitality, as a result of the atomic mass of cobalt-56 is 55.939839 amu. Determinethe complete quantity of vitality, Edecay, launched from the radioactive decay of all thenickel to cobalt.2(b) The radioactive vitality shouldn't be launched unexpectedly, however over a number of days. The rateof radioactive vitality launch is the optical luminosity, with L(t) = L0 e?t/? , and? = eight.764 days. Use your outcomes from (a), and the truth that Edecay =R ?0L(t) dtto calculate the preliminary optical luminosity L0 in items of L.three