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SBU PHY 521 - Exam 2

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1PHY 521 – StarsExam #2 – due 17 May 20101. The equation of radiative transfer for coherent i sotropic scattering for a plane-parallel gray atmosphere isµdI (µ, τ)dτ= I (µ, τ) − J (τ) .Assume a solution of the formI (µ, τ) = ab + τ + µ +c1 + dµe−fτ,where (a, b, c, d, f) are constants. Find the mean intensity J(τ), the flux F (τ ), and thepressure integral K(τ). Show that the condition of radiative equil ibrium in the atmosphere,F (τ) = F =constant, al so implies that K(τ) = J(τ)/3, a = 3F/4 and d i s infinitesimallysmall but finite.The exact sol ution of the gray atmosphere problem can be writtenJ (τ ) =34F [τ + q (τ) ] ,where q(τ ) is the Hopf function. The exact solution has q(0) = 1/√3 and q(∞) = 0.710446.Find the constants (b , c, d, f ) that result in radiative equilibrium and the correct Hopffunction in the limits of τ = 0 and τ = ∞.For this model, how much brighter would the center of the stellar disk appear comparedto the limb of a star?Can this model generally satisfy the condition that there is no incoming radiation atthe stellar surface?2. The problem concerns the cooling of white dwarfs. Suppose that the conductiv-ity in a white dwa rf interior i s so large, because of electron degeneracy, that the star isnearly isothermal with temperature T . However, there is an atmosphereand so the visiblesurface of the star has an effective temperature Teff6= T . Approxima te the effect of theatmosphere throughTeff= KT7/8,where K is a constant. Assume the mass M and radius R of the white dwarf remainconstant as it cools. Although the pressure in the white dwarf interior is dominated bydegenerate electrons, the heat capacity in the interior is dominated by ions. Approximatethe total heat capacity byCV=32NkBwhere N is the number of ions in the interior. Also assume that the ions are all C12.The white dwarf cools primarily by thermal emission of radiation from the surface.What is the slope d ln L/d ln Teffof the cooling trajectory in the Hertzsprung-Russel dia-gram? How does this compare to the slope for lower Main Sequence stars?2Suppose that white dwarfs have been born at a constant rate over the age of the GalaxyTG. What is the smallest luminosity a nd effective temperature a white dwarf with massof 1 M⊙and radius 1000 km would have at the present time? Finally, sketch the behaviorof the rela tive number of white dwarfs in our Galaxy in a logarithmic luminosity intervalas a function of luminosity.3. Consider a binary star with masses M01and M2initially in a circular orbit wit hsemi-major axis a0. A supernova occurs in star 1 resulting in an immediate mass lossof ∆M = M01− M1. If the supernova remnant receives no kick, i.e., its spatial velocityimmediately after the explosion is the same as before, what is the maximum mass loss thatcan occur yet still result in a bound binary? Express your answer i n terms of a fraction ofthe total initial mass M = M01+ M2.In some neutron star binaries, it is obvious that explosions have occurred in whichmore mass loss occurred than predicted above, yet the binaries are still bound? How couldthis happen?4. In this problem consider a n explosion that occurs w ithin a star of uniform density ρ,uniform temperature T , and mass M . Immediately after the shock wave from the explosionreaches the surface, the stellar gas is so hot that it is radiation-pressure dominated. Energydiffuses out of the star by radiative transport , and the opacity is due mostly to electronscattering. The total energy i mpart ed t o the matter is of order ESN= εrV , where εris theenergy density of radiation and V is the initial stellar volume 4πR30/3 with R0the initialstellar ra dius. Assuming the stellar structure radiative transfer equati on holds throughoutthe star, show that the luminosity at the stellar surface isL ∝ESNMR0κwhere κ is the opacity.Estimate the ratio ESN/M for a Type Ia supernova.Assuming that this rat io is about the same for a gravitational collapse (Ty pe II) su-pernova, why is the initial luminosity of a Type Ia supernovae much, much fainter than aType II?At late ti mes, both types of supernovae exhibit light curves that exponentially decaywith a time constant of about 80 days. Why is this?5. Assume the core of a red giant star is isothermal, with T ≃ 107K and a coremass Mcore≃ 1 M⊙. Assume the pressure in the core is due to an ideal gas. Beyond itsboundary a thin layer produces essenti a lly a ll the star’s luminosity. Also assume t he redgiant’s envelope is radiative and use the equations of hydrostatic equilibrium, the ideal gaslaw, and radiative transfer to show that the stellar radius is approximat ely 10 00


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SBU PHY 521 - Exam 2

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