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Unformatted text preview: Introduction to Astrophysics 0321.3108 Exercise 4 1. For radiation we know that the pressure is P = u/ 3 where u is the radiation energy density. Assume that u is a function of the temperature only, i.e. u = u ( T ) (this fact was discovered by Kirchoff). a. Show that from the second law of thermodynamics T ds = d ( uV ) + P dV, (1) (where V is the volume) that it follows that u ( T ) = aT 4 where a is an undetermined constant. This thermodynamic proof was first given by Boltzmann (prior to Planck’s discovery of his formula). Hint: Find what are the derivatives of the entropy with respect to the volume and with the pressure, i.e., ( ∂s ∂V ) P and ( ∂s ∂P ) V . Than use the symmetry of second the derivatives. b. Show that the entropy per unit volume is: s = 4 3 aT 3 . 2. Kelvin-Helmholtz . a. When the Sun evolved to become a main sequence star it contracted slowly because of gravity (but always close to hydrostatic equilibrium). The inner temperature rose from 30 , 000 K to 6 ×...
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