Astro 160: The Physics of Stars

where k is now the boltzman constant. Substituting this expression into the above equation yields( ) 15/2L ∝ caR −1/2 M 11/2 Gmpkthis gives us an order of magnitude estimate of the luminosity of a star with mass M and radius R.(b). If all stars have roughly the same central temperature, and are supported by gas pressure, what isthe mass-luminosity scaling (proportianality) relationship for stars?we now know that the luminosity scales asL ∝ M 11/2 R −1/2we can find the relationship between the mass M and the radius R of a star by using hydrostaticequilibrium.dPdr= − GMr 2 ρP c ∝ M R ρρT ∝ M R ρM ∝ Rsince T is constant, substituting this into the luminosity relationship yieldsL ∝ M 5(c). Give a quantitative argument as to whether free-free opacity dominates electron scattering opacityin stars more massive that the sun or in stars less massive that the sun.We can solve this problem by looking at the defenition for the opacity in free-free absorption, whichcan be written as with T constantκ ∝ ρ ρ ∝ M R 3 M ∝ Rthus we findκ ∝ 1 M 2this expression tells us that the lower the mass of the star the higher the opacity, thus in lower massstars the free-free opacity dominates.Problem # 2The central density and temperature of the sun are ρ c ≃ 150 g cm 3 and T c ≃ 1.5 × 10 7 K. For theconditions at the center of the sun, answer the following questions. Assume that the sun is composedsolely of ionized hydrogen.(a). What is the mean free path of an electron due to electron-electron Coulomb collisions? What isthe typical time between collisions?2