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Outline – Stellar Evolution I. Review Stellar Evolution (Quick) II. Compact Objects & White Dwarfs III. Lane-Emden Equation IV. Chandrasekar Limit V. Mixing Length Theory A Census of the Stars Faint, red dwarfs (low mass) are the most common stars. Bright, hot, blue main-sequence stars (highmass) are very rare. Giants and supergiants are extremely rare. Stellar Evolution High mass star (> 8 M ) burn fuel faster, are brighter, shorter life SuperNova Medium mass - (0.4 to 4 M) burn fuel moderatley, live long PN + WD Low mass - (< 0.4 M) burn fuel slowly - live long long time! Sun Energy Transport Structure Many Open Questions on stellar interior models A-stars with X-rays; Models for low mass stars (BDs) O & B-stars with Spots 8 Msun Inner convective, outer radiative zone 4 Msun Inner radiative, outer convective zone CNO cycle dominant PP chain dominant Lane-Emden Equation Use as a simple model of a star with polytropes in Hydrostatic equilibrium -or- Think of it as Poisson’s Eq for a gravitational potential of a Newtonian self gravitating, spherically symmetric, polytropic fluid. Solutions of Lane-Emden equation for n=0 to 5 Mass Functions – The Eddington Solution Red – standard solar model Blue = n=3 polytrope Black – linear density law Solar & n=3 Agree very well! Solutions of Lane-Emden equation n = 0, the density of the solution as a function of radius is constant, ρ(r) = ρc. This is the solution for a constant density incompressible sphere. n = 1 to 1.5 approximates a fully convective star, i.e. a very cool late-type star such as a M, L, or T dwarf. n = 3 is the Eddington Approximation. There is no analytical solution for this value of n, but it is useful as it corresponds to a fully radiative star, which is also a useful approximation for the Sun. n > 5, the binding energy is positive, and hence such a polytrope cannot represent a real star. Analytical solutions of Lane-Emden equation for n=0,1,5 Expansion onto the Giant Branch Expansion and surface cooling during the phase of an inactive He core and a H- burning shell Sun will expand beyond Earth’s orbit! Red Giant Evolution He-core gets smaller & denser & hotter 4 H → He He GMm r Core - gravitational contraction makes more heat H-burning shell keeps dumping He onto the core. At shell, density and gravity higher, must burn faster to balance More E, star expands He fusion through the “Triple-Alpha Process” the next stage of nuclear burning can begin in the core: 4He + 4He 8Be + g 8Be + 4He 12C + g 3 types of Compact Objects White Dwarfs < 1.4 Msun (Chandrasekar limit) Neutron Stars > 1.4 Msun < 3 Msun Black Holes > 3 Msun White Dwarfs - size of Earth (6000 km radius) Neutron Stars - size of small city (10 km radius) Black Holes - smaller* still than a city (< 10 km radius)! (* really depends on mass) Sizes of Stars & Stellar Remnants (1) Sizes of Stars Pauli’s Exclusion Principle: No 2 electrons can have the same Quantum mechanical state! Sun WD BD Sizes of Stars & Remnants WD White Dwarfs (2) The more massive a white dwarf, the smaller it is! R ~ M-1/3 Pressure becomes larger, until electron degeneracy pressure can no longer hold up against gravity. WDs with more than ~ 1.4 solar masses can not exist! Chandrasekhar Limit = 1.4 Msun White Dwarfs (3) The more massive a white dwarf, the smaller it is! R ~ M-1/3 Summary of Stellar Evolution Mixing Length Theory The mixing length is conceptually analogous to the concept of mean free path in thermodynamics: a fluid parcel will conserve its properties for a characteristic length, , before mixing with the surrounding fluid. Prandtl described that the mixing length: “may be considered as the diameter of the masses of fluid moving as a whole in each individual case; or again, as the distance traversed by a mass of this type before it becomes blended in with neighbouring masses... “ Prandt 1925