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Lecture 12 Stellar structure equations Convection A bubble of gas that is lower density than its surroundings will rise buoyantly From the ideal gas law: if gas is in approximate pressure equilibrium (i.e. not expanding or contracting) then pockets of gas that are hotter than their surroundings will also be less dense. P kT mH Convection Convection is a very complex process for which we don’t yet have a good theoretical model The first law of thermodynamics dU dQ dW For an ideal, monatomic gas: 3k CV 2mH CP 5 CV 3 The first law of thermodynamics dU dQ dW For an adiabatic process (dQ=0): From the ideal gas law 5 3 1 P V P T / 1 for ideal, monatomic gas • In a stellar partial ionization zone, where some of the heat is being used to ionize the gas. • In isothermal gas Polytropes A polytrope is a gas that is described by the equation of state: P 5/3 4/3 1 For an adiabatic, monatomic ideal gas For radiative equilibrium, or degenerate matter For isothermal gas Convection Assume that the bubble rises in pressure equilibrium with the surroundings. What temperature gradient is required to support convection? dT 1 T dP 1 dr P dr Using the ideal gas law and the equation for hydrostatic equilibrium: dT 1 mH GM r 1 dr k r2 Convection Compare the temperature gradient due to radiation: with that required for convection: When will convection dominate? Simulation of convection at solar surface dT 3 Lr dr 64 r 2T 3 dT 1 mH GM r 1 dr k r2 Observations of granulation on solar surface Static Stellar structure equations Hydrostatic equilibrium: Mass conservation: dT 3 Lr dr 64 r 2T 3 Radiation dP GM r dr r2 dM r 4r 2 dr or Equation of state: P Energy generation: Polytrope kT mH dLr 4r 2 dr P dT 1 mH GM r 1 dr k r2 Convection Break Derivation of the Lane-Emden equation 1. Start with the equation of hydrostatic equilibrium dP GM r dr r2 2. Substitute the equation of mass conservation: dM r 4r 2 dr 3. Now assume a polytropic equation of state: 4. Make the variable substitution: 1 d 2 d 1 / 1 x x 2 dx dx x P K , c r K n 1 n 1 2 c 4G 1 1 The Lane-Emden equation So we have arrived at a fairly simple differential equation for the density structure of a star: 1 d 2 d n x x 2 dx dx x , c r n n 1 1 K n 1 n 1 2 c 4G This equation has an analytic solution for n=0, 1 and 5. This corresponds to =∞, 2 and 1.2 1 n=0,1,2,3,4,5 (left to right) x Stellar structure equations For the polytropic solution, we can easily find the temperature gradient, using the ideal gas law and polytropic equation of state. This is equal to the adiabatic temperature gradient: dT 1 mH GM r 1 dr k r2 Finally, to determine the luminosity of the star we use the equation dLr 4r 2 dr Where the energy generation depends on density, temperature and chemical composition.