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Lecture 12

Lecture 12. Stellar structure equations. Convection. A bubble of gas that is lower density than its surroundings will rise buoyantly

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Lecture 12

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  1. Lecture 12 Stellar structure equations

  2. 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.

  3. Convection • Convection is a very complex process for which we don’t yet have a good theoretical model

  4. The first law of thermodynamics • For an ideal, monatomic gas:

  5. The first law of thermodynamics • For an adiabatic process (dQ=0): • From the ideal gas law • 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

  6. Polytropes • A polytrope is a gas that is described by the equation of state: • For an adiabatic, monatomic ideal gas • For radiative equilibrium, or degenerate matter • For isothermal gas

  7. Convection • Assume that the bubble rises in pressure equilibrium with the surroundings. What temperature gradient is required to support convection? • Using the ideal gas law and the equation for hydrostatic equilibrium:

  8. Convection • Compare the temperature gradient due to radiation: • with that required for convection: • When will convection dominate? • Observations of granulation on solar surface • Simulation of convection at solar surface

  9. Static Stellar structure equations • Hydrostatic equilibrium: • Equation of state: • Mass conservation: • Energy generation: • Polytrope • or • Radiation • Convection

  10. Break

  11. Derivation of the Lane-Emden equation • 1. Start with the equation of hydrostatic equilibrium • 2. Substitute the equation of mass conservation: • 3. Now assume a polytropic equation of state: • 4. Make the variable substitution:

  12. The Lane-Emden equation • So we have arrived at a fairly simple differential equation for the density structure of a star: • n=0,1,2,3,4,5 • (left to right) • This equation has an analytic solution for n=0, 1 and 5. This corresponds to g=∞, 2 and 1.2

  13. 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: • Finally, to determine the luminosity of the star we use the equation • Where the energy generation e depends on density, temperature and chemical composition.

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