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Understanding Stellar Structure: Convection, Gas Laws, and the Lane-Emden Equation

This lecture explores the fundamental principles of stellar structure, focusing on convection processes in stars, the application of the ideal gas law, and the implications of hydrostatic equilibrium. We examine how buoyancy affects gas bubbles of varying densities in a star, delve into the complex nature of convection, and introduce the Lane-Emden equation, which models the density structures of stars under different polytropic conditions. Observations of solar surface granulation and simulations illustrate the practical aspects of these theoretical concepts.

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Understanding Stellar Structure: Convection, Gas Laws, and the Lane-Emden Equation

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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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