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 In a stellar partial ionization zone, where some of the heat is being used to ionize the gas. In isothermal gas For an adiabatic process (dQ=0): From the ideal gas law for ideal, monatomic gas

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

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: Simulation of convection at solar surface Observations of granulation on solar surface When will convection dominate?

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

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: This equation has an analytic solution for n=0, 1 and 5. This corresponds to  =∞, 2 and 1.2 n=0,1,2,3,4,5 (left to right)

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  depends on density, temperature and chemical composition.

14. Thermodynamics Convection is the transport of heat: thus we need to understand the basic laws of thermodynamics. The first law of thermodynamics: energy conservation Change in internal energy Heat added Work done on the surroundings For a mass element dm: The internal energy U is a state function: it depends only on the current state of the gas (not the processes leading up to that state)  The change dU is independent of the process for that change, whereas dQ and dW are not.