1. Stellar Atmospheres: Hydrostatic Equilibrium 1 Hydrostatic Equilibrium Particle conservation

2. Stellar Atmospheres: Hydrostatic Equilibrium 2 Ideal gas P P+dP r dr dA (pressure difference * area)

3. Stellar Atmospheres: Hydrostatic Equilibrium 3 Ideal gas In stellar atmospheres: log g is besides T eff the 2nd fundamental parameter of static stellar atmospheres Type log g Main sequence star Sun Supergiants White dwarfs Neutron stars Earth 4.0.... 4.5 4.44 0.... 1 ~8 ~15 3.0

4. Stellar Atmospheres: Hydrostatic Equilibrium 4 Hydrostatic equilibrium, ideal gas buoyancy = gravitational force: example:

5. Stellar Atmospheres: Hydrostatic Equilibrium 5 Atmospheric pressure scale heights Earth: Sun: White dwarf: Neutron star:

6. Stellar Atmospheres: Hydrostatic Equilibrium 6 Effect of radiation pressure 2nd moment of intensity 1st moment of transfer equation (plane-parallel case)  

7. Stellar Atmospheres: Hydrostatic Equilibrium 7 Effect of radiation pressure Extended hydrostatic equation In the outer layers of many stars: Atmosphere is no longer static, hydrodynamical equation Expanding stellar atmospheres, radiation-driven winds

8. Stellar Atmospheres: Hydrostatic Equilibrium 8 The Eddington limit Estimate radiative acceleration Consider only (Thomson) electron scattering as opacity (Thomson cross-section) number of free electrons per atomic mass unit Pure hydrogen atmosphere, completely ionized Pure helium atmosphere, completely ionized Flux conservation: 

9. Stellar Atmospheres: Hydrostatic Equilibrium 9 The Eddington limit Consequence: for given stellar mass there exists a maximum luminosity. No stable stars exist above this luminosity limit. Sun: Main sequence stars (central H-burning) Mass luminosity relation: Gives a mass limit for main sequence stars Eddington limit written with effective temperature and gravity Straight line in ( log T eff,log g )-diagram

10. Stellar Atmospheres: Hydrostatic Equilibrium 10 The Eddington limit Positions of analyzed central stars of planetary nebulae and theoretical stellar evolutionary tracks (mass labeled in solar masses)

11. Stellar Atmospheres: Hydrostatic Equilibrium 11 Computation of electron density At a given temperature, the hydrostatic equation gives the gas pressure at any depth, or the total particle density N : N N massive particle density The Saha equation yields for given ( n e,T ) the ion- and atomic densities N N. The Boltzmann equation then yields for given ( N N,T ) the population densities of all atomic levels: n i. Now, how to get n e ? We have k different species with abundances  k Particle density of species k :

12. Stellar Atmospheres: Hydrostatic Equilibrium 12 Charge conservation Stellar atmosphere is electrically neutral Charge conservation electron density=ion density * charge Combine with Saha equation (LTE) by the use of ionization fractions: We write the charge conservation as Non-linear equation, iterative solution, i.e., determine zeros of use Newton-Raphson, converges after 2-4 iterations; yields n e and f ij, and with Boltzmann all level populations

13. Stellar Atmospheres: Hydrostatic Equilibrium 13 Summary: Hydrostatic Equilibrium

14. Stellar Atmospheres: Hydrostatic Equilibrium 14 Summary: Hydrostatic Equilibrium Hydrostatic equation including radiation pressure Photon pressure: Eddington Limit Hydrostatic equation  N Combined charge equation + ionization fraction  n e  Population numbers n ijk (LTE) with Saha and Boltzmann equations

15. Stellar Atmospheres: Hydrostatic Equilibrium 15 3 hours Stellar Atmospheres… …per day… …is too much!!!