Stellar Atmospheres Hydrostatic Equilibrium Particle conservation 1 Stellar

Stellar Atmospheres: Hydrostatic Equilibrium Particle conservation 1

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

Stellar Atmospheres: Hydrostatic Equilibrium Ideal gas In stellar atmospheres: log g is besides Teff the 2 nd 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 3

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

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

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

Stellar Atmospheres: Hydrostatic Equilibrium 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 7

Stellar Atmospheres: Hydrostatic Equilibrium 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: 8

Stellar Atmospheres: Hydrostatic Equilibrium 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 Teff, log g)-diagram 9

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

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

Stellar Atmospheres: Hydrostatic Equilibrium 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 ne and fij, and with Boltzmann all level populations 12

Stellar Atmospheres: Hydrostatic Equilibrium Summary: Hydrostatic Equilibrium 13

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