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Hydrostatic Equilibrium

Hydrostatic Equilibrium. Particle conservation. dr. dA. r. Ideal gas. P+dP. P. (pressure difference * area). Ideal gas. In stellar atmospheres: log g is besides T eff the 2nd fundamental parameter of static stellar atmospheres. Hydrostatic equilibrium, ideal gas.

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Hydrostatic Equilibrium

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  1. Hydrostatic Equilibrium Particle conservation

  2. dr dA r Ideal gas P+dP P (pressure difference * area)

  3. Ideal gas • In stellar atmospheres: • log gis besidesTeffthe 2nd • fundamental parameter of • static stellar atmospheres

  4. Hydrostatic equilibrium, ideal gas • buoyancy = gravitational force: • example:

  5. Atmospheric pressure scale heights • Earth: • Sun: • White dwarf: • Neutron star:

  6. Effect of radiation pressure  • 2nd moment of intensity • 1st moment of transfer equation (plane-parallel case) 

  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. 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. 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

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

  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: • 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:

  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 ne and fij, and with Boltzmann all level populations

  13. Summary: Hydrostatic Equilibrium

  14. 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

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