Model Construction The atmosphere connects the star to the outside world. All energy generated in the star has to pass through the atmosphere which itself usually does not produce additional energy. The photosphere is the region of the atmosphere where most of the radiation escapes from the star.

What needs to be done?

Parameters There are many ways to construct model atmospheres. Using a fixed optical depth grid helps avoid pre-specifying the physical extension of the atmosphere. There are many ways to construct model atmospheres. Using a fixed optical depth grid helps avoid pre-specifying the physical extension of the atmosphere. Minimum independent parameters: Minimum independent parameters: Effective temperature T eff Gravity g(r) = G M / r 2 Mass, Radius or Luminosity L= 4πR 2  T eff 4 Abundances of all elements  i = n i / n T

Hydrostatic Equilibrium When mass loss is negligible, the total gas pressure in the atmosphere is: dP/dr = -g(r)  dP/dr = -g(r)  With the optical depth: d  = -  dr = -(  +  ) dr d  = -  dr = -(  +  ) dr where , ,  are the extinction, absorption and scattering coefficients, we get: dP/d  = g(r)  /  dP/d  = g(r)  / 

Energy Conservation In plane-parallel geometry, we have: F rad + F conv = ∫ F d =  T eff 4 = cte F rad + F conv = ∫ F d =  T eff 4 = cte Each volume element has emission = absorption: ∫  (J - S ) d = 0 ∫  (J - S ) d = 0 with J the mean intensity (direction averaged) S the source function (simplest: B (T) ) S the source function (simplest: B (T) ) The energy conservation determines essentially the T(  ) structure!

Model Flow Chart Départ avec: T(  )= grey model (T 4 =3/4 T eff 4 (  +2/3)) P out = dyne/cm 2 15 to 30 iterations Spectrum: ∫F rad d =  T eff 4 > 30,000 pts > 30,000 pts UV  sbmm  = 0.01 Å

Opacities Absorption and scattering coefficients ∑  i j n i j ∑  i j n i j j: ionization stage i: energy level within each ionization stage  i j : cross-section (cm 2 ) n i j : population density (cm -3 ) ∑ over all elements, processes, ionization stages, level.  i j from QM, measurements

LTE TE = thermodynamic Equilibrium = detailed balance of all process = detailed balance of all process = state described by P gas,T = state described by P gas,TIf: - Collisions dominate radiation - Radiation field is Planckian - No scattering of radiation  Local Thermodynamic Equilibrium (LTE) Not the case in exospheres of all stars and planets (radiation dominates) and in lines such as the Lyman series of hydrogen (scattering is important).

Comparison of Opacity Calculations A75 AJR 83 AF 94 Phoenix Equation of state Super- saturation ratio Decoupled gas & dust Gas & dust in equilibrium Molecular opacity Straight mean 2x10 5 lines + straight mean water 3x10 7 lines ~8x10 8 lines Dust opacity 1 species Rayleigh 3 species Mie 4 species CDE 31 species EMT # of frequencies ,00025,000

CO & CH 4 are dominant molecules CH 4 CO

Beware of extrapolating polynomials beyond their intended temperature range

The role of atomic and molecular opacity increases at lower temperatures

H 2 O Abundance

Temperature Dependence of H 2 O Opacity

Sources of H 2 O opacities Lab. ‘70s Empirical ‘90s Theoretical ‘90s Empirical ‘02

Line density varies among different molecules

TiO only exists over a narrow temperature range

Temperature Dependence of TiO Opacity

Even scarce molecules can affect model spectra

Line density is also important in the visual spectrum

Hydrides can be important in dwarfs FeH abundance and spectrum

Conclusions Models rely upon only a few basic equations and several simplifying assumptions (hydrostatic eq., energy eq., LTE), valid only for the photospheres objects (Gas giant planets, brown dwarfs, stars older than 1 Myr). Improvements over the past 15 yrs: computer capacities  better opacities ! Complete atmosphere course online:

References