Model Model ConstructionConstruction

The atmosphere connects the star to The atmosphere connects the star to the outside world. All energy the outside world. All energy generated in the star has to pass generated in the star has to pass through the atmosphere which itself through the atmosphere which itself usually does not produce additional usually does not produce additional energy.energy.

The photosphere is the region of the The photosphere is the region of the atmosphere where most of the atmosphere where most of the radiation escapes from the star. radiation escapes from the star.

What needs to be done?What needs to be done?

ParametersParameters

There are many ways to construct There are many ways to construct model atmospheres. Using model atmospheres. Using a fixed a fixed optical depth gridoptical depth grid helps avoid pre- helps avoid pre-specifying the physical extension of specifying the physical extension of the atmosphere. the atmosphere.

Minimum independent parameters:Minimum independent parameters:Effective temperature Effective temperature TTeffeff

Gravity Gravity g(r) = G M / rg(r) = G M / r22

Mass, Radius or Luminosity Mass, Radius or Luminosity L= 4πRL= 4πR2 2 TTeffeff

44

Abundances of all elements Abundances of all elements i i = n= nii / n / nTT

Hydrostatic EquilibriumHydrostatic EquilibriumWhen mass loss is negligible, the total When mass loss is negligible, the total gas pressure in the atmosphere is:gas pressure in the atmosphere is:

dP/dr = -g(r) dP/dr = -g(r) With the optical depth:With the optical depth:

dd = - = - dr = -( dr = -( + + ) dr) dr

where where , , , , are the extinction, are the extinction, absorption and scattering coefficients, absorption and scattering coefficients, we get:we get:

dP/ddP/d = g(r) = g(r) / /

Energy ConservationEnergy ConservationIn plane-parallel geometry, we have:In plane-parallel geometry, we have:

FFradrad + F + Fconv conv = ∫ F= ∫ F dd = = T Teffeff4 4 = cte= cte

Each volume element has Each volume element has emission = emission = absorptionabsorption::

∫ ∫ (J(J - S- S )) dd = 0 = 0withwith JJ the mean intensity (direction the mean intensity (direction averaged)averaged) SS the source function (simplest:the source function (simplest: BB(T) (T) ))

The energy conservation determines The energy conservation determines essentially the T(essentially the T()) structure!structure!

Model Flow ChartModel Flow Chart

Départ avec:Départ avec:

T(T()= grey model )= grey model (T(T44=3/4 T=3/4 Teffeff

4 4 ((+2/3))+2/3))

PPoutout= 10= 10-4-4 dyne/cm dyne/cm22

15 to 30 iterations15 to 30 iterations

Spectrum:Spectrum:

∫∫FFrad rad dd = = T Teffeff44

> 30,000 pts> 30,000 pts

UV UV sbmm sbmm

= 0.01 Å= 0.01 Å

OpacitiesOpacities

Absorption and scattering Absorption and scattering coefficientscoefficients ∑ ∑ ii

jj n niijj

jj: ionization stage: ionization stageii: energy level within each ionization : energy level within each ionization stagestageii

jj: cross-section (cm: cross-section (cm22))nnii

jj: population density (cm: population density (cm-3-3))∑ ∑ over all elements, processes, over all elements, processes, ionization stages, level.ionization stages, level. ii

jj from QM, measurementsfrom QM, measurements

LTELTETE = thermodynamic Equilibrium TE = thermodynamic Equilibrium = detailed balance of all process= detailed balance of all process = state described by P= state described by Pgasgas,T ,T If:If:- Collisions dominate radiation- Collisions dominate radiation- Radiation field is Planckian- Radiation field is Planckian- No scattering of radiationNo scattering of radiation

Local Thermodynamic Equilibrium (LTE)Local Thermodynamic Equilibrium (LTE)

Not the case in exospheres of all stars and Not the case in exospheres of all stars and planets (radiation dominates) and in lines planets (radiation dominates) and in lines such as the Lyman series of hydrogen such as the Lyman series of hydrogen (scattering is important).(scattering is important).

Comparison of Opacity Comparison of Opacity CalculationsCalculations

A75A75 AJR 83AJR 83 AF 94AF 94 PhoenixPhoenixEquation Equation of stateof state

Super-Super-saturatisaturation ratioon ratio

Decoupled Decoupled gas & gas & dustdust

Decoupled Decoupled gas & gas & dustdust

Gas & Gas & dust in dust in equilibriequilibriumum

MoleculaMolecular opacityr opacity

Straight Straight meanmean

2x102x1055 lines + lines + straight straight mean mean waterwater

3x103x1077 lineslines

~8x10~8x1088 lineslines

Dust Dust opacityopacity

1 1 speciesspecies

RayleigRayleighh

3 species3 species

MieMie4 species4 species

CDECDE31 31 speciesspecies

EMTEMT

# of # of frequenciefrequenciess

5050 900900 9,0009,000 25,00025,000

CO & CHCO & CH44 are dominant are dominant moleculesmolecules

CHCH44

COCO

log T

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

log P

-8

-6

-4

-2

0

2

4

6

-15

-10

-10

-5

-5

-5

-5

-5

-5

-5

-10

-10

-10

-10

-15

-15

-15

log T

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

log P

-8

-6

-4

-2

0

2

4

6

-5

-5

-5

-10

-10

-10

-10

-15

-15

-15

-15

-15

Beware of extrapolating Beware of extrapolating polynomials beyond their intended polynomials beyond their intended

temperature rangetemperature range

log T

2.4 2.6 2.8 3.0 3.2 3.4 3.6

difference log K

p

-6

-5

-4

-3

-2

-1

0

1

CH4

2.4 2.6 2.8 3.0 3.2 3.4 3.6

difference log K

p

-1

0

1

2

3

4

5

6

CO

Tsuji (1973)/JANAFSharp & Huebner (1990)/JANAF

T = 10000 K

(μ )m

1 2 3 4 5

log

ν

-14

-12

-10

-8

-6

-4

-2

Atoms

= 8000 T K

log

ν

-18-16-14-12-10-8-6

Molecules

= 6000 T K

log

ν

-18-16-14-12-10-8-6

= 4000 T K

(μ )m

1 2 3 4 5

log

ν

-20-18-16-14-12-10-8-6

& H H-

The role of The role of atomic and atomic and molecular molecular opacity opacity increases at increases at lower lower temperaturestemperatures

HH22O AbundanceO Abundance

log T

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

log P

-8

-6

-4

-2

0

2

4

6-5

-5

-5

-5-10

-10

-10

-10

-15

-15

-15

-15

Temperature Temperature Dependence of HDependence of H22O O

OpacityOpacity

(μm)

1 2 3 4 5

log

κ

-24

-22

-20

-18

-16

-14

-12

-10

1000 K

2000 K

3000 K

500 K

Sources of HSources of H22O opacitiesO opacities

Lab. ‘70sLab. ‘70s

Empirical ‘90sEmpirical ‘90s

Theoretical ‘90sTheoretical ‘90s

Empirical ‘02Empirical ‘02

Line density variesLine density variesamong different moleculesamong different molecules

CO

(μ )m

2 4 6 8 10

log

ν

-30

-25

-20

-15

-10

-5

H2O

(μ )m

2 4 6 8 10

log

ν

-30

-25

-20

-15

-10

-5

TiO only exists over a TiO only exists over a narrow temperature rangenarrow temperature range

log T

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

log P(i) (dynes cm

-2)

-8

-6

-4

-2

0

2

4

6

-8.0-7.5

-7.5

-7.0

-7.0

-7.0-7.5-8.0

Temperature Temperature Dependence of TiO Dependence of TiO

OpacityOpacity

(μm)

0.5 1.0 1.5 2.0 2.5 3.0

log

κ /n

i

-26

-24

-22

-20

-18

-16

1000 K

2000 K

3000 K

Temperature Temperature Dependence of TiO Dependence of TiO

OpacityOpacity

(μm)

0.5 1.0 1.5 2.0 2.5 3.0

log

κ

-30

-25

-20

-15

-10

1000 K

2000 K

3000 K

Even scarce molecules can Even scarce molecules can affect model spectraaffect model spectra

(μm)0.6 0.8 1.0 1.2 1.4

log F

λ

4.0

4.5

5.0

5.5

Teff = 3100 K

log L/Lsun = 3.0

no TiO

λ (μm)0.6 0.8 1.0 1.2 1.4

log F

λ

4.0

4.5

5.0

5.5

no VO

Line density is also Line density is also importantimportant

in the visual spectrumin the visual spectrum

FeH

(μ )m

0.4 0.6 0.8 1.0 1.2 1.4

log

ν

-20

-18

-16

-14

-12

-10

-8

-6

TiO

(μ )m

0.4 0.6 0.8 1.0 1.2 1.4

log

ν

-20

-18

-16

-14

-12

-10

-8

-6

Hydrides can be important in Hydrides can be important in dwarfsdwarfs

FeH FeH abundance and abundance and

spectrumspectrum

log T

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

log P

-8

-6

-4

-2

0

2

4

6

-15

-15

-15

-15

-15

-15

-15 -10

-10

-10

-10

(μm)

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

log

κ λ

-20

-18

-16

-14

-12

-10

-8

-6

ConclusionsConclusionsModels rely upon only a few basic Models rely upon only a few basic equations and several simplifying equations and several simplifying assumptions (hydrostatic eq., energy assumptions (hydrostatic eq., energy eq., LTE), valid only for the eq., LTE), valid only for the photospheres objects (Gas giant photospheres objects (Gas giant planets, brown dwarfs, stars older than planets, brown dwarfs, stars older than 1 Myr).1 Myr).

Improvements over the past 15 yrs: Improvements over the past 15 yrs: computer capacities computer capacities better better opacities !opacities !

Complete atmosphere course online:Complete atmosphere course online:

http://www.hs.uni-hamburg.de/http://www.hs.uni-hamburg.de/~stcd101/~stcd101/

ReferencesReferences