1. The Model Photosphere (Chapter 9)
Basic Assumptions
Hydrostatic Equilibrium
Temperature Distributions
Physical Conditions in Stars – the dependence of T(t), Pg(t), and Pe(t) on effective temperature and luminosity

2. Basic Assumptions in Stellar Atmospheres
Local Thermodynamic Equilibrium
Ionization and excitation correctly described by the Saha and Boltzman equations, and photon distribution is black body
Hydrostatic Equilibrium
No dynamically significant mass loss
The photosphere is not undergoing large scale accelerations comparable to surface gravity
No pulsations or large scale flows
Plane Parallel Atmosphere
Only one spatial coordinate (depth)
Departure from plane parallel much larger than photon mean free path
Fine structure is negligible (but see the Sun!)

3. Hydrostatic Equilibrium
Consider an element of gas with mass dm, height dx and area dA
The upward and downward forces on the element must balance:
PdA + gdm = (P+dP)dA
If r is the density at location x, then
dm= r dx dA
dP/dx = g r
Since g is (nearly) constant through the atmosphere, we set
g = GM/R2 P x
gdm
x+dx
P+dP
dP/dx = g r

4. In Optical Depth dP/dtn = g/kn Since dtn=kn rdx and dP=g rdx
CLASS PROBLEM:
Recall that for a gray atmosphere,
For k=0.4, Teff=104, and g=GMSun/RSun2, compute the pressure, density, and depth at t=0, ½, 2/3, 1, and 2. (The density r and pressure equal zero at t=0 and k =1.38 x erg K-1)
dP/dtn = g/kn

5. Estimate Teff, log g, & DepthPe Pg
k 5000
1.0E-5
6896
2.56E+1
4.79E+3
2.69E-1
5.0E-4
6971
1.16E+2
7.20E+4
1.06E 0
2.0E-3
7049
2.09E+2
1.75E+5
1.81E 0
1.0E-2
7179
4.27E+2
4.74E+5
3.46E 0
4.0E-2
7379
8.93E+2
1.08E+6
6.56E 0
8.0E-2
7556
1.42E+3
1.58E+6
9.62E 0
2.0E-1
7925
3.40E+3
2.48E+6
1.79E+1
5.0E-1
8601
8.90E+3
3.58E+6
4.01E+1
8.0E-1
9114
1.74E+4
4.16E+6
6.67E+1
1.60E 0
10182
5.31E+4
4.93E+6
1.58E+2
3.0E 0
11481
1.53E+5
5.49E+6
3.89E+2
6.0E 0
13228
4.47E+5
5.94E+6
1.13E+3

6. Wehrse Model, Teff=10000, log g=8Pe Pg
k5000
2.00E-03
7925
5.80E+02
8.82E+04
3.74E+00
6.00E-03
8064
9.80E+02
1.71E+05
5.48E+00
1.00E-02
8129
1.24E+03
2.33E+05
7.14E+00
2.00E-02
8208
1.70E+03
3.54E+05
9.38E+00
6.00E-02
8414
3.06E+03
6.78E+05
1.55E+01
1.00E-01
8571
4.34E+03
9.01E+05
2.06E+01
2.00E-01
8920
7.70E+03
1.28E+06
3.28E+01
5.00E-01
9600
2.03E+04
1.89E+06
7.16E+01
7.00E-01
10040
3.07E+04
2.13E+06
1.01E+02
8.00E-01
10223
3.68E+04
2.22E+06
1.18E+02
1.00E+00
10544
4.98E+04
2.37E+06
1.53E+02
1.60E+00
11377
9.77E+04
2.66E+06
2.83E+02
2.00E+00
11831
1.37E+05
2.78E+06
3.95E+02
3.00E+00
12759
2.41E+05
2.96E+06
7.15E+02
4.00E+00
13476
3.50E+05
3.07E+06
1.09E+02
6.00E+00
14278
5.01E+05
3.23E+06
1.67E+02
8.00E+00
15413
7.41E+05
3.32E+06
2.71E+02

7. In Integral Form - The differential form: x Pg½
(where k0 is kn at a reference wavelength, typically 5000A)
Then integrate:

8. Procedure Guess at Pg(tn) Guess at T(tn)
Do the integration, computing kn at each level from T and Pe
This gives a new Pg(tn)
Interate until the change in Pg(tn) is small

9. The T(t) Relation In the Sun, we can use to get the T(t) relation
Limb darkening or
The variation of kn with wavelength
to get the T(t) relation
Limb darkening can be described from:
We have already considered limb darkening in the gray case, where

10. The Solar Limb Darkening

11. The Solar T(t) Relation
So one can measure In(0,q) and solve for Sn(tn)
Assuming LTE (and thus setting Sn(tn)=Bn(T)) gives us the T(t) relation
The profiles of strong lines also give information about T(t) – different parts of a line profile are formed at different depths.

12. The T(t) Relation in Other Stars
Use a gray atmosphere and the Eddington approximation
More commonly, use a scaled solar model:
Or scale from published grid models
Comparison to T(t) relations iterated through the equation of radiative equilibrium for flux constancy suggests scaled models are close

13. Comparing T(t)’s at Teff=4000, log g=2.25

14. T(t) vs. gravity Kurucz models at 5500K
Depart at depth, similar in shallow layers

15. Temperature vs. Metallicity

16. DON’T Scale Pg(t)!
Models at 5000 K

17. Temperature Pressure Relation with Metallicity

18. Gas Pressure vs. Metallicity

19. Electron Pressure vs. Metallicity

20. Computing the Spectrum
Now can compute T, Pg, Pe, k at all t (Pe=NekT)
Does the model photosphere satisfy the energy criteria (radiative equilibrium)?
Compute the flux from
Express In in terms of the source function Sn, and adopt LTE (Sn =B(T))