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

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!)

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

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 10-16 erg K-1) dP/dtn = g/kn

Estimate Teff, log g, & Depth Pe 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

Wehrse Model, Teff=10000, log g=8 Pe 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

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

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

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

The Solar Limb Darkening

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.

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

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

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

Temperature vs. Metallicity

DON’T Scale Pg(t)! Models at 5000 K

Temperature Pressure Relation with Metallicity

Gas Pressure vs. Metallicity

Electron Pressure vs. Metallicity

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))