The Model Photosphere Chapter 9

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 (PdP)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
xdx
PdP
dP/dx g r
4
In Optical Depth

• Since dtnkn rdx
• and dPg rdx
• CLASS PROBLEM
• Recall that for a gray atmosphere,
• For k0.4, Teff104, and g2GMSun/RSun2,
  compute the pressure, density, and depth at t0,
  ½, 2/3, 1, and 2. (The density r and pressure
  equal zero at t0 and k 1.38 x 10-16 erg K-1)

dP/dtn g/kn
5
Estimate Teff, log g, depth
6
Wehrse Model, Teff10000, log g8
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
• Limb darkening or
• The variation of In 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 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.

11
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

12
Comparing T(t)s at Teff4000, log g2.25
13
T(t) vs. gravity
14
Temperature vs. Metallicity
15
DONT Scale Pg(t)!
Models at 5000 K
16
Temperature Pressure Relation with Metallicity
17
Gas Pressure vs. Metallicity
18
Electron Pressure vs. Metallicity
19
Computing the Spectrum

• Now can compute T, Pg, Pe, k at all t (PeNekT)
• 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))