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
P
x
gdm
x+dx
P+dP
dP/dx = g r
• 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
In Optical Depth
• Since dtn=kn rdx
• and dP=g rdx
dP/dtn = g/kn
CLASS PROBLEM:
• Recall that for a gray atmosphere,
3
2
4
T  Teff (t  )
4
3
4
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)
Estimate Teff, log g, & Depth
t 5000
T
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
t5000
T
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
1
2
Pg dPg  Pg
1
2
g
k0
dP
 g / kn
dt n
dt 0
(where k0 is kn at a reference wavelength,
typically 5000A)
• Then integrate:
2
3
1
2/3


1/ 2
 3 logt 0 t0 Pg

t 0 Pg 2
3
  g
Pg (t 0 )   g 
d log t0 

 2 0 k 0 dt0 
 2  k 0 log e



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
– Limb darkening or
– The variation of kn with wavelength
to get the T(t) relation
• Limb darkening can be described from:

In (0, )   Sn e
0
tn sec
secdtn  a  b cos
• We have already considered limb darkening
in the gray case, where
Sn  a  btn
The Solar Limb Darkening

In (0, )   Sn e
0
tn sec
secdtn  a  b cos
The Solar T(t) Relation
• So one can measure In(0,) 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:
Teff*
T (t )* 
T (t ) Sun
Teff Sun
• 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
7500
6500
6000
Scaled HM
Bell et al.
5500
5000
4500
4000
3500
Optical Depth
10
6
2
1
0.
4
0.
1
3000
0.
00
1
0.
00
5
0.
02
Temperature
7000
T(t) vs.
gravity
Kurucz
models at
5500K
Depart at
depth,
similar in
shallow
layers
Temperature vs. Metallicity
DON’T Scale Pg(t)!
9000
8000
Models at 5000 K
Temperature (K)
Log g = 1.0
7000
Log g = 2.0
Log g = 3.0
6000
Log g = 4.0
5000
4000
3000
1.00E+01
1.00E+02
1.00E+03
1.00E+04
Gas Pressure
1.00E+05
1.00E+06
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

Fn  2  2 In sin  cosd
0
• Express In in terms of the source
function Sn, and adopt LTE (Sn =B(T))