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Model Photospheres
I. What is a photosphere?
II. Hydrostatic Equilibruium
III.Temperature Distribution in the Photosphere
IV. The Pg-Pe-T relationship
V. Properties of the Models
VI. Models for cool stars
I. What is a Stellar Atmosphere?
• Transition between the „inside“ and „outside“ of the star
• Boundary between the stellar interior and the interstellar medium
• All energy generated in the core has to pass through the
atmosphere
• Atmosphere does not produce any energy
• Two basic parameters:
• Effective temperature. Not a real temperature but the
temperature needed to produce the observed flux via 4pR2T4
• Surface Gravity – log g (although g is not a dimensionless
number). Log g in stars range from 8 for a white dwarf to 0.1 for a
supergiant. The sun has log g = 4.44
What is a Photosphere?
• It is the surface you „see“ when you look at a star
• It is where most of the spectral lines are formed
What is a Model Photosphere?
• It is a table of numbers giving the source function and the
pressure as a function of optical depth. One might also list the
density, electron pressure, magnetic field, velocity field etc.
• The model photosphere or stellar atmosphere is what is used
by spectral synthesis codes to generate a synthetic spectrum of
a star
Real Stars:
1. Spherical
2. Can pulsate
3. Granulation, starspots, velocity fields
4. Magnetic fields
5. Winds and mass loss
Our model:
1. Plane parallel geometry
2. Hydrostatic equilibrium and no mass loss
3. Granulation, spots, and velocity fields are
represented by mean values
4. No magnetic fields
II. Hydrostatic equilibrium
dA
P +dP
P + dP
P
r + dr
A
dr
r
dm
P
M(r)
Gravity
The gravity in a thin shell should be balanced by the outward gas
pressure in the cell
Fp = PdA –(P + dP)dA = –dP dA
dM
FG = –GM(r) r2
Pressure Force
Gravitational Force
P +dP
r
M(r) = ∫ r(r) 4pr2 dr
A
0
dr
dM = r dA dr
Both forces must balance:
dm
FP + FG = 0
–dP dA + –G r(r)M(r) dA dr
r2
P
=0
Gravity
The pressure in this equation is the total pressure supporting the small
volume element. In most stars the gas pressure accounts for most of
this. There are cases where other sources of pressure can be
significant when compared to Pg.
Other sources of pressure:
1. Radiation pressure: PR =
2. Magnetic pressure:
4s T4
3c
= 2.52 ×10–15 T4 dyne/cm2
B2
PM =
8p
3. Turbulent pressure: ~ ½rv2 v is the root mean square velocity of
turbulent elements
Footnote: Magnetic pressure is what is behind the emergence of
magnetic „flux ropes“ in the Sun
Pphot
B2
Ptube+
8p
In the magnetic flux tube the magnetic field provides partial pressure
support. Since the total pressure in the flux tube is the same as in
the surrounding gas Ptube < Pphot. Thus rtube < rphot and the flux tube
rises due to buoyancy force.
Pressures
Teff
(K)
Sp
Pg
(dynes/cm2)
PR
(dynes/cm2)
B
(Gauss)
v
4000
K5 V
1 × 105
0.6
1584
7.5
8000
A6 V
1 × 104
10
501
10.6
12000
B8 V
3 × 103
52
274
13.0
16000
B3 V
3 × 103
165
274
15.0
20000
B0 V
5 × 103
403
354
16.7
B is for magnetic pressure = Pg
v is velocity that generates pressure equal to Pg according to ½rv2
We are ignoring magnetic fields in generating the photospheric models.
But recall that peculiar A-type stars can have huge global magnetic
fields of several kilogauss in strength. In these atmospheric models one
has to treat the magnetic pressure as well.
dP
=
–G r(r)M(r) dr
r2
In our atmosphere
GM/r2 = g (acceleration of gravity)
F
x increases inward so no negative sign
P
dx
P + dP
dA
r
F + dF
dP = grdx
The weight in the narrow column is just
the density × volume × gravity
dtn = knrdx
gravity
dP = g
dtn kn
One way to integrate the hydrostatic equation
Pg½ dPg = Pg½ g dk0
k0
k0 is a reference wavelength (5000 Å)
to
Pg(t0) =
(
3 g
2 0∫
log to
Pg(t0) =
(
⅔
Pg½
dt0
k0
t0½ Pg½
3 g
∫ k log e dlog t0
2 –∞
0
⅔
(
2 P (t ) =
3 g 0 0∫
gPg½
dt0
k0
(
to
3/2
Integrating on a logarithmic optical depth scale gives better precision
Numerical Procedure
• Guess the function Pg(t0) and perform the numerical integration
• New value of Pg(t0) is used in the next iteration until convergence is
obtained
• A good guess takes 2-3 iterations
Problem: we must now k0 as a function of t0 since k0 appears in
the integrand. kn is dependent on temperature and electron
pressure. Thus we need to know how T and Pe depend on t0.
III. Temperature Distribution in the Solar
Photosphere
Two probes of depth:
q
• Limb Darkening
ds
The increment of path
length along the line of
sight is ds = dx sec q
• Wavelength dependence of the
absorption coefficient
Limb darkening is due to the decrease of the continuum source
function outwards
∞
In (0) =
0
∫Sne–tnsec q sec q dtn
The exponential extinction varies as tnsec q, so the position of the unit
optical depth along the line of sight moves upwards, i.e. to smaller t.
Limb Darkening
Temperature profile
of photosphere
Temperature
Bottom of photosphere
10000
8000
6000
4000
q2
q1
z=0
tn =1 surface
Top of photosphere
z
dz
The path length dz is
approximately the same
at all viewing angles, but
at larger the optical depth
of t=1 is reached higher
in the atmosphere
Solar limb
darkening as a
function of
wavelength in
Angstroms
Solar limb
darkening as a
function of position
on disk
At 1.3 mm the solar atmosphere exhibits limb brightening
Horne et al. 1981
In radio waves one is
looking so high up in the
atmosphere that one is
in the chromosphere
where the temperature
is increasing with heigth
Temperature
Temperature profile of photosphere
and chromosphere
10000
8000
chromosophere
6000
4000
z=0
z
Limb darkening in
other stars
Use transiting planets
No limb darkening
transit shape
At the limb the star has less flux than is expected, thus the planet blocks less light
The depth of the light curve gives you the Rplanet/Rstar, but
the „radius“ of the star depends on the limb darkening,
which depends on the wavelength you are looking at
q
To get an accurate measurement of the planet radius you
need to model the limb darkening appropriately
If you define the radius at
which the intensity is 0.9 the
full intensity:
At l=10000 Å, cos q=0.6,
q=67o, projected disk radius
= sin q = 0.91
At l=4000 Å, cos q=0.85,
q=32o, projected disk radius
= sin q = 0.52 → disk is only
57% of the „apparent“ size at
the longer wavelength
The transit duration depends
on the radius of the star but
the „radius“ depends on the
limb darkening. The duration
also depends on the orbital
inclination
When using different data sets to look for changes in the transit duration
due to changes in the orbital inclination one has to be very careful how you
treat the limb darkening.
Possible inclination changes in TrEs-2?
Evidence that transit duration has decreased
by 3.2 minutes. This might be caused by
inclination changes induced by a third body
But the Kepler Spacecraft does not show this effect.
One possible explanation is that this study had to combine
different data sets taken at different wavelength band passes
(filters). But the limb darkening depends on wavelength. At
shorter wavelengths the star „looks“ smaller.
The only star for
which the limb
darkening is well
known is the Sun
In the grey case had a linear source function:
Sn = a + btn
Using:
∞
In (0) =
0
∫Sne–tnsec q sec q dtn
In (0) = a + b cosq
This is the Eddington-Barbier relation which says that at cos q = tn
the specific intensity on the surface at position equals the source
function at a depth tn
Limb darkening laws usually of the form:
Ic = Ic(0) (1 – e + e cos q)
e ≈ 0.6 for the solar case, 0 for A-type stars
Ic(0) continuum
intensity at disk center
Rewriting on a log scale:
∞
In (0) =
–tnsec q
∫S
e
n
0
d log tn
tn sec q log e
Contribution function
No light comes from the
highest and lowest layers,
and on average the surface
intensity originates higher in
the atmosphere for positions
close to the limb.
Sample solar
contribution functions
Wavelength Variation of the Absorption Coefficient
Since the absorption coefficient depends
on the wavelength you look into different
depths of the atmosphere. For the Sun:
• See into the deepest layers at 1.6mm
• Towards shorter wavelengths kn increases
until at l = 2000 Å it reaches a maximum.
This corresponds to a depth of formation at
the temperature minimum (before the
increase in the chromosphere)
Solar Temperature distributions
Best agreements are
deeper in the atmosphere
where log t0 = –1 to 0.5
Poor aggreement is
higher up in the
atmosphere
Temperature Distribution in other Stars
The simplest method of obtaining the temperature distribution in other
stars is to scale to a standard temperature distribution, for example the
solar one.
T(t0) = S0T‫(סּ‬t0)
In the grey case:
¼
T(t) = ¾ [t + q(t)] Teff
Teff
T(t) =
T‫סּ‬
T‫(סּ‬t)
eff
In the grey case the scaling
factor is the ratio of
effective temperatures
Scaled solar models agree well (within a few percent) to calculations using
radiative equilibrium. They also agree well when applying to giant stars.
Numerically it was easier to use scaled solar models in the past. Now, one
just uses a grid of models calculated using radiative equilibrium
IV. The Pg–Pe–T Relation
When solving the hydrostatic pressure equation we start with an initial
guess for Pg(t0). We then require that the electron pressure Pe(t0) = Pe(Pg)
in order to find k0(t0) = k0(T,Pe) for the integrand. The electron pressure
depends on the temperature and chemical composition.
N1j
N0j
=
Fj(T)
Pe
N1j = number of ions per unit volume of the jth element
See Saha equation from 2nd
lecture
N0j = number of neutrals
F(T) = 0.65
u1
u0
T
5/2
10–5040I/kT
Neglect double ionization. N1j = Nej, the number of electrons per unit
volume that are contributed by the jth element.
Fj(T)
Pe
=
Nej
N0j
=
Nej
Nj – Nej
The total number of jth element particles is Nj = N1j + N0j. Solving for Nej
Fj(T)/Pe
Nej = N
j 1 + F (T)/P
j
e
The pressures are:
Pe = Sj NejkT
Pg = S (Nej + Nj)kT
j
Taking ratios:
Pe
Pg
Pe
Pg
=
S Nj
=
S NejkT
S (Nej + Nj)kT
Fj(T)/Pe
1 + Fj(T)/Pe
Fj(T)/Pe
Using the number
abundance Aj = Nj/NH
NH = number of hydrogen
S Nj 1 + 1 + F (T)/P
j
e
S Aj
Pe =
Pg
Fj(T)/Pe
1 + Fj(T)/Pe
Fj(T)/Pe
S Aj 1 + 1 + F (T)/P
j
e
This is a transcendental equation in Pe that has to be solved
iteratively. F(T) are constants for such an iteration. Pe and Pg are
functions of t0. This equation is solved at each depth using the first
guess of Pg(t0).
log t
1
0
–1
–2
–3
–4
For the cooler models the
temperature sensitivity of the
electron pressure is very large
with d log Pe/d log T ≈ 12 since the
absorption coefficient is largely
due to the negative hydrogen ion
The absorption coefficient kn is
largely due to the negative
hydrogen ion which is
proportional to Pe so the opacity
increases very rapidly with depth.
Hydrogen dominates at high temperatures and when it is fully ionized
Pg ≈ 2Pe
At cooler temperatures Pe ~ Pg½
Where does the later come from? Assume the photosphere is made
of single element this simplifies things:
Pe = P g
F(T)/Pe
1 + 2 F(T)/Pe
Pe = F(T)Pg – 2F(T)Pe = F(T)(Pg– 2Pe)
2
Pg >> Pe in cool stars
2
Pe ≈ F(T)Pg
Completing the model
Pg(t0) =
(
t0½ Pg½
3 g
dlog t0
∫
2 –∞ k0 log e
⅔
(
log to
We can now can compute this
• Take T(t0) and our guess for Pg(t0)
• Compute Pe(t0) and k0(t0)
• Above equation gives new Pg(t0)
• Iterate until you get convergence (≈ 1%)
• Can now calculate geometrical depth and surface flux
The Geometric Depth
We are often interested in the geometric depth scale (i.e.
where the continuum is formed). This can be computed from
dx = dt0/k0r
t0
x(t0) = ∫
1
k0(t0)r(t0)
dt0
0
The density can be calculated from the pressure ( P = (r/m)KT )
r = NH (hydrogen particles per cm3) x SAjmj grams/H particle)
where mj is the atomic weight of the jth element
NH =
N – Ne
S Aj
=
Pg – Pe
kT S Aj
log t0
x(t0) =
∫
–∞
S AjkT(t0)t0
d log t0
k0(t0)S Ajmj[Pg(t0)t0 – Pe(t0)] d log e
A more interesting form is to integrate on a Pg scale with dPg = rgdx
1
x(t0) = g
Pg
∫
–∞
S AjkT(p) dp
p
S Ajmj
The thickness of the atmosphere is inversely proportional to the surface
gravity since T(Pg) depends weakly on gravity
This makes physical sense if you
recall the scale height of the
atmosphere:
Scale height H = kT/mg
Computation of the Spectrum
The spectrum
∞
Fn(0) = 2p
∫ S (t ) E (t )dt
n n
2 n
n
0
It is customary to integrate on a log t scale
∞
Fn(0) = 2p
∫ S (t ) E (t )
n
n
–∞
Flux contribution function
2
n
kn(t0)t0 d log t0
k0(t0) d log e
Flux Contribution Functions as a Function of
Wavelength
Flux at 8000 Å originates
higher up in the atmosphere
than flux at 5000 or 3646 Å
But cross the Balmer jump and
the flux dramatically increases.
This is because there is a sharp
decrease in the opacity across
the Balmer jump.
Flux Contribution Functions as a Function of
Effective Temperature
T= 10400 K
T= 8090 K
T= 4620 K
A hotter star produces more flux, but this originates higher up in the
atmosphere
Computation of the Spectrum
There are other techniques for computing the flux → Different integrals.
Integrating flux equation by parts:
∞
∫
Fn(0) = 2p Sn(tn) E2(tn)dtn
0
∞
Fn(0) = pSn(0) +
∫
0
d Sn(tn) E (t )dt
3 n
n
dtn
The flux arises from the gradient of the source function. Depths
where dS/dt is larger contribute more to the flux
V. Properties of Models: Pressure
d log T/d log Pg = 0.4
Temperature
Convection gradient
Relationship between pressure and temperature for models of
effective temperatures 3500 to 50000 K. The dashed line marks
where the slope exceeds 1–1/g ≈ 0.4 and implies instability to
convection
Cannot scale
T(Pg), unlike
T(t0)!!!
Teff = 8750 K
Effects of gravity
dlogPg
dlog g
dlogPg
dlog g
0.62
=
=
0.85
Increasing the gravity increases all pressures. For a given T
the pressure increases with gravity
Pg(t0) =
(
t0½ Pg½
3 g
dlog t0
∫
2 –∞ k0 log e
⅔
(
log to
Pg ≈ C(T) g ⅔ since pressure dependence in the
integral is weak. So dlog P/dlog g ~ 0.67
In general Pg ~ gp
In cool models p ranges from 0.64 to 0.54 in going from deep to
shallow layers
In hotter models p ranges from 0.85 to 0.53 in going from deep to
shallow layers
Recall Pe ≈ Pg½ in cool stars → Pe ≈ constant g⅓
Pe ≈ 0.5 Pg in hot stars → Pe ≈ constant g⅔
Properties of Models: Chemical Composition
Gas pressure
Electron pressure
• In hot models hydrogen takes over as electron donor and the
pressures are indepedent of chemical composition
• In cool models increasing metals → increasing number of electrons
→ larger continous absorption → shorter geometrical penetration in
the line of sight → gas pressure at a given depth decreases with
increasing metal content
Qualitatively:
Using SAj for the sum of the metal abundances
Pe
1
Pg
S Aj
=
=
Pe
NH
Pg
S Nj
Pe
PH
Pg
S N jkT
kT
kT
≈
Pe
S N jkT
Since PH, the partial pressure of Hydrogen dominates the gas pressure
SNj = S(N1 + N0)j, the number of element particles is the sum of ions
and neutrals and Pe=NekT = SN1jkT for single ionizations
Pe
1
Pg
S Aj
≈
S N 1j
S (N 1 + N0)j
In the solar case metals are ionized SN1j >> N0j
dPg =
Pe
1
Pg
S Aj
g
dk0 =
k0
≈
g
Pek0/Pe dk0
1
g
= P SA k /P
g
j 0 e
k0 is dominated by the negative hydrogen ion, so k0/Pe is independent of Pe
Integrating:
2
1
½Pg =
S Aj
t0
g
∫ k /P
0
dt0
e
0
g and T are constants
–½
Pg =c0 (S Aj)
½
Pe =c0 (S Aj)
For metals being neutral: SN1J << S(N1 + N0)j can show
–⅓
Pg =c0 (S Aj)
⅓
Pe =c0 (S Aj)
Properties of Models: Effective Temperature
Note scale change of ordinate
• In hotter models opacity increases dramatically
• More opacity → less geometrical penetration to reach the same optical
depth
• We see less deep into the stars → pressure is less
• But electron pressure increases because of more ionization
This is seen in the models
If you can see down to an optical depth of t ≈ 1, the higher the
effective temperature the smaller the pressure
Properties of Models: Effective Temperature
For cool stars on can write:
Pe ≈ C eWT
At high temperatures
the hydrogen (ionized)
has taken over as the
electron donor and the
curves level off
A grid of solar models
Temperature
Log t
Depth (km)
Depth (km)
The mapping between optical depth and a real depth
Log t
Amplitude (mmag)
Why do you need to know the geometric depth?
Wavelength (Ang)
In the case the pulsating roAp stars, you want to know where the high amplitude
originates
Pgas
Log (Pressure)
Pe ≈Pg½
Pelectron
Log t
Note bend in Main Sequence at the low temperature end. This is
where the star becomes fully convective
VI. Models for Cool Objects
Models for very cool objects (M dwarfs and brown dwarfs) are
more complicated for a variety of reasons, all related to the
low effective temperature:
1. Opacities at low temperatures (molecules, incomplete
line lists etc.) not well known
2. Convection much stronger (fully convective)
3. Condensation starts to occur (energy of condensation,
opacity changes)
4. Formation of dust
5. Chemical reactions (in hotter stars the only „reactions“
are ionization which is give by ionization equilibrium)
Much progress in getting more complete line lists for water as well
as molecules. Models have gotten better over time, but all models
produce a lack of flux (over opacity) in the K-band.
M8V
1997
1995
2001
Allard et al. 2010
1971
Dust Clouds
The cloud composition according to equilibrium chemistry
changes from:
Zirconium oxide (ZrO2)
Perovskite and corundum (CaTiO3, Al2O3)
Silicates: forsterite (Mg2SiO4)
Salts: (CsCl, RbCl, NaCl)
Ices: (H2O, NH3, NH4SH)
M → L → T dwarfs
At Teff < 2200 K the cloud layers become optically thick enough to
initiate cloud convection.
Intensity variation due cloud formation and granulation.
Teff = 2600 K : No dust formation
Teff = 2200 K : dust has
maximum optical thick density
Teff = 1500 K: Dust starts to settle
and gravity waves causing regions
of condenstation
Allard et al. 2010
Of course these models do not include rotation and Brown Dwarfs can
have high rotation rates. Jupiter is as a good approximation as to what
a brown dwarf atmosphere really looks like
Teff = 2900 K, log g=5-0 model compared to GJ 866
Infrared
3mm
8000 Å
Most cool star
models have a use a
more complete line
lists for molecules,
including water, and
also include dust in
the atmosphere
Optical
4000 Å
10000 Å
Discrepancies are
due to missing
opacities
Comparison of Models
NextGen:
overestimating Teff
Ames-Cond/Dusty:
underestimating Teff
BT-Settl: Using Asplund
Solar abundances
Stars with
„normal“ opacities
Condensation
Dust clouds
Allard et al. 2010
Now days researchers just download models from webpages. Kurucz model
atmospheres have become the „industry standard“, and are continually being
improved. These are used mostly for stars down to M dwarfs. The Phoenix
code is probably more reliable for cool objects.
• Kurucz (1979) models - ApJ Supp. 40, 1
R.L. Kurucz homepage: http://kurucz.harvard.edu
• The PHOENIX homepage
P.H. Hauschildt:
http://www.hs.uni-hamburg.de/EN/For/ThA/phoenix/index.html
• Holweger & Müller 1974, Solar Physics, 39, 19 – Standard Model
• Allens Astrophysical Quantities (Latest Edition by Cox)
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1.47117179E-02 3325.9 4.649E-02 1.165E+07 7.271E-05 3.535E-03 0.000E+00 0.000E+00 0.000E+00
1.92498763E-02 3393.3 6.080E-02 1.503E+07 7.447E-05 2.979E-03 0.000E+00 0.000E+00 0.000E+00
2.51643328E-02 3425.0 7.950E-02 1.949E+07 7.579E-05 2.870E-03 0.000E+00 0.000E+00 0.000E+00
3.29103373E-02 3462.1 1.040E-01 2.526E+07 7.737E-05 2.733E-03 0.000E+00 0.000E+00 0.000E+00
4.30172811E-02 3502.2 1.359E-01 3.272E+07 7.906E-05 2.600E-03 0.000E+00 0.000E+00 0.000E+00
5.62014022E-02 3540.2 1.776E-01 4.238E+07 8.090E-05 2.473E-03 0.000E+00 0.000E+00 0.000E+00
7.33634752E-02 3576.1 2.318E-01 5.487E+07 8.294E-05 2.359E-03 0.000E+00 0.000E+00 0.000E+00
9.56649214E-02 3609.9 3.023E-01 7.100E+07 8.521E-05 2.267E-03 0.000E+00 0.000E+00 0.000E+00
1.24571186E-01 3643.2 3.936E-01 9.176E+07 8.778E-05 2.219E-03 0.000E+00 0.000E+00 0.000E+00
1.61931251E-01 3676.7 5.117E-01 1.184E+08 9.071E-05 2.191E-03 0.000E+00 0.000E+00 0.000E+00
2.10049977E-01 3710.3 6.637E-01 1.525E+08 9.409E-05 2.159E-03 0.000E+00 0.000E+00 0.000E+00
2.71775682E-01 3744.2 8.588E-01 1.959E+08 9.802E-05 2.123E-03 0.000E+00 0.000E+00 0.000E+00
3.50602645E-01 3779.0 1.108E+00 2.511E+08 1.026E-04 2.087E-03 0.000E+00 0.000E+00 0.000E+00
4.50760286E-01 3814.6 1.424E+00 3.209E+08 1.080E-04 2.071E-03 0.000E+00 0.000E+00 0.000E+00
5.77317822E-01 3851.4 1.824E+00 4.086E+08 1.142E-04 2.062E-03 0.000E+00 0.000E+00 0.000E+00
7.36336604E-01 3889.2 2.327E+00 5.186E+08 1.216E-04 2.038E-03 0.000E+00 0.000E+00 0.000
Sample grid
from Kurucz