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3. Transport of energy: radiation
specific intensity, radiative flux
optical depth
absorption & emission
equation of transfer, source function
formal solution, limb darkening
temperature distribution
grey atmosphere, mean opacities
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No sinks and sources of energy in the atmosphere
à all energy produced in stellar interior is transported through the atmosphere
à at any given radius r in the atmosphere:
F is the energy flux per unit surface and per unit time. Dimensions: [erg/cm2/sec]
The energy transport is sustained by the temperature gradient.
The steepness of this gradient is dependent on the effectiveness of the energy transport through the different atmospheric layers.
24 ( ) .r F r const Lπ = =
Energy flux conservation
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Mechanisms of energy transport
a. radiation: Frad (most important)
b. convection: Fconv (important especially in cool stars)
c. heat production: e.g. in the transition between solar cromosphere and corona
d. radial flow of matter: corona and stellar wind
e. sound waves: cromosphere and corona
Transport of energy
We will be mostly concerned with the first 2 mechanisms: F(r)=Frad(r) + Fconv(r). In the outer layers, we always have Frad >> Fconv
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The specific intensity
Measures of energy flow: Specific Intensity and Flux
The amount of energy dEν transported through a surface area dA is proportional to dt (length of time), dν (frequency width), dω (solid angle) and the projected unit surface area cos θ dA.
The proportionality factor is the specific Intensity Iν(cosθ)
Intensity depends on location in space, direction and frequency
d E º = I º ( c o s µ ) c o s µ d A d ! d º d t ( [ I º ] : e r g c m ¡ 2 s r ¡ 1 H z ¡ 1 s ¡ 1 )
I = c 2 I º
( f r o m I d = I º d º a n d º = c / )
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Invariance of the specific intensity
The area element dA emits radiation towards dA’. In the absence of any matter between emitter and receiver (no absorption and emission on the light paths between the surface elements) the amount of energy emitted and received through each surface elements is:
d E º = I º ( c o s µ ) c o s µ d A d ! d º d t d E 0 º = I 0 º ( c o s µ 0 ) c o s µ 0 d A 0 d ! 0 d º d t
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Invariance of the specific intensity
energy is conserved: d E º = d E 0 º and d ! = p r o j e c t e d a r e a
d i s t a n c e 2 = d A
0 c o s µ 0
r 2
d ! 0 = d A c o s µ r 2
I º = I 0 º
In TE: Iν = Bν"
Specific intensity is constant along rays - as long as there is no absorption and emission of matter between emitter and receiver
d E º = I º ( c o s µ ) c o s µ d A d ! d º d t d E 0 º = I 0 º ( c o s µ 0 ) c o s µ 0 d A 0 d ! 0 d º d t and
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s o l i d a n g l e : d ! = d A
r 2
T o t a l s o l i d a n g l e = 4 ¼ r 2
r 2 = 4 ¼
d A = ( r d µ ) ( r s i n µ d Á )
! d ! = s i n µ d µ d Á
d e fi n e µ = c o s µ
d µ = ¡ s i n µ d µ
d ! = s i n µ d µ d Á = ¡ d µ d Á
Spherical coordinate system and solid angle dω
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Radiative flux
How much energy flows through surface element dA?
dEν ~ Iν cosθ dω
à integrate over the whole solid angle (Ω = 4π):
¼ F º =
Z 4 ¼
I º ( c o s µ ) c o s µ d ! =
Z 2 ¼ 0
Z ¼ 0
I º ( c o s µ ) c o s µ s i n µ d µ d Á
“astrophysical flux”
Fν is the monochromatic radiative flux. The factor π in the definition is historical.
Fν can also be interpreted as the net rate of energy flow through a surface element.
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Radiative flux
The monochromatic radiative flux at frequency ν gives the net rate of energy flow through a surface element.
dEν ~ Iν cosθ dω à integrate over the whole solid angle (Ω = 4π):
We distinguish between the outward direction (0 < θ < π/2) and the inward direction (π/2 < θ < π), so that the net flux is:"
¼ F º = ¼ F + º ¡ ¼ F ¡
º =
=
Z 2 ¼ 0
Z ¼ / 2 0
I º ( c o s µ ) c o s µ s i n µ d µ d Á +
Z 2 ¼ 0
Z ¼ ¼ / 2
I º ( c o s µ ) c o s µ s i n µ d µ d Á
¼ F º =
Z 4 ¼
I º ( c o s µ ) c o s µ d ! =
Z 2 ¼ 0
Z ¼ 0
I º ( c o s µ ) c o s µ s i n µ d µ d Á
Note: for π/2 < θ < π −> cosθ < 0 à second term negative !!
“astrophysical flux”
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Total radiative flux
Integral over frequencies ν ◊"
"Z 1
0
¼ F º d º = F r a d
Frad is the total radiative flux.
It is the total net amount of energy going through the surface element per unit time and unit surface."
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Stellar luminosity
At the outer boundary of atmosphere (r = Ro) there is no incident radiation
à Integral interval over θ reduces from [0,π] to [0,π/2].
¼ F º ( R o ) = ¼ F + º ( R o ) =
Z 2 ¼ 0
Z ¼ / 2 0
I º ( c o s µ ) c o s µ s i n µ d µ d Á
This is the monochromatic energy that each surface element of the star radiates in all directions
If we multiply by the total stellar surface 4πR02
à monochromatic stellar luminosity at frequency ν
and integrating over ν"
◊ total stellar luminosity "
4 ¼ R 2 o · ¼ F º ( R o )
4 ¼ R 2 o · Z 1
0
¼ F + º ( R o ) d º = L ( L u m i n o s i t y )
= Lν
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Observed flux
What radiative flux is measured by an observer at distance d? à integrate specific intensity Iν towards observer over all surface elements note that only half sphere contributes
¼ F + º
E º =
Z 1 / 2 s p h e r e
d E = ¢ ! ¢ º ¢ t
Z 1 / 2 s p h e r e
I º ( c o s µ ) c o s µ d A
i n s p h e r i c a l s y m m e t r y : d A = R 2 o s i n µ d µ d Á
! E º = ¢ ! ¢ º ¢ t R 2 o
Z 2 ¼ o
Z ¼ / 2 0
I º ( c o s µ ) c o s µ s i n µ d µ d Á
à because of spherical symmetry the integral of intensity towards the observer over the stellar surface is proportional to πFν
+, the flux emitted into all directions by one surface element !!
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Observed flux
Solid angle of telescope at distance d:
F o b s º = r a d i a t i v e e n e r g y
a r e a · f r e q u e n c y · t i m e =
R 2 o d 2
¼ F + º ( R o )
This, and not Iν, is the quantity generally measured for stars. For the Sun, whose disk is resolved, we can also measure Iν (the variation of Iν over the solar disk is called the limb darkening)
unlike Iν, Fν decreases with increasing distance
¢ ! = ¢ A / d 2
+
E º = ¢ ! ¢ º ¢ t R 2 o ¼ F + º ( R o )
flux received = flux emitted x (R/d)2
R 1
0 F o b s
º , ¯ d º = 1 . 3 6 K W / m 2
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Mean intensity, energy density & radiation pressure
Integrating over the solid angle and dividing by 4π:
J º = 1
4 ¼
Z 4 ¼
I º d ! mean intensity
energy density
radiation pressure (important in hot stars)
u º = r a d i a t i o n e n e r g y
v o l u m e =
1
c
Z 4 ¼
I º d ! = 4 ¼
c J º
p º = 1
c
Z 4 ¼
I º c o s 2 µ d !
p r e s s u r e = f o r c e
a r e a =
d m o m e n t u m ( = E / c )
d t
1
a r e a
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Moments of the specific intensity
0th moment
1st moment (Eddington flux)
2nd moment
J º = 1
4 ¼
Z I º d ! =
1
4 ¼
Z 2 ¼ 0
d Á
Z 1 ¡ 1
I º d µ = 1
2
Z 1 ¡ 1
I º d µ
H º = 1
4 ¼
Z I º c o s µ d ! =
1
2
Z 1 ¡ 1
I º µ d µ = F º 4
K º = 1
4 ¼
Z I º c o s 2 µ d ! =
1
2
Z 1 ¡ 1
I º µ 2 d µ =
c
4 ¼ p º
for azimuthal symmetry
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Convention: τν = 0 at the outer edge of the atmosphere, increasing inwards
Interactions between photons and matter absorption of radiation
Iν"
ds"
Iνo Iν(s) s Over a distance s:
I º ( s ) = I o º e ¡
s R 0
· º d s
¿ º : =
s Z 0
· º d s optical depth (dimensionless)
or: dτν = κν ds
loss of intensity in the beam (true absorption/scattering) microscopical view: κν=n σν"
d I º = ¡ · º I º d s
· º : a b s o r p t i o n c o e ± c i e n t
[ · º ] = c m ¡ 1
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optical depth
I º ( s ) = I o º e ¡ ¿ º
i f ¿ º = 1 ! I º = I o º e ' 0 . 3 7 I o º
The quantity τν = 1 has a geometrical interpretation in terms of mean free path of photons :!¯ s
photons travel on average for a length before absorption
¯ s
We can see through atmosphere until τν ~ 1
optically thick (thin) medium: τν > (<) 1
The optical thickness of a layer determines the fraction of the
intensity passing through the layer
¿ º = 1 =
¯ s Z o
· º d s
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photon mean free path
What is the average distance over which photons travel?
< ¿ º > =
1 Z 0
¿ º p ( ¿ º ) d ¿ º expectation value
probability of absorption in interval [τν,τν+dτν]
= probability of non-absorption between 0 and τν and absorption in dτν"
- p r o b a b i l i t y t h a t p h o t o n i s n o t a b s o r b e d : 1 ¡ p ( 0 , ¿ º ) = I ( ¿ º )
I o = e ¡ ¿ º
- p r o b a b i l i t y t h a t p h o t o n i s a b s o r b e d : p ( 0 , ¿ º ) = ¢ I ( ¿ )
I o =
I o ¡ I ( ¿ º ) I o
= 1 ¡ I ( ¿ º ) I o
t o t a l p r o b a b i l i t y : e ¡ ¿ º d ¿ º
- p r o b a b i l i t y t h a t p h o t o n i s a b s o r b e d i n [ ¿ º , ¿ º + d ¿ º ] : p ( ¿ º , ¿ º + d ¿ º ) = d I º I ( ¿ º )
= d ¿ º
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photon mean free path
< ¿ º > =
1 Z 0
¿ º p ( ¿ º ) d ¿ º =
1 Z 0
¿ º e ¡ ¿ º d ¿ º = 1
mean free path corresponds to <τν>=1
i f · º ( s ) = c o n s t : ¢ ¿ º = · º ¢ s ! ¢ s = ¯ s = 1
· º (homogeneous
material)
Z x e ¡ x d x = ¡ ( 1 + x ) e ¡ x
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Principle of line formation
observer sees through the atmospheric layers up to τν ≈ 1
In the continuum κν is smaller than in the line à see deeper into the atmosphere
T(cont) > T(line)
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radiative acceleration
In the absorption process photons release momentum E/c to the atoms, and the corresponding force is:
The infinitesimal energy absorbed is:
The total energy absorbed is (assuming that κν does not depend on ω):
π Fν"
f o r c e = d f p h o t = m o m e n t u m ( = E / c ) d t
d E a b s º = d I º c o s µ d A d ! d t d º = · º I º c o s µ d A d ! d t d º d s
E a b s =
1 Z 0
· º
Z 4 ¼
I º c o s µ d ! d º d A d t d s = ¼
1 Z 0
· º F º d º d A d t d s
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radiative acceleration
d f p h o t = ¼
c
1 R 0
· º F º d º
d t d A d t d s = g r a d d m ( d m = ½ d A d s )
g r a d = ¼
c ½
1 Z 0
· º F º d º
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emission of radiation
ds"
dA"dω"
dV=dA ds"
energy added by emission processes within dV
d E e m º = ² º d V d ! d º d t
² º : e m i s s i o n c o e ± c i e n t
[ ² º ] = e r g c m ¡ 3 s r ¡ 1 H z ¡ 1 s ¡ 1
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The equation of radiative tranfer
If we combine absorption and emission together:
d E a b s º = d I a b s
º d A c o s µ d ! d º d t = ¡ · º I º d A c o s µ d ! d t d º d s
d E a b s º + d E e m
º = ( d I a b s º + d I e m º ) d A c o s µ d ! d º d t = ( ¡ · º I º + ² º ) d A c o s µ d ! d º d t d s
d E e m º = d I e m
º d A c o s µ d ! d º d t = ² º d A c o s µ d ! d º d t d s
d I º = d I a b s º + d I e m º = ( ¡ · º I º + ² º ) d s
d I º d s
= ¡ · º I º + ² º differential equation describing the flow of radiation through matter
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The equation of radiative tranfer Plane-parallel symmetry
d x = c o s µ d s = µ d s
d
d s = µ
d
d x
µ d I º ( µ , x ) d x
= ¡ · º I º ( µ , x ) + ² º
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The equation of radiative tranfer Spherical symmetry
angle θ between ray and radial direction is not constant
d
d s =
d r
d s
@
@ r + d µ
d s
@
@ µ
@
@ µ = @ µ
@ µ
@
@ µ = ¡ s i n µ
@
@ µ
= ) d d s
= µ @
@ r +
s i n 2 µ
r
@
@ µ = µ
@
@ r +
1 ¡ µ 2 r
@
@ µ
µ @ @ r I º ( µ , r ) + 1 ¡ µ 2
r @ @ µ I º ( µ , r ) = ¡ · º I º ( µ , r ) + ² º
¡ r d µ = s i n µ d s ( d µ < 0 ) ! d µ d s
= ¡ s i n µ
r
d r = d s c o s µ ! d r d s
= c o s µ ( a s i n p l a n e ¡ p a r a l l e l )
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The equation of radiative tranfer Optical depth and source function
In plane-parallel symmetry: optical depth increasing
towards interior:
µ d I º ( µ , x ) d x
= ¡ · º ( x ) I º ( µ , x ) + ² º ( x )
µ d I º ( µ , ¿ º ) d ¿ º
= I º ( µ , ¿ º ) ¡ S º ( ¿ º )
S º = ² º · º
Observed emerging intensity Iν(cos θ,τν= 0) depends on µ = cos θ , τν(Ri) and Sν "
"
The physics of Sν is crucial for radiative transfer "· º = d ¿ º
d s ¼ ¢ ¿ º
¢ s ¼ 1
¯ s
S º = ² º · º ¼ ² º · ¯ s
τ = 1 corresponds to free mean path of photons
source function Sν corresponds to intensity emitted over the free mean path of photons
source function
dim [Sν] = [Iν]
¡ · º d x = d ¿ º
¿ º = ¡ x Z
R o
· º d x
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The equation of radiative tranfer Source function: simple cases
a. LTE (thermal absorption/emission)
S º = ² º · º
= B º ( T ) Kirchhoff’s law photons are absorbed and re-emitted at the local temperature T
Knowledge of T stratification T=T(x) or T(τ) à solution of transfer equation Iν(µ,τν)
independent of radiation field
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The equation of radiative tranfer Source function: simple cases
b. coherent isotropic scattering (e.g. Thomson scattering)
the absorption process is characterized by the scattering coefficient σν, analogous to κν:"
d I º = ¡ ¾ º I º d s
ν = ν’"
and at each frequency ν: " d E e m º = d E a b s º
incident = scattered"
Z 4 ¼
² s c º d ! =
Z 4 ¼
¾ º I º d !
² s c º
Z 4 ¼
d ! = ¾ º
Z 4 ¼
I º d !
d E e m º =
Z 4 ¼
² s c º d !
d E a b s º =
Z 4 ¼
¾ º I º d !
² s c º ¾ º
= 1
4 ¼
Z 4 ¼
I º d !
S º = J º completely dependent on radiation field
not dependent on temperature T
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30
The equation of radiative tranfer Source function: simple cases
c. mixed case
S º = ² º + ² s c º · º + ¾ º
= · º
· º + ¾ º
² º · º
+ ¾ º
· º + ¾ º
² s c º ¾ º
S º = ² º + ² s c º · º + ¾ º
= · º
· º + ¾ º B º +
¾ º · º + ¾ º
J º
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31
Formal solution of the equation of radiative tranfer
we want to solve the equation of RT with a known source function and in plane-parallel geometry
multiply by e-τν/µ and integrate between τ1 (outside) and τ2 (> τ1, inside)"
d
d ¿ º ( I º e
¡ ¿ º / µ ) = ¡ S º e ¡ ¿ º / µ
µ
linear 1st order differential equation
µ d I º d ¿ º
= I º ¡ S º
h I º e
¡ ¿ º µ
i ¿ 2 ¿ 1
= ¡ ¿ 2 Z ¿ 1
S º e ¡ ¿ º
µ d t º µ
check, whether this really yields transfer equation above"
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32
Formal solution of the equation of radiative tranfer
intensity originating at τ2 decreased by exponential factor to τ1 contribution to the intensity by
emission along the path from τ2 to τ1 (at each point decreased by the exponential factor)
integral form of equation of radiation transfer
Formal solution! actual solution can be complex, since Sν can depend on Iν"
h I º e
¡ ¿ º µ
i ¿ 2 ¿ 1
= ¡ ¿ 2 Z ¿ 1
S º e ¡ ¿ º
µ d t º µ
I º ( ¿ 1 , µ ) = I º ( ¿ 2 , µ ) e ¡ ¿ 2 ¡ ¿ 1
µ +
¿ 2 Z ¿ 1
S º ( t ) e ¡ t ¡ ¿ 1
µ d t
µ
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33
Boundary conditions
a. incoming radiation: µ < 0 at τ2 = 0
usually we can neglect irradiation from outside: Iν(τ2 = 0, µ < 0) = 0
solution of RT equation requires boundary conditions, which are different for incoming and outgoing radiation
I i n º ( ¿ º , µ ) =
0 Z ¿ º
S º ( t ) e ¡ t ¡ ¿ º
µ d t
µ
I º ( ¿ 1 , µ ) = I º ( ¿ 2 , µ ) e ¡ ¿ 2 ¡ ¿ 1
µ +
¿ 2 Z ¿ 1
S º ( t ) e ¡ t ¡ ¿ 1
µ d t
µ
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34
Boundary conditions
b. outgoing radiation: µ > 0 at τ2 = τmax à ∞
We have either
or
I º ( ¿ m a x , µ ) = I + º ( µ ) finite slab or
shell
l i m ¿ ! 1
I º ( ¿ , µ ) e ¡ ¿ / µ = 0 semi-infinite
case (planar or spherical) Iν increases less rapidly than the exponential
I o u t º ( ¿ º , µ ) =
1 Z ¿ º
S º ( t ) e ¡ t ¡ ¿ º
µ d t
µ
I º ( ¿ º ) = I o u t º ( ¿ º ) + I i n º ( ¿ º )
and at a given position τν in the atmosphere:
I º ( ¿ 1 , µ ) = I º ( ¿ 2 , µ ) e ¡ ¿ 2 ¡ ¿ 1
µ +
¿ 2 Z ¿ 1
S º ( t ) e ¡ t ¡ ¿ 1
µ d t
µ
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35
Emergent intensity
from the latter à emergent intensity
τν = 0, µ > 0
I º ( 0 , µ ) =
1 Z 0
S º ( t ) e ¡ t µ d t
µ
intensity observed is a weighted average of the source function along the line of sight. The contribution to the emerging intensity comes mostly from each depths with τ/µ < 1."
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36
Emergent intensity
suppose that Sν is linear in τν (Taylor expansion around τν = 0):
Eddington-Barbier relation"
emergent intensity I º ( 0 , µ ) =
1 Z 0
( S 0 º + S 1 º t ) e ¡ t µ d t
µ = S 0 º + S 1 º µ
I º ( 0 , µ ) = S º ( ¿ º = µ )
we see the source function at location τν = µ "
the emergent intensity corresponds to the source function at τν = 1 along the line of sight
S º ( ¿ º ) = S 0 º + S 1 º ¿ º Z
x e ¡ x d x = ¡ ( 1 + x ) e ¡ x
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37
Emergent intensity
µ = 1 (normal direction):
I º ( 0 , 1 ) = S º ( ¿ º = 1 )
µ = 0.5 (slanted direction):
I º ( 0 , 0 . 5 ) = S º ( ¿ º = 0 . 5 )
in both cases: ∆τ/µ ≈ 1
spectral lines: compared to continuum τν/µ = 1 is reached at higher layer in the atmosphere
à Sνline < Sν
cont
à a dip is created in the spectrum
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38
Line formation
simplify: µ = 1, τ1=0 (emergent intensity), τ2 = τ"
Sν independent of location
I º ( 0 ) = I º ( ¿ º ) e ¡ ¿ º + S º
¿ º Z 0
e ¡ t d t = I º ( ¿ º ) e ¡ ¿ º + S º ( 1 ¡ e ¡ ¿ º )
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39
Line formation
Optically thick object: I º ( 0 ) = I º ( ¿ º ) e ¡ ¿ º + S º ( 1 ¡ e ¡ ¿ º ) = S º
Optically thin object: I º ( 0 ) = I º ( ¿ º ) + [ S º ¡ I º ( ¿ º ) ] ¿ º
τ à ∞
exp(-τν) ≈ 1 - τν"
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40
independent of κν, no line (e.g. black body Bν)
Iν = τν Sν = κν ds Sν"
e.g. HII region, solar corona
enhanced κν"
I º ( 0 ) = I º ( ¿ º ) + [ S º ¡ I º ( ¿ º ) ] ¿ º
e.g. stellar absorption spectrum (temperature decreasing outwards)
e.g. stellar spectrum with temperature increasing outwards (e.g.
Sun in the UV)
From Rutten’s web notes
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41
Line formation example: solar corona
Iν = τν Sν"
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42
The diffusion approximation
At large optical depth in stellar atmosphere photons are local: Sν à Bν
Expand Sν (= Bν) as a power-series:
S º ( t ) = 1 X n = 0
d n B º d ¿ n º
( t ¡ ¿ º ) n / n !
In the diffusion approximation (τν >>1) we retain only first order terms:
B º ( t ) = B º ( ¿ º ) + d B º d ¿ º
( t ¡ ¿ º )
I o u t º ( ¿ º , µ ) =
1 Z ¿ º
[ B º ( ¿ º ) + d B º d ¿ º
( t ¡ ¿ º ) ] e ¡ ( t ¡ ¿ º ) / µ d t
µ
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43
The diffusion approximation
Substituting:
At τν = 0 we obtain the Eddington-Barbier relation for the observed emergent intensity.
It is given by the Planck-function and its gradient at τν = 0.
It depends linearly on µ = cos θ.
1 Z 0
u k e ¡ u d u = k !
I i n º ( ¿ º , µ ) = ¡ ¿ º / µ Z 0
[ B º ( ¿ º ) + d B º d ¿ º
µ u ] e ¡ u d u
I o u t º ( ¿ º , µ ) =
1 Z 0
[ B º ( ¿ º ) + d B º d ¿ º
µ u ] e ¡ u d u = B º ( ¿ º ) + µ d B º d ¿ º
I o u t º ( ¿ º , µ ) =
1 Z ¿ º
[ B º ( ¿ º ) + d B º d ¿ º
( t ¡ ¿ º ) ] e ¡ ( t ¡ ¿ º ) / µ d t
µ
t ! u = t ¡ ¿ º µ
! d t = µ d u
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44
Solar limb darkening
I º ( 0 , µ ) I º ( 0 , 1 )
= B º ( 0 ) + d B º
d ¿ º µ
B º ( 0 ) + d B º d ¿ º
from the intensity measurements à Bν(0), dBν/dτν"
B º ( t ) = B º ( 0 ) + d B º d ¿ º
t = a + b · t = 2 h º 3
c 2 1
e h º / k T ( t ) ¡ 1
T(t): empirical temperature stratification of solar photosphere
center-to-limb variation of intensity
diffusion approximation:
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45
Solar limb darkening
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46
Solar limb darkening: temperature stratification
I º ( 0 , µ ) =
1 Z 0
S º ( t ) e ¡ t µ d t
µ
exponential extinction varies as -τν /cosθ"
From Sν = a + bτν:"
I º ( 0 , µ ) = S º ( ¿ º = µ ) Sν!
I º ( 0 , c o s µ ) = a º + b º c o s µ
R. Rutten, web notes
Unsoeld, 68
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47
Eddington approximation
In the diffusion approximation we had:
B º ( t ) = B º ( ¿ º ) + d B º d ¿ º
( t ¡ ¿ º )
0 < µ < 1
-1 < µ < 0
τ >> 1
I i n º ( ¿ º , µ ) = ¡ ¿ º / µ Z 0
[ B º ( ¿ º ) + d B º d ¿ º
µ u ] e ¡ u d u
I o u t º ( ¿ º , µ ) = B º ( ¿ º ) + µ d B º d ¿ º
I ¡ º ( ¿ º , µ ) = B º ( ¿ º ) + µ
d B º d ¿ º
we want to obtain an approximation for the radiation field – both inward and outward radiation - at large optical depth
à stellar interior, inner boundary of atmosphere
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48
Eddington approximation
J º = 1
2
Z 1 ¡ 1
I º d µ = B º ( ¿ º )
H º = F º 4
= 1
2
Z 1 ¡ 1
µ I º d µ = 1
3
d B º d ¿ º
K º = 1
2
Z 1 ¡ 1
µ 2 I º d µ = 1
3 B º ( ¿ º )
= ¡ 1
3
1
· º
d B º d x
= ¡ 1
3 · º
d B º d T
d T
d x
flux Fν ~ dT/dx diffusion: flux ~ gradient (e.g. heat conduction)
K º = 1 3 J º Eddington approximation
With this approximation for Iν we can calculate the angle averaged momenta of the intensity
à simple approximation for photon flux and a relationship between mean intensity Jν and Kν
à very important for analytical estimates
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49
J º = 1
2
1 Z ¡ 1
I º d µ = 1
2
1 Z 0
I o u t º d µ + 1
2
0 Z ¡ 1
I i n º d µ
J º = 1
2
2 4
1 Z 0
1 Z ¿ º
S º ( t ) e ¡ ( t ¡ ¿ º ) / µ d t
µ d µ ¡
0 Z ¡ 1
¿ º Z 0
S º ( t ) e ¡ ( t ¡ ¿ º ) / µ d t
µ d µ
3 5
s u b s t i t u t e w = 1 µ ) d w
w = ¡ 1
µ d µ w = ¡ 1
µ ) d w
w = ¡ 1
µ d µ
µ < 0 µ > 0
After the previous approximations, we now want to calculate exact solutions for tha radiative momenta Jν, Hν, Kν. Those are important to calculate spectra and atmospheric structure Schwarzschild-Milne equations
I o u t º ( ¿ º , µ ) =
1 Z ¿ º
S º ( t ) e ¡ t ¡ ¿ º
µ d t
µ
I i n º ( ¿ º , µ ) =
0 Z ¿ º
S º ( t ) e ¡ t ¡ ¿ º
µ d t
µ
J º = 1
2
2 4 1 Z 1
1 Z ¿ º
S º ( t ) e ¡ ( t ¡ ¿ º ) w d t
d w
w +
1 Z 1
¿ º Z 0
S º ( t ) e ¡ ( ¿ º ¡ t ) w d t
d w
w
3 5
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50
Schwarzschild-Milne equations
J º = 1
2
2 4 1 Z ¿ º
S º ( t )
1 Z 1
e ¡ ( t ¡ ¿ º ) w d w
w d t +
¿ º Z 0
S º ( t )
1 Z 1
e ¡ ( ¿ º ¡ t ) w d w
w d t
3 5
> 0 > 0
J º = 1
2
1 Z 0
S º ( t )
1 Z 1
e ¡ w | t ¡ ¿ º | d w
w d t =
1
2
1 Z 0
S º ( t ) E 1 ( | t ¡ ¿ º | ) d t
Schwarzschild’s equation
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51
Schwarzschild-Milne equations
w h e r e
E 1 ( t ) =
1 Z 1
e ¡ t x d x
x =
1 Z t
e ¡ x
x d x
i s t h e fi r s t e x p o n e n t i a l i n t e g r a l ( s i n g u l a r i t y a t t = 0 )
E x p o n e n t i a l i n t e g r a l s
E n ( t ) = t n ¡ 1 1 Z t
x ¡ n e ¡ x d x
E n ( 0 ) = 1 / ( n ¡ 1 ) , E n ( t ! 1 ) = e ¡ t / t ! 0
d E n d t
= ¡ E n ¡ 1 ,
Z E n ( t ) = ¡ E n + 1 ( t )
E 1 ( 0 ) = 1 E 2 ( 0 ) = 1 E 3 ( 0 ) = 1 / 2 E n ( 1 ) = 0
Gray, 92
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52
Schwarzschild-Milne equations
Introducing the Λ operator:
J º ( ¿ º ) = ¤ ¿ º [ S º ( t ) ]
Similarly for the other 2 moments of Intensity:
¤ ¿ º [ f ( t ) ] = 1
2
1 Z 0
f ( t ) E 1 ( | t ¡ ¿ º | ) d t
Milne’s equations
Jν, Hν and Kν are all depth-weighted means of Sν"
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53
Schwarzschild-Milne equations
the 3 moments of Intensity:
Jν, Hν and Kν are all depth-weighted means of Sν"
◊ the strongest contribution comes from the depth, where the argument of the exponential integrals is zero, i.e. t=τν"
Gray, 92
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54
The temperature-optical depth relation Radiative equilibrium
The condition of radiative equilibrium (expressing conservation of energy) requires that the flux at any given depth remains constant:
4 ¼ r 2 F ( r ) = 4 ¼ r 2 · 4 ¼
1 Z 0
H º d º = c o n s t = L
In plane-parallel geometry r ≈ R = const 4 ¼
1 Z 0
H º d º = c o n s t
and in analogy to the black body radiation, from the Stefan-Boltzmann law we define the effective temperature:
4 ¼
1 Z 0
H º d º = ¾ T 4 e ®
F ( r ) = ¼ F =
1 Z 0
Z 4 ¼
I º c o s µ d ! d º = ¼
1 Z 0
F º d º = 4 ¼
1 Z 0
H º d º
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55
The effective temperature
The effective temperature is defined by:
It characterizes the total radiative flux transported through the atmosphere.
It can be regarded as an average of the temperature over depth in the atmosphere.
A blackbody radiating the same amount of total energy would have a temperature T = Teff.
4 ¼
1 Z 0
H º d º = ¾ T 4 e ®
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56
Radiative equilibrium
Let us now combine the condition of radiative equilibrium with the equation of radiative transfer in plane-parallel geometry:
µ d I º d x
= ¡ ( · º + ¾ º ) ( I º ¡ S º )
1
2
1 Z ¡ 1
µ d I º d x
d µ = ¡ 1
2
1 Z ¡ 1
( · º + ¾ º ) ( I º ¡ S º ) d µ
Hν"
d
d x
2 4 1
2
1 Z ¡ 1
µ I º d µ
3 5 = ¡ ( · º + ¾ º ) ( J º ¡ S º )
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57
Radiative equilibrium
Integrate over frequency:
d
d x
1 Z 0
H º d º = ¡ 1 Z 0
( · º + ¾ º ) ( J º ¡ S º ) d º
const
1 Z 0
( · º + ¾ º ) ( J º ¡ S º ) d º = 0 s u b s t i t u t e S º = · º
· º + ¾ º B º +
¾ º
· º + ¾ º J º
1 Z 0
· º [ J º ¡ B º ( T ) ] d º = 0
4 ¼
1 Z 0
H º d º = ¾ T 4 e ® in addition:
0 @
1 Z 0
· º J º d º = a b s o r b e d e n e r g y
1 A
0 @
1 Z 0
· º B º d º = e m i t t e d e n e r g y
1 A
T(x) or T(τ)
at each depth:
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58
Radiative equilibrium
1 Z 0
· º [ J º ¡ B º ( T ) ] d º = 0 4 ¼
1 Z 0
H º d º = ¾ T 4 e ®
T(x) or T(τ)
The temperature T(r) at every depth has to assume the value for which the left integral over all frequencies becomes zero. à This determines the local temperature.
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59
Iterative method for calculation of a stellar atmosphere: the major parameters are Teff and g
T(x), κν(x), Bν[T(x)],P(x), ρ(x) Jν(x), Hν(x)
R 1 0 · º ( J º ¡ B º ) d º = 0 ?
4 ¼ R 1 0
H º d º = ¾ T 4 e ® ?
∆T(x), ∆κν(x), ∆Bν[T(x)], ∆ρ(x)
equation of transfer
a. hydrostatic equilibrium
b. equation of radiation transfer
d P d x
= ¡ g ½ ( x )
µ d I º d x
= ¡ ( · º + ¾ º ) ( I º ¡ S º )
c. radiative equilibrium 1 Z 0
· º [ J º ¡ B º ( T ) ] d º = 0
d. flux conservation
4 ¼
1 Z 0
H º d º = ¾ T 4 e ®
e. equation of state P = ½ k T
µ m H
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Grey atmosphere - an approximation for the temperature structure
We derive a simple analytical approximation for the temperature structure. We assume that we can approximate the radiative equilibrium integral by using a frequency-averaged absorption coefficient, which we can put in front of the integral. 1 Z 0
· º [ J º ¡ B º ( T ) ] d º = 0 ¯ ·
1 Z 0
[ J º ¡ B º ( T ) ] d º = 0
W i t h : J =
1 Z 0
J º d º H =
1 Z 0
H º d º K =
1 Z 0
K º d º B =
1 Z 0
B º d º = ¾ T 4
¼
J = B 4 ¼ H = ¾ T 4 e ®
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Grey atmosphere
We then assume LTE: S = B.
From
and a similar expression for frequency-integrated quantities
and with the approximations S = B, B = J:
J º ( ¿ º ) = ¤ ¿ º [ S º ( t ) ] = 1
2
1 Z 0
S º ( t ) E 1 ( | t ¡ ¿ º | ) d t
Milne’s equation
!!! this is an integral equation for J(τ) !!!"
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62
The exact solution of the Hopf integral equation
Milne’s equation J(τ) = Λτ[J(t)] à exact solution (see Mihalas, “Stellar Atmospheres”)
J(τ) = const. [ τ + q(τ)], with q(τ) monotonic
Radiative equilibrium - grey approximation
1 p 3
= 0 . 5 7 7 = q ( 0 ) · q ( ¿ ) · q ( 1 ) = 0 . 7 1 0
J(τ) = B(τ) = σ/π T4(τ) = const. [ τ + q(τ)] with boundary conditions à T4(τ) = ¾ T4
eff [τ + q(τ)]
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63
A simple approximation for T(τ)
0th moment of equation of transfer (integrate both sides in dµ from -1 to 1)
µ d I
d x = ¡ ¯ · ( I ¡ B ) d H
d ¯ ¿ = J ¡ B = 0 ( J = B ) H = c o n s t =
¾ T 4 e ®
4 ¼
1st moment of equation of transfer (integrate both sides in µdµ from -1 to 1)
d K
d ¯ ¿ = H =
¾ T 4 e ®
4 ¼ K ( ¿ ) = H ¯ ¿ + c o n s t a n t
From Eddington’s approximation at large depth: K = 1/3 J
µ
2 dI
dx
= (µI µB)
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64
Grey atmosphere – temperature distribution
T 4 ( ¿ ) = 3 ¼ H
¾ ( ¿ + c ) H =
¾
4 ¼ T 4 e ®
T 4 ( ¿ ) = 3
4 T 4 e ® ( ¿ + c ) T4 is linear in τ"
Estimation of c
1 Z
0
t s E n ( t ) d t = s !
s + n
1/3 1/2
H º ( ¿ = 0 ) = 1
2 3 H
2 4 1 Z 0
t E 2 ( t ) d t + c
1 Z 0
E 2 ( t ) d t
3 5
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65
Grey atmosphere – Hopf function
T 4 ( ¿ ) = 3
4 T 4 e ® ( ¿ +
2
3 )
based on approximation K/J = 1/3 T = Teff at τ = 2/3, T(0) = 0.84 Teff
Remember: More in general J is obtained from
T 4 ( ¿ ) = 3
4 T 4 e ® [ ¿ + q ( ¿ ) ] q ( ¿ ) : H o p f f u n c t i o n
Once Hopf function is specified à solution of the grey atmosphere (temperature distribution)
1 p 3
= 0 . 5 7 7 = q ( 0 ) · q ( ¿ ) · q ( 1 ) = 0 . 7 1 0
H ( 0 ) = H = 1
2 H ( 1 +
3
2 c ) ! c =
2
3
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66
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67
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68
Mean opacities: flux-weighted
1st possibility: Flux-weighted mean
allows the preservation of the K-integral (radiation pressure)
Problem: Hν not known a priory (requires iteration of model atmospheres)
¯ · =
1 R 0
· º H º d º
H
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69
Mean opacities: Rosseland
2nd possibility: Rosseland mean
to obtain correct integrated energy flux and use local T
d K º d x
! 1
3
d B º d x
= 1
3
d B º d T
d T
d x
K º ! 1
3 J º , J º ! B º a s ¿ ! 1
1
¯ · R o s s =
1 R 0
1 · º
d B º ( T ) d T
d º
1 R 0
d B º ( T ) d T
d º
large weight for low-opacity (more transparent to radiation) regions
1
¯ · =
1 R 0
1 · º
d K º d x
d º
d K d x
dKºdx = ¡·ºHº
1R0
1·ºdKºdx dº = ¡
1R0Hºdº
1 Z 0
1
· º
d K º d x
d º = ¡ H ) ( g r e y ) ) 1
¯ ·
d K
d x = ¡ H
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70
Mean opacities: Rosseland
at large τ the T structure is accurately given by
T 4 = 3
4 T 4 e ® [ ¿ R o s s + q ( ¿ R o s s ) ] Rosseland opacities used
in stellar interiors
For stellar atmospheres Rosseland opacities allow us to obtain initial approximate values for the Temperature stratification (used for further iterations).
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71
grey: q(τ) = exact grey: q(τ) = 2/3
non-grey numerical
T4 vs. τ
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72
T vs. log(τ)
non-grey numerical
grey: q(τ) = exact grey: q(τ) = 2/3
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73
Iterative method for calculation of a stellar atmosphere: the major parameters are Teff and g
T(x), κν(x), Bν[T(x)],P(x), ρ(x) Jν(x), Hν(x)
R 1 0 · º ( J º ¡ B º ) d º = 0 ?
4 ¼ R 1 0
H º d º = ¾ T 4 e ® ?
∆T(x), ∆κν(x), ∆Bν[T(x)], ∆ρ(x)
equation of transfer
a. hydrostatic equilibrium
b. equation of radiation transfer
d P d x
= ¡ g ½ ( x )
µ d I º d x
= ¡ ( · º + ¾ º ) ( I º ¡ S º )
c. radiative equilibrium 1 Z 0
· º [ J º ¡ B º ( T ) ] d º = 0
d. flux conservation
4 ¼
1 Z 0
H º d º = ¾ T 4 e ®
e. equation of state P = ½ k T
µ m H