Transcript Hydrostatisches Gleichgewicht

Stellar Atmospheres: Radiative Equilibrium
Radiative Equilibrium
Energy conservation
1
Stellar Atmospheres: Radiative Equilibrium
Radiative Equilibrium
Assumption:
Energy conservation, i.e., no nuclear energy sources
Counter-example: radioactive decay of Ni56 Co56 Fe56 in
supernova atmospheres
Energy transfer predominantly by radiation
Other possibilities:
Convection e.g., H convection zone in outer solar layer
Heat conduction e.g., solar corona or interior of white dwarfs
Radiative equilibrium means, that we have at each location:
Radiation energy absorbed / sec
=
Radiation energy emitted / sec
integrated over all
frequencies and
angles
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Stellar Atmospheres: Radiative Equilibrium
Radiative Equilibrium

Absorption per cm2 and second:
 d  dv (v) I
4
Emission per cm2 and second:
v
0

 d  dv (v)
4
0
Assumption: isotropic opacities and emissivities
Integration over d then yields

 dv (v) J
0
v


0
0
  dv (v)    (v) J v  S v dv  0
Constraint equation in addition to the radiative transfer
equation; fixes temperature stratification T(r)
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Stellar Atmospheres: Radiative Equilibrium
Conservation of flux
Alternative formulation of energy equation
In plane-parallel geometry: 0-th moment of transfer equation
dH v
  J v  Sv 
dt

Integration over frequency, exchange integration and
differentiation:


d
H v dv     J v  S v  dv  0

dt 0
0

 H   H v dv  const 
0
 4
Teff
4
 4  dK v
0 H v dv  4 Teff  0 d dv
because of radiative equilibrium
for all depths. Alternatively written:




0
d ( fv J v )
 4
dv 
Teff
d
4
(1st moment of transfer equation)
(definiton of Eddington factor)
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Stellar Atmospheres: Radiative Equilibrium
Which formulation is good or better?
I Radiative equilibrium: local, integral form of energy
equation
II Conservation of flux: non-local (gradient), differential form
of radiative equilibrium
I / II numerically better behaviour in small / large depths
Very useful is a linear combination of both formulations:


 d ( f v J v )

A      J v  S v dv  B   
dv  H   0
0

 0 d

A,B are coefficients, providing a smooth transition between
formulations I and II.
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Stellar Atmospheres: Radiative Equilibrium
Flux conservation in spherically symmetric geometry
0-th moment of transfer equation:
1  2
r H v   S v  J v 
2
r r





  r 2  H v dv   r 2   S v  J v dv  0
r  0
0




1
r  H v dv  const 
L
2
16
0
2
because L  16 2 R 2 H
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Stellar Atmospheres: Radiative Equilibrium
Another alternative, if T de-couples from radiation field
Thermal balance of electrons
Q H  QC  0

QffH  4 ne  N j   ff, j (v, T ) J v dv
j
0

3

  hv kT
2
hv
C
Qff  4 ne  N j   ff, j (v, T ) J v  2 e
dv
c 
j

0

 vlk 
Q  4  nl   bf, lk (v) J v 1  dv
v 

l ,k
0
H
bf

 vlk
Q  4  nk   bf, lk (v) J v 1 
v

l ,k
0
C
bf
QcH  ne  nm qlm (T )hvlm
2hv 3   hv kT

dv
 J v  2 e
c 

l ,m
QcC  ne  nl qlm (T )hvlm
l ,m
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Stellar Atmospheres: Radiative Equilibrium
The gray atmosphere
Simple but insightful problem to solve the transfer equation
together with the constraint equation for radiative equilibrium
Gray atmosphere:   
Moments of transfer equation
dH v
dK v
 J v  Sv  II 
 H v with    dt
d
d
Integration over frequency
I 
I 
dH
 J S
d
 II 
Radiative equilibrium
 I  J  S
dK
H
d
 (J
v
 Sv )dv    ( J v  Sv )dv  J  S  0
and because of conservation of flux
  II 
dH
0
d
d 2K
dK

0

K

c


c
from
II
follows
c

 H , c2 see below


1
2
1
2
d
d
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Stellar Atmospheres: Radiative Equilibrium
The gray atmosphere
Relations (I) und (II) represent two equations for three
quantities S,J,K with pre-chosen H (resp. Teff)
Closure equation: Eddington approximation
K  1 3J  S  J  3K  3H  3c2
 III 

Source function is linear in 
Temperature stratification?
In LTE:
S ( )  B(T ( )) 
insert into  III  :
with H 
 4
T

 4
T  3H  3c2

 4
Teff we get:
4
 4
3
T ( ) 
 Teff4   3c2

4
 IV 
c2 is now determined from boundary condition ( =0)
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Stellar Atmospheres: Radiative Equilibrium
Gray atmosphere: Outer boundary condition
Emergent flux:

1
H (0)   S ( ) E2 ( )d  with S from III 
20


1
  3H   3c2 E2 ( )d 
20



3
  H    E2 ( )d   c2  E2 ( )d 
2 0
0


with  t l En (t )dt 
0
H ( 0) 
from (IV):

l!
1
and E2 (t ) 
e t  tE1 (t )
ln
2 1


3 1
1 
2
H

c

c

H
2
2

2 3
2 
3
3 
2
2

 T 4  Teff4    , S  3H    (from III)
4 
3
3

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Stellar Atmospheres: Radiative Equilibrium
Avoiding Eddington approximation
Ansatz:
J ( )  3H (  q ( )) generaliza tion of III 
q ( )  Hopf function
J ( ) 
3 4
Teff (  q ( ))
4

Insert into Schwarzschild equation:
J ( )  S  J
integral equation for J

1
   q( )      q( ) E1     d  (*) integral equation for q, see below
20
Approximate solution for J by iteration (“Lambda iteration“)
J (1)  3H (  2 3)
i.e., start with Eddington approximation
2 1
1


J ( 2 )  J (1)  3H (  2 3)   3H    E 2 ( )  E3 ( ) 
3 3
2


(was result for linear S)
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Stellar Atmospheres: Radiative Equilibrium
At the surface
  0 , E2 (0)  1 , E3 (0) 
1
2
2 1 1

J ( 2 )  3H       3H   0.583
3 3 4

exact: q(0)=0.577….
At inner boundary    , E2 ()  0 , E3 ()  0
2

J (2)  3H    
3

Basic problem of Lambda Iteration: Good in outer layers, but
does not work at large optical depths, because exponential
integral function approaches zero exponentially.
Exact solution of (*) for Hopf function, e.g., by Laplace
transformation (Kourganoff, Basic Methods in Transfer Problems)
Analytical approximation (Unsöld, Sternatmosphären, p. 138)
q ( )  0.6940  0.1167 e 1.972
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Stellar Atmospheres: Radiative Equilibrium
Gray atmosphere: Interpretation of results
Temperature gradient
d 4
dT 3 4
T  4T 3
 Teff
d
d 4
The higher the effective temperature, the steeper the
dT
~ Teff4
temperature gradient.
d
dT
dT The larger the opacity, the steeper the (geometric) temperature

dt
d gradient.
Flux of gray atmosphere
LTE: Sv  Bv (T ( ))


1
1
H v ( )   Bv (T ( )) E2 (t   )dt   Bv (T ( )) E2 (  t )dt
2
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with   hv kTeff , T Teff  3 4 (  q( )) 
1/ 4
H  d  H v dv and H  
4
 p ( )  hv kT   p ( )
Teff4



H v dv
E2 (t   )
E2 (  t )
4 kTeff
4 k 4 3 
 H  ( ) / H 

Hv  3 2   
dt  
dt 
H d  Teff4 h
hc
exp(

p
(

))

1
exp(

p
(

))

1
0


1 2 hv3 4 k  3 k 3
2 c 2  h h3 v 3
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Stellar Atmospheres: Radiative Equilibrium
Gray atmosphere: Interpretation of results
Limb darkening of total radiation
I(  0, )  S(  )  B(T(  )) 

 4
 4 3
2
T (  )  Teff






4
3
I(0, )   2 / 3 2
3

 (1  cos )
I(0,1) 1  2 / 3 5
2
i.e., intensity at limb of stellar disk smaller than at center by
40%, good agreement with solar observations
Empirical determination of temperature stratification
measure I (  0,  )  S (   )  S ( )  B(T ( ))  T

Observations at different wavelengths yield different Tstructures, hence, the opacity must be a function of
wavelength
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Stellar Atmospheres: Radiative Equilibrium
The Rosseland opacity
Gray approximation (=const) very coarse, ist there a good
mean value  ? What choice to make for a mean value?
gray
transfer equation
0-th moment
1st moment
dI
   (S  I )
dz
dH
  (S  J )  0
dz
dK
 H
dz
non-gray
dI v

  (v)( S v  I v )
dz
dH v
  (v)( S v  J v )
dz
dK v
  (v) H v
dz
For each of these 3 equations one can find a mean , with
which the equations for the gray case are equal to the
frequency-integrated non-gray equations.
Because we demand flux conservation, the 1st moment
equation is decisive for our choice:
 Rosseland mean of opacity
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Stellar Atmospheres: Radiative Equilibrium
The Rosseland opacity


1 dK v
1 dK
dv 
 (v) dz
 R dz
0
 H v dv  const  
0

1 dK v
dv

 (v) dz
1
0

with Eddington approximat ion K  1 / 3 J and LTE J  B :
dK
R
dz

1 dBv
dv

 (v) dz
dBv dBv dT
1
dB d   4  4 3 dT
0
with

and
  T 
T
dB
R
dz
dT dz
dz dz  
dz
 
dz

1 dBv
dv

 (v) dT
1
0

4 3
R
T

Definition of Rosseland mean of opacity
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Stellar Atmospheres: Radiative Equilibrium
The Rosseland opacity
The Rosseland mean
1
R
is a weighted mean
of opacity 1 with weight function dBv
 (v )
dT
Particularly, strong weight is given to those frequencies,
where the radiation flux is large.
The corresponding optical depth is called Rosseland depth
z
 Ross ( z )    R ( z )dz 
0
For  Ross  1 the gray approximation with  R is very good,
i.e.
3
T 4 ( Ross )  Teff4 ( Ross  q( Ross ))
4
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Stellar Atmospheres: Radiative Equilibrium
Convection
Compute model atmosphere assuming
• Radiative equilibrium (Sect. VI)  temperature stratification
• Hydrostatic equilibrium
 pressure stratification
Is this structure stable against convection, i.e. small
perturbations?
• Thought experiment
Displace a blob of gas by r upwards, fast enough that no heat
exchange with surrounding occurs (i.e., adiabatic), but slow
enough that pressure balance with surrounding is retained (i.e.
<< sound velocity)
18
Stellar Atmospheres: Radiative Equilibrium
Inside of blob
outside
T  Tad  Tad (r  r )
T  Trad  Trad (r  r )
   ad   ad (r  r )
   rad   rad (r  r )
r
T (r ),  (r )
T (r ),  (r )
ad (r  r )   rad (r  r )  further buoyancy, unstable
ad (r  r )   rad (r  r )  gas blob falls back, stable
d ad   d  rad unstable
 

dr   dr  stable
k
with ideal gas equation p=
T and pressure balance adTad = radTrad
AmH
i.e.
dTad
dr
  dTrad
 
  dr
unstable

 stable
Stratification becomes unstable, if temperature gradient dTad dr
19
rises above critical value.
Stellar Atmospheres: Radiative Equilibrium
Alternative notation
Pressure as independent depth variable:
AmH
p
hydrostatic equation: dp    g eff dr  
g eff dr
k
T
kT
 dr   dp
AmH g eff p
(ideal gas)
AmH
AmH
dT
dT T
d (ln T )

g eff

g eff
dr
k
dp p
k
d (ln p )
d (ln Tad )   d (ln Trad ) unstable
 

d (ln p )   d (ln p )  stable
Schwarzschild criterion
Abbreviated notation
d (ln Tad )
d (ln Trad )
;  rad 
d (ln p )
d (ln p )
 ad   rad stable
 ad 
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Stellar Atmospheres: Radiative Equilibrium
The adiabatic gradient
dQ  0 (no heat exchange)
dQ  dE  pdV (1st law of thermodynamics)
dE  cV dT internal energy  cV dT  pdV  0 (*)
Internal energy of a one-atomic gas excluding effects of
ionisation and excitation
3
3
E  NkT  cV  Nk
2
2
But if energy can be absorbed by ionization:
3
cV  Nk
2
Specific heat at constant pressure
cp 
Q
T

p const
 cp  cV  Nk
dE
dV
p
dT
dT
 cV  p
p const
d ( NkT p)
Nk
 cV  p
dT
p
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Stellar Atmospheres: Radiative Equilibrium
The adiabatic gradient
Ideal gas: pV  NkT  Vdp  pdV  NkdT   cp  cV  dT
dT 
Vdp  pdV
cp  cV
(**)
from(*) with (**)  cV
Vdp  pdV
 pdV  0
cp  cV
/pV

cp  cV
cV
dp dV dV cp  cV


0
p V
V
cV
dp dV cp

0
p V cV
cp
cV
d (ln V )  d (ln p )
definition:  :
cp
cV
d (ln V )
1

d (ln p )

22
Stellar Atmospheres: Radiative Equilibrium
The adiabatic gradient
needed:
d (ln T )
d (ln p ) ad
T  pV / Nk
ln T  ln p  ln V  ln( Nk )
d (ln T )
d (ln V )
 1
d (ln p )
d (ln p )
d (ln T )
1  1
 1 
d (ln p )


 1
 ad 

 rad
 1

stable

Schwarzschild criterion
23
Stellar Atmospheres: Radiative Equilibrium
The adiabatic gradient
• 1-atomic gas
cV  3 2 Nk
 5 3
cp  cV  Nk  5 2 Nk
ad  2 5  0.4
• with ionization   1 ad  0 convection starts   effect
• Most important example: Hydrogen (Unsöld p.228)
 ad 
2   x  x 2   5 2  EIon kT 
5   x  x 2   5 2  EIon kT 
2
2
f (T )
f (T )
 f (T ) 
with ionization degree x  
 
 
2N
N
 2N 

24
Stellar Atmospheres: Radiative Equilibrium
The adiabatic gradient
ad 
2   x  x 2   5 2  EIon kT 
5   x  x 2   5 2  EIon kT 
2
2
x
f (T )
f (T )
 f (T ) 
 


2N
2
N
N


25
Stellar Atmospheres: Radiative Equilibrium
Example: Grey approximation


   
2
d (ln T ) d  ln   3  
1


d
4d
4   2 
3
T 4 ( )  3 Teff4   2
4
3
4 ln T  ln 3 Teff4  ln   2
4
3
hydrostatic equation:
dp g

d 
Ansatz:   Ap b
( here a mass absorption coefficient)
dp g
1 b 1 g
g
1

integrate 
p  


d A
b 1
A
Ap b 1 (b  1)
d (ln p ) 1 dp 1 g
g
1




d
p d p Ap b Ap b 1 (b  1)
d ln T d
(b  1)
 rad 

d ln p d 4   2
3
 rad becomes large, if opacity strongly increases with depth (i.e. exponent b large).
 pb


The absolute value of  is not essential but the change of  with depth (gradient)
 rad large (> ad ): convection starts,  -Effekt
26
Stellar Atmospheres: Radiative Equilibrium
Hydrogen convection zone in the Sun
-effect and -effect act together
Going from the surface into the interior: At T~6000K ionization of
hydrogen begins
ad decreases and  increases, because a) more and more
electrons are available to form H and b) the excitation of H is
responsible for increased bound-free opacity
In the Sun: outer layers of atmosphere radiative
Video
inner layers of atmosphere convective
In F stars: large parts of atmosphere convective
In O,B stars: Hydrogen completely ionized, atmosphere radiative;
He I and He II ionization zones, but energy transport by
convection inefficient
27
Stellar Atmospheres: Radiative Equilibrium
Transport of energy by convection
Consistent hydrodynamical simulations very costly;
Ad hoc theory: mixing length theory (Vitense 1953)
Model: gas blobs rise and fall along distance l (mixing length).
After moving by distance l they dissolve and the surrounding
gas absorbs their energy.
l   H (r ) H = pressure scale height
 mixing length parameter
 =0.5
2
Gas blobs move without friction, only accelerated by buoyancy;
detailed presentation in Mihalas‘ textbook (p. 187-190)
28
Stellar Atmospheres: Radiative Equilibrium
Transport of energy by convection
Again, for details see Mihalas (p. 187-190)
For a given temperature structure
 compute Fconv ( r )
 flux conservation including convective flux
Frad (r ) 
 4
Teff  Fconv (r )

iterate
 new temperature stratification T ( r )
with  ad     rad
29
Stellar Atmospheres: Radiative Equilibrium
Summary: Radiative Equilibrium
30
Stellar Atmospheres: Radiative Equilibrium
Radiative Equilibrium:


 d ( f v J v )

A      J v  S v dv  B   
dv  H   0
0

 0 d

Schwarzschildt Criterion:
d (ln Tad )   d (ln Trad ) unstable
 

d (ln p)   d (ln p)  stable
Temperature of a gray Atmosphere
3 4 
2
T  Teff   
4
3

4
31