Stellar Atmospheres: Hydrostatic Equilibrium
Hydrostatic Equilibrium
Particle conservation
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Stellar Atmospheres: Hydrostatic Equilibrium
Ideal gas
dr
dA
P+dP
P
r
P
k
 T
AmH
P  pressure
  mass density
A  atomic weight
forces acting on volume element:
dV  dAdr
dm   dV
GM r dm
GM r 


dAdr
2
2
r
r
buoyancy:
dFg  
dFP  dPdA
(pressure difference * area)
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Stellar Atmospheres: Hydrostatic Equilibrium
Ideal gas
In stellar atmospheres:
M r  M  mass of atmosphere negligible
r  R
thickness of atmosphere  stellar radius
 dFg  
with g :
GM  
dAdr  g  dAdr
2
R
GM 
surface gravity
2
R
-2
usually written as log( g / cm s )
log g is besides Teff the 2nd
fundamental parameter of
static stellar atmospheres
Type
log g
Main sequence star
Sun
Supergiants
White dwarfs
Neutron stars
Earth
4.0 .... 4.5
4.44
0 .... 1
~8
~15
3.0
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Stellar Atmospheres: Hydrostatic Equilibrium
Hydrostatic equilibrium, ideal gas
buoyancy = gravitational force:
dFP  dFg  0
dPdA  g  dAdr  0
dP
  g  (r )
dr
A(r )mH
dP
eliminate  (r ) with ideal gas equation:
 g
P( r )
dr
kT (r )
example:
T (r )  T  const , A(r )  A  const (i.e., no ionization or dissociation)
AmH
AmH
dP
1 dP
 g
P(r ) 
 g
dr
kT
P dr
kT
solution:
P(r)  P(r0 )e  ( r  r0 ) gAmH / kT
P(r)  P(r0 )e  ( r  r0 ) / H
H :
kT
pressure scale height
gAmH
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Stellar Atmospheres: Hydrostatic Equilibrium
Atmospheric pressure scale heights
Earth:
Sun:
A  28 (N 2 )

T  300 K  H  9 km
log g  3 
H
kT
gAmH
A  1 (H) 

T  6000 K  H  180 km
log g  4.44
White dwarf:
A  0.5 (H   ne )

T  15000 K  H  0.25 km

log g  8

Neutron star:
A  0.5 (H   ne )

T  106 K
 H  1.6 mm !
log g  15 
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Stellar Atmospheres: Hydrostatic Equilibrium
Effect of radiation pressure
2nd moment of intensity
4
PR (v) 
Kv
c
1st moment of transfer equation (plane-parallel case)


dK v
 Hv
d (v)
dPR
4
H v with d (v)   (v)dr

c
d (v)
dPR 4
 (v ) H v

c
dr
integration over frequencies:

dPR 4
 (v) H v dv


c 0
dr
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Stellar Atmospheres: Hydrostatic Equilibrium
Effect of radiation pressure
Extended hydrostatic equation

dP
dP
4
 g  (r )  R  g  (r ) 
 (v) H v dv

dr
dr
c 0
 g eff (r )  (r )
definition: effective gravity

4 1
g eff (r ) : g 
 (v) H v dv  g  g rad

c  (r ) 0
(depth dependent!)
In the outer layers of many stars:

geff  0 i.e. g rad
4 1

 (v) H v dv  g

c  (r ) 0
Atmosphere is no longer static, hydrodynamical equation
Expanding stellar atmospheres, radiation-driven winds
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Stellar Atmospheres: Hydrostatic Equilibrium
The Eddington limit
Estimate radiative acceleration
Consider only (Thomson) electron scattering as opacity
 (v)   e (Thomson cross-section)
q  number of free electrons per atomic mass unit
Pure hydrogen atmosphere, completely ionized
q 1
Pure helium atmosphere, completely ionized
q  2 / 4  0.5


4

1
4

q
4 q e
e
g rad


n
H
dv


H
dv

H
e e
v
e
v


c ne mH / q 0
c mH 0
c mH

Flux conservation: H  Teff4
4
e
g rad
4 q e  4
M 1 q e 1 4 R 2Teff4
e 

Teff G 2 
c m H 4
R
c m H 4 G
g
M
q e
L/L
L
e 
 104.51 q
4 cm H G M
M /M

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Stellar Atmospheres: Hydrostatic Equilibrium
The Eddington limit
Consequence: for given stellar mass there exists a maximum
luminosity. No stable stars exist above this luminosity limit.
Lmax L  104.51 1 q  M M
e  1
Sun:
Main sequence stars (central H-burning)
3
L / L   M / M   M max  180M
Mass luminosity relation:
Gives a mass limit for main sequence stars
Eddington limit written with effective temperature
and gravity e  1015.12 qTeff4 / g  1
 15.12  log q  4 log Teff  log g  0
Straight line in (log Teff,log g)-diagram
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Stellar Atmospheres: Hydrostatic Equilibrium
The Eddington limit
Positions of analyzed
central stars of planetary nebulae
and
theoretical stellar evolutionary tracks
(mass labeled in solar masses)
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Stellar Atmospheres: Hydrostatic Equilibrium
Computation of electron density
At a given temperature, the hydrostatic equation gives the
gas pressure at any depth, or the total particle density N:
Pgas  NkT
N  Natoms  Nions  ne  N N  ne
NN massive particle density
The Saha equation yields for given (ne,T) the ion- and atomic
densities NN.
The Boltzmann equation then yields for given (NN,T) the
population densities of all atomic levels: ni.
Now, how to get ne?
We have k different species with abundances k
Particle density of species k:
N k   k N N   k  N  ne  , and it is
K
N
k 1
k
 NN
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Stellar Atmospheres: Hydrostatic Equilibrium
Charge conservation
Stellar atmosphere is electrically neutral
Charge conservation electron density=ion density * charge
K
jk
ne   j  N jk , N jk  density of j-th ionization stage of species k
k 1 j 1
jk-1
Combine with Saha equation (LTE)
N jk

by the use of ionization fractions: f jk 
Nk
We write the charge conservation as
K
jk
K
jk
k 1
j 1
neΦlk (T)

l j
jk
jk-1
1   neΦlk (T)
m 1 l  m
ne   j  N k f jk (ne , T )    k ( N  ne ) j  f jk (ne , T )
k 1 j 1
K
jk
k 1
j 1
ne  ( N  ne )  k  j  f jk (ne , T )  F (ne )
Non-linear equation, iterative solution, i.e., determine zeros of
F (ne )  ne  0
use Newton-Raphson, converges after 2-4 iterations;
yields ne and fij, and with Boltzmann all level populations 12
Stellar Atmospheres: Hydrostatic Equilibrium
Summary: Hydrostatic Equilibrium
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Stellar Atmospheres: Hydrostatic Equilibrium
Summary: Hydrostatic Equilibrium
Hydrostatic equation including radiation pressure

dP
dP
4
 g  (r )  R  g  (r ) 
 (v) H v dv

dr
dr
c 0
Photon pressure: Eddington Limit
Hydrostatic equation  N
Combined charge equation + ionization fraction  ne
 Population numbers nijk (LTE) with Saha and Boltzmann
equations
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