Hydrostatisches Gleichgewicht
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Transcript Hydrostatisches Gleichgewicht
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
dFg
dAdr
2
2
r
r
buoyancy:
(pressure difference * area)
dFP dPdA
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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
GM
dFg
dAdr g dAdr
2
R
with g :
GM
surface gravity
2
R
-2
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
usually written as log( g / cm s )
log g is besides Teff the 2nd
fundamental parameter of
static stellar atmospheres
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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
geff (r ) (r )
definition: effective gravity
4 1
geff (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
104.51 q
4 cmHG 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 104.51 1 q M M
e 1
Sun:
Main sequence stars (central H-burning)
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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 NN 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:
Nk 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
n Φ
e
l j
jk jk-1
lk
(T)
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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