Transcript Hydrostatisches Gleichgewicht

Stellar Atmospheres: Non-LTE Rate Equations
The non-LTE Rate Equations
Statistical equations
1
Stellar Atmospheres: Non-LTE Rate Equations
Population numbers
LTE: population numbers follow from Saha-Boltzmann
equations, i.e. purely local problem
ni*  ni* (T , ne )
Non-LTE: population numbers also depend on radiation field.
This, in turn, is depending on the population numbers in all
depths, i.e. non-local problem.
ni  ni (T , ne , J )
The Saha-Boltzmann equations are replaced by a detailed
consideration of atomic processes which are responsible for
the population and de-population of atomic energy levels:
Excitation and de-excitation
by radiation or collisions
Ionization and recombination
2
Stellar Atmospheres: Non-LTE Rate Equations
Statistical Equilibrium
Change of population number of a level with time:
= Sum of all population processes into this level
- Sum of all de-population processes out from this level
d
ni   n j Pji  ni  Pij
dt
j i
j i
d
ni
dt
  n j Pji
j i
 ni  Pij
j i
One such equation for each level
Rij
The transition rate Pij comprises radiative rates
and collision rates Cij
In stellar atmospheres we often have the stationary case:
d
ni  0 hence
dt
n P
j i
j
ji
 ni  Pij for all levels i
j i
These equations determine the population numbers.
3
Stellar Atmospheres: Non-LTE Rate Equations
Radiative rates: bound-bound transitions
Two alternative formulations:
a) Einstein coefficients
Bij B ji A ji
ij ()
b) Line absorption coefficients
advantage a): useful for analytical expressions with simplified
model atoms
advantage b): similar expressions in case of bound-free
transitions: good for efficient programming
Number of transitions ij induced by intensity I in frequency
interval d und solid angle d
ni Bijv Iv dvd / 4
(absorbed Energy / hv)
Integration over frequencies and
angles yields


ni Rij  ni Bij   v J v dv
Or alternatively
0

with  ij (v)  Bij (v)hv / 4 ni Rij  ni 4 
0
 ij (v)
hv
J v dv
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Stellar Atmospheres: Non-LTE Rate Equations
Radiative rates: bound-bound transitions
In analogy, number
of stimulated
emissions:


n j R ji  n j B ji   v J v dv  n j Bij
0
gi
 v J v dv
g j 0

g i  ij
n j R ji  n j 4
J v dv

g j 0 hv
Number of spontaneous emissions:
3
2hv
 v dv
2
c
0
n j R ji  n j  A ji v dv  n j B ji 
0

g i  ij 2hv 3
n j R ji  n j 4
dv
g j 0 hv c 2
Total downwards rate:


gi  ij  2hv 3
n j R ji  n j  Rji  Rji   n j 4

J

 dv
v
2

g j 0 hv  c

n
n j R ji  n j  i
n
 j
*

 ni
R

n
 ji
j 
n

 j
*


 ij  2hv3
  hv kT
dv
  4  
 2  Jv  e
hv  c

0

5
Stellar Atmospheres: Non-LTE Rate Equations
Radiative rates: bound-free transitions
Also possible: ionization into excited states of parent ion
Example C III:
Ground state
2s2 1S
Photoionisation produces C IV in ground state
2s 2S
C III in first excited state
2s2p 3Po
Two possibilities:
Ionization of 2p electron → C IV in ground state
2s 2S
Ionization of 2s electron → C IV in first excited state
2p 2P
C III two excited electrons, e.g. 2p2 3P
Photoionization only into excited C IV ion
2p 2P
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Stellar Atmospheres: Non-LTE Rate Equations
Radiative rates: bound-free transitions
Number of photoionizations = absorbed energy in d, divided by
photon energy, integrated over frequencies and solid angle


  ni pv I v ddv  ni Rij  ni 4 
0
0
 ij (v)
hv
J v dv

Number of spontaneous recombinations:


2hv3
h
n
n
(v)
F
(v)
d

v
d
v

n
R

n
4

n
(v)
G
(v
)
dv
j ji
j
0  j e
0 e c 2
m
2hv
m  hv kT  ni
n j R ji  n j 4  ne (v) 2 pv e

c
h
0
 nj

 ni
n j R ji  n j 
n
 j
3

*
 1 h
dv
 n (v) m
 e
*


 ij (v) 2hv3  hv kT
4 
e
dv
2

hv
c
0

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Stellar Atmospheres: Non-LTE Rate Equations
Radiative rates: bound-free transitions
Number
of induced recombinations

  n j ne (v)G(v) I v dvdv  n j R ji  n j 4  ne (v)G(v) J v
0
0
h
dv
m
*
m  hv kT  ni  1
h
n j R ji  n j 4  ne ( v) pv e
J
dv
v


h
0
 n j  ne ( v) m

*

 ni 
 ij (v)


n j R ji  n j   4 
J v e  hv kT dv
hv
0
 nj 
Total recombination rate
*
*

 ni 
 ni 
 ij (v)  2hv 3
  hv kT




 2  J v e
n j   R ji  n j   4 
dv
n
n
hv
c


0
 j
 j
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Stellar Atmospheres: Non-LTE Rate Equations
Radiative rates
Upward rates:
ni Rij with

Rij  4 
0
Downward rates:
Remark: in TE we have
 ni
nj
n
 j
 ij (v)
hv
J v dv
*

 R ji with



 ij (v)  2hv3
  hv kT
 2  J v e
R ji  4 
dv
hv  c

0
ni  ni 
J v  Bv  R  R 
  
nj  nj 
*
ij
*
*
ji
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Stellar Atmospheres: Non-LTE Rate Equations
Collisional rates
Stellar atmosphere: Plasma, with atoms, ions, electrons
Particle collisions induce excitation and ionization
Cool stars: matter mostly neutral  frequent collisions with
neutral hydrogen atoms
Hot stars: matter mostly ionized  collisions with ions
become important; but much more important become
electron collisions
1/ 2
v electron  ion mass 


v ion
 electronmass 
1/ 2
 mH A 

 
 me 
 43 A
Therefore, in the following, we only consider collisions of
atoms and ions with electrons.
10
Stellar Atmospheres: Non-LTE Rate Equations
Electron collisional rates
Transition ij (j: bound or free), ij (v) = electron collision
cross-section, v = electron speed
Total number of transitions ij:

ni Cij  ni ne   ij ( v) f ( v) vdv  ni ne ij (T )
v0
minimum velocity necessary for excitation (threshold)
f ( v)dv velocity distribution (Maxwell)
In TE we have therefore
v0
n *iCij  n *jC ji
Total number of transitions ji:
*
 ni 
n jC ji  n j   Cij
 nj 
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Stellar Atmospheres: Non-LTE Rate Equations
Autoionization and dielectronic recombination
negative
0
ion I, e.g. d
He I
ion II, e.g.
He II
c
ionization energy
positive
Energy
b
b bound state, d doubly excited state, autoionization level
c ground state of next Ion
d  c: Autoionization. d decays into ground state of next
ionization stage plus free electron
c  d  b: Dielectronic recombination. Recombination via a
doubly excited state of next lower ionization stage. d autoionizes again with high probability: Aauto=1013...1014/sec!
But sometimes a stabilizing transition d  b occurs, by
which the excited level decays radiatively.
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Stellar Atmospheres: Non-LTE Rate Equations
Complete rate equations
For each atomic level i of each ion, of each chemical element
we have:
ni  Pij   n j Pji  0
j i
In detail:
j i
ni    Rij  Cij 
 j i

 nj 
     Rij  C ji  

j i  ni 
*
excitation and ionization
rates out of i
de-excitation and recombination
*
 ni 
 n j    R ji  Cij 
j i
 nj 
 n j  R ji  C ji 
j i
0
de-excitation and recombination
rates into i
excitation and ionization
20
Stellar Atmospheres: Non-LTE Rate Equations
Closure equation
One equation for each chemical element is redundant, e.g.,
the equation for the highest level of the highest ionization
stage; to see this, add up all equations except for the final
one: these rate equations only yield population ratios.
We therefore need a closure equation for each chemical
species:
Abundance definition equation of species k, written for
example as number abundance yk relative to hydrogen:
yk
populationnumbersof species k


 populationnumbersof hydrogen
21
Stellar Atmospheres: Non-LTE Rate Equations
Abundance definition equation
Notation:
Population number of level i in ionization stage l : nl,i
LTE levels
do not appear explicitly in
the rate equations;
populations depend on
ground level of next
ionization stage:
nl*,i  ne nl 1,1 l ,i (T )
NLTE levels
E0
22
Stellar Atmospheres: Non-LTE Rate Equations
Abundance definition equation
Notation:
NION number of ionization stages of chemical element k
NL(l) number of NLTE levels of ion l
LTE(l) number of LTE levels of ion l
LTE ( l )
 NL (l )
 NL ( H ) LTE ( H ) *

* 

  nl ,i   nl ,i   yk   ni   ni  n protons  
l 1  i 1
i 1
i 1

 i 1

LTE ( l )
LTE ( H )
NION NL ( l )
 NL ( H )






n

n
n

(
T
)

y
n

n
1

n

(
T
)

k   l ,i
protons 
e 
i
  l ,i l 1,1 e  li 
l 1  i 1
i 1
i 1



 i 1
NION
Also, one of the abundance definition equations is redundant,
since abundances are given relative to hydrogen (other
definitions don‘t help)  charge conservation
23
Stellar Atmospheres: Non-LTE Rate Equations
Charge conservation equation
Notation:
Population number of level i, ion l, element k: nkli
NELEM number of chemical elements
q(l)
charge of ion l
NELEM NION
 
k 1
l 1
NELEM NION
 
k 1
l 1
LTE ( l , k )
 NL (l , k )
* 
q(l )   nkli   nkli
  ne 
i 1
 i 1

LTE ( l )
 NL (l , k )

q(l )   nkli  nk ,l 1,1ne   kli (T )  ne
i 1
 i 1

24
Stellar Atmospheres: Non-LTE Rate Equations
Complete rate equations: Matrix notation
Vector of population numbers
n   n1 , n2 ,
, nNLALL  NLALL= total number of NLTE levels
An  b rate equation in matrix notation
One such system of equations per depth point
Example: 3 chemical elements
Element 1: NLTE-levels: ion1: 6, ion2: 4, ion3: 1
Element 2: NLTE-levels: ion1: 3, ion2: 5, ion3: 1
Element 3: NLTE-levels: ion1: 5, ion2: 1, hydrogen
Number of levels: NLALL=26, i.e. 26 x 26 matrix
25
Stellar Atmospheres: Non-LTE Rate Equations
x
x
n1
n2
x
ion 1
Ionization into excited states
x
x
x
x
x
x
x
z
x
1
x
z
ion 2
abundance def.
1
1 1 1 1
1
1
1
 y1
x
ion 1
x
 y1
z
 y2
x
z
x
x
0
x
z
x
x
LTE contributions
x
x
x
x
x
x abundances
x
z
x
1
x
z  y2
x
ion 2
abundance def.
1
1
1
1
1 1
H
charge conservtn. q11
q11
z q12
q12
z q21
q21
z q22
q22
z
x
0
0 0 0
n25
np
0
0
0
0
0
0
0
0
0
0
0
0
0
=
0
0
0
0
0
0
0
0
0
0
0
0
ne
26
Stellar Atmospheres: Non-LTE Rate Equations
Elements of rate matrix
For each ion l with NL(l) NLTE levels one obtains a submatrix with the following elements:
  R ji  C ji 
j  i lower left

  n *
Aij    i   R ji  Cij 
j  i upper right


  nj 

*
k
  nm 
     Rim  Cmi     Rim  Cim  j  i diagonal
m 1
 m<i  ni 
i  1 NL(l ) j  1 NL(l ) k k highest level in parent ion,
into which ion l can ionize; does not have to be  NL(l )  1 !
27
Stellar Atmospheres: Non-LTE Rate Equations
Elements corresponding to abundance definition eq.
Are located in final row of the respective element:
i
NION
 NL(l )
l 1
NION

j  1  NL(l ) except of ground state of excited ions
1
l 1

Aij  1  ne   l 1,m
j  ground state of excited ions

m  LTE ( l 1)

j   NLALL - NL( H )  NLALL  1
  yk


  y 1  n


e
m  j  NLALL
 k
m  LTE ( H )



28
Stellar Atmospheres: Non-LTE Rate Equations
Elements corresponding to charge conservation eq.
Are located in the very final row of rate matrix, i.e., in
i  NLALL
q (l )


Aij  
q(l )  q(l  1)   l 1,m

m  LTE ( 1)
j  1 NLALL, except of
ground state of excited ions
else
Note: the inhomogeneity vector b (right-hand side of statistical equations)
contains zeros except for the very last element (i=NLALL):
electron density ne (from charge conservation equation)
29
Stellar Atmospheres: Non-LTE Rate Equations
Solution by linearization
The equation system An  b is a linear system for n and
can be solved if, ne , T , J v are known. But: these quantities are
in general unknown. Usually, only approximate solutions within
an iterative process are known.
Let all these variables change by  ne ,  T ,  J v e.g. in order to
fulfill energy conservation or hydrostatic equilibrium.
Response of populations  n on such changes:
Let An  b with actual quantities


And A   A  n   n    b   b  with new quantities ne , T , J v
Neglecting 2nd order terms, we have:
An  b   n  n A   b
30
Stellar Atmospheres: Non-LTE Rate Equations
Linearization of rate equations
Needed: expressions for:  A,  b
J discretized in NF frequency points
One possibility:
 A
A
T
T 
A
ne
NF
A
k 1
J k
 ne  
 Jk
If in addition to n the variables ne , T , J k are introduced as
unknowns, then we have the
 Method of Complete Linearization
Other possibility: eliminates J k from the equation system by
expressing J k through the other variables ne , T :
J k  f  n, T , ne 
As an approximation one uses
(and iterates for exact solution)
J kd ~ Skd (n, T , ne )
31
Stellar Atmospheres: Non-LTE Rate Equations
Linearization of rate equations
 A
A
T
T 
A
ne
NF
A
k 1
S k
 ne  
 Sk
NLALL
S k
S k
S k
 Sk 
T 
 ne  
 nj
T
ne
j 1 n j
Method of approximate -operators (Accelerated Lambda Iteration)
analogous,  b :
NF
b
b
b
b 
T 
 ne  
 Sk   0,
T
ne
k 1 S k
, 0,  ne 
32
Stellar Atmospheres: Non-LTE Rate Equations
Linearization of rate equations
NF
 A Sk 
 A
An  b   n A  T  
n  n


T

S

T
k 1
k


NF
 A Sk 
 A
 ne  
n  n


n

S

n
k 1
e
k
e 


NLALL
 n
j 1
j
 NF  A Sk 
  n

 k 1 Sk n j 
Linearized equation for response  n as answer on changes
 ne ,  T ,  J v
NF
n
 A Sk
Sk n j
Expressions k 1
show the complex coupling of all
variables. A change in the radiation field and, hence, the source
function at any frequency causes a change of populations of all
levels, even if a particular level cannot absorb or emit a photon
at that very frequency!
33
Stellar Atmospheres: Non-LTE Rate Equations
Linearization of rate equations
In order to solve the linearized rate equations we need to
compute these derivatives:
   
,
,
,
ne T n j Sk
with respect to A,b,Sk
All derivatives can be computed analytically!
Increases accuracy and stability of numerical solution. More
details later.
34
Stellar Atmospheres: Non-LTE Rate Equations
LTE or NLTE?
When do departures from LTE become important?
LTE is a good approximation, if:
1) Collisional rates dominate for all transitions
Rij  Cij  Pij  Rij  Cij  Cij
*
n 
because
= i 
C ji  n j 
solution of rate equations  LTE
Cij
2) Jv =Bv is a good approximation at all frequencies
Rij  R ji
*
n n 
 i  i 
nj  nj 
solution of rate equations  LTE
35
Stellar Atmospheres: Non-LTE Rate Equations
LTE or NLTE?
When do departures from LTE become important?
LTE is a bad approximation, if:
1) Collisional rates are small Cij ~ ne / T
ne , T   Cij 
Rij ~ T  ,  >1
T   Rij 
2) Radiative rates are large
3) Mean free path of photons is larger than that of electrons
Example: pure hydrogen plasma
z ~ 1/ nH (density of neutral H)
Saha: nH ~ ne n pT
ne , T  
3/ 2 E / kT
e
T 3/ 2 E / kT
 z ~
e
ne n p
z 
Departures from LTE occur, if temperatures are high and
densities are low
36
Stellar Atmospheres: Non-LTE Rate Equations
LTE or NLTE?
37
Stellar Atmospheres: Non-LTE Rate Equations
DO with log g= 7.5
DAO with log g= 6.5
LTE or NLTE?
40
Stellar Atmospheres: Non-LTE Rate Equations
Summary: non-LTE Rate Equations
41
Stellar Atmospheres: Non-LTE Rate Equations
Complete rate equations
For each atomic level i of each ion, of each chemical element
we have:
ni  Pij   n j Pji  0
j i
In detail:
j i
ni    Rij  Cij 
 j i

 nj 
     Rij  C ji  

j i  ni 
*
excitation and ionization
rates out of i
de-excitation and recombination
*
 ni 
 n j    R ji  Cij 
j i
 nj 
 n j  R ji  C ji 
j i
0
de-excitation and recombination
rates into i
excitation and ionization
42