Transcript pcaws4 5659

Scattering Polarization
in the Solar Atmosphere
R. Casini
High Altitude Observatory
National Center for Atmospheric Research
High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR)
The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Research
under sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer.
19 May 2005
Polarized Radiation
• Origin: symmetry-breaking processes of the Atom-Photon
interaction
(e.g., anisotropic illumination, deterministic magnetic and/or
electric fields, anisotropic collisions)
Zeeman effect
Circular
polarization
Linear
polarization
Polarized Radiation
• Description: 4 independent parameters
– i) coherency matrix (a.k.a. polarization tensor )
 E x* E x
 *
E E
 y x
E x* E y 

*
E y E y 
Jones calculus
1 I Q

2 U  iV
– ii) Stokes parameters
I  intensity


Q,U  linear polarizati on  Mueller calculus
V  circular polarizati on 
U  iV 

I Q 
Polarized Radiation
• Operational definition of Stokes parameters
For a l/4 retarder

S ( ,  ) 
1
I  (Q cos 2  U sin 2 ) cos 2(    )  V sin 2(    )
2
Polarized Radiation
• Polarized radiation tensors
3
J ( )   
K
Q
i 0
dk K
TQ (i, k ) Si ( , k )
4
( K  0,1,2 ; Q   K ,  , K )
( S 0 , S1 , S 2 , S3 )  ( I , Q, U ,V )  Stokes vector
 K
TQ (i, k )  geometric tensors
Irreducible spherical tensors
transformation
J QK ( ) S    DQKQ ( RSS  ) J QK ( ) S
Q
conjugation
J QK ( )*  (1)Q J KQ ( )
Polarized Radiation
Example: Unpolarized radiation from the quiet-sun photosphere
Only two non-vanishing components:
1
 0
1
 J 0 ( )   d I ( ,  )
2 1


1
1
2
2
 J ( ) 
d

(
1


) I ( ,  )
0


2 2 1
(   cos  )
J 02 ( )
w( )  2 0
 anisotropy factor
J 0 ( )

1
 w( )  1
2
Atomic Polarization
Gas of atoms subject to:
•
Anisotropic and/or polarized illumination
•
External fields
•
Collisions
1. Atomic system not in a “pure state”
2. Population imbalances and quantum interferences
between atomic levels
Atomic Polarization
• Density operator
 A   p (i )  (i )  (i )
i
 p (i )  statistical weights

  (i )  C (i )   possible states of the atom




Atomic eigenstates
or some other complete basis
• Density matrix
     A     p (i ) C(i )*C(i )
i
Atomic Polarization
• Irreducible spherical components of the density matrix
If
  JM
(e.g., Zeeman effect)
J
M
QK ( J , J )   (1) J  M 2 K  1 
MM 
transformation
J
K
  JM , J M 
 M  Q 
( K  J  J  ,, J  J ; Q   K ,, K )
QK ( J , J ) S    DQKQ ( RSS  )* QK ( J , J ) S
Q
conjugation
Otherwise
   CJM JM
JM
QK ( J , J )*  (1) J  J 'Q  KQ ( J , J )
(e.g., Paschen-Back effect, Stark effect)
Atomic Polarization
Example: Multi-level atom  QK ( J , J )   JJ  QK ( J )
1
 JM , JM

2J 1 M
1.
Population:
2.
Orientation: 01 ( J ) 
3
 M JM , JM
J ( J  1)( 2 J  1) M
3.
Alignment:
5
[3M 2  J ( J  1)]  JM , JM

J ( J  1)( 2 J  1)( 2 J  1)( 2 J  3) M
 00 ( J ) 
02 ( J ) 
QK ( J ), Q  0  (M , M ) coherences
Atomic Polarization
Ex. 1: Positive orientation in a level
Ex. 2: Positive alignment in a level
J1
M 1
2
M 1
2
2
M  1
J 1
M 0
M  1
M  1
Ex. 3: Orientation and alignment in a level
J 1
M 0
M  1
 Presence of net polarization in the re-emitted radiation
(even in the absence of external fields)
Time evolution of the system
•
Liouville’s equation
Atom
i
d
ρ(t )  [ H (t ), ρ(t )]
dt
t

•
1
ρ(t )  ρ(t0 )   dt H (t ),  (t ) ,
i t 0
Radiation
 
 (t0 )   A (t0 )   R (t0 )
Evolution equation for expectation values
1
d
O(t )  Tr {O (t ) ρ(t )}  Tr {O(t ) [ H (t ), ρ(t )]}
dt
i
t
1
ρ(t )  ρ(t0 )   dt H (t ),  (t )
i t0
Perturbative expansion
OR (t )  ak (t )ak (t )
OA (t )  c (t )c (t )
Resonance Scattering
• Atom-Photon interaction to 2nd order of perturbation
1st order 
2nd order 
n  virtual state
Resonance Scattering
• Restriction: Non-coherent scattering
Scattering as the succession of
1st-order absorption and re-emission
• Complete Re-Distribution in frequency
The atom loses memory of the incident photons,
and the re-emitted photons are statistically
re-distributed in frequency
Flat-Spectrum Approximation
Resonance Scattering
•
Restriction: Non-coherent scattering
Scattering as the succession of
1st-order absorption and re-emission
•
Two-step solution
i.
Determine the excitation state of the atomic system
consistently with the ambient radiation field (Statistical
Equilibrium Problem)
ii.
Calculate the scattered radiation consistently with the
excitation state of the atomic system (Radiative Transfer
Problem)
Statistical Equilibrium
functions of the
incident radiation
 RA,E,S  Relaxation rates

TA,E,S  Transfer rates
d
   i       RA,E,S ( ,  )   
dt
  
  TA ( , l l)  l l   TE,S ( , u u )  u u
 l l
 u u
Radiative Transfer
in stationary regime
1d
1 
d

I kk'    c    I kk'  I kk'
c dt
c  t
ds

  I kk' K( a ) (k,k' ;h,h' )  K( s ) (k,k' ;h,h' )  J (k,k' )


hh'
Absorbtion matrix
Function of  l l
Stimulation matrix
Function of  u u
Emission vector
Function of  u u
Resonance Scattering
Self-consistency loop
(L-iteration)
non-LTE of the 2nd kind
QK ( J , J )
Si ( , k )
Resonance Scattering
Difficulties
1.
The Statistical Equilibrium problem grows rapidly with the
complexity of the atomic system (very large sparse matrices)
Possible strategy: weak-anisotropy approximation
2.
The Radiative Transfer problem requires the solution of a
set of 4 coupled ODEs
Possible strategy: Diagonal Elements Lambda Operator (DELO)
3.
No guarantee of convergence of the self-consistency loop
(maybe with the exception of the simplest atomic models,
with appropriate initialization)
Possible strategy: ?????
Atom 0-1
 J 1
J 0
d K
(1) K Q
K
K
Q (1)  i B QQ (1)   A10 Q (1) 
B01J KQ (10 )  00 (0)



dt
3


RE
TA
K Q K
(

1
)
J Q (10 )
B
1
d K
K
01
 Q (1)  0   Q (1) 

3 A10
dt
1  iQ B
A10
Classical analogy in the 3D harmonic oscillator with forcing term
B 
B
A10
 Hanle - effect parameter
 B ~ 1  max. depolariz ation rate
 B  1  complete coherence relaxatio n
d  10
 ( M  1)
 ( M  1)
2
 ( M  0)
2
A10
B
B
2
Atom 0-1
 J 1
J 0
 QK ( J )
 (J )  0
0 ( J )
K
Q
 02 (1) 
1
2 2
(3 cos 2B  1) w(10 )
d  10
Atom 1-0
J 0
 J 1
d K
1
 Q (1)  i B Q  QK (1) 
 K 0 Q 0 A01  00 (0)  RA
dt
3

TE

1
1  (1) K  K  K r
K K r 2
RA  B10 
K

1
d  10
3(2 K  1)( 2 K   1)( 2 K r  1)
K  Kr 
Q  K
  (1)  Q
1
1  QQr

• Hanle effect of the lower level ~
• Non-linear dependence on

w( 01 )
B
B10 J 00 ( 01 )
K  K r  Kr
 J Qr ( 01 )  QK (1)
 Q Qr 
Atom 1-0
J 0
 J 1
d  10
Atom 1-1
 Ju
1
 Jl
1
d  18
Atom 1-1
 Ju
1
 Jl
1
d  18
Atomic polarization and Radiative transfer
  0,   2
0-1 or 1-0 w/o atomic pol.
(Zeeman effect)
1-0 with atomic pol.
B  10 G, B  45,  B  0,
0-1 with atomic pol.
Homogeneous slab
Atomic polarization and Radiative transfer
  0
2 3P2,1,0
1
2
Homogeneous slab
He I
B  10 G, B  30,  B  0,   1
  90
  90
d  405
0
2 3S1
  0
Atomic polarization and Radiative transfer
  0
2 3P2,1,0
0
1
2
He I
Homogeneous slab
  90
2 3S1
  90
  0
Trujillo Bueno et al., Nature 415, 403 (2002)
Atomic polarization in Na I
F
3
2
1
0
2
1
D2
2
1
 3 2P3 2
 3 2P1 2
D1



d  640
3 2S1 2
Atomic polarization in Na I
D2
F
 3 2P3 2
3
2
1
0
 3 2P1 2
2
D2
2
1
5896 Å
1
D1



d  640
D1
3 2S1 2
Atomic polarization in Na I
F
3
2
1
0
2
1
2
1
 3 2P3 2
 3 2P1 2



d  640
3 2S1 2
Alignment-to-Orientation transfer
F
3
2
1
0
2
1
 3 2P3 2
 3 2P1 2
When quantum interferences between
FS and/or HFS levels are important
i   i JF , J 'F '  i B 
 N ( KQ, JFJ F ; K QJ F J F )
K Q J F J F 
2
1



3 2S1 2
diagonal
depolarization
KK coupling
alignment-to-orientation
d  640
Atomic orientation in H I
HAO Advanced Stokes Polarimeter
March 2003
THEMIS heliographic telescope
September 2003
Spectro-polarimetric observations of
H in solar prominences (off the limb)
Atomic orientation in H I
Spectro-polarimetric simulations
with FS and HFS
THEMIS heliographic telescope
September 2003
Maximum net circular polarization
1 order of magnitude too small
for typical prominence fields
(less than ~ 100 G)
d  2080
Atomic orientation in H I
0 Vcm 1
Catalytic effect of small
electric fields on H I
atomic orientation

Enhanced net circular
polarization in H
d  1416
1 Vcm 1 || B
vertical magnetic field, forward scattering
Catalytic effect of small
electric fields on H I
atomic orientation
Inclinations of
random-azimuth,
1 V cm-1 fields
H

present also for
isotropic electric fields
Isotropic
Prominence
d  1416 w/o HFS
d  5664 with HFS
E field
B fields
Only
B
Conclusions
•
Spectro-polarimetric observations reveal the complexity of the atomic
processes underlying resonance scattering (atomic coherences, FS and
HFS effects, magnetic and electric fields, alignment-to-orientation transfer)

The local problem can already become numerically very intensive
•
Points to focus on:
a.
Improve speed in the construction of the Statistical Equilibrium matrix
b.
Invent new strategies to accelerate convergence of the iterative
scheme for atoms of arbitrary complexity and general illumination
conditions