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J. Sánchez Almeida
Instituto de Astrofísica de Canarias
Magnetometry: set of techniques and procedures to determine the physical properties of a magnetized plasma (magnetic field and more ...)
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Main Constraints:
– No in-situ measurements are possible; inferences have to be based on interpreting properties of the light.
– Interpretation not straightforward. The resolution elements of the observations are far larger than the magnetic structures
(or sub-structure)
Needed Tools:
– Radiative transfer for polarized light
– Instrumentation: telescopes and polarimeters
– Inversion techniques (interpreting the polarization through many
simplifying assumptions)
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Purpose: –To give an overview of all ingredients that must be considered, and to
illustrate the techniques with examples taken form recent research.
–It is not a review since part of the techniques used at present are not
covered (not even mentioned). Explicitly
–Devoted to the magnetometry of the photosphere.
– No proxi-magnetometry (jargon for magnetic field
measurements which are no based on polarization)–No extrapolations of photospheric magnetic fields to
the Corona)– No in-situ measurements (solar wind)
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Summary – Index (1):
– Stokes parameters, Jones parameters, Mueller matrixes and Jones matrixes– Equation of radiative transfer for polarized light– Zeeman effect– Selected properties of the Stokes profiles, ME solutions, etc.
– Polarimeters, including magnetographs– Instrumental Polarization
Instrumentation:
Radiative Transfer for Polarized Radiation.
Inversion Techniques:– General ingredients– Examples, including the magnetograph equation
Examples of Solar Magnetometry:– Kitt Peak Synoptic maps– Line ratio method– Broad Band Circular Polarization of Sunspots – Quiet Sun Magnetic fields
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Summary – Index (2):
– Hanle effect based magnetometry– Magnetometry based on lines with hyperfine structure– He 1083nm chromospheric magnetometry– Polarimeters on board Hinode
Advanced Solar magnetometry.
goto end
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Stokes parameters, Jones parameters, Mueller Matrixes and Jones Matrixes
– The light emitted by a point source is a plane wave
– Monochromatic implies that the EM fields describe
elliptical motions in a plane
– The plane is quasi-perpendicular to the direction of
propagation
– Quasi monochromatic implies that the ellipse
changes shape with time
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x
y
0
)](cos[)(
)](cos[)(
)(Re
ReRe),( )(
twttA
twttA
tEe
EeeEetre
yy
xxiwt
jj
twwiiwt
jj
tiw jj
Quasi-monochromatic means that the ellipse change with time
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time (t)
ex(t)
Frequency (1/t)
w/2
1/
2/w = 10-15 s, in the visible (5000 A)
: coherency time, for which the ellipse keeps a shape
= 10-8 s, electric dipole transition in the visible
= 5 x 10-10 s, (multimode) He-Ne Laser
= 5 x 10-10 s, high resolution spectra (
Integration time of the measuremengts: 1 s (<< << 2/w),
ellipse changes shape some 108 -109 times during the measurement
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)(Re),( tEetre iwt
)(
)(
tE
tE
J
JJ
y
x
y
x
J Jones Vector, complex amplitude of the electric field in the
plane perpendicular to the Line-of-Sight (LOS). It completely describes the radiation field, including its polarization.
Consider the effect of an optical system on the light. It just transfoms
outin JJ
Most known optical systems are linear (from a polarizer sheet to a
magnetized atmosphere)
inout JmJ
: m Jones Matrix (Complex 2x2 matrix)
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T
yx
dttfT
tf
teteI
0
22
1)()(
)()( (T: integration time of the measurement)
The polarization of the light can be determined using intensity detectors (CCDs, photomultipliers, etc.) plus linear optical systems.
m
outJ
inJ
VMUMQMIMIout 14131211
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VMUMQMIMIout 14131211
*
*
Im
Re
yx
yx
yx
yx
JJV
JJU
JJQ
JJI
2
2
22
22
*Z Zis the complex conjugate of
Stokes Parameters, that completely characterize the properties of the light from an observational point of view
ijM describes the properties of the optical system
yyyx
xyxx
mm
mmm
**
**
Im
Re
/
/
yyyxxyxx
yyyxxyxx
yyxyyyxx
yyxyyyxx
mmmmM
mmmmM
mmmmM
mmmmM
14
13
2222
12
2222
11
2
2
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(Some) Properties of the Stokes Parameters
– Two beams with the same Stokes parameters cannot be
distinguished
– Which kind of polarization is coded in each Stokes parameter?
– The Stokes parameters of a beam the combines two
independent beams is the sum of the Stokes parameters of the
two beams
– Any polarization can be decomposed as the incoherent
superposition of two fully polarized beams with opposite
polarization states
– A global change of phase of the EM field does not modify the
Stokes parameters
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(Some) Properties of the Linear Optical Systems
– Only seven parameters characterize the change of polarization
produced by any optical system. A Jones matrix is characterized by 4
complex numbers (8 parameters) minus an irrelevant global phase. – The modification of the Stokes parameters produced by one of
these systems is linear
inout SMS
V
U
Q
I
S
44434241
34333231
24232221
14131211
MMMM
MMMM
MMMM
MMMM
M
Stokes vector
Mueller Matrix
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–The Mueller matrix contain redundant information. It has 16 elements, but only seven of them are independent. The relationships bewteen the elements are not trivial, though.
– The Mueller matrix becomes very simple if the optical element is weakly polarizing, i.e., if
2221
1211
10
01
aa
aam with 1ija then
IQUV
QIVU
UVIQ
VUQI
M
1000
0100
0010
0001
2211
2112
2112
2112
2112
2211
2211
aa
aa
aa
aa
aa
aa
aa
Q
U
V
V
U
Q
I
Im
Im
Re
Im
Re
Re
Re
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- Mueller Matrix for an optical system producing selective absorption
IQUV
QIVU
UVIQ
VUQI
M
1
0
0
0
V
Q
U
aV
aU
aQ
aI
V
U
Q
I
a
a
a
a
V
U
Q
IStokes Vector de type of
absorbed light
Change of amplitude produced by the selective OS
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linear polarizer transmitting the vibrations in the x-axis
0
0
1
1
a
a
a
a
V
U
Q
I
1000
0100
001
001
M
0
0
1
V
U
Q
I
Then for unpolarized input light one ends up with
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- Mueller Matrix for an optical system producing selective retardance
IQUV
QIVU
UVIQ
VUQI
M
1
aV
aU
aQ
V
U
Q
I
V
U
Q
0
0
0
0
a
a
a
a
V
U
Q
IStokes Vector de type of
polarization that is retarded
Change of phase produced by the selective OS
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- The Mueller matrix of a series of optical systems is the product of the individual matrixes. The order does matter
j
jMM
if the chain is formed by weakly polarizing optical systems, then the order of the different elements is irrelevant
i
i
i
i
i
i MMMM 11 )(
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Equation of Radiative Transfer for Polarized Light
zobserver
line-of-sight
layer of atmosphere
S+S S
zzS
SzM
zSS emi
i
1
V
U
Q
I
S
zzSem
Emission produced by the layer
zMz i Mueller matrix of i-th process changing the polarization
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zzS
SzM
zSS emi
i
1
z
SS
zM
zS em
i
i
aiiaiiaiiaii
aiiaiiaiiaii
aiiaiiaiiaii
aiiaiiaiiaii
IQUV
QIVU
UVIQ
VUQI
i
IQUV
QIVU
UVIQ
VUQI
zzM
zzAii change of amplitude
zzPii change of phase
taiaiaiai VUQI Stokes vector of the selective absorption + retardance
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emIQUV
QIVU
UVIQ
VUQI
V
U
Q
I
V
U
Q
I
dzd
?
?
?
?
iaiiV
iaiiQ
iaiiV
iaiiI
VzP
QzP
VzA
IzA
.
.
.
.
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0
0
0
B
IQUV
QIVU
UVIQ
VUQI
em
?
?
?
?
Emission term ? Simple assuming emitted radiation field is in LTE (Local Thermodynamic Equilibrium). In TE
0
dzd
0
0
0
B
V
U
Q
I
and with B the Planck function
then
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V
U
Q
BI
V
U
Q
I
dzd
IQUV
QIVU
UVIQ
VUQI
iaiiV
iaiiI
VzA
IzA
.
.
iaiiV
iaiiQ
VzP
QzP
.
.
Radiative transfer equation for polarized light in any atmosphere whose emission is produced in LTE
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linear polarizer transmitting the vibrations in the x-axis
0
0
1
1
a
a
a
a
V
U
Q
I
There is just one i which absorbs
V
U
Q
I
zA
V
U
Q
I
dzd
1000
0100
0011
0011
and no emission (B=0)
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V
U
Q
I
zA
V
U
Q
I
dzd
1000
0100
0011
0011
inout ))0 QIQIQIdzd (()(
0))2 0
2
inout
LdzzA
eQIQIQIzAQIdzd
(()()(
0UU2 0
2
inout
LdzzA
eUzAUdzd
inoutV
U
Q
I
V
U
Q
I
0000
0000
0011
0011
21 Typical Mueller matrix of a linear
polarizer
],[ Lz 0
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Zeeman EffectPurpose: work out the ´s and´s in the absorption matrix in the case of a magnetized atmosphere
– Electric dipole transitions– Hydrogen-like atoms– Linear Zeeman effect
Work out contributions to the change of polarization due to:
2) Continuum absorption
1) Spectral line absorption
Assumptions:
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The wave function characterizing eigenstate of theses Hydrogen-like atoms can be written down as
tEi
iM eertr ),(),,,( 0
where M is the magnetic quantum number and E is the energy of the level.
r
The electric dipole of the corresponding distribution of charges will be
dvrqdvolume
2
Spectral line absorption
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dvrqdvolume
2
When you have a transition between states b (initial) and f (final), the wave function is a linear combination of the two states
ffbb tctc )()(
tcc
tcc
ff
bb
when 1 and 00
when 0 and 10
)(
)(
dvrcqcddvolume
fbfb ** Re 20
dvreecqcdvolume
MMi
fb
tEE
i
fbfb
fb )(** Re 000 2
0d
constant over the period of the wave
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dvreecqcddvolume
MMifb
tEE
i
fbfb
fb )(** Re 000 2
0
1
20
1
21
0
0
ier
ier
rrr ii
sinsincos
cos
cossin
cossin
drdeide
drdeide
drdede
tEE
iMMi
tEE
iMMi
tEE
iMMi
fb
fb
fb
fb
fb
fb
0
1
0
1
1
0
0
2
0
1
2
0
1
2
0
)(
)(
)( 002
0
pdeip
Which leads to the selection rules for E-dipole transitions
each one associated with a polarization
1 ,0 M
r
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y
z
x
observer
We are interested in the projection in the plane perpendicular to the line of sight (x-y plane)
sin
cos
0
dd y
0
0
1
dd x
a) For M=0
0
0
0
1
1
0
0
iwt
x ed Re
)cos(sin
sin
cosRe wted iwty
1
1
0
0
x
y
)(td
0
0
1
1
2
sin
V
U
Q
I
There are only three types of polarization
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b) For M=Mb- Mf=+1
)cos(Re wtied iwtx
0
0
1
0
1
)sin(cos
sin
cosRe wtied iwty
0
0
1
cos
sin
cos
2
0
12
2
V
U
Q
I
b) For M=Mb- Mf=-1
1
x
y
)(td
cos
1
x
y
)(td
cos
cos
sin
cos
2
0
12
2
V
U
Q
I
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If the atom is in a magnetized atmosphere, the energy of each Zeeman sublevel is different, which produces a change of resonance frequency of the transitions between sublevels depending on M,
w
w0
B=0
w
w0
B=B0w
0Bw
Associated to each transition there is a absorption profile plus a retardance profile
w0w0w w0
w0w
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In short: for an electric dipole atomic transition, only three kinds of polarizations can be absorbed. They just depend on M (with M the difference of magnetic quantum numbers between the lower and the upper levels)
x
observer
B
y
x
y M=0
x
y
M=-1
Bw
w0w0w
cos
x
y
M=+1
w0w0w
absorption retardance
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Continuum Absorption
Although, no details will be given, it is not difficult to show that the continuum absorption has a characteristic polarization for selective absorption of the order of (Kemp 1970),
)/( kGBV
U
Q
I
a
a
a
a
510
0
0
1
– For the solar magnetic fields (1kG magnetic field strengths), the
continuum absorption is unpolarized unless you measure degrees
of polarization of the order of 10-5.– In white dwarfs, B ~ 106 G, leading to large continuum
polarization (~ 1%)
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Radiative Transfer Equation in a Magnetized Atmosphere
The equation is generated considering the four types of polarization that are possible
V
U
Q
BI
V
U
Q
I
dzd
IQUV
QIVU
UVIQ
VUQI
cos2
2sinsin
2cossin
cos1
22
cos2
2sinsin
2cossin
cos1
22
0
2sin
2cos
1
sin2
0
0
0
1
2
2
2
2
2
2
2 lllc
V
U
Q
I
kkk
same for ´s with replacing ´s with ´s
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x
observer
B
y
V
U
Q
BI
V
U
Q
I
dzd
IQUV
QIVU
UVIQ
VUQI
22
222
222
122
2
2
22
lV
lU
lQ
lcI
k
k
k
k
sinsin
cossin
)cos(sin
22
222
222
2
2
lV
lU
lQ
k
k
k
sinsin
cossin
Unno-Rachkovsky Equations
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Zeeman triplet
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general Zeeman pattern
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effect of a change of macroscopic velocity
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effect of a change of magnetic field strength
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weak magnetic field strength regime
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Selected Properties of the Stokes Profiles
Stokes Profiles representation of the four stokes parameters as a function of wavelength within a spectral line
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Stokes Profiles
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1.- Symmetry with respect to the central (laboratory) wavelength of the spectral line. If the macroscopic velocity is constant along the atmosphere, then
I() = I(- )Q() = Q(- )U() = U(- )V() = -V(- )
wavelength - laboratory wavelength of the spectral line corrected by the macroscopic velocity
No proof given, but it follows from the symmetry properties of the ´s and ´s of the absorption matrix
these symmetries disappear the velocity varies within the resolution elements (asymmetries of the Stokes profiles)
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Symmetries and asymmetries Stokes Profiles
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2.- Weak Magnetic Field Approximation,
the width of the absorption and retardance coefficients of the various Zeeman components are much smaller than their Zeeman splittings
if is the Zeeman splitting of a Zeeman triplet, and D is the width of the line, it can be shown that (e.g., Landi + Landi 1973)
)/(
)/(
)/(
)/(
DBVV
DBUU
DBQQ
DBIII
2
22
22
220
)/(
)/(
)/(
DBVV
DBUU
DBQQ
2
22
22
then to first order in ( / D )
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V
U
Q
BI
V
U
Q
I
dzd
IV
IV
VI
VI
00
00
00
00
00 UQdz
Ud
dz
Qdn
n
n
n
Since there is no polarization at the bottom of the atmosphere
))(()(
BVIdz
VIdVI
)cos(
)(cos)(
Blc
BlcVI
kkd
dkk
(a)
(b)
)cos( BlcVI kk
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I+V and I-V follow to equations that are identical to the equation for unpolarized light except that the absorption is shifted by cos B
)()cos()(
BVIkkdz
VIdBlc
If the longitudinal component of the magnetic field is constant then cos B is constant and I+V and I-V are identical except for a shift
I-VI+V
2 cos B
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BB
BB
BB
d
dI
d
dfVIVIV
fVIVII
d
dfffVI
d
dfffVI
cos)(
cos)(
)()(
)()()(
cos)(
)()cos(
cos)(
)()cos(
2
12
1
BddI
V cos
)()(
Magnetograph equation: the Stokes V signal is proportional to the longitudinal component of the magnetic field
V cos B
0
observer
B
cosB
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The previous argumentation is based on the assumption that the Zeeman pattern is a triplet (one component, one + component and one - component). If the pattern is more complex but the magnetic field is weak, one can repeat theargumentation to show that everything remains the same except that the full Zeeman pattern has to be replaced by a equivalent Zeeman triplet whose splitting is
BgeffB
effg Is the so-called effective Landé factor, and it equals one for the classical Zeeman effect
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4.- Stokes profiles of an spatially unresolved magnetic structure (2-component magnetic atmosphere).
resolution element
non-magnetic magnetic
VVVV
UUUU
QQQQ
III
magnonobs
magnonobs
magnonobs
magnonobs
)1(
)1(
)1(
)1(
filling factor, i.e., fraction of resolution element filled by magnetic fields
:area total
area red
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VV
IIIIII
obs
magnonmagnonobs
) (if )1(
Effect on the magnetograph equation cos
)()( B
d
dICV
cos with )(
)( BBBd
dIV effeff
obsobs
Bsd
observer
pixelpixel
eff dssdB /
B
Magnetic flux density
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4.- Milne-Eddington solution of the Radiative Transfer Equation for Polarized Light (RTEPL).
Assumptions: all those needed to get an analytic solution of the of the radiative transfer equations for polarized light
Importance: Used for measuring magnetic field properties
RTEPL: first order linear differential equation. Admits an analytic formal solution of the coefficients are constant (basic maths)
V
U
Q
BI
V
U
Q
I
dzd
IQUV
QIVU
UVIQ
VUQI
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V
U
Q
BI
V
U
Q
I
dz
d
IQUV
QIVU
UVIQ
VUQI
V
U
Q
I
S
0
0
0
1
1
IQUV
QIVU
UVIQ
VUQI
c
K
1
dzd c
)( 1SKS
Bd
d
continuum optical depth
Compact form of the RTEPL
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)( 1SKS
Bd
d
Assumptions:
the ratio line to continuum absorption coefficient does not depend on optical depth
The source function depends linearly on continuum optical depth
depth opticalith constant w / cl
10 BBB Broadening of the line constant (both Doppler and damping)
Magnetic field vector constant with depth
… all them together lead to constant absorption matrix
depth opticalith constant w K
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constant and both with , solutionstry 1 100 SSSSS
)( 1SKS
Bd
d
)()( 11001 1SK1SKS
BB
1S01SK
1111 )( BB
1SKS1SKS 1
010001 )( BB
1K1SS 10
10)0( BB
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Milne-Eddington solutions of the RTEPL (e.g., Landi Degl´Innocenti, 1992)
222222222
21
21
21
222210
)()(
/)()(
/)()(
/)()(
/)(
VVUUQQVUQVUQII
VVUUQQVUQQUIVI
VVUUQQUQVVQIUI
VVUUQQQVUUVIQI
VUQII
BV
BU
BQ
BBI
Free parameters: 1. Magnetic field strength2. Magnetic field azimuth3. Magnetic field inclination4. B0
5. B1
6. Macroscopic velocity7. Doppler broadening8. Damping 9. Strength of the spectral line
IDL
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5.- 180o azimuth ambiguity (exact)
x
observer
B
y
x
observer
o180
B
y
These two magnetic fields produce the same polarization, therefore, one cannot distinguish them from the polarization that they generate.
IDL
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6.- Stokes V reverses sign upon changing the sign of the magnetic field component along the line-of-sight (approximate).
cos)180cos( and cos since o V
x
observer
B
yx
o180
observer
B
y
)()180( o VV IDL
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7.- Q=U = 0 for longitudinal magnetic fields. V=0 for transverse magnetic fields. (Approximate.)
x
0
observer
B
yx
o90
observer
B
y
IDLQ=U=0
V=0
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Polarimeters
– Modulation package– Intensity detector– Calibration package– Instrumental polarization
Basic elements:
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VpMUpMQpMIpMpI jjjjjout~)(~)(~)(~)()( 14131211
optics modulator (pj)
calibration optics
telescope+ optics
optics
V
U
Q
I
V
U
Q
I
~
~
~
~
Intensity detector
out
out
out
out
V
U
Q
I
V
U
Q
I
V
U
Q
I
Telescope
Matrix
Mueller
~
~
~
~
)(
)(
)(
)(
)(
~
~
~
~
4
3
2
1
1
pI
pI
pI
pI
pM
V
U
Q
I
out
out
out
out
kij
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Modulation package
Optical system whose Mueller matrix can be (strongly) varied upon changing a set of control parameters.
Usually the last element is an optical element that fixes the polarization state of the exit beam, but this is not always the case.
Example
fixed linear polarizer rotating retarder (/4)
VUQI
V
U
Q
I
out
out
out
out
~)sin(~)cos()sin(~)(cos~ 2222
0
0
1
1
2
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Intensity detector for example a CCD
Calibration package
Optical system whose exit polarization is known. It allows to determine the (linear) relationship bewteen the intensities measured by the intensity detector and the input polarization.
2
22
2
12
sin
cossin
cos
out
out
out
out
V
U
Q
I
Examplefixed linear polarizerrotating retarder (/4)
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Instrumental Polarization
Ideally, one would like to place calibration optics in front of the optical system used to measure, including the telescope. Unfortunately, this is not possible (there are not high precision polarization optics with the size of a telescope). This causes that the solar polarization is modified (by the telescope etc.) before we can calibrate the system: instrumental polarization.
It is an important effect
(mostly) produced by oblique reflections (e.g. folding mirrors, and windows (stress induced birefringence of the vacuum windows)
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GCT Obs. Teide
SPh, 134, 1
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Techniques to overcome the instrumental polarization
a) carring out the analysis (the calibration) in the optical axis of the telescope (before the optical system loses axi-symmetry). Specially designed telescopes like THEMIS (Obs. Teide).
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b) modeling (and correcting for) the Mueller matrix of the telescope.
The theoretical expression for the Mueller matrixes of all individual optical elements forming the telescope are known (given the geometry the light path, complex refractive indexes of the mirrors, specific retardances of the windows, and the like). It is possible to write down a theoretical Mueller matrix than can be confronted with observations. One can use this Mueller matrix to correct the measurements
j
jMM
Teslecope
matrix
Mueller
V
U
Q
I
V
U
Q
I
Telescope
Matrix
Mueller
~
~
~
~
V
U
Q
I
V
U
Q
I
~
~
~
~1
Telescope
Matrix
Mueller
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Instrumental Polarization: removing I V crosstalk
V
U
Q
I
V
U
Q
I
Telescope
Matrix
Mueller
~
~
~
~
VMUMQMIMV 44434241
~
V and ,,UQIsince
VMIMV
IMI
4441
11
~
~
cc
cc
IMV
IMI
41
11
~
~
cc
c
c
cc
I
V
M
MI
I
IVV
IIII
11
44~/~
~~~
/~
/~
at continuum wavelengths V=0
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CCD
(longitudinal) Magnetograph
2 states modulator/4-plate + linear
polarizer
V
U
Q
I
Narrow-band color
filter
then2
1
)()(
)()(
VICtI
VICtI
out
out
)()()()(
)()(
)()(
21
21
21
21
IV
and
tItItItI
tItII
tItIV
outout
outout
outout
outout
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Magnetogram : just an image of Stokes V in the wing of a spectral line.
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Order of magnitude of the degree of polarization to be expected in the various solar magnetic structures (for a typical photospheric line used in magnetic studies):
regionsnetwork -interin 10
regionsnetwork in 1
regions plagein 10
sunspotsin 30
%.
%
%
%
IV
IV
IV
IV
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Instrumental Polarization: Seeing Induced Crosstalk
Important bias of any high angular resolution observation, although it is easy to explain in magnetograph observations.
If the two images whose difference should render Stokes V are not taken strictly simultaneously (within a few ms, the time scale that characterizes atmospheric turbulence variations) then Stokes I Stokes V
)()()(
)()()(
222
111
xVxItI
xVxItI
out
out
2with
)
210
00210
212121
/)(
()()(
)()()()()()(
xxx
xVxVttdtxd
xI
xVxVxIxItItI outout
(Lites 1987)
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Seeing Induced Crosstalk
How to solve the problem?
1. Using high frequency modulation, so that the atmosphere is frozen during a modulation cycle. (ZIMPOL like.)
2. Using simultaneous spatio-temporal modulation. Preferred technique in ground based observations.
3. Applying image restoration before demodulation. (SST approach.)
4. Going to space (e.g. Hinode), but then you have jitter from the satellite.
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Techniques to deduce physical properties of the magnetic atmosphere upon the interpretation of the polarization that it produces.
Ingredients:
model atmosphere (assumptions on the properties of
atmosphere whose magnetic field will be inferred)
polarized spectral synthesis code
fitting technique (e.g., 2 minimization techniques)
All solar magnetic fields measurements (magnetometry) need, and are based on, these ingredients and assumptions. Frequently the assumptions are implicit and people tend to think that they do not exit. The inferred magnetic field depends, sometimes drastically, on the asumptions.
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Longitudinal magnetograph
It is just an image showing the degree of circular polarization in the flank of spectral line.
– If the solar atmosphere where the polarization is produced has a discrete number of magnetic component– If the magnetic field of this component does not vary, neither along the line-of-sight nor across the line-of-sight– If the temperature and pressure of the atmosphere does not depend on the magnetic field– If the velocities is constant in the resolution element
Model atmosphere:
Synthesis Code:
– Multi component atmosphere– Weak magnetic field approximation
Fitting technique:
– No sophistication; one observable and one free parameter
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resolution
icomponents
iiii
i
icomponents
iiii
resolution
components
iii
sdBd
dICBf
ddI
C
ddI
BfCVfdsxVV
)(cos
)(
cos)(),()(
#
##
resolution
i
resolution
components
iii sdIIfdsxII )()(),()(
#
resolution
resolution
sd
sdB
cIV
)()()(
d
IdCc i )(ln
)( A calibrated magnetograph gives the longitudinal component of the magnetic flux density (mag flux per unit surface)
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Milne-Eddington fitting technique
– If the solar atmosphere where the polarization is produced has two components: one magnetic and one non-magnetic–If the magnetic field of this component does not vary, neither along the line-of-sight nor across the line-of-sight– If the line to continuum absorption coefficient ratio does not vary with height in the atmosphere– If the source function varies linearly with continuum optical depth
Model atmosphere:
Synthesis Code:
– Milne Eddington analytic solution of the radiative transfer equations for polarized light
Fitting technique:
– Non-linear least squares minimization
(e.g. Skumanich & Lites 1987)
data
syntheticobserved StokesStokes22
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Input model atmosphereB,,, ...
new atmosphere B,,, ...giving a smaller 2
,,,
22
2
B
,,,
V
U
Q
I
V
U
Q
I
B
V
U
Q
I
synthesis
Observed I,Q,U & V
2 small
enough?
2 m
inim
izat
ion
B,,, ...
NONO
YESYESobserved
B,,
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Sunspot observation
Skumanich & Lites 1987
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MISMA inversion code
– complex, having many different magnetic fields, velocities, temperatures, etc.
Model atmosphere:
Synthesis Code:
– numerical solution of the radiative transfer equations for polarized light
Fitting technique:
– Non-linear least squares minimization
data
syntheticobserved StokesStokes22
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Observations
Synthetic
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PCA inversions
(PCA: principal component analysis)
Important, since they are extremely fast, and so, they are bound to become popular in the next future.
For example, they may allow to process, on line, the huge data flux produced by the new synoptic magnetographs (e.g., SOLIS, see http://solis.nso.edu)
It belongs to the class of Prêt-à-porter inversions as opposed to the classical Taylor-made inversions.
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Which synthetic profiles are closest to the
observed profiles?
If # i are the closest onesthen
Pre-computed data base
nnnnnnn
iiiiiii
VUQIBn
VUQIBi
VUQIB
,,,),,( # model
,,,),,( # model
,,,),,( #1 model 1111111
Prêt-à-porter inversions
iiiBB ,,,, observed
Obser ved I,Q,U & V
VUQI ,,, Observed
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Eigenfaces
Reconstructed faces
Rees et al., 2000
# of eigenfaces used in the reconstruction
i
ii eigenface eeigen valuface
Fitting technique for PCA:
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Rees et al. (2000)
eigenvalueth -i :
reigenvectoth -i :
vectorStokes :
i
i
iii
s
e
S
seS
Only a few eigenvalues are needed to characterize the Stokes profiles
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Forward modeling (which is an inversion technique!!!)
– Resulting from the solutions of the MHD equations under ´realistic´ solar conditions.
Model atmosphere:
Synthesis Code:
– numerical solution of the radiative transfer equations for polarized light
Fitting technique:
– Not well defined (yet?) The synthetic spectra have to reproduce the observed spectra in some statistical sense.
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Tu
rbu
len
t D
ynam
o S
imu
lati
ons
by
Cat
tan
eo &
Em
onet
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1´´ seeing
clu
ste
r an
aly
sis
cla
ssifi
catio
n
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The case of the large magnetic flux concentrations
Observed
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¨A measurement process is regarded as precise if the dispersion of values is regarded as small. A measurement process is regarded as accurate if the values cluster closely about the correct value¨ (definition; e.g., Cameron 1960)
Caveats to keep in mind:
– The simplest the model atmosphere in which the inversion code is based, the higher the precision of the measurement (e.g., no problems of uniqueness in magnetographic observations).
– However precision is not the aim of solar magnetometry; accuracy is more important since it is more difficult to achieve.
– It makes no sense oversimplifying the model atmospheres to end up with magnetic field determinations that are very precise but very inaccurate.
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Applications of the tools and techniques developed in the notes to specific problems of solar physics.
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Understanding Real Magnetograms, e.g., Kitt Peak Synoptic Maps
README_1
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Jones et al., 1992, Solar Phys. 139, 211
README_2
Coelostat Instrumental polarization
Noise 7G
max @flux solar 21
Mx 1024 SuraceSolar G 7 23 .
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Line Ratio Method, or the field strength of the network magnetic concentrations
The network magnetic concentrations have very low flux density (say, less than 100 G) but a large magnetic field strength similar to that of sunspots (larger than 1 kG).
network
This fact is known thanks to the so-called line-ratio method (Stenflo 1973)
Pre-line-ratio-method situation (late 60´s and early 70´s): magnetograms of a network region taken using different spectral lines showed inconsistent results.
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This is due to the fact that in network regions the magnetograph equation is not valid, implying network magnetic field strength of kG even though the magnetograms show a flux density of a few hundred G.
Stenflo took simultaneous magnetograms in two selected lines,
Fe I 5247 (geff=2.)Fe I 5250 (geff=3.)
These two lines are almost identical if there no magnetic field in the atmoshere (same log(gf) same excitation potential, same element and ionization state), however, they have (very) different magnetic sensitivity.
field magnetic no is thereif )()( 52505247 II
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If weak field (sub-kG):
d
dIkBV
d
dIkBV
z
z
)(3/)(
)(2/)(
205250
205247
13/)(
2/)(
5250
5247
V
V
,...)(13/)(
2/)( 2
5250
5247
fBV
Vz
If strong field (sub-kG):
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Line ratio obseved in network
kG 1zB0zB
Fe I 5247 Fe I 5250
resolution element
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Broad Band Circular Polarization of Sunspots (BBCP)
Clues on the fine-scale structure of the Sunspot´s magnetic fields
Observational facts:
– Sunspots produce (large) Broad-Band circular polarization (
V/I10-3 ,Illing et al. 1974a,b)– It is produced by the individual spectral lines in the band-pass
(i.e., it is not continuum polarization: Makita 1986)– It is maximum produced in to the so-called neutral line, where
the magnetic field is supposed to be perpendicular to the line-of-
sight. (Makita 1986.)– In the neutral line Stokes V is never zero but shows the cross-
over effect
IDL
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Broad Band Imaging - Polarimetry
d )Filter( )Signal(Signal
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neutral line
Sun
sunspot
neutral line
solar limb
solar center
we
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typical resolution element
a) The BBCP is produced by gradients along the line-of-sight, i.e., the magnetic field, velocity etc. change in the sunspot over scales of less than 150 km, i.e., much smaller than the resolution element of typical observations (1” or 1000 km). Why?
0gradient LOS nofor since
LOS thealong gradients are thereunless 0
BBCP
widthband
i
resolution widthband
ii
widthband resolutionii
widthband
dV
dVf
dVfdV
)(
)(
)()(
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b) it is produced by gradients of inclination along the LOS. They are present since Stokes V is never zero in the neutral line (i.e., there is no point where the magnetic field is perpendicular to the line-of-sight).
SA & Lites, 1992, ApJ, 398, 359
Cross-over effect, Grigorjev and Kart, 1972, SPh, 22, 119
Sto
kes
V
0 and 0 then 90 andconstant is if o VdzdV
B
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150 km
750 km Resolution element
c) The BBCP cannot be due to smooth well-organized vertical variations of magnetic fields inclination.
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BB zzB
1
0Sanchez Almeida (2005)
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150 km
750 km
Resolution element
The BBCP has to be due to very intermitent variations of magnetic field inclinations.
This is a general feature of the magnetic fields in the penumbrae of sunspots that is inferred from the (careful) interpretation of the circular polarization that it produces, despite the fact that we do not resolve the fine-scale structuring of the magnetic field
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Quiet Sun Magnetic Fields
Cancellation of polarization signals in complex (tangled) magnetic fields
1B
2B
Q2 = -Q1 Q1+Q2 = Qobs = 02B
1B
V2 = -V1 V1+V2 = Vobs = 0
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This kind of cancellation seems to take place in the quiet Sun
Size of a Network cell (25000 km)
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Turbulent Dynamo Simulations by Cattaneo & Emonet
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Effect of insufficient angular resolution
1” seeingoriginal
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Variation of the Flux Density in the simulations with the angular
resolution and the sensitivity of the synthetic magnetograms.
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1”x1”
Do
mín
gue
z C
erd
eña
et a
l. (0
3)Inter-Network Quiet Sun
angular resolution mag. 0.5”sensitivity 20 GVTT (obs. Teide), speckle reconstructedUnsigned flux density 20 G
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1.6 G
12 G
12 G x SolarSurface = 7x1023 Mx = solarflux@max
Rabin et. al. 2001
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How can we measure the properties of the quiet Sun magnetic fields?
Need to use inversion techniques whose model atmospheres allow for the complications that the quiet Sun field has:
Different polarities in the resolution element (different magnetic field inclinations in the resolution element) Different magnetic field strength in the resolution element …
Quite Sun fields: matter of active research
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Techniques and methods employed in the recent literature on solar magnetometry. Used by specialist groups.
Model dependent but with substantial potential.
No realistic inversion techniques exist so far.
– Hanle effect based magnetometry– Magnetometry based on lines with hyperfine structure– He 1083nm chromospheric magnetometry– Polarimeters on board Hinode
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dvreecqcddvolume
MMifb
tEE
i
fbfb
fb )(** Re 000 2
0
1
20
1
21
0
0
ier
ier
rrr ii
sinsincos
cos
cossin
cossin
drdeide
drdeide
drdede
tEE
iMMi
tEE
iMMi
tEE
iMMi
fb
fb
fb
fb
fb
fb
0
1
0
1
1
0
0
2
0
1
2
0
1
2
0
)(
)(
)( 002
0
pdeip
Which leads to the selection rules for E-dipole transitions
each one associated with a polarization
1 ,0 M
r
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A weak magnetic field splits the Zeeman sublevels but … it is weaker than the natural width of the lines.
The eigenstates involved in the transition are not pure states but combinations of them …Various frequencies are excited at the same time, and they add coherently.
w0-w w0w
Hanle Effect Based Magnetometry
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wtiwtiiwt
y
xeee
d
d
21Re UU
In the case that two eigenstates contribute to the dipolar emergent radiation, the resulting electric dipole is .
1. Since non-monochromatic, the radiation is always partly polarized (Hanle effect is said to depolarize)
2. Modifies the state of polarization with respect to the case Δw=0 (Hanle effect rotates the plane of polarization.)
3. Purely non-LTE effect, since the integration of many atoms emitting at random times lead to the incoherent superposition of the two polarization states U1 and U2, and have no effect. In the coherency matrix representation,
wtiyx
wtiyxyxyxyx
wtiyyyyy
wtixxxxx
eUUeUUUUUUJJ
eUUUUJ
eUUUUJ
2*12
2*21
*22
*11
*
2*21
2
2
2
1
2
2*21
2
2
2
1
2
Re2
Re2
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)sin(
)cos()cos(0 wt
wtwtU
d
d
y
x
Textbook case: describes linearly polarized in the x axis at t=0.
0/
)2sin(/
)2cos(/
2
0
IV
wtIU
wtIQ
UI
w
w2
unpolarized
w
: coherency time
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We
atom
Sun
0B
Sun
0B
Sun
non-magnetic scattering Hanle effect
2w
Bw
For Hanle effect to depend on the field strength (and so to be a useful tool),
)s10/( g
nm) /500( G 70 ||
8eff
2
B
Hanle signals even if tangled fields
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Faurobert et al. (2001)
observed
modelled) (known 0 B
Sr I 4607Å Hanle depolarization
Hanle saturation at some 50 G
depolarizing collisions are critical for a proper modeling
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general Zeeman pattern
Magnetometry Based on Lines With Hyperfine
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Magnetometry Based on Lines With Hyperfine Structure
Hyperfine Structure: due to the interaction between the electron angular momentum and the nuclear angular momentum.
Old theory by Landi Degl’Innocenti (1975), but recently recovered and used for actual observations by López Ariste et al. (2002, ApJ, 580, 519).
What would be a single line becomes a blend of lines. They now undergo regular Zeeman effect, with their π and σ± components. Hundreds of components show up.
When the HFS splitting and the Zeeman splitting become comparable, Zeeman pattern depends on the magnetic field strength (it is not the independent superposition of the Zeeman patterns of the independent components).
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σ
Landi Degl’Innocenti (1975)
π
López Ariste et al. (2002)
Stokes V changes shape when the field is several hundred G … good diagnostic tool for hG field strengths.
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Despite the apparent complexity, the HFS patterns present several regularities (Landi Deg’Innocenti 1975)
π and σ components are normalized to one (there is no net circular polarization).
When the magnetic field is weak enough, the Stokes V signal follow the weak magnetic field approximation.
BddI
V cos
)()(
The centers of gravity of the π and σ components is independent of the HFS.
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He I 1083nm Chromospheric Magnetometry
The need for a simple but quantitative diagnostic of upper chromospheric magnetic fields is keenly felt (Rüedi et al. 1995, 293, 252).
Popular in chromospheric magnetometry.
It is a bend of 3 He I lines sharing the same lower level (19.79 ev).
Optically thin. Bend modeled using ME profiles given line strengths and Zeeman splittings. Need incomplete Pashen-Back effect to carry out the calculations.
Entirely formed in the chromosphere in standard 1D model atmospheres (Fontenla et al. 1993). Formed by recombination.
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Rüedi et al. (1995)
Incomplete Pashen-Back effect required for a proper analysis (Socas-Navarro et al. 2004)
blend of 3 linesME fitME fit
Creates NCP by saturation
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Hinode, satellite ideal for polarimetry. 50 cm diffraction limited optical telescope (λ/D~0.26’’ @ 6302 Å)
Polarimeters on board Hinode
Open data policy! Every one is welcome to use them
Launched, end of 2006
Japanese (ISAS), in cooperation with US (NASA) and Europe (PPARC, ESA).
Hinode European Data Center here in Oslo.
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SOT-SP
SOT-FG
SOT: Solar Optical Telescope
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GOTO Summary -- Index:
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Selected referencesref_magnetometry.pdf
Exercises on solar magnetometry
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Sutterlin et al, 1999, DOT, G-band, speckle reconstructed
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Volume averaged in one pixel of a typical photospheric observation
The cartoon shows the right scale for the horizontal and vertical smearing
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SST, Scharmer et al. 20020.12 arcsec, spatial resolution
1´´ x 1´´
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Point Source
kk
/
A
B
Observers A and B receive exactly the same signal, which is constant in the plane perpendicular to k
constantrk
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sc
zz
yy
xx
zz
yy
xx
zz
yy
xxrkwti
AwtAwt
A
A
A
wt
A
A
A
wt
wtA
wtA
wtA
rkEetre
sincos
sin
sin
sin
sin
cos
cos
cos
cos
)cos(
)cos(
)cos(
)(Re),( )(
Monochromatic means plane Elliptical Motion
z
y
x
iz
i
y
ix
eA
eA
eA
E
z
y
x
z
y
x
rk
rk
rk
)( 1te
)( 2te
)( 3te 321 ttt
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)(Re),( )( rkEetre rkwti
Inserting monochromatic solutions of the kind
into the wave equation derived from the Maxwell equations ,one finds
1 LEE //||
:
:
:
:||
L
E
E
Component in the direction of
Transverse component
Characteristic scale for the variation of
Wavelength
k
E
wc /2
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0
0
0
1
V
U
Q
Ix
y
)(te
If Jx(t) and Jy(t) vary at random, then the light Unpolarized Light
2222 IVUQ
p is the degree of polarization> p=0 represents unpolarized light> p=1 corresponds to fully polarized light
x
y
)(te
Monochromatic wave
1222
pI
VUQIn general
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x
y
)(te
0
0
1
1
V
U
Q
I
x
y
)(te
0
0
1
1
x
y
)(te
0
1
0
1
x
y
)(te
0
1
0
1
1
0
0
1x
y
)(te
1
0
0
1x
y
)(te
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21
*22
*11
*22
*11
*
21*
21
22
21
22
Im2Im2Im2Im2
Re2
VVJJJJJJJJJJV
UUJJU
QQJJQ
IIJJI
yxyxyxyxyx
yx
yx
yx
Re
ReReRe),(),(),(
21
21
2
2
1
1
2
2
1
121
yy
xxiwt
y
x
y
xiwt
y
xiwt
y
xiwt
JJ
JJe
J
J
J
Je
J
Je
J
Jetretretre
21
21
yy
xx
y
x
JJ
JJ
J
J
*22
*12
*21
*11
*2
*121
*yxyxyxyxyyxxyx JJJJJJJJJJJJJJ
0*12
*21 yxyx JJJJ
*22
*11
*yxyxyx JJJJJJ
(because the two beams are incoherent)
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V
U
Q
pI
pp
V
U
Q
pI
pp
V
U
Q
I
21
21
IVUQp /222
x
y
Decomposition of any polarization in two fully polarized beams
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x
y
y
x
J
JJ1
*
*
2
x
y
J
JJ
*
*
22
22
Im2
Re2
yx
yx
yx
yx
JJV
JJU
JJQ
JJI
VJJV
UJJU
QJJJJ
IJJ
xy
xy
yxxy
xy
*
*
2222
22
Im2
Re2
The Jones vectors of these two beams are orthogonal
1J
2J
0*21 JJ
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yyyx
xyxx
mm
mmm
**
**
**
**
**
**
**
**
**
**
**
**
Re
Im
Im
Im
Im
Re
Re
Re
Im
Re
/
/
Im
Re
/
/
yyxxyxxy
yyxxyxxy
yyxyyxxx
yyxyyxxx
yyxxyxxy
yyxxyxxy
yyxyyxxx
yyxyyxxx
yyyxxyxx
yyyxxyxx
yyxyyyxx
yyxyyyxx
yyyxxyxx
yyyxxyxx
yyxyyyxx
yyxyyyxx
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
mmmmM
44
43
42
41
34
33
32
31
24
23
2222
22
2222
21
14
13
2222
12
2222
11
2
2
2
2
ijm
From Jones matrix
to Mueller matrix
ijM
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1U
2U
1 ; 0 21*21 UUUU
: , 21 UU
For any selective absorption, this set is a base of complex 2D vectors (e.g., the Jones vector)
For any polarization with Jones vector J
2*21
*1 )( ))(1( UUJUUJJmJout
2*21
*1 )( )( UUJUUJJ
The OS just changes the Jones vector as
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y
x
yyx
yxx
yyyxx
yxyxx
y
xyyxx
J
J
UUU
UUU
UJUUJ
UUJUJ
U
UUJUJUUJJm
2
11*1
*11
2
12
11*1
*11
2
1
1
1*1
*11
*1
)(
)( )(10
01
2
11*1
*11
2
1
2221
1211
10
01
yyx
yxx
UUU
UUU
aa
aam
0Im
0Im
0Re
Im2Im/
Re2Re/
Re/
)1( Re/
2211
2112
2112
1*112112
1*112112
1
2
1
2
12211
1
2
1
2
12211
aa
aa
aa
VUUaa
UUUaa
QUUaa
IUUaa
Q
U
V
yxV
yxU
yxQ
yxI
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weak magnetic field approximation
)( B )( B
)()()( 2 BB
)()( BB
dd
B
)(2 B Zeeman shift
2
22
2
dd
dd B
BB
)()()()(
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Band-pass of typical magnetogram observations
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continuum
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References
•Kemp 1970, ApJ, 162, 169, in connection with the continuumpolarization in a magnetic field•Sanchez Almeida