The best lines to measure
temperatures, velocities and
magnetic fields in the solar
photosphere
Daniel Cabrera Solana (IAA, CSIC)
Luis R. Bellot Rubio (KIS)
Jose Carlos del Toro Iniesta (IAA, CSIC)
Introduction
1. Observations
τ5 ≡
R z0
z
χc,5000 dz
Introduction
2. Retrieving information from the observations
-Inversions of the radiative transfer equation
Westendorp Plaza, C. et al. (2001)
Initial model
atmosphere
Observed Stokes
profiles
Spectral
synthesis
Lest-squares fit
Perturbed model
atmosphere
Final model
atmosphere
Field strength [G]
Motivation of the work
Main problem
Obtain information about the thermal, magnetic and
dynamic structure of the solar photosphere from the
observations
How to select the best lines to do it?
Motivation of the work
• There are no criteria to decide which line is best suited
to the particular problem one is dealing with.
• There is a lack of a simple formulation capable of
describing the sensitivity of spectral lines to the
various atmospheric parameters.
• Multiwavelength polarimetry. (UV, Visible, IR)
• The importance of a proper selection of spectral lines
has been emphasized by recent advances in
instrumentation. (IMAX (SUNRISE), HMI (SDO))
• Rather frequent misconceptions appear in the
literature about the sensitivities of given lines.
Sensitivity of spectral lines to
atmospheric parameters
1. Response functions (RFs)
Ruiz Cobo & del Toro Iniesta (1994)
The sensitivity of Stokes profiles to perturbations of the atmospherical
physical quantities are given by the so-called response functions (RFs)
X
δI(0)
= = R(τ
R(τ
) δx(τ ) ∆τ
δI(0)
c )cδx(τc )c∆τc c
τc
Velocity
Normalized RF of Stokes I to vLOS
Fe I 630.2 nm
δvLOS (τc )
Sensitivity of spectral lines to
atmospheric parameters
2. Sensitivities to constant pertubations
The sensitivity of the lines will be characterized by the maximum
value of their integrated RFs
Normalized RF of Stokes I to vLOS
RF(λmax )
δxi (τc ) = δx
δxi (τc ) = δx
δI(0) =
X
τc
R(τc ) δx(τc ) ∆τc
δI(0) = R 0 δx
R0 =
X
τc
R(τc ) ∆τc =
δI(0)
δx
Sensitivity of spectral lines to
atmospheric parameters
5. Set of commonly used lines
Species
Wavelength [nm]
geff
Species
Wavelength [nm]
geff
Ni I
491.20
0
Ni I
676.78
1.5
Fe I
524.71
2.0
Fe I
709.04
0
Fe I
525.02
3.0
Fe II
722.45
0
Fe I
537.96
1.0
Si I
1062.76
1.8
Ti II
538.10
0.9
Fe I
1089.63
1.5
Fe I
557.61
0
Fe I
1142.23
2.0
Fe I
569.15
0
Fe I
1221.33
2.5
Fe II
614.93
1.3
Fe I
1558.83
1.5
Fe I
617.33
2.5
Fe I
1564.85
3.0
Fe I
630.25
2.5
Fe I
630.35
1.5
Ti I
630.38
0.9
RF calculated by
SIR.
Sensitivity of spectral lines to
atmospheric parameters
3. Weak line model
T, vLos
−(λ − λ0 − λLOS )2
I(λ) = A2 + A0 exp [
]
2A21
B: Strong field Regime
V (λ) =
A00
−(λ − λ0 ± λB )2
cos γ exp [
]
2(A01 )2
B: Weak field Regime
V (λ) = λB cos γ
∂I
∂λ
2A1
A0
Sensitivity of spectral lines to
atmospheric parameters
4. Analitycal calculation
Sensitivity to vLOS
−(λ − λ0 − λLOS )2
I(λ) = A2 + A0 exp [
]
2A21
1. Differenciating Stokes I with respect to vLOS:
Rv0 LOS ,1 (λ)
∂I
2A0 λ0
−(λ − λ0 − λLOS )2
(λ − λ0 − λLOS ) × exp [
≡
=
]
∂vLOS
A21 c
A21
2. Which has extrema at:
λext = λ0 + λLOS ±
A1
(2)1/2
3.The analytical sensitivity is given by:
Rv0 LOS ,1 (λmax ) = −(2)1/2 e−1/2
A0 λ0
A1 c
Sensitivity to line of sight velocity
Rv0 LOS ,1 (λmax ) =
∂I
A0
λ0
∝
∂vLOS
A1
•Our simple analytical model provides an
excellent description of the sensitivity to
velocity.
•Usually, visible lines are more sensitive
to velocity than the infrared:
-This is due to their greater shape ratio.
(smaller thermal width Æ narrower profiles)
-The shape ratio turns out to be dominant for
the line behaviour.
-The stronger and the narrower the line, the
greater sensitivity!.
Quiet Sun
Penumbra B=0
Umbra B=0
•You can use this relationship to
compare the sensitivities of different
lines with a simple estimate of the line
depth and width.
Sensitivity to line of sight velocity
Rv0 LOS ,1 (λmax ) =
∂I
A0
λ0
∝
∂vLOS
A1
2A1
A0
2A1
Quiet Sun
Penumbra B=0
Umbra B=0
A0
Sensitivity to magnetic field strength
0
RB,4
(λmax )
∂V
A00 2
=
∝ geff cos γ 0 λ0
∂B
A1
0
RB,4
(λmax ) =
∂V
A0 2
λ
∝ geff cos γ
∂B
A1 0
Strong field Regime
Weak field Regime
Quiet Sun Æ B(2000G,30º)
Quiet Sun Æ B(200G,0º)
Penumbra Æ B(1500G,70º)
Penumbra Æ B(200G,0º)
Umbra Æ B(2000G,30º)
Umbra Æ B(200G,0º)
Sensitivity to temperature
Response of Stokes I to temperature perturbations
1 ∂ ln A0 /∂T
∂I(λaext )
∂A2
∂ ln A1
= RT0 ,1 (λaext ) =
+ 2A0 (
) × exp [−1 +
]
PT =
∂T
∂T
∂T
2 ∂ ln A1 /∂T
Quiet Sun Æ B=0
Penumbra Æ B=0
Umbra Æ B=0
Sensitivity to temperature
Response of W to temperature perturbations
0
PW = RTW =
(∂A1 /∂T )(1/A0 ) + (∂A0 /∂T )(1/A1 ) A0 A1 ∂ ln A2
∂W
= −(2π)1/2
)
(
)(
∂T
A2
A2
∂T
Quiet Sun Æ B=0
Penumbra Æ B=0
Umbra Æ B=0
Conclusions
•We can determine how much a given line reacts to
velocity, and magnetic field perturbations with a
simple estimate of A0 and A1 of the line.
•The sensitivity of the lines to velocity and magnetic
field increases with the sharpness (A0/A1) of the
profiles.
•We are able to reproduce the sensitivities of I and
W to temperature perturbations with the model
parameters and their derivatives.
•The sensitivity of Stokes I to temperature is
determined mainly by the variations of the source
function which are smaller at longer wavelengths.
Visble lines have a higher sensitivity to T.
Conclusions
•Lines considered as insensitive to temperature, like
Fe I 1564.85 and Fe I 557.61, display larger changes
of W than lines presumed to have bigger sensitivity
to T as Fe I 630.25.
•Hence we learn which line is more suited to
measure a given atmospheric parameter.
•The deduced relationships for weak lines do not
depend on the model atmosphere. They are
universal.
•Our results are of practical interest both for the
design of new instruments (IMAX, HMI)
HMI and for a
better exploitation of existing facilities (TIP).
TIP
The best lines of the set
•Lines with better response to velocity:
Fe I 524.7, Fe I 525.02, Ni I 676.70
•Lines suited for measurements of line bisectors (VLOS):
-Lines with geff = 0
-Lines with low response to temperature and high response to VLOS
Fe I 709.04, Ni I 491.21, Fe I 569.1, Fe I 722.4
•Lines with better response to B:
Fe I 1142.23, Fe I 525.02, Fe I 1564.85
•Lines with better response to temperature:
FeI 557.6, Fe I 525.02, FeI 524.7
•Lines with a good response to all the parameters (suited for
inversions):
FeI 525.02, Fe I 617.3, Fe I 1142.23