Sn

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From Stokes profiles to the models of solar
atmosphere
Jan Jurcák Astronomical Institute of the AS CR
Ondrejov, 26th February 2009
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Contents

• Introduction to the Stokes profiles
• Inversion code (SIR)
• Problems with inversion
• Models of solar atmosphere is sunspot penumbra

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Stokes profiles

• Vector of four spectral profiles describing the
  full polarisation state of the light.
• I - intensity of unpolarised light
• Q - intensity of linearly polarised light
• U - intensity of linearly polarised light
• V - intensity of circularly polarised light

Definition of Stokes profiles
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Stokes profiles

We will consider the polarisation induced by the
Zeeman effect only. The presence of the magnetic
field causes and also changes the polarisation of
the light beam.
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Examples of synthetic Stokes profiles
Constant plasma parameters with height LOS
velocity 0.5 km/s Field strength 800 G Field
azimuth 165 deg Inclination change from 0 to
180 deg
LOS
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Examples of synthetic Stokes profiles
Constant plasma parameters with height LOS
velocity 0.5 km/s Field strength 800 G Field
inclination 90 deg Azimuth change from 0 to
360 deg
LOS
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Inversion code (SIR)
What are the inversion codes?
Programs that numerically (or analytically)
solves the radiative transfer equation and
numerically determines the values of plasma
parameters in the atmosphere to obtain the best
match between the observed profiles and the
synthetic profiles.
Assumptions
Local Thermodynamic Equilibrium (LTE) It implies
that there is no polarized light in the source
function. The atomic populations are computed by
means of Saha and Boltzmann equations
(collisional rates are high enough, complete
redistribution on scattering). Hydrostatic
equilibrium Allows us to determine the electron
pressure from the stratification of temperature
using an equation of state of an ideal gas.
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Radiative Transfer Equation (RTE)
Light beam Stokes vector
Source function vector, Planck function in case
of LTE
Propagation matrix
Absorption Dichroism Dispersion
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Formal solution of RTE
Evolution operator
It can be shown that evolution operator must
verify this differential equation.
This equation is also the reason, why there is no
general analytical solution for RTE.
Solution in the form of attenuation exponential
is valid only in specific cases, because the
matrices do not commute in general.
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Stokes Inversion based on Response functions
This inversion code does not assume simple form
of propagation matrix and thus there is no
analytical solution for the RTE. Although, the
LTE is assumed, the source function is not
linearly dependent on optical depth.
Concept of Response functions (linearization of
the RTE)
We suppose that a small perturbation of the
physical parameters of the model atmosphere will
propagate linearly to small changes in the
observed Stokes spectrum.
Model atmosphere, where represent all
the physical quantities characterizing the
propagation matrix and source function
(temperature, magnetic field vector, LOS
velocity, etc.).
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Response functions (RF)
Consider a small perturbation that
induce small changes in K and S that, to a first
order of approximation, can be written in the
following form.
This will lead to a small changes in the Stokes
vector and modification of the RTE
Neglecting the second order terms and taking the
RTE into account we get
Introducing the effective source function as we
get the RTE for the Stokes profile perturbations
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Response functions
As the RTE for perturbations is formally
identical to RTE itself, the solution must be
formally the same
The integrand is a contribution function to the
perturbations. This leads to the definition of
the response function
The solution of RTE for perturbations can be thus
rewritten in the form
RFs behave the same way as partial derivatives of
the spectrum with respect to the physical
quantities. Within linear approximation, RFs
give the sensitivities of the emerging Stokes
profiles to perturbations of plasma parameters
in the atmosphere.
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Response functions
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Inversion scheme
In this scheme is supposed one-component model
of atmosphere height-independent
macroturbulent velocity (MAC) point spread
function of the instrument (IPS) stray-light
with a fraction s
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Merit function and its minimization
We want to change the model of atmosphere
by to move in the space of free parameters
closer to the global minimum. Close to the
minimum, the new merit function can be
approximated by the Tailor series of the old one
partial derivative of the merit function
curvature matrix containing the second partial
derivatives of the merit function one
half of the Hessian matrix
close to the minimum
far from the minimum
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Error estimation
The error is proportional to the merit function
and disproportional to the response function.
The better we fit the observed profiles, the
smaller are the errors of retrieved plasma
parameters. Plasma quantities that have at some
optical depth little influence on the emerging
Stokes profiles (their response function is close
to zero at this particular optical depth) will
have large uncertainties.
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Recipe of the Marquardt method (inversion code)
1) Evaluate merit function with an
initial guess of the model atmosphere 2) Take a
modest value for , say 10-3 3) Solve
equation for and evaluate 4) If
greater than , we were
too far from the minimum. Therefore,
increase significantly (by a factor of 10)
and go back to step number 3. 5)
If smaller than
, we were close to the minimum, so that
we have to decrease significantly to refine
the step. Update the trial solution
as new and go back to step number 3. 6)
To stop, wait until the merit function decreases
negligibly (say 0.1) several times. Then
we are either satisfied with the obtained model
of atmosphere or increase the number of
free parameters and continue to step number 3.
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Concept of nodes
The size of curvature matrix is important
in the inversion process and is the same as the
number of free parameters nmr n - number of
optical depth grid points m - physical
quantities varying with depth r - constant
physical quantities
Therefore, we can use successive approximation
cycles in each of which the number of free
parameters increases as the minimum of merit
function is approached more and more. 1 node -
constant with height 2 nodes - linear dependence
with height etc. The nodes are distributed
equidistantly and the values of plasma parameters
in between them are obtained according to the
specification in the control file (splines,
linearly).
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Profile inversion (set up)
Arbitrary set plasma parameters and corresponding
synthetic Stokes profiles.
Initial guess of atmosphere and corresponding
synthetic Stokes profiles.
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Profile inversion (1st inversion)
Arbitrary set plasma parameters and corresponding
synthetic Stokes profiles.
Results of inversion with 1 node for all
parameters
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Profile inversion (2nd inversion)
Arbitrary set plasma parameters and corresponding
synthetic Stokes profiles added noise in the
order of 10-3 Ic.
Results of inversion with 1 node for all
parameters
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Profile inversion (3rd inversion)
Arbitrary set plasma parameters and corresponding
synthetic Stokes profiles added noise in the
order of 10-3 Ic.
Results of inversion with 3 nodes for most
parameters
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Umbral profile
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Umbral profile
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Umbral profile
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Umbral profile
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Summary of the inversion code
It is not a difficult task to use the inversion
code SIR. It is possible to learn the basics in
one day. There is an unpublished guidebook called
Inversions of Stokes profiles with SIR that can
help with almost all problems. It is much more
difficult to interpret the results. Not always is
the best fit of the observed profiles the
solution closest to the reality. Be careful
especially about unreasonable high number of
nodes. You cannot invert the observed profiles
blindly and accept the obtained stratifications
of plasma parameters without analyzing the actual
quality of the fits. There might be some
systematic errors that can be misinterpreted. It
is useful to have some idea (knowledge) of the
expected stratifications of plasma parameters.
For example, if they should be constant with
height or change dramatically.
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Models of solar atmosphere is sunspot penumbra
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SIR/GAUSS
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P1 profiles and models
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P2 profiles and models
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P3 profiles and models
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Comparison of profiles and resulting models
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Models of solar atmosphere is sunspot penumbra
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Models of solar atmosphere is sunspot penumbra
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Conclusions

• The bright penumbral filaments are (at least in
  the innermost penumbra) structures located around
  the continuum layer with weaker magnetic field
  compared with the surrounding plasma. The
  magnetic field is nearly horizontal and the LOS
  velocity reach values around 4 km/s there.
• At higher layers, the plasma parameters reaches
  values comparable with the surrounding umbra and
  cannot be distinguished.
• Since we cannot see any possible lower boundary
  of the weak field region, we cannot yet figure
  out which of the proposed models of the penumbral
  fine structure is correct.

Mitaka, NAOJ, 26.1. 2007