The determination of the plasma parameters

in the sunspot photosphere using the

inversion code

Contents

• Introduction to the inversion code

• Derivation of the reponse functions

• The inversion process

• Error estimation

• Examples of response functions

• Results

Inversion code SIR(Stokes Inversion based on Response

functions)• one-dimensional code

• local termodinamic equilibrium assumed

• one-component model atmosphere

Ruiz Cobo, B. & del Toro Iniesta, J.C. 1992, ApJ, 398,543

Response functions

• RTE for polarised light

• Formal solution of RTE

• Both the propagation matrix and the source function are dependent on the atmospheric parameters. The integrand is so-called contribution function, which inform us how the different atmospheric layers contribute to the observed spectrum.

Response functions

• If we change the input parameters, the propagation matrix and the source functions change at first approximation as

• The asociated changes in the intensities follow the transfer equation. If we neglect the second order terms, we gain

• We define effective source function as

Response functions• The last equation was formally identical to RTE and therefore the solution must be formally the same

• The integrand is now the contribution function to theperturbations of Stokes profiles. If we define the responsefunction

we gain

Inversion process

Error estimation• The error of the plasma parameter is proportional to the merit function (quality of the fit) and to the inverse of the response function

• The basic assumtion required for the derivation of this equationis, that the plasma parameters are independent, so changes of one parameter do not influence the remaining ones.

Relative response function of Stokes I

Relative response function of Stokes Q

Relative response function of Stokes U

Relative response function of Stokes V

Averaged models of

atmosphere

Absolute response

functions to temperature

Absolute response

functions to magnetic

field strength

Absolute response

functions to velocity

Individual profiles and corresponding stratification of plasma parameters

Maps of magnetic field configuration(Area 1, horizontal cuts)

Maps of magnetic field configuration(Area 2, horizontal cuts)

The canopy scheme