From Stokes profiles to the models of solar atmosphere

1. From Stokes profiles to the models of solar atmosphere Jan Jurčák Astronomical Institute of the AS CR Ondřejov, 26thFebruary 2009

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

3. 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:

4. 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.

5. 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

6. 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

7. 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.

8. Radiative Transfer Equation (RTE) Light beam Stokes vector Source function vector, Planck function in case of LTE Propagation matrix Absorption Dichroism Dispersion

9. 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.

10. 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.).

11. 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

12. 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.

13. Response functions

14. 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

15. 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

16. 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.

17. 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.

18. Concept of nodes The size of curvature matrix is important in the inversion process and is the same as the number of free parameters: nm+r 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).

19. Profile inversion (set up) Arbitrary set plasma parameters and corresponding synthetic Stokes profiles. Initial guess of atmosphere and corresponding synthetic Stokes profiles.

20. Profile inversion (1st inversion) Arbitrary set plasma parameters and corresponding synthetic Stokes profiles. Results of inversion with 1 node for all parameters

21. 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

22. 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

23. Umbral profile

24. Umbral profile

25. Umbral profile

26. Umbral profile

27. 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.

28. Models of solar atmosphere is sunspot penumbra

29. SIR/GAUSS

30. P1 profiles and models

31. P2 profiles and models

32. P3 profiles and models

33. Comparison of profiles and resulting models

34. Models of solar atmosphere is sunspot penumbra

35. Models of solar atmosphere is sunspot penumbra

36. 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