Determinación de los parámetros físicos de estructuras magnéticas solares

Abstract

The Sun is the nearest star for us. It is an almost perfect sphere of magnetized plasma which emits radiation constantly and it is fundamental for the existence and development of life on Earth but at the same time, it can be a threat because we are dependent on electricity and eventually a solar storm could destroy our infrastructures, leaving us without access to technology, so it is important to understand how the Sun is and predict its behaviour. In this Master thesis, we will focus on obtaining the thermodynamic and magnetic conditions from the polarized light, that is to say from the Stokes parameters. We will use not only simulated conditions but real data from the solar photosphere, which is the nearest layer to the Sun’s surface. In order to do this, we will talk about the Stokes parameters and about the Radiative Transfer Equation for full Polarized Light. After that, we will solve this equation under the Milne-Eddington approximation and we will explain its analytical solution. However, this isn’t the only way to obtain a solution. In fact, nowadays this equation is solved numerically because when we consider the dependence of parameters with optical depth there is not analytical solution although there are several programs to study the low atmosphere of the Sun like SIR, which we will explain how it works and its advantages and disadvantages. To make this possible, once we have seen how analytical solution of the Radiative Transfer Equation for full Polarized Light are, we will explain the method to synthesize under the Milne-Eddington approximation or in other words, how to obtain the Stokes parameters from 9 free parameters. These parameters are S0 and S1 which are the associated coefficiens to the source function. η is the ratio between the absorption on the line and on the continuum. a is the damping adimensionalized parameter whereas wm is the line of sight velocity of the medium. Respect to ∆λD, B, θ and χ, they are the Doppler broadening in wavelength units, the intensity of the magnetic field in Gauss, the inclination of the magnetic field respect to the line of sight in degrees and the azimuth of the magnetic field also in degrees. We will particularize for the spectral line of Fe I at 6301.5 ˚A and maintaining an initial set constant, we will change each parameter to observe and explain what happens. Using this method, we will study what occurs when we vary the intensity of magnetic field, the inclination of the magnetic field respect to the line of sight and its azimuth. After that, we will build up the response functions. Under the Milne-Eddington hypothesis, we will show you that the response functions are analytical and are the first derivative of Stokes parameters respect to the free parameters of the MilneEddington approximation. In any case, we will also check these analytical response functions by using a central difference method. Nevertheless, the response functions, numerical or analytical, tell us how Stokes parameters change because of linear perturbations on the parameters which we use for synthetizing the Stokes profiles. Now, in order to achieve the inversion of an observed Stokes profile under the Milne-Eddington approximation, one must introduce to the program a set of MilneEddington parameters to synthesize the Stokes parameters, compare between the observed and synthetic Stokes profiles and change iteratively the parameters until a good fit is achieved. As the Radiative Transfer Equation for full Polarized Light is not lineal, we have to do it iteratively until the difference between the observed and synthesized Stokes profiles are minimum or specifically, until χ 2 is minimum. Fortunately, there are algorithms to make non-linear fits from which LevenbergMarquardt algorithm is selected for being standard and giving excellent results. As a test, we can substitute the observed profile by a synthetic Stokes profile plus a noise. Applying our inversion code to this profile, we can check the behaviour of our code. In practice, we will use first a synthetic Stokes profile without noise to obtain the 9 free parameters and later, the same profile but noise is added. However, in both cases, we will simulate the spectral line of Fe I at 6301.5 ˚A. In addition to this, we are going to apply our Milne-Eddington inversion code on real spectropolametric data, which are Stokes profiles versus wavelength. Particularly, we are going to use it on a granule near to the active region NOAA 10953 which was observed by HINODE in 2007. But now, we are going to invert the line of Fe I at 6302.49 ˚A instead of the line of Fe I at 6301.5 ˚A because the first one is more sensitive to magnetic field as we will see. At the same time, we are going to compare our results from Milne-Eddington inversion code with those from SIR, which doesn’t have problems to invert multiple spectral lines simultaneously, and we will explain why both programs agree with the values of magnetic field or the inclination of magnetic field. Here, we will see the fundamental disadvantage of maintaining the quantities constant with optical depth because a Milne-Eddington inversion code can’t explain the asymmetries on Stokes profiles or the ascent and the descent of plasma in photospheric layers of granules. In addition to this, our Milne-Eddington inversion code can be used only for one spectral line whereas SIR inverts multiple spectral lines at the same time. Last of all, we are going to obtain the maps of temperature, microturbulence, velocity along the line of sight, intensity of magnetic field, inclination and azimuth of magnetic field for a section of the before mentioned active region, which includes the quiet Sun, and a part of a sunspot with its penumbra and its umbra. Those will be obtained with SIR and we will discover that the values are concordant with the literature. Also we will obtain the map of temperatures at a top layer and we will check that this quantity decreases with height, which is expected in the solar photosphere and we will relate the intensity of magnetic field to the temperature for the different structures which appear on the section of the before mentioned active region.