The solar atmosphere is heated by energy generated in the solar core by thermonuclear fusion. This energy is transported towards the surface firstly by radiation (in the radiation zone) and next by convection (in the convection zone). However, the low-lying photo-sphere (6000K) is significantly cooler than the solar corona (1 MK) above. This suggests that an additional source of energy exists. The problem of how the solar corona is heated is not fully solved. An important questions still remain open: which structures or mecha-nisms have the highest influence for the heating? How is the heating generated?
Several mechanisms of the coronal heating were proposed. They are usually catego-rized into two groups: DC (Direct Current) and AC (Alternating Current). The DC pro-cesses are related to the stresses and reconnection, in other words they are dependent on the magnetic field. Some example are stress-induced reconnection (Sturrock and Uchida 1981) and stress-induced turbulence (Heyvaerts and Priest 1992). The AC processes are related to wave heating processes, for example, the Alfvénic resonance (Hollweg 1991) and MHD turbulence (Inverarity and Priest 1995). Based on these mechanisms, several models try to explain the source of the coronal heating Klimchuk (2006).
The solar spectrum is a key source of information about the physical condition of the solar atmosphere. Spectroscopy is used in diagnostics of the chemical composition as well as temperature and density. The Doppler effect provides information about the velocity of the plasma motions that are parallel to the line-of-sight. The Zeeman effect allows to study the magnetic field is based on polarimetry in spectral lines. In this section we describe measurements of the photospheric magnetic field (Sect. 3.1) as well as some theory of the line formation in the chromosphere (Sect. 3.2) and optically thin plasma medium, the transition region and the solar corona (Sect. 3.3).
3.1 Magnetic field in photosphere
The magnetic field in the solar atmosphere covers a wide range of sizes, from small-scale structures such as magnetic bright dots to the large-scale such as coronal holes. The plasma in the solar atmosphere is governed by the magnetic field. Therefore, the analysis of magnetic fields is necessary to understand phenomena and evolution of structures in the solar atmosphere. The magnetic field measurement in the solar atmosphere is based on methods such as Zeeman effect, Hanle effect or gyroresonance. Proxies of magnetic field strength can be also obtained in an indirect way, for example, from the measurement the intensity of the Ca iiK line. In this chapter, we focus on the Zeeman effect because all magnetograms presented in the thesis are based on this effect.
In 1896 Pieter Zeeman discovered that a static magnetic field causes a splitting in the Fraunhofer D line (sodium doublet at 5890 Å and 5896 Å) into several components (Zeeman 1897). Theses components are polarized which allows to use a polarimetric method and measure the magnetic field.
On the atomic level, the Zeeman effect is due to the removal of degeneracy from atomic levels through the interaction between the external field and the atomic magnetic moment (Shore 2002). The following description is based on Stix (2004). When taking an analytical approach to that effect, first, we define the state of an atom in term of the orbital angular momentum (L), the spin angular momentum (S), total angular momentum (J) and the magnetic quantum number (MJ). For a weak magnetic fields (B) that satisfies the LS coupling, the change of the energy (EJ M) of the atom is:
EJ M = EJ +µ0gMJB. (3.1)
In Eq. 3.1,EJ is the energy of the atomic level in absence of the magnetic field,µ0 = e}/(2mc) is the Bohr magneton (m is the electron mass rest, eis the elementary charge,
cis the speed of light), g is the Landé factor. The Landé factor for each state with LS condition is:
g= 1+ J(J+1)+S(S +1)−L(L+1)
2J(J+1) (3.2)
We now consider two cases of the spin angular momentumS =0 andS , 0 that cor-respond to the normal Zeeman effect and the anomalous Zeeman effect, respectively. In normal Zeeman effectgis one and transition∆MJ = −1,0,1. The spectral line splits into a triplet consisting of two shiftedσcomponents (related to∆MJ = −1,1) and unshiftedπ components (related to∆MJ =0). The difference of wavelengths between the peak of the line center and the peak of theσcomponent (∆λ, eq. 3.3) is proportional to a wavelength (λ) of the center peak and the magnetic field (B):
∆λ∼ λ2B (3.3)
Figure 3.1: The characteristic profile of the the Stokes parameter of the absorption line with longitudinal magnetic field (left) and transverse magnetic field (right). I0is the pro-file with no magnetic field. I is the intensity profile with magnetic field that V is the difference of left and right circular polarization. Qis difference between linear polariza-tions (Stix 2004).
The intensity profile across the spectral line depends on the geometrical configuration of the line-of-sight (LOS) and the magnetic field. In the longitudinal Zeeman effect, the magnetic field is parallel to LOS, then only theσ components are visible. They have a circular polarization with an opposite sign. Theπ component disappears (see left panel of Fig. 3.1). This description is valid for emission and absorption lines. In transverse Zeeman effect, the magnetic field is perpendicular to the LOS then one π and two σ components are observed, all with linear polarization (see right panel of Fig. 3.1). In the case of emission line, theσcomponents are perpendicular toBandπis parallel toB(the opposite for absorption lines).
The normal Zeeman effect is really rare. The normal case is the anomalous Zeeman effect, whenS , 0 andgis calculated from Eq. 3.2 with proper quantum numbers. The
polarization rules for normal and anomalous Zeeman effect are the same. In the observa-tions of the magnetic field in the solar photosphere heavily used lines are Fei(6173 Å) or Nii(6768 Å).