This subsection discusses the formation of chromospheric lines. To this end, a brief in-troduction to the radiative transfer equation is presented. Then the concept of the Local Thermodynamic Equilibrium (LTE) is discussed. Finally, chromospheric lines are de-scribed, especially Mgiih&k. A more detail introduction to the radiative theory is pro-vided by Rybicki and Lightman (1979). A description of the line formation in the solar chromosphere can be found e.g. Stix (2004).
3.2.1 The introduction to radiative transfer theory
The intensity,Iν [W sr−1 m−2 Hz−1], is one of the key quantities in the study of radiative theory. It is defined (eq. 3.4) as the amount of energydE crossing an area dA within a frequency rangedν in a time dt. The inclination (θ) between the line of sight and the normal ofdAis accounted for by a factor cosθ.
Iν = dE
cosθdA dt dΩdν (3.4)
Emission and absorption are responsible for increase and decrease of the intensity.
The added intensity (eq. 3.5) is proportional to the spontaneous emission coefficient, jν
[W sr−1m−3Hz−1], and the path between source and observer (s):
dIν(s)= jν(s)ds. (3.5)
The specific intensity is reduced by absorption. The specific intensity subtracted (eq. 3.6) from the beam is proportional to the intensity, the absorption coefficient,α[m−1], and a distance between source and observer (s):
dIν(s)=−Iν(s)α(s)ds. (3.6)
The ratio between emission and absorption is called the source function,Sν[W sr−1m−2Hz−1], (eq. 3.7):
Sν = jν
αν. (3.7)
The relation between added and subtracted specific intensity is described by the radia-tive transfer equation (eq. 3.8), whereµ= cosθ:
µdIν
ds =−ανIν+ jν. (3.8)
Usually in astrophysics we use the optical depthτinstead of the path lengths. These two quantities are related by:
dτν =−ανds. (3.9)
The optical depth is dimensionless. The medium withτ > 1 is called optically thick (opaque), while forτ <1 the medium is optically thin (transparent). The transfer equation (eq. 3.8) can be rewritten in terms of the optical depth and the source function:
µdIν
dτ = Iν−Sν. (3.10)
The formal solution of this transfer equation is:
Iν(τν, µ)= Iν(τ0, µ)e−(τ0νµ−τν) + 1 µ
Z τ0ν
τν
S(τ0ν)e
−(τ0 ν−τν)
µ dτ0ν, (3.11) where isτ0ν the optical depth at the reference level.
AssumingS =constandµ=1 we obtain:
Iν(τν)= Iν(0)e−τν +Sν(1−e−τν). (3.12) The solution of the transfer equation depends on the optical thickness. Two extreme cases are presented below:
• optical thick medium: τ7→ ∞ ⇒Iν = Sν;
• opitical thin medium: τ7→0⇒ Iν = Iν(τν)+[Sν−Iν(τν)]τν.
The solution of the transfer equation for an optically thick medium is independent of the optical depth (e.g. for the black-body radiation). In an optically thin medium (τ < 1) the relation between Iν(τν) andSν determine the emission and absorption under the following conditions:
• emission line: Iν(τν)<SνorIν(τν)= 0;
• absorption line: Iν(τν)> Sν.
The information in this subsection is based on the Rybicki and Lightman (1979) and Stix (2004).
3.2.2 Local Thermodynamic Equilibrium (LTE):
Before we apply the radiative transfer equation to describe line formation in the chro-mosphere we must discuss the thermal properties of the solar atchro-mosphere, especially the concept of Local Thermodynamic Equilibrium (LTE).
Thermodynamic equilibrium (TE) is a state where the entire medium has the same temperature. In other words, a single value of temperature is sufficient to characterize the medium. At that temperature, particles have a Maxwellian velocity distribution at that temperature, ionization and excitation states satisfy Saha and Boltzmann equations for that temperature and the radiation field is a thermal black-body radiation which is described by the Planck function (eq. 3.13):
Bν = 2hν3/c2
exp(hν/kT)−1, (3.13)
again with the same temperature.
In the solar atmosphere the temperature changes from approximately 6000K in the photosphere to more than 1 MK in the solar corona. This implies that solar atmosphere is not in thermal equilibrium. However, locally this is the case, at least in the photosphere.
This state is called the Local Thermodynamic Equilibrium (LTE). In LTE the source func-tion is expressed by Planck funcfunc-tion (eq. 3.14):
Sν = Bν. (3.14)
In the rarefied medium, the mean free path of particles is larger than a distance of the temperature changes. This state is called non-Local Thermodynamic Equilibrium (non-LTE or N(non-LTE). The N(non-LTE condition usually exists in the core of the strong lines.
Otherwords, in the LTE condition temperatures of radiation (Trad), excitation (Tex), ionization (Tion) and gas temperature (Tgas) are the same, the excitation and ionization states satisfy the Boltzmann and Saha equation. In the non-LTE condition at least one of temperatures is different than rest, for example in the solar corona the excitation tem-perature (Tex) are not with equilibrium with gas (Tgas) and ionization (Tion) temperature.
Additionally, in the solar corona the states of ionization of the atom does not satisfy the Saha equation.
Information presented in this subsection is based on Stix (2004).
3.2.3 The line formation
The photospheric emission covers the spectral range from the UV (λ > 1600Å) to the IR (λ < 100µm). The usual approximation is a black-body radiation, with a temperature at 5778 K. The solar photosphere is optically thick and satisfies the LTE conditions, however in this layer the temperature decrease with height. Therefore, the Earth and the space-based observer sees that cooler material cover the underlying hotter part of the solar pho-tosphere. In that condition, the cooler material absorbs photons of some wavelength from the photospheric continuum emission and creates absorption lines (Fraunhofer lines).
In the chromosphere, the temperature increases with height in a result of non-thermal heating. In that condition, one of the strongest chromospheric line, Hα(6563 Å), is formed.
The Hαemission is the main source of the reddish light observed immediately before or after the maximal phase of the total solar eclipse. The spectroscopic observation of the solar disk in Hα line shows a deep absorption profile, however above the solar limb it turns into emission. This suggests that the chromosphere is mostly transparent for the photospheric radiation, but the Hαline is still opaque and creates an absorption profile in solar disk observations.
The CaiiH&K (3969 Å and 3934 Å respectively) and Mgiih&k (2803 Å and 2796 Å respectively) lines play an important role in diagnostics of the solar chromosphere. They are strong, collisional controlled, resonance lines with similar spectral profiles showing an emission reversal. However, the chemical elements that create these lines have diff er-ent abundances 1, magnesium (logMg=7.60± 0.04) has larger abundance than calcium (logCa=6.34 ± 0.04) (Asplund et al. 2009). This implies that Mg ii is formed higher
1The abundance is defined as logX=log(NX/NH)+12, where the hydrogen abundance is logH=12 (Asplund et al. 2009).
Figure 3.2: The average spectrum of the Mg ii h&k line observed by IRIS (Kerr et al.
2015). On the reversal profile of the quiet Sun are marked the intensity minimum (k1), the emission peak or emission maxima (k2) and the line center or central deep (k3). The k1 and k2 points located on the left or on the right respect to the k3 points are called violet (thereafter v) and red (thereafter r), respectively.
in the chromosphere than Ca ii(Leenaarts et al. 2013b). Figure 3.2 presents a profile of Mg ii k line with the emission reversal. When the cooler medium covers the medium with higher temperature the observer sees the absorption profile as in Mg ii wings. For the Mg ii spectral line the emission created at higher temperatures is closer to the line center. Therefore, intensity minima (k1) correspond to the temperature minimum at the upper photosphere and the lower chromosphere. In the chromosphere the temperature increases with height, therefore the cooler photospheric medium is covered by the hotter chromospheric medium. The absorption in this cooler material can be so significant that radiation from hotter layers begins to be visible as a clear emission profile. Moving up towards higher temperature the intensity increases. This corresponds with the emission profile between k1 and k2. Moving further up the temperature continuously increases and the density decreases so LTE conditions are not satisfied. The line center (k3) originates under such non-LTE conditions.
The Mg iiand Ca iilines are used to investigate the thermal properties of the upper photosphere (k1), mid chromosphere (k2) and upper chromosphere (k3) as well as to study the plasma velocity in the chromosphere. They are also used as a proxy of solar and stellar activity.