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Astron. Astrophys. 333, 1130-1142 (1998)

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2. The H [FORMULA] absorption profile

The starting point of our theory is to model the photospheric incident radiation around the H [FORMULA] line by a continuum from which we substract a gaussian profile to fit the atlas data by Delbouille et al. (1973 ).

Such a profile is written as (see Fig. 1)

[EQUATION]

where the constants are fixed by the data available in the atlas quoted (see Fig. 1).

[FIGURE] Fig. 1. Plot of the observed solar spectrum between 6540 Å and 6580 Å (full line). The continuum around H [FORMULA] is arbitrarily set to 1000. Dotted line: fit to the observed spectrum by means of a gaussian profile (see text).

The result of the fit brings to

[EQUATION]

and to

[EQUATION]

corresponding to a line width at half maximum of 1.41 Å .

2.1. The density matrices of the problem

Taking into account the particular form of the profile, we use Eq. (1) of Sahal-Bréchot et al. (1992 ) to obtain the density matrix of the incident photons.

We recall that an atom having an individual velocity [FORMULA] absorbs, in its atomic frame, at the frequency [FORMULA] such that

[EQUATION]

where [FORMULA] is the unit vector of the ray of light whose direction is specified by the angles ([FORMULA]) (see Fig. 2.1), and [FORMULA] is the atomic frequency of the unperturbed transition corresponding to an upper level u and a lower level l which are assumed to be infinitly sharp (coherent redistribution in the atomic frame).

[FIGURE] Fig. 2. Coordinates of the scattering atom A located at the height h ; Oz is the preferred direction of incident radiation; AZ is the line of sight; Ax and AX are identical and perpendicular to the scattering plane ZAz ; [FORMULA] is the scattering angle and equal to [FORMULA] if (case of figure) A is in the plane of the sky, then [FORMULA] and [FORMULA]. The incident light comes from the spherical cap limited by the angle [FORMULA] ; each point P of the cap emits a ray [FORMULA] whose coordinates are the angles [FORMULA] in the frame Axyz. The atom A has the velocity [FORMULA] and the mean velocity of the ensemble is [FORMULA] ; [FORMULA] and [FORMULA] are the unitary vectors of the directions [FORMULA] and AZ. The angle [FORMULA] is the angle between the normal to the surface of the Sun and the incident light direction at point [FORMULA]. The direction of the Stokes parameter [FORMULA] and [FORMULA] is that of a totally polarized radiation along Ax or AX. The direction of polarization is given by the angle [FORMULA] with AX or Ax. The angle [FORMULA] lies in the plane of the sky (xAz).

The density matrix now reads

[EQUATION]

where [FORMULA] is the limb darkening coefficient, and where the index K runs from 0 to 2. [FORMULA] is a unitary matrix characterizing the angular behaviour of an unpolarized radiation beam propagating in the direction ([FORMULA]) (see Eq. (43) of Sahal-Bréchot et al. 1986 ).

Let R be the distance of the atom to the center of the Sun, and [FORMULA] the solar radius; then the angle [FORMULA] is defined by the equation (see Fig. 2.1)

[EQUATION]

The scalar product [FORMULA], depending on ([FORMULA]), leads to the appearance of coherences [FORMULA], [FORMULA], [FORMULA] in [FORMULA]

As we shall see later on, the Stokes parameters [FORMULA] of the H [FORMULA] line are obtained once the density matrix [FORMULA] of the reemitted photons is specified in the observer's frame AXYZ of Fig. 2.1.

In order to obtain the frequency dependence of the total scattered radiation we take into account the contributions of all atoms contained in a unit of volume of radiating matter. Due to the Doppler effect, each frequency [FORMULA] of the scattered radiation corresponds to an atom having a velocity [FORMULA] such that

[EQUATION]

We need to perform the average of the density matrix [FORMULA] over the atomic distribution of velocities projected onto a plane perpendicular to the line of sight, i.e., over [FORMULA] and [FORMULA], and to multiply by the density [FORMULA] of the H atoms (Sahal-Bréchot et al. 1998 ). The distribution of the velocities is (Sahal-Bréchot et al. 1992 , Sahal-Bréchot et al. 1998 )

[EQUATION]

where [FORMULA] or [FORMULA] is the velocity vector of the stream of radiating H atoms, and where [FORMULA] and k are, respectively, the atomic mass of Hydrogen and the Boltzmann constant.

We write the velocity distribution [FORMULA] in units of frequency using the following transformations

[EQUATION]

also

[EQUATION]

and thus

[EQUATION]

A temperature [FORMULA] K (corresponding to the value [FORMULA] s-1) is a reasonable assumption for the atomic radiators in the spicule.

We define, as usual, the polarization degree [FORMULA] and the angle of rotation [FORMULA] of the polarization direction by the formulae

[EQUATION]

where the positive Q direction is defined as the tangent to the solar limb.

2.2. The equations of statistical equilibrium

We use the theory of atom-radiation interaction in the density matrix formalism given by Bommier (1977), and in particular Eq. (III-39) of that paper (see also Eq. (36) of Bommier & Sahal-Bréchot 1978 ).

Neglecting coherences between level [FORMULA] and [FORMULA] (with [FORMULA]) and neglecting the possible presence of a magnetic field, the equations assume the form given in 7, where [FORMULA] is the Einstein coefficient for spontaneous de-excitation between the levels [FORMULA] and where [FORMULA] is given by Eq. (3).

We set

[EQUATION]

looking for solutions of a stationary system. In these equations we take into account that the radiation field in the Ly [FORMULA] and Ly [FORMULA] is isotropic, because these lines are opticaly thick in the solar chromosphere, so that

[EQUATION]

Our further assumption is to neglect any velocity effect on the profile of the Ly [FORMULA] and Ly [FORMULA] lines for the radiating atom. If, for instance, we consider a velocity of 60 km/s for the atom, the shift in wavelength is of the order of 0.2 Å for both lines. At this distance from line center the profiles are nearly constant (Gouttebroze et al. 1978 ).

We also neglect the central dip of the profiles that would affect the radiator for smaller velocities.

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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