Astron. Astrophys. 340, 579-592 (1998)

2. Basic theory

2.1. General considerations

Our purpose is to calculate the Stokes parameters of a spectral line emitted by a moving ensemble of atoms which scatter the same line (resonance fluorescence) emitted by an underlying surface. As for example, we will treat the case of the O vi 103.2 nm line emitted by an ensemble of atoms located at a point A in the corona, at a height h above the solar surface. They have a macroscopic velocity and scatter (absorb and reemit) the same line emitted by the underlying transition region of the sun (cf. Fig. 3). Owing to the anisotropy of the incident radiation which comes from the spherical cap limited by the angle , the reemitted line is linearly polarized; in the absence of any perturbing effect, the direction of the linear polarization of the scattered line should be parallel to the solar limb, except at very low heights ( in units of solar radius) where it becomes radial if the limb-brightening is taken into account (Sahal-Bréchot et al. 1986).

Fig. 3. Coordinates of the scattering atom A located at the distance r (in units of solar radius) from the center of the sun O; h is the height above the limb (); 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; is the scattering angle; if it is equal to , which is the case of this figure where the atom is in the plane of the sky, then , and . The incident light comes from the spherical cap limited by the angle , each point P of the cap emits a ray referred by the angles () in the frame Axyz. The atom A has the velocity ; and are the unitary vectors of the directions and AZ.

In Sahal-Bréchot et al. (1986), the modification of the polarization of the O vi 103.2 nm line by a magnetic field (the Hanle effect) was studied. However, the Doppler effect due to the macroscopic velocity of the scattering atoms and which induces an angular dependence of the incident radiation intensity was outside the scope of that paper (cf. the end of Sect. 2.1. of Sahal-Bréchot et al. 1986). In the present one the Hanle effect is not taken into account but the Doppler effect is introduced. As we will show in the following, this angular dependence of the incident radiation can also modify the intensity and polarization parameters of the scattered line.

As far as possible, we will use the same notations as Sahal-Bréchot et al. (1992b) and Sahal-Bréchot et al. (1986) which will be referred to as Paper I hereafter.

The density-matrix formalism that we use is based on the pioneer works by Fano (1957) and Cohen-Tannoudji (1962); then it has been reinforced (Cohen-Tannoudji 1977, Omont 1977, Cohen-Tannoudji et al. 1988).

This basic density-matrix formalism was extended to a complex medium for the first time for solar prominences studies by Bommier (1977) and by Bommier & Sahal-Bréchot (1978): multilevel atom coupled to an anisotropic beam of photons not perfectly directive and to a magnetic field, the strength and direction of which are unknown and may "a priori" be arbitrary. In addition, excitation by isotropic collisions has been included in Paper I.

In that formalism, rederived from the basic equations of quantum electrodynamics by Bommier & Sahal-Bréchot (1991), the coupling of an ensemble of independent identical radiating atoms with an incident anisotropic radiation field, a magnetic field and collisions (which can be also anisotropic) has been treated in a non-phenomenological manner. In that work the atoms have been assumed at rest and their levels infinitely sharp (i.e. the natural and collisional broadening of the lines is negligible).

The theory has been recently revisited by Bommier (1997a, 1997b), who has included line-broadening effects, through a perturbation development of the matter-radiation interaction up to orders higher than 2. Since line-broadening effects are neglected in the present paper, it can be deduced from Bommier (1997a) that effects of orders higher than 2 will vanish in the present case: consequently the lowest non-zero order (order 2) of the matter-radiation interaction will be the only one to play a role as in Bommier & Sahal-Bréchot (1991).

The problem can be reduced in a first step to the study of one atom only interacting with the ensemble (the "bath") of perturbers. The bath is composed of photons and colliding particles which act independently: this is the impact approximation. The coupling with the particles of the bath leads to the "master equation" for the atomic density matrix, and then to the statistical equilibrium equations at the stationary state, the solution of which being the "populations" of the Zeeman states of the atom and the "coherences" between these Zeeman states (see for instance Bommier 1996 for definitions). These equations can be considered as a generalization, convenient for polarization studies, of the usual non-LTE statistical equilibrium equations for the atomic populations.

In a second step, the polarization matrix characterizing the intensity and polarization of the emitted photons in the optically thin case is obtained through a trace of the emissivity operator over the states of the upper level of the atomic line studied (Bommier 1977; formula (37) of Bommier & Sahal-Bréchot 1978):

where is the angular frequency of the radiation, c the velocity of light, the atomic dipole and the atomic density matrix.

Bommier (1997a) has shown that the above expression (1) is valid only within the second order expansion of the matter-radiation interaction.

In the third step, the Stokes parameters are obtained through a projection of that polarization matrix on a plane perpendicular to the line-of-sight and through adequate linear combinations of its matrix elements. Then the Stokes parameters of the emitted line which have thus been obtained for one atom only are summed over the total number of atoms in the element of volume (Bommier 1977 and Bommier & Sahal-Bréchot 1978 for the optically thin case).

In the final step the integration of these Stokes parameters over the line-of-sight is performed.

Bommier (1991, 1997a, 1997b) has studied the optically thick case and has rederived the radiative transfer equations in the presence of a magnetic field for the Stokes parameters through the density-matrix formalism (cf. also Landi Degl'Innocenti 1983, 1984).

In all these papers the ensemble of atoms has been assumed at rest: thus no average over the atomic velocities has been performed.

In the present paper we will extend the formalism to an assembly of moving atoms having an anisotropic distribution of velocities with a macroscopic velocity .

The present formalism will be limited to the optically thin case, which is suitable for the lines of the solar wind acceleration region. We will continue to assume that the absorption and emission profiles are infinitely sharp in the atomic rest frame. This means that the intrinsic widths of the lines are very small compared to the frequency distribution variation of the incident radiation spectrum. This is valid in the physical conditions of the solar corona, because the density is very low and thus the collisional widths are very small. Consequently the intrinsic widths are restricted to the natural ones which are also very small.

2.2. Introduction of the atomic motion in the master equation for the radiating atom

The hamiltonian of the total system (radiating atoms A in interaction with a bath of particles consisting of colliding particles C, and photons R, and in the presence of a magnetic field ) is given by

where is that of the interaction of the atom with the magnetic field , that of the photons, and that of the colliding perturbers. is that of the interaction atom-radiation (or atom-photons) and that of the collisional interaction.

We will include in (weak field limit). The atomic wave-functions are unchanged and the internal energies become those of the Zeeman states

The eigenstates of the atom at rest are denoted as

The total system being isolated and reversible, its evolution can be described by a density matrix , solution of the Schrödinger (or Liouville equation for the density matrix) equation

The evolution of the subsystem A is described by the density matrix , defined by a trace over all the states of the bath

Within the usual language of classical mechanics, this trace corresponds to an average over all the particles of the bath.

As shown by Cohen-Tannoudji (1977), (cf. also Cohen-Tannoudji et al. 1988), the subsystem A can be described by the reduced density matrix . Yet we have now to derive the time evolution of , which is not given by a Liouville equation, because we have introduced some irreversibility in the above process: A is no longer an isolated system. The time evolution of is now given by a "master equation" and describes the evolution of a "mean" atom. In the corresponding classical picture, the time evolution of the classical equivalent to is given by the "kinetic equation" (Oxenius 1986). It can be shown that the kinetic equation for the radiating atoms or molecules reduces to the so-called "statistical equilibrium equations" at the steady state.

We have to solve the master equation in a fixed reference frame, the so-called laboratory frame. In this frame, the atomic velocity operator is denoted by and the atomic mass by . Following Smith et al. (1971) and Nienhuis (1976, 1977), we introduce the atomic motion in the atomic hamiltonian by adding the translation hamiltonian to the atomic hamiltonian at rest .

The atomic hamiltonian in the laboratory frame is

with

where is the atomic momentum operator. in the non relativistic limit. The corresponding eigenstate of is the plane wave , with eigenvalue

The eigenstates of take the form

and the corresponding eigenvalues are

We assume that collisional and radiation interactions do not modify the state of translation of the atom: this implies that the recoil of the atom during an interaction and the radiation pressure effects can be neglected. At the considered temperatures, the atomic momentum is much larger than the atomic electron one because of the high ratio. It is also much larger than the photon momentum , being the angular frequency and c the velocity of light. Consequently, the momentum transfer during the interaction process is negligible and the atomic velocity is unchanged (Cohen-Tannoudji et al. 1988, p. 286). Apart from the recoil effect, the atomic velocity can change under the effect of collisions (Oxenius 1986, Sahal-Bréchot et al. 1992a, Landi Degl'Innocenti 1996): this change can be determined by calculating the differential cross-section (Balança & Feautrier 1998, Balança et al. 1998). In fact, owing to the relative mass effect, only elastic collisions with particles having about same masses or higher masses than the radiating atom are to be considered. In solar and coronal conditions, these collisional rates are very weak and the atom has time to emit a photon or to be excited or deexcited by any process before its velocity could change (cf. Sahal-Bréchot et al. 1992a).

Consequently the atomic velocity is conserved, and the matrix elements of H can be considered as diagonal in .

Therefore, as for the external states , the off-diagonal terms (with ) will be completely decoupled from the diagonal terms which represent the density of probability (with respect to the external states ) for the density matrix (with respect to the internal states ) of an atom having the velocity . We can denote this density matrix as , the elements of which are given by

Moreover, there will be no coupling between the reduced density matrices and for different velocities and .

For the internal states , will be normalized to the distribution of atomic velocities :

with the normalization condition

Then, within the impact approximation, the interactions with the radiation (the photons) R and the colliding particles C (which are the electrons of the corona in the present paper) are decoupled (cf. Bommier & Sahal-Bréchot 1991 and earlier papers). Using the no-back reaction approximation and the impact approximation, as well as the fact that the evolution of the reduced atomic density matrix is markovian, i.e., it depends only on the instant t and not on the past history of , one obtains the master equation for (cf. Bommier & Sahal-Bréchot 1991 and Sahal-Bréchot et al. 1992a for the atom at rest), which describes the evolution of the atom coupled to the bath in the laboratory frame (cf. also Cohen-Tannoudji 1977 and related papers, Omont 1977, Bommier 1997a). The coarse grained approximation leads to achieve the average over the states of the bath over a time interval s much larger than the mean duration of a collision (resp. for the radiation) and much smaller than the mean interval between two successive collisions (resp. , inverse of the lifetime). We obtain

Owing to the fact that the interactions of the particles of the bath B are decoupled, we can write

S is the scattering matrix calculated for a complete interaction with one colliding perturber (subscript C) or with one photon (subscript R). The density matrix of the photons and the density matrix of the colliding perturbers take place for averaging over all the particles, i.e. over the states of the bath B (radiation R and collisions C that are decoupled).

Then we obtain the projected master equation averaged over the states of the bath (cf. also Cohen-Tannoudji et al. 1988, p. 486):

where is the Bohr frequency and and are Kronecker symbols.

2.3. Coupling with the colliding particles

We show now that the collisional interaction is not affected by the atomic motion for the processes studied in the present paper: electrons are very much faster than the radiating ions which can thus be assumed at rest. This would not have been the case if we were studying collisions between heavy particles such as protons and hydrogen atoms for instance (cf. Sahal-Bréchot et al. 1996). Herewith the collisional S matrix has to be calculated within the usual framework of the theory of collisions. The colliding perturbers are isotropic maxwellian electrons at a temperature . Owing to that isotropy, the collisional coupling coefficients in the master equation link only the Zeeman populations , . In addition the coronal electronic densities are so low that depolarizing collisions are negligible: the photon is emitted before any occurrence of a collision. The only collisional processes that are to be taken into account are excitation from the lower level towards the upper level of the considered transition. Deexcitation by collisions is negligible. The velocity that takes place in the calculation of the cross-section is the relative one which can be considered as equal to the electron velocity .

Thus the problem is rather straightforward. We obtain after integration over the Maxwell isotropic distribution of electron velocities at the temperature and multiplication by the electron density

In order to conclude this paragraph, the collisional cross-sections and their average over the electron distribution of velocities are not affected by the atomic motion. The effect of the atomic motion only appears in the matrix elements of the atom-radiation field interaction that we will study in the following.

2.4. Coupling with the radiation field

Ignoring collisions now, the total hamiltonian H of the system is that of the subsystem atom+radiation and can be written as

where is the hamiltonian of the radiation field and is the atom-radiation interaction

where is the atomic dipole which can be calculated in the basis of the atom at rest (Smith et al. 1971), is the electric field of the radiation and is the atomic position operator in the laboratory frame.

Using the second quantization formalism, the electric field can be developed in plane waves as

for the mode , defining the direction of unitary vector , the angular frequency and the polarization (polarization vector ). and are the operators of creation and annihilation of photons for the radiation propagating in the direction with the polarization . The normalisation chosen is such that

where is the specific intensity.

By expanding the dipole moment and the polarization vector over the standard basis , , of the laboratory frame (see Bommier & Sahal-Bréchot 1991 or Bommier 1997a for definition), we obtain

The atom-radiation interaction contains the atomic motion through the factors .

The developments made by Bommier & Sahal-Bréchot (1978, 1991) for the atom at rest can now be rewritten for a moving atom by including these factors.

By using, in interaction representation,

it can be shown that the results of Bommier & Sahal-Bréchot (1991), obtained for an atom at rest at , can be applied to the present case by replacing any radiation angular frequency which appears in the terms of that calculation by , being the corresponding wave vector.

Due to the Markov approximation which leads to limit the perturbation expansion of the matter-radiation interaction to the lowest non-zero order (the second one), the absorption and emission line profiles are infinitely sharp: this is valid if the intrinsic line-widths are very small compared to the frequency distribution variation of the incident radiation spectrum. All this is valid in the solar corona.

Thus the scattering is monochromatic in the atomic rest frame. Following Hummer (1962) and Mihalas (1978, p. 412), this corresponds to Case I of the redistribution process (Doppler redistribution).

Therefore, as the line profile is now given by a -function

the only change is to replace the angular atomic frequency (frequency corresponding to the energy difference between the levels u (upper) and l (lower) which are assumed infinitely sharp) by

in the laboratory frame. This is expected: the atom absorbs the radiation at its eigenfrequency in its own `comoving' frame (the atomic frame), and due to the Doppler effect, this corresponds to the angular frequency

of the incident radiation in the laboratory frame.

We are now able to write the system of statistical equilibrium equations leading to the populations and coherences for the atom having the velocity , interacting with an incident flux of radiation characterized by its polarization matrix . We will solve this system in the atomic frame.

The first step is to obtain the elements of the polarization matrix which enter the coupling coefficients of the elements of the master equation leading to the atomic density matrix .

2.4.1. Calculation of the components of the polarization matrix of the incident radiation

We will use the same notations as those of previous papers (Paper I and Sahal-Bréchot et al. 1992b).

We will assume hereafter that the incident line profile is the same for all the points of the solar surface: we will neglect inhomogeneities and take only into account the limb-brightening effect. Inclusion of inhomogeneities of the incident radiation was studied in Paper I and is outside the scope of the present paper.

The atomic frame (the comoving frame) Axyz and the line-of-sight frame (the laboratory frame) AXYZ are defined in the following manner (cf. Fig. 3):

• AZ is the line-of-sight oriented towards the observer, its unitary vector is denoted by .

• Az is the axis of symmetry of the system atom A + incident photons, that is the vertical to the surface of the Sun. It is oriented towards the outside.

• The scattering plane contains the vertical Az and the line-of-sight AZ.

• The perpendicular to the scattering plane defines the common axis Ax and AX. Since there are "a priori" two possible senses for Ax (or AX), the angle between Az and AZ is oriented. The sense of Ax (or AX) gives the sign of which is thus not a polar angle.

• Ay and AY follow.

The polar angle and the azimuth define the direction of the atomic velocity in the atomic frame. Likewise the polar angle and the azimuth define the direction of the velocity of the ensemble of atoms in the atomic frame.

The polarization matrix of the incident photons which was obtained for the atomic frequency only in Paper I is now frequency-dependent through the Doppler effect.

The radiation incoming from an element of volume centered on a point P of the solar surface in the direction (unitary vector ) is characterized by the angles and in the atomic Axyz frame. , , is the angular distribution of the local intensity at point A for the frequency (erg cm^-2 s^-1 sr^-1 Hz^-1) and incoming from point P:

and (cf. Fig. 3)

The polarization matrix is obtained through an angular average over all the directions of incoming radiation as follows:

The matrix characterizes the angular behaviour of a dipolar unpolarized radiation incoming in the direction and is normalized to unity (Eq. (43) of Paper I). It can be expanded in multipoles terms over the basis of the irreducible tensorial operators relative to the quantization axis Az which is also adapted to the present problem:

The results are the same as in Paper I, except that is changed into . In fact, in Paper I, was the azimuth of point P, referred in cylindrical coordinates in the atomic frame Axyz. In the present paper, is the azimuth of the direction in the atomic frame.

Thus, using relations (23) and (24), the polarization matrix of the incident radiation becomes velocity-dependent and we can write

The angular dependence of on the angles and leads to appearance of coherences (, , ) in the matrix elements of (in the atomic Axyz frame). This will lead to a rotation of polarization of the scattered line because the cylindrical symmetry about the Az axis is broken. We can also verify that the coherences vanish when the atomic velocity is directed along the vertical (i.e., ), because the cylindrical symmetry is recovered.

Formula (28) is general and does not assume anything on the incident radiation. Thus we have obtained a general expression for the elements of the polarization matrix of the incident radiation that enter the statistical equilibrium equations that have to be solved for each atomic velocity . In fact, it is also not restricted to the two-level atom.

2.4.2. Expression of the statistical equilibrium equations for the atomic density-matrix and for a two-level atom

We refer to Eq. (5) of Paper I in zero-magnetic field, and we use Eq. (28) of the present paper for the polarization matrix of the incident radiation. We obtain for a two-level atom (u = upper and l = lower level) in the atomic reference frame in the basis of the irreducible tensorial operators

In the above equation k can be equal to 0 (population component) or 2 (alignment component). and are the Einstein cofficients for spontaneous emission and absorption, the collisional excitation coefficient (obtained from Eq. (14)). is the Kronecker symbol: in fact the collisional processes are isotropic and cannot create alignment and we recall that they are also insensitive to the atomic velocity.

The symbol between brackets is a "6j" coefficient. Stimulated emission is negligible and is not introduced. Likewise collisional deexcitation is negligible. The only relevant processes are excitation by collisions and radiation followed by spontaneous emission.

2.4.3. Expression of the polarization matrix of the reemitted photons

Following Eq. (49) of Paper I and using the present equation (28), the polarization matrix of the photons reemitted by an atom having the velocity can now be written. being expressed in number of emitted photons per second, per unit of volume and per unit of distribution of atomic velocities , we obtain in the Axyz atomic comoving reference frame

where is the number of radiating ions per unit of volume.

is the population of the lower level l (the ground state), is that of the excited state u:

The normalization condition for the internal states (Eq. (10)) leads to

for the two-level atom. Since the population of the excited level is very small compared to that of the ground state, (case of the solar corona), the preceding Eq. (33) reduces to

being the distribution of atomic velocities.

is defined in Paper I, according to Bommier & Sahal-Bréchot (1982). It can be expressed in terms of "6j" coefficients as

It can be noticed that is equal to the coefficient

defined in Landi Degl'Innocenti (1984) where these coefficients are tabulated. The notation of Stenflo (1994, pp. 184-187 and p. 91) for is . We have (Paper I):

In order to obtain the polarization matrix of the reemitted photons in the line-of-sight fixed frame AXYZ (the laboratory frame), we have to rotate the coordinates system and to use the proper atomic eigenfuctions including translation states.

Firstly, we have to perform a rotation through the angle about the Ax-axis (or AX). In fact we will modify Eq. (50) of Paper I which was obtained for . The resulting polarization matrix is denoted by .

The Euler angles of the rotation are

We obtain after calculations

The other components are not written because they do not enter the expression of the following Stokes parameters.

Secondly, the atom reemits the line at the frequency in the comoving atomic frame. We introduce the translation motion dependence in the atomic wavefunctions of the laboratory frame.

This is equivalent to write that, due to the Doppler effect, the observer sees a line emitted at the shifted frequency

along the line-of-sight AZ characterized by the direction . We have

and

Then, the Stokes parameters of the line emitted at the frequency in the AZ direction by the atoms having the given projected velocity can be obtained.

Firstly, for the atoms having the velocity ,, we have (Paper I)

in number of photons per steradian.

Secondly, we have to sum over all the atoms having the given projected value for obtaining the frequency distribution of intensity and polarization of the reemitted radiation. Therefore we have to integrate over the atomic velocities in the XAY plane perpendicular to AZ. We obtain

and

This is the general formula giving the Stokes parameters, as a function of frequency, of the reemitted line (in number of photons per unit of volume of matter, per unit of time, per unit of frequency and per unit of solid angle in the direction ) for any atomic velocity distribution. In fact, the formal formula (40) is not restricted to the two-level atom.

The degree of linear polarization p and the angle of the polarization direction with respect to the AX direction follow (cf. Fig. 2):

© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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