## 2. Basic theory## 2.1. General considerationsOur 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
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, 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 atomThe hamiltonian of the total system (radiating atoms 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 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 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 Consequently the atomic velocity is conserved, and the matrix
elements of 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) Owing to the fact that the interactions of the particles of the
bath
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 particlesWe 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 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 fieldIgnoring collisions now, the total hamiltonian 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 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 radiationWe 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) -
*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 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 The results are the same as in Paper I, except that
is changed into . In
fact, in Paper I, was the azimuth of point
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 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 atomWe 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 ( In the above equation 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 photonsFollowing 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 where is the number of radiating ions per unit of volume. is the population of the lower level
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 Firstly, we have to perform a rotation through the angle
about the 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 and Then, the Stokes parameters of the line emitted at the frequency
in the 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
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 © European Southern Observatory (ESO) 1998 Online publication: November 9, 1998 |