Doppler redistribution of anisotropic radiation and resonance polarization in moving scattering media

Transcription

1 Astron. Astrophys (1998) Doppler redistribution of anisotropi radiation and resonane polarization in moving sattering media I. Theory revisited in the density matrix formalism S. Sahal-Bréhot V. Bommier and N. Feautrier ASTRONOMY AND ASTROPHYSICS Département Atomes et Moléules en Astrophysique Unité Assoiée au CNRS n 812 Observatoire de Paris-Meudon F Meudon Frane ( Sylvie.Sahal-Brehot@obspm.fr Veronique.Bommier@obspm.fr Niole.Feautrier@obspm.fr) Reeived 29 May 1998 / Aepted 22 September 1998 Abstrat. Under the light of reent developments of the theory of matter-radiation interation in the presene of magneti field applied to non-lte spetropolarimetry in astrophysis we have revisited the theory of anisotropi resonane line sattering in moving media by means of the density-matrix formulation. This has led us to present a theoretial method of determination of the matter veloity field vetor in the solar wind aeleration region. The example of the O vi 13.2 nm line has been hosen for putting this theory into operation. It has been observed by the ultraviolet spetrograph Sumer of Soho in different regions of the solar wind aeleration region; it is partially formed by resonane sattering of the inident underlying transition region radiation whih ompetes (and an predominate) with eletron ollisional exitation at the low densities whih prevail at these high altitudes. The theory whih is developed hereafter not only shows that this line is shifted and its intensity dimmed by the Doppler effet due to the matter veloity field of the solar wind but also predits that it is linearly polarized owing to the anisotropy of the inident radiation field; its two linear polarization parameters degree and diretion of polarization are sensitive to the matter veloity field vetor. Our results show that the interpretation of polarimetri data assoiated to the shift and the Doppler-dimming effet may offer a method of diagnosti of the omplete veloity field vetor provided that the partial anisotropy of the inident radiation field be taken into aount. In fat suh a diagnosti is urrently missing. Yet its interest is ruial to understand various problems in astrophysis suh as stellar winds and espeially the aeleration mehanisms of the solar wind. It is also essential for a dynamial modelling of solar strutures. Key words: atomi proesses line: formation line: profiles polarization Sun: orona Sun: solar wind Send offprint requests to: S. Sahal-Bréhot Correspondene to: Sylvie.Sahal-Brehot@obspm.fr 1. Introdution Aurate knowledge of the physial onditions whih prevail in the solar orona and in the solar wind aeleration region is ruial for understanding the physial mehanisms whih are at the origin of oronal heating mass and energy transport and aeleration of the solar wind. In partiular determination of both magneti and matter veloity field vetors is essential sine their oupling must be taken into aount in magnetohydrodynamial modelling. Determination of salar quantities suh as temperatures and densities an be ahieved through usual spetrosopi methods based on interpretation of the frequeny dependene of the intensity of suitable lines or of the ontinuum of the observed radiation field. Yet determination of vetorial quantities is more omplex: in fat the omplete information on strengths and diretions is ontained in the polarization parameters of the reeived radiation. Intensity transports a part of the information only. Consequently for ahieving a omplete vetor diagnosti polarimetri measurements are required and theoretial methods of spetropolarimetri diagnostis leading to all the Stokes parameters of adequate lines must be developed. During the twenty past years methods of determination of vetor magneti fields in astrophysis have been developed (Bommier 1977; Bommier & Sahal-Bréhot 1978 and further papers) leading to interpretation of the linear polarization parameters of lines affeted by the Hanle effet in terms of magneti field vetors in solar prominenes (Leroy et al ; Bommier et al a 1986b; Bommier et al. 1994; Bommier & Leroy 1997). This has been arried out with the help of the quantum theory of the matter-radiation interation within the density matrix formalism (Fano 1957; Cohen-Tannoudji ; Cohen-Tannoudji et al. 1988). The entirely new vetorial data whih have been obtained for solar prominenes have hanged our knowledge on their magneti struture; they have shown the interest of having developed a method apable of determining the omplete vetor field and not only its projetion on the line-of-sight as a result of onventional Zeeman studies. Likewise it would be ertainly interesting to also develop a method of determination of the matter veloity field

2 58 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I x X va Ι(ν) Ι(ν) Diretion of inident unpolarized radiation ν ν=ν (1+v A / ) z A v Az z ν =ν (1+v A / ) Z Diretion of Z sattered radiation Ι (ν ) ν v AZ Diretion of polarization of sattered radiation Fig. 1. Resonane sattering in the perfetly diretive ase: A is the sattering atom with veloity v A. Az is the diretion of inident radiation line. Its intensity I(ν)is frequeny-dependent. AZis the line-of-sight. The atom absorbs and reemits the inident light at the eigenfrequeny ν in its atomi frame at rest. Due to the Doppler effet in the laboratory frame the atom absorbs the frequeny ν = ν (1 + v Az /) and reemits the line in the AZ diretion at the frequeny ν = ν (1 + v AZ /). Thus the sattered line is shifted and dimmed through I(ν)/I(ν ). The shift is sensitive to v AZ and the dimming to v Az. The sattered line is linearly polarized along the perpendiular to the sattering plane but the degree of polarization does not depend on the atomi veloity: for a sattering at right angles it is equal to 1 for a normal Zeeman triplet and to.428 for a J =3/2 J =1/2 line. vetor. Usual methods based on the interpretation of the Doppler shift of spetral lines yield only the omponent along the lineof-sight. The first attempts for determining one more omponent of the matter veloity field of the solar aeleration region have been arried out by Gabriel (1971) and Bekers & Chipman (1974) and then by Kohl & Withbroe (1982) and Withbroe et al. (1982a 1982b). Following Gabriel et al. (1971) who observed high in the orona during an elipse the Lyα line of hydrogen formed by resonane sattering of the inident Lyα hromospheri radiation they remarked that a number of other interesting lines of the transition region should also be observed high in the orona: they should be partially formed by resonane sattering of the inident transition region radiation. In partiular Li -like ion lines should be expeted and the O vi 13.2 nm line should be one of the most intense: in fat these ions have broad abundane urves and thus a suffiient amount of ions an remain at oronal temperatures allowing their detetion. Besides these ions may also be frozen in within several solar radii (Bame et al. 1974; Withbroe et al. 1982a 1982b; Kohl & Withbroe 1982) whih enhanes the probability of finding them high in the orona. The EUV spetrometer Sumer of the spatial Solar and Heliospheri Observatory Soho of ESA-NASA offers a new and important opportunity for suh observations (Wilhelm et al ; Lemaire et al. 1997; Hassler et al. 1997). The interest of deteting these resonane sattered lines lies in the fat that they should be affeted by the veloity field of these oronal ions: qualitatively the moving ion absorbs the inident radiation somewhere in the inident line wing beause of the Doppler effet and the absorbed intensity is smaller. This leads to a derease of the sattered line intensity whih is alled the Doppler dimming effet (Hyder & Lites 197). Thus the intensity of the sattered line will be sensitive to the projetion of the veloity field on the diretion from the sattering ion towards the region of inident radiation (the vertial to the surfae of the sun in average). The sattered line being also shifted by the Doppler effet (leading to the determination of the projetion of the veloity field on the line-of-sight) Kohl & Withbroe (1982) and Withbroe et al. (1982a 1982b) suggested that the interpretation of the Doppler shift assoiated to that of the Doppler dimming should offer an opportunity of inreasing our knowing on the veloity field of the solar wind aeleration region. They based their analysis on the basi theory of resonane sattering for an inident radiation perfetly diretive the moving sattering atoms having an anisotropi Maxwell distribution of veloities with a hydrodynamial ensemble veloity. The diagnosti an then give two informations (f. Fig. 1): the veloity field omponent V z along the diretion of the inident radiation and that along the line-of-sight V Z. However the omponent perpendiular to the sattering plane is not attained and the omplete vetor diagnosti annot be ahieved. The aim of the present paper is to show that the omplete information on the three omponents of the matter veloity field vetor is ontained in the three first Stokes parameters (I Q U: intensity and linear polarization) of the oronal sattered line sensitive to the Doppler dimming effet provided that the omplete geometry of the sattering would be taken into aount (i.e. the inident radiation is partially and not perfetly diretive f. Fig. 2).

3 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I 581 Y O SUN Line of sight Z Fig. 2. The diretion of polarization of the reemitted line: OAz is the preferred diretion of inident radiation. (a) Without veloity field the diretion of polarization of the reemitted line is most often perpendiular to the sattering plane zaz (i.e. along Ax (or AX)). It an be in the sattering plane (along AY ) in the ase of a strong limb brightening and a sattering atom very lose to the limb. (b) In the presene of a veloity field the polarization degree and polarization diretion an be modified (rotation of angle φ). π/2 φ π/2 (b) X φ x φ > φ < In Set. 2 we will fous on the formalism whih is general and an be applied to a variety of astrophysial problems where anisotropies of resonane sattering our. In Set. 3 we will apply the formalism to the two-level atom and we will give the expression of the Stokes parameters of a oronal line formed by resonane sattering of the same line originating from the transition region and having a Doppler profile. Other line-broadening mehanisms will be negleted. Doppler redistribution of radiation will be only onsidered. Quantitative results will be presented in the next paper on the example of the O vi 13.2 nm line whih should be one of the most intense lines that are sensitive to the Doppler dimming effet in the solar wind aeleration region (f. also Sahal- Bréhot et al. 1992b; Sahal-Bréhot & Chouq-Bruston 1994 where preliminary results have already been given). 2. Basi theory 2.1. General onsiderations Our purpose is to alulate the Stokes parameters of a spetral line emitted by a moving ensemble of atoms whih satter the same line (resonane fluoresene) emitted by an underlying surfae. As for example we will treat the ase of the O vi 13.2 nm line emitted by an ensemble of atoms loated at a point A in the orona at a height h above the solar surfae. They have a marosopi veloity V and satter (absorb and reemit) the same line emitted by the underlying transition region of the sun (f. Fig. 3). Owing to the anisotropy of the inident radiation whih omes from the spherial ap limited by the angle α max the reemitted line is linearly polarized; in the absene of any perturbing effet the diretion of the linear polarization of the sattered line should be parallel to the solar limb exept at very low heights (h <.7 in units of solar radius) where y (a) A z it beomes radial if the limb-brightening is taken into aount (Sahal-Bréhot et al. 1986). In Sahal-Bréhot et al. (1986) the modifiation of the polarization of the O vi 13.2 nm line by a magneti field (the Hanle effet) was studied. However the Doppler effet due to the marosopi veloity of the sattering atoms and whih indues an angular dependene of the inident radiation intensity was outside the sope of that paper (f. the end of Set of Sahal-Bréhot et al. 1986). In the present one the Hanle effet is not taken into aount but the Doppler effet is introdued. As we will show in the following this angular dependene of the inident radiation an also modify the intensity and polarization parameters of the sattered line. As far as possible we will use the same notations as Sahal- Bréhot et al. (1992b) and Sahal-Bréhot et al. (1986) whih 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 reinfored (Cohen-Tannoudji 1977 Omont 1977 Cohen- Tannoudji et al.1988). This basi density-matrix formalism was extended to a omplex medium for the first time for solar prominenes studies by Bommier (1977) and by Bommier & Sahal-Bréhot (1978): multilevel atom oupled to an anisotropi beam of photons not perfetly diretive and to a magneti field the strength and diretion of whih are unknown and may a priori be arbitrary. In addition exitation by isotropi ollisions has been inluded in Paper I. In that formalism rederived from the basi equations of quantum eletrodynamis by Bommier & Sahal-Bréhot (1991) the oupling of an ensemble of independent idential radiating atoms with an inident anisotropi radiation field a magneti field and ollisions (whih an be also anisotropi) has been treated in a non-phenomenologial manner. In that work the atoms have been assumed at rest and their levels infinitely sharp (i.e. the natural and ollisional broadening of the lines is negligible). The theory has been reently revisited by Bommier (1997a 1997b) who has inluded line-broadening effets through a perturbation development of the matter-radiation interation up to orders higher than 2. Sine line-broadening effets are negleted in the present paper it an be dedued from Bommier (1997a) that effets of orders higher than 2 will vanish in the present ase: onsequently the lowest non-zero order (order 2) of the matter-radiation interation will be the only one to play a role as in Bommier & Sahal-Bréhot (1991). The problem an be redued in a first step to the study of one atom only interating with the ensemble (the bath ) of perturbers. The bath is omposed of photons and olliding partiles whih at independently: this is the impat approximation. The oupling with the partiles of the bath leads to the master equation for the atomi density matrix and then to the statistial equilibrium equations at the stationary state the solution of whih being the populations of the Zeeman states of the atom and the oherenes between these Zeeman states (see for instane Bommier 1996 for definitions). These equations an

4 582 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I Y Line of sight O y Z P β x X β v v A α v A n α n θ (sattering angle) α max ζ z Fig. 3. Coordinates of the sattering atom A loated at the distane r (in units of solar radius) from the enter of the sun O; h is the height above the limb (h = r 1); Oz is the preferred diretion of inident radiation; AZ is the line-of-sight; Ax and AX are idential and perpendiular to the sattering plane zaz; θ =(Az AZ) is the sattering angle; if it is equal to 9 whih is the ase of this figure where the atom is in the plane of the sky then AY = Az and AZ = Ay. The inident light omes from the spherial ap limited by the angle α max eah point P of the ap emits a ray PAζ referred by the angles (α β) in the frame Axyz. The atom A has the veloity v A(v Aα vβ v); n and n are the unitary vetors of the diretions Aζ and AZ. be onsidered as a generalization onvenient for polarization studies of the usual non-lte statistial equilibrium equations for the atomi populations. In a seond step the polarization matrix φ haraterizing the intensity and polarization of the emitted photons in the optially thin ase is obtained through a trae of the emissivity operator over the states of the upper level of the atomi line studied (Bommier 1977; formula (37) of Bommier & Sahal-Bréhot 1978): φ = 4ω3 3 3 Tr A D ρ A D ] (1) where ω is the angular frequeny of the radiation the veloity of light D the atomi dipole and ρ A the atomi density matrix. Bommier (1997a) has shown that the above expression (1) is valid only within the seond order expansion of the matterradiation interation. In the third step the Stokes parameters are obtained through a projetion of that polarization matrix on a plane perpendiular to the line-of-sight and through adequate linear ombinations of its matrix elements. Then the Stokes parameters of the emitted line whih 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-Bréhot 1978 for the optially thin ase). In the final step the integration of these Stokes parameters over the line-of-sight is performed. Bommier ( a 1997b) has studied the optially thik ase and has rederived the radiative transfer equations in the presene of a magneti field for the Stokes parameters through the density-matrix formalism (f. also Landi Degl Innoenti ). In all these papers the ensemble of atoms has been assumed at rest: thus no average over the atomi veloities has been performed. In the present paper we will extend the formalism to an assembly of moving atoms having an anisotropi distribution of veloities with a marosopi veloity V. The present formalism will be limited to the optially thin ase whih is suitable for the lines of the solar wind aeleration region. We will ontinue to assume that the absorption and emission profiles are infinitely sharp in the atomi rest frame. This means that the intrinsi widths of the lines are very small ompared to the frequeny distribution variation of the inident radiation spetrum. This is valid in the physial onditions of the solar orona beause the density is very low and thus the ollisional widths are very small. Consequently the intrinsi widths are restrited to the natural ones whih are also very small Introdution of the atomi motion in the master equation for the radiating atom The hamiltonian of the total system (radiating atoms A in interation with a bath of partiles onsisting of olliding partiles C and photons R and in the presene of a magneti field B) is given by H = H A + H magn + H R + H C + V AR + V AC (2) where H magn is that of the interation of the atom with the magneti field B H R that of the photons and H C that of the olliding perturbers. V AR is that of the interation atom-radiation (or atom-photons) and V AC that of the ollisional interation. We will inlude H magn in H A (weak field limit). The atomi wave-funtions are unhanged and the internal energies beome those of the Zeeman states E i + µ B g i B. The eigenstates of the atom at rest are denoted as α i J i M i. The total system being isolated and reversible its evolution an be desribed by a density matrix ρ solution of the Shrödinger (or Liouville equation for the density matrix) equation i dρ ] dt = H ρ. (3)

5 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I 583 The evolution of the subsystem A is desribed by the density matrix ρ A defined by a trae over all the states of the bath ρ A = Tr RC ρ]. (4) Within the usual language of lassial mehanis this trae orresponds to an average over all the partiles of the bath. As shown by Cohen-Tannoudji (1977) (f. also Cohen- Tannoudji et al. 1988) the subsystem A an be desribed by the redued density matrix ρ A (t). Yet we have now to derive the time evolution of ρ A (t) whih is not given by a Liouville equation beause we have introdued some irreversibility in the above proess: A is no longer an isolated system. The time evolution of ρ A (t) is now given by a master equation and desribes the evolution of a mean atom. In the orresponding lassial piture the time evolution of the lassial equivalent to ρ A (t) is given by the kineti equation (Oxenius 1986). It an be shown that the kineti equation for the radiating atoms or moleules redues to the so-alled statistial equilibrium equations at the steady state. We have to solve the master equation in a fixed referene frame the so-alled laboratory frame. In this frame the atomi veloity operator is denoted by v A and the atomi mass by m A. Following Smith et al. (1971) and Nienhuis ( ) we introdue the atomi motion in the atomi hamiltonian H A by adding the translation hamiltonian H tr to the atomi hamiltonian at rest H. The atomi hamiltonian in the laboratory frame is H A = H + H tr (5) with H tr = p2 A (6) 2m A where p A is the atomi momentum operator. p A = m A v A in the non relativisti limit. The orresponding eigenstate of H tr is the plane wave p A with eigenvalue 1 2 m AvA 2. The eigenstates of H A take the form u i = α i J i M i p A = α i J i M i p A (7) and the orresponding eigenvalues are E i m Av 2 A. (8) We assume that ollisional and radiation interations do not modify the state of translation of the atom: this implies that the reoil of the atom during an interation and the radiation pressure effets an be negleted. At the onsidered temperatures the atomi momentum is muh larger than the atomi eletron one p e = m e v e beause of the high m A /m e ratio. It is also muh larger than the photon momentum k = ω/ ω being the angular frequeny and the veloity of light. Consequently the momentum transfer during the interation proess is negligible and the atomi veloity is unhanged (Cohen-Tannoudji et al p. 286). Apart from the reoil effet the atomi veloity an hange under the effet of ollisions (Oxenius 1986 Sahal- Bréhot et al. 1992a Landi Degl Innoenti 1996): this hange an be determined by alulating the differential ross-setion (Balança & Feautrier 1998 Balança et al. 1998). In fat owing to the relative mass effet only elasti ollisions with partiles having about same masses or higher masses than the radiating atom are to be onsidered. In solar and oronal onditions these ollisional rates are very weak and the atom has time to emit a photon or to be exited or deexited by any proess before its veloity ould hange (f. Sahal-Bréhot et al. 1992a). Consequently the atomi veloity is onserved and the matrix elements of H an be onsidered as diagonal in p A. Therefore as for the external states p A the off-diagonal terms p A ρ A (t) p A (with p A /= p A ) will be ompletely deoupled from the diagonal terms p A ρ A (t) p A whih represent the density of probability (with respet to the external states p A ) for the density matrix (with respet to the internal states α i J i M i ) of an atom having the veloity v A.Wean denote this density matrix as ρ A (v A t) the elements of whih are given by α i J i M i p A ρ A (t) α i J i M i p A (9) = α i J i M i ρ A (v A t) α i J i M i. Moreover there will be no oupling between the redued density matries ρ A (v A t) and ρ A (v A t) for different veloities v A and v A. For the internal states α i J i M i ρ A (v A t) will be normalized to the distribution of atomi veloities F (v A ): Tr A ρ A (v A t)] = F (v A ) (1) with the normalization ondition F (v A )d 3 v A =1. (11) Then within the impat approximation the interations with the radiation (the photons) R and the olliding partiles C (whih are the eletrons of the orona in the present paper) are deoupled (f. Bommier & Sahal-Bréhot 1991 and earlier papers). Using the no-bak reation approximation and the impat approximation as well as the fat that the evolution of the redued atomi density matrix ρ A (v A t) is markovian i.e. it depends only on the instant t and not on the past history of ρ A one obtains the master equation for ρ A (v A t) (f. Bommier & Sahal-Bréhot 1991 and Sahal-Bréhot et al. 1992a for the atom at rest) whih desribes the evolution of the atom oupled to the bath in the laboratory frame (f. also Cohen-Tannoudji 1977 and related papers Omont 1977 Bommier 1997a). The oarse grained approximation leads to ahieve the average over the states of the bath over a time interval s muh larger than the mean duration τ of a ollision (resp. τ r for the radiation) and muh smaller than the mean interval Γ 1 between two sues- inverse of the lifetime). We obtain ( ) dρa (v A t) dt sive ollisions (resp. Γ 1 r dρ A (v A t) dt = 1 i H Aρ A (v A t)] + B

6 584 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I Owing to the fat that the interations of the partiles of the bath B are deoupled we an write dρ A (v A t) = dt i 1 ] H A ρ A (v A t) ( ) ( ) dρa (v + A t) dρa (v + A t) dt dt = i 1 ] H A ρ A (v A t) SR ρ A (v A t) ρ R S s Tr R + 1 s Tr C R R ρ ] A (v A t) ρ R SC ρ A (v A t) ρ C S 1 C ρ ] A (v A t) ρ C = at the steady state. C (12) S is the sattering matrix alulated for a omplete interation with one olliding perturber (subsript C) or with one photon (subsript R). The density matrix of the photons ρ R and the density matrix ρ C of the olliding perturbers take plae for averaging over all the partiles i.e. over the states of the bath B (radiation R and ollisions C that are deoupled). Then we obtain the projeted master equation averaged over the states of the bath (f. also Cohen-Tannoudji et al p. 486): d dt i ρ A (v A t) i = iω ii i ρ A (v A t) i + (13) j ρ A (v A t) j S 1 j i S ij δ ij δ i j. jj where ω ii is the Bohr frequeny and δ ij and δ i j are Kroneker symbols Coupling with the olliding partiles We show now that the ollisional interation is not affeted by the atomi motion for the proesses studied in the present paper: eletrons are very muh faster than the radiating ions whih an thus be assumed at rest. This would not have been the ase if we were studying ollisions between heavy partiles suh as protons and hydrogen atoms for instane (f. Sahal-Bréhot et al. 1996). Herewith the ollisional S matrix has to be alulated within the usual framework of the theory of ollisions. The olliding perturbers are isotropi maxwellian eletrons at a temperature T e. Owing to that isotropy the ollisional oupling oeffiients in the master equation link only the Zeeman populations i ρ A (t) i j ρ A (t) j. In addition the oronal eletroni densities are so low that depolarizing ollisions are negligible: the photon is emitted before any ourrene of a ollision. The only ollisional proesses that are to be taken into aount are exitation from the lower level towards the upper level of the onsidered transition. Deexitation by ollisions is negligible. The veloity that takes plae in the alulation of the ross-setion σ (i j v e ) is the relative one v e v A whih an be onsidered as equal to the eletron veloity v e. B Thus the problem is rather straightforward. We obtain after integration over the Maxwell isotropi distribution of eletron veloities f (v e ) at the temperature T e and multipliation by the eletron density N e S 1 C S ]j i C ] ij C = δ jj δ ii N e v e f(v e )σ(i j v e )dv e = δ jj δ ii N e α(i j T e ). (14) In order to onlude this paragraph the ollisional rosssetions and their average over the eletron distribution of veloities are not affeted by the atomi motion. The effet of the atomi motion only appears in the matrix elements of the atomradiation field interation that we will study in the following Coupling with the radiation field Ignoring ollisions now the total hamiltonian H of the system is that of the subsystem atom+radiation and an be written as H = H A + H R + V AR (15) where H R is the hamiltonian of the radiation field and V AR is the atom-radiation interation V AR = D E (r A ) (16) where D is the atomi dipole whih an be alulated in the basis of the atom at rest (Smith et al. 1971) E (r A ) is the eletri field of the radiation and r A is the atomi position operator in the laboratory frame. Using the seond quantization formalism the eletri field an be developed in plane waves as E(r A )= d 3 k hω i (2π) 3 λ (17) ] a (kλ) e λ e ik r A a (kλ) e λ e ik r A for the mode k(k ˆk) ˆk defining the diretion of unitary vetor n the angular frequeny ω = k and the polarization λ (polarization vetor e λ ). a (kλ) and a (kλ) are the operators of reation and annihilation of photons for the radiation propagating in the n diretion with the polarization λ. The normalisation hosen is suh that a (kλ) a (kλ) = 2 hν 3 I ν(ν ˆk λ) (18) where I ν (ν ˆk λ) is the speifi intensity. By expanding the dipole moment D and the polarization vetor e λ over the standard basis e + e e of the laboratory

7 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I 585 frame (see Bommier & Sahal-Bréhot 1991 or Bommier 1997a for definition) we obtain V AR = i d 3 k hω 1 (2π) 3 λ p= 1 a (kλ)( 1) p D p e λ ] p e ik r (19) A ] a (kλ) D p e λ ] p e ik r A. The atom-radiation interation ontains the atomi motion through the fators e ±ik r A. The developments made by Bommier & Sahal-Bréhot ( ) for the atom at rest an now be rewritten for a moving atom by inluding these fators. By using in interation representation r A (τ) =e ih Aτ/ r A e ih Aτ/ =e ip2 A τ/2m A r A e ip2 A τ/2m A = r A + p A m A τ = r A + v A τ (2) it an be shown that the results of Bommier & Sahal-Bréhot (1991) obtained for an atom at rest at r A = an be applied to the present ase by replaing any radiation angular frequeny ω whih appears in the e ±iωτ terms of that alulation by (ω k v A ) k being the orresponding wave vetor. Due to the Markov approximation whih leads to limit the perturbation expansion of the matter-radiation interation to the lowest non-zero order (the seond one) the absorption and emission line profiles are infinitely sharp: this is valid if the intrinsi line-widths are very small ompared to the frequeny distribution variation of the inident radiation spetrum. All this is valid in the solar orona. Thus the sattering is monohromati in the atomi rest frame. Following Hummer (1962) and Mihalas (1978 p. 412) this orresponds to Case I of the redistribution proess (Doppler redistribution). Therefore as the line profile is now given by a δ-funtion δ ω (ω k v A )] (21) the only hange is to replae the angular atomi frequeny ω (frequeny orresponding to the energy differene between the levels u (upper) and l (lower) whih are assumed infinitely sharp) by ω +(k v A )=ω 1+ (v ] A n) (22) in the laboratory frame. This is expeted: the atom absorbs the radiation at its eigenfrequeny ω in its own omoving frame (the atomi frame) and due to the Doppler effet this orresponds to the angular frequeny ω 1+ (v ] A n) of the inident radiation in the laboratory frame. We are now able to write the system of statistial equilibrium equations leading to the populations and oherenes for the atom having the veloity v A interating with an inident flux of radiation haraterized by its polarization matrix ϕ.we will solve this system in the atomi frame. The first step is to obtain the elements of the polarization matrix whih enter the oupling oeffiients of the elements of the master equation leading to the atomi density matrix ρ A (v A ) Calulation of the omponents of the polarization matrix ϕ of the inident radiation We will use the same notations as those of previous papers (Paper I and Sahal-Bréhot et al. 1992b). We will assume hereafter that the inident line profile is the same for all the points of the solar surfae: we will neglet inhomogeneities and take only into aount the limb-brightening effet. Inlusion of inhomogeneities of the inident radiation was studied in Paper I and is outside the sope of the present paper. The atomi frame (the omoving frame) Axyz and the lineof-sight frame (the laboratory frame) AXY Z are defined in the following manner (f. Fig. 3): AZ is the line-of-sight oriented towards the observer its unitary vetor is denoted by n. Az is the axis of symmetry of the system atom A + inident photons that is the vertial to the surfae of the Sun. It is oriented towards the outside. The sattering plane ontains the vertial Az and the line-ofsight AZ. The perpendiular to the sattering plane defines the ommon axis Ax and AX. Sine there are a priori two possible senses for Ax (or AX) the angle θ between Az and AZ is oriented. The sense of Ax (or AX) gives the sign of θ whih is thus not a polar angle. Ay and AY follow. The polar angle α v and the azimuth β v define the diretion of the atomi veloity v A in the atomi frame. Likewise the polar angle α V and the azimuth β V define the diretion of the veloity V of the ensemble of atoms in the atomi frame. The polarization matrix ϕ of the inident photons whih was obtained for the atomi frequeny only in Paper I is now frequeny-dependent through the Doppler effet. The radiation inoming from an element of volume entered on a point P of the solar surfae in the diretion PAζ (unitary vetor n) is haraterized by the angles α and β in the atomi Axyz frame. J(α β ν) is the angular distribution of the loal intensity at point A for the frequeny ν (erg m 2 s 1 sr 1 Hz 1 ) and inoming from point P : ν = ν 1+ (v ] A n) (23) and (f. Fig. 3)

8 586 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I (v A n) =v A os α os α v (24) + sin α sin α v (os β os β v + sin β sin β v )]. The polarization matrix is obtained through an angular average over all the diretions of inoming radiation as follows: αmax ϕ (ν) = 4π 1 sin α dα M (α β) J (α β ν). 2π dβ (25) The matrix M (α β) haraterizes the angular behaviour of a dipolar unpolarized radiation inoming in the diretion Pζ and is normalized to unity (Eq. (43) of Paper I). It an be expanded in multipoles terms over the basis of the irreduible tensorial operators T k q relative to the quantization axis Az whih is also adapted to the present problem: M (α β) = 2 k k= q= k M k q (α β) T k q. (26) The results are the same as in Paper I exept that β is hanged into β + π. In fat in Paper I β was the azimuth of point P referred in ylindrial oordinates in the atomi frame Axyz. In the present paper β is the azimuth of the PAζ diretion in the atomi frame. M (α β) = 1 3 M 1 (α β) =M 1 ±1 (α β) = M 2 (α β) = 1 2 ( 3 os 2 α 1 ) (27) 6 M 2 1 (α β) = M 2 1 (α β) ] = 1 sin 2α ( os β + i sin β) 4 M 2 2 (α β) = M 2 2 (α β) ] = 1 4 sin2 α (os 2β i sin 2β). Thus using relations (23) and (24) the polarization matrix of the inident radiation beomes veloity-dependent and we an write αmax ϕ k q (v A )= 4π 1 sin α dα M k q (α β) J (α β v A ). 2π dβ (28) The angular dependene of (v A n) on the angles α and β leads to appearane of oherenes (k =2 q = ±1 ±2) in the matrix elements of ϕ (v A ) (in the atomi Axyz frame). This will lead to a rotation of polarization of the sattered line beause the ylindrial symmetry about the Az axis is broken. We an also verify that the oherenes vanish when the atomi veloity is direted along the vertial (i.e. α v =) beause the ylindrial symmetry is reovered. Formula (28) is general and does not assume anything on the inident radiation. Thus we have obtained a general expression for the elements of the polarization matrix of the inident radiation that enter the statistial equilibrium equations that have to be solved for eah atomi veloity v A. In fat it is also not restrited to the two-level atom Expression of the statistial equilibrium equations for the atomi density-matrix and for a two-level atom We refer to Eq. (5) of Paper I in zero-magneti field and we use Eq. (28) of the present paper for the polarization matrix of the inident radiation. We obtain for a two-level atom (u = upper and l = lower level) in the atomi referene frame in the basis of the irreduible tensorial operators A (J u J l ) JuJu ρ A ] k q (v A)= J lj l ρ A ] (v A) N e α (J l J u T e ) 2Jl +1 2J u +1 δ (k ) +B (J l J u ) ϕ k q (v A ) 3 2J l +1( 1) Ju+J l+k+1 { }] 1 1 k. J u J u J l (29) In the above equation k an be equal to (population omponent) or 2 (alignment omponent). A (J u J l ) and B (J l J u ) are the Einstein offiients for spontaneous emission and absorption N e α (J l J u T e ) the ollisional exitation oeffiient (obtained from Eq. (14)). δ (k ) is the Kroneker symbol: in fat the ollisional proesses are isotropi and annot reate alignment and we reall that they are also insensitive to the atomi veloity. The symbol between brakets is a 6j oeffiient. Stimulated emission is negligible and is not introdued. Likewise ollisional deexitation is negligible. The only relevant proesses are exitation by ollisions and radiation followed by spontaneous emission Expression of the polarization matrix of the reemitted photons Following Eq. (49) of Paper I and using the present equation (28) the polarization matrix φ (v A ) of the photons reemitted by an atom having the veloity v A an now be written. φ (v A ) being expressed in number of emitted photons per seond per unit of volume and per unit of distribution of atomi veloities F (v A ) we obtain in the Axyz atomi omoving referene frame φ (v A )=N A n l (v A ) 1 N e α (J l J u ) 3 +N A n l (v A ) B(J l J u )ϕ (v A ) φ 2 (v A )=N A n l (v A ) B(J l J u ) ul ϕ 2 (v A ) (3) φ 2 1 (v A )= φ 2 1] (va ) = N A n l (v A ) B(J l J u ) ul ϕ 2 1 (v A )

9 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I 587 φ 2 2 (v A )= φ 2 2] (va ) = N A n l (v A ) B(J l J u ) ul ϕ 2 2 (v A ); where N A is the number of radiating ions per unit of volume. n l (v A ) is the population of the lower level l (the ground state) n u (v A ) is that of the exited state u: n l (v A )= 2J l +1 J lj l ρ A ] (v A) (31) n u (v A )= 2J u +1 JuJu ρ A ] (v A). (32) The normalization ondition for the internal states (Eq. (1)) leads to n l (v A )+n u (v A )=F (v A ) (33) for the two-level atom. Sine the population of the exited level is very small ompared to that of the ground state (ase of the solar orona) the preeding Eq. (33) redues to n l (v A )=F (v A ) (34) F (v A ) being the distribution of atomi veloities. ul is defined in Paper I aording to Bommier & Sahal- Bréhot (1982). It an be expressed in terms of 6j oeffiients as { } { } J u J u J l ul =3(2J u +1) = J u J u J { } 2. (35) l 1 1 J u J u J l It an be notied that ul is equal to the oeffiient ] 2 W 2 (J J )= w (2) J J defined in Landi Degl Innoenti (1984) where these oeffiients are tabulated. The notation of Stenflo (1994 pp and p. 91) for ul is W 2. We have (Paper I): ul = 1 2 for J u = 3 2 and J l = 1 2 ul =1 for J u =1and J l = ul = for J u = 1 2 and J l = 1 2 ul = for J u =and J l =1 In order to obtain the polarization matrix of the reemitted photons in the line-of-sight fixed frame AXY Z (the laboratory frame) we have to rotate the oordinates system and to use the proper atomi eigenfutions inluding translation states. Firstly we have to perform a rotation through the angle θ about the Ax-axis (or AX). In fat we will modify Eq. (5) of Paper I whih was obtained for θ = π/2. The resulting polarization matrix is denoted by Φ(v A ). The Euler angles of the rotation are ( π ) 2 θ+π. 2 We obtain after alulations Φ (v A )=φ (v A ) Φ 2 (v A )= 2 1 ( 3 os 2 θ 1 ) φ 2 (v A ) 32 sin 2θImφ 2 1 (v A ) Φ 2 2 (v A )= 38 sin 2 θφ 2 (v A ) 32 sin 2 θreφ 2 2 (v A ) (36) 1 2 sin 2θImφ2 1 (v A )+isinθreφ 2 1 (v A ) + 1 ( os 2 θ ) Reφ 2 2 (v A )+iosθimφ 2 2 (v A ). The other omponents are not written beause they do not enter the expression of the following Stokes parameters. Seondly the atom reemits the line at the frequeny ν in the omoving atomi frame. We introdue the translation motion dependene in the atomi wavefuntions of the laboratory frame. This is equivalent to write that due to the Doppler effet the observer sees a line emitted at the shifted frequeny ν = ν 1+ (v A n ] ) along the line-of-sight AZ haraterized by the diretion n. We have (v A n )=v AZ (37) = v A (os θ os α v sin θ sin α v sin β v ) and ν v AZ ν = ν. (38) Then the Stokes parameters of the line emitted at the frequeny ν in the AZ diretion by the atoms having the given projeted veloity v AZ an be obtained. Firstly for the atoms having the veloity v A (v AX v AY v AZ ) we have (Paper I) I (v A )= 3 8π 2 3 Φ (v A )+ 2 6 Φ 2 (v A ) ] Q (v A )= 3 8π 2ReΦ2 2 (v A ) (39) U (v A )= 3 8π 2ImΦ2 2 (v A ) V (v A )= in number of photons per steradian. Seondly we have to sum over all the atoms having the given v AZ projeted value for obtaining the frequeny distribution of intensity and polarization of the reemitted radiation. Therefore we have to integrate over the atomi veloities in the XAY plane perpendiular to AZ. We obtain I(ν )dν =dv AZ I (v A )dv AX dv AY Q(ν )dν =dv AZ Q (v A )dv AX dv AY (4)

10 588 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I U(ν )dν =dv AZ U (v A )dv AX dv AY V (ν )dν = and dv AZ = dν. ν This is the general formula giving the Stokes parameters as a funtion of frequeny of the reemitted line (in number of photons per unit of volume of matter per unit of time per unit of frequeny and per unit of solid angle in the diretion n ) for any atomi veloity distribution. In fat the formal formula (4) is not restrited to the two-level atom. The degree of linear polarization p and the angle φ of the polarization diretion with respet to the AX diretion follow (f. Fig. 2): Q2 + U p = 2 (41) I Q os 2φ = Q2 + U (42) 2 sin 2φ = U Q2 + U 2. (43) 3. Appliation to the O vi line 13.2 nm of the solar orona We begin by giving the expression of the polarization matrix of the inident radiation J(α β ν) whih is that of the same O vi line but emitted by the underlying transition region. Negleting inhomogeneities of the solar surfae and using the same notations as Paper I we an write J(α β ν) as J (α β ν) =J (α) j i (ν) (44) where j i (ν) is the inident normalized profile. J (α) =I C f (α) (45) where f (α) is the limb-brightening oeffiient (f () =1) and I C is the speifi intensity emitted from the enter of the disk in the line studied and integrated over the inident line profile. The inident line profile is a gaussian Doppler profile with a linewidth ν Di j i (ν) = 1 1 exp π ν Di and ϕ k q (v A )= I αmax C 4π 1 ν Di 1 π exp ( ) ] 2 ν ν (46) ν Di f(α) sin α dα ( ) ] 2 ν (v A n) ν Di 2π dβ M k q (α β). (47) Then we assume that the atomi veloity distribution an be written as F (v A )d 3 v A =d 3 v A ( ma 2πkT ) 3/2 exp m A 2kT v A V 2] (48) whih orresponds to a Maxwellian distribution at a temperature T with an hydrodynamial drift of veloity V (V X V Y V Z ) or (V x V y V z ). This drift represents the veloity of the solar wind. We an write eah subdistribution being normalized to 1 F (v A )d 3 v A = F X (v AX )dv AX F Y (v AY )dv AY F Z (v AZ )dv AZ with F Z (v AZ )dv AZ =dv AZ ( ma 2πkT ) 1/2 exp m A 2kT (v A Z V Z ) 2] (49) (5) and analogous expressions for the two other distributions. The integrations over v AX and v AY an be ahieved analytially by means of elementary methods and are not detailed here. The Stokes parameters I (ν) Q (ν) U (ν) of the frequeny distribution of intensity and polarization of the reemitted line at the frequeny ν are given in Appendix A: Eq. (A1). They are expressed in number of photons emitted per steradian per unit of volume per unit of time and per unit of interval of frequeny. This equation has to be applied with ul =1/2 (Li -like resonane line 2p 2 P 3/2 2s 2 S 1/2 ) and ul =for the other omponent of the doublet (2p 2 P 1/2 2s 2 S 1/2 ). Without limb-brightening f (α) =1 and sin α max = R R (1 + h). After integration over the sattered line-profile one obtains the global Stokes parameters of the sattered line I Q U (in number of photons emitted per steradian per unit of volume and per unit of time). They are given in Appendix A: Eq. (A8). These formulae show that the three first Stokes parameters (intensity and linear polarization parameters) are sensitive to the three omponents of the vetor veloity of the drift. The order of magnitude of the sensitivity to these omponents will be the objet of the next paper (f. also Sahal-Bréhot et al. 1992b and Sahal-Bréhot & Chouq-Bruston 1994 for preliminary results) Asymptoti limit: perfetly diretive ase At great distanes from the limb (i.e. h>5 in units of solar radius) the results orresponding to a perfetly diretive inident radiation (Hyder & Lites 197; Bekers & Chipman 1974) are reovered. By taking the limit α the integration of Eq. (A1) over α and β beomes analyti. The sattered line has also a gaussian profile and the frequeny dependene is the same for all the Stokes parameters: they are thus haraterized by 5 parameters w d p s φ whih are defined in the following.

11 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I 589 Writing the intensity of the purely diretive inident radiation as ( ) ] 2 ν ν I i (ν) =I exp with I = I C 1 π νdi ν Di (w d s p -φ) V 3 V'1 (w -d s p -φ) x X V (w d s p φ) V' 2 (w -d s p φ) we obtain a sattered intensity in the AZ diretion ( ) ] 2 ν ν d I s (ν) =I exp w O Sun A z with a width νd w = ν Ds (1 2 s νd 2 i + νd 2 s where ν Ds = ν 2kT m A a shift d = ν ) os 2 θ (51) ( ν 2 ) D V Z s νd 2 i + νd 2 V z os θ (52) s a dimming 2 ν I = exp s ] = exp V z (53) I νd 2 i + νd 2 s whih haraterizes the dimming of the sattered line intensity with respet to the sattered intensity in the absene of veloity field a degree of polarization p = Q I and φ = = 3 ul sin 2 θ 4 ul +3 ul os 2 θ (54) whih orresponds to a diretion of polarization perpendiular to the sattering plane (U =and Q>). In partiular for a J u =3/2 J l =1/2 line and for θ = ±π/2 we obtain the well-known result p =.428. (55) Therefore at great distanes from the surfae of the sun the analysis of the sattered line intensity gives only two omponents of the veloity field: its projetion on the diretion of the inident radiation V z and its projetion on the line-of-sight V Z through the interpretation of the dimming and of the shift. The other Stokes parameters do not bring any additional information. The third omponent of the veloity field V X orv x i.e. the perpendiular to the sattering plane annot be determined. Consequently the vetorial diagnosti annot be performed with this method for a perfetly diretive inident radiation. y Line of sight Z V 2 (w d s p φ) V' (w -d s p φ) V1 (w d s p -φ) V' 3 (w -d s p -φ) Fig. 4. The symmetries and degeneraies of the solution for the diagnosti of the veloity field vetor. Oz is the preferred diretion of inident radiation 3.2. Symmetries of the problem The following disussion is restrited to a sattering atom in the plane of the sky (θ = ±π/2). We show that the results for π/2 <α V <π and for π/2 <β V < 2π an be dedued from those of the first quadrant ( <α V <π/2 and <β V <π/2). The symmetries of the problem are illustrated on Fig Symmetry due to the hange of the diretion of propagation of the inident light into the opposite sense For the sattering atom this is equivalent to { αv π α V β V π β V and the line-of-sight is onserved. For θ = ±π/2 (the present figure is drawn for θ = π/2) this orrespond to a symmetry with respet to the line-of-sight AZ. As the reoil of the atomi motion due to absorption or emission of light is negligible the atom is not sensitive to the sense of propagation of the light and two opposite diretions of propagation have the same effet. Consequently the Stokes parameters of the sattered line will be the same for two veloity field vetors whih are symmetrial with respet to the line-of-sight. This is the fundamental degeneray whih exists in every polarimetri diagnosti and whih has been abundantly disussed in Hanle effet studies (Bommier & Sahal-Bréhot 1978 and further papers); V V 2 on Fig. 4. In fat if the sattering atom was not in the plane of the sky (θ /= ±π/2) the two orresponding veloity field vetors would not be symmetrial with respet to the line-of-sight.

12 59 S. Sahal-Bréhot et al.: Doppler redistribution and resonane polarization. I Symmetry with respet to the preferred diretion of the inident radiation { αv α V β V β V + π. The harateristis of the sattered radiation are the same for two opposite lines of sight the only hanges are the sign of the shift d of the sattered line d d and that of the polarization rotation: I (apart from the sign of the shift) and Q are unhanged and the sign of U is hanged φ φ ; this symmetry exists also in Hanle effet studies. V V 3 on Fig Symmetry with respet to the enter of sattering A { αv π α V β V β V + π The only hange is the sign of the shift d d 4. Conlusion We have rederived the basi theory of resonane sattering for an inident radiation partially diretive and frequeny-dependent the moving sattering atoms having an anisotropi distribution of veloities. We have shown that the three first Stokes parameters of the reemitted line are sensitive to the three omponents of the marosopi veloity field. Therefore the interpretation of these Stokes parameters may offer a method for deriving the omplete veloity field vetor. The formalism is general and be applied to a variety of astrophysial problems where anisotropies of resonane sattering our. Then we have applied the formalism to the two-level atom and we have given the expression of the Stokes parameters of the O vi 13.2 nm line of the solar orona formed in the solar wind aeleration region. This line whih has a Doppler profile is formed by resonane sattering of the same line originating from the transition region and whih has also a Doppler profile. The first numerial results of this formalism have been given by Sahal-Bréhot et al. (1992b) and Sahal-Bréhot & Chouq- Bruston (1994). They will be analysed and disussed in the next paper in view of future interpretation of solar data obtained by Sumer-Soho. Aknowledgements. The authors are indebted to the referee Maro Landolfi for a ritial reading of the manusript and helpful omments. This symmetry does not exists in Hanle effet studies beause V is a real (polar)-vetor whereas the magneti field B is an axial-vetor; V V on Fig Symmetry with respet to the sattering plane { αv α V ; β V π β V this symmetry is in fat the produt of the three preeding symmetries: the result is that I and Q are unhanged and that the sign of U is hanged: the rotation of polarization is hanged in its opposite and the shift is unhanged; V V 1 on Fig. 4.