J. O. Stenflo^{1,2}
1 - Institute of Astronomy, ETH Zentrum, 8092 Zurich, Switzerland
2 -
Faculty of Mathematics & Science, University of Zurich, 8057 Zurich, Switzerland
Received 15 July 2004 / Accepted 9 September 2004
Abstract
The Sun's spectrum is linearly polarized by coherent scattering processes. Here we develop the theory for the formation of the polarized continuum, identify the relevant physical mechanisms, and clarify their relative roles. The polarized photons are produced by scattering at neutral hydrogen in its ground state (Lyman scattering), and to a smaller degree by scattering at free electrons (Thomson scattering). The polarized photons are diluted by the unpolarized photons from the H^{-} opacity and radiative absorption from the Balmer bound-bound and bound-free transitions. Due to pressure broadening of the Balmer lines from the statistical Stark effect the polarized Balmer jump is shifted from the series limit to substantially longer wavelengths. In the second part of the paper the Atlas of the Second Solar Spectrum that covers 3161-6995 Å for disk position
(where
is the cosine of the heliocentric angle) is used to extract the empirical values of the continuum polarization with the help of a model for the behavior of the depolarizing lines. The empirically determined continuum polarization lies systematically lower than the values that have been predicted for
Å from radiative-transfer modelling. The Balmer jump is found to be shifted as expected from pressure-broadening theory. Through scaling of the relative center-to-limb variations obtained from radiative-transfer theory with the empirically determined values (valid for )
we finally obtain the semi-empirical function that describes the variation of the continuum polarization with both wavelength and disk position .
The empirically determined continuum polarization can be used to constrain model atmospheres as well as to fix the zero point of the polarization scale in observations of the scattering polarization and the Hanle effect.
Key words: polarization - scattering - Sun: photosphere - atomic processes - techniques: polarimetric - radiation mechanisms: general
The Second Solar Spectrum is characterized by a polarized continuous background, on which a rich variety of both intrinsically polarizing and depolarizing lines are superposed. While the depolarizing lines have the appearance of "absorption'' lines, the polarizing lines look like "emission'' lines. Many lines however belong to an intermediate category if they are only weakly polarizing, or if their line polarization is partially depolarized by magnetic fields via the Hanle effect. The appearance of the Second Solar Spectrum actually varies with the phase of the solar cycle, since the Hanle effect depolarization is more pronounced at activity maximum due to the larger amounts of hidden magnetic flux in the Sun's atmosphere (cf. Berdyugina et al. 2002). To properly interpret the line polarization and the Hanle effect one needs to know the level of the continuum polarization background and understand how this background is formed (cf. Stenflo et al. 1998).
Unfortunately the level of the continuous polarization cannot presently be determined as precisely as the relative polarization variations in the spectrum (for which a polarimetric precision of 10^{-5} is routinely achieved). Slight asymmetries in the bidirectional charge shifts within the demodulating ZIMPOL detector and instrumental polarization cause zero-point offsets of the polarization scale. These effects can be calibrated, but currently not to the same polarimetric precision as that of the relative polarization scale. For practical purposes the zero point offset therefore has to be treated as an unknown free parameter, while the relative variations are well determined. If the level of the continuous polarization were known from other investigations, then this knowledge could be applied to fix the zero point of the polarization scale for any polarimetric recording (that includes a portion of the continuum). The present paper aims at finding the functional dependence of the continuum polarization with respect to both wavelength and disk position (represented by , the cosine of the heliocentric angle).
Various attempts were made in the 1970s to measure the continuum polarization and its center-to-limb variation (Leroy 1972; Mickey & Orral 1974; Wiehr 1975). They have been reviewed by Leroy (1977). Although these broad-band and non-imaging observations allowed a general verification of theoretical concepts for the origin of the continuum polarization that had been developed in the pioneering works of Débarbat et al. (1970) and Dumont & Pecker (1971), they are not of sufficient accuracy, wavelength coverage and resolution to allow a good quantitative comparison with detailed radiative-transfer modelling. Such modelling has provided us with theoretical values for the continuum polarization over the wavelength range 4000-8000 Å (Fluri & Stenflo 1999), but the theory has not been applied below 4000 Å. The UV is the most interesting wavelength region, since the continuum polarization increases steeply with decreasing wavelength, and there is interesting physics occuring around the Balmer jump.
With the availability of the new Atlas of the Second Solar Spectrum, which covers the wavelength range 3161-6995 Å in three volumes (Gandorfer 2000, 2002, 2004), we now have a comprehensive data set with which we may in principle extract the values for the continuum polarization. Since however the zero point of the polarization scale is unknown in the Atlas, the extraction procedure needs to make use of a model for the behavior of the depolarizing lines. A similar approach was applied to the first survey of the Second Solar Spectrum more than two decades ago (Stenflo et al. 1983a,b), but the polarimetric accuracy at that time did not allow the scale of the continuum polarization to be determined with much confidence, in contrast to the present situation.
The present data set covers a wide wavelength range, where many different physical mechanisms come into play. As previous theoretical work (Débarbat et al. 1970; Fluri & Stenflo 1999) has focused on the radiative-transfer aspects of the problem, we find a need to elucidate the underlying physics in a more systematic and transparent way. Therefore we develop in the first part of the paper the theory that governs the formation of the polarized continuum, identify the various contributing mechanisms, and clarify their respective roles. In the second part we extract the empirical values of the continuum polarization from the Atlas data and compare them with the theoretical predictions, to examine to what extent some physics is missing. The empirical values are finally connected with radiative-transfer modelling to obtain the semi-empirical function that gives the continuum polarization for all wavelengths and disk positions .
The oscillating electric vector
induces an oscillating dipole moment d_{q} per unit volume. The stationary oscillatory solution of Eq. (1) is
The non-magnetic refractive index n is given by
The excited oscillators emit dipole radiation. From the classical expression for dipole radiation one can derive an expression for the relation between d_{q} and the electric vector E_{q} of the emitted radiation. Together with Eq. (2) one then gets a relation between the exciting electric field and the scattered electric field E_{q}, which, when transformed to a basis of linear polarization vectors (which is the natural basis for measuring devices), can be expressed in terms of the Jones scattering matrix. From the tensor product of two Jones scattering matrices one obtains the Mueller scattering matrix that describes how the incident Stokes vector is transformed to the scattered Stokes vector . The normalization condition for the Mueller scattering matrix leads to the definition and expression for the scattering cross section . For details of all these derivations we refer to a relevant monograph (Stenflo 1994).
The result is
Inserting Eqs. (3) and (4) in Eq. (5) we obtain
It is interesting to note that Eq. (6) contains one resonant and one non-resonant term, which have exactly the same form as in the Kramers-Heisenberg scattering theory. In quantum electrodynamics these two terms are interpreted in terms of time ordering of the Feynman diagrams. Thus the resonant term represents absorption followed by emission, while the non-resonant term represents the seemingly strange process of emission followed by absorption. Since the classical theory is not a perturbation theory, it produces automatically both terms without the need for any intriguing time-ordering interpretations.
If the oscillating medium would have resonances at several frequencies other than , which were coherently excited by the same driving electric field of the incident radiation, then we would get a linear superposition of the Jones matrices for the various resonances. Due to the bilinear multiplications between the Jones matrices when forming the Mueller matrix, we would then get interference terms between scattering amplitudes belonging to different resonant frequencies. We will see explicitly in the quantum-mechanical formulation how such interferences must occur because the intermediate state is a mixed quantum state (linear superposition of all possible intermediate states).
In the limit of vanishing damping constant we obtain
The phenomenological use of transition rates given by the Einstein coefficients is conceptually simple and a convenient way to find the magnitude of the scattering cross section, but this approach cannot properly deal with quantum interferences or polarization. Still, because of its simplicity and transparency, it helps to illuminate the physics.
This approach conceptually views scattering as a two-step process: radiative excitation, followed by spontaneous emission. Let us label the initial, intermediate, and final states by indices i, e, and f, respectively. To make the treatment more general, we allow i and f to be different (Raman scattering). i=f then represents the special case of Rayleigh scattering. The incident frequencies are (or ), the scattered frequencies .
The coefficient of radiative absorption is, for a given resonant transition ,
Of all the radiatively excited atoms, only the fraction
that represents spontaneous emission from level e to level f constitutes the scattering transition under consideration.
The scattering cross section
represents the product of the absorption cross section and the branching ratio. Thus
where
is the resonant frequency of the absorption transition (while
represents the emission transition). With Eqs. (11) and (13) this gives
In standard non-LTE theory,
is (in the frame of the atom) taken to be the Lorentz profile:
If we define
The correspondence with the classical theory is now manifest. The classical theory corresponds to an transition without electron spin, where , f_{ie}=f_{fe}=1, , g_{f}=1, and g_{ e}=3 (the three m states of the excited P state, or, classically, the three spherical vector components of the oscillation, with ). Inserting these values in Eq. (15), we retrieve precisely the classical scattering cross section, as required by the correspondence principle.
The phenomenological treatment in terms of transition rates gives us the correct Raman scattering cross section for a single scattering transition, but it does not include the quantum interferences between all the possible excited states that are involved in scattering from i to f, and it cannot deal with the polarizability of a scattering transition. For this we need the quantum-mechanical perturbation theory. For the scattering cross section this theory gives the same results as Eqs. (15)-(20), except that we need to sum over all the possible excited states before forming the product of the profile functions.
Let us for convenience define a generalized profile function as follows:
Let us generalize the scattering cross section
by introducing ,
where
is identical to our previous
and the polarizability W_{2} can be expressed as
Index K represents the 2K-multipole. K=0 relates to the unpolarized intensity, K=2 to the atomic alignment. The circumstance that
is identical to our previous scattering cross section
implies that
If several different types of opacity sources are present, then the effective polarizability of the medium is obtained by summing over the various opacity and alignment contributions labeled with summation index j:
The amount of continuum polarization in the emergent radiation depends on the effective polarizability of Eq. (27) and on the scattering geometry, in particular on the anisotropy of the incident radiation field and the viewing angle. Since the relative magnitudes of the various opacity sources depend on temperature and density, is height dependent. As the radiation anisotropy is also height dependent and the medium optically thick (multiple scattering), the full radiative transfer problem needs to be solved numerically to obtain the continuum polarization correctly.
It is however possible to obtain good estimates of the amount of scattering polarization by using the idealization of the last scattering approximation. Since the polarization amplitudes that we are dealing with are very small (a fraction of one percent), the polarization of the incident radiation at the last scattering event can be disregarded. It is thus assumed that the polarization of the emergent radiation is created in a single scattering event (the last one) rather than through multiple scattering. The only thing we then need to know, besides , is the anisotropy of the incident radiation field that the last scattering event sees.
Our next idealization is to let this anisotropy be given by the observed limb darkening function at the considered wavelength. This is not quite correct, since the observed limb darkening refers to zero optical depth, whereas most of the observed photons originate at line-of-sight optical depth unity, or vertical optical depth , where is the cosine of the heliocentric angle. This difference however becomes small at the extreme limb, where goes to zero, since there the radiation comes from the top of the atmosphere (in the plane-parallel approximation). As varies with height, we have to choose a value that is representative for the height (around optical depth ) from where the bulk of the continuum photons come.
Thus, at least for order of magnitude estimates, we expect the continuum polarization to scale with the product
,
where k_{G} is a geometric depolarization factor that depends on the viewing angle and describes the dilution of the polarization due to the angular integration over the incident radiation. For pure
scattering we would have k_{G}=1. In the last scattering approximation we obtain k_{G} by multiplying the Rayleigh phase matrix with an unpolarized Stokes vector and integrating over all incident angles. As shown in Stenflo (1982), we then obtain
While this idealized treatment provides a good description of how the anisotropy and the continuum polarization vary with wavelength, it does not give a good description of the center-to-limb variation of the polarization. One reason for this is that the scattering opacity is confined to an optically thin layer, most of which lies above the non-polarizing region where the bulk of the continuum intensity originates. Therefore the relative proportion of polarized scattered photons to all photons scales with the optical thickness in the viewing direction of the scattering layer, i.e., in proportion to . The relative center-to-limb variation of the scattering polarization is therefore expected to have the approximate analytical form , and will be given in greater detail in Sect. 4.4 below. At the extreme limb, where the scattering layer becomes optically thick, our idealization for k_{G} may be more valid, but it is clear that we should use it with caution, as a tool to elucidate the basic physical mechanisms without having to enter into the realm of radiative transfer computations.
Figure 1: Wavelength dependence of the anisotropy factor k_{G} for . Note that the effective Balmer jump occurs at substantially longer wavelengths than the actual series limit (marked by the vertical line). | |
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The by far dominating contribution to the continuum opacity at optical wavelengths on the Sun comes from hydrogen. We can distinguish between unpolarized and polarizing opacity contributions. Unpolarized contributions come from the H^{-} opacity and from bound-bound and bound-free radiative absorption that is not part of a scattering process. The polarizing contributions are due to scattering from initial levels n=1 (Lyman scattering) and n=2 (Balmer scattering). In addition, there are contributions from Thomson scattering at free electrons.
Lyman scattering at visible wavelengths occurs in the dispersion wings of the Lyman series lines, very far from the respective resonance frequencies. The opacity in the visible is attenuated by the great distance from these resonances, and is approximately proportional to (cf. Eq. (9)), where is the center of gravity of the resonances in the Lyman series. In contrast, Balmer scattering in the visible occurs in the midst of the contributing resonances, which are distributed across the visible range. The Balmer scattering contribution is however attenuated by the population factor N_{n=2}/N_{n=1}. Quantitative comparison of these attenuation factors (see below) shows that the Balmer scattering contribution to the visible continuum is much smaller than the Lyman contribution.
To evaluate the expressions for the scattering opacity and polarizability in hydrogen, we need the oscillator strengths of all the possible transitions, both the bound-bound and bound-free. For bound-bound transitions in hydrogen between lower level
and upper level n, the absorption oscillator strength is
The bound-free case requires special considerations. A discrete continuation of Kramers' formula into the domain of continuum frequencies is obtained through substitution of quantum number n with imaginary quantum number ik, where k is a real-valued integer:
(34) |
To properly treat the bound-free contributions beyond the series limit (for
), we need to transform the discrete representation of Eq. (39) (in which k is an integer) into a continuous representation. Explicitly, the sum of
over the excited levels n_{ e} in Eqs. (23) and (25) may be generalized to include not only the bound-bound but also the bound-free transitions by continuing the sum to go not only over
,
but also over
.
The bound-free sum
then needs to be transformed into an integral through the substitution
As the state only consists of the state, the initial and final states i and f can only be different if the final state has . This is only possible for photons shortwards of Ly, where the solar spectrum is orders of magnitude weaker than in the visible. For this reason only the Rayleigh scattering transitions and are of any practical relevance for scattering from .
Figure 2: Lyman scattering opacity per hydrogen atom, in units of the Thomson scattering cross section per electron. The dashed curve represents the analytic expression of Eq. (46), which is an excellent approximation above 2000 Å. Below the Lyman limit the scattering behaves as if the excited electron were free (like Thomson scattering), while the cross section for radiative ionization (dash-dotted curve) is many orders of magnitude larger. | |
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For Rayleigh scattering we have according to Eqs. (23) and (41)
For the continuum range below the Lyman limit at Å, we need to replace the sum in Eq. (45) with the integral in Eq. (42) times (cf. Eq. (41)).
The results for the scattering opacity
per atom (i.e., for N=1) are plotted in Fig. 2 over the wavelength range 0-3000 Å, in units of the Thomson scattering cross section
.
The dashed curve for wavelengths above the Lyman limit is a simple analytical approximation
given by Baschek & Scholz (1982):
To obtain well defined peak amplitudes for the bound-bound transitions, we have to choose the values for the Doppler widths and damping constants for the various bound-bound transitions. Here we have adopted a Doppler broadening of 2 km s^{-1} for all the resonances. The choice of damping constant for the Lyman lines will be defined in the context of a discussion of the behavior of the Balmer bound-bound absorption near the series limit, which we will do in Sect. 3.6 below. The detailed pressure broadening mechanisms of the hydrogen lines are quite complex, but we do not need to go into these details here, since our focus is on the contributions to the continuum, which take place in the distant dispersion wings. The damping constant plays no role for the scattering opacity in the dispersion wings, as seen from Eqs. (44) and (45).
Figure 3: Main contributors to the Sun's continuum opacity in the visible and near UV. The lower solid curve represents the Balmer scattering cross section, while the upper solid curve gives the cross section for bound-bound radiative absorption. Pressure broadening due to the statistical Stark effect is accounted for to describe convergence of the bound-bound opacities to the bound-free ones. Comparison is made with the Lyman scattering cross section (dashed), the Thomson scattering cross section (horizontal solid line), the H^{-} opacity (dotted), and the bound-free opacity for radiative ionization (dash-dotted). The relative contributions of these opacity sources depend on the electron densities and pressures and the relative level populations of hydrogen. The values chosen are typical for solar conditions. | |
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Below the Lyman limit we have for comparison also plotted in Fig. 2 the coefficient for radiative ionization. According to Eq. (11) the total (integrated) contribution per atom from a discrete transition to the absorption coefficient is
We see from Fig. 2 that the cross section for scattering via continuum states is almost identical to the Thomson scattering cross section at free electrons. This is to be expected, since when the continuum state is not a bound state, the electron will tend to behave like a free particle while in this state. It is also natural to expect the cross section for radiative ionization to be very much larger than the scattering cross section, since once the electron has become free, it is unlikely to return back to the initial state, as it would have to do in a scattering transition. The small likelihood that this will occur is automatically included in the present theory.
Another important feature that is the result of the coherent nature of the scattering process is the occurrence of very deep and narrow minima in the scattering opacity between the bound-bound resonant wavelengths. They are the result of quantum interference between the excited states of different n quantum number. We note in Eq. (45) that the scattering amplitude of each transition is negative for and positive for . Between the resonances there is therefore cancellation between positive and negative contributions as the intermediate states are superposed coherently. It is because of these quantum cancellations that we get the deep, narrow dips in the scattering opacity. For the non-coherent process of radiative absorption, such dips do not occur (as we will see in Fig. 3 below).
Only for a fraction of the emission processes in a Balmer line the excited state has been radiatively excited by that Balmer transition. Conversely, radiative absorption in a Balmer line is followed by Balmer emission in only a fraction of the cases. Therefore the Balmer scattering opacity is considerably smaller than the Balmer coefficient for radiative absorption. While this distinction is important for the Balmer case, it was not needed when considering Lyman scattering at visible wavelengths, since excitation from the ground state to a virtual state that lies far below the first resonance has no other decay channel than spontaneous emission back to the ground state, i.e., a Rayleigh scattering transition.
As we will see, the Balmer scattering opacity does not contribute significantly to the continuum polarization problem. The coefficient of bound-bound radiative absorption, on the other hand, becomes important as we get close to the Balmer series limit, across which it transforms itself from bound-bound to bound-free unpolarized opacity in a continuous manner, without any real discontinuity, because of the increasing collisional damping, which smears the resonances into a quasi-continuum near the series limit.
While the Lyman transitions have two fine-structure components, the Balmer transitions (with ) have 7: , , , , , , , which can be combined to form Raman scattering transitions whenever . Since the fine-structure splitting is small, we can for our continuum polarization problem neglect it by using the principle of spectroscopic stability and letting the electron spin be zero. In this simplified case, Balmer scattering can only occur in the form of three different Rayleigh scattering transitions, all of which have i=f: , , and .
The application of the general theory to the Balmer case is done like in the Lyman case, although now we have to deal with the oscillator strengths of the individual fine structure components rather than with the total oscillator strength. In Fig. 3 we have plotted as the solid curve with the many resonances in the bottom part of the diagram the results for the Balmer scattering opacity per atom in units of the Thomson scattering cross section, after scaling it with the Boltzmann factor , where is the frequency of Lyman (since it defines the energy separation between the first two levels of hydrogen). Here we will use T=5740 K (the Sun's effective temperature), which gives . A more proper scaling would require radiative-transfer calculations, but for our order of magnitude comparison between the Balmer and Lyman opacity contributions, the exact choice is not important here.
As we see from a comparison with the Lyman contribution at visible wavelengths, obtained from expression (46) and plotted as the dashed curve in Fig. 3, the Balmer contributions to the continuum are at least 3 orders of magnitude smaller than the Lyman contributions, except in the vicinity of the Balmer resonant frequencies, where the Balmer contributions naturally dominate.
As in the case of Lyman scattering, the amplitudes of near the line cores are determined with the assumption that the Doppler broadening is 2 km s^{-1}, and that the damping constant varies as described in Sect. 3.6 below.
The coefficient for radiative absorption for bound-bound transitions
is given by Eq. (11). With Eq. (14) for B_{ie} and expressing the profile function in terms of a Voigt function H(a,v), we obtain
Normalizing with the Thomson scattering cross section to make the expression dimensionless, multiplying with the relative level population factor N_{2}/N_{1} to refer to the population N_{1} of the ground state, and assuming a Doppler velocity of 2 km s^{-1}, we obtain
The coefficient of radiative absorption as computed from Eq. (56) with the choice of damping constants as described in the next subsection is plotted in Fig. 3 as the solid resonant curve in the upper part of the diagram. Note how this curve does not have the deep dips between the resonances that the scattering opacity has, since these dips are exclusively due to quantum interferences, which only affect the scattering problem. When approaching the series limit the amplitude of the resonant oscillations of the curve goes to zero, which leads to a smooth and continuous transition to the coefficient for radiative ionization (bound-free case). The way in which the series limit is approached depends on the details of the pressure broadening of high-lying hydrogen levels and will be discussed in the next subsection.
For comparison we have in Fig. 3 also given the Thomson scattering cross section and the H^{-} absorption coefficient . To obtain from we have applied as scaling factor an estimate of the ratio N_{ e}/N_{1} between the electron density and the hydrogen population density in its ground state. Based on model atmosphere calculations we adopt as a characteristic number . We see that the Thomson scattering contribution to the continuum is 2-3 orders of magnitude larger than the Balmer scattering contribution, but typically one order of magnitude smaller than the Lyman contribution (with the difference decreasing with wavelength), and generally smaller than the Balmer bound-bound absorption coefficient, in particular as we approach the series limit.
Finally we make a comparison with the H^{-} opacity, which we have taken from Chandrasekhar & Breen (1946), applied to an electron pressure of P_{ e}=0.395 (SI units) and a temperature of 5740 K. It has been plotted as the dotted curve in Fig. 3. H^{-} is the dominating source of continuum opacity, typically two orders of magnitude larger than the Lyman contribution.
The level of the scattering opacity outside the immediate cores of the resonances is independent of the assumption for the damping constant , as seen from Eqs. (23) and (21). In contrast, for the coefficient of bound-bound radiative absorption the levels of the distant line wings scale with , as seen from Eqs. (11) and (16). Note also that the wings of the Voigt profile H(a,v) in Eq. (56) scale with . The coefficient for bound-bound absorption is therefore a sensitive function of the line-broadening processes. Doppler broadening affects only the line cores and is unimportant in the wings, where collisional or pressure broadening dominate.
Pressure broadening increases greatly as we approach the series limit. This smearing leads to a merging of the crowded high-level resonances into a quasi-continuum before the ionization limit is reached, resulting in a gradual approach to the ionization limit rather than a discontinuous jump. It is beyond the scope of the present paper to model in any quantitative detail how this quasi-continuum is formed. The aim here is to try to clarify the main physics involved and make a relatively crude approximate treatment that can be used to compare with the observations and thereby to verify that the relevant physical mechanisms have been identified.
Pressure broadening of the hydrogen lines is normally described in terms of the statistical Holtsmark theory, according to which the broadening is due to the Stark effect splitting of the lines from the statistically fluctuating electric fields of the surrounding ions and electrons. The resulting Holtsmark shape of the absorption coefficient in the line wings is found to vary with
(Unsöld 1955). As this is steeper than the
decrease of the unbroadened, quantum-mechanical dispersion wings, the ordinary opacity from the dispersion wings will take over at large distances from the resonances. Since for the continuum problem we are only interested in the behavior far from the resonances, it is a reasonable approximation for our purposes to describe the lines in terms of Voigt profiles with a damping constant ,
which leads to a
behavior in the wings. For
we use
Following Inglis & Teller (1939) and Unsöld (1955),
According to Unsöld (1955),
The damping parameter a in the Voigt function is
,
or, using the Doppler formula,
In the range below the Balmer limit we plot in Fig. 3 as dot-dashed the bound-free cross section for radiative ionization, like in Fig. 2. We see that this bound-free cross section is reached well before the series limit by the rapid convergence of the bound-bound cross section oscillations. We also note that the Balmer absorption opacity becomes comparable to the H^{-} opacity near the Balmer limit and is larger there than the Lyman scattering opacity (dashed line in Fig. 3). The wavelength where the bound-bound radiative absorption coefficient surpasses the Lyman opacity lies approximately 140 Å above the series limit and is the place where the effective "Balmer jump'' transition in the scattering polarization may be expected to begin. A wavelength shift of order 100 Å of the apparent Balmer jump is indeed seen in the observational data, as we will see below.
In the present simplified treatment we have not accounted for collisional ionization, which increases with n, and photo-ionization, which decreases with n. For high levels collisional ionization will dominate over bound-bound radiative absorption and couple effectively to the continuum state. This may speed up the transition to a quasi-continuum and thereby both shift and steepen the effective Balmer jump. The quantitative details of this complex subject is however outside the scope of the present paper.
As we have seen in Sect. 2.4, the intrinsic polarizabilities
of the individual scattering transitions need to be combined, weighted according to their relative opacity contributions, to obtain the effective polarizability
of the medium. With
A more correct way to add the opacity contributions in a weighted way is to attach
to the scattering transitions, and
to the pure absorption transitions if they are treated in LTE. This kind of weighting is automatically done in the formalism of radiative transfer. Here, in our phenomenological treatment that bypasses radiative transfer, we make the simplifying LTE-type assumption that
and
are of the same magnitude, which allows us to simply write
We base our empirical determination of the continuum polarization on the Atlas of the Second Solar Spectrum that has been compiled in three voumes by Achim Gandorfer. Atlas Vol. I covers 4625-6995 Å (Gandorfer 2000), Vol. II covers 3910-4630 Å (Gandorfer 2002), while the still unpublished Vol. III covers 3161-3913 Å. The scattering polarization (Stokes Q/I) in the solar spectrum has been recorded in seemingly non-magnetic solar regions near the limb, always at a limb distance that corresponds to (which is 5 arcsec inside the extreme limb). The positive Q direction is defined as the direction parallel to the nearest solar limb, i.e., perpendicular to the radius vector from disk center. All the recordings have been made with the Zurich Imaging Polarimeter ZIMPOL (cf. Povel 1995). For Volumes I and II of the Atlas ZIMPOL was used at IRSOL (Istituto Ricerche Solari Locarno), for Vol. III that covers the UV portion ZIMPOL was used at the McMath-Pierce facility of the National Solar Observatory (Kitt Peak).
While the three atlases provide very high accuracy in terms of relative polarization values, the zero point of the polarization scale cannot be determined with comparable precision due to instrumental effects. On spectral scales less than about 10 Å the relative spectral variations in Q/I with the polarized line profiles are very well determined, but the Q/I zero-point offset is floating around on spectral scales larger than 10 Å. In the published atlases the Q/I zero point has been fixed by fitting the polarized continuum to the theoretical value derived from the radiative-transfer theory of Fluri & Stenflo (1999). This procedure of relying on theory however cannot be used if we want to obtain an empirical determination of the continuum polarization from the atlas data. For such a determination we have to regard the zero point of the polarization scale as an entirely unknown quantity that needs to be determined in the same context together with the value for the continuum polarization. The adopted procedure for doing this will be outlined in the next subsection.
The richly structured Second Solar Spectrum contains, like the ordinary intensity spectrum, a multitude of spectral lines superposed on a background continuum. In the polarized spectrum we can distinguish between two types of lines: depolarizing and intrinsically polarizing lines. For the depolarizing lines the unpolarized line opacity dilutes the polarizing continuum opacity such that the continuum polarization level gets depressed. In Q/I such lines therefore have the appearance of absorption lines, similar to the corresponding absorption lines in the intensity spectrum. The intrinsically polarizing lines, on the other hand, appear like "emission'' lines in Q/I if they polarize more than the continuum. The Second Solar Spectrum contains a mixture of both types of lines. Since the amount of intrinsic line polarization is modified by the Hanle effect through the presence of hidden, turbulent magnetic fields (Stenflo 1982), and since these fields vary with time, the relative proportion of absorption- and emission-like lines in the Second Solar Spectrum varies with phase of the solar cycle (cf. Berdyugina et al. 2002).
Let us assume that we have been able to find a spectral region that only contains purely depolarizing lines that have no intrinsic line polarization. Let p=Q/I be the "true'' polarization (without zero-point error) and
the corresponding level of the continuum (outside the spectral lines). Then we may model the relative line depth in Q/I in terms of the relative line depth in intensity I with the following one-parameter model:
While a statistical analysis of the first atlas survey of the Second Solar Spectrum (based on observations made in 1978, in Stenflo et al. 1983a,b) suggested that a value of would provide an adequate description of the depolarizing lines, later work with ZIMPOL indicated that a considerably lower value, like , would be more appropriate for the particular spectral windows that were considered (Stenflo et al. 1998). More recently the theoretical mechanisms behind the formation of the depolarizing line profiles have been identified, and a parameter survey has been performed (Fluri & Stenflo 2003). It is found that the depolarizing line depth is a sensitive function of height of formation, and that this function is entirely different for lines formed by pure absorption as compared with lines formed by unpolarized scattering. In terms of the simplified model of Eq. (68), one may say that the value of the parameter depends on the details of line formation and may vary considerably from one line to the other. Theory does not allow us to find a value of that applies to all lines, there will be a substantial scatter between the lines. Therefore Eq. (68) only has validity in a statistical sense for an ensemble of spectral lines, and it is prudent to treat as a free parameter that is poorly known.
Figure 4: Illustration of a 2-parameter fit to the observed Q/I with the model given by Eq. (71), assuming that . The thick solid lines represent the observations (taken from the currently unpublished UV portion of Gandorfer's Atlas of the Second Solar Spectrum), while the thin curves represent the model. The determined level of the continuum polarization is given by the horizontal line in the bottom panel. | |
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Since the zero point of the polarization scale is unknown and floating around on scales larger than 10 Å, the observed polarization
is the sum of the true polarization p and the value p_{0} of the true zero point on the chosen polarization scale:
The parameter f only presents a certain problem in the UV, where the spectrum is very crowded, and no clean continuum window is available. Here the level of the intensity continuum is chosen manually to be consistent with other spectral atlases. This choice of fixes the value of parameter f and leaves us with three free model parameters, which can be determined by an iterative least squares fitting method. Although such a 3-parameter fit generally converges to a unique solution, the goodness of the fit depends only weakly on parameter . The fit procedure becomes numerically much more robust if we fix the value of and only seek to determine the two free parameters (the continuum polarization) and p_{0} (the zero point of the polarization scale).
The procedure is illustrated in Fig. 4, where we for clarity have selected a spectral region that seems to be devoid of intrinsically polarizing lines. The thick solid lines represent the observations, while the thin solid lines illustrate the model. In this example we have assumed ad hoc that the value of is 0.6, and then done a 2-parameter fit to the observed Q/I curve. The so obtained value of the continuum polarization is given as the horizontal line in the lower panel (Q/I). The thin curve in the upper panel ( ) represents . This curve, when scaled with the value of , becomes the thin fit curve in the lower panel. We see that the fit is excellent, which means that the value of is very accurately determined, provided that the value of has been chosen correctly.
Note that the thick solid curves are slightly thicker around 3304.3 Å, since there is the overlap region of two separate spectral recordings that have been pieced together to cover the whole spectral window shown in Fig. 4. The almost imperceptible broadening of the curves in the overlap region illustrates the nearly perfect reproducibility of every single wiggle of the spectral recordings in both and Q/I, which indicates that practically all the tiny Q/I wiggles are solar features and not noise.
The excellent match that we see between the thick and thin curves in the Q/I diagram would be nearly as perfect also for other values of , so the quality of the fit cannot be used as a good criterion for the selection of . The only significant change in the fit diagram is that the Q/I continuum level and zero point would be placed differently for other values of . With the help of Fig. 4 we can however understand and get a feeling for how the extracted value of depends on . If were larger, for instance unity, the thick and thin curves in the panel would coincide. This would in the Q/I diagram have the effect both that the continuum level would be lowered to lie closer to the observed curve, and at the same time that the zero point of the scale would be lowered as well to be somewhat farther from the observed curve. These effects partly compensate each other in producing a value for , but for the special case of Fig. 4 the lowering of the continuum is somewhat larger than the lowering of the zero point, so that the net effect is a reduction of . This net effect is however relatively small.
In spectral regions with little line blanketing and weak absorption lines, however, which is the situation in the red part of the spectrum, an increase of from 0.6 to 1.0 will have the opposite net effect: The zero level will be lowered more than the continuum level, so that the value of will increase with increasing . In summary we see that the dependence of the extracted value of on the assumed value of is opposite between the UV and the red parts of the spectrum, since it is related to the degree of line blanketing, but that on average the dependence is relatively modest. This rather weak dependence on implies that our method of finding the empirical value of the continuum polarization should be rather robust in spite of the uncertainties in our knowledge about the formation of the depolarizing lines.
Figure 5: Continuum polarization , as determined from the Atlas of the Second Solar Spectrum using the model of Eq. (68). Asterisks represent model fits with , filled circles fits with , open circles fits with . The vertical lines give the resonance wavelengths of the first ten Balmer lines, plus the series limit as the left-most vertical line. The three solid curves represent second-order polynomial fits to the values: a central curve as the most likely representation, and two outer curves that indicate the approximate lower and upper limits. | |
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We have seen in the preceding subsection that the fit procedure works well, provided that the spectral fit region exclusively contains depolarizing lines. The spectral region in Fig. 4 with only depolarizing lines that we have used to illustrate the method is however not typical for the Second Solar Spectrum. Usually, intrinsically polarizing lines are abundant everywhere in the spectrum. Their presence would invalidate the model fit, since the model is based on Eq. (68), which of course has no validity for lines with intrinsic polarization. We therefore have to take great care to exclude intrinsically polarizing lines before applying the fit procedure.
Intrinsic line polarization comes with all kinds of amplitudes and profile shapes and is therefore difficult to identify and isolate except in the cases when the line polarization clearly dominates over the continuum polarization with profile features that look like "emission'' lines in Q/I. Due to the highly individualistic behavior of the spectral features in the Second Solar Spectrum we have refrained from trying to find some automated computer algorithm that could decide whether or not intrinsic line polarization is present at a given wavelength, and instead resorted to a manual procedure. Thus we have done careful visual inspection of the Second Solar Spectrum with educated judgement based on many years of experience in dealing with this spectrum, and so defined hundreds of "exclusion windows'' of various widths throughout the spectrum. The data inside these exclusion windows are skipped by the least squares fit procedure because of their "contamination'' with intrinsic line polarization.
To determine the continuum polarization as a function of wavelength we divide the spectrum into a sequence of partially overlapping segments. Each segment consists of approximately 660 points, with the wavelength monotonically increasing but with the exclusion windows skipped. If no exclusion window is present, 660 points corresponds to about 6 Å, otherwise the segment as expressed in Å is wider. For each segment and p_{0} are determined by the iterative least squares fit, using widely different start values for the iterations to verify the uniqueness of the fit. Due to the uncertainty in the proper choice of to use for the fit, we make three sets of fits throughout the whole spectrum, for the three widely different values 0.3, 0.6, and 1.0. It is highly unlikely that the statistically "true'' value of would lie outside this very wide range. Trials with 3-parameter fits, using as a free parameter, result in a wide distribution of values, but the bulk of the distribution is bracketed by the extreme values of and 1.0 that we have chosen here.
The results of these fits are presented in Fig. 5, where the values for each spectral segment are plotted (in logarithmic scale) vs. the average wavelength of the segment. Three different symbols are used for the three values: asterisks for , filled circles for , and open circles for . To reduce the scatter of the points we have in Fig. 5 applied median smoothing with a running window that increases in width from 11 segments in the UV to 21 segments in the red. The corresponding window width in Å depends on the distribution of the exclusion windows but is of order 100 Å in the UV and 200 Å in the red. This is smaller than the scale over which varies significantly.
We notice a big gap in the data points between 3730 and 4090 Å. The reason for this is not the absence of data but the circumstance that this region is completely swamped with strongly polarizing lines, which invalidate our procedure to find the continuum polarization. The polarization from the Ca II K 3933 and H 3968 Å lines dominates the spectrum over more than 200 Å. In the lower portion of the gap the spectrum is crowded with polarizing lines of the CN molecule as well as a number of strong iron lines, while in the upper portion of the gap there are some very strongly polarizing lines of Fe I and Sr II.
For reference, vertical lines have been drawn in Fig. 5 to mark the location of the Balmer series lines (up to H_{10}) as well as the Balmer limit at 3646 Å (the left-most vertical line). The most notable feature here is a very clear and conspicuous Balmer jump, which however does not occur at the series limit but at considerably longer wavelengths (nearly 80 Å longer), as expected from pressure broadening of the highly excited hydrogen levels, as explained in Sect. 3.6.
We notice in Fig. 5 a slight but systematic dependence of the fit values on the assumed value of parameter . At longer wavelengths the asterisks lie systematically above the open circles, while in the UV the opposite situation is the case. An explanation for this wavelength variation of the dependence was given at the end of the preceding subsection.
In the UV the main source of scatter of the points is due to the systematic dependence of the results. At higher wavelengths, however, other sources of scatter dominate, much because of the presence of non-excluded lines with weak and inconspicuous intrinsic polarization. Figure 5 may give the misleading impression that the noise gets larger with increasing wavelength. However, we have to remember that the polarization in the figure is given with a logarithmic scale, and that the amplitude decreases by two orders of magnitude, from about 1% to about 0.01%, as we go from the UV to the red part of the spectrum. It is actually remarkable that it is at all possible to extract the minute values of the continuum polarization from the atlas data at these long wavelengths. That this can be done is testimony to the high quality of the atlas data.
In the following we will not try to distinguish between the various sources of scatter of the data points. Thus we will not try to treat the data referring to the three different symbols differently, but will consider the scatter of all the symbols as representative of the uncertainty in the determination of the continuum polarization . Since we know from theory that is expected to vary only slowly with wavelength (except near the Balmer jump) in a lin-log representation like Fig. 5, it is sufficient to describe the dependence of on wavelength in terms of a second-order polynomial. However, because of the Balmer jump, we have to use a different second-order polynomial below and above the Balmer jump, as well as a special, steep polynomial in the region 3650-3730 Å where the Balmer jump is found to take place.
Because of the large scatter of the values we should not attach significance to the rapid fluctuations of the data points, like to the apparent peak near 5800 Å, since these fluctuations are most likely due to the statistical nature of the extraction method used and to ubiquitous contamination from intrinsically polarizing spectral lines. Instead we have found it convenient to do smoothing by choosing second-order polynomials by hand (rather than by automated fits). Three such sets of curves have been selected and are shown in Fig. 5. The two outer curves represent what we consider to be the approximate lower and upper limits to the empirical values of , between which the bulk of the data points fall, while the central curve is exactly in the middle between the two limit curves and represents our best estimate for . Theoretical modelling should aim at reproducing the middle curve, within the tolerance region defined by the outer curves.
Figure 6: Comparison between observations and theory. The shaded area represents the empirically allowed region, defined as the region between the two outer polynomial curves in Fig. 5. The vertical line marks the wavelength of the Balmer series limit. The dashed curve has been obtained from the radiative-transfer theory of Fluri & Stenflo (1999). The thick, solid curve is based on the last scattering approximation, Eq. (72), with the anisotropy and the opacities given by Figs. 1 and 3. Note in particular the large displacement of the Balmer jump with respect to the series limit. | |
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At the time when the radiative-transfer theory for the formation of the continuum polarization was developed (Fluri & Stenflo 1999), there did not exist adequate empirical data with which the theoretical values could be compared in a meaningful way. However, now we are in a position to make such a comparison. In Fig. 6 the empirically allowed region, defined as the region between the outer polynomial curves in Fig. 5, is given as the shaded area. The solution of the polarized transfer equation with a numerically given model atmosphere, as obtained by Fluri & Stenflo (1999), is shown by the dashed curve. This radiative-transfer theory has only been applied above 4000 Å and therefore does not address the interesting region around the Balmer jump.
Although the empirical and theoretical curves behave in similar ways, it is notable that the dashed curve lies systematically above the empirically allowed region for wavelengths below about 5700 Å. Although the source of this discrepancy cannot be identified at the present time, it is likely to be related to the choice of model atmosphere. The radiative-transfer problem needs to be revisited, both to extend the theory to shorter wavelengths and to fit the empirical data by modifying the model atmosphere. Our continuum polarization data may serve as a qualitatively new type of constraint on models of the solar atmosphere.
For comparison, the thick, solid curve in Fig. 6 represents the values for
that we get when we bypass radiative-transfer theory and use the last scattering approximation as explained in Sect. 2.5. What is plotted is simply
The shape and position of the Balmer jump in in Fig. 6 appear similar to the Balmer jump in Fig. 1 of the anisotropy factor k_{G} alone. Since represents the product of k_{G} and , the question arises which of these two factors is most responsible for the Balmer jump in Fig. 6. To answer this question we have replaced k_{G} in Eq. (72) with a straight, slanted line without any Balmer jump. The influence on the shape and magnitude of the Balmer jump in is insignificant. If we remove the bound-bound Balmer opacity contributions, the Balmer jump appears as a sharp discontinuity exactly at the location of the series limit, regardless of whether we remove or retain the observed Balmer jump in Fig. 1 from k_{G}. These tests conclusively show that the Balmer jump in Fig. 6 comes almost exclusively from a corresponding Balmer jump in , while the influence of the Balmer jump in k_{G} on is a second-order effect that plays a subordinate role.
The similarity of the Balmer jumps in and k_{G} points to a common origin: the wavelength variation of . This ratio becomes significant already at considerably longer wavelengths than the formal Balmer limit. Due to the increasing pressure broadening the bound-bound opacity converges to the bound-free opacity long before the series limit is reached.
It is to be noted that the empirically determined Balmer jump appears to be more abrupt and discontinuous than can be accounted for by our simplified theory, as seen from Figs. 5 and 6. The observed shape of the Balmer jump may therefore serve as a constraint on refined theories for the pressure broadening of atomic energy levels in stellar atmospheres. Unfortunately it is difficult to determine the detailed shape of the Balmer jump from polarimetric observations, since it occurs near the edge of the excluded region in Fig. 5, which is filled with intrinsically polarizing lines.
Through model fitting of the data in the Atlas of the Second Solar Spectrum we have obtained the function that represents the continuum polarization at the particular center-to-limb distance that is defined by (where is the cosine of the heliocentric angle). To connect this result to other values we need the help of radiative-transfer calculations that have established the relative shape of the center-to-limb variations.
Figure 7: Overview of the functional behavior of the empirically determined continuum polarization . The two upper panels give the wavelength variation of for , in logarithmic and linear scales. The curves are the same as the central polynomial curve in Fig. 5. The two bottom panels give the center-to-limb variation of for Å, in logarithmic and linear scales. | |
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According to the radiative-transfer theory of Fluri & Stenflo (1999) the results of the numerical calculations can be closely approximated by the following semi-analytical formula for the continuum polarization
as a function of
and of wavelength :
Expressed in terms of the above functions, our empirically determined continuum polarization
becomes
The two lower panels in Fig. 7 show the center-to-limb variation of the continuum polarization for Å, both with a logarithmic and a linear scale for . They illustrate the steepness of the center-to-limb variation, whose shape is given by function of Eq. (74).
A number of different physical processes contribute to the formation of the polarized continuum at visible wavelengths. With the exception of Thomson scattering at free electrons they all have to do with absorption or scattering by hydrogen. The only processes that contribute with polarized photons are Lyman scattering at hydrogen in its ground state and Thomson scattering. Lyman scattering is the larger of the two, although the scattering takes place so far from the Lyman resonances. The polarized photons get diluted by the unpolarized photons from the other opacity sources. The dominating opacity is from H^{-}, but radiative absorption in the bound-bound and bound-free Balmer transitions becomes important as we approach the Balmer series limit. The Balmer jump that results from these Balmer absorptions is shifted from the series limit towards longer wavelengths due to pressure broadening of the high atomic levels from the statistical Stark effect. Our simplified treatment of this pressure broadening can approximately reproduce the shift but not the shape (abruptness) of the observed Balmer jump.
The polarization amplitude scales with the anisotropy of the radiation field. This anisotropy increases rather steeply as we go towards shorter wavelengths, and it exhibits a Balmer jump that is also shifted to longer wavelengths with respect to the series limit. The Balmer jump in the continuum polarization is however determined almost exclusively by the Balmer jump in the effective, intrinsic polarizability , not by the Balmer jump in the anisotropy factor (k_{G}). The Balmer jump in occurs when the bound-bound radiative absorption in the Balmer lines overtakes the Lyman scattering opacity and starts to become comparable to the H^{-} opacity. The detailed modelling of the effective Balmer jump will remain a challenge for future pressure broadening theories and of general interest for the physics of stellar atmospheres.
The empirical value of the continuum polarization and its wavelength variation over the range 3161-6995 Å could be determined for disk position from Gandorfer's three volumes of the Second Solar Spectrum. Since the zero point of the polarization scale is basically unknown in these observations, it had to be determined together with the continuum polarization via a model for the behavior of the depolarizing lines in the Second Solar Spectrum. Parameter studies show that the results for the continuum polarization are not very sensitive to the details of this model. Still we have explored how the model uncertainties affect the results and added this influence to the general noise to obtain an estimate of the accuracy (as a function of wavelength) with which the continuum polarization has been determined. It is surprising that it is possible to determine rather well even in the red part of the spectrum, where the polarization amplitude is as low as 10^{-4}. However, the range between 3730 and 4090 Å could not be used because it is dominated by strong line polarization, in particular from the CN molecule, from the Ca II K and H lines, and from strongly polarizing lines of Fe I and Sr II.
The empirically determined is found to lie systematically lower than the values previously obtained from radiative-transfer modelling (for Å). The radiative-transfer theory needs to be revisited to identify the origin of this discrepancy. Potentially the observed continuum polarization may be used to constrain the model atmospheres, in a different way as compared with the constraints imposed by spectral intensity data.
The main strength of the present data set is the large wavelength range over which the wavelength variation of could be determined. The main weakness is the considerable range of uncertainty in the values, much due to the statistical nature of the extraction procedure and the ubiquitous contamination from intrinsic line polarization. Therefore the constraints on model atmospheres that the current data set provides are crude and need to be much improved upon in future work.
Although current radiative-transfer modelling thus fails to properly reproduce the polarization amplitudes, it is likely that the relative shape of the center-to-limb variation is predicted well by such modelling. If we use the so predicted relative shape functions and scale them with the observed values of the continuum polarization for , we obtain a semi-empirical representation of that covers all wavelengths and values. This is of great value to have as a tool in the reduction of any polarimetric observations of the Second Solar Spectrum, since the absolute position of the zero point of the polarization scale cannot be determined in the observations with a precision that can come close to the relative polarimetric precision. We are now in a position to overcome this problem. With our new tool we may fit the observed continuum to the semi-empirical values for at the respective wavelength and position and thereby be able to determine the absolute polarization scale with much improved precision.
Acknowledgements
I am grateful to Hans Martin Schmid for illuminating discussions, and to Achim Gandorfer for providing me with the complete data set for the three volumes of his Atlas of the Second Solar Spectrum. The observations for this atlas were done with the ZIMPOL polarimetric equipment at IRSOL (Istituto Ricerche Solari Locarno) for Volumes I and II, and at the National Solar Observatory/Kitt Peak for Volume III. The engineering group at ETH Zurich (Peter Povel, Peter Steiner, Urs Egger, Frieder Aebersold, Stefan Hagenbuch) built the ZIMPOL system and provided the technical support, financially supported by the Swiss Nationalfonds, grant No. 20-64945.01. IRSOL has been financially supported by the canton of Ticino, the city of Locarno, ETH Zurich, and the Swiss Nationalfonds. NSO is one of the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation. I also like to thank the anonymous referee for thoughtful suggestions.