Heights of formation of non-magnetic solar lines suitable for velocity studies

Heights of formation of lines that do not exhibit Zeeman splitting are calculated using an LTE, partial non-LTE, and full non-LTE approach. Non-magnetic (g=0) lines are valuable for velocity investigations in quiet-Sun magnetic field regions, and a knowledge of their formation heights is useful for obtaining three dimensional velocity profiles in these regions. Presently at Sacramento Peak Observatory. Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

Asymmetries of the solar Ca ii lines

A theoretical study of the influence of propagating acoustic pulses in the solar chromosphere upon the line profiles of the Ca ii resonance and infrared triplet lines has been made. The major objective has been to explain the observed asymmetries seen in the cores of the H and K lines and to predict the temporal behavior of the infrared lines caused by passing acoustic or shock pulses. The velocities in the pulses, calculated from weak shock theory, have been included consistently in the non-LTE calculations. The results of the calculations show that these lines are very sensitive to perturbations in the background atmosphere caused by the pulses. Only minor changes in the line shapes result from including the velocities consistently in the line source function calculations. The qualitative changes in the line profiles vary markedly with the strength of the shock pulses. The observed differences in the K line profiles seen on the quiet Sun can be explained in terms of a spectrum of pulses with different wave-lengths and initial amplitudes in the photosphere.     

Eu III oscillator strengths and europium abundances in Ap stars

We present our calculations of the spectrum and oscillator strengths for the 4f⁷?(4f⁶5d+4f⁶6s) Eu III transitions. The calculations were performed with Cowan's RCN-RCG-RCE codes in the single-configuration approximation. A comparison of computed level lifetimes with experimental data for three levels shows that the scale of theoretical oscillator strengths could be overestimated by a factor of 3. The theoretical oscillator strengths of red Eu III lines are two orders of magnitude smaller than their astrophysical oscillator strengths derived by Ryabchikova et al. (1999) from the condition of ionization balance. The new oscillator strengths were tested by analyzing the Eu abundance using Eu II and Eu III lines in the spectra of hot peculiar stars (α² CVn is a typical representative) and cool peculiar stars (β CrB is a typical representative). First, we computed non-LTE corrections, which proved to be significant for α² CVn. We also analyzed the Eu II λ6645.11-Å line as well as ultraviolet and optical Eu III lines. We show that the new oscillator strengths together with the non-LTE corrections allow the contradiction between the Eu abundances derived by Ryabchikova et al. (1999) separately from optical Eu II and Eu III lines in α² CVn to be resolved. The new Eu abundance, log(Eu/N _tot)=?6.5, also faithfully describes the blended near-ultraviolet resonance Eu III lines. Using the new Eu III oscillator strengths to analyze the spectrum of the cool Ap star β CrB, we found a significant deviation of the n(Eu II)/n(Eu III) ratio from its equilibrium value. For a chemically homogeneous model atmosphere, to obtain the observed intensity of the Eu III λ 6666.35-Å line, the Eu abundance must be increased by two orders of magnitude compared to that required to describe the Eu II λ 6645.11-Å line. We discuss the possibility of explaining the observed intensities of Eu II and Eu III lines in the spectrum of β CrB by the presence of an inhomogeneous atmosphere with Eu concentrated in its uppermost layers. In such atmospheres, the role of non-LTE effects becomes dominant.     

Analysis of the HST ultraviolet spectra for T Tauri stars: RY Tau and HD 115043

The ultraviolet spectra of the stars RY Tau and HD 115043 from the Hubble Space Telescope are analyzed. RY Tau belongs to the classical T Tauri stars, while HD 115043 is a young (t~3×10⁸ years), chromospherically active star. The most intense emission lines were identified, and their fluxes were measured. Low-resolution spectra of RY Tau and HD 115043 in the wavelength range 1160–1760 Å exhibit almost the same set of emission lines. However, first, the luminosity of RY Tau in these lines is approximately a factor of 300 higher than that of HD 115043, and, second, the relative line intensities differ greatly. The intensity ratio of the C IV λ1550, Si IV λ1400, and NV λ1240 doublet components is close to 1: 2 in the spectra of both stars. Judging by the continuum energy distribution, the spectral type of RY Tau is later than that of HD 115043. Synchronous flux variability in the C IV λ1550 and He II λ1640 lines in a time of ~20 min was detected in RY Tau. The flux rise in these lines was accompanied by a redshift of the intensity peak in the profiles by~50 km s^?1. Intermediate-resolution spectra are used to study line profiles in the spectrum of RY Tau. In particular, the profiles of (optically thin) Si III]λ1892 and C III]λ1909 lines were found to be asymmetric and about 300 km s^?1 in width. The (optically thick) C IV λ1550 doublet lines have similar profiles. The Mg II λ2800 doublet lines are also asymmetric, but their shape is different: they consist of a broad (?750 km s^?1 at the base) emission component on which an interstellar absorption line shifted from the line symmetry center by about 20 km s^?1 is superimposed. The intensity ratio of the Mg II λ2800 doublet components is?1.4. Whether there are molecular hydrogen lines in the spectrum of RY Tau is still an open question. It is shown that the emission lines in the ultraviolet spectrum of RY Tau cannot originate in a hydrostatically equilibrium chromosphere. It is argued that quasi-steady accretion of circumstellar matter is responsible for the emission.     

The center-to-limb variations of four Ca i lines in the photospheric spectrum at λ6500

We study the center-limb (CL) variation of the average profiles of four Ca i lines near 6500 and compare these observations with synthetic data obtained from several line formation models having different thermal structures, line parameters, LTE and non-LTE conditions, and micro and macroturbulence values, to assess the formation characteristics of our Ca i lines in the solar photosphere.Comparison of numerical results with observations indicates that non-LTE is indispensable to fit the CL variation of the central residual intensity for the line 6493, and anisotropic microturbulence is indispensable to improve the CL behavior of the equivalent widths for all lines. The Ca i line analysis favors a cool photospheric model, but this cannot be disentangled clearly from the effects of non-LTE and small-scale velocity fields on the grounds of the present line formation models.     

Observational restrictions on sodium and aluminium abundance variations in evolution of the galaxy

In this paper we construct and analyze the uniform non-LTE distributions of the aluminium ([Al/Fe]-[Fe/H]) and sodium ([Na/Fe]-[Fe/H]) abundances in the sample of 160 stars of the disk and halo of our Galaxy with metallicities within ?4.07 ≤ [Fe/H] ≤ 0.28. The values of metallicity [Fe/H] and microturbulence velocity ξ _turb indices are determined from the equivalent widths of the Fe II and Fe I lines. We estimated the sodium and aluminium abundances using a 21-level model of the Na I atom and a 39-level model of the Al I atom. The resulting LTE distributions of [Na/Fe]-[Fe/H] and [Al/Fe]-[Fe/H] do not correspond to the theoretical predictions of their evolution, suggesting that a non-LTE approach has to be applied to determine the abundances of these elements. The account of non-LTE corrections reduces by 0.05–0.15 dex the abundances of sodium, determined from the subordinate lines in the stars of the disk with [Fe/H] ≥ ?2.0, and by 0.05–0.70 dex (with a strong dependence on metallicity) the abundances of [Na/Fe], determined by the resonance lines in the stars of the halo with [Fe/H] ≤ ?2.0. The non-LTE corrections of the aluminium abundances are strictly positive and increase from 0.0–0.1 dex for the stars of the thin disk (?0.7 ≤ [Fe/H] ≤ 0.28) to 0.03–0.3 dex for the stars of the thick disk (?1.5 ≤ [Fe/H] ≤ ?0.7) and 0.06–1.2 dex for the stars of the halo ([Fe/H] ≤ ?2.0). The resulting non-LTE abundances of [Na/Fe] reveal a scatter of individual values up to Δ[Na/Fe] = 0.4 dex for the stars of close metallicities. The observed non-LTE distribution of [Na/Fe]-[Fe/H] within 0.15 dex coincides with the theoretical distributions of Samland and Kobayashi et al. The non-LTE aluminium abundances are characterized by a weak scatter of values (up to Δ[Al/Fe] = 0.2 dex) for the stars of all metallicities. The constructed non-LTE distribution of [Al/Fe]-[Fe/H] is in a satisfactory agreement to 0.2 dex with the theoretical data of Kobayashi et al., but strongly differs (up to 0.4 dex) from the predictions of Samland.     

Formation of Zr I and II lines under non-LTE conditions of stellar atmospheres

The formation of Zr I and Zr II lines in stellar atmospheres under non-LTE conditions has been considered for the first time. A model zirconium atom has been composed using 148 Zr I levels, 772 Zr II levels, and the ground Zr III state. Non-LTE calculations have been performed for model atmospheres with T _eff = 5500 and 6000 K, log g = 2.0 and 4.0, [M/H] = −3, −2, −1, 0. In the entire investigated range of parameters, the Zr I levels are shown to be underpopulated relative to their LTE populations in the line formation region. In contrast, the excited Zr II levels are overpopulated, while the ground state and lower excited levels of Zr II retain their LTE populations. Since the non-LTE effects cause the Zr I and Zr II spectral lines being investigated to weaken, the non-LTE corrections to the abundance derived from Zr I and Zr II lines are positive. For Zr II lines, they increase with decreasing metallicity and surface gravity up to 0.34 dex for the model with T _eff = 5500, log g = 2.0, and [M/H] = −2. The non-LTE effects depend weakly on temperature. The non-LTE corrections for Zr I lines reach 0.33 dex for solar-metallicity models. Zr I and Zr II lines in the solar spectrum have been analyzed. The non-LTE zirconium abundances derived from lines in the two ionization stages are shown to agree between themselves within the error limits, while the LTE abundance difference is 0.28 dex. The zirconium abundance in the solar atmosphere (averaged over Zr I and Zr II lines) is log ɛ_Zr,⊙ = 2.63 ± 0.07.     

Non-LTE effects in Al I lines

We present the theoretical analysis of the Al I line formation in the spectra of late-type stars ignoring the assumption of local thermodynamic equilibrium (LTE). The calculations were based on the 39-level aluminum atom model for one-dimensional hydrostatic stellar atmosphere models with the parameters: T _eff from 4000 to 9000 K, log g = 0.0–4.5, and metallicity [A] = 0.0;–1.0;–2.0;–3.0;–4.0. The aluminum atom model and the method of calculations were tested by the study of line profiles in the solar spectrum. We refined the oscillator strengths and Van-der-Vaals broadening constants C ₆ of the investigated transitions. We conclude that the Al I atom is in the overionization state: the 3p level is underpopulated in the line formation region. This leads to the line weakening, as compared with the LTE results. The overionization effect becomes more pronounced with increasing temperature and decreasing metallicity. We show that the use of various atomic data (ionization cross-sections) for the low levels of Al I does not change the behavior of non-LTE deviations, whereas the value of these deviations varies essentially. For nine selected Al I lines we calculated the grids of theoretical non-LTE corrections (ΔX _NLTE = logɛ _NLTE − log ɛ _LTE) to the Al abundances determinedwith the LTE assumption. The non-LTE corrections are positive and significant for the stars with temperatures T _eff > 6000 K. These corrections weakly depend on log g, and increase with declining stellar metallicity.     

Semi-empirical solar line blanketing

The line blanketing procedure described in an earlier paper has been tested by a series of blanketed solar model atmosphere calculations incorporating methodological and data variations. The results of those tests show the method to be reasonably insensitive to data uncertainties and to certain assumptions employed. The method was also checked by calculating several blanketed stellar atmospheres for comparison with blanketed atmospheres computed by other investigators. Except at the highest atmospheric levels the comparisons of the model structures were good and the blanketing method appears to be validated.Central solar intensities were computed for two semi-empirical solar models and for a theoretical model using the present blanketing method. The results are compared with the observations of Labs and Neckel (1968) in the region 3300 Å to 6500 Å. It is found the blanketing improves agreement of the model predictions with observations in a substantial way, particularly at short wavelengths. Limb-darkening predictions with blanketing were also made for these models at four wavelengths and compared with observations. The blanketing generally produces an important improvement in the comparisons; however, the amount of blanketing to be included for limb-darkening is uncertain.The temperature distribution of the blanketed theoretical model was compared with semi-empirical solar models and a blanketed model by Athay (1970) including non-LTE effects. Over a large range in optical depth the agreement is reasonably good; at small optical depths (₀ 0.01) large divergences are seen which may be due to the effects of non-LTE and to the neglect of strong lines in the present blanketing method. In addition to the good structural comparisons, the quality of the blanketed theoretical model in terms of comparison with central intensity and limb darkening is nearly as good as the semi-empirical models. We conclude that theoretical model atmospheres of solar type or nearly solar type computed with current constant-flux programs and with blanketing included are of high quality.Possible improvements in the blanketing method of importance in some instances are suggested; these include the use of a picket-fence procedure, the inclusion of strong lines, and the consideration of non-LTE effects. Further applications to the solar UV region and to stellar atmospheres are suggested.Publications of the Goethe Link Observatory, Indiana University, No. 131.     

Spectro-Polarimetric Observations and Non-Lte Modeling of Ellerman Bombs

Ellerman bombs are bright emission features observed in the wings of Hα, usually in the vicinity of magnetic concentrations. Here we show that they can also be detected in the Ca II infrared triplet lines, which are easier to interpret and therefore allow for more detailed diagnostics. We present full Stokes observations of the 849.8 and 854.2 nm lines acquired with the new spectro-polarimeter SPINOR. The data show no significant linear polarization at the level of 3 × 10⁻⁴. The circular polarization profiles exhibit measureable signals with a very intricate pattern of peaks. A non-LTE analysis of the spectral profiles emerging from these features reveals the presence of strong downflows (∼10 {km s⁻¹}) in a hot layer between the upper photosphere and the lower chromosphere. Visiting Astronomers, National Solar Observatory, operated by the Association of Universities for Research in Astronomy, Inc. (AURA), under cooperative agreement with the National Science Foundation. The National Center for Atmospheric Research (NCAR) is sponsored by the National Science Foundation, USA.     

High-resolution optical spectroscopy of the sharp-lined B-type star HD 83206

Very-high-resolution ( R ∼160 000) spectroscopic observations are presented for the early B-type star, HD 83206. Because it has very sharp metal lines, this star affords an opportunity to test theories of model atmospheres and line formation. Non-LTE model atmosphere calculations have been used to estimate the atmospheric parameters and absolute metal abundances (C, N, O, Mg and Si); an LTE analysis was also undertaken to investigate the validity of this simpler approach and to estimate an iron abundance. For the non-LTE calculations, there is excellent agreement with observations of the Balmer lines H α and H δ and the lines of Si  ii and Si  iii for atmospheric parameters of T _eff≃21 700±600 K and log  g ≃4.00±0.15 dex. The agreement is less convincing for the LTE calculations, and a higher gravity is deduced. Careful comparison of the metal line profiles with non-LTE calculations implies that the projected rotational and microturbulent velocities have maximum values of ≃5 and ≃2 km s⁻¹, respectively. The latter value is smaller than has often been adopted in LTE model atmosphere analyses of main-sequence stars. Non-LTE absolute metal abundances are estimated, and a comparison with those for normal B-type stars (deduced using similar non-LTE techniques) shows no significant differences. A comparison of the abundances deduced using non-LTE and LTE calculations implies systematic differences of 0.1–0.2 dex, showing the importance of using a non-LTE approach when accurate absolute abundances are required. Its location in the Hertzsprung–Russell diagram and normal metal abundance lead us to conclude that HD 83206 is probably a main-sequence B-type star. As such, it is among the sharpest-lined young B-type star discovered to date.     

The effects of partial redistribution on facular K line profiles

We present theoretical Ca ii K-line profiles and filtergram contrasts for several recent models of solar faculae. The line profiles vary greatly between models and between complete and partial frequency redistribution non-LTE calculations for any given model. The filtergram contrasts are relatively insensitive to the line formation theory which greatly simplifies the calculation for comparison with observations. All of the models considered exhibit K-line contrasts smaller than the mean value observed by Mehltretter.Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

Influence of departures from LTE on oxygen abundance determination in the atmospheres of A-K stars

We have performed non-LTE calculations for O I with a multilevel model atom using currently available atomic data for a set of parameters corresponding to stars of spectral types from A to K. Departures from LTE lead to a strengthening of O I lines, and the difference between the non-LTE and LTE abundances (non-LTE correction) is negative. The non-LTE correction does not exceed 0.05 dex in absolute value for visible O I lines for main-sequence stars in the entire temperature range. For the infrared O I 7771 Å line, the non-LTE correction can reach ?1.9 dex. The departures from LTE are enhanced with increasing temperature and decreasing surface gravity. We have derived the oxygen abundance for three A-type mainsequence stars with reliably determined parameters (Vega, Sirius, HD 32115). For each of the stars, allowance for the departures from LTE leads to a decrease in the difference between the abundances from infrared and visible lines, for example, for Vega from 1.17 dex in LTE to 0.14 dex when abandoning LTE. In the case of Procyon and the Sun, inelastic collisions with HI affect the statistical equilibrium of OI, and agreement between the abundances from different lines is achieved when using Drawin’s classical formalism. Based on the O I 6300, 6158, 7771-5, and 8446 Å lines of the solar spectrum, we have derived the mean oxygen abundance log ? = 8.74 ± 0.05 using a classical plane-parallel model solar atmosphere and log ? _+3D = 8.78 ± 0.03 by applying the 3D corrections taken from the literature.     

Mass motion in the solar chromosphere

A high resolution profile of the solar Oi 1304.9 Å line has been measured from rocket spectrograms. The profile is nearly flat-topped, showing only a slight solar reversal after instrument effects and absorption due to atomic oxygen in the earth's atmosphere have been allowed for. A theoretical analysis of this line, under the assumption of non-LTE conditions and a homogeneous, spherically symmetric chromosphere, predicts a rather deep solar reversal. The theoretical profile may be made consistent with the observed profile if mass motion is present in the chromospheric region where the line is formed. A Gaussian distribution of up and down velocities with a root mean square velocity of about 7 km/sec gives best agreement between the predicted and observed profile. This result is consistent with the conclusion made from a study of high resolution profiles of solar lines in the visible spectrum that mass vertical velocities increase with height above the photosphere.     

Influence of departures from LTE on calcium,titanium, and iron abundance determinations in cool giants of different metallicities

We have performed statistical equilibrium calculations for Ca I–Ca II, Ti I–Ti II, and Fe I–Fe II by taking into account the nonequilibrium line formation conditions (the non-LTE approach) in model atmospheres of giant stars with effective temperatures 4000 K ≤ T _eff ≤ 5000 K and metal abundances ?4 ≤ [Fe/H] ≤ 0. The dependence of departures from LTE on atmospheric parameters has been analyzed. We present the non-LTE abundance corrections for 28 Ca I lines, 42 Ti I lines, 54 Ti II lines, and 262 Fe I lines and a three-dimensional interpolation code to obtain the non-LTE correction online for an individual line and specified atmospheric parameters.     

Non-LTE modeling of narrow emission components of He and Ca lines in optical spectra of classical T Tauri stars

Using LTE calculations of the structure of T Tauri stellar atmospheres heated by radiation from an accretion shock (Dodin and Lamzin 2012), we have calculated the spectrum of the hot spot emerging on the stellar surface by taking into account non-LTE effects for He I, He II, Ca I, and Ca II. Assuming the pre-shock gas density N ₀ and velocity V ₀ to be the same at all points of the accretion stream cross section, we have calculated the spectrum of the star+circular spot system at various N ₀, V ₀, and parameters characterizing the star and the spot. Using nine stars as an example, we show that the theoretical optical spectra reproduce well the observed veiling of photospheric absorption lines as well as the profiles and intensities of the so-called narrow components of He II and Ca I emission lines with an appropriate choice of parameters. The accreted gas density in all of the investigated stars except DK Tau has been found to be N ₀ > 10¹² cm^?3. We have managed to choose the parameters for eight stars at a calcium abundance in the accreted gas ξ _Ca equal to the solar one, but we have been able to achieve agreement between the calculations and observations for TW Hya only by assuming ξ _Ca to be approximately a factor of 3 lower than the solar one. The estimated parameters do not depend on interstellar extinction, because they have been determined from the spectra normalized to the continuum level. The calculated intensity of Ca II lines has turned out to be lower than the observed one, but this contradiction can be eliminated by assuming that, in addition to the accreted gas with a high density N ₀, a more rarefied gas also falls onto the star. The theoretical equivalent widths and relative intensities of the subordinate He I lines disagree significantly with the observations. This is apparently because non-LTE effects should be taken into account when calculating the structure of the upper layers of the hot spot, the accuracy of the cross sections for collisional processes from upper levels is insufficient, and the spot inhomogeneity should probably be taken into account.     

Non-LTE Effects of Helium Lines in Late-B and A Stars

A helium model atom that includes 55 He I levels and the He II ground level in a detailed consideration has been constructed to investigate the departures from local thermodynamic equilibrium (LTE) in the formation of helium lines in stars with effective temperatures from 9300 to 20 000 K. For eight stars with effective temperatures from 9380 to 17 500 K the helium abundance has been determined from He I lines. The neutral helium lines in B stars cannot be described under LTE conditions using the common helium abundance. Furthermore, the profiles of several lines cannot be described in terms of the LTE approach at all. In contrast, a satisfactory coincidence of the theoretical and observed profiles for the entire set of helium lines observed in a wide spectral range can be achieved using virtually the same helium abundance by taking into account the departures from LTE. The LTE and non-LTE helium abundances can differ by up to a factor of 2–3, depending on the stellar parameters. The higher the stellar temperature, the stronger the departures from LTE. As a rule, the lines in the blue spectral region are less affected by non- LTE effects. In the atmospheres of six stars the helium abundance corresponds, within the error limits, to the present-day solar value. A helium underabundance is observed in the atmospheres of Sirius and HD 72660 classified as hot Am stars.     

Comoving frame calculations of spectral lines formed in rapidly expanding media with the partial frequency redistribution function for zero natural line width

Comoving frame calculations have been used to compute the spectral lines formed in rapidly expanding spherical media. We have employed the angle-averaged partial frequency redistribution functionR _I with a two-level atom model in non-LTE atom approximation. A linear velocity law increasing with radius has been employed with maximum velocity at Τ=0 being set equal to 30 mean thermal units. It is found that one obtains almost symmetric emission line profiles at large velocities similar to those found in quasars.     

Spectral Diagnostics of the Magnetic Field Orientation in a Prominence Observed with SOHO/SUMER

During several campaigns focused on prominences we have obtained coordinated spectral observations from the ground and from space. The SOHO/SUMER spectrometer allows us to observe, among others, the whole Lyman series of hydrogen, while the Hα line was observed by the MSDP spectrograph at the VTT. For the Lyman lines, non-LTE radiative-transfer computations have shown the importance of the optical thickness of the prominence – corona transition region (PCTR) and its relation to the magnetic field orientation for the explanation of the observed line profiles. Moreover, Heinzel, Anzer, and Gunár (2005, Astron. Astrophys. 442, 331) developed a 2D magnetostatic model of prominence fine structures that demonstrates how the shapes of Lyman lines vary, depending on the orientation of the magnetic field with respect to the line of sight. To support this result observationally, we focus here on a round-shaped filament observed during three days as it was crossing the limb. The Lyman profiles observed on the limb are different from day to day. We interpret these differences as being due to the change of orientation of the prominence axis (and therefore the magnetic field direction) with respect to the line of sight. The Lyman lines are more reversed if the line of sight is across the prominence axis as compared to the case when it is aligned along its axis.     

The polarization-free approximation applied to multi-level non-LTE radiative transfer

The polarization-free (POF) approximation (Trujillo Bueno and Landi Degl'Innocenti, 1996) is capable of accounting for the approximate influence of the magnetic field on the statistical equilibrium, without actually solving the full Stokes vector radiative transfer equation. The method introduces the Zeeman splitting or broadening of the line absorption profile I in the scalar radiative transfer equation, but the coupling between Stokes I and the other Stokes parameters is neglected. The expected influence of the magnetic field is largest for strongly-split strong lines and the effect is greatly enhanced by gradients in the magnetic field strength. Formally the interaction with the other Stokes parameters may not be neglected for strongly-split strong lines, but it turns out that the error in Stokes I obtained through the POF approximation to a large extent cancels the neglect of interaction with the other Stokes parameters, so that the resulting line source functions and line opacities are more accurate than those obtained with the field-free approach. Although its merits have so far only been tested for a two-level atom, we apply the POF approximation to multi-level non-LTE radiative transfer problems on the premise that there is no essential difference between these two cases. Final verification of its validity in multi-level cases still awaits the completion of a non-LTE Stokes vector transfer code.For two realistic multi-level cases (CaII and MgI in the solar atmosphere) it is demonstrated that the POF method leads to small changes, with respect to the field-free method, in the line source functions and emergent Stokes vector profiles (much smaller than for a two-level atom). Real atoms are dominated by strong ultraviolet lines (only weakly split) and continua, and most lines with large magnetic splitting (in the red and the infrared) are at higher excitation energies, i.e. they are relatively weak and unable to produce significant changes in the statistical equilibrium. We find that it is generally unpredictable by how much the POF results will differ from the field-free results, so that it is nearly always necessary to confirm predictions by actual computations.The POF approximation provides more reliable results than the field-free approximation without significantly complicating the radiative transfer problem, i.e. without solving any extra equations and without excessive computational resource requirements, so that it is to be preferred over the field-free approximation.