L. Neven

A SYSTEMATIC METHOD FOR THE ANALYSIS OF HIGH-RESOLUTION FRAUNHOFER LINE PROFILES.

A fraunhofer line profile depends on various parameters, partly related to the photospheric structure (T, Pg, Pe, vconv, vturb), partly to the atom or ion involved (such as oscillator strength, energy levels), partly also resulting from the interaction of the relevant kind of particles with the photosphere, and the photospheric radiation field. In this paper we shall mainly pay attention to the determination of: the macroturbulent (convective) velocities, vconv (τ); the damping constant γ (τ); the abundance, Ael; the distribution function ϕ(vconv, τ) of the convective velocities at each depth τ; the source function, S (τ); the microturbulent velocities, vturb (τ).The particular difficulty with these unknowns is that they are, as a rule, coupled in the resulting line profiles, that is: the shapes and intensities in these profiles are determined by the combined influence of these unknowns (together with the other above-given parameters).In this paper we describe a method to determine these six unknowns empirically by separating them, in analysing accurate high-resolution observations of line profiles of a multiplet. The unknown functions and quantities are consecutively determined in the above given succession. For each determination another, appropriate part of the line profile is used. In some cases the influence of the mutual coupling of the various parameters cannot be completely eliminated, and an iterative method has to be used.The method is summarized in Table II and section 2, and is further explained in sections 3 to 8. It is applied to an infrared Ci multiplet. The main results are the following:

Damping constants for infrared fraunhofer lines

Empirical values of solar damping constants and their variation with optical depth were derived according to a method developed earlier by the authors. The damping constants refer to six infrared multiplets (24 lines). The average optical depths range from τ0 = 0.5 to 2.2. Corresponding theoretical damping constants were computed, mainly on the basis of Van der Waals damping, and with the help of detailed computations of the mean square radii of the atomic levels by Van Rensbergen. The empirical values are systematically larger than the theoretical ones, with factors ranging between 1.8 and 4.9. Some speculations about the source of this discrepancy are given.

On the occurrence of convective motions in the upper photosphere

We have examined whether the motion field in the photosphere in the range of optical depths 0.25< τ0< 0.6 is dominated by thermal convection or by vibrations. The observed asymmetries of infrared Fraunhofer lines indicate the presence of motions, and the fact that the asymmetry is zero for lines of low excitation and increases with the excitation potential shows that these motions are chiefly convective in this part of the photosphere: upward moving elements appear to be hotter than downward moving ones.Assuming furthermore that the photosphere can be described by a three-column model, with temperature differences as given by Edmonds (1967), we find that in the range of optical depths given above, where ΔT seems to vary between 80 and 160 °K, average convective velocities of 2.3 to 3.2 km/sec should occur. This result is in numerical agreement with (a) a previous one by the present authors (1967) derived from the variation of line asymmetry with depth in lines of one multiplet, (b) a finding by Lambert and Mallia (1968) deduced from absolute wavelength measurements of Fraunhofer lines, and (c) a recent result of Beckers (1968) found from a comparison of two granulation pictures obtained simultaneously with a narrow-band filter centred on the two wings of a faint line.

On the applicability of Goldberg and Unno's method to the determination of microturbulent velocities in an atmosphere with convection

The method of Goldberg and Unno for the determination of microturbulent velocities in a stellar atmosphere is only applicable if there are no macroturbulent or convective motions.If such motions occur, as in the solar photosphere, the derived results are false and may lead to misinterpretations such as an increase of the microturbulent velocity with depth or anisotropic microturbulence.

The empirical determination of line source functions, β L -values, and the microturbulent and convective velocity components as functions of depth in the photosphere-chromosphere transition region

An empirical method for determining line source functions, previously applied by us to the cores of infrared lines has now been extended to the whole line profile and was applied to centre-limb observations of sixteen lines of five infrared multiplets, mainly of high excitation potential (Table I). The present investigation was performed in two steps. In the first part of the paper approximate values are derived for the depth dependence of the four functions named in the title of this paper, where βL is the ratio between the actual and the LTE population of the lower level of the transitions involved. In the second part of the paper we use these empirically derived functions to compute the line profiles. From the remaining differences between observed and computed profiles, corrections are derived to the four functions. The main results are: (a) Convective velocities: see Table IV.(b)(Micro-)turbulent velocities: see Figure 8. Between τ5 = 10-4 and 10-1: 〈υτ〉 ≈ 1.4 km s-1, which is an upper limit since an unknown contribution of macroscopic motions could not be separated, (c) Line source functions: see Figures 9, 15 and 16. The source functions are close to the black-body function for τ5≳ 10-3, slight deviations occur in higher levels. The interesting behaviour of the Caii source function near τ5 = 10-5 should be noted. (d) Non LTE-functions: first approximations for the functions log βL (τ5) were derived empirically in the first part, and are shown in Figure 10; the second approximation shows them to be too large and the real values seem to be closer to one-half or one-third of these functions.

Source Functions in the Cores of Infrared Fraunhofer Lines

If the profiles of all Fraunhofer lines were formed according to the mechanism of pure absorption (the L.T.E.-hypothesis) the source functions Sλ(τ) deduced empirically from the central intensities of these lines should all be identical, and equal to the blackbody function Bλ(τ). In order to examine this hypothesis the source functions Sλ(τ) were deduced empirically according to a method applied earlier by us (De Jager and Neven, 1967a) for some twenty infrared lines for which accurate centre-limb observations were available (De Jager and Neven, 1967b). The lines belonged to five different multiplets.

The empirical determination of the damping constants of Fraunhofer lines

Abstract A systematic method is developed to determine the damping constant, γ, and its variation with optical depth, τ, in the solar photosphere, using the observed profiles of wings of Fraunhofer lines and their center-to-limb variation. The method is applied to the 3s3P°−3p3D multiplet of carbon at 10,700A. The resulting γ(τ) curve with its mean errors is shown in Fig. 2. This result can be explained by the usual assumption γ(τ) = γH, He(τ)+γe(τ), where γH, He is the contribution to the damping constant by collisional broadening by neutral particles; γe is due to the quadratic Stark effect of passing electrons. The “observed” values of the function γH,He(τ) agree perfectly with theoretical predictions. No theoretical predictions can yet be made for γe(τ) but the empirically derived values for this function are similar to values found from laboratory measurements for some lines of other elements, and look acceptable.

The radiation field in photospheric models for extreme supergiants

On the basis of assumed photospheric temperature models for 36 extreme supergiants (logge-values of 1, 0.5 and 0;Teranging from approx. 3700–33 000 K) photospheric fluxesS(τλ) were computed for 36 wavelengths ranging from 100 Å to 60 000 Å. The hot models are in perfect radiative equilibrium; the cooler show deviations up to 10%, sometimes even larger. Only in the relatively deep parts of the photospheres (τ5≳1) the radiation field at each geometrical level can be characterized by one unique radiation temperature; for smaller τ5-values there are large deviations from local thermal equilibrium. The influence of deviations from local thermodynamical equilibrium on the fluxes is briefly examined, and appears small but for the shortest wavelengths. In tables and graphs we give for these models πF(γ)-values, integrated fluxes, effective temperatures, coloursU, B andV, and the Balmer discontinuityD.

Stellar atmospheric velocity fields: The Beta Cephei variables γ Peg and β Cep

The shape parameters of a number of selected ultraviolet lines in BUSS-spectra of the Beta Cephei stars γ Peg and β Cep have been analyzed to determine the principal parameters of the atmospheric velocity field. We find for both stars a fairly high value (∼5 km s−1) for the microturbulent line-of-sight velocity component, which confirms an earlier result based on lower resolution UV spectra. Macroturbulent and rotational velocities are virtually zero in the atmosphere of γ Peg; for β Cep we findvrotsini=40 km s−1.

The photospheric velocity field of Procyon

BUSS observations of the profiles of two well observed spectral lines in the ultraviolet spectrum of αCMi (Procyon; F5 IV–V) are analysed with a Fourier transform method in order to determine values of various parameters of the velocity field of the upper photosphere. We find a microturbulent line-of-sight velocity componentLμ = 0.9 ± 0.4 km s−1, a macroturbulent velocity componentLM = 5.3 ± 0.2 km s−1, and a rotational velocity componentvR sini=10.0±1.2 km s−1. In these calculations a single-moded sinusoidal isotropic macroturbulent velocity function was assumed. The result appears to be sensitive to the assumed shape of the macroturbulence function: for an assumed Gaussian shape the observations can be described withvR sini=4 km s−1 andLM = 11.6 ± 2.7 km s−1. A comparison is made with other results and theoretical predictions.