* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download On the interpretation of stellar disc observations in terms of diameters
Survey
Document related concepts
Dyson sphere wikipedia , lookup
Theoretical astronomy wikipedia , lookup
Perseus (constellation) wikipedia , lookup
Corvus (constellation) wikipedia , lookup
International Ultraviolet Explorer wikipedia , lookup
Stellar kinematics wikipedia , lookup
Astronomical seeing wikipedia , lookup
Astronomical spectroscopy wikipedia , lookup
Transcript
Mon. Not. R. Astron. Soc. 321, 347±352 (2001) On the interpretation of stellar disc observations in terms of diameters M. Scholz1,2w 1 2 Institut fuÈr Theoretische Astrophysik der UniversitaÈt Heidelberg, Tiergartenstr. 15, D-69121 Heidelberg, Germany Chatterton Department of Astronomy, School of Physics, University of Sydney, Sydney, NSW 2006, Australia Accepted 2000 September 8. Received 2000 August 16; in original form 2000 May 17 A B S T R AC T A primitive two-layer model atmosphere may illustrate the formation of different types of brightness distributions on stellar discs. Similar types of distributions are found in realistic atmospheres, some of which are very hard to interpret in terms of a diameter-type quantity. Assigning a disc size to observational data and relating this size to an optical-depth diameter is extremely difficult in these cases. In particular, naive interpretation of disc observations in the light of absorption features in the Wien part of the Planck function and of disc observations of geometrically very extended configurations may yield spurious results. Key words: methods: miscellaneous ± stars: fundamental parameters ± stars: general. 1 INTRODUCTION The brightness distribution on the stellar disc is the only directly observable source of information about the size of a star. This is true both for stars that have a compact (or plane-stratified) atmosphere, the geometric extension of which is small compared to the total dimensions of the star, and for stars that have a geometrically extended atmosphere. Strictly compact atmospheres do not exist in nature because no gaseous object has a real edge, but the compact-atmosphere concept is often a reasonable approximation if some caution is taken regarding thin outer atmospheric layers like, e.g., the solar chromosphere and corona lying above the compact solar photosphere. Geometrically extended, either static or moving, atmospheres occur in various parts of the Hertzsprung±Russell (HR) diagram (cf. Baschek, Scholz & Wehrse 1991), and different sensible definitions of the diameters of such stars can be given. Common definitions found in the literature have been summarized and discussed by Baschek et al. (1991). The relevant diameter to be considered in the present context is the `intensity diameter', which is defined in terms of the shape of the centre-to-limb variation (CLV) of emitted intensity across the star's disc and which is in some way related through radiation transport to the `optical-depth diameter' of the star, 2r tr 1; conventionally used in model construction. Here, we assume spherical symmetry of the star r distance from the centre, tr radial optical depth) and, since both the CLV and the optical depth are wavelength-dependent quantities, the respective diameters depend on wavelength l . `The' radius and the corresponding effective temperature of an extendedatmosphere model star given in the literature often refer to some mean extinction coefficient and resulting mean optical depth like, e.g., the conventional Rosseland radius r tr; Ross 1 (cf. Baschek et al. 1991; Scholz 1997). w E-mail: [email protected] q 2001 RAS It is the purpose of this paper to clarify why, under certain circumstances, a `non-perfect' observer is in principle unable to measure a star's diameter, and how one may proceed in cases of very unfavourable brightness distributions. 2 C O M PAC T T W O - L AY E R AT M O S P H E R E The basic problems of deriving a star's diameter from observation of the stellar disc may be outlined by studying the properties of a primitive two-layer model atmosphere. The atmosphere is assumed to be strictly compact and to have a sharp edge. The inner hot and the outer cool layers are homogeneous and radiate as Planck emitters with temperatures T1 and T2, respectively. The transparency of the outer layer is wavelength-dependent and is expressed in terms of its radial optical thickness tl x=ll ; which is the radial geometrical thickness x in units of the mean free path ll 1= kl r) of photons of wavelength l kl monochromatic extinction coefficient per mass unit, r density: This primitive model star has a well-defined radius R given by the distance of its atmosphere from the star's centre. If u is the angle between the line of sight and the radius vector, the intensity at the position R sin u on the disc seen by the observer is I l R sin u Bl T 1 exp 2tl =cos u Bl T 2 1 2 exp 2tl =cos u 1 p for sin u , 1 (centre: cos u 1 2 sin2 u 1; edge: cos u 0: At the edge, the emitted intensity drops discontinuously to zero, which is a non-physical artefact of the cos u 0 singularity of any strictly plane-parallel stratification. Equation (1) shows immediately that there are several different types of CLV of normalized intensity I l R sin u=I l 0: Since T 1 . T 2 , we have Bl T 1 . Bl T 2 at any l . The intensity emitted towards the observer is composed of a Bl T 1 2 Bl T 2 component diminished by an exp 2tl =cos u factor that decreases 348 M. Scholz towards the cos u 0 limb, plus a constant Bl (T2) background component of photons originating from the outer cool layer. For vanishing t l , the observer sees a uniform disc (UD) of `continuum' photons. For tl . 0; he or she sees a limb-darkened disc in the light of some `absorption feature', caused by the centreto-limb decrease of the first component. 2.1 Strong difference of Planck functions Case A: Bl T 1 @ Bl T 2 : This case occurs in a realistic atmosphere for steep temperature gradient and/or, usually more importantly, for l in the Wien part of the Planck function(s). (A1) For small to moderate radial optical thickness t l , the photons originating from the inner hot layer dominate the emitted intensity at any position sin u on the disc except at the very edge, cos u < 0; where the intensity drops steeply to the tiny, nonmeasurable background level of cool-layer photons. The CLV inflection point is just inside the limb of the star. The intensity discontinuity at the very edge is virtually invisible. The brightness on the disc appears slightly to moderately limb-darkened. (A2) For large radial optical thickness t l , the flood of hot-layer photons still dominates the brightness of the central disc portion but is reduced quickly towards increasing sin u because of the high efficiency of the exp 2tl =cos u factor for large t l . Compared to case A1, the steep intensity drop to the low background level of cool-layer photons at the limb is replaced by a smooth CLV shape whose inflection point moves towards the centre of the disc with increasing t l . The intensity discontinuity at the edge is barely visible. The disc shows strong limb darkening and the variation of intensity from the centre to the limb has a Gaussian-like appearance. (A3) For very large optical thickness t l , the few photons that do pass from the inner hot layer get lost in the constant Bl (T2) intensity component, and the disc brightness becomes more and more uniform as t l increases. Cases A1 and A2 illustrate the formation of an unsaturated and an almost saturated absorption feature, respectively, in the spectrum that is formed by diminishing the number of emerging continuum photons in a semi-opaque layer while photons from this layer contribute only a small fraction of the observed intensity and observed flux. If the star were observed in different spectral features with successively increasing strength at about the same wavelength, an observer who cannot spatially resolve the disc and tries to interpret lunar-occultation or interferometric data naively in terms of, e.g., a uniform disc would see the star first shrink A1 ! A2 from its real diameter 2R to a much smaller value and then expand again A2 ! A3 towards 2R when observed features approach full saturation. If the observer used model-predicted CLVs for data reduction, he or she would obtain the real diameter, but would depend very sensitively on the correct prediction of the CLV in the Gaussian-like case A2. Fig. 1 shows an extreme example of case A. Temperatures are T 1 3000 K and T 2 1000 K; the wavelength is 0.7 mm, which is in the Wien part of both Planck functions, and the ratio of the two Planck functions is Bl T 2 =Bl T 1 , 1026 : The normalized intensity CLV is given for tl 0; 0.1, 1, 10, 15 and 20, where tl 0 and 20 yield a box-shaped UD of 3000-K and 1000-K photons, respectively. For tl 1; the background level of coollayer photons is just 0.0003 per cent and only shows up at the very edge above sin u , 0:997: The tl 10 case is a typical example of a Gaussian-like CLV shape with a 2 per cent admixture of Figure 1. Normalized intensity distribution at l 0:7 mm on the disc of the compact two-layer model with T 1 3000 K and T 2 1000 K: The abscissa is sin u where u is the angle between the line of sight and the radius vector at the distance R from the star's centre. The radial optical thickness of the outer cool layer is tl 0 (box-shaped UD), 0.1 (dashdotted curve), 1 (dotted), 10 (full), 15 (dashed) and 20 (UD). The tripledot-dashed curve is for l 1:0 mm; T 2 1000 K; tl 10 or [0.7 mm, 1250 K, 10]. See text. cool-layer background photons seen above about sin u , 0:8: The additional tl 10 CLV plotted in Fig. 1 for l 1:0 mm; T 2 1000 K or for [0.7 mm, 1250 K] (both curves are indistinguishable within the plotting accuracy) demonstrates the extreme sensitivity to wavelength or temperature changes in the Wien part of the Planck function. 2.2 Moderate difference of Planck functions Case B: Bl (T1) and Bl (T2) are of the same order of magnitude. This case occurs in a realistic atmosphere for flat temperature gradient and, usually more importantly, for l in the nearmaximum or the Rayleigh±Jeans part of the Planck function(s). Except for very small and very large optical thickness t l , the intensity is a mixture of hot-layer and cool-layer photons with slight to moderate limb darkening of the disc brightness. The observer is able to measure the position of the outer cool layer in the light of an absorption feature, and this measurement is not overly sensitive to the determination or to the model-predicted knowledge of the detailed shape of the CLV. Fig. 2 shows an example of case B. Temperatures are T 1 3000 K and T 2 1500 K; the wavelength is 2 mm, which is in the near-maximum part of both Planck functions, and the ratio of the two Planck functions is Bl T 2 =Bl T 1 , 0:08: The normalized intensity CLV is given for tl 0; 0.1, 1, 3, 5 and 10, where tl 0 and 10 yield a box-shaped UD of 3000-K and 1500-K photons, respectively. For tl 0:1 and 1, the 9 per cent and 20 per cent level of cool-layer background photons is only reached close to the edge. q 2001 RAS, MNRAS 321, 347±352 Interpreting stellar discs in terms of diameters Figure 2. Same as Fig. 1 at 2 mm with 3000 K and 1500 K for tl 0 (boxshaped UD), 0.1 (dash-dotted curve), 1 (dotted), 3 (full), 5 (dashed) and 10 (UD). 3 S P H E R I C A L LY E X T E N D E D T W O - L AY E R AT M O S P H E R E A primitive extended counterpart of the compact two-layer model atmosphere may be constructed by lifting the outer cool layer to a higher position r 2 . r 1 ; whereas one has r 2 < r 1 in Section 2. The two layers are still assumed to be internally compact so that the space is transparent between the inner hot and the outer cool `shell' in this simple model picture. The angle u now refers to the position R r 2 of the outer layer, and the intensity variation Il (R sin u ) on the disc measured by the observer is given by equation (1) for sin u , r 1 =r 2 and by I l R sin u Bl T 2 1 2 exp 22tl =cos u 2 for sin u . r 1 =r 2 as the hot-layer emission term Bl (T1) in equation (1) is replaced by the inward emission Bl T 2 [1 2 exp 2tl =cos u from the outer cool layer seen on the star's far side. Monochromatic r tr;l 1 optical-depth radii are either equal to r2 for tl . 1 or equal to r1 for tl , 1: Owing to the assumption of strictly plane-parallel stratification of the two layers, this model exhibits non-physical intensity discontinuities at both sin u r 1 =r 2 and the disc's edge sin u 1: In principle, the various cases discussed for the compact twolayer model also occur for this extended configuration and show similar properties. Since, however, the contribution of hot-layer photons abruptly vanishes beyond the position sin u r 1 =r 2 on the inner disc, the portion of the disc that solely emits photons of the outer cool layer is now substantially larger. In fact, there is even a slight limb brightening for cos u ! 0; according to equation (2). As lunar-occultation and interferometric diameter observations imply area integrations, the chance of seeing cool-layer photons and of measuring the geometrical position of their origin is noticeably enhanced. Yet, in the equivalent to case A1 of Section q 2001 RAS, MNRAS 321, 347±352 349 Figure 3. Normalized intensity distribution on the disc of the extended two-layer model with T 1 3000 K at r1 r2 =2 and T2 at r2. The abscissa is sin u where u is the angle between the line of sight and the radius vector at the distance r2 from the star's centre. Intensity distributions are shown for l 0:7 mm; T 2 1000 K; tl 1; dotted curve], [0.7 mm, 1000 K, 10; full], [2 mm, 1500 K, 0.1; dash-dotted] and [2 mm, 1500 K, 5.0; dashed]. See text. 2.1, the intensity CLV is so strongly dominated by the hot-layer contribution on the inner disc that the non-perfect observer would not be able to measure r2, even though the observation is done in the light of the absorption feature. Only extremely high observational accuracy or spatial resolution of the disc would enable one to filter out the tiny contribution of cool-layer photons indicating the geometrical position r2. This situation still holds in case A2 while t l increases and the flank of the core of the brightness distribution migrates inward until the level of coollayer photons has risen to about the per cent order and area integration effects become efficient. Fig. 3 illustrates four different cases. The extended counterpart of compact case A1 with l 0:7 mm; T 1 3000 K; T 2 1000 K; tl 1 is an almost uniform disc with radius r1 and no observable intensities beyond this distance from the disc's centre. The extended variant A2 with [0.7 mm, 3000 K, 1000 K, 10] exhibits a strongly limb-darkened `continuum disc' plus a 2 per cent extended intensity wing between r1 and r2. The extended variant B with [2 mm, 3000 K, 1500 K, 0.1] is an almost uniform inner disc plus an adjacent tail with limb brightening from r1 (2 per cent intensity level) to r2 (9 per cent) that covers the threefold area of the inner disc. Intensities of the extended variant B with [2 mm, 3000 K, 1500 K, 5.0], which are dominated by coollayer photons, look hardly different for the compact and the extended two-layer model. 4 R E A L I S T I C M O D E L AT M O S P H E R E S The basic features of the primitive two-layer picture are also found in realistic multi-layer models of stellar atmospheres and, certainly, in the atmospheres of real stars. Reconstruction of limb darkening of stellar discs from observations is difficult and 350 M. Scholz prone to inaccuracies from various sources, but there are a few reconstructed intensity CLVs available in the literature (cf. the listing of Scholz 1997), and limb darkening of the Sun has been studied in great detail. 4.1 CLV inflection points The non-physical intensity discontinuities occurring in Section 2 as a result of the assumption of strictly plane-parallel stratification do not show up in a realistic stellar atmosphere, but steep intensity declines are common features both at the edge of very compact or slightly extended and at inner disc positions of significantly extended atmospheric configurations. Since, even in a very compact configuration, rays along the line of sight are affected by curved geometry near the very limb of the disc and eventually pass through the thin gas of the outermost stellar atmosphere, an edge discontinuity converts into a steep intensity decrease with an inflection point. Also, high-layer absorption of the type of case A1 of Section 2.1 may lead to a sudden exponential intensity drop through the exp 2tl =cos u factor when cos u gets very close to zero. The (essentially coinciding) positions of these inflection points of monochromatic intensity CLVs define a unique intensity radius (Baschek et al. 1991) of a very-compact-atmosphere star and are conventionally used for determining the photospheric radius of the Sun. As such inflection points are not observable on stellar discs with presently available techniques, CLV predictions from compactatmosphere models and corresponding conventional CLV approximations (e.g. Hestroffer 1997; Scholz 1997) implying an edge discontinuity are perfectly suited for interpretation of measurements of diameters of most stars with compact atmospheres. It should be pointed out, however, that the position of the CLV inflection point is physically not directly related to the position of a specific near-surface layer of the star. Case A2 of Section 2.1 shows that, in principle, an inflection point may occur on the inner disc of a compact-atmosphere star, and monochromatic CLV inflection points calculated from just slightly extended realistic M giant model atmospheres (e.g. Watanabe & Kodaira 1979; Scholz & Takeda 1987; Hofmann & Scholz 1998) sometimes are poorly correlated with each other or with the position of the corresponding tr;l 1 layer. A very flexible CLV approximation that avoids edge discontinuities and may also be used for describing limb darkening with an inner-disc inflection point has been proposed by Hestroffer (1997). In extended configurations, monochromatic CLV inflection points have no special relevance. They may or may not mark the `typical' depth of formation of a spectral feature. Broad Gaussianlike intensity curves may occur (Section 4.2), and multiple inflection points are to be expected in some cases (Section 4.4). 4.2 Gaussian-like limb darkening Intensity CLVs which exhibit a wing structure that can be approximated with fair accuracy by a Gauss-type exponential are common in stars. For instance, solar limb intensities are often modelled by and, indeed, closely resemble a Gaussian decline (e.g. Pierce & Slaughter 1977; Chollet & Sinceac 1999). Brightness distributions with fairly steep Gaussian-like flanks are predicted at various wavelengths of both continuum and absorption features by moderately to strongly extended model atmospheres of, e.g., non-Mira and Mira M giants (e.g. Watanabe & Kodaira 1979; Scholz & Takeda 1987; Hofmann & Scholz 1998; Hofmann, Scholz & Wood 1998; Hofmann et al. 2000) and supernovae (e.g. HoÈflich 1990; Karovska & Nisenson 1992). Observed examples are CLV reconstructions from lunar-occultation data of M giants (Bogdanov & Cherepashchuk 1990, 1991) and HST data of a Ori (Gilliland & Dupree 1996). In most, though not in all, cases, the tr;l 1 point lies on the flank of the intensity curve, a monochromatic radius can be sensibly defined and measured, and the numerical value of the observed diameter does not depend very sensitively upon the CLV shape adopted for data reduction. In very extended atmospheric configurations, however, both model predictions (e.g. Scholz & Takeda 1987; HoÈflich 1990; Hofmann et al. 1998, 2000) and fits of accurate interferometric Mira data (e.g. Haniff, Scholz & Tuthill 1995; Hofmann et al. 2000) reveal the occurrence of very pronounced Gaussian-like CLV shapes with extended wings at some wavelengths. These brightness distributions are poorly correlated with the position of layers of photon generation. There is no conspicuous portion of the intensity curve that might reasonably define an intensity radius, and the tr;l 1 point usually lies in the outermost wing of the distribution. Deducing a radius from lunar-occultation or interferometric data strongly depends on model assumptions. Jacob et al. (2000) reported an extreme case of modelled limb darkening for which the Gaussian-like CLV is so strongly peaked towards the disc's centre that naive first-glance interpretation of observed visibilities makes the star appear smaller in the light of a strong TiO molecular band than in the light of nearby nearcontinuum features. This situation is understandable in terms of case A2 of the two-layer model (cf. the full curves in Figs 1 and 3), since the normalized brightness distribution on the disc that the observer measures is dominated by deep-layer photons. He or she does not primarily see photons originating from the absorbing high layers that would indicate the position of these layers, but rather observes a geometrical projection effect as the dominating contribution of deep-layer photons is efficiently cut off towards the disc's limb by absorption in high layers and as the line of sight passes by deeper layers. Jacob et al. (2000) demonstrated the sensitivity of this type of CLV shape to the temperature changes one expects from the primitive two-layer model. Substantial changes of such CLV shapes with varying Mira phase (fig. 2 of Scholz 1997) also show the strong stratification dependence. Hence, disc observations at wavelengths of such features are more suited for probing the quality of an atmospheric model stratification than for measuring the distance of the absorbing layer from the star's centre. From the discussion of case A2 of Section 2, it is obvious that observing at longer wavelengths can dramatically increase the chance of accurately determining the position of an absorbing layer. Note that absorption features may appear shallower at longer wavelengths as a result of the behaviour of the Planck function but may still probe the same layers as a deeper feature in the Wien part of the Planck function. 4.3 Position of t r,l 1 layer As mentioned before, tr;l 1 optical-depth diameters that are the relevant radius-type quantities in modelling are often not closely related to intensity diameters that may be defined in terms of the observed shape of the intensity distribution. Since r tr;l r 2 r 0 =ll r 0 dr 0 1 rs q 2001 RAS, MNRAS 321, 347±352 Interpreting stellar discs in terms of diameters 351 gives the position of the layer that is located one mean photon free path ll below the star's `surface', rs, this is usually assumed to be the layer around which photons seen by the observer are produced. In fact, this assumption is well justified in most cases and the position of the tr;l 1 point is sensibly related to measurements of the disc size in such cases. The above examples given for the two-layer picture demonstrate, however, that there may be remarkable deviations from that rule. The radial intensity contribution function may be so strongly peaked towards larger t r,l that, in an extended configuration, the geometrical position of the layers that emit the bulk of photons is appreciably closer to the star's centre than that of the absorbing layers producing an unsaturated spectral depression but contributing only a few photons. So to speak, tr;l 1 gives the typical penetration depth of photons shot from space towards the star rather than the origin of photons actually seen by an observer. 4.4 Two-component CLV appearance Observing the continuum diameter of a spherically extended stellar atmosphere means measuring the width of the bright core of the stellar disc in the light of an absorption-free continuum window. As continuous absorption by H, H2 and H2 2 are strongly increasing functions of T(t r) in middle- to late-type stars, the dll =dr gradient is steep and the optical scaleheight 2dr=d ln tr;l is small in deep layers, and the continuum-forming region of the stellar atmosphere usually is very compact even in very extended atmospheric configurations (cf. Baschek et al. 1991; Scholz 1997). Hence, the bright core has a steep flank and a well-defined width. Since, however, space between these layers and high atmospheric layers is not empty, there must be some low-level photon emission even in the absence of substantial non-continuous absorption or scattering. The resulting CLV has a low-level intensity tail that is slowly decreasing towards large r or even shows a flat intermediate maximum, roughly equivalent to the [2 mm, 3000 K, 1500 K, 0.1] example in Fig. 3. Published (and available unpublished) brightness distributions predicted by extended M-type Mira model atmospheres (e.g. Scholz & Takeda 1987; Scholz 1997; Hofmann et al. 1998) show a wide variety of low-level CLV extensions ranging from virtually zero to a few per cent of the central intensity depending sensitively on the details of stratification, extinction coefficients and geometrical dimensions, as well as on wavelength. Slight limb brightening also occurs. Similar tails are predicted by supernova models (e.g. HoÈflich 1990; Karovska & Nisenson 1992), including examples of stunning limb brightening. Fig. 4 shows the CLV in infrared H and K bandpasses predicted by the very extended P74200 Mira model of Hofmann et al. (1998). It shows a two-component structure. The core width marks the position of the continuum layers, and the tr;l 1 point lies on the steep flank of this core. Because of the large disc area, however, which emits photons of the low-level tail component (here essentially produced by water absorption), the resulting visibility that an interferometric observer would have to evaluate is strongly affected by this component. Data reduction by means of, e.g., a UD or similar one-component CLV would significantly overestimate the `continuum size' of this star unless observations extend to high spatial frequencies. Very accurate visibility data would show the distortion of the central maximum produced by the flat outer-disc intensity component. Inspection of the UD fits of Hofmann et al. (1998) shows that model-, phase- and cycle-dependent variations of infrared UD diameters with q 2001 RAS, MNRAS 321, 347±352 Figure 4. Normalized intensity distributions and corresponding visibilities in H and K bandpasses predicted by the near-maximum Mira model P74200 (curves `with tail') of Hofmann et al. (1998). The abscissa of the intensity plot is in units of the parent star's Rosseland radius. Crosses give the position of the tr 1 filter radii (H: 1.04; K: 1.11) after the definition of Scholz & Takeda (1987). If the outer intensity tail were not present (curves `without tail'), the observer would be able to derive these radii quite accurately without knowledge of the full intensity distribution by assuming, e.g., a uniform disc. wavelength, partly resulting from two-component CLV structures, are quite common in Mira models. Sample cases demonstrating the diversity of such structures will be given in the belowmentioned dust study of Bedding et al. Thus, unexpectedly large (e.g. Perrin et al. 1999; see also Scholz & Wood 2000) and wavelength-dependent (e.g. Tuthill, Monnier & Danchi 1998, 1999; Thompson, Creech-Eakman & van Belle 2000) continuum diameters found in Miras on the basis of the UD or other simple CLV assumptions have to be interpreted with due caution. 352 M. Scholz Whilst Rayleigh scattering in the blue spectrum of late-type stars is known to reduce noticeably limb darkening (e.g. Scholz & Takeda 1987), the role of scattering with respect to such tail-type structures has not yet been explored. Molecular scattering as suggested by Perrin et al. (1999) might, if present, affect either the full brightness distribution or only the outer disc. Dust scattering, which is currently being investigated by Bedding et al. (in preparation), would be a high-layer phenomenon of probably small to modest influence upon the intensity CLV. 4.5 Impure bandpasses An actual observation is not monochromatic, but rather refers to a bandpass of non-zero width. It is obvious from the above examples that the numbers of photons originating in deep, continuum-forming layers and in high, line-forming layers may be so extremely different that stellar disc observations using impure bandpasses, i.e. bandpasses including both continuum and absorption features of the spectrum, may be completely dominated by continuum contributions. Even tiny transmission wings of a filter centred on an absorption feature may be penetrated by enough deep-layer photons that they dominate the brightness distribution. The observer of an extended-atmosphere star would measure the continuum radius of the star or collect some mixture of deep-layer and high-layer photons that is hardly interpretable in terms of a diameter-type quantity. Still more dangerous is the interpretation of disc observations in certain molecular bands whose lines are densely spaced but not really blended. A deep band feature may be seen in the spectrum to be due to line blanketing of most of the continuum, whereas the overwhelming portion of observed photons is emitted in the remaining gaps between lines and dominates the normalized intensity CLV. Photons emitted by the lines lead to a faint CLV tail structure that is hardly accessible to current observational techniques. The geometrical position of formation of such picket-fence type molecular bands cannot be measured unless extremely high observational accuracy or spatial resolution of the disc reveals the extent of this tail. Discussion of molecular band structures is outside the goal of this paper, but inspection of molecular data and/or high-resolution control of line-blanketed features selected for diameter measurement is strongly recommended. 5 CONCLUSION Deriving stellar diameters from observations of the star's disc is a straightforward procedure in many cases, and insufficient knowledge of limb darkening results in small to, at worst, modest errors only. There are, however, some important cases for which deducing the size of the disc from non-perfect observations is very difficult or even impossible. This happens preferentially, though not exceptionally, in stars with spherically extended atmospheres where diameters are l -dependent and may be very poorly correlated with observational brightness data. Both diameters measured in absorption features and near-continuum diameters may be affected. In particular, observations of absorption features in the Wien part of the Planck function and observations of stars with very extended atmospheres must be interpreted with great caution. Though most theoretical data are presently available for latetype giants and supergiants, as these are readily accessible to present-technique observations, similar problems are known to occur in supernovae and have to be expected for other extended configurations (cf. Baschek et al. 1991). Interpretation of future diameter observations of these objects in terms of a naive UD assumption may lead to spurious results. AC K N O W L E D G M E N T S I am indebted to colleagues at the Chatterton Department of Astronomy for their hospitality during a research visit in 2000 February to April, in particular to Tim Bedding for critical comments on the manuscript and to Andrew Jacob for technical help. Rainer Wehrse kindly read the final text version. REFERENCES Baschek B., Scholz M., Wehrse R., 1991, A&A, 246, 374 Bogdanov M. B., Cherepashchuk A. M., 1990, SvA, 34, 393 Bogdanov M. B., Cherepashchuk A. M., 1991, SvA, 35, 392 Chollet F., Sinceac V., 1999, A&AS, 139, 219 Gilliland R. L., Dupree A. K., 1996, ApJ, 463, L29 Haniff C. A., Scholz M., Tuthill P. G., 1995, MNRAS, 276, 640 Hestroffer D., 1997, A&A, 327, 199 HoÈflich P., 1990, A&A, 229, 191 Hofmann K.-H., Scholz M., 1998, A&A, 335, 637 Hofmann K.-H., Scholz M., Wood P. R., 1998, A&A, 339, 846 Hofmann K.-H., Balega Y., Scholz M., Weigelt G., 2000, A&A, 353, 1016 Jacob A. P., Bedding T. R., Robertson J. G., Scholz M., 2000, MNRAS, 312, 733 Karovska M., Nisenson P., 1992, A&A, 253, 210 Perrin G. et al., 1999, A&A, 345, 221 Pierce A. K., Slaughter C. D., 1977, Sol. Phys., 51, 25 Scholz M., 1997, in Bedding T. R., Booth A. J., Davis J., eds, Proc. IAU Symp. 189, Fundamental Stellar Properties. Kluwer, Dordrecht, p. 51 Scholz M., Takeda Y., 1987, A&A, 186, 200 (erratum: 196, 342) Scholz M., Wood P. R., 2000, A&A, 362, 1065 Thompson R. R., Creech-Eakman M., van Belle G. T., 2000, BAAS, 195, 45.02 (complete text available from [email protected]) Tuthill P. G., Monnier J. D., Danchi W. C., 1998, in Bradley P. A., Guzik J. A., eds, ASP Conf. Ser. Vol. 135, A Half Century of Stellar Pulsation Interpretations. Astron. Soc. Pac., San Francisco, p. 322 Tuthill P. G., Monnier J. D., Danchi W. C., 1999, in le Bertre T., Lebre A., Waelkens C., eds, Proc. IAU Symp. 191, Asymptotic Giant Branch Stars. Astron. Soc. Pac., San Francisco, p. 331 Watanabe T., Kodaira K., 1979, PASJ, 31, 61 This paper has been typeset from a TEX/LATEX file prepared by the author. q 2001 RAS, MNRAS 321, 347±352