Download On the interpretation of stellar disc observations in terms of diameters

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 ˆ 1†; conventionally used in model
construction. Here, we assume spherical symmetry of the star …r ˆ
distance from the centre, tr ˆ 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 ˆ 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 ˆ x=ll ; which is the
radial geometrical thickness x in units of the mean free path
ll ˆ 1=…kl r) of photons of wavelength l …kl ˆ monochromatic
extinction coefficient per mass unit, r ˆ 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† ˆ Bl …T 1 † exp…2tl =cos u†
‡ Bl …T 2 †‰1 2 exp…2tl =cos u†Š
…1†
p
for sin u , 1 (centre: cos u ˆ …1 2 sin2 u† ˆ 1; edge: cos u ˆ 0†:
At the edge, the emitted intensity drops discontinuously to zero,
which is a non-physical artefact of the cos u ˆ 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 †Š
component diminished by an exp…2tl =cos u† factor that decreases
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M. Scholz
towards the cos u ˆ 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 ˆ 3000 K and T 2 ˆ 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 ˆ 0; 0.1, 1, 10, 15 and 20, where
tl ˆ 0 and 20 yield a box-shaped UD of 3000-K and 1000-K
photons, respectively. For tl ˆ 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 ˆ 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 ˆ 0:7 mm on the disc of
the compact two-layer model with T 1 ˆ 3000 K and T 2 ˆ 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 ˆ 0 (box-shaped UD), 0.1 (dashdotted curve), 1 (dotted), 10 (full), 15 (dashed) and 20 (UD). The tripledot-dashed curve is for ‰l ˆ 1:0 mm; T 2 ˆ 1000 K; tl ˆ 10Š or [0.7 mm,
1250 K, 10]. See text.
cool-layer background photons seen above about sin u , 0:8: The
additional tl ˆ 10 CLV plotted in Fig. 1 for ‰l ˆ 1:0 mm; T 2 ˆ
1000 KŠ 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 ˆ
3000 K and T 2 ˆ 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 ˆ 0; 0.1, 1, 3, 5 and 10, where
tl ˆ 0 and 10 yield a box-shaped UD of 3000-K and 1500-K
photons, respectively. For tl ˆ 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 ˆ 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 ˆ 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† ˆ Bl …T 2 †‰1 2 exp…22tl =cos u†Š
…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†Š from the outer cool layer seen on the
star's far side.
Monochromatic r…tr;l ˆ 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 ˆ
r 1 =r 2 and the disc's edge sin u ˆ 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 ˆ 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 ˆ 3000 K at r1 ˆ 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 ˆ 0:7 mm; T 2 ˆ 1000 K; tl ˆ 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 ˆ 0:7 mm; T 1 ˆ 3000 K; T 2 ˆ
1000 K; tl ˆ 1Š 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
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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 ˆ 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 ˆ 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 ˆ 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 ˆ 1 layer
As mentioned before, tr;l ˆ 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† ˆ 2 ‰r 0 =ll …r 0 †Š dr 0 ˆ 1
rs
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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 ˆ 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 ˆ 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 ˆ 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 (ˆcurves `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 ˆ 1 filter radii (H: 1.04; K: 1.11) after the definition
of Scholz & Takeda (1987). If the outer intensity tail were not present
(ˆcurves `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.
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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.
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This paper has been typeset from a TEX/LATEX file prepared by the author.
q 2001 RAS, MNRAS 321, 347±352