Mon. Not. R. Astron. Soc. 325, 1487–1496 (2001)
The influence of atmospheric dust on limb darkening of M-type Mira
variables
T. R. Bedding,1P† A. P. Jacob,1 M. Scholz1,2P† and P. R. Wood3
1
Chatterton Department of Astronomy, School of Physics, University of Sydney, NSW 2006, Australia
Institut für Theoretische Astrophysik der Universität Heidelberg, Tiergartenstr. 15, D-69121 Heidelberg, Germany
3
Research School of Astronomy and Astrophysics, Australian National University, Cotter Road, Weston Creek, ACT 2611, Australia
2
Accepted 2001 March 29. Received 2001 January 19; in original form 2000 October 27
A B S T R AC T
We explore the occurrence of dust in M-type Mira atmospheres and its effect on limb
darkening under schematic assumptions about dust temperatures and dust particle properties.
Dust particles that are thermodynamically coupled to the surrounding gas may form and may
affect limb darkening, though only by very little in infrared continuum bandpasses. Dust
particles that assume the equilibrium temperature given by the mean intensity of the radiation
field only form under rare circumstances. Unexpectedly large or wavelength-dependent
infrared continuum radii observed by interferometry are unlikely to be caused by atmospheric
dust, except possibly near 1 mm; however, radius measurements may be significantly affected
by molecular band contamination.
Key words: stars: atmospheres – circumstellar matter – stars: variables: other – dust,
extinction – infrared: stars.
1
INTRODUCTION
Dynamic models of M-type Mira variables show that temperatures
in the outermost atmospheric layers may drop to around 1000 K or
less when effective temperatures are low and effects of
atmospheric extensions are strong (Bessell et al. 1989; Bessell,
Scholz & Wood 1996, hereafter BSW96; Hofmann, Scholz &
Wood 1998, hereafter HSW98). This raises the question of whether
condensation of dust particles may occur in those layers at
distances of, say, 3–5 stellar (continuum) radii. In a recent study
based on a hydrostatic approximation of the density stratification of
the Mira atmosphere, Lobel et al. (2000) estimated dust to form
around 3:5–4 stellar (continuum) radii. Other, more schematic,
models of circumstellar dust envelopes of very cool oxygen-rich
Miras (e.g. Danchi & Bester 1995; Lorenz-Martins & Pompeia
2000) also assumed inner envelope radii of this order of magnitude.
Since, however, this dust was not embedded in the temperature and
density stratification of a realistic, dynamic model atmosphere,
these models do not allow us to reach conclusions on the properties
of the atmospheric radiation field, which we address here.
It is beyond the scope of this paper to model the complex
processes of formation and destruction of dust particles in a
dynamic stellar atmosphere in which outflow and inflow motions
P
E-mail: [email protected] (TRB);
[email protected] (MS)
†Authors listed alphabetically. Please send enquiries and reprint requests to
M. Scholz.
q 2001 RAS
vary and shock fronts pass within time-scales of a few months.
Rather, we intend to check under very schematic assumptions the
extent to which dust, if present, could modify the stratification of
outermost atmospheric layers enough to influence the brightness
distribution on the stellar disc. In particular, we investigate whether
unexpectedly large or wavelength-dependent diameters measured
by interferometry in infrared near-continuum bandpasses (Tuthill,
Monnier & Danchi 1998, 1999; Perrin et al. 1999; Thompson,
Creech-Eakman & van Belle 2000; Weigelt et al. 2000) may be
explained in terms of a dust-generated two-component appearance
of limb darkening (cf. Scholz 2001). This issue is also relevant to
the question of the pulsation mode of Miras, given that most
measured angular diameters appear too large to be reconciled with
fundamental mode pulsation indicated by MACHO (massive
compact halo object) observations (Wood et al. 1999) and pulsation
velocities (Scholz & Wood 2000).
2
MODEL ASSUMPTIONS
We selected from the M-type Mira models of BSW96 and HSW98
a set of models that exhibit a variety of properties with respect to
their atmospheric extension and their near-surface temperature and
density stratification. These are dynamical models whose atmospheric density stratification is determined by the propagation of a
pulsation shock front. Material accelerated by the shock falls back
after some time, so yielding a velocity stratification. Occasionally,
a subsequent shock front of the next pulsation cycle enters the
lower photosphere while its predecessor is still effective in the
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T. R. Bedding et al.
Table 1. M-type Mira models.
Model
D27760
D28760
E8380
P71800
P73200
P74200
M97600
a
Reference
Modea
P
(d)
M
(M()
Rp
(R()
rs
(Rp)
Phase
L
(L()
R
(Rp)
Teff
(K)
BSW96
BSW96
BSW96
HSW98
HSW98
HSW98
HSW98
f
f
o
f
f
f
f
330
330
328
332
332
332
332
1.0
1.0
1.0
1.0
1.0
1.0
1.2
236
236
366
241
241
241
260
3.0
3.0
3.0
5.0
5.0
5.0
5.0
1.5
2.0
1.0
0.5
1.0
2.0
1.0
2210
4560
6750
1650
5300
4960
4910
0.91
1.04
1.09
1.20
1.03
1.04
1.19
2710
3030
2620
2160
3130
3060
2750
f ¼ fundamental, o ¼ 1st overtone.
uppermost layers. The resulting density stratification has a
discontinuity at the shock front position but a flat gradient behind
and in front of the shock, leading to the strong geometrical
extension of the atmosphere of the Mira star as compared to that of
a non-Mira giant. The non-grey temperature stratification of
BSW96 and HSW98 models is computed in local thermodynamic
and radiative equilibrium. The influence of the very thin hightemperature shock layer is neglected (see, e.g., arguments given by
Scholz & Wood 2000). The low high-layer temperatures predicted
by the models result from a combination of dilution and so-called
peaking (¼ continuum-forming layers seen under shrinking solid
angle towards upper layers) of radiation and of efficient blanketing
by strong molecular bands in the upper atmosphere. Further details
of approximations, limitations and shortcomings of these models
are discussed in BSW96 and HSW98 and should be kept in mind.
From a preliminary set of models, we chose for more detailed
investigation seven models listed in Table 1. They represent very
different typical structures of the high atmosphere in which dust
may form. Since observations are usually done near maximum
brightness, we focused our attention on near-maximum phase
models, but also included two near-minimum models. Stellar radii,
R, are Rosseland radii (cf. Baschek, Scholz & Wehrse 1991)
referring to the position of the tRoss ¼ 1 layer of the star’s
atmosphere, and all radius quantities are in units of the Rosseland
radius, Rp, of the non-pulsating parent star of the Mira variable (see
BSW96, HSW98). The effective temperature, T eff / L 1=4 R 21=2
ðL ¼ luminosityÞ is defined in terms of the Rosseland radius, R.
The ‘surface’ radius rs of the dust-free Mira model atmospheres is
chosen so that the condition tl ! 1 is fulfilled at all wavelengths
that are relevant to radiative energy transport and to considered
spectral properties.
The injection of dust particles into the upper atmosphere was
done with the following schematic assumptions, which do not
account for the dynamic phase-to-phase evolution of the Mira
atmosphere and are only meant to give a very schematic and coarse
picture of expected dust effects:
(i) We assumed that dust forms instantaneously when the local
kinetic gas temperature in near-surface layers falls below the
equilibrium condensation temperature of the dust species. We
assumed that 30 per cent of the available heavy-element material
condenses into grains, but we did not consider the corresponding
metal depletion of the gas resulting, in particular, in reduced line
blanketing by TiO molecules (cf. Gail & Sedlmayr 1998). In
addition, we performed some calculations with 70 per cent
condensation to assess the sensitivity of limb darkening to this
parameter and to intensify mild dust effects. For the solar-type
element mixture adopted here, 100 per cent condensation in
chemical equilibrium corresponds to a dust-to-gas mass ratio of
about 1 per cent. In the non-grey radiation transport calculations,
dust extinction at wavelength l was considered to be composed of
a pure absorption coefficient per unit mass, kld, and an isotropic
and coherent scattering coefficient per unit mass, sld, to be added
to the atomic and molecular coefficients. The dust coefficients per
unit mass were assumed to show the same properties as the
chemical equilibrium coefficients with 100 per cent condensation,
i.e. the same l dependence, independence of local temperature and
density, and simple scaling by a factor of 0.3 and 0.7 for the cases
of 30 and 70 per cent condensation.
(ii) Calculations were done for two extreme cases. In the first
(henceforth called D1), dust particles were thermodynamically
fully coupled to the surrounding gas and had the local kinetic gas
temperature T, T d ðrÞ ¼ TðrÞ. In the second case (D2), they were
totally decoupled from the surrounding gas and had the radiative
equilibrium temperature given by
ð
kld {Bl ½T d ðrފ 2 J l } dl ¼ 0;
where Bl and Jl are the Planck function and the mean intensity,
respectively, and all dust species were summed up in kld of an
‘averaged particle’ for simplicity. Dust temperatures Td in the D2
case turned out to be appreciably higher than gas temperatures, so
that D2 particles cannot, in fact, condense in atmospheric regimes
in which gas temperatures are only moderately below condensation
temperatures. Therefore, they only show up in Mira model
atmospheres with very low effective temperature and very strong
geometric extension.
(iii) All dusty models were re-iterated to radiative equilibrium,
taking into account both gas and additional dust contributions to
absorption and scattering coefficients. Wavelength integrations
entering the energy equation were carried out from 0.28 to 30 mm
with Planck-type flux extensions to zero and infinity. In the D1
case, the dust absorption, which is a ‘quasi-grey’ slow function of l
in the relevant optical to near-infrared wavelength range,
counteracts the cooling effects of molecular band absorption so
that near-surface temperatures tend to be significantly higher than
those of the dust-free model. In the D2 case, dust temperatures can
only fall below condensation temperatures in very high layers, if at
all, and only slightly affect the temperature structure of the
uppermost atmosphere. In both cases the occurrence of dust
particles is typically confined to a substantially smaller regime of
the atmosphere than one would expect naively from the
temperature stratification of the dust-free configuration.
(iv) In all calculations, the dynamical density stratification of the
dust-free atmosphere was retained in the dusty models, i.e. the
strong driving effect of radiative outward acceleration on the grains
was not taken into account. Lobel et al. (2000) assumed dustgenerated gas velocities of the order of #5 km s21 in the upper
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layers of a model of o Cet. In a fully consistent description,
pulsation models including a realistic mixture of radiatively
accelerated dust particles would have to be followed over several
cycles in order to obtain the dynamical structure of the upper
atmosphere. Furthermore, we adopted the same ‘surface’ radius rs
for the dusty atmosphere as in the dust-free model, although this
meant in a few models that the tl ! 1 condition at rs was not
fulfilled at wavelengths of strong molecular bands or towards the
blue spectral region because of noticeable additional dust opacity.
We chose not to extend the models into the circumstellar regime
although, of course, there is no physical boundary between
atmospheric and circumstellar matter, and the occurrence of
atmospheric dust would further blur this distinction.
(v) The dust was composed of amorphous corundum, Al2O3
(absorption and scattering coefficients: Koike et al. 1995), and
amorphous silicate, MgFeSiO4 (Dorschner et al. 1995). Spherical
Mie particles with a Mathis– Rumpl – Nordsieck size distribution
(Mathis, Rumpl & Nordsieck 1977) were assumed. Corundum is
commonly considered to be only a seed particle of the grain
condensation process rather than an abundant grain [e.g. Kozasa &
Sogawa (1997), but see, however, objections given by Gail &
Sedlmayr (1998)]. In any case, including it in the present
exploratory calculations appears reasonable because it condenses
at the highest temperature in the oxygen-rich element mixture of an
M-type star and provides some kind of ‘extreme-case’ scenario. At
typical gas pressures in high layers of an M-type Mira atmosphere,
Pg , 1022 –1025 dyn cm22 , corundum condenses around 1400 K
whereas the silicate grains only form at about 250 K below that
(Gail 1998). For numerical reasons, the onset of dust extinction in
the atmospheric model was smeared out over a range of 100 K
below the condensation temperature. If one assumes that all silicon
and aluminium atoms are bound in silicate and corundum grains,
respectively, in chemical equilibrium, then about 95 per cent of the
dust mass is silicate and about 5 per cent is corundum. This results
in absorption and scattering ratios of the order of 10 to 100, and
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whenever both dust species are present, effects upon the star’s
spectrum including the appearance of the strong spectral features
above 10 mm produced by both particles are dominated by silicate.
Clearly, real dust will contain a mixture of numerous components
(e.g. Alexander & Ferguson 1994; Lorenz-Martins & Pompeia
2000), but these two species should be well suited for exploring
typical effects.
3
3.1
EFFECT OF DUST ON MODELS
Temperature stratification
Fig. 1 shows the temperature stratification of the upper layers of a
typical sample model (P74200). Gas temperatures of the dust-free
upper atmosphere fall well below the adopted maximum
condensation temperature of about 1400 K above 3.2Rp at gas
pressures in the 1021.7 to 1023.6 dyn cm22 range. (Note that the
temperature decline in this extremely extended model atmosphere
is not yet close to the asymptotic T / r 21=2 behaviour in uppermost
layers, and that a temperature cut-off at u ¼ 5040 K/ T ¼ 6:8 was
chosen because of deficiencies of the equation of state used in the
BSW96 and HSW98 models at very low temperatures.)
When D1 grains are injected that attain the kinetic temperature
of the gas particles and the model is iterated to radiative
equilibrium with this additional opacity source, surface temperatures rise to about 1000 K. This means that, on the one hand, the
range of the upper atmosphere in which dust particles (in particular
silicate) can condense, and the direct effects of dust absorption and
scattering upon the brightness distribution on the stellar disc and
upon the spectrum, turn out to be smaller than initially expected.
On the other hand, however, the warming of the upper atmosphere
may substantially affect the spectral appearance of deep absorption
features and the shape of the centre-to-limb variation (CLV) of
intensity in the light of such features. Figs 2 and 3 show that this
warming effect depends on the details of the atmospheric structure.
Figure 1. Temperature and gas pressure in the near-maximum model P74200. The dust condensation is 30 per cent (see text). The radius is in units of Rp. For
the model containing D1 dust (full thermodynamical coupling, see text), the temperature of dust grains condensing below about 1400 K is identical to the gas
temperature (dotted line). For the model containing D2 dust (no thermodynamical coupling), the gas temperature is almost indistinguishable from that in the
dust-free model (solid line) and not shown here. The gas temperature is cut off at 740 K (see text).
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Figure 2. Same as Fig. 1 for model M97600. The dust condensation is 30 per cent. For the model containing D2 dust, the gas temperature is almost
indistinguishable from that in the dust-free model and not shown here. The gas temperature is cut off at 740 K.
Figure 3. Same as Fig. 1 for model D28760. The dust condensation is 30 per cent. D2 dust cannot condense in the atmosphere of this model. The gas
temperature is cut off at 740 K.
In the near-maximum model M97600 (Fig. 2), which has a higher
mass and lower effective temperature than P74200, gas
temperatures are raised over a very large portion of the atmosphere
through the effects of D1 dust extinction on radiative equilibrium.
Still, effects upon the spectrum tend to be less pronounced than for
P74200 because pressures, densities and, hence, optical depths in
the outer atmosphere are noticeably smaller. Similarly, strong
warming by dust seen in the more compact model D28760 (Fig. 3)
comprises a relatively small, low-density volume and only results
in quite modest spectral effects.
In contrast D2 particles, which are thermodynamically
decoupled from the gas, have such high grain temperatures that
they can barely survive in the radiation field of this model star.
They only show up near the very ‘surface’ of the model and their
opacity does not have a noticeable effect on the gas temperature
stratification or on spectral features. In fact, the very cool and very
extended near-minimum model P71800 is the only one of our
model set of Table 1 in which D2 dust temperatures are below
1400 K in a sufficiently large volume, r . 2:5Rp , to have
appreciable effects on gas temperatures and spectral features. If
D2 dust can partially survive a few months while temperatures
slowly rise from minimum to maximum phase, dust effects may be
more important in the D2 case than predicted by our nearmaximum models P73200 and P74200. In the warmer and more
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compact near-minimum model D27760, however, D2 dust is not
abundant or efficient. Note also significant uncertainties in the
upper layers of the P71800 model, which result from inadequate
accuracy of the state equation and the extinction coefficients at
very low temperatures (BSW96, HSW98). In the moderately
extended atmospheres of the near-maximum models D28760
(Fig. 3) and E8380, calculated temperatures of D2 grains do not fall
below the condensation temperature.
Increasing the D1 dust content by a factor of 70/30, i.e.,
Table 2. Bandpasses (box-shaped, with limiting wavelengths as
given).
Name
l1
(mm)
l2
(mm)
Spectral feature
blue
0.700
0.712
0.818
0.845
1.04
H
K
L*
0.400
0.698
0.711
0.816
0.844
1.0375
1.483
1.995
3.499
0.410
0.702
0.713
0.820
0.846
1.0425
1.783
2.395
4.090
near-continuum
near-continuum (TiO contamination)
strong TiO
near-continuum
moderate TiO
near-continuum
near-continuum
near-continuum
near-continuum
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increasing the dust absorption and scattering coefficients per unit
mass by this factor as described in point (i) of Section 2, leads to
only modest temperature changes in the upper layers of the
D28760, P73200 and P74200 test models. Surface temperatures
rise by 30, 60 and 80 K, respectively. Also, limb darkening and
spectral characteristics discussed below do not change strongly.
Thus, the adopted dust abundance, i.e. 0.3 or 0.7 per cent of the
mass condensed in dust grains, should yield fairly representative
results.
3.2
Limb darkening
We studied the behaviour of the intensity CLV in nine different
bandpasses listed in Table 2, being mainly interested in the effect of
dust extinction on continuum diameters measured in infrared
bandpasses. Whilst D2 dust does not produce noticeable effects on
limb darkening in our infrared bandpasses, as expected from the
discussion in Section 3.1, the injection of D1 particles in the upper
atmosphere leads to a very typical two-component shape of the
CLV. This has a low-intensity tail feature, which can readily be
understood in terms of added Planck function contributions
(Scholz 2001), modified by scattering. In Fig. 4, we show this
effect in the M97600 model. Although the intensity level of the
Figure 4. Centre-to-limb variations of intensity (left) and visibility curves (right) for model M97600 both without dust (upper) and with D1 dust (lower) in
infrared bandpasses 1.04 (full line), H (dotted), K (dashed) and L* (dot-dashed). The radius is in units of Rp and spatial frequency is in arbitrary units. The dust
condensation is 30 per cent (see text).
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dust-generated CLV tail is very low at 1.04 mm, it influences
noticeably the corresponding visibility, which an interferometric
observer would have to interpret in terms of, e.g., a uniform-disc
(UD) diameter. Depending on the spatial frequency at which the
UD fit is performed, the observer would deduce a slightly larger or
smaller diameter than the true 1.04 continuum diameter given by
the position of the steep CLV flank (intensity diameter) or the
almost coinciding position of the t1:04 ¼ 1 layer (optical-depth
diameter; cf. Baschek et al. 1991; Scholz 2001). In the H, K and L*
bandpasses, the dust tail effects do not show up unless spatial
frequencies beyond the first visibility minimum are accessible.
Fig. 4 also shows that there is a competing effect resulting from
weakly wavelength-dependent continuous opacity plus, more
importantly, molecular band contamination that makes the fitted
UD continuum diameter grow slightly with increasing l from H to
K to L*. A very pronounced example is shown in Fig. 5, where we
plot infrared brightness distributions and visibilities for the model
P74200 without dust and with increased dust abundance [70 per
cent, see point (ii) in Section 2]. The dust-free atmosphere shows
short CLV tails of substantial strength in H, K and L* mainly due to
water absorption. As a consequence, visibilities are distorted and
deduced UD diameters grow significantly towards longer
wavelengths (see the UD fits of this model and other models in
table 4 of HSW98 for conspicuous examples). Only the UD
diameter fitted in the 1.04 bandpass is close to the true continuum
diameter. After injecting D1 dust, the CLV tails extend still further
at a low intensity level and resulting H, K and L* visibilities are still
more distorted. At 1.04 mm, however, dust produces an efficient
CLV tail, which makes the 1.04 UD diameter grow substantially
when visibility fits are done at small to medium spatial frequencies.
Fig. 6 shows the equivalent situation for the model P73200 (70 per
cent dust condensation), which represents the near-maximum
phase of the previous cycle of the same Mira as P74200 in Fig. 5.
Here, CLV tails in the dust-free model are much weaker and dustgenerated tails are more efficient, to such an extent that 1.04 UD
diameter may look larger than those in H, K and L* bandpasses at
small spatial frequencies. In Fig. 7, a grid of UD visibilities,
superimposed on the 1.04 mm visibilities of Fig. 6, demonstrates
this effect.
Inspection of our other test models of Table 1 shows that effects
of D1 dust in the 1.04 to L* bandpasses are of a similar order of
magnitude in the first overtone model E8380 as in M97600; are
negligible and just perceptible in the models of the D series near
maximum (D28760) and near minimum (D27760), respectively;
and are very pronounced, as expected, in the near-minimum model
atmosphere P71800. In all models, tails or tail-type protrusions of
the pure continuum CLV produced by molecular band contamination (in H, K and L*, chiefly water) are present, sometimes
Figure 5. Same as Fig. 4 for model P74200 (except 70 per cent dust condensation).
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Figure 6. Same as Fig. 4 for model P73200 (except 70 per cent dust condensation).
Figure 7. Visibility curves in the 1.04 bandpass for model P73200 shown in
Fig. 6 (full lines) without dust and with D1 dust, and for uniform discs with
angular diameters 1 (rightmost dotted line), 2; …; 10 (leftmost) in arbitrary
units. If the visibility of the dusty model is fitted by a UD at spatial
frequency 0.3 and 0.5, it appears about 30 per cent larger and 20 per cent
smaller, respectively, than the UD diameter of the dust-free model. UD fits
at smaller spatial frequencies drastically overestimate the star’s 1.04 mm
continuum size.
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conspicuously. They affect the shape of the visibility curve in a
similar way as dust: the central maximum becomes narrower and
often shows a pronounced two-component appearance.
It is obvious from this study that: (i) the effects of molecular
contamination cannot easily be disentangled from those of D1 dust
opacity in 1.04 to L* bandpasses; and (ii) both effects strongly
depend on stellar parameters, on phase and even on the considered
pulsation cycle of the Mira variable. We may still draw a few
general conclusions. First, a systematic increase of UD diameters
with increasing wavelength from J to H to K to L, as reported by
Tuthill et al. (1998, 1999) for W Hya and o Cet and by Thompson
et al. (2000) for X Del (H, K ), may hardly be blamed on dust but
more likely on molecules contaminating the pure continuum. The
wavelength dependence found inside the K band by Thompson
et al. (2000) for numerous M-type Miras is presumably also due to
molecular (in particular water) absorption, as indicated by a few
sample test calculations. Similarly, the unexpectedly large K-band
continuum radius of R Leo measured by Perrin et al. (1999) is
unlikely to be due to dust absorption and scattering. Secondly, the
20 per cent increase of the 1.04 near-minimum UD diameter of
R Cas (Hofmann et al. 2000) as compared to the K-band nearmaximum UD diameter (Weigelt et al. 2000) may well be affected
by dust opacity, provided that dust particles of type D1 do exist in
the star’s cool atmosphere. Thirdly, dust is most probably not the
cause of the discrepancy between evidence for fundamental mode
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pulsation of M-type Mira variables from MACHO observations
(Wood et al. 1999) and pulsation velocities (Scholz & Wood 2000)
and for first overtone pulsation from the period – radius relation
(e.g. Haniff, Scholz & Tuthill 1995; van Belle et al. 1996; van
Leeuwen et al. 1997).
The optical to near-infrared range below 1 mm is not the main
topic of this investigation. It is much more heavily affected by the
occurrence of dust because of the strong increase of dust opacity
towards smaller wavelengths, as well as the strong reaction of
spectral features in the Wien part of the Planck function to the
warming of the upper atmosphere by D1 dust. Effects on limb
darkening are generally significant and sometimes even extreme,
and they depend strongly on stellar parameters, phase and cycle.
Fig. 8 shows CLV and visibility curves in the blue to 0.845
bandpasses of Table 1 for the P73200 model (70 per cent dust
condensation). We see the pronounced changes of brightness
distributions through dust injection, generally again in the form of
tail-type extensions towards the edge of the stellar disc. For the
blue bandpass, where dust opacity is largest (and which is
extremely sensitive to dust-induced stratification effects), and for
the 0.712 bandpass, where dust opacity is added to strong TiO
absorption, intensities do not approach zero at large r, which means
that the adopted position of the star’s ‘surface’ should be pushed
upward to fulfil the tl ! 1 surface condition. Monochromatic
diameters based on the UD approximation or on CLVs predicted by
dust-free models would require significant corrections. The
extreme limb darkening effect seen in the 0.712 bandpass (central
maximum of visibility broader in the 0.712 strong-TiO bandpass
than in the 0.700 near-continuum bandpass; see Jacob et al. 2000)
occurring in two dust-free models of Table 1 (E8300, M97600)
disappears in the dusty counterparts because of its temperature
sensitivity and the increase of opacity. D28760 is an example of a
model whose optical limb darkening is only slightly affected by D1
dust, with very low to low intensity tails of CLVs appearing that are
strongest in the blue. The three P series models of Table 1 are the
only ones in which dust particles of type D2 produce noticeable
effects, namely a weak CLV tail in the blue for the near-maximum
models and significant CLV changes with tails in all bandpasses for
the near-minimum model.
It would appear from these model predictions that observing
visibilities at optical and near-infrared wavelengths may provide a
tool for proving or excluding the presence of dust in upper
atmospheric layers of Mira variables. One has to be aware,
however, that the effects of dust upon the CLV are so pronounced
below 1 mm and depend so strongly on details of dust extinction
that a model based on the coarse assumptions of Section 2 is not
suited for quantitative discussion. However, the primitive
calculations presented here show that, if dust grains can form in
Figure 8. Same as Fig. 6 for model P73200 (70 per cent dust condensation) but in optical and near-infrared bandpasses blue (full lines), 0.700 (dotted), 0.712
(dashed), 0.818 (dot-dashed) and 0.845 (triple-dot-dashed).
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Figure 9. Spectrum of the M97600 model calculated both without dust (thin line) and with D1 dust (thick line; 30 per cent dust condensation).
the atmosphere, strong effects on disc brightness distributions and
on interferometric diameter measurements must be expected at
short wavelengths.
3.3 Spectra
In Fig. 9, we compare the spectra of a typical sample model,
M97600, calculated without dust and with atmospheric D1 dust.
Differences are strong below about 1 mm and become weaker and
negligible towards the far-infrared. We must be aware, however,
that one does not only see the direct effects of large dust opacity at
shorter wavelengths but also the effects caused by changes of the
temperature stratification. These can be quite substantial in the
Wien part of the Planck function. The warming of the upper layers
due to dust substantially affects the appearance of strong molecular
bands but may also affect the so-called ‘continuum windows’,
which are indeed all contaminated by molecular band absorption
below 1 mm in the optical to near-infrared wavelength region.
Hence, defining and determining from observations reddening of
the continuum by dust absorption is not a trivial procedure.
Determination of effective temperatures from fitting modelled to
observed spectra (cf. examples given by Lobel et al. 2000) has to be
tackled with great caution if dust is presumed to be present in the
upper Mira atmosphere. Further discussion of spectral effects of
atmospheric dust absorption would require a more sophisticated
dust model, as well as inclusion of the effects of circumstellar dust.
4
CONCLUSIONS
This study shows that continuum diameters of M-type Mira
variables measured by interferometry in infrared bandpasses are
not substantially affected by atmospheric dust extinction. Even if
dust were thermodynamically fully coupled to the surrounding gas
(case D1), its influence upon limb darkening of the disc is small
and often negligible. If it were decoupled from the gas (case D2), as
is most likely in outermost atmospheric and in circumstellar layers,
it cannot form in a sufficiently large range of the upper atmosphere
to produce significant effects. One would have to extend the
atmospheric model far into the circumstellar regime above, say,
5Rp to find an appreciable amount of dust.
Generally, the brightness distribution in infrared continuum
bandpasses is much more affected by molecular band contamination than by dust extinction in the near-maximum Mira models
studied here. Molecules, in particular water, may generate tail-type
extensions of the CLV and seriously impede continuum diameter
measurements. Only in the 1.04 bandpass may D1 dust noticeably
q 2001 RAS, MNRAS 325, 1487–1496
distort the brightness distribution on the disc in some cases and
may feign a larger continuum radius.
The situation is somewhat different at optical and near-infrared
wavelengths below 1 mm, as well as at all wavelengths of nearminimum Mira models. There, atmospheric dust may change
significantly not only the stellar spectrum but also the brightness
distribution in various bandpasses, as demonstrated in Fig. 8.
Whenever dust is suspected to affect limb darkening in infrared
bandpasses, extending observations to short wavelengths may
provide discriminating clues. However, effects sensitively depend
on dust properties as well as on model properties, i.e. on
atmospheric parameters, and would require individual analysis on
the basis of models that are specifically tailored to the stellar
parameters, pulsation phase and pulsation cycle of observed Miras.
AC K N O W L E D G M E N T S
We are indebted to H.-P. Gail for discussing and providing data on
dust properties. This work was supported by the Australian
Research Council.
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This paper has been typeset from a TEX/LATEX file prepared by the author.
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