Astron. Astrophys. 324, 185–195 (1997)
ASTRONOMY
AND
ASTROPHYSICS
Model atmospheres of cool, low-metallicity stars:
the importance of collision-induced absorption
Aleksandra Borysow1,2 , Uffe Gråe Jørgensen1 , and Chunguang Zheng2
1
2
Niels Bohr Institute, Copenhagen University Observatory, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark
Physics Department, Michigan Technological University, Houghton, MI 49931, USA
Received 29 July 1996 / Accepted 12 December 1996
Abstract. We have extended our data base of collision induced
absorption (CIA) in the high-temperature regime applicable
to stellar atmospheres. Improvements of existing data include
computation of series of hot-bands and extension of the spectral
wings much further away from the maxima than hitherto. Computation of new bands includes the first and the second overtone
bands of H2 -H2 . We apply these data to an extensive grid of
oxygen-rich model atmospheres with scaled solar metallicities
in order to investigate for which range of fundamental stellar
parameters (i.e., effective temperature, gravity, and chemical
composition) CIA has a significant impact on the atmospheric
structure. Besides the CIA due to H2 -H2 and H2 -He pairs, our
models include complete molecular line data for TiO, H2 O, CN,
CH, and SiO.
For stellar models with low effective temperatures, high
gravity, and low metallicity, the atmospheric structure and the
emergent spectrum are completely dominated by the effects of
CIA. For our test-models of lowest effective temperature and
lowest metallicity (Teff = 2800 K and Z = 10−3 Z ) the effect
of CIA is pronounced even for sub-giant stars with log(g) = 2.0.
For dwarf models with Z = 10−3 Z and log(g) = 5.0 the effects
are visible in the overall flux distribution for effective temperatures as high as 4000 K, and for models with Teff = 2800 K
and log(g) = 5.0 CIA has effects on the spectrum of stars with
metallicities as high as 0.1 Z .
Key words: molecular data – molecular processes – stars: atmospheres – stars: late-type – stars: low-mass – infrared: stars
1. Introduction
In homonuclear molecules, such as H2 , there is no change of
dipole moment during rotation or vibration, and such molecules
are therefore unable to absorb (or emit) dipole radiation. However, during transient interactions of such non-polar molecules,
short-lived “super-molecular” species, as for example H2 -H2 ,
Send offprint requests to: Aleksandra Borysow1
are formed, a temporary dipole moment is induced, and a relatively weak dipole absorption becomes possible.
Herzberg (1952) was the first to bring collision induced absorption (CIA) into an astrophysical context, and suggested this
mechanism as being responsible for some hitherto unexplained
absorption bands in the spectrum of Uranus. By comparing laboratory H2 spectra with spectra of Uranus and interpreting the
bands as due to CIA, Herzberg derived the first limits on the
pressure of the thick Uranian cloud top, and produced one of
the first pieces of spectroscopic evidence that hydrogen was the
primary constituent in Uranus. Today CIA processes are part of
the standard modelling of the low-temperature, dense gases in
the atmosphere of the giant planets and of Titan and Venus.
Linsky (1969) developed a simplified analytic description of
a few bands of H2 -H2 and H2 −He CIA as a function of frequency
and temperature, and was the first to point out that the pseudocontinuous character of the CIA could cause it to have a large
effect also in stellar atmospheres. Here it will block the energy
which would otherwise have escaped in the transparent spectral
regions between the absorption lines. This will be particularly
true when polar absorbers (e.g., H2 O, H− , TiO, etc) are depleted,
as for example in cool stars of low metallicity.
In the same year, Tsuji (1969) considered molecular opacities in cool stellar atmospheres, including various sources of
continuous opacity. Independently, using his own estimation of
CIA intensity, he reached the same conclusion that CIA is an
important source of opacity at high pressures, which can not be
neglected.
In the following decade CIA in stellar atmospheres attracted
more attention. Shipman (1977) computed the first white dwarf
model atmosphere composed of pure H2 , which included Linsky’s data. He found that at a temperature of 4000 K CIA contributes essentially all the opacity at the wavelengths where the
flux is emitted. At higher temperatures other opacity sources
were dominant. Also Mould & Liebert (1978) have included
CIA due to H2 –H2 and H2 –He pairs, to compute new white
dwarf model atmospheres, again using Linsky’s data. They reiterated the importance of CIA, but no specific results were
shown.
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A. Borysow et al.: Impact of CIA on stellar atmospheres
Palla (1985) pointed out, for the first time, that, depending
upon gas density, CIA may be an essential, and often even the
dominant source of opacity, in primordial protostars at temperatures between 1000 and 7000 K. In Palla’s work, atmospheres
composed of hydrogen and helium were considered and Linsky’s (1969) data were again input.
One year later, Stahler et al. (1986), using Linsky’s data,
confirmed Palla’s findings: the source of opacity due to collision
induced absorption cannot be neglected.
Based on the then existing quantum mechanical CIA models due to Borysow and collaborators (Borysow 1994 and references therein; see also summary below), Lenzuni et al. (1991)
and Saumon et al. (1994) studied the impact of CIA on hypothetical zero-metallicity, primordial proto-stars and zero-metallicity
brown dwarfs, and found very large effects.
We have now extended these computations of CIA due to
H2 -H2 and H2 -He into the higher temperature regime appropriate for stellar atmospheres, and studied their impact on realistic
atmospheric models of existing stars. In Sect. 2 we summarize
the existing quantum mechanical CIA data and in Sect. 3 we
present the result of our new calculations. In Sect. 4 we compare our results with previous data, where available, and give
some warnings about the limitations of our data and of previous
data. In Sect. 5 we describe the impact of our data on stellar
model atmospheres, and quantify in which region of the HR
diagram (and metallicity) the collision induced absorption is
important for the stellar atmospheric structure and for the analysis of photometric and spectroscopic data.
Our data are applicable to stars of arbitrary chemical composition, but in this paper we will restrict the discussion to oxygenrich stars. We cover the region in fundamental stellar parameters
which represents brown dwarfs, all types of K and M dwarfs,
as well as the metal-poor sub-giants and giants that could be
found in the Galactic halo, in globular clusters, and in metalpoor galaxies such as the Fornax dSph and the SMC. In later
papers we will discuss carbon dwarfs and white dwarfs, which
will require additional input data. Our CIA data do not include
H2 −H pairs, which could also contribute significantly to the
opacity, and we have restricted the inclusion of bound−bound
molecular line transitions to those molecules for which complete, high-quality quantum mechanical computations exist. In
particular, it may be desirable to include additional diatomic
hydrides in the discussion. In future work we hope to be able to
extend the data toward a higher degree of completeness. Also
dust formation may be of relevance for the very coolest objects.
Table 1. Population probability of the ground vibrational state of H2
at temperatures between 1000 and 7000 K.
Temperature(K)
1000
2000
3000
4000
5000
6000
7000
P (v = 0)
0.993
0.944
0.848
0.750
0.667
0.595
0.535
erate intermolecular distances), and on the isotropic, effective
H2 –H2 interaction potential by Ross et al. (1983). We have upgraded the dipole data to meet the demands of the high temperature predictions of CIA. The model accounts for hot bands
involving ∆(v1 ) = ∆(v2 ) = 0 vibrational transitions with
v1 , v2 = 0, 1, 2, 3, which are populated at temperatures below
7000 K. Variables v1 and v2 denote vibrational quantum numbers of each hydrogen molecule.
2.2. H2 -H2 fundamental band
The existing model of the fundamental band (Borysow &
Frommhold 1990) is available at temperatures from 600 to
5000 K. It includes, however, only one vibrational transition,
i.e. 0 → 1, although at high temperatures also hot bands (with
∆v = 1) corresponding to the same frequency range are expected to be present. We tried to correct for this fact, although it
is apparent that an updated model is necessary. An extrapolation
of the models to temperatures other than those they are designed
for can give very undependable results. We have therefore chosen to use the results for 5000 K to represent opacities of the
0 → 1 transition also at 6000 and 7000 K. However, in order to
account for the missing transitions (giving rise to the hot bands)
with ∆(v) = v 0 − v = 1, corresponding to v → v 0 = v + 1, with
v > 0, (we account, in fact, globally for two cases: ∆(v1 ) = 1,
∆(v2 ) = 0 and ∆(v1 ) = 0, ∆(v2 ) = 1, both bands are identical,
due to the symmetry), we rescaled the intensities of the 0 → 1
transition by dividing them by the probability of the population
of v = 0 state at each temperature (see P (v), Table 1).
Such a procedure is correct if we assume that the shape
and the intensity of the unweighted (by the P(v) ) translational,
rotational, and vibrational transitions with ∆v=1 are identical.
Whereas it is not generally the case, we think that this procedure
significantly improves the data over those available from the
existing program.
2. CIA opacities from existing quantum mechanical sources
2.3. H2 -He roto-translational band
2.1. H2 -H2 roto-translational band
The existing model of the roto-translational (RT) band (Zheng
& Borysow 1995b) in H2 –H2 has been designed specifically
for astrophysical applications. It provides absorption intensities at temperatures from 600 to 7500 K. We based our model
on the available induced dipole functions due to Meyer et al.
(1989), designed to work well at low temperatures (i.e. at mod-
The existing model of the 0 → 0 band predicts CIA intensities of this band at temperatures below 3000 K (Borysow et al.
1988). In the absence of any newer data we have applied similar rescaling procedure as we did for the fundamental band of
H2 -H2 . Again, initially we have computed intensities at all temperatures up to 3000 K, and then copied the results for 3000 K
to all higher temperatures. Next, in order to account for the hot
A. Borysow et al.: Impact of CIA on stellar atmospheres
bands, we divided those intensities by P (v = 0). We expect quite
large uncertainties associated with this procedure, but we note
that the far infrared intensities, corresponding to the ∆(v) = 0
band, are less important than those in the near infrared. In addition, the H2 -He intensities are expected to matter much less
that those due to H2 -H2 , on account of lower abundance of helium in the stellar atmospheres. Also, CIA usually matters less
at high temperatures, so the inaccuracy of the high temperature
predictions is less relevant. A better CIA model of this band is
certainly being called for.
2.4. H2 -He fundamental band
The models of all relevant vibrational transitions corresponding
to the fundamental band frequency region are available for the
H2 -He complex. The models are applicable for all temperatures
up to 7000 K, and involve ∆(v) = 1 transitions, including the
0 → 1 transition (Borysow et al. 1989) and many hot bands
(Borysow & Frommhold 1989).
2.5. H2 -He higher overtones
Models of several vibrational transitions corresponding to
∆(v) = 2, 3 are available (Borysow & Frommhold 1989) at
temperatures up to 7000 K. We have incorporated all of them
into our opacity code.
3. New H2 –H2 CIA opacities
No quantum mechanical models have been available for H2 –
H2 CIA spectra in the first and second hydrogen overtone bands.
Below, we summarize our attempts in making such models,
which we applied to our present studies. Our models are applicable at temperatures from 1000 to 7000 K.
For stars with effective temperatures between 3000 and
4000 K, the maximum of the black body radiation is emitted
between about 10000 and 14000 cm−1 . The centre of the first
overtone CIA band of H2 –H2 falls around the frequency ∼ 8000
cm−1 , and that of the second overtone band, at ∼ 12000 cm−1 .
The second overtone band is extremely weak compared to the
first overtone (< 3% for peak intensity). At the highest temperatures (∼ 7000 K), the peaks of the second overtone band are
even smaller than the far wings of the first overtone band at the
same frequencies. Thus we have extended the wings of the first
overtone band up to 25000 cm−1 , and predict that overtones
higher than the second are not important. Since our analytical
models are not designed to work well in the far wings (roughly
at intensities < 1% peak intensity), the uncertainties of the extended wings of the first overtone band may be substantial.
3.1. The first overtone band
Ab initio induced dipole functions (Meyer et al. 1993) and an
“effective” isotropic H2 –H2 potential (Ross et al. 1983) were
used as input to produce the lowest three semi-classical spectral
moments. These moments were then used to model the H2 –
H2 CIA spectra by the model lineshapes, used successfully to
187
model the low (T < 500 K) temperature spectra of this band
(Zheng & Borysow 1995a).
At low temperatures only the ground vibrational state, v = 0,
is populated, but at the high temperatures of importance here,
the higher vibrational states are also significantly populated.
We account for the initial vibrational states of two interacting
hydrogen molecules v1 , v2 = 0, 1 and 2. In this band, there are
two different kinds of vibrational transitions present: (i) single
transition, ∆(v1 ) = 2, ∆(v2 ) = 0 (or ∆(v1 ) = 0, ∆(v2 ) = 2);
and (ii) double transitions: ∆(v1 ) = ∆(v2 ) = 1. Both kinds of
vibrational transitions fall in the same frequency region centred
around 8000 cm−1 .
Dipolar terms (Meyer et al. 1993) λ1 λ2 ΛL = 2023, 0223,
2021, 0221 and 2233 have been included for the single vibrational transition, and λ1 λ2 ΛL= 2023, 0223, 0001, 2021, 0221
and 0445 for the double vibrational transitions. The other terms
were found to contribute less than 2% of the total intensity.
The induced dipole functions also depend on the rotational
states of the two interacting molecules. We account for such
dependencies only for the largest, λ1 λ2 ΛL = 2023 and 0223,
terms (for both single and double transitions), which contribute
more than 70% to the total intensity. Since these dependencies
are available only for the lowest three J values (Meyer et al.
1993), we have scaled the entire β(R) functions according to
their long range asymptotic forms, in the same way as described
previously (Zheng & Borysow 1995b):
√
β2023 (R) → 3 < v1 J1 |α|v10 J10 >< v2 J2 |Q|v20 J20 > /R4 (1)
and
√
β0223 (R) → − 3 < v2 J2 |α|v20 J20 >< v1 J1 |Q|v10 J10 > /R4 ,(2)
with R being the H2 –H2 intermolecular distance. We have
computed the matrix elements of the electronic polarizability hvJ|α(r)|v 0 J 0 i and quadrupole moment hvJ|Q(r)|v 0 J 0 i of
the hydrogen molecule, for all needed J values, based on the
functions α(r) (Kolos & Wolniewicz 1967), Q(r) (Poll & Wolniewicz 1978), and the H-H potential V (r) (Kolos & Wolniewicz
1965, 1968, 1975), with r being the H-H internuclear distance.
It is difficult to predict the accuracy of our model. Over the
frequency range where the spectral intensities are larger than 1%
of the peak intensity, the accuracy mainly depends on our selection of the “effective” isotropic H2 –H2 potential (Ross et al.
1983); the word “effective” is being used to indicate that it effectively accounts for the anisotropy, and the v-dependence of
the H2 –H2 interaction and is expected to describe accurately
H2 -H2 interactions at high temperature. According to the discussion in our previous work (Zheng & Borysow 1995b), we
expect the accuracy to be better than 50% at this frequency
range. However, the uncertainty becomes less predictable in
the far wings where the spectral intensities fall below 1% of
those of the peak. Among various reasons we name the use of
the model lineshapes, which become less predictable in the far
wings, the dependence of the far wings on the dipole functions
at short range (largely uncertain), and the use of the effective
intermolecular potential rather than the real one. We expect the
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A. Borysow et al.: Impact of CIA on stellar atmospheres
Fig. 1. CIA opacities of H2 –H2 at temperatures between 1000 and
7000 K.
far wings to be accurate within an order of magnitude. Further
work on this band is in progress (Zheng, Fu, Borysow 1996, in
preparation).
3.2. The second overtone band
The prediction of the second overtone band H2 –H2 CIA spectra
presents an even greater challenge, due to the lack of detailed
knowledge of the induced dipole functions. Vibrational transitions with ∆(v1 ) = 0, 1, 2, 3 and ∆(v2 ) = 3 − ∆(v1 ) were
included (each of them falling into the same frequency region
centred around 12000 cm−1 ), and both interacting hydrogen
molecules were assumed to be in the initial ground vibrational
state (v1 , v2 = 0).
The ab initio dipole functions are not available for this band.
Since the λ1 λ2 ΛL = 2023 and 0223 terms contribute more than
70% to the total intensity in the first overtone band, we included
only λ1 λ2 ΛL = 2023 and 0223 in our model of the second overtone band, considering all other (unknown) terms to be small.
The asymptotic form of those dipole functions is well known,
and is given by Eqs 1 and 2. However, at high temperatures (i.e.
at high collisional energies, probing the short intermolecular
distances), the importance of the electronic overlap increases.
In order to correct for this unknown short-range behaviour, we
have scaled the known dipole functions for the v1 = 0 → 1 and
v2 = 0 → 1 transitions (of the first overtone band) so that we
matched the long-term dipole magnitudes given by Eqs 1 and
2 for the second overtone. Next, we proceeded with the same
modelling procedure as we have used for the first overtone.
The assumption of v1 , v2 = 0 introduces additional inaccuracy, which we consider minor, in view of other uncertainties
involved, like the choice of the H2 –H2 potential, or scaling of
the dipole functions. We estimate that the model for this band
is accurate within an order of magnitude.
3.3. The combined quantum mechanical CIA data
Figs. 1 and 2 show the total CIA opacities of H2 –He and H2 –
H2 plotted at frequencies from 0 to 20000 cm−1 , at temperatures
Fig. 2. CIA opacities of H2 –He at temperatures between 1000 and
7000 K.
from 1000 to 7000 K. At lower temperatures separate vibrational bands are identifiable, but with increasing temperature
the spectral bands broaden and result in one featureless continuum. Both H2 –H2 and H2 –He opacities show the maximum
of intensity at frequencies around 5000 cm−1 . We observe a
very wide range of intensities over the frequency band from 0
to 20000 cm−1 .
4. Comparison with previous data
Although quantum mechanical computations of CIA have been
available at high temperatures for some time, they have been
used in stellar studies only recently. In more widespread use
are Linsky’s (1969) analytical expressions for CIA transitions
– sometimes applied in combination with a limited set of quantum mechanical results. Linsky’s data include estimates of the
roto-translational band of H2 -H2 at temperatures from 600 to
3000-4000 K, the fundamental roto-vibrational band at temperatures up to 3000 K, and its first overtone band for temperatures
up to 4000 K. No hot bands or higher overtones are included.
Linsky’s treatment of the 0 → 1 roto-vibrational band has been
superseded by the ab initio computations of Patch (1971). As already stressed by these authors, the adopted lineshapes are, however, modelled by highly ‘ad hoc’ analytical functions that are
designed to reproduce the desired absorption intensities within
strictly limited ranges of temperatures and frequencies – let’s
say where the intensity of each band falls off by one order of
magnitude compared to its peak value. In this sense Linsky’s
data have not been intended for use for wavenumbers above ν
≈ 10 000 cm−1 (i.e., λ <
∼ 1 µm), or for temperatures outside the
range described above. In addition, Linsky’s opacities given for
H2 -He rely on simple rescaling of the absorption coefficients
of H2 -H2 , which is now known to give quite incorrect results
(compare Figs. 1 and 2, this paper).
Although the data by Linsky and Patch were, hence, intended for a rather limited range in frequency and temperature, simple programming of their analytical expressions will
of course give answers also outside this range, and great cau-
A. Borysow et al.: Impact of CIA on stellar atmospheres
2µm
1µm
0.5µm
10-5
Opacity [cm-1 amagat-2]
total CIA H2-H2 opacity
10-6
at T = 4000 K
10-7
10-8
our results
10-9
Linsky/Patch
10-10
10-11
0.0
0.5
1.0
1.5
Wavenumber [104cm-1]
2.0
Fig. 3. Comparison of our results for the H2 −H2 CIA opacities with
those due to Linsky and Patch.
tion should obviously be taken in the interpretation of results
based on such “extrapolations”. For example, the default continuum opacities in recent versions (e.g., Jørgensen et al. 1992,
Plez et al. 1992) of the widely used marcs code include Linsky’s roto-translational band and Patch’s fundamental band (the
first overtone is missing). The expression for these two bands
should be used only between 0 and ≈ 9 000 cm−1 . In default
computations they will, however, be included at any frequency
range, and any temperature demanded by the program. In Fig. 3
we demonstrate the difference between our new quantum mechanical CIA opacities as applied in the marcs code and the
Linsky/Patch-continuum CIA opacities as given by the marcs
code, when computed at 4000 K at the frequency range from
0 to 20 000 cm−1 . It is easily seen that at higher frequencies,
overlapping well with the black body radiation flux, the data
by Linsky and Patch overestimate the realistic opacities by as
much as three orders of magnitude. In fact, numerical experiments showed that the Linsky/Patch data may often give artificially larger effects on the model atmosphere, because of such
an erroneous extrapolation, than will application of the more
realistic quantum mechanical CIA, in spite of the fact that the
values of the Linsky/Patch data are considerably lower around
the maximum CIA absorption intensity. We would like to draw
the attention of all users of analytical CIA expressions in any
atmospheric code to this point; they should examine carefully
their inputs, and have in mind the comparison in Fig. 3 and the
limitations in analytical expressions alluded to above.
We would also like to extend this warning to all other cases
of indiscriminate use of all analytical opacity models which are
designed to reproduce certain values within a given (and tested)
range. As another example, we mention the use of the model of
roto-translational CIA for H2 pairs by Borysow et al. (1985), designed to model CIA intensities at temperatures up to 300 K, but
189
instead used (for example in the work by Lenzuni et al. 1991) to
model CIA opacities at 3000 K. The user needs to have in mind
that whereas authors who publish their opacity data can make
certain that the selected analytical, multi-parametric spectral
lineshapes reproduce well their quantum mechanical computations within certain range of temperatures and frequencies, there
is no attempt made, and in fact it is highly unlikely, that the same
lineshapes will give correct results at temperatures different by
one order of magnitude from those tested. Therefore, such “extrapolation” procedures will inadvertently lead to unpredictable
results and should be strictly avoided.
Our Figs. 1 and 2 above show a very large range of intensities, up to five orders of magnitude for H2 –H2 and 3–5 orders
for H2 -He. As we mentioned above, analytical lineshapes used
by us to model spectral lineshapes reproduce real profiles well
within the 1:100 range of intensities. Their far wing behaviour is
rather unpredictable, though on some occasions they were seen
to perform impressively well over extended frequency/intensity
range. We need to remind the reader that whereas at shorter
frequencies the spectral bands overlap each other, making the
far wings more irrelevant in the presence of the next, more intense band, the intensities at frequencies higher than those due
to the first and the second overtones (ν̃ > 15000 cm−1 ) may
be prone to more uncertainties. Thus, even though great care
has been placed on making the present models as realistic as
possible under current circumstances, it needs to be understood
that uncertainties of such models are also unavoidable.
5. Impact of CIA on stellar model atmospheres and synthetic
spectra
We have included the CIA data described above in the computation of a grid of photospheric models. The aim is to identify the
range of the fundamental stellar parameters (Teff , log(g), and
metallicity) within which CIA affects the stellar atmosphere,
and to quantify the effect CIA has on such models. The model
atmosphere code we use is an updated version (Jørgensen et al.
1992, Helling et al. 1996) of the marcs code (Gustafsson et al.
1975). This version of the program assumes hydrostatic equilibrium, spherical geometry (applied when appropriate), and line
blanketing by molecules treated by the opacity sampling technique. Line opacities were included for a total of approximately
20 million molecular lines of H2 O (from Jørgensen & Jensen
1993), TiO (from Jørgensen 1994), CO (from Goorvitch &
Chackerian 1994), SiO (from Langhoff & Bauschlicher 1994),
CN (from Jørgensen & Larsson 1990), and CH (from Jørgensen
et al. 1996). Spectra were computed as the emergent flux of
the model computation as well as in a separate synthetic spectrum program which allows us to study the contribution of each
opacity source separately.
We expect the effect of CIA to be largest for stars of low
effective temperature (corresponding to a small number density of free electrons and a large abundance of molecular hydrogen), high gravity (corresponding to high density in the atmosphere and therefore a large number density of H2 -H2 and
H2 -He “pairs”), and low metallicity (corresponding to a small
190
A. Borysow et al.: Impact of CIA on stellar atmospheres
3.0
Teff logg C/O logZ/Zo
2800 5.0 0.43 0.01
Flux
2.5
continuum
b-b molecular lines
b-b + CIA bands
2.0
1.5
1.0
0.5
0.0
0.5
1.0 1.5 2.0 2.5
Wavelength (µm)
3.0
Fig. 4. The contributions of various opacity sources to the spectrum of
a stellar model with Teff = 2800 K, log(g) = 5.0, and Z = 10−2 Z – a
typical main sequence member of a globular cluster. The model computation includes opacities of continuum sources, molecular lines, and
CIA. The spectra are computed based on the continuum alone (upper,
convolving curve), continuum + molecular (b−b) lines, and continuum
+ molecular lines + CIA. Only the latter is consistent with the underlying model atmosphere, but the difference between the three spectra
illustrates the relative contribution of the three sources of opacity.
amount of other absorbers). Therefore we started the computations with low-metallicity dwarf star models, such that our grid
of stars will span the coolest and most metal-deficient main sequence stars of our Galaxy, as for example cool M dwarf stars
in low-metallicity globular cluster or in the Galactic halo. In the
“corner” of our grid (Teff = 2800 K, log(g) = 5.0, Z = 10−3 Z )
CIA was found to be by far the dominant contributor to the
opacity at all wavelengths longer than approximately 1 µm and
for all depths – even in the surface layers (corresponding to τross
= 10−4 in this case).
Next, we varied (compared to the choice in our standard
model) the fundamental stellar parameters until we saw no more
significant effect of CIA in the emergent flux spectrum. In this
way we defined the region of interest for CIA inclusion. In
Fig. 4 we show the contribution of different absorbers to the
synthetic spectrum of a typical model in our grid, representing
a typical M dwarf in a globular cluster. The figure shows the
continuum, the molecular spectrum, and the spectrum including
both molecular lines and CIA, respectively. The latter spectrum
is the only one of the three which is internally consistent with
the underlying model in the sense that it includes all the opacity
sources which are also included in the corresponding model
atmosphere computation. Comparison of the three spectra only
illustrates the contribution of the different species to the total
emergent spectrum. It is seen that even at such relatively high
metallicities as adopted here (Z = 0.01 Z ), CIA is the main
contributor to the infrared stellar spectrum. Its presence induces
such a strong continuum depression that the bands longward
of 1.5 µm (due mainly to water) virtually disappear from the
spectrum. The strongest molecular features left in the spectrum
are the TiO bands, the water bands shortward of 1.5 µm, and the
G band (around 4300Å) due to CH.
The results of our analysis of the stellar models in the whole
range of fundamental parameters where we found CIA to be
of importance are summarized in Figs. 5 to 7. Fig. 5 illustrates
the effect of varying the gravity, whereas Fig. 6 and Fig. 7 show
how the relative importance of CIA, continuum absorption, and
molecular line absorption change when we vary the metallicity
and the effective temperature, respectively.
In each of the Figs. 5 to 7, the first column shows the emergent flux spectra computed based on models respectively with
and without CIA included in the opacity. Here all the spectra
and the underlying model atmospheres are internally consistent,
in the sense that the spectrum based on the atmospheric structure computed without CIA also itself is without CIA, and vice
versa. Hence these are the complete, self-consistent spectra one
would predict based on models respectively with and without
CIA included in the computations.
Each row of plots in Figs. 5 to 7 represents a given set of
fundamental parameters. Whereas the first column illustrates the
results of models respectively with and without CIA, the two
next columns concentrate only on the models with CIA included
in the model calculations. The plots in column two correspond
to the depth in the atmosphere where τross = 0.01, whereas plots
in column three correspond to depths where τross = 1.0. Both
columns show the total opacity (in units of cm2 per gram of
stellar material) due to CIA (dotted lines), due to the sum of all
other continuum sources (dashed lines), and due to the sum of
all bound-bound molecular line transitions (full drawn lines).
It is seen from the spectra that the major effect of the CIA is
to absorb energy in the infrared and to re-emit it at visible wavelengths. For the model shown in Fig. 4, this re-distribution of
the flux is so pronounced that the infrared (J−H, H−K) colours
change from (0.6, 0.3) to (0.3, −0.1) when CIA is included in
the model atmosphere and the corresponding synthetic spectrum. The B−V colour, on the other hand, is only very little
affected. The effect of CIA on the emergent infrared spectrum
is still substantial even for the cool low-metallicity sub-giant
model (i.e., log(g) = 2.0) in Fig. 5. The impact of CIA is also
seen in the Z = 0.1 Z cool dwarf model in Fig. 6, and in the
spectrum of the Teff = 4200 K low-metallicity dwarf model of
Fig. 7. These models define the boundary-region of the grid of
models where CIA seems to be important for the spectrum. The
effect described for Fig. 4, that the infrared molecular bands
“disappear” when CIA is included in the computations (i.e.,
that the flux distribution becomes very “smooth” at the shown
resolution), is seen in several of the flux diagrams of Figs. 5 to
7. The intensity of the G-band due to CH, on the other hand,
is either unchanged or even increased (due to the increased reemitted flux in the blue spectral region) when CIA is included
in the opacities.
For the cooler, low-metallicity, dwarf models, the impact of
CIA on the spectral distribution is seen to be dramatic, and it
is much larger than the impact of other molecular species. It is
caused by the relative heating and cooling effect which CIA has
on the atmospheric structure. General features of atmospheric
A. Borysow et al.: Impact of CIA on stellar atmospheres
τ-ross = 1.0
3
-2
2
-4
1
-6
0
-8
0
log(g)=4.0
τ-ross = 0.01
τ-ross = 1.0
-2
2.0
-4
1.5
1.0
-6
0.5
0.0
3.0
log(g)=2.0
τ-ross = 0.01
τ-ross = 1.0
flux
2.5
-8
0
-2
2.0
-4
1.5
1.0
-6
0.5
0.0
2.5
log(opacity)
flux
2.5
log(opacity)
3.0
log(g)=1.0
τ-ross = 0.01
τ-ross = 1.0
-8
0
-2
2.0
flux
0
log(opacity)
τ-ross = 0.01
1.5
-4
1.0
log(opacity)
flux
log(g)=5.0
191
-6
0.5
0.0
-8
0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0
Wavelength (µm)
0.5 1.0 1.5 2.0 2.5 3.0
Fig. 5. Showing the effect of varying g for models with Teff = 2800 K, metallicity Z = 10−3 Z , and C/O = 0.43 (the solar value). The first column
of plots in this figure shows the emergent flux from models with (highest peak) and without (lowest peak) CIA included in the opacity. The two
next columns show log10 of the opacity (in units of cm2 per gram of stellar material) due to various species (for those models from column one
where CIA is included) at τross = 0.01 (second column of plots) and at τross = 1.0 (third column), respectively. Dotted lines correspond to the
opacity of CIA, dashed lines represent the sum of the opacities due to all continuum sources other than CIA, and the full drawn lines represent
the molecular line opacity. The different rows of plots in the figure correspond to models of different gravity, decreasing from log(g) = 5.0 in
the uppermost row of plots to log(g) = 4.0, 2.0 and 1.0, respectively, in the rows below.
192
A. Borysow et al.: Impact of CIA on stellar atmospheres
4
Z/Zo=0.0001
τ-ross = 0.01
τ-ross = 1.0
0
-6
0
-8
0
τ-ross = 0.01
τ-ross = 1.0
3
-2
2
-4
1
-6
0
-8
0
3.0
Z/Zo=0.01
τ-ross = 0.01
τ-ross = 1.0
2.5
-2
flux
2.0
-4
1.5
1.0
0.5
0.0
3.0
2.5
-6
Z/Zo=0.1
τ-ross = 0.01
τ-ross = 1.0
-8
0
-2
2.0
flux
log(opacity)
flux
Z/Zo=0.001
log(opacity)
1
-4
1.5
1.0
log(opacity)
flux
-4
2
log(opacity)
-2
3
-6
0.5
0.0
-8
0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0
Wavelength (µm)
0.5 1.0 1.5 2.0 2.5 3.0
Fig. 6. Showing the effect of varying Z. Same as Fig. 5, but for models of Teff = 2800 K, log(g) = 5.0, and Z increasing from Z = 10−4 Z in
the uppermost row of plots to Z = 10−3 Z , Z = 0.01 Z , and Z = 0.1 Z in the lower rows.
heating and cooling and the corresponding flux re-distribution
are described in standard text-books on stellar atmospheres (e.g.,
Mihalas 1978) and they have been the subject of detailed discussion in several papers (e.g., Gustafsson & Olander 1979, Scholz
& Wehrse 1994, Gustafsson & Jørgensen 1994). The particularly large impact of CIA is due to its substantial pressure
dependence which results in a backwarming in the very deep
atmospheric layers where the continuum is formed (as opposed
to the effect of, for example, H2 O, where the main effect is in the
more shallow atmospheric layers above the continuum forming
region). The heating-cooling balance is shown for two models
(of log(g)=5.0, C/O = 0.43, and log(Z/Z )=10−3 , and with Teff
= 2800 K and Teff = 3800 K, respectively) in Fig. 8. The Teff
= 2800 K model with CIA shows a cooling as large as almost
A. Borysow et al.: Impact of CIA on stellar atmospheres
τ-ross = 1.0
3
-2
2
-4
1
-6
0
-8
0
Teff = 3400
τ-ross = 0.01
τ-ross = 1.0
flux
4
-4
3
2
-6
1
0
10
8
Teff = 3800
τ-ross = 0.01
τ-ross = 1.0
-8
0
-2
6
-4
4
-6
2
flux
0
14
12
10
8
6
4
2
0
log(opacity)
-2
5
log(opacity)
6
flux
0
log(opacity)
τ-ross = 0.01
Teff = 4200
τ-ross = 0.01
τ-ross = 1.0
-8
0
-2
-4
log(opacity)
flux
Teff = 2800
193
-6
-8
0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0
Wavelength (µm)
0.5 1.0 1.5 2.0 2.5 3.0
Fig. 7. Showing the effect of varying the effective temperature. Same as Fig. 5, but for models of log(g) = 5.0, Z = 10−3 Z and Teff increasing
from Teff = 2800 K in the uppermost row of plots to Teff = 3400 K, 3800 K, and 4200 K, respectively, in the lower rows.
500 K (compared to the corresponding model without CIA) in
the layers around log(τross ) = −2. A corresponding heating of
200 to 300 K results in the deeper layers where log(τross ) >
∼ 0
and where most of the continuum is formed. This substantial
heating increases the continuum flux at the shorter wavelengths
and gives rise to the marked shortward shift of the wavelengths
of the emergent flux spectrum. The corresponding model with
Teff = 3800 K shows much less surface cooling and backwarm-
ing from CIA than do the Teff = 2800 K model, and, as is already
seen in Fig. 7, the corresponding change in the emergent flux
(and the frequency of the flux-maximum) is affected much less.
Expressed in terms of the gas pressure, the effect of CIA on
the Teff = 2800 K model is, for a given temperature, to lower
the gas pressure by about one order of magnitude (in the deeper
atmospheric layers), or for a given gas pressure to increase the
temperature with approximately 1000 K.
194
A. Borysow et al.: Impact of CIA on stellar atmospheres
Temperature [K]
4000
logg C/O logZ/Zo
5.0 0.43 0.001
3500
3000
2500
T=2800, CIA
ditto, no CIA
T=3800, CIA
ditto, no CIA
2000
1500
-6
-4
-2
0
log(τross )
2
Fig. 8. Effect on the T−τross structure of including CIA in models of
log(g) = 5.0, Z/Z = 10−3 , C/O = 0.43, and with Teff = 2800 K and Teff
= 3800 K, respectively.
A deeper understanding of the reason for the spectral
changes can be obtained by comparing the various contributions to the total opacities at different depths of the atmosphere,
as is shown in the columns two and three of Figs. 5 to 7. In each
of the three figures, the contribution of CIA relative to the continuum sources (mainly H− ) decreases “downward” along the
figure (i.e., for decreasing gravity, increasing metallicity, and
increasing effective temperature, respectively), until CIA is no
longer the dominant opacity source. For such models, inclusion
of CIA will no longer change the shape of the emergent flux
from the shape of the continuum spectrum, although the total
opacity of the CIA can still be appreciable at various depths.
The contribution of the molecular line opacities relative to
the CIA and the continuum opacity can also be seen from Figs. 5
to 7, although the curve representing the line opacities must be
interpreted a bit more cautiously, because the plotted molecular
line opacities are necessarily the average of the opacities over a
given wavelength range, inside which the value of the true line
opacity may fluctuate very much up and down. Qualitatively, it
is seen that the molecular features show up in the spectrum when
(and where) the average molecular opacity is large compared to
the continuum and CIA opacities. It is also, qualitatively, seen
from these plots that when the CIA opacity is much larger than
the (average) molecular line opacity (such that the sum of the
two opacities is almost identical to the CIA opacity alone), then
the spectrum appears smooth and featureless. It can be difficult observationally to recognize the contribution of the CIA
to such a spectrum, unless the colours and/or the flux distribution over a substantial wavelength range is compared with
the predicted synthetic flux distribution. As opposed to models computed without CIA, it is nevertheless easily seen from
Figs. 5 to 7 that the predicted low-resolution spectra of stars
over a considerable region in low-metallicity HR diagrams will
be almost featureless throughout the infrared.
By comparing columns two and three in each row of the figures (corresponding to a given model) it is seen that the collision
induced opacity increases with increasing depth, which is due
to the increased number density (of H2 and He). The UV/blue
continuum opacity (due to bound-free edges of neutral atoms) is
almost independent of optical depth (and choice of fundamental
stellar parameters) as long as both the metallicity and effective
temperature are relatively low. The visible and infrared continuum opacities, on the other hand, increase markedly with
increasing depth in the model, because of the increasing number of free electrons available to form H− . The molecular line
opacity (e.g., per unit mass of stellar material) is proportional
to the number of molecules (of a given type) in the gas. This
number will in general be a function of temperature as well as of
gas pressure. The higher pressures at larger depths will favour
larger molecules, but at the same time the increasing temperature will act in the opposite direction. In the infrared, where
the opacity of water dominates the line opacity, the net effect
is seen to be an almost unchanged opacity, with the individual bands, however, being considerably broader at large depths
due to the increased contribution of hot bands and high excitation lines at high temperatures. The relative distribution between
polyatomic molecules (mainly H2 O) and diatomics (mainly TiO
and CH) is very sensitive to temperature, the diatomics being
strongly favoured at the larger optical depths where the temperature is highest. In particular, it is seen that the contribution of CH
(around the G-band at 4300 Å) increases dramatically relative
to the water-bands with increasing depths. For low metallicities, the opacity of CH increases in importance relative to TiO
(because it contains only one “metal”) as is seen from Fig. 6.
Also, CH is more important relative to TiO at high effective
temperatures than at low (Fig. 7), whereas the dependence on
gravity is less pronounced (Fig. 5).
6. Conclusions
We have computed the absorption strength at frequencies corresponding to a number of collision-induced transitions in the
H2 -H2 complex, which are of relevance at the physical conditions that prevail in stellar atmospheres and for which quantum
mechanical results did not exist previously. We have also updated the existing data for H2 -H2 and H2 -He with estimates of
the intensity in the far spectral wings, and with data for additional overtones and hot bands, so that the whole set of data is
more complete for stellar atmosphere computations.
We have applied the new data set in computations of a grid of
cool, low-metallicity stellar atmospheres, in order to investigate
the impact of CIA processes on such models. We found that CIA
is the dominant opacity source in the coolest, low-metallicity
main sequence stars. The effect on the flux distribution is pronounced for dwarfs of low metallicity and Teff <
∼ 4000 K, for
<
all dwarfs with Z <
∼ 0.1 Z as long as Teff is low enough (Teff ∼
3000 K), and for dwarfs and sub-giants if both the metallicity
and the effective temperature are low enough.
This group of stars includes cool white dwarfs, metal-poor
brown dwarfs, M dwarfs in the halo and in globular clusters,
sub-giants in the most metal-poor globulars and in metal-poor
external galaxies such as the SMC and the Fornax dSph, and
A. Borysow et al.: Impact of CIA on stellar atmospheres
metal-poor carbon dwarfs. For analysis of the structure, colours
and spectra of such stars inclusion of CIA is essential.
For several of the collision-induced transitions much more
accurate modelling than we have presented here is, however,
still required, and there exists no laboratory work for comparison with the the quantum mechanical results at elevated temperatures. For the H2 -H CIA transitions, which might likewise be
of importance for stellar atmospheres, neither computations nor
laboratory measurements exist. Although our results represent
the most detailed study of the CIA phenomena in stars, we point
out that they are still very limited and preliminary compared to
what is known about atoms and dipole transitions in single polar
molecules and their role for the radiative transfer in stars.
Acknowledgements. We acknowledge the support of NATO Collaborative Research Grant 941197. The necessary computing capacity was
due to support from the Carlsberg Foundation. AB acknowledges support from the Danish Research Academy and the Danish Natural Science Research Council. CZ acknowledges partial support from NASA,
Planetary Atmospheres Division. Valuable comments from Hollis R.
Johnson are greatly appreciated.
References
Borysow A. 1994, In: U.G. Jørgensen (ed.), Molecular Opacities in the
Stellar Environment, Springer Verlag, p. 209
Borysow A., Frommhold L. 1989, ApJ 341, 549
Borysow A., Frommhold L. 1990, ApJL 348, L41
Borysow A., Frommhold L., Moraldi M. 1989, ApJ 336, 495
Borysow J., Trafton L., Frommhold L., Birnbaum G. 1985, ApJ 296,
644
Borysow J., Frommhold L., Birnbaum G. 1988, ApJ 326, 509
Goorvitch D., Chackerian C.Jr. 1994, ApJS 91, 483
Gustafsson B., Jørgensen U.G. 1994, A&AR 6, 19
Gustafsson B., Olander N. 1979, Physica Scripta 20, 570
Gustafsson B., Bell R.A., Eriksson K., Nordlund Å. 1975, A&A 42,
407
195
Helling Ch., Jørgensen U.G., Plez B., Johnson H.R. 1996, A&A 315,
194
Herzberg G. 1952, ApJ 115, 337
Jørgensen U.G. 1994, A&A 284, 179
Jørgensen U.G., Jensen P. 1993, J. Mol. Spectr. 161, 219
Jørgensen U. G., Larsson M. 1990, A&A, 238, 424
Jørgensen, U. G., Johnson, H. R., Nordlund, Å. 1992, A&A 261, 263
Jørgensen U.G., Larsson M., Iwamae A., Yu B. 1996, A&A 315, 204
Kolos W., Wolniewicz L. 1965, J. Mol. Spectra 43, 2429
Kolos W., Wolniewicz L. 1967, J. Chem. Phys. 46, 1426
Kolos W., Wolniewicz L. 1968, J. Chem. Phys. 49, 404
Kolos W., Wolniewicz L. 1975, J. Mol. Spectra 54, 303
Langhoff S.R., Bauschlicher C.W.Jr. 1994, In: U.G. Jørgensen (ed.),
Molecular Opacities in the Stellar Environment, Springer Verlag,
p. 310
Lenzuni P., Chernoff D.F., Salpeter E.E. 1991, ApJS 76, 759
Lenzuni P., Saumon D. 1992, Rev. Mexicana Astron. Astrof. 23, 223
Linsky J.L. 1969, ApJ 156, 989
Meyer W., Frommhold L., Birnbaum G. 1989, Phys, Rev. 39A, 2434
Meyer W., Borysow A., Frommhold L. 1993, Phys. Rev. A 47, 4065
Meyer W., Frommhold L., Birnbaum G. 1989, Phys, Rev. 39A, 2434
Mihalas D. 1978, Stellar Atmospheres, Freeman
Mould J., Liebert J. 1978, ApJ 226, L29
Palla F. 1985, In: G. H. F. Diercksen et al. (eds.), Molecular Astrophysics, D. Reidl Publ. Co., p. 687
Patch R.W. 1971, JQSRT 11, 1331
Plez B., Brett J.M., Nordlund Å. 1992, A&A 256, 551
Poll J. D., Wolniewicz L. 1978, J. Chem. Phys. 68, 3053
Ross M., Ree F. H., Young D. A. 1983, J. Chem. Phys. 79, 1487
Saumon D., Bergeron P., Lunine J.I., Hubbard W.B., Burrows A. 1994,
ApJ 424, 333
Scholz M., Wehrse R. 1994, In: U.G. Jørgensen (ed.), Molecular Opacities in the Stellar Environment, Springer Verlag, p. 49
Shipman H. L. 1977, ApJ 213, 138
Stahler S.W., Palla F., Salpeter E.E. 1986, ApJ 302, 590
Tsuji T. 1969, In: S.S.Kumar (ed.), Low Luminosity Stars, (New York:
Gordon and Breach Science Publ.), p. 457
Zheng C., Borysow A. 1995a, ApJ 441, 960
Zheng C., Borysow A. 1995b, Icarus 113, 84
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