model atmospheres of cool, low-metallicity stars: the importance...

Astron. Astrophys. 324, 185–195 (1997) ASTRONOMYAND

ASTROPHYSICS

Model atmospheres of cool, low-metallicity stars:the importance of collision-induced absorptionAleksandra Borysow1,2, Uffe Grae Jørgensen1, and Chunguang Zheng2

1 Niels Bohr Institute, Copenhagen University Observatory, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark2 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 inducedabsorption (CIA) in the high-temperature regime applicableto stellar atmospheres. Improvements of existing data includecomputation of series of hot-bands and extension of the spectralwings much further away from the maxima than hitherto. Com-putation of new bands includes the first and the second overtonebands of H2-H2. We apply these data to an extensive grid ofoxygen-rich model atmospheres with scaled solar metallicitiesin order to investigate for which range of fundamental stellarparameters (i.e., effective temperature, gravity, and chemicalcomposition) CIA has a significant impact on the atmosphericstructure. Besides the CIA due to H2-H2 and H2-He pairs, ourmodels include complete molecular line data for TiO, H2O, CN,CH, and SiO.

For stellar models with low effective temperatures, highgravity, and low metallicity, the atmospheric structure and theemergent spectrum are completely dominated by the effects ofCIA. For our test-models of lowest effective temperature andlowest metallicity (Teff = 2800 K and Z = 10−3 Z�) the effectof 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 effectsare visible in the overall flux distribution for effective temper-atures as high as 4000 K, and for models with Teff = 2800 Kand log(g) = 5.0 CIA has effects on the spectrum of stars withmetallicities as high as 0.1 Z�.

Key words: molecular data – molecular processes – stars: at-mospheres – stars: late-type – stars: low-mass – infrared: stars

1. Introduction

In homonuclear molecules, such as H2, there is no change ofdipole moment during rotation or vibration, and such moleculesare therefore unable to absorb (or emit) dipole radiation. How-ever, 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 rela-tively weak dipole absorption becomes possible.

Herzberg (1952) was the first to bring collision induced ab-sorption (CIA) into an astrophysical context, and suggested thismechanism as being responsible for some hitherto unexplainedabsorption bands in the spectrum of Uranus. By comparing lab-oratory H2 spectra with spectra of Uranus and interpreting thebands as due to CIA, Herzberg derived the first limits on thepressure of the thick Uranian cloud top, and produced one ofthe first pieces of spectroscopic evidence that hydrogen was theprimary constituent in Uranus. Today CIA processes are part ofthe standard modelling of the low-temperature, dense gases inthe atmosphere of the giant planets and of Titan and Venus.

Linsky (1969) developed a simplified analytic description ofa few bands of H2-H2 and H2−He CIA as a function of frequencyand temperature, and was the first to point out that the pseudo-continuous character of the CIA could cause it to have a largeeffect also in stellar atmospheres. Here it will block the energywhich would otherwise have escaped in the transparent spectralregions between the absorption lines. This will be particularlytrue when polar absorbers (e.g., H2O, H−, TiO, etc) are depleted,as for example in cool stars of low metallicity.

In the same year, Tsuji (1969) considered molecular opac-ities in cool stellar atmospheres, including various sources ofcontinuous opacity. Independently, using his own estimation ofCIA intensity, he reached the same conclusion that CIA is animportant source of opacity at high pressures, which can not beneglected.

In the following decade CIA in stellar atmospheres attractedmore attention. Shipman (1977) computed the first white dwarfmodel atmosphere composed of pure H2, which included Lin-sky’s data. He found that at a temperature of 4000 K CIA con-tributes essentially all the opacity at the wavelengths where theflux is emitted. At higher temperatures other opacity sourceswere dominant. Also Mould & Liebert (1978) have includedCIA due to H2–H2 and H2–He pairs, to compute new whitedwarf model atmospheres, again using Linsky’s data. They re-iterated the importance of CIA, but no specific results wereshown.

186 A. Borysow et al.: Impact of CIA on stellar atmospheres

Palla (1985) pointed out, for the first time, that, dependingupon gas density, CIA may be an essential, and often even thedominant source of opacity, in primordial protostars at temper-atures between 1000 and 7000 K. In Palla’s work, atmospherescomposed of hydrogen and helium were considered and Lin-sky’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 collisioninduced absorption cannot be neglected.

Based on the then existing quantum mechanical CIA mod-els due to Borysow and collaborators (Borysow 1994 and refer-ences therein; see also summary below), Lenzuni et al. (1991)and Saumon et al. (1994) studied the impact of CIA on hypothet-ical zero-metallicity, primordial proto-stars and zero-metallicitybrown dwarfs, and found very large effects.

We have now extended these computations of CIA due toH2-H2 and H2-He into the higher temperature regime appropri-ate for stellar atmospheres, and studied their impact on realisticatmospheric models of existing stars. In Sect. 2 we summarizethe existing quantum mechanical CIA data and in Sect. 3 wepresent the result of our new calculations. In Sect. 4 we com-pare our results with previous data, where available, and givesome warnings about the limitations of our data and of previousdata. In Sect. 5 we describe the impact of our data on stellarmodel atmospheres, and quantify in which region of the HRdiagram (and metallicity) the collision induced absorption isimportant for the stellar atmospheric structure and for the anal-ysis of photometric and spectroscopic data.

Our data are applicable to stars of arbitrary chemical compo-sition, but in this paper we will restrict the discussion to oxygen-rich stars. We cover the region in fundamental stellar parameterswhich represents brown dwarfs, all types of K and M dwarfs,as well as the metal-poor sub-giants and giants that could befound in the Galactic halo, in globular clusters, and in metal-poor galaxies such as the Fornax dSph and the SMC. In laterpapers we will discuss carbon dwarfs and white dwarfs, whichwill require additional input data. Our CIA data do not includeH2−H pairs, which could also contribute significantly to theopacity, and we have restricted the inclusion of bound−boundmolecular line transitions to those molecules for which com-plete, high-quality quantum mechanical computations exist. Inparticular, it may be desirable to include additional diatomichydrides in the discussion. In future work we hope to be able toextend the data toward a higher degree of completeness. Alsodust formation may be of relevance for the very coolest objects.

2. CIA opacities from existing quantum mechanical sources

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 specificallyfor astrophysical applications. It provides absorption intensi-ties at temperatures from 600 to 7500 K. We based our modelon the available induced dipole functions due to Meyer et al.(1989), designed to work well at low temperatures (i.e. at mod-

Table 1. Population probability of the ground vibrational state of H2

at temperatures between 1000 and 7000 K.

Temperature(K) P (v = 0)1000 0.9932000 0.9443000 0.8484000 0.7505000 0.6676000 0.5957000 0.535

erate intermolecular distances), and on the isotropic, effectiveH2–H2 interaction potential by Ross et al. (1983). We have up-graded the dipole data to meet the demands of the high temper-ature predictions of CIA. The model accounts for hot bandsinvolving ∆(v1) = ∆(v2) = 0 vibrational transitions withv1, v2 = 0, 1, 2, 3, which are populated at temperatures below7000 K. Variables v1 and v2 denote vibrational quantum num-bers 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 to5000 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 ex-pected to be present. We tried to correct for this fact, although itis apparent that an updated model is necessary. An extrapolationof the models to temperatures other than those they are designedfor can give very undependable results. We have therefore cho-sen to use the results for 5000 K to represent opacities of the0 → 1 transition also at 6000 and 7000 K. However, in order toaccount for the missing transitions (giving rise to the hot bands)with ∆(v) = v′ − v = 1, corresponding to v → v′ = v + 1, withv > 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 → 1transition by dividing them by the probability of the populationof v = 0 state at each temperature (see P (v), Table 1).

Such a procedure is correct if we assume that the shapeand 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 proceduresignificantly improves the data over those available from theexisting program.

2.3. H2-He roto-translational band

The existing model of the 0 → 0 band predicts CIA intensi-ties of this band at temperatures below 3000 K (Borysow et al.1988). In the absence of any newer data we have applied sim-ilar rescaling procedure as we did for the fundamental band ofH2-H2. Again, initially we have computed intensities at all tem-peratures up to 3000 K, and then copied the results for 3000 Kto all higher temperatures. Next, in order to account for the hot

A. Borysow et al.: Impact of CIA on stellar atmospheres 187

bands, we divided those intensities byP (v = 0). We expect quitelarge uncertainties associated with this procedure, but we notethat the far infrared intensities, corresponding to the ∆(v) = 0band, are less important than those in the near infrared. In ad-dition, the H2-He intensities are expected to matter much lessthat those due to H2-H2, on account of lower abundance of he-lium in the stellar atmospheres. Also, CIA usually matters lessat high temperatures, so the inaccuracy of the high temperaturepredictions is less relevant. A better CIA model of this band iscertainly being called for.

2.4. H2-He fundamental band

The models of all relevant vibrational transitions correspondingto the fundamental band frequency region are available for theH2-He complex. The models are applicable for all temperaturesup to 7000 K, and involve ∆(v) = 1 transitions, including the0 → 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) attemperatures up to 7000 K. We have incorporated all of theminto 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 appli-cable at temperatures from 1000 to 7000 K.

For stars with effective temperatures between 3000 and4000 K, the maximum of the black body radiation is emittedbetween about 10000 and 14000 cm−1. The centre of the firstovertone CIA band of H2–H2 falls around the frequency∼ 8000cm−1, and that of the second overtone band, at∼ 12000 cm−1.The second overtone band is extremely weak compared to thefirst overtone (< 3% for peak intensity). At the highest temper-atures (∼ 7000 K), the peaks of the second overtone band areeven smaller than the far wings of the first overtone band at thesame frequencies. Thus we have extended the wings of the firstovertone band up to 25000 cm−1, and predict that overtoneshigher than the second are not important. Since our analyticalmodels are not designed to work well in the far wings (roughlyat intensities < 1% peak intensity), the uncertainties of the ex-tended 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) wereused as input to produce the lowest three semi-classical spectralmoments. These moments were then used to model the H2–H2 CIA spectra by the model lineshapes, used successfully to

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 interactinghydrogen molecules v1, v2 = 0, 1 and 2. In this band, there aretwo different kinds of vibrational transitions present: (i) singletransition, ∆(v1) = 2, ∆(v2) = 0 (or ∆(v1) = 0, ∆(v2) = 2);and (ii) double transitions: ∆(v1) = ∆(v2) = 1. Both kinds ofvibrational transitions fall in the same frequency region centredaround 8000 cm−1.

Dipolar terms (Meyer et al. 1993) λ1λ2ΛL = 2023, 0223,2021, 0221 and 2233 have been included for the single vibra-tional transition, and λ1λ2ΛL= 2023, 0223, 0001, 2021, 0221and 0445 for the double vibrational transitions. The other termswere found to contribute less than 2% of the total intensity.

The induced dipole functions also depend on the rotationalstates of the two interacting molecules. We account for suchdependencies only for the largest, λ1λ2ΛL = 2023 and 0223,terms (for both single and double transitions), which contributemore than 70% to the total intensity. Since these dependenciesare available only for the lowest three J values (Meyer et al.1993), we have scaled the entire β(R) functions according totheir long range asymptotic forms, in the same way as describedpreviously (Zheng & Borysow 1995b):

β2023(R) →√

3 < v1J1|α|v′1J ′1 >< v2J2|Q|v′2J ′2 > /R4 (1)

and

β0223(R) → −√

3 < v2J2|α|v′2J ′2 >< v1J1|Q|v′1J ′1 > /R4,(2)

with R being the H2–H2 intermolecular distance. We havecomputed the matrix elements of the electronic polarizabil-ity 〈vJ |α(r)|v′J ′〉 and quadrupole moment 〈vJ |Q(r)|v′J ′〉 ofthe hydrogen molecule, for all needed J values, based on thefunctions α(r) (Kolos & Wolniewicz 1967), Q(r) (Poll & Wol-niewicz 1978), and the H-H potentialV (r) (Kolos & Wolniewicz1965, 1968, 1975), with r being the H-H internuclear distance.

It is difficult to predict the accuracy of our model. Over thefrequency range where the spectral intensities are larger than 1%of the peak intensity, the accuracy mainly depends on our se-lection of the “effective” isotropic H2–H2 potential (Ross et al.1983); the word “effective” is being used to indicate that it ef-fectively accounts for the anisotropy, and the v-dependence ofthe H2–H2 interaction and is expected to describe accuratelyH2-H2 interactions at high temperature. According to the dis-cussion in our previous work (Zheng & Borysow 1995b), weexpect the accuracy to be better than 50% at this frequencyrange. However, the uncertainty becomes less predictable inthe far wings where the spectral intensities fall below 1% ofthose of the peak. Among various reasons we name the use ofthe model lineshapes, which become less predictable in the farwings, the dependence of the far wings on the dipole functionsat short range (largely uncertain), and the use of the effectiveintermolecular potential rather than the real one. We expect the


Fig. 1. CIA opacities of H2–H2 at temperatures between 1000 and7000 K.

far wings to be accurate within an order of magnitude. Furtherwork on this band is in progress (Zheng, Fu, Borysow 1996, inpreparation).

3.2. The second overtone band

The prediction of the second overtone band H2–H2 CIA spectrapresents an even greater challenge, due to the lack of detailedknowledge of the induced dipole functions. Vibrational tran-sitions with ∆(v1) = 0, 1, 2, 3 and ∆(v2) = 3 − ∆(v1) wereincluded (each of them falling into the same frequency regioncentred around 12000 cm−1), and both interacting hydrogenmolecules were assumed to be in the initial ground vibrationalstate (v1, v2 = 0).

The ab initio dipole functions are not available for this band.Since the λ1λ2ΛL = 2023 and 0223 terms contribute more than70% to the total intensity in the first overtone band, we includedonly λ1λ2ΛL = 2023 and 0223 in our model of the second over-tone 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 intermoleculardistances), the importance of the electronic overlap increases.In order to correct for this unknown short-range behaviour, wehave scaled the known dipole functions for the v1 = 0 → 1 andv2 = 0 → 1 transitions (of the first overtone band) so that wematched the long-term dipole magnitudes given by Eqs 1 and2 for the second overtone. Next, we proceeded with the samemodelling procedure as we have used for the first overtone.

The assumption of v1, v2 = 0 introduces additional inaccu-racy, which we consider minor, in view of other uncertaintiesinvolved, like the choice of the H2–H2 potential, or scaling ofthe dipole functions. We estimate that the model for this bandis 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 and7000 K.

from 1000 to 7000 K. At lower temperatures separate vibra-tional bands are identifiable, but with increasing temperaturethe spectral bands broaden and result in one featureless con-tinuum. Both H2–H2 and H2–He opacities show the maximumof intensity at frequencies around 5000 cm−1. We observe avery wide range of intensities over the frequency band from 0to 20000 cm−1.

4. Comparison with previous data

Although quantum mechanical computations of CIA have beenavailable at high temperatures for some time, they have beenused in stellar studies only recently. In more widespread useare Linsky’s (1969) analytical expressions for CIA transitions– sometimes applied in combination with a limited set of quan-tum mechanical results. Linsky’s data include estimates of theroto-translational band of H2-H2 at temperatures from 600 to3000-4000 K, the fundamental roto-vibrational band at temper-atures up to 3000 K, and its first overtone band for temperaturesup to 4000 K. No hot bands or higher overtones are included.Linsky’s treatment of the 0 → 1 roto-vibrational band has beensuperseded by the ab initio computations of Patch (1971). As al-ready stressed by these authors, the adopted lineshapes are, how-ever, modelled by highly ‘ad hoc’ analytical functions that aredesigned to reproduce the desired absorption intensities withinstrictly limited ranges of temperatures and frequencies – let’ssay where the intensity of each band falls off by one order ofmagnitude compared to its peak value. In this sense Linsky’sdata have not been intended for use for wavenumbers above ν≈ 10 000 cm−1 (i.e., λ<∼ 1 µm), or for temperatures outside therange described above. In addition, Linsky’s opacities given forH2-He rely on simple rescaling of the absorption coefficientsof 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, in-tended for a rather limited range in frequency and tempera-ture, simple programming of their analytical expressions willof course give answers also outside this range, and great cau-


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tion should obviously be taken in the interpretation of resultsbased on such “extrapolations”. For example, the default con-tinuum opacities in recent versions (e.g., Jørgensen et al. 1992,Plez et al. 1992) of the widely used marcs code include Lin-sky’s roto-translational band and Patch’s fundamental band (thefirst overtone is missing). The expression for these two bandsshould be used only between 0 and ≈ 9 000 cm−1. In defaultcomputations they will, however, be included at any frequencyrange, and any temperature demanded by the program. In Fig. 3we demonstrate the difference between our new quantum me-chanical CIA opacities as applied in the marcs code and theLinsky/Patch-continuum CIA opacities as given by the marcscode, when computed at 4000 K at the frequency range from0 to 20 000 cm−1. It is easily seen that at higher frequencies,overlapping well with the black body radiation flux, the databy Linsky and Patch overestimate the realistic opacities by asmuch as three orders of magnitude. In fact, numerical experi-ments showed that the Linsky/Patch data may often give artifi-cially larger effects on the model atmosphere, because of suchan erroneous extrapolation, than will application of the morerealistic quantum mechanical CIA, in spite of the fact that thevalues of the Linsky/Patch data are considerably lower aroundthe maximum CIA absorption intensity. We would like to drawthe attention of all users of analytical CIA expressions in anyatmospheric code to this point; they should examine carefullytheir inputs, and have in mind the comparison in Fig. 3 and thelimitations in analytical expressions alluded to above.

We would also like to extend this warning to all other casesof indiscriminate use of all analytical opacity models which aredesigned to reproduce certain values within a given (and tested)range. As another example, we mention the use of the model ofroto-translational CIA for H2 pairs by Borysow et al. (1985), de-signed to model CIA intensities at temperatures up to 300 K, but

instead used (for example in the work by Lenzuni et al. 1991) tomodel CIA opacities at 3000 K. The user needs to have in mindthat whereas authors who publish their opacity data can makecertain that the selected analytical, multi-parametric spectrallineshapes reproduce well their quantum mechanical computa-tions within certain range of temperatures and frequencies, thereis no attempt made, and in fact it is highly unlikely, that the samelineshapes will give correct results at temperatures different byone order of magnitude from those tested. Therefore, such “ex-trapolation” procedures will inadvertently lead to unpredictableresults and should be strictly avoided.

Our Figs. 1 and 2 above show a very large range of intensi-ties, up to five orders of magnitude for H2–H2 and 3–5 ordersfor H2-He. As we mentioned above, analytical lineshapes usedby us to model spectral lineshapes reproduce real profiles wellwithin the 1:100 range of intensities. Their far wing behaviour israther unpredictable, though on some occasions they were seento perform impressively well over extended frequency/intensityrange. We need to remind the reader that whereas at shorterfrequencies the spectral bands overlap each other, making thefar wings more irrelevant in the presence of the next, more in-tense band, the intensities at frequencies higher than those dueto the first and the second overtones (ν > 15000 cm−1) maybe prone to more uncertainties. Thus, even though great carehas been placed on making the present models as realistic aspossible under current circumstances, it needs to be understoodthat uncertainties of such models are also unavoidable.

5. Impact of CIA on stellar model atmospheres and syntheticspectra

We have included the CIA data described above in the computa-tion of a grid of photospheric models. The aim is to identify therange of the fundamental stellar parameters (Teff , log(g), andmetallicity) within which CIA affects the stellar atmosphere,and to quantify the effect CIA has on such models. The modelatmosphere 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 equilib-rium, spherical geometry (applied when appropriate), and lineblanketing by molecules treated by the opacity sampling tech-nique. Line opacities were included for a total of approximately20 million molecular lines of H2O (from Jørgensen & Jensen1993), TiO (from Jørgensen 1994), CO (from Goorvitch &Chackerian 1994), SiO (from Langhoff & Bauschlicher 1994),CN (from Jørgensen & Larsson 1990), and CH (from Jørgensenet al. 1996). Spectra were computed as the emergent flux ofthe model computation as well as in a separate synthetic spec-trum program which allows us to study the contribution of eachopacity source separately.

We expect the effect of CIA to be largest for stars of loweffective temperature (corresponding to a small number den-sity of free electrons and a large abundance of molecular hy-drogen), high gravity (corresponding to high density in the at-mosphere and therefore a large number density of H2-H2 andH2-He “pairs”), and low metallicity (corresponding to a small


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Fig. 4. The contributions of various opacity sources to the spectrum ofa stellar model with Teff = 2800 K, log(g) = 5.0, and Z = 10−2 Z� – atypical main sequence member of a globular cluster. The model com-putation includes opacities of continuum sources, molecular lines, andCIA. 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 under-lying model atmosphere, but the difference between the three spectraillustrates the relative contribution of the three sources of opacity.

amount of other absorbers). Therefore we started the computa-tions with low-metallicity dwarf star models, such that our gridof stars will span the coolest and most metal-deficient main se-quence stars of our Galaxy, as for example cool M dwarf starsin 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 theopacity at all wavelengths longer than approximately 1 µm andfor 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 moresignificant effect of CIA in the emergent flux spectrum. In thisway we defined the region of interest for CIA inclusion. InFig. 4 we show the contribution of different absorbers to thesynthetic spectrum of a typical model in our grid, representinga typical M dwarf in a globular cluster. The figure shows thecontinuum, the molecular spectrum, and the spectrum includingboth molecular lines and CIA, respectively. The latter spectrumis the only one of the three which is internally consistent withthe underlying model in the sense that it includes all the opacitysources which are also included in the corresponding modelatmosphere computation. Comparison of the three spectra onlyillustrates the contribution of the different species to the totalemergent spectrum. It is seen that even at such relatively highmetallicities as adopted here (Z = 0.01 Z�), CIA is the maincontributor to the infrared stellar spectrum. Its presence inducessuch a strong continuum depression that the bands longwardof 1.5µm (due mainly to water) virtually disappear from thespectrum. The strongest molecular features left in the spectrum

are the TiO bands, the water bands shortward of 1.5µm, and theG band (around 4300A) due to CH.

The results of our analysis of the stellar models in the wholerange of fundamental parameters where we found CIA to beof importance are summarized in Figs. 5 to 7. Fig. 5 illustratesthe effect of varying the gravity, whereas Fig. 6 and Fig. 7 showhow the relative importance of CIA, continuum absorption, andmolecular line absorption change when we vary the metallicityand the effective temperature, respectively.

In each of the Figs. 5 to 7, the first column shows the emer-gent flux spectra computed based on models respectively withand without CIA included in the opacity. Here all the spectraand the underlying model atmospheres are internally consistent,in the sense that the spectrum based on the atmospheric struc-ture computed without CIA also itself is without CIA, and viceversa. Hence these are the complete, self-consistent spectra onewould predict based on models respectively with and withoutCIA included in the computations.

Each row of plots in Figs. 5 to 7 represents a given set offundamental parameters. Whereas the first column illustrates theresults of models respectively with and without CIA, the twonext columns concentrate only on the models with CIA includedin the model calculations. The plots in column two correspondto the depth in the atmosphere where τross = 0.01, whereas plotsin column three correspond to depths where τross = 1.0. Bothcolumns show the total opacity (in units of cm2 per gram ofstellar material) due to CIA (dotted lines), due to the sum of allother continuum sources (dashed lines), and due to the sum ofall bound-bound molecular line transitions (full drawn lines).

It is seen from the spectra that the major effect of the CIA isto absorb energy in the infrared and to re-emit it at visible wave-lengths. For the model shown in Fig. 4, this re-distribution ofthe flux is so pronounced that the infrared (J−H, H−K) colourschange from (0.6, 0.3) to (0.3, −0.1) when CIA is included inthe model atmosphere and the corresponding synthetic spec-trum. The B−V colour, on the other hand, is only very littleaffected. The effect of CIA on the emergent infrared spectrumis still substantial even for the cool low-metallicity sub-giantmodel (i.e., log(g) = 2.0) in Fig. 5. The impact of CIA is alsoseen in the Z = 0.1 Z� cool dwarf model in Fig. 6, and in thespectrum of the Teff = 4200 K low-metallicity dwarf model ofFig. 7. These models define the boundary-region of the grid ofmodels where CIA seems to be important for the spectrum. Theeffect 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 shownresolution), is seen in several of the flux diagrams of Figs. 5 to7. The intensity of the G-band due to CH, on the other hand,is either unchanged or even increased (due to the increased re-emitted flux in the blue spectral region) when CIA is includedin the opacities.

For the cooler, low-metallicity, dwarf models, the impact ofCIA on the spectral distribution is seen to be dramatic, and itis much larger than the impact of other molecular species. It iscaused by the relative heating and cooling effect which CIA hason the atmospheric structure. General features of atmospheric


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flux

log(g)=2.0

τ-ross = 0.01

-8

-6

-4

-2

0

log(

opac

ity)

τ-ross = 1.0

0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

flux

log(g)=1.0

0.5 1.0 1.5 2.0 2.5 3.0 Wavelength (µm)

τ-ross = 0.01

0.5 1.0 1.5 2.0 2.5 3.0

-8

-6

-4

-2

0

log(

opac

ity)

τ-ross = 1.0

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 columnof plots in this figure shows the emergent flux from models with (highest peak) and without (lowest peak) CIA included in the opacity. The twonext columns show log10 of the opacity (in units of cm2 per gram of stellar material) due to various species (for those models from column onewhere CIA is included) at τross = 0.01 (second column of plots) and at τross = 1.0 (third column), respectively. Dotted lines correspond to theopacity of CIA, dashed lines represent the sum of the opacities due to all continuum sources other than CIA, and the full drawn lines representthe molecular line opacity. The different rows of plots in the figure correspond to models of different gravity, decreasing from log(g) = 5.0 inthe uppermost row of plots to log(g) = 4.0, 2.0 and 1.0, respectively, in the rows below.


0

1

2

3

4flu

x Z/Zo=0.0001

τ-ross = 0.01

-8

-6

-4

-2

0

log(

opac

ity)

τ-ross = 1.0

0

1

2

3

flux

Z/Zo=0.001

τ-ross = 0.01

-8

-6

-4

-2

0

log(

opac

ity)

τ-ross = 1.0

0.00.5

1.0

1.5

2.0

2.5

3.0

flux

Z/Zo=0.01

τ-ross = 0.01

-8

-6

-4

-2

0

log(

opac

ity)

τ-ross = 1.0

0.5 1.0 1.5 2.0 2.5 3.00.00.5

1.0

1.5

2.0

2.5

3.0

flux

Z/Zo=0.1

0.5 1.0 1.5 2.0 2.5 3.0 Wavelength (µm)

τ-ross = 0.01

0.5 1.0 1.5 2.0 2.5 3.0

-8

-6

-4

-2

0

log(

opac

ity)

τ-ross = 1.0

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� inthe 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-distributionare described in standard text-books on stellar atmospheres (e.g.,Mihalas 1978) and they have been the subject of detailed discus-sion in several papers (e.g., Gustafsson & Olander 1979, Scholz& Wehrse 1994, Gustafsson & Jørgensen 1994). The partic-ularly large impact of CIA is due to its substantial pressuredependence which results in a backwarming in the very deep

atmospheric layers where the continuum is formed (as opposedto the effect of, for example, H2O, where the main effect is in themore shallow atmospheric layers above the continuum formingregion). 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


0

1

2

3flu

x Teff = 2800

τ-ross = 0.01

-8

-6

-4

-2

0

log(

opac

ity)

τ-ross = 1.0

01

2

3

4

5

6

flux

Teff = 3400

τ-ross = 0.01

-8

-6

-4

-2

0

log(

opac

ity)

τ-ross = 1.0

0

2

4

6

8

10

flux

Teff = 3800

τ-ross = 0.01

-8

-6

-4

-2

0

log(

opac

ity)

τ-ross = 1.0

0.5 1.0 1.5 2.0 2.5 3.002468

101214

flux

Teff = 4200

0.5 1.0 1.5 2.0 2.5 3.0 Wavelength (µm)

τ-ross = 0.01

0.5 1.0 1.5 2.0 2.5 3.0

-8

-6

-4

-2

0

log(

opac

ity)

τ-ross = 1.0

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 increasingfrom 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) inthe layers around log(τross) = −2. A corresponding heating of200 to 300 K results in the deeper layers where log(τross) >∼ 0and where most of the continuum is formed. This substantialheating increases the continuum flux at the shorter wavelengthsand gives rise to the marked shortward shift of the wavelengthsof the emergent flux spectrum. The corresponding model withTeff = 3800 K shows much less surface cooling and backwarm-

ing from CIA than do the Teff = 2800 K model, and, as is alreadyseen 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 onthe Teff = 2800 K model is, for a given temperature, to lowerthe gas pressure by about one order of magnitude (in the deeperatmospheric layers), or for a given gas pressure to increase thetemperature with approximately 1000 K.


-6 -4 -2 0 2 log(τ

ross )

1500

2000

2500

3000

3500

4000T

empe

ratu

re [K

]

T=2800, CIAditto, no CIAT=3800, CIAditto, no CIA

logg C/O logZ/Zo 5.0 0.43 0.001

Fig. 8. Effect on the T−τross structure of including CIA in models oflog(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 spectralchanges can be obtained by comparing the various contribu-tions 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 eachof the three figures, the contribution of CIA relative to the con-tinuum sources (mainly H−) decreases “downward” along thefigure (i.e., for decreasing gravity, increasing metallicity, andincreasing effective temperature, respectively), until CIA is nolonger the dominant opacity source. For such models, inclusionof CIA will no longer change the shape of the emergent fluxfrom the shape of the continuum spectrum, although the totalopacity of the CIA can still be appreciable at various depths.

The contribution of the molecular line opacities relative tothe CIA and the continuum opacity can also be seen from Figs. 5to 7, although the curve representing the line opacities must beinterpreted a bit more cautiously, because the plotted molecularline opacities are necessarily the average of the opacities over agiven wavelength range, inside which the value of the true lineopacity may fluctuate very much up and down. Qualitatively, itis seen that the molecular features show up in the spectrum when(and where) the average molecular opacity is large compared tothe continuum and CIA opacities. It is also, qualitatively, seenfrom these plots that when the CIA opacity is much larger thanthe (average) molecular line opacity (such that the sum of thetwo opacities is almost identical to the CIA opacity alone), thenthe spectrum appears smooth and featureless. It can be diffi-cult observationally to recognize the contribution of the CIAto such a spectrum, unless the colours and/or the flux distri-bution over a substantial wavelength range is compared withthe predicted synthetic flux distribution. As opposed to mod-els computed without CIA, it is nevertheless easily seen fromFigs. 5 to 7 that the predicted low-resolution spectra of starsover a considerable region in low-metallicity HR diagrams willbe almost featureless throughout the infrared.

By comparing columns two and three in each row of the fig-ures (corresponding to a given model) it is seen that the collision

induced opacity increases with increasing depth, which is dueto the increased number density (of H2 and He). The UV/bluecontinuum opacity (due to bound-free edges of neutral atoms) isalmost independent of optical depth (and choice of fundamentalstellar parameters) as long as both the metallicity and effectivetemperature are relatively low. The visible and infrared con-tinuum opacities, on the other hand, increase markedly withincreasing depth in the model, because of the increasing num-ber of free electrons available to form H−. The molecular lineopacity (e.g., per unit mass of stellar material) is proportionalto the number of molecules (of a given type) in the gas. Thisnumber will in general be a function of temperature as well as ofgas pressure. The higher pressures at larger depths will favourlarger molecules, but at the same time the increasing temper-ature will act in the opposite direction. In the infrared, wherethe opacity of water dominates the line opacity, the net effectis seen to be an almost unchanged opacity, with the individ-ual bands, however, being considerably broader at large depthsdue to the increased contribution of hot bands and high excita-tion lines at high temperatures. The relative distribution betweenpolyatomic molecules (mainly H2O) and diatomics (mainly TiOand CH) is very sensitive to temperature, the diatomics beingstrongly favoured at the larger optical depths where the tempera-ture is highest. In particular, it is seen that the contribution of CH(around the G-band at 4300 A) increases dramatically relativeto the water-bands with increasing depths. For low metallici-ties, 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 effectivetemperatures than at low (Fig. 7), whereas the dependence ongravity is less pronounced (Fig. 5).

6. Conclusions

We have computed the absorption strength at frequencies cor-responding to a number of collision-induced transitions in theH2-H2 complex, which are of relevance at the physical condi-tions that prevail in stellar atmospheres and for which quantummechanical results did not exist previously. We have also up-dated the existing data for H2-H2 and H2-He with estimates ofthe intensity in the far spectral wings, and with data for addi-tional overtones and hot bands, so that the whole set of data ismore complete for stellar atmosphere computations.

We have applied the new data set in computations of a grid ofcool, low-metallicity stellar atmospheres, in order to investigatethe impact of CIA processes on such models. We found that CIAis the dominant opacity source in the coolest, low-metallicitymain sequence stars. The effect on the flux distribution is pro-nounced for dwarfs of low metallicity and Teff

<∼ 4000 K, forall 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 metallicityand the effective temperature are low enough.

This group of stars includes cool white dwarfs, metal-poorbrown dwarfs, M dwarfs in the halo and in globular clusters,sub-giants in the most metal-poor globulars and in metal-poorexternal galaxies such as the SMC and the Fornax dSph, and


metal-poor carbon dwarfs. For analysis of the structure, coloursand spectra of such stars inclusion of CIA is essential.

For several of the collision-induced transitions much moreaccurate modelling than we have presented here is, however,still required, and there exists no laboratory work for compari-son with the the quantum mechanical results at elevated temper-atures. For the H2-H CIA transitions, which might likewise beof importance for stellar atmospheres, neither computations norlaboratory measurements exist. Although our results representthe most detailed study of the CIA phenomena in stars, we pointout that they are still very limited and preliminary compared towhat is known about atoms and dipole transitions in single polarmolecules and their role for the radiative transfer in stars.

Acknowledgements. We acknowledge the support of NATO Collabo-rative Research Grant 941197. The necessary computing capacity wasdue to support from the Carlsberg Foundation. AB acknowledges sup-port from the Danish Research Academy and the Danish Natural Sci-ence 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 theStellar Environment, Springer Verlag, p. 209

Borysow A., Frommhold L. 1989, ApJ 341, 549Borysow A., Frommhold L. 1990, ApJL 348, L41Borysow A., Frommhold L., Moraldi M. 1989, ApJ 336, 495Borysow J., Trafton L., Frommhold L., Birnbaum G. 1985, ApJ 296,

644Borysow J., Frommhold L., Birnbaum G. 1988, ApJ 326, 509Goorvitch D., Chackerian C.Jr. 1994, ApJS 91, 483Gustafsson B., Jørgensen U.G. 1994, A&AR 6, 19Gustafsson B., Olander N. 1979, Physica Scripta 20, 570Gustafsson B., Bell R.A., Eriksson K., Nordlund A. 1975, A&A 42,

407

Helling Ch., Jørgensen U.G., Plez B., Johnson H.R. 1996, A&A 315,194

Herzberg G. 1952, ApJ 115, 337Jørgensen U.G. 1994, A&A 284, 179Jørgensen U.G., Jensen P. 1993, J. Mol. Spectr. 161, 219Jørgensen U. G., Larsson M. 1990, A&A, 238, 424Jørgensen, U. G., Johnson, H. R., Nordlund, A. 1992, A&A 261, 263Jørgensen U.G., Larsson M., Iwamae A., Yu B. 1996, A&A 315, 204Kolos W., Wolniewicz L. 1965, J. Mol. Spectra 43, 2429Kolos W., Wolniewicz L. 1967, J. Chem. Phys. 46, 1426Kolos W., Wolniewicz L. 1968, J. Chem. Phys. 49, 404Kolos W., Wolniewicz L. 1975, J. Mol. Spectra 54, 303Langhoff 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, 759Lenzuni P., Saumon D. 1992, Rev. Mexicana Astron. Astrof. 23, 223Linsky J.L. 1969, ApJ 156, 989Meyer W., Frommhold L., Birnbaum G. 1989, Phys, Rev. 39A, 2434Meyer W., Borysow A., Frommhold L. 1993, Phys. Rev. A 47, 4065Meyer W., Frommhold L., Birnbaum G. 1989, Phys, Rev. 39A, 2434Mihalas D. 1978, Stellar Atmospheres, FreemanMould J., Liebert J. 1978, ApJ 226, L29Palla F. 1985, In: G. H. F. Diercksen et al. (eds.), Molecular Astro-

physics, D. Reidl Publ. Co., p. 687Patch R.W. 1971, JQSRT 11, 1331Plez B., Brett J.M., Nordlund A. 1992, A&A 256, 551Poll J. D., Wolniewicz L. 1978, J. Chem. Phys. 68, 3053Ross M., Ree F. H., Young D. A. 1983, J. Chem. Phys. 79, 1487Saumon D., Bergeron P., Lunine J.I., Hubbard W.B., Burrows A. 1994,

ApJ 424, 333Scholz M., Wehrse R. 1994, In: U.G. Jørgensen (ed.), Molecular Opac-

ities in the Stellar Environment, Springer Verlag, p. 49Shipman H. L. 1977, ApJ 213, 138Stahler S.W., Palla F., Salpeter E.E. 1986, ApJ 302, 590Tsuji T. 1969, In: S.S.Kumar (ed.), Low Luminosity Stars, (New York:

Gordon and Breach Science Publ.), p. 457Zheng C., Borysow A. 1995a, ApJ 441, 960Zheng C., Borysow A. 1995b, Icarus 113, 84

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