Astron. Astrophys. 364, 137-156 (2000)

4. Influence of stellar parameters on synthetic spectra

In this section the effect of stellar parameters on the absorption by CO, SiO, OH and H₂O is discussed. We mainly focus on these molecules since they are the most prominent absorbers for oxygen-rich giants in the wavelength range considered here (2.38-12 µm) (see, e.g., Fig. 2 and Fig. 3 in Decin et al. (1997) and Fig. 4 in this paper). The goal of this part of the study is to learn how the synthetic spectrum will change when one of the parameters is changed within its uncertainties, the only exception being the stellar mass and the ratio due to their small influence. A simple equation for any chemical compound is not easily obtained, nor is a unique scenario able to explain all different behaviours, but some patterns do emerge. These results are then useful for the determination of the origin of the discrepancies seen between ISO-SWS data and synthetic spectra.

Since the strength of molecular (and atomic) lines is proportional to the ratio of line to continuous absorption coefficient, , approximations for this ratio are sought for. The approximate ratio is referred to as and equations for some molecules have been deduced according to the approach of Kjægaard et al. (1982) (see Appendix B).

The ratio has been studied for the surface flux using the following parameters:

= 3650 K-3850 K-4050 K

g = 1.00-1.50-2.00

z = -0.15-0.00-0.15

M = 1.5 -10 -15

= 1.0 -2.0

= 10-4.26

(C) = 8.24-8.54

(N) = 8.26-8.56

(O) = 8.83-9.13

All computed spectra are compared with the synthetic spectrum with parameters: = 3650 K, g = 1.00, z = 0.00, M = 1.5 , = 2.0 , = 10, (C) = 8.24, (N) = 8.26, (O) = 8.83. If not specified these stellar parameters are used. In Fig. 2 the overall results of a change = 200 K (a), g = 0.50 (b), M = 13.5 (c), z = 0.15 (d), = 1.0 (e), = 5.84 (f), (C) = 0.30 (g), (N) = 0.30 (h), (O) = 0.30 (i) are shown. The results mentioned and discussed in this study apply only to models with similar stellar parameters.

Fig. 2. Division of the synthetic spectrum with parameters = 3650 K, g = 1.00, M = 1.5 , z = 0.00, = 2.0 , = 10, (C) = 8.24, (N) = 8.26, (O) = 8.83 by the synthetic spectrum with the same parameters but with (a) = 3850 K, (b) g = 0.50, (c) M = 15 , (d) z = 0.15, (e) = 1 , (f) = 4.26, (g) (C) = 8.54, (h) (N) = 8.56 and (i) (O) = 9.13. Prominent molecular bands are specified in the plot. Plot (a) has been divided by the factor to compensate for the higher flux of the spectrum with an effective temperature of 3850 K.

4.1. Models

The models and corresponding synthetic spectra have been computed by using the MARCS-code (Gustafsson et al. 1975). Since 1975, this code has undergone some modifications, the most important ones being the replacement of the Opacity Distribution Function (ODF) technique by the Opacity Sampling (OS) technique, the possibility to use a spherically symmetric geometry for extended objects and major improvements of the line and continuous opacities (Plez et al. 1992).

The common assumptions of spherical or plane-parallel stratification in homogeneous stationary layers, hydrostatic equilibrium and LTE were made. Energy conservation was required for radiative and convective flux, where the energy transport due to convection was treated through a local mixing-length theory (Henyey et al. 1965). The mixing-length l was chosen as 1.5 , with the pressure scale height, which is a reasonable quantity to simulate the temperature structure beneath the photosphere (Nordlund & Dravins 1990). Turbulent pressure was neglected. The reliability of these assumptions is discussed by Plez et al. (1992).

The synthetic spectra were generated using the TurboSpectrum program described by Plez et al. (1993), and further updated. The program treats the chemical equilibrium for hundreds of molecules with a consistent set of partition functions and dissociation energies. Solar abundances from Anders & Grevesse (1989) have been assumed, except for the iron abundance, (Fe), which is in better agreement with the meteoritic value.

The continuous opacity sources considered here are H^-, H, Fe, (H+H), , , He I, , He^-, C I, C II, , , C^-, N I, N II, N^-, O I, O II, O^-, CO^-, H₂O^-, Mg I, Mg II, Al I, Al II, Si I, Si II, Ca I, Ca II, H₂(pr), He(pr), , , , where `pr' stands for `pressure induced' and `sc' for `scattering'.

For the line opacity in the SWS range (2.38-12 µm) a database of infrared lines including atoms and molecules has been prepared. For the atomic lines the data listed by Hirata & Horaguchi (1995) have been used, for CO those by Goorvitch (1994), for SiO those by Langhoff & Bauschlicher (1993), for H₂O those by Jorgensen (1994) and Ames (Partridge & Schwenke 1997), OH lines by Sauval (Melen et al. 1995), Schwenke (1997) and Goldman et al. (1998), NH lines by Sauval (Grevesse et al. 1990; Geller et al. 1991) and CH lines by Sauval (Melen et al. 1989; Grevesse et al. 1991) and CN lines by Plez (private communication). The dissociation energy for CN was taken to be 7.76 eV. An exhaustive discussion on the accuracy and completeness of infrared spectroscopic line lists can be found in Decin (2000). From this study, a preference emerged for the H₂O line list by Ames and the OH line list by Goldman. To remove the small difference between the originally `vacuum' wavelengths for the ISO observations and `air' wavelengths in spectroscopic linelists above 200 nm, Edlen's formula (Edlen 1966) was extended to the infrared.

4.2. Effect of changing the effective temperature

When the temperature increases, fewer molecules are formed resulting in a smaller line absorption coefficient. At 2.3 µm, is primarily due to H^- free-free absorption, thus its value depends directly on the electron pressure . Increasing the temperature causes more ionization events and thus more free electrons, but at the same time the H^- ion itself is less easily formed. Fig. 3a. shows that, for the models under consideration, this latter effect is the most important one and so decreases when the effective temperature is increased from 3650 K to 3850 K. Moving inwards from the outer photosphere, the line absorption coefficient of all molecules (not H^-) reaches its maximum at the location where the effect of an increasing temperature overtakes the effect of a higher density. The partial pressure p of H^-, being proportional to p(H I)*, keeps on rising because of the increase of the number of free electrons at higher temperature. When the line-to-continuum contrast has to be known, one has to consider the partial pressure of the molecule divided by (p(H I)*) (see Fig. 2 and Fig. 3b). The simple statement that there are weaker absorption bands for higher temperatures does not always hold: the approximate ratio of CO, SiO, OH and H₂O decreases, but for CN the lower continuous opacity compensates for the lower line opacity (see Fig. 3b). Furthermore, the relative population of rotational levels strongly depends on T (the maximum population occurs at a -value and a lower T corresponds to a lower -value). Therefore a given line (J) intensity could be increased or decreased according to its J-value only.

Fig. 3a and b. Comparison between the partial pressure of several molecules for the models with = 3650 K (full line) and = 3850 K (dashed line) and with g = 1.00, M = 1.5 , z = 0.00, = 2.0 , = 10, (C) = 8.24, (N) = 8.26, (O) = 8.83. In plot (a) the partial pressure of the molecules is given, while in plot (b) the partial pressure of the molecules is divided by p(H I)*.

4.3. Effect of changing the gravity and the mass

Although g and M are closely related, the effect of changing the mass is smaller than that of changing the gravity, because the first one only affects the extension and the latter one also changes the pressure structure of the atmosphere. This is clearly visible in Fig. 2. The assumption of hydrostatic equilibrium links the gas pressure to the surface gravity. In a cool star, the gas and electron pressure ( and ) can be written as:

Due to the higher extension of the atmospheres considered here, this value of p has a larger range compared to the value discussed by Gray (1992).

When the gravity increases, the electron pressure increases rapidly (Eq. 2), resulting in a higher H^- free-free absorption. Other effects of increasing the gravity are higher number densities supporting the molecular formation and the fact that molecules can be formed till higher temperatures, both increasing . The final result is that the approximate absorption coefficient can either increase or decrease depending on the relative change of and .

Using the approximate relations according to Kjærgaard et al. (1982) (see Appendix B), the conclusion is reached that an increase of gravity leads to a decrease of the strength of the CO, NH and CN lines and to an increase of the strength of the H₂O lines, with no dependence for the OH and SiO lines. The increasing gravity does lead to a strengthening of the H₂O lines and also of the OH lines while it diminishes the ratio of CO, NH, SiO and CN (Fig. 2). From the approximate relations for no quantitative conclusion on the parameter dependence can thus be made.

An increase in mass yields the inverse (but smaller) effect on the partial pressure (and so on the line absorption coefficient ) as an increase in gravity.

4.4. Effect of changing the metallicity

For the computations with a higher metallicity, [atom/H] has been increased from 0.00 dex to 0.15 dex for all atoms except for H, He, C, N and O. Because an increase in metal abundance leads to a decrease in gas pressure at each optical depth (as increases), the partial pressures of the molecules CO, OH, N₂, H₂O and CN consequently decrease, while p(SiO) and p(TiO) are somewhat higher because of the higher abundance of Si and Ti respectively. An increase in the overall metal abundance increases the amount of electrons () and as a consequence also the value of formed by H^-. This increase of dominates the changes in the spectrum: all molecular features become fainter at increased metallicities (with C, N and O kept constant) (Fig. 2). This can also be seen in the equations for (see appendix B) for the extreme case in which all carbon is present in the form of CO with  K. The strong metallicity dependence of H₂O in the approximate relation for may be well visible, though this increase (with decreasing z) is small for weak H₂O bands.

4.5. Effect of changing the microturbulent velocity

The impact of changing the microturbulent velocity is most important for the CO lines with a large equivalent width. When a line is saturated, increasing widens the wavelength range covered by the absorption and reduces the saturation, thus increasing the total absorption. In the line center of an unsaturated line, a smaller microturbulence corresponds to a higher absorption coefficient at that frequency (see Fig. 2).

Decreasing the microturbulent velocity from 2 to 1 diminishes the CO first overtone absorption by maximum 10% and the CO fundamental by up to 6%, the SiO first overtone by up to 4%, and the SiO fundamental by up to 3%, the fundamental band of OH by up to 3% and the ro-vibrational stretching modes ( and ) and bending mode () of H₂O by up to 0.5%.

4.6. Effect of changing the isotopic ratio

Red giant atmospheres display altered C, N, and O abundances and isotopic ratios (in particular ), due to dredge-up episodes. Suppose that the ratio is changed from 10 (case I) to 4.26 (case II). If a ¹²CO line is saturated in case I, its equivalent width will decrease very few from case I to case II. If we assume the ¹³CO line to be very weak in case I, its equivalent width will increase very much from case I to case II by a factor nearly equal to the new/old isotopic ratio. Therefore it is understandable that a decrease in will consequently cause (CO) to increase, although this increase may be very small if the CO bands are weak (Fig. 2).

4.7. Effect of changing (C), (N) or (O)

When the ratio C/O increases (but remains 1), the structure of the atmosphere is affected through changes in line blanketing. More CO will be formed and less oxygen will be left (after CO formation) to form oxygen-based molecules. Hence, as illustrated in Fig. 2, the total absorption of CO will increase, while it will decrease for SiO, OH and H₂O. Increasing the oxygen abundance gives roughly the reverse effect, but the total CO absorption hardly changes. The largest effect of an increasing nitrogen abundance is an increase of the NH and CN line strengths.

© European Southern Observatory (ESO) 2000

Online publication: December 15, 2000
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