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Astron. Astrophys. 357, 61-65 (2000)

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2. The CO index

2.1. Spectroscopic and photometric definitions

The CO index was originally defined as the magnitude difference between a relatively narrow filter ([FORMULA] µm) centered at 2.3 µm, which includes the first four band-heads of [FORMULA]=2 CO roto-vibrational transitions, and a similarly narrow filter centered at 2.2 µm (Baldwin et al. 1973). The central wavelength of the CO filter was then increased to 2.36 µm and slightly different filter parameters were adopted by different groups. A comprehensive database of CO photometric measurements was produced in the 70-80's. These include measurements of field stars (e.g. McWilliam & Lambert 1984), Galactic globular clusters (Frogel et al. 1983), young stellar clusters in Magellanic Clouds (Persson et al. 1983) and old spheroidal galaxies (Frogel et al. 1978). These data are still considered a fundamental benchmark for verifying the predictions of stellar evolutionary models.

The spectroscopic CO index was defined by KH86 who measured the (2,0) band-head at 2.29 µm from medium resolution ([FORMULA]) spectra of a sample of field stars. The strength of this band is unequivocally defined as the ratio between the fluxes integrated over narrow wavelength ranges centered on the line and nearby continuum, i.e. 2.2924-2.2977 and 2.2867-2.2919 µm, and expressed in terms of magnitudes. The same quantity is sometimes given in terms of equivalent width (e.g. Origlia et al. 1993) and the numbers are simply related by

[EQUATION]

where [FORMULA] is the spectroscopic index and [FORMULA] is the equivalent width of the (2,0) band-head.

The relationship between spectroscopic and photometric indices is not obvious because they are based on measurements of different quantities which have different behaviours on the stellar physical parameters, e.g. [FORMULA] also depends on the 12C/13C ratio (see McWilliam & Lambert 1984). Nevertheless, an empirical correlation between the two indices is generally adopted following from observations of giant stars in the field. This yields (see Fig. 3 of KH86)

[EQUATION]

To complicate the scenario further, other definitions of the spectroscopic index exist in the literature. These are based on spectroscopic measurements of equivalent widths over wavelength ranges much broader than those used by KH86 and similar to that adopted in the photometric definition. The most popular of these intermediate indices is that proposed by Doyon et al. (1994) which measures the equivalent width over the 2.31-2.40 µm range relative to a continuum which is extrapolated from shorter wavelengths. The specific advantage of this definition is that it allows measurements of [CO] even from spectra of relatively poor quality, while proper measurement of the spectroscopic index requires high s/n spectra with resolution [FORMULA].

These "broad spectral indices" are calibrated using measurements of field stars and can be therefore converted to photometric indices using empirical relationships similar to Eq. (2). To avoid confusion we will hereafter express measured and predicted quantities in terms of [FORMULA] with additional comments on how it was obtained or scaled from spectroscopic measurements.

2.2. CO index and stellar parameters

In general, the strength of the CO features depends on the following stellar parameters (for a more detailed discussion see Sect. 4.1 of Origlia et al. 1993)

  • Effective temperature, [FORMULA], which sets the CO/C relative abundance.

  • Surface gravity, g, which influences the H-/H equilibrium and hence modifies the total gas column density of the photosphere. In cool stars with CO/C[FORMULA]1 the column density of CO scales as [FORMULA] and, therefore, the lines become more opaque when the gravity decreases.

  • Microturbulent velocity, [FORMULA], which determines the Gaussian width of the lines and, therefore, the equivalent width of saturated lines (the CO transitions are semi-forbidden and have very weak Lorentzian wings).

  • Metallicity and carbon relative abundance, which define the value of C/H. For a given set of [FORMULA] and g, all CO line opacities scale linearly with the carbon abundance C/H.

The well known correlation between CO index and spectral type of giant stars is due to a complex combination of the effects of the first three parameters. The variation of [FORMULA] is important only up to early K stars where most of the carbon is already in the form of CO. Therefore, in later spectral types the variation of [FORMULA] has virtually no direct effect on [CO], i.e. the CO index is not a thermometer for cool stars .

The steady deepening of the CO features along the K4 III-M7 III sequence is driven by the decrease of surface gravity and increase of microturbulent velocity. The variation of surface gravity, [FORMULA] from early K to late M giants (McWilliam & Lambert 1984), follows from the fact that field red giants are stars with similar mass and quasi-solar metallicities taken at different phases of their evolution on the RGB. Thus M III stars, being cooler and more luminous, have a lower surface gravity than K giants.

The value of [FORMULA] cannot be directly related to the stellar parameters but, based on detailed spectroscopic observations, is found to increase when the bolometric luminosity increases. To a first approximation, [FORMULA] scales linearly with [FORMULA] and, therefore, late M giants have microturbulent velocities about 0.8 km/s higher than early K III stars (see e.g. Tsuji 1986, 1991).

The effect of microturbulent velocity becomes particularly prominent in red supergiants and accounts for the fact that class I stars have much stronger [CO] than giants of similar temperatures and gravities (see e.g. Tsuji et al. 1994, McWilliam & Lambert 1984). The derived values of [FORMULA] scale [FORMULA] linearly with [FORMULA], i.e. a behaviour similar to that found in giant stars. Therefore, for practical purposes, the variation of [CO] in red giants and supergiants can be reproduced adopting an empirical, linear relationship between [FORMULA] and bolometric luminosity, namely:

[EQUATION]

An indirect test (and confirmation) of this equation comes from integrated spectral analysis of old and metallic stellar systems (Origlia et al. 1997) and young clusters in Magellanic Clouds (Oliva & Origlia 1998). The derived values of average microturbulent velocities, namely [FORMULA] and [FORMULA] km/s for old and young systems, respectively, are in good agreement with those derived from spectral synthesis models adopting Eq. (3).

The effect of carbon abundance is relatively unimportant in field stars, which span a relatively narrow range of metallicities, but becomes evident in stellar clusters of low metallicity which are characterized by very weak CO features (see the left panel of Fig. 3). However, it should be kept in mind that the nice correlation between [CO] and metallicity of Fig. 3 also reflects the metallicity dependence of the temperature of giant stars, i.e. lower metallicity clusters have weaker [CO] not only because their giant stars have less carbon, but also because they are warmer than those in more metallic stellar systems (e.g. Origlia et al. 1997).

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
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