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Appendix A: CO data – figures and tables

All ¹²CO and ¹³CO data of the sample and the soft-parabola fits (Sect. 2.3.1) to these data are shown in Figs. A.1 and A.2. The most important line parameters – main-beam temperature at the line centre and integrated intensity – and the β parameter from the soft-parabola fitting procedure as described in Sect. 2.3.1 are given in Tables A.1 and A.3. References for, or remarks on, the data are listed in Tables A.2 and A.4 for the ¹²CO and ¹³CO data respectively.

	Fig. A.1 Profiles of 12CO with fitted soft-parabola profiles.
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	Fig. A.2 Profiles of 13CO with fitted soft–parabola profiles.
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Appendix B: Dereddening

Since the effective temperatures of late-type stars are well-determined by the intrinsic V−K colour (Sect. C.1), we need a way to derive this colour accurately. The magnitudes of both the V and K bands – at 0.550 μm and 2.2 μm, respectively – are reddened due to interstellar and circumstellar contributions. The data were corrected for interstellar extinction using the dust maps of Schlegel et al. (1998), which give the interstellar reddening at infinity, E_B−V,∞, for a given set of galactic latitudes b and longitudes l. This value was transformed into interstellar reddening at a given distance d using (B.1)following Feast et al. (1990). The resulting value for E_B−V was combined with R_V = 3.1 and A_K = 0.112A_V (Schlegel et al. 1998) to give interstellar extinction coefficients for the V and K bands.

Circumstellar extinction was not accounted for, even though it is assumed to be significant for high mass-loss rate LPV targets. Knapp et al. (1998) use the mass-loss rate Ṁ to determine the circumstellar extinction in the K band. Since the aim of the paper is to establish empirical relations between stellar parameters and Ṁ, this method can not be applied to the data for the sake of consistency. Knapp et al. (2003) suggest a formalism to calculate the circumstellar extinction in the K band, A_K,CS, as a function of S₁₂/S_2.2, the ratio of flux densities at 12 μm and 2.2 μm, respectively. This method is only valid for low optical depths (Knapp et al. 2003). When applied to the sample, unrealistic values for A_K,CS as high as several tens (LP And) or hundreds (CW Leo) of magnitudes are indeed obtained for the objects with presumed high Ṁ values. To decide, however, whether or not this correction should be applied for a given target seems arbitrary. For these reasons we did not correct for circumstellar reddening.

Appendix C: Basic stellar parameters

In this appendix we describe the methods that we used to obtain the effective temperature, stellar luminosity and distance towards objects. The results are given in Table 6 in the paper.

C.1. Effective temperature

	Fig. C.1 Different calibration relations linking the (dereddened) broad-band colour (V−K)0 and Teff. Relations were taken from di Benedetto (1993); Bessell et al. (1998); Bergeat et al. (2001); Levesque et al. (2005).
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Since broad-band near-infrared colours are very good temperature criteria, being relatively independent of stellar gravities and abundances, they are often used to determine effective temperatures. The V−K colour is a good T_eff-indicator for cool, late-type stars, e.g. GKM giants and supergiants (di Benedetto 1993; Bessell et al. 1998; Bergeat et al. 2001; Levesque et al. 2005). For hotter A through K stars the V−I colour is more suitable (Bessell et al. 1998).

In all T_eff versus V−K relations the Johnson-Cousins-Glass photometric system is used. K magnitudes for the sample stars that were obtained in the systems of 2MASS (K_s) and SAAO (K_SAAO) were transformed via (C.1)following (Bessell 2005). Since the accuracy of the K_SAAO-measurements (~0.03 mag; Olivier et al. 2001) is almost a factor 10 better than that of K_s-data (~0.2 mag), preference was given to the SAAO-measurements when both were available. The errors on the magnitudes are expected to be small relative to the amplitude in the K band caused by pulsations of the stars. The error bars on the T_eff values determined here, were calculated assuming an error on (V−K)₀ of 1 mag, leading to rather conservative temperature estimates.

In all equations listed (V−K)₀ is the intrinsic V−K broad-band colour with (1) both the V and K magnitudes at maximum light, or (2) both at their mean value if maximum light was not available, and (3) corrected for interstellar reddening in the way described in Sect. B.

C.1.1. Long-period variables: Miras, semi-regulars and irregular variables

Bessell et al. (1998) defined a T_eff versus V−K calibration for KM giants (luminosity class III), given by (C.2)This relation is valid for 1.5 mag ≤ (V−K)₀ ≤ 10 mag (see Fig. C.1). However, Bessell et al. (1998) mention that their static model atmospheres cannot adequately represent the M giants cooler than 3000 K, since most of these are long-period variables (LPVs) with shock waves traversing and greatly extending their atmospheres.

di Benedetto (1993) presented a temperature calibration in which T_eff is derived from angular-diameter measurements based on Michelson interferometry. In case of variable stars, measurements at maximum light were used. The effective temperature of GK stars of luminosity classes II-III-IV-V is described by (C.3)for 1.4 mag ≤ (V−K)₀ ≤ 3.7 mag, for FGKM supergiants by (C.4)for (V−K)₀ ≥ 0.6 mag and for M-type giants of type IV, Miras and carbon stars by Eq. (C.4) for 3.7 mag ≤ (V−K)₀ ≤ 10 mag (see Fig. C.1). The constants a and b in Eqs. C.3 and C.4, and the validity ranges in terms of (V−K)₀ are listed in Table C.2.

For oxygen-rich LPVs, except for OH/IR stars, the effective temperature was derived via Eq. (C.4) (di Benedetto 1993). The uncertainty on the temperature would then be of the order of 1 % according to di Benedetto (1993).

Bergeat et al. (2001) focussed on carbon-rich objects and derived a relation for T_eff versus V−K given by Eqs. C.5, C.6 and C.7. The notation T_eff,d denotes that the relation is based on angular-diameter measurements, T_eff is the final temperature adopted for their sample stars. These T_eff values are estimated to be accurate to within an error margin of  ± 140 K (Bergeat et al. 2001). Because this T_eff-calibration is based on a sample of 54 carbon-rich stars, preference was given to this calibration, rather than to the one by di Benedetto (1993), in case of carbon stars. The validity range was adopted to be 4.99 ≤ (V−K)₀ ≤ 8.31 mag, consistent with the minimum and maximum in Table 5 of Bergeat et al. (2001).

It is clearly visible from Fig. C.1 that the carbon stars have lower predicted effective temperatures than the oxygen-rich stars. This is in agreement with the results of Marigo (2002), showing a clear dichotomy in T_eff between both chemical types of AGB stars. This dichotomy is caused by different molecular opacities in stellar atmospheres with differing C/O abundance ratios.

When using these T_eff estimators, one should be aware of the effect of circumstellar reddening on the V and K band data. This reddening is stronger for objects with higher values of Ṁ and depends on the chemistry of the central star (Knapp et al. 1998, 2003). As mentioned in Appendix B, the data were not corrected for circumstellar reddening.

For oxygen-rich targets with (V−K)₀ outside the valid ranges for the above mentioned relations, the temperature corresponding to the spectral type listed in Table 6 was used (Table C.1; Marigo et al. 2008). An error bar of 500 K was assumed for these values. For the carbon-rich targets with (V−K)₀ outside the valid ranges the effective temperatures found in literature were used. References are given in Table 6.

C.1.2. OH/IR stars

In the case of OH/IR stars the photosphere is not easily probed since the CSE is optically thick and consequently the infrared sources often do not have optical counterparts. Therefore, the V−K colour is no longer a meaningful diagnostic of T_eff. For these objects an effective temperature of 2750 ± 750 K was assumed.

C.1.3. Red supergiants

The T_eff-calibration of Bessell et al. (1998) is not valid for red supergiants (RSGs) because of their low log g-values. Levesque et al. (2005) derived a T_eff-scale for RSGs in the Milky Way given by Eq. (C.8). This relation between the broad-band colour V−K and the effective temperature is consistent with the results from their grid of marcs models and is valid for 2.9mag ≤ (V−K)₀ ≤ 8.0mag. Adopting this new T_eff-calibration for RSGs yields a better agreement between evolutionary tracks and observationally determined locations of RSGs in the HR-diagram (see Massey et al. 2008, Fig. 3). This T_eff-scale for RSGs closely follows the relation of Bessell et al. (1998) defined for KM giants (see Fig. C.1), but since Eq. (C.8) was derived especially for RSGs, preference was given to this relation over Eq. (C.2). It is given by (C.8)Levesque et al. (2005) expect that the effective temperatures for M-type supergiants were obtained with a precision of 50 K.

For RSGs with (V−K)₀ < 2.9 or (V−K)₀ > 8.0, the temperature was determined using the scale of T_eff versus spectral type – in case of galactic targets – presented by Levesque et al. (2005).

C.1.4. Yellow hypergiants

The adopted T_eff-calibration of di Benedetto (1993) for FGKM targets of luminosity class I seems in accordance with the results presented by de Jager (1998) and Oudmaijer et al. (2009) for the yellow hypergiants (YHGs) IRC +10420 and AFGL 2343. Since both objects are known to exhibit large T_eff-variations on time scales of a few decades (Oudmaijer et al. 2009), this calibration is somewhat uncertain.

C.1.5. Alternative methods

Haniff et al. (1995) derive effective temperatures from angular diameters and bolometric fluxes obtained through fits to optical and infrared photometry. The infrared flux method (IRFM), presented by Blackwell & Shallis (1977), consists of determining the ratio of the bolometric flux to the flux in a selected photometric band and comparing these results to predictions from model atmospheres. Since the goal of this paper is not to individually model all sample targets, we did not pursue these methods, nor did we perform full fits of the spectral energy distributions (SEDs). The latter would require detailed radiative transfer analysis for each star.

C.2. Luminosity

The luminosities of Miras, OH/IR stars, semi-regular pulsators of types a and b and red supergiants were determined through empirical period-bolometric magnitude relations presented in the literature (Feast et al. 1989; Groenewegen & Whitelock 1996; Whitelock et al. 1991; Barthès et al. 1999; Whitelock & Feast 2000; Yeşilyaprak & Aslan 2004). These relations are then used to calculate luminosities adopting the solar bolometric magnitude M_bol, ⊙   = 4.75 mag.

	Fig. C.2 P−L ⋆  relations taken from Groenewegen & Whitelock (1996); Feast et al. (1989); Whitelock et al. (1991); Yeşilyaprak & Aslan (2004); Barthès et al. (1999), and Whitelock et al. (2000). Dashed lines indicate extrapolations of the P−L ⋆  relations from the literature to shorter or longer periods of pulsation. See text for details.
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C.2.1. Miras and OH/IR stars

Feast et al. (1989) derived period-luminosity, P−L _⋆ , relations for oxygen-rich and carbon-rich Miras in the LMC with periods ranging from 115 up to 420 days. Groenewegen & Whitelock (1996) revised the P−L _⋆ relation for carbon-rich Miras and extended the period range up to periods of 520 days. Whitelock et al. (1991) developed a similar relation for OH/IR targets with periods of pulsation exceeding 1000 days. These relations – and their extrapolations to shorter or longer pulsation periods – are shown in Fig. C.2.

The relation derived by Feast et al. (1989) for oxygen-rich Miras was assumed to be valid also for galactic oxygen-rich Miras and was adjusted with a distance modulus of 18.5 mag. It is given by (C.9)with P in days.

Whitelock et al. (1991) mention a rather large scatter of 0.39 mag on their derived P−L _⋆ relation for OH/IR variables with periods ranging up to 2000 days. The large range in predicted luminosities associated with this scatter has been made visible in the figure. Note that the extrapolated relation for oxygen-rich Miras derived by Feast et al. (1989) matches the higher luminosity edge derived for OH/IR targets very well. Whitelock et al. (1991) also point out that there are suggestions that the luminosities of OH/IR sources with P > 1000 days probably follow this extrapolated P−L _⋆ relation that was established for oxygen-rich Miras by Feast et al. (1989). Because of these arguments and the strong evolutionary connection between oxygen-rich Miras and OH/IR stars (see Sect. 2.1) we adopt the P−L _⋆ relation of Feast et al. (1989) for oxygen-rich Miras, given in Eq. (C.9), for the determination of luminosities for all oxygen-rich Miras and OH/IR targets, regardless of their period of pulsation. Relation C.9 was also used in determining the luminosity of S-type Miras in the sample.

Luminosities of carbon-rich Miras were determined via (C.10)following Groenewegen & Whitelock (1996). This relation predicts luminosities substantially higher than does the relation derived by Feast et al. (1989). Preference was given to the former since the sample of Groenewegen & Whitelock (1996) is almost three times as large as the one used by Feast et al. (1989) (54 versus 20 objects). For longer-period carbon-rich Miras, this P−L _⋆ relation was extrapolated, similar to what was done in the oxygen-rich case.

C.2.2. Semi-regular pulsators: SRa & SRb

Many semi-regular type a (SRa) objects differ from Miras only by their smaller amplitudes in the V band (ΔV < 2.5 mag). Objects classified as semi-regular b (SRb) variables have rather poorly defined periodicity, show a superposition of multiple pulsation periods, or have alternating intervals of periodic and slow irregular changes.

Semi-regulars and Miras are linked in terms of evolution, since SRs of types a and b are thought to be the progenitors of Miras (Whitelock & Feast 2000; Yeşilyaprak & Aslan 2004). Therefore some type of P−L _⋆ relation, analogous to those for Miras is probably justified for these LPVs.

Yeşilyaprak & Aslan (2004) presented period-magnitude relations for all magnitudes in the Johnson system and the IRAS system for SRa and SRb stars. To determine the bolometric magnitude, and hence the luminosity, only the J and K period – magnitude relations were used Yeşilyaprak & Aslan mention a scatter of 0.63 mag and 0.85 mag for the J and K relations, respectively. The bolometric correction on the K magnitude was calculated via (C.13)taken from Whitelock et al. (2000), where (C.14)Relations (C.11) and (C.12) were however only adopted in estimating the luminosities for SRa variables, and not for the SRb variables in the sample. Barthès et al. (1999) presented P−(J−K) relations for long-period variables, including a large number of SRb type stars. The sample of Barthès et al. (1999) is divided in four groups, and the SRb type stars are largely split up in two of those – Groups 2 and 3, corresponding to younger and older kinematic properties respectively. Their Group 2 is far less luminous than their Group 3 and preference is given to the relations for their Group 3 – Eqs. (C.15) and (C.16) – since these estimates are in best accordance with luminosities derived in the literature for the SRb objects in the sample. Bolometric corrections were again derived via Eq. (C.13). The uncertainties on calculated luminosities only take into account the uncertainties on the coefficients in Eqs. C.9 through C.16. The error bars on the pulsation periods are assumed to be small compared to the errors on the coefficients in the above relations.

C.2.3. Red supergiants

The luminosity of the RSGs in the sample was determined as a function of period of pulsation and effective temperature. Equation (C.17) gives the relation between the K magnitude and the pulsation period for RSGs in the milky way galaxy as presented by Dambis (1993). This relation was completed with the bolometric correction on the K band magnitudes given by Levesque et al. (2005) (Eq. (C.18)). This method leads to luminosities ranging from  ~10⁴ L_⊙ up to  ~510⁵ L_⊙. It is given by

C.2.4. Yellow hypergiants

No general period-luminosity relations can be established for these targets because of their very strong variability. The latter is reflected in the multiple motions from blue to red and vice versa in the HR diagram (de Jager 1998; Oudmaijer et al. 2009). For both AFGL 2343 and IRC +10420, the two YHGs in the sample, luminosities were taken from de Jager (1998), yielding values for log L _⋆ /L_⊙ of 5.30 and 5.80, respectively.

C.2.5. Alternative methods

More recently, Barthès et al. (1999) and Knapp et al. (2003) derived P−K relations for oxygen-rich LPVs, including Miras and semi-regulars of types a and b. Using these relations in determining the luminosities of Miras would introduce extra uncertainties, since bolometric corrections for K band magnitudes would have to be derived. This is the main reason why the above mentioned P−M_bolrelations of Feast et al. (1989) and Groenewegen & Whitelock (1996) were used for Miras.

C.3. Distance

Distances towards the sample stars have been determined in various ways. (1) OH-maser phase lag distances were used for most OH/IR stars, since these provide a very accurate distance estimate (van Langevelde et al. 1990). (2) Hipparcos-data were used when available with a relative error lower than 50 %. (3) values in the literature derived from SED fitting were used and assumed to have uncertainties of 50 %, unless stated otherwise in the original papers.

C.3.1. Alternative method

An alternative method to derive distances is to compare observed magnitudes with absolute magnitudes predicted through formalisms as presented in Sect. C.2. The relations that could be used for distance determination from magnitudes for Miras are Since the K magnitude is less sensitive to metallicity than the bolometric magnitude m_bol, it is a better distance indicator (Whitelock & Feast 2000, and references therein). The strong variability in the K band of most sample stars, however, gives rise to large uncertainties on absolute magnitudes. An additional constraint to the application of this method to determine distances is that the circumstellar material can redden the K band magnitudes significantly.

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