© ESO, 2011

1. Introduction

The red supergiant stars (hereafter RSG) are massive stars evolving beyond the main sequence and passing a fraction of their lives in the cool upper region of the Hertzsprung-Russell (H-R) luminosity-temperature diagram. In the solar neighbourhood, the initial mass of RSGs is about 8 to 25 M_⊙, their effective temperature is for most objects 3000 to 4000 K, and their luminosity is between 2 × 10⁴ and  ~6 × 10⁵ L_⊙. Several important works have recently revisited the basic properties of these stars in the Galaxy and in other systems like the Magellanic Clouds and M31 (Massey 1998; Levesque et al. 2005, hereafter LM05; Massey et al. 2008; Levesque 2009, and references therein). In general, a fairly good agreement between recent observations and the predictions of evolutionary models has been established. However, one of the important processes occuring at the surface of these stars, mass loss, is still poorly known. The radii of RSGs are huge, typically 500−1500 R_⊙. Consequently, the gravity of these stars is very small, about 50 000 times smaller than that of the Sun. The small gravity favours ejection of matter to the interstellar medium, and indeed some RSGs lose mass at a high rate, 10^-6 to 10^-4  M_⊙ yr^-1.

Mass loss has several consequences. First, a significant fraction of the stellar mass is lost, which is important for the subsequent evolution of the star. Secondly, the ejected circumstellar matter enriches the ISM in dust, e.g. diverse types of silicates (see for example Verhoelst et al. 2009) and even carbonaceous matter (Sylvester et al. 1994, 1998). One can note also that some types of core-collapse supernovae have RSG progenitors (Smartt et al. 2008; Josselin & Lançon 2009), so that the density and radial distribution of the RSG winds play an important role in the expansion and evolution of the supernova remnant (Chevalier et al. 2006; Smith et al. 2009; van Loon 2010).

Despite the importance of RSG mass loss, a quantitative physical model of the winds has yet to be established. Like for other mass-losing cool giants, atmospheric motions such as pulsation, convection, and shocks occur (e.g. Kiss et al. 2006; Josselin & Plez 2007), and this may produce out-moving gas sufficiently cool for dust to form. Then, radiation pressure on the grains may be sufficient to drive the outflow (see for example Habing et al. 1994). Although this scenario is plausible, some important details are still unclear. For example, the condensation of the Fe-rich silicate grains, which are crucial to drive the wind, is not well understood (Woitke 2007). When dust is incompletely formed or absent, other mass-loss mechanisms have to be found, but none is completely satisfactory (e.g. Bennett 2010, and references therein).

Because no physical model exists, diverse empirical parametrizations are used in stellar evolution codes for estimating Ṁ  as a function of basic stellar characteristics, i.e. mass, luminosity, and effective temperature. An important example is the empirical law built by de Jager et al. (1988, hereafter dJ88). This law was based on observational estimates of Ṁ  for stars located over the whole H-R diagram, including 15 RSGs. It is used in the Cambridge stellar evolution code with a (Z/Z_⊙)^0.5 metallicity scaling (Stancliffe & Eldridge 2009). The dJ88 law was also used in past evolution models of the Geneva group (described in, e.g., Maeder & Meynet 1989; Eggenberger et al. 2008). However, in the most recent Geneva code, the dJ88 prescription is used only for yellow supergiants with T_eff between 5000 K and 8000 K. For smaller T_eff, a power-law in L, Ṁ ∝ L^1.7, is adopted with no metallicity scaling.

The goal of this paper is to investigate whether this de Jager formulation still agrees with more recently obtained mass-loss rates. For example, mass-loss rates of RSGs were estimated with IRAS observations of the circumstellar dust (Jura & Kleinman 1990, hereafter JK90; Josselin et al. 2000, hereafter JB00) or from ISO spectra (Verhoelst et al. 2009). Several radio millimeter observations of CO from RSGs have also been made since 1988 (JB00; Woodhams 1993). The question is whether these more recent measurements support the dJ88 prescription.

In the following, Sect. 2 recalls the prescription of de Jager and compares it with a few others. In Sect. 3 we compare the dJ88 rate with measured mass-loss rates of RSGs, and we consider three different cases: i) the RSGs of the solar neighbourhood and Ṁ estimated with their infrared excess at 60 μm; ii) the eight cases of these RSGs for which circumstellar gas has been directly observed and used to derive Ṁ; iii) the RSGs of the Magellanic Clouds. In Sect. 4 we discuss our results and conclude in Sect. 5.

2. Comparison of de Jager rates with other prescriptions

In the paper of dJ88, the authors collected a large number of mass-loss rates determined for 271 Galactic stars located over the whole H-R diagram. The mass-loss rates could be matched by a single empirical formula in which the parameters are the effective temperature T_eff and the luminosity L. Hereafter, we call this formula “the de Jager rate”. According to dJ88, the accuracy of this formula is about 0.45 in log Ṁ, i.e. about a factor of 3. Because the de Jager formula comprises 20 Chebycheff polynomia, the dependence of Ṁ  on T_eff and L cannot be seen clearly. Therefore, we show in Fig. 1 the situation for RSGs. The de Jager rate is plotted for two different effective temperatures, 3500 and 4000 K, since the majority of Galactic RSGs have their T_eff between these values (LM05). More precisely, in the list of 74 RSGs of LM05 (with a M-type classification), only α Her has a lower temperature, with 3450 K, but it is not a massive star, M ≈ 2.1 M_⊙  according to Schröder & Cuntz (2007). Figure 1 shows that Ṁ  is not very dependent on T_eff in the de Jager formulation, and that Ṁ  only depends on L.

Among the 271 stars (hot or cool, luminous or not) that are at the basis for the dJ88 prescription, there are only 15 Galactic RSGs, for which dJ88 compiled T_eff, L and an observational estimate of Ṁ, or an average of estimates. These compiled data are listed in Appendix A together with results of this work, which we discuss below. It can be noted that the T_eff are significantly cooler than those recently determined by LM05, when available. In addition, about half of the estimated Ṁ are based on the early paper of Gehrz & Woolf (1971). So, we think it appropriate to reexamine the validity of the dJ88 prescription with a larger sample of objects (40 are considered in this work) and more recent data.

There exists a second formulation published later by Nieuwenhuijzen and de Jager (1990, hereafter NJ90). The stellar mass is taken into account, and the dependence of Ṁ  on stellar mass, luminosity, and temperature is more explicit. Nieuwenhuijzen & de Jager (1990) find:

It can be seen in the above formula that Ṁ depends very little on M, because for M = 9 M_⊙ and 25 M_⊙, the second term is equal to 0.15 and 0.22 respectively. This represents a difference of only 17 percent on Ṁ , and consequently this mass term can be considered to be constant and equal to 0.18. Then, a more explicit formulation is

The rate of NJ90 is plotted in Fig. 1 for T_eff = 3750 K. For other temperatures, this rate does not change much: for effective temperatures of 3500 or 4000 K, the NJ90 line in Fig. 1 would be shifted up or down by only 0.05 in the y-axis. This NJ90 rate is not very different from the de Jager rate: the largest difference is about a factor of 2 at L ~ 10⁵. Also Nieuwenhuijzen & de Jager note that for cool evolved stars the dJ88 rate is more accurate than the NJ90 one in fitting the Ṁ  observational measurements (their Table 2).

We also plotted in Fig. 1 the rate adopted in the recent Geneva models (Hirschi, private communication). This rate is represented by a continuous straight line located very close to, and just below the NJ90 line. This Geneva rate is practically identical to the NJ90 one, but there is no dependency on T_eff. The Geneva rate is Ṁ = 4.7 × 10^-6(L/10⁵)^1.7.

Fig. 1 Comparison of several prescriptions for the RSG mass-loss rates (unit  M⊙ yr-1) plotted against luminosity (in L⊙). The de Jager prescription is represented by the two curved lines (solid line Teff = 4000 K, dotted line Teff = 3500 K).

Figure 1 also shows the Reimers law (Reimers 1975; Kudritzki & Reimers 1978)

with L, R, and M in solar units. Because we restrict our survey to RSGs, this relation can be simplified: R is equal to L^0.5 × (T_eff/5770)^-2 and the factor depending on T_eff does not vary much (2.72 and 2.08 for T_eff = 3500  and 4000 K respectively). In addition, for the RSGs studied here, there is an approximate relation between mass and luminosity. For M = 9 M_⊙, L ≈ 2.5 × 10⁴ L_⊙, and for M = 25 M_⊙, L  ≈ 3 × 10⁵ L_⊙ (see, for example, Fig. 8 of Massey et al. 2009, lower panels). We find M ≈ 0.14   L^0.41. The Reimers relation for an average temperature T_eff = 3750 K is

This relation is plotted in Fig. 1, and obviously the Reimers rate is close to the dJ88 rate (within a factor of 2 for L ≤ 2 × 10⁵ L_⊙), but the latter is significantly stronger for higher luminosities. The Reimers lines corresponding to other temperatures would be 0.060 (in log) above and 0.056 below the plotted line, for 3500 and 4000 K respectively. This shows that the dependence on temperature is quite small compared to the observational uncertainty on measured mass-loss rates considered below (typically a factor 2 or 3).

There exist other mass-loss prescriptions. Feast (1992) noted that there is a very good correlation between the pulsation period P of 15 RSGs of the Large Magellanic Cloud and their luminosities L based on JHK and IRAS data and estimated by Reid et al. (1990). The mass-loss rates published by Reid et al. for these stars also correlate fairly well with P. A relation linking Ṁ and L is derivable and given in Eq. (12) of Salasnich et al. (1999). This relation (plotted in Fig. 1) can be written as

A prescription based also on the LMC RSGs and the work of Reid et al. (1990) is used by Vanbeveren et al. (1998; see also Vanbeveren et al. 2007). It is labelled “Vanb.” in Fig. 1. This rate agrees with the Feast rate for L ~ 10⁵ L_⊙, but its slope is lower. Its expression is

Another mass-loss prescription has been proposed by Salasnich et al. (1999), who adopt the Feast relation, but take a gas-to-dust mass ratio that increases with luminosity. Typically, this ratio would be 200 for AGB stars, but could increase to 1000 for RSGs with a luminosity of 10⁵ L_⊙. Consequently, the Salasnich prescription implies very high mass-loss rates. For L ~ 10⁵ L_⊙, this prescription gives 10^-4  M_⊙ yr^-1, about a factor of 50 above the de Jager rate. The Salasnich law is

Finally, van Loon et al. (2005) derived a mass-loss law for AGB stars and RSGs, but it cannot be applied if the circumstellar envelope is not very dusty. Nevertheless, the rate of van Loon et al. (2005) is interesting to compare with the de Jager rate, and this will be done with more details in the discussion. In the next section, we compare the de Jager rate with modern mass-loss measurements.

3. Comparison of the de Jager rate with observations

3.1. The 60-μm excess emission of RSGs in the solar neighbourhood

The RSGs of the solar neighbourhood are key targets for investigating mass loss, either because they are close to us like Betelgeuse or Antares, or because the membership to an OB association gives a distance indication and confirms the object to be a massive star. These RSGs have been originally listed by Humphreys (1978). Jura & Kleinman (1990) investigated mass loss of the most luminous RSGs. They consider a list of 21 objects more luminous than 10⁵ L_⊙  and estimate mass loss with the IRAS 60 μm flux. Josselin et al. (2000) consider 65 RSGs, present photometric JHKL measurements of most of them, and detect the CO millimeter emission in several cases. One of their conclusions is a large scatter of Ṁ for a given luminosity, a property that was also noted by JK90. Levesque et al. (2005) have revised effective temperatures, distances, and interstellar extinction for 74 RSGs.

A first list of RSGs was built from the list of 74 objects of LM05, supplemented with 21 objects from JB00. For these 95 sources, we collected optical (UBVI) and near-infrared photometry (JHKL), preferentially from Lee (1970). Data from Simbad, 2MASS, JB00 and Gezari et al. (1993) were also used. At mid-infrared wavelengths, IRAS measurements were adopted. Then, for most of the stars, photometry from the U band to the 60 μm band is available. Almost all these stars belong to OB associations, and the distance moduli and interstellar absorption given by LM05 were adopted. When a star was not examined in the work of LM05, we took the distance modulus of the association from Humphreys (1978) and the reddening was derived from the averaged reddening of the OB-type stars of the association. For several objects (e.g., VY CMa, VX Sgr), the distance and absorption come from recent, individual determinations.

From this primary list of 95 objects, we are interested in those presenting evidence of mass loss. For that purpose, we selected objects presenting an infrared excess, measured by the ratio of 12 μm flux to the dereddened K-band flux. Following JK90, only stars with f₁₂/f_K larger than 0.1 were selected. In addition, we imposed f₆₀ ≥ 2 Jy, in order to work with reliable 60 μm fluxes. This resulted in a list of 39 objects for which the photometric data are given in Appendix B. The characteristics of these 39 objects are given in Table 1. An additional remarkable RSG, NML Cyg, will be examined in the next subsection.

In order to compare the mass-loss rate of these RSGs with the de Jager prescription, we estimate Ṁ  from the infrared excess at 60 μm with the formulation of Jura and collaborators (JK90):

where Ṁ_60 μm  is in  M_⊙ yr^-1, V₁₅ is the wind expansion velocity normalized to 15 km s^-1, D_kpc is the distance in kpc, f₆₀ is the IRAS flux at 60 μm in Jy (assumed to dominate the photospheric flux at this wavelength), λ₁₀ is equal to the mean wavelength λ_m of the object spectral energy distribution, normalized to 10 μm, and L₄ is the luminosity in units of 10⁴ L_⊙. This formula is valid for a gas-to-dust mass ratio of 200, which corresponds to complete condensation of refractory elements for solar abundances, and we adopt this ratio here.

The wind speed in Jura’s formula is known for only a few RSGs (see Table 2). For all others, we choose to scale the value for this speed with the luminosity, and we adopt V ∝ L^0.35 following predictions of dust-driven wind modelling (Habing et al. 1994). For the RSGs of the solar neighbourhood, we find that V ≃ 20 × (L/10⁵)^0.35 km s^-1 roughly represents the observed values. For the LMC RSGs considered below, we find V ≃ 14 × (L/10⁵)^0.35 km s^-1 (see Appendix C for details).

Concerning the luminosity and mean wavelength, these quantities are calculated by integrating over the spectral energy distribution. We use the UBVIJHKL magnitudes and IRAS fluxes and correct the magnitudes for interstellar reddening following the extinction law of Cardelli et al. (1989). Between the central wavelengths of these spectral bands, interpolation is done linearly on the relation of log(λf_λ) as a function of log(λ). The accuracy of the derived luminosities (listed as L_phot in Table 1) depends on several factors. Once the distance is fixed with the distance modulus, factors like photometric uncertainties, variability or errors on E_B−V introduce uncertainty. For example, adopting JHK magnitudes smaller by 0.2 mag results a luminosity higher by  ≲11%, and a decrease of Ṁ by  ≲7%. Similarly, increasing E_B−V by 0.25 mag implies an increase in L by  ≲40%, and a decrease in Ṁ by  ≲16%. These variations are smaller than the accuracy of Jura’s formula.

In Table 1 we included the luminosities from LM05 (column L_Lev for the 26 objects originating from their study). Many determinations of luminosity agree very well between their and our studies. However, there are also a few significant discrepancies, up to a factor of 5 for V354 Cep. Because we use the same distance and reddening and almost the same photometry, the discrepancy must originate from the method for deriving L, but we could not trace exactly where the differences come from. Consequently, we used our luminosities for plotting Ṁ versus L (Fig. 2).

It can be seen in Fig. 2 that if we omit four exceptions labelled 1, 2, 3, and 7, the majority of points are clustered near the de Jager prescription line. For log L < 5.2, the de Jager line slightly overestimates the mass-loss rates (by a factor  ~1.5), while for log L > 5.2, the de Jager line passes below the stars PZ Cas and VY CMa, but above KW Sgr and VX Sgr, and decreasing the dJ88 rate by 1.5 would not improve the situation.

The stars α Ori, α Sco, and μ Cep (labelled 1, 2, and 3 respectively) present a low Ṁ here, but observations of their circumstellar gas suggest that they are plausibly deficient in grains (see next section). We note that our estimate of Ṁ  for μ Cep, based on the 60 μm flux, agrees well with that of de Wit et al. (2008) derived from imaging at 24 μm, after appropriate scaling of the outflow velocity.

There is also one star at very high luminosity: EV Car (labelled 7), with L = 6.7 × 10⁵ L_⊙  and Ṁ = 1.3 × 10^-5  M_⊙ yr^-1. The huge luminosity is owing to the distance modulus given by Humphreys (1978), (m − M) = 13.3, corresponding to 4.2 kpc. This puts the star far beyond the Car OB1 association. If we had taken the distance of this association (2.5 kpc, adopted by JK90), we would have found L = 2.4 × 10⁵ L_⊙  and Ṁ = 4.6 × 10^-6  M_⊙ yr^-1, and the location of the star in Fig. 2 would be very close to the de Jager line. But to our knowledge, there is no observational evidence for EV Car to be member of Car OB1.

Fig. 2 Comparison of de Jager rate (plotted for Teff = 3500 K and 4000 K) with mass-loss rates of 39 RSGs listed in Table 1. The mass-loss rates are based on the study of dust and obtained with Jura’s formula. The labels indicate a few wellknown RSGs as follows: 1: α Ori; 2: α Sco, 3: μ Cep, 4: S Per, 5: PZ Cas, 6: VY CMa, 7: EV Car (at 4.2 kpc); 8: VX Sgr. The small cross represents EV Car if a smaller distance of 2.5 kpc is adopted. At the upper left corner, error bars represent uncertainties of  ±30 percent in luminosity, and a factor of  ±2 in Ṁ.

3.2. Circumstellar gas of RSGs in the solar neighbourhood

Only a few RSG winds have been detected through observation of the circumstellar gas. Methods of detection include measurements of the H i 21cm emission (details on observations and mass-loss determination can be found in Bowers & Knapp 1987), millimetric emission of CO (Olofsson 2004; Ramstedt et al. 2008), submillimeter emission of C i (Keene et al. 1993; Huggins et al. 1994), emission of scattered resonance lines of Na i and K i (Mauron 1990; Guilain & Mauron 1996; Gustafsson et al. 1997), and observation of the H ii region around Antares (Kudritzki & Reimers 1978; Reimers et al. 2008).

For eight stars we collected in Table 2 the mass-loss rates obtained with these methods, giving detailed reference for the adopted parameters (D and E _B−V) used to derive luminosities. When Ṁ  was obtained in original papers with a different distance, we assumed Ṁ ∝ D². The expansion velocities V are from Huggins et al. (1994) for α Ori, Woodhams (1993) for S Per and PZ Cas, and from Kemper et al. (2003) for VY CMa, VX Sgr, NML Cyg. Finally, for μ Cep, the situation is unclear (for details see Bernat 1981; the discussion in Le Borgne & Mauron 1989; and Kemper et al. 2003) and we adopted 20 km s^-1.

Several remarks have to be made on individual objects. For the most observed star (α Ori), there is good agreement among the rates based on H i, C i and K i (Bowers & Knapp 1987; Huggins et al. 1994; Glassgold & Huggins 1986; Plez & Lambert 2002). When the distance given by Hipparcos is adopted (parallax 7.63  ±  1.64 mas, giving 130 pc  ± 30 pc), one obtains Ṁ ≈ 1 to 2 × 10^-6  M_⊙ yr^-1. The distance proposed by Harper et al. (2008) is 197 ± 45 pc and would imply Ṁ ~ 3 × 10^-6  M_⊙ yr^-1. For this star, the interstellar extinction A_V is not very well determined, being between 0.3 and 0.8 mag according to Lambert et al. (1984). We have chosen the average of these values implying L = 56   000 L_⊙, but the main uncertainty comes from the error on parallax, and L is known within  ±50 percent.

In the case of VX Sgr, the distance has recently been measured by Chen et al. (2007) with SiO maser proper motions (D = 1.57 ± 0.27 kpc), and this distance means that the star belongs to the Sgr OB1 association. The average interstellar extinction to the OB stars of this association is E _B−V = 0.5 (from data of Humphreys 1978), and the luminosity is 3.4 × 10⁵ L_⊙.

For VY CMa, a distance of 1.14 ± 0.10 kpc has been obtained by Choi et al. (2008) through astrometric monitoring of H₂O masers. This distance is smaller than the one often used. For example, Decin et al. (2006) have adopted 1.50 kpc for their detailed study of circumstellar CO lines. The ejection rate obtained by Decin et al. is highly variable. The episode with the highest mass-loss rate (~3 × 10^-4  M_⊙ yr^-1) has a duration of only 100 yr. The average mass-loss rate is about 7 × 10^-5  M_⊙ yr^-1. Scaled to the distance of 1.14 kpc, the average rate becomes 4 × 10^-5  M_⊙ yr^-1. The new smaller distance means that the star is less luminous than previously believed. We find L = 2.9 × 10⁵ L_⊙. Because the star is extremely red (the flux averaged wavelength is around 8 μm, compared to  ~2   μm for the majority of objects), this luminosity does not depend much on the choice of the extinction: for E _B−V in the range 0 to 1, the luminosity is between 2.7 and 3.5 × 10⁵ L_⊙.

The points representing these eight stars with measured gas mass-loss rates as a function of luminosity are shown in Fig. 3. The error bars correspond to  ±30 percent in luminosity and a factor of  ±2 of uncertainty on the mass-loss rate. The actual uncertainties on Ṁ are probably higher than that but difficult to quantify. The points are not very far from the de Jager line. Betelgeuse and Antares agree for Ṁ ≈ 10^-6  M_⊙ yr^-1 at a luminosity of  ~65 000 L_⊙. S Per is a factor of 4 above the line and μ Cep is a factor of 10 below the line. It can be seen that on average the de Jager prescription agrees reasonably well with measurements of the gas mass-loss rates.

Fig. 3 Mass-loss rates (from observations of the gas) versus luminosity for the eight RSGs of Table 2. The lines are the de Jager rates for 4000 K (solid line) and 3500 K (dotted line).

Because Ṁ_60   μm  was used in the previous section, we can compare it for these stars with the mass-loss rates derived from observation of the gas Ṁ_gas. This is done in Fig. 4 and Table 2. Figure 4 shows that the agreement between the two rates is satisfactory for some stars (S Per, NML Cyg, VX Sgr), but there are also differences of a factor of  ~4 for VY CMa and μ Cep. These differences may arise because different regions of the envelope are probed when comparing different techniques. For α Ori and α Sco, Ṁ_60   μm is too low by a factor of  ~10, plausibly due to incomplete dust formation. There is a trend in Fig. 4 that for low Ṁ, grain condensation may be incomplete, but obviously many additional sources should be examined with independent estimates of the dust mass-loss rate and the gas mass-loss rate to confirm this trend.

Fig. 4 Mass-loss rates from observation of the gas versus mass-loss rate from observation of the dust (the 60 μm emission) for the eight RSGs of Table 2.

3.3. Infrared measurements in the Large Magellanic Cloud

The mass-loss rate of RSGs in the Large Magellanic Cloud (LMC) has been investigated in several works, with various data and various methods. All works are based on the circumstellar dust emission. Reid et al. (1990) used IRAS fluxes and Jura’s formula. van Loon et al. (2005) exploited MSX, ISO and IRAS fluxes (their Table 3) and analysed the spectral energy distribution with the dust radiative transfer code DUSTY, which provides Ṁ. More recently, Groenewegen et al. (2009) exploited Spitzer results in addition to all previously obtained data. In particular, Spitzer data include photometry from 3.6 to 24 μm and spectroscopy from 5.2 to 37 μm. Groenewegen et al. (2009) fit the data also with dust radiative transfer models. (The work of Bonanos et al. (2010) also treats mass loss of RSGs in the Magellanic Clouds and is considered in the Discussion.)

The mass-loss rates given in the works cited above cannot be compared directly, because their assumed parameters, such as the expansion velocity of the wind or the gas-to-dust mass ratio are different. We assume here that the LMC RSGs have a metallicity lower than that of the Sun so that complete condensation of refractory metals leads to a gas-to-dust mass ratio ψ = 500. We also assume that all RSGs have a wind velocity v(L) depending on the luminosity L as found in Appendix C i.e. v(L) = 14(L/10⁵)^0.35 km s^-1. Consequently, we multiplied the Ṁ  of Reid et al. (1990) by 2.5  ×  v(L)/15 (because Jura’s formula used by Reid et al. supposes ψ = 200 and a wind speed of 15 km s^-1). We multiplied the Ṁ  of van Loon et al. (2005) by v(L)/v′ where v′ is their published wind velocity provided by DUSTY. Finally, we multiplied by 2.5 × v(L)/10 the Ṁ  of Groenewegen et al. (2009) because they adopt ψ = 200 and a wind speed of 10 km s^-1.

The results are given in Tables 3−5. As far as possible, we did not include AGB stars in these tables. More precisely, we followed Groenewegen et al. (2009) by considering that all objects with L ≳ 1.2 × 10⁵ L_⊙ are genuine RSGs, and that objects with L between  ~5 × 10⁴ and 1.2 × 10⁵ L_⊙ can be considered as RSGs only if their amplitude of variation in the optical I band was less than 0.45 mag. This selection implied that seven objects in the Groenewegen sample were kept (marked with “s” after their names in the first column of Table 5) and nine were rejected.

Tables 3−5 include names, J2000 coordinates, luminosities, mass-loss rates (scaled as explained above and in units of 10^-6  M_⊙ yr^-1). The last column in Tables 3 and 4 is the ratio of the Ṁ  from Reid et al. (or from van Loon et al.) and the Ṁ  from Groenewegen et al. (2009) for objects belonging to both samples.

Figure 5 shows these scaled Ṁ  as a function of L from the three works mentioned above. We also show the de Jager rate for 4000 and 3500 K exactly as for previous plots. It is obvious that there is a difference between the recent rates from Groenewegen sample on one hand (the median rate is 2.3 × 10^-6  M_⊙ yr^-1, 30 RSGs represented by circles), and, on the other hand, the Reid sample (median 9.0 × 10^-5  M_⊙ yr^-1; 15 crosses) or the van Loon sample (median 3.6 × 10^-5  M_⊙ yr^-1; 10 small squares). One can also consider the objects in common (see Tables 3 and 4): these objects indicate that the Reid mass-loss rates are  ~35 times higher than those of Groenewegen, and that the van Loon mass-loss rates are  ~10 times higher than those of Groenewegen. One notes however that for WOH-G64 the Ṁ  from van Loon and Groenewegen agree reasonably well (within a factor 1.6). In contrast, for SP77-46-44 (alias W60 A27), they disagree by a factor 37.

There are two assumptions in Reid et al. (1990) that can explain (at least in part) why their rates are so high. First, there is probably an overestimate of the flux averaged wavelength: Reid et al. (1990) take λ₁₀ = 1, which is not reached even by NML Cyg (see Table 2). For the optical RSGs making the Reid sample, λ₁₀ is likely close to 0.2 only. Secondly, Reid et al. (1990) overestimate the f₆₀/f₂₅ IRAS flux ratio used to obtain f₆₀. They take 0.50 instead of  ~0.20. A ratio of 0.23 ± 0.03 is reached only by dusty Galactic RSGs (VX Sgr: 0.19, VY CMa: 0.22, PZ Cas: 0.24, NML Cyg: 0.26). Consequently the Reid et al. mass-loss rates have to be decreased by a factor  ~12.

Concerning the van Loon rates, the reasons why they are generally higher than those of Groenewegen are not clear. With Spitzer data, small infrared excesses become measurable and are more accurate, while this was not the case with MSX, ISO, or IRAS data available for the van Loon et al. work. Therefore, there may be a bias in favour of large Ṁ in the van Loon sample compared to the Groenewegen sample. Another possibility is that the higher spatial resolution of Spitzer allows to avoid source confusion and contaminating sources. This may be in particular the case for SP77-46-44. For this source, the IRAS fluxes at 12 and 25 μm are 0.26 and 0.18 Jy, while Spitzer photometry (from Bonanos et al. 2009) indicates 0.19 Jy at 12 μm (through interpolation from the 8 μm and 24 μm fluxes) and 0.11 Jy at 25 μm. So these fainter Spitzer fluxes suggest a smaller infrared excess and may go some way to explain the large discrepancy for this object.

Finally, if we consider the rates by Groenewegen et al. (2009) and their positions with respect to the de Jager lines (see Fig. 5), one can note a rather good agreement for 12 objects with luminosities lower than 1.6 × 10⁵ L_⊙(with the exception of W60-A72 at L = 115   000 L_⊙, Ṁ = 8.4 × 10^-5  M_⊙ yr^-1; in Fig. 5 this object is the circle located in the middle of Reid’s plus signs). But at higher luminosities, the Groenewegen rates are below the dJ88 lines by an average factor  ~8 (13 objects), with the exception of WOH-G64 where the de Jager rate is  ~10 times lower than observed. It is also remarkable that at high luminosities, two objects from the van Loon sample are quite close to the dJ88 line.

Fig. 5 Mass-loss rates plotted as a function of luminosity for the LMC RSGs. Three samples are plotted: Reid et al. (1990) (15 small crosses), van Loon et al. (2005) (10 small squares), and Groenewegen et al. (2009) (30 circles). The de Jager rates are represented as in previous figures by solid and dotted lines for 4000 K and 3500 K, respectively.

4. Discussion

The goal of this paper is to see whether the de Jager prescription is still applicable in view of new results obtained since 1988 and concerning the RSGs and their winds. One important fact is that for only eight RSGs in the solar neighbourhood the direct detection of the circumstellar gas is available. In all other cases, only dust can be seen. We find that for these eight RSGs, the mass-loss rate estimated from observations of the gas is in most cases within a factor of 4 of the de Jager prescription for L < 2 × 10⁵ L_⊙ (Fig. 2). For higher luminosity it is a factor of 10. These deviations have to be considered in light of the following characteristics of the RSG winds. The winds are changing with time (timescale  ~10²−10⁴ years), and the rate of mass loss and the expansion velocity vary. This is proved by high-resolution spectroscopy of the circumstellar absorption lines, which display discrete velocity components (on μ Cep, Bernat 1981), by direct imagery (α Ori, Plez & Lambert 2002) or by study of the CO millimetric lines (VY CMa, Decin et al. 2006). Also, the winds are often not spherically symmetrical, as revealed by some CO line profiles (PZ Cas, Woodhams 1993) or direct imagery (HST imaging of NML Cyg, Schuster et al. 2006). Given these properties, the measurements of Ṁ  with often limited data and simple modelling are necessarily poor in accuracy and do not reflect an average mass-loss rate. Therefore, the agreement with the de Jager prescription seems acceptable.

For other, much more numerous RSGs, the gas is not detectable and only dust can be seen through its infrared radiation. When Jura’s formula is adopted for deriving Ṁ, the resulting Ṁ also agree with de Jager lines, except for some rare stars like μ Cep, which has a very low dust content. In general, the agreement is within a factor of 4 (Fig. 2). Part of this deviation may be caused by a natural scatter of Ṁ  at a given luminosity (see the characteristics of winds mentioned above) and another part may be owing to the use of Jura’s formula.

4.1. Verification of the Jura’s formula

In order to check Jura’s formula, we have considered the work by Verhoelst et al. (2009) who exploit ISO-SWS spectra (in addition to the full spectral energy distribution) and use state-of-the-art dust modelling. In particular, they include several types of grains such as melilite, olivine, alumine, carbon and FeO. Together with results concerning the dust condensation sequence in RSGs, they derive mass-loss rates. We took into account their assumptions on distance, expansion velocity and gas-to-dust mass ratio and we applied appropriate scaling, so that we can compare their results to ours (Ṁ_60   μm in Table 2). We find that their Ṁ are 0.57, 0.88, 1.20 and 1.9 times ours, for α Ori, S Per, PZ Cas and μ Cep, respectively. In conclusion, the agreement with Jura’s formula is at most a factor of 2, which is very good, and there is no particular reason to have doubts on this formula.

4.2. Comparison with other mass-loss prescriptions

In Sect. 2, we mentioned in Fig. 1 a few mass-loss prescriptions. After our analysis of previous sections, we can reconsider this point. Concerning the RSGs in the solar vicinity, all observed mass-loss rates (from dust or gas observations) are well below the rate by Salasnich et al. (1999). For example, α Ori and α Sco are located at L ≈ 60   000   L_⊙, and their Ṁ are  ~1.5 × 10^-6  M_⊙ yr^-1. The Salasnich rate would give 3.4 × 10^-5  M_⊙ yr^-1, 20 times larger. Because they are based on mass-loss rates of Reid et al. which we have shown are overestimated, the Salasnich rate and the Feast rate are too large. The Reimers rate is as good as the dJ88 rate for objects with L ≲ 2 × 10⁵ L_⊙, but this is not the case for higher luminosity objects like NML Cyg or VY CMa.

4.3. Comparison with the van Loon formalism

It is appropriate to consider here the mass-loss formulation of van Loon et al. (2005). These authors derived a prescription for mass-loss rates of AGB stars and red supergiants. It is based on the study of evolved stars in the LMC, but is also valid for Galactic AGB stars and RSGs. An important feature of this prescription is that it is applicable only to dusty stars. Unfortunately, the dusty character of a circumstellar envelope is not possible to be derived from the basic stellar parameters, i.e. L, M and T_eff. Nevertheless, one can compare the results of this prescription to the de Jager rate. van Loon et al. (2005) derived effective temperatures from spectral classifications. Luminosities and mass-loss rates were determined by modelling the spectral energy distribution. The van Loon prescription is:

This prescription is shown in Fig. 6 along with the de Jager lines. Both are drawn for T_eff of 4000, 3500 and 3000 K. van Loon et al. (2005) have found that their recipe overestimates mass-loss rates for Galactic “optical” RSGs. We can see that clearly in Fig. 6, where the van Loon lines are well above the de Jager lines on which we found that most of the Galactic RSG are located (Figs. 2 and 3). However, the de Jager rate and the van Loon rate fairly agree with each other for the highest luminosities. Also, it is worth noting that the van Loon et al. prescription offers a significant spread of Ṁ for a given luminosity. The spread allowed by the deJ88 prescription is clearly very small.

We have collected in Table 6 the mass-loss rates for the eight stars possessing an Ṁ_gas based on observation of the gas (from Table 2). In this table, the T_eff values are taken from Lambert et al. (1984) for α Ori, from LM05 for S Per, PZ Cas, and μ Cep, from van Loon et al. (2005) for VY CMa, NML Cyg and VX Sgr, and from Kudritzki & Reimers (1978) for α Sco. Column 5 of this table gives the de Jager rates, and Col. 6 gives the van Loon rates (in Col. 6, the numbers in parenthesis concern “optical” RSGs and are just given for information). The ratios between van Loon rates and the observed rates are 2.8, 5.1, 2.2, 0.85 and 7.2 for S Per, PZ Cas, VY CMa, NML Cyg, and VX Sgr, respectively. By considering the median of these numbers, the van Loon rates are  ~2.8 times too large on average. For comparison, the ratios between de Jager rates and observed rates for the same stars are 0.20, 0.78, 0.45, 0.15 and 2.9. The median is 0.45 so that, the de Jager rates are  ~2.2 times too small on average. In conclusion, the de Jager and van Loon rates are almost equally accurate (or inaccurate), but one (de Jager) underestimates and the other (van Loon) overestimates the observed rates. The de Jager formalism is closer to the observed rates for non-dusty RSGs, although it is poor in the case of μ Cep (off by an order of magnitude).

Fig. 6 Mass-loss rate prescription of van Loon et al. (2005) and de Jager (1988) with Ṁ plotted as a function of luminosity and for three values of Teff. The de Jager lines are solid, dotted and dashed for 4000, 3500 and 3000 K respectively.

4.4. The behaviour of the de Jager rate with effective temperature

Figure 6 shows in particular the temperature dependance for the van Loon and de Jager prescriptions. It is surprising that for log L ≳ 4.5, the Ṁ of de Jager gets smaller for decreasing T_eff. We cannot explain this behaviour in view of the data concerning the 15 RSGs that dJ88 used when deriving their formula (these data are listed in Appendix A). In particular, VY CMa has the largest mass-loss rate of their small sample (2.4 × 10^-4~ M_⊙ yr^-1) and its temperature adopted by dJ88 is low, T_eff = 2840 K. In addition, the weight assigned to this star by dJ88 is considerably larger (60) than the weight of any other RSG in the sample (weights  ~1−4). Yet, the de Jager rate decreases for cooler temperatures. A possible explanation of this peculiar T_eff behaviour is the fact that the fit of the data for the dJ88 rate was made globally on all stars over the H-R diagram and not just “locally” on the RSGs.

4.5. Is the scatter of Ṁ at a given luminosity due to effective temperature?

An interesting question is whether the deviation of the measured Ṁ from the dJ88 rate is correlated with effective temperature. Thanks to the work of LM05, T_eff is known for the 27 first RSGs of Table 1 (from V589 Cas to TZ Cas). T_eff is also known for the 8 stars of Table 6. We adopt the infrared-based Ṁ from Table 1, but prefer when possible Ṁ_gas from Table 6. Figure 7 (panel a) plots the ratio (in log) of measured mass-loss rate to the de Jager rate. It shows a slight tendency of Ṁ decreasing with increasing T_eff. The standard deviation is 0.410 when all 31 stars are considered and it can be seen that 6 stars (of 31) have ratios larger than 4 or smaller than 1/4. Changing the de Jager rate by adding a factor would slightly reduce this scatter. Values of α between 6 and 10 are suitable. We show in panel b the resulting situation for (T_eff/3600)^-10. The standard deviation is 0.367 for all objects included. It is 0.296 if μ Cep and V396 Cen are omitted, and all other objects are within a factor of 4 of agreement. V396 Cen has a poor quality 60 μm flux, coded F in the IRAS point source catalog, and presents an anormally low f₆₀/f₂₅ flux ratio. Therefore its position in Fig. 7 is plausibly explained by an underestimated 60 μm flux.

Fig. 7 Panel a): discrepancies between measured mass-loss rates (MLR) and mass-loss rates predicted by the de Jager prescription, plotted versus Teff, for Galactic red supergiants. The horizontal dotted lines indicate discrepancies by factors 4 and 1/4. Panel b): same as panel a) when the de Jager rate is multiplied by (Teff/3600)-10.

4.6. The RSGs in the Magellanic Clouds and the dependence on metallicity

We have seen that the mass-loss rates in the LMC from Groenewegen et al. do not follow fully the de Jager rate. For L between 63 000 and 160 000, there is reasonable agreement, but at larger L , the de Jager rate is off (too large) by a factor of  ~8. Recently, Bonanos et al. (2010) have also considered the mass-loss rates of RSGs in the LMC and SMC. Their sample of RSGs (~100 sources in the LMC, 34 in the SMC) is taken from a list of luminous objects having optical spectral classification and photometry from 0.3 to 24 μm obtained during the Spitzer SAGE survey of the Magellanic Clouds (Meixner et al. 2006). Bonanos et al. (2010) estimate mass-loss with a Ṁ versus the K −  [24]  relation derived from model dusty envelopes.

From the data plotted in their Fig. 17, which show important scatter at a given luminosity, we derive median mass-loss rates in several bins in L. The 5 bins for the LMC have respectively 15, 24, 28, 25 and 5 data points. The 4 bins for the SMC have 6, 10, 11 and 7 data points. The resulting Ṁ versus L relations are shown in Fig. 8 (histograms). Both the SMC and LMC data suggest that Ṁ increases with luminosity if log L is less than  ~5.3, as already concluded by Bonanos et al. (2010). It can be noted that the LMC bin with the highest luminosity suggests that Ṁ could saturate, but the low number of points makes it difficult to be sure of that (see these 5 points at the highest luminosity in Fig. 17 of Bonanos et al. 2010). However, we note that this is quite consistent with our Fig. 5 and its high-L, low-Ṁ group of stars, as mentioned above.

In order to investigate how the RSG mass-loss rates depend on metallicity, we can compare the median rates obtained above with the dJ88 prescription scaled by various Z values. Here we adopt Z(LMC) = Z_⊙/2, and Z(SMC) = Z_⊙/5. Figure 8 shows the de Jager lines for warm effective temperatures which are suitable for the Clouds. It shows that, for the SMC, the observed rates are between the de Jager rate scaled by Z^0.5 and Z, and a Z^0.7 scaling would fit the data quite well. As for the LMC, the situation is less clear because the metallicity decrease is only a factor 2 with respect to solar. If we omit the most luminous bin, a Z^0.7 scaling is not unreasonable, but a Z^0.5 is also possible.

Fig. 8 Mass-loss rates versus luminosity for Magellanic Clouds. The (median) data of Bonanos et al. (2010) are represented by histograms (see text). The solid and dotted lines show the dJ88 rates for 3500 and 4000 K scaled for various metallicity dependences, as indicated in the panels.

5. Conclusions

The de Jager prescription used in evolution codes for RSGs was established in 1988 when the mass-loss rates of stars located over all the H-R diagram were fitted as a function of L and T_eff. The fit was not specifically made for this type of stars, and the goal of this work was to compare the de Jager rate to estimates of Ṁ based on data acquired after 1988. Our main conclusions are the following:

• 1.

  When Jura’s formula is adopted to get an observationalestimate of Ṁ  based on the IRAS infrared excess at 60 μm and supposing a gas-to-dust ratio of 200 (corresponding to complete condensation of refractory material), it is found that the Galactic RSGs are satisfactorily distributed around the de Jager line. The deviation from de Jager rate is in general less than a factor of 4.

• 2.

  Only eight Galactic RSGs have their circumstellar gas detected, i.e. through detection of H i or CO emission, or other methods permitting to estimate Ṁ from observation of the gas. For these stars, Ṁ agrees reasonably well with the de Jager rate. The most discrepant cases are found at high luminosity, where the de Jager prescription underestimates Ṁ of NML Cyg by a factor of 7, but overestimates Ṁ for μ Cep by a factor of 10 (see Table 3).

• 3.

  The Spitzer data from Bonanos et al. (2010) show that the mass-loss rates of RSGs in the Small Magellanic Cloud agree on average with the de Jager rate scaled for metallicity by (Z/Z_⊙)^0.7.

• 4.

  A similar behaviour is less clear with RSGs of the Large Magellanic Cloud. For luminosities lower than 1.6 × 10⁵ L_⊙, the median mass-loss rates roughly follow the de Jager prescription and its increasing trend with luminosity. These measured rates are also roughly compatible with the Z^0.7 scaling found for the SMC. However, for higher luminosities, one finds that the mass-loss rates level off at a rate of  ~3 × 10^-6  M_⊙ yr^-1 and are smaller by a factor of  ~8 than the (unscaled) dJ88 rate (with the exception of WOH-G64).

• 5.

  Finally, although the dJ88 prescription presents some odd aspects that can be criticized (e.g., its behaviour with T_eff), it passes several tests fairly satisfactorily (listed 1, 2, 3, and part of 4 above). Therefore, we think it more appropriate to keep it unchanged in the stellar evolution models, but to apply a Z^0.7 factor for metallicity dependence. The LMC mass-loss saturation issue needs to be clarified. We hope to further contribute to that subject in the future.

Acknowledgments

We thank our referee, J. Th. van Loon, for many remarks that considerably improved the paper. N.M. warmly thanks Olivier Richard for his help concerning computers, and Tim Kendall and Astrid Peter for their help concerning English. N.M. also thanks Raphael Hirschi for discussions about the Geneva code. We acknowledge the use of the database produced by the Centre de Données de Strasbourg (CDS). This research was partly supported by the French National Research Agency (ANR) through program number ANR-06-BLAN-0105.

References

Online material

Appendix A: List of red supergiants considered by de Jager et al. (1988)

The list of RSGs considered by dJ88 are given in Table A.1. The data from dJ88 are the stellar effective temperature T₁ (K), the luminosity L₁ (L_⊙) and the mass-loss rate Ṁ₁ in  M_⊙ yr^-1. The columns T₂ , L₂ and Ṁ₂ give the same quantities as derived in this work. Many mass-loss rates used by dJ88 (Ṁ₁) came from the exploitation and analysis of infrared flux measurements between 3.5 and 11.5 μm (Gehrz & Woolf 1971). The mass-loss rates of Hagen (1982) are derived from flux measurements at 20 and 25 μm. The mass-loss rates of Sanner (1976) are based on high-resolution spectral profiles of strong resonance lines. For objects with Ṁ₁ based on several individual determinations (from four to nine determinations for μ Cep and α Ori, respectively), see the details in de Jager et al. (1988).

Appendix B: Photometric data of the studied RSGs

Table B.1 presents the magnitudes in the UBVIJHK filters and the IRAS fluxes at 12, 25, and 60 μm that were used to calculate the flux averaged wavelength λ_m, and the luminosities L. If a magnitude in the above filters was not available, we used the following relations, which were established from the sample data:

Appendix C: Wind expansion velocities

Figure C.1 shows the wind expansion velocities of several RSGs plotted versus luminosity. The Galactic RSG data are from Table 2 of this paper. The LMC RSG data are from Marshall et al. (2004, their Table 1 for luminosities and their Table 2 for expansion velocities). The error bars represent typical uncertainties on luminosity,  ±30% for the Galaxy and  ±20% for the LMC. The straight lines have the slope predicted by Habing et al. (1994) for dust-driven winds and were used in Jura’s formula for red supergiants lacking a wind velocity measurement.

Fig. C.1 Wind expansion velocity, in km s-1, as a function of luminosity (in L⊙), for Galactic RSGs (lower panel), and LMC RSGs (upper panel). The straight line for Galactic RSGs corresponds to V = 20   (L/105)0.35 in km s-1. The straight line for LMC RSGs corresponds to V = 14   (L/105)0.35.

All Tables

All Figures

	Fig. 1 Comparison of several prescriptions for the RSG mass-loss rates (unit  M⊙ yr-1) plotted against luminosity (in L⊙). The de Jager prescription is represented by the two curved lines (solid line Teff = 4000 K, dotted line Teff = 3500 K).
In the text

Fig. 2 Comparison of de Jager rate (plotted for Teff = 3500 K and 4000 K) with mass-loss rates of 39 RSGs listed in Table 1. The mass-loss rates are based on the study of dust and obtained with Jura’s formula. The labels indicate a few wellknown RSGs as follows: 1: α Ori; 2: α Sco, 3: μ Cep, 4: S Per, 5: PZ Cas, 6: VY CMa, 7: EV Car (at 4.2 kpc); 8: VX Sgr. The small cross represents EV Car if a smaller distance of 2.5 kpc is adopted. At the upper left corner, error bars represent uncertainties of  ±30 percent in luminosity, and a factor of  ±2 in Ṁ.

In the text

	Fig. 3 Mass-loss rates (from observations of the gas) versus luminosity for the eight RSGs of Table 2. The lines are the de Jager rates for 4000 K (solid line) and 3500 K (dotted line).
In the text

	Fig. 4 Mass-loss rates from observation of the gas versus mass-loss rate from observation of the dust (the 60 μm emission) for the eight RSGs of Table 2.
In the text

	Fig. 5 Mass-loss rates plotted as a function of luminosity for the LMC RSGs. Three samples are plotted: Reid et al. (1990) (15 small crosses), van Loon et al. (2005) (10 small squares), and Groenewegen et al. (2009) (30 circles). The de Jager rates are represented as in previous figures by solid and dotted lines for 4000 K and 3500 K, respectively.
In the text

	Fig. 6 Mass-loss rate prescription of van Loon et al. (2005) and de Jager (1988) with Ṁ plotted as a function of luminosity and for three values of Teff. The de Jager lines are solid, dotted and dashed for 4000, 3500 and 3000 K respectively.
In the text

	Fig. 7 Panel a): discrepancies between measured mass-loss rates (MLR) and mass-loss rates predicted by the de Jager prescription, plotted versus Teff, for Galactic red supergiants. The horizontal dotted lines indicate discrepancies by factors 4 and 1/4. Panel b): same as panel a) when the de Jager rate is multiplied by (Teff/3600)-10.
In the text

	Fig. 8 Mass-loss rates versus luminosity for Magellanic Clouds. The (median) data of Bonanos et al. (2010) are represented by histograms (see text). The solid and dotted lines show the dJ88 rates for 3500 and 4000 K scaled for various metallicity dependences, as indicated in the panels.
In the text

	Fig. C.1 Wind expansion velocity, in km s-1, as a function of luminosity (in L⊙), for Galactic RSGs (lower panel), and LMC RSGs (upper panel). The straight line for Galactic RSGs corresponds to V = 20   (L/105)0.35 in km s-1. The straight line for LMC RSGs corresponds to V = 14   (L/105)0.35.
In the text