7.1. Matter Injection for M 3 and M 22
A straightforward estimate of the rate at which stars are evolving off the main-sequence can be made by counting the number of horizontal branch stars in an annulus where crowding is not severe. A comparison of the total cluster light with the light in this annulus will then give the total number of horizontal-branch stars, which coupled with a knowledge of the horizontal-branch lifetime gives the rate of evolution. Tayler & Wood (1975) give figures for this present rate for M 3 and M 22. These are reproduced, in a different format, in Table 6.
Table 6. Mass of gas and dust injected into the clusters since the last Galactic plane crossing. T - time since last Galactic plane crossing. The figures for the rate at which stars are evolving off the main sequence comes from Tayler & Wood (1975)
The amount of mass lost by each star is the difference between its mass on the main-sequence and that of its final white dwarf state. The present mass of stars leaving the main-sequence is about 0.8 (see e.g. Bergbusch & VandenBerg 1992). Richer et al. (1995) have determined a mass of 0.5 for the white dwarfs in M 4, giving a mass loss per star of about 0.3 .
Over the time since the last plane crossing, the rate of evolution off the main-sequence, the mass of such stars, the time taken to reach the white dwarf stage, and the mass of the resultant white dwarfs are expected to have been relatively unchanged. For example, over 100 Myr, the mass of stars evolving off the main-sequence changes by (e.g. Bergbusch & VandenBerg 1992). Thus, with little error, the present values can be taken as average values
The amount of dust injected into the clusters may be calculated if the division into gas and dust of the matter ejected by stars is known. The dust-to-gas ratio will depend on the mass and metallicity of the stars and their exact evolutionary stage when the mass-loss occurs. A good review of this is given by Gehrz (1989). Following the discussion in Sect. 2.3, we assume a typical value of for the outflow of evolved stars, but scaled by the metallicity of the GCs relative to solar (cf. KGC95). We take scaling factors of 0.038 for M 3 and 0.027 for M 22, based on Kraft et al. (1995) and Brown & Wallerstein (1992) respectively. The total masses of gas and dust injected into the clusters are given in Table 6.
7.2. The conversion from flux to dust mass
Assuming that any dust emission at millimetre wavelengths is optically thin, we can write
where is the flux density, the beamsize, the optical depth and the Planck function at frequency and dust temperature . Assuming a standard model for the distribution of stars in a GC, and that the dust is distributed in the cluster like the stars, the expected dust temperature is in the 40...80 K range, depending on factors such as metallicity and grain emissivity (Angeletti et al. 1982). We assume three values of here, the expected 40 K and 80 K, and also 20 K, to explore the possibility that the dust may be cold, as discussed by Forte et al. (1992).
Here a is grain radius, the bulk density of grain material, the absorption efficiency of the grain material and the angular diameter of the beam.
Table 7. 3 upper limits on optical depths and dust masses
The deduced dust mass is clearly sensitive to the dust absorption efficiency assumed. In this context we first note that the dust in the Galactic interstellar medium has suffered considerable processing by supernova shocks, mantle formation and polymerization etc. (Whittet 1992). As the dimensions of a GC are typically pc, and the environment experienced by an interstellar grain in a GC is substantially different to that experienced by a grain in the Galactic interstellar medium, it seems plausible that the interstellar dust in GCs will bear little resemblance to 'standard' Galactic interstellar dust and may more likely resemble the pristine 'stardust' ejected by stars into the GC. Indeed, the typical distance between stars in the core of a GC is pc, comparable with the dimensions of the circumstellar dust shell of an evolved star. Thus although we are searching for interstellar dust in GCs it may be argued that we should use the absorption efficiency appropriate to circumstellar dust. Unfortunately however the absorption efficiency of circumstellar dust at millimetre wavelengths is generally very poorly known, and is usually based on extrapolation from shorter wavelengths (e.g. Hoare 1990), or on millimetre observations of young (Gear, Robson & Griffin 1988) or highly evolved (Knapp, Sandell & Robson 1993) objects. Extrapolation requires assumptions about the index of the dust, defined by . For grains in the Galactic interstellar medium (e.g. Wright et al. 1991; Fischer et al. 1995), , whereas for circumstellar dust (Knapp et al. 1993).
In their study of 47 Tuc Gillett et al. (1988) effectively assumed ( in µm) - appropriate to 'dirty silicate' - and extrapolated to 100 µm assuming ; extrapolating to 1100 µm gives for 0.1 µm grains. KGC95 used the well-known Hildebrand (1983) opacity, corresponding to ( in µm), significantly lower than the values used by Gillett et al. (1988). Extrapolating the Hildebrand opacity to 1000 µm assuming gives for 0.1 µm grains. LR90 used Draine & Lee's (1984) optical constants for 1 µm silicate grains at the IRAS wavelengths; the Draine-Lee values for µm are (silicate) and (graphite) for 0.1 µm grains (see also Draine 1985). The Draine-Lee values however were tailored to describe the optical properties of Galactic interstellar dust and, for reasons discussed above, may not be appropriate to GC grains. It may be argued that more reliable values follow from millimetre observations of evolved circumstellar envelopes and recently Knapp et al. (1993) have deduced (silicates) and (graphite) at 1100 µm, and unit index. We therefore calculate the dust masses using two values of , namely at 1100 µm (based on the millimetre observations and on 'dirty silicates [Gillett et al. 1988]), and at 1100 µm, based on the Draine (1985) values for silicate grains; we assume a index of unity.
The resulting upper limits on the dust mass are listed in Table 7, for three possible values of grain temperature, and the two assumed values of ; in each case a grain radius µm has been assumed. The limits on the dust mass derived from the lower for M 3 in Table 7 may be compared with that obtained by KGC95, . The tightest limits found by LR90 were (M 3) and (M 22) for 1 µm grains and . On the other hand, our preferred value of is similar to that used by Gillett et al. (1988) in their study of 47 Tuc, and our dust mass limits are comparable with their detection of in 47 Tuc. However the in 47 Tuc and our upper limits are all increased by a factor with the Draine (1985) values of but, for reasons already outlined above, we suggest that the higher values of are more likely to be appropriate to GC dust.
We have used the upper limits on IR flux for M 3 and M 22 from LR90 and KGC95, converted them (where appropriate) to flux limits and thence to upper limits on mass, using the same (40 K, 0.1 µm, 3.5 g cm-3 respectively) and
From LR90 we take the HCON1 flux limits that lead to the tightest upper limits on dust mass, which are for the 60 µm IRAS fluxes; we use the 100 µm data from KGC95. We then determine the limits on dust mass per unit area and per unit beam, so that upper limits obtained by the various methods can be directly compared on an equivalent basis. The IRAS beam is taken to be rectangular, at 60 µm for the HCON data (LR90), and at 100 µm (KGC95); in all cases the dust is assumed to be uniformly spread over the beam. The mass of dust per unit area and the mass of dust per beam follow directly from Eq. (1). The corresponding equivalent mass limits are given in Table 8.
Table 8. Equivalent upper limits on dust mass
It is apparent that the upper limits on dust mass from the present work are comparable with or, in the case of M 22, better than those deduced from IRAS data. On the other hand, while the upper limits per unit area and per unit beam derived from the present data are (again in the case of M 22) close to those obtained from the IRAS data, the corresponding IRAS limits for M 3 are much tighter than they are for the JCMT data. Comparison of the expected dust masses in Table 6 - calculated for the first time on the basis of reliable GC orbits - and the upper limits listed in Table 7 shows that we are now close either to detecting the dust in GCs, or to pushing the dust limits well below that expected on the basis of standard injection models. The present work also shows the considerable potential for pushing the mass limits down when our planned ISO observations of GCs are carried out, and when new sensitive bolometer arrays [such as SCUBA on the JCMT (Cunningham & Gear 1990)] become available on millimetre telescopes.
7.3. Distribution of dust
The chop throw of means that the reference positions were at radii comparable to the radii inside which half the cluster light is contained, as given in Table 1. If the dust distribution is smooth and concentric with the distribution of stars in the cluster, as in the models of Angeletti et al. (1982), and if the dust is distributed either like the stars or on a more compact scale, then the emission at the reference positions would be well below that at the cluster centres and should have been detectable. Thus for both clusters the size of the chop should have been adequate.
However, the projected beams of the JCMT (radii of 0.33 pc for M 3 and 0.11 pc for M 3) were significantly smaller than the scales of the light distribution, so if the dust is distributed like the stars then much of the dust will be outside our beam. However, there are arguments to suggest that any dust will be concentrated to the cluster core, see e.g. Angeletti et al. (1982). Thus while it is possible that there is more dust in the cluster than our upper limits imply, this is not necessarily the case.
If there is dust in these clusters but it does not lie at the cluster centres, as proposed by Forte et al. (1992) for NGC 6624, then again this search would miss it. If by chance the dust was located in the reference position instead of at the cluster core, a negative signal would be found; however no such negative signals were obtained.
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998