1. Introduction

     We have obtained accurate relative abundances for a sample of 21 mildly metal-poor field stars from the analysis of high resolution, high signal-to-noise spectra (Jehin et al. 1999a). Looking for correlations between the element abundances, we found that the -elements and the iron-peak elements are well correlated with each other, and the abundances of the rapid neutron-capture elements (r-process elements) are well correlated with those of the -elements, which is in agreement with the generally accepted idea that those elements are produced during the explosion of massive stars. For the slow neutron-capture elements (s-process elements), we find that the stars can be separated into two subpopulations. For those in PopIIa, the abundances of the s-process elements vary little while that of the elements increases up to a maximum value. The stars in PopIIb show a large range in their s-process elements abundances, while they show a constant and maximum value for the abundances of the elements. We called this behavior the "two-branches diagram". To explain this result, we have developed the EASE scenario, which links the metal-poor field stars to the globular clusters. The observations and the EASE scenario are described in more details in Jehin et al. (1999a, 1999b).

     One crucial piece of this scenario consists in explaining how unevolved stars (PopIIb) can get enriched in s-process elements while the elements abundances remain constant. The s-process elements are mainly produced in asymptotic giant branch (AGB) stars, where they are brought to the surface through dredge-up processes. The AGB stars lose a large fraction of their mass through stellar winds or superwind events, releasing the s-process elements enriched gas in the interstellar medium. Main sequence stars can accrete this matter, thereby enriching their surface abundances in those elements.

     The idea that the gas ejected by the AGB stars in globular clusters can be accreted by other stars in the cluster is not an entirely new one. Observations of globular clusters show that they contain much too little gas or dust, compared to what is lost by their AGB stars, and globular cluster stars also show many abundance anomalies. Many authors have been intrigued by the fate of the gas in globular clusters, and among them, Scott & Rose (1975), Faulkner & Freeman (1977), VandenBerg & Faulkner (1977), VandenBerg (1978), and Scott & Durisen (1978), and accretion has already been suggested as a plausible mechanism to explain abundance anomalies (D'Antona et al. 1983, Faulkner 1984, Faulkner & Coleman 1984, Smith 1996).

     In order for accretion to be efficient, the gas density should be as high as possible, the relative velocity of the accreting star with respect to the gas should be low, and the time during which the accretion takes place should be long.

2. Mass Lost by AGB Stars

     When a star reaches the AGB phase it will lose a large fraction of its mass. We assume here that those stars lose their mass instantaneously, so that the rate of mass injection in the cluster is given by

(1)

where M_ej is the mass ejected by a star reaching the AGB, and dN/dt is the rate at which the cluster stars reach the AGB. It can be rewritten in terms of the mass spectrum dN/dM and the rate dM/dt at which stars leave the main sequence:

(2)

We assume a power-law mass spectrum,

(3)

where is the power-law index of the initial mass spectrum, M_l and M_u are its lower and upper mass limits, and M_cl,0 is the globular cluster's initial mass. The rate dM/dt can be derived from the following equation, relating the mass M of a star (in units of 1M) to its lifetime on the main sequence t (in years) (Bahcall & Piran 1983)

(4)

     For the mass ejected by each star as it reaches the AGB, we fitted the results from Weidemann & Koester (1983):

(5)

where the masses are again expressed in units of 1M.

     Finally, we have

(6)

where t₁ is the time at which the highest mass star (M = M_u) reaches the AGB.

     Using typical values for the parameters, M_u = 10M, M_l = 0.1M, and  = 2.35 (the Salpeter's IMF), we obtain that about 20 % of the cluster's initial stellar mass is returned to the cluster as gas in 10 Gyrs. Most of the gas is ejected into the ISM within the first Gyr.

(1)

3. Fate of the Gas in Globular Cluster

     Smith (1996) has derived criteria for the fate of the ejected gas in globular clusters. By comparing the stellar-ejecta speed to the cluster's escape speed, he obtained the following criterion for retention of the gas in the cluster:

(7)

where M_cl is the mass of the cluster (in units of 1M), is the cluster's effective radius (in parsec), r_c and r_t are the cluster's core and tidal radii, f_c is the fraction of the initial ejecta energy lost via radiative cooling and v₁₀ is the stellar-ejecta speed in units of 10 km/s. He also shows that many Galactic globular clusters obey this relation, using present-day parameters. Therefore, in tightly bound globular clusters, the gas released by the AGB stars can be retained in the cluster's center, forming a central reservoir of gas. The gas will accumulate in the cluster's center between passages through the galactic plane, at which time the cluster will be swept clean of the gas.

     If the cluster is not dense enough to retain the gas, it can escape from the cluster either through a continuous wind, when the cluster crossing time for individual ejecta shells is long compared to the average time between individual mass-loss episodes, or through stochastic mass loss events. The criterion for establishing a continuous wind is

(8)

where ₁₉ = (dM/dt)/M is the inverse characteristic time of mass loss in units of 10^-19s^-1.

4. Accretion of Gas in Globular Clusters

     The accretion rate is determined by the Bondi's formula (Bondi, 1952):

(9)

where M_s is the mass of the accreting star, _g is the unperturbed gas density, v_rel is the relative velocity of the star with respect to the gas, c_s is the sound speed in the gas, and is a parameter of order 1.

4.1. Accretion from a Reservoir of Gas

     If we assume that the accretion takes place from a reservoir of gas, and that the density is homogeneous within the gas reservoir, we have

(10)

where M_g is the total (integrated) mass of gas ejected by the AGB stars in the globular cluster, M_a is the mass of gas which has been re-accreted by cluster stars, M_esc is the mass of gas lost from the cluster when crossing the galactic plane, and r_g is the radius of the gas reservoir. The mass of the accreting star, M_s = M_s,0 + M_s,a, where M_s,0 is its initial mass and M_s,a is the mass it has accreted. A given star will only spend a fraction of its lifetime inside the gas reservoir, as its orbit in the cluster will take it out of the gas reservoir. We therefore write

(11)

The total amount of gas which is accreted by the cluster stars is obtained by integrating the mass accreted by each star over the mass spectrum:

(12)

where M_l is the lower limit of the mass spectrum dN/dM, and M_TO is the turn-off mass.

     Assuming that the radius r_g of the gas reservoir is given by the core radius of the cluster, and taking the ratio of the number of stars in the core to the total number of stars in the cluster for the parameter , we find that stars in large and tightly bound clusters (47Tuc, M15,~...) can re-accrete up to 90 % of the ejected gas. Individual stars in the cluster can therefore accrete a significant fraction of their initial mass.

4.2. Accretion from a wind

     Smith (1996) has shown that accretion from a wind is much less efficient than accretion from a central reservoir. Indeed, even though in this case we have = 1, the gas density will be several orders of magnitude lower, leading to a much smaller accretion rate.

4.3. Accretion during Close Encounters

     Main sequence stars can also accrete matter during close encounters with mass-losing AGB stars. In this case, the accretion time will be much smaller, but the gas density will be much larger. Here, it will be given by

(13)

where is the AGB star mass loss rate, v_g is the ejected gas velocity, and r (r > R_AGB) is the distance to the star. Replacing _g into Eq. (9), we have

(12)

For an order of magnitude result, we use r ~ b (b is the impact parameter), v_rel > c_s, and the interaction time t ~ b/v_rel. We get

(12)

which gives for , v_g ~ 10 km/s, v_rel ~ 10 km/s and b = 1au. This result should be multiplied by the number of encounters the main sequence star will have with AGB stars during its lifetime. We expect however the accretion through this process to be much less efficient than the accretion from a central reservoir of gas.

5. Conclusions

     We have investigated different gas accretion processes in globular clusters, namely, accretion from a central reservoir of gas, accretion from a cluster wind, and accretion during close encounters with mass-losing stars. By far the most promising process is the first one.

     We argue that accretion of gas by globular cluster stars has to take place, at least to some degree, and that it could be large in some cases. Furthermore, even a small amount of accretion will affect the surface composition of main sequence stars. The amount of accretion will depend on many parameters, such as the stellar initial mass, velocity, and orbit in the cluster, the cluster's core radius, concentration, mass, and orbit in the Galaxy, the AGB ejecta speed, etc... so that the accretion will be highly variable from star to star.

Acknowledgements

     This work has been supported by the Pôle d'Attraction Interuniversitaire P4/05 (SSTC, Belgium) and by FRFC F6/15-OL-F63 (FNRS, Belgium).

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