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Mon. Not. R. Astron. Soc. 316, 689±698 (2000) The eccentricities of the barium stars Amanda I. Karakas,1,2w Christopher A. Tout2,1 and John C. Lattanzio1,2 1 2 Department of Mathematics and Statistics, Monash University, Clayton, Victoria 3168, Australia Institute of Astronomy, The Observatories, Madingley Road, Cambridge CB3 0HA Accepted 2000 March 17. Received 2000 March 10; in original form 1998 August 7 A B S T R AC T We investigate the eccentricities of barium (Ba ii) stars formed via a stellar wind accretion model. We carry out a series of Monte Carlo simulations using a rapid binary evolution algorithm, which incorporates full tidal evolution, mass loss and accretion, and nucleosynthesis and dredge-up on the thermally pulsing asymptotic giant branch. We follow the enhancement of barium in the envelope of the accreting main-sequence companion and dilution into its convective envelope once the star ascends the giant branch. The observed eccentricities of Ba ii stars are significantly smaller than those of an equivalent set of normal red giants but are nevertheless non-zero. We show that such a distribution of eccentricities is consistent with a wind accretion model for Ba ii star production with weak viscous tidal dissipation in the convective envelopes of giant stars. We successfully model the distribution of orbital periods and the number of observed Ba ii stars. The actual distribution of eccentricities is quite sensitive to the strength of the tides, so that we are able to confirm that this strength is close to, but less than, what is expected theoretically and found with alternative observational tests. Two systems ± one very shortperiod but eccentric, and one long-period and highly eccentric ± still lie outside the envelope of our models, and so require a more exotic formation mechanism. All our models, even those which were a good fit to the observed distributions, overproduced the number of high-period barium stars, a problem that could not be solved by some combination of the three parameters: tidal strength, tidal enhancement and wind accretion efficiency. Key words: accretion, accretion discs ± methods: analytical ± methods: numerical ± stars: AGB and post-AGB ± stars: chemically peculiar ± stars: formation. 1 INTRODUCTION Barium or Ba ii stars were first identified by Bidelman & Keenan (1951) as a class of chemically peculiar Population I red giants with an observed frequency of about 1 per cent of the total population of G and K giants. They exhibit unusually strong absorption lines of Ba ii and Sr ii, as well as enhanced CN and CH bands. Detailed abundance analyses of the Ba ii stars reveal that the heavy elements produced by the s-process are enhanced by factors of 2±30 with respect to normal giants (Malaney & Lambert 1988), whereas carbon is overabundant by a factor of 3 in the most extreme cases (Lambert 1988; Barbuy et al. 1992). The discovery that all observed Ba ii stars belong to spectroscopic binary systems (McClure, Fletcher & Nemec 1980; McClure 1984; Jorissen & Mayor 1988; McClure & Woodsworth 1990) with eccentricities significantly lower than those of a sample of spectroscopically normal G and K giants (McClure & Woodsworth 1990) is evidence that the large s-process overabundances [email protected] q 2000 RAS might be the result of accretion, from a former asymptotic giant branch (AGB) companion, of material rich in heavy elements (Webbink 1986; Boffin & Jorissen 1988), rather than the result of internal nucleosynthesis. According to the mass transfer hypothesis, the former AGB stars would now be optically invisible white dwarfs (McClure 1984; Brown et al. 1990; Luck & Bond 1991; McClure 1997). Several investigations, with the International Ultraviolet Explorer (IUE) satellite, that looked for characteristic ultraviolet (UV) excesses have found white dwarfs, but there have also been some negative results. McClure & Woodsworth (1990) explain these by suggesting that the s-processed material was transferred a very long time ago so that the white dwarfs have cooled down below detectable levels. Mass functions derived from observations also support the mass transfer hypothesis. If it is assumed that the barium stars have masses in the range 1.0± 3.0 M(, then the mass functions indicate that their companions would have masses near 0.6 M(, similar to those expected of white dwarfs (McClure & Woodsworth 1990). Jorissen et al. (1998) also find Ba ii stars with masses in the range 1.3±2.1 M( if they assume that the companions have masses of about 0.6 M(. 690 A. I. Karakas, C. A. Tout and J. C. Lattanzio Further evidence for the mass transfer scenario comes from Bergeat & Knapik (1997), who find that Ba ii stars are spread over five luminosity classes, from the main sequence to the supergiant branch, with a tendency for concentration in the region of class III giants. This is consistent with the original classification of Ba ii stars as chemically peculiar G- and K-type giants with large overabundances of s-process elements. It also deals with one of the first criticisms of the mass transfer phenomenon, that no mainsequence barium stars exist. In fact, the dwarf barium stars were first observed by Bidelman (1985) and further investigated by North & Duquennoy (1991) and North, Berthet & Lanz (1994). A recent study by Bergeat & Knapik (1997) finds that six or seven of their sample of 37 barium stars are on or near the main sequence and identifies these as the pre-Ba/CH stars found in the simulations of Han et al. (1995). Two distinct mass transfer processes are possible. The first is Roche lobe overflow (hereafter RLOF; Iben & Tutukov 1985; Webbink 1986), where the thermally pulsing asymptotic giant branch (TP-AGB) companion of the pre-barium star has in the past filled or very nearly filled its Roche lobe. Because of the nonzero eccentricities of the Ba ii stars, Webbink (1986) ruled out such a process particularly with either common-envelope (CE) or contact-binary evolution (McClure & Woodsworth 1990). The second mechanism suggested by Boffin & Jorissen (1988) to circumvent the problem of rapid orbital circularization is the transfer of mass by a stellar wind. In this case the system always remains detached, and eccentric orbits can survive the mass transfer process (McClure & Woodsworth 1990). We test this hypothesis further, by including tidal evolution in a rapid binary evolution algorithm developed by Tout et al. (1997), modified to include stellar wind accretion and nucleosynthesis. In the wind accretion scenario, a binary system always remains detached, the radius of the primary star always being less than its corresponding Roche lobe radius. For this analysis of wind accretion we monitor the stellar R and Roche lobe RL radii of both the primary and the secondary star in our models to ensure that R , RL always remains satisfied. In the event of R > RL ; we stop the calculation, as that binary system no longer contributes to the number of barium star systems produced from wind accretion alone, though of course it might still end up as a Ba ii star. We ignore these systems primarily because we are interested in wind accretion. Very few barium stars have zero eccentricities, so we cannot expect many to result from RLOF. This is borne out by the simulations of Han et al. (1995) with significantly enhanced mass loss (which we shall show is necessary), where the fraction of barium stars formed from RLOF and CE is less than 1 per cent of all the Ba ii stars. We have synthesized a series of Monte Carlo populations that reproduce the observed orbital period and eccentricity distribution of Ba ii stars and their observed frequency compared with normal giants. The rapid binary evolution algorithm is based on fitting formulae derived from full evolution calculations for single stars and presently only models Population I stars. Formerly Boffin & Jorissen (1988), Jorissen & Boffin (1992), Han et al. (1995) and Theuns, Boffin & Jorissen (1996) investigated the formation of Ba ii and CH stars using the wind accretion scenario. Han et al. (1995) also investigated stable RLOF and CE evolution as channels in the production of Ba ii and CH stars, even though both these tend to circularize orbits completely and reduce periods to well below 80 d, the observed lower limit for barium star orbital periods. There are, however, a small fraction in circular orbits with low periods that could have formed from either RLOF or CE evolution. We find that all but the shortest-period barium star, HD 77247, with a period of 80 d and eccentricity of 0.09 (Jorissen et al. 1998), and one long-period system, HD 123949, with a period of 9200 d and eccentricity of 0.97, can be explained adequately within the stellar wind accretion scenario. A third system, which is a little more awkward to fit than most, BD 388 118, is a triple hierarchical system, and it is not clear which star is responsible for the barium enhancement (Jorissen et al. 1998). We carry out the population synthesis in a similar way to that Figure 1. Observed (e, P) diagram (left) and eccentricity distributions (right). The (e, P) diagram contains a sample of barium stars from Jorissen et al. (1998). In this sample, there are 62 barium stars with known orbital periods and eccentricities (large black dots) and eight with lower-limit estimates given for the orbital period and unknown eccentricity (open stars). The barium stars with unknown eccentricity have been arbitrarily assigned a high eccentricity of about 0.9 to distinguish them from the other stars. The large filled triangle represents the star BD 388 118 a b; which has been excluded from the sample of barium stars used later to test the models. For comparison we plot a sample of 213 spectroscopically normal giant binaries from Boffin et al. (1993) that have G and K spectral types as do the barium stars (small grey dots). We also include a shaded region between 104 and 105 d to represent barium stars with no evidence for binary motion. It is quite probable that these stars have periods greater than 104 d if they are binaries. Note that the eccentricity histogram on the right has the number of normal giants scaled to that of the Ba ii stars. q 2000 RAS, MNRAS 316, 689±698 The eccentricities of the barium stars described by Han et al. (1995), but improve on the estimate of the barium enhancement in the TP-AGB phase of the primary1 and dilution into the atmosphere of the pre-Ba ii stars and add a tidal evolution model. Han et al. (1995) examined the distribution of orbital periods and mass functions as well as the frequency of Ba ii/CH star production from population-synthesis models, so we compare our results (except for eccentricities) to theirs and to observations. As no previous population-synthesis models have addressed the problem of barium star eccentricities, we can only compare with observations. We discuss observed period and eccentricity distributions of barium stars in Section 2. Section 3 contains the details of the mass loss applied, the wind accretion model, the nucleosynthesis model and the Monte Carlo simulation parameters. The tidal evolution model is discussed in Section 4, results and discussion are presented in Section 5, and Section 6 concludes the paper. 2 O B S E RVAT I O N S Radial velocity variations over a decade led to the conclusion that all Ba ii stars are in binary systems with orbital periods typically longer than 100 d but less than 10 yr (McClure et al. 1980; McClure 1984; McClure & Woodsworth 1990). Jorissen et al. (1998) have recently reported Ba ii stars with periods well outside this range. Fig. 1 shows the observed eccentricity±period (e, P) diagrams and eccentricity distributions of Ba ii stars from Jorissen et al. (1998) and a similar set of normal giants from Boffin, Cerf & Paulus (1993). The sample of barium stars includes both `mild' and `strong' according to the classification system devised by Warner (1965). Warner's classification scheme for the barium Ê line strength to indicate stars uses an estimate of the Ba ii 4554-A the extent of abundance anomalies on a scale of 1 to 5, with higher numbers indicating higher overabundances. Strong barium stars have a Ba index of 3, 4 or 5, and the mild or weak barium stars have an index of either ,1, 1 or 2. The sample of spectroscopically normal giant binaries have MK spectral classifications indicating giant luminosity and G or K spectral type as the barium stars. The average eccentricities are 0.17 for the barium stars and 0.23 for the normal giants, while a Kolmogorov±Smirnov test reveals that the probability of the barium star eccentricities being chosen from the same distribution as those of the normal giants is about 1023. Because lower eccentricities are much easier to measure than higher eccentricities, selection affects the shape of the observed eccentricity distributions, and this must be accounted for when comparing the observed distributions with the simulated distributions produced by our models. This means that any statistical tests conducted as a comparison with the observed eccentricity distributions will favour models that produce more lowereccentricity barium stars than higher ones. This is not the case for periods, however. We note that all but two of the 37 known strong barium stars have known orbital periods, and all but three of a sample of mild barium stars taken by Jorissen et al. (1998) have known orbital periods. Of the sample of 70 mild and strong barium stars investigated by Jorissen et al. (1998), only eight do not have well-determined eccentricities. We conclude that selection effects are minor for periods ,104 d. 1 From here on, `primary' always refers to the mass-losing star, while `secondary' refers to the accreting or pre-Ba ii star. q 2000 RAS, MNRAS 316, 689±698 3 691 M O D E L PA R A M E T E R S Formulae for the computation of stellar evolution as a function of initial mass, age and current mass (reduced by mass loss) are to be found in detail in the paper by Tout et al. (1997). The formulae are good empirical fits to the detailed stellar models described in that paper. They give the radius, luminosity, stellar type and core mass, for giants throughout the evolution. In the following sections we describe the additional components that we must add to this model specifically for this work. 3.1 Mass loss Throughout this work we use the well-known mass-loss formula of Reimers (1975), _ R =M( yr21 4 10213 hR R=R( L=L( ; M M=M( 1 where R, L and M, respectively, are the radius, luminosity and mass of the mass-losing star, and h R is the Reimers parameter, which we set to unity here. Observations indicate that formulae that concentrate mass loss towards the tip of the AGB may be more appropriate, and we comment further on such prescriptions when we discuss our results. Tout & Eggleton (1988) added a term to Reimers mass loss to allow for tidal effects of a companion star and to account for the formation of certain Algol-like binaries, " ( )# R 6 1 21 _ _ 2 M TE =M( yr M R 1 B min ; 6 ; RL 2 where RL is the radius of the Roche lobe around the mass-losing star, and the constant B is a parameter determining the size of the increase. When B 0; equation (2) reduces to that of Reimers (1975). Tout & Eggleton (1988) required B 104 to obtain massloss rates high enough to fit the observed parameters of the RS CVn system Z Her, but we find a milder enhancement is sufficient for Ba ii stars. 3.2 Stellar wind accretion A fraction of the mass lost in a stellar wind from the primary can Ç 2 for the be accreted by the secondary star. The accretion rate M secondary is estimated by Boffin & Jorissen (1988), and we use ( ) 1 GM 2 2 aacc _ 1; _ M M 2 2max 1; p 3 1 2 e2 V 2wind 2a2 1 v2 3=2 where v V orb =V wind ; Vwind is the wind velocity from the primary _ 1 , 0 the mass-loss rate from the primary, M1 and M2 the star, M primary and secondary mass respectively, e the orbital eccentricity, a the semimajor axis and Vorb the orbital velocity given by p V orb G M 1 M 2 =a; 4 where G is the gravitational constant. The parameter a acc is an accretion efficiency, varied between 0.05 to 1.5 in our models to examine which value leads to the best fit to the observed period distribution. The appropriate value of a acc for Bondi±Hoyle accretion is 1.5, but Theuns et al. (1996) suggest that a value as low as 0.05 may be more appropriate for stellar wind accretion in the case of the formation of the barium stars following detailed hydrodynamic simulations. In the case of barium star formation, the binary motion strongly disturbs the shape of the accretion 692 A. I. Karakas, C. A. Tout and J. C. Lattanzio column as V orb < V wind when r < a; where r is the radius of the accretion column and a the semimajor axis of the binary system. We take the wind velocity Vwind to be the escape velocity from the primary r GM 1 ; 5 V wind V esc R1 where R1 is its radius. This is the most uncertain quantity in equation (3), because it is hard to determine accurately through observations. Decreasing a acc is equivalent to increasing the ratio of the kinetic energy of the wind to the escape energy. We evaluate the change in orbital separation and eccentricity due to wind accretion according to the formalism of Huang (1956) and find _1 a_ 1 2 _ 1 e2 M 2 2 6 M2 a M M M2 1 2 e2 and e_ _ 2 1 2 , 0; 2M e M M2 7 where M M 1 M 2 ; if we assume that the change in momentum of the accreting star is given by M 2 dv2 2dM 2 V; where d v2 is the change to the instantaneous velocity of the secondary star and V is the total instantaneous velocity of the system. Both equations (6) and (7) take account of changes to the separation and eccentricity caused by the accretion of more matter at periastron than at apastron in the case of an already eccentric orbit. These formulae assume that the momentum and angular momentum lost by the accreting component, due to the drag it experiences in moving through the primary's wind, is lost from the system in the escaping wind. The corresponding change, due to mass transfer, in the instantaneous relative velocity v of the two stars is then m_ 2 v_ 2 v; 8 M2 Ç 2 is the instantaneous accretion rate. They differ from the where m formulae derived by Theuns et al. (1996) and also those derived by Eggleton, Kiseleva & Hut (1998), both of which can lead to growing eccentricity in certain circumstances. The latter implicitly assume that any momentum lost by star 2 is somehow returned to star 1. This is difficult to envisage when the stars are well _ 1 j; so we prefer the assumption that we separated and m_ 2 ! jM make here. In any case eccentricity changes that are due to formula (7) are small compared with those due to tides in the case of barium star formation. A more careful examination of the slow wind approximation (Snellgrove & Gair, private communication) leads to e_ _2 1 1 ; 9 2M e M 2M 2 and _1 a_ M 2 2 a M1 1 e2 2 2 e2 M M2 _2 M ; 1 2 e2 10 but using these instead of equations (6) and (7) does not affect the results. 3.3 Nucleosynthesis model We include the effects of TP-AGB evolution with a synthetic evolution model that evaluates composition changes at each thermal pulse. The core mass at the first pulse (in solar units) is taken from Lattanzio (1989) to be 0:53 2 1:3 log Z Y 2 0:2; M pulse c Z > 0:01; 11 where Y is the mass fraction of helium and Z is the metallicity at the first thermal pulse. Barium stars are Population I objects with a solar abundance of Z , 0:02 and Y , 0:28; so that M pulse c 0:562 M( for all initial masses. We take the interpulse period to be a constant, tip 104 yr; 12 for all stars. Although this assumption is crude, it is acceptable because the interpulse period is not critical to our calculations. We tested this by varying the interpulse period between 105 and 5 103 yr and found no appreciable difference in the models. The same number of barium stars were produced, though the Kolmogorov±Smirnov (KS) test results and the average eccentricities were slightly but not significantly different. In the population syntheses we are only interested in the production of s-process elements, particularly barium, so we do not include nucleosynthesis and dredge-up prior to the thermally pulsing phase. We assume that third dredge-up begins when the core mass is higher than a critical value M min c : The exact value of M min is still a matter of debate for stars with solar metallicity, but c for the present calculations we set ( 0:61 M( ; M 0 < 3:0 M( ; min 13 Mc 0:795 M( ; M 0 . 3:0 M( ; where M0 is the initial total mass of the star. The value for M 0 < 3:0 M( is consistent with low-mass stellar models of Lattanzio (1989) and Straniero et al. (1997), while that for M 0 . 3:0 M( is from Frost (1997). Between pulses on the AGB the core mass will grow during the interpulse period according to tip dM c DM c dt; 14 dt 0 with the rate of core mass growth in M( yr21 to a good approximation given by dM c =dt 9:555 10212 LH ; X 15 where LH is the luminosity provided by hydrogen burning (in solar units) and X the hydrogen abundance (by mass) in the envelope. If there is dredge-up M c . M min c ; then a fraction of DMc will be dredged up, DM dredge lDM c ; 16 where l is the dredge-up efficiency parameter. For M 0 , 3:0 M( ; calculations of Straniero et al. (1997) indicate a linear relation between l and M0, while for M 0 > 3:0 M( ; calculations by Frost (1997) favour a constant l : ( 0:2 0:0866M 0 ; M 0 , 3:0 M( ; 17 l 0:9; M 0 > 3:0 M( : These values of l are not continuous. This reflects the lack of reliable AGB stellar models for stars around 3 and 4 M( and the uncertainties in finding a consistent numerical treatment for dredge-up (see Frost & Lattanzio 1996 for a review). Now if M c . M min c ; the core mass Mc will also decrease in size as a result q 2000 RAS, MNRAS 316, 689±698 The eccentricities of the barium stars of dredge-up, so that over an interpulse phase the core mass increases by DMc, then decreases by DMdredge, while the envelope mass Menv will increase by DMdredge but decrease by DMc and by mass loss (Groenewegen & de Jong 1993). Barium is produced in the intershell zone under radiative conditions (Straniero et al. 1995) where the logarithm of the barium overabundance with respect to the solar value reaches an approximately constant value of Bashell =Fe < log Bashell =Ba( < 2:6 18 (Gallino, private communication). The mass of barium added to the envelope during dredge-up is then DM Ba Bashell DM dredge : 19 The actual abundance of barium in the primary's envelope between the ith and the i 1th dredge-up episode is Ba i 2 1M env i 2 1 DM Ba i Ba i ; M env i 2 1 DM dredge i 20 where Ba i 2 1 is the barium mass fraction in the envelope just before the ith dredge-up episode and Ba 0 Ba( : We compute the abundance of barium in the envelope following dredge-up after each thermal pulse, and this is the abundance of barium in the wind of the primary, a fraction of which is accreted by the secondary. The mass transferred from the primary to the secondary between the ith and the i 1th dredge-up episode is Macc. Over the life of the star (in dredge-up episodes) this sums to give P aP total accreted mass of M acc ni1 M acc i; of which n i1 M acc iBa i is barium. The pre-Ba ii star polluted by the TP-AGB star evolves to become a red giant with a maximum envelope mass Menv,2. The accreted barium is diluted into the envelope of the secondary at its deepest extent, which has a mass of M env;2 < M 2 M acc 2 M c;bgb ; 21 where Mc,bgb is the core mass of the secondary star at the beginning of the giant branch. Subsequently the surface barium abundance remains constant for the red giant branch (RGB) lifetime at n P Bafinal i1 M acc iBa i M env;2 2 M acc Ba( M env;2 ; 22 where the entire AGB envelope has been lost in the wind before the n 1th pulse which shows dredge-up. 3.4 Monte Carlo simulation parameters For each Monte Carlo simulation we model 1000 000 pairs of stars with initial separations, masses and eccentricities determined randomly from the following distributions. The initial separation distribution is flat in log a, amax X ; 23 a amin amin from Eggleton, Fitchett & Tout (1989), where X is a random number between 0 and 1, amax 104 au and amin 0:1 au: We are not interested in very close systems that interact before the primary has left the giant branch, but note that they will affect the q 2000 RAS, MNRAS 316, 689±698 693 overall fraction of Ba ii stars to a small extent. We give the initial distribution of eccentricities a linear slope to approximate the spread of observed red giant eccentricities between 0 and 0.99. Such a linear slope results from the generating function p e 0:99 1 2 X ; 24 where X is a random number uniformly distributed between 0 and 1. Most observed normal giants with periods in the same range as the barium stars should have eccentricities very close to their initial eccentricities because separations remain too large for tidal circularization to become effective. The initial mass distribution is generated according to the formula of Eggleton et al. (1989), M 0:19X ; 1 2 X0:75 0:032 1 2 X0:25 25 where X is again a random number uniformly distributed between 0 and 1, which leads to a mass function similar to that of Miller & Scalo (1979). The masses of the two stars in each binary system were each chosen at random with equation (25). Duquennoy & Mayor (1991) note that this is acceptable for long-period binaries, but for tight binaries the hypothesis of random association is not likely to be valid. The binary evolution algorithm always makes the heavier of the two stars to be star 1, the mass-losing star, thus setting the lower-mass binary component to be star 2, the massgaining star. The mass range chosen for both the primary and the secondary is 0:08 < M 1 ; M 2 < 8 M( : Stars below 0.08 M( are brown dwarfs, while those above 8 M( are rare and will not experience thermal pulses. We evolve each pair of stars from the zero-age main sequence (ZAMS) to the age of the Galaxy, T gal 1:2 1010 yr: 4 T I DA L E VO L U T I O N M O D E L Tidal forces tend to drive a binary system to its lowest energy state for a given angular momentum. This, if stable, is one in which the orbit is circular, the spins of the stars are synchronized with the orbital motion, and the spin axes are parallel to the orbital axis (Hut 1980). When a star, particularly a giant, fills its Roche lobe, it is assumed, and confirmed by observations, that its orbit is already circular and that its spin is synchronized with the orbit. During Roche lobe overflow, tidal forces are strong enough to maintain this circular, synchronous state, and it is indeed the fact that the majority of barium stars have eccentric orbits that leads us to reject Roche lobe overflow as the dominant formation mechanism. After mass transfer, although neither star need remain synchronous, there is no force within the binary that can drive it eccentric. Under some conditions the binary system may develop a circumbinary accretion disc instead of the standard internal accretion disc. According to calculations by Artymowicz et al. (1991) the development of such an accretion disc could provide the mechanism to drive the eccentricity of the orbit upwards. In the standard case, however, the resonant-tidal mechanism proposed would be rapidly damped by stellar tides and can be ignored. A more thorough treatment would include the effects of circumbinary accretion discs on orbital properties if we know how many binaries end up with such discs as opposed to standard internal accretion discs. As this fraction is largely unknown, we do not include the effects of circumbinary accretion discs in our models. Although the orbits of barium stars are eccentric they still tend to be more circular than those of ordinary giant binaries (Fig. 1). 694 A. I. Karakas, C. A. Tout and J. C. Lattanzio This is a fortuitous situation because we can conclude that tidal circularization has begun but not proceeded to its conclusion. If our model for the formation of barium stars is correct, then we must be able to include the effects of tides and recover the distribution of barium star eccentricities alongside their other properties. The time-scale on which tidal circularization takes place is given by Verbunt & Phinney (1995) or Rasio et al. (1996) as 8 1 f 0 M env R ; 26 q 1 q tcirc tconv M a where Menv is the mass of the convective envelope, q M 1 =M 2 the mass ratio, M M 1 M 2 the total mass of the system, a the semimajor axis and t conv the time-scale on which the largest convective cells turn over. From mixing length theory this can be approximated by 1=3 M env R2 tconv < ; 27 12L where L is the stellar luminosity. Rasio et al. split the factor f into ( ) Ptid 2 0 f f min ;1 ; 28 2tconv where Ptid is the tidal pumping time-scale given by 1 Vspin 1 ; 2 2p Ptid Porb 29 with Vspin the spin angular velocity of the star on which the tides are raised. The second part of the factor takes account of the fact that convective cells that turn over in a time longer than the tidal pumping time-scale cannot contribute to the viscous damping of the tides (Goldreich & Keeley 1977). Because convective turnover time-scales are about a year whereas the orbital periods of barium stars are typically a few years, this effect is not large, but it is important for the closest systems and those with large eccentricity in which the stellar spin pseudosynchronizes with the periastron orbital angular velocity. The remaining factor f we are free to vary in the absence of a complete understanding of tidal damping. It has been found both theoretically (Zahn 1977) and observationally (Verbunt & Phinney 1995) to be about 1. Because formula (26) is only valid for small eccentricity and nearly synchronous spin, we use the formalism of Hut (1981), which is valid for all e and Vspin on the assumption that damping leads to a constant lag time between the tide raised and the tidal force that raises it. Comparison of Hut's equation (10) with our equation (26) for small eccentricity and spin±orbit synchronism reveals that Hut's k 2 f 0 M env ; T 21 tconv M 30 and we use this in his equations (10) and (11) to find de/dt and dV/ dt as a result of the tides. We calculate the change in semimajor axis a by conserving the total (spin and orbital) angular momentum of the system. We also take into account spin-down of an expanding star by conserving the spin angular momentum Jspin of a star as it expands with the approximation J spin M env k2g R2 Vspin ; 31 Table 1. Selection of results: Column 1 gives the model number. Column 2 gives the wind enhancement parameter B. Column 3 shows the tidal strength factor f defined in Section 4; f 1 represents tides at the expected strength. Column 4 gives the value of the accretion efficiency parameter a acc. Column 5 shows the weighted percentage of Ba ii stars Ba > 0:2 to normal giants. Column 6 gives the Kolmogorov±Smirnov (KS) probability of association between the distribution of eccentricities from the model and the observed distribution of barium star eccentricities from Jorissen et al. (1998). Column 7 gives the KS probability of association between the distribution of orbital periods from the model and the observed distribution of periods from Jorissen et al. (1998). Column 8 shows the percentage of model barium stars that have periods between 104 and 105 d. Finally, column 9 shows the percentage of model barium stars that have periods greater than or equal to 105 d. The observed sample of barium star periods used for the KS tests also contains stars with only lower limits given for orbital period and unknown eccentricity. Note that the KS tests were only performed on simulated barium stars from each distribution that have orbital periods below 10 000 d to match the observed range of periods. Model number B f a acc Weighted per cent Ba ii stars KS prob. eccentricities P < 104 d KS prob. periods P < 104 d Per cent with 104 , P , 105 d Per cent with P > 105 d 1 2 3 4 5 6 7 8 9 10 11 12 13 14 2 103 2 103 2 103 1:5 103 2 103 1:5 103 2 103 2 103 1 103 7:5 102 1 103 2 103 1 103 5 102 0.1 0.25 0.1 0.2 0.1 0.1 0.05 0.075 0.075 0.075 0.05 0.05 0.05 0.05 1 0.5 0.5 0.2 0.2 0.2 0.2 0.1 0.075 0.075 0.075 0.05 0.05 0.05 2.68 2.46 2.51 2.02 2.19 2.04 2.22 1.85 1.39 1.24 1.41 1.40 1.13 0.852 0.17 0.17 0.12 0.15 0.15 0.20 0.028 0.15 0.32 0.44 0.11 0.24 0.42 0.61 0.036 0.055 0.057 0.052 0.071 0.047 0.1 0.027 0.12 0.13 0.13 0.054 0.021 0.031 33.7 32.5 32.3 30.6 29.5 30.9 29.2 29.0 32.8 35.5 32.8 26.3 30.9 37.8 17.4 13.1 12.9 8.15 7.61 8.07 7.51 2.26 1.21 1.26 1.20 0.135 0.082 0.103 15 16 17 18 19 7:5 102 102 0 7:5 102 7:5 102 0.075 0.075 0.075 0.01 1.0 0.01 0.075 0.075 0.075 0.075 0.0106 0.647 0.464 1.39 1.16 0.305 0.347 0.0348 3:89 1024 7:2 10211 3:1 1023 7:6 1023 1:2 1023 0.17 0.057 20 50 57.8 34.5 34.4 0 1.78 2.09 1.35 1.11 q 2000 RAS, MNRAS 316, 689±698 The eccentricities of the barium stars k2g 695 with the ratio of the radius of gyration to the stellar radius 0:1: It turns out that a significant fraction of the system's angular momentum is required by the spin of a giant that is close to filling its Roche lobe, so that incorporation of tidal effects not only allows us to follow the eccentricity evolution of barium stars but also leads to a significant, and hitherto ignored, shrinkage of their orbital separation, which in turn affects the efficiency of mass transfer via stellar wind. We calculate tidal effects throughout the giant evolution of all our binary stars so that tides are also operating when the contaminated barium stars ascend the first giant branch themselves. However, any further circularization at this stage is negligible because the systems are now much wider because of their earlier mass loss. Any time a barium star is produced in the 1:2 1010 yr timespan we note how long it spends on the red giant branch and as a core helium-burning star. We do the same any time either star in the binary pair becomes a normal red giant or core helium-burning star. From each run we then weight the final percentage of barium stars to the number of normal red giants according to the length of time each star has spent on the RGB (plus core helium burning) normalized to the time a 1 M( star would spend in the same phase. With a constant star formation rate, the effect of weighting the statistics by the normalized lifetimes means that the final percentage will not be an integrated value but a value representing the fraction of barium stars to red giants in the galaxy at the present time. Note that we are calculating 5 number model Ba ii stars number model RG binaries R E S U LT S A N D D I S C U S S I O N We evolved over 600 population syntheses models, each containing 1000 000 binary systems, varying B, f and a acc to find where the highest KS probabilities occur for both the period and eccentricity distributions. Note that Table 1 does not give a full list of the populations modelled but rather a selection of our best models, with the exception of the last five models, which are included to allow us to narrow the range of the three parameters. Our simulated populations contain, on average, about 2000 barium stars. We tested our models with the KS statistical test to calculate a probability of association between our model distributions of orbital period and eccentricity and the observed distributions. The observed eccentricity distribution was taken from a sample of 62 strong and weak Ba ii stars given in Jorissen et al. (1998). To compare model orbital period distributions, we drew a sample of observed orbital periods from the same set of stars used to test the eccentricity distribution but also included eight stars that had only lower limits given on orbital period and unknown eccentricities. In both cases, we only performed the KS tests on model Ba ii stars with periods up to 10 000 d to match the observed range of barium star periods. We note that there are a small number of barium stars observed with no evidence for binary motion, and these stars could be very high-period barium stars P > 104 d: However, the data are consistent with an upper limit to barium star orbital periods of about 10 000 d. In Jorissen et al.'s (1998) sample, about 20 per cent of the mild barium stars have no evidence for binary motion or only lower limits for the orbital periods and no eccentricity measurements. 32 and comparing with number observed Ba ii stars number observed RG stars: 33 These two fractions are not equal because number RG binaries ± number RG stars: 34 However we expect 0:4 , number RG binaries ,1 number RG stars 35 (Eggleton et al. 1989), so that our percentages are overestimates of the observed percentage of barium stars to normal giants. In fact, if we take model 10, which has a simulated percentage of 1.24 per cent, as an example, we get the correct fraction of 1 per cent if number RG binaries 0:81: number RG stars 36 Table 1 shows that most models overproduced the fraction of barium stars to red giants compared with the observed fraction of 1 per cent. B < 2 103 produced barium stars at a higher frequency, up to 2.68 per cent in the case of model 1 compared with lower values of B. All are therefore consistent with a reasonable binary fraction amongst all stars. Fig. 2 shows the (e,P) diagrams for models 1 and 10. Comparing these results with the observed barium star (e,P) diagram in Fig. 1, we see that the observations are consistent with Figure 2. Simulated (e,P) diagram for the barium stars produced using (a) f 0:1; B 2000 and aacc 1:0 (model 1) and (b) f 0:075; B 7:5 102 and aacc 0:075 (model 10). We have assumed that an overabundance of Ba > 0:2 is necessary for spectroscopic detection. q 2000 RAS, MNRAS 316, 689±698 696 A. I. Karakas, C. A. Tout and J. C. Lattanzio some of the models from Table 1 using viscous tidal evolution in convective envelopes and the wind accretion model for Ba ii star production. Note that our favoured value for B is at least five times smaller than that originally proposed by Tout & Eggleton (1988) and that most models required reductions in f of at least four (preferably, at least ten) times smaller than the expected tidal strength. Fig. 2(b) is the best fit to the observed eccentricity and period distributions according to KS probabilities calculated from the observed distributions. Fig. 3 shows the model eccentricities and period distributions obtained for model 10 plotted against the observations for comparison. Note that in this diagram the numbers have been scaled to the number of observed Ba ii stars. The model eccentricity histogram follows the observed distribution quite closely, but the model period histogram is significantly different from the observed distribution in Fig. 3 for periods greater than 104 d. For this diagram, we binned all model barium stars with periods greater than 104 d into one bin, which accounts for 36.8 per cent of the total number of barium stars for this model. We are comparing this number to a hypothetical upper limit of 20 per cent of all observed barium stars having periods between 104 and 105 d. While our best model almost entirely covers the observed range in eccentricity and orbital period and gives the highest KS probabilities, the area covered by the simulated distribution in the (e,P) plane is not the same as the area covered by the observed distribution. By altering B, f and a acc we can actually correct this discrepancy as shown in Fig. 2(a), which covers more of the observed distribution but has a higher overproduction of highperiod systems. The period distributions were highly dependent on a acc and B while mostly independent of the tidal strength f. Reducing a acc from 1 to 0.05 greatly reduced the number of high-period barium systems, while increasing B from zero to 104 increased the number of low-period, zero-eccentricity systems. It is for these reasons that B must be greater than zero but less than 104 together with a low value of a acc to match the observed distribution of barium star orbital periods. The eccentricity distributions were found to be highly sensitive to changes in the tidal strength f and moderately dependent on B and a acc, as Table 1 and Fig. 2 demonstrate. There is not a unique set of the three parameters, B, f and a acc, that give a significantly better fit to the observations than other combinations of parameters. This is demonstrated by Table 1, where every model is a reasonable fit to the observed distributions (for P , 104 d according to the KS probabilities with model 10 chosen as the best fit because the KS probability for both the period and eccentricity distributions were the largest for the combination of parameters such that B 7:5 102 ; f 0:075 and aacc 0:075: From running a large number of models we can rule out extreme values of the three parameters, f, B and a acc, with some confidence in their application to the modelling of barium star formation. Notably, we can rule out strong and very weak tidal strengths, so that 0:01 , f , 1:0: From Table 1 we require a tidal wind enhancement of 102 , B < 2 103 : For accretion efficiency, we find applicable values close to that expected by Theuns et al. (1996), where 0:01 , aacc , 0:2: Models with aacc > 0:2 can still produce barium stars that fit the observed eccentricity distribution but produce too many high-period barium star systems and so can be ruled out. From observations we expect no more than 20 per cent of (mild) barium stars to have periods between 104 and 105 d and 0 per cent of barium stars to have periods greater than 105 d. Only model 15 adequately satisfied these criteria, though this model's low KS probabilities ruled it out as a good fit to the observed distributions. Another motivation for choosing model 10 as the best model was that the number of high-period barium stars (over 105 d) was effectively zero (<1 per cent) owing to its low value of a acc; models with a acc larger than 0.2 produced too many barium stars with periods greater 105 d, up to 17.4 per cent in the case of model 1, where a acc is equal to 1. All models except model 15 produced 26±37 per cent of barium stars between 104 and 105 d, greater in all cases than the upper limit of 20 per cent expected from observations. If we use high KS probabilities and low numbers of high-period barium stars as criteria in choosing which models fit the observations, then only models 9, 10 and 11 from Table 1 can be considered good fits. But all models, even those that were good fits to the observed distributions, produce too many barium stars in the range 104 , P , 105 d: We could not solve this problem by reducing the wind accretion efficiency parameter or by some Figure 3. Histograms showing the distribution of (a) eccentricities and (b) periods for model 10 (full line), which was the best fit to all three parameters with f 0:075; B 750 and aacc 0:075; compared with the observed distributions (dashed line). This diagram illustrates the small visual difference that exists between the observed and model eccentricity distributions. For (b), all model barium stars with periods greater than 104 d have been put in one bin, while a hypothetical upper limit of 20 per cent of all observed barium stars has been binned between 104 and 105 d (shaded region) for comparison. Even with this hypothetical percentage of observed high-period barium stars, the two distributions look quite different. Note that the numbers in (a) have been scaled to the number of observed barium stars given in Jorissen et al. (1998) with known eccentricities, and the numbers in (b) have been scaled to the number of barium stars with known periods or upper limit period estimates in Jorissen et al. (1998). q 2000 RAS, MNRAS 316, 689±698 The eccentricities of the barium stars 697 Figure 4. The left-hand diagram is a plot of the orbital period versus the average D(38±41) colour index from Jorissen et al. (1998) (large black dots), which has been shown to give a reasonable measure of the heavy-element overabundances. The right-hand diagram is a plot of the correlation between the average barium overabundance (large black dots) and the orbital period found for the best model, model 10, which has f 0:075; B 750 and aacc 0:075: In both figures, the actual abundances for each observed and model barium star has also been plotted (small dots). Note that we set a spectroscopic detection threshold of Ba , 0:2 in our models to distinguish barium stars from normal giants. This detection limit is the cause of the apparent hump in average barium abundances near 104 d for model 10. Reducing the detection limit by a factor of 10 completely eliminates the hump altogether so the average trend for the model follows the observed. Even though the two distributions are plots of different quantities, they are in essence plotting the same thing ± an indication of the level of heavy-element overabundances in barium stars. Figure 5. Diagram showing the distribution of masses for the barium stars and white dwarf companions from model 10. The average mass of the barium stars for this model was 1.63 M( and the average mass of the companion 0.84 M(. special combination of the three parameters, f, B and a acc. This may indicate that there is a fundamental problem with the wind accretion model as a comprehensive formation mechanism in the production of barium star binary systems, or that some aspect of our modelling or parametrizations gives an inadequate description of the stellar physics. Our detailed modelling of the barium enhancement allows us to plot the barium overabundances for each star against orbital period to see if any correlation exists in the models. The left-hand diagram in Fig. 4 is a plot of the [P,D(38±41)] diagram from Jorissen et al. (1998), and the right-hand plot is an example of the correlation found for all models between the orbital period and the average barium overabundance, for the best model, model 10. The quantity D(38±41) is believed to give quite a good measure of the heavy-element overabundances observed in the barium stars. Unfortunately there are no complete samples of quantities directly associated with barium star chemical peculiarities, such as s-process abundances. Though we are comparing two different q 2000 RAS, MNRAS 316, 689±698 quantities, both are representative of the heavy-element overabundances in barium stars. We find that the model distributions illustrated by the right-hand plot of Fig. 4 are similar to the observed [P,D(38±41)] diagram from Jorissen et al. (1998), particularly in the tendency for greater enhancement among shorter-period systems. As explained in Fig. 4 the hump is simply a result of artificial selection effects, which disappear when the detection limit is reduced. Models that use tidally enhanced stellar winds were found to have more strong barium low-period stars than models that use Reimers mass loss alone. This is because stars that experience a tidally driven wind will lose more matter before growing large enough to fill their Roche lobes and consequently enable the secondary to accrete more heavily polluted matter and thus end up with higher barium overabundances. Lastly we plot the mass distribution of the simulated barium stars and their companions from model 10 in Fig. 5. We can see that the barium stars produced by the model (all models produced 698 A. I. Karakas, C. A. Tout and J. C. Lattanzio a similar distribution) span a large range, from 0.9 M( up to about 6 M( with an average mass (for model 10) of about 1.63 M( for the barium stars, which is consistent with observations. Decreasing the size of the wind enhancement parameter decreases slightly the mass of the barium stars in the models. The companion stars also span a fairly large range from 0.6 M( up to 1.3 M( with an average mass of about 0.8 M( for all models. 6 CONCLUSIONS The eccentricity distribution of barium stars is consistent with the stellar wind accretion and tidal evolution models for the production of barium stars, though we require weaker tides than expected to model the eccentricity distribution adequately. The shape of the eccentricity distributions is strongly dependent on the strength of the tides used in the models and also to some extent on the mass loss and on the efficiency of wind accretion. It was found that to fit the observations a decrease of at least four times the expected tidal strength was required. A tidally enhanced wind produces the best fit to the observations, for both the eccentricity and the period distributions, but we find that this enhancement should not be as strong as originally proposed by Tout & Eggleton (1988). Even so, the stellar wind accretion model is found to be unable to account for the lowest-period system and the highesteccentricity system. The period distribution was found to be sensitive to changes in efficiency in wind accretion for long-period systems and changes in the wind tidal enhancement parameter for the short-period systems. Changing tidal strength had little effect on the period distributions. Models that fit well the observed eccentricity and period distributions below 10 000 d are prone to produce too many high-period barium stars, a problem we could not reconcile by reducing the efficiency of wind accretion or any other combination also involving tidal wind enhancement and tidal strength. AC K N O W L E D G M E N T S We thank Alain Jorissen for helpful comments and advice. CAT is pleased to thank the IAU Commission 38 for a travel grant, Monash University for hospitality, and the UK PPARC for an advanced fellowship. AIK is very grateful for a Monash Graduate Scholarship, and to the British Council for providing a partial postgraduate bursary for travel to the University of Cambridge. JCL wishes to thank the IoA for hospitality, and PPARC for travel assistance. This research was partially funded by the Australian Research Council. REFERENCES Artymowicz P., Clarke C. J., Lubow S. H., Pringle J. E., 1991, ApJ, 370, L35 Barbuy B., Jorissen A., Rossi S. C. F., Arnould M., 1992, A&A, 262, 216 Bergeat J., Knapik A., 1997, A&A, 321, L9 Bidelman W. P., 1985, AJ, 90, 341 Bidelman W. P., Keenan P. C., 1951, ApJ, 114, 473 Boffin H. M. J., Jorissen A., 1988, A&A, 205, 155 Boffin H. M. J., Cerf N., Paulus G., 1993, A&A, 271, 125 Brown J. A., Smith V. V., Lambert D. L., Duthchover E. Jr, Hinkle K. H., Johnson H. R., 1990, ApJ, 99 Duquennoy A., Mayor M., 1991, A&A, 248, 485 Eggleton P. P., Fitchett M. J., Tout C. A., 1989, ApJ, 354, 387 Eggleton P. P., Kiseleva L. G., Hut P., 1998, ApJ, 499, 853 Frost C. A., 1997, PhD thesis, Monash Univ. Frost C. A., Lattanzio J. C., 1996, ApJ, 473, 383 Goldreich P., Keeley D. A., 1977, ApJ, 211, 934 Groenewegen M. A. T., de Jong T., 1993, A&A, 267, 410 Han Z., Eggleton P. P., Podsiadlowski P., Tout C. A., 1995, MNRAS, 277, 1442 Huang S. S., 1956, AJ, 61, 49 Hut P., 1980, A&A, 92, 167 Hut P., 1981, A&A, 99, 126 Iben I. Jr, Tutukov A. V., 1985, ApJS, 58, 661 Jorissen A., Boffin H. M. J., 1992, in Duquennoy A., Mayor M., eds, Binaries as Tracers of Stellar Formation. Cambridge Univ. Press, Cambridge, p. 110 Jorissen A., Mayor M., 1988, A&A, 198, 187 Jorissen A., Van Eck S., Mayor M., Udry S., 1998, A&A, 332, 877 Lambert D. L., 1988, in Cayrel de Strobel G., Spite M., eds, Proc. IAU Symp. 132, The Impact of Very High S/N Spectroscopy on Stellar Physics. Davis Press, Schenectady, NY, p. 563 Lattanzio J. C., 1989, ApJ, 347, 989 Luck R. E., Bond H. E., 1991, ApJS, 77, 515 McClure R. D., 1984, PASP, 96, 117 McClure R. D., 1997, PASP, 109, 536 McClure R. D., Woodsworth A. W., 1990, ApJ, 352, 709 McClure R. D., Fletcher J. M., Nemec J. M., 1980, ApJ, 238, L35 Malaney R. A., Lambert D. L., 1988, MNRAS, 235, 695 Miller G. E., Scalo J. M., 1979, ApJS, 41, 513 North P., Duquennoy A., 1991, A&A, 244, 335 North P., Berthet S., Lanz T., 1994, A&A, 281, 775 Rasio F. A., Tout C. A., Lubow S. H., Livio M., 1996, ApJ, 470, 1187 Reimers D., 1975, in Baschek B., Kegel W. H., Traving G., eds, Problems in Stellar Atmospheres and Envelopes. Springer, Berlin, p. 229 Straniero O., Gallino R., Busso M., Chieffi A., Raiteri C. M., Limongi M., Salaris M., 1995, ApJ, 440, L85 Straniero O., Chieffi A., Limongi M., Busso M., Gallino R., Arlandini C., 1997, ApJ, 478, 332 Theuns T., Boffin H. M., Jorissen A., 1996, MNRAS, 280, 1264 Tout C. A., Eggleton P. P., 1988, MNRAS, 231, 823 Tout C. A., Aarseth S. J., Pols O. R., Eggleton P. P., 1997, MNRAS, 291, 732 Verbunt F., Phinney E. S., 1995, A&A, 293, 709 Warner B., 1965, MNRAS, 129, 263 Webbink R. F., 1986, in Leung K.-C., Zhai D. S., eds, Critical Observations versus Physical Models for Close Binary Systems. Gordon & Breach, New York, p. 403 Zahn J.-P., 1977, A&A, 57, 383 This paper has been typeset from a TEX/LATEX file prepared by the author. q 2000 RAS, MNRAS 316, 689±698