Download The eccentricities of the barium stars

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 ˆ 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 ˆ 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 ˆ 0; equation (2) reduces to that of Reimers
(1975). Tout & Eggleton (1988) required B ˆ 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 ˆ 2max 1; p
…3†
1 2 e2 V 2wind 2a2 …1 ‡ v2 †3=2
where v ˆ 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 ˆ 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 ˆ V esc ˆ
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
ˆ2
2
…6†
M2
‡
a
M
M M2
…1 2 e2 †
and
e_
_ 2 1 ‡ 2 , 0;
ˆ 2M
e
M M2
…7†
where M ˆ M 1 ‡ M 2 ; if we assume that the change in momentum
of the accreting star is given by M 2 dv2 ˆ 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_ ˆ 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†
ˆ 2M
e
M 2M 2
and
_1
a_
M
ˆ2
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
ˆ 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
ˆ
c
0:562 M( for all initial masses. We take the interpulse period to be
a constant,
tip ˆ 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 ˆ
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 ˆ
dt;
…14†
dt
0
with the rate of core mass growth in M( yr21 to a good
approximation given by
dM c =dt ˆ 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 ˆ 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†
lˆ
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 =FeŠ < log…Bashell =Ba( † < 2:6
…18†
(Gallino, private communication). The mass of barium added to
the envelope during dredge-up is then
DM Ba ˆ Bashell DM dredge :
…19†
The actual abundance of barium in the primary's envelope
between the ith and the …i ‡ 1†th dredge-up episode is
Ba…i 2 1†M env …i 2 1† ‡ DM Ba …i†
Ba…i† ˆ
;
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† ˆ 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 ‡ 1†th 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 ˆ niˆ1 M acc …i†; of which
n
iˆ1 M acc …i†Ba…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 ˆ
iˆ1
M acc …i†Ba…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 ‡ 1†th 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 ˆ amin
amin
from Eggleton, Fitchett & Tout (1989), where X is a random
number between 0 and 1, amax ˆ 104 au and amin ˆ 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 ˆ 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),
Mˆ
0:19X
;
…1 2 X†0:75 ‡ 0:032…1 2 X†0:25
…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 ˆ
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
ˆ
;
…26†
q…1 ‡ q†
tcirc tconv M
a
where Menv is the mass of the convective envelope, q ˆ M 1 =M 2
the mass ratio, M ˆ 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 ˆ f min
;1 ;
…28†
2tconv
where Ptid is the tidal pumping time-scale given by
1
Vspin 1
;
ˆ
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
;
ˆ
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 ˆ 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 ˆ 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
…‰BaŠ > 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
ˆ
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†
ˆ 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 ˆ 0:1; B ˆ 2000 and aacc ˆ 1:0 (model 1) and (b) f ˆ 0:075; B ˆ 7:5 102 and
aacc ˆ 0:075 (model 10). We have assumed that an overabundance of ‰BaŠ > 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 ˆ 7:5 102 ; f ˆ 0:075 and aacc ˆ 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 ˆ 0:075; B ˆ 750 and aacc ˆ 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 ˆ 0:075; B ˆ 750 and aacc ˆ 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 ‰BaŠ , 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