Monte Carlo simulations of the orbital elements and abundances of barium stars

CHINESE ASTRONOMY AND ASTROPHYSICS PERGAMON

Chinese

Astronomy

and Astrophysics

27 (2003)

292-302

Monte Carlo Simulations of the Orbital Elements and Abundances of Barium Stars+* ZHANG Boll2 NIU Ping’ PENG Qiu-he4

SHI Wei-bin’ ‘Department

of Physics, Hebei Normal

LIU Jun-hong3

University, Shijiazhuang

050016

2Nationul Laboratory of Heavy Ion Accelerator, Lanzhou 730000 3 Department of Physics, Shijiazhung Teachers’ College, Shijiazhuang 050041 4Department of Astronomy, Nanjing University, Nanjing 210093

We have carried out a series of Monte Carlo simulations to study the distributions of the orbital elements of normal red giant binary systems and barium stars with the wind accretion model under the condition of total angular momentum conservation. Since barium star systems have evolved from normal red giant binary systems, their distributions of orbital eccentricities and periods exhibit the characteristics of the final orbits of binaries after mass accretion. Our calculations show that in the process of wind accretion and in the masslosing stage, the system gets bigger, and its orbital period increases, while the orbital eccentricity does not vary much. This can explain the various features in the distributions of the orbital elements of normal red giant binary systems and barium stars, as well as features in the distribution of the heavy-element abundances of barium stars.

Abstract

Key words: stars: chemically peculiar post ~ AGB ~ binaries - stars: winds

stars: abundaces -

stars: AGB and

1. INTRODUCTION

The s-element-rich AGB stars can be divided into two categories[l]: intrinsic AGB stars and extrinsic AGB stars. The intrinsic AGB stars, including the MS, S and C (N-type) stars t Supported by National Received 2002-07-08; * A translation of Acta

Natural revised Astron.

0275-1062/03/$-see front matter DOI: 10,1016/SO275-1062(03)00069-9

Science Foundation version 2002-10-28 Sin. Vol. 44, No. @ 2003

Elsevier

2, pp.

Science

116-125,

2003

B. V. All rights

reserved

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with observable “Tc, are commonly called TP-AGB stars, their s-element overabundances are caused by the ongoing nucleosynthesis and the third dredge-up. The extrinsic AGB stars, including various G, K-type barium stars and cool S, C stars in which ggTc is not observed, are binary systems whose heavy-element overabundance is caused by accretion, i.e., in the process of mass transfer they obtain the s-element-rich matter from their former companions12-51 (former AGB stars, progenitors of current white dwarfs (WDs)). Such mass transfer took place 1 x lo6 years previously, so the element ggTc produced in the original TP-AGB stars hass almost totally decayed. The study of the barium stars is of very great significance for the theory of binary evolution and the theory of heavy-element nucleosynthesis. In 1956, S. S. Huang[‘jl discussed the variations of the orbital parameters of a binary system undergoing mass-loss. In 1988, Boffin et a1.131calculated quantitatively the variations of the orbital parameters of barium stars caused by wind accretion using the available equations of the variations of orbital parameters, and estimated their heavy-element overabundances. Later, some authors calculated again the heavy-element overabundances of barium stars by similar methods171, and explained the relationship between their s-element overabundances and their orbital periods. Although the wind accretion theory has given some significant results, the condition of tangential momentum conservation adopted in the model was contradicted by the calculated result. In addition, the calculation of the overabundances uses the ‘step-process” model: it is assumed that at some time the heavy-element abundance of the star in solar units, changes suddenly from 1 to some value f (specific to the element), and is then kept constant till the end of AGB stage. But in fact, after the beginning of the third dredge-up, the overabundance factors of the AGB star will change with the number of dredge-ups. It is only after multiple dredge-ups that the C/O ratio in the outer envelope of the AGB star reaches 1, and the star becomes a C star181. The heavy-element overabundances of barium stars should be the result of successive pulses of mass accretion and mixing. In 1995, Busso et al.fgl discussed the heavy-element overabundances of barium stars by diluting (i.e., decreasing by a certain factor) the calculated heavy-element abundances of intrinsic AGB stars, but the effects of mass accretion and of orbital parameters were not considered. Han et al.[iOl studied in detail the different formation mechanisms of barium stars, and four were suggested: wind accretion, wind exposure, stable Roche lobe overflow, and common envelope ejection, then the authors established a complete model for the formation of barium stars. Jorissen et a1.151analyzed the orbital elements of a large sample of binary systems, and pointed out that for long-period systems the accreting mechanism is wind accretion and for short-period systems it is matter-filled Roche lobe overflow. In the study of barium stars the eccentricity-period (e, log P) diagram of binary systems is very important, for it shows the final orbital features of binary systems after accretion151. It is commonly accepted that the barium star binaries evolved from normal red giant binaries. The (e, 1ogP) diagrams of these two kinds of binary systems showed that the mean orbital eccentricity is clearly smaller in the barium star systems than in the normal red giant systems[5]. In 2000, Karakas et al.[‘1l studied the distributions of the orbital elements of barium stars. Because they again adopted the model with tangential momentum conservation, their results, while giving a general explanation of the distributions of the orbital elements of barium stars, failed to do so for those barium star systems with high eccentricities, as well as the distribution of the orbital parameters of normal red giants.

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For a binary system with mass-loss due to wind, it is more reasonable to start from the condition of conservation of the total angular momentum of the system. In 2000, Liang et a1.[12] and Liu et a1.[131 derived again the equations for the variations of the orbital elements for the wind accretion model of binary systems under the condition of total angular momentum conservation, and calculated self-consistently the heavy-element overabundances of barium stars by combining the wind accretion model with the updated model of intrinsic TP-AGB nucleosynthesis, for successive pulses of mass accretion and mixing. On the basis of the above works we have restudied the variations of the orbital elements of barium stars in wind accretion and the the distribution of their heavy-element abundances.

2. OBSERVATIONAL

MATERIAL

If barium star systems have evolved from normal red giant binaries, the mass accretion model should be able to explain the following observational data: (1) The distributions of the orbital eccentricities and periods of red giant binary systems and of barium star systems[51. The envelope line of the maximal eccentricity of barium star systems is approximately parallel to that of normal red giant binary systems, with a displacement in the direction of long periods. The maximum eccentricities of both the normal red giant binary systems and barium star systems can be as high as 0.9. (2) The fact that for normal red giant binary systems, no circular orbit exists when P > 350d; and for barium star systems, no circular orbit exists when P > 2000d. (3) The mass function distributions of both the normal red giant binary systems and barium star systems[141. (4) The s-process-element abundance-period relationship of the barium starsf71.

3.1

The

Model

and

3. MODEL

AND

MAIN

Parameters

for

Intrinsic

PARAMETERS AGB

Stars

3.1.1 Core Mass The C-O core mass of the AGB star at the beginning of the thermal pulse is taken from Groenewegen et al.I1’l. As successive pulses take place, the burning H/He shells are pushed outward, and the C-O core mass A4, is increased outward[l’l. The C-O core mass at the end of the thermal pulse is taken from Jarrod et a1.[16]. The variation of the core mass affects directly the mass dredged up each time to the surface and the mass-loss due to wind. 3.1.2 h/lass-loss due to wind The two major features of AGB stars are the thermal pulses and mass loss. In the course of evolution, mass loss due to wind will lead to a decrease of the envelope mass. The mass loss rate was given by Reimers[171, and based on it Busso et a1.[181gave the following expression: izd, = 4.65 x 1O-5q x 1.175°.31s x

(111, - o.5)‘%“~088

(1) ’ 1 in which, MI is the mass of the AGB star, M, is the core mass (they are in units of solar mass hIa), and 2 is the initial metal abundance. S takes the value 1 when Ml > 1.175, [email protected]&5”

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otherwise t,alces 0, and the other two parameters are a =2.2, 77=3.0[1”l. The mass loss per pulse is AM1 = -ti,At.

(2)

In the calculations the time interval between two pulses, At, is taken as At = lo4 yr[“l. 3.1.3 Dilution factor The dilution factor (after the third dredge-up) is the ratio of the mass dredged up to the surface by each dredge-up, AMTDU, to the mass of the outer envelope, Me,,,, i.e.,

f=

AMTDU Menv

+

(3)

AMTDU

The mass dredged up during each thermal pulse is AMTDIJ = ~AM,[~OI, and t,he dredge-up efficiency X is[‘g,“ll:

A= 0.2 + 0.0866M1,o 0.9 )

) Ml,0 < 3.OM@ Ml,0 2 3.0Ma.

(4

3.1.4

Overabundance factors of AGB stars The barium overabundance in the He-shell depends on the initial mass Ml,o. For lowmass AGB stars, the main neutron source is 13C(a, n)160; because of its longer time-scale, the neutron irradiation is larger, so the overabundance factor is greater. But for AGB this has a shorter stars of intermediate mass, the main neutron source is 22Ne(a,n)25Mg; time-scale, and the neutron irradiation is less, so the overabundance factor is also less[221231: 3.0 < log(Ba,k,,ll/Bao) 1.6 5 log(Ba,t,,ll/Bao) 3.2

Wind

Accretion

Model

and

the

5 4.0, Ml,0 < 3.0Mo 5 2.6, Ml,0 2 3.0Mo. Main

Parameters

For the wind accretion model, we assume that the binary system system, that the interior structures of the two stars are not affected by the evolutionary model for a single star is still usableL31. In consideration momentum conservation, the equations of the variations of the orbital and eccentricity e are[‘“,‘“]: AA A

AMI +AM,

= -q-E”)3

2(1 -$)T

1 M,(AM, MI(MI

Ml

+ AMz) + M2)

+

442

u -+ 7)0,1,

is always a detached each other, and that of the total angular semi-major axis ,4,

1 (5)

.(2(l-e’)i-l]A$~~, 1

2

(6)

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in which Ml and Mz are respectively the masses of the intrinsic and extrinsic AGB stars. The Bondi-Hoyle (B-H) accretion rate15>241is adopted in the mass accretion equation of barium stars:

(7) in which cr is the constant accretion efficiency, Vej is the wind velocity, taken to be 15 km/s171 in the calculations, and V&b = pizjizi is the mean orbital velocity. In the calculations, 15 percent of the Bondi-Hoyle accretion rate is taken as the actual accretion rate171. As soon as the initial condition is given, for a given mass AMI ejected by the primary star in a pulse, the mass AM2 accreted by the secondary barium star can be obtained from Eqs. (5147). 3.3

Parameters for Monte Carlo Simulations (1) The initial mass of the primary star is taken asl25l:

Ml,0 = 0.3( &)“”

.

In our simulations, the primary’s mass is randomly selected in the range between 1.5 Ma and 8.0 Ma. Xi (as well as all Xi below) is the uniformly distributed random variable in (OJ). (2) The distribution

of the initial mass ratio

q

(= Ml,o/M2,0)

is taken ;t~l~~l:

from which the initial mass of the companion M2,o (1.0 MO < MZ,O < 8.0 Ma) is determined. (3) The initial eccentricity distribution isl”l: e = 0.99(1 - a) Considering the observed eccentricity

distribution

(10)

of the red giant binary systems, we take:

e < 0.5(logP - 1.25) ,

(11)

and for system with log P > 2.6, we take e > 0.03151. (4) The initial period distribution is1251: p = 5.104 j&&)3,3

>

(12)

in which ILI = Ml,0 + M~,o, and from Eq.(ll) we have log P > 2e + 1.25. For the progenitor binary systems of the barium star systems, their period distribution will be restricted indirectly by the abundance condition of barium stars ([Ba/Fe]> 0.2) as well. (5) The overabundance factor in the He-shell of the AGB star According to the discussion in section 3.1.4, the barium-overabundance factor in the AGB star’s He-shell is:

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MI,~ < 3.OM,

3+x5,

log(B%dB%) = (6) The mass function

Astronomy

297

1.6 + X5, Ml,o _> 3.OM,.

(13)

f(m) = sin3ecrnl+ m2)2,

(14)

is1141:

in which 0 is the orbital inclination, 8 = Xs1r/2, i.e., taken randomly in the O”-90” range Cm2 > ml). The calculations are performed in two steps. First, we select randomly an initial mass of the primary star (Eq.(8)), an initial mass-ratio (Eq.(9)), an initial eccentricity (Eq.( 10)) and an initial orbital period (Eq.( 12)), and so fix the initial core mass (refer to the Fig.1 in paper [15]). According to Eqs.(3), (4), and (13) we calculate the heavy-element overabundance on the surface of the intrinsic AGB star for each pulse, calculate the mass-loss by wind by Eqs.( 1) and (2), and the total number of the pulses is determined by the stripping-off of all the mass of the AGB star envelope. Afterwards, combining Eqs. (5)-(7) for the wind accretion model we calculate the variations of the orbital elements and the accreting mass for each pulse, and via the successive pulse accretion and mixing calculate the heavy-element overabundance of the barium star. In the calculations the initial envelope mass Men”,0 of the barium star is taken to be131: M env,0 = (0.6Mz,o + 0.925Mz,,, - 0.425)/2.

(15)

The overabundance factor gi for nuclear element i of the barium star is given by the following formula:

n=l,m Si

=

Me,, +

C

M,“,,

’

(16)

n=l,m

in which Me,, is the envelope mass of the extrinsic star, M.& is the wind-accreted mass of the extrinsic star during the n-th pulse of the intrinsic star (the current WD), f: is the overabundance factor for the nuclear element i of the intrinsic star during the n-th pulse, m is the total number of the pulses experienced by the intrinsic star, and gp is the overabundance factor for the nuclide i of the extrinsic star before mass accretion. In the above formulae it is assumed that the accreting matter has been mixed sufficiently with the convective envelope of the extrinsic star.

4. RESULTS

AND

ANALYSIS

By using the Monte Carlo method, the initial mass of the primary star, the initial mass-ratio distribution, the initial eccentricity and orbital period are randomly selected for lo5 binary systems, and their evolution is then followed according to the wind accretion model. Fig. 1 shows the eccentricity-period distribution (e, log P) for the G-K red giant binaries with masses restricted to (1.5 n1a < Ml < 8.0 Ma, Mz > 1.0 hfa) and with companions

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satisfying the abundance condition of barium stars ([Ba/Fe]> 0.2). Because of the above restrictions the number of the binary systems is greatly reduced (in the figure the actual number is 800). We note that the calculated results have covered the observed distribution of all G-K red giant binaries, and that circular orbit does not exist for periods longer than 350 days. Fig.1 can .be considered as the (e, 1ogP) distribution of the progenitor binary systems of barium star systems, i.e., the initial distribution of barium star systems. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 I Fig. 1 stars

1.5 (e,

2 log P)

2.5

3

3.5

logp diagram of the

4 G-K

0

4.5 red

giant

Fig. 2 normal

0.2 The G-K

0.4 Eccantrictiy distribution

red

giant

0.6 0.8 of G-K stars of eccentricities

for

the

stars

For comparisons in detail, Fig.2 gives the histogram for the distribution of the eccentricities of G-K red giant binaries corresponding to Fig.1, in which the dashed line represents and the full line represents the distribution given by the the observational distribution[5], model. The two distributions are consistent with each other within the errors by the KS test at 97% confidence level. Fig.3 presents the histogram for the distribution of the periods of G-K red giant binaries corresponding to Fig.1, in which the dashed line represents the observational distribution[51, and the full line represents the distribution given by the model. By the KS test the two are mutually consistent within the errors at 99.9% confidence level. Fig.4 shows the mass function distribution of G-K red giant binaries, in which the dashed line represents the observational distribution[141, and the full line represents the distribution given by the model; the two are consistent at 99% confidence level. Fig.5 shows the eccentricity-period distribution (e, 1ogP) of barium star systems ([Ba/Fe]>_ 0.2) d erived from the wind accretion model. For the same reason as mentioned for Fig.1, the number of barium star systems is much reduced (in the figure the actual number is 800), but the results obtained have covered the all observational values of barium stars. The maximum orbital eccentricity can reach 0.9 for both the normal red giant binaries and barium star systems. And when the period is longer than 2000 days, no binary systems exist with circular orbits. This is basically in agreement with the observations. For a more detailed comparison, Fig.6 displays the histogram for the distribution of the eccentricities of barium stars, in which the dashed line represents the observational distributionL51, and the full line represents the distribution given by the model. The two are consistent within the observing errors by the KS test at 67.5% confidence level. The rather low confidence level is caused by the small number of observed barium star samples. In addition, for the short-period systems the tidal effect should be taken into consideration, and this will make the orbits circular. Fig.7 shows the histogram for the distribution

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30 -

b

20 __/

---

I I

6-

I

1.5 I.5

Fig. 3 G-K

2

The

2.5

3 3.5 4 logp of the periods

distribution

giant

5

4.5

of normal

stars

0.08

Fig. 4

The

distribution

normal

G-K

giant

20--18lfj-

0.; 0.8 0.7 0.6 05 0.4 0.3 0.2 0.1 0 1.5

2

Fig. 5

2.5 The

3

3.5 4 4.5 fogp (e, log P) diagram of barium

2 0

5 stars

/

10 -

,- - -,

,t5-

”

I

J I.5

/ /

I I

--2

2.5

0

Fig. 6

The stars

3

3.5

4

E

10

z’

6 4

2 4.5

5

Fig. 7

The

distribution

mass

function

of

0.4 0.6 0.8 Eccentrictiy of Ba stars distribution of the eccentricities

1

0.2

of

l-Ba /-model

2 80 : 0

0.08

0.16

1

0.24

f(m)

logp stars

of the

stars

14 I2

20 IS-

0.24

0.16

I ‘-----,

barium

I

8 2 z

--r-r

420 0

s50

299

292-302

20 _ 18 --/ l6’ l4I ti I2 ZIOz’ 8-

25 -

-E 15 s IO 10 -

27 (2003)

of the

periods

of barium

Fig. 8

The

barium

stars

distribution

of the

mass

function

of

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j 1jj 400

3oo-

i

-0

1

Fig. 9 The

2

mass

3 4 5 6 Ba-star masses distribution of barium

7

200-

8

stars

Fig.

0.5

0.6

0.7

10

The

mass

0.8 0.9 1 1.1 White-dwarf masses distribution

1.2

of white

1.3

1.4

dwarfs

of the periods of the barium stars, in which the dashed line represents the observational distribution15], and the full line represents the distribution of the model. The KS test (at 99.9% confidence level) indicates that the model is quite consistent with the observational result within the error range. From Figs.2,3, 6 and 7 we can find that mass loss in the course of wind accretion will lead to an increase in the orbital semi-major axis and the orbital period, but only to small variations in the eccentricity. The mean orbital eccentricity is smaller in the barium stars than in the normal red giant binaries, and this is in accordance with the observations[261. Fig. 8 gives the mass function distribution, in which the dashed line represents the observational distributionl141, the full line represents the distribution given by the model: they are mutually consistent at 97% confidence level. Figs. 9 and 10 present separately the histograms of mass distributions for the barium stars ([Ba/Fe]> 0.2) and the white dwarfs (the companions of the barium stars). It is found that when the mass of the barium star is rather small, the envelope mass is small. This is favorable for overabundance, so more barium stars are generated. The masses of barium stars are mainly concentrated in the range l-3 M 0; the smaller the initial mass of the AGB star, the higher is the overabundance factor, and the greater the number of barium stars generated. This is basically in accordance with the result of Karakasll’l. 1.6

--

1.4

.:.

+.

J . -i’

1

2

3

4

5

6

‘ogp Fig. 11 periods

‘I‘he

distribution

of the barium

of the stars

abundances

and

Fig. 11 shows the relationship between the Ba abundances on the surface of barium stars and the orbital periods. The dark dots represent the simulated Ba abundances of barium stars, the observational results are also indicated17]. We can see that the simulated results have covered all the observational results. The shorter the orbital period of the binary system, the larger the mass accretion, and the higher the heavy-element abundance. This is characteristic of wind accretion. In the figure those objects satisfying the condition [Ba/Fe]> 0.2 are barium stars.

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CONCLUSIONS

In this paper, we adopted the wind accretion model with the condition of total angular momentum conservation and used the Monte Carlo method to study the variations and distributions of the orbital elements of barium stars. The calculations demonstrate that in the course of wind accretion, in the mass-losing stage of the wind the orbital semi-major axis increases, the orbital period increases, but the eccentricity does not vary significantly. These can interpret the following observational facts: (1) The mean orbital eccentricity is smaller in the barium star systems than in the normal red giant binaries of the same orbital period. (2) The maximum orbital eccentricity can reach 0.9 in both the normal red giant binaries and barium star systems. (3) The envelope line of maximal eccentricity of the barium star systems is approximately parallel to that of the normal red giant binary systems, with a displacement in the long-period direction. (4) For the normal red giant binary systems, circular orbits do not exist when P > 350 d; for the barium star systems, when P > 2000 d, in common with the CH and S binary systems. Because barium star systems have evolved from normal red giant binary systems, the distributions of the orbital eccentricities and periods of barium star systems exhibit the features of the final orbits of binary systems after mass accretion. The model adopted in this paper can also explain well the following observational results: (1) The distributions of orbital eccentricities and periods of red giant binary systems, and those of barium star systems151; (2) The mass function distribution of normal red giant binary systems and that of barium star systemsl’41; (3) The s-process-element abundance-period relationship of barium stars171. ACKNOWLEDGEMENT We sincerely thank Professor Han Zhan-wen Observatory for his valuable help.

of Yunnan

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