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Astronomy Reports, Vol. 47, No. 10, 2003, pp. 826–830. Translated from Astronomicheski˘ ı Zhurnal, Vol. 80, No. 10, 2003, pp. 896–901. Original Russian Text Copyright c 2003 by Tutukov, Fedorova. Rotational Mass Loss by Be Stars A. V. Tutukov and A. V. Fedorova Institute of Astronomy, Moscow, Russia Received March 27, 2003; in nal form, May 8, 2003 AbstractWe have carried out a numerical study of rotational mass loss by rapidly rotating Be stars assuming preservation of rigid-body rotation during their main-sequence evolution. Evolutionary models are computed for stars with solar chemical composition and initial masses of 3, 10 and 30 M . As a result of their rapid initial rotation, these stars can lose one to four percent of their initial mass during the main- sequence stage. The amount of mass lost increases with the initial mass of the star. The matter lost by Be stars can form gasdust disks with masses comparable to the masses of planets, which, in principle, makes possible the formation of planetary systems around such stars. c 2003 MAIK Nauka/Interperiodica. 1. INTRODUCTION The classical topics of modern observational as- tronomy include Be stars, in which the usual spectra of B stars are superposed with emission lines, indi- cating the presence of a large amount of gas in the vicinity of the star. This gas forms a circumstellar gaseous disk. Emission lines can be found in the spectra of main-sequence (MS) stars of almost all spectral types. For instance, Oe, Ae, Fe, Ge, and Me stars are known. Thus, the presence of circumstellar disks is, to various extents, characteristic of almost all MS stars. For this reason, we will assume that Be stars are MS stars with rotationally supported circumstellar disks. The masses of Be stars are con- ned to the range 318 M . The fraction of B stars with emission lines in their spectra is 20% [1, 2]. It is possible that this fraction depends on the abun- dance of metals. For the stars of the Small Magellanic Cloud, this fraction increases to 40% [2], for reasons that remain unclear. At least for some Be stars, it is likely that the for- mation of their circumstellar envelopes is due to mass loss. The mass-loss rates of several Be stars have been estimated. For X Persei with T e= 31 000 K, log g =4, and R =9 R the mass-loss rate is 5 × 10 9 M /yr [3]. The star’s parameters suggest that the mass of X Persei is (1015) M . Assum- ing that the MS lifetime of such a star is 10 7 yr, we can estimate the total amount of mass lost by the star in the course of its MS evolution, 0.05 M . It is evident that such mass loss does not signicantly inuence the evolution of the star. Let us consider the main approaches to studies of the Be-star phenomenon. Some Me, Ge, Fe, Ae, and Be stars may be young objects that are accreting the remains of their protostellar envelopes. This is relevant rst and foremost in connection with Her- big Ae and Be stars. It is interesting that, in this case, the protostellar material can fall into the close vicinity (within several stellar radii) of the accreting star. Later, due to viscosity, the outer parts of the initially compact disk expand and attain the size of the protoplanetary cloud, 100 AU [4]. One example of such a star is the Aе star HD 34282 [5] with a mass of 2 M , which has a gasdust disk with a mass of 0.1 M and a radius of 800 AU. Such disks are typical of young Ae and Be stars [6]. The peculiarities of the chemical compositions of circumstellar disks suggest that planets may be present in at least some of them, such as the disk of HD 160546 [7]. Naturally, the outer parts of accretiondecretion disks not only survive until the central star arrives on the main se- quence, but continue to exist over the entire lifetime of the star, providing, in principle, the possibility of forming planetary systems. Let us continue to consider the formation of cir- cumstellar gaseous disks. About 50% of stars are members of close binary systems [9]. Over the Hubble time, the components of binaries with masses greater than 1 M expand and transfer some fraction of their matter to their companions. This matter forms an accretion disk, as in classical Algols. Examples of such binaries containing Be stars are provided by the semi-detached systems V 360 Lac (B3e + F9IV) [10] and HD 50123 (B6Ve + KIII) [11]. The orbital periods of Be stars in known binary systems lie in the range 5400 day [12]. Up to 30% of Be stars have various types of degenerate dwarfs as companions [13]. One example is γ Cas, which is a Be + WD system [14]. Some fraction of Be stars may have neutron stars or black holes as companions [13]; one example is LS1 + 61 303 [15]. It is interesting that the variability of some Be stars may be due to the orbital motion 1063-7729/03/4710-0826$24.00 c 2003 MAIK Nauka/Interperiodica

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Astronomy Reports, Vol. 47, No. 10, 2003, pp. 826–830. Translated from Astronomicheskiı Zhurnal, Vol. 80, No. 10, 2003, pp. 896–901.Original Russian Text Copyright c© 2003 by Tutukov, Fedorova.

Rotational Mass Loss by Be Stars

A. V. Tutukov and A. V. FedorovaInstitute of Astronomy, Moscow, Russia

ReceivedMarch 27, 2003; in final form, May 8, 2003

Abstract—We have carried out a numerical study of rotational mass loss by rapidly rotating Be starsassuming preservation of rigid-body rotation during their main-sequence evolution. Evolutionary modelsare computed for stars with solar chemical composition and initial masses of 3, 10 and 30 M�. As a resultof their rapid initial rotation, these stars can lose one to four percent of their initial mass during the main-sequence stage. The amount of mass lost increases with the initial mass of the star. The matter lost by Bestars can form gas–dust disks with masses comparable to the masses of planets, which, in principle, makespossible the formation of planetary systems around such stars. c© 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION

The classical topics of modern observational as-tronomy include Be stars, in which the usual spectraof B stars are superposed with emission lines, indi-cating the presence of a large amount of gas in thevicinity of the star. This gas forms a circumstellargaseous disk. Emission lines can be found in thespectra of main-sequence (MS) stars of almost allspectral types. For instance, Oe, Ae, Fe, Ge, and Mestars are known. Thus, the presence of circumstellardisks is, to various extents, characteristic of almostall MS stars. For this reason, we will assume thatBe stars are MS stars with rotationally supportedcircumstellar disks. The masses of Be stars are con-fined to the range 3–18 M�. The fraction of B starswith emission lines in their spectra is ∼20% [1, 2].It is possible that this fraction depends on the abun-dance of metals. For the stars of the Small MagellanicCloud, this fraction increases to∼40% [2], for reasonsthat remain unclear.

At least for some Be stars, it is likely that the for-mation of their circumstellar envelopes is due to massloss. The mass-loss rates of several Be stars havebeen estimated. For X Persei with Teff = 31000 K,log g = 4, and R = 9 R� the mass-loss rate is∼5 × 10−9 M�/yr [3]. The star’s parameters suggestthat the mass of X Persei is ∼(10–15)M�. Assum-ing that the MS lifetime of such a star is ∼107 yr,we can estimate the total amount of mass lost by thestar in the course of its MS evolution, ∼0.05 M�. Itis evident that such mass loss does not significantlyinfluence the evolution of the star.

Let us consider the main approaches to studiesof the Be-star phenomenon. Some Me, Ge, Fe, Ae,and Be stars may be young objects that are accretingthe remains of their protostellar envelopes. This is

1063-7729/03/4710-0826$24.00 c©

relevant first and foremost in connection with Her-big Ae and Be stars. It is interesting that, in thiscase, the protostellar material can fall into the closevicinity (within several stellar radii) of the accretingstar. Later, due to viscosity, the outer parts of theinitially compact disk expand and attain the size of theprotoplanetary cloud, ∼100 AU [4]. One example ofsuch a star is the Aе star HD 34282 [5] with a massof ∼2 M�, which has a gas–dust disk with a mass of∼0.1 M� and a radius of ∼800 AU. Such disks aretypical of young Ae and Be stars [6]. The peculiaritiesof the chemical compositions of circumstellar diskssuggest that planets may be present in at least someof them, such as the disk of HD 160546 [7]. Naturally,the outer parts of accretion–decretion disks not onlysurvive until the central star arrives on the main se-quence, but continue to exist over the entire lifetimeof the star, providing, in principle, the possibility offorming planetary systems.

Let us continue to consider the formation of cir-cumstellar gaseous disks. About 50% of stars aremembers of close binary systems [9]. Over theHubbletime, the components of binaries with masses greaterthan ∼1 M� expand and transfer some fraction oftheir matter to their companions. This matter formsan accretion disk, as in classical Algols. Examples ofsuch binaries containing Be stars are provided by thesemi-detached systems V 360 Lac (B3e +F9IV) [10]andHD 50123 (B6Ve+KIII) [11]. The orbital periodsof Be stars in known binary systems lie in the range5–400 day [12]. Up to∼30% of Be stars have varioustypes of degenerate dwarfs as companions [13]. Oneexample is γ Cas, which is a Be + WD system [14].Some fraction of Be stars may have neutron starsor black holes as companions [13]; one example isLS1+ 61◦303 [15]. It is interesting that the variabilityof some Be stars may be due to the orbital motion

2003 MAIK “Nauka/Interperiodica”

Page 2: Rotational mass loss by Be stars

ROTATIONAL MASS LOSS BY Be STARS 827

of the binaries to which they belong and quasi pe-riodic activity of their mass-losing companions, asis observed for FY CMa [16]. It is evident that, af-ter the cessation of mass transfer, the relic gaseousdisks around the accreting components of close bi-naries rapidly dissipate. However, they can exist forcomparatively long times in wide binaries and maybe transformed into protoplanetary disks if they haveenough time.

Let us consider two other possible mechanisms forthe formation of circumstellar gaseous disks aroundrapidly rotating stars with radiative envelopes. Anal-ysis of the initial semi-major orbital axes of closebinaries as a function of the primary masses indicatesthat, in the course of their accretion as they are form-ing, the semi-major axes of young binaries follow an

a ∼ M1/31 law, where M1 is the mass of the primary.

Since the radii of MS stars with M1 � 1.5 M� in-

crease as M2/31 , some of the closest MS binaries with

M1 � 10 M� merge, forming configurations with twonuclei [13]. No systems of this kind have been ob-served as yet; it may be that they should be searchedfor among early-type contact binaries with periodsclose to or slightly shorter than one day. The coales-cence of the nuclei in such “binuclear” stars shouldresult in the outward diffusion of angular momentumand the formation of a circumstellar decretion disk.Observationally, such a system would be manifest asa B star with emission lines in its spectrum.

The last possible mechanism for the formation ofcircumstellar gaseous disks, which is the topic of thepresent study, is the diffusion of the angular momen-tum of a star that evolves along the main sequencewhile maintaining rigid-body rotation [17]. Qualita-tive estimates have shown that, for a star with a radia-tive envelope, the ratio of the angular velocity and thebreak-up velocity increases as the star evolves alongthe main sequence. For sufficiently rapid initial rota-tion, this should lead to continuous mass loss duringthe star’s evolution from point A of its evolutionarytrack to point B and further to point C (see Fig. 1).The mass-loss rate is especially high in the B to Cportion of the track. When the star begins to expandafter leaving the main sequence, rotational mass lossterminates. In this simple model, we cannot excludethe possibility that some excess angular momentumis stored by the young star in its inner dense layers,which can rotate faster than the envelope. This effectcan enhance the total mass-loss by a factor of a few.

Observations of Be stars provide some evidence infavor of this last explanation for their mass loss. Thestudy of Be stars in extremely young clusters [18] hasshown that these stars are concentrated toward earlysubtypes, characteristic of stars that are leaving themain sequence. The more detailed study [19] showed

ASTRONOMY REPORTS Vol. 47 No. 10 2003

A BC

A

A

C

C

B

B

30

M

10

M

3

M

4.7 4.6 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.81.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0log

L

/

L

log

T

eff

[K]

Fig. 1. Evolutionary tracks of stars in the Hertzsprung–Russell diagram. The solid curves show sections of trackscorresponding to the phase of rotational mass loss. Thedashed curves show sections of tracks without mass loss.The numbers by the tracks indicate the initial massesof the stars. The letters mark points A, B, and C of theevolutionary tracks.

that Be stars are present only in clusters with agesof (1.3–2.5) × 107 yr, which corresponds to the MSlifetimes of stars with masses ∼10 M�. Thirty-twostars in the cluster h and χ Persei have masses ofthe same order of magnitude [20]. Finally, the detailedstudy of the rotation of 1092 main-sequence B starsled Abt et al. [21, 22] to conclude that the rigid-body rotation of the stars was preserved right up tothe stages when the stellar radii expand to about fourtimes their Zero Age Main Sequence (ZAMS) radii.Thus, we can indeed consider a massive (�2 M�)star to rotate as a rigid body in the course of its MSevolution, even though its radius may grow by a factorof two. The mechanism that sustains the rigid-bodyrotation has not been clearly established; it may beassociated with the magnetic field of the star.

2. COMPUTATION RESULTS

We can numerically study mass loss by rapidlyrotating stars using the law of conservation of an-gular momentum, Iω = const, where I is the star’smomentum of inertia and ω is the angular velocityof its rigid-body rotation. To estimate the equatorialmass loss, we assume that the matter lost by the starcarries away a specific angular momentum equal toω2crR, where R is the instantaneous radius of the star

and ωcr is the break-up angular velocity. The resulting

Page 3: Rotational mass loss by Be stars

828 TUTUKOV, FEDOROVA

1.9 2.0 2.1 2.2 2.3–11

–10

–9

–8

log

L

/

L

log

dM

/

dt

[

M

/year]

Fig. 2. Dependence of the rate of rotational mass loss onstellar luminosity for a star with initial mass 3 M�. Thetriangle corresponds to point A, the filled circle to pointB, and the open circle to point C.

3.7 3.8 3.9 4.0 4.1–9.0

–8.0

–7.0

–6.0

log

L

/

L

log

dM

/

dt

[

M

/year]

–6.5

–7.5

–8.5

4.2

Fig. 3. Same as Fig. 2 for a star with initial mass 10 M�.

equation for themass-loss rate of a star rotating at thestability limit is

dM

dt=

I

R2

(d ln I

dt− 3

2d ln R

dt

)(1 − I

2MR2

)−1

.

(1)

This equation is valid only when dM/dt < 0 andthe angular velocity exceeds the break-up velocity.These conditions hold for a star evolving on the main

5.0 5.1 5.2 5.3 5.4

–5.5

–4.5

log

L

/

L

log

dM

/

dt

[

M

/year]

–5.0

–6.0

–6.5

–7.0

–7.55.5

Fig. 4. Same as Fig. 2 for a star with initial mass 30 M�.

0.4 0.5 0.6 0.7 0.8

0.02

0.04

ω

/

ω

cr

M

/ M

0.03

0.01

01.00.9

30

M

10

M

3

M

Fig. 5. Fraction of mass lost by a star as a function ofthe initial ratio of the stellar angular velocity to the break-up velocity assuming rigid-body rotation. The numbersindicate the initial masses of the stars.

sequence if its rotation on the ZAMS was sufficientlyfast [17].

Our analysis is only approximate, since our mod-els are one-dimensional. Moreover, we assume thatequatorial mass loss begins only when the star at-tains the break-up velocity, while, in reality, mass lossprobably starts earlier, due to radiation pressure andpossible pulsations. Therefore, our results should be

ASTRONOMY REPORTS Vol. 47 No. 10 2003

Page 4: Rotational mass loss by Be stars

ROTATIONAL MASS LOSS BY Be STARS 829

considered only preliminary numerical estimates ofthe rate of rotational mass loss.

We carried out computations for MS stars withsolar chemical composition (X = 0.70, Y = 0.28,Z = 0.02) and initial masses of 3, 10, and 30 M�.Evolutionary tracks were computed for theMS phaseand further, to the termination of the rotational mass-loss. For comparison, we computed evolutionarytracks for stars with constant mass. Figure 1 presentsthe tracks of mass-losing stars and stars with con-stant mass in the Hertzsprung–Russell diagram. Wecan see that the rotational mass loss is evolutionarilyinsignificant and does not appreciably change thetracks of the stars.

Figures 2, 3, and 4 show the dependence of therate of rotational mass loss on the stellar luminosity.We can clearly see that the mass-loss rate is relativelysmall in the first stage of evolution, from the ZAMSto point B, but grows appreciably in the next evo-lutionary stage, from point B to point C. Note thatthe estimated mass-loss rate for X Persei given in theIntroduction (∼5 × 10−9 M�/yr [3]) is in fairly goodagreement with our estimate of the mass-loss rate fora star with initial mass 10 M� in the first, longest,phase of evolution (Fig. 3).

In Fig. 5, we show the dependence of the fractionof mass lost by the star on the ratio of the initial andbreak-up angular velocities. The minimum values ofthis ratio for which rotational mass loss occurs are0.78, 0.75, and 0.47 for stars with initial masses of3, 10, and 30 M�, respectively. Note that nonuniformrotation of a star on the ZAMS can significantly re-duce the initial rotational velocity necessary for mass-loss. Figure 5 clearly shows that the fraction of masslost due to rotation increases with increasing stellarmass. This can be explained by the properties of thecores of MS stars of different masses. The higher themass of the star, the higher the mass of the con-tracting helium core in the final phases of the star’sMS evolution. This results in a stronger reductionof the star’s moment of inertia and, hence, in theacceleration of its rotation.

3. CONCLUSIONS

Our computations have shown that a Be star canlose 1–4% of its initial mass in the course of its MSevolution if the initial velocity of its rotation was closeto the break-up velocity and the rotation itself wasrigid-body. Let us consider some consequences ofrotational mass loss. It is obvious that the matter lostby the star due to rotation initially forms a compactgaseous disk. This disk will expand in the courseof time due to friction and will probably partiallyevaporate [4, 23, 24]. Observational studies of severalBe stars has revealed the formation and expansion

ASTRONOMY REPORTS Vol. 47 No. 10 2003

of gaseous rings around them [23, 25]. By analogywith the accretion–decretion disks observed aroundHerbig Ae and Be stars [26], which are occasionallyeclipsed by dust clouds when the line of sight coin-cides with the orbital plane [27], it seems likely thatthe formation of dust is possible in the outer parts ofthe disk as it expands and cools. If the mass of thecentral star does not exceed ∼10M�, its lifetime isalso sufficiently long for the formation of large planetsat the edge of the dust-forming region [9].

We can estimate the masses of the extendedgaseous decretion disks around the stars underinvestigation, md, using the condition of conserva-tion of angular momentum. As a result, we obtainmd = ∆M(R/A)1/2, where ∆M ≈ 0.03M is thetotal amount of matter lost by the star, R is thestar’s radius, and A is the radius of the disk. Ifwe assume that A ≈ 100 AU [5], then md/M ≈3 × 10−4; i.e., when M ≈ (2–20)M�, the mass ofthe disk is comparable to the masses of solar andextrasolar planets [4]. If we assume that one conditionfor the formation of a planet is the condensationof dust at T ≈ 1500 K, the position of the closestplanet becomes a function of the mass of centralstar [9]. For L/L� ≈ (M/M�)4, the distance to thedust-condensation zone is A ≈ 0.1(M/M�)2, andmd/M� ≈ 0.01(M/M�)1/3, if the stellar radius isR/R� ≈ (M/M�)2/3. Thus, in this case as well,the mass of the gaseous disks is comparable to themasses of planets.

Generally speaking, judging from the positions ofmassive planets in extrasolar systems, dust formationmay not be a necessary condition for the formationof planets. In this case, given the low viscosity of thedisk material, we cannot rule out the development ofgravitational instability in the compact, dense disksand the formation of brown dwarfs [4, 28] with proba-ble masses of≈ 0.01 M�.

When estimating the role of rotational mass lossby MS stars, we should bear in mind other possi-ble mechanisms for the loss of mass and angularmomentum. First and foremost, this concerns massloss via the stellar wind, which is most efficient formassive stars. Conti [29] and Stothers [30] provide acompilation of observational estimates of mass-lossrates by OB stars, which can be approximated by theanalytical expression

MOB = 10−20

(LOB

L�

)2.5

M�/yr. (2)

Since LOB ≈ 102(MOB/M�)2L� [31], and theMS lifetimes of OB stars are τMS ∼ 3×

Page 5: Rotational mass loss by Be stars

830 TUTUKOV, FEDOROVA

107(MOB/M�)−1/2 yr [31], the total mass loss in theMS stage of a star’s evolution does not exceed

∆MOB

MOB� 3 × 10−8(MOB/M�)4.5. (3)

Comparing this mass loss to the amount of masslost due to rotation for the most massive stars(∼0.1MOB), we find that mass loss due to the stellarwind is negligible compared to rotational mass losswhen MOB � 30M�. This justifies our neglect ofstellar-wind mass loss in this study.

It is interesting that two rapidly rotating Ae starsthat have been studied in detail, ν Cyg and κ UMa,display a luminosity excess of one magnitude com-pared to ZAMS stars [32]. A special analysis [32] hasshown that these stars are not young; i.e., they are notHerbig–Haro objects. Hence, these stars have hadtime to evolve to some extent. This result is consis-tent with our conclusions that rotational mass lossby Ae and Be stars becomes most efficient for starsare leaving the main sequence (Figs. 2–4) and thatrotational mass loss occurs in the vicinity of point Cof the evolutionary tracks, even for stars that rotatedon the ZAMS with velocities comprising 50–70% ofthe break-up velocity (Fig. 5).

Other possibilities for enhancing the rotationalmass loss of rapidly rotating stars and supportingtheir circumstellar gaseous disks appear if we con-sider the possible presence of a close companion tothe Be star [33]. If the rotation of the star is notsynchronized with the orbital motion, the presence ofa close companion may facilitate the mass loss andmodify the shape of the Be-star envelope.

ACKNOWLEDGMENTS

The authors thank A.A. Boyarchuk for discus-sions of problems associated with Be stars. This workwas supported by the program “Leading ScientificSchools” (grant 00-15-96722), the State Scienceand Technology Program “Astronomy,” the Programof the Presidium of the Russian Academy of Sciences“Nonstationary Phenomena in Astronomy,” and theRussian Foundation for Basic Research (projectcode 03-02-16254).

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Translated by L. Yungel’son

ASTRONOMY REPORTS Vol. 47 No. 10 2003