Vibrational instability of Population III very massive main-sequence stars due to the $\varepsilon$-mechanism

0.18 MB·English

by  Takafumi Sonoi

Mon.Not.R.Astron.Soc.000,1–5(2002) Printed5January2012 (MNLATEXstylefilev2.2)
Vibrational Instability of Population III Very Massive
ε
Main-Sequence Stars due to the -Mechanism
Takafumi Sonoi1⋆ and Hideyuki Umeda1
2 1Department of Astronomy, Graduate School of Science, The Universityof Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, Japan 113-0033
1
0
2

n
a ABSTRACT
J
Very massive stars are thought to be formed in the early Universe because of a lack

4 of cooling process by heavy elements, and might have been responsible for the later
evolutionoftheUniverse.Wehadaninterestinvibrationalstabilityoftheirevolution
]
andcarriedoutthelinearnonadiabaticanalysisofradialandnonradialoscillationsfor
R
population III very massive main-sequence stars with 500−3000M⊙. We found that
S onlytheradialfundamentalmodebecomesunstableduetotheε-mechanismforthese
.
h stars. The instability appears just after the CNO cycle is activated and the nuclear
p energy generation rate becomes large enough to stop the pre–main-sequence contrac-
- tion, and continues during the early stage of the core hydrogen burning. Besides, we
o
roughly estimated amount of mass loss due to the instability to know its significance.
r
t
s Key words: stars: evolution – stars: massive – stars: oscillations (including pulsa-
a
tions) – stars: mass-loss – stars: Population III
[

1
v
4
8 1 INTRODUCTION excited mainly by the κ-mechanism of the metallic absorp-
8 tion ratherthan theε-mechanism.
The first-generation stars born in the early Universe are
0
. thoughttohaveno,orfewheavyelements.Theymightplay
1
an important role for chemical evolution of the early Uni-
0
verse.Theinitialmassfunctionforthefirstgenerationstars In the zero-metallicity case, however, the metallic ab-
2
hasnotbeendefinitivelydeterminedyet.However,alackof sorption never occurs. Baraffe et al. (2001) analyzed the
1
heavy elements leads to a deficiency of the coolants in the stability of population III very massive stars and found
:
v star-formation stage and hence to an expectation that very that the fundamental mode becomes unstable due to the
Xi massive stars might have been formed (e.g. Bromm et al. ε-mechanism for stars with 120−500M⊙.
1999;Abel et al. 2002; Omukai& Palla 2003).
r
a Ledoux(1941)suggested thatverymassivestarsmight
experience vibrational instability in the core hydrogen-
Population III very massive stars might have released
burning stage. In case of very massive stars, influence of
a significant amount of heavy elements by supernova ex-
radiation pressuremakes theeffective ratio of specific heat,
plosions and contributed to the chemical evolution of the
γ, close to 4/3. Then pulsation amplitude in the core be-
early Universe. The stars with & 300M⊙ experience the
comescomparable withthat intheenvelope.Thissituation
core-collapse supernova explosion at the end of their life,
isfavorable forinstability duetotheε-mechanismsinceex-
while those with 130−300M⊙ the pair-instability super-
citation by this mechanism in the core can exceed flux dis-
novaexplosion(Ohkuboet al.2006).Inaddition,theformer
sipation in the envelope. Many authors have analyzed the
stars leave intermediate mass black holes, for which several
stabilityofverymassivestarsandproposedthecriticalmass
candidateshavebeen found as ultraluminous X-raysources
above which the instability occurs. With the pure electron
(Feng& Soria 2011). Then, the evolution of the very mas-
scatteringopacitytheradialfundamentalmodebecomesun-
sive stars is very important for the above phenomena, and
stable in case of & 60M⊙ (Schwarzschild & H¨arm 1959), have been investigated by several groups (e.g. Klapp 1983,
whilewiththerecentOPALopacitytable(Rogers & Iglesias
1984; Heger et al. 2003; Marigo et al. 2003; Ohkuboet al.
1992) the critical mass is 121M⊙ for the solar metallicity 2006; Bahena & Klapp 2010). In this paper, we focused on
(Stothers1992).Inthelattercase,thefundamentalmodeis
thevibrationalstabilityoftheirevolution,andextendedthe
Baraffe et al.’sanalysistowardthemoremassivestars.That
is,weconstructedthestellarmodelswith>500M⊙andcar-
ried out the linear nonadiabatic analysis against radial and
⋆ E-mail:[email protected] nonradial oscillations.
(cid:13)c 2002RAS
2 T. Sonoi and H. Umeda
Table 1.Propertyofzero-metallicityZAMSmodels. -1yr 1 3000Msun
M R logTeff logL logTc Mcc -40 1000Msun
(M50⊙0) (1R0⊙.4) 5(K.0)4 (7L.1⊙0) 8(K.1)7 (M44⊙5) π2) 1 0.5 500Msun
1000 14.7 5.06 7.46 8.19 898 σ/(I 0
3000 25.6 5.07 7.99 8.21 2731 -
M2WUmceceEddeQaandUo&oteIpNsLtecoIdomBnRovetIcaotUiv(M2eH0c0eoM2nr,eyOem2y0Da-0tsE5ys.)pLeaSndcoOdhekubdoeveetloapl.ed(200b6y) -1-1ε (erg g s)c 111100003456 εc ρc/<ρ> R/Rsu5n00Msun 369111258 ρρ/<>, R/Rcsun
to calculate the stellar evolution. We constructed zero- 102
104 105 106
metallicity models with 500, 1000 and 3000 M⊙ duringthe Age (yr)
core hydrogen-burning stage. These stars were calculated
without considering mass loss. Overshooting is not taken Figure 1. Top: variation of the growth rate of the radial fun-
into account in this study. damental mode for the 500, 1000 and 3000M⊙ stars. The open
For the zero-metallicity stars, the only possible way of trianglescorrespondtotheZAMSmodels,shownintable1,while
starting hydrogen burning is the pp-chain no matter how theopencirclescorrespondtothemostunstableones,forwhich
thepropertyofoscillationsisshownintable2.Bottom:variation
massive the star is. Hence, the temperature near the stel-
lar centre is much higher than in the case of metal-riched of the nuclear energy generation rate at the centre εc and the
ratio of the central density to the average density in the whole
stars. During the pre–main-sequence stage, gravitational
starρc/hρifor500M⊙.
contraction releases energy so that the energy equilibrium
in the whole star can be kept. When the central tempera-
turereaches∼107 K,thepp-chainisactivatedandbecomes
part of σ, σR, represents the oscillation frequency, and the
to release enough energy to stop the gravitational contrac-
imaginary partof σ,σI,givesthegrowth rateorthedamp-
tion for stars with . 20M⊙. For the more massive stars, ing rate, depending on its sign. We used a Henyey-type
however, the gravitational contraction cannot be stopped
code developed by Sonoi & Shibahashi (2012) to solve this
only by the pp-chain burning (Marigo et al. 2001). When
eigenvalue problem. Note that for stability analysis related
the central temperature reaches ∼ 108 K, the triple alpha
to the ε-mechanism, it is important to evaluate the tem-
reactionproducesenough12CtoactivatetheCNOcycleand
perature and density dependences of the nuclear reactions
thenthecontractionstops.Thus,themainenergysourcefor
in the oscillation time scale. The detail is introduced by
corehydrogen-burningofpopulationIIImassivestarsisthe
Sonoi & Shibahashi (2012).
CNOcycle.Table1showspropertyoftheequilibriummod-
We adopted the “frozen-in convection” approximation,
els for the ZAMS stage, which we define as the time when
in which perturbations of theconvectiveflux are neglected.
thegravitationalcontractionstops.Thecentraltemperature
The very massive stars have the large convective core and
ishardlydifferentamongthedifferentmassmodelsandcor-
the convection-pulsation interaction might affect the vibra-
reponds to the occurence of the triple alpha reaction. Due
tional stability. But the uncertainty in the physical process
to the large nuclear energy generation, the convective core
remains still now, and hencewe neglected it for simplicity.
isverylargeandoccupies≃90percentofthestellarmass.
4 RESULTS
3 LINEAR NONADIABATIC ANALYSIS OF
RADIAL/NONRADIAL OSCILLATIONS 4.1 Variation of stability with stellar evolution
We carried out a linear nonadiabatic analysis of radial Asdescribed above, thegravitational contraction continues
(l = 0) and nonradial (l = 1, 2) oscillations. Any eigen- while only the pp-chain works. When the CNO cycle is ac-
function of linear oscillations of stars is expressed in terms tivated,thecontractionstopssincetheenergygenerationof
of a combination of a spatitial function and a time varying the CNO cycle is much higher than the pp-chain and can
function. The latter is expressed by exp(iσt), where σ de- maintain theenergy equilibrium without thecontraction.
notes the eigenfrequency. The former, the spatial part, is a Suchhighenergygenerationinducesthevibrationalin-
radialfunctionfortheradialoscillations, whiledecomposed stability duetotheε-mechanism;thenuclearenergy gener-
intoasphericalharmonicfunction,whichisafunctionofthe ationincreasesatshrinkingphaseinpulsations.Thismakes
colatitudeandtheazimuthalangle,andaradialfunctionfor expansioninthepulsationsstrongerandthenuclearenergy
thenonradial oscillations. generation drops more. This cyclical behaviour makes the
For the radial (non-radial) oscillations, equations gov- star act as a heat engine and the pulsation amplitude will
erning the radial functions lead to a set of fourth-order grow.
(sixth-order) differential equation, of which coefficients are Fig.1showsthevariationofthegrowthrate,−σI/(2π),
complex, including terms of frequency σ. Together with of theradial fundamental mode for 500, 1000 and 3000M⊙
proper boundary conditions, this set of equations forms a stars and theprofiles of the equilibrium models for 500M⊙
complex eigenvalue problem with an eigenvalue σ.The real star with the stellar evolution. The positive value of the
(cid:13)c 2002RAS,MNRAS000,1–5
Instability of Pop III Very Massive Stars 3
Table 2.Propertyofoscillationsforthemodelsmarkedwiththeopencirclesinfigure1.
Periods(h) Growthtime-scaleτg (yr)
l Mode 500M⊙ 1000M⊙ 3000M⊙ 500M⊙ 1000M⊙ 3000M⊙
0 F 4.31 6.17 10.4 1.63×104 1.22×104 8.99×103
1H 1.70 2.13 2.98 −6.81×10 −2.90×10 −1.97×10
2H 1.24 1.59 2.28 −2.35 −7.12×10−1 −5.45×10−1
1 g1 10.3 11.1 12.7 −9.20×104 −1.54×105 −3.51×105
p1 1.84 2.17 2.71 −4.42×102 −8.31×102 −7.37×102
2 g1 6.32 6.77 7.64 −6.31×104 −1.02×105 −2.09×105
f 2.13 2.53 3.19 −2.40×103 −3.55×103 −4.46×103
p1 1.53 1.80 2.37 −8.21×10 −1.88×102 −2.13×102
8×10) 68 /drK 00..46 12HHF l=0
E) ( 4 de 0.2
π 2
4 0
W/( 0 0.6 g2 l=1
-1s)1100-256 411...995×××111000563 yyyrrr de/drK 00 ..024 gpp112
-1erg g 110034 500Msun /drK 00..46 gg21f pp12 l=2
ε (102 de 0.2
101 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Mr/M r/R
Figure2.WorkintegralW fortheradialfundamentalmode(top) Figure3.Profilesofdek/dr≡σ2ξr2ρr2,whichisproportionalto
and nuclear energy generation rate ε (bottom) for 500M⊙. The thelocalkineticenergydensityoftheoscillationsandnormalized
R
winotrhkeinwtehgoralelisstanrorEm.alizedwiththeoscillationenergyofthemode masoRd0esdfeokr/tdhred5r0/0RM=⊙1m,oofdrealdmiaalr(kle=dw0)itahndthneoonp-reandciairlc(lle=in1F,i2g).
1.Thedottedverticallinedenotesthetopoftheconvectivecore.
growth rate means instability, while the negative value sta-
bility. As the nuclear energy generation abruptly increases Lamb frequency. The gravity waves of the g-modes cannot
just before the ZAMS stage, the growth rate does as well propagate in the central nuclear burning region due to the
and becomes positive. large convectivecore (Fig. 3).
Fig.2showstheworkintegralforthefundamentalmode Theinstabilityofthefundamentalmodecontinuesdur-
and the nuclear energy generation rate for 500M⊙. Before ing the early stage of thecore hydrogen burning. The most
the abrupt increase in the nuclear energy generation rate, unstablemodels,markedwiththeopencirclesinFig.1,are
the ε-mechanism does not work efficiently in the core. But atage≃5×105yrandthegrowthtime-scaleis∼104yr.As
aftertheincrease,excitationbytheε-mechanismcausesthe pointed out by Baraffe et al. (2001), the growth time-scale
instability. in the case of theε-mechanism is relatively long. And since
Table2showstheresultsofthestabilityanalysisforthe the central temperature does not increase efficiently from
models marked with the open circles in Fig. 1. The growth 500 to 3000M⊙, theeffect of thenuclear energy generation
time-scale is defined as τg ≡ −2π/σI, that is, the e-folding and thegrowth rate does either.
time of the amplitude. The positive values of the growth After the activation of the CNO cycle, the stellar ra-
time-scale mean the instability as well as the growth rate. dius expands. With this, the density contrast between the
The radial fundamental mode becomes unstable since the innerand outerparts of thestar, or theratio of thecentral
amplitudeisrelativelylargeinthecoreandexcitationbythe density to theaverage density in thewhole star, ρc/hρi, in-
ε-mechanism thereexceeds fluxdissipation in theenvelope. creases. This causes theamplitudein theenvelopebecomes
Ontheotherhand,theotherradialmodesarevibrationally much larger than in the core. Fig. 2 shows that in such a
stable since the amplitude is much larger in the envelope stagefluxdissipation intheenvelopeexceedstheexcitation
than in thecore (Fig. 3). bytheε-mechanism,whichworksefficientlyinthecore.The
Althoughnon-radialmodesofl=1and2werealsoan- instability disappears at age ≃1×106 yr,or Xc ≃0.4. Ac-
alyzed,nounstablemodeswerefound.Theacousticwavesof cordingtoBaraffe et al.,thedurationoftheunstablephase
thenon-radialp-modescanpropagateinalessdeepinterior becomes longer as the stellar mass increases from 120 to
than those of the radial oscillations since they are reflected 500M⊙. In this mass range, the stellar structure may be-
at the depth where the oscillation frequency is equal to the comemorefavourabletotheε-mechanisminstabilityasthe
(cid:13)c 2002RAS,MNRAS000,1–5
4 T. Sonoi and H. Umeda
stellar mass increases. In the mass range analyzed in this
Table 3.Propertyofmasslossduetotheinstabilityofthefun-
study, on the other hand, the stellar structure and hence damentalmode.
the duration of the unstable phase do not change substan-
tially with stellar mass. (MM⊙) τ(uynr)s (Md⊙My/rd−t1) (∆MM⊙) (MM⊙cc)
Compared with the result of Baraffe et al., the dura- 500 1.2×106 2.0×10−5 2.0×10 3.4×102
tion of the unstable phase for 500M⊙ is longer, while the 1000 1.1×106 6.8×10−5 6.1×10 6.9×102
maximum value of the growth rate is lower. In their anal- 3000 1.0×106 3.1×10−4 2.3×102 2.0×103
ysis, around the ZAMS stage, the growth rate initially has
Fromlefttorightthetableentriesread:stellarmass,durationof
a positive and extremely high value, and then shows the
vibrationallyunstablephase,mass-lossrateforthemodelmarked
rapidvariation with timeduetotheadjustmentofthestel-
withtheopencirclesinFig.1,totalamountofthemasslossand
lar structure when the initial CNO is produced (see their
convectivecoremassattheevolutionarystagewhentheinstabil-
fig.4).Inouranalysis,ontheotherhand,thegrowthrateis
itydisappears.
initially negative and becomes positive just after the onset
of the CNO cycle. This implies a difference in the methods
Forthestarswith500−3000M⊙,theconvectivecoreoccu-
to calculate evolutionary models.
pies≃90percentofthestellarmassattheZAMSstage(ta-
ble 1), but monotonically decreases during the core hydro-
genburningstage.Then,themassfractionoftheconvective
4.2 Mass loss
corebecomeslessthan70percentattheevolutionarystage
Appenzeller (1970a,b) performed non-linear analysis of vi- when the instability disappears (table 3). Thustheamount
brational instability of very massive main-sequence stars of the mass loss in our result is not enough to trigger the
against radial pulsation. It was demonstrated that recur- above situation. For the more massive stars, however, the
ringshockwavesintheouterlayersresultsintheformingof convectivecoreandamount ofmass lossareexpectedtobe
acontinuouslyexpandingshellaroundthestarwhentheve- larger.Besides,althoughtheinstabilitydisappearsduetoin-
locity amplitudeexceeds thesound velocity.Eventually the creaseinthedensitycontrastasdiscussedinsection4.1,the
outermost layers of the shell reach the escape velocity and strongermasslosscouldmaintainthevibrationalinstability
mass loss occurs. In addition, due to the shock dissipation, by limiting the density contrast, like models of Wolf-Rayet
the amplitude cannot increase significantly above the value stars shown in Maeder (1985). Then, it is worth analyzing
at which theshell is formed. if theabove situation occurs for stars with >3000M⊙.
Baraffe et al. (2001) performed linear stability analy-
sis of the population III very massive stars and estimated
mass loss due to the vibrational instability against the ra-
5 CONCLUSIONS
dial fundamental mode. Although the absolute value of the
amplitudecannotbedeterminedinthelinearanalysis,they We have analyzed the vibrational stability of the main-
set thevelocity amplitudeto thesound velocity at thestel- sequencestageofpopulationIII500−3000M⊙ starsagainst
larsurface.Besides,theyassumedthatthepulsationenergy the radial and non-radial modes. We found that only the
gainedthroughtheheat-enginemechanismwastransformed radial fundamental mode becomes unstable due to the ε-
into thekinetic energy of theescaping matter. mechanism.Theinstabilityappearsjustafterthepre–main-
Althoughnon-linearanalysisisactuallynecessarytoin- sequence contraction stops and the nuclear energy genera-
vestigate the pulsational mass loss, we estimated the mass- tion becomes large enough to maintain the energy equilib-
lossrateinthesamemethodasBaraffe et al.Themassloss rium, while it disappears in the middle part of the core hy-
rate was evaluated for each unstable model and integrated drogen burning stage as the density contrast between the
withtime.Inthisstudy,theequilibriummodelsevolvewith- innerandouterpartsofthestarsbecomeslarger.Theother
outmasslossalthoughtheirmassshouldbeactuallyreduced radialandnon-radialmodesneverbecomeunstablesincethe
by following themass loss rate at each evolutionary stage. amplitude in the envelope is much larger than in the core.
Theresultsareshownintable3.∆M denotesthetotal Unfortunately, the result contains uncertainty due to the
mass lost during the unstable stage. The one for 500M⊙ is large convectivecore.Improvementinourknowledgeabout
roughly consistent with the result of Baraffe et al., 25M⊙. the convection-pulsation interaction is required to obtain a
Besides,intheirresult,theratioofthemass-lossamountto definiteconclusion on thestability analysis.
thetotalstellarmassincreasesfrom120to500M⊙.Extrap- To measure the significance of the instability, we esti-
olating from theirresult, significant mass loss was expected mated the amount of mass loss due to the instability by
to occur for stars with > 500M⊙. Such a tendency is also adopting the Baraffe et al. (2001) method, although non-
foundinourresult;4percentfor500M⊙,while7.7percent linear analysis was actually necessary. We found that the
for 3000M⊙. Such mass loss would to some extent cause amount is less than 10 per cent of the whole stellar mass.
differencefrom theevolution withoutthemassloss, but,by Thus, we concludethat mass loss dueto thisinstability, by
itself,mightnothavesignificantinfluenceonthequalitative itself, is not significant for the later evolution. However, we
later scenario of thestars. note that Marigo et al. (2003) suggested that the radiative
The very massive stars have significantly large convec- mass loss is supposed to siginificantly affect the evolution
tive cores. If the mass loss is so strong that the stellar sur- of the population III very massive stars with > 750M⊙.
facewouldbecomeclosetothetopoftheconvectivecore,the Ekstr¨om et al.(2008)showedthatrotationalmodelsofpop-
CNO-elementsproducedbythetriplealphareactionandthe ulation III stars favourthesurface enrichment of theheavy
CNO cycle will reach the surface by the convective mixing. elements due to rotational mixing processes, which might
Thiscouldcausetheradiativemasslosstobecomeefficient. trigger stronger radiative mass loss. Then, the effect of the
(cid:13)c 2002RAS,MNRAS000,1–5
Instability of Pop III Very Massive Stars 5
vibrational instability should be combined with the above
effects for more exact investigation of the evolution.
ACKNOWLEDGMENTS
The authors are grateful to the anonymous referee for
his/her useful comments for improving thisLetter. The au-
thors are also grateful to K. Omukai for his helpful ad-
vicesforstartingthisstudy.Thisresearchhasbeenfinanced
by Global COE Program “the Physical Sciences Frontier”,
MEXT, Japan.
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