Table Of ContentAstrophysicsandSpaceScience
DOI10.1007/s•••••-•••-••••-•
Semiconvective Mixing in Low-Mass Stars
Silva Aguirre, V. • Ballot, J. • Serenelli, A. •
Weiss, A.
0
1
0
2
n
a
J
(cid:13)c Springer-Verlag••••
1
1
Abstract Mixing processes such as convection, over- One of the mixing processes that has been extensively
shooting and rotational mixing have long been known studied in massive stars evolution is that of semicon-
]
R to affect the evolutionary properties of low-mass stars. vection. During the hydrogen-burning phase the ra-
S While modeling a 1.2 M⊙ star, we encountereda semi- diation pressure makes the core expand. Opacities
h. convectiveregionoutsidethefullyconvectivecore,rem- are increased outside the core by electron scattering,
p iniscent of the well-known situation for massive stars. where a chemical discontinuity appears as a result
- Inthisstudywefocusonlow-massstarspresentingcon- of this core expansion. Since the pioneer work of
o
r vective cores and, by applying different prescriptions Schwarzschild& Ha¨rm(1958), severalauthors havein-
st for the determinationof the convectiveboundaries and vestigated the occurrence of semiconvective mixing in
a using different mixing descriptions for the dynamical massive stars and its effects on stellar evolution (e.g.
[
processesinthecore,welookfortheeffectsofsemicon- Stothers 1970; Stothers & Chin 1975; Langer 1985).
1 vectivemixinginthe interiorstructureofthe starsand Although semiconvection was initially thought to oc-
v its observable quantities. With this purpose, we have curonlyinmassivestars,itwasalsofoundinlow-mass
5
constructed different sets of evolutionary models using stars as a consequence of a discontinuity in the molec-
0
6 astellarevolutioncode(GARSTEC),andanalyzedthe ular weight produced either by convective core expan-
1 models looking for imprints of these processes. sion due to the increasing importance of the CNO cy-
1. cle over the pp chain (Mitalas 1972; Saio 1975), or by
Keywords Stars: interiors–convection–semiconvec-
0 the retreating convective core leaving behind a chemi-
0 tion caldiscontinuity(Faulkner & Cannon1973),bothcases
1
producing higher opacities outside the convective core.
:
v Weaddresstheissueofthedeterminationofconvective
1 Introduction
i
X boundaries and the treatment of mixing zones which
r It has long been known that different mixing processes presentagradientinthemolecularweightbytheinclu-
a instellarcoresstronglyinfluencetheoverallevolutionof sion of the Ledoux criterion for convective instability,
stars. Thetreatmentofconvectivelyunstablezonesand and a diffusive approach for semiconvective mixing in
overshootingregionsin evolutionarymodels affects the our stellar evolution code.
outcomingluminosity,temperature,andmain-sequence
lifetime of the stars, among other consequences.
2 Mixing Prescriptions
SilvaAguirre,V.
The convectivezoneboundariesaredefined eitherwith
MaxPlanckInstitute forAstrophysics,Garching,Germany
the Schwarzschild criterion (Schwarzschild & Ha¨rm
Ballot,J.
1958) or the Ledoux criterion (Ledoux 1947). For the
LATT/ObservatoireMidi-Pyr´en´ees,Toulouse,France lattercase,the Ledouxtemperature gradientisdefined
Serenelli,A. as
MaxPlanckInstitute forAstrophysics,Garching,Germany
Weiss,A.
β
MaxPlanckInstitute forAstrophysics,Garching,Germany ∇L =∇ad+ 4−3β∇µ, (1)
2
where ∇ = d ln µ/d ln P, and β is the ratio of gas (Angulo et al. 1999). We computed models without
µ
pressure to total pressure. If the Ledoux criterion is diffusion and a He content of Y=0.25.
applied, a zone is considered convectively unstable if Several models were computed to test the different
∇L < ∇rad, while in the case of Schwarzschild crite- mixing prescriptions and the convective boundary def-
rion this reduces to ∇ad <∇rad. Convective zones can inition. Within a convective zone the mixing is per-
be mixed instantaneously or as a diffusive process us- formed instantaneously,while in a semiconvectivezone
ing the convective velocity estimated from the mixing- or overshooting region the mixing is carried out as ex-
length theory. plainedinSect. 2. Calculationsweremadeforboththe
The inclusion of molecular weight gradients in the cri- Schwarzschild and the Ledoux criterion for the defini-
terion for convection allows us to identify semicon- tionoftheconvectivezones,withandwithoutincluding
vective zones, which are defined as the regions where extra mixing due to overshooting and semiconvection.
∇ad < ∇rad < ∇L. Once a semiconvective region has Weexploretheeffectsofthisprocessesinmodelsstart-
been found, the temperature gradient is calculated us- ing from the pre-main-sequence phase and evolved up
ingtheprescriptiongivenbyLanger(1983,1985),anda to hydrogen exhaustion in the core.
time-dependent diffusive mixing in the semiconvective InFig. 1,wepresenttheconvectivecoreevolutiondur-
zoneisperformed. Boththediffusioncoefficientandthe ing the main-sequence lifetime for three models of 1.2,
temperaturegradientinthis prescriptiondepend onan 1.5 and 2.0 M⊙. The left panel shows the case where
efficiency parameterofsemiconvection(αsc). We stress we applied the Schwarzschild criterion for convective
the fact that unlike this study, previous efforts to treat boundaries determination, while the right panel shows
semiconvectioninlow-massstars(e.g.Crowe & Mitalas the case where the Ledoux criterion was applied. In
1982)consideredthemixinginthesemiconvectivelayer both cases, solid lines depict the size of the convec-
tobeperformedbyadjustingthecompositionuntilcon- tive core, while the dotted lines representthe extent of
vective neutrality accordingto the Schwarzschildcrite- the semiconvective region. No mixing was performed
rion is reached, making the hydrogen discontinuity in throughout the semiconvective zones (αsc = 0), which
the boundary of the convective core disappear. do appear in models for both convective criteria. We
Convectiveovershootingisalsoconsideredinourstudy choose these models as they represent cases of inter-
anditisimplementedasadiffusiveprocessconsistingin est: one model where the development of the convec-
anexponentialdecayoftheconvectivevelocitieswithin tive core depends on the convective boundary defini-
the radiative zone. The diffusion constant is given by tionused(1.2M⊙),onemodelwithgrowingconvective
core during the main-sequence lifetime (1.5 M⊙), and
one model with a receding convective core during the
−2z
D(z)=D0 exp , (2) H-burning phase (2.0 M⊙). However, it is important
(cid:18)ξH (cid:19)
p to keep in mind that the effects of semiconvection are
washedout when overshootingis included as described
where ξ corresponds to an efficiency parameter cal-
in Sect. 2, at least for the calibrated value of the effi-
ibrated with open clusters, H is the pressure scale
p
ciency parameter ξ, no matter which criterion is used
height, z is the distance from the convective border,
to define the boundary of the convective regions. For
and the constant D0 is derived from MLT-convective
the cases where it is present, the convective core dis-
velocities. Theextentoftheovershootingregionislim-
appearswhenthecentralhydrogenisexhausted,which
ited in the case of small convective cores.
corresponds to the end of the main-sequence phase.
3 Models In this study, we will focus mainly onthe 1.5 M⊙ case,
as it corresponds to the model which has the most ex-
tended semiconvective zone of all the considered ones.
ForourmodelcalculationsweusedtheGarchingStellar
In Fig. 2 we present the main-sequence evolutionary
EvolutionCode(GARSTEC,Weiss & Schlattl(2008)).
models for this mass value, including now semiconvec-
The input physics consideredincludes the OPALequa-
tive mixing. As can be seen, already with a very small
tion of state (Rogers et al. 1996) complemented with
value for the efficiency parameter of semiconvection,
the MHD equation of state for low temperatures
thisextramixingmakestheconvectivecoregrowmuch
(Hummer & Mihalas 1988), Ferguson’s opacities for
closer to the size of the Schwarzschild criterion case.
low temperature (Ferguson et al. 2005), OPAL opac-
ities for high temperatures (Iglesias & Rogers 1996), This result is expected, as large values of αsc imply
the Grevesse & Sauval (1998) solar mixture, and the fastermixing, and inthe limit whenαsc →∞the mix-
ingisperformedinstantaneouslyandtheSchwarzschild
NACRE compilation for thermonuclear reaction rates
Semiconvection inLow-MassStars 3
Fig. 1 Left panel: Models computed using the Fig. 2 Convective core evolution for models of 1.5 M⊙.
Schwarzschild criterion for convective boundary definition. Black lines present the cases of Ledoux criterion with no
Right panel: Models computed using the Ledoux criterion semiconvective mixing (dots) and Schwarzschild criterion
for convective boundary definition. Solid lines depict the (solid). Blue lines show the models where additional mix-
convective core evolution, while dotted lines show the ex- ing was included: semiconvection with αsc = 0.001 (dots),
tent of semiconvective zones. No semiconvective mixing or αsc =1 (dashed),and overshooting (solid).
overshooting was applied. See text for details
almost at the end of its main-sequence lifetime, which
limit should be recovered. As was already mentioned,
isnotthe casefortheothermodels consideredbeing in
weconsiderthemain-sequencelifetimetoendwhenhy-
a less advanced evolutionary stage.
drogenisexhaustedinthe center,whichcoincideswith
the center of the star becoming radiative. It is inter-
esting to note that in our models, the lifetime on the
main-sequence is reduced when the Ledoux criterion
and semiconvective mixing are applied to the models
4 Conclusions and Future Prospects
withrespecttotheusualSchwarzschildcriterion,which
is an opposite result with respect to previous findings
Inthepresentstudy,wehaveexploredtheeffectsofdif-
(e.g.Crowe & Mitalas1982). Thisisbecauseinthepre-
ferent convective boundary determinations and mixing
vious implementations of semiconvection for low-mass
prescriptionsonthe internalstructure andevolutionof
stars, the mixing through that region recovering con-
low-mass stars. We have done so considering mainly
vective neutrality gave a fresh input of hydrogen fuel
the case ofa 1.5M⊙ star,for whichwe haveshownthe
supply to the core and extended the H-burning phase.
differenceintheinternalstructureresultingasaconse-
This is not the case when a diffusive, time-dependent
quence of the applied physics, although the position in
mixingisconsideredforthesemiconvectiveregionasin
the Hertzsprung-Russel diagram remains very similar.
our models.
In an upcoming study (Silva Aguirre et al. 2010), we
willfurtherinvestigatethe effects ofthese processesfor
InFig. 3wepresentaHertzsprung-Russelldiagramfor
different masses and metallicities, as well as changes in
the 1.5 M⊙ models considered. Markedwith diamonds the efficiency parametersof semiconvectivemixing and
aremodels whichhaveverysimilarpositions inthis di-
overshooting, and the inclusion of diffusive mixing. It
agram,butthemixingprescriptionsappliedmaketheir
is also important to keep in mind that semiconvection
evolutionarystageinthe main-sequencedifferent. This
willplayanimportantroleintheevolutionofastarde-
is shown in Fig. 4, where the hydrogen profiles are
pendingonthetransitionbetweenthepp chainandthe
plotted for the selected models. The sharp chemical
CNO cycleasthemainsourceofenergyproduction(for
compositiongradientsareclearlyseen, signatureofthe
agivenmetallicity). This transitionis critically depen-
location of the convective core boundary. It is inter-
dent on the value of the (N14+p) cross section, which
esting to note that for the model computed with the
has been subject of substantial reductions in the past
Ledoux criterion and no mixing in the semiconvection
years (e.g. Marta et al. 2008).
region, the location in the HRD corresponds to a star
4
Fig. 3 Hertzsprung-Russelldiagram for the1.5 M⊙ mod-
els. BlacklinespresentthecasesofLedouxcriterionwithno
semiconvective mixing (dots), and Schwarzschild criterion
(solid). Blue lines show the models where additional mix-
ing was included, semiconvection with αsc = 0.001 (dots),
αsc =1 (dashed), and overshooting (solid).
Although accurate stellar parameters are of course of
vital importance, asteroseismology can also be used as
atooltodisentangletheobservationaldegeneracybyfo-
cusing on the structural differences among the models
produced by the mixing processes applied. It has been
shown that both p-modes (Popielski & Dziembowski
2005) and g-modes (e.g. Miglio et al. 2008) are sensi-
tive to changes of the molecular weight in the interior
ofstars,andcouldinprinciplebeappliedtodistinguish
the type of mixing that has occurred in the stellar in- Fig. 4 Hydrogen profiles for the 1.5 M⊙ models with
Xc∼0.25, marked with diamonds in Fig. 3. Black lines
terior.
present the cases of Ledoux criterion with no semiconvec-
tivemixing(dots),andSchwarzschildcriterion(solid). Blue
linesshowthemodelswhereadditionalmixingwasincluded,
semiconvection with αsc = 0.001 (dots), αsc = 1 (dashed),
and overshooting (solid).
Semiconvection inLow-MassStars 5
References
Angulo,C.,Arnould,M.,Rayet,M.,etal.1999,Nuc.Phys,
A656, 3
Crowe, R.A., & Mitalas, R. 1982, A&A,108, 55
Faulkner, D.J., & Cannon, R.D.1973, ApJ,180, 435
Ferguson, J.W., Alexander, D.R., Allard, F., et al. 2005,
ApJ, 623, 585
Grevesse, N., & Sauval, A.J. 1998, Sp.Sc. Rev.,85, 161
Hummer,D.G.,&Mihalas,D.1988,Astrophys.J.,331,794
Iglesias, C.A., & Rogers, F.J. 1996, Astrophys.J., 464, 943
Langer, N.1983, A&A,126, 207L
Langer, N.1985, A&A,145, 179
Ledoux, P.1947, AJ, 105, 305
Marta, M. et al. 2008, Phys. Rev.C, 78, 022802 (R)
Miglio, A., Montalban, J., Noels, A., & Eggenberger, P..
2008, MNRAS,386, 1487
Mitalas, R. 1972, ApJ, 177, 693
Popielski,B.L.,&Dziembowski,W.2005,ActaAstron.,55,
177
Rogers, F.J., Swenson, F.J., & Iglesias, C.A. 1996, ApJ,
456, 902
Saio, H.1975, Sc. Rep.Tohoku Univ.Ser. I 58, 9
Schwarzschild, M., & H¨arm, R. 1958, ApJ, 128, 348
SilvaAguirre,V.,Ballot,J.,Serenelli,A.,&Weiss,A.2010,
in preparation.
Stothers, R. 1970, MNRAS,151, 65
Stothers, R., Chin, C.W. 1975, ApJ, 198, 407
Weiss, A., & Schlattl, H. 2008, Astrophys. Space Sci., 316,
99
ThismanuscriptwaspreparedwiththeAASLATEXmacrosv5.2.
