Convective envelopes in rotating OB stars

0.2 MB·English

by  André Maeder

Astronomy & Astrophysics manuscript no. 9007 (cid:13)c ESO 2008
February 2, 2008
Letter to the Editor:
Convective envelopes in rotating OB stars
Andr´e Maeder, Cyril Georgy, Georges Meynet
GenevaObservatory CH–1290 Sauverny,Switzerland
email: [email protected]
8 email: [email protected]
0 email: [email protected]
0
2 Received / Accepted

n
ABSTRACT
a
J
Aims. Westudy theeffects of rotation on theouter convectivezones of massive stars.
7 Methods. We examine the effects of rotation on the thermal gradient and on the Solberg–Hoiland term by analytical
developmentsand by numerical models.
] Results. Writing the criterion for convection in rotating envelopes, we show that theeffects of rotation on the thermal
h
gradient are much larger and of opposite sign to the effect of the Solberg–Hoiland criterion. On the whole, rotation
p
- favors convection in stellar envelopes at the equator and to a smaller extent at the poles. In a rotating 20 M⊙ star at
o 94%ofthecriticalangularvelocity,therearetwoconvectiveenvelopes,withthebiggeronehavingathicknessof13.2%
r of the equatorial radius. In the non-rotating model, the corresponding convectivezone has a thicknessof only 4.6% of
t theradius. Theoccurrence of outer convection in massive stars has many consequences.
s
a
Key words.stars: evolution - convection - rotation
[

1
v 1. Introduction – An outer convective envelope may make a dynamo and
8
contribute to some chromospheric activity generating
1 It is generally considered that the Cowling model applies
an X–ray emission from OB–stars.
0 to massive OB stars: i.e., a convective core surrounded by
1 – Aconvectiveenvelopetransportschemicalelementsand
alargeradiativeenvelope.However,longsincestellarmod-
. angular momentum.
1 els have shown that massive stars have an outer convec-
– Theoccurrenceofouterconvectionmaymodifythevon
0 tive envelope encompassing several percent of the stellar
Zeipel theorem (von Zeipel 1924).
8 radius (Maeder 1980). Also, Langer (1997) has shown that
0 an Eddington factor Γ=κL/(4πcGM) tending toward1.0
Acloserinvestigationisjustified.Westartbyananalytical
:
v implies convection. Our aim is to show that fast rotation approach (Sect. 2), and finish by two–dimensional models
Xi amplifies the size of the convectiveenvelope in OB stars as of 20 M⊙ rotating envelopes (Sect. 3).
well as to develop anisotropic convective envelopes.
r Various limits can be considered about the effects of
a
rotationandhighluminosityonthestellarstability(Langer
2. Convection in rotating stars
1997; Maeder & Meynet 2000): the Γ–limit, which is the
EddingtonlimitforΓ→1;theΩ–limit,whichisreachedby In a rotating star of mass M, luminosity L, and angular
starsatrotationalbreak–upwithasmallornegligibleeffect velocityΩ(supposedtobeshellular,i.e.constantonshells),
of the Eddington factor Γ; the ΩΓ– limit, which applies to thetotalgravityisthesumofthegravitational,centrifugal,
stars where both luminosity and rotation play significant and radiative accelerations:
roles. We show that not only the stars at the Γ–limit, but
also the stars at the ΩΓ–limit and at the Ω–limit, have g =g +g =g +g +g . (1)
tot eff rad grav rot rad
amplified external convective zones.
The occurrence of outer convective envelopes in OB Thevectorg hasbothradialandtangentialcomponents,
eff
stars and their anisotropic structure lead to many astro- the radial component at colatitude ϑ is
physical consequences:
– Convection generates acoustic modes that may allow g =−GMr 1− Ω2r3 sin2ϑ , (2)
asteroseismic observations of OB stars. eff,r r2 (cid:18) GMr (cid:19)
– Convectivemotionsmayplayaroleindrivingmassloss
by stellar winds. where r is the radius at colatitude ϑ. The radiative accel-
– For stars close to the critical rotation, convective mo- eration is directed outward
tions lower the effective break–up velocities.
1 κ(ϑ)F
g = ∇P = , (3)
Send offprint requests to: Andr´eMaeder rad ρ rad c
2 Andr´eMaeder, Cyril Georgy, Georges Meynet:Convection in OB Stars
where F is the flux. On an isobaric surface, F is given by
the von Zeipel theorem (von Zeipel 1924),
L(P)
F =− g , (4)
eff
4πGM
⋆
Ω2
with M (r)=M 1− , (5)
⋆ r
(cid:18) 2πGρ (cid:19)
m
and L(P) is the luminosity on the isobar, ρ the inter-
m
nal average density. In baroclinic stars, there are other
terms(Maeder1999),howevertheyaresmallandneglected
here. The flux is proportional to the effective gravity g .
eff
The effective mass M is the mass reduced by the cen-
⋆
trifugal force. Let us note that one has Ω2/(2πGρ ) ≈
m
(4/9)(v/v )2 ≈ (16/81)ω2, where ω = Ω/Ω , v is
crit crit
the rotation velocity at the level considered and v =
crit
(2/3)[GM/R (ϑ=0)]1/2.
crit
Fig.1. The ratio β = P /P as a function of the stellar
gas
2.1. Effect of rotation on the thermal gradient masseson the zero–agemain–sequence with Z =0.02.The
continuous line shows the value at the stellar centers, the
Formally the Solberg–Hoiland criterion is to be considered
dotted line the maximum value of β inside these models.
in a rotating star, as in Sect. 2.2. However, the radiative
The long–dashed line indicates the minimum value of β
gradient ∇ is also modified by rotation, an effect gen-
rad permitted by equilibriumconditions (Chandrasekhar1984;
erally not accounted for in Schwarzschild’s criterion. The
Mitalas 1997).
local flux and the equation of hydrostatic equilibrium are
F =−χ∇T and ∇P =̺geff , (6) – In the absence of rotation, expression (8) is equivalent
to Langer’s result (Langer 1997). The right–hand side
with χ = 4acT3/(3κ̺). Radiation pressure is included in
of Eq. (10) is always smaller than 1.0, thus if Γ → 1,
P,the totalpressure.The localradiativegradientbecomes
the criterion is satisfied. Convectionis present in layers
in a rotating star,
close to the Eddington limit.
– Inarotatingstar,inequality(10)ismoreeasilysatisfied.
dT dn P 3 κL(P)P
∇ = = , (7) Thus, rotation favors convection in stellar envelopes.
rad dn dP T 16πacG M (r)T4
⋆ – The occurrence of convection depends on both κ(ϑ)
and β(ϑ). Equatorial ejection is always favored, even
wherethederivativesarecomputedalongadirectionnper-
for electron scattering opacity caused by the higher β.
pendiculartothe isobars.ExceptL(P),theterms arelocal
– When the centrifugal force can be derived from a po-
and thus have to be taken at a given (r, ϑ). We ignore the
tential(conservativecase),thetemperatureanddensity
horizontal thermal gradient and take L(P) as constant in
are constant on isobars and so that Γ and β are also
theenvelope.With(5)andtheexpressionoftheEddington
constant on isobars. In that case, rotation also favors
factor Γ, we get
convection as can be seen from Eq. (9).
Γ
∇ = , (8) Onecanwonderwhetherrotatingstarsthatareneither
rad
4(1−β) 1− Ω2 at the critical nor at the Eddington limit may develop a
2πGρm
(cid:16) (cid:17) convectiveenvelope. The ratio β decreaseswith mass (Fig.
where β = P /P is the ratio of gas pressure to the total 1), while the Eddington factor Γ increases, e.g. Γ = 2.5×
pressure, thusgP/(aT4) = 1/[3(1−β)]. The adiabatic gra- 10−5,0.0047,0.021,0.098,0.239,0.343,0.544 for 1, 5, 9, 20,
dient ∇ is 40, 60, and 120 M⊙ stars on the ZAMS. The parameter β
ad
is at a minimum in the stellar centers, reaches a maximum
8−6β in the envelope, and is zero at the stellar surface. One has
∇ = . (9)
ad 32−24β−3β2 the following relation between the maximum β–values in
the outer layers and Γ,
As T varies with ϑ, β also varies with colatitude, and we
1
write β(ϑ). Thatβ(ϑ) is higheratthe equatorfavorsequa- β = 1.0−sΓ or Γ = (1−β), (11)
torial convection. The criterion for convective instability s
∇rad >∇ad becomes with s = 0.72±0.01 between 20 M⊙ and 120 M⊙. Using
Eq. (11), one eliminates Γ from criterion (10) and gets
Γ(ϑ)
> 4[1−β(ϑ)]∇ , (10)
(cid:16)1− 2πΩG2ρm(cid:17) ad s 1−1 Ω2 > 4 32−82−4β6−β 3β2 . (12)
2πGρm
where the ϑ–dependence of Γ comes only through κ(ϑ) (cid:16) (cid:17)
(Maeder & Meynet 2000). Eq. (10) has various interesting This relationindicates abovewhichvalueofω there iscon-
consequences: vection for a given value of β at the maximum (for lower
Andr´eMaeder, Cyril Georgy, Georges Meynet:Convection in OB Stars 3
β in the envelope,the inequality is evidently satisfiedmore 3. Numerical models
easily). For example, for β → 1.0, the inequality becomes
Ω2 > 0.132 or (v/v ) > 0.54. This is an approxima- We do some 2-D models of the outer regions of a 20 M⊙
2πGρm crit fast–rotating star with X = 0.70 and Z = 0.020 (Fig. 2).
tion owing to the simplified relation adopted and because
Ateachlatitudeweintegratetheequationsofthestructure
the various parameters also vary with depth. However, it
forthecorrespondingeffectivegravityandT oftheRoche
eff
shows that rotating massive stars not even at the critical model of the given rotation, also taking the effect of the
limit may have enhanced convection.
reduced mass into account. In the envelope, we suppose
that Ω is a constant as a function of depth (the problem is
2.2. The Solberg–Hoiland criterion conservative). We first consider only the effect of rotation
on the thermal gradient and then the complete Solberg–
A fluid element displaced in a rotating star is also sub- Hoiland criterion to see the differences.
ject to the restoring effect of angular momentum conserva-
tion.This leadsto the Solberg–Hoilandcriterionfor stabil-
ity (Kippenhahn & Weigert 1990), which is (for constant 3.1. Effects of rotation on the thermal gradient
mean molecular weight µ)
Without rotation, a 20 M⊙ model at the end of the MS
∇ad−∇rad+∇Ωsinϑ>0 (13) evolution has two outer convective zones. The first one is
very close to the surface and is due to an increase of the
with ∇ = HP 1 d(Ω2̟4) , (14) opacity caused by partial He ionization. It extends from
Ω g δ ̟3 d̟ r/R = 0.992 to 0.999, i.e. only 0.7% of the radius, and
grav
containsaverysmallfractionofthetotalstellarmass(2.5·
where ̟ = rsinϑ is the distance to the rotation axis and
10−9). The second one is deeper, between r/R = 0.915 to
δ = −(∂ln̺/∂lnT) . The quantity ∇ depends on the
P Ω 0.962(4.7%ofthetotalradius),andcontainsafraction7.4·
distribution of the specific angular momentum j = ̟2Ω,
10−7ofthetotalmass.Bothconvectivezonesareassociated
whichresultsfromtransportprocesses.As j decreasesout-
with opacity enhancements.
ward,∇ generallyhas a stabilizing effect. Let us consider
Ω
For fast rotation, with a ratio Ω/Ω = 0.94, where
the two extreme cases for Ω(r): crit
Ω is the critical angular velocity, the convective lay-
– 1. Constant specific angular momentum: a distribution crit
Ω ∼ r−2 may result from the Rayleigh–Taylor instability. ers are shown in Fig. 2. They are more extended than
without rotation. The thin upper convective zone extends
Thisdistributionisalsosometimesconsideredinconvective
from r/R = 0.987 to 0.999 at the pole, i.e., over 1.2% of
regions, with the argument that the plumes rapidly redis-
the stellar radius, and it contains 1.3 · 10−8 of the stel-
tribute the angular momentum. If so, ∇ = 0, and one is
Ω
lar mass. The deeper convective zone covers the regionbe-
brought back to Schwarzschild’s criterion.
tween r/R = 0.836 to 0.936 (i.e. 10.0% of the polar ra-
–2. Constant angular velocity: this assumption is also used
dius), and its mass fraction is 2.8·10−6. At the equator,
inconvectiveregions,withtheargumentthatturbulentvis-
we have the following sizes for the two convective layers:
cosity favors solid rotation. If so, ∇ simplifies to
Ω
between r/R = 0.958 and r/R = 0.988 for the first one
∇ =4 Ω2 HP = 4Ω2 P . (15) (3.0% of the equatorial radius) and between r/R = 0.727
Ω
g δ g ̺ g δ and r/R =0.862 for the deeper one (13.5% of the radius).
grav grav eff
The included masses are the same as at the pole.
Wecansimplifythisexpressionfurther.Intheouterlayers,
as long as κ≈ const. and g ≈ const, at an optical depth In agreement with Sect. 2.1, convection is more ex-
eff
τ one has P ≈ (g /κ)τ. This gives tendedintherotatingmodelthaninthe non–rotatingone.
eff
Fig.2showsthatcontrarilytotheclassicalCowlingmodel,
Ω2 Ω2R3 τ
massive rotating stars have a large–size convective enve-
∇ ≈ 4 τ ≈4 . (16)
Ω
ggrav̺κδ (cid:18)GM (cid:19) (cid:20)̺κRδ(cid:21) lope. Interesting is that, if we look at the structure of the
Theterminthefirstparenthesisisω2,whilethatinsquare envelopeofthestarasafunctionofthepressure,weseethat
this structure is independent of the colatitude. This is ex-
bracketsisjusttheratio(R−r)/R(assumingδ =1),which
pected since we have supposed Ω constant in the envelope.
is small in the envelope.The Solberg–Hoilandcriterionbe-
Inthatcase,asrecalledabove,Γandβ areconstantoniso-
comes in this approximation,
bars and the extensions of the convective zones, expressed
Γ > 4(1−β) ∇ +ω2 R−r sinϑ ,(17) in term of differences of pressure between the bottom and
ad
1− Ω2 (cid:18) (cid:20) R (cid:21) (cid:19) the top of the convective zone are the same at the pole
(cid:16) 2πGρm(cid:17) and at the equator. The spatial extensions are, however,
whereasabovethevariousquantitiesarelocalones.Atlow greater in the equatorial region than in polar ones. This
rotation, the Solberg–Hoiland term ∇ is negligible with comes from the variations in the spatial gradients of the
Ω
respect to the other terms. At high rotation for constant pressure and temperature with the colatitude imposed by
Ω,itisnotnegligible,butingeneralsmallerthantheother the hydrostatic equilibrium (gradients of pressure have to
terms because the convective zone lies very close to the balance ρg ). Another result of the constancy of pressure
eff
surface and the term (R−r/R) is small. andtemperature(andthusdensity)onanisobaristhatthe
Since the actual rotation laws are likely between the radiativegradientisalsoconstantonthisisobar,exceptfor
twoextremecasesΩ(r)=const.andΩ∼r−2,weconclude the change in the effective mass as given by M∗, which is
that the main effect of rotation on convection in stellar lowerthanM inrotatingstars.Thus,forrotatingstarsthe
envelopes is not the inhibiting effect due to the Solberg– radiative gradient is larger and convection is favored not
Hoiland criterion,but the effect of rotationon the thermal only atthe equator,but at eachcolatitude, comparedwith
gradient (Eq. 10), which enhances convection. the non–rotating model.
4 Andr´eMaeder, Cyril Georgy, Georges Meynet:Convection in OB Stars
12
1.4 10
12
1.2 10
12
1.0 10
11
8.0 10
11
6.0 10
11
4.0 10
11
2.0 10
0
0 11 11 12 12
2.0 10 6.0 10 1.0 10 1.6 10
Fig.2. 2–D model of the external convective zones and of
the convective core (dark areas) in a model of 20 M⊙ with
X = 0.70 and Z = 0.020 at the end of MS evolution with
fast rotation ( ω = Ω/Ω = 0.94). The axes are in units
crit
of cm.
Fig.3. ∇ −∇ (solid line) and ∇ sinϑ (short–dashed
ad rad Ω
line) at the equator in a 20 M⊙ model with Ω/Ωcrit =0.94
The mass loss rate in the considered model is 6.2 × at the end of the MS. The long–dashed line indicates the
10−7 M⊙ yr−1. This means that within one year about 4 zero level.
times all the matter in the thin convective zone is lost in
the stellar winds! Thus, the matter carried by the winds
sign with respect to the Solberg–Hoiland criterion, so that
is continuously passing through the superficial convective
rotation favors convection instead of inhibiting it. The in-
zones in a dynamical process.
crease of the convective zone occurs mainly at the equator
and also a bit at the poles. In a fast–rotating 20 M⊙ PopI
3.2. Solberg–Hoiland criterion star,there aretwoequatorialzonescoveringatotalof16%
of the stellar radius at the equator.
WealsocomputedtheenvelopestructurewiththeSolberg–
Thereareseveralconsequencesoftheesresultstobeex-
Hoilandcriterionforconvection.Thevaluesof(∇ −∇ )
ad rad amined in future. The outer convective motions may lower
and ∇Ωsinϑ are shown in Fig. 3 for a 20 M⊙ model at the escape velocity as well as the critical rotation veloc-
the equator, i.e. where the effect of the Solberg–Hoiland
ity. The matter accelerated in the winds continuously goes
criterion is the strongest. The solid line shows the dif-
through the convective zone in a dynamical process, sug-
ference between the adiabatic and the radiative gradient
gestingthatconvectionplaysaroleinacceleratingthestel-
(identical to the values obtained in the model computed
lar winds and in producting the clumps in the winds. The
with the Schwarzschild criterion). The short-dashed line
convectivepistonsgenerateacousticwavesofperiodsofsev-
shows the Solberg-Hoiland term. The limits of the con-
eral hours to a few days. The density is very low, and it is
vective zones when the Soilberg-Hoiland criterion is used
thuslikelythatconvectioninjectsoscillationsintothewind
are inside the regions where (∇ − ∇ ) is negative.
ad rad rather than into the interior.
Indeed, the limits are where (∇ −∇ )+∇ sinϑ = 0.
ad rad Ω
Convectivezonesarethussmallerthanthoseobtainedwith
theSchwarzschildcriterion.Wefindthefollowingvaluesfor References
the extension of the convective zones at the equator when
theSolberg-Hoilandcriterionisused:betweenr/R=0.960 Chandrasekhar,S.1984,Rev.ofModernPhys.,56,137
Kippenhahn,R.&Weigert,A.1990,StellarStructureandEvolution
and r/R=0.988 for the superficial one (2.8% of the equa-
(StellarStructureandEvolution,XVI,468pp.192figs.. Springer-
torial radius) and between r/R = 0.727 and r/R = 0.859
Verlag Berlin Heidelberg New York. Also Astronomy and
for the deeper one (13.2% of the total radius), i.e. values AstrophysicsLibrary)
slightly smaller than but very similar to those obtained in Langer, N. 1997, in ASP Conf. Ser., Vol. 120, Luminous Blue
themodelcomputedwiththe Schwarzschildcriterion.This Variables: Massive Stars in Transition, ed. A. Nota & H. Lamers,
83
numericalexampleconfirmsthattheSolberg–Hoilandterm
Maeder,A.1980, A&A,90,311
∇Ωsinϑhasaverylimitedinfluenceinstellarenvelopes,as Maeder,A.1999, A&A,347,185
discussed in Sect. 2.3. Maeder,A.&Meynet,G.2000,A&A,361,159
Mitalas,R.1997,MNRAS,285,567
vonZeipel,H.1924,MNRAS,84,665
4. Conclusions
In stellar envelope of rotating stars, the effects of rotation
on the thermal gradient arestrongerand with the opposite