Differential rotation in rapidly rotating early-type stars. I. Motivations for combined spectroscopic and interferometric studies

Abstract

Since the external regions of the envelopes of rapidly rotating early-type stars are unstable to convection, a coupling may exist between the convection and the internal rotation. We explore what can be learned from spectroscopic and interferometric observations about the properties of the rotation law in the external layers of these objects. Using simple relations between the entropy and specific rotational quantities, some of which are found to be efficient at accounting for the solar differential rotation in the convective region, we derived analytical solutions that represent possible differential rotations in the envelope of early-type stars. A surface latitudinal differential rotation may not only be an external imprint of the inner rotation, but induces changes in the stellar geometry, the gravitational darkening, the aspect of spectral line profiles, and the emitted spectral energy distribution. By studying the equation of the surface of stars with non-conservative rotation laws, we conclude that objects undergo geometrical deformations that are a function of the latitudinal differential rotation able to be scrutinized both spectroscopically and by interferometry. The combination of Fourier analysis of spectral lines with model atmospheres provides independent estimates of the surface latitudinal differential rotation and the inclination angle. Models of stars at different evolutionary stages rotating with internal conservative rotation laws were calculated to show that the Roche approximation can be safely used to account for the gravitational potential. The surface temperature gradient in rapid rotators induce an acceleration to the surface angular velocity. A non-zero differential rotation parameter may indicate that the rotation is neither rigid nor shellular underneath the stellar surface. Comment: Paper in press

Full text

Page 1

arXiv:1012.1707v1 [astro-ph.SR] 8 Dec 2010
Astronomy & Astrophysics manuscript no. 15691
December 9, 2010
c ? ESO 2010
Differential rotation in rapidly rotating early-type stars. I.
Motivations for combined spectroscopic and interferometric
studies
J. Zorec1, Y. Fr´ emat2, A. Domiciano de Souza3, O. Delaa3, P. Stee3, D. Mourard3, L. Cidale4,5,⋆, C. Martayan6, C.
Georgy7, and S. Ekstr¨ om7
1Institut d’Astrophysique de Paris, UMR 7095 du CNRS, Universit´ e Pierre & Marie Curie, 98bis bd. Arago, 75014 Paris, France
2Royal Observatory of Belgium, 3 av. Circulaire, 1180 Brussels, Belgium
3Laboratoire Fizeau, UNS-OCA-CNRS UMR6203, Parc Valrose, 06108 Nice Cedex 02, France
4Facultad de Ciencias Astron´ omicas y Geof´ ısicas, Universidad Nacional de La Plata, Paseo del Bosque S/N, La Plata, Buenos Aires,
Argentina
5Instituto de Astrof´ ısica de La Plata, (CCT La Plata - CONICET, UNLP), Paseo del Bosque S/N, La Plata, Buenos Aires, Argentina
6European Organization for Astronomical Research in the Southern Hemisphere, Alonso de Cordova 3107, Vitacura, Santiago de
Chile, Chile
7Observatoire de Gen` eve, Universit´ e de Gen` eve, 51 Chemin des Maillettes, CH-1290 Sauverny, Suisse
Received ..., ; Accepted ...,
ABSTRACT
Context. Since the external regions of the envelopes of rapidly rotating early-type stars are unstable to convection, a coupling may
exist between the convection and the internal rotation.
Aims. We explore what can be learned from spectroscopic and interferometric observations about the properties of the rotation law in
the external layers of these objects.
Methods. Using simple relations between the entropy and specific rotational quantities, some of which are found to be efficient at
accounting for thesolar differential rotationin theconvective region, we derived analytical solutions that represent possible differential
rotations in the envelope of early-type stars. A surface latitudinal differential rotation may not only be an external imprint of the inner
rotation, but induces changes in the stellar geometry, the gravitational darkening, the aspect of spectral line profiles, and the emitted
spectral energy distribution.
Results. By studying the equation of the surface of stars with non-conservative rotation laws, we conclude that objects undergo
geometrical deformations that are a function of the latitudinal differential rotation able to be scrutinized both spectroscopically and by
interferometry. The combination of Fourier analysis of spectral lines with model atmospheres provides independent estimates of the
surface latitudinal differential rotation and the inclination angle. Models of stars at different evolutionary stages rotating with internal
conservative rotation laws were calculated to show that the Roche approximation can be safely used to account for the gravitational
potential. The surface temperature gradient in rapid rotators induce an acceleration to the surface angular velocity. Although a non-
zero differential rotation parameter may indicate that the rotation is neither rigid nor shellular underneath the stellar surface, still
further information, perhaps non-radial pulsations, is needed to determine its characteristics as a function of depth.
Key words. Stars: early-type; Stars: rotation; Stars: spectroscopy; Stars: interferometry
1. Introduction
1.1. Review of observational approaches
One of the most enduring unknowns in stellar physics has been
the inner distribution of the angular momentum in a star. In
the past few decades, significant progress has been made in de-
scribing theoretically the evolution of rotating stars. This has re-
quired an understanding of numerous hydrodynamic and mag-
netic instabilities triggered by the rotation, as well as the mixing
processes of chemical elements unleashed by these instabilities
(Tassoul 1978, 2000; Zahn 1983, 1992; Maeder & Eenens 2004;
Maeder 2009). However, apart from the Sun, reliable observa-
tional information about the internal rotation of stars remains
scarce or non-existent.
Send offprint requests to: J. Zorec: e-mail: zorec@iap.fr
⋆Member of the Carrera del Investigador Cient´ ıfico, CONICET,
Argentina
Nevertheless, many attempts have been made to obtain in-
formationon the internal rotation from detailed studies of: a) the
position of stars in the HR diagram;b) the evolutionof the Vsini
parameter during the main sequence (MS) phase; c) the shape of
absorption lines, whose characteristics can depend upon the ro-
tational law in layers close to the stellar surface; d) the global
stellar geometry described with interferometric data.
We briefly review these efforts:
a) The most numerous efforts among those just mentioned
are the statistical analysis of photometric data on the rotational
spread of the MS, which can be described by (Roxburgh et al.
1966; Maeder 1968; Collins & Sonneborn 1977; Collins et al.
1991)
∆MV= k(n)Vn,
(1)
where ∆MV is the deviation in absolute magnitude from the
zero-rotation MS, V is the true equatorial rotational velocity,
1

Page 2

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
and k(n) is a constant whose value depends on the power n.
When n = 2, k(n = 2) is on the order of ko10−5mag/(km s−1)2,
so that for ko ? 1 the deviations may indicate that the in-
ternal rotation is uniform, while for ko ? 1 the internal ro-
tation can be differential (Cotton & Smith 1983). This type
of analysis found that stars do not seem to rotate uniformly.
However, owing to the measurement uncertainties and dif-
ficulties in defining the MS of zero rotation, the available
data could not provide any firm evidence of a particular law
of non-uniform rotation (Strittmatter & Sargent 1966; Golay
1968; Maeder 1968; Maeder & Peytremann 1970; Smith 1971;
Smith & Worley 1974; Moss & Smith 1981). Furthermore, us-
ing detailed model atmospheres for differentially rotating stars,
Collins & Smith (1985) concluded that photometry alone can
place albeit rather weak constraints on the degree of differential
rotation within the stars.
b) Depending upon the internal angular momentum redistri-
bution and evolutionary rearrangements of the inertial momen-
tum, the surface equatorial rotational velocity of stars changes
accordingly.Thus,the studyofthe variationinthe truerotational
velocity, V, as a function of time was studied by several authors
using the ratio
RLC=
?Vsini?LC
?Vsini?ZAMS
=
?V?LC
?V?ZAMS,
(2)
where ?Vsini?LCis the average of the Vsini parameters of stars
with in principle the same mass and luminosity class (LC),
?Vsini?ZAMS is the average of Vsini for stars with the same
mass, but located near the zero-age-main sequence (ZAMS).
These ratios were compared with similar ones predicted theo-
retically for stars evolving as rotators in two different and ex-
treme ways. On the one hand, the stars were assumed to evolve
all their way as uniform rotators, which implies that the angu-
lar momentumis entirely redistributedat each evolutionarystep.
On the other hand, it was assumed that each stellar layer con-
served its initial specific angular momentum, i.e., the stars did
not undergo any redistribution of its internal angular momen-
tum. Since in many cases the observed ratios RLCwere found
to be situated in-between the two extreme theoretical predic-
tions, it was suggested that stars should be differential rota-
tors. However, these studies could not provide any information
about the characteristics of the internal rotational law (Sandage
1955; Danziger & Faber 1972; Zorec et al. 1987; Zorec 2004).
Somewhat related to this category of inquiries is the study of
the evolution of the total angular momentum of B and Be stars
carried by Zorec et al. (1990), who concluded that these objects
should undergo some internal angular momentum redistribution
to explain the observed evolution of the Vsini parameters.
c1) The study of the absorption line profiles of MS B-type
stars found some evidence of possible surface differential rota-
tion. The angular velocity in the surface of stars was assumed to
depend on the colatitude angle, θ, as
Ω(θ) = Ωo[1 − S × (1 − R(θ)sin2θ)] ,
(3)
where Ωois the equatorial angular velocity, R(θ) is the equation
ofthestellar surface,S is theparameterthattestifies to thediffer-
ential rotation. Using stellar models that are more or less gravity
darkened,Stoeckley (1968a) and Stoeckley & Buscombe (1987)
found that in most cases S < 0, which suggested that the an-
gular velocity tends to increase from the equator to the pole.
Nevertheless, a dependence of the surface angular velocity on
the latitude Ω(θ) could be due either to an actual differential
rotation present under the stellar surface, or simply to zonal
atmospheric currents, which could appear in rapidly rotating
early-type stars, as speculated by Cranmer & Collins (1993). In
Sect. 2.4, we recall that an acceleration of the angular velocity
towards the equator,i.e. S >0, can be promotedby a temperature
gradient induced by the gravity darkening effect.
c2) The possibility of detecting surface differential rota-
tion by means of the Fourier analysis of spectral line profiles
was discussed by Huang (1961), Gray (1977), Bruning (1981),
Garcia-Alegre et al. (1982), and Reiners & Schmitt (2002).
Evidence of surface differential rotation in late-type stars with
Vsini < 50 km s−1were given by Reiners & Schmitt (2003b),
Reiners & Schmitt (2004), and Reiners & Royer (2004), but no
differential rotation for late-type stars with Vsini > 50 km s−1
and A-type stars with Vsini > 150 km s−1were reported by
Reiners & Schmitt (2003a) and Gray (1977), respectively. It is
possible that modest differential rotation is difficult to detect
with the Fourier transform technique in slowly rotating A-type
stars, because the rotational broadening is not large compared
with the broadening caused by other mechanisms such as ther-
mal turbulence and pressure effects (Gray 1977). However, in
those cases where there is some evidence of differential rota-
tion, the parameter S cannot be differentiated from the unknown
inclination angle factor sini. Nevertheless, its sign indicates ac-
celeration of the angular velocity towards the equator. To our
knowledge,the Fouriertechniquefordifferentialrotationhas not
yet been applied to early-type stars.
c3) Ando (1980) proposed a method to probe the inner an-
gular velocity of stars based on the use of the rotational split-
ting of non-radial oscillations. However, owing to uncertainties
in the identification of pulsation modes and rough determina-
tions of stellar fundamental parameters, particularly their evolu-
tionary stage, this method has not yet been able to be applied
with reliable success. Nevertheless, from the analysis of pulsa-
tion modes derived from photometric variations (Dupret et al.
2004) and COROT data (Degroote et al. 2009), constraints on
the internal rotation of β Cep stars have been inferred.
d)Inthepastfewyears,interferometricmethodshavehelped
provide remarkable insights into not only the rotational distor-
tion of stars (Domiciano de Souza et al. 2003; van Belle et al.
2004), but also the induced gravity darkening effect by means
of imaging techniques (McAlister et al. 2005; Aufdenberg et al.
2006; van Belle et al. 2006; Zhao et al. 2009). New instruments
with higher spectral resolutions of up to the 10000 attained by
VLTI/AMBER in the J and K bands and an angular resolution
of about 1 mas in the K band (Petrov et al. 2007), or spectral
resolution reaching 30000 and angular resolutions as high as 0.3
mas in the visible using the VEGA/CHARA interferometric ar-
ray (Mourard et al. 2009), will not only probably enable us to
determine with greater detail than in previous studies the global
geometryofstars deformedandgravitydarkenedbytherotation,
but also carry out differential interferometry.
A method based on differential interferometry that re-
quires high spectral and spatial resolution was presented by
Domiciano de Souza et al. (2004b,c,a) to distinguish observa-
tionally the parameter controlled by the degree of the surface
differential rotation from the inclination angle factor sini.
2

Page 3

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
1.2. Aims of the present attempt
Most theoretical predictions about the evolution of rotating
early-type stars and the mixing of chemical species triggered
by the instabilities set up by the rotation, come from calcula-
tions performed in the framework of two significant assump-
tions: a) the global rotational energy stored by the stars in the
ZAMS is lower than the limit allowed by the rigid rotation in the
critical regime; b) the internal angular velocity undergoes an in-
stantaneous “shellular” redistribution at each evolutionary step.
However, Clement (1979) using a cylindrical (conservative) dis-
tributionof the angularvelocity,andMaeder et al. (2008) basing
their calculation on a “shellular” distribution, showed in a more
detailedway that inrapidlyrotatingearly-typestars theenvelope
layers beneath the surface, may have wider convective zones
in radius than in non-rotating stars. In the Sun, only the layers
unstable to convection rotate differentially with a non-shellular
pattern. This motivates the inquiry of whether in massive and
intermediate mass stars some coupling may also exist between
convection and rotation beneath their surface. In this case, the
characteristics of the rotational law in the external stellar layers
should differ from those currently assumed in the above evoked
stellar models.
As demonstrated by many authors, the global geometry of
a star depends not only on the total amount of angular mo-
mentum stored by the star, but also on its internal distribu-
tion (Bodenheimer 1971; Zorec 1986; Smith & Collins 1992;
Uryu & Eriguchi 1994, 1995). This geometry mostly relies on
the stellar surface rotation, which acts as an imprint of its prop-
erties in the layers beneath the surface. In this case, we should
notexcludetheresultingmixingofchemicalelementsinthestel-
lar atmosphere being more or less dependent on the characteris-
tics of the external rotational law, upon which the description
of the stellar structure, based on the abundance determination
of chemical elements, should also rely. Therefore, to provide
new information and/or constraints to test the global assump-
tions currently made to calculate models of stellar structure with
rotation and thus help deepen our understanding of the proper-
ties of early-type fast rotators, we might ask: 1) what can be
deduced, using first principles, about the properties the rotation
laws can have beneath the surface as a consequence of the cou-
pling between rotation and convection; 2) what are the parame-
ters needed to characterize these stars that may be accessible to
observations; 3) whether the combined interpretations of spec-
troscopic and interferometric data of rapidly rotating early-type
stars enable us to determine these parameters.
In this attempt, the most interestinginformationmight be the
indicationof some differentialrotation in the stellar surfaces and
the sign of its latitudinal gradient. Both pieces of data can be ob-
tained, as much as possible, in a consistent way by taking into
account the stellar geometrical deformationproducedby this ro-
tation and the concomitantgravitational darkeningeffect that re-
sponds to possible non-conservativerotation laws.
The present paper is organized as follows. In Sect. 2, we use
first principles to infer possible rotation laws in the convective
layers beneath the stellar surface of early-type rapid rotators.
Sect. 3 presents the equation of the stellar surface of stars with
non-conservativerotationlaws. A discussion of the gravitydark-
ening effect for non-conservative rotation laws is presented in
Sect. 4. The discussion about the validity of the Roche approx-
imation in representing the gravitational potential is presented
in Sect. 5. This discussion is based on 2D models of rotating
stars where the evolutionary stages are taken into account in a
simplified way. We briefly comment on the determination of the
rotational profile in the stellar envelope in Sect. 7. In Sect. 6, we
summarize the attainable information on rapidly rotating early-
type stars with external differential rotation from the combined
analysis of spectroscopic and interferometric data. Our conclu-
sions are presented in Sect. 8.
2. Rotational law in the stellar envelope
2.1. The angular velocity distribution beneath the stellar
surface
The effects of rotation are generally introduced in the struc-
ture equations of rotating stars by replacing the spherical strat-
ification of non-rotating star-models by a rotationally distorted
stratification, which keeps the whole calculation problem to one
dimension (Kippenhahn & Thomas 1970; Endal & Sofia 1976;
Pinsonneault et al.1990;
1995; Chaboyer et al. 1995; Meynet & Maeder 1997). This pro-
cedureis justified if the internal differentialrotationhas a shellu-
lar distribution law because it comes from theoretical inferences
made by Zahn (1983, 1992), which rely on the assumption that
the horizontal turbulence is much stronger than the vertical one.
The limiting case of a shellular rotational profile is rigid rota-
tion. From the calculations by Maeder & Meynet (2005), it fol-
lows that magneticfields created by the Pitts & Tayler instability
(Pitts & Tayler 1985; Spruit 1999, 2002) can lock the stellar lay-
ers to each other and force the star to evolve as a rigid rotator.
However, Zahn et al. (2007) concluded that the dynamo action
can be less efficient as previously expected (Spruit 1999, 2002)
and that the magnetic fields created contribute little or less to the
transport of the angular momentum. In addition, if the magnetic
field in early-type rapid rotators is created, it could perhaps has
some effect on their rotational profile in the convective parts of
the envelope as in the Sun (Balbus 2009; Balbus et al. 2009).
The differential rotation in the surface of the Sun is a direct
consequence of the differential rotation in the convective layers
beneaththesurfacelayers.Therefore,it isimportanttostress that
this differential rotation in depth is not shellular, in spite of the
strong turbulent viscosity that accordingto the above-mentioned
theoreticalassumptions, might otherwisecause shellular-like ro-
tation.
Maeder et al. (2008) demonstrated that rotation does not in-
hibitconvectionas couldbe thoughtfromSolberg-Høiland’ssta-
bility criterion, but it changes the thermal gradient so as to en-
hance convection. Hence, in rapidly rotating massive stars the
two external convection zones, associated with increased opaci-
ties due to He- and Fe-ionization, respectively, are both consid-
erablyenlargedin depthso that theentireconvectivezonecovers
a non-negligibleexternal region, which ranges from some 1/8 of
the stellar radius in the pole to nearly 1/4 in the equator.
However, the angular momentum distribution in the convec-
tive regions remains a puzzling question. To account for it in
the stellar model calculations, two extreme approximationshave
beenused:a)rigidrotation,angularvelocityΩ=constant,which
is supposedly promotedby the turbulent viscosity (Maeder et al.
2008); b) constant specific angular momentum j (j = Ω̟2; ̟
is the distance to the axis of rotation), possibly due to the re-
distribution produced by the convective plumes (Tayler 1973).
Nevertheless, the solar convective regions, characterized by sig-
nificant turbulence, are the only ones with significant differen-
tial rotation (Schou et al. 1998), even though the solar rotational
profile in the convective region, revealed by the helioseismolog-
ical data, does not fall between these two extreme possibilities
(Deupree 2001). Global insights into the rotational law in the
Fliegner & Langer
3

Page 4

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
12.08 12.1 12.12 12.14 12.16 12.18 12.2 12.22 12.24
log R (cm)
-20
-15
-10
-5
0
(1/ )dln P
-/dln r
equator
pole
M = 20M
Fig.1. AdiabaticgradientdlnPρ−γ/dlnr in thetwo rotationally
enlarged convective regions in the envelope of a 20M⊙star. The
gradients are shown as a function of the logarithm of the stellar
radius in the polar and equatorial directions.
convective regions could be obtained by exploring the solutions
to the thermal wind balance relation under imposed conditions
between the entropy and rotation. This procedure successfully
explains the Solar rotation law in the convectiveregions (Balbus
2009; Balbus et al. 2009). However,we pay attention here to our
dealing with rapidly rotating stars. Detailed physical justifica-
tionsoftheassumptionsmadein thepresentpaperarepostponed
to future contributions.
2.2. Rotation inferred from the baroclinic balance relation
The curl of the time-independent momentum equation of an
inviscid, axisymmetric rotating star with negligible magnetic
fields, yields the baroclinic balance relation. Using the cylindri-
cal coordinates (̟,φ,z), this balance condition reads (Tassoul
1978)
1
̟3
∂j2
∂z
=1
ρ2(∇P × ∇ρ).ˆ eφ,
(4)
where j is the specific angular momentum (j = Ω̟2), P and
ρ are pressure and density, respectively, and ˆ eφis the azimuthal
unit vector.Takinginto accountthe equationof hydrostaticequi-
librium
1
ρ∇P = geff,
(5)
where geffis the effective gravity, and using for the specific en-
tropy S the expression
S =
k
γ − 1lnPρ−γ+ constant,
(6)
where k is the Boltzmann constant and γ is the ratio of specific
heats at constantpressureandconstantvolumeper unitmass, the
wind equation in Eq. (4) can be rewritten as
1
̟3
∂j2
∂z=1
CP(∇S × geff).ˆ eφ,
(7)
where CPis the constant pressure specific heat.
We can attempt a discussion of Eq. (4) by seeking solutions
for the stellar internal rotation under at least three different con-
ditions where for the moment the effects carried by the merid-
ional circulation are neglected: a) marginal stability imposed
by the Solberg-Høiland criterion; b) state enforced by parallel
surfaces of specific entropy and specific angular momentum; c)
frame where the specific entropy parallels the local specific ki-
netic rotational energy.
a)The Solberg-Høilandstability criterionstates that “a baro-
clinicstarinpermanentrotationis dynamicallystableagainstax-
isymmetric perturbations if two conditions are satisfied: (i) the
specific entropy S increases outwards, and (ii) on each surface S
= constant, the specific angular momentum increases from the
pole to the equator”. The second condition is written mathemat-
ically as (Tassoul 1978)
− gz[∇j2× ∇S] ≥ 0,
(8)
where gzis the z−componentof the effective gravity.Makingthe
ansatz for strict marginal equilibrium, we have
− gz[∇j2× ∇S] = 0.
(9)
Equation (9) suggests then that surfaces j2= constant and
S = constant should be parallel, S = S(j2), i.e. a displaced
fluid element in baroclinic turbulence retains both entropy and
angular momentum. We do not consider here the conditions that
might render possible the balance implied by Eq. (9).
b) The assumption that the surfaces of constant specific en-
tropy and constant angular velocity coincide, i.e. S = S(Ω2),
brings another alternative solution to the baroclinic equilibrium
equation (4). This coupling can be enforced by magnetic fields
(Balbus 2009), although hydrodynamic constraints in the Sun
can justify it entirely (Balbus et al. 2009).
c) Only as an extrapolation to the conditions S = S(j2) and
S =S(Ω2) maywe also considerS =S(̟2Ω2),since two energy-
related quantities are parallel: specific entropy S and specific ro-
tational kinetic energy ǫΩ=̟2Ω2.
We now use H to represent in turn j2, Ω2, and ǫΩ. We have
then S = S(H ) and ∇S = (dS/dH )∇H . Rewriting Eq. (7)
in spherical coordinates (r,θ,φ) and knowing that in these coor-
dinates the r− and θ−components of the effective gravity geff
are
gr = −∂ΦG
∂r
∂ΦG
∂θ
+ Ω2(r,θ)r sin2θ ,
gθ = −1
r
+ Ω2(r,θ)rsinθcosθ ,
(10)
we readily obtain
∂H
∂r(1−αHH )−tanθ
r
∂H
∂θ
[1−αHH +αHGMF] = 0 ,(11)
where we have used the notations
αH =
1
CP
dS
dH
(12)
and
F =

rsin2θ if H = j2
1
r3sin2θ
1
r
if H = ǫΩ
if H = Ω2
(13)
4

Page 5

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
The characteristic equation of Eq. (11) is
dr
1 − αHH
= −
dθ
tanθ
r[(1 − αHH ) + αHGMF].
(14)
As for the assumption S =S(H ) made here, H and S are con-
stant along the characteristic curves of Eq. 11, and the differen-
tial equation Eq. (14) integrates immediately to give
1
r2sin2θ
rsin2θ = −1
=
1
rΥJ+CJ
if H = j2
rΥΩ+ CΩ
= CǫΩexp(ΥǫΩ
if H = Ω2
1
r2sin2θ
r
) if H = ǫΩ
(15)
withtheCH as theintegrationconstants(whereH has different
meanings). Once the function giving the angular velocity Ωs(θ)
in the stellar surface is specified, the iso-rotation curves can be
obtained easily everywhere inside the star using the relations in
Eq. (15). Each iso-rotation curve depends on an integration con-
stant CH obtained as CH=CH(ΥH,θs), where θsis the colat-
itude angle at which a given iso-rotation contour intersects the
stellar surface. The constants ΥH are defined as
ΥH = −
?2αHGM
1 − αH
?
.
(16)
Since we consider convective regions where ∇S ≤ 0, we find
that ΥH ≥ 0. We note that depending on the meaning of H ,
the expressions ΥH = ΥJRe, ΥΩ/R2
constants of the order of unity (Reis the stellar radius at the
equator). Defining
eand ΥǫΩare dimensionless
AH = −1
γ
?dlnPρ−γ
dlnr
??
dlnr
dlnH
?
,
(17)
which in convection regions is AH ≥ 0, we have
ΥH = 2
?GM
R3
eΩ2
??
AH
1 − AH
?
.
(18)
From the model of a rapidly rotating star with M = 20M⊙
given in Maeder et al. (2008), we derived the adiabatic gradi-
ent ∇ = dlnPρ−γ/dlnr shown in Fig. 1. This quantity is shown
as a function of stellar radius in the polar and equatorial direc-
tions.ThevalueofΥH dependsalsoontheadoptedvalueforthe
gradient dlnr/dlnΩ2that represents the differential rotation.
For the Sun, the term dlnr/dlnΩ2comes from seismology data
and its value ranges from 1 to 10. However, until now in early-
type stars, the factor dlnr/dlnΩ2has been a perfectlyunknown
quantity. For a rough estimate, we may assume that it can be at
leastofthesameorderofmagnitudeas intheSun.Similarvalues
are also obtained in the models calculated by Meynet & Maeder
(2000). Using the values of the gradient∇ shown in Fig. 1, it can
be shown that the ratio AH/(1 − AH) is nearly constant over
the entire convective region and that depending on the assumed
value of dlnr/dlnΩ2, it changes from 0 to 1. Hence, we have
that
0 ? ΥH ? 2(GM/R3
eΩ2),
(19)
where 0.5 ? (R3
comparison, we note that in the convective region of the Sun, it
is |∇| ≃ 10−6and (GM/R3
eΩ2/GM) ? 1 for rapid rotators. For the sake of
eΩ2) ≃ 105(Balbus 2009).
Examples of isorotation curves obtained from Eq. (15) are
shown in Fig. 2, where we used stellar external contours whose
rotationaldeformationis characterizedbythe ratioofcentrifugal
togravitationalaccelerationat the equatorηo= Ω2
and a Maunder surface differential rotation law with parame-
ter α = 0.3. The equation of the stellar surface with latitudi-
nal differential rotation is discussed in some detail in Sect. 3.
Solutions similar to those shown in Figs. 2a and 2b were ob-
tained by Balbus (2009) for the Sun. This author assumed that
∂P/∂r ≫ ∂S/∂θ and ∂S/∂θ ≫ ∂P/∂r, which according to our
expressions are for the limits αJj2→ 0 and αΩΩ2→ 0 that suit
slowly rotating objects like the Sun.
eR3
e/GM = 0.8
2.3. Comments on the rotational profiles obtained
The rotational profiles Ω(r,θ) in the convective layers beneath
the surface of early-typerapidly rotatingstars were inferred here
usingthemarginalconditionoftheSolberg-Høilandstabilitycri-
terion against axisymmetricperturbations.The derivedsolutions
are the consequence of a dominant thermal wind balance where
the entropy is: a) a function of the specific angular momentum,
S =S(j2); b) a function of the angular velocity, S =S(Ω2), and
c) a function of the specific rotational kinetic energy, S =S(ǫΩ).
The first case was widely studied in the literature (Tassoul 1978,
references therein), and a number of independent numerical at-
tempts to obtain baroclinic stars inexorably ended up with re-
sults close to the category shown in Figs. 2a1, i.e. conservative-
like (Uryu & Eriguchi 1994, 1995). Two-dimensional, implicit
hydrodynamic numerical simulations by Deupree (1998) and
Deupree (2001) showed that in convective regions Ω tends to
adopt a cylindrical-like profile rather than a shellular or Ω =
constant form. Intermediate solutions to the classes shown in
Figs. 2b1and b2are compatible with the seismological data of
the Sun relative to its convective regions. The rotational profile
obtained by Espinosa Lara & Rieutord (2007) for fully radiative
stars curiouslyentersintothecategoryofsolutionsshownin2c2.
We note that for all cases shown in Fig. 2 we have used constant
values of ΥH all over the star. However, they are valid only for
the convective regions, which lie roughly above the shaded sec-
tor and where, as already noted, ΥH is fairly uniform.
According to a strong claim that shear instabilities widely
prevail over all other mechanisms that can act to redistribute the
angular velocity, most if not all models of stellar evolution of
early-type stars with rotation were carried out by assuming that
Ω has a shellular nature. This strongly simplifies the numeri-
cal aspect of the study, but does not necessarily preclude that
other distributions of Ω might coexist with the shellular one,
or even dominate in wide domains of the stellar interior. The
S = S(Ω2)−case with ΥΩ > 1.6 clearly depicts shellular-like
rotation over a wide interior stellar region.
Althoughthe present discussion should not be considered an
argument in favor of non-conservative-like profiles of the an-
gular velocity in the envelopes of early-type rapid rotators, the
success that the displayed arguments do have in explaining the
solar rotational profile in the convective layers, strongly moti-
vate us to address the study of the surface differential rotation in
rapidly rotating early-type stars. Whatever the observational or
theoretical indication that ∂Ω/∂θ ? 0 can exist in the surface of
rapidly rotating early-type stars, they would immediately imply
that profiles of non-shellularrotationexist, perhapsof some type
similar to that in Fig. 2. They would also immediately enable us
to achieveclearer understandingsof the transportof angularmo-
mentuminthestars andtherelatedmixingprocessesofchemical
elements in the stellar surface layers.
5

Page 6

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
0.0
0.2
0.4
0.6
0.8
1.0
z/Rp
J= 0.3
(a1)
J= 1.6
(a2)
0.0
0.2
0.4
0.6
0.8
1.0
z/Rp
= 0.3
(b1)
= 1.6
(b2)
0.00.2 0.40.60.8
x/Rp
1.0 1.21.4
0.0
0.2
0.4
0.6
0.8
1.0
z/Rp
= 0.3
(c1)
0.00.20.4 0.60.8
x/Rp
1.0 1.21.41.6
= 1.6
(c2)
Fig.2. Ω(r,θ)/Ωe= constant curves inside the stellar envelope of a model star with a global shape generated with an equatorial
rotational parameter ηo= 0.8 and a surface differential rotation represented by a Maunder relation with parameter α = +0.3. All
curves were generated with parameters ΥJ,Ω,ǫΩ= 0.3 and 1.6. The iso-rotationcurves in (a1) and (a2) correspondto S = S(j2); those
in (b1) and (b2) are for S = S(Ω2); the curves in (c1) and (c2) obey the condition S = S(̟2Ω2). The solutions are supposed to be
valid in the convective zones above the shaded domain.
2.4. Induced gradient in Ω(θ) in the surface of early-type
rapid rotators
In rapidly rotating stars, there is a strong latitudinal temperature
gradient, which can induce a latitudinal gradient on the surface
angular velocity Ωs. This effect can be estimated from Eq. (4)
by calculating the latitudinal variation in the temperature over
an isobar. Giving to the surface temperature a functional form
T = T(P,θ), where P is the pressure and θ the colatitude, we
obtain
∇T(P,θ) =∂T
∂θ
where ˆ eθis the colatitude unit vector and r is the radial spher-
ical coordinate. Making use of the equation of state P ∝ ρT,
Eq. (20) and the r−componentof Eq. (5), Eq. (4) transformsinto
(Espinosa Lara & Rieutord 2007)
∂P∇P +∂T
ˆ eθ
r,
(20)
̟∂Ω2
∂z
= −gr
r
?∂lnT
∂θ
?
P
,
(21)
6

Page 7

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
where the notation (...)Pindicates that the partial θ−derivative
is along a barotropic surface. As in rapidly rotating stars, it is
∂lnT/∂θ<0 and gr<0, from Eq. (21) it appears that ∂Ω/∂z< 0,
whichmeans that Ω shouldincreasefromthe poleto the equator.
Althoughin rapidrotatorsthereareconvectivezonesinthe enve-
lope, the dominant radiation flux can still be estimated using the
diffusion approximation Frad= −χ∇T, where χ is the coefficient
of radiative conductivity. For simplicity, we adopt von Zeipel’s
approximation (von Zeipel 1924) to represent the stellar surface
temperature distribution as a function of the colatitude
T(θ) = Tp
?g
gp
?1/4
,
(22)
where Tpis the polar temperatureof the star, g=
withgrandgθgivenbyEq.(10),andgpis thevalueofthegravity
in the pole. Using the Roche approximation for the gravitational
potential, Eq. (21) written in spherical coordinates becomes
?
(gr)2+(gθ)2?1/2
∂lnΩ2
∂r
−1
−tanθ
r
[1−(1−ηr)f1(θ,ηr)]∂lnΩ2
∂θ
=
rf1(θ,ηr)f2(θ,ηr) (23)
with the following definitions
f1(θ,ηr) =
1
4
?2 − ηr(1 + ηr)sin2θ
sinθcosθ
Ω2r3
GM.
?
1 − ηrsin2θ
1 − ηr(2 − ηr)sin2θ
?
f2(θ,ηr) =
?∂lnr
∂θ
+ 2 − ηr
ηr =
(24)
The characteristic equationdescribing the dependenceof the
surface angular velocity with θ is then
−
dθ
tanθ
r[1−(1−ηr)f1(θ,ηr)]= −
dlnΩ2
1
rf1(θ,ηr)f2(θ,ηr),
(25)
which is solved by numerical integration. Figure 3 shows
the θ−dependence of the angular velocity given as ∆Ω(θ) =
[Ω(θ,ηr) − Ω(40o,ηr)]/Ω(40o,ηr), which was obtained from
Eq. (25) for different values of η. In this calculation, the gradi-
ent ∂lnr/∂θ in the stellar surface was estimated using the Roche
potential of a rigid rotator characterized by average rotational
parameters η. In Fig. 3, we see that in general it is ∂Ω/∂θ > 0,
in particular for ηo > 0.7 near the equator. This acceleration
seems to be similar in character to the one previously calcu-
lated by Espinosa Lara & Rieutord (2007). As η → 0, the curves
∆Ω(θ) → 0.
3. Equation of the stellar surface
3.1. The angular velocity distribution in the stellar surface
Since the shape of a rapidly rotating star depends explicitly on
its surface angular velocity distribution, we adopt a law to study
thepossible effectsit canproduce.InSect. 2.4, wehaveseen that
the temperature gradient induced by the gravity darkening effect
introduces a small acceleration of the angular velocity from the
0.4 0.60.81.01.21.41.6
(radians)
-0.04
-0.03
-0.02
-0.01
0.0
0.01
0.02
( )/ (40o)
= 1.00
= 0.95
= 0.90
= 0.80
= 0.70
Fig.3. Normalized surface angular velocity Ωsas a function of
the colatitude θ and the rotational parameter η.
poletowardsthe equator,whichis notsimpleto representanalyt-
ically. The physical properties of the layers beneath the surface
could perhaps enforce a stronger surface angular velocity gradi-
ent than inferred in Sect. 2.4. If in rapidly rotating massive stars
rotational profiles of the class obtained in Sect. 2.2 actually ex-
isted, a first approach to describing their surface rotation could
rely on the use of a solar-like surface angular velocity. The so-
lar surface differential rotation depends on the colatitude θ as
(Snodgrass 1984)
Ω⊙(θ)
2π
= 451.5− 65.3cos2θ − 66.7cos4θ nHz.
(26)
This expression carries three coefficients that need to be deter-
mined empirically. At the moment, it seems unrealistic to use a
similar expression for other stars since their surface cannot be
resolved.However,within the errors smaller than 1.6%, Eq. (26)
can be reduced to the simplest one
Ω⊙(θ)
2π
= 459.3(1− 0.29cos2θ) nHz,
(27)
where there are only two quantities to determine from observa-
tions. A first inquiryaboutthe differentialrotation on the surface
of massive and intermediate mass stars can then be justified by
using the simplified expression (Maunder formula)
Ω(θ) = Ωo(1 + αcos2θ) .
(28)
In Eq. (28), α < 0 indicates that the pole rotates slower than
the equator and vice-versa if α > 0. If massive and intermediate
mass stars had surface rotations similar to the Sun, their differ-
ential rotation parameter would be expected to be at least of the
order of α ∼ −0.3.
In general, to study stars with unresolvedsurfaces we should
probably use expressions of the type
Ω(θ) = Ωo[1 + αf(θ)],
(29)
where the function f(θ) needs to be somehow specified in ad-
vance.
7

Page 8

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
z/Rp
o= 0.7
(a)
7 :
= -0.6
6 :
= -0.4
5 :
= -0.2
4 :
= 0.0
3 :
= 0.2
2 :
= 0.4
1 :
= 0.6
1
3
5
7
0.0 0.20.40.6 0.8
x/Rp
1.0 1.21.4 1.6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
z/Rp
o= 0.9
(b)
7 :
= -0.6
6 :
= -0.4
5 :
= -0.2
4 :
= 0.0
3 :
= 0.2
2 :
= 0.4
1 :
= 0.6
1
3
5
7
Fig.4. Shape of stars having average surface rotational parame-
ters ηo= 0.7 (a) and 0.9 (b), and latitudinal differential rotations
given by Eq. (28) for several values of the parameter α. The stel-
lar shape for α = 0 is indicated by a dashed line to more clearly
show the effect of α in other cases.
3.2. Formulation of the equation of the stellar surface
The shape of a rapidly rotating star is generally described us-
ing the total potential of a self-gravitating system with a conser-
vative rotation law, i.e., an internal cylindrical angular velocity,
Ω=Ω(̟), of which the rigid rotation is a particular case. When
a star has a non-conservativeinternal rotational law Ω=Ω(̟,z),
as used in models of stellar evolution and/or suggested by the
developments given in Sect. 2.2, it is no longer possible to de-
fine a rotational potential. However,accordingto Maeder (2009)
the surface of a star with a non-conservative rotation law can be
identified as the region where an arbitrary displacement ds does
not imply any work done by the effective gravity geff
geff.ds = 0 .
(30)
The effective gravity geffwritten in cylindrical coordinates has
the form
geff= −∇ΦG+ Ω2(̟,z)̟.ˆ e̟,
(31)
where ΦG is the gravitational potential. Using the function
Ψ(̟,z) defined as
Ψ(̟,z) = ΦG−1
2Ω2(̟,z)̟2,
(32)
Table 1. Radii ratios Re/Rpas a function of parameters ηoand α
ηo
α=0.6 0.40.20.0-0.2-0.4 -0.6
0.4
0.5
0.6
0.7
0.8
0.9
1.317
1.388
1.456
1.522
1.585
1.645
1.275
1.339
1.401
1.462
1.520
1.577
1.236
1.293
1.349
1.404
1.458
1.512
1.200
1.250
1.300
1.350
1.400
1.450
1.167
1.211
1.255
1.300
1.346
1.392
1.138
1.176
1.215
1.255
1.296
1.340
1.114
1.146
1.179
1.215
1.253
1.293
the effective gravity can be expressed in the form
geff= −[∇Ψ(̟,z) +1
2̟2∇Ω2] ,
(33)
so that the condition given in Eq. (30) can be rewritten as
dΨ(̟,z) +1
2̟2(∇Ω2.ds) = 0.
(34)
The function Ψ(̟,z) can be a potential if and only if it is
∂Ω/∂z = 0. It has been shown by Meynet & Maeder (1997) that
in stars with shellular internal rotational profiles, the surfaces
Ψ(̟,z)= constantare parallelto isobarsurfaces.However,since
dΨ(̟,z) is not an exact differential, the integration of Eq. (34)
depends on the chosen path. To define the shape of a star, it
seems natural to integrate Eq. (34) over a meridian curve, from
the pole (spherical coordinate θ = 0) towards an arbitrary point
Rs(θ), where Rs(θ) represents the function describing the stellar
surface. Thus, by virtue of Eq. (32) we obtain
ΦG(θ) −1
2Ω2(̟,z)̟2+1
2
?θ
0
̟2(∇Ω2.ds) = ΦG(0) ,
(35)
which is in principle the sought equation to calculate the shape
of a star having non-conservative rotational laws. Nevertheless,
since the integration indicated in Eq. (35) depends on the angu-
lar velocity distribution defined only over the stellar surface, the
followingchangeinthecoordinatesdescribingthestellar surface
̟s(θ) = R(θ)ssinθ ;
zs(θ) = Rs(θ)cosθ ,
(36)
readily transforms ∇Ω2(̟,z).ds into a total differential that en-
ables us to integrate Eq. (35) by parts and derive the sought re-
lation
ΦG(θ) −1
2
?θ
0
Ω2
s(θ)
?d̟2
dθ
?
dθ = ΦG(0) .
(37)
In Eq. (37), we wrote Ω2
velocity concerns only the stellar surface. From Eq. (37), we can
see that the stellar shape depends on both the rotational profile
in the surface Ωs(θ) and the gravitational potential ΦG, which
in turn depends on the internal rotational law by means of the
centrifugally distorted density distribution in the stellar interior.
However, this last dependence is of second order or negligible,
as will be shown in Sect. 5.4.
sto indicate explicitly that the angular
3.2.1. Shape of stars with a Maunder rotation law in the
surface
According to the discussion in Sect. 2.4 and the simplified solar
surface rotational law (28), the differential rotation parameter in
8

Page 9

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
fast rotating early-type stars is expected to be α < 0. However,
because of the still poorly known physical characteristics of the
convective external envelope layers of these objects, and pos-
sible effects linked to magnetic fields, we cannot exclude that
α > 0 in some objects. In this respect, we note that hydrody-
namic calculations in stars with 8.75M⊙carried out by Deupree
(1998) showthat the rotationlaw in the convectivecore becomes
of conservative type Ω ∼ ̟−0.7(̟ being the distance from the
rotation axis). If convection in the envelope forced it to rotate
in a similar way, we should expect an external imprint of this
law given by an α > 0. Many calculations of the stellar structure
with fast differential rotation have been done in the past assum-
ing that Høilnad’s criterion for dynamical stability is satisfied
and that radiative viscosity has significant effects on the sur-
face layers. The resulting rotation law implies that the specific
angular momentum increases with the mass contained in cylin-
ders with radius ̟ (Tassoul 1978), which also leads to rotation
laws that imply α > 0. However, the final shape of the rotation
law in the envelope must certainly be the consequence of the in-
terplay amongmany hydrodynamicand magneto-hydrodynamic
instabilities, whose very final combined effect is still highly un-
known. We adopt then the surface rotational law with the func-
tional form given by Eq. (27), and assume that the parameter α
can imply either equator-wardor polar-wardaccelerated angular
velocities.
In a first step, we separate the effects produced on the exter-
nal stellar shape by the surface differential rotation from those
induced by the internal rotation on the mass distribution. This
can simply be done, as we justify in Sect. 5.4, by using a Roche
approximation for the gravitational potential
ΦG(θ) = −GM/Rs(θ) .
(38)
Equation (37) can then be given the form
Rs(θ)
Re
=
1
1 + ηo[I(π/2)− I(θ)],
(39)
where we have written
I(θ) =1
2
?θ
0
?Ωs(θ)
Ωo
?2?d̟2
dθ
?
dθ .
(40)
Sincein relation(39)the functionofthestellar surfaceRs(θ) also
appears in the integrand in terms of ̟ by (36), Rs(θ) is obtained
by iteration. We start the iteration with a first estimate of I(π/2)
based on the shape of a star with a rigid rotation characterized
by the rotational parameter ηo
ηo=Ω2
oR3
GM
e
,
(41)
so that
R(o)
s (θ)
Re
=
1
1 +1
2ηo
?
1 −
?
R(o)
s(θ)
Re
?2
sin2θ
? .
(42)
Itis alsoobviousthatinrelation(39)theequatorial-to-polarradii
ratio Re/Rpis the solution of the iteration, which finally gives
Re
Rp
= 1 + ηoI
?π
2,Re
Rp,α
?
.
(43)
We recall that for rigid rotation in the Roche approximation we
have
?Re
Rp
rigid
?
= 1 +1
2ηo.
(44)
In Table 1, we present the ratios of equator-to-polar radii
Re/Rpas a function of ηoand α used to obtain the stellar shapes
shown in Fig. 4, which shows the shapes of stars computed with
equatorial acceleration ratios ηo = 0.7 and 0.9, whose surface
angular velocity is given by Eq. (28), where the parameter α has
several positive and negative values. In this figure, we can see
that the stellar shape is a function of α that sensitively differs
from that of homologous rigid rotators with the same parame-
ter ηo. We note that for Ω(θ) with α > 0 (acceleration from the
equator towards the pole), we always find that
Re(ηo,α)
Rp(ηo,α)>
?Re(ηo)
Rp(ηo)
?
rigid
.
(45)
For modest α > 0 (recalling that in the Sun, it is α ≃ −0.3),
we can have Re/Rp> 1.5 even for ηo < 1, although the stel-
lar surface has no polar dimples as in models with high rota-
tional energy content (See Sect. 3.2.2). This is due both to a
rotational stretching of Reand to the concomitant shrinkage of
the polar radius Rp, where Ωpole> Ωequator. When Ωs(θ) is ac-
celerated from the pole towards the equator (α < 0), we have
Re(ηo,−|α|)/Rp(ηo,−|α|) < (Re(ηo)/Rp(ηo))rigid. Here, the equa-
tor/pole radii ratios respond mainly to the rotational stretching
of Re.
We note that in stars where α?0, the Vsini parameter does
not only depend on the rotation in the equator. A straightfor-
ward interpretation of the quantity Vsini can produce an incor-
rect value of the acceleration ratio ηo. Since we are dealing here
with fast rotators (ηo? 0.5), the formation of spectral lines used
to determine the Vsini needs to be treated properly by taking
into account the surface velocity fields, the stellar deformation,
and the gravity darkening effect, all dependent on ηoand α. We
also note that if the interferometric measurements produce radii
ratios Re/Rp> 1.5, it does not necessarily mean that the star is
in a state of “supercritical” rotation. From these comments and
the results shown in Table 1, we clearly see that the study of the
surface differential rotation prefigures the need for two sources
of information: a) spectroscopy, which can help us to estimate
α and the inclination angle factor sini; b) interferometry, which
provides information related to the stellar geometrical deforma-
tion and incidentally with α. We briefly discuss these issues in
Sects. 6.1 and 6.2.
3.2.2. Stars with polar dimples
Owing to the centripetal acceleration, the fraction of the grav-
itational force that must be supported by the pressure gradient
increases towards the center of the stars. This means that the
pressure gradient is more centrally concentrated in a rotating
star than a non-rotating one. The rotation tends then to decrease
the moment of inertia, which is accompanied by an increase in
the density gradient towards the center. The gravitational attrac-
tion towards the pole is thus stronger, which produces a flatten-
ing of the pole or a dimple. In models of stars with high dif-
ferential rotations in depth that enables the star to have a sig-
nificant content of rotational energy, i.e. when the energy frac-
tion τ = K/|W| (K = rotational kinetic energy; W = gravita-
tional potential energy)becomes τ ? 0.03, the polar regionis ei-
ther strongly flattened or hollower(Bodenheimer1971; Clement
9

Page 10

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
0.0 0.51.0 1.52.0
x/Rp
2.5 3.03.5 4.0
0.0
0.5
1.0
1.5
2.0
2.5
z/Rp
= 5.0
o= 0.9
o= 0.8
o= 0.7
o= 0.6
o= 0.5
o= 0.4
o= 0.3
Fig.5. Shape of model stars having the surface angular velocity
profile given by relation (46) for β = 5.0 and several values of
ηe.
1974, 1979; Zorec 1986; Jackson et al. 2005). Similar dimples
can also be produced by a surface angular velocity accelerated
enough towards the polar latitudes. To illustrate this effect, we
adopt the following angular velocity profile, which is sometimes
used in the literature to model internal conservative (cylindrical)
laws (Jackson et al. 2005) for stars with high rotational energies
(0.02 ? τ ? 0.22)
Ωs(θ) =
Ωo
1 + β[Rs(θ)/Re]2sin2θ
,
(46)
where Rs(θ) is the equation of the stellar surface and β is a free
parameter. A relation similar to Eq. (46) may suit the rotational
profiles shown in Fig. 2a if the angular velocity is accelerated
from the equator to the pole. Figure 5 shows the shape of model
stars with a surface angular velocity profile given by expression
(46) for β = 5.0 and several values of ratio ηo in the equator
calculated with Eq. (39). It can be shown that using the Roche
approximation and the rotational law in Eq. (46), the ratio of
equatorial to polar radii is then given by
Re
Rp
= 1 +ηo
2(1 + β) .
(47)
Writing R2
polar region to appear is dz/dx ≥ 0 at x = 0. The condition to
obtain a hollowed polar region is then given by
s(θ) = x2+ z2, the condition for a flattened/hollowed
ηo(1 + β)2≥
?
1 + ηo
?1 + β
2
??3
.
(48)
Makingu=Re/Rp=1+ηo(1+β)/2,relation(48) becomesu3/(u−
1)≥2(1+β).Since it is [u3/(u−1)]minfor u=1.5, condition (48)
requires that β≥2.375 independently of the value of ηo.
Since the simplest equation to represent the rotationally de-
formed surfaces shown in Fig. 5 is
Rs(θ)
Re
= 1 −
?
1 −Rp
Re
?
cosθ ,
(49)
an alternative expression to relation (28) for the surface angular
velocities accelerated from the equator to the pole could be
Ωs(θ) =
Ωo
Rp
Re
1 + β
?
1 −
?
1 −
?
cosθ
?2sin2θ
,
(50)
Table 2. Models of stars with rigid rotation
ZAMS
M = 5M⊙,
Rp/R⊙
t/tMS= 0.011,
Re/Rp
Ωcr= 1.92 × 10−4
J/M
×10−17
Ω/Ωcr
ρc
Veq
ηo
K/|W|
×102
0.00
0.50
0.60
0.70
0.80
0.90
0.95
0.98
0.99
1.00
19.498
19.688
19.773
19.875
19.996
20.136
20.213
20.262
20.279
20.295
2.644
2.611
2.598
2.583
2.560
2.539
2.531
2.526
2.521
2.517
1.000
1.050
1.076
1.111
1.160
1.239
1.311
1.387
1.422
1.512
0 0.000
2.091
2.515
2.944
3.378
3.819
4.043
4.179
4.225
4.270
0.000
0.096
0.147
0.216
0.311
0.466
0.606
0.755
0.824
0.996
0.000
0.172
0.248
0.339
0.444
0.564
0.630
0.671
0.686
0.700
183
224
268
316
377
418
455
470
503
TAMS
M = 5M⊙,
t/tMS= 1.000,
Ωcr= 4.79 × 10−5
0.00
0.50
0.60
0.70
0.80
0.90
0.95
0.98
0.99
1.00
27.479
27.527
27.548
27.573
27.672
27.636
27.654
27.666
27.670
27.682
6.454
6.432
6.424
6.411
6.397
6.382
6.375
6.370
6.369
6.367
1.000
1.044
1.068
1.101
1.149
1.228
1.296
1.369
1.410
1.510
00.000
0.952
1.145
1.340
1.536
1.735
1.836
1.896
1.917
1.936
0.000
0.088
0.135
0.200
0.294
0.448
0.582
0.723
0.804
0.996
0.000
0.024
0.034
0.046
0.061
0.077
0.086
0.092
0.094
0.096
112
137
165
196
234
260
282
293
317
ZAMS
M = 15M⊙,
t/tMS= 0.013,
Ωcr= 1.34 × 10−4
0.00
0.50
0.60
0.70
0.80
0.90
0.95
0.98
0.99
1.00
5.781
5.847
5.877
5.914
5.956
6.006
6.034
6.051
6.058
6.064
4.895
4.825
4.785
4.742
4.698
4.647
4.629
4.603
4.596
4.591
1.000
1.052
1.079
1.115
1.166
1.246
1.321
1.385
1.421
1.517
00.000
6.510
7.839
9.184
10.551
11.947
12.660
13.095
13.240
13.387
0.000
0.099
0.151
0.220
0.319
0.475
0.619
0.743
0.814
0.994
0.000
0.268
0.386
0.528
0.692
0.880
0.984
1.050
1.072
1.095
237
290
346
409
485
540
580
600
644
TAMS
M = 15M⊙,
t/tMS= 1.000,
Ωcr= 2.76 × 10−5
0.00
0.50
0.60
0.70
0.80
0.90
0.95
0.98
0.99
1.00
9.226
9.236
9.240
9.245
9.251
9.259
9.263
9.265
9.266
9.267
13.346
13.306
13.289
13.268
13.241
13.212
13.197
13.189
13.186
13.183
1.000
1.045
1.068
1.101
1.149
1.228
1.297
1.369
1.410
1.511
00.000
2.268
2.728
3.192
3.661
4.136
4.377
4.522
4.571
4.619
0.000
0.088
0.135
0.200
0.294
0.447
0.581
0.722
0.802
0.996
0.000
0.022
0.032
0.044
0.057
0.072
0.081
0.087
0.088
0.090
133
163
196
233
279
310
336
350
378
Note: ρcis given in gcm−3; Ωcris given in s−1; Veqis given in kms−1; the units of J/M
are cm2s−1; t is age and tMSis the MS life span
would require us to infer the ratio Rp/Refrom interferometry,
which according to Eq. (47) is related to ηoand β, and β derived
fromspectroscopy.However,beforeusingexpressionsofthetype
given by either Eq. (46) or (50), it would perhaps be preferable
to await observational confirmation that stellar shapes similar
to those depicted in Fig. 5 actually exist.
4. The gravity darkening effect in stars with
non-conservative rotation laws
Owing to the radiative equilibrium that prevails in the atmo-
sphere of massive and intermediate-mass stars, the emerging
bolometric radiative flux F is proportional to the temperature
gradient ∇T in the outer layers of the stellar envelope. Since
these layers are also in hydrostatic equilibrium, their pressure
gradient ∇P ∝ ∇T is balanced by the surface effective gravity
geff. We have then the following phenomenological relation that
describes the principle of von Zeipel’s relation
F ∝ ∇T ≡ ∇P ∝ geff.
(51)
10

Page 11

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
As F ∝ T4
point r in the stellar surface, it follows the known von Zeipel’s
relation (von Zeipel 1924)
eff, where Teffis the effectivetemperatureat a given
Teff= C(r) × geff1/4.
(52)
However, according to Poincar´ e-Wavre’s theorem (Tassoul
1978) only when the rotational law in the external layers is con-
servative do the constant equipotential, density, and tempera-
ture surfaces coincide to consider that C(r) = C = constant.
Otherwise, as for rotational laws discussed in Sect. 2.2, C(r) is
not constant with the colatitude θ. Its detailed expression for the
non-conservative case with shellular rotation was obtained by
Maeder (1999).
An equivalent relation to Eq. (52) for spectroscopic and in-
terferometric analysis can be given the form
Teff= C × geffβGD,
(53)
where βGD = 0.25 + δ, so that δ masks the variation in
C(r) over the stellar surface. The particular notation βGD
used for the gravity darkening power in relation (53) is in-
troduced to avoid confusions with the β parameter that ap-
pears in the internal rotation law (46). Two-dimensional (2D)
models of rotating stars and some observations suggest that
δ ? 0 [c.f. Lucy (1967); Lovekin et al. (2006); van Belle et al.
(2006); Monnier et al. (2007); Zhao et al. (2009)]. We insist on
δ not being in principle the function of taking into account the
change in the stellar surface from radiative to possible convec-
tive equilibrium-dominated layers because of the strong change
in Teff with θ, as seems to be understood in Aufdenberg et al.
(2006) and Zhao et al. (2009), but mainly the non-conservative
nature of the rotational law in the external layers.
5. Models of rotating stars
To study the effect of the centrifugally distorted internal mass
distribution on the gravitational potential in the surface of a star,
to test the validity of the Roche approximation, it is enough to
calculate models of stellar structure where only the global dy-
namical aspects induced by the rotation are considered. To this
end, we differentiate the primary effects produced by the rota-
tion from those induced by the evolution. The primary thermo-
dynamic effects due to the stellar evolution are taken into ac-
count by making use of the relations between the pressure and
density calculatedwith one-dimensionalmodels of stellar evolu-
tion without rotation. We assume thus that the changes produced
on the barotropic relation P = P(ρ) at a given evolutionary stage
of a star by the several instabilities and the diffusion of chemical
elements unleashed through the stellar evolution by the rotation,
have second order effects on the establishment of the dynamical
equilibrium of the rotating star. In principle, nothing prevents us
from using the barotropic relations derived with models of stel-
lar evolutionwith shellular rotation,but the results will probably
not be more reliable. They all are calculated for low energy ra-
tios τ = K/|W|, which do not correspond to the model inputs
regarding the rotational law and energy tested in the present at-
tempt.
Since we are not interested in the precise description of all
non-linear time-dependent phenomena associated with the vis-
cosity and internal flows in rotating stars, and because the total
energy carried by the meridional circulation is small, our mod-
els are axisymmetric, steady state, and circulation free. Owing
to these assumptions and Poincar´ e-Wavre’s theorem (Tassoul
1978), our model stars behave as barotropes. We then adopt in-
ternal rotational laws of conservative form, Ω=Ω(̟), where ̟
is thedistanceto the rotationaxis. Accordingto the discussionin
Sect.5.2, the conservative laws are expected to produce stronger
stellar geometricaldeformationsfora givenamountofrotational
kinetic energy than the homologous non-conservativeones.
For conservative rotation laws, the gravitational potential
Φ(̟,z) and the density distribution ρ(̟,z) in the rotating star
are simultaneous solutions to the hydrostatic equilibrium equa-
tion
ρ−1∇P = ∇Φ + j2̟−3ˆ e̟
(54)
and the Poisson equation
∆Φ = 4πGρ,
(55)
where (̟,φ,z) are the cylindrical coordinates with z containing
the rotationaxis, ˆ e̟is the unit vectorperpendicularto the z-axis,
P is the pressure, and j = Ω̟2is the specific angular momen-
tum.
Equations(54) and (55) are solved with the adoptedcomple-
mentary barotropic relation
P = P(ρ) .
(56)
The P=P(ρ) relations used in this work have a two-component
polytropic character
P = aργa+ bργb,
(57)
where the constants a, b, γa, and γbwere adjusted so as to: a)
reproduce the pressure Pcand the density ρcin the center of the
non-rotatingstar of givenmass and evolutionarystage; b) ensure
a continuous distribution of the pressure-density relation at the
radius of the stellar core; c) obtain the correct stellar mass at the
stellar radiusas tabulatedbySchaller et al. (1992) for1-Devolu-
tionary models for the initial metallicity Z = 0.02. The function
(57) is continued in the stellar atmosphere by another pressure-
density relation calculated by Castelli & Kurucz (2003) for stel-
lar atmospheres as a function of the parameters (Teff,logg).
The first order effects due to the stellar evolution are thus
accounted for by the pressure-density relations at the center of
the star and the ∂P/∂ρ gradients. An additional term in relation
(57) could also in principle take into account the presence of the
convectiveregions in the envelopeinducedby the rapid rotation,
but this was not done in the present approach. The only rota-
tional effect on the P=P(ρ) relation consideredhere is by means
of the mass-compensation effect (Sackmann 1970), which in-
creases the density ρcat the center of the star. To this end, we
iterated ρcuntil the nominal stellar mass M was attained. This
iteration also implied that the central pressure Pcwas changed
in accordance.
ThegravitationalpotentialΦ(̟,z)is obtainedby solvingthe
Poisson equation given in Eq. (55) with the cell-method adapted
by Clement (1974) to stellar structure calculations. The den-
sity distribution ρ(̟,z) is derived from the integrated form of
Eq. (54)
?ρ
ρc
dP
ρ
= Φ(̟,z) − Φc+
?̟
0
Ω2(̟)̟d̟ ,
(58)
where Φcis the gravitationalpotential at the stellar center. Given
a rotational law Ω(̟) and a barotropic relation in Eq. (56), the
common solution to equations (54) and (55) is performed over
11

Page 12

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
the entire space. The iteration of Φ and ρ is stopped when the
highest density difference in the (̟,z)-space is max(δρ/ρ)<∼
10−6. In our iterations, the virial relation δ = [2(K + U) −
W]/|W| = 0, where K = kinetic energy, U = internal energy,and
W = total gravitational potential energy,is verified to better than
δ ≈ 2×10−4in theZAMS models,andδ ≈ 6×10−3bytheTAMS
models. Since in the framework of conservative rotational laws,
the surfaces of constant pressure, density, and total potential are
parallel, the rotationally distorted shape of our models is defined
by the total equipotential surface that contains the polar ‘photo-
spheric’ radius Rp. This radius is identified by the layer whose
density satisfies the model-atmosphere relation τRoss(ρ)=2/3 in
the stellar atmosphere models of Castelli & Kurucz (2003). The
local effective temperature in the pole needs to satisfy also the
gravity darkening effect. We modified accordingly the effective
temperaturegivenby Schaller et al. (1992) for the givenmass M
using von Zeipel’s approximation (von Zeipel 1924). The trans-
formation to the rotation-dependent effective temperature was
performedfollowingthe proceduregiveninFr´ emat et al.(2005).
5.1. Rigid rotation
The simplest conservative rotational law is that of rigid rotation.
The modelsof stars with rigid rotationwerecalculated assuming
that at eachevolutionaryphasetheyexperienceinstantaneousto-
tal redistribution of their internal angular momentum. The char-
acteristics of these models are given in Table 2. In this table,
t/tMSis the fractional age of the star, where tMSis the time that
a non-rotating star of mass M spends on the main sequence, Ωcr
is the critical angular velocity, Ω/Ωcrrepresents the angular ve-
locity for which the model was calculated, ρcis the core den-
sity of the rotating object, Re/R⊙is the equatorial radius of the
model star in solar units and Re/Rpthe equatorial-to-polar radii
ratio, Veqis the equatorial linear velocity in kms−1, J/M is the
total specific angular momentum, η = Ω2R3
centrifugal to the gravitational acceleration in the equator, and
K/|W| is the ratio of the kinetic rotational energy (K) to the ab-
solute value of the gravitational potential energy (W). Some of
these models are shown in Fig. 6, where in all cases the iso-
density surfaces are for the same ρ/ρc density ratios. Similar
models were also calculated for other masses and age ratios
t/tMS. In spite of the simple approach used to calculate them,
the radius ratios compare very closely with those calculated by
Ekstr¨ om et al. (2008) for the same K/|W| energyratios. We note,
however, that the models obtained by these authors cannot be
compared directly with ours because the distribution of the in-
ternal rotational velocity is not the same. Starting from a quasi-
rigid rotation in the ZAMS, Ekstr¨ om et al. (2008) accounted for
a consistent evolution of the angular momentum distribution in-
side the star throughout the calculated stellar evolution span.
e/GM is the ratio of
5.2. Shellular differential rotation
Zahn et al. (2010) studied the shapes of stars with internal shel-
lular differential rotation and concluded that in 7M⊙ stars the
radius ratio Re/Rpat critical equatorial rotation can be enlarged
from1% to 4%, dependingon the internal rotationalenergycon-
tent and the evolutionary phase. However, it can easily be seen
that when two rotation laws are described with the same func-
tion and both imply the same central to surface angular velocity
ratios Ωcenter/Ωsurface, but one is shellular and the other cylindri-
cal, the law that is shellular may have a weaker effect on the
global internal density distribution than the cylindrical one. For
r
1
p
Fc
x
z
Fig.8. Schematic comparison of the centrifugal force produced
by a shellular and a cylindrical rotation law when both are de-
scribed by the same analytical function
simplicity, we change the independent variable P (pressure) in
the shellular law by the radius r. The comparison is depicted in
Fig. 8: the specific centrifugal force acting at a point p in the
stellar interior is
Fshellular
c
Fcylindrical
c
= Ω2(r)̟
= Ω2(̟,θ)̟
(59)
for shellular and cylindrical angular velocity distribution laws,
respectively (̟ =
r sinθ is the distance to the rotation
axis). At the equator, if we assume that Ωcylindrical(̟,π/2) =
Ωshellular(r,π/2), for a given point p(z,̟), it will be
Ω(p)cylindrical> Ωshellular(p) ,
Fcylindrical
c
(p) > Fshellular
c
(p) .
(60)
This holds in particular for the points in the stellar surface.
However, since it has been shown in Sect. 3.2 that the exter-
nal geometrical deformation of the star is mainly a function
of the angular velocity law on the surface, cylindrical rotation
laws will carry stronger geometrical deformations than shellular
ones for a similar total kinetic energy. Moreover, in Sect. 2.2 we
have anticipated that strong cylindrical components of the an-
gular velocity distributions in the external layers probably exist.
It is then important to determine the order of magnitude of the
quadrupole factor γ in rotating stars using conservative internal
rotationallaws. To this end, in the next section we calculate two-
dimensional models of rotating stars with conservative internal
rotational laws.
5.3. Cylindrical differential rotation
The models with differential rotation were calculated using the
same barotropicrelations P=P(ρ) as for the solid-bodyrotation.
We adopted the internal angular velocity distributed as
Ω(̟) =
Ωo
1 + β(̟/Re)n,
(61)
12

Page 13

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
0.0 0.51.0 1.52.0 2.53.0
x/R
0.0
0.5
1.0
1.5
2.0
2.5
z/R
a1
= 0
8: 10-12
7: 10-7
6: 10-5
5: 7.10-4
4: 0.07
3: 0.2
2: 0.5
1: 0.8
/c
1
23
4
5
6
78
0.00.51.0 1.5 2.02.5 3.03.5
x/R
a2
= 0.85
c
0.0 0.51.01.5 2.02.5 3.03.5
x/R
a3
=
c
012345678
x/R
0
1
2
3
4
5
6
z/R
b1
= 0
012345678
x/R
b2
= 0.85
c
0123456789
x/R
b3
=
c
Fig.6. Rigid rotators. a: Iso-density surfaces in model stars of M = 5M⊙in the ZAMS rotating at (a1) Ω = 0; (a2) Ω = 0.85Ωc; (a3)
Ω = Ωc= 1.92 × 10−4s−1; b: Iso-density surfaces in stars of M = 5M⊙in the TAMS rotating at (b1) Ω = 0; (b2) Ω = 0.85Ωc; (b3)
Ω = Ωc= 4.79×10−5s−1.The iso-density surfaces are labeled with the correspondingdensity ratios ρ/ρc, which are the same in all
panels of the figure.
012345
x/R
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
z/R
a1
=4.0
o/
cr=4.0
8
9: 10-9
/c
1
2
3
4
5
67
9
01234560
x/R
a2
= 6.0
o/
cr=7.0
123456
x/R
a3
= 8.0
o/
cr=9.6
051015 20
x/R
z/R
b1
= 4.0
o/
cr=4.0
010 203040
x/R
506070
b2
= 6.0
o/
cr=7.0
0102030 40
x/R
50607080
b3
= 8.0
o/
cr=8.5
Fig.7. Differential rotators. a: Iso-density surfaces in model stars of M = 15M⊙in the ZAMS having the internal rotation law (61),
whose parameters Ωo/Ωcrand β are indicated. Ωcris for the critical rigid rotation. The iso-density surfaces are labeled with the
corresponding density ratios ρ/ρc, which are the same as in Fig. 6 and for all panels of this figure. b: Iso-density surfaces in model
stars of M = 15M⊙in the TAMS having the internal rotation law (61), whose parameters Ωo/Ωcrand β are indicated. In each panel,
the ordinates are in the same scale as the abscissas, but they differ from one to the other panel.
13

Page 14

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
Table 3. Models of stars with internal differential rotation
ZAMS
M = 5M⊙, Ωcr= 1.92 × 10−4
Re/R⊙
Re/Rp
ZAMS
M = 15M⊙, Ωcr= 1.34 × 10−4
Re/R⊙
Re/Rp
Ωo/Ωcr
ρc
Rav/R⊙
Veq
J/M
×10−17
K/|W|
Ωo/Ωcr
ρc
Rav/R⊙
Veq
J/M
×10−17
K/|W|
β = 2
β = 2
1.0
2.0
3.0
3.5
3.9
19.996
21.593
24.741
27.451
32.121
2.671
2.821
3.451
4.813
9.933
2.711
2.967
3.938
5.799
11.652
1.052
1.269
1.947
3.143
6.998
117
225
290
255
152
3.326
6.723
10.437
12.898
17.775
0.0044
0.0175
0.0389
0.0532
0.0683
1.0
2.0
3.0
3.5
4.0
5.945
6.482
7.580
8.529
10.930
4.915
5.092
5.828
7.330
16.655
4.986
5.383
6.666
8.938
20.092
1.047
1.265
1.878
2.809
7.455
151
294
396
381
216
9.920
20.091
31.115
38.020
58.322
0.0064
0.0250
0.0553
0.0752
0.1035
β = 4
β = 4
1.0
2.0
3.0
4.0
5.0
6.2
19.870
21.022
23.095
26.378
31.538
46.905
2.639
2.667
2.731
2.906
3.400
6.692
2.656
2.729
2.867
3.185
3.899
7.790
1.023
1.138
1.353
1.728
2.489
6.456
70
139
202
250
269
181
2.824
5.642
8.458
11.318
14.450
22.434
0.0033
0.0130
0.0285
0.0489
0.0736
0.1128
1.0
2.0
4.0
5.5
6.2
6.4
5.899
6.273
8.085
11.219
14.319
16.610
4.875
4.861
4.955
5.774
7.599
9.221
4.915
4.981
5.455
6.887
9.291
11.247
1.025
1.133
1.697
2.843
4.558
5.958
91
181
341
396
349
304
8.220
16.443
32.810
45.878
56.026
64.434
0.0046
0.0179
0.0663
0.1163
0.1457
0.1588
β = 6
β = 6
1.0
2.0
4.0
5.0
7.0
8.35
19.796
21.022
23.095
28.193
40.196
67.403
2.634
2.626
2.631
2.677
3.160
5.514
2.652
2.670
2.768
2.907
3.641
6.446
1.022
1.095
1.426
1.727
2.969
6.938
50
100
195
235
275
196
2.482
4.946
9.761
12.103
17.048
25.828
0.0027
0.0104
0.0392
0.0586
0.1047
0.1483
1.0
3.0
5.0
7.0
8.0
8.4
5.874
6.681
8.684
12.871
16.731
19.300
4.856
4.726
4.544
4.618
5.171
5.767
4.889
4.869
4.941
5.417
6.299
7.106
1.010
1.221
1.693
2.664
3.689
4.564
65
196
324
424
430
408
7.103
21.149
34.451
46.715
53.793
58.318
0.0035
0.0303
0.0770
0.1336
0.1648
0.1796
β = 8
β = 8
1.0
3.0
5.0
7.0
9.0
10.0
19.746
21.808
26.464
34.953
50.578
66.836
2.626
2.580
2.519
2.529
2.873
3.572
2.640
2.630
2.669
2.788
3.343
4.219
1.009
1.164
1.495
2.070
3.305
4.962
39
118
195
264
292
264
2.227
6.610
10.776
14.665
18.741
22.199
0.0022
0.0190
0.0492
0.0878
0.1313
0.1575
1.0
2.0
4.0
6.0
8.0
9.6
5.857
6.092
7.143
9.316
13.338
18.901
4.857
4.790
4.551
4.229
3.980
4.078
4.884
4.837
4.722
4.610
4.573
4.940
1.011
1.066
1.304
1.733
2.373
3.327
51
102
209
319
428
484
6.293
12.548
24.704
35.796
45.301
52.380
0.0029
0.0112
0.0425
0.0869
0.1363
0.1764
TAMS
M = 5M⊙, Ωcr= 4.79 × 10−4
TAMS
M = 15M⊙, Ωcr= 2.76 × 10−4
β = 2
β = 2
1.0
2.0
3.0
3.5
27.626
28.085
29.015
29.954
6.639
7.556
12.077
27.310
7.708
7.915
13.648
30.655
1.051
1.280
2.334
5.467
71
132
137
78
1.692
3.482
5.633
7.475
0.0008
0.0031
0.0072
0.0104
1.0
2.0
3.0
3.5
9.255
9.357
9.572
9.982
13.564
15.546
25.858
66.541
13.777
16.370
25.259
73.661
1.049
1.285
2.411
6.333
86
160
162
81
4.119
8.521
14.021
19.758
0.0007
0.0030
0.0073
0.0107
β = 4
β = 4
1.0
3.0
4.0
5.0
5.5
5.7
27.601
28.632
29.723
31.729
34.006
36.217
6.535
7.844
10.200
19.386
42.148
85.056
6.564
8.169
11.025
20.867
44.030
88.533
1.028
1.384
1.984
4.110
9.199
19.108
43
110
116
81
43
22
1.541
4.829
6.805
9.748
13.124
16.649
0.0006
0.0059
0.0107
0.0177
0.0221
0.0246
1.0
2.0
3.0
4.0
5.0
5.5
9.249
9.331
9.480
9.732
10.236
11.028
13.363
14.200
16.212
21.574
44.869
129.78
13.483
14.476
16.970
23.300
48.918
136.36
1.026
1.128
1.387
2.021
6.602
9.820
52
99
133
138
87
35
3.753
7.644
11.882
16.979
25.481
39.665
0.0006
0.0025
0.0058
0.0108
0.0183
0.0240
β = 6
β = 6
1.0
3.0
4.0
5.0
6.0
7.0
27.584
28.440
29.275
30.516
32.471
36.522
6.497
7.200
8.086
9.898
14.334
32.588
6.515
7.383
8.447
10.573
15.413
33.300
1.021
1.232
1.494
2.007
3.190
7.737
31
83
100
103
88
48
1.428
4.396
6.025
7.883
10.297
14.903
0.0006
0.0051
0.0091
0.0145
0.0216
0.0308
1.0
3.0
4.0
5.0
6.0
7.0
9.246
9.437
9.626
9.914
10.393
11.634
13.246
14.815
16.812
20.975
31.844
91.051
13.307
15.268
17.571
22.325
34.264
95.665
1.013
1.238
1.498
2.043
3.419
10.725
37
72
121
123
99
42
3.479
10.791
14.905
19.763
26.539
43.227
0.0006
0.0050
0.0091
0.0148
0.0223
0.0334
β = 8
β = 8
1.0
3.0
5.0
6.0
8.0
9.0
27.571
28.311
29.992
31.368
36.726
47.921
6.474
6.933
8.280
9.791
20.408
88.304
6.496
7.037
8.640
10.361
21.171
88.136
1.018
1.173
1.610
2.081
5.150
24.993
24
67
94
96
65
18
1.337
4.082
7.099
8.858
14.213
24.175
0.0005
0.0045
0.0127
0.0185
0.0345
0.0453
1.0
3.0
5.0
6.0
7.0
8.5
9.243
9.408
9.790
10.112
10.602
12.570
13.239
14.238
17.301
20.816
28.159
86.558
13.327
14.472
18.070
22.169
30.082
90.257
1.006
1.164
1.624
2.148
3.159
11.410
29
82
113
113
99
41
3.260
10.008
17.656
22.314
28.363
49.916
0.0005
0.0049
0.0128
0.0189
0.0267
0.0438
Note: ρcis given in gcm−3; Ωcris given in s−1; Veqis given in kms−1; the units of J/M are cm2s−1; Rav/R⊙is the radius of a sphere having the same volume as the rotatio-
nally deformed object
where β > 0 is a free parameter and Ωois the angular veloc-
ity at the axis of rotation. For simplicity, we considered n = 2.
Nevertheless, it can be shown that all models with n≤2 obey the
Solberg-Høilandstability criterion,and that at whateverdistance
̟ the energy ratio τ(̟) is K(̟)/|W(̟)| ? 0.138. This ensures
that no region in the stellar interior is unstable to the secular in-
stability that can carry it to a three-axial Jacoby ellipsoidal con-
figuration.
In Table 3, we indicate the characteristics of the calculated
models. The entries have the same meaning as in Table 2. The
higher K/|W| values given in Table 3 are about the highest
we could obtain for the given masses, ages, and β parameters.
Figure 7 shows some of these calculated models.
Using the same algorithm, we can obtain models that ap-
proachmorecloselythelimitofdynamicalstability K/|W| ∼ 1/4
if the objects are on the ZAMS and have masses higher than
M = 15M⊙. However, Clement (1979) in his analysis of the sec-
ularstability ofdifferentiallyrotatingobjects alreadypointedout
the unstable character of model stars with K/|W| ? 0.1.
5.4. The Roche approximation
With the models thus calculated, which give us the shape of a
rotationally distorted star, we can test to what degree the gravi-
tational potential ΦGin the stellar surface may deviate from the
simplecentral-fieldexpressionusedintheRocheapproximation.
14

Page 15

Zorec et al.: Spectroscopy and interferometry of early-type differential rapid rotators
Table 4. Quadrupole factors in stars with rigid rotation at differ-
ent angular velocity ratios Ω/Ωc
Ω/Ωc=
Phase
ZAMS
TAMS
ZAMS
TAMS
0.50.7 0.9
γ
0.951.0
M/M⊙
5.0
5.0
15.0
15.0
0.0012 0.0018
0.0003 0.0007
0.0015 0.0025
0.0004 0.0007
0.0026
0.0016
0.0037
0.0018
0.0028
0.0020
0.0040
0.0023
0.0030
0.0025
0.0043
0.0028
Hubbard et al. (1975) showed that the gravitational potential
of rotating centrally condensed objects can be given by a multi-
pol expansion
ΦG= −GM
R(θ)
1 −
∞
?
n=1
?Ro
R(θ)
?2n
J2nP2n(cosθ)
,
(62)
where Rois the radius of the rotationally undistorted star, J2nare
the zonal harmonic coefficients, and P2n(cosθ) are the Legendre
polynomials. To derive relation (62), Hubbard et al. (1975) as-
sumed that the density can be expanded in Legendre polynomi-
als. The J2ncoefficients are then obtained as integrals over the
volumeof the axisymmetricobject which generallyproduceval-
ues so that J2> 0, J4< 0, J6> 0, J8< 0, and so on. Since we
are not interested in developing a detailed theory of the gravita-
tional potential of rotating stars, but only to test the validity of
the Roche approximation, we do not calculate the volume inte-
grals over the rotationally deformed star, but use the harmonic
coefficients coefficients as mere parameters giving a quantita-
tive indication of the deviation from the central-field form of
the gravitational potential. We determine the quantities J2n in
expression (62) by simply fitting the gravitation potential pre-
dicted by the 2D models of rotating stars calculated in Sects. 5.1
and 5.3 using a least squares method. In this algebraic estimate
of J2n, their orderof magnitudeare preserved,but not always the
expected sign.
In relation (62), we retain only the quadrupole moment to
estimate the effects in rigid rotators
ΦG= −GM
R(θ)
1 −
?Ro
R(θ)
?2
J2P2(cosθ)
.
(63)
The shape of the surface thus becomes
Rs(θ)
Re
=
1 − γ
?Rs(θ)
2ηo
Re
?2P2(cosθ)
1 −
?Rs(θ)
?
1 +1
2γ
?
+1
?
Re
?2sin2θ
? ,
(64)
where we have introduced the notation
γ = J2
?Ro
Re
?2
.
(65)
ThelargestdifferencebetweentheestimategivenbyEq.(64)
to the ratio Rs(θ)/Re and that producedbythe Roche approxima-
tion in Eq. (44), is expected for θ=0. Thus according to (64) we
have
Re
Rp
=1 +1
1 − γ
2(ηo+ γ)
?
Re
Rp
?2
.
(66)
Since it is always true that Re/Rp≥ 1, relation (66) implies that
for whatever 0 ≤ ηo ≤ 1 and γ ? 0 the equator to polar radii
ratio is slightly larger than obtained from the sheer Roche ap-
proximation.
In Table 4, we indicate the quadrupolefactors γ for 5M⊙and
15M⊙stars in the ZAMS and TAMS evolutionary phases ob-
tained by fitting relation (64) with surface gravitational potential
obtainedwith the model calculation.Inthis table, we see that the
smaller J2the moreevolvedis the star, simply because the star is
more centrally condensed such that the central-field approxima-
tion for ΦGholds better. Zahn et al. (2010) calculated the same
factorsforrigidandshellulardifferentialrotators.Reducingtheir
J2estimatesby(Rp/Re)2tobeabletoapproachtheratio(Ro/Re)2
more closely than using the average stellar radius for Ro, we
see that for the ZAMS and TAMS epochs of rigid critical ro-
tators we obtain the same values for the harmonic coefficient J2
than Zahn et al. (2010). Having then γ ? 0.004, from (44) we
see that neglecting the quadrupole term in (64), in rigid rota-
tors at critical rotation the Re/Rpcan be underestimated by less
than 2%. For shellular rotators, Zahn et al. (2010) found that at
critical equatorial rotation the flattening changes by 4% because
Ωcenter/Ωsurface= 4 in the ZAMS and is smaller near the TAMS.
Finally, we note that within approximation (64), the actual
ratio of the centrifugal to gravity acceleration at the equator
should be
ηo
1 +3
ηJ=
2γ.
(67)
In stars with conservative differential rotation laws of the type
given in Eq. (61) that imply rotational energies τ=K/|W| > τcr,
the deformations of the internal mass distribution and the devi-
ations to the central-field gravitational potential are expected to
be stronger than in stars with rigid rotation. To test the use of the
Rocheapproximationinthesecases,we calculatedthezonalhar-
monic coefficients γ2n=J2n(Ro/Re)2from n=1 to 4. Table 5 lists
the zonal multipolar coefficients γ2nobtained by fitting relation
(62) with the calculated gravitational potential for several val-
ues of the coefficient β and angular velocity ratio Ωo/Ωcr, where
Ωois the angular velocity at the rotation axis, and Ωcr is the
critical angular velocity of an homologous star (same mass and
evolutionary stage) behaving as a rigid rotator. We insist on the
algebraic nature of the γ2ncoefficients, since they were derived
as fitting parameters to model-calculated gravitational potential.
In this Table, we also give the fractional deviations of the radius
vectors ∆R(θ)/R(θ) = [R(θ)model− R(θ)γ2n]/R(θ)model× 104for
different colatitude angles θ, where R(θ)modelis obtained with
the models calculated in Sect. 5.3, while the R(θ)γ2nvalues re-
fer to those estimated with the total potential that includes the
gravitational potential given by Eq. (62). A quick inspection of
Table 5 shows that in most cases a fairly precise description of
the stellar surface can be obtained by consideringonly γ2and γ4
in the gravitational potential given by Eq. (62). Nevertheless, in
all cases the Roche approximationcan be valid to better than 5%
for ratios Ωcenter/Ωsurface ≃ 4. The approximation works better
the more evolved the star is and the higher the rotational energy
parameter τ.
6. Attainable stellar parameters
Accordingto Vogt-Russell’s theorem,“the complete structure of
a star in hydrostatic and thermal equilibrium is uniquely deter-
mined by the total mass M and the run of chemical composition
throughout the star, provided the structure equations are func-
tion of local parameters” (Cox & Giuli 1968). Then, for a given
15