We take advantage of interferometric measurements of Achernar to inquire on its internal rotational law. The reinterpretation of interferome-tric data and the use of fundamental parameters corrected for gravitational darkening effects and models of 2D-models of internal stellar structures, lead us to the conclusion that the star could not be a rigid, near critical, rotator but a differential rotator with the core rotating $\sim3$ times faster than the surface

Domiciano de Souza, Armando

Fremat, Yves

Vakli, Farok

Zorec, Juan

arXiv

Domiciano De Souza, Armando

HAL-UNICE

ACHERNAR CAN BE A DIFFERENTIAL ROTATOR
Juan Zorec, Armando Domiciano de Souza, Yves Fremat, Farok Vakli
To cite this version:
Juan Zorec, Armando Domiciano de Souza, Yves Fremat, Farok Vakli. ACHERNAR CAN
BE A DIFFERENTIAL ROTATOR. Societe Francaise d’Astronomie et d’Astrophysique, 2005,
France. 2005. <hal-00008527>
HAL Id: hal-00008527
https://hal.archives-ouvertes.fr/hal-00008527
Submitted on 7 Sep 2005
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SF2A 2005
F. Casoli, T. Contini, J.M. Hameury, and L. Pagani (eds)
ACHERNAR CAN BE A DIFFERENTIAL ROTATOR
Zorec, J.1, Domiciano de Souza, A.2, Fre´mat, Y.3 and Vakili, F.2
Abstract. We take advantage of interferometric measurements of Achernar
to inquire on its internal rotational law. The reinterpretation of interferome-
tric data and the use of fundamental parameters corrected for gravitational
darkening effects and models of 2D-models of internal stellar structures, lead
us to the conclusion that the star could not be a rigid, near critical, rotator
but a differential rotator with the core rotating ∼ 3 times faster than the
surface.
1 Introduction
Achernar (α Eri, HD 10144) is the brightest Be star in the sky and as such it
has deserved detailed observations. Of particular interest are the λ 2.2 µm inter-
ferometric observations carried out by Domiciano de Souza et al. (2003), which
together with spectroscopic and spectrophotometric measurements enable us to
inquire on what internal rotational law can account for its geometrical deforma-
tion.
2 Method and models
Achernar is an active Be star, even during a quiescent or apparent non-emission
phase. The star had a small Hα emission at the epoch of interferometric measure-
ments (Vinicius et al. 2005), which imnplies a flux excess at λ 2.2 µm of rougly 20%
produced by a circumstellar disc. We may assume, however, that interferometry
carried in the stellar polar directions are free from disc perturbations. The actual
stellar equatorial radius can then be estimated requiring apparent area conserva-
tion: R2sph = R
app
pole(i)×Requat, where Rsph is the radius of the equivalent circular
stellar disc derived by comparing measured fluxes with model atmospheres. The
relation between the apparent i-dependent interferometric polar radius Rapppole(i)
and the true stellar polar radius Rpole is given by:
1 Institut d’Astrophysique de Paris, UMR7095 CNRS, Universite´ Pierre & Marie Curie
2 Laboratoire Univ. d’Astroph. de Nice (LUAN), UMR 6525 CNRS
3 Royal Observatory of Belgium
c© EDP Sciences 2005
2 SF2A 2005
Rapppole(i)/Requat = {1−
[
1− (Rpole/Requat)
2]2 sin2 i}1/2 (2.1)
where i is the inclination angle. The use of the V sin i parameter corrected for
gravitational darkening effects enables us to discuss the stellar rotational distortion
as a function of the internal rotation law and of the amount of stored angular
momentum. Inspired by model predictions of rotating stars by Meynet & Maeder
(2000), we adopt the internal rotational law given by:
Ω(r)/Ωcore = 1− p.e
−a.rb (2.2)
where p determines the Ωcore/Ωsurf ratio; b determines the steepness of the drop
from Ωcore to Ωsurf ; a depends on the size or the core. 2D-models of stellar interiors
were used to determine the changes of the polar and equatorial radii as a function
of the energy ratio τ = kinetic energy/|gravitational potential energy| for a star
with mass M = 6.7± 0.4M⊙ near the end of the MS phase (Vinicius et al. 2005).
3 Results
The use of (2.1) and Requat/R⊙ against τ enables us to compare the observed
(Requat/R⊙, Requat/Rpole, V sin i) with the model predicted ones and determine:
p = 0.624± 0.001
τ = 0.014± 0.001
η = 0.69± 0.07
Ve = 308± 16km/s
i = 52o ± 4o
Ωcore/Ωsurf = 2.7


(3.1)
with η = R3eΩ
2
e/GM = ratio of centrifugal to gravity accelerations. The force ratio
is η < 1 which indicates that the stellar equator is not at critical rotation. We
note that for a plain rigid rotation approximation, Achernar would have Ω/Ωcrit =
0.8, which corresponds to ηrigid = 0.66.
It is interesting to note that the energy ratio τ = 0.014 of our differential
rotator is quite similar to the value of critical rigid rotators: τc = 0.015. In the
rigid rotation approximation the star would have τr = 0.011, which implies that
Achernar is possibly storing 30% more rotational energy that it would have if it
were at rigid rotation.
References
Domiciano de Souza, A., Kervella, P., Jankov, S. et al. 2000, A&A, 407, L47
Meynet, G., Maeder. A. 2000, A&A, 361, 101
Vinicius, M.M.F., Zorec, J, Leister, N.V., Levenhagen, R.S. 2005, A&A, submitted


ACHERNAR CAN BE A DIFFERENTIAL ROTATOR

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ACHERNAR CAN BE A DIFFERENTIAL ROTATORJuan Zorec, Armando Domiciano de Souza, Yves Fremat, Farok VakliTo cite this version:Juan Zorec, Armando Domiciano de Souza, Yves Fremat, Farok Vakli. ACHERNAR CANBE A DIFFERENTIAL ROTATOR. Societe Francaise d’Astronomie et d’Astrophysique, 2005,France. 2005. <hal-00008527>HAL Id: hal-00008527https://hal.archives-ouvertes.fr/hal-00008527Submitted on 7 Sep 2005HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00008527, version 1 - 7 Sep 2005SF2A 2005F. Casoli, T. Contini, J.M. Hameury, and L. Pagani (eds)ACHERNAR CAN BE A DIFFERENTIAL ROTATORZorec, J.1, Domiciano de Souza, A.2, Fre´mat, Y.3 and Vakili, F.2Abstract. We take advantage of interferometric measurements of Achernarto inquire on its internal rotational law. The reinterpretation of interferome-tric data and the use of fundamental parameters corrected for gravitationaldarkening effects and models of 2D-models of internal stellar structures, leadus to the conclusion that the star could not be a rigid, near critical, rotatorbut a differential rotator with the core rotating ∼ 3 times faster than thesurface.1 IntroductionAchernar (α Eri, HD 10144) is the brightest Be star in the sky and as such ithas deserved detailed observations. Of particular interest are the λ 2.2 µm inter-ferometric observations carried out by Domiciano de Souza et al. (2003), whichtogether with spectroscopic and spectrophotometric measurements enable us toinquire on what internal rotational law can account for its geometrical deforma-tion.2 Method and modelsAchernar is an active Be star, even during a quiescent or apparent non-emissionphase. The star had a small Hα emission at the epoch of interferometric measure-ments (Vinicius et al. 2005), which imnplies a flux excess at λ 2.2 µm of rougly 20%produced by a circumstellar disc. We may assume, however, that interferometrycarried in the stellar polar directions are free from disc perturbations. The actualstellar equatorial radius can then be estimated requiring apparent area conserva-tion: R2sph = Rapppole(i)×Requat, where Rsph is the radius of the equivalent circularstellar disc derived by comparing measured fluxes with model atmospheres. Therelation between the apparent i-dependent interferometric polar radius Rapppole(i)and the true stellar polar radius Rpole is given by:1 Institut d’Astrophysique de Paris, UMR7095 CNRS, Universite´ Pierre & Marie Curie2 Laboratoire Univ. d’Astroph. de Nice (LUAN), UMR 6525 CNRS3 Royal Observatory of Belgiumc© EDP Sciences 20052 SF2A 2005Rapppole(i)/Requat = {1−[1− (Rpole/Requat)2]2 sin2 i}1/2 (2.1)where i is the inclination angle. The use of the V sin i parameter corrected forgravitational darkening effects enables us to discuss the stellar rotational distortionas a function of the internal rotation law and of the amount of stored angularmomentum. Inspired by model predictions of rotating stars by Meynet & Maeder(2000), we adopt the internal rotational law given by:Ω(r)/Ωcore = 1− p.e−a.rb (2.2)where p determines the Ωcore/Ωsurf ratio; b determines the steepness of the dropfrom Ωcore to Ωsurf ; a depends on the size or the core. 2D-models of stellar interiorswere used to determine the changes of the polar and equatorial radii as a functionof the energy ratio τ = kinetic energy/|gravitational potential energy| for a starwith mass M = 6.7± 0.4M⊙ near the end of the MS phase (Vinicius et al. 2005).3 ResultsThe use of (2.1) and Requat/R⊙ against τ enables us to compare the observed(Requat/R⊙, Requat/Rpole, V sin i) with the model predicted ones and determine:p = 0.624± 0.001τ = 0.014± 0.001η = 0.69± 0.07Ve = 308± 16km/si = 52o ± 4oΩcore/Ωsurf = 2.7(3.1)with η = R3eΩ2e/GM = ratio of centrifugal to gravity accelerations. The force ratiois η < 1 which indicates that the stellar equator is not at critical rotation. Wenote that for a plain rigid rotation approximation, Achernar would have Ω/Ωcrit =0.8, which corresponds to ηrigid = 0.66.It is interesting to note that the energy ratio τ = 0.014 of our differentialrotator is quite similar to the value of critical rigid rotators: τc = 0.015. In therigid rotation approximation the star would have τr = 0.011, which implies thatAchernar is possibly storing 30% more rotational energy that it would have if itwere at rigid rotation.ReferencesDomiciano de Souza, A., Kervella, P., Jankov, S. et al. 2000, A&A, 407, L47Meynet, G., Maeder. A. 2000, A&A, 361, 101Vinicius, M.M.F., Zorec, J, Leister, N.V., Levenhagen, R.S. 2005, A&A, submitted

ACHERNAR CAN BE A DIFFERENTIAL ROTATOR

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