The ambition of this project is to set out a two-dimensional stellar evolution code, which fully takes into account the effects of rotation, at any rate and in a self-consistent way. The difficult, but important point is that rotating stars are spheroidal and are never in hydrostatic equilibrium. They are pervaded by flows everywhere, even in the stably stratified radiative regions. These flows are essentially convective flows in thermally unstable regions (convection zones) and baroclinic flows in the radiative regions. These latter flows are grosso modo a differential rotation and a meridional circulation, with likely some small-scale turbulence. Radiative regions of non-rotating stars experience very little mixing (only due to microscopic phenomena). Rotation induces flows and therefore some mixing, well-known as "rotational mixing". This is a key feature of the evolution of rotating stars. Besides, these stars also oscillate, and astronomers would like to get from the oscillation frequencies some good constraints on the structure of these stars. The foregoing reasons and many others motivated us to construct a two-dimensional model of rotating stars that is kind enough to enlight us on the numerous questions we are wondering about...

The project is lead by Michel Rieutord. The project has been strongly supported by the Programme National de Physique Stellaire (PNPS), the Action spécifique pour la Simulation Numérique en Astrophysique (ASSNA), and the CNRS in general with a two-year post-doc position in 2005-2007.

The ESTER project as such was first funded by ANR in 2009-2014. The main result of this first part was the release of steady models for main sequence early-type stars (mass above 2 solar masses).

The ESTER project is now funded under the ANR-ESRR project (Evolution Stellaire en Rotation Rapide 1/2017--1/2021), which is focusing on secular time evolution of stars and applications to the interpretation of interferometry and asteroseismology data.

The ESTER project is contributing to the PLATO mission by delivrering the first 2D models of solar-type stars.

- The first ESTER Workshop: 10-11-12 June 2014 in Toulouse: see the web page for all the details of the workshop.
- Kick-off meeting of the ANR-ESRR project: 2-3 March 2017 in Toulouse.
A short summary of the meeting here.
Talks have been given by:
- Michel Rieutord Le point sur les modeles ESTER et les challenges du projets
- Bertrand Putigny Les aspects 'Domain specific Language' du projet ESTER
- Daniel Reese Les derniers developpements du code TOP pour l'asterosismologie des étoiles en rotation rapide
- Damien Gagnier Initial conditions of early type stars reaching critical rotation during the main sequence"
- Armando Domiciano de Souza L'interprétation des données d'interférométrie stellaire
- Kévin Bouchaud
*Quelques notes sur la reconstruction du spectre continu de l'étoile theta Sco*

The ESTER code is, at the moment (January 2019), computing the steady state of an isolated early-type star (mass larger than 2 solar masses). The only convective region computed as such is the core where isentropy is assumed. It provides the user with solutions of the partial differential equations, for the pressure, density, temperature, angular velocity and meridional velocity for the whole volume. The angular velocity (differential rotation) and meridional circulation are computed consistently with the structure as a result of the driving by the baroclinic torque.

Input parameters include the mass of the star, its equatorial angular velocity scaled by the critical one, the mass-fraction of hydrogen in the core. Opacity may be computed either analytically from Kramers-like formulae or from OPAL tables (Grevesse and Noels 1993 mixture), or a smoother interpolation from Houdek. Same as for the EOS which can be chosen between analytics (ideal gas+ radiation) and OPAL tables. Other parameters control the numerics.

The code uses spectral methods, both radially and horizontally, with spherical harmonics and Chebyshev polynomials. The iterations follow Newton's algorithm.

The code is object-oriented, written in C++; a python suite allows an easy visualization of the results. While running, Python graphs are displayed to show the evolution of iterations.

The code is a public code under GNU General Public License. Its first public version was hosted on the web as a Google-code and can still be downloaded at the ESTER code, first release The new version has been moved to gitHub and can be downloaded at ESTER. Please refer to the Wiki pages for the installation.

Inspired by our colleagues developing the Pencil Code, we, the ESTER-project community, ask that in publications and presentations the use of the code (or parts of it) be acknowledged with reference to the web site http://userpages.irap.omp.eu/~mrieutord/ESTER.html or (equivalently) to http://ester-project.github.io/ester/. As a courtesy to people involved in the development of the program, we suggest to give appropriate reference to one or several of the following papers (listed here in temporal order):

- Espinosa Lara F. and Rieutord M.
(2007), ``The dynamics of a
fully radiative rapidly rotating star enclosed within a spherical box",
in
*Astron. Astrophys.*, vol. 470, p. 1013-1022 - Rieutord M. and Espinosa Lara F. (2009),
``On the dynamics of a radiative rapidly rotating star", in
*Comm. Asterosismology*, vol. 158., pp.~99-103 - Rieutord M. and Espinosa Lara F. (2013),
``Ab initio modelling of steady rotating stars", in
*Seismology for studies of stellar rotation and convection*, Edts Goupil et al.,*Lecture Notes in Physics*, vol. 865, p. 49, Springer - Espinosa Lara F. and Rieutord M.
(2013), ``Self-consistent 2D models of fast rotating early-type stars",
in
*Astron. Astrophys.*, vol. 552, A35 - Rieutord M., Espinosa Lara F. and Putigny B. (2016),
``An algorithm for computing the 2D structure of fast rotating stars",
in
*J. Computational Phys.*, vol. 318, 277-304

The history of modelling rotating stars in two-dimensions can be found
here:
Modeling rapidly rotating stars in * Proceedings of SF2A Annual meeting
*, p. 501-506

A first investigation of the baroclinic flows in stars:
Rieutord M. (2006),
The dynamics of the radiative envelope of rapidly rotating stars. I. A
spherical Boussinesq model, ref. *Astron. Astrophys.*, vol. 451,
p. 1025-1036

A first "realistic" model of a completely radiative star
encapsulated in bounding sphere may be found in Espinosa
Lara F. and Rieutord M. (2007),
The dynamics of a fully radiative rapidly rotating star enclosed within a
spherical box, ref. *Astron. Astrophys.*, vol. 470, p. 1013-1022

The same but now with a spheroidal boundary which spouses an
isobar surface; in **Rieutord M. and Espinosa Lara F.
(2009),**
On the dynamics of a radiative rapidly rotating star, in * Comm.
Asterosismology*, vol. 158., pp.~99-103

We show here how the gravity darkening (i.e. the latitude variation
of the surface brightness) can be modelled by a single parameter (the
ratio ω of actual rotation to the critical one) instead of two
in previous models (ω and β an exponent, controlling the
relation between effective temperature and effective gravity); see
**Espinosa Lara F. and Rieutord M. (2011),**
Gravity darkening in rotating stars, in * Astron. Astrophys.*,
vol. 533, p. A43

The preceding results have been extended to binary stars and it was
shown in passing that the old result of Lucy (1967) that gravity
darkening exponent in low-mass stars is β=0.08, cannot be used to
model actual gravity darkening on stars with variable surface gravity.
This exponent is only valid for spherically symmetric models and comes
from the surface opacity law. See **Espinosa Lara F. and Rieutord M.
(2012),**
Gravity darkening in binary stars, in * Astron. Astrophys.*,
vol. 547, p. A32

A synthesis of the results dating October 2011 may be visualized in this pdf presentation given by F. Espinosa Lara at Santa Barbara conference on Asteroseismology KITP Conference

A first detailed account on the ESTER code: the physics of the
model, its mathematical description, its numerical technique and the
first results. These models give the characteristics of stars with mass
in the range 3 to 20 solar masses, rotating up to 98% of the break-up
velocities. These models describe steady-state solutions of the 2D
stellar structure solution including for the first time a
self-consistent determination of the differential rotation and the
associated meridian circulations; see
**M. Rieutord and F. Espinosa Lara (2012),**
Ab initio modelling of steady rotating star in
*Seismology for studies of stellar rotation and
convection*, Edts Goupil et al., *Lecture Notes in Physics*,
vol. 835, p. 49, Springer.
and ** Espinosa Lara F. and Rieutord M. (2013)**, Self-consistent
2D-models of fast rotating early-type stars, in * Astron. Astrophys.*,
vol. 552, A35

Some illustrations of the first results:

A representation of the discretization grid: the star is shaded and split into several layers; the outer white region is an outer vacuum domain where only the gravitational potential is computed. | The shape of a n=3/2 polytrope rotating close to the break-up velocity. | The same for a n=3 polytrope; note that as this polytrope is more centrally condensed, its inner regions are less flattened. |

On the top row we see the meridional circulation and differential rotation as predicted by a simplified Boussinesq model of a massive star (see Rieutord 2006 for details). In the bottom row we show the computed differential rotation of a purely radiative star, with realistic physics (see Espinosa Lara and Rieutord 2007). |

Left: Differential rotation in a 5 solar mass star rotating at 70% of the critical angular velocity. Abundances are solar. Note the similarity with the fully radiative star. Inside the convective the rotation is cylindrical (more details in Rieutord and Espinosa Lara 2012). Right: The associated meridian circulations. The cells attached to the convective core are the trace of the Stewartson layers when no viscosity is present in the bulk of the fluid and at finite numerical resolution. Spatial resolution: 400 Chebyshev polynomials+64 spherical harmonics. |

Left: Gravity darkening for an intermediate mass star rotating at 90% of the critical angular velocity. Abundances are solar. Right: The differential rotation of a 30 solar mass ZAMS star rotating at 98% of the critical angular velocity. The ratio of equatorial radius to polar radius is 1.56. |

The Be star (around 6 solar mass) Achernar modeled by ESTER as it is viewed in the sky. |

More to come soon ...