ESTER Code: 2D Stellar Evolution in Rotation

• ESTER is a two-dimensional stellar structure code that models the steady-state hydrodynamics of rapidly rotating stars, accounting for oblate geometries and latitude-dependent surface properties.
• It employs advanced numerical methods like spectral discretization with Chebyshev polynomials and spherical harmonics, using Newton iterations to ensure rapid convergence.
• The code predicts observable features such as gravity darkening, differential rotation, and asteroseismic signatures while being validated against traditional 1D models.

The Evolution STEllaire en Rotation (ESTER) code is the first open-source, two-dimensional stellar structure code developed to calculate, from first principles, the steady-state hydrodynamic structure and large-scale flows of rapidly rotating stars. Unlike traditional one-dimensional stellar evolution codes, ESTER solves for the full axisymmetric 2D structure, enforcing self-consistency between the distorted oblate geometry, the latitude-dependent surface parameters, and the internal flows including differential rotation and meridional circulation. It is designed for massive and intermediate-mass stars where rotational effects are essential, and its application spans model benchmarking, interpretation of interferometric and photometric data, and direct prediction of observational signatures in fast-rotating early-type stars (White et al., 24 Sep 2025).

1. Mathematical Formalism and Physical Content

ESTER computes the steady-state solution to the full set of hydrodynamical stellar structure equations in two dimensions. The core equations include:

• Poisson’s equation for gravity: $\nabla^2 \phi = 4\pi G \rho$
• Steady-state momentum equation including centrifugal and viscous effects:

$$\rho\, \mathbf{v}\cdot\nabla \mathbf{v} = -\nabla p -\rho\nabla\phi + \mathbf{F}_v$$

• Mass continuity: $\nabla\cdot(\rho \mathbf{v}) = 0$
• Energy (or entropy) conservation, including radiative and convective (if present) terms:

$$F_{\mathrm{heat,tot}} = -\chi \nabla T + F_\mathrm{conv}$$

The code replaces the spherically symmetric treatment with an oblate, spheroidal coordinate grid to accommodate the deformation from rapid rotation. Centrifugal effects are incorporated by modifying the effective gravitational potential: $$\nabla P = -\rho\,\nabla\left(\Phi_{\rm grav} - \frac{1}{2}\Omega^2 s^2\right)$$ where $s$ is the distance to the rotation axis and $\Omega$ is the local angular velocity (Mombarg et al., 16 Jan 2024). Thermodynamic and opacity parameters are taken from established tabulations (e.g., OPAL), and microphysics such as the equation of state and energy generation/reactions are treated as in advanced 1D codes.

2. Numerical Methods and Spectral Discretization

ESTER employs spectral element methods well-suited to smooth (infinitely differentiable) solutions required for high-precision 2D modeling. The domain is subdivided into multiple spheroidal shells (“multi-domain” approach), and fields are expanded as:

• Chebyshev polynomials radially (on Gauss–Lobatto points)
• Spherical harmonics latitudinally (on Gauss–Legendre grids)

The nonlinear system is solved primarily through Newton iterations, with the Jacobian of the whole set (including mapping parameters due to moving free boundaries) constructed for each step. Linear systems within Newton’s method are handled with an efficient hybrid strategy: block LU preconditioning followed by Conjugate Gradient Squared (CGS) iterative solvers, typically requiring only two or three CGS cycles for convergence (Rieutord et al., 2016).

Oblate geometry is implemented via nonlinear coordinate mapping, mapping the computational $(\zeta,\theta)$ grid onto the physical $(r,\theta)$ coordinates so that the surface ($\zeta=1$) matches the actual distorted stellar surface. The mapping ensures regularity at the center and provides domain boundaries for sharp transitions in microphysics (e.g., at the convective core boundary).

3. Large-Scale Flows and Differential Rotation

One defining feature of ESTER is the direct calculation of differential rotation and meridional circulation arising from baroclinicity (i.e., misalignment of pressure and density gradients). These flows are solved self-consistently via the axisymmetric momentum equation, for which the angular momentum evolution can be written (in the presence of viscosity) as: $$s^2\Omega + \mathbf{v}\cdot \nabla (s^2\Omega) = \frac{1}{\rho} \nabla\cdot\left(\rho\,\nu\, s^2\,\nabla\Omega\right)$$ where $\nu$ is a phenomenological viscosity encapsulating unresolved turbulent stresses. The envelope generally exhibits latitudinal shear (shellular rotation), while the convective core approaches cylindrical rotation, and Stewartson layers (thin shear zones) form near the tangent cylinder to the core (Rieutord et al., 2012, Rieutord, 2013).

These flows enable ESTER to model the redistribution of angular momentum, the advective and diffusive mixing of chemical species, and the emergence of global-scale circulations observed or inferred in early-type stars.

4. Modeling Gravity Darkening, Oblateness, and Observable Signatures

A primary astrophysical application of ESTER is the prediction of surface observables arising from rapid rotation:

• Gravity Darkening: The code naturally produces latitude-dependent effective temperatures and fluxes. Unlike the classical von Zeipel law ($T_{\rm eff} \propto g_{\rm eff}^{1/4}$), ESTER finds deviations, particularly in the equatorial region where the flux can be underestimated by up to a factor of three if the traditional power law is applied (Rieutord, 2013, White et al., 24 Sep 2025).
• Oblateness/Shape: The equatorial bulge is computed self-consistently, with flattening $(1 - R_p/R_e)$ values reaching 0.2–0.3 for near-critical rotators. Radii and radiative fluxes can be extracted at any latitude.
• Spectro-interferometry: The local $T_{\rm eff}$, radius, and gravity can be mapped on the surface and convolved with stellar atmosphere models (e.g., PHOENIX) to synthesize intensity profiles, line shapes, and interferometric visibilities (Bouchaud et al., 2019, Lazzarotto et al., 2023).
• Asteroseismology: By feeding 2D structure (especially the rotation profile and oblateness) into pulsation codes, mode frequencies and rotationally split multiplets can be directly modeled, including for $\delta$ Scuti and $\beta$ Cephei stars (Reese et al., 2020, Mombarg et al., 2023).

The code’s output includes all global and local parameters needed for forward modeling of multi-technique stellar data.

5. Comparison to 1D Codes and Model Validation

ESTER has been systematically benchmarked against state-of-the-art 1D evolutionary codes such as MESA:

• In the non-rotating or slow-rotation regime, 1D and 2D models give very similar core density, temperature, and luminosity profiles (discrepancies typically within a few percent).
• At high rotational velocities, 1D codes (even those using “shellular” approximations) cannot capture the true latitudinal variation or large-scale flow patterns. ESTER’s more luminous and hotter core by up to ~5% compared to 1D codes is attributed to its more physically accurate treatment of rotational distortion and internal flows (White et al., 24 Sep 2025).
• Gravity darkening, differential rotation, and meridional flow predictions made by ESTER have been verified observationally in interferometrically resolved stars such as Altair, Vega, and others (Bouchaud et al., 2019).
• For rapidly rotating massive stars, 2D evolutionary tracks computed with ESTER show that only low- to intermediate-mass stars can reach critical rotation during MS evolution, a prediction consistent with the fraction and mass dependence of Be star phenomena (Mombarg et al., 16 Jan 2024).

6. Code Distribution, Installation, and Student Usage

ESTER is distributed as open-source software (e.g., version 1.1.0rc2), with documentation on installation and operation tailored for university clusters and student users (White et al., 24 Sep 2025). The recommended workflow includes:

1. Compilation with the appropriate Fortran/C/C++ compilers, linking FFT and HDF5 libraries, with configuration for local cluster architecture.
2. Construction of 1D models for initialization.
3. Progressive increase of input rotation rate to build up converged 2D models.
4. Use of utility programs for converting model outputs into standard visualization formats for analysis with tools like ParaView.

The code provides sample input templates for physical and numerical parameters (mass, hydrogen content, angular velocity, opacities, EOS tables), as well as guidance on troubleshooting and further model refinement. Regular comparisons against established codes (MESA) serve to verify physical consistency and numerical stability.

7. Benchmarking Observables and Future Applications

ESTER enables the exploration of stellar phenomena inaccessible to 1D models:

• Quantitative modeling of gravity darkening, radius, and effective temperature as a function of latitude and rotation rate.
• Prediction of observable surface signatures (including the curvature of convective zones at the equator in intermediate-mass stars) (White et al., 24 Sep 2025).
• Construction of large-scale model grids across stellar mass and rotation, informing population studies of cluster stars and the origin of Be stars (Mombarg et al., 16 Jan 2024).
• Support for inversion techniques in asteroseismology using accurate 2D rotational kernels (Reese et al., 2020).

Ongoing extensions of ESTER now include full time-dependent evolution in 2D, with coupling to hydrogen burning, angular momentum redistribution, and chemical evolution along the main sequence (Mombarg et al., 2023). Benchmarks show reproducibility of detailed asteroseismic properties and internal rotation gradients as observed in rapidly rotating early-type pulsators.

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The ESTER code advances stellar modeling by enforcing self-consistent 2D hydrodynamics, accurate treatment of rotation and internal flows, and by providing a framework for interpreting high-resolution observations of fast rotators. Its open-source availability, comprehensive documentation, and alignment with observational diagnostics position it as a reference tool for the astrophysical modeling of rapidly rotating stars (White et al., 24 Sep 2025).