Source-linked AI summary
ASTEC -- the Aarhus STellar Evolution Code
J. Christensen-Dalsgaard
TL;DR
ASTEC addresses the need for stellar models usable with observed oscillation frequencies and asteroseismic fitting. It combines stellar evolution with adiabatic oscillation calculations and flexible microphysics, and has proved useful with reasonably reliable solar results within standard solar modelling. Important limitations remain in the nuclear network, diffusion treatment, convective-core calculations, and general usability.
Problem
ASTEC addresses the need for stellar models that can be compared with observed oscillation frequencies and used in asteroseismic fitting.
Method
ASTEC combines stellar evolution and adiabatic oscillation calculations in a flexible code with modular microphysics and a single-subroutine fitting interface.
Results
ASTEC has proved useful in applications, with solar results described as reasonably reliable within standard solar modelling.
Takeaways & Limitations
Integrated frequency computation makes ASTEC usable within larger fitting calculations, while its microphysics flexibility supports varied stellar-model applications.
Takeaways & Limitations
Further development is required because the nuclear network is limited, diffusion is fully coupled only for helium, convective-core treatment has serious problems, and general release is impractical.
Abstract
from arXiv · showhide
The Aarhus code is the result of a long development, starting in 1974, and still ongoing. A novel feature is the integration of the computation of adiabatic oscillations for specified models as part of the code. It offers substantial flexibility in terms of microphysics and has been carefully tested for the computation of solar models. However, considerable development is still required in the treatment of nuclear reactions, diffusion and convective mixing.
1 Introduction
ASTEC began in 1974 as a solar-stability modelling effort and evolved toward helioseismic modelling and broader stellar applications. Its asteroseismic version integrates stellar evolution and adiabatic-frequency calculations in a single callable package, though the full code remains too complex for general release.
- The code was further developed with more realistic physics as helioseismic data increased in quality and extent, including diffusion and settling.
- Extensions support evolution of stars other than the Sun, including convective cores, core overshoot, and red-giant evolution.
- For asteroseismic fitting, a subroutine version combines evolution and adiabatic oscillation-frequency computation through a simple calling structure.
- The combined package is relatively straightforward to install and runs on various platforms, but its complexity makes general release inadvisable.
- ASTEC originated around 1974 to improve solar equilibrium models and was soon extended to compare models with observed solar oscillation frequencies.
2 Equations and numerical scheme
ASTEC formulates stellar evolution on a logarithmic mass-fraction mesh and solves coupled structure, composition, and diffusion equations through an implicit Henyey-type numerical scheme. Adaptive mesh redistribution, timestep control, and Newton-Raphson iteration support evolving stellar models, with different time discretizations selected for differing timescales.
- 2.1 Formulation of the equations: ASTEC uses x = log10 q as the independent variable, where q is the mass fraction interior to a point relative to the photospheric mass.
- 2.2 Boundary conditions: Boundary conditions are imposed at the inner and outer mesh boundaries, with the photosphere defined by T = Teff and an atmospheric pressure matching condition.
- 2.1 Formulation of the equations: The dependent variables include radius, pressure, temperature, luminosity, elemental abundances, and diffusion-related variables.
- 2.3 Numerical scheme: The scheme combines stellar-evolution equations with diffusion equations and supports transformations between thermodynamic variable sets.
- 2.3 Numerical scheme: The equations are discretized on a mesh across successive timesteps and solved with the Newton-Raphson-Kantorovich, or Henyey, scheme.
- 2.3 Numerical scheme: Time-centred differences are typically used for slow processes, whereas fully implicit treatment is used for short-timescale energy-equation derivatives.
- 2.3 Numerical scheme: Each timestep iterates linearized corrections until convergence, while mesh redistribution and timestep selection respond to structural and compositional changes.
3 Microphysics
ASTEC is modular and offers flexible microphysics, including multiple equation-of-state and opacity options, nuclear-reaction treatments, and diffusion prescriptions. The paper identifies important scope limitations: the nuclear network is restricted, diffusion is coupled consistently with nuclear evolution only for helium, and some approximations become questionable outside solar-like conditions.
- ASTEC’s modular design makes replacing equation-of-state and opacity routines relatively straightforward and permits alternative dependent-variable sets.
- 3.1 Equation of state: The code includes the Eggleton equation of state and interpolated MHD and OPAL table options, with thermodynamically consistent features in the original formulation.
- OPAL opacity tables form the basis of the opacity calculation, while interpolation and localized opacity modifications support sensitivity studies.
- 3.3 Nuclear reactions: Nuclear reaction rates use standard reaction-integral approximations with electron screening and a specified treatment of 7Be electron capture.
- 3.3 Nuclear reactions: The nuclear network is relatively limited, with several pp-chain and CNO-cycle species treated by equilibrium assumptions and no pre-main-sequence evolution calculation.
- 3.4 Diffusion and settling: Diffusion and settling use Michaud–Proffitt approximations, but heavy elements are approximated as fully ionized 16O and selective radiative levitation is not included.
- 3.4 Diffusion and settling: Diffusion and settling are coupled consistently to nuclear evolution only for helium; diffusion is neglected for the remaining network elements.
4 Treatment of convection
ASTEC treats convective composition, core evolution, and overshoot through configurable approximations. Convective envelopes are enforced as chemically uniform, while convective-core treatment remains a concern because composition discontinuities and incomplete diffusion treatment complicate the calculation.
- Convective envelopes are made chemically uniform by using a very high diffusion coefficient in convective regions.
- The treatment of convective cores remains an area of active development, partly because diffusion of all elements is not properly handled.
- The convective-core formulation uses a reaction rate averaged over the core and includes terms sensitive to composition discontinuities and diffusion.
- Growing convective cores can produce a hydrogen-abundance discontinuity at the core edge when diffusion is neglected.
- Overshoot options allow fully mixed regions extending from convective cores or envelopes over distances defined using αovHp or αov min(rcc,Hp).
- Overshoot regions may be treated as either adiabatically or radiatively stratified, with a more elaborate envelope prescription also available.
5 Implementation details
ASTEC supports detailed model output and integrates stellar evolution with adiabatic oscillation calculations for asteroseismic fitting, but its complexity limits general release.
- ASTEC can output detailed models as emdl, amdl, or gong files, with emdl files retaining calculation variables and input parameters.
- A single subroutine call can perform the evolution and adiabatic oscillation calculations while passing intermediate products internally to fitting codes.
- The code is distributed as a complete tar package with setup and build support, and has been implemented on multiple platforms.
- Its complexity and inadequate documentation make release for general use unrealistic.
6 Further developments
ASTEC’s treatment of convective-core composition and boundaries remains under development, with alternative hydrogen gradients affecting pulsational properties and serious diffusion-related failures.
- ASTEC has significant deficiencies, including a restricted nuclear network, incomplete diffusive treatment, and unresolved convective-region boundaries.
- The usual convective-core treatment creates a discontinuous hydrogen profile, whereas an alternative gradient sets ∇rad −∇ad = 0 in the core’s outer parts.
- The alternative boundary treatment changes only a small core region, so global effects are modest but pulsational effects may influence g modes or interface modes near the boundary.
- A convergence issue leaves ∇rad −∇ad positive at the identified convective boundary.
- Poor semiconvection treatment and mesh resolution can produce much larger frequency differences between oscillation codes, especially for g-mode-like modes near the core edge.
7 Concluding remarks
ASTEC has supported solar helioseismic applications and diverse development efforts, but increasingly precise asteroseismic data require continued testing and development.
- ASTEC has proved useful in several applications, and its solar results appear reasonably reliable within standard solar modelling.
- Application to increasingly accurate and detailed asteroseismic data will require further development.
- Testing through the ESTA collaboration and HELAS Coordination Action is described as valuable for further development.
- The paper acknowledges contributions from many researchers to ASTEC’s long-term development.
A Treatment of diffusion and settling
ASTEC computes diffusion and settling using coefficients and velocities formulated for trace elements, with collision integrals and hydrogen-abundance gradients incorporated explicitly.
- The diffusion and settling expressions are largely based on Michaud and Proffitt, with minor modifications.
- The diffusion coefficient is specified within the adopted cgs-unit formalism.
- For trace element k, the formulation uses its charge and atomic mass together with collision integrals involving hydrogen and helium.
- Hydrogen-abundance gradients enter trace-element diffusion velocities through an expression in terms of the transformed abundance variable ˜Y1.
B Mixing-length formulation
Convective-region temperature gradients are computed with mixing-length theory in the Gough formulation, using geometrical parameters and a mixing length tied to the pressure scale height. The solution for the auxiliary variable Y uses asymptotic forms in extreme-A regimes with very small relative error.
- Convective-region gradients use Vitense mixing-length theory in the formulation given by Gough.
- The mixing length is set to ℓ = αMLHp, while η and Φ are constant geometrical quantities related to convective-cell aspect ratio.
- The computation adopts η = 2/9 and Φ = 2, yielding the Böhm-Vitense expressions.
- The general solution is formulated through Y, defined as the positive root of the relevant equation, with asymptotic expressions available for large and small A.
- The asymptotic expressions are used for A ≥ 15 and A ≤ 10^-5, where their relative differences from the full solution are below 5×10^-7.