White Dwarf Evolution Code (WDEC)

• WDEC is a Fortran-based computational tool that builds self-consistent static models of cooling white dwarfs and computes their adiabatic nonradial pulsation spectra.
• It integrates state-of-the-art microphysics, using flexible modular structures and iterative relaxation methods to achieve high numerical convergence and detailed interior profiles.
• The code enables comprehensive parameter space grids and precise asteroseismic fitting to infer white dwarf internal structure, envelope stratification, and cooling rates.

The White Dwarf Evolution Code (WDEC) is a Fortran-based computational tool designed to construct self-consistent static models of cooling white dwarfs and to compute their linear, adiabatic nonradial pulsation spectra. Widely adopted in the field of white dwarf asteroseismology, WDEC balances computational speed, robust numerical relaxation, and physically detailed microphysics with a flexible modular structure. Modern versions incorporate state-of-the-art microphysics inputs, specialized boundary treatments, and interfacing capabilities for external stellar evolution codes such as MESA. WDEC supports both detailed parameter-space grid explorations and direct asteroseismic fitting, making it an essential instrument for probing internal white dwarf structure and evolutionary cooling.

1. Historical Context and Development

Development of WDEC began in the late 1960s, spearheaded by Don Lamb, who unified the Rochester interior evolution code (Kutter & Savedoff 1969) with white dwarf envelope models by Gilles Fontaine. The foundational instrument paper appeared in 1975 (Lamb & van Horn). The code’s role in white dwarf asteroseismology was established in the 1970s and 1980s (notably by Don Winget), continuing with major maintenance and modularization at the University of Texas at Austin (notably by Wood) (Bischoff-Kim et al., 2018). Milestones included implementation of improved mesh partitioning routines and the coupling of oscillation modules to genetic algorithms for parameter searches (Metcalfe et al. 2000, 2003).

Modern revisions (e.g., Bischoff-Kim & Montgomery 2018) restructured WDEC into a Fortran 90 modular package, replaced original microphysics with MESA-linked routines, and introduced simplified input/output processes optimized for contemporary computer systems. WDEC has been released as open-source on GitHub and archived on Zenodo (Bischoff-Kim et al., 2018).

2. Underlying Physical and Numerical Framework

WDEC constructs fully consistent, static, spherically symmetric models by solving the standard 1D stellar structure equations in Lagrangian mass coordinate $m$:

• Hydrostatic equilibrium:

$$\displaystyle \frac{dP}{dr} = - \frac{G m(r) \rho(r)}{r^2}$$

• Mass conservation:

$\displaystyle \frac{dm}{dr} = 4\pi r^2 \rho$

• Energy conservation:

$$\displaystyle \frac{dL}{dr} = 4\pi r^2 \rho \varepsilon$$ Typically, $\varepsilon$ includes neutrino and axion cooling, but nuclear burning is neglected in degenerate cores.

• Energy transport:

The radiative gradient is $$\displaystyle \frac{dT}{dr} = -\frac{3 \kappa \rho L}{16\pi a c T^3 r^2}$$; convective regions use mixing-length theory (MLT), with $\alpha$ calibrated via non-linear and 3D hydrodynamical studies.

EOS and opacity inputs are taken directly from MESA modules, providing consistency across OPAL, SCVH, PC, and HELM tables, with compositional- and phase-continuity at the core-envelope interface. Neutrino emission in the core is fully considered (photoneutrino, pair, plasmon, bremsstrahlung, recombination sources); axion emission is optional (Bischoff-Kim et al., 2018).

The code employs a relaxation scheme (Newton–Raphson-type) to iterate over discretized mesh points (typically 100–1000+) until global model variables converge, with standard thresholds $\sim10^{-6}$ in $P$, $T$, and $L$. Adaptive meshing refines zoning at chemical interfaces and near the base of convection zones (Bischoff-Kim et al., 2018, Chen et al., 2013).

3. Chemical Profiles, Diffusion, and Core Interface

Chemical composition profiles are a central aspect of WDEC’s configuration. The core is typically initialized as C/O (or pure-He for appropriate cases) with envelope stratification defined by user-specified helium and hydrogen layer masses ($\log(M_{\mathrm{He}}/M_*)$, $\log(M_{\mathrm{H}}/M_*)$).

For enhanced realism, WDEC imports detailed core composition profiles from MESA evolutionary tracks. This is accomplished by:

• Extracting C (and O) abundance as a function of mass fraction $q=m/M_*$.
• Splicing MESA C/O profiles onto the WDEC grid, typically by piecewise-linear interpolation between MESA anchor points (Chen et al., 2013, Chen, 2020).
• Excluding configurations involving three simultaneous core species (e.g., He/C/O); if present, He is eliminated within the core region for compatibility (Chen, 2020).

Element diffusion is a critical feature for the evolution of the helium and hydrogen transition zones. WDEC integrates the Burgers equations (multi-species force-balance equations for a plasma), using the approach of Thoul, Bahcall & Loeb (1994), and, in some studies, updates the Coulomb interaction to include Debye screening rather than a pure $1/r$ potential. This enhancement impacts the diffusion coefficients $D_i$ and composition profiles near interfaces (Chen, 2020).

Boundary conditions on composition are set by requiring the mass fractions at the interface match the input layer masses. For equilibrium solutions, the system of discretized Burgers equations is solved simultaneously with the structure equations using iterative relaxation methods (Chen et al., 2013, Chen, 2020).

4. Oscillation Calculations and Asteroseismic Fitting

Once a cooling sequence reaches the desired $T_{\mathrm{eff}}$, the fixed model structure is used to compute the adiabatic, linear nonradial pulsation modes.

WDEC integrates the linearized system of four first-order ODEs for the Lagrangian displacements ($\xi_r$, $\xi_h$), Eulerian pressure perturbation ($\delta P$), and gravitational potential perturbation ($\delta \Phi$), following the formalisms of Dziembowski or Unno and implemented via modules originating from C. Hansen or S. Li (Bischoff-Kim et al., 2018, Chen et al., 2017, Chen, 2020). The surface boundary condition imposes $\delta P=0$ at the photosphere; regularity is enforced at the center.

Eigenvalues (periods $P_{k,\ell}$) for modes of spherical degree $\ell=1,2$ are identified via root-finding in frequency space, often using shooting or relaxation algorithms with fine frequency tolerance ($\sim10^{-6}$ rad s$^{-1}$). The Brunt–Väisälä frequency $N^2(r)$ and Lamb frequency $L_\ell(r)$ are computed to characterize mode propagation cavities (Chen et al., 2013).

Asteroseismic fitting proceeds by constructing dense grids in the parameter space ($M_*$, $T_{\mathrm{eff}}$, $\log(M_{\mathrm{H}}/M_*)$, $\log(M_{\mathrm{He}}/M_*)$), and for each model comparing computed periods with observed mode periods. Fitting statistics include:

• Mean absolute error:

$$\displaystyle \phi = \frac{1}{n}\sum_{i=1}^{n}\left|P_{\mathrm{mod}}^i-P_{\mathrm{obs}}^i\right| [1310.7464]$$

• Root-mean-square error:

$$\displaystyle \sigma_{\mathrm{RMS}} = \sqrt{\frac{1}{n} \sum_{i=1}^n (P_{\mathrm{obs},i}-P_{\mathrm{cal},i})^2 } [2005.06407]$$

Best-fit models minimize these statistics over the full parameter grid; more sophisticated searches (GA, Levenberg–Marquardt) have been utilized in earlier work but are not universally employed.

WDEC also supports calculation of rates of period change $\dot{P}(k)$ by tracking the same mode in models separated by a fixed temperature interval and dividing period change by the cooling age difference (Chen et al., 2017).

5. Grids, Inputs, and Computational Workflow

WDEC is designed for rapid generation and analysis of large model grids suitable for asteroseismic exploration and inversion. Key configurable parameters and grid ranges (as used in recent studies) include:

Parameter	Typical Range	Step Size
$M_*/M_\odot$	$0.56$–$0.80$	$0.005$–$0.01$
$T_{\mathrm{eff}}$ (K)	$10\,800$–$12\,800$	$10$–$200$
$\log(M_{\mathrm{He}}/M_*)$	$-4.0$ to $-2.0$	$0.1$–$0.5$
$\log(M_{\mathrm{H}}/M_*)$	$-10.0$ to $-4.0$	$0.1$–$1.0$
Mesh points (per model)	$\sim 300$–$1000+$	—
Convergence threshold	$|dX| < 10^{-6}$ (varies by variable)	—

Parameter files (Fortran namelist format) specify physical parameters, mesh resolution, composition profiles, and convection treatment. Code builds via standard GNU make and can be scripted for high-throughput grid runs (Bischoff-Kim et al., 2018).

Output files report radial profiles ($r, m(r), P, T, \rho, L, N^2$, abundances), full pulsation spectra, eigenfunctions, and diagnostic logs. No time-dependent nuclear burning or mass loss is modeled; all models are “static” snapshots at chosen $T_{\mathrm{eff}}$ (Chen, 2020).

6. Impact, Applications, and Current Limitations

WDEC is central to asteroseismic inferences of white dwarf structure, including envelope stratification, core composition, convection parameters, and cooling rates. It has enabled:

• Quantitative fits to observed $g$-mode period spectra of pulsating DAV and DBV stars.
• Precise determination of hydrogen and helium envelope masses in ZZ Ceti stars (Bischoff-Kim et al., 2018).
• Seismological inference of core C/O profiles via MESA integration (Chen et al., 2013, Chen, 2020, Chen et al., 2017).
• Computation of theoretical $\dot{P}$ values that match long-baseline O–C measurements for benchmark targets such as G117–B15A and R548 (Chen et al., 2017).

Limitations include the inability to model three-species (He/C/O) cores, lack of full evolutionary mass-loss coupling, and the static nature (no time-stepping of thermal or diffusive evolution except in envelope/interface layers). The accuracy of interface diffusion is sensitive to the assumed Coulomb potential, with studies demonstrating improved fits when screened/Debye potentials are included (Chen, 2020). Mesh requirements and convergence properties necessitate care at extremes of composition or temperature.

7. Code Availability and Community Adoption

WDEC is openly available at https://github.com/kim554/wdec, accompanied by a user manual and example parameter files. Source code and documentation are periodically updated, reflecting new microphysics modules and structural improvements (Bischoff-Kim et al., 2018). The code base is widely utilized for observationally-driven parameter studies, benchmarking, and as a reference implementation for testing novel microphysics or oscillation analysis methods.

WDEC’s role as a rapid, robust generator of static white dwarf structural and pulsation models ensures its continued utility for time-domain space missions (e.g., Kepler, K2) and ground-based global campaigns, enabling precise reconstruction of white dwarf internal stratification and evolution.