Theory and modelling

This part documents the physics and the numerical methods of each module, in roughly the order in which they appear in the solution loop.

• The coupled problem and the solution loop
  • The two physical structure equations
  • Energy: radiative equilibrium
  • The solution loop
  • Convergence criteria
• Radiative transfer
  • Method overview
  • Geometry: impact parameters and tangent rays
  • Formal solution on a ray (Feautrier)
  • The moment system for \(J_\nu\)
  • Taylor vs. spline discretisation
  • Two flux definitions and why they coexist
  • Outputs of the RT module
  • The linearised moment operator
• Radiative-equilibrium temperature correction
  • Why two temperatures
  • Unsöld–Lucy correction
  • The decisive subtlety: the ratio form of the constraint
  • Full-linearisation (Newton) correction
  • Local RE alone is not enough: the flux-constancy term
  • Two-phase corrector and coupled-loop stability
  • Anderson acceleration
  • Accuracy achieved
• Gas-phase chemistry (FastChem)
  • Interface
  • Carbon depletion coupling
  • Equation of state
• Dust formation (Gail & Sedlmayr)
  • The moment equations
  • Nucleation rate
  • Carbon depletion in a single causal sweep
  • Outputs and dust opacity
  • Coupling to the hydrodynamics
• Opacities (transport coefficients)
  • Gas opacities
  • Spectral grid and sampling
  • Dust opacities
• Wind hydrodynamics
  • The radiative acceleration
  • The Melia \(\Phi\) transform
  • Critical point and the mass-loss eigenvalue
  • Shooting solution (default)
  • The grey (Lucy) starting model
  • Optional: the Henyey global-Newton structure solver
  • Prescribed vs. eigenvalue mass-loss rate
• Numerical methods and robustness
  • Fixed-point structure and convergence tests
  • Adaptive per-layer under-relaxation
  • The per-step cap
  • Anderson acceleration
  • Movable (adaptive) radial grid
  • Reproducibility
  • Performance notes

Notation

Throughout, \(r\) is the radial coordinate, \(R_\star\) the stellar reference radius, and a subscript \(\star\) denotes a stellar quantity. The radiation field is described by its frequency-dependent moments — mean intensity \(J_\nu\), Eddington (first) flux \(H_\nu\), and second moment \(K_\nu\) — and the variable Eddington factor \(f_\nu=K_\nu/J_\nu\). The extinction (total), absorption and scattering coefficients are \(\chi_\nu=\kappa_\nu+\sigma_\nu\). The Planck function is \(B_\nu(T)\). Dust quantities use the Gail & Sedlmayr moments \(K_j\) (with \(K_3\) the condensed volume per hydrogen nucleus) and the nucleation rate \(J_\star\).

Equation numbers such as (2.59) refer to the diploma thesis of D. Kitzmann, which is the primary reference for the radiative-transfer scheme.