Version 15 June 2007 compiled: 11-11-2009 - Hamburger Sternwarte

• deviations from “local thermodynamic equilibrium”, i.e., the Saha-Boltzmann equation for the
atomic level populations, are allowed for important species as, e.g., H, He, Li, CNO, S, Si, Mg,
Ca, Ti, Co, Fe, for a total of more than 3700 NLTE level and more than 37000 primary NLTE
lines,
• local thermodynamic equilibrium for another ca. 40 elements and their ions.
• local chemical equilibrium for up to ≈ 229 molecules and some of their ions.
1.3 Most important model parameters
• exponent nρ of the density law, typically nρ ≈ 10 . . . 20 for supernovae and nρ ≈ 3 for novae;
• gravity (for stellar models)
• outer pressure Pout or outer density ρout;
• radius R0 at an optical depth of unity, i.e. rout has to be determined so that
� rout
R0
κ dr = 1;
• total radiative luminosity L0 or, equivalent, the “effective temperature” Teff;
• abundances ɛi for all considered elements,
• for NLTE calculations: number of levels of each model atom;
• for line calculations: number of lines to be included;
1.4 Numerical methods
• 1-D Newton or Brent’s method with respect to the electron pressure for the solution of the
coupled (generalized) Saha-Boltzmann equations for all elements (no molecules),
• Multi-D Newton method for the solution of the coupled (generalized) Saha-Boltzmann and
molecular dissociation equations for the ’big’ EOS,
• operator splitting/approximate Λ-operator iteration (OS/ALI) method for the solution of the
special relativistic radiative transfer equation for all wavelength points,
• rate operator formalism for multi-level direct NLTE,
• multi-D Newton-Raphson method (DOME), hybrid or modified Unsöld-Lucy method (OS/ALI)
for the solution of the energy conservation equation in the Lagrangian frame,
• Bulirsch-Stoer method for the solution of auxiliary ordinary differential equations, e.g., for the
energy equation dL/dr = f(r) and for the computation of the radial grid according to
dr/dτ = −1/κ,
• Shooting method for the computation of the outer radius Rout.
1.5 Model equations
• The time independent, special relativistic equation of radiative transfer, with the assumptions:
– spherical symmetry,
– time independence (∂/∂t ≡ 0),
Spherically symmetric, special relativistic equation of radiative transfer
– partial integro-differential equation,
– telegrapher’s equation: boundary value problem in r and initial value problem in λ as long
as the velocities are monotonically increasing.
The equation of radiative transfer
3