Numerical Model Atmospheres (Hubeny & Mihalas 16, 17) Equations Hydrostatic Equilibrium Temperature Correction Schemes

Summary: Basic Equations Corresponding State Parameter Radiative transfer Mean intensities, Jν Radiative equilibrium Temperature, T Hydrostatic equilibrium Total particle density, N Statistical equilibrium Populations, ni Charge conservation Electron density, ne

Physical State Recall rate equations that link the populations in each ionization/excitation state Based primarily upon temperature and electron density Given abundances, ne, T we can find N, Pg, and ρ With these state variables, we can calculate the gas opacity as a function of frequency

Hydrostatic Equilibrium Gravitational force inward is balanced by the pressure gradient outwards, Pressure may have several components: gas, radiation, turbulence, magnetic μ = # atomic mass units / free particle in gas

Column Density Rewrite H.E. using column mass inwards (measured in g/cm2), “RHOX” in ATLAS Solution for constant T, μ (scale height):

Gas Pressure Gradient Ignoring turbulence and magnetic fields: Radiation pressure acts against gravity (important in O-stars, supergiants)

Temperature Relations If we knew T(m) and P(m) then we could get ρ(m) (gas law) and then find χν and ην Then solve the transfer equation for the radiative field (Sν = ην / χν ) But normally we start with T(τ) not T(m) Since dm = -ρ dz = dτν / κν we can transform results to an optical depth scale by considering the opacity

ATLAS Approach (Kurucz) H.E. Start at top and estimate opacity κ from adopted gas pressure and temperature At next optical depth step down, Recalculate κ for mean between optical depth steps, then iterate to convergence Move down to next depth point and repeat

Temperature Distributions If we have a good T(τ) relation, then model is complete: T(τ) → P(τ) → ρ(τ) → radiation field However, usually first guess for T(τ) will not satisfy flux conservation at every depth point Use temperature correction schemes based upon radiative equilibrium

Solar Temperature Relation From Eddington-Barbier (limb darkening) τ0 = τ(5000 Å)

Rescaling for Other Stars Reasonable starting approximation

Temperature Relations for Supergiants Differences small despite very different length scales

Including line opacity or line blanketing Other Effects on T(τ) Including line opacity or line blanketing Convection

Temperature Correction Schemes “The temperature correction need not be very accurate, because successive iterations of the model remove small errors. It should be emphasized that the criterion for judging the effectiveness of a temperature correction scheme is the total amount of computer time needed to calculate a model. Mathematical rigor is irrelevant. Any empirically derived tricks for speeding convergence are completely justified.” (R. L. Kurucz)

Some T Correction Methods Λ iteration scheme Not too good at depth (cf. gray case)

Some T Correction Methods Unsöld-Lucy method similar to gray case: find corrections to the source function = Planck function that keep flux conserved (good for LTE, not non-LTE) Avrett and Krook method (ATLAS) develop perturbation equations for both T and τ at discrete points (important for upper and lower depths, respectively); interpolate back to standard τ grid at end (useful even when convection carries a significant fraction of flux)

Some T Correction Methods Auer & Mihalas (1969, ApJ, 158, 641) linearization method: build in ΔT correction in Feautrier method Matrices more complicated Solve for intensities then update ΔT