Understanding Numerical Model Atmospheres: Equations & Temperature Corrections

1. Numerical Model Atmospheres (Gray 9) EquationsHydrostatic EquilibriumTemperature Correction Schemes

2. Summary: Basic Equations

3. 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

4. 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

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

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

7. 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 / κνwe can transform results to an optical depth scale by considering the opacity

8. 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

9. 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

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

11. Rescaling for Other Stars Reasonable starting approximation

12. Temperature Relations for Supergiants • Differences smalldespite very different length scales

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

14. 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)

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

16. Some T Correction Methods • Unsöld-Lucy methodsimilar 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)

17. 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