Ivan Hubeny University of Arizona

1. FROM COMPLETE LINERIZATION TO ALI AND BEYOND(how a somewhat younger generation built upon Dimitri’s work) Ivan Hubeny University of Arizona Collaborators: T. Lanz, D. Hummer, C. Allende-Prieto, L.Koesterke, A. Burrows, D. Sudarsky

2. Introduction • Stellar atmosphere (accretion disk “atmosphere”) = the region from where the photons escape to the surrounding space (and thus can be recorded by an external observer) • Radiation field is strong - it is not merely a probe of the physical state, but an important energy (momentum) balance agent • Radiation in fact determines the structure, yet its structure is probed only by radiation (exception: solar neutrinos, a few neutrinos from SN 1987a) • Most of our knowledge about an object (a star) hinges on an understanding of its atmosphere (all basic stellar parameters) • Unlike laboratory physics, where one can change a setup of the experiment to separate various effects, we do not have this luxury in astrophysics: we are stuck with an observed spectrum • We should better make a good use of it!

3. Motto: One picture is worth 1000 words, but one spectrum is worth 1000 pictures!

4. The Numerical Problem A model stellar atmosphere is described by a system of highly-coupled, highly non-linear set of equations • Radiative Equilibrium Temperature • Hydrostatic Equilibrium Mass density • Charge Conservation Electron density • Statistical Equilibrium NLTE populations ~ 100,000 levels • Radiative Transfer Mean Intensities ~ 200,000 frequencies The number of unknowns and cost of computing a model atmosphere increases quickly with the complexity of the atmospheric plasma.

5. Complete Linearization • Auer & Mihalas 1969, ApJ 158, 641: one of the most important papers in the stellar atmospheres theory in the 20th century • Discretize ALL the structural equations (I.e., differentials to differences; integrals to quadrature sums) • Resulting set of non-linear algebraic equations solve by the Newton-Raphson => “linearization” • Structure described by a state vector at each depth: • {J1, …, JNF, N, T, ne, n1, …, nNL} • J - mean intensities in NF frequency points; • N - total particle number density; T - temperature; ne - electron density • n - level populations of NL selected levels (out of LTE) • Resulting in a block-tridiagonal system of NDxND outer block matrix (ND=depths) with inner matrices NN x NN, where NN=NF + NL + 3 • Computer time scales as (NF+NL+3)3 x ND x Niterations • => with such a straightforward formulation, one cannot get to truly realistic models

6. Why a linearization? • A global scheme is needed because: • An intimate coupling between matter and radiation -- e.g., the transfer equation needs opacities and emissivities to be given, which are determined through T, ne and level populations; these in turned are determined by rate equation, energy balance, hydrostatic equilibrium, which all contain radiation field ==> a pathologically implicit problem (Auer) • If one performs a simple iteration procedure (e.g. Lambda iteration - iterating between the radiation field and level populations), the convergence is too slow to be of practical use - essentially because a long-range interaction of the radiation compared to a particle mean-free-path • But a straightforward global scheme is extremely costly, and fundamentally limited for applications • What is needed: something that takes into account the most important part of the coupling explicitly (globally), while less important parts iteratively

7. Two ways of reducing the problem • Use of form factors: iterating on a ratio of two similar quantities instead on a single quantity (ratio of two similar quantities may change much slower that the quantities itself) • Classical and most important example - Variable Eddington Factors technique - Auer & Mihalas 1970, MNRAS 149, 65 • Solving moment equations for RT instead of angle-dependent RT • There are two moment equations for three moments, J, H, K • The system is closed by calculating a form factor f=K/J (VEF) separately (by an angle-dependent RT), and keeping it fixed in the subsequent iteration of the global system of structural equations • Works well also in radiation hydro and multi-D (Eddington tensor) • Use of adequate preconditioners (= “Accelerated Lambda Iteration”)

8. Accelerated Lambda Iteration Transfer equation Formal solution Rate equation (def of S) ==> Ordinary Lambda Iteration: Accelerated Lambda Iteration: and iterate as:

9. Another expression of ALI Define FS = Formal Solution - uses an old source function Ordinary Lambda Iteration Accelerated Lambda Iteration acceleration operator

10. Iterative solution: acceleration • It may not be efficient to determine the next iterate solely by means of the current residuum - slow convergence • The rescue: to use information from previous iterates • Ng acceleration - residual minimization • Generally: Krylov subspace methods - using subspace spanned by (r0, M r0, M2r0, …) • Krylov subspace generally grows as we iterate • In other words: instead of using current residual, new iterate is obtained using a pseudo-residual, which is chosen to be orthogonal to the currently built Krylov subspace • Several (many) variants of the Krylov subspace method • We selected GMRES (Generalized Minimum Residual) method, and/or Ng method • A reformulated, but equivalent scheme ORTHOMIN(k) (Orthogonal minimization) • One can truncate the orthogonalization process to k most recent vectors

11.    (future) 

12. OIV SXI NLTE line blanketing: level grouping • Individual levels grouped into superlevels according to • Similar energies • Same parity (Iron-peak elements) Assumption:Boltzmann distribution inside each superlevel FeIV

13. NLTE line blanketing: lines & frequencies Fe III Transition 1-13 Absorption cross-section OS Sorted cross-section ODF

14. Hybrid CL/ALI method • Hubeny & Lanz 1995, ApJ 439, 875 • Essentially a usual linearization, but: • mean intensity in most frequencies treated by ALI • mean intensity in selected frequencies (cores of the strongest lines, just shortward of Lyman continuum, etc.) linearized • ==> convergence almost as fast as CL • ==> computer time per iteration as in pure ALI (very short)

15. Rybicki modification - Formulated by Rybicki 1971, JQSRT 11, 589 for a two-level atom - Suggested extension for LTE model atmospheres by Mihalas 1978 (SA2) - Implemented for cool atmospheres by Hubeny, Burrows, Sudarsky 2003 original Rybicki Outer structure: depths Inner structure: state parameters (intensities) Block tri-diagonal Inner matrices diagonal + added row(s) Execution time scales: -- linearly with ND -- cubically with NF ! Outer structure: intensities Inner structure: depths Block diagonal + added row(s) Inner matrices tri-diagonal Execution time scales: -- linearly with NF ! -- cubically with ND (only once)

16. TLUSTY/CoolTLUSTY • Physics • Plane-parallel geometry • Hydrostatic equilibrium • Radiative + convective equilibrium • Statistical equilibrium (not LTE) • Computes model stellar atmospheres or accretion disks • Possibility of including external irradiation (extrasolar planets) • Computes model atmospheres or accretion disks • Numerics • Hybrid CL/ALI method (Hubeny & Lanz 1995) • Metal line blanketing - Opacity Sampling, superleves • Rybicki solution (full CL) in CoolTlusty (LTE) • Range of applicability: 50 K - 109 K, with a gap 3000-5500 K • CoolTLUSTY - for brown dwarfs and extrasolar giant planets: • Uses pre-calculated opacity and state equation tables • Chemical equilibrium + departures from it • Effects of clouds • Circulation between the day and night side (EGP) ------------------------ filled within the last month

17. OSTAR 2002; BSTAR 2006 GRIDS Lanz & Hubeny, ApJS 146, 417; 169,83

18. OSTAR2002 & BSTAR2006 • OSTAR2002 • 680 metal line-blanketed, NLTE models • 12 values of Teff - 27,500 - 55,000 K (2500 K step) • 8 log g’s • 10 metallicities: 2, 1, 1/2, 1/5, 1/10, 1/30, 1/50, 0.01, 0.001, 0 x solar • H, He, C, N, O, Ne, Si, P, S, Fe, Ni in NLTE • ~1000 superlevels, ~ 107 lines, 250,000 frequencies • BSTAR2006 • 1540 metal line-blanketed, NLTE models • 16 values of Teff - 15,000 - 30,000 K, step 1000 K • 6 metallicities: 2,1, 1/2, 1/5, 1/10, 0 x solar • Species is in OSTAR + Mg, Al, but not Ni • ~1450 superlevels, ~107 lines, 400,000 frequencies

19. Temperature structure for various metallicities

20. Comparison to Kurucz models 50,000 K 40,000K 30,000 K

21. Comparison to Kurucz Models Teff = 25,000 log g = 3

22. Do stellar atmosphere structural equations have always a unigue solution?Well, not always…Bifurcation with strong external irradiation! Hubeny, Burrows, Sudarsky 2003

23. Thermal Inversion: Water in Emission (!) Strong Absorber at Altitude (in the Optical) Hubeny, Burrows, & Sudarsky 2003 Burrows et al. 2007 OGLE-Tr-56b

24. Burrows, Hubeny, Budaj, Knutson, & Charbonneau 2007

25. Another Dimitri’s legacy: Mixed-frame formalismMihalas & Klein 1982, J.Comp.Phys. 46, 92 • Fully Laboratory (Eulerian) Frame • l.h.s. - simple and natural • r.h.s. - complicated, awkward, possibly inaccurate • Fully Comoving (Lagrangian) Frame • r.h.s. - simple and natural • l.h.s. - complicated • difficult in multi-D, difficult to implement to hydro • BUT: very successful in 1-D with spectral line transfer (CMFGEN, PHOENIX) • Mixed Frame • combines advantages of both • l.h.s. - simple • r.h.s. - uses linear expansions of co-moving-frame cross-sections => also simple (at least relatively) • BUT: cross-sections have to be smooth functions of energy and angle • not appropriate for photon transport (with spectral lines), but perfect for neutrinos! • elaborated by Hubeny & Burrows 2007, ApJ 659,1458 (2-D, anisotropic scattering) r.h.s. lives in the comoving frame l.h.s. lives in the lab frame

26. Application of the ideas of ALI in implicit rad-hydro Hubeny & Burrows 2007 example: the energy equation backward time differencing - implicit scheme intensity at the end of timestep - expressed through an approximate lambda operator lLinearizarion of the source function moments of the specific intensity at the end of timestep

27. Conclusions and Outlook 1) 1-D STATIONARY ATMOSPHERES • Thanks to standing on the shoulders of giants (Mihalas, Auer, Hummer, Rybicki, Castor, …), this is now almost done - last 2 decades (fully line-blanketed NLTE models - photospheres, winds) • Remaining problems: • Despite of heroic effort of a few brave individuals (OP, IP, OPAL), there is still a lack of needed atomic data (accurate level energies, collisional rates for forbidden transitions, data for elements beyond the iron peak, etc.) • For cool objects - a lack of molecular data (hot bands of methane, ammonia, etc.) • Level dissolution and pseudocontinua (white dwarfs) -- Can convection be described within a 1-D static picture? -- Technical improvements in the modeling codes (more efficient formal solvers; even more efficient iteration procedure - Newton-Krylov; multigrid schemes; AMR; etc.) 2) 3-D SNAPSHOT OF HYDRO SIMULATIONS (i.e. with radiation-hydro split) • Existed for the last decade, but simplified (one line, few angles) • NLTE simplified • Now: one is in the position to do NLTE line-blanketing in 3-D! 3) FULL 3-D RADIATION HYDRO • Many talks at this meeting • Decisive progress expected in the near future

28. Dimitri, we all salute you!