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

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!

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

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 withthe complexity of the atmospheric plasma.

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

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

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”)

Accelerated Lambda Iteration

Transfer equation Formal solution

Rate equation (def of S)

==>

Ordinary Lambda Iteration:

Accelerated Lambda Iteration:

and iterate as:

Another expression of ALI

Define

Ordinary Lambda Iteration

Accelerated Lambda Iteration

FS = Formal Solution - uses an old source function

acceleration operator

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, M2 r0, …)

– 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

(future)

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

O IV

S XI

Fe IV

Fe IIITransition 1-13

Absorption cross-section

OSOS

Sorted cross-section

ODFODF

NLTE line blanketing: lines & frequencies

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)

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: depthsInner structure: state parameters (intensities)Block tri-diagonalInner matrices diagonal + added row(s)Execution time scales:-- linearly with ND-- cubically with NF !

Outer structure: intensitiesInner structure: depthsBlock diagonal + added row(s)Inner matrices tri-diagonalExecution time scales:-- linearly with NF !-- cubically with ND (only once)

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

OSTAR 2002; BSTAR 2006 GRIDSLanz & Hubeny, ApJS 146, 417; 169,83

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

Temperature structure for various metallicities

Comparison to Kurucz models

50,000 K

40,000 K30,000 K

Comparison to Kurucz Models

Teff = 25,000

log g = 3

Do stellar atmosphere structural equations have always a unigue solution?

Well, not always…Bifurcation with strong external irradiation!

Hubeny, Burrows, Sudarsky 2003

Strong Absorber at Altitude (in the Optical)

Thermal Inversion: Water in Emission (!)

Hubeny, Burrows, & Sudarsky 2003Burrows et al. 2007

OGLE-Tr-56b

Burrows, Hubeny, Budaj, Knutson, & Charbonneau 2007

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)

l.h.s. lives in the lab frame r.h.s. lives in the comoving frame

Application of the ideas of ALI in implicit rad-hydroHubeny & 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

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

Dimitri, we all salute you!