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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.

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Ivan Hubeny University of Arizona

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Ivan hubeny university of arizona

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



  • 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

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

Ivan hubeny university of arizona

The Numerical Problem

A model stellar atmosphere is described by a system of highly-coupled,

highly non-linear set of equations

  • Radiative EquilibriumTemperature

  • Hydrostatic EquilibriumMass density

  • Charge ConservationElectron density

  • Statistical EquilibriumNLTE populations~ 100,000 levels

  • Radiative TransferMean Intensities~ 200,000 frequencies

    The number of unknowns and cost of computing a model atmosphere increases quickly with

    the complexity of the atmospheric plasma.

Complete linearization

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

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

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

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

Another expression of ALI


FS = Formal Solution - uses an old source function

Ordinary Lambda Iteration

Accelerated Lambda Iteration

acceleration operator

Iterative solution acceleration

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

Ivan hubeny university of arizona


Nlte line blanketing level grouping



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


Ivan hubeny university of arizona

NLTE line blanketing: lines & frequencies


Transition 1-13




Sorted cross-section


Hybrid cl ali method

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

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



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)

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

Ivan hubeny university of arizona


Lanz & Hubeny, ApJS 146, 417; 169,83

Ostar2002 bstar2006

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

Ivan hubeny university of arizona

Temperature structure for various metallicities

Ivan hubeny university of arizona

Comparison to Kurucz models

50,000 K


30,000 K

Comparison to kurucz models

Comparison to Kurucz Models

Teff = 25,000

log g = 3

Ivan hubeny university of arizona

Do stellar atmosphere structural equations have always a unigue solution?Well, not always…Bifurcation with strong external irradiation!

Hubeny, Burrows, Sudarsky 2003

Ivan hubeny university of arizona

Thermal Inversion: Water in Emission (!)

Strong Absorber at Altitude (in the Optical)

Hubeny, Burrows, & Sudarsky 2003

Burrows et al. 2007


Ivan hubeny university of arizona

Burrows, Hubeny, Budaj, Knutson, & Charbonneau 2007

Another dimitri s legacy mixed frame formalism mihalas klein 1982 j comp phys 46 92

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

Application of the ideas of ali in implicit rad hydro

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

Conclusions and outlook

Conclusions and Outlook


  • 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!


  • Many talks at this meeting

  • Decisive progress expected in the near future

Dimitri we all salute you

Dimitri, we all salute you!

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