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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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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!

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

  • 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


FS = Formal Solution - uses an old source function

Ordinary Lambda Iteration

Accelerated Lambda Iteration

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, 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




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


NLTE line blanketing: lines & frequencies


Transition 1-13




Sorted cross-section


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



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)


  • 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


Lanz & 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


30,000 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

Thermal Inversion: Water in Emission (!)

Strong Absorber at Altitude (in the Optical)

Hubeny, Burrows, & Sudarsky 2003

Burrows et al. 2007


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)

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

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


  • 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!

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