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### FROM COMPLETE LINERIZATION TO ALI AND BEYOND(how a somewhat younger generation built upon Dimitri’s work)

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

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!

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.

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

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

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

NLTE line blanketing: lines & frequencies

Fe III

Transition 1-13

Absorption

cross-section

OS

Sorted cross-section

ODF

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: 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

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

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

OGLE-Tr-56b

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

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

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