Radiation Transport in

1
Radiation Transport in Rapidly
Expanding Envelopes (Problems,
fixes and limitations) P. Hoeflich
(Dept. of Astronomy, Florida State University,
Tallahassee) I) Introduction
II) Formalism and Implementation III)
Some Examples - Applications
- Limitations Problems
2
Radiation Transport in Rapidly
Expanding Envelopes (Problems,
fixes and limitations) P. Hoeflich
(Dept. of Astronomy, Florida State University,
Tallahassee) I) Introduction
II) Formalism and Implementation III)
Some Examples - Applications
- Limitations Problems
3
Radiation Transport in Rapidly
Expanding Envelopes
Incomplete list of collaborators Baade (ESO),
Dominguez et al.(Grenada), Fesen
Co(Dartmouth), Gamezo Co(NRL), Khokhlov Co
(Chicago), Baron (Oklahoma), Hamuy Co(CTIO),
Kevin/Suntzeff/Wang(Texas AM), Quimby, Straniero
Co(Terramo), Thielemann et al. (Basel),
Wheeler(UT), FSU (Gerardy, Mandeau, Michell,
Pelluchi, Penny, Sadler) Contributors almost
of all of you
4

Typical properties of objects (e.g. thermonuclear
SN) Velocities 1,000 to 30,000 km/sec (fully
or piece-wise monotonic) gt narrow lines and
complete redistribution Densities 1E-17 to
1E-7g/ccm (up to 1E9 g/ccm for a WD) Large
Size 1E8 to 1E16 cm to days Excitation and
Ionization Mechanisms Thermal, photons
(electron-transitions), gamma-rays and hydro.
Chemestry (depth dependent) H/He
(SNII,SNIb...) or C,O, Si, S, Ca, Fe, Co, Ni-rich
(SNIa, novae,...)

5
Typical Structure of a Type Ia SN at Maximum
Light Example DD-Model for SN1994D
I
Photospheric region
- NLTE up to center - gamma-deposition at the
photosphere gt LL(r)
log(density) T
gamma-deposition
Tau
6
Time dependence - Changing background/hydro -
Radiation transport terms - Occupation
numbers gt Initial value problem (no spatial
boundary problem)
7
Problems of Radiation Transport - Combined
system of hydro, radiation transport, and rate
equations of different type. - Coupling of
frequency and spatial grid - Resolution of
photosphere - Grid-imprint in multi-dimensional
radiation transport a) Error in speed of light
(bad) b) Directional dependence of c (very
bad) c) Error accumulates with time -
Discretization error - practical purpose
(dimensionality of the problem) Equations may be
unstable!
8
Example of HYDRA Thermonuclear Explosion
(Hydrodynamics/PPM)
a) 1-D Lagrangian (sphericalfront tracking) b)
3-D Eulerian (cartesian, adaptive mesh) c) Free
expansion
MC gamma-ray transport a) 1-D spherical b) 3-D
(given cartesian grid)
Nuclear network a) NSE b) Full network decays

Master module/ switch
Radiation transport (3 modules) a1)Spherical,
comoving Rybicki scheme (MKH75, 76,81)
for spherical LCs and atmospheres a2) Formal
integration of RT in observes frame
(spherical) b) Variable Eddington Tensor solver
(implicit) for given factors b1) 1-D spherical
energy b2) 3-D cartesian energy c) Monte
Carlo Scheme c1) for Eddington tensor 3-D,
relativistic to calculate from diffusion
approximation c2) flux polarization spectra
stationary transport (with given T and
occupation numbers)
EOS a) 1E10lt ??lt1 g/ccm b) 1g/ccm gt ?
Statistical equations for ionization and level
population
LTE
Opacities
9
Example Light Curves, Flux and Polarization
Spectra
(Hydrodynamics/PPM)
a) 1-D Lagrangian (sphericalfront tracking) b)
3-D Eulerian (cartesian, adaptive mesh) c) Free
expansion
MC gamma-ray transport a) 1-D spherical b) 3-D
(given cartesian grid)
Nuclear network a) NSE b) Full network decays

Master module/ switch
Radiation transport (3 modules) a1)Spherical,
comoving Rybicki scheme (MKH75, 76,81)
for spherical LCs and atmospheres a2) Formal
integration of RT in observes frame
(spherical) b) Variable Eddington Tensor solver
(implicit) for given factors b1) 1-D spherical
energy b2) 3-D cartesian energy c) Monte
Carlo Scheme c1) for Eddington tensor
elements 3-D, relativistic to calculate
from diffusion approximation c2) flux and
polarization spectra stationary transport
( given T(x) population numbers)

EOS a) 1E10lt ??lt1 g/ccm b) 1g/ccm gt ?
Statistical equations for ionization and level
population
LTE
Opacities
10
General Concepts 1) ALI for coupling of
equations 2) Implicit 'equivalent 2-level"
approach for optical thick continua 3) Level
merging/superlevels 4) Implicit time dependence
of ionization by bound-free transitions 5)
Narrow line limit 6) Frequency groups 7)
Monochromatic moment equations for Radiation
Transport 8) Tensor elements via Monte Carlo
- Stationary solution with 'old' time steps -
MultiGroupTimeEvolution
11
Iteration Scheme with ALI
a) Rate equations b) particle conservation
(2-3)
Charge conservation
model iteration (2-5 between T correction)
a) Opacities b) Implicit and explicit source
functions c) Approximate Lambda Operator
T-correction
Radiation transport(with ALI)
Temperature by Hydro including integrated
RT-terms (with ALI)
Typically, between 5 and 150 iterations are
needed for convergence

12
Opacities in rapidly expanding envelopes (Op
-gt d?d?? Individual opacity

and
Photons travel in both the spatial and frequency
space !!! Assumption (valid for small
distances)
(??,r1)
(vo,r1)
D
Photon can interact with envelope in very well
defined region if the intrinisic line with is
small
path of photon
position of line in frequency space
(?o,ro)
?????????????
?

for rate equations
energy eq.
13
Time dependent rate equations Ansatz reduce the
time dependent eq. the the time-independent
case

(1)

(2) and solve by
standard methods Assumption tscale(b-f) gtgt
tscale(bb), normalized (2)-(1) gt i.e. a ODE
of first order (which has an analytic solution as
initial boundary problem)
with
If relaxation terms dominate we
use Charge conservation is obtained
iteratively with REM For (ngt2) ionization
stages it can be solved as n linear equations or,
alternatively, as an equation of
order n
14
Adaptive Mesh Refinement for Radiation Tr. Why?
- discretization error in explosion because
grid is optimized for hydro (e.g.?Mconst.) -
errors are small for large optical depths
(diffusion) but large at small tau. Example
500 ... 1000 depths needed - reason radiation
field changes from isotropic to un-isotropic
at decoupling region. gt AMR Issue
Photon freeze-out Problem 'Photosphere' is not
related to a local physical property (rho, v, T
etc.) AMR by a Monte Carlo Torch

Solution Shoot photons from outside and see
where it interacts Recipes - number of test
photons 1E6 ... 1E7 Spherical case n(AMR)1E3
n(? (repr.)), 3D case n(AMR)5E5
n(?(repr.)) - divide a cell neighbors by 2 if
actual count exceeds average by about 10. -
dezone only after 10 to 20 steps (Rem. cavities
H2002)

torch
15
Influence of Level-Locking and
3D-Envelopes
- qualitatively ok but ... some discrepancies
in particular below 5000 A due to a) locking of
levels b) frequency resolution c) for net-rates
below 1E-3, deviations from diffusion is set to
zero
16
Some Problems (or lame excuses for our approach)
- full non-LTE problem (including small
net-rates)/accuracy - time-dependent radiation
transport and populations - c may see the
imprint of a grid - high scattering
(-gtthermalization lengths) - photon decoupling
region varies with frequency and time -
frequency, depth and time space couple -
stability of the system
Grid MC
?? -- - - -
- - -
? ?
17
RT via Variable Eddington Tensor Method
Assumption Order O(v/c)
Buchler(79/83,JQSRT 22/83,293/395)
Drop 1/c iterate
(corr)
Tensor
Rem. Acceleration neglected over RT-time_scales,
or dominated by hydro. Rem.2 GMRES (Generalized
Minimal Residual), JordanGauss Seidel,
Bi-conjugate gradient, ...
18
The Energy Equation
-
19
Other Problems Long characteristics Hard to
manage in 3D Short characteristics Accuracy
decreases with resolution /discretization error
may either fixed grid or angle


fixed grid rays or fixed angle
2 cells
8 cells
20
Eddington Tensor Elements via Monte Carlo (in
practice experimenting) - Time dependence via
frequency groups - 'old' time steps from
'previous models gt limit on time step
determines accuracy in c - Calculation of MC in
every time-step using previous time-steps to
reduce noise (and instabilities) - Photon
'noise' enters geometry only - Adjustment of
isotropy factor to requirement in rates
(meaning more photons or smoothing) - Set net
rates to 0 if in 'photon' noise
21

• Thumbnail Sketch of Thermonuclear SNe
• .
• SNe Ia are thermonuclear explosions of White
  Dwarfs
• SNe Ia are homogeneous because nuclear physics
  determines the W D structure, and the explosion
• The total energy production is given by the
  total amount of burning
• The light curves are determined by the amount
  of radioactive 56Ni
• The progenitor evolution and explosion go
  through several phases of stellar
  amnesia
  
  
  
  
  gt Homogeneity does not imply a unique
  explosion scenario !!! gt Revolution in
  observations allows to probe physics of SN !!!

22
Explosion Scenarios for Type Ia Supernovae
Initial WD Deflagration
phase(2...3sec) Detonation phase
(0.2...0.3 sec)
preexpansion of the WD hardly
any time for further expansion
C/O
C/O
Si/S
C/O
Ni
Ni
MERGER
Deflagration Energy transport by heat
conduction over the front, v ltltv(sound)
gt ignition of unburned fuel
(C/O) Detonation ignition of unburned fuel by
compression, v v(sound)
23
Radial/v-Structures of 3-D Deflagration and DD
Models (from Gamezo et al. 2002/2003, Science)
Deflagration -no radially stratified -about 1/3
of WD remains unburned - E(kin) 4-7E50
erg DDT - radially stratified - almost entire WD
is burned , detonation almost wipes out
signatures of deflagration outcome mainly
depends on amount of burning not details
24
Delayed detonation models for various transition
densities rho(tr) M(MS) 3 Mo Z 1.E-3
solar rho(c) 2E9 g/ccm with rho(tr)8, 16, 25
g/ccm
25
Comparison of LCs with Observations
The brightness decline relation and
colors?

Phillips et al. 2003

Phillips (2003)? dm(B)-gtdm(V) Garnavich (2002)?
- Generic Brightness decline relation is an
opacity effect (Hoeflich etal 96,Mazzali et al.
2001) - Small spread requires similar explosion
energies (?0.5mag for all scenarios H. et al.
96)? - Within DD models, relation can be
understood as change of burning before DDT
26
Progenitor Signatures on SNIa Light Curves
C/O profile of the WD depends on MS mass and
metallicity of WD Accretion Rate
gt Central density at explosion changes electron
capture
Differential change in B and V light-curves

Observations Theory (predicted)
MS-masses 1 to8 Mo rhoc
16E9g/cm3
27
Polarization as Tool to Decipher the 3D Structure
of Type Ia SNe
SN2004dt 12 days after the explosion with VLT
VLT vs. Model
(Hoeflich, 1995)?
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Example Time dependent NLTE spectra IR-Analysis
of SN1999by (as followed from explosion without
tuning)
30
IR-Analysis of SN1999by (as followed from
explosion without tuning)
31
Final Discussions and Conclusions
- Hybrid Schemes between VET-methods and MC for
tensor elements for rapidly expanding
envelopes - MC is suited to determine the I
distribution for given structures, - VET
provides the accuracy to solve rate equations, T
and time-dependence - ALI decouples equations of
hydro, radiation transport and rate equations -
Equivalent-two level approach helps for thick
continua to solve 'globally' - Nature helps with
lines to decouple frequency and space (high
velocities) - Error control and approximations
are critical - May be applicable to other
regimes and applications
Applications Some examples for time-dependent
and Multi-D problems
32
Multi-Dimensional Effects (Off-center DDT)
Approach spherical deflagration to mimic little
mixing and DDT at DM0.5 Polarization by
'cover-up' Color-coding of mass fractions
blue lt0.1 -gt green0.4 -gt redgt0.7
C-burning (O,Ne,Mg) incomplete Si-burning
NSE
blue0




Observer
Observer

Observer
O Si
56Ni
Photosphere

Line of constant wavelength
line of constant
wavelength line of constant
wavelength
- Layers along black line are contribute to the
same wavelength gt asymmetric covering/P
requires chemical gradients along iso-spheres
- O- is unpolarized because abundance is
similar in burned and unburned region Rem.
Parallel-shift correspond to frequency shift
because vr Remark2

33
Final Discussions and Conclusions 1) ALI is a
key for NLTE problems which invoke radiation
hydro - explicit perturbation theory (Cannon
1973, Scharmer, 1984) - Two cases for the
construction of the rates a) S(transition)
dominates S(nu) -gt correction terms have the form
of net rates b) otherwise extrapolation of
current rate by previous iterations or dominating
transition - equivalent 2-level" approach for
optical thick continua from the ground state
and lines (in some instances). - partial
linearization for the energy and statistical
equations - Concept of leading elements (i.e.
updated influence of the 'most important'
elements) - Locking of departure coefficients
within multiplets - stabilization of the
iteration scheme by construction of global photon
redistribution fct. 2) The solution for the
time-dependent problem can be reduced to the
time-independent c. 3) Moment equations ALI
for S are suitable for coupling of the energy
(hydro) and R.T. 4) Method of long
characteristics is useful for radiation hydro (or
light goes around corners)
Example Analysis of the subluminous SN1999by 1)
Consistent treatment of hydro and spectral
analysis is useful 2) Subluminous S N e Ia can
be understood as explosion of M(Ch) - WD 3) The
asphericity of SN1999by and the subluminous
nature may be connected (rotation) 4) Detailed
analysis allows to probe the microphysical
properties of nuclear burning fronts 5) SN1999by
may be the first evidence for an extended
smoldering phase prior of a WD
34
3-D Structure for a deflagration model
(H2002) WD Mch, rho(c)2E9 g/ccm
Day 01
Day 21
Contour 2 of maximum Ni
deposition - Energy deposition is highly
aspherical early on but spherical later on
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Approach to Polarization Calculations - 2 or 3D
geometry for entire envelope (no inner
boundary) - Monte Carlo method after convergence
(of occupation numbers and energy eq.) -
Mechanism for P a) Thomson scattering b)
Line scattering - Scattering matrix is
approximated by a linear combination of Rayleigh
and isotropic phase matrix (Hamilton
1947, Domke Hubeney, 1988) gt complete
redistribution in frequency space
- Thermalization is treated by branching ratio
according to the equivalent 2-level approach (see
above) - Size of polarization is
given by the angular momentum j of the lower
state and Delta J (Chandrasekhar,
1960) - Large optical depth of
an individual line is treated by n-scatterings
at the location of interaction (H95)

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Modular structure allows for code verification a)
Atomic level populations (Example for maximum
light, Hoeflich 95) Ion Superlevels b-b for
rate eq. for radiation transport CI
27 123
242 OI 43
129 506 MgII
20 60
153 Si II 35
212 506 Ca II
41 195
742 Ti II 62
75 592 Fe
II 137 3120
7293 Co II 84
1355 5396 Ni
II 71 865
3064 SUM 520
6134 18650
gt (about 20,000 transitions) - Adjoining
ionization stages are 3 to 5 super-levels - about
1E6 additional lines with equivalent-two level
approach selected out of a line list of
40,000,000 (Kurucz) - bf- and ff-cross sections
- for LC and aspherical spectra, the multiplets
have been 'merged' to super-superlevels C)
Typical, global discretization (H02) Spherical
atmospheres 95 spatial points (40AMR) , 2 to
3E4 frequencies Aspherical "
69/69/(69) " " (lt10000AMR) , 1-3,000
frequency groups (nar.line) LC and gamma's
911 " " (100AMR)
, 3000-5000 " " ( " )
Remark Level-coupling must be taken into account
in energy budget !!!
40
Basic Equations for the Radiation-Hydro Problem
a) Radiation transport to determine the
intensity I(x,y,z,dx,dy,dz,?,t) b) Statistical
equations for level population A
n(el,ion,lev,x,y,z) b b1) Rate equations
with
PC(T,?,ne)R RR(J,T) b2) Particle
conservations and charge
conservation c) Hydro equations d) Energy
equation (radiation, EOS, nuclear decay, hydro)
E? E(i) e.g. Separation of global and local
equations and iterative coupling (ALI) DIM n(x)
n(y) n(z) n(freq) SUM(el) SUM(ion(el))
(levels(ion)1) 1E6 1E4 1E3 (with ALI)
gt SUM(el) SUM(ion(el) (levels(ion1) 10
3 (20...200) 6E2 ... 6e3

with OpOp(x,y,z,?/dx,?/?y,?z,d/d?),
??(x,y,z,??????????????),
SS(?,T,n,d???????? or, implicit, S e BB(?,T)
(1-?) J(x,y,z,t)
41
Implicit formulation of source functions for
Lyman lines (Athay)
with
with gt and
(with narrow line limit, integral vanishes)
42
Explicit forms used Source function Emissivity
Opacity Formal definition of
redistribution functions ? and f (over the entire
or parts of the wavelength range)

Remark red. function are determined
numerically for each model iteration and used to
damp the frequency redistribution (if
needed), and for parallization.
43
Line Blanketing Effect in the Narrow Line
Limit Individual opacity Effect of multiple
lines / effective opacity (similar Karp et al.
1977 but with integration boundaries according
cell boundaries) with
and
Rem. For large optical depths
(Hoeflich 1990)
44
The Accelerated Lambda Operator (Canon 73,
Scharmer 84, Olson 87,Hoeflich 87,..) Iterative
method with m being the model interation
Construction of ? in 2-stream approximation
(Op-gtd/dd) and using intensity like variables u
(I I- ) (Rem Probability for
interactiointeraction is W exp(-?t?) ) with
discretization with gt For
optical/thin cores, we use
and
,respectively.
?
d
Problem Approximation and convergence are grid
dependent In practice Redefine grid/average
over several zones (?thermalization height) for
non-converged systems
45
Line Blanketing Effect in the Narrow Line
Limit Individual opacity Effect of multiple
lines / effective opacity (similar Karp et al.
1977 but with integration boundaries according
cell boundaries) with
and
Rem. For large optical depths
(Hoeflich 1990)
46
Charge conservation To allow for a separation of
the rate equations of different elements, the
electron density is solved iteratively by
Partial linearization for rate equations and
particle conservation (in the usual form)
with
gt
47
Matrix elements for the rate equations
P(?-rays)
Separation of two cases S is/ is not dominated
by a given transition with identical 0-order
terms
P(?-rays)
Remark In principle, energy by hard radiation
produces non-thermal electrons
direct ionization In practice,
distribution of the energy according to the
element abundances Z is very close
to reality
48
Case A 1st order correction If a leading
element dominates (marked by dom) otherwise
second(or higher) order extrapolation (by rotated
parabulae) with
49
Case B substitute
in rate equation gt gt for 1st order
correction terms in form of net rates
Rem. Same trick also applies to the coupling
with hydro
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Basic Approach I) Separation of variables
II) Analytic solutions for separated equations
whenever possible (e.g. time dependence)
III) Coupling of modules by Accelerated Lambda
Iteration

- Development driven by astronomical problems
(explosions, light curves and flux and
polarization spectra for SNe) - FORTRAN -
highly modular - Parallel (shared
distributed) - Not designed from scratch a)
use of well tested modules (individual history)
b) designed to work but not for beauty c) user
interfaces - dynamical task allocation
52
Charge conservation To allow for a separation of
the rate equations of different elements, the
electron density is solved iteratively by
Partial linearization for rate equations and
particle conservation (in the usual form)
with
gt
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c
Part 1Search for the Signatures of DDT with
Subaru IR at 300 days after maximum of 2003du



At
Comparison with DD-model of rho2E9g/ccm -
mixing of neutron rich isotopes as expected
from 3D - spherical model Central 56Ni hole
3000 km/sec 56Ni wind 9000
km/sec - pretty symmetric profile gt not much of
rising plumes Fe-line 'off-set' by about
300km/sec
55
Part 2 Search for the Isotopic Composition
SN2003hv at about 390 days after the explosion


Similarities to SN2003du Flat topped profile
(v2500km/sec)-gt central hole in 56Ni
distribution Differences - Very asymmetric line
wings. Blue vs. red edge (vlt1000km/sec vs. 3000
km/sec)
56
Polarization as Tool to Decipher the 3D Structure
of Type Ia SNe
SN2006X (Patat et al.)? (times are relative to
maximum)?
DDT model (Hoeflich et al. 2002)?
Ca
39 days
57
III) Direct Imaging SN1885, the Great Supernova
in M31 HST Observations of the Remnant

Based on HST observations and a paper by R.
Fesen (Dartmouth) P. Hoeflich (Austin) , A.
Hamilton (Boulder), M. Hammel (Dartmouth),
Gerardy (London), A. Khokhlov (Chicago), C.
Wheeler
58

The Supernovae Remnant SN1885 with Hubble (Fesen,
Hoeflich, Hamilton, et al. 2006)?
Ca II Round with 11000 km/sec radius, slight
ring-like structure Ca I Part of ring and
off-center by 3000 km/sec towards the bulge FeII
Similar to CaII but slightly smaller FeI
Off-center as Ca I All Clumps of about 1100
km/sec
59
Influence of Mixing and Alternative Scenarios
Case a) Mixing of the layers with deflagration
burning Case b) Mixing of all Ca-rich layers Case
c) Mixing of all burned layers as in
3D-deflagrations Rem Mixing may be due to late
time instabilities (56Ni)?
MM


M

Alternative Models SNIa.1) Pure deflagration
models are ruled out for SN1885. SNIa.2) Confined
delayed detonation models are unlikely because
they show high velocity Fe and Ca (which has
not been seen). SNIa.3) Mergers may be ok but
those have no high density region with electron
capture. II) Core collapse SN Unlikely because
their strong asymmetry and chemistry


60
S-Andromeda in the UV (a case for STIS)
Evolution with time
with a STIS-like instrument
Influence of slit
size
MM


M