Three-Dimensional and Time-Dependent
Effects in Spectral Irradiance Modeling
Han Uitenbroek
National Solar Observatory/Sacramento Peak
Sunspot NM, USA
2012 SORCE Science Meeting, Annapolis, Maryland, Sep. 18–19
Modeling full-disk spectral irradiance, while accounting
for fine structure and dynamics of the solar atmosphere
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Overview
• Radiative transfer
• The dynamic three-dimensional solar atmosphere
• Semi-empirical modeling with hydrostatic atmospheres
• Non-linear spectral averaging
• Processes that are slower than dynamical time scales
• Conclusions
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Radiative Transfer
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Equation of radiative transfer and source function
How do we determine the emergent radiation of a given plasma, and how
do we extract physical properties from that radiation field?
'
$
Equation of radiative transfer:
dI
= −χI + η = −χ(I − S);
ds
S = η/χ
Integral form, the formal solution:
Z
∞
I=
&
S(τ )e−τ dτ ;
dτ = −χds
0
Uitenbroek, NSO/SP
%
Three-Dimensional and Time-Dependent Effects
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Source function of the Mg ii k Line
S, J [J m−2 s−1 Hz−1 sr−1]
10−7
Stotal
Sactive
Sbackgr
BPlanck
J
10−8
279.3 279.4 279.5 279.6 279.7 279.8
λ[nm]
10−9
10−10
0.0001
Uitenbroek, NSO/SP
0.0010
0.0100
0.1000
1.0000
−2
Column Mass [kg m ]
Three-Dimensional and Time-Dependent Effects
10.0000
100.0000
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The Solar Atmosphere
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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The Inhomogeneous and Dynamic Solar Atmosphere
• The solar photosphere and chromosphere are complicated highly
structured and dynamic environments shaped by the interplay between
hydrodynamics, magnetic fields and the flow of radiation.
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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The Inhomogeneous and Dynamic Solar Atmosphere
• The solar photosphere and chromosphere are complicated highly
structured and dynamic environments shaped by the interplay between
hydrodynamics, magnetic fields and the flow of radiation.
• To properly understand these environments and model their spectrum
requires full three-dimensional Radiation Magneto-HydroDynamics
(RMHD).
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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The Inhomogeneous and Dynamic Solar Atmosphere
• The solar photosphere and chromosphere are complicated highly
structured and dynamic environments shaped by the interplay between
hydrodynamics, magnetic fields and the flow of radiation.
• To properly understand these environments and model their spectrum
requires full three-dimensional Radiation Magneto-HydroDynamics
(RMHD).
• To model the full-disk integrated spectrum, which is required for
Spectral Irradiance modeling is not possible now, nor in the near future.
So suitible approximations will have to be made.
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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The Inhomogeneous and Dynamic Solar Atmosphere
• The solar photosphere and chromosphere are complicated highly
structured and dynamic environments shaped by the interplay between
hydrodynamics, magnetic fields and the flow of radiation.
• To properly understand these environments and model their spectrum
requires full three-dimensional Radiation Magneto-HydroDynamics
(RMHD).
• To model the full-disk integrated spectrum, which is required for
Spectral Irradiance modeling is not possible now, nor in the near future.
So suitible approximations will have to be made.
• So far we rely on semi-empirical hydrostatic modeling. The question is
if this is adequate.
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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The Quiet Solar Photosphere with Granulation
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Transition from Photosphere to Magnetically Dominated
Chromosphere
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Equations of Magneto-HydroDynamics
'
$
Mass:
∂ρ
= −∇ · (ρu)
∂t
&
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
%
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Equations of Magneto-HydroDynamics
'
$
Mass:
∂ρ
= −∇ · (ρu)
∂t
&
%
'
$
Momentum:
∂ρu
= −∇ · (ρuu) − ∇P − ρ∇Φ − ∇ · τvisc
∂t
&
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
%
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Equations of Magneto-HydroDynamics
'
$
Mass:
∂ρ
= −∇ · (ρu)
∂t
&
%
'
$
Momentum:
∂ρu
= −∇ · (ρuu) − ∇P − ρ∇Φ − ∇ · τvisc
∂t
&
%
'
$
Internal energy:
∂e
= −∇ · (eu) − P ∇ · u + Qrad + Qvisc + Qcond
∂t
&
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
%
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After all the hard work
Courtesy: Mats Carlsson
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Semi-Empirical Modeling
with Hyrdostatic
Atmospheres
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Hydrostatic equilibrium
'
$
Momentum:
∂ρu
= −∇ · (ρuu) − ∇P − ρ∇Φ − ∇ · τvisc
∂t
&
%
'
$
For small velocities, large scale horizontal structure,
no magnetic field, and constant gravity:
P ∂ ln P
∂Φ
−
=
= gz
ρ ∂z
∂z
P = P0e−z/HP ;
&
Uitenbroek, NSO/SP
HP =
P
ρgz
Three-Dimensional and Time-Dependent Effects
%
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Identification of solar regions
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Different models for different (average) regions
Fontenla et al., SRPM
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What does Semi-Empirical Modeling Entail?
• Semi-empirical model are inversions of the (usually spatially and
temporally avereged spectrum of a given solar region. They are
Multi-parameter fits to this spectrum, with temperature as function of
column mass as free parameters.
• Additional constraints are often imposed like hydrostatic equilibrium,
and smoothness and monotonicity.
• Often, sophisticated radiative transfer is used to solve equation of state
(electron density), but no further physical laws are implemented to
explain the derived thermodynamic structure.
• Question: Can a time-dependent and/or spatially inhomogeneous
atmosphere be represented by a series of such models?
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Effects that may compromise 1-D hydrostatic irradiance
modeling
• Non-linear spectral averaging of inhomogeneities and/or temporal
variations
• Physical processes on time scales that are longer than the dynamical
time scales
• Lateral radiative transfer
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Non-Linear Averaging of
Spectral Quantities
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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A Snapshot of a 3-D Convection Simulation
8
x [arcsec]
6
4
2
0
0
2
4
x [arcsec]
6
8
10
200
8
T [103 K]
z [km]
400
0
6
−200
0
Uitenbroek, NSO/SP
2
4
x [arcsec]
Three-Dimensional and Time-Dependent Effects
6
8
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The τ = 1 Level almost follows Isotherms
Source function
2•10−8
4•10−8
6•10−8
8•10−8
z [km]
400
200
0
−200
0
Uitenbroek, NSO/SP
1000
2000
3000
x [km]
Three-Dimensional and Time-Dependent Effects
4000
5000
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Second Derivative of Planck Function is always Positive
$
'
hIλixy
= hBλ(T )iτλ=1
t ≡ ∆T /T
1 2 d Bλ(t) hBλi = hBλ(t)i + t
2
dt2 2
&
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
3
+ O( t ),
t=0
%
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Even simple Continua don’t Match
4•10−8
intensity [J m−2 s−1 Hz−1 sr−1]
500 nm
3•10−8
2•10−8
1•10−8
3−D
1−D
0
0.0
0.2
0.4
0.6
0.8
1.0
µ
Uitenbroek, NSO/SP
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Concentration of Diatomic Molecules
'
$
nA n B
2πmAB kT
=
nAB
h2
nA n B
6= Φ(< T >)
nAB
3/2
e
−D/kT
UA(T )UB (T )
= Φ(T )
QAB (T )
&
%
For small concentrations and small ∆T :
'
$
nAB
&
Uitenbroek, NSO/SP
eD/kT ≡ Cφ(T )
2
3
1 2 d φ(t) t C
+ O( t );
2
dt2 = C
h∆nAB i =
D
kT
3/2
t=0
Three-Dimensional and Time-Dependent Effects
t ≡ ∆T /T
%
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Concentration of CO Molecules
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Lateral Radiative Transfer
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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The Solar Ca ii 854.2 nm Line
Average Intensity [J m−2 s−1 Hz−1 sr−1]
10−7
disk center
10−8
853.0
Uitenbroek, NSO/SP
853.5
854.0
854.5
Wavelength [nm]
Three-Dimensional and Time-Dependent Effects
855.0
855.5
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Source function and equivalent two-level atom
'
$
S = B + (1 − )J
&
%
S, J [J m−2 s−1 Hz−1 sr−1]
10−6
10−7
853.9 854.0 854.1 854.2 854.3 854.4
λ[nm]
10−8
10−9
0.0001
Uitenbroek, NSO/SP
Stotal
Sactive
Sbackgr
BPlanck
J
0.0010
0.0100
0.1000
1.0000
Column Mass [kg m−2]
Three-Dimensional and Time-Dependent Effects
10.0000
100.0000
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Two-dimensional Source Function of Ca ii 854.2 nm
854.190 [nm]
2000
853.9 854.0 854.1 854.2 854.3 854.4
λ[nm]
z [km]
1500
1000
500
Source Function [J m−2 s−1 Hz−1 sr−1]
10−7
10−8
0
0
2000
4000
6000
8000
10000
x [km]
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Lateral Radiative Transfer Modifies the Source Function
1.5
log Source function [J m−2 s−1 Hz−1 sr−1]
−6.8
−7.0
1.0
z [Mm]
−7.2
−7.4
0.5
−7.6
0.0
−7.8
−8.0
4.5
Uitenbroek, NSO/SP
5.0
x [Mm]
Three-Dimensional and Time-Dependent Effects
5.5
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Fluxtube Appears Larger in the Line Core
10000
3.0•10−8
Intensity [J m−2 s−1 Hz−1 sr−1]
8000
2.5•10−8
x [km]
6000
4000
2.0•10−8
2000
1.5•10−8
0
854.1
Uitenbroek, NSO/SP
854.2
lambda [nm]
Three-Dimensional and Time-Dependent Effects
854.3
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Processes that are Slower
than Dynamical Time
Scales
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Hydrodynamic simulation of Chromosperic Dynamics
Carlsson & Stein 1995, 440, L29
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Dynamic Hydrogen ionization
Carlsson & Stein 2002, ApJ 572, 626
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Ionization Fraction in Initial Radiative Equilibrium Model
Uitenbroek, NSO/SP
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Hydrogen Ionization Rates
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Hydrogen Ionization Time Scales
Uitenbroek, NSO/SP
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Dynamic Hydrogen ionization
Uitenbroek, NSO/SP
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The Semi-Empirical Model Leads to a very Different
Conclusion
Uitenbroek, NSO/SP
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Conclusions
• Effects of Non-linear spectral averaging, lateral radiative transfer, and
hysteresis are not represented by semi-empirical hydrostatic models, but
are important for the solar spectrum.
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
yw w w wy p 7
Conclusions
• Effects of Non-linear spectral averaging, lateral radiative transfer, and
hysteresis are not represented by semi-empirical hydrostatic models, but
are important for the solar spectrum.
• Interpretation of the solar spectrum in terms of such hydrostatic models
mis-represents the physics of the solar atmosphere, and puts us at risk
of mis-predicting parts of the spectrum that are not used to construct
the models.
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Conclusions
• Effects of Non-linear spectral averaging, lateral radiative transfer, and
hysteresis are not represented by semi-empirical hydrostatic models, but
are important for the solar spectrum.
• Interpretation of the solar spectrum in terms of such hydrostatic models
mis-represents the physics of the solar atmosphere, and puts us at risk
of mis-predicting parts of the spectrum that are not used to construct
the models.
• Much more three-dimensional spectral modeling is needed to assess the
severity of these effects.
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Thank you
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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Hydrostatic model FAL C (average quiet Sun)
20
temperature [1000 K]
15
10
5
0
0.00001
0.00010
0.00100
0.01000
0.10000
log column mass [kg m−2]
1.00000
10.00000
Back
Uitenbroek, NSO/SP
Three-Dimensional and Time-Dependent Effects
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