Trent Schindler
Trent Schindler
Radiative Transfer and Molecular Lines
Sara Seager
MIT
Sagan Workshop 2009
Lecture Contents
• Overview of Equations for PlanetaryAtmospheres
• Radiative Transfer• Thermal Inversions• Molecular Line Primer
Planet Atmosphere Equations
!
dP(r)
dr= "
Gm(r)#(r)
r2
!
P = nkT!
Eout = Ein,* + Ein,planet!
dI(s,",µ,t)
ds= #(s,",µ,t) $%(s,",t)I(s,",µ,t)
Energy transport
Consv. of Energy(in each layer)
Hydrostatic Eq.
Ideal Gas Law
Chemical EquilibriumNo simple equation
Want to derive: Flux, T, P, ρ, chemical composition
Atmosphere Model Uncertainties• Only known inputs
– Surface gravity– Radiation from star
• Unknowns:• Atmospheric circulation• Clouds
– Particle size distribution,composition, and shape
– Fraction of gas condensed– Vertical extent of cloud
• Chemistry– Elemental abundances– Nonequilibrium– Photochemistry
• Opacities• Internal Luminosities
Lodders 2005
Lecture Contents
• Overview of Equations for PlanetaryAtmospheres
• Radiative Transfer• Thermal Inversions• Molecular Line Primer
Definition of Intensity
Intensity is a ray ofinfinitesimally small area,frequency interval, solidangle, and time interval. Theintensity is traveling in thedirection of photonpropagation.
1D Radiative Transfer Equation
I = intensity, ε = emissivity, k = absorption coefficient
Change in Intensity = gains to the beam - losses from the beam
!
dI(s,",t)
ds= #(s,",t) $%(s,",t)I(s,",t)
Optical Depth τ
!
d" = #$ds
• A measure of transparency• Think of an object in a fog• When the object is immediately in front of you, the intervening
fog has τ = 0• As the object moves away, τ increases• τ is frequency-dependent• We use τ as a vertical distance scale
1D Radiative Transfer Equation
• Equation from before
• Divide by κ
• Substitute τ and useKirchoff’s Law
!
dI(z,",t)
dz= #(z,",t) $%(z,",t)I(z,",t)
dI(z,",t)
%(z,",t)dz=#(z,",t)
%(z,",t)$ I(z,",t)
$dI(&,",t)
d&= B(&,",t) $ I(&,",t)
1D Radiative Transfer Solution
!
dI(",#,t)e$"
d"$ I(",#,t)e$" = $B(",#,t)e$"
dI(",#,t)e$"
d"$ I(",#,t)e$"
%
& '
(
) *
0
+
, d" = $B(",#,t)e$"d"0
+
,
I("+,#,t)e$"=$+ $ I("
0,#,t)e$"= 0
= $B(",#,t)e$"d"0
+
,
I("0,#,t) = B(",#,t)e$"d"
0
+
,
Emergent intensity is like a black body with “bites” taken out of it.
Earth’s Thermal EmissionSpectrum
Pearl and Christensen 1997
Emergent intensity is like a black body with “bites” taken out of it.
!
I("0,#,t) = B(",#,t)e$"d"
0
%
&
Lecture Contents
• Overview of Equations for PlanetaryAtmospheres
• Radiative Transfer• Thermal Inversions• Molecular Line Primer
Origin of Absorption andEmission Lines
Constant Temperature
Assume that the deepestlayer of a planetaryatmosphere is a black body.
For an isothermal planetatmosphere, the spectrumemerging from the top of theatmosphere will also be ablack body
Constant Temperature
!
I(0,",t) = B(T,",t)
...
I3(z,",t) = B5(z,",t)
dI3(z,",t)
dz= 0
dI3(z,",t)
dz= #3(z,",t) $%3(z,",t)I4 (z,",t)
dI4 (z,",t)
dz= 0,I4 (z,",t) = B5(z,",t)
dI4 (z,",t)
dz= #4 (z,",t) $%4 (z,",t)I5(z,",t)
I5(z,",t) = B5(z,",t)!
"(z,#,t) =$(z,#,t)B(z,#,t)
Rea
d go
ing
up
Constant Temperature
• No absorption or emission lines!• Mathematically, constant T means B(τ,ν,t) = constant
and can be removed from integrand!
I("0,#,t) = B(",#,t)e$"d"
0
%
&
!
e"#d#
0
$
% =1
I(#0,&,t) = B(T,&)
Origin of Absorption andEmission Lines
Decreasing T with IncreasingAltitude
Think of this situation as coolerlayers of gas overlying hotterlayers. According to theconceptual picture law, anabsorption spectrum will result.
Decreasing T with IncreasingAltitude
!
"(z,#,t) =$(z,#,t)B(z,#,t)
!
I("0,#,t) = B(",#,t)e$"d"
0
%
&Take two values of κ at neighboringfrequencies.κ1 is in the band center andκ2 is in the neighboring “continuum”;κ1 is >> κ2.Consider an optical depth τFrom optical depth definition τ = κs,s1 is shallower than s2.And T1 < T2, and B1 < B2.Therefore, I1 < I2 namely the intensityis lower in the band center than thecontinuum, generating an absorptionline.
Decreasing T with IncreasingAltitude
Do you expect emission or absorptionlines if the temperature is increasingwith altitude?
Think of the simple conceptual pictureof where emission lines come from.
Earth
Hanel et al. 2003
Lecture Contents
• Overview of Equations for PlanetaryAtmospheres
• Radiative Transfer• Thermal Inversions• Molecular Line Primer
Atmosphere Processes
Kiehl and Trenberth, 1997
Earth’s Thermal EmissionSpectrum
Pearl and Christensen 1997
Molecular absorption controls energy balance.Which molecule is Earth’s strongest greenhouse gas?CO2, N2, H2O, N2O, or CH4
Turnbull et al. 2007
Line-By-Line Opacities
Burrows and Sharp 2007
Absorption Coefficient
!
k" = Sf (" #"0)
k is in units of [m2/molecule]S is the line strengthf is the (normalized) broadening functionν0 is is the wavenumber of an ideal,monochromatic line
For a given molecule ata given frequency
!
k"#$
$
% dv =S Normalized
*Note: κν= nkν in units of [m-1], where n is the number density for agiven molecule
Summary
!
k" = Sf (" #"0)
S ~ Rij
2
!
f " #"0( )
#$
$
% d" =1
!
Bfi =1
4"
8" 3# fi
3hc$ f
* µ $i( )dV%2
& f
*&id'%2
!
N j
N=
g je"E j /KT
gie"Ei /KT
i
#
We want to find the absorption coefficient for a given transition
Line strength is related to the square oftransition probability, itself originatingfrom quantum mechanics
Selection rules for permitted lines
The energy level population ratio in LTEis defined by Boltzmann statistics. g isthe degeneracy.
!
"# = k#B# Emission coefficient in LTE (Kirchoff’s Law)
Line broadening includes Doppler broadeningand collisional broadening.
Summary
!
k" = Sf (" #"0)
S ~ Rij
2
!
f " #"0( )
#$
$
% d" =1
!
Bfi =1
4"
8" 3# fi
3hc$ f
* µ $i( )dV%2
& f
*&id'%2
!
N j
N=
g je"E j /KT
gie"Ei /KT
i
#
We want to find the absorption coefficient for a given transition
Line strength is related to the square oftransition probability, itself originatingfrom quantum mechanics
Selection rules for permitted lines
The energy level population ratio in LTEis defined by Boltzmann statistics. g isthe degeneracy.
!
"# = k#B# Emission coefficient in LTE (Kirchoff’s Law)
Line broadening includes Doppler broadeningand collisional broadening.
H Atom Energy Levels
Molecular Energy Levels
• Electrons travel much fasterthan nuclei
• We may assume that theelectronic energy depends onlyon the positiions of the nucleiand not on their own velocities
• The potential energy distributionis then a function of theinternuclear distances alone
• The potential energy has aminimum at locations where theattractive binding forces balancerepulsive internuclear forces.
Molecular Energy Levels
• Which molecules haverotational-vibrationaltransitions?
• Must have an electric dipoleto interact with radiation
• Electric dipole: a differencebetween the center ofcharge and the center ofmass
• Homonuclear molecules donot have a dipole moment
Brent Iversonhttp://www.chem.tamu.edu/organic/Spring99/dipolemoments.html
Rotational Energy LevelsRotation of a diatomic molecule• approximate the molecule astwo masses m1 and m2 at a fixedseparation r1 and r2• the molecule rotations about anaxis perpendicular to the linejoining the nuclei
!
E =1
2m
ivi
2
i
" =1
2I# 2
=L2
2I
ω is the rotation frequency
I is the moment of inertia = Σimiri2
L is the angular momentum L = Σimi ω ri
Linear diatomic molecules can be approximated by a classicalrigid rotor. The classical expression for energy isTotal energy = Kinetic energy + Potential energy.A classical rigid rotor has no potential energy.
Rotational Energy LevelsIn classical theory E and ωcan assume any valueIn quantum mechanics, therigid rotor can only exist indiscrete energy states.
To find the energy states,we would use the time-independent Schrodingerequation (with theHamiltonian expressed interms of the angularmomentum operator).
A pure rotational spectrumhas energies at microwavefrequencies.
!
H" =L2
2I" = E"
H" =J J +1( )h2
2I"
!
EJ
=h2
2IJ(J +1)
!
B =h2
2I
J =0, 1, 2, …
!
NJ
N0
= (2J +1)e"BJ (J +1)/KT
Vibrational Energy Levels
• In a linear diatomicmolecule all vibratorymotion takes place alongthe line joining the atoms.
• Think of the atoms inperiodic motion with respectto the center of mass.
• As the molecule vibrates,rotational modes are alsoexcited. Vibration androtation occur together.
The potential energy can beapproximated by the parabolic formof the simple harmonic oscillator.
Similar to a simple harmonicoscillator but with quantized energylevels at equally spaced values.
Vibrational Energy Levels
µ is the reduced massk denotes the normal modesC is the bond force constant
Energies ~ infrared
!
E =1
2m1v1
2 +1
2m2v2
2 +V (x)
V (x) =1
2Cx
2
E =p
2µ+
1
2Cx
2
...Associate with the QM operator ...
Ev = v +1/2( )h1
2"
C
µ
#
$ %
&
' (
Evk= h)0 vk +1/2( )
vk = 0, 1, 2K
!
"0
=C
µ
Rotational Vibrational Transitions
ΔJ = -1 P branchΔJ = 1 R branch
Simulated intensity of rotational-vibrationaltransitions of the CO molecule for the v = 1-0band.
The vibrational transitions are alwaysaccompanied by rotational transitions.
Rotational Vibrational Cross-Sections for CO
From the VPL website
v = 1-0
v = 2-0
Non-Linear Molecules
3N-6 vibrational modes, where N is the number of atoms
http://en.wikipedia.org/wiki/Water_absorption
Water VaporCrossSections
From the VPL website
Summary• Overview of Equations for Planetary
Atmospheres
• Radiative Transfer– 1D equation: change of intensity is the sum of gains to
and losses from the beam• Thermal Inversions
– Absorption lines for no thermal inversion– Emission lines indicate thermal inversion
• Molecular Line Primer– Molecular vibrational and rotational lines (which appear in
the IR spectrum) were outlined for a linear molecule
Planet Atmosphere Equations
!
dP(r)
dr= "
Gm(r)#(r)
r2
!
P = nkT!
Eout = Ein,* + Ein,planet!
dI(s,",µ,t)
ds= #(s,",µ,t) $%(s,",t)I(s,",µ,t)
Energy transport
Consv. of Energy(in each layer)
Hydrostatic Eq.
Ideal Gas Law
Chemical EquilibriumNo simple equation
Want to derive: Flux, T, P, ρ, chemical composition
Summary• Overview of Equations for Planetary
Atmospheres
• Radiative Transfer– 1D equation is change of intensity is the sum of gains
to and losses from the beam• Thermal Inversions
– Absorption lines for no thermal inversion– Emission lines indicate thermal inversion
• Molecular Line Primer– Molecular vibrational and rotational lines (which appear in
the IR spectrum) were outlined
A New Text Book
Exoplanet Atmospheres: PhysicalProcesses
Sara Seager
Princeton University PressShould appear by end of 2009