Radiative Transfer and Molecular Lines Sagan Workshop...

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



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



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


!

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


• 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.


!

"(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.


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



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.

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

Radiative Transfer and Molecular Lines Sagan Workshop...

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