Analysis of High Resolution Propane Spectra and the Interpretation of Titan’s Infrared Spectra For ASTR 498: Independent Study Course

Valerie Klavans

June 2, 2011

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Table of Contents

Glossary .............................................................................................................................................. 3

1. Background .......................................................................................................................... 4

1.1 Objectives .................................................................................................................................. 4

1.2 Titan’s Atmosphere .................................................................................................................... 4

1.3 Propane in Titan’s Atmosphere ................................................................................................... 6

1.4 Vibration/Rotation Spectrum of Propane ................................................................................... 7

1.5 Radiative Transfer .................................................................................................................... 11

2. Laboratory Data Analysis .................................................................................................... 12

2.1 Acquisition of Laboratory Data ................................................................................................. 12

2.2 Characterizing Spectra .............................................................................................................. 13

2.3 Line Modeling ........................................................................................................................... 15

2.4 Results ..................................................................................................................................... 16

3. Radiative Transfer Modeling of Titan’s Atmosphere ............................................................ 18

3.1 Sample Calculations .................................................................................................................. 18

4. Summary/Conclusion .......................................................................................................... 23

Appendix ................................................................................................................................ 25

References ............................................................................................................................. 27

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Glossary

(C3H8) Propane – Spectra of this hydrocarbon were analyzed in this project

(C2H6) Ethane – This hydrocarbon was used in the Radiative Transfer section of this project as an aid in investigating the interaction of radiation with an infrared active molecule having well established spectroscopic parameters.

CIRS – The Composite Infrared Spectrometer is one of the instruments on board the Cassini orbiter that records the infrared spectrum of targets in the Saturnian system and provides details on their chemical composition and thermal structure

ESA – The European Space Agency played a large part in the Cassini-Huygens Mission

FTS – The Fourier Transform Spectrometer is an instrument that collects spectra. The McMath-Pierce FTS is a modified version of the classic Michelson type interferometer. This instrument was used by Dr. D. E. Jennings in obtaining the spectra used in this project.

GEISA 2009 – Gestion et Etude des Informations Spectroscopiques Atmosphériques is a spectroscopic database (similar to HITRAN) for the management and study of atmospheric spectroscopic information. The current distribution was released in 2009.

HIPWAC – Heterodyne Instrument for Planetary Wind And Composition probes planetary atmospheres such as Venus, Mars, Jupiter, Saturn, Titan, and Neptune

HITRAN 2008 – HIgh Resolution TRANsmission is a compilation of spectroscopic parameters used widely for analysis of atmospheric constituents (similar to the GEISA database). The current version was released in 2008.

KPNO – Kitt Peak National Observatory hosts the McMath-Pierce spectrometer used in obtaining the propane spectra discussed herein.

LMM – The Levenberg-Marquardt Method is a technique for model parameter optimization.

NASA – National Aeronautics and Space Administration was one of three partners (along with ESA and the Italian Space Agency – Agenzia Spaziale Italiana) in the Cassini-Huygens Mission

𝝂𝟐𝟏, 922 cm-1 band – Vibrational mode of propane that was primarily analyzed in this project

VPL – The Virtual Planetary Laboratory is a project with the goal of understanding the characteristics and environments of extrasolar planets. The project’s publicly accessible resources include compilations of spectra and spectroscopic parameters for molecules of interest in planetary science.

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1. Background

1.1 Objectives

The goals of this project are to study the following topics:

• Background research o Titan’s atmosphere o Propane molecule o Atmospheric radiative transfer

• Analysis of laboratory propane spectra

• Radiative transfer modeling of simulated Titan atmosphere

1.2 Titan’s Atmosphere

Titan’s atmosphere, dominated by nitrogen, includes a range of trace species such as hydrocarbons, nitriles, and carbon dioxide (those of interest are summarized in Table 1). Due to differences in gravity, Titan’s atmosphere is more extended than Earth’s. Earth’s atmosphere extends to about 50 km while Titan’s atmosphere extends another order of magnitude to roughly 600 km. In addition, Titan has a greenhouse effect as well as an anti-greenhouse effect. Its greenhouse comes from its nitrogen, methane and hydrogen. Titan’s atmosphere is so dense that molecules collide and acquire induced dipole moments.1 These molecules are primarily N2-N2, N2-CH4, and N2-H2 pairs (dimers) in the troposphere. When these molecules collide, their symmetry is perturbed and they can absorb radiation. They are transparent in the visible but opaque in the thermal infrared, which allows shortwave solar radiation to reach the surface but does not let longwave thermal radiation to escape. Titan’s anti-greenhouse is a result of its optically thick organic haze. It absorbs at shortwave solar radiation but is transparent in the longwave thermal infrared. Thus, it regulates the shortwave energy reaching the surface, creating a cooling mechanism.2

In the Radiative Transfer section of this project, tests were made by varying certain parameters for Titan’s troposphere and stratosphere, such as the abundance of major gases, and altering the thermal profile in those regions (see section 3. Radiative Transfer). This paper will only cover Titan’s atmosphere from the troposphere through the stratosphere. The surface temperature of Titan is ~93 K and decreases through the troposphere until the tropopause where the temperature reaches ~71 K (Figure 1). Convection occurs in the troposphere mixing the hotter atmosphere near the surface with the cooler atmosphere closer to the tropopause. Then the temperature increases into the stratosphere, where heating takes place due to aerosol absorption, reaching a maximum temperature at ~170 K. The abundances of most chemical species increase in the stratosphere, and they condense out of the gas phase near the tropopause. In the stratosphere, haze particles reflect and absorb some sunlight, leading to a well-defined stratopause maximum in temperature. Some sunlight reaches the surface, but the greenhouse effect in the troposphere raises the temperature by ~13 K. Examples of trace species abundance profiles in Titan’s atmosphere are shown in Figure 2.

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Figure 1: Titan’s thermal profile in the troposphere and stratosphere. The tropopause is well-defined around 100 mbar separating the troposphere (negative slope) with the stratosphere (positive slope). The stratopause is well-defined at 0.100 mbar. The two gases used in this project are ethane (C2H6) and ethylene (C2H4). Their profiles used this project are reresentative (not measured) as shown in this figure.

Figure 2: Vertical profiles for major hydrocarbons inferred from limb data from CIRS (blue lines, 15°S and red lines, 30°N), and the model simulations (black lines). The model simulations correctly reproduce the results from CIRS nadir and limb data.3

Pres

sure

(mba

r)

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Table 1: Composition of Titan’s atmosphere4

and mixing ratios of trace gases.2

1.3 Propane in Titan’s Atmosphere

The Cassini-Huygens mission is a joint NASA and ESA mission studying Saturn and its natural satellites. The Cassini orbiter’s 12 instruments have been returning cutting-edge information on the Saturnian system since arriving in 2004. One such instrument is the Composite Infrared Spectrometer (CIRS). The CIRS instrument records the infrared spectrum of targets in the Saturnian system and provides details on their chemical composition and thermal structure. CIRS infrared mapping of Titan’s atmosphere has confirmed the presence of many chemical species and detected new isotopic varieties.

Titan is one of the prime targets of the Cassini-Huygens mission. Titan’s atmosphere, dominated by nitrogen, includes a range of trace species such as hydrocarbons, nitriles, and CO2 (Figure 3). One such hydrocarbon is propane (C3H8). Propane is a critical species spectroscopically, as it has 21 active IR bands covering broad regions of the mid-infrared, and so its ubiquitous signature may potentially mask weaker signatures of other undetected species with important roles in Titan’s chemistry. Thus, modeling the C3H8 contribution is of current scientific interest. Unfortunately, C3H8 line atlases for some spectral regions (bands near 869, 922, 1054, and 1157 cm-1) have not yet been analyzed and therefore cannot be used for research. When available, such line atlases will be the basis for modeling and subtracting the C3H8 signal from the CIRS spectral data to reveal, or constrain, the signature of underlying chemical species. In this project, the 922 cm-1 propane band was inspected and fitted with the goal of producing a line atlas of spectroscopic parameters (e.g., frequency and intensity) and then to use it for a Titan radiation model prediction.

The analysis of C3H8 will allow us to better understand the composition of Titan’s atmosphere and planetary atmospheres in general. In the long run, projects like this will help us better understand the

Composition of Titan's Atmosphere

Constituent Formula Equator North Pole Nitrogen N2 0.95-0.984 0.95-0.984 Methane CH4 0.045-0.015 0.045-0.015 Hydrogen H2 0.001 0.001 Acetylene C2H2 3.25x10-6 5.10x10-6 Ethylene C2H4 1.35x10-7 5.50x10-7 Ethane C2H6 8.00x10-6 1.15x10-5 Propyne C3H4 5.90x10-9 2.40x10-8 Propane C3H8 5.00x10-7 6.95x10-7

Diacetylene C4H2 1.32x10-9 2.30x10-8 Benzene C6H6 3.00x10-10 4.20x10-9

Hydrogen Cyanide HCN 1.13x10-7 9.70x10-7 Cyanoacetylene HC3N 5.00x10-10 4.60x10-8 Carbon Dioxide CO2 1.11x10-8 1.40x10-8

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interaction of hydrocarbons and other species in Titan’s atmosphere in photochemical reactions and its methanological cycle. This understanding can shed light on Titan’s weather, along with the feasibility of hydrocarbon rain and lakes on Titan’s surface. Titan is the only other body in the Solar System besides Earth that has flowing liquids on its surface, which could support life. One tantalizing conjecture is that life may even exist within these liquid hydrocarbons.

Figure 3: CIRS model spectral data at an altitude of 125 km5

1.4 Vibration/Rotation Spectrum of Propane

The chemical bonds in polyatomic molecules such as propane can be approximated by a number of masses connected by springs. The movements of each mass (atom) connected by springs (bonds) are comparable to the vibrational and rotational movements of a molecule. Therefore, the motion of each molecule is harmonic, where each oscillator (atom) has its own characteristic frequency and spring constant (from the strengths of bonds). The expected number of modes of vibration for propane is (3x11)–6 = 27 (see Table 2). This number signifies that the 11 atoms in the propane molecule may stretch, bend (scissoring, twisting, rocking or wagging), or exhibit torsion in 27 different ways. A simpler molecule that illustrates vibrational modes is the molecule of water (H2O). As shown schematically in Figure 4, water has two stretching modes (𝜈1, 𝜈3) and two bending modes (𝜈2). 6 An example of a torsion can be seen in the ethane (C2H6) molecule, more similar to propane (Figure 5).

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(a)

(b)

(c)

Figure 4: The normal modes of vibration for H2O7

(a) Symmetric stretching mode: 𝝂1 = 3657 cm-1 and (b) Anti-symmetric stretching mode: 𝝂3 = 3756 cm-1 (c) two bending modes on the x and y axes, the scissoring and rocking modes: 𝝂2 = 1595 cm-1 (rocking mode is not shown)

Figure 5: Sample torsional mode of the ethane molecule8

Rotation of a molecule occurs when a molecule rotates about an axis with a non-zero moment of inertia. The energy of rotational transitions is much lower than that of vibrational transitions.

Propane is a much more complex molecule than ethane or water. It has many more atoms which increases the number of degrees of freedom in which to vibrate. Table 2 shows the normal modes of vibration for propane. Each one corresponds to a certain vibration of the propane molecule. For example, Figure 6 shows the stretching and bending (in-plane: scissoring and rocking; out-of-plane: twisting and wagging) modes of vibration for a molecule containing a CH2 group. Note that the figure does not show the CH3 modes given in Table 2.

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Table 2: Normal Vibration Modes of Propane from VPL9

* Infrared-inactive modes

Normal Vibration Modes of Propane

Fundamental Frequencies Vibration

ν1 2977 cm-1 CH3 stretch

ν2 2962 cm-1 CH3 deform ν3 2887 cm-1 CH2 stretch ν4 1476 cm-1 CH3 deform ν5 1462 cm-1 CH2 scissor ν6 1392 cm-1 CH3 deform ν7 1158 cm-1 CH3 rock ν8 869 cm-1 CC stretch ν9 369 cm-1 CCC deform ν10

* 2967 cm-1 CH3 stretch ν11

* 1451 cm-1 CH3 deform ν12

* 1278 cm-1 CH2 twist ν13

* 940 cm-1 CH3 rock ν14

* 216 cm-1 torsion ν15

* 2968 cm-1 CH3 stretch ν16 2887 cm-1 CH3 stretch ν17 1464 cm-1 CH3 deform ν18 1378 cm-1 CH3 deform ν19 1338 cm-1 CH2 wag ν20 1054 cm-1 CC stretch ν21 922 cm-1 CH3 rock ν22 2973 cm-1 CH3 stretch ν23 2968 cm-1 CH2 stretch ν24 1472 cm-1 CH3 deform ν25 1192 cm-1 CH3 rock ν26 748 cm-1 CH2 rock ν27 268 cm-1 torsion

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Figure 6: Normal vibration modes (stretching and bending) for the CH2 (methylene) group10

Figure 7: Propane molecular structure11

showing one CH2 (methylene) group surrounded by two CH3 (methyl) groups

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1.5 Radiative Transfer

In order to understand propane’s role in Titan’s spectrum, one must understand radiative transfer. This is the process by which electromagnetic radiation propagates through a medium, in this case, the atmosphere of Titan. By studying reflected, absorbed, and emitted radiation, one can gain insight into the properties of planetary atmospheres. In particular, infrared spectroscopy tells us about the physical and chemical traits of the emitting or absorbing material.12

The Schwarzschild Equation explains the transmission of radiation of a particular frequency through an absorbing medium. The change in intensity after passing through a thin layer which absorbs a fraction of incident radiation (first term) and contributes emission (second term) is given by:

𝑑𝐼 = – 𝐼𝜅𝜌𝑑𝑧 + 𝐵𝜅𝜌𝑑𝑧

where:

• 𝐼 = radiation intensity at any given height (erg/cm2/cm-1/sr)

• 𝜅 = absorption coefficient (cm-2/g)

• 𝜌 = density of the absorbing/emitting gas (g/cm-3)

• 𝑧 = thickness of absorbing layer (cm)

• 𝐵 = Planck function, the source function:

𝐵 =2𝜋ℎ𝑐2𝜈3

𝑒ℎ𝑐𝜈/𝑘𝑇 − 1

where:

• 𝜈 = wavenumber of the line being considered (cm-1)

• 𝑇 = absolute temperature (K)

• 𝑐 = speed of light = 2.998 × 1010 cm/s

• ℎ = Planck’s constant = 6.626 × 10-27 cm2 g / s

When the Schwarzschild Equation is integrated on both sides:

�𝑑𝐼

(𝐼 − 𝐵)= −𝑘𝜌𝑧 + 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

ln(𝐼 − 𝐵) = −𝑘ρ𝑧 + constant

When 𝑧 = 0, 𝐼 = 𝐼0, therefore, ln(𝐼0 − 𝐵) = constant

ln �𝐼 − 𝐵𝐼0 − 𝐵

� = −𝑘𝜌 𝑧

where, 𝐼0 = incident radiation intensity

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Simplifying the previous equation:

𝐼 = 𝐼0 𝑒𝑥𝑝(−𝑘ρ𝑧) + 𝐵[1 – 𝑒𝑥𝑝(−𝑘ρ𝑧)]

(1)

where 𝑘𝑝𝑧 = 𝜏, the optical depth. Rewriting the above equation,

𝐼 = 𝐼0𝑒−𝜏 + 𝐵(1 – 𝑒−𝜏)

When 𝜏 ≪ 1 (optically thin medium) 𝐼 = 𝐼0(1 − 𝜏) + 𝐵(𝜏)

When 𝜏 ≫ 1 (optically thick case) 𝐼 = 𝐵

In equation (1), the formal solution of the Schwarzschild Equation, the two terms take into account that an atmosphere not only absorbs but also emits radiation. The first term of this equation is an expression of the Beer-Lambert Law, which relates the absorption of incident light to the medium in which it travels. Also, the second term is the source function (i.e. emission) modulated by the opacity of the medium.13

2. Laboratory Data Analysis

2.1 Acquisition of Laboratory Data

C3H8 spectra taken at the McMath-Pierce 1-meter Fourier Transform Spectrometer (FTS) at Kitt Peak National Observatory (KPNO) by Jennings et al.14

We used two different data sets taken on April 26, 1984 (“warm data”) and on May 15, 1989 (“cold data”). (They were named “warm” and “cold” data for their differing observational ambient temperatures.) For the warm data, the laboratory conditions are as follows. The 2400 cm gas cell was filled with propane, at a pressure of .558 torr and observations were made at an ambient temperature of 293 K. For the cold data, the 30 cm gas cell was filled with propane, at a pressure of 3.2 torr and observations were made at an ambient temperature of 175 K.

were used in this analysis. (See Appendix A for all propane data taken.) The spectra were obtained at full-resolution of 0.005 cm-1 and the raw interferograms were already converted to calibrated spectra (by application of a Fourier Transform, followed by radiometric and wavelength calibration).

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2.2 Characterizing Spectra

There were no empty cell data available for either warm nor cold data (gas plus cell data). Usually subtraction of cell data from gas plus cell data results in only gas data with a smooth, linear baseline (Figure 8). An example region on an expanded scale is shown in Figure 9.

Figure 8: Transmittance shows a smooth, linear baseline for C3H8 IR spectrum: 720-1250 cm-1

Figure 9: Sample C3H8 spectral region 929.627 to 930.55 cm-1

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Therefore, in order to obtain a flat baseline, attempts were made to model the shape of the baseline (Figure 10). Initially, a spline was used to model the structures in the baseline. However, this method did not prove sucessful because there were at least two visible fringes (instrumental interference) and the overall shape of the baseline also had a component from the strength of the lines themselves.

In Figure 10a, the gas plus cell data is shown in white near the 𝜈21 band. The gas plus cell data was smoothed in an attempt to manually subtract the overall baseline shape, shown in red. After attempting this method, a manual fit of the fringes was then attempted, shown in Figure 10b. This also did not prove sucessful because there were two fringes or possibly more present that could not be modeled easily. A sine wave function was used to model the underlying fringe (shown in red):

𝐼 = 𝐴 𝑠𝑖𝑛(𝑘1𝜆 + 𝜙1) 𝑠𝑖𝑛(𝑘2𝜆 + 𝜙2)

(a)

(b)

Figure 10: (a) First baseline subtraction technique – smoothing. (b) Second baseline subtraction technique – modeling the fringe patterns. In an attempt to model and then subtract them, another fringe pattern appeared. In both images: white – original spectrum, red – one modeled fringe

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After these attempts were made, it was found that regions (~.25 cm-1 or smaller) showed to have very little variance in baseline. Therefore, .25 cm-1 regions were fitted sequentially.

Then, the spectral region around the 922 cm-1 band was inspected. Selected regions that did not have strong spectral lines, and therefore presumably dominated by noise, were used to estimate the noise level. The noise level was determined by the standard deviation (assuming Gaussian noise):

𝜎 = �∑ (𝑥𝑖 − 𝑥0)2𝑁𝐷𝑖=1𝑁𝐷 − 1

2.3 Line Modeling

As an initial test of the fitting procedures, allene (CH2CCH2) was fit in a region of known spectroscopic parameters from 1002.4 to 1002.8 cm-1 (𝜈9 999 cm-1). Figure 11 shows the comparison between previously published results10 and initial guesses for the allene lines generated by an automated algorithm which was based on the first order differential of the spectra.

(a)

(b)

Figure 11: (a) Figure from VPL Molecular Spectroscopic Database15

of allene (C3H4) (b) Testing my peak–finding technique using allene. The dotted lines refer to initial guesses of the peaks of individual allene lines. The solid lines are actual data.

In our initial fitting we approximated the shape of the spectral lines with a Gaussian line shape:

𝐼𝐼0

= 𝑒−(𝜈−𝜈0)2

𝛿𝜈2

where 𝐼 = intensity, 𝐼0 = amplitude, 𝜈 = frequency, 𝜈0 = linecenter, and 𝛿𝜈 = half-width. The example region shown in Figure 9 spans the 929.627 to 930.55 cm-1 interval and contains 𝑁𝐷 = 208 data points.

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Each spectral line is assigned two parameters which characterize the line shape: transition frequency (𝜈0𝑖) and amplitude {𝐼0𝑖} for the 𝑖𝑡ℎ line. The half-width, 𝛿𝜈, is assumed to be common for all lines. The automated line guessing program identifies lines deeper than 3s (where s is the noise level) and reports the center frequency and amplitude for each. For the spectral range in Figure 12a, the algorithm found

𝑁𝐿 = 26 spectral lines. Manual inspection shows ~23 are acceptable. Figure 12c shows the guesses without the manual rejections.

(a)

(c)

(b)

Figure 12: (a) Plot of linecenters and amplitude guesses for region: 929.627 to 930.55 cm-1. (b) Gaussian multi-line fit for region: 929.627 to 930.55 cm-1. Top has initial guesses with data points. Bottom shows the LMM Gaussian fit with data points and residuals below. (c) Table of fit results.

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These guesses were fed as initial values to a non-linear parameter optimization program to determine solutions for the set of line shape parameters: half-width, intensity, and an offset correction. This algorithm is based on the Levenberg-Marquardt method (LMM) for parameter optimization (see Appendix B) and is implemented in an IDL code called curvefit. As a part of this analysis we developed IDL based codes for parsing of the data, initial guess generation, and interfacing to the standard routines.

2.4 Results

Fitting of large regions, several wavenumbers or more, resulted in a poor fit. Thus, it was found by trial-and-error that smaller sections on the scale of a quarter of a wavenumber should be fit in a larger pipeline procedure to make the fitting process accurate and more efficient. Figure 13 shows a large region of the 𝜈21 band modeled. For the region, 925.0 to 932.2 cm-1, the fitting program was able to mimic the shape of the baseline. Towards 925.0 cm-1, the lines become deeper because this is part of the R branch of the 922 cm-1 band. Although the baseline is fixed (the offset 𝑂𝑆 is constrained), the strength of the lines create the cascading shape. However, smaller sections were not fit properly as shown in the residuals below the fit region.

Figure 13: Gaussian fit for region: 925.0 to 932.2 cm-1. The 929.627 to 930.55 cm-1 region (Figure 9) is a subsection of this larger region. The subsection was not being fitted in the larger region. Therefore, it was fitted for a smaller range. The smaller the range, the more acceptable the fit.

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From the converged solutions to the Gaussian fit, the line amplitudes can be used to determine the line strengths (𝑆). Transmittance due to the gas is given by

𝑇𝜈 = 𝑒−𝜏𝜈

and 𝜏𝜈 is the optical depth: 𝜏𝜈 = 𝜅𝜌𝛥𝑧

where 𝜅 = opacity (cm2/g), 𝜌 = density (g/cm3), 𝛥𝑧 = path length (cm). 𝜅𝜌 can be approximated using 𝑆𝑝, where 𝑆 = line intensity (cm-1 atm-1) and 𝑝 = pressure (atm). For an optically thin line, the calibrated

intensity of the 𝑖𝑡ℎ line is given by

𝑆𝑖 =𝐼0𝑖𝑝𝛥𝑧

and the line intensities can be found. These can be used to model the emission of chemical species from each layer in an atmosphere.

Therefore, an IDL pipeline of procedures were made to make initial line lists for the 𝜈21 and 𝜈20 bands. It converted amplitudes to calibrated intensities and made an initial line list in the GEISA 2009 and HITRAN 2008 formats.

3. Radiative Transfer Modeling of Titan’s Atmosphere

3.1 Sample Calculations

The radiative transfer code (CoDAT) used in this analysis is based on the formal solution to the Schwarzschild Equation discussed in the section 1.5 Radiative Transfer and is from the members of HIPWAC16. For the purposes of understanding the radiation transport in Titan, I assumed a representative Titan thermal profile (Figure 1) including the troposphere through the stratosphere. Two gases were used in this code: ethane and ethylene. The surface temperature was set to ~93 K. The line atlas that was used as the source for the molecular parameters was HITRAN 200817

Figure 14 shows a small portion of the simulated Titan spectrum with ethane (one line). It is using an un-altered Titan thermal profile and un-altered abundances for the hydrocarbons. The sample calculations in this section were performed in order to understand how changing the thermal profile and abundances of hydrocarbons present in the atmosphere affects the resultant spectra.

.

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Figure 14: Spectrum using representative Titan thermal and abundance profiles

When setting the thermal profile as isothermal in the stratosphere of Titan’s atmosphere to 80 K, one will see an absorption spectrum. This is because the temperature of the surface (93 K) is higher than the stratosphere (Figure 15a). Kirchoff’s laws tell us that a source viewed through a cool gas will produce an absorption spectrum. The stratosphere acts as this cool gas. When the stratospheric temperature exceeds the surface temperature one will see an emission spectrum, as when the thermal profile is isothermal at 110K in the stratosphere. Not much change occurs when the thermal profile is altered in the lower troposphere because the temperatures are too cold and the vapor pressure is too low (Figure 15b). C2H4 and C2H6 condense out and rain to the surface. However, one will notice that the wings of the lines grow when the tropospheric temperature is increased. This is due to the tropospheric temperatures reaching sufficient values to appear in emission, and these regions being dominated by collisions that broaden the line shape more than at higher altitudes (which have lower pressures). I judiciously selected the abundance so that the lines are barely saturating in the isotherm region. Figure 16 demonstrates the effect of increasing the isotherm temperature: the brightness increases with temperature, and the brightness temperature of the line matches the isotherm temperature.

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(a)

(c)

(b)

Figure 15: (a) Plot of thermal profile altered isothermally for 80K in the stratosphere. (b) Plot of thermal profile altered isothermally for 110K in the stratosphere. (c) Resultant spectrum after altering the thermal profiles isothermally. Red line = 80K Isotherm. Blue line = 110K Isotherm. Assuming surface temperature of 93 K.

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(a)

(c)

(b)

(d)

Figure 16: Plots (a)-(c) Thermal profiles altered isothermally for 140, 170, and 200K in the stratosphere. (d) Resultant spectrum after altering the thermal profiles isothermally. Red line = 140K Isotherm. Green line = 170K Isotherm. Blue line = 200K Isotherm.

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In addition to altering the thermal profile, one can alter the abundances of molecules in the atmosphere. Figure 17a shows the result of changing the abundance of ethane to .1, 1 and 10 times the original abundance. The wings of the line as well as the amplitude grow with increased abundance. In Titan’s atmosphere, these hydrocarbons would exist mainly in the stratosphere, as they freeze out in the troposphere. This change in abundance simulates what could happen in a change of seasons when certain portions of Titan receives different amounts of sunlight, resulting in the destruction or production of a certain species.

In Figure 17b more lines were added to the spectrum to simulate a more realistic Titan atmosphere. Just as in the single-line example, the wings and amplitude of the lines increased with increased abundance. Note the saturation that occurs in the strong line at 851.5 cm-1, while the weaker lines remain optically thin and therefore their amplitude continues to increase with increasing abundance.

Figure 17a: Plot of varying abundances of ethane in the model Titan atmosphere. As the abundance increases, the lines widen and the wings grow. (red = .1x original abundance, green = 1x, blue = 10x)

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Figure 17b: Plot of varying abundances of ethane in the model Titan atmosphere. As the abundance increases, the lines widen and the wings grow. This plot would be equivalent to figure 15, but this plot shows more than one line to further illustrate the growth of the wings.

4. Summary/Conclusion

The purpose of this project was to study Titan’s atmosphere, the propane molecule, and atmospheric radiative transfer to understand the analysis of laboratory propane spectra with the application of radiative transfer modeling of a simulated Titan atmosphere. This was achieved by researching how Titan’s atmosphere behaves, the basics of vibrations and rotations of simple molecules, and the theory behind radiative transfer. It was also achieved by analyzing propane spectra over a number of years through fitting individual propane lines in certain bands. Then, understanding and altering a model of Titan’s atmosphere brought all the knowledge gained full circle.

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For future work, we could use the calculated intensities to model propane spectra of Titan from Cassini/CIRS. Also, the Gaussian approximations for the individual lines could be replaced by Voigt lineshapes. This will take into account Doppler and pressure broadening of lines (derived from temperature and pressure measurements from the observing conditions).

This project is just the beginning of creating line atlases for the C3H8 bands. The 𝜈8, 869 cm-1; 𝜈20, 1054 cm-1; and 𝜈7, 1157 cm−1 bands of C3H8 still need to be analyzed. The atlases will be needed to subtract the C3H8 signal from the infrared spectrum of Titan from CIRS. Subtracting the 𝜈21 band may result in the detection of propylene (C3H6) at 912 cm-1. Ultimately, these line atlases will be used to search for more chemical species in Titan’s stratosphere. Then, we will analyze other molecular species for the same goal, e.g. allene.

In addition, more radiative transfer modeling using the propane line lists along with other hydrocarbons present in Titan’s atmosphere may lead to a better understanding of the dynamics as well as constituents in Titan’s atmosphere.

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Appendix A

Figure 18: Table of propane spectra taken at KPNO from 1983-1989

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Appendix B

The Levenberg-Marquardt Method (LMM) is a standard technique used in fitting non-linear models. Its purpose is to minimize a statistic, chi-squared, which estimates the deviation of the model from the data and find a solution for the model parameters. It should be noted that the LMM finds only a local minimum not a global minimum.18

In the LMM, the degree of the model fit is characterized by Chi squared

𝜒2 = �(𝐷𝑖 − 𝑀𝑖)2

𝜎𝑖2

𝑁𝐷

𝑖=1

where 𝐷𝑖, 𝑀𝑖, 𝜎𝑖, are the data, model, and error values, respectively, for the 𝑖𝑡ℎ element. The spectral model for the lines is a function of the set of parameters {𝑃𝑗}: 𝐺𝑘 (𝜈𝑖) = 𝐺𝑘 ({𝑃𝑗}, 𝜈𝑖) In the LMM, the

parameter values are iteratively changed to search for a minimum in 𝜒2. In our case, the model consists of a complement of Gaussian functions that represent these spectral lines. The bandshape is represented by the contribution of this set of lines and is given by

𝑀𝑖 = 𝑂𝑆 −�|𝐼0𝑘|𝑒−(𝜈−𝜈0𝑘)2

𝛿𝜈2

𝑁𝐿

𝑘=1

The vector of parameters (𝑂𝑆 , 𝛿𝜈,{𝜈0𝑖}, {𝐼0𝑖}) has 2𝑁𝐿+2 elements, where 2𝑁𝐿 refers to the parameters {𝜈0𝑖}, {𝐼0𝑖} characterizing each of 𝑁𝐿 spectral lines and the +2 refers to the offset correction term (𝑂𝑆) and half-width (𝛿𝜈) which is common to all lines. For the example in Figure 6, the total number of derived parameters is 54 (2𝑁𝐿 + 2 = 2(26) + 2). The number of data points per line is ~8 (𝑁𝐷/𝑁𝐿 = 208/26).

27

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