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Title: Synergistic Study of Hydrocarbon Photochemistry in the Laboratory and Planetary
Atmospheres
Article Type: Special Issue: Outer Planets V-Coustenis
Keywords: Photochemistry; Hydrocarbon kinetics; Laboratory; Outer planets and their satellites;
Jupiter; Titan
Corresponding Author: Dr. Mao-Chang Liang, Ph.D.
Corresponding Author's Institution: Academia Sinica
First Author: Mao-Chang Liang, Ph.D.
Order of Authors: Mao-Chang Liang, Ph.D.; Daven Henze; Mate Adamkovics; Emily F Chu;
Kristie Boering; Yuk L Yung
Abstract: A synergistic study of hydrocarbon photochemistry in the laboratory and planetary
atmospheres has been carried out using the Caltech/JPL KINETICS photochemical model and
laboratory measurements from Adamkovics and Boering (2003). The laboratory simulations provide
the data for the time-evolution of gaseous species such as H2, C2H2, C2H4, C2H6, C3H4, C4H2
and C4H10 during UV irradiation of CH4. We apply forward and adjoint models to analyze the
experiments. Different photochemical schemes (e.g., Moses et al. 2000, 2005) are compared and
modified to reproduce the laboratory results. We first test the full sensitivity of the model results to
all chemical kinetics using the adjoint model and show that the abundances of C2H2, C2H4, C2H6,
and C4H10 can be well reproduced while that of C4H2 is underestimated by 1-2 orders of
magnitude. The abundance of C3H4 is underestimated with Moses et al.' (2000) kinetics but
overestimated with Moses et al.' (2005) kinetics. This suggests a major gap in our understanding of
chemical pathways to higher hydrocarbons. We next examine higher order hydrocarbon chemistry
(>C2). In this study, we assume that all rate coefficients for the chemistry of C1 and C2
hydrocarbons remain invariant in the adjoint optimization. Better agreement is achieved, but
complete agreement remains elusive. Further laboratory measurements are urgently needed to
constrain the pathways. The implications for modeling the atmospheres of Titan and the giant
planets (e.g., Jupiter) are discussed.
Manuscript
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Synergistic Study of Hydrocarbon Photochemistry in the Laboratory and Planetary
Atmospheres
Mao-Chang Lianga,b,*, Daven Henzec, Mate Adamkovicsd, Emily F. Chue, Kristie
Boeringf, and Yuk L. Yungg
a
Research Center for Environmental Changes, Academia Sinica, Taipei, Taiwan.
b
Graduate Institute of Astronomy, National Central University, Jhongli, Taiwan.
c
NASA Goddard Institute for Space Studies and the Earth Institute, Columbia University,
New York, New York, USA.
d
Department of Astronomy, University of California at Berkeley, Berkeley, USA.
e
Departments of Chemistry and of Earth and Planetary Science, University of California,
Berkeley, USA
e
Department of Chemistry, University of California at Berkeley, Berkeley, USA.
g
Division of Geological and Planetary Sciences, California Institute of Technology,
Pasadena, USA.
* To whom all correspondence should be addressed. E-mail: mcl@rcec.sinica.edu.tw
TO BE SUBMITTED TO PSS, NOVEMBER 6, 2008
1
Abstract
A synergistic study of hydrocarbon photochemistry in the laboratory and planetary
atmospheres has been carried out using the Caltech/JPL KINETICS photochemical model
and laboratory measurements from Adamkovics and Boering (2003). The laboratory
simulations provide the data for the time-evolution of gaseous species such as H2, C2H2,
C2H4, C2H6, C3H4, C4H2 and C4H10 during UV irradiation of CH4. We apply forward and
adjoint models to analyze the experiments. Different photochemical schemes (e.g., Moses
et al. 2000, 2005) are compared and modified to reproduce the laboratory results. We first
test the full sensitivity of the model results to all chemical kinetics using the adjoint
model and show that the abundances of C2H2, C2H4, C2H6, and C4H10 can be well
reproduced while that of C4H2 is underestimated by 1-2 orders of magnitude. The
abundance of C3H4 is underestimated with Moses et al.’ (2000) kinetics but
overestimated with Moses et al.’ (2005) kinetics. This suggests a major gap in our
understanding of chemical pathways to higher hydrocarbons. We next examine higher
order hydrocarbon chemistry (>C2). In this study, we assume that all rate coefficients for
the chemistry of C1 and C2 hydrocarbons remain invariant in the adjoint optimization.
Better agreement is achieved, but complete agreement remains elusive. Further laboratory
measurements are urgently needed to constrain the pathways. The implications for
modeling the atmospheres of Titan and the giant planets (e.g., Jupiter) are discussed.
2
1. Introduction
The thermodynamically stable form of carbon in the solar giant planets is CH4. However
in the mesosphere of giant planets and Titan, the molecule undergoes photofragmentation by absorption of solar UV photons. As a result, photochemical processes
dominate in the upper stratosphere, mesosphere and lower thermosphere. In the giant
planets, CH4 is the most abundant molecule next to H2; the volume mixing ratio is ~3103
on Jupiter, 4.510-3 on Saturn, 210-5 on Uranus, 810-3 on Neptune, and 210-3 on
Titan. The resulting hydrocarbon abundances (mainly C2H6) are about three orders of
magnitude lower than that of CH4 (e.g., Yung and DeMore 1999). Higher order
hydrocarbons can condense to form aerosols (tholins), which affect the optical and
thermal properties of the atmosphere. Of the known atmospheres, Titan has the richest
hydrocarbon chemistry because of the lower abundance of H2, which plays an important
role of recycling hydrocarbons back to their parent molecule CH4. In this paper, we focus
primarily on Jupiter and Titan; the former has high H2 abundance (volume mixing ratio
~1) and the latter is low (~0.1%).
Methane is dissociated mainly by ultraviolet photons with wavelengths between Ly-
and ~140 nm, where there is no shielding by H2 or N2. As a result, higher order
hydrocarbons are formed. For example, one consequence of the CH4 photolysis is 2CH4 +
h  C2H6 + 2H (or H2) (e.g., Yung and DeMore 1999). On giant planets such as Jupiter,
H and H2 are retained in the atmospheres by gravity. On satellites such as Titan, H and H2
can escape readily (e.g., Yelle et al. 2006), resulting in low hydrogen abundance in the
atmosphere, and hence, formation of higher hydrocarbons provides an irreversible sink
for CH4. This implies that there must be a source of CH4 in/at the surface of Titan and
deposits of complex hydrocarbon/nitrogen compounds at the surface (Yung et al. 1984).
Though the current hydrocarbon kinetics schemes have been accepted for years, there
remain major uncertainties. For example, heterogeneous chemistry has been included
only recently (e.g., Liang et al. 2007; Zhou et al. 2008); prior to this only gas-phase
chemistry was considered. One reason is the lack of laboratory experiments that are
3
appropriate for planetary conditions. Another reason is that the current kinetics seems to
be satisfactory for the chemical processes in the atmospheres of giant planets (e.g., Moses
et al. 2000a,b; Moses et al. 2005a,b), though not for Titan (e.g., Lebonnois et al. 2001;
Liang et al. 2007), suggesting incompleteness in the chemical kinetics. In this paper, we
carefully analyze time-resolved laboratory experiments (Adamkovics and Boering 2003);
forward and adjoint models are used to assess various hydrocarbon schemes via
simulation and assimilation of the experimental results. Applications and implications for
giant planets and their satellites are presented and discussed.
2. Hydrocarbon Kinetics and Models
2.1 Laboratory Simulation
A first attempt to monitor the time-evolution of hydrocarbons and aerosols in the
laboratory was made by Adamkovics and Boering (2003). The results potentially provide
a unique tool for validating not only chemical pathways but also their associated rate
coefficients. Pure CH4 gas at 70 torr (93 mbar) was irradiated with 8.80.81015 photons
s-1 vacuum UV light (Figure 1) for ~70 hours. The mixing ratios of seven gaseous species
(H2, C2H2, C2H4, C2H6, C3H4, C4H2, and C4H10) and aerosol scattering were reported, but
the aerosol chemical composition was not determined. The net production rates were
9.92.21011, 2.80.51010, 5.59.4109, 8.62.51010, 2.51.2109, 6.65.0108,
1.30.51010, 7.53.11011 cm-3 s-1, respectively. The UV flux reaching the sample was
not constant in time, decreasing as UV-shielding aerosols formed on the window of the
UV lamp. Though the actual UV flux over time was not known, it was estimated by
matching the production of H2, which is chemically inert, yielding an empirical
attenuation coefficient of 0.0158 hr-1 (Adamkovics and Boering 2003). The major and
ultimate source of H2 is the photolysis of CH4, which requires photons with wavelengths
≤140 nm. As in Adamkovics and Boering (2003), we assume that the attenuation factor is
constant with wavelength.
2.2 Forward Model
4
The one-dimensional Caltech/JPL photochemical model is used to simulate the laboratory
hydrocarbon formation described above. Two complete sets of hydrocarbon chemistry
(Moses et al. 2000, 2005) are evaluated in this work. The Moses et al.’s (2000) set is
selected to be our reference chemistry, which includes 41 species (H, H2, C, CH, 1CH2,
3
CH2, CH3, C2, C2H, C2H2, C2H3, C2H4, C2H5, C2H6, C3, C3H, C3H2, C3H3, CH3C2H,
CH2CCH2, C3H5, C3H6, C3H7, C3H8, C4H, C4H2, C4H3, C4H4, C4H5, 1-C4H6, 1,2-C4H6,
1,3-C4H6, C4H8, C4H9, C4H10, C6H, C6H2, C6H3, C6H6, C8H2, and C8H3) and ~250
chemical reactions. The Moses et al.’s (2005) set contains 4 more species, namely, C5H3,
C5H4, C6H4, and C6H5. These two sets yield similar results for simple hydrocarbons but
may differ significantly for higher hydrocarbons (see Moses et al. 2005; Liang et al.
2007).
Since most of the UV radiation is absorbed by molecules in the top few cm in the
reaction cell, the model grid (58 points) is chosen carefully to ensure that the UV
attenuation is calculated correctly. The model has a vertical dimension of 36 cm,
temperature of 295 K, and pressure of 93 mbar (70 torr). Molecules are allowed to diffuse
freely in the model box. We fix CH4 abundance at 2.31018 cm-3 (93 mbar). The model
box is impermeable to all species. The intensity of the UV source is scaled as a function
of time (see section 2.1), and additional attenuation is caused by molecular absorption.
2.3 Adjoint Model
An adjoint model is an efficient method for calculating the sensitivity of a model
response with respect to numerous model parameters. For purposes of inverse modeling,
the response can be defined as a temporally discretized cost function,
J
1
1
n
n
(Hc n  c nobs )T S1
(p  pa )T S1

obs (Hc  c obs ) 
a (p  p a ) .
2 n 
2
(1)
where c nobs is the vector of observed concentrations at discrete time steps n, cn is the
vector of model estimated concentrations, H is the observational operator that maps from
the model space to the observation space,  is the domain over which the discrete
5
observations are available, p is a vector of model parameters, pa is the initial (a priori)
estimate of these model parameters, Sobs is the observational error covariance matrix, and
Sa is the parameter error covariance matrix. In the present application, the model
parameters are the rate coefficients listed partially in Table 2, and the observed species
are H2, C2H2, C2H4, C2H6, C3H4, C4H2 and C4H10. The cost function thus represents the
square of the difference between the model estimates and the observations, weighted by
the uncertainties in the observations, with an additional penalty term which increases as
the estimated model parameters depart from their initial values, also weighted by the
parameters’ uncertainty. In the reference cases, the parameter uncertainties are all
assumed to be infinite and uncorrelated, the penalty term thus is negligible. The
relaxation of this assumption will be discussed later.
The adjoint model is used to calculate the sensitivity of the cost function with respect to
the model parameters. These sensitivities are themselves gradients of the cost function.
These gradients are used with the L-BFGS-B quasi-Newtonian optimization routine
(Byrd et al., 1995; Zhu et al., 1994) to iteratively seek the set of parameters (rate
coefficients) that minimizes the cost function.
A brief derivation of adjoint sensitivities is presented here; a more detailed treatment of
adjont modeling for chemical kinetics is given in the review of Wang et al. (2001). The
forward model of the chemical kinetics can be viewed as a discrete operator which
advances the concentration vector from time step n to step n + 1.
cn+1 = Fn(cn,p).
(2)
In the present work, the Kinetic Pre-Processor (KPP) (Sandu et al., 2003; Damian et al.,
2002; Daescu et al., 2003) is used to build the forward model operator by parsing a
symbolic list of the chemical reactions, and solving the resulting system of coupled
differential equations using a 3rd-order Rosenbrock solver.
For simplicity, we will consider a cost function where there are observations at every
time step, i.e.   {n  1,...,N} . It is first necessary to consider the sensitivities of the cost
6
function with respect to the species concentrations at any given time step n, nc , which
can be expressed as
nc   c J 
n
N
g n 
 c
n
,
(3)
n  n
n
n
where g n  (Hc n  c nobs )T S1
obs (Hc  c obs ) .
The right hand side of Eq. (3) can be expanded using the chain rule,
n 1 n  T g n  g n
    (Fc )  n   n
c
c
n  n 1n  n
N
n
c
(4)
where Fcn is the Jacobian around a single time step n,
c n 1 F n (c n )

 Fcn .
n
n
c
c
The products in Eq. (4) are most efficiently evaluated in reverse order (Giering and
Kaminski, 1998). By initializing the adjoint variable at step N as
Nc  H T (Hc N  c Nobs )S obs
and evaluating the following iterative formula from time step n = N to n = 1,
n1 T n
n1
c  (Fc ) c 
g n1
.
c n1
(5)
The resulting adjoint variable at step n = 0 is the sensitivity of the cost function with
respect to the initial concentrations,
0c   c J .
0
The sensitivities with respect to the model parameters,  p , are thus derived from nc as
N
 p   (Fpn1 )T nc  (p  pa )T Sa ,
(6)
n1
where
c n 1 F n (c n )

 Fpn .
p
p
The product of the adjoint vector with the Jacobians of the forward model operator,
(Fcn1 )T nc , is evaluated using KPP generated code that is automatically constructed from
the numerical algorithm used to solve the governing equations of forward model, Eq. (2).
The resulting sensitivities are thus discrete model sensitivities. Updates to KPP described
in Appendix B of Henze et al. (2007) are used to similarly calculate the discrete
7
sensitivities with respect to the reaction rate parameters, (Fpn )T nc . The overall CPU
requirement of evaluating the adjoint equations (5) and (6), is about twice that of running
the forward model. Therefore, the sensitivity of the cost function with respect to all rate
coefficients is calculated in about three times the amount required for the forward model
alone. For the present application, this is ~100 times more efficient than evaluating these
sensitivities through successive forward model evaluations (i.e., finite difference
sensitivities).
3. Results
3.1 Forward Model
The one-dimensional photochemical results are presented in Figures 2 and 3. Both
baseline cases (black curves) underestimate the abundances of C2H2, C2H4, C3H4, and
C4H2 by more than an order of magnitude and overestimate that of C2H6 and C4H10. A
proposed modification to the rate coefficients (Table 1) provides better but not
completely satisfactory fits to the measurements. Abundances that are too high can be
reduced and explained by introducing aerosol particle production, which is observed in
the experiments (Adamkovics and Boering 2003).
Figure 2 shows the results of the forward model simulation with the chemical scheme of
Moses et al. (2000). The final abundance of modeled H2 is lowered by 15% compared
with the observed value, suggesting that the attenuation factor we adopted is biased too
high. The previously derived factor (Adamkovics and Boering 2003) was based on an old
photodissociation coefficient of CH4 (e.g., Yung and DeMore 1999), but we note that the
difference is not significant, as compared with experimental errors. In the following
discussion, we will focus only on those whose deviations are off by an order of
magnitude or more. The baseline model underestimates the abundance of C2H2 by nearly
a factor of 10. An attempt made to bring the abundance up is to modify the four reactions
(cases 2-5) shown in Table 1. The exercise demonstrates that the responses of species’
8
concentrations to changing chemical reaction rates are not linear. For example, consider
case 5 for C4H2; while the combined changes to abundances inferred from the individual
tests would be an increase of 81, the actual increase incurred by all changes
simultaneously is ~1500. Case 6 is used to see how the atomic hydrogen affects the
concentrations of the hydrocarbons. Such nonlinearity greatly reduces the possibility of
finding an optimized set of chemical reactions that can well reproduce the measurements
by tuning each individual reaction manually. To achieve the optimization, the adjoint of
the forward model is developed and is used to obtain the full sensitivity of the model
results with respect to “all” chemical reactions (see below).
The modifications shown in Table 1 are selected to enhance hydrocarbon abundances by
reducing C2H6, the most abundant gaseous species next to CH4 and H2; the major
pathway to form C2H6 is 2CH3 + M  C2H6 + M, where M is a third molecule (mostly
CH4 here). There are some other species that are abundant but not reported such as C3H6
(volume mixing ratio of 10-5), C3H8 (710-3), C4H4 (10-5), 1-C4H6 (210-4), 1,2-C4H6
(410-6), 1,3-C4H6 (810-5), C 4H8 (310-4), and C6H6 (610-5); the least abundant species
among observed ones is C4H2 which has a modeled volume mixing ratio of 210-7; the
above values are all taken from the base case. Results using this modified scheme are the
red lines in Figures 2 and 3. This suggests that further analysis of the experimental results
is urgently needed to better calibrate the hydrocarbon chemistry.
Figure 3 shows the forward model results with Moses et al.’s (2005) new compilation.
The new chemistry produces even less H2; the underestimation factor is 45%. The
unattenuated (by aerosols) column-mean photolysis rate coefficients of CH4 for the
Moses et al.’s (2000) and Moses et al.’s (2005) cases are 4.510-7 and 4.310-7 s-1,
respectively. The modeled species with volume mixing ratios larger than 10-7 are H2
(310-2), C2H2 (510-5), C2H4 (210-5), C2H6 (210-2), C3H4 (910-6), C3H6 (310-6),
C3H8 (310-3), C4H4 (10-7), 1-C4H6 (310-5), 1,2-C4H6 (210-6), 1,3-C4H6 (310-5), C4H8
(210-4), C4H10 (10-3), C5H4 (10-7), and C6H6 (10-4). The hydrogen production is
accompanied by the production of hydrocarbons; the production of three hydrocarbons
(C2H6, C3H8, and C4H10) contributes >80% of hydrogen production. From the above
9
values, the formation of higher hydrocarbons is largely suppressed: reduced by a factor of
2 and 3 for C3H8 and C4H10, respectively. The same modifications (Table 1) are also
applied to this set of the chemistry, and the results are shown by the red curves. We see
that the improvement is not satisfactory. Moreover, the production of C4H10 (the most
abundant species next to CH4, H2, C2H6, and C3H8) is too slow during first ~20 hours,
compared with the Figure 2 set. This is caused primarily by inefficient production of
C2H6, which can undergo photosensitized dissociation to form C4H10 (e.g., Yung and
DeMore 1999):
2(C2H2 + h  C2H + H)
2(C2H + C2H6  C2H2 + C2H5)
C2H5 + C2H5 + M  C4H10 + M
Net:
2C2H6  C4H10 + M
Though we can choose a new list of chemical reactions and modify them, it is inefficient
and time-consuming. A more sophisticated method demonstrated below will be used to
optimize all reactions.
To explain the over-production of hydrocarbons and to provide a first order estimate of
the aerosols produced in the experiments, we manually set a conversion rate for the three
most abundant hydrocarbon species: C2H6, C3H8, and C4H10.
C2H6 + wall  Aerosols
C3H8 + wall  Aerosols
C4H10 + wall  Aerosols
Here we assume that the aerosols are produced heterogeneously on the chamber wall. The
rate coefficient k (s-1) is estimated to be inversely proportional to the transverse time  =
l2/D, where D is the molecular diffusion coefficient and l is the chamber size which is
about 30 cm:
k = / = 210-5 s-1,
where  is the proportionality constant and is assumed to be 1%, a value that is in
concordance with the adsorption coefficient (adsorption on dust grain surfaces) derived
by Liang et al. (2007). The modeled results are shown by the dashed curves in Figure 2
10
and 3. The derived C-C bound mixing ratios for the two kinetics compilations (Moses et
al. 2000 and 2005) are 3.410-2 and 2.610-2, respectively, which are consistent with the
values derived by Adamkovics and Boering (2003). A major problem is that the
production of other less abundant hydrocarbons is also significantly reduced. The
problem cannot be resolved until we learn more about the aerosol production pathways
and higher order hydrocarbon kinetics.
3.2 Adjoint Model
To obtain the full set of sensitivities of the mismatch between the observed and modeled
concentrations with respect to chemical reactions, the adjoint model is applied. The
results are shown in Figures 4 and 5 (obtained by using the Moses et al.’s (2000) and the
Moses et al.’s (2005) kinetics, respectively) and the optimized rate coefficients are
summarized in Table 2 (models A-D). We see that the agreement between the model
(blue curves) and experiments is greatly improved; the cost function is reduced by more
than two orders of magnitude. A good convergence is arrived in ~50 iterations (see
bottom panel), demonstrating the efficiency of the method for searching for optimized
rate coefficients. We tested the adjoint calculation up to >1000 iterations and no
significant reduction in cost function was observed. Similarly good optimization is
achieved by fitting H2, C2H2, C2H6, and C4H10 only (not shown here), which have higher
measurement accuracy and precision. Though the agreement is largely improved, the
optimized rate coefficients may not be reasonable. Some modified rate coefficients are
far beyond the measurement errors, suggesting that major chemical pathways are missing.
We stress that the unreasonably large modifications for some chemical reactions imply
that reactions schemes involving the species in those reactions are incomplete. Thus,
while the optimized values themselves for the rate coefficients should not be taken
seriously, this method certainly provides a good guideline for searching for possible
missing chemistry and motivates further laboratory measurements.
4. Implications for Jupiter and Titan
11
We apply the adjoint optimized rate coefficients to Jupiter and Titan. The key chemical
difference between the two objects is that the former has substantially more hydrogen
than the latter. The amount of hydrogen in the atmosphere determines the efficiency of
hydrocarbons recycling back to methane; the more hydrogen the more efficient the
recycling processes. As a result, about one-half of methane photolysis produces species
with orders of magnitude more higher-than-C2 hydrocarbons (including aerosols and
hazes) in the atmosphere of Titan (see, e.g., Liang et al. 2007 and references therein).
Since the mixing ratio of hydrogen in the atmosphere of Titan is similar to the one in the
laboratory measurements described above, we expect the optimization is more applicable
to Titan than to Jupiter. Figures 6-9 demonstrate cases for Titan and Jupiter with the two
hydrocarbon kinetics sets.
We compare the model results with those obtained from observations. Seven hydrocarbon
species are retrieved in the neutral atmospheres of Jupiter and Titan: C2H2, C2H4, C2H6,
C3H4, C3H8, C4H2, and C6H6. In this work, we focus on the region below the
thermosphere. We first apply the rate coefficients obtained by optimizing all chemical
reactions (models A and C), followed by cases optimizing >C2 reactions (models B and D)
only, that is only those that involve C3 and higher hydrocarbons are used in the adjoint
optimization. Two hydrocarbon kinetics schemes (Moses et al. 2000, 2005) are tested.
The latter contains more complete chemistry for higher order chemistry. The results are
summarized in Figures 6-9.
The models with standard (unmodified) chemistry are shown by the black curves. This
type of model can reproduce most of the observations fairly well, including the absolute
concentrations and their scale heights, for the atmospheres of Jupiter and Titan. C2H4 is
so far the most outstanding exception in our models, where we underestimate its
abundance by an order of magnitude in the stratosphere of Titan. In general the current
models of Titan (Figures 6 and 8) yield results consistent with those of Wilson and
Atreya (2004), but perform better than that of Lebonnois et al. (2001). Detailed
comparisons such as for meridional variations will be carried out in a latter work (Liang
12
and Yung 2008). Jupiter can be reasonably represented by current models (Figures 7 and
9), but tend to be better modeled by models with the Moses et al.’s (2005) chemistry. The
major improvements are due primarily to the updates of higher order hydrocarbon
chemistry. Here we devote the remaining paragraphs to a comparative study of the two
hydrocarbon kinetics sets, their chemical sensitivity, applications to planetary
atmospheres, and implications for missing chemistry.
An optimized configuration of the kinetics is shown by the blue curves. All reactions are
varied by the adjoint optimization method. The configuration is obtained by reproducing
the laboratory measurements (Figures 4 and 5). Though applying the configuration to
Titan and Jupiter can produce more C2H4 than the unoptimized models, such an
optimized set of chemistry in general is at other times poorer at producing other species
(Figures 6-9). This poorer representation of the optimized set with independent data from
Titan and Jupiter implies that the inverse solution based on the lab data was poorly
constrained.
The major difference between Jupiter and laboratory box experiments/Titan is the molar
fraction of hydrogen in the system. The degree of hydrogenation determines the
efficiency of hydrocarbons cycling back to CH4. The hydrogenation of C2 species is
through the following reactions:
C2H2 + H + M  C2H3 + M
C2H3 + H2  C2H4 + H
C2H4 + H + M  C2H5 + M
C2H5 + H  2CH3
CH3 + H + M  CH4 + M.
The second reaction is the bottleneck of the process and has a strong and negative
temperature dependence (Knyazev et al. 1996; Weissman and Benson 1988; see Moses et
al. 2005 for detailes). At room temperature the rate coefficient is 10-17-10-20 cm3 s-1; the
higher value (Weissman and Benson 1988) is used in the Moses et al. (2005) case. This
explains that the Moses et al. (2005) chemistry provides a better fit of C2H6 to the
observations than the Moses et al. (2000) chemistry (see the black curve of Figure 9
13
versus that of Figure 7), because C2H6 is produced by the reaction 2CH3 + M  C2H6 +
M. The high efficiency of the hydrogenation is then expected for extrasolar “hot Jupiters”
(see Liang et al. 2004). Therefore in low temperature environments such as the
atmospheres of outer solar planets and their satellites, the abundance of hydrogen plays a
crucial role in the hydrogenation process. As a result, the C1-C2 hydrocarbon chemistry is
less coupled with the >C2 chemistry in the atmosphere of Jupiter, but they are strongly
coupled on Titan, which is similar to the case we had for the laboratory experiments. The
reasonable simulation (Gladstone et al. 1996; Moses et al. 2000; Moses et al. 2005; Liang
et al. 2005) of C1 and C2 species on giant planets implies that the current C1 and C2
chemistry schemes are satisfactory and suggests that uncertainties in the rate coefficients
of these chemical reactions and pathways are much less than for the larger hydrocarbons.
In order to reflect the fact that the C1-C2 chemistry and rate coefficients are relatively
better known a priori in inverse modeling, the inverse modeling is repeated, but this time
the uncertainties in the rate coefficients for C1-C2 chemistry are assumed to be very small.
The consequence is that the optimization only affects rate coefficients for >C2 chemistry.
The results for optimizing >C2 chemistry are shown by the red curves in Figures 4-9. In
general, these new sets of optimizations greatly improve the agreement of models and
observations/experiments. One can see that the agreement obtained with the Moses et
al.’s (2005) chemistry is generally better than that with Moses et al.’s (2000). The major
difference between the two is that there are about 200 new high order hydrocarbon
chemical reactions (>C2) added in Moses et al. (2005). There are 3 conclusions we can
draw from this sensitivity study. First, some high order chemical reactions were compiled
erroneously and/or incompletely. This can be inferred from post-optimized rate
coefficients whose values equal to zero (see Table 2). Second, optimization of the entire
set of rate coefficients is not justified; at the very least a coarse delineation between
relatively certain (C1, C2) and uncertain (>C2) reactions is warranted. Third, there are too
few constraints (laboratory experiments) for the adjoint optimization. The laboratory
experiments at different pressures, temperatures, and initial concentrations of H2 and CH4
will be useful for constraining the kinetics.
14
5. Discussion
In the above calculations, we did not explicitly consider error covariance terms for
observed species, nor rigorous estimates of errors for individual parameters. With
additional constraints in the cost function from measurement uncertainty, better matches
to the species with higher accuracy observations (i.e., H2, C2H2, C2H6, and C4H10) are
achieved. The other three species (C2H4, C3H4, and C4H2) are underestimated by more
than two orders of magnitude. It is caused by poor sensitivity in the measurements and as
a result, the cost function has little sensitivity to these three species. We next test the
effect of penalty constraints. The purpose of the penalty is to set a limit to the space of
the parameters we are adjusting. In one case we investigated, the rate coefficients are
allowed to vary only by less than a factor of ~20. Though significant reduction (by a
factor of 100) in cost function is achieved, there is no noticeable improvement for
planetary applications compared with the baseline kinetics. This further suggests that
there are significant missing reactions in higher order hydrocarbon chemistry.
It should be noted that the results of Adamkovics and Boering (2003) were preliminary
and that additional experiments are underway. It is possible that deconvolution of the
measured mass fragmentation patterns by mass spectrometry to yield the relative
hydrocarbon concentrations in the gas mixture was difficult under the conditions
employed. Indeed, several aliquots of the gas mixture from the reaction chamber were
run offline using gas chromatography followed by mass spectrometry, and propane was
detected. For all other species, however, the offline gc-ms comparison resulted in lower
mixing ratios. Thus the pattern of the model-measurement discrepancies that the C2H2,
C2H4, C3H4, and C4H2 species are underpredicted by the model while the C2H6, C3H8, and
C4H10 species are overpredicted cannot easily be explained by such a deconvolution error
and still suggests that, despite these uncertainties, that missing chemistry is involved.
15
Acknowledgements
This research was supported in part by NSC grant 97-2628-M-001-001 to Academia
Sinica, NASA grant NNX07AI63G to the California Institute of Technology, and NASA
grants NNX08AE69G to UC Berkeley.
16
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25
Table 1: Modified chemical reactions. All of the modeled abundances of the 6
hydrocarbons are taken at the end of the simulation and their values are referenced to the
laboratory values which are then normalized. “Modification” refers to the scaling factor
of the value of the rate coefficient of the reaction.
Case
0
Reaction
lab result
Modification
[C2H2]
1
[C2H4]
1
[C2H6]
1
[C3H4]
1
[C4H2]
1
[C4H10]
1
1
H2 profile normalized
none
0.05
0.27
8.33
3.8×10-3
1.6×10-5
2.78
-3
-5
2
CH3 + C2H3 → CH4 + C2H2
10
0.16
0.35
8.00
5.8×10
2.2×10
2.78
3
CH3 + C2H5 → CH4 + C2H4
10
0.06
0.38
7.92
5.0×10-3
2.7×10-5
2.78
-5
4
CH3 + C3H5 → CH4 + C3H4
100
0.05
0.27
8.33
0.02
1.7×10
2.78
5
CH3 + C4H5 → CH4 + C4H4
100
0.05
0.30
8.25
4.2×10-3
4.8×10-4
2.78
-4
5
6
2H + M → H2 + M
10
0.28
0.78
6.67
0.02
6.6×10
4.17
7
combination of cases 2-5
combined
1
1.57
6.67
0.11
0.02
3.61
26
Table 2: Rate coefficients modified by the adjoint model. Only those modified by more
than 20% are shown here. Models A-B and C-D are for the kinetics of Moses et al.
(2000) and Moses et al. (2005), respectively. Models A and C are obtained by optimizing
all chemical reactions and models B and D are obtained by only optimizing >C2
chemistry.
Model
Reactions
R5
R6
R7
R8
R9
R11
R13
R14
R17
R18
R19
R25
R26
R28
R29
R33
R34
R35
R36
R37
R38
R39
R40
R41
R42
R47
R49
R50
R51
R52
R53
R54
R62
R64
R65
R66
R67
R68
R69
R70
R71
R72
R73
R74
R78
R79
R80
CH4 + h
CH4 + h
CH4 + h
CH4 + h
CH4 + h
C2H2 + h
C2H4 + h
C2H4 + h
C2H6 + h
C2H6 + h
C2H6 + h
CH3C2H + h
CH3C2H + h
CH2CCH2 + h
CH2CCH2 + h
C3H6 + h
C3H6 + h
C3H6 + h
C3H6 + h
C3H6 + h
C3H6 + h
C3H8 + h
C3H8 + h
C3H8 + h
C3H8 + h
C4H4 + h
1-C4H6 + h
1-C4H6 + h
1-C4H6 + h
1-C4H6 + h
1-C4H6 + h
1-C4H6 + h
1,3-C4H6 + h
1,3-C4H6 + h
1,3-C4H6 + h
1,3-C4H6 + h
C4H8 + h
C4H8 + h
C4H8 + h
C4H8 + h
C4H8 + h
C4H8 + h
C4H8 + h
C4H10 + h
C4H10 + h
C4H10 + h
C4H10 + h
A
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
CH3 + H
1
CH2 + H2
1
CH2 + 2 H
3
CH2 + 2 H
CH + H + H2
C2 + H2
C2H2 + H2
C2H2 + 2 H
C2H4 + H2
C2H4 + 2 H
C2H2 + 2 H2
C3H3 + H
C3H2 + H2
C3H3 + H
C3H2 + H2
C3H5 + H
CH3C2H + H2
CH2CCH2 + H2
C2H4 + 1CH2
C2H3 + CH3
C2H2 + CH4
C3H6 + H2
C2H6 + 1CH2
C2H5 + CH3
C2H4 + CH4
C4H2 + H2
C4H4 + 2H
C3H3 + CH3
C2H5 + C2H
C2H4 + C2H + H
C2H3 + C2H + H2
2 C2H2 + H2
C4H5 + H
C3H3 + CH3
C2H4 + C2H2
2C2H3
1,3-C4H6+ 2 H
C3H5 + CH3
CH3C2H + CH4
CH2CCH2 + CH4
C2H5 + C2H3
2C2H4
C2H2 + 2 CH3
C4H8 + H2
C2H6 + C2H4
2C2H5
C2H4 + 2CH3
-0.03
2.9
1.76
2.06
1.35
-0.43
1.23
-2.47
----------4.94
1.51
0.72
0.6
-1.26
2.15
0.14
0
1.26 × 10-5
1.26
--0.77
-1.49
0.48
--0.74
--3.22
1.39
---
B
-----------0.68
-0.00
2.13
2.79
1.78
2.41
1.52
0.01
0.03
7.46
1.84
-1.97
1.22
2.44
0.20
--0.03
--0.00
1.70
1.35
0.12
0.32
-0.67
4.96
4.33
1.06 × 10-4
7.12
-2.33
--
C
2.97
0.57
1.34
0.65
1.32
1.88
5.15
1.68
0.65
0.55
0.66
1.24
--1.91
0
0.48
0.62
-2.05
-6.09
2.32
1.73
1.76
0.72
-1.8
0.52
0.64
1.21
1.37
0.27
1.26
--0.12
1.9
--0.12
-1.77
3.66
2.09
---
D
-----------0.51
0.64
1.58
2.46
2.51
1.34
2.32
-0.05
0.52
8.43
1.57
0.37
-6.03 × 10-4
1.90
3.40
0.03
0.19
1.24
1.71
1.74 × 10-4
0.19
4.60
3.85
1.43
0.40
1.80
4.63
6.47
3.00
0.00
13.81
3.90
3.35
0.77
References
19, 59, 60
19
19 (33)
19 (33)
19
4, 19, 41, 42, 45, 47
1, 19
1, 19
19
19
19
19, 61, 62
19, 61, 62
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
27
R85
R184
R186
R187
R190
R192
R194
R195
R196
R197
R198
R200
R201
R203
R204
R205
R206
R207
R208
R209
R210
R211
R212
R213
R214
R215
R216
R217
R219
R223
R224
R225
R230
R245
R251
R259
R261
R262
R266
R270
R271
R275
R276
R278
R281
R282
R283
R286
R287
R289
R290
R291
R292
R293
R294
R296
R297
R298
R299
R302
R303
R305
R306
R314
R316
R319
C6H6 + h
H + 3CH2
H + CH3 + M
H + CH4
H + C2H2 + M
H + C2H3 + M
H + C2H4 + M
H + C2H5
H + C2H5
H + C2H5 + M
H + C2H6
H + C3H3 + M
H + C3H3 + M
H + CH3C2H
H + CH3C2H + M
H + CH2CCH2
H + CH2CCH2 + M
H + C3H5
H + C3H5
H + C3H5
H + C3H5 + M
H + C3H6
H + C3H6
H + C3H6 + M
H + C3H7
H + C3H7
H + C3H7 + M
H + C3H8
H + C4H2 + M
H + C4H4 + M
H + C4H5
H + C4H5 + M
H + C6H2 + M
CH + CH4
CH + C2H6
1
CH2 + H2
1
CH2 + CH4
1
CH2 + CH4
1
CH2 + C2H2
2 3CH2
3
CH2 + CH3
3
CH2 + C2H2
3
CH2 + C2H2
3
CH2 + C2H3
3
CH2 + C2H5
3
CH2 + C3H2
3
CH2 + C3H3
CH3 + H2
2 CH3 + M
CH3 + C2H3
CH3 + C2H3
CH3 + C2H3 + M
CH3 + C2H5
CH3 + C2H5 + M
CH3 + C3H2
CH3 + C3H3 + M
CH3 + C3H5
CH3 + C3H5
CH3 + C3H5 + M
CH3 + C3H7
CH3 + C3H7 + M
CH3 + C4H5
CH3 + C4H5 + M
C2H + CH4
C2H + C2H2
C2H + C2H4
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
3 C2H2
CH + H2
CH4 + M
CH3 + H2
C2H3 + M
C2H4 + M
C2H5 + M
2 CH3
C2H4 + H2
C2H6 + M
C2H5 + H2
CH3C2H + M
CH2CCH2 + M
CH3 + C2H2
C3H5 + M
CH3C2H + H
C3H5 + M
CH3C2H + H2
CH2CCH2 + H2
CH3 + C2H3
C3H6 + M
C3H5 + H2
CH3 + C2H4
C3H7 + M
C3H6 + H2
C2H5 + CH3
C3H8 + M
C3H7 + H2
C4H3 + M
C4H5 + M
C4H4 + H2
1-C4H6 + M
C6H3 + M
C2H4 + H
C3H6 + H
CH3 + H
3
CH2 + CH4
2 CH3
C3H3 + H
C2H2 + 2 H
C2H4 + H
C3H2 + H2
C3H3 + H
C2H2 + CH3
C2H4 + CH3
C4H3 + H
C4H4 + H
CH4 + H
C2H6 + M
CH4 + C2H2
C3H5 + H
C3H6 + M
CH4 + C2H4
C3H8 + M
C2H2 + C2H3
1,2-C4H6 + M
CH4 + CH3C2H
CH4 + CH2CCH2
C4H8 + M
CH4 + C3H6
C4H10 + M
CH4 + C4H4
PROD + M
C2H2 + CH3
C4H2 + H
C4H4 + H
0.57
3.06
2.1
0
0.53
1.73
0.02
1.28
0.53
5.76 × 10-18
-------1.27
1.3
0.54
0.23
-1.31
0.16
-1.34
1.72
1.78
-0.7
2.21
1.29
-1.42
-0.71
3.2
3.19
-0.24
2.05
--1.66 × 10-3
3.09 × 10-3
---2.33
1.26
-0.16
-4.88
----1.48
-1.31
0.19
0.02
0.78
1.23
--
-------------0.72
1.21
-1.38
---0.63
0.66
0.52
0.25
0.69
0.00
0.00
1.73
1.21
---0.69
-1.55
------1.21
1.26
--------3.11
-4.20
----1.75
3.18
0.00
0.55
2.30
----
-0
3.31
0
1.74
1.34
0.08
0.47
0.72
0.04
1.42
--1.36
2.51
0.59
-1.95
2.04
---2.35
0.02
--0.75
1.22
-------1.96
2.13
0.06
0.76
0
3.28
-0.08
-2.97
0.56
0.31
1.35
0.04
0
0
1.37
-2.13
1.45
0.79
0.05
0.5
0
-2.03
0.61
-2.28
0.05
--
-----------2.56
1.76
0.40
5.21
-1.22
--0.07
0.00
1.36
3.33 × 10-4
--2.87
-6.24
-1.64
1.44
1.77
--1.76
-------1.28
--1.48
----2.05
4.66
-11.70
1.58
-2.65
3.98
2.01
2.14
4.21
2.08
3.03
-0.23
1.29
29, 32, 38
9
12, 32 (33)
40 (2)
3, 20, 23 (3, 33)
16, 30, 32 (33)
3
46
48
32 (33, 46)
2
22, 32 (33)
32 (33)
(33)
32, 55 (33)
(33)
32, 52 (33)
32
32
32 (33)
21, 32 (33)
50
50
32, 50
49
49
32, 34 (33, 34)
49
32, 35, 58 (33)
32, 43 (33)
32 (33)
19, 32 (19, 33)
32 (33)
5 (13)
5 (13)
11, 26
7
7
(33)
2
2
8
8
32
32
(33)
(33)
40
2, 28 (33)
16
(33)
16, 32 (33)
2
19, 46 (33, 46)
(33)
32, 56 (33)
50 (33)
50 (33)
18, 32 (24, 33)
49
27, 49
32
32 (33)
36
39
37
28
R323
R328
R329
R330
R331
R333
R334
R338
R339
R340
R344
R345
R346
R347
R350
R351
R352
R353
R382
R383
R393
C2H + C2H6
C2H3 + H2
C2H3 + CH4
C2H3 + C2H2
C2H3 + C2H2 + M
2 C2H3 + M
C2H3 + C2H4
C2H3 + C2H5 + M
C2H3 + C3H2
C2H3 + C3H3
2 C2H5
2 C2H5 + M
C2H5 + C3H2
C2H5 + C3H3
C3H2 + C2H2 + M
C3H2 + C2H2 + M
2 C3H3 + M
C3H2 + C2H3
C3H3 + C3H5
C4H5 + H2
C4H5 + C2H2
C6H5 + C2H2
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
→
C2H2 + C2H5
C2H4 + H
C2H4 + CH3
C4H4 + H
C4H5 + M
1,3-C4H6+ M
1-C4H6 + H
C4H8 + M
C3H3 + C2H2
CH3C2H + C2H2
C2H6 + C2H4
C4H10 + M
C3H3 + C2H4
CH3C2H + C2H4
prod + M
C5H4 + M
C6H6 + M
C3H3 + C2H2
2 CH3C2H
1-C4H6 + H
C6H6 + H
PROD + H
---0.62
0.64
1.4
1.25
1.28
---1.48
---
-----0.00
0.09
1.80× 10-3
---0.21
---
0.26
3.23
0.59
0.08
0.17
1.24
1.91
--0.73
1.31
--1.47
1.93
---1.28
0.06
1.39
0.00
0.28
0.00
-0.36
0.29
1.11 × 10-4
37 (14)
(25, 33, 53)
(48)
17
32, 53
16, 32 (‘’)
17
32, 48 (33, 48)
32 (33)
(33)
2
19, 27 (19, 33)
32
(33)
0.75
1.31
0.78
--
32 (33)
---1.69 × 10-3
1.3
--
-0.76
-0.19
---
1.25
-1.35
1.75
1.37 × 10-4
0.73
2.09
--3.72
0.48
--
31, 32 (33)
(33)
(33)
53
54 (54)
(57)
References: 1. Balko et al. (1992), 2. Baulch et al. (1992), 3. Baulch et al. (1994), 4. Bénilan et al. (1995), 5. Berman and Lin (1983), 6. Böhland et al. (1985a), 7.
Böhland et al. (1985b), 8. Böhland et al. (1986), 9. Boullart and Peeters (1992), 10. Brachhold et al. (1988), 11. Braun et al. (1970), 12.Brouard et al.
(1989), 13. Canosa et al. (1997), 14. Ceursters et al. (2001), 15. Durán et al. (1988), 16. Fahr et al. (1991), 17. Fahr and Stein (1988), 18. Garland and
Bayes (1990), 19. Gladstone et al. (1996), 20. Gordon et al. (1978), 21. Hanning-Lee and Pilling (1992), 22. Homann and Wellmann (1983), 23.
Hoyermann et al. (1968), 24. Kinsman and Roscoe (1994), 25. Knyazev et al. (1996), 26. Langford et al. (1983), 27. Laufer et al. (1983), 28.
MacPherson et al. (1983), 29. Malkin (1992), 30. Monks et al. (1995), 31. Morter et al. (1994), 32. Moses et al. (2000), 33. Moses et al. (2005), 34.
Munk et al. (1986), 35. Nava et al. (1986), 36. Opansky and Leone (1996a), 37. Opansky and Leone (1996b), 38. Pantos et al. (1978), 39. Pedersen et
al. (1993), 40. Rabinowitz et al. (1991), 41. R. Wu, personal communication 1997, 42. Satyapal and Bersohn (1991), 43. Schwanebeck and Warnatz
(1975), 44. Seakins et al. (1993), 45. Segall et al. (1991), 46. Sillesen et al. (1993), 47. Smith et al. (1991), 48. Tsang and Hampson (1986), 49.
Tsang(1988), 50. Tsang (1991), 51. Vakhtin et al. (2001), 52. Wagner and Zellner (1972), 53. Weissman and Benson (1988), 54. Westmoreland et al.
(1989), 55. Whytock et al. (1976), 56. Wu and Kern (1987), 57. Yu et al. (1994), 58. Yung et al. (1984) , 59. Mordaunt et al. (1993), 60. Heck et al.
(1996), 61, Seki and Okabe (1992), 62, Payne and Stief (1972).
29
Figure file
Figures
Figure 1: UV intensity (photons per unit area, time, and wavelength) in arbitrary unit.
Laboratory and solar UV spectra are depicted by the solid and dashed curves,
respectively.
1
Figure 2: Volume mixing ratios of H2, C2H2, C2H4, C2H6, C3H4, C4H2, and C4H10 as a
function of time in hour. The spectra are normalized by factors of 0.06, 0.002, 0.002,
0.02, 0.0002, 0.00007, and 0.003, respectively. Laboratory results are marked by black
dots (original resolution) and green curve (one-hour smoothing). Model results before
(black curve) and after (red curve) adjusting rate coefficients are shown for comparison.
Aerosol model is shown by the dashed curve. See text. The model chemistry is taken
from Moses et al. (2000).
2
Figure 3: Same as Figure 2 but with chemistry from Moses et al. (2005). The black curve
of C4H2 is below the plotting range. The spectra are normalized by factors of 0.06, 0.002,
0.002, 0.02, 0.0002, 0.00007, and 0.001, respectively.
3
Figure 4: (a) The designation of black dots and green curve. The spectra are normalized
by factors of 0.06, 0.002, 0.002, 0.02, 0.0002, 0.00007, and 0.003, respectively. The
chemistry is taken from Moses et al. (2000). The black curves are the model results
before optimization. Models with optimizing all (model A) and >C2 (model B) chemistry
are shown by the blue and red curves, respectively. The cost functions for both cases are
reduced by a factor of 100 (b). The cost functions are normalized. The cost functions of
the two cases are similar. See text.
(a)
4
(b)
5
Figure 5: The designation of black dots and green curve. The spectra are normalized by
factors of 0.06, 0.002, 0.002, 0.02, 0.0002, 0.00007, and 0.001, respectively. The
chemistry is from Moses et al. (2005). The black curves are the model results before
optimization. Models with optimizing all (model C) and >C2 (model D) chemistry are
shown by the blue and red curves, respectively. The cost function is reduced by a factor
of 250. See text.
6
Figure 6: Titan application. Moses et al.’s (2000) kinetics is used. The black curves are
the model results before optimization. Models with optimizing all (model A) and >C2
(model B) chemistry are shown by the blue and red curves, respectively. Symbols denote
observed values (Coustenis et al. 1989, 1991; Flasar et al. 2005 ; Vinatier et al. 2007).
7
Figure 7: Jupiter application. Moses et al.’s (2000) kinetics is used. The black curves are
the model results before optimization. Models with optimizing all (model A) and >C2
(model B) chemistry are shown by the blue and red curves, respectively. Symbols denote
observed values (Orton and Aumann 1977; Owen et al. 1980; Festou et al. 1981; Clarke
et al. 1982; Gladstone and Yung 1983; Wagener et al. 1985; Noll et al. 1986; Kostiuk et
al. 1987; McGrath et al. 1989; Livengood et al. 1993; Bezaed et al. 1995; Morrissey et al.
1995; Edgington et al. 1998; Sada et al. 1998; Betremieux and Yelle 1999 ; Fouchet et al.
2000 ; Yelle et al. 2001; Moses et al. 2005 ; Greathouse et al. 2008).
8
Figure 8: Titan application. Moses et al.’s (2005) kinetics is used. Symbols denote
observed values (see Figure 6). The black curves are the model results before
optimization. Models with optimizing all (model C) and >C2 (model D) chemistry are
shown by the blue and red curves, respectively.
9
Figure 9: Jupiter application. Moses et al.’s (2005) kinetics is used. Symbols denote
observed values (see Figure 7). The black curves are the model results before
optimization. Models with optimizing all (model C) and >C2 (model D) chemistry are
shown by the blue and red curves, respectively.
10