% Chris April 21, 1997

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\def \lya {Lyman--$\alpha \,$}

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\begin{document}

\title{Estimates of Deuterium Lyman-$\alpha$ Airglow in the Thermosphere 
of Jupiter}

\author{C. D. Parkinson}
\affil{Department of Earth and Atmospheric Science, York University, North 
York, Ontario, e-mail: chris@nimbus.yorku.ca}

\author{J. C. McConnell}
\affil{Department of Earth and Atmospheric Science, York University, North 
York, Ontario}

\author{L. Ben Jaffel}
\affil{Institut d'Astrophysique de Paris, Paris, France}

\author{G. R. Gladstone}
\affil{Southwest Research Institute,\\ San Antonio, TX 78238}

\begin{abstract}

New calculations are presented showing airglow estimates of Deuterium~\lya
in Jupiter's thermosphere. 

\end{abstract}

\section{Introduction}

The D/H abundance has important implications for the evolution of the
solar system and until recently the only inferences of the ratio on
the giant planets has been by spectroscopy. 
Recently {\it in-situ}
measurements have been made by the Galileo probe mass spectrometer (GPMS) 
[Neimann et al., 1996]. There has
also been a tentative detection of the deuterium \lya line emission
($\lambda_{\rm lab}\sim 1215.34$ \AA: Strigabov and Sventitsii, 1968)
from the dawn limb of Jupiter using the Goddard High Resolution
Spectrometer on board the Hubble Space Telescope (HST) on the 5th
July, 1993 [Ben Jaffel et al., 1994].  This new
technique utilises the fact the weak deuterium emission is expected to
be limb brightened due to the optically thin nature of the line.

In this note we address the problem of the abundance and airglow
intensities of D in the thermosphere making using of photochemical and
radiative transfer models.  Our perspective is to assume that the mixing
ratio of D to H$_2$ in the deep atmosphere is given by that of
HD/H$_2$ and is well determined by the GPMS instrument [Niemann et
al., 1996]. We investigate the impact of eddy diffusion, and
temperature profile on the D abundance and on the expected D emission.

As we illustrate below, the D distribution is sensitive to that of H; this requires
that the modelling process must also produce a valid H atom distribution.  To do this we
assume that the H~\lya emergent intensities (airglow) can be used to adequately characterise 
the H distribution.  The intensity of \lya is not a tight constraint given the uncertainties 
in the measurements, the solar \lya flux and our knowledge of the emission processes.  However,
Ben Jaffel et al. [1993] have used line shape measurements to further constrain H distributions
and excitation mechanisms.  The details of the source of the H~\lya airglow remains 
problematical. Nevertheless, we have adopted the approach of Clarke et al. [1991] and 
Ben Jaffel et al. [1993].  The H~\lya airglow may be represented by two regions, a non-bulge 
region where the solar maximum sub-solar intensity is $\sim$~11~kR and a bulge region 
where the sub-solar intensity is $\sim$~18~kR for solar maximum conditions 
[Sandel et al., 1980; Clarke et al., 1991; McGrath, 1991, Ben Jaffel et al., 1993]. 
We assume that in the non-bulge region intensities are due to an
H column of $\sim$~3.7$\times 10^{17}$ molecules cm$^{-2}$ excited by the solar
\lya line.  Following Ben Jaffel et al. [1993] we will assume that for the bulge region the 
excess intensity is due to a column of $\sim$~2$\times 10^{15}$ molecules
cm$^{-2}$ of `hot' hydrogen which may be represented by an effective
temperature $\sim$~6,050~K or a turbulent velocity, V$_{t}$,
$\sim$~10~km/s.

Possible excitation mechanisms of \lya airglow include the solar \lya,
photoelectrons, H$_3^+$ recombination [Shemansky, 1985], the
interplanetary medium (IPM) \lya [McConnell et
al., 1980].  We assume that 1~kR of \lya is due to excitation by the IPM
[e.g., McConnell et al., 1980] and that the remainder is due to excitation 
by the solar line.  In the following pages we describe the chemical and radiative 
transfer models we have used; more specifically we have included discussions relating 
the temperature profile, eddy diffusion coefficient and solar flux and line shape to 
these models. 

%   H Lyman alpha = 1215.670 A
%   D Lyman alpha = 1215.340 A


\section{Model Description}

\subsection{General}

It is our aim to model both H Lyman-$\alpha$ and D Lyman-$\alpha$ so we use
a chemical model to produce an atmosphere with number densities of key species as
a function of temperature and altitude.  We use this atmosphere in conjunction with a 
radiative transfer model [Gladstone, 1982; 1988] to quantify estimates of the 
deuterium Ly-$\alpha$ airglow in the thermosphere of Jupiter.  These models and 
some of the parameters they are dependent upon are described below. 

\subsubsection {Temperature}

There are few direct measurements of the temperature profile in the upper atmosphere
and our knowledge of the heating process is certainly insufficient to provide information
in the geographical and diurnal variation of thermospheric temperatures.  For our calculations
we have adopted two quite different temperature profiles.  One is based
on the stellar and solar occulatations of the Voyager~1 ultraviolet
spectrometer at Jupiter encounter in 1979, a time of solar maximum.
The exospheric temperature is set to 1000~K based on the Jovian solar
occultation analysis of G.~Smith (see Figure~2 in McConnell et al.
[1982]).  At heights $\sim$1600~km above the 1~bar level the H$_2$
density is relatively well constrained to be $\sim$10$^8$ cm$^{-3}$.  At the lower levels the
temperature and background densities are constrained by the stellar
occultation results of Romani [1996]. This we call
our temperature profile A.  Recently Hubbard et al.
[1995] have reported analyses of the occultation of the star SAO~78505 by the
mesosphere of Jupiter. They find a mean temperature of $\sim$~176~K around
2~$\mu$bars.  Using this information and an analysis of
Jupiter's UV airglow by Liu and Dalgarno [1996] together with a
reanalysis of the UVS occultation data, Yelle et al. [1996] present
quite a different picture of Jupiter's thermosphere. A comparison of this temperature
profile is made with results from the Galileo Probe measurements to determine the
thermal structure of Jupiter (Seiff et al [1996]).  Seiff et al. [1997]
re-analysed the Galileo Probe data and the updated results are much more similar
to those of Yelle et al. [1996] when exospheric temperatures of 800K or 900K are
considered.  We regard temperature profile B to be the updated Seiff et al. [1997]
temperature profile where the exospheric temperature is 900K.  
Both temperature profiles are displayed in Figure~\ref{T}.  
Since the temperature profiles are so different we have done the calculations 
for both to estimate the sensitivity of the D~\lya airglow.  

For all calculations modeling the bulge region, we regard the
temperature to be increasing from the background to 6,050~K extending over a vertical
section where an H column of $\sim$~2$\times 10^{15}$~cm$^{-2}$ is assumed.  As previously
mentioned, this temperature corresponds to V$_{t} \sim$~10~km/s.  
The relationship between V$_{t}$ and temperature is taken from
\begin{equation}
V^{2}_{t} = {2 k T_{tot} \over m}
\label{velocity}
\end{equation}
where $T_{tot} = T_{hot} + T_{neutral}$, $T_{neutral}$ is the background temperature,
$T_{hot}$ is the temperature corresponding to the hot H column, $k$ is Boltzmann's constant 
and $m$ is the mass of H.  This is clearly an 
oversimplification of the physical situation, but it should yield reasonable
results for our `numerical experiments'.  An investigation into the effect of the location
of the `hot' region versus altitude was carried out.  This will be discussed in a later section. 

\subsection{Chemical Model}

The species densities shown in this paper were calculated by 
solving the continuity equation for each species, $i$,
\begin{equation}
{\partial n_i \over \partial t} + {\partial \phi_i \over \partial z} =
P_i - L_i n_i 
\label{cty}
\end{equation}
where the vertical flux, $\phi_i$, is given by
\begin{equation}
\phi_i = \phi^K_i + \phi^D_i.
\end{equation}
The eddy flux, $\phi^K_i$,
\begin{equation}
\phi^K_i = -K({\partial n_i \over\partial z} + ({1\over H_{av}} + {(1+\alpha_i)\over T}
{\partial T \over \partial z})n_i)
\label{K}
\end{equation}
represents the vertical flux that parameterizes macroscopic
motions, such as the large scale circulation and gravity waves, while  $\phi^D_i$
\begin{equation}
\phi^D_i = -D_i({\partial n_i \over \partial z} + {(1+\alpha_i)\over T}
{\partial T \over \partial z} + {n_i\over H_i})
\label{D}
\end{equation}
is the vertical flux carried by molecular diffusion. The species
number density is given by $n_i$, $P_i$ is the chemical production
rate (cm$^{3}$ s$^{-1}$) and $L_i$ is the loss frequency (sec$^{-1}$) at  
altitude $z$ and time $t$ (e.g., Chamberlain and Hunten, 1987). 
The temperature is given by $T$ and 
$D_i$ and $K = K(z)$ are respectively the molecular and eddy diffusion 
coefficients. The species and atmospheric scale heights are denoted by
$H_i$ and $H_{av}$ respectively. In these calculations we have neglected 
the effects of the thermal diffusion coefficient, $\alpha_i$. 

The photochemical
model used for these calculations contain reactions involving H, CH$_{x}$,
C$_{2}$H$_{x}$, and C$_{2}$.  Additional reactions involving higher
order hydrocarbons of the form C$_{3}$H$_{x}$ and C$_{4}$H$_{y}$ have
not been included here for simplicity since the main focus of this study
is deuterium Lyman-$\alpha$ and they should have little impact.  
The details of the basic model are given in the appendix A.

The atmosphere used is assumed to contain a helium mole fraction of
0.136$\pm$0.004 in the mixed region [von Zahn and Hunten, 1996],
2.1($\pm$0.15)$\times 10^{-3}$ for the methane mole fraction [Niemann
et al., 1996]. As noted above, until recently, the deuterium abundance
on the outer planets has been determined spectroscopically, viz., 
$(D/H)_{Jupiter} = 2.0^{+0.6}_{-0.6} \times 10^{-5}$ [Gautier
and Owen, 1989]; $(D/H)_{Saturn} = 1.6^{+1.6}_{-1.0} \times 10^{-5}$ 
[Courtin et al., 1984; De Bergh, 1986]; $(D/H)_{Uranus} = 7.2^{+7.2}_{-3.6} \times 10^{-5}$ 
[De Bergh, 1986]; $(D/H)_{Neptune} = 12^{+12}_{-8} \times 10^{-5}$ [De Bergh et al., 1990]. 
However, recent
measurements using the GPMS yielded a value
of 11($\pm$3)$\times 10^{-5}$ for HD/H$_2$ at Jupiter [Niemann et al. 1996].
We have adopted this value for the calculations.

From our evaluation of the chemistry of D atoms in the Jovian
thermosphere the main source of D appears to be the reaction
\begin{equation}
\rm H + HD \rightarrow H_2 + D   \ \ \ k_1 = 7.9\times 10^{-11}exp(-4327/T) cm^{3}s^{-1}
\label{dreaction1}
\end{equation}
Compared to this source, CH$_3$D photolysis to CH$_3$ + D is minor.
The main loss for D is 
\begin{equation}
\rm D + H_2 \rightarrow  HD + H \ \ \ \ k_2 = 1.6\times 10^{-10}exp(-3875/T) cm^{3}s^{-1}
\label{dreaction2}
\end{equation}
We have also included the following reactions as sinks for D
\begin{equation}			
\rm H + D + M \rightarrow  HD + M  
\label{dreaction3}
\end{equation}
\begin{equation}			
\rm D + CH_3 + M \rightarrow CH_3D + M 
\label{dreaction4}
\end{equation}
with rates given by the analogous H reactions. However, they are less important.
The reactions for H and HD are listed in Table~1.  For rate data
we adopt the ratio of k$_1$/k$_2$ given by Trotman-Dickenson and Milne
[1967] and the value of k$_2$ by Rozenshtein et al. [1985].

The top of the chemical model is within the ionosphere.  The ion molecule reactions that
generate H have not been included directly in the chemical model.  Rather, the impact of the
ionosphere has been allowed for by adopting a downward flux, $\phi_{Top}$(H) at the top of 
the model.  Normally, H is produced by ionization of H$_2$ by photons and photoelectrons

\begin{equation}
H_2 + h\nu \rightarrow H_{2}^{+} + e
\label{dreaction5}
\end{equation}
\begin{equation}
H_2 + e \rightarrow H_{2}^{+} + 2e
\label{dreaction6}
\end{equation}
followed by reaction with H$_2$ to produce H$_{3}^{+}$
\begin{equation}
H^{+}_{2} + H_2 \rightarrow H_{3}^{+} + H
\label{dreaction7}
\end{equation}
and recombination of H$_{3}^{+}$
\begin{equation}
H_{3}^{+} + e \rightarrow 3H
\label{dreaction8}
\end{equation}
or 
\begin{equation}
H_{3}^{+} + e \rightarrow H + H_{2}
\label{dreaction9}
\end{equation}
In our case, we assume that every photon gives rise to four H atoms.  For solar maximum EUV
fluxes capable of ionizing H$_2$ this implies a minimum H source of about 4$\times 10^{9}$
H atoms cm$^{-2}$ s$^{-1}$ at Jupiter [eg. Gladstone et al., 1996].  
However, to obtain the requisite H column of $\sim$ 3.7$\times 10^{17}$
cm$^{-2}$ above the CH$_4$ absorbing layer to provide the \lya airglow, we find
it necessary to supply an H flux of 8$\times 10^{9}$ cm$^{-2}$ s$^{-1}$  
at the top of the model for temperature profile A and 
12$\times 10^{9}$ cm$^{-2}$ s$^{-1}$  for temperature profile B, which fall within the range
of realistic values tested by Gladstone et al. [1996].  
Although the solar H \lya flux will actually vary over a period of 27 Jovian days, 
the diffusion time constant for H at the homopause in the Jovian atmosphere is similar 
in value [Lean 1987;1991].
Hence, we employ a constant value for the solar H \lya flux for the photochemical 
model calculations.
%It should be pointed out that although
%a constant value for the solar H \lya flux was used for the photochemical model calculations, 
%the diffusion time constant for H in the Jovian atmosphere is about 27 Jovian days and so the value
%for the solar H \lya flux will actually vary over a period of this length [Lean 1987;1991]. 
%This willbe discussed in more detail below.        


The corresponding calculations for the He 584~\AA \ line were done to ensure consistency and
yield sub-solar brightnesses consistent with the analyses of Vervack et al. [1996].

Assuming that the above H flux comes from the ionosphere then it implies a concomitant
D flux, $\phi_{Top}(D)$.  This we estimate by scaling the H flux by the HD mixing ratio at
the top of the model divided by 2 since we assume that when H$_2$ is photoionized
it will react with HD to produce a single D atom.  Likewise, if HD is ionized 
it will produce a single D atom. Thus,
\begin{equation}
\phi_{Top}(D) = \phi_{Top}(H) / 2 \times f_{HD}
\label{phitopD}
\end{equation}
as a standard where $f_{HD}$ is the HD/H$_{2}$ mixing ratio at the model top.  
We have also increased $\phi_{Top}(D)$ by a factor of 10 while leaving $\phi_{Top}(H)$
at the value quoted above to investigate the sensitivity to this assumption.
We assume that HD is mixed well below the homopause and that its thermospheric
distribution is controlled by diffusion from the lower atmosphere and chemistry. 
At the bottom we assume that D and H are in photochemical steady state (PCSS).

From the reaction sequence it is clear that the D
distribution is intimately related to the H abundance in the
thermosphere. Thus we have used a combination of modelling and measurements
to constrain the H distribution in the thermosphere.  First we extract
the H column by means of the disk H~\lya. The vertical profile of the H
distribution is provided by the photochemical diffusion model which
will clearly be sensitive to the temperature profile.


\subsubsection{Eddy Diffusion}

The nominal value for the eddy diffusion coefficient at the homopause, $K_h$, 
that we adopt is based on a reanalysis of
the Voyager He~584 airglow data (Vervack et al., 1995) which yielded a
value of $K_h$ $\sim$ 2$^{+2}_{-1} \times 10^6$ cm$^2$ s$^{-1}$.  Molecular
diffusion coefficients are taken from Mason and Marrero (1970) and
Atreya (1986) where applicable.


\subsection{Radiative Transfer}

We have applied a resonance scattering model to the D/H problem that uses the Feautrier 
technique to solve the equation of radiative transfer assuming partial frequency 
redistribution [Gladstone, 1982, 1988].  For calculations of intensities on the planetary disk, 
we assume a plane-parallel atmosphere.  For a zero order approximation we include only D
atoms excited by the solar H~\lya line and a CH$_{4}$ absorber to calculate the 
D~\lya emission.  However, as can be seen below, this is clearly much too simple an 
approximation to be very useful.

The H~\lya and D~\lya excitation should properly be considered as a coupled problem 
since H frequencies can excite the D \lya line and vica versa.  For a first order 
approximation we assume that the radiative transfer calculations for D are done 
with solar H~\lya excitation and CH$_{4}$ absorption is included.  For the calculation 
of the source function on the planetary disk for D we do not include the scattering 
in the far wings of the planetary H~\lya line as we assume 
that at these wavelengths the photons scatter coherently [e.g. Mihalas, 1970] and so 
remain to interact with D \lya. Thus the source function for D~\lya is calculated 
ignoring the presence of H atoms.  However, for the calculations of the intensities 
as viewed from outside the planet we have included the attenuation of the D~\lya source 
function due to the optical thickness blue wing of the H~\lya line.  

For terminator viewing we assume similar conditions except that 90$\deg$
is used to calculate the initial source function, S$_0$.  The assumption of 
spherical geometry is used to obtain final limb intensities, I$_l$. 
The source function is calculated assuming a plane-parallel atmosphere with
a 90$\deg$ solar zenith angle and spherical geometry. Further, we also assume that
H~\lya absorption operates for the calculation of S$_0$ as well as I$_l$
since for the limb geometry we anticipate that photons will be scattered out
of the solar beam.  

For a second order approximation, we consider D and H to be coupled in the 
following sense.  Calculations for 
D~\lya and H~\lya intensities are performed together, centred and with symmetry about 
the D~\lya line using only solar H~\lya as the excitation source.  We consider angle-averaged 
frequency redistribution for D~\lya but assume that H~\lya scatters coherently since it occurs
in the wing.  This approximation is limited by the lack of allowance of photons scattering
from the core of the H~\lya line into the wings and vicinity of D~\lya.  

%In all other respects, these two latter calculations are as described for the decoupled case. 
%(xxxxx Thesis?: except possibly using a locus of source functions at the terminator along the 
%line of sight through the atmosphere instead of just a single source function calculated 
%for the plane parallel case, which is not adequate for spherical geometry).  
A more detailed discussion regarding the equation of radiative transfer is presented 
in appendix B.   

\subsection{Solar Flux and Line Shape}

The solar \lya line shape is important for the excitation of both H
and D~\lya.  It has been measured by Lemaire et al. [1978] (xxxxxCHECK D~\lya against Lemaire
solar line since this approx. wasn't designed for D~\lya and so may not be a good approx.)
and we use the analytic approximation to the line shape function suggested by
Gladstone [1988]

\begin{equation}
S(x) = {\pi F \over 2 \sqrt{\pi} x_{dis} }(e^{-((x-x_{off} - x_r)/x_{dis})^2}  
+ e^{-((x+x_{off} - x_r)/x_{dis})^2}) 
\label{shape}
\end{equation}
where $\pi F$ is the total flux in a given line incident on the upper boundary of 
the atmosphere. $x$ is the wavelength value from line center, x$_{off}$ is the offset
of the gaussian curves from line center, x$_{dis}$ is a measure of the width of the line 
(dispersion), all measured in Doppler units, and   
\begin{equation}
x_r = \Delta + \frac{\lambda R\Omega cos\theta sin\Phi}{c}
\label{shape1}
\end{equation}
accounts for the planetary rotation.  $\lambda$ is the line centre frequency of interest, 
$\Delta$ is the frequency difference between
H~\lya and D~\lya, $\theta$ is the planetary solar latitude, and
$\Phi$ is the planetary solar longitude.  We have ignored the 3$\deg$ inclination of
Jupiter's rotation axis.  Values for $\pi F$, x$_{off}$, x$_{dis}$, 
the width in angstroms of an standard doppler unit at a reference temperature of 500K, and
$\Delta$ are listed in Table~2.	 The line shape used is shown in Figure~\ref{line}.

For D~\lya we assume that it is excited solely by the solar H~\lya line. 
We have not directly included contributions from the IPM \lya.  
%For the former possible source the 1/e half width is about 3~DU (Doppler Units)
%while the maximum possible doppler shift is 11.2~DU at 1000K [McConnell
%et al., 1980] (<--xxxxxDo we need this remark? It was included only to give some scope 
%on the IPW parms).  
We will discuss the impact of the planetary line
later.  Figure~\ref{line} also shows the relative location of the D~\lya line
for $\Phi$ = 0$\deg$, and $\pm$~90$\deg$.  Although the impact of planetary
rotation is small for H~\lya  it is significant for D~\lya since it is on
the steep side of the exciting solar line.  As can be seen in Figure~\ref{line}
there is a factor of about 1.6 variation in the solar flux from D from the day to night 
terminator.

Solar ultraviolet irradiance variations have been discussed by Lean [1987; 1991] 
who notes that the mechanisms for the variability are not completely understood
nor adequately determined experimentally.  For instance, a variation of $\sim$ 36~\% 
was observed during one month in July during 1982.  One unresolved aspect 
of solar UV irradiance variation is that of the line shape of H~\lya solar line 
and its variability with solar activity.  Lean [1987] illustrates that there are differences 
in the widths of two profiles presented [Meier and Prinz, 1970; Lemaire et al. 
[1978] for a ``quiet'' sun.  The differences in the widths of the two profiles are quite clear
and cannot be accounted for by the simple subtraction of the film background from the
Meier and Prinz profile.  These differences need to be resolved owing to the importance
of the profile for calculations of \lya absorption in a wide variety of phenomena in our 
solar system. 
An additional confusing factor is that the Atmospheric Explorer E (AE-E) and Solar 
Mesosphere Explorer (SME) data sets do not overlap and appear to disagee [Lean, 1987;1991].  
This increases the difficulty of
obtaining a value for the solar flux as determined by various solar EUV irradiance 
models [Tobiska, 1991, 1994; Richards et al., 1994] as discussed by Bush and Chakrabarti
[1995] and Parkinson et al. [1996].  Certainly, the total solar flux appears to have an 
upper bound of $\sim$ 5$\times 10^{11}$ photons cm$^{-2}$ s$^{-1}$ for solar 
maximum conditions at 1 A.U..  We adopt the value of 4.4$\times 10^{11}$ 
photons cm$^{-2}$ s$^{-1}$ 
for the solar maximum as a standard for our calculations in keeping with Parkinson
et al. [1996].

Table~2 gives a list of the important parameters used in these
calculations.  The values of the pararameters x$_{off}$ and x$_{dis}$ have been
adjusted to allow more grid points in the region of interest in the wings of the line.
%[Helium stuff also????xxxxx Thesis!?]

\section{Results and Discussion}


Figure~\ref{T}a shows three temperature profiles considered for the mesophere
and thermosphere of Jupiter.  Sample bulge temperature profiles are shown in Figure~\ref{T}b
and represent a `hot' column of $\sim$~2$\times 10^{15}$ molecules cm$^{-2}$ of hydrogen.
Both figures are for the standard case of K$_{h}$ = 2 $\times 10^{6}$ cm$^2$ s$^{-1}$.  
An investigation into the effect of the location of the `hot' region versus altitude was 
carried out and showed little impact on the results.  In addition, halving or doubling the the 
column amount of `hot' H changed the results by less than 7\%.  These results were expected 
since the deuterium is much lower in the atmosphere than where the `hot' H resides.  It is seen
that 

Figure~\ref{line}a shows the calculated H~\lya  \ emergent planetary line profile (non-bulge) 
for a number of cases using the temperature profile A from Figure~\ref{T}a. 
Included is a case from Gladstone [1988] for comparative purposes.
In this paper, we are mainly interested in the H~\lya \ behaviour at the terminator. 
Figure~\ref{line}b shows the D~\lya \ planetary line profile including cases where 
doppler shifting of the line occurs due to planetary rotation.  The cases examined 
are disc center where no doppler shifting happens ($\Phi$ = 0$\deg$) and both 
limbs ($\Phi$ = $\pm$90$\deg$) where maximum doppler shifting effects occur.  
All of these cases are for the zero order approximation.  
Also included are the H~\lya solar line and two cases of the H~\lya planetary lines
at the terminator, viz., the emergent planetary line and one further down in the atmosphere.
Figure~\ref{line}c is the same as Figure~\ref{line}b, but is for the bulge case.

We can develop a simple approximation to the D profile, in terms of H density profile
as follows.  We assume that the HD density profile (cf. Figure~\ref{atmos-AB})
is in diffusive equilibrium and that the D density profile may be calculated assuming 
PCSS between reactions (\ref{dreaction1}) and (\ref{dreaction2}) so that
\begin{equation}
[D] = {k_1 \over k_2} \times f_{HD} \times [H]
\label{density1}
\end{equation}
where $f_{HD}$ is the HD mixing ratio given by $f_{HD} = R e^{-\int {dz \over H(1)}}$,
where H(1) is the scale height of a species of unit mass and R is the HD/H$_2$ mixing ratio 
in the homosphere while we can approximate $k_{1}/k_{2} \sim 0.5 e^{-452/T}$.  Thus the
D/H ratio is given by
\begin{displaymath}
{[D] \over [H]} = 0.5 \times R \times e^{-452/T} e^{-\int {dz \over H(1)}} 
\label{ratio}
\end{displaymath}
and above the homopause will be much smaller than in the lower atmosphere.  
Figures~\ref{test}a,b show the calculated D/H ratio from the model (standard case) and 
the ratio from the calculated from equation~\ref{ratio} for each of the temperature 
profiles A and B.  Thus equation~\ref{ratio} provides an adequate approximation for 
the D profile, if the H profile is known at least in the region where most of the D 
atoms reside.

Figure~\ref{time} gives the chemical time constants $\tau_{C}$ = 1/L 
(see equation~\ref{cty}) and the diffusion time constant $\tau_{diff}$ = 
${H^{2}_{ave} \over (D + K) }$ versus height for both D and HD.  The chemical 
time constants for HD are longer than the diffusion time constant.  
Thus HD is to close approximation in diffusive equilibrium.  For D, it is only in 
diffusive equilibrium above $\sim$~1300 km.  Below this altitude chemical time constants
are shorter and PCSS is an adequate approximation.  This explains why equation~\ref{ratio}
represents such a good approximation for the ${[D] \over [H]}$ ratio. 

Figures~\ref{labela}a,b show the D~\lya intensity across the disk for temperature profiles 
A and B for the antibulge case and further illustrate differences between these temperature 
profiles.

Figures~\ref{labelc}a,b show D~\lya line integrated profiles at the terminator 
(bulge and antibulge) at various altitudes for temperature profiles A and B. 
Figure~\ref{labeld} shows D~\lya line profiles at the terminator for an optically 
thick case in the bulge region. 

Figure~\ref{eddy} shows the emergent D~\lya intensities at the terminator for fixed $\Phi$(H)  
and various values of K.  Included are curves for temperature profiles A and B with and without 
absorption due to H atoms.  The emissions using temperature profile A are brighter 
for all values of K than those for temperature profile B for the absorption case whereas
the opposite is true for the case with no H absorption.  
Also, the plots with H absorption have increasing intensity as K increases while the 
cases without absorption are relatively flat.  Examining the histograms of H and D columns above
$\tau_{CH_{4}}$=1 for both atmospheres corresponding to temperature profiles A and B as shown
in figures~\ref{barcharts} a and b sheds some light on this.  As expected, we see that the 
H column densities decrease for increasing K, 
however, the D column densities only decrease slightly for increasing K values for temperature 
profile A and not at all for temperature profile B.  This result is expected when we consider 
that increasing the eddy diffusion coefficient mixes more CH$_{4}$ into the upper atmosphere 
while decreasing the amount of H present higher up in this region 
but minimally affecting the amount of deuterium which is present somewhat lower in this 
part of the atmosphere.  Additionally, even though 
the H column densities for each temperature profile are matched very closely for 
K = 2 $\times 10^{6}$ cm$^2$ s$^{-1}$ and are very similar the other values
of K, there is a marked difference for the corresponding D column densities for the respective 
temperature profiles (as we expect recalling equation~(\ref{ratio}).  So it is seen that 
eddy diffusion minimally affects the column amounts of deuterium. However, the column amounts 
of deuterium are very sensitive to the temperature profile considered when the H column 
density is held constant for each K.


\section{Conclusions}

State conclusions here...

Differences between the decoupled and ``first order'' case are major.  Further 
calculations utilising the fully coupled case may/may not shed further light on the problem.  
The fully coupled case entails a) symmetric and centred on H~\lya, b) more freqency 
points around the D~\lya line, c) frequency shift the D~\lya line, viz., 
$\sigma_{D} \phi(x_{H} + \Delta)$, and d) utilise solar and planetary H~\lya in the 
calculations.  This case will be examined in a later work.

\appendix

\section{Appendix A: Chemical Model}

The solar fluxes and absorption cross-sections used in the
calculations are listed in Table~A1.  The wavelength interval of
integration is from 1000 \AA\ to 2000 \AA\ divided into 50 \AA\ bins
with the exception of a separate bin at 1215.67 \AA.  Wavelengths
below 1000 \AA\ are not included since the corresponding solar
radiation is significantly absorbed by molecular hydrogen in the upper
atmosphere.  Tables A2 to A4 give the photodissociation reactions,
two-body reactions and three-body reactions, respectively.  

We have only employed chemistry up to and including C$_2$H$_x$ species
for these simulations. We have used the rate constant data assembled
by Romani et al. (1993) and modified by Romani (1996). A comprehensive
set of rate data has also been assembled by Gladstone et al. (1996)
and a comparison of the results from the 2 data sets has been
presented by Parkinson (1998). We have found it difficult to reconcile
our calculated C$_2$H$_2$ to C$_2$H$_6$ ratio in the $\mu$bar pressure
region from our simulations with those presented by Gladstone et al.
(1996). A similar problem has been noted by Romani (1996).  Our
results are more comparable to those of Romani (1996). We note that
the study of Gladstone et al. (1996), as did Romani (1996), included
hydrocarbon chemistry up to C$_4$ species, although this should not
have a large impact on the ratio at these altitudes. In the $\mu$bar
pressure region an important difference in the two compilations is in
the pressure dependence of the CH$_3$ + H $\rightarrow$ CH$_4$ and
CH$_3$ + CH$_3$~$\rightarrow$ C$_2$H$_6$ recombination reactions.

Romani (1996) has updated his reaction scheme using recent
measurements of rate constants and products for the reactions
\begin{equation} 
\rm C_{2}H_{3} + H_{2} \rightarrow C_{2}H_{4} + H \label{eqn:A} 
\end{equation} 
and 
\begin{eqnarray} 
\rm C_{2}H_{3} + H &\rightarrow& \rm H_2 + C_2H_2 \ \ \ \ (a) 
\label{eqn:B} \\ 
\rm &\stackrel{\rm M}{\rightarrow}& \rm C_2H_4 \ \ \ \ \ 
\ \ \ \ \ \ (b) \nonumber 
\end{eqnarray} 
The measurements of Fahr et al. (1995) at room temperature combined
with an estimate of the temperature dependence from Tsang and Hampson
(1986) provide a rate constant for equation~(\ref{eqn:A}) that is much
slower than used earlier by Romani et al. (1993) and Gladstone et al.
(1996) and indicates that it will not be of importance under most
conditions in the lower stratospheres of the outer planets.  In
addition, recent measurements by Monks et al.  (1995) on
reaction~(\ref{eqn:B}) show that channel~(b) is important. Romani
(1996) has integrated information on the rate from Monks et al.
(1995), Heinemann et al. (1988), and Fahr et al. (1995) to estimate
the rate constants for the abstraction and recombination channels for
reaction~(\ref{eqn:B}).

Equation~(\ref{cty}) is solved using a finite central difference
approximation for the vertical derivatives and the species densities
are solved semi-implicitly using a simple tridiagonal solver. For
these applications we have assumed a steady state exists and so have
driven the solution so that ${\partial n_i \over \partial t}
\rightarrow 0$.

\section{Appendix B: Radiative Transfer Model}
The one-dimensional, non-linear integro-differential equation of radiative transfer 
may be written as
\begin{equation}
\mu \frac{d}{dz} I(z;\mu,x) = -\epsilon(z,x) \left\{ I(z;\mu,x) -
S(z;\mu,x) \right\} \label{RTdiff1}
\end{equation}
with
\begin{eqnarray}
S(z;\mu,x) & = & \left\{\frac{\varpi_{\circ}n_{ts}(z)}{2}
\int_{-\infty}^{+\infty}dx' 
\bar{\sigma}_{s}(z,x') \right. \nonumber \\
 &   & \int_{-1}^{+1}d\mu' r(z;\mu,x;\mu',x') I(z;\mu',x') \nonumber \\
 &   &    \nonumber \\
 &   & + \ \frac{\varpi_{\circ}n_{ts}(z)}{4\pi}
\int_{-\infty}^{+\infty}dx' \bar{\sigma}_{s}(z,x')
r(z;\mu,x;-\mu_{\circ}',x') \nonumber \\
 &   & \left. \ \ \ \pi F(x')e^{\left[-\tau(z,x')/\mu_{\circ}\right]} \right\} / \epsilon(z,x) \label{source1}
\end{eqnarray}
where

$z \equiv$ height; 

$\mu \equiv$ cosine of the zenith angle; 

$x \equiv$ frequency in Doppler units from line center and is equal to
\( (\nu - \nu_{\circ})/\Delta \nu_{D} \) where $\nu_{\circ} \equiv $ line
center freqency, \( \Delta \nu_{D} = \frac{\nu_{\circ}}{c}\sqrt{2kT/m} \),
$T \equiv$ temperature, and $m \equiv$ mass of scattering particle; 

$I(z;\mu,x) \equiv$ specific intensity or radiance
(photons cm$^{-2}$ s$^{-1}$ sr$^{-1} \Delta \nu_{D}^{-1}$);

\( \epsilon(z,x) = n_{a}(z)\sigma_{a} + n_{s}(z) \sigma_{s}(z,x) f_{osc}
\equiv \) total extinction per unit path length;

\( \sigma_{s}(z,x) = \sigma_{\circ}(z)\Phi(a(z),x) =
\frac{\pi e^{2}}{m_{e} c} \frac{\Phi(a(z),x)}{\sqrt{\pi} \Delta \nu_{D}(z) f_{osc}} \);

\( f_{osc} \equiv \) oscillator strength for the line;

\( n_{s}(z) \equiv \) number density of scatterers (cm$^{-3}$) for the line;

$\sigma_{a} \equiv$ absorber cross-section;

\( n_{a}(z) \equiv \) number density of absorbers (cm$^{-3}$) for the line;

\( \pi F(x) \equiv \) solar flux (photons cm$^{-2}$ s$^{-1} \Delta x$)
for the line;

$\frac{V(z,x)}{4\pi} \equiv$ total volume production rate of an isotropic
internal source.

$\Phi(a,x)$ is the normalised Voigt function [Mihalas, 1978]
and $a$ is the ratio of the natural width of the line to the Doppler width and
may vary with altitude.

In the decoupled case, we only use angle averaged partial redistribution (AAPR) 
for the particular line we are considering, viz., 
H~\lya or D~\lya.  However, for the ``higher order'' coupled cases, we use AAPR for D~\lya and 
monochromatic scattering (MS) for H~\lya simultaneously.  Thus we have

\begin{eqnarray}
S(z;\mu,x) & = & \left\{ \frac{n_{D}(z)}{2}
\int_{-\infty}^{+\infty}dx' \sigma_{D}(z,x') \int_{-1}^{+1}d\mu' r(z;\mu,x;\mu',x') I(z;\mu',x') 
\right.  \nonumber \\
% &   & \left\dot \int_{-1}^{+1}d\mu' r(z;\mu,x;\mu',x') I(z;\mu',x') \right. \nonumber \\
 &   &    \nonumber \\
 &   & + \left. \frac{n_{H}(z)}{2} \int_{-\infty}^{+\infty}dx' 
\sigma_{H}(z,x') \delta(x - x') \int_{-1}^{+1}d\mu' I(z;\mu',x') \right. \nonumber \\
% &   & \left\dot \int_{-1}^{+1}d\mu' I(z;\mu',x') \right. \nonumber \\
 &   &    \nonumber \\
 &   & + \left. \frac{1}{4\pi} \int_{-\infty}^{+\infty}dx' 
\left( n_{D}(z)\sigma_{D}(z,x')r(z;\mu,x;-\mu_{\circ}',x') +  n_{H}(z)\sigma_{H}\delta(x - x') \right) 
 \right. \nonumber \\
 &   & \left.  \ \ \ \pi F(x')e^{\left[-\tau(z,x')/\mu_{\circ}\right]} \right\} / \epsilon(z,x)  \label{source2}
\end{eqnarray}
where 
\( \epsilon(z,x) = n_{H}(z)\sigma_{H}(x) + n_{CH_{4}}(z)\sigma_{CH_{4}} 
+ n_{D}(z)\sigma_{D}(z,x) \)
and 
\( \tau(z,x) = N_{H}(z)\sigma_{H}(x) + N_{CH_{4}}(z)\sigma_{CH_{4}} 
+ N_{D}(z)\sigma_{D}(z,x)\).

$N_{i}$ is the column amount of species, $i$, and we have assumed 
$\sigma_{CH_{4}}$ is independent of frequency.


\acknowledgments
J. McConnell wishes to thank the Natural Sciences and Engineering Research
Council of Canada for continuing support. NATO xxxxx. We wish to thank 
Erik Griffioen and Yves Rochon for useful assistance.

\newpage

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\newpage



\newpage

% Figure Captions

\begin{figure}
\caption{(a)Temperature profiles adopted for the mesosphere and thermosphere
of Jupiter. Curve~A is based on an early analysis of the Voyager occultation
results [Festou et al., 1981] while Curve B is taken from Yelle
et al. [1996]. See text for details. 
(b) same as (a) except bulge region shown included.  See text for details.}
\label{T}
\end{figure}

\begin{figure}
\caption{Line profile of the solar Lyman-$\alpha$ line (xxxxxshow this)adopted as a standard
case for these calculations based on Gladstone's [1988] approximation of the profile 
of Lemaire et al. [1978]. Also shown are the positions of D
Lyman-$\alpha$ for solar longitudes of 0 and $\pm$~90. +90 corresponds to blue wavelength shifting and -90 red wavelength shifting}
\label{line}
\end{figure}

\begin{figure}
\caption{The model atmospheres used in the calculations consisting
of H$_{2}$, HD, CH$_{4}$, H and D. For (a) the temperature profile A and  
(b) temperature profile B shown in Figure~1. (c) is the same as (a) except
the D flux is multiplied by a factor of 10.}
\label{atmos-AB}
\end{figure}

\begin{figure}
\caption{Calculated D/H ratio from the model and from equation~X }
%\ref{ratio} }
\label{test}
\end{figure}

\begin{figure}
\caption{Chemical and Diffusion time constants for D and HD, $\tau_{D}, \tau_{HD}, \tau_{diff}$ 
for the terminator, temperature profiles A and B}
\label{time}
\end{figure}

\begin{figure}
\caption{Intensity across the disk for temperature profiles A and B -- AntiBulge}
\label{labela}
\end{figure}

%\begin{figure}
%\caption{Intensity across the disk for temperature profiles A and B -- Bulge}
%\label{labelb}
%\end{figure}

\begin{figure}
\caption{(a) D~\lya Line Integrated Profile at the terminator for various altitudes--Antibulge,
(b) D~\lya Line Integrated Profile at the limb for various altitudes--Bulge}
\label{labelc}
\end{figure}

\begin{figure}
\caption{D~\lya Line Terminator Profiles for an optically thick case at 1.011R$_J$ 
(or z$_{graz}$ = 7.135 $\times 10^{7}$ cm)-- Bulge case}
\label{labeld}
\end{figure}

\begin{figure}
\caption{Emergent D~\lya at the terminator for fixed $\Phi$(H) as a function of K, for 
temperature profiles A and B}
\label{eddy}
\end{figure}

\begin{figure}
\caption{(a) H column density versus Eddy diffusion coefficient K for temperature profiles A and B,
         (b) D column density versus Eddy diffusion coefficient K for temperature profiles A and B
all cases are for columns above $\tau_{CH_{4}}$ = 1 }
%\ref{ratio} }
\label{barcharts}
\end{figure}

\newpage
% Tables

%\begin{planotable}{rll}
%\tablewidth{20pc}
%\tablecaption{Table of Gas-Phase Chemical Reactions}
%\tablenum{1}
%   % Note the \tablenum{} command above.  Since this manuscript
%   % includes an appendix, a \tablenum command is needed or the 
%   % table caption will appear "Table B1. Relevant Information"
%\tablehead{
%\colhead{\ }  &      \colhead{Reaction} & \colhead{Ref} }
%%\tablecomments{References: 
%%1, {\it Demore et al.,} [1994];
%%2, {\it Put some reference here} }
%\startdata
%\label{T1}
%1&$\rm O+O_2 \longrightarrow O_3$& 1\nl
%2&$\rm O+O_3 \longrightarrow 2O_2$& 1 \nl
%\end{planotable}


\small
\begin{planotable}{rl}
\tablewidth{15pc}
\tablecaption{Main deuterium reactions.}
\tablenum{1}
% Note the \tablenum{} command above.  Since this manuscript
   % includes an appendix, a \tablenum command is needed or the 
   % table caption will appear "Table B1. Relevant Information"
\tablehead{
\colhead{Reaction }  & \colhead{ }  }
%\tablenotetext{\star}{Reactions and quantum yields are from Yung et al (1984) unless otherwise specified.} 
\startdata
%HD + h$\nu$ & $\rightarrow$ H + D \nl 
D + H$_{2}$ & $\rightarrow$ DH+H \nl 
H + HD & $\rightarrow$ H$_{2}$ + D \nl 
D + H + M & $\rightarrow$ DH + M \nl
D + CH$_{3}$ +M & $\rightarrow$ CH$_{3}$D + M \nl
\end{planotable}

\begin{planotable}{rl} 
\tablewidth{25pc} 
\tablecaption{Standard parameters used} 
\tablenum{2} 
% % Note the \tablenum{} command above.  Since this manuscript 
% % includes an appendix, a \tablenum command is needed or the 
% % table caption will appear "Table B1.  Relevant Information" 
\tablehead{ \colhead{Parameter } & \colhead{Value}  } 
%\tablecomments{References: 
%1,{\it Demore et al.,} [1994]; 
%2, {\it Put some reference here} }
\tablenotetext{\star}{sdu $\equiv$ standard doppler unit}
\startdata 
\label{T1} 
H \lya & 1215.67~\AA\ \nl 
D \lya & 1215.34~\AA\ \nl
$\pi F$ & 4.4$\times 10^{11}$ photons cm$^{-2}$ s$^{-1}$ \nl
1 sdu$^{\star}$ = $\Delta~\nu_{D}$ @ 500K & 0.116~\AA \nl
x$_{off}$ & 18.9 sdu (or 0.219 \AA) \nl
x$_{dis}$ & 18.5 sdu (or 0.215 \AA) \nl
$\Delta$ & 28.4 sdu (or 0.33 \AA) \nl
oscillator strength, $f_{osc}$ & 0.4162 \nl
K$_h$ & 2$^{+2}_{-1} \times 10^6$ cm$^2$ s$^{-1}$ \nl
\end{planotable}
\normalsize

\vfill

\newpage

\small
\begin{planotable}{lllllll}
\tablewidth{35pc}
\tablecaption{Integrated solar flux at 1 A.U. and cross sections.}
%\tablenum{7}
% Note the \tablenum{} command above.  Since this manuscript
   % includes an appendix, a \tablenum command is needed or the 
   % table caption will appear "Table B1. Relevant Information"
\tablehead{
%Wavelength$^{a}$ & 
%Integrated Flux$^{b}$ &
%& & & 
%Cross sections (cm$^{2}$)} & & \nl
\colhead{$\lambda (\AA)$} & 
\colhead{Integrated Flux} &
\colhead{H$_{2}^{c}$} & 
\colhead{CH$_{4}^{d}$} &  
\colhead{C$_{2}$H$_{2}^{e}$} &  
\colhead{C$_{2}$H$_{4}^{d}$} &  
\colhead{C$_{2}$H$_{6}^{f}$} }
\tablenotetext{\star}{with units ph cm$^{-2}$ s$^{-1}$ $\Delta~\lambda ^{-1}$}
\tablenotetext{a}{Values denote the initial wavelengths of 
50 \AA intervals except for 1200 and 
Ly-$\alpha$ at 1216 with respective 
intervals of 1200-1210 plus 1220-1250 and 1210-1220} 
\tablenotetext{b}{Solar fluxes at 1 au integrated over the corresponding
wavelength interval; Strobel (1973) and Hinterreger (1970)
for 850-1150, Rottman (1981) for 1200-1850,
and Mount et al. (1980) for 1900-2000}
\tablenotetext{c}{Atreya (1986) and Strobel (1969)}
\tablenotetext{d}{Strobel (1973 and 1969)}
\tablenotetext{e}{Hamai and Hirayama (1979)} 
\tablenotetext{f}{Mount and Moos (1978)} 
\startdata
%850 & 1.5(10) & 5.0(-19) & 4.2(-17) & 2.8(-17) & 5.0(-17) & 6.5(-17) \nl
%900 & 1.3(10) & 7.4(-20) & 4.2(-17) & 2.8(-17) & 5.0(-17) & 6.5(-17) \nl
%950 & 3.6(9) & 4.8(-20) & 5.6(-17) & 2.8(-17) & 5.0(-17) & 5.8(-17) \nl
1000 & 6.2(9) & 0.0 & 3.6(-17) & 2.7(-17) & 5.0(-17) & 5.0(-17) \nl
1050 & 6.2(9) & 0.0 & 3.0(-17) & 2.5(-17) & 4.0(-17) & 4.7(-17) \nl
1100 & 1.2(9) & 0.0 & 2.2(-17) & 2.5(-17) & 1.9(-17) & 4.0(-17) \nl
1150 & 1.9(9) & 0.0 & 1.8(-17) & 1.4(-17) & 1.6(-17) & 3.0(-17) \nl
1200 & 1.4(10) & 0.0 & 1.9(-17) & 1.4(-17) & 2.2(-17) & 2.2(-17) \nl
1216 & 4.4(11) & 0.0 & 1.6(-17) & 2.8(-17) & 2.5(-17) & 2.0(-17) \nl
1250 & 6.7(9) & 0.0 &  1.8(-17) & 1.5(-17) & 2.8(-17) & 2.1(-17) \nl
1300 & 2.5(10) & 0.0 & 1.7(-17) & 6.3(-17) & 1.7(-17) & 1.8(-17) \nl
1350 & 1.4(10) & 0.0 & 7.2(-18) & 3.0(-17) & 2.0(-17) & 1.1(-17) \nl
1400 & 1.8(10) & 0.0 & 1.1(-18) & 4.0(-18) & 1.3(-17) & 6.7(-17) \nl
1450 & 2.7(10) & 0.0 & 2.2(-20) & 1.0(-17) & 1.1(-17) & 2.4(-18) \nl
1500 & 5.2(10) & 0.0 & 8.9(-23) & 3.0(-17) & 1.8(-17) & 9.4(-19) \nl
1550 & 7.1(10) & 0.0 & 8.8(-24) & 5.6(-19) & 2.4(-17) & 9.8(-20) \nl
1600 & 9.7(10) & 0.0 & 6.0(-24) & 6.7(-19) & 3.0(-17) & 1.1(-21) \nl
1650 & 2.1(10) & 0.0 & 0.0 & 9.3(-19) & 3.6(-17) & 0.0 \nl
1700 & 3.7(11) & 0.0 & 0.0 & 1.2(-18) & 2.6(-17) & 0.0 \nl
1750 & 6.3(11) & 0.0 & 0.0 & 1.1(-18) & 1.5(-17) & 0.0 \nl
1800 & 8.3(11) & 0.0 & 0.0 & 7.4(-19) & 1.6(-18) & 0.0 \nl
1850 & 1.3(12) & 0.0 & 0.0 & 3.7(-19) & 3.7(-19) & 0.0 \nl
1900 & 1.9(12) & 0.0 & 0.0 & 1.5(-19) & 5.0(-20) & 0.0 \nl
1950 & 2.6(12) & 0.0 & 0.0 & 1.1(-19) & 1.0(-20) & 0.0 \nl
2000 & 3.4(12) & 0.0 & 0.0 & 5.0(-20) & 1.0(-21) & 0.0 \nl
\end{planotable}
\normalsize

\vfill

\small
\begin{planotable}{lllccl}
\tablewidth{35pc}
\tablecaption{Hydrocarbon photodissociation reactions.$^{\star}$}
%\tablenum{3}
% Note the \tablenum{} command above.  Since this manuscript
   % includes an appendix, a \tablenum command is needed or the 
   % table caption will appear "Table B1. Relevant Information"
\tablehead{
\colhead{ No.}  & \colhead{Reaction } &  & 
\colhead{$\phi^{\dagger}_{Lyman-\alpha}$} & 
\colhead{$\phi^{\dagger}_{other \lambda}$} & \colhead{Ref.} }
\tablenotetext{\dagger}{Quantum yield} 
\tablenotetext{\star}{Reactions and quantum yields are from Yung et al (1984) unless otherwise specified.} 
\tablenotetext{a}{Mentall and Gentieu (1970)}
\tablenotetext{b}{Gladstone (1982)} 
\tablenotetext{c}{Okabe (1983)} 
\tablenotetext{d}{Gladstone et al.  (1996)}
\startdata
R1 & $H_{2}$+$h\nu$ & $\rightarrow 2H$ & 1.00 & 1.00 & a,b \nl 
R2 & $CH_{4}$+$h\nu$ & $\rightarrow$  $^{3}CH_{2}$+$2H$ & 0.05 & 0.00 & d \nl 
R3 & & $\rightarrow$  $^{1}CH_{2}$+$H_{2}$ & 0.41 & 0.90 & d \nl 
R4 & & $\rightarrow CH$+$H$+$H_{2}$ & 0.05 & 0.00 & d \nl 
R5 & $C_{2}H_{2}$+$h\nu$ & $\rightarrow C_{2}H$+$H$ & $\lambda < 1500$\AA, 0.30
& $\lambda >1500$\AA, 0.06 & c\nl 
R6 & & $\rightarrow C_{2}$+$H_{2}$ &  0.10 & 0.10 & \nl 
R7 & $C_{2}H_{4}$+$h\nu$ & $\rightarrow C_{2}H_{2}$+$H_{2}$ & 0.51 & 0.51 & \nl 
R8 & & $\rightarrow C_{2}H_{2}$+$2H$ & 0.49 & 0.49 & \nl 
R9 & $C_{2}H_{6}$+$h\nu$ & $\rightarrow C_{2}H_{4}$+$H_{2}$ & 0.13 & 0.56 & \nl 
R10 &  & $\rightarrow C_{2}H_{4}$+$2H$ & 0.30 & 0.14 & \nl 
R11 &  & $\rightarrow C_{2}H_{2}$+$2H_{2}$ & 0.25 & 0.27 & \nl 
R12 &  & $\rightarrow CH_{4}$+$^{1}CH_{2}$ & 0.25 & 0.02 & \nl 
R13 &  & $\rightarrow 2CH_{3}$ & 0.08 & 0.01 & \nl 
%R14 & $HD$+$h\nu$ & $\rightarrow H$ + $D$ & 0.00 & 0.00 & \nl 
\end{planotable}
\normalsize

\vfill
 
\small
\begin{planotable}{lllll}
\tablewidth{35pc}
\tablecaption{Hydrocarbon two-body reactions.$^{\star}$}
%\tablenum{4}
% Note the \tablenum{} command above.  Since this manuscript
   % includes an appendix, a \tablenum command is needed or the 
   % table caption will appear "Table B1. Relevant Information"
\tablehead{
\colhead{ No.} & \colhead{Reaction } &  & 
\colhead{ Rate constant, $k$ $^{\dagger}$ } & \colhead{Reference} } 
\tablenotetext{ijk}{CHECK THESE REFERENCES}
\startdata
R14 & $CH$+$H_{2}$ & $\rightarrow ^{1}CH_{2}$+$H$ & 2.38x10$^{-11}e^{-1760/T}$ & i \nl 
R15 & $CH$+$CH_{4}$ & $\rightarrow C_{2}H_{4}$+$H$ & 3.0x10$^{-11}e^{-200/T}$ & \nl 
R16 & $^{1}CH_{2}$+$H_{2}$ & $\rightarrow$ $^{3}CH_{2}$+$H_{2}$ & 1.26x10$^{-11}$ & \nl 
R17 & & $\rightarrow CH_{3}$+$H$ & 9.24x10$^{-11}$ & \nl 
R18 & $^{1}CH_{2}$+$CH_{4}$ & $\rightarrow$ $^{3}CH_{2}$+$CH_{4}$ & 1.20x10$^{-11}$ & \nl 
R19 & & $\rightarrow 2CH_{3}$ & 5.90x10$^{-11}$ & \nl 
R20 & $^{3}CH_{2}$ + $^{3}CH_{2}$ & $\rightarrow C_{2}H_{2}$+$2H$ & 2.10x10$^{-10}e^{-408/T}$ & \nl 
R21 & $CH_{3}$+$^{3}CH_{2}$ & $\rightarrow C_{2}H_{4}$+$H$ & 7.0x10$^{-11}$ & \nl 
R22 & $C_{2}$+$H_{2}$ & $\rightarrow C_{2}H$+$H$ & 1.77x10$^{-10}e^{-1469/T}$ & \nl 
R23 & $C_{2}$+$CH_{4}$ & $\rightarrow C_{2}H$+$CH_{3}$ & 5.05x10$^{-11}e^{-297/T}$ & \nl 
R24 & $C_{2}H$+$C_{2}H_{2}$ & $\rightarrow C_{4}H_{2}$+$H$ & 4.0x10$^{-11}$ & \nl 
R25 & $C_{2}H$+$H_{2}$ & $\rightarrow C_{2}H_{2}$+$H$ & 5.6x10$^{-11}e^{-1443/T}$ & \nl 
R26 & $C_{2}H$+$CH_{4}$ & $\rightarrow C_{2}H_{2}$+$CH_{3}$ & 9.0x10$^{-12}e^{-250/T}$ & \nl 
R27 & $C_{2}H_{3}$+$H$ & $\rightarrow C_{2}H_{2}$+$H_{2}$ & 6.0x10$^{-12}$ & \nl 
R28 & $C_{2}H_{3}$+$H_{2}$ & $\rightarrow C_{2}H_{4}$+$H$ & 2.6x10$^{-13}e^{-2646/T}$ & j,k \nl 
R29 & $C_{2}H_{3}$+$CH_{3}$ & $\rightarrow C_{2}H_{2}$+$CH_{4}$ & 6.5x10$^{-13}$ & \nl 
R30 & $2C_{2}H_{3}$ & $\rightarrow C_{2}H_{4}$+$C_{2}H_{2}$ & 1.8x10$^{-11}$ & \nl 
R31 & $C_{2}H_{5}$+$C_{2}H_{3}$ & $\rightarrow C_{2}H_{6}$+$C_{2}H_{2}$ & 8.0x10$^{-13}$ & \nl 
R32 & & $\rightarrow 2C_{2}H_{4}$ & 8.0x10$^{-13}$ & \nl
R33 & $C_{2}H_{5}$+$H$ & $\rightarrow 2CH_{3}$ & 7.95x10$^{-11}$ & \nl 
R34 & $C_{2}H_{5}$+$CH_{3}$ & $\rightarrow C_{2}H_{4}$+$CH_{4}$ & 3.3x10$^{-11}T^{-0.5}$ & \nl 
R35 & $2C_{2}H_{5}$ & $\rightarrow C_{2}H_{6}$+$C_{2}H_{4}$ & 1.2x10$^{-11}e^{-540/T}$ & \nl 
R36 & $D$ + $H_{2}$ & $\rightarrow DH$+$H$ & 1.6x10$^{-10}e^{-3875/T}$ & \nl 
R37 & $H$ + $HD$ & $\rightarrow H_{2}$+$D$ & 7.91x10$^{-11}e^{-4327/T}$ & \nl 
\end{planotable}
\normalsize

\vfill
 
\small
\begin{planotable}{lllll}
\tablewidth{35pc}
\tablecaption{Hydrocarbon three-body reactions.$^{\star}$}
%\tablenum{5}
% Note the \tablenum{} command above.  Since this manuscript
   % includes an appendix, a \tablenum command is needed or the 
   % table caption will appear "Table B1. Relevant Information"
\tablehead{
\colhead{ No.}  & \colhead{Reaction } &  & \colhead{ Rate constants $^{\dagger}$} & 
\colhead{Reference} } 
\tablenotetext{\star}{Reactions and rates are from Yung et al. (1984) unless otherwise specified.} 
\tablenotetext{\dagger}{Most of the three-body rate coefficients, in cm$^{6}$s$^{-1}$, are set as 
$k$=$k_{o}k_{\infty}/(k_{o}M+k_{\infty})$. $k_{o}$ is listed first above $k_{\infty}$ }  
\tablenotetext{o}{Trainor et al. (1973)}
\tablenotetext{p}{Prather et al. (1978)} 
\tablenotetext{q}{Van den Berg (1969)}
\tablenotetext{r}{Van den Berg (1970)}
\startdata
R38 & $2H$+$M$ & $\rightarrow H_{2}$+$M$ & 1.5x10$^{-29}T^{-1.3}$ & o,p \nl 
 & & & n/a  & \nl 
R39 & $CH_{3}$+$H$+$M$ & $\rightarrow CH_{4}$+$M$ & 3.8x10$^{-28}e^{-20/T}  T<200^{\circ}$ & \nl
 & & & 5.8x10$^{-29}e^{-355/T}  T>200^{\circ}$ & \nl 
 & & & 1.0x10$^{-9}T^{-0.4}$ & \nl 
R40 & $2CH_{3}$+$M$ & $\rightarrow C_{2}H_{6}$+$M$ & 8.76x10$^{-7}e^{-1390/T}T^{-7.03}$ & q \nl 
 & & & 1.5x10$^{-7}e^{-329/T}T^{-1.18}$ & \nl 
R41 & $C_{2}H_{2}$+$H$+$M$ & $\rightarrow C_{2}H_{3}$+$M$ &
6.4x10$^{-25}T^{-2}e^{-1200/T}$ \nl
 & & & 3.8x10$^{-11}e^{-1374/T}$ & \nl 
R42 & $C_{2}H_{4}$+$H$+$M$ & $\rightarrow C_{2}H_{5}$+$M$ & 
2.15x10$^{-29}e^{-349/T}$ \nl
 & & & 4.39x10$^{-11}e^{-1087/T}$ & \nl 
R43 & $D$+$H$+$M$ & $\rightarrow DH$+$M$ & 
1.5x10$^{-29}T^{-1.3}$ \nl
% & & & n/a  & \nl 
R44 & $D$+$CH_{3}$+$M$ & $\rightarrow CH_{3}D$+$M$ & 3.8x10$^{-28}e^{-20/T}  T<200^{\circ}$ & \nl
 & & & 5.8x10$^{-29}e^{-355/T}  T>200^{\circ}$ & \nl 
 & & & 1.0x10$^{-9}T^{-0.4}$ & \nl 
\end{planotable}
\normalsize

\vfill 

 
\end{document}