# Methane, Carbon Monoxide, and Ammonia in Brown Dwarfs and Self-Luminous
Giant Planets

Kevin J. Zahnle  NASA Ames Research Center, MS-245-3, Moffett Field, CA 94035;
<EMAIL_ADDRESS>Mark S. Marley  NASA Ames Research Center, MS-245-3,
Moffett Field, CA 94035<EMAIL_ADDRESS>
###### Abstract

We address disequilibrum abundances of some simple molecules in the
atmospheres of solar composition brown dwarfs and self-luminous extrasolar
giant planets using a kinetics-based 1D atmospheric chemistry model. Our
approach is to use the full kinetics model to survey the parameter space with
effective temperatures between 500 K and 1100 K. In all of these worlds
equilibrium chemistry favors CH4 over CO in the parts of the atmosphere that
can be seen from Earth, but in most disequilibrium favors CO. The small
surface gravity of a planet strongly discriminates against CH4 when compared
to an otherwise comparable brown dwarf. If vertical mixing is like Jupiter’s,
the transition from methane to CO occurs at 500 K in a planet. Sluggish
vertical mixing can raise this to 600 K; but clouds or more vigorous vertical
mixing could lower this to 400 K. The comparable thresholds in brown dwarfs
are $1100\pm 100$ K. Ammonia is also sensitive to gravity, but unlike CH4/CO,
the NH3/N2 ratio is insensitive to mixing, which makes NH3 a potential proxy
for gravity. HCN may become interesting in high gravity brown dwarfs with very
strong vertical mixing. Detailed analysis of the CO-CH4 reaction network
reveals that the bottleneck to CO hydrogenation goes through methanol, in
partial agreement with previous work. Simple, easy to use quenching relations
are derived by fitting to the complete chemistry of the full ensemble of
models. These relations are valid for determining CO, CH4, NH3, HCN, and CO2
abundances in the range of self-luminous worlds we have studied but may not
apply if atmospheres are strongly heated at high altitudes by processes not
considered here (e.g., wave breaking).

##  1\. Introduction

Disequilibrium chemistry has been known in Jupiter’s atmosphere for several
decades (Prinn and Barshay, 1977; Bézard et al., 2002) and has been expected
and suspected in brown dwarf atmospheres from the time of their discovery
(Fegley and Lodders, 1996; Noll et al., 1997; Saumon et al., 2000). The most
famous disequilibrium is an overabundance of CO relative to CH4. This occurs
in Jupiter and brown dwarfs when CO is dredged up from deep, hot layers of the
atmosphere more quickly than chemical reactions with ambient hydrogen can
convert it to CH4. It is to be expected that similar processes take place in
young, self-luminous extrasolar giant planets. But the apparent paucity of
methane in the atmospheres of planets with effective temperatures comparable
to those of methane-rich T-type brown dwarfs was underpredicted and has been
met with surprise.

Among field brown dwarfs methane appears—by definition—at the L to T-type
transition where their near-infrared colors turn to the blue at effective
temperatures near 1200 K (Kirkpatrick, 2005). The first directly imaged
planets, however, were found to have effective temperatures below 1200 K and
yet their near-infrared spectra were devoid of signs of methane in K band
(e.g., Barman et al., 2011a). The best example is HR8799c: despite an
effective temperature near 1100 K, in both low and high resolution
spectra—particularly the high resolution spectrum taken by Konopacky et al.
(2013)—methane has gone missing.

Surface gravity is the defining difference between extrasolar giant planets
and brown dwarfs, and to date is the only proven difference, although there
are great hopes for metallicity. Field brown dwarfs (hereafter BDs) have high
surface gravities ($g$ of order $10^{5}$ cm/s2) and therefore very compressed
scale heights. Extrasolar giant planets (EGPs) have modest surface gravities
($g$ of order $10^{3}$ cm/s2) and extended scale heights (e.g., Burrows et
al., 1997; Saumon & Marley, 2008)111We do not intend to wade into the
nomenclature battles here. For our purposes companion objects below $\sim
13\,\rm M_{J}$ are planets.. A BD and a self-luminous EGP of the same
composition and the same effective temperature will have similar optical
depths as a function of temperature, at least in the absence of clouds; i.e.,
the function $T(\tau)$ is roughly the same. However, because $p\propto\tau g$,
the BD has a much higher pressure at a given optical depth than does the EGP,
and a BD is much cooler than the EGP at a given pressure. Because lower
temperatures and higher pressure favor CH4 in its struggle with CO, it has
been pointed out that CH4 will be more easily seen and CO less easily seen at
lower gravity in BDs (Hubeny and Burrows, 2007). Hence Barman et al. (2011a)
suggested that the even lower gravity of HR8799b might be why no methane is
seen in it. We agree. We will verify that the dependence on $g$ is strong and
leads to qualitatively different outcomes for BDs and self-luminous EGPs, a
result implicit in previous work but far from fully appreciated.

Previous studies of carbon speciation in BDs and EGPs have mostly focused on a
few particular objects (Saumon et al., 2006; Geballe et al., 2009; Barman et
al., 2011a, b; Moses et al., 2011; Line et al., 2011), or on hot highly
irradiated EGPs (Moses et al., 2011; Visscher, 2012), or used one of several
quench approximations culled from the literature (Fegley and Lodders, 1996;
Saumon et al., 2000; Lodders and Fegley, 2002; Hubeny and Burrows, 2007), or
various combinations of the above (Cooper and Showman, 2006; Visscher and
Moses, 2011; Moses et al., 2013a, b). The basic idea is that CH4 and CO will
often be seen in disequilibrium because the chemical reactions that would
enforce equilibrium don’t have time enough to take place while the gas is cool
(Prinn and Barshay, 1977). The disequilibrium composition that results is
described as “frozen-in” or “quenched.”

Quenching has been widely used to quantify discussions of CO-CH4 and N2-NH3
disequilibria in a wide range of astrophysical problems, dating at least back
to Prinn and Barshay (1977)’s study of CO in Jupiter. Earlier discussions of
quenching can be found with respect to the N2-O2-NO system that is important
in thunderbolts (Chameides et al., 1979), meteor entry and rocket rëentry
(Park and Menees, 1978), and explosions in Earth’s atmosphere (Zel’dovich and
Raizer, 1967). In the quench approximation, the disequilibrium composition
that one can observe is approximated by the equilibrium composition when the
relevant chemical reaction time scale $t_{\rm chem}$ equals the relevant
cooling timescale, which if due to mixing can be written $t_{\rm mix}$ (Prinn
and Barshay, 1977). There are many different prescriptions for defining
$t_{\rm chem}$ and $t_{\rm mix}$ that we will discuss below. The history of
quench schemes for jovian planets, exoplanets, and brown dwarfs has been
comprehensively recounted in a series of recent papers by Moses and colleagues
(Moses et al., 2010, 2011; Visscher and Moses, 2011).

Here we do something different. We use a 1D chemical kinetics code coupled to
$p$-$T$ profiles from a detailed 1D atmospheric structure code to compute a
galaxy of chemical compositions in a wide range of possible brown dwarfs and
cooling EGPs. The code explicitly includes reverses of all reactions so that,
in the absence of atmospheric physics, the chemical composition would relax to
equilibrium at every height. Our strategy is to find the apparent quench
points in all the models and analyze these for their systematic properties. We
then fine tune our results by using them in quench approximations. Our
objective is to describe the emergent properties of the chemical network as a
whole. Our strategy differs from previous work that seeks to determine the one
key rate-limiting step in a network of reactions, which is then treated as the
effective reaction rate for the network as a whole.

We limit the study to self-luminous cooling worlds for which insolation is not
(yet) thermally important. This includes brown dwarfs, free-floating planets,
and young directly-imaged planets. This category includes most of the
exoplanets for which good data can be obtained in the present or in the near
future. What this limitation means for carbon speciation is that we are
concerned only with the conversion of CO to CH4. Unlike Line et al. (2011),
Visscher and Moses (2011), and Visscher (2012), we do not address the kinetic
inhibition against oxidizing CH4 to CO. The latter is an issue in strongly
irradiated planets that are warm at high altitudes where, if vertical mixing
is fast and the temperature not too hot, CH4 and its photochemical products
will be overabundant (Line et al., 2010; Miller-Ricci Kempton et al., 2011;
Morley et al., 2013). Methane oxidation could be an issue for BDs and cooling
EGPs if the higher parts of their atmospheres are strongly heated by wave-
breaking processes not taken into account in our radiative-convective model,
but we do not further address this possibility here.

Quenching in the nitrogen system has also been the subject of many studies
over the years (Abelson, 1966; Chameides and Walker, 1981; Prinn and Fegley,
1987; Fegley and Lodders, 1994, 1996; Saumon et al., 2006; Moses et al., 2010,
2011; Line et al., 2011). As with methane, the underabundance of ammonia in
brown dwarfs cold enough to favor it has been attributed to disequilibrium
chemistry (Saumon et al., 2006). The visibility of ammonia has been made the
distinguishing characteristic of the Y dwarf, the newest, coldest, and
possibly last member of the stellar spectral sequence (Cushing et al., 2011).
The NH3-N2-HCN system differs from CO and CH4 in interesting ways that lead to
significantly different behavior.

Finally, for completeness, we address quenching of CO2, a gas that can be
relatively easy to observe from a space-based observatory and, because its
abundance is sensitive to metallicity, can be relatively telling. CO2 is not
always thought of as a species subject to quenching (but see Prinn and Fegley
(1987)). In the H2-rich, UV-poor worlds that are the subject of this study,
CO2 quenching does take place.

##  2\. Overview of CO and CH4

Figure 1 illustrates the most important chemical pathways between CO and CH4
in a warm H2-rich atmosphere. The path from CO to CH4 climbs over three energy
barriers, the first between CO and formaldehyde (H2CO), the second between
formaldehyde and methanol (CH3OH), and the third between methanol and methane.
The three barriers can be thought of as reducing the C$\equiv$O triple bond to
a double bond, reducing the C=O double bond to a single bond, and splitting C
from O entirely.

Figure 1.— Major chemical pathways linking CO and CH4 in an H2-rich
atmosphere. Reactions from left to right are with H2 or H. Key intermediate
molecules are formaldehyde (H2CO) and methanol (CH3OH). Other intermediates
(HCO, H2COH, CH3O, CH3) are short-lived free radicals. Where the species are
plotted gives a rough indication of the energetics. Energy barriers correspond
to breaking C-O bonds: from triple to double, from double to single, and from
single to freedom. Relative magnitudes of reaction rates are indicated by
arrow thickness for conditions near the quench point in a particular model
(red star on Fig 4) that gives roughly equal amounts of CO and CH4. At higher
temperatures and lower pressures the plot as a whole would tilt to the left,
with carbon pooling in CO. At lower temperatures and higher pressures the plot
as a whole tilts to the right and carbon pools in CH4. The energy barriers
pose greater obstacles when the gas is colder.

In photochemistry the first barrier is unimportant because atomic hydrogen is
generated in abundances that vastly exceed equilibrium (Liang et al., 2003).
Kinetic barriers to adding H to CO to make HCO or adding H to HCO to make H2CO
are insignificant. The third barrier is unimportant in photochemistry because
incident UV radiation readily splits CH3OH into CH3 and OH, which is quickly
followed by adding photochemical H to CH3 to make CH4. In photochemistry the
energy to overcome the first and third barriers comes from UV photons.

By contrast, the middle barrier is not easily overcome by photochemistry.
Formaldehyde is readily photolyzed but the products are either CO or HCO;
i.e., regress that leaves the C=O bond unbroken. Meanwhile successive 3-body
additions of photochemical H to H2CO to make CH3OH face considerable kinetic
barriers, save at high temperatures where CH4 is not favored. The historic
focus of the planetary literature was therefore on reactions that go from
formaldehyde to methanol. Prinn and Barshay (1977) suggested ${\rm H}_{2}+{\rm
H}_{2}{\rm CO}\rightarrow{\rm CH}_{3}+{\rm OH}$, an ambitious reaction that
jumps two hurdles at once, as the rate-limiting step, while Yung et al (1988)
suggested that ${\rm H}+{\rm H}_{2}{\rm CO}+{\rm M}\rightarrow{\rm CH}_{3}{\rm
O}+{\rm M}$ (where ${\rm M}$ represents a third body) would be the bottleneck.
Both schemes have been widely used, for Jupiter (Bézard et al., 2002), EGPs
(Cooper and Showman, 2006; Line et al., 2011), and BDs (Hubeny and Burrows,
2007; Barman et al., 2011a).

Without the photochemical assist, any of the three barriers could potentially
block the reaction. Recently, Moses et al. (2011) concluded that the rate-
limiting reaction is the one that breaks the C-O single bond. They report that
the bottleneck is between methanol and methane, ${\rm CH}_{3}{\rm OH}+{\rm
M}\rightarrow{\rm CH}_{3}+{\rm OH}+{\rm M}$ when methane is abundant, or goes
through a free radical ${\rm H}+{\rm H}_{2}{\rm COH}\rightarrow{\rm
CH}_{3}+{\rm OH}$ when methane is scarce. We will concur with the former but
not the latter.

##  3\. The photochemical model

Standard 1D codes simulate atmospheric chemistry by computing the gains and
losses of chemical species at different altitudes while accounting for
vertical transport. Vertical transport is parameterized as a diffusive process
with an “eddy diffusion coefficient,” denoted $K_{zz}$ [cm2/s]. Volume mixing
ratios $f_{i}$ of species $i$ are obtained by solving continuity

$N{\partial f_{i}\over\partial
t}=P_{i}-L_{i}Nf_{i}-{\partial\phi_{i}\over\partial z}$ (1)

and force (flux)

$\phi_{i}=b_{ia}f_{i}\left({m_{a}g\over kT}-{m_{i}g\over
kT}\right)-\left(b_{ia}+K_{zz}N\right){\partial f_{i}\over\partial z}$ (2)

equations. In these equations $N$ is the total number density (cm-3);
$P_{i}-L_{i}Nf_{i}$ represent chemical production and loss terms,
respectively; $\phi_{i}$ is the upward flux; $b_{ia}$, the binary diffusion
coefficient between $i$ and the background atmosphere $a$, describes true
molecular diffusion; and $m_{a}$ and $m_{i}$ are the molecular masses of $a$
and $i$. We implement molecular diffusion of a heavier gas through H2 by
setting $b_{ia}=6\times 10^{19}\left(T/1400\right)^{0.75}$ cm-1s-1—appropriate
for CO—for all the heavy species. This is a reasonable approximation for
present purposes, as our concern is with quenching at altitudes well below the
homopause. The physical meaning of $b_{ia}$ is the ratio of the relative
thermal velocities of the two species to their mutual collision cross section.
In the present circumstances, the relative thermal velocity is effectively
that of H2, so the only important source of variation in $b_{ia}$ stems from
the different diameters of the molecules. The code has been used by Zahnle et
al. (2009) to address hot Jupiters and by Miller-Ricci Kempton et al. (2011)
and Morley et al. (2013) to address H2-rich hot Neptunes and super-Earths.

Temperature and pressure profiles are imported from the output of detailed
radiative-convective models that are described elsewhere (Saumon & Marley,
2008). The models are cloudless and of solar metallicity. Adding cloud opacity
would make the $p$-$T$ profiles hotter and less favorable to CH4. The
cloudless model is something of a best case for methane.

Treating vertical transport as diffusion is a necessary evil in a 1-D code. We
will regard $K_{zz}$ as a mildly constrained free parameter. In free
convection, $K_{zz}$ has been estimated from mixing length theory (Gierasch &
Conrath, 1985),

$K_{zz}={1\over 3}H\left({R_{\rm gas}F_{\rm conv}\over\mu\rho
C_{p}}\right)^{1/3},$ (3)

where $R_{\rm gas}$ is the universal gas constant, $F_{\rm conv}$ the
convective heat flux, $\mu$ the mean molecular weight (dimensionless), $\rho$
the density, and $C_{p}$ the heat capacity of the gas. The mixing length is
approximated by the scale height. An upper bound is obtained from Eq 3 by
equating $F_{\rm conv}$ with $\sigma T_{\rm eff}^{4}$,

$K_{zz}<2.5\times 10^{10}\left({T_{\rm eff}\over
600}\right)^{8/3}\left({1000\over g}\right){\rm~{}~{}cm}^{2}/{\rm s}.$ (4)

These large values of $K_{zz}$ might be regarded with some skepticism. If
applied to Earth’s troposphere, Eq 3 predicts $K_{zz}>10^{7}$ cm2/s, which is
two orders of magnitude too high. Even in the present context, $F_{\rm conv}$
can be much smaller than $\sigma T_{\rm eff}^{4}$ (or even fall to zero in
places), because a considerable fraction of energy transport is by radiation.
Depending on how well one tolerates theory, one might expect $K_{zz}$ to be of
the order of $10^{9}$-$10^{11}$ cm/s2 in the convecting zones of self-luminous
EGPs, and $100\times$ smaller for BDs with $g\\!=\\!10^{5}$ cm/s2.

It is likely that $K_{zz}$ would be much smaller in stratified gas above the
convecting regions. On Earth, $K_{zz}$ drops by two orders of magnitude at the
tropopause. Hubeny and Burrows (2007) reduce $K_{zz}$ by several orders of
magnitude in the stratosphere. Simulations by Freytag et al. (2010) support
this expectation. The tradeoff is between simplicity (constant $K_{zz}$) and
realism (a vertical discontinuity in $K_{zz}$). The latter choice would
introduce two more free parameters, the value of $K_{zz}$ in the stratosphere
(ill constrained) and the altitude of the discontinuity (reasonably well
constrained, but results could be sensitive to this). A two layer model also
makes interpreting the numerical results in terms of a quench approximation
less straightforward because there can be more than one quench point. For this
study we choose to treat $K_{zz}$ as constant with height, but vary it over a
range wide enough to encompass all likely values.

The chemical system used here comprises 366 forward chemical reactions and 32
photolysis reactions of 64 chemical species made of H, C, N, O, and S. The
most important missing species is probably methylamine, CH3NH2. Reaction rates
when known are selected from the publicly available NIST database
(http://kinetics. nist.gov/kinetics). Although many of the important reactions
have been measured in both directions in the lab (e.g., both ${\rm
CH}_{4}+{\rm H}\rightarrow{\rm CH}_{3}+{\rm H}_{2}$ and ${\rm CH}_{3}+{\rm
H}_{2}\rightarrow{\rm CH}_{4}+{\rm H}$ have been heavily studied), in general
one direction is much better characterized than the other. Thermodynamic data
(enthalpies and entropies) are usually better known over a wider range of
temperatures. Thus it is better to complement each specific reaction with its
exact reverse, with the forward and reverse rates linked self-consistently by
the thermodynamic data of the species involved. This way the transition from
equilibrium to kinetically-controlled abundances is automatic. Equilibrium is
reached when all the important forward and reverse reactions are fast compared
to changes in the state variables.

How this is done is fully explained by Visscher and Moses (2011). What follows
is a telegraphic summary. For reactions of the form ${\rm A}+{\rm
B}\rightarrow{\rm C}+{\rm D}$ with forward reaction rate $k_{f}$, the reverse
rate (i.e., the rate for ${\rm C}+{\rm D}\rightarrow{\rm A}+{\rm B}$) is
$k_{r}=k_{f}\exp{\left(-\Delta G/RT\right)}$, where $\Delta G$, the Gibbs free
energy, is obtained from enthalpies and entropies of the of the reactants and
products, $\Delta
G=H_{A}+H_{B}-H_{C}-H_{D}-T\left(S_{A}+S_{B}-S_{C}-S_{D}\right)$. For
associative reactions of the form ${\rm A}+{\rm B}\rightarrow{\rm AB}$, the
rate for the reverse reaction (dissociation of AB) is
$k_{r}=k_{f}\left(kT/P_{\circ}\right)\exp{\left(-\Delta G/RT\right)}$, where
$P_{\circ}=10^{6}$ dynes/cm2 is one atmosphere. Similarly, the associative
reverse of a dissociative reaction is given by
$k_{r}=k_{f}\left(P_{\circ}/kT\right)\exp{\left(-\Delta G/RT\right)}$.

Thermodynamic data as a function of temperature are available for atoms and
most small molecules from NIST in the form of empirical Shomate equation fits
for enthalpy and entropy (http://webbook.nist.gov/chemistry/form-ser). Zero
point data for HS are corrected by Lodders (2004). For CH3OH and C2H6 we use
heat capacities as a function of temperature to derive Shomate equation fits
for enthalpy and entropy. Unfortunately, many of the more exotic free radicals
are not listed in the publicly available NIST databases. For many of these we
use estimates given in Burcat and Ruscic (2005)’s widely available gray
literature compilation. For NNH we follow Haworth et al. (2003), and for N2H2
and N2H3 we follow Matus et al. (2006). Inaccurate thermodynamic data for free
radicals are not a problem for equilibrium calculations of well-characterized
abundant species because poorly-characterized free radicals are never
abundant. On the other hand, poorly-characterized species do pose a problem in
disequilibrium kinetics because net reaction rates depend on the uncertain
abundances of free radicals, which are determined by their thermodynamic
properties.

The lower boundary is set deep enough that all species are in thermodynamic
equilibrium. We find that pressures above 300 bars and temperatures above 2000
K usually suffice, with the limiting species being N2. The upper boundary
condition is zero flux for all species. We place the upper boundary at
$\sim\\!10^{-6}$ bars, as higher altitudes require additional physics and
chemistry (ion chemistry, thermospheric heating) that go beyond the scope of
this work. Steady state solutions are found by integrating the time-dependent
chemical and transport equations through time using an overcorrected fully
implicit backward-difference method. Most models take a few minutes to run to
steady state from arbitrary initial conditions on a vintage laptop computer,
although some particular cases can be more challenging.

Photolysis significantly affects the composition of atmospheres at very high
altitudes even when the incident UV flux is small. In our models we have set
the incident UV flux to 0.1% that at Earth. Photolysis plays almost no part at
the higher pressures germane to quenching and is not further discussed here.
We have not included the reverses of photolysis reactions (radiative
attachment, e.g. ${\rm OH}+{\rm H}\rightarrow{\rm H}_{2}{\rm O}+h\nu$) in
detailed balancing. Recent work has shown that radiative attachment can be
important in hydrocarbon growth in planetary atmospheres when the resulting
molecule is complex enough that radiative relaxation can be effective (Vuitton
et al., 2012).

Figure 2.— Methane and CO in an exemplary lukewarm EGP. The planet is cloud-
free, has an effective radiating temperature of 600 K, $g=10^{3}$ cm/s2, and
receives insignificant insolation. The $p$-$T$ profile of the atmosphere is
compared to the $p$-$T$ curve (gold dot-dash) corresponding to equilibrium
$f_{{\rm CH}_{4}}=f_{\rm CO}$. Methane is thermodynamically favored when $T$
is to the left of the $f_{{\rm CH}_{4}}=f_{\rm CO}$ curve, CO is favored to
the right. Computed CO and CH4 mixing ratios are shown from three 1D kinetics
models that differ only in vertical eddy mixing: $K_{zz}=10^{10}$ cm2/s
(solid), $K_{zz}=10^{7}$ cm2/s (dashed), and $K_{zz}=10^{4}$ cm2/s (dots).
Deep in the atmospheres CO and CH4 are in equilibrium. High in the atmospheres
CO and CH4 are unreactive and $f_{\rm CO}$ and $f_{{\rm CH}_{4}}$ would be
constant but for molecular diffusion. The quench points are obvious and well-
defined in all three models.

###  3.1. Two examples

Figure 2 shows how $K_{zz}$ effects chemistry on a tepid ($T_{\rm eff}=600$
K), cloud-free, relatively low mass ($g=10^{3}$ cm/s2) extrasolar planet of
solar composition. Insolation is insignificant. Three different $K_{zz}$ are
compared. (i) Strong vertical mixing—$K_{zz}=10^{10}$ cm2/s—suppresses CH4 and
maintains $f_{\rm CO}\gg f_{{\rm CH}_{4}}$ to very high altitudes. CO and CH4
are in chemical equilibrium below the quench point at $\sim\\!4$ bars and 1450
K. (ii) A more Jupiter-like $K_{zz}=10^{7}$ cm2/s raises the quench point to
1.5 bars and 1280 K, which is better for CH4. (iii) Weak mixing,
$K_{zz}=10^{4}$ cm2/s, raises the quench point to 0.6 bars and 1000 K, which
is cool enough, barely, to fall into the CH4 field, so that $f_{{\rm
CH}_{4}}>f_{\rm CO}$ in this planet’s photosphere. The effect of molecular
diffusion (in which heavy molecules sink through H2) is apparent above
$3\times 10^{-6}$ bars at $K_{zz}=10^{7}$ cm2/s and above $3\times 10^{-3}$
bars at $K_{zz}=10^{4}$ cm2/s.

Figure 3.— Three models with three different surface gravities. The plot shows
$p$-$T$ profiles and CO and CH4 abundances. The planets are 600 K, self-
luminous, cloudless, of solar composition, with $K_{zz}=10^{7}$ cm2/s. Surface
gravities are $g=10^{3}$ cm/s2 (solid), $g=10^{4}$ cm/s2 (dashed), and
$g=10^{5}$ cm/s2 (dots). Higher gravity models are colder at a given pressure
and are more favorable to CH4.

At a given effective temperature, surface gravity determines whether the
atmosphere is CO or CH4 dominated. Figure 3 shows CO and CH4 abundances in
three worlds that differ only in their surface gravities. The atmospheres are
cloud-free and of elementally solar composition. The models shown here have
effective radiating temperatures of 600 K and a Jupiter-like $K_{zz}=10^{7}$
cm2/s. Surface gravities range from a Saturn-like $g=10^{3}$ cm/s2 to a brown-
dwarf-like $g=10^{5}$ cm/s2. The computed CO/CH4 ratio is sensitive to surface
gravity. This is because the $p$-$T$ profile is displaced to higher pressures
and lower temperatures when gravity is higher. Quenching at higher pressures
and lower temperatures favors CH4 over CO. The result is that the 600 K planet
($g=10^{3}$ cm/s2) has abundant CO and relatively little CH4, while the 600 K
brown dwarf is methane-rich with only a trace of CO.

###  3.2. The ensemble of models

Figure 4.— An overview of the ensemble of models. There are six panels for six
different combinations of gravity and metallicity. Relative CO and CH4
abundances above the quench point are indicated by the areas of the
semicircles, CO black and following, CH4 green and preceding. In general,
higher gravities, cooler temperatures, and lower metallicity favor CH4 vs. CO.
Different metallicities can give the same CO/CH4 ratio at the same $T_{\rm
eff}$ and $g$ with modest and probably unmeasurable differences in $K_{zz}$.
Blue squares denote the specific cases in Figures 2 and 3. Red circles denote
the 16 cases used in Fig 7 to illustrate quenching. Orange diamonds denote the
two models used in Fig 8 to compare reaction rates. The red star is the guide
to Fig 1. In all cases, equilibrium $f_{{\rm CH}_{4}}>f_{\rm CO}$.

We use the 1D chemical kinetics models to explore the phase space defined by
$500<T_{\rm eff}<1100$ K, $10^{4}<K_{zz}<10^{11}$ cm2/s, and $10^{3}<g<10^{5}$
cm/s2. Models are run to steady state. Ensembles were computed at solar
($m\\!=\\!1$) and thrice solar ($m\\!=\\!3$) metallicities, but $p$-$T$
profiles were not adjusted to take into account the altered chemical
composition. The CO and CH4 abundances of all the models are summarized in
Figure 4. Low $K_{zz}$ models can be numerically challenging when the gas is
very hot. We include two models that did not reach a true steady-state on
Figure 4: $T_{\rm eff}=600$ K, $g=10^{3}$ cm/s2, $m\\!=\\!1$, and
$K_{zz}=10^{4}$ and $10^{5}$ cm2/s. These models reached persistent near
steady states, but were held back by undiagnosed features in the sulfur
chemistry. They are included here as part of our exploration of conditions
that can produce CH4 at low gravity.

##  4\. Determining the quench point for CO and CH4

Here we want to determine the quench conditions that correspond to the
asymptotic CO and CH4 abundances in the complete kinetics models shown in
Figure 4. Our purpose in doing so is to quantify the emergent quench behaviour
of the kinetics models, and if possible describe this behaviour with a simple
equation or set of equations. That it is possible to do this is the subject of
Figure 5.

Figure 5.— CO-hydrogenation timescales as a function of quench point
temperature $T_{q}$ and quench pressure $p_{q}$ for the ensemble of solar
composition kinetics models. Quench conditions $T_{q}$ and $p_{q}$ are
determined numerically from the full kinetics models using the entire network
of chemical reactions. How this is done is explained in the text. The quench
timescale used here is a mixing timescale, $t_{q}=t_{\rm mix}\equiv
H_{q}^{2}/K_{zz}$, where $H_{q}$ is the pressure scale height at the quench
point. The relation between the true chemical timescale and $t_{q}$ is
arbitrary to a factor $L^{2}/H_{q}^{2}$, where $L$ refers to the true mixing
length. We do not plot $t_{q}$ itself (for which the points are scattered),
but rather $t_{q}\times p_{q}$ (left) and $t_{q}\times p^{2}_{q}$ (right),
which resolve into orderly arrays. Left. A simple exponential provides a good
fit to $t_{q}\times p_{q}$ for all but the coldest $T_{q}$. Right. A different
simple exponential can be fit to $t_{q}\times p^{2}_{q}$ for the coldest
$T_{q}$. Lines show fits.

First, we determine the CO-CH4 quench point in each atmosphere in Figure 4 by
finding the lowest altitude in each atmosphere where the CO and CH4 mixing
ratios depart measurably from equilibrium. Deep in the atmosphere CH4 and CO
are in chemical equilibrium with H2 and H2O. Thus the product

$K_{\rm test}={p{\rm CH}_{4}\\!\cdot\\!p{\rm H}_{2}{\rm O}\\!\over\\!p{\rm
CO}\\!\cdot\\!\left(p{\rm H}_{2}\right)^{3}}$ (5)

equals the corresponding equilibrium constant

$K_{{\rm CH}_{4}\cdot{\rm CO}}=5.24\times 10^{-14}\exp{\left(27285/T\right)}.$
(6)

In Eqs 5 and 6 (and all succeeding expressions of similar form), pressures are
in atmospheres and temperatures in Kelvins.

In the regions of a convective atmosphere relevant to the present
investigation, equilibrium contours of constant CH4/CO are always shallower
than the adiabat, hence the equilibrium CH4 mixing ratio always decreases with
increasing depth. To show this (and its limits), use the adiabatic relation
$T^{\,\gamma}\propto p^{\,\gamma-1}$, where $\gamma=C_{p}/C_{v}$ has its usual
meaning as the ratio of heat capacities. The temperature gradient along the
adiabat (constant entropy $S$) is

$\left({\partial T\over\partial p}\right)_{\\!S}={\gamma-1\over\gamma}{T\over
p}={R\over C_{p}}{T\over p}.$ (7)

For a temperate cosmic mix of H2 (84%) and He (16%), we can expect H2 to have
$C_{p}\approx 3.5R$ and He to have $C_{p}=2.5R$, and hence $R/C_{p}\approx
0.30$. Warmer temperatures excite hydrogen’s vibrational modes, for which
$C_{p}\approx 4.5R$ and $R/C_{p}\approx 0.24$. Detailed calculations show that
$R/C_{p}$ decreases monotonically from $0.29$ at 1000 K to $0.25$ at 2100 K to
$0.20$ at 3000 K as dissociation (another energy sink) becomes important.

Equation 7 needs to be compared to contours of equilibrium chemistry. Defining
$\xi\equiv{\rm CH}_{4}/{\rm CO}$, we can write the equilibrium (Eqs 5 and 6)
in the form

$\xi\\!\cdot\\!\left(f_{\rm O}-{f_{\rm C}\over 1+\xi}\right)=p^{2}f_{{\rm
H}_{2}}^{3}Ae^{B/T}$ (8)

where $f_{{\rm H}_{2}}\approx 0.84$ is the H2 mixing ratio, $f_{\rm O}\approx
f_{{\rm H}_{2}{\rm O}}+f_{\rm CO}$ is the total oxygen mixing ratio, and
$f_{\rm C}\approx f_{{\rm CH}_{4}}+f_{\rm CO}$ is the total carbon mixing
ratio. These are generally very good approximations for $T<2500$ K. The
quantity in parentheses is $f_{{\rm H}_{2}{\rm O}}$. For contours of constant
$\xi$, the left hand side of Eq 8 is constant. Taking derivatives, the
temperature gradient is

$\left({\partial T\over\partial p}\right)_{\\!\xi}={2T\over B}\,{T\over
p}\approx 0.11{T\over 1500}\,{T\over p}.$ (9)

Thus $\left({\partial T/\partial p}\right)_{\xi}<\left({\partial T/\partial
p}\right)_{S}$ for temperatures lower than about 2900 K, by which point
several of the assumptions made in deriving this relation have begun to break
down. Even with $K_{zz}$ as high as $10^{11}$ cm2/s, quenching in the CO-CH4
system takes place at temperatures less than 2200 K. Hence we can always
presume that CO is most abundant and CH4 least abundant at the lower boundary
of our model. It follows that we can determine the quench conditions
(temperature $T_{q}$, pressure $p_{q}$) by working up from the lower boundary
and finding the lowest altitude at which the product $K_{\rm test}$ is
appreciably smaller than the equilibrium constant $K_{{\rm CH}_{4}\cdot{\rm
CO}}$. In practice we searched for the lowest altitude where $K_{\rm
test}<0.9K_{{\rm CH}_{4}\cdot{\rm CO}}$.

We also wish to determine a quench timescale $t_{q}$. In principle, the right
way to do this is to determine the true chemical reaction timescale from the
sum of all relevant reaction rates at the quench point. In general this
information is surprisingly hard to extract from a model, because the
breakdown of equilibrium appears as small differences between much larger
forward and reverse reaction rates in many reactions. This approach is
practical only if one already knows exactly which species and reactions are
key. Once known, for $t_{q}$ to be useful in a quench approximation one then
needs a comparably fine-grained description of the cooling time implicit in
$K_{zz}$ and the particular chemical reactions that establish $t_{q}$.

Much easier, and more useful, is to define a quench timescale for the system
as a whole in terms of a relevant mixing time, $t_{q}=t_{\rm
mix}=L^{2}/K_{zz}$ where $L$ is a characteristic length scale. The simple
choice is to set $L=H_{q}$, the scale height at $T_{q}$. This definition is
arbitrary to within a multiplicative factor because the true mixing length $L$
is not in general equal to $H_{q}$. But this definition of $t_{q}$ is
precisely what is needed for constructing a new quench approximation that
assumes $t_{\rm mix}=H_{q}^{2}/K_{zz}$ and, as seen in Figure 5, it leads to a
very simple description of the overall behaviour of the full network of
chemical reactions.

Smith (1998) demonstrated that $L$ is often considerably shorter than $H$,
mostly because the key reactions determining $t_{q}$ are much more sensitive
to temperature than to pressure. Smith (1998) developed a five step iterative
algorithm for determining $L$ and presented it in the form of a “recipe” to be
used in quenching calculations, which is especially useful when the system is
controlled by a single reaction whose rate is known. Several workers have
employed Smith’s recipe. Unfortunately, the range of mixing times that results
can be very great. For example, Visscher and Moses (2011) list values for $L$
between $0.1\,H$ and $0.7\,H$ for the CO-CH4 system, the equivalent of a
factor of 50 in $t_{\rm mix}$ and therefore equivalent to a factor of 50 in
the inferred value of $K_{zz}$, itself a parameter of easy virtue. The
resulting uncertainty frustrates intercomparison between models. Our position
is that the most useful approximations are simple approximations. We therefore
stick with the old rule $L=H$ to define $t_{\rm mix}$, rather than iterate
between $L$ and $T_{q}$. How $L$ is defined is important for comparing to
previous work that tries to estimate $t_{\rm chem}$ from particular reaction
rates, but it is simply a scaling factor for us.

Irrespective of how $L$ is defined, we find that we can fit the quench time
$t_{q}$ to a simple functional form that spans all our models. We start from
the expectation that ${\rm CO}\rightarrow{\rm CH}_{4}$ will depend on
temperature and pressure, perhaps the C/O ratio, and possibly on metallicity.
It is reasonable to hope that the net reaction has an Arrhenius-like rate

$t_{q}=A\;p^{-b}m^{-c}\exp{\left(B/T\right)},$ (10)

where possible pressure and metallicity dependencies are written as power
laws.

Our hope is nicely borne out by Figure 5, which plots $t_{q}$ for the CO-CH4
reaction for our complete roster of solar metallicity models. All but the
coldest quench points are described by a rather well-defined Arrhenius
expression with $b\\!=\\!1$. The Arrhenius fit (for $m\\!=\\!1$) is

$t^{\prime}_{q1}=3.0\times
10^{-6}p^{-1}\exp{\left(42000/T\right)}{\rm~{}sec,}$ (11)

where $p$ is in bars. The time scale for models with $m\\!=\\!3$ (not shown)
is roughly half of this.

The prime is placed on $t_{q1}^{\prime}$ because, by using $K_{\rm
test}<0.9K_{eq}$ rather than $K_{\rm test}<K_{eq}$, we have overestimated the
quench time. To correct for this—and to verify that a quench scheme works—we
compare the predictions of the quench scheme with the predictions of the
ensemble of models. Fits are shown in Figure 6. These use

$t_{q1}=1.5\times
10^{-6}\,p^{-1}m^{-0.7}\exp{\left(42000/T\right)}{\rm~{}sec.}$ (12)

The methods of Figure 6 and Figure 5 are complementary. A satisfactory fit on
Figure 6 is more sensitive to the $A$ factor (overall rate) and the
metallicity (Eq 10), whilst the fit on Figure 5 is more sensitive to the $B$
factor (temperature dependence) and pressure.

Figure 6.— CO (left) and CH4 (right) mixing ratios predicted by the quenching
approximation $t_{\rm chem}=t_{\rm mix}$ (y-axis) plotted against the actual
asymptotic mixing ratio determined from the full ensemble of 1-D kinetics
model (x-axis). A perfect approximation would adhere to the line. Open red
symbols are for $m\\!=\\!1$ and solid black symbols are for $m\\!=\\!3$ using
Eq 12.

The half-dozen models that are ill fit by Eq 11 are fit instead by a second
Arrhenius relation

$t_{q2}=40\,p^{-2}\exp{\left(25000/T\right)}{\rm~{}sec.}$ (13)

where we have given the iterated rate. The models fit by Eq 13 are those with
very low quench temperatures $T_{q}$ and very little CO. We have no such
models with $m\\!=\\!3$.

The high and low temperature fits combine harmonically to give a general
description of reaction times in the CO-CH4 system,

$t_{\rm CO}=\left({1\over t_{q1}}+{1\over t_{q2}}\right)^{-1}.$ (14)

Equation 14 when used with the timescale $t_{\rm mix}=H^{2}/K_{zz}$ works for
all of our models for CO and CH4 and we expect should work generally for
compositions not too far from solar.

###  4.1. Analysis

Figure 7.— Quenching behavior of the three intermediate reactions (a, b, c)
and of the ${\rm CO}\leftrightarrow{\rm CH}_{4}$ system as a whole (d).
Results are shown from 16 models chosen to provide a representative sample of
the ensemble (marked by red circles in Fig. 4). Panels (a,b) show that ${\rm
CO}\leftrightarrow{\rm H}_{2}{\rm CO}$ and ${\rm H}_{2}{\rm
CO}\leftrightarrow{\rm CH}_{3}{\rm OH}$ appear to be equilibrated in all
models at $T>900$ K. Panel (c) shows that equilibration of ${\rm CH}_{3}{\rm
OH}$ and ${\rm CH}_{4}$ is more difficult. In detail the quench points for
${\rm CH}_{3}{\rm OH}\leftrightarrow{\rm CH}_{4}$ appear to be the same as the
quench points for ${\rm CO}\leftrightarrow{\rm CH}_{4}$ as shown in panel (d)
for the same 16 models. Evidently the bottleneck is between ${\rm CH}_{3}{\rm
OH}$ and ${\rm CH}_{4}$, in accord with conclusions reached by Moses et al
(2011).

We examine the ensemble of models for quench points pertinent to the reaction
intermediates, to determine where in the CO-CH4 reaction network quenching
first takes place. We consider the CO-H2CO, H2CO-CH3OH, and CH3OH-CH4
equilibria. We analyzed 16 models taken arbitrarily from the ensemble. The
chosen ones are marked on Fig 4 with scarlet circles. We find, to within our
ability to determine this, that the CH3OH-CH4 reaction quenches at the same
temperature as the CO-CH4 reaction as a whole. This is shown by panels (c) and
(d) of Figure 7. By contrast the CO-H2CO and H2CO-CH3OH reactions stay near
equilibrium for temperatures above 900 K (Figures 7a and 7b). Hence as a
parcel cools equilibrium first breaks down between CH3OH and CH4. This is in
accord with conclusions reached by Moses et al. (2011) and Visscher and Moses
(2011).

Figure 8.— The four most important reactions for quenching in our model CO-CH4
system. All four reactions yield ${\rm CH}_{3}+{\rm OH}$ as products. Solid
curves show rates of the forward reaction that creates CH3, dashed curves show
the reverse reaction that destroys CH3, and wide dots the difference (the net
production of CH3), which is equivalent to net loss of CO. Black dot-dash
lines are the altitudes where CH4 and CO reach their final quenched values.
Left: A warm, strongly-mixed ($T_{\rm eff},K_{zz},g=1100,10^{10},10^{4}$),
model dominated by CO. Disequilibrium reactions are fastest at 210 km (at 1700
K) but full quenching is delayed until 270 km (at 1360 K); the distance
corresponds to a bit more than a pressure scale height. Right: A cool,
moderately-mixed ($T_{\rm eff},K_{zz},g=500,10^{7},10^{5}$) BD dominated by
CH4. Disequilibrium reactions are fastest at 7 km but quenching is at 9 km
(the scale height is 5 km).

Figure 8 examines in detail the key reactions that break the CO-CH4
equilibrium. The important reactions form the methyl radical CH3 from species
such as CH3OH in which the C-O bond is still intact. Figure 8 illustrates two
cases, one a cool ($T_{\rm eff}=500$ K) brown dwarf with a preference for CH4
($f_{\rm CO}=1.4\times 10^{-5}$, $f_{{\rm CH}_{4}}=4.7\times 10^{-4}$), the
other a warmer ($T_{\rm eff}=1100$ K), smaller world that favors CO ($f_{\rm
CO}=4.8\times 10^{-4}$, $f_{{\rm CH}_{4}}=1.0\times 10^{-5}$). Both models
assume solar metallicity ($m\\!=\\!1$). In both cases the most important
reaction is simple thermal decomposition of methanol, ${\rm CH}_{3}{\rm
OH}+{\rm M}\rightarrow{\rm CH}_{3}+{\rm OH}$, although the reaction of
methanol with atomic hydrogen is nearly as fast in the colder, higher gravity
case. The reaction ${\rm H}_{2}{\rm COH}+{\rm H}$, which Moses et al. (2011)
report as the most important in their CO-rich models, does not appear to be
very important in ours. We do not claim that our system is better; we merely
point out that abundances and reaction rates of furtive species like ${\rm
H}_{2}{\rm COH}$ are highly uncertain.

###  4.2. Comparisons with previous work

To put our results in perspective, we compare them to some quench
approximations seen in the literature. Published quench approximations begin
by identifying a particular forward reaction as the bottleneck and then
calculate CO loss time scales from the chosen reaction’s rate; variety lies in
the reactions that are chosen and the reaction rates adopted.

We begin with the classic Prinn and Barshay (1977) prescription, used by
Hubeny and Burrows (2007) as their “slow” case. Prinn and Barshay (1977)
suggested

${\rm H}_{2}+{\rm H}_{2}{\rm CO}\rightarrow{\rm CH}_{3}+{\rm OH}$ (15)

as a rate-limiting reaction, with reaction rate $k_{f}=2.3\times
10^{-10}e^{-36200/T}$ cm3/s. R15 is a reaction that jumps over two energy
barriers in Figure 1, which makes it look like it should be relatively
unlikely. Figure 8 shows that R15 is modestly important in our system, but we
use a newer slower rate estimated by Jasper et al (2007). The time constant
for CO loss by reaction R15 using Prinn and Barshay (1977)’s rate is
approximated by the forward reaction

$t_{\rm chem}^{-1}={-1\over\left[{\rm CO}\right]}{\partial\left[{\rm
CO}\right]\over\partial t}={k_{f}\left[{\rm H}_{2}\right]\left[{\rm H}_{2}{\rm
CO}\right]\over\left[{\rm CO}\right]}{\rm~{}sec}^{-1}$ (16)

where the notation $\left[{\rm CO}\right]$ refers to number density [cm-3]. To
evaluate $t_{\rm chem}$ for R15, we put formaldehyde in equilibrium with CO
and H2,

${p{\rm H}_{2}{\rm CO}\over p{\rm H}_{2}\\!\cdot\\!p{\rm CO}}=K_{{\rm
H}_{2}{\rm CO}}=3.3\times 10^{-7}\exp{\left(1420/T\right)},$ (17)

to obtain

$t_{\rm chem}={1.3\times 10^{16}\exp{\left(34780/T\right)}\over p\,Nf_{{\rm
H}_{2}}^{2}}{\rm~{}sec}.$ (18)

The number density $N$ is related to the pressure $p$ in bars by
$NkT\\!=\\!10^{6}\,p$. The predictions of Eq 18 are compared to results from
our complete models in Figure 9. Equation 18 works quite well in a quench
approximation to our full model, especially for cases where CH4 is predicted
to be abundant, despite its being based on the wrong reaction with the wrong
rate. As the Prinn and Barshay (1977) approximation has been widely used for a
very long time, it is valuable to see that it seems to work rather well. When
compared to Eq 12, Eq 18 has a stronger pressure dependence ($p^{-2}$ vs.
$p^{-1}$), a weaker temperature dependence ($T\\!\cdot\\!e^{34780/T}$ vs.
$e^{42000/T}$), and no dependence on metallicity.

Figure 9.— Comparison of some quench approximations (y-axes) to the actual CO
and CH4 mixing ratios computed by the full 1D model (x-axes). A perfect
approximation would adhere to the line. Only results for solar metallicity
($m\\!=\\!1$) are shown. Purple plusses follow the widely applied Prinn and
Barshay (1977) model with $L=H$. Red diamonds use the channel through CH3O
suggested by Yung et al. (1988) but using a more recent estimate for the
reaction rate and reducing the mixing length to $L=0.14\,H$; this is not the
same as Yung et al. (1988)’s algorithm. Green squares implement Line et al.
(2011)’s CH3O channel with $L\\!=\\!H$. Black triangles implement the methanol
decomposition channel favored for low $f_{\rm CO}$ by Moses et al. (2011) and
Visscher and Moses (2011). These are computed with mixing length $L=0.14\,H$.
Blue circles represent our attempt to implement the Moses et al. (2011) H2COH
channel with $L=0.14\,H$.

More recent discussions of the CO-CH4 quench approximation omit enough details
that they can be challenging to reproduce. We attempt to do so here because
the comparisons are illuminating. Yung et al. (1988) suggested that the rate-
limiting step is the 3-body reaction

${\rm H}+{\rm H}_{2}{\rm CO}+{\rm M}\rightarrow{\rm CH}_{3}{\rm O}+{\rm M}.$
(19)

Hubeny and Burrows (2007) use R19 as their “fast” rate with $L\\!=\\!H$.
Barman et al. (2011a) also use this rate, but following Smith (1998) they set
the mixing length $L$ to unspecified values between 10% to 20% of the scale
height, which effectively slows the “fast” rate by a factor of 25-100 compared
to $L\\!=\\!H$. Cooper and Showman (2006) implemented the CH3O channel with a
different reaction rate while following Smith’s “recipe.” We use the
asymptotic high pressure rate

$k_{\infty}=8.0\times 10^{-10}\exp{\left(-3160/T\right)}$,

a very fast rate which we obtained by reversing the first order rate for ${\rm
CH}_{3}{\rm O}\rightarrow{\rm H}+{\rm H}_{2}{\rm CO}$ given by Rauk et al.
(2003). If we presume that H and H2CO are both in equilibrium, we can use the
hydrogen equilibrium constant $K_{{\rm H}\cdot{\rm
H}_{2}}=1155\,e^{-26916/T}=p{\rm H}\cdot p{\rm H}_{2}^{-0.5}$ to obtain

$t_{\rm chem}^{-1}={k_{\infty}\left[{\rm H}\right]\left[{\rm H}_{2}{\rm
CO}\right]\over\left[{\rm CO}\right]}=k_{\infty}\,p^{1.5}\,f_{{\rm
H}_{2}}^{1.5}\left(10^{6}\over kT\right)K_{{\rm H}_{2}{\rm CO}}K_{{\rm
H}\cdot{\rm H}_{2}}.$ (20)

Evaluated,

$t_{\rm chem}=4.5\times 10^{-10}\,Tp^{-1.5}f_{{\rm
H}_{2}}^{-1.5}\exp{\left(28656/T\right)}.$ (21)

Results of using Eq 21 are shown as red diamonds on Fig 9 with $L=0.14H$,
which is comparable to what Barman et al. (2011a) use (the fit would look
worse with $L=H$). These stand out with too much CH4 and too little CO,
because R19 is not a true bottleneck.

Line et al. (2011) moved the bottleneck to ${\rm CH}_{3}{\rm O}+{\rm
H}_{2}\rightarrow{\rm CH}_{3}{\rm OH}+{\rm H}$. To compute a rate constant, we
use $p{\rm CH}_{3}{\rm O}=K_{{\rm CH}_{3}{\rm O}}\\!\cdot\\!p{\rm
H}_{2}\\!\cdot\\!p{\rm H}\\!\cdot\\!p{\rm CO}$, where $K_{{\rm CH}_{3}{\rm
O}}=3.0\times 10^{-11}e^{10582/T}$ is the relevant equilibrium constant (the
large number of significant digits in these equilibrium constants do not imply
accuracy). To complete the equation we also assume that H and H2 are in
equilibrium. We use the reaction rate given by Line et al. (2011), which
results in

$t_{\rm chem}={1.2\times 10^{29}\exp{\left(45720/T\right)}\over T^{4}Npf_{{\rm
H}_{2}}^{2}}{\rm~{}~{}sec}.$ (22)

Results are shown as green squares on Figure 9 using $L=H$, as Line et al.
(2011) do. Because it predicts too much CO and too little CH4, $t_{\rm chem}$
given by Eq 22 is too slow.

Based on detailed analysis of the reactions in a complete 1D model similar to
our own, Moses et al. (2011) concluded that the bottleneck is associated with
breaking the C-O bond. They propose several key channels. One channel is
thermal decomposition of methanol,

${\rm CH}_{3}{\rm OH}+{\rm M}\rightarrow{\rm CH}_{3}+{\rm OH}+{\rm M}.$ (23)

Visscher and Moses (2011) and Moses et al. (2011) state that this is more
important when CH4 is abundant. Another channel, which they state is more
important when ${\rm CO}\gg{\rm CH}_{4}$, is

${\rm CH}_{2}{\rm OH}+{\rm H}\rightarrow{\rm CH}_{3}+{\rm OH}.$ (24)

The rate of R24 depends on the uncertain thermodynamic properties of ${\rm
CH}_{2}{\rm OH}$. In the high pressure limit, methanol decomposition R23 is a
first order reaction,

${\partial\left[{\rm CH}_{3}{\rm OH}\right]\over\partial
t}=-k_{\infty}\left[{\rm CH}_{3}{\rm OH}\right].$ (25)

Jasper et al (2007) estimate $k_{\infty}$ from theory

$k_{\infty}=6.251\times
10^{16}\left(300/T\right)^{0.6148}\exp{\left(-46573/T\right)}{\rm~{}sec}^{-1}.$

The methanol equilibrium constant

$K_{{\rm CH}_{3}{\rm OH}}={p{\rm CH}_{3}{\rm OH}\over p{\rm CO}\cdot p{\rm
H}_{2}^{2}}=1.1\times 10^{-13}e^{13000/T}$

relates $\left[{\rm CH}_{3}{\rm OH}\right]$ to $\left[{\rm CO}\right]$. The
reaction timescale is then

$\displaystyle t_{\rm chem}$ $\displaystyle=$
$\displaystyle\left(k_{\infty}p{\rm H}_{2}^{2}\,K_{{\rm CH}_{3}{\rm
OH}}\right)^{-1}$ (26) $\displaystyle=$ $\displaystyle 4.4\times
10^{-6}\,T^{0.6148}\exp{\left(33573/T\right)}\,p^{-2}f_{{\rm
H}_{2}}^{-2}{\rm~{}~{}sec}.$

Equation 26 has a weaker dependence on $T$, a stronger dependence on $p$ than
we find for the ensemble, and no dependence on metallicity.

Figure 9 illustrates quenching using Eq 26 as black triangles. For the
comparison we take $L=0.14\,H$. This approximates what Moses et al. (2011) may
be using. Equation 26 does well, especially in cases where ${\rm CO}\ll{\rm
CH}_{4}$ at quenching, which is the regime for which Moses et al. (2011)
report that Eq 26 applies. Equation 26 does better if $L\\!>\\!0.14\,H$. The
match is best for CO with $L=0.2\,H$, and better for CH4 with
$L\\!\approx\\!0.5\,H$.

To reconstruct a simple form for the H2COH channel, we need to estimate the
equilibrium abundance of H2COH. We use $p{\rm H}_{2}{\rm COH}=K_{{\rm
H}_{2}{\rm COH}}\cdot p{\rm H}_{2}\cdot p{\rm H}\cdot p{\rm CO}$ with $K_{{\rm
H}_{2}{\rm COH}}=1.0\times 10^{-12}e^{15843/T}$. We assume that H and H2 are
in equilibrium. Jasper et al. (2007) list a fast rate, $k=2.8\times 10^{-10}$
cm3/s, for the reaction of H and H2COH. Assembling the parts, we obtain

$t_{\rm chem}={2.7\times 10^{15}\exp{\left(38000/T\right)}\over Npf_{{\rm
H}_{2}}^{-2}}{\rm~{}~{}sec}.$ (27)

Results are shown as blue circles on Figure 9. Equation 27 has a similar $T$
dependence to what we find for the system as a whole, but the overall rate is
about 3 orders of magnitude slower using $L=0.14\,H$, or about ten times
slower using $L=H$.

Disagreement between our model and Moses et al over the H2COH channel is also
apparent in Figure 8. The H2COH radical plays a much more modest role in our
1-D code than it does in Moses et al. (2011)’s. The same is true for CH3O. Why
this should be so is likely related to differing guesses of H2COH’s and CH3O’s
ill-known thermodynamic properties.

There are some more subtle differences between our model and previous quench
models that should be noted. One is that we find $t_{\rm chem}\propto p^{-1}$
by fitting to the ensemble, whilst quench models that attempt to identify the
one key forward reaction tend to get $t_{\rm chem}\propto p^{-2}$. This
difference is the reason why quench prescriptions in Figure 9 that do well
with $f_{\rm CO}\ll f_{{\rm CH}_{4}}$ miss the mark somewhat with $f_{{\rm
CH}_{4}}\ll f_{\rm CO}$. Figure 6 that uses $t_{q}\propto p^{-1}$ shows fine
agreement at both ends of the scale.

Another difference is that we see a well-defined metallicity dependence when
fitting to the ensemble. The metallicity dependence probably stems from the
reverse reaction, which as noted is usually ignored. As equilibrium breaks
down, the reaction slows in both directions, more quickly in the reverse
direction than in the forward direction. This behavior is shown clearly in
Figure 8, where forward, reverse, and net reaction rates are plotted. It is
the net reaction, the difference between the forward and reverse reactions,
that defines the retreat from equilibrium, not the forward reaction alone. For
example, the time scale for the methanol decomposition channel is

$t^{-1}_{\rm net}={-1\over\left[{\rm CO}\right]}{\partial\left[{\rm
CO}\right]\over\partial t}={k_{f}\left[{\rm CH}_{3}{\rm
OH}\right]\over\left[{\rm CO}\right]}-{k_{r}\left[{\rm
CH}_{3}\right]\left[{\rm OH}\right]\over\left[{\rm CO}\right]}.$ (28)

The reverse reaction is quadratic in metallicity, whilst [CO] is quadratic in
metallicity only for $f_{\rm CO}\ll f_{{\rm CH}_{4}}$, which suggests that it
is through the reverse reaction that metallicity enters $t_{\rm CO}$. The same
consideration applies for all the important reactions in Figure 8, as the
reverse reaction in every case is between CH3 and OH.

##  5\. Quenching in the N2-NH3 system

The approach we used to search for quenching of CO and CH4 does not work well
for N2 and NH3. Indeed, Figure 10—the analog to Fig 4 for N2 and NH3—shows no
sign that quenching plays any role in the N2-NH3 reaction, although doubtless
quenching occurs.

Figure 10.— Nitrogen. The six panels are as in Fig. 4. Relative abundances of
NH3 (red, east) and N2 (blue, west) are indicated by area. HCN abundances are
multiplied by 5 to make them visible at lower $g$ and superposed in gold on
N2. Higher gravities and cooler temperatures favor NH3. The NH3/N2 ratio is
insensitive to $K_{zz}$ and only weakly sensitive to $m$, which suggests that
NH3 can be a proxy for $g$. HCN fares best at high $g$ and high $K_{zz}$. The
HCN/NH3 ratio is sensitive to $K_{zz}$ and only weakly sensitive to
metallicity. The two boxed models are ilustrated in Figure 11.

Two things get in the way. Of less importance, a third species—HCN—becomes
non-negligible at high temperatures. HCN also quenches at high temperature,
which leaves the nitrogen system with three possibly distinct quench points:
NH3-N2, NH3-HCN, and N2-HCN. Although HCN is never very abundant in
equilibrium, it is often abundant enough that its decomposition can increase
the NH3 abundance by more than 10% after the NH3-N2 reaction has quenched.
When confused with the second, greater obstacle, HCN can make it difficult to
pinpoint where quenching occurs.

Figure 11.— Two examples of quenching in nitrogen. The particular models are
indicated by boxes on Figure 10. Volume mixing ratios of N2, NH3, and HCN are
shown for two different $K_{zz}$ pertinent to brown dwarfs. The relative
insensitivity of NH3 and N2 to $K_{zz}$ is a consequence of contours of
constant ${\rm NH}_{3}/{\rm N}_{2}$ being nearly parallel to the adiabat
$p(T)$. This is illustrated here by showing three relevant contours of
constant $\xi\equiv{\rm NH}_{3}/{\rm N}_{2}$. Both $\xi$ and $p(T)$ are
plotted against the temperature axis (top). In the full models $\xi=0.28$ and
$\xi=0.24$ for $K_{zz}=10^{10}$ cm2/s and $K_{zz}=10^{7}$ cm2/s, respectively.

The greater obstacle is that curves of constant NH3/N2 are nearly parallel to
the adiabat at the pressures and temperatures where quenching occurs. This is
illustated in Figure 11. Consequently the NH3/N2 ratio in a parcel remains
close to equilibrium well after quenching has taken place, which makes the
NH3/N2 ratio indifferent to quenching (Saumon et al., 2006). To show this,
write the N2-NH3 equilibrium in the form

$K_{{\rm NH}_{3}\cdot{\rm N}_{2}}=5.90\times
10^{-13}\exp{\left(13207/T\right)}={p{\rm NH}_{3}^{2}\over p{\rm
N}_{2}\\!\cdot\\!p{\rm H}_{2}^{3}}.$ (29)

Define $\xi\equiv{\rm NH}_{3}/{\rm N}_{2}$. If we approximate the total mixing
ratio of N by $f_{\rm N}\approx f_{{\rm NH}_{3}}+2f_{{\rm N}_{2}}$ (including
HCN makes this very complicated), we can write

$\xi^{2}\left({f_{\rm N}\over 2+\xi}\right)=p^{2}f_{{\rm H}_{2}}^{3}Ae^{B/T}.$
(30)

With context-obvious substitutions, the temperature gradient for contours of
constant $\xi$ is

$\left({dT\over dp}\right)_{\xi}={2T\over B}\,{T\over p}\approx 0.23{T\over
1500}\,{T\over p}.$ (31)

The temperature gradient in Eq 31 is parallel to the adiabat (Eq 7) at 1750 K,
and nearly parallel for $1500<T<2000$ K. For temperatures initially greater
than 1750 K, the equilibrium ratio of NH3/N2 decreases as the parcel cools,
reaching a minimum at $\sim$1750 K where
$\left({dT/dp}\right)_{\xi}=\left({dT/dp}\right)_{S}$. If the parcel remains
in chemical equilibrium, NH3/N2 will increase again as it cools further. This
then is how we explain the insensitivity of NH3/N2 to $K_{zz}$ seen in Figure
10: the NH3/N2 ratio computed by the full kinetics model is near the
equilibrium value at the temperature where contours of NH3/N2 are parallel to
the adiabat, which is also the minimum equilibrium value in the atmosphere.
For the kinds of atmospheres considered in this paper, it appears that the
amount of ammonia in the visible parts of the atmosphere will be comparable to
the minimum equilibrium abundance computed along the adiabat.

On the other hand, because our results are insensitive to quenching, it is not
difficult to find a non-unique chemical reaction time scale for the nitrogen
system that predicts $f_{{\rm NH}_{3}}$ well. A chemical reaction time scale
that works well in a quenching scheme that sets the reaction rate time
$t_{{\rm NH}_{3}}$ equal to the mixing time $t_{\rm mix}=H^{2}/K_{zz}$ is

$t_{{\rm NH}_{3}}=1.0\times
10^{-7}p^{-1}\exp{\left(52000/T\right)}{\rm~{}sec.}$ (32)

This particular choice presumes that the energy barrier is set by ${\rm
N}_{2}+{\rm H}_{2}\rightarrow{\rm NNH}+{\rm H}$. This choice of $t_{{\rm
NH}_{3}}$ is not unique. Almost any plausible choice of Arrhenius parameters
that gives roughly the same time scale as Eq 32 at 10 bars and 1750 K will
work just as well. We have no information to constrain dependence on $p$ or
$m$.

Figure 12.— Ammonia predicted by the quenching approximation $q({\rm NH}_{3})$
(y-axes) plotted against the actual asymptotic mixing ratio $f({\rm NH}_{3})$
determined from the ensemble of 1-D kinetics models (x-axes). A perfect
approximation would adhere to the line. Open red symbols are for $m\\!=\\!1$
and solid black symbols are for $m\\!=\\!3$. Left. Using Eq 32. Right. Open
red diamonds $m\\!=\\!1$ and filled black circles $m\\!=\\!3$ are based on
Lodders and Fegley (2002); open green triangles and purple crosses are based
on Line et al. (2011).

Figure 12 shows how our expression compares to some expressions in the
literature. The chemical reaction time recommended by Lodders and Fegley
(2002) is equivalent to

$t_{\rm chem}={1.2\times 10^{8}\exp{\left(81515/T\right)}\over
p\\!\cdot\\!N\\!\cdot\\!f_{{\rm H}_{2}}^{2}}{\rm~{}~{}sec}.$ (33)

Equation 33 is based on the possible reaction ${\rm N}_{2}+{\rm
H}_{2}\rightarrow{\rm NH}_{2}+{\rm N}$. Figure 12 shows that Eq 33 used with
$L=H$ gives as good a fit to our models as Eq 32. The extreme temperature
dependence of Eq 33 appears justified.

Line et al. (2011) look for quenching in the reaction ${\rm H}_{2}+{\rm
N}_{2}{\rm H}_{2}\rightarrow 2{\rm NH}_{2}$. Using rates given by Line et al.
(2011), and using $p{\rm N}_{2}{\rm H}_{2}=K_{{\rm N}_{2}{\rm
H}_{2}}\\!\cdot\\!p{\rm N}_{2}\\!\cdot\\!p{\rm H}_{2}$ with $K_{{\rm
N}_{2}{\rm H}_{2}}=3.8\times 10^{-6}e^{-25738/T}$, we get

$t_{\rm chem}={1.3\times 10^{11}\,T^{0.93}\exp{\left(46400/T\right)}\over
N\,p\,f_{{\rm H}_{2}}^{2}}{\rm~{}~{}sec}$ (34)

as our best effort to reproduce their chemical time scale for NH3
equilibration with N2. Figure 12 shows that Eq 34 agrees well with the
predictions of our kinetics model when $m\\!=\\!3$, but sometimes predicts
more NH3 than we find for $m\\!=\\!1$. Disagreement is limited to cases with
$K_{zz}<10^{7}$ cm2/s, which means that Eq 34 is relatively fast at the lowest
quench temperatures. It is possible that the temperature dependence of Eq 34
is not steep enough, or that the bottleneck involves N2 rather than N2H2, or
that the highly uncertain thermodynamic parameters of NNH are being treated
differently between models.

###  5.1. HCN-NH3-N2

The approach we used for CO and CH4 works moderately well for HCN and NH3 and
less well for HCN and N2.

The equilibrium between HCN and CH4 and NH3,

$K_{{\rm HCN}\cdot{\rm CH}_{4}}={p{\rm CH}_{4}\\!\cdot\\!p{\rm NH}_{3}\over
p{\rm HCN}\\!\cdot\\!p{\rm H}_{2}^{3}}=3.0\times
10^{-14}\exp{\left(33460/T\right)},$ (35)

closely resembles the parallel equilibrium Eq 6 between CO and CH4. For the
cool objects in which CH4 is abundant at depth, the strong temperature
dependence of $K_{{\rm HCN}\cdot{\rm CH}_{4}}$ means that HCN fares best with
respect to NH3 at high temperature when parcels move up or down along an
adiabat.

For warmer worlds where CO and N2 are dominant at depth, the most informative
equilibrium is with CO, N2, H2, and H2O, all of which are nearly constant when
nearly all the C is in CO. The formal reaction is $2{\rm CO}+{\rm N}_{2}+3{\rm
H}_{2}\leftrightarrow 2{\rm HCN}+2{\rm H}_{2}{\rm O}$, and the corresponding
equilibrium constant is

$K_{{\rm HCN}\cdot{\rm N}_{2}}={p{\rm CO}^{2}\\!\cdot\\!p{\rm
N}_{2}\\!\cdot\\!p{\rm H}_{2}^{3}\over p{\rm HCN}^{2}\\!\cdot\\!p{\rm
H}_{2}{\rm O}^{2}}=4.278\times 10^{11}\exp{\left(-528/T\right)},$ (36)

for which the temperature dependence is weak at quenching where $T_{q}\approx
2000$ K. Equation 36 indicates that $f_{\rm HCN}\propto p$. Hence, in both the
cool and the warm limits HCN increases with depth in deep atmospheres. Thus to
identify HCN quenching it suffices to start from the bottom and find the
altitude where the HCN abundance stops decreasing.

We looked for quench points defined against Eq 35, Eq 36, and a third
equilibrium with CO and NH3,

$K_{{\rm HCN}\cdot{\rm CO}}={p{\rm CO}\\!\cdot\\!p{\rm NH}_{3}\over p{\rm
HCN}\\!\cdot\\!p{\rm H}_{2}{\rm O}}=0.5025\exp{\left(6339/T\right)}.$ (37)

Quenching with N2 ($K_{{\rm HCN}\cdot{\rm N}_{2}}$) gives an indifferent fit
to an Arrhenius relation, as might be expected given the insensitivity of Eq
36 to $f_{\rm HCN}$ and the high thermal stability of N2. The other two
equilibria both give plausible Arrhenius-like fits, although far from perfect
(Figure 13).

Figure 13.— HCN quenching time scales determined from the ensemble of models
with respect to NH3 and CH4 (red $m\\!=\\!1$ and black $m\\!=\\!3$) and CO
(green $m\\!=\\!1$ and violet $m\\!=\\!3$). Symbol areas are proportional to
$f_{\rm HCN}$. Left. The time scale plotted is $t_{q}\propto p_{q}^{-1}$.
Right. High $T_{q}$ results fit better to the Arrhenius form with time scale
$t_{q}\propto p_{q}^{-0.5}$.

A direct fit to the chemical time scale derived from the equilibrium $K_{{\rm
HCN}\cdot{\rm CH}_{4}}$ for the full ensemble of models is

$t^{\prime}_{\rm q1}=1.6\times
10^{-4}p^{-1}m^{-0.7}\exp{\left(37000/T\right)}{\rm~{}~{}sec,}$ (38)

whilst the corresponding fit to $K_{{\rm HCN}\cdot{\rm CO}}$ is

$t^{\prime}_{\rm q2}=1.3\times
10^{-4}p^{-1}m^{-1}\exp{\left(34500/T\right)}{\rm~{}~{}sec.}$ (39)

At quench temperatures greater than 1600 K, the pressure dependence is better
described by $t^{\prime}_{\rm q}\propto p^{-0.5}$, and the $T$ dependence is
stronger with an Arrhenius $B$-factor of order 46000 K. The stronger
temperature dependence for high $T_{q}$ suggests that reactions with N2 with
its high activation energy are becoming important. The higher $T_{q}$ cases
correspond to higher $f_{\rm HCN}$.

As noted above, the quench approximation is not very sensitive to the details
of $t_{q}$, and that is the case here as well. Computed quenched abundances
using $t_{q}=t_{\rm mix}=H^{2}/K_{zz}$ provide a good approximation to the HCN
mixing ratios computed by the full model with

$t_{\rm HCN}=1.5\times
10^{-4}p^{-1}m^{-0.7}\exp{\left(36000/T\right)}{\rm~{}sec.}$ (40)

This expression seems to work well for all cases we have considered (Fig 14).

Figure 14.— HCN mixing ratios predicted by various quenching approximations
(y-axes) plotted against the actual asymptotic HCN mixing ratio as computed by
the full ensemble of 1-D kinetics models (x-axes). A perfect approximation
would adhere to the line. Left. Quench predictions using Eq 40. Open red
symbols are for $m\\!=\\!1$ and solid black symbols are for $m\\!=\\!3$.
Right. Quench approximations using Eq 41 with $L=H$ (open green circles,
$m\\!=\\!1$; filled black circles, $m\\!=\\!3$) and using Eq 42 with
$L=0.14\,H$ (open red diamonds, $m\\!=\\!1$; filled blue circles,
$m\\!=\\!3$).

Fegley and Lodders (1996) treat HCN destruction as controlled by direct
reaction of HCN with H2 to make NH and CH2. The corresponding chemical time
scale is

$t_{\rm chem}={9.3\times 10^{7}\exp{\left(70456/T\right)}\over Nf_{{\rm
H}_{2}}^{2}}{\rm~{}sec.}$ (41)

This has a very steep temperature dependence. Quench approximations using Eq
41 with $L=H$ are shown in Figure 14. This approximation predicts much more
HCN (a higher quench temperature) than we find in our 1-D models, a result
consistent with the steep temperature dependence of Eq 41.

Moses et al. (2010) presume that HCN destruction is controlled by reaction of
H2 with the H2CN radical. This is a much faster reaction. To convert their
discussion into a reaction time requires defining the H2CN equilibrium
abundance. We write $p{\rm H}_{2}{\rm CN}=K_{{\rm H}_{2}{\rm
CN}}\\!\cdot\\!p{\rm HCN}\\!\cdot\\!p{\rm H}$ with $K_{{\rm H}_{2}{\rm
CN}}=1.0\times 10^{-6}e^{14240/T}$. This is likely not the same as what Moses
et al. (2010) use. Atomic and molecular hydrogen are also assumed to be in
equilibrium. Other pertinent information is given in Moses et al. (2010). The
reaction time scale that results is

$t_{\rm chem}={8.3\times 10^{20}\exp{\left(23358/T\right)}\over
T^{1.941}Np^{0.5}f_{{\rm H}_{2}}^{1.5}}{\rm~{}~{}sec.}$ (42)

Figure 14 shows that Eq 42 used in a quench approximation predicts much less
HCN than we compute in our 1D models. This means that reactions destroying HCN
are occurring at relatively low temperatures. This fits with the relatively
weak temperature dependence of $t_{\rm chem}$ in Eq 42.

The comparison of models may be frustrated in part by a hole in our model. We
did not include methylamine (CH3NH2), which Moses et al. (2010) argue plays
the same role in hydrogenation of HCN at low temperatures and high pressures
that methanol plays for CO. Their scheme is plausible but almost entirely
hypothetical because it passes through several free radicals that must exist
but about which little else is known. Our omission of a CH3NH2 channel implies
that Eq 40 overestimates HCN, especially in cool worlds. Another issue
undermining comparison between models is that we have not implemented Moses et
al. (2010)’s full quench scheme: Moses et al. (2010) require that HCN quench
with respect to already quenched abundances of CH4 and NH3. But at the
temperatures at which HCN might actually be abundant enough to be detectable
in EGPs and BDs, our prescription should work and is easy to use.

##  6\. CO2

In principle CO2 is also subject to quenching (Prinn and Fegley, 1987), with
caveats. First, because CO2 quenches at a lower temperature than CO, it
quenches with respect to the disequilibrium (quenched) abundance of CO.
Second, in practice, CO2 can be much enhanced by photochemistry if the world
in question is subject to significant stellar irradiation. Under such
conditions quenching is a poor guide. But for solitary brown dwarfs and
planets in wide orbits it should do fine.

Figure 15.—  Left. CO2 quenching time scales determined from the ensemble of
models. Here quenching refers to disequilibrium between CO2 and CO and H2O; CO
and H2O are themselves not in equilibrium because they have already quenched.
The time scale plotted is $t_{q}\times p_{q}^{0.5}$. Right. $f_{{\rm CO}_{2}}$
predicted by the quenching approximation (y-axis) plotted against the actual
asymptotic mixing ratio determined from the full ensemble of 1-D kinetics
model (x-axis). A perfect approximation would adhere to the line. Open red
symbols are for $m=1$ and solid black symbols are for $m\\!=\\!3$.

The approach is similar to that used for CO and CH4 above. Equilibrium between
CO2, CO, H2, and H2O can be approximated by

$K_{{\rm CO}_{2}}={p{\rm CO}\\!\cdot\\!p{\rm H}_{2}{\rm O}\over p{\rm
CO}_{2}\\!\cdot\\!p{\rm
H}_{2}}=18.3\exp{\left(-2376/T-\left(932/T\right)^{2}\right)}.$ (43)

The equilibrium product Eq 43 is evaluated using the quenched values of $p{\rm
CO}$ and $p{\rm H}_{2}{\rm O}$, beginning at the altitude where CO and CH4
quench, and then extending to all higher altitudes. CO2 quenching is pinned at
the altitude where the equilibrium defined by Eq 43 breaks down. Figure 15
shows the results of doing so for the ensemble of models. Figure 15 is noisy
because the deviations from equilibrium are modest and can go in either
direction. It is interesting that the quenching timescale $t_{q}$ varies
inversely with the square root of the quench pressure $p_{q}$, and that unlike
the CO-CH4 system there is no discernible dependence on metallicity. The
temperature dependence in the Arrhenius-like relation $t_{q}\cdot
p^{0.5}_{q}\propto\exp{\left(38000/T\right)}$ is similar to the other cases we
have looked at involving CO, which is also notable.

Figure 15 shows that CO2 abundances in the ensemble of models are well
approximated by a quench model provided that the appropriate disequilibrium CO
and H2O mixing ratios are used. A chemical reaction timescale that works well
for CO2 quenching is

$t_{{\rm CO}_{2}}=1.0\times
10^{-10}\,p^{-0.5}\exp{\left(38000/T\right)}{\rm~{}~{}sec}$ (44)

where $p$ is in bars. The results shown in Fig 15 are rather insensitive to
the Arrhenius $A$ factor in $t_{{\rm CO}_{2}}$, which perhaps is to be
expected given the weak temperature dependence of the equilibrium constant Eq
43 compared to the very strong temperature dependence of Eq 44.

##  7\. Detectability

Whether or not a molecule can be detected depends on the abundance and opacity
of the species in question and on the opacities of other molecules and clouds.
For reference, Figure 16 shows absorption cross sections at 650 K and 1 bar
pressure for ${\rm H}_{2}{\rm O}$, $\rm NH_{3}$, $\rm CH_{4}$, HCN, and CO.
These can be compared to illustrative column densities shown in Figure 17. The
latter are integrated upward from the 1, 0.1, and 0.01 bar pressure level,
typical near-IR photospheric pressures for planets or brown dwarfs with
gravities of $10^{5}$, $10^{4}$, and $10^{3}\,\rm cm\,s^{-2}$, respectively.
For CO, CH4, CO2, and HCN we plot column densities for only one value of
$K_{zz}$ for each $(g,T_{\rm eff})$ pair. We arbitrarily select
$K_{zz}=10^{13}/g$, a high value of $K_{zz}$ but consistent with Eq 3, for the
illustration. For N2 and NH3 we plot column densities for all $K_{zz}$ to
emphasize how little these depend on mixing.

While only a complete model spectrum can definitively predict the visibility
of each molecule given a set of assumptions, we can use Figure 16 together
with Figure 17 to make some generalizations. We defer the task of properly
including our new chemical network into a complete model atmosphere to the
future.

Figure 16.— Absorption cross sections of molecules of interest here, computed
at 650 K and 1 bar pressure, for wavelengths between 1 and 4 microns. UCL
opacities are provided by the University College London and the Hitemp
opacities are from the HITRAN opacity database (Rothman et al., 2003).

Figure 17.— A sequence of models that illustrates how column densities of
important carbon-bearing species change with temperature $T_{\rm eff}$ and
gravity $g$. $K_{zz}$ is restricted to $10^{13}/g$ for CO, CH4, CO2, HCN. All
$K_{zz}$ are plotted for N2 and NH3. Columns are integrated above
$p=10^{-5}\\!\cdot\\!g$ bars. Small symbols represent $g\\!=1000$, mid-sized
symbols $g\\!=\\!10^{4}$, and big symbols $g\\!=\\!10^{5}$. Left. CO (black
circles), CH4 (green diamonds), CO2 (blue squares). Right. N2 (blue diamonds),
NH3 (red circles), HCN (gold squares).

The absorption cross section per molecule for the CO band head at $2.3\,\rm\mu
m$ is about $10^{-21}\,\rm cm^{2}$ per molecule. We know from model
comparisons to data (e.g., Cushing et al., 2008) that by $T_{\rm
eff}=1100\,\rm K$ at $g=10^{5}$ this band head produces only a slight spectral
feature. Consulting Figure 17 and scaling appropriately we expect, all else
being equal, that CO would remain detectable down to $T_{\rm eff}\sim 900\,\rm
K$ at $g=10^{4}$ and 500 K at $g=10^{3}$. This temperature is considerably
cooler than expected by the conventional wisdom.

We likewise can consider the appearance of the $2.2\,\rm\mu m$ methane band
which by definition first appears at the T0 spectral type at $T_{\rm eff}\sim
1200\,\rm K$. Figure 17 suggests that a comparable gravity-adjusted column
abundance of methane will not appear in a Saturn-like $g=10^{3}$ planet until
$T_{\rm eff}\sim 650\,\rm K$. The directly imaged companion GJ 504 b (Kuzuhara
et al., 2013; Janson et al., 2013) provides a test of this reasoning. The mass
and effective temperature of GJ 504 b have been estimated at $\sim 5\,\rm
M_{J}$ and 600 K. Its gravity should be $\sim\\!6000\,\rm cm\,s^{-2}$. By the
reasoning above methane would be expected and indeed has been detected (Janson
et al., 2013).

Methane is far more detectable in the $3.3\,\rm\mu m$ fundamental band where
the absorption cross section is over an order of magnitude larger. $\rm
CH_{4}$ has been detected here as early as spectral type L5 (Noll et al.,
2000) in field dwarfs. Likewise spectra over this wavelength range would
better help to constrain the arrival of methane at low gravity in the directly
imaged planets. Indeed Skemer et al. (2013) report a possible detection of
this same feature in HR 8799 c, suggesting a trace abundance in the upper
atmosphere.

Another test of these ideas may be provided by field T or Y dwarfs with
unusual colors not predicted by models. Faint, red objects in particular would
be good candidates for the low gravity, methane-poor objects we predict. For
example the cool Y dwarf WISE 1828+2650 (Cushing et al., 2011), is both faint
and unusually red in $J-H$ color (Beichman et al., 2014). While Cushing et al.
did detect methane in this object, chemical equilibrium models do a very poor
job fitting the available photometry (Beichman et al., 2014). Near-IR spectra
should be obtained of all such color outliers to search for methane spectral
features.

We note that some searches for planetary companions to young stars (e.g., Liu
et al., 2010) as well as surveys for ‘planetary-mass’ T-dwarfs in young
stellar clusters (e.g., Parker & Tinney, 2013) have employed differential-
methane band imaging. In this technique (Tinney et al., 2005) two images of a
target are taken, one in a filter that matches the $1.7-\,\rm\mu m$ $\rm
CH_{4}$ band and one which probes the entire H band. When the two images are
differenced, methane-bearing objects stand out as they are dark in the $\rm
CH_{4}$ filter. Our conclusions here suggest that such techniques must be used
with caution as methane may simply not yet be present in planetary-mass
objects even at effective temperatures below 1000 K.

Similar arguments can be made for the appearance of other spectral features of
interest. We expect that $\rm NH_{3}$ will appear at effective temperatures
about $200\,\rm K$ cooler in planets compared to field brown dwarfs. HCN, with
a cross section of $10^{-20}\,\rm cm^{2}$ per molecule at $1.55\,\rm\mu m$ is
unlikely to be detectable in most objects as the computed column abundances
are less than $10^{18}\,\rm cm^{-2}$. HCN’s prospects are poor at 3.0 microns
despite a high cross-section unless the C/O ratio is higher than solar and
water’s abundance reduced. Water’s cross section is 100-fold smaller, but with
solar abundances its column ($1-2\times 10^{21}$ cm-2) is 1000 times what HCN
can reach at its best. Carbon dioxide has an absorption cross section
$10^{-17}\,\rm cm^{2}$ per molecule at $4.2\,\mu\rm m$, and with abundances
approaching $10^{18}\,\rm cm^{-2}$ we expect it to be detectable at around 900
to 1100 K in field brown dwarfs. Indeed the AKARI space telescope discovered
$\rm CO_{2}$ features in several late L and early T dwarfs (Yamamura et al.,
2010). Judging by Figure 17, we would predict $\rm CO_{2}$ to be detectable to
500 K and cooler in the lowest mass planets.

##  8\. Conclusions

We use a reasonably complete 1-D chemical kinetics code to survey the
parameter space that encompasses atmospheres of cool brown dwarfs and warm
young extrasolar giant planets. Our model contains only gas phase chemistry of
small molecules containing H, C, N, O, and S. We use realistic $p$-$T$
profiles for cloudless atmospheres with effective temperatures between 500 and
1100 K and surface gravities between $10^{3}$ cm/s2 to $10^{5}$ cm/s2.
Vertical transport is described by an eddy diffusivity $K_{zz}$ that we vary
over a wide range. Our objective is to describe carbon and nitrogen
speciation, especially at lower (planetary) surface gravities. Overviews of
what we found are presented in Figure 4 for carbon and Figure 10 for nitrogen.

We find that carbon in cloudless brown dwarfs is predominantly in the form of
methane at 900 K for $g=10^{5}$ cm/s2. The small surface gravity of planets
strongly discriminates against CH4 when compared to an otherwise comparable
brown dwarf. If vertical mixing is comparable to Jupiter’s, methane first
predominates over CO in planets cooler than 500 K. Sluggish vertical mixing
can raise the transition to 600 K; clouds or more vigorous vertical mixing
could lower it to 400 K.

The detectability of specific molecular features in a spectrum depends on the
strength of the molecular absorption cross sections as well as the gaseous
abundance. Nevertheless a natural prediction of our model is that there will
be cool planets with no methane observed in the H or K spectral bands. The
refractory behavior of CO in low gravity objects is likely at least partially
responsible for the lack of cool planets found by the NICI survey, which
relied upon the methane absorption H band to identify planets (Liu et al.,
2010).

Ammonia is also sensitive to gravity, but unlike methane and CO, ammonia is
insensitive to mixing, which makes it a proxy for gravity. We did not explore
temperatures low enough to determine the transition from N2 to ammonia in
planets, but it is nearly as abundant as N2 at 500 K in brown dwarfs, which is
broadly consistent with the observed properties of the Y dwarfs (Cushing et
al., 2011) for which $\rm NH_{3}$ is seen in H band. On the other hand,
ammonia persists as an abundant minor species to rather high temperatures and
this is consistent with it being readily detected in mid-IR spectra of
T-dwarfs (Cushing et al., 2006). HCN might become interesting in high gravity
brown dwarfs if vertical mixing is very vigorous.

When expressed in terms of quenching parameters, nearly all our results for CO
and CH4 can be reduced to a simple equation in Arrhenius form that is easy to
use but not easy to interpret. From a strictly practical perspective, what we
find is close to what one gets using Prinn and Barshay (1977)’s algorithm in
its original form (a fortuitous accident of using too high a rate for the
wrong reaction). The apparent simplicity is somewhat surprising given the
complexity of the chemistry involved and the many different ways the system
has been described in the literature. Nor is our result quite what we
expected. First, we find that the timescale $t_{\rm CO}$ for hydrogenation of
CO is shorter for higher metallicity. Because the forward reactions that
destroy CO have no metallicity dependence, we infer that metallicity
dependence enters through the reverse reaction (oxidation of CH4). By
definition the reverse reaction is as fast as the forward reaction while in
equilibrium, but it falls off more quickly than the forward reaction as the
temperature drops. The important forward reactions are linear in metallicity
whilst the important reverse reactions are quadratic in metallicity.

Second, we find that $t_{\rm CO}$ is inversely proportional to pressure.
Analyses based on isolating a limiting reaction predict a stronger dependence
on pressure, typically $p^{-2}$. The weaker $p^{-1}$ pressure dependence is
expected for a reaction involving CO itself, such as

${\rm H}_{2}+{\rm CO}\rightarrow{\rm HCO}+{\rm H}.$ (45)

It is interesting that R45 actually does have the highest $\Delta G$ value in
the CO hydrogenation sequence, and therefore the highest energy barrier, and
that the magnitude of $\Delta G$ of R45 is what we infer in Eq 14. But it is
also quite clear from Figure 7 that equilibrium breaks down between CH3OH and
CH4 well before if breaks down between CO and CH3OH, as was found by Moses et
al. (2011). Reactions involving CH3OH imply a $p^{-2}$ pressure dependence,
because the equilibrium abundance of CH3OH with respect to CO goes as the
square of the H2 pressure. The system behaves—both in the 40,000 K energy
barrier and the $p^{-1}$ pressure dependence— as if the overall rate depends
on the entry reaction R45 rather than on the rate of R23 in which equilibrium
actually breaks down. The apparent paradoxes might be resolved through a more
complete analysis of the reverse reactions, or they may result from systematic
biases stemming from our treating the mixing length as equal to a scale
height, rather than as the complex function of many variables that it probably
is (Smith, 1998).

For nitrogen and ammonia we were unable to recover an emergent Arrhenius-like
behavior from the full system. On the other hand, it was easy to devise a good
quench approximation, and we also found that some published quench
approximations (e.g., Fegley and Lodders, 1994) work very well. For HCN, we
met with mixed success. We did find emergent Arrhenius-like behaviors, and we
developed a new quench approximation that works quite well for our models. On
the other hand there does not yet appear to be a consensus on what the quench
chemistry of HCN actually is. We found a wide scatter of different outcomes
when comparing different published quench schemes. We think our quench
approximation should work well for warm objects where HCN is predicted to be
relatively abundant but, because we did not consider hypothetical
hydrogenation channels through methylamine (Moses et al., 2010), our models
may overestimate $f_{\rm HCN}$, especially in cooler objects.

There are many effects that our models do not address. None of these effects
have much sway over our quench approximations, but they have much to do with
what might actually be present in a real atmosphere.

Clouds are likely the most important. Adding infrared opacity to an atmosphere
is effectively equivalent to reducing the gravity. This is because opacity
raises the $p$-$T$ profile (i.e., the adiabat) to a lower pressure for a given
effective temperature. A cloudy atmosphere favors CO and N2 over CH4 and NH3,
other things equal. Increasing metallicity also increases opacity to the
detriment of methane and ammonia.

Another issue for real worlds is that $K_{zz}$ may be much smaller in higher
altitude radiative regions of the atmosphere than it is in the convecting
region. The radiative stratosphere is the part of the atmosphere where
$K_{zz}$ the modeling parameter is most suspect. For N2 and NH3 pronounced
vertical structure in $K_{zz}$ will not matter much. For CO and CH4, sharply
lower values of $K_{zz}$ in the stratosphere might be important, because if
$K_{zz}$ is low enough, the atmosphere can have a second quench point in the
stratosphere. Such behavior no problem to a full kinetics model, but it can
make implementing quench approximations more complicated.

Photochemistry induced by irradiation from a nearby star can deplete NH3 and
increase CO2 and HCN. Lightning and/or impact shocks in cool NH3\- and
CH4-rich atmospheres can generate CO and HCN (Chameides and Walker, 1981). In
warmer objects, catalysts that may be present in metallic clouds (here we mean
real metals) will lower the quench point and thus will favor NH3, CH4, and CO4
(Prinn and Fegley, 1987).

All of these ideas will be put to the test by the directly imaged planets that
are expected to be discovered in the coming months and years. Constraining the
composition of their atmospheres will no doubt be a rewarding endeavour.

## Acknowledgments

The authors thank Richard Freedman, Caroline Morley, Julianne Moses, Didier
Saumon, and Channon Visscher for many insightful discussions and occasional
course corrections. The authors thank the NASA Planetary Atmospheres Program
for support of this work.

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