# Uranus and Neptune as methane planets: producing icy giants from refractory
planetesimals

Uri Malamud1, Morris Podolak2, Joshua I. Podolak3, Peter H. Bodenheimer4
1Physics Department, Technion - Israel institute of Technology, Haifa, Israel
2School of the Environment and Earth Sciences, Tel Aviv University, Ramat
Aviv, Tel Aviv, Israel
3D. E. Shaw Group, New York, NY
4Astronomy and Astrophysics Department, University of California Santa Cruz,
CA 95064, USA

## Abstract

Uranus and Neptune are commonly considered ice giants, and it is often assumed
that, in addition to a solar mix of hydrogen and helium, they contain roughly
twice as much water as rock. This classical picture has led to successful
models of their internal structure and has been understood to be compatible
with the composition of the solar nebula during their formation (Reynolds and
Summers, 1965; Podolak and Cameron, 1974; Podolak and Reynolds, 1984; Podolak
_et al._ , 1995; Nettelmann _et al._ , 2013). However, the dominance of water
has been recently questioned (Teanby _et al._ , 2020; Helled and Fortney,
2020; Podolak _et al._ , 2022). Planetesimals in the outer solar system are
composed mainly of refractory materials, leading to an inconsistency between
the icy composition of Uranus and Neptune and the ice-poor planetesimals they
accreted during formation (Podolak _et al._ , 2022). Here we elaborate on this
problem, and propose a new potential solution. We show that chemical reactions
between planetesimals dominated by organic-rich refractory materials and the
hydrogen in gaseous atmospheres of protoplanets can form large amounts of
methane ’ice’. Uranus and Neptune could thus be compatible with having
accreted refractory-dominated planetesimals, while still remaining icy. Using
random statistical computer models for a wide parameter space, we show that
the resulting methane-rich internal composition could be a natural solution,
giving a good match to the size, mass and moment of inertia of Uranus and
Neptune, whereas rock-rich models appear to only work if a rocky interior is
heavily mixed with hydrogen. Our model predicts a lower than solar hydrogen to
helium ratio, which can be tested. We conclude that Uranus, Neptune and
similar exoplanets could be methane-rich, and discuss why Jupiter and Saturn
cannot.

##  1 Introduction

The first models of Uranus and Neptune were constructed based on the premise
that they were formed in the outer parts of a primitive proto-planetary
nebula, where the elemental abundances were considered solar. Thus, assuming
chemical equilibrium, early studies used calculations of the expected relative
molecular abundances to gauge the potential composition of the planets that
formed in this region (Reynolds and Summers, 1965; Podolak and Cameron, 1974).
Ices, such as H2O, CH4, and NH3 were collectively found to be about twice as
abundant as rocky materials. Hereafter, we refer to rocks as those refractory
minerals in which the ratio of C to silicate-forming elements such as Si, Fe,
Mg or Ca is less than 0.1, whereas in refractory organics (CHONs) this ratio
is larger than 10 (Fomenkova _et al._ , 1992).

These early models typically considered a layered structure, with a rock core,
an ice mantle, and an envelope rich in hydrogen and helium (Podolak and
Reynolds, 1984; Podolak _et al._ , 1995). Rocks were modeled mainly as silica
(SiO2), but sometimes mixed with other molecules (e.g., MgO, Fe, Ni) and water
was considered the most abundant and therefore primary ice. The gas envelope
was modeled mainly as a H+He solar mix with an admixture of H2O, CH4, and NH3.
Later developments highlighted the role of a more complex internal structure
including material mixing and gradients (Helled _et al._ , 2020; Helled and
Fortney, 2020).

There is increasing evidence however that the constituent building blocks of
the outer planets might be refractory-rich, rather than water-rich,
challenging common convention with respect to Uranus and Neptune. We summarize
these arguments below, indicating that refractory materials in the outer solar
system could be 3-6 times more abundant than water.

Intermediate-sized Kuiper Belt objects (KBOs) indicate a trend of increasing
bulk density with mass (Brown, 2012). It was shown by means of several
thermophysical evolution models (see appendix A for detailed references) that
the trend may arise solely from differences in their internal porosity, due to
thermal and mechanical processing. Assuming that KBOs have all formed in the
same region and thus have a similar basic composition (Kenyon _et al._ ,
2008), these models have shown that a rock/water mass ratio of $\sim$3 fits
the observed trend.

Large KBOs like Pluto are of such considerable size, that they contain very
little residual porosity. Thus, its measured bulk density of 1854 kg m-3
(Stern _et al._ , 2015) directly translates into a similarly rock-rich
composition. Other large KBOs like Eris and Haumea often have even higher
densities, which imply yet richer rocky compositions (although these are
typically associated with further mantle-striping modifications via giant
collisions (Lissauer and Stewart, 1993; Leinhardt _et al._ , 2010; Barr and
Schwamb, 2016)).

Comets are nowadays regarded as highly refractory-rich bodies. There is a
growing body of evidence that strongly suggest comets have refractory to ice
mass ratios of _at least_ 3, and potentially even twice that value (Rotundi
_et al._ , 2015; Fulle _et al._ , 2016, 2017; Fulle _et al._ , 2019; Choukroun
_et al._ , 2020).

Polluted white dwarfs are astrophysical mass spectrometers. Their atmospheres
probe the composition of accreted exo-Solar planetary debris, inferring the
bulk chemical composition of the pollutants (Hollands _et al._ , 2018;
Harrison _et al._ , 2018, 2021). The typical finding is of almost purely rocky
debris (Jura and Xu, 2012; Jura and Young, 2014; Pietro Gentile Fusillo _et
al._ , 2017). It must be born in mind however that planetesimals can originate
from different regions of the disk, devolatilize before their host star
becomes a white dwarf (Malamud and Perets, 2016, 2017a, 2017b), and the
progression of accretion can complicate the association between pollution and
pollutant (Brouwers _et al._ , 2022). Despite these complications, relatively
rare few individual cases point to polluting debris which probably originate
in the outer parts of their systems and which contain a significant mass
fraction of water (Farihi _et al._ , 2013; Raddi _et al._ , 2015; Xu _et al._
, 2017). Yet even in these outlier cases the water fraction is 26%-38%,
showing that a large rock/water mass ratio might be a ubiquitous feature of
outer planetary disks.

These studies all suggest that the planetesimals accreted during the formation
of Uranus and Neptune should contain _at least_ 2-3 as much mass in refractory
material as in ices, leading to the question of why models of Uranus and
Neptune seem to imply they are water-rich. This inconsistency, or composition
tension problem, motivates our work, and there could only be two diametrically
opposed solutions: (I) ice-poor building blocks can somehow lead to the
formation of ice-rich planetary interiors, or (II) the internal structure
models of Uranus and Neptune could be compatible with a large rock/ice mass
ratio. We identify a novel path for enabling the former, transforming ice-poor
building blocks into an ice giant by invoking chemistry. At the same time we
discuss past literature that shows rock-rich interiors require a fine-tuned
solution. Under ordinary assumptions, we demonstrate that the rock/ice mass
ratio is very severely limited.

##  2 Expected composition of outer solar system planetesimals

If small bodies in the outer solar system are indeed rich in refractory
material, what do we expect this material to be composed of? We start by
considering the detailed composition of solids that contributed to the
accretion phase of the outer giant planets. However, as we have very little
direct information about the mineralogy of nearby Kuiper belt solids, we
assume instead accretion of solids with comet-like composition, since comets
similarly originated in the outer solar system. This assumption was also made
in some prior studies of Uranus and Neptune (Hubickyj _et al._ , 2005;
Iaroslavitz and Podolak, 2007).

Present-day comets typically contain tiny amounts of CO and other
hypervolatiles, relatively little water, and are _primarily_ composed of
refractory materials (Bockelée-Morvan and Biver, 2017; Fulle _et al._ , 2017;
Fulle _et al._ , 2019; Choukroun _et al._ , 2020). Observations of various
proto-planetary discs indirectly imply that when sufficiently far out in the
disc, carbon is depleted from the gas phase and instead enters into solid
grains locked inside planetesimals (McClure, 2019; Tabone _et al._ , 2023).
Below we however discuss _direct_ observations of cometary dust from our own
solar system, pointing towards a composition rich in carbon-bearing organic
refractories. We show that between 1/3-1/2 of the mass of their refractory
material is contained in organic rather than silicate material:

Comet P/Halley was explored by Giotto, a flyby mission in 1986 (Keller _et
al._ , 1986), showing that cometary dust is extremely rich in organics
relative to typical chondrites, and more similar to dust in dense molecular
clouds. High-velocity collisions of dust particles with metal targets on the
spacecraft provided a source of ions for mass-spectrometric analysis (Kissel
_et al._ , 1986; Lawler and Brownlee, 1992). Particles containing higher
fractions of organics were relatively more abundant closer to the nucleus
(Fomenkova, 1999), since organic components are more easily dissociated in the
coma (Fomenkova _et al._ , 1994; Wooden _et al._ , 2007), providing an
extended source for various gaseous species (Cottin and Fray, 2008). Studies
also discuss the detailed distribution of carbon among different compounds
(Fomenkova, 1999). Overall, the mass ratio of rocks to organics in dust is
between 2 and 1 (Fomenkova and Chang, 1993). I.e, carbon-rich organics
constitute as 33-50% of the dust mass.

Comet 67P/Churyumov-Gerasimenko was explored by ESA’s Rosetta mission, which
was the first to successfully rendezvous with a comet nucleus, follow it in
its orbit and dispatch a lander to the surface. The first spectrospcopic
detection of organics on the surface of a comet was made by the Rosetta VIRTIS
near-IR spectrometer instrument. Carboxylic acid was suggested as a primary
organic material on the nucleus (Capaccioni _et al._ , 2015; Quirico _et al._
, 2016). The Rosetta COSIMA instrument is a secondary ion mass spectrometer
equipped with a dust collector, a primary ion gun, and an optical microscope
for determining the position and characteristics of the dust particles (Le Roy
_et al._ , 2012). Many organic compounds were identified (Le Roy _et al._ ,
2015), and approximately 45% of the refractories in comet 67P/C-G could be in
organics (Bardyn _et al._ , 2017).

Comet C/2013 US10 (Catalina), a dynamically new Oort cloud comet with a close-
Earth approach, was determined to have a 47% amorphous carbon mass fraction
(Woodward _et al._ , 2021).

Ultra-carbonaceous Micrometeorites (UCAMMs) represent a small fraction of
interplanetary dust particles (IDPs) reaching the Earth, and are exceedingly
carbon rich with respect to other types of IDPs (Thomas _et al._ , 1993,
1994), as well as meteoritic samples (Dartois _et al._ , 2018). They are
recovered in the antarctic continent in a relatively pristine state (Duprat
_et al._ , 2010). The bulk fraction of organic refractory material in UCAMMs
ranges between 22-58% (Dobrică _et al._ , 2012). A recent review (Wooden _et
al._ , 2017) pieces together a line of arguments that advocate in favor of a
cometary origin for UCAMMs: (a) their crystalline and amorphous silicates are
similar to those in comet dust (Wooden, 2008); (b) high S and low Si
abundances in glass with embedded metal and sulfides promote a cometary origin
(Bradley and Ishii, 2008); (c) their fluffy textures appear minimally altered
by heating during relatively low-velocity atmospheric entry usually associated
with cometary origin (Duprat _et al._ , 2010); (d) they have D/H ratio
enrichment expected in cometary dust (Duprat _et al._ , 2010), including that
analyzed by Rosetta (Paquette _et al._ , 2021); (e) they are globally
unequilibrated, in a similar manner as other cometary materials (e.g from
Stardust samples (Brownlee _et al._ , 2007)). All this taken together
indicates a cometary-like nature for UCAMMs. We also note briefly that many
similar arguments also place the origin of relatively carbon-rich chondritic
anhydrous IDPs in comets (Engrand _et al._ , 2023).

Spitzer remote observations of cometary dust revealed a wide range of
amorphous carbon mass fractions spanning 10-90 %, with a mean value of 54%,
based on a large set of a few dozen comets (Harker _et al._ , 2023). This work
supports a new paradigm in which most comets have a carbonaceous content of
$\sim$50%.

In Appendix B we provide a note about why Stardust, the only comet sample
return mission to date, could not reliably provide an assessment of the bulk
organic fraction. However, enough evidence has accumulated as portrayed above,
in order to establish carbon-rich organics as major constituents of the solids
that accreted to form the outer planets. We will later discuss why such solids
can be transformed into copious quantities of methane in Uranus’ and Neptune’s
bulk interiors, during their growth.

##  3 Expected composition of Uranus and Neptune

As noted in the introduction, classical models of Uranus and Neptune assumed
they were ice-rich, based on the expected composition of a solar-composition
gas in equilibrium at low temperature. Models constructed based on this
assumption were found to be consistent with the observed physical properties
of these planets. If, as we argued in the previous section, the planetesimal
building blocks have a high refractory content, why do Uranus and Neptune
appear to be water rich? In order to investigate this composition tension
problem, we explore below the various possibilities for their composition
using a statistical approach.

We have previously developed (Podolak _et al._ , 2022) and here further
modified a statistical computer code that creates random, monotonic density
distributions, which fit a given mass, radius, and moment of inertia. For
Uranus and Neptune the gravitational moments J2 and J4 have also been measured
(Jacobson, 2009, 2014). However, in order to compute these moments for a given
density profile, the rotation period of the body must be known, and for both
Uranus and Neptune there is some ambiguity in this matter (Helled _et al._ ,
2011). In addition, the computation of these moments is extremely time-
consuming, so we have chosen, instead, to fit the moment of inertia (MOI)
only. While less accurate than using J2 and J4, the difference in the derived
density distribution is not significant for our purpose (Podolak _et al._ ,
2022).

By integrating the equation of hydrostatic equilibrium we can find the
pressure for each density. Podolak _et al._ (2022) describe several algorithms
for associating a temperature and composition with each pressure-density
point. Here we use a modified version of the ’inward’ algorithm. Briefly, we
assume that at any point the material is composed of a mixture of two adjacent
components from the following set of six materials, ordered according to
intrinsic density: Fe, SiO2, CO, H2O, CH4, and a solar mix hydrogen/helium gas
(hereafter called _envelope_). Thus SiO2 can mix with either Fe or CO (or H2O
for models which do not include carbon compounds), but not with anything else.
In addition, the mass fraction of the denser component cannot decrease going
inwards. Podolak _et al._ (2022) did not consider CO and CH4, and we now allow
for these species as well due to the importance of carbon in outer solar
system composition. Starting with a given surface temperature and pressure, we
compute the mass fractions of envelope and CH4 (or H2O for models which do not
include carbon compounds). Continuing inwards, we try to minimize the mass
fraction of the denser material while keeping the temperature constant. If
there is no mix that matches under those conditions, we increase the
temperature while keeping the same composition. The ’inward’ algorithm tries
to maximize the rock to water ratio while minimizing the central temperature.
Nevertheless, the central temperatures that are generated can sometimes be
quite high, and we discard those models where it exceeds $2.5\times 10^{4}$ K,
which we consider to be unlikely based on models of heat transport in these
planets (Podolak _et al._ , 2019; Vazan and Helled, 2020).

For computing random models with Fe (iron), SiO2 (silica) and H2O (water) we
use the quotidian equation of state (More _et al._ , 1988; Vazan _et al._ ,
2013). For H+He (envelope) material we use SCvH equation of state (Saumon _et
al._ , 1995) as modified in (Chabrier _et al._ , 2019). For CO (carbon-
monoxide) we use the quotidian equation of state Podolak _et al._ (2023). For
CH4 (methane) we use the well-known SESAME equation of state Lyon and Johnson
(1992). For mixing of different materials we use the additive volume law.

By generating hundreds of thousands of models, we can explore the space of
allowable internal compositions and structures. While we cannot say which
internal composition is more likely, we can in fact place meaningful limits on
the composition.

###  3.1 Water-rich Models

Podolak _et al._ (2022) investigated Uranus models with no carbon and with a
MOI factor of 0.23. They found that, for central temperatures below $2.5\times
10^{4}$ K there were no models with a rock/water mass ratio of 2 or more,
although some models came quite close. Thus one model, with a rock/water mass
ratio of 1.90 had 6.63 $M_{\oplus}$ of rock, 3.49 $M_{\oplus}$ of water, and
4.41 $M_{\oplus}$ of envelope. The central temperature was
$T_{\mathrm{c}}=1.77\times 10^{4}$ K. A second model, with a rock/water mass
ratio of 1.85 had 6 $M_{\oplus}$ of rock, 3.25 $M_{\oplus}$ of water, and 5.29
$M_{\oplus}$ of envelope. For this model $T_{\mathrm{c}}=1.14\times 10^{4}$ K.
A feature of many of these relatively rock-rich models is that they reach much
higher central temperatures compared to water-rich models.

Newly generated results in this study confirm previous results (Podolak _et
al._ , 2022) (and are therefore not shown here explicitly), indicating that
the rock/water mass ratio of both Uranus and Neptune is severely restricted
when typical materials are considered, which include Fe, SiO2, H2O, and H+He
envelope gas. The exact upper limit value depends sensitively on which moment
of inertia factor was assigned to Uranus and Neptune. Whereas classical models
considered a MOI factor of 0.23 for both planets (de Pater and Lissauer,
2015), the most recent studies (Neuenschwander and Helled, 2022) suggest that
0.22 is probably a more appropriate value for Uranus and 0.24 for Neptune. We
now considered these newer values as lower and upper limits, respectively.
Using these new values, the rock/water mass ratio obtained statistically is
likewise never more than 2, and actually rarely exceeds 1. In contrast, the
water/rock mass ratio has no upper bound, and there are a multitude of water-
rich models with far lower central temperatures. These results are consistent
with past studies, which almost never found large rock/ice mass ratios.

###  3.2 Rock-rich Models

It should be noted that, perhaps, more rock-rich interiors are possible using
future algorithms which will consider other material mixing options. Whereas
classic layered models usually did not consider mixing at all, we have
considered mixing of multiple adjacent materials, while some studies (Helled
_et al._ , 2011; Vazan and Helled, 2020) also considered mixing of SiO2 and
H+He, or triple mixing of SiO2, H2O and H+He (Vazan _et al._ , 2020). Despite
these various options, neither Vazan and Helled (2020) nor any other study has
ever reached a rock/water mass ratio larger than 2, with one exception.
(Helled _et al._ , 2011) managed to accomplish that by suggesting a mixture of
pure SiO2 with a very large fraction (15%) of H+He. While a rock-rich interior
is therefore technically possible, recent discussion regarding the question of
miscibility (Nettelmann _et al._ , 2016; Miguel and Vazan, 2023) supports the
adjacent mixing of SiO2 and H2O, while mixing of SiO2 and hydrogen (And also
Fe and hydrogen, see Wahl _et al._ (2013); Komabayashi (2021)) remains
empirically under-explored and probably requires high temperatures. While our
conservative choice of adjacent mixing only is a general limitation, other
modes of mixing are perhaps more uncertain and require fine-tuned conditions
to obtain a ’desirable’ outcome that leads to rock-rich interiors.

###  3.3 Models with Carbon Monoxide

When one considers ices in the outer solar system, one typically neglects ices
which are not water, since water is by far the most abundant ice in present
day comets (Bockelée-Morvan and Biver, 2017). However, CO is the second-most
abundant volatile in molecular clouds (Tanaka _et al._ , 1994; Chiar _et al._
, 1995), and in the early solar system, we expect to have a large fraction of
condensed carbon monoxide ice, even with respect to water (Lisse _et al._ ,
2022, and the references therein). At a later stage, CO ice in the region of
Uranus and Neptune is expected to sublimate (Lisse _et al._ , 2022), and
eventually disperse along with the gaseous proto-planetary disc (Ercolano and
Pascucci, 2017). A possible way around this, is if CO ice survives inside
comets that are placed into distant Oort cloud orbits early enough in order to
avoid complete loss of CO ice (Davidsson, 2021). C/2016 R2 might be a prime
example of a rare, extremely hyper-volatile rich, yet water and dust poor
comet, belonging to this category (McKay _et al._ , 2019). Such comets provide
an intriguing indication that the solar system may have indeed began with
abundant CO, as hypothesized above.

Assuming that CO was indeed prevalent when Uranus and Neptune formed, and also
ignoring chemistry and additional concerns, which are addressed later in the
paper, we explore the space of possible compositions for Uranus and Neptune
given two ices - water and carbon monoxide. If the CO can replace the H2O in
these models, it should raise the rock/water mass ratio above 2. For
comparison purposes, we use an identical numerical setup as in our previous
study of randomly generated models (Podolak _et al._ , 2022), only now instead
of 4 materials (iron, silica, water and H+He gas), we investigate 5 materials,
with carbon monoxide as an additional ice. We consider both Uranus and Neptune
and account for MOI factors in the range 0.22-0.24, as previously outlined.

Using the inward algorithm to match 100000 randomly generated density
profiles, we show the interior models of Uranus and Neptune with CO in Figure
1. Each random model is depicted by a single pixel, whose color corresponds to
the central temperature. The mass fraction of envelope in the planet is shown
as a function of the water/rock mass ratio. We note that by rock we refer to
the high-Z components SiO2+Fe, and by ice we refer to H2O+CO (distinguishing
ice/rock from water/rock). Since here the water/rock mass ratio extends to
very large values, we obtain a better view when the x-axis is in logarithmic
scale and truncated at a water-dominated mass ratio of 10.

---
(a) Uranus MOI factor = 0.22

---
(b) Uranus MOI factor = 0.23

---
(c) Neptune MOI factor = 0.23

---
(d) Neptune MOI factor = 0.24

Figure 1: Mass fraction of envelope vs. water/rock mass ratio (logarithmic).
Each point denotes a random interior model, color coded by its central
temperature. The dashed vertical line indicates equal fractions of water and
rock. The right (left) dotted vertical line indicates a water/rock mass ratio
of 2 (0.5).

The dashed vertical line denotes equal proportions of water and rock. The
right (left) dotted vertical line indicates a water/rock mass ratio of 2
(0.5). Models with zero rock ($\infty$ on the x-axis) are not shown. As can be
seen, much of the model parameter space is in the region of large water/rock
mass ratios. As was the case for the models without CO, there are no models
with a water/rock mass ratio less than 0.5, that is, the rock/water mass ratio
is never more than 2, and rarely exceeds 1. The reason for this is that, in
addition to replacing some H2O, the CO predominantly replaces SiO2, as can be
seen in Figure 2.

---
(a) Before adding CO

---
(b) After adding CO

Figure 2: An example of a random rock-rich model of Uranus, and the internal
delineation of its composition (normalized), before and after adding CO ice as
a fifth material. This model has a MOI factor of 0.22, central temperature of
12000 K and water/rock mass ratio of 0.7.

CO has a density between that of water and SiO2. Therefore the compositional
layers transition smoothly from water to CO and then SiO2. If the same density
profile is fit first without CO ice and then including it, CO can replace
either water or SiO2. In the models CO tends to replace more of the SiO2, than
the water. Thus, in Figure 2, we see the mass fraction of each component as a
function of radius for a particular Uranus model, originating from the top
left of Figure 1, panel (a). Here the MOI is 0.22, the central temperature
reaches 12000 K and the water/rock mass ratio is 0.7. Comparing the model with
CO (right panel) to that without (left panel) we see that the CO tends to
replace the SiO2 in preference to the water. Thus, simply adding CO to models
is not a good solution for the composition tension problem of Uranus and
Neptune.

###  3.4 Models with Methane

We also consider a third ice, methane, as motivated by the following
arguments. The CH4 abundance in the upper atmosphere of Uranus and Neptune is
observed to be far greater than solar (Atreya _et al._ , 2020). This should
not be regarded as indicative of the _bulk_ abundance of methane, since these
measurements only probe the atmosphere at the $\sim$1 bar level. Nevertheless,
early studies did in fact consider models where CH4 was mixed into the entire
envelope with far greater abundances than solar (Podolak, 1976; Podolak and
Reynolds, 1981), or with an icy layer that consisted primarily of H2O and CH4
(Hubbard and Macfarlane, 1980). This was motivated by the idea that nebular
carbon should condense to form CH4 and accrete onto Uranus and Neptune in
large amounts. At around the same time, however, the view was shifting away
from condensation of solid CH4 as a result of kinetic inhibition (Lewis and
Prinn, 1980). Instead, at sufficiently low temperatures, the expectation was
that the large majority of carbon would condense directly to CO (Lewis and
Prinn, 1980).

Later it was briefly suggested that, instead, CO might be chemically
transformed in the atmosphere of the gas giants and convert to CH4 (Podolak
and Reynolds, 1984) when reacting with hydrogen, altering their bulk
composition. However, this hypothesis did not develop further since. Here we
extend the idea that CH4 could be a key volatile in the outer planets. As
already outlined in Section 2, while CO is almost negligible and therefore has
a very small potential contribution for generating methane in typical
contemporary planetesimals, refractory organics contribute to a very
significant portion of their mass. These refractory organics may also
chemically transform into copious amounts methane when they accrete in vast
quantities onto the planet’s atmosphere, and are capable of altering its bulk
composition.

We use the same setup as in the previous section, but now replace carbon
monoxide with methane. When we incorporate CH4 as our second icy material, the
results are very different. Randomly generated interior models of Uranus and
Neptune with CH4 are shown in Figure 3. In these models the water/rock mass
ratio now spans the range 0.1-10. As noted above, CO as an additional ice
tends to replace SiO2. CH4, on the other hand, tends to replace water, so the
rock/water mass ratio can be significantly larger.

---
(a) Uranus, MOI factor = 0.22

---
(b) Uranus, MOI factor= 0.23

---
(c) Neptune, MOI factor = 0.23

---
(d) Neptune, MOI factor = 0.24

Figure 3: Mass fraction of envelope vs. water/rock mass ratio (logarithmic).
Each point denotes a random interior model, color coded by its central
temperature. The dashed vertical line indicates equal fractions of water and
rock. The right (left) dotted vertical line indicates a water/rock mass ratio
of 2 (0.5).

Is methane stable to pressure-induced dissociation in Uranus’ and Neptune’s
interior? Methane has a density between that of envelope H+He gas and water.
Therefore the compositional layers transition smoothly from envelope to
methane and then water. Figure 4 shows the mass fractions of the different
materials, the density, and the pressure as a function of radius for one of
Neptune’s random models. In this case the MOI factor is 0.24, the central
temperature reaches 13700 K and the water/rock mass ratio is 2.

---
(a) Compositional delineation

---
(b) Density profile

---
(b) Pressure profile

Figure 4: An example of a typical random water-rich model of Neptune

In panel (a) we see the transition from methane-dominated to water-dominated
composition, at a distance of about 15000 km from the planet’s center
($\sim$60% of the total radius). The corresponding pressure at this point is
shown in panel (c) to be just under 200 GPa, which is less than the 300 GPa
required for complete methane dissociation (Benedetti _et al._ , 1999). A
certain fraction of methane may reside in the pressure range 100-300 GPa,
which might be enough to partially dissociate some of it into other
hydrocarbons (Ancilotto _et al._ , 1997; Benedetti _et al._ , 1999; Gao _et
al._ , 2010; Sherman _et al._ , 2012), but should not result in complete
dissociation into pure carbon and hydrogen. In addition, since methane and
water can be mixed in the transition layer, it should be noted that pressures
up to 150 GPa can also give rise to a high-pressure form of methane clathrate
hydrate (Schaack _et al._ , 2019). We have examined a number of random models
in detail, and the results are similar. Typically, only the deepest submersed
methane is just marginally able to dissociate due to high pressure. We thus
consider our assumption of using the methane equation of state as judicious,
and we expect little difference in density at these pressures before and after
dissociation.

Finally, in Figure 5, we show the methane mass fraction as a function of the
water/rock mass ratio. The methane mass fraction probed by the space of
possible solutions is at least 10%. In models that also have a large
rock/water mass fraction (left of the vertical dashed line) it is necessarily
larger, upwards of 20%. Since the envelope mass fraction in these models also
has large values (Figure 3), the combination of large envelope and even larger
CH4 mass fractions means that the other materials - H2O, SiO2, and Fe, are
collectively only a relatively small fraction of the planet’s mass. Thus, in
models that have a large rock/water mass fraction, specifically, the methane
fraction might even be larger than that of H2O, SiO2, and Fe combined. This is
a difficulty because CH4 is certainly not that prevalent in the contemporary
solar system, and is normally a very small fraction compared to water
(Bockelée-Morvan and Biver, 2017). Thus, for methane to participate in solving
the composition tension problem, a large methane fraction must be justified.
We suggest that organic-rich refractories are sufficiently abundant in outer
solar system planetesimals to drive chemical reactions in the atmospheres of
Uranus and Neptune, which, during the phase of planet growth, may produce the
required methane. These processes are discussed next.

---
(a) Uranus, MOI factor = 0.22

---
(b) Uranus, MOI factor = 0.23

---
(c) Neptune, MOI factor = 0.23

---
(d) Neptune, MOI factor = 0.24

Figure 5: Mass fraction of CH4 vs. water/rock mass ratio (logarithmic). Each
point denotes a random interior model, color coded by its central temperature.
The dashed vertical line indicates equal fractions of water and rock. The
right (left) dotted vertical line indicates a water/rock mass ratio of 2
(0.5).

##  4 Growth of the Outer Planets and Accretion of Solids

We consider a growing planet in the outer solar system, which starts accreting
mass onto an embryo-sized core (the incipient core mass in the range 0.05-0.5
M⊕ (Brouwers _et al._ , 2018; Bodenheimer _et al._ , 2018)). Classical models
emphasized subsequent growth due to accretion of large planetesimals. These
planetesimals could fall directly onto the core (Podolak _et al._ , 1988)
however they could also dissolve in the atmosphere due to thermally
(Iaroslavitz and Podolak, 2007; Brouwers _et al._ , 2018) or mechanically
(Mordasini _et al._ , 2015) driven ablation. As core mass accumulates,
additional atmosphere is collected (Pollack _et al._ , 1996). Energy release
is added to the luminosity of the planet and thus internal temperatures
increase during growth (Pollack _et al._ , 1996).

Planetesimals can be large enough so that thermal ablation alone will not
erode a significant fraction of their volume (Valletta and Helled, 2019).
However, ram-pressure can fragment large planetesimals which are
insufficiently strong. In that case they break up and become pancaked,
decelerating much more effectively while passing through the atmosphere, and
quickly lose their kinetic energy before reaching the core. If they are
sufficiently large, exceeding about 10-100 km, this outcome might be avoided
due to high self-gravity, and direct impacts onto the core are possible
(Mordasini _et al._ , 2015; Valletta and Helled, 2019).

A second accretion process that was discussed in the literature involved
pebble accretion (Lambrechts _et al._ , 2014; Bitsch _et al._ , 2015; Pessah
and Gressel, 2017; Ormel _et al._ , 2021). Pebbles are small mm-dm sized
objects, and as such they are more susceptible to gas drag. The planet’s
effective cross section for accretion is much larger. The ablation of the
pebbles in the atmosphere is also greatly enhanced (Brouwers _et al._ , 2018;
Ormel _et al._ , 2021). In reality, the accretion of Uranus and Neptune might
have involved both pebbles and planetesimals, and thus we examine both
options.

Pebbles are much more easily affected by thermal heating due to their small
size and the increase in ambient temperature as they plunge deeper into the
atmosphere. They are also affected by frictional heating, when a fraction of
the kinetic energy lost through friction is converted into heat. Frictional
heating is more dominant in the cold outer regions of the atmosphere,
particularly for fast impacting pebbles. As a result, they quickly and fully
evaporate and dissolve into the atmosphere (Brouwers _et al._ , 2018). Despite
the fact that ablation is more efficient for pebbles than for planetesimals,
both types of objects are viable sources for dispersing high-Z materials into
the atmosphere (Brouwers _et al._ , 2018), and we consider them by accounting
for a range of object sizes: 10 m, 100 m, 1 km, 10 km and 100 km. Solid
objects below 10 m are eroded so effectively that we need not consider them
explicitly. The exact details of ablation of both small and large
planetesimals depend sensitively on material properties and especially the
atmospheric density profile, which changes during the planet’s growth.

We can get an idea of the relevant density profile, as well as the
temperature-pressure (T-P) profiles, using a newly updated code (Stevenson _et
al._ , 2022) implemented here for the first time for a growing Uranus. This
code solves the equations of mass conservation, hydrostatic equilibrium,
energy conservation, and energy transport assuming spherical symmetry. It
includes the effects of mass and energy deposition by accreting solids and
gases, and equations of state of water, SiO2 and iron. It also includes
opacity due to dust grains, the ablation and breakup of (here only 100 km
mono-sized) solid planetesimals in the gaseous envelope, treatment of
compositional gradients and heating by the host star which drives mass loss.
Using this code, we obtain snapshots of the density/temperature/pressure
profiles of the atmosphere, at three different times after the start of the
evolution: 1.35, 2.06 and 2.65 Myr.

We then use these atmospheric density profiles as input to an ablation code
(Podolak _et al._ , 1988), which includes the detailed effects of frictional
gas drag, heating by ambient thermal radiation, and the ensuing evaporation
(sublimation) erosion derived from the energy input. Following the
trajectories of accreted planetesimals of various sizes, we can determine the
characteristic temperature-pressure conditions during ablation. For the
parameters we selected, we find that small sub-km planetesimals are strongly
slowed by atmospheric drag and ablate as a consequence. The larger
planetesimals are correspondingly less affected by drag and are therefore able
to maintain the speed necessary to break up deeper in the atmosphere, due to
ram pressure. As already mentioned, these results depend on the assumed
material properties. Since the code is unable to account for the intricacies
of planetesimal intra-porosity or material mixing, we employ a similar
approach to other studies (Iaroslavitz and Podolak, 2007; Brouwers _et al._ ,
2018; Brouwers and Ormel, 2020; Ormel _et al._ , 2021) and simplify matters by
considering non-porous planetesimals with a uniform density of 2000 kg m-3,
and the material thermodynamic/cohesion properties of water, which might
represent a mean value for a mixture of silicates, organics, water and other
high-volatility ices. Given these assumptions, the temperature-pressure
conditions during ablation are provided in Table 1. Note that ablation is a
gradual process and therefore it occurs for a range of T-P conditions. To
simplify matters, for sub-km planetesimal we listed a single T-P value that
matches the moment during which half of the planetesimal mass has eroded. For
larger planetesimals, the listed value matches the moment of ram-pressure
breakup when most of the mass is instantantly delivered to the atmosphere.

Table 1: Temperature and pressure conditions during ablation of planetesimals, when, for sub-km planetesimals, half of the planetesimal mass has eroded; and for the larger planetesimals ram-pressure breakup occurs. Time \ Size | 10 m | 100 m | 1 km | 10 km | 100 km
---|---|---|---|---|---
1.35 Myr | 300 K, 90 Pa | 500 K, 1 kPa | 500 K, 2 kPa | 500 K, 2 kPa | 2300 K, 700 kPa
2.06 Myr | 290 K, 40 Pa | 470 K, 500 Pa | 950 K, 20 kPa | 950 K, 20 kPa | 1900 K, 400 kPa
2.65 Myr | 230 K, 20 Pa | 410 K, 400 Pa | 800 K, 20 kPa | 880 K, 30 kPa | 1800 K, 900 kPa

Even if the local conditions are sometimes insufficient for chemistry to occur
at first, a necessary outcome of planetary growth is that the conditions
change with time, namely: (a) previously too shallow deposits are increasingly
buried within an externally growing atmosphere, and thus the conditions might
be expected to turn in favor of chemistry at a later point; and (b) our
chemical reactants are also generally expected to sink deeper into the
atmosphere as a result of condensation. These aspects, in addition to possible
convective mixing are discussed next.

###  4.1 Compositional Stratification During Accretion

Once high-Z materials are present in the atmosphere, they are expected to form
a compositional gradient inside the planet (Helled and Stevenson, 2017;
Lozovsky _et al._ , 2017; Brouwers _et al._ , 2018; Bodenheimer _et al._ ,
2018; Valletta and Helled, 2019; Vazan and Helled, 2020; Brouwers and Ormel,
2020; Ormel _et al._ , 2021). Indeed there is some seismic and gravity-based
evidence for a deep compositional gradient in Saturn (Fuller, 2014; Mankovich
and Fuller, 2021) and Jupiter (Wahl _et al._ , 2017; Debras and Chabrier,
2019).

Assuming that high-Z vapor in the atmosphere obeys the ideal gas law, an
assumption which was found to be judicious in giants such as Uranus and
Neptune (Iaroslavitz and Podolak, 2007), then compositional stratification can
be explained as follows. At a given layer, and for a given high-Z material
dissolved in the atmosphere by ablation, its local abundance determines its
exact partial pressure in the gas. If the partial pressure of each high-Z
material exceeds its saturated vapor pressure, which is a function of the
ambient temperature in that layer, then this means that the high-Z material is
in excess of a saturated state, and it must condense, releasing latent heat
during the phase transition. Condensed matter descends towards the core if it
is heavier than the ambient gas, losing gravitational energy on its way. This
so-called ’rainout’ process leads to a compositional gradient. Since the
temperature increases with deepening layers, each layer is able to contain
more vapor of the same material (the saturated vapor pressure increases with
the temperature). At the boundary of the solid silicate core and the overlying
saturated atmosphere, if the latter cannot absorb all the (continually)
ablated high-Z material, the rest of it adds to the solid core. This allows
the solid core to grow even without direct impacts (Iaroslavitz and Podolak,
2007; Leconte _et al._ , 2017; Bodenheimer _et al._ , 2018; Brouwers _et al._
, 2018; Brouwers and Ormel, 2020). The rainout process implies that high-Z
materials increase inward according to Z. Given differences in their mean
molecular weights, saturated vapor pressures and the temperature gradient in
the atmosphere, one expects a stratified structure to emerge, just as it does
in our growth and evolution code (Figures 6-8).

Figure 6: Growth of Uranus after 1.35 Myr, showing the mass fraction of iron
(red), silica (green), water (orange), and H+He (blue) on the right axis and
the temperature (black) on the left axis, as a function of distance from the
center. The arrows mark the T-P conditions where the composition changes from
100% H+He to 0% H+He. Figure 7: Growth of Uranus after 2.06 Myr, showing the
mass fraction of iron (red), silica (green), water (orange), and H+He (blue)
on the right axis and the temperature (black) on the left axis, as a function
of distance from the center. The arrows mark the T-P conditions where the
composition changes from 100% H+He to 0% H+He. Figure 8: Growth of Uranus
after 2.65 Myr, showing the mass fraction of iron (red), silica (green), water
(orange), and H+He (blue) on the right axis and the temperature (black) on the
left axis, as a function of distance from the center. The arrows mark the T-P
conditions where the composition changes from 100% H+He to 0% H+He.

Compositional gradients also provide the physical justification for our
statistical random model code, which is currently implemented based on the
idea of adjacent-material mixing and stratification. Convective mixing might
in principle also hinder the rainout process and thus mix all of the various
materials together. However, it has been shown previously that the formation
of compositional gradients generally tend to prevent this from happening,
stabilizing the layers against large-scale convective motion (Leconte _et al._
, 2017; Vazan and Helled, 2020). Note that the rainout process will occur for
any high-Z material, both the original materials introduced into the
atmosphere by planetesimal ablation, or their chemical products, invoked by
our chemical model below. Thus, even if the local conditions are initially
incompatible for the chemistry we invoke, condensible species can still
descend in the atmosphere until the required conditions are satisfied.

###  4.2 Chemistry During Accretion

Here we describe a simple chemistry model to facilitate the construction of an
icy planet from ice-poor pebbles or planetesimals. In Section 1 we have shown
that accreted comet-like solids are mostly refractory, and in Section 2 we
have shown that 1/3-1/2 of these refractories are carbon-rich organics. By
some estimates, aliphatic and aromatic carbons (consisting entirely of C and
H) constitute more than 75% by mass of the bulk organic matter (Derenne and
Robert, 2010) for meteroites, indicating that their mass is dominated by
carbon. If we make a similar assumption for outer solar system refractories,
graphite (pure C) might be taken to represent refractory organics in a
simplified model. As additional validation of this assumption, we show in
appendix A that KBOs and not merely comets, could have densities that are
compatible with such a carbon-rich material. This implies that the solids
which contributed material to Uranus and Neptune during their formation could
be similar to the solids that were incorporated into the Kuiper belt, and
therefore could be compatible with our model.

Our outer solar system planetesimals are represented by the following
simplified components: Fe2SiO4 (fayalite) represents the rocky mineral
component; C (graphite) represents the organic (or idealized carbon-rich)
refractory component; The icy components are H2O and CO. While there are many
possible mineralogies to choose from, the one that was chosen above has a
specific intention. By adding hydrogen, we can chemically separate Fe2SiO4
into Fe, SiO2 and H2O, while graphite will convert to CH4, and CO to CH4 and
H2O, and therefore all of these chemical products (iron,silica,water,methane
and envelope) are standard materials that are most frequently used in
canonical models of gas giants, and have well tested equations of state. We
maintain flexibility in the model, by varying the ratios among the reactants.

We consider three idealized chemical reactions:

$\displaystyle fayalite+hydrogen\rightarrow iron+silica+water$ (1)
$\displaystyle Fe_{2}SiO_{4}+2H_{2}=Fe_{2}+SiO_{2}+2H_{2}O$

$\displaystyle graphite+hydrogen\rightarrow methane$ (2) $\displaystyle
C+2H_{2}=CH_{4}$

$\displaystyle carbon~{}monoxide+hydrogen\rightarrow methane+water$ (3)
$\displaystyle CO+3H_{2}=CH_{4}+H_{2}O$

We first discuss the relevance of these equations based on empirical evidence,
from direct laboratory experiments. For the first reaction, some experiments
show that the temperatures required in order to chemically decompose fayalite
in the presence of molecular hydrogen, producing iron, silica and water, are
in the range 550-670 K for pressures of 3.3 GPa and 1.4 GPa, respectively
(Efimchenko _et al._ , 2019). These authors showed that full decomposition
takes 1-3 hours, and also that it requires a sufficient ratio of hydrogen to
fayalite (Efimchenko _et al._ , 2021). However, different experiments
(Massieon _et al._ , 1993) show that fayalite decomposition also occurs at far
lower pressures, as long as the temperature is sufficiently high (e.g., 1070 K
is required at a pressure of merely 1.4$\times 10^{4}$ Pa). A similar range of
temperatures between 570-770 K and pressures spanning 0.5-3 GPa are likewise
reported in lab experiments for our second chemical reaction, synthesizing
methane from graphite and molecular hydrogen (Peña-Alvarez _et al._ , 2021),
although an exact reaction timescale is currently unavailable (private
correspondence with authors). For our third reaction, methane and water are
produced via hydrogenation of carbon monoxide, as known for more than a
century (Sabatier and Senderens, 1902). This process, also known as carbon
monoxide methanation (Johnson and Nuth, 2015), generally occurs in the lab at
temperatures in excess of about 450 K and at modest pressures (around
atmospheric pressure, or 1$\times 10^{5}$ Pa, but the reaction would likewise
occur somewhat lower or higher pressures as well (Gao _et al._ , 2012),
usually in the presence of various catalyzers). The idealized reaction
timescale (without catalyzers) at a temperature of 777.5 K and pressure of
1.1$\times 10^{4}$ Pa is estimated at around one hour (Podolak and Reynolds,
1984).

These laboratory experiments show that the original refractories invoked by
our model are readily transformed through chemical reactions. However, the
fayalite decomposition literature above demonstrated how a very wide range of
temperature conditions led to the same experimental outcomes by greatly
varying the pressure. This presents a problem since lab experiments are
practically limited to a narrow set of conditions and can only partially probe
the full parameter space of T-P conditions relevant to our problem. That being
the case, we also investigated these reactions via theoretical chemical
equilibrium calculations. Theory allows us to both validate experimental
results, and also enable a full investigation of the entire range of T-P
conditions relevant to Uranus and Neptune during their growth.

For chemical equilibrium calculations we utilize the widely used STANJAN code
(Reynolds, 1986), which provides the chemical equilibrium state of an ideal
gas mixture. The list of elements that can be employed in STANJAN is quite
extensive, however Fe is not one of the included elements in the code.
Therefore, of our three reactions, only that of fayalite decomposition could
not be accounted for with STANJAN. Fortunately, the fayalite decomposition
experiments robustly cover a much wider range of T-P conditions compared to
the other two reactions, spanning over 5 orders of magnitude in pressure
(Massieon _et al._ , 1993; Efimchenko _et al._ , 2019). As a first step we
validate the two other experimental results for CO methanation (Johnson and
Nuth, 2015), and C methanation (Peña-Alvarez _et al._ , 2021). We find that
STANJAN reproduces the same results as the experiments, when we input the same
T-P conditions into the code. We therefore gain confidence in its predictive
capabilities, while we plan to use more elaborate codes in future studies.

In order to investigate methanation for the entire range of T-P conditions
shown in Table 1, we consider the elements C, H and O. Of course, since Fe and
Si are excluded from STANJAN, the chemical equilibrium resulting in these
calculations gives an incomplete picture, and is only designed to obtain a
partial view of the chemistry. Using these elements we can account for the
five species that participate in Equations 2 and 3, namely: H2, C, CO, H2O and
CH4. We require the relative fraction of each species in a state of
equilibrium at constant pressure and temperature, but also for completion we
look at several other potential chemical products which might stably form at
the appropriate conditions: CO2 (carbon dioxide), C2H6 (ethane) and CH3OH
(methanol). Since we must neglect fayalite, we assume as a heuristic approach,
that our input planetesimals consist of twice as much C (40% mass fraction) as
H2O (20% mass fraction), and equal mass fraction of H2O and CO (20%). Our
input mass fraction for H2 (20%) is selected arbitrarily in order to provide
sufficient hydrogen for full chemical transformation of C, H2O and CO. We note
that our primary concern is not to explore various realizations of the input
composition, but rather to investigate if the chemistry goes in the direction
of methane or not. For this purpose, small changes in the initial composition
do not qualitatively alter the outcomes. We demonstrate the effect of the
temperature and pressure state on the chemistry in Table 2, where we show the
relative mass fractions of the various chemical products, given a range of
conditions.

Table 2: The relative mass fractions of the products in our chemical reactions, given a range of temperature and pressure conditions experienced by dissociated high-Z materials during ablation, rainout or subsequent cooling of the planet. The initial mass fractions are shown in the parentheses. T-P state \ Species | H2 | C | CO | H2O | CH4 | CO2 | C2H6 | CH3OH
---|---|---|---|---|---|---|---|---
| (0.2) | (0.4) | (0.2) | (0.2) |  |  |  |
300 K, 100 Pa | 0.024 | 0 | 1E-17 | 0.329 | 0.648 | 3E-12 | 3E-12 | 8E-21
300 K, 10 kPa | 0.024 | 0 | 1E-21 | 0.329 | 0.648 | 3E-16 | 3E-12 | 8E-21
500 K, 1 kPa | 0.026 | 0 | 4E-05 | 0.319 | 0.643 | 0.012 | 1E-07 | 9E-13
500 K, 100 kPa | 0.024 | 0 | 5E-09 | 0.329 | 0.648 | 2E-06 | 1E-07 | 1E-12
750 K, 10 kPa | 0.087 | 0 | 0.148 | 0.093 | 0.5 | 0.171 | 6E-06 | 8E-10
750 K, 1 MPa | 0.025 | 0 | 1E-03 | 0.314 | 0.641 | 0.017 | 3E-05 | 1E-08
1000 K, 100 kPa | 0.129 | 4E-29 | 0.478 | 0.012 | 0.37 | 0.011 | 4E-05 | 6E-09
1000 K, 10 MPa | 0.04 | 4E-32 | 0.03 | 0.267 | 0.611 | 0.052 | 4E-04 | 7E-07
1000 K, 1 GPa | 0.024 | 0 | 2E-05 | 0.328 | 0.647 | 6E-05 | 7E-04 | 2E-06
2000 K, 1 MPa | 0.134 | 5E-10 | 0.511 | 2E-06 | 0.351 | 3E-07 | 4E-03 | 1E-09
2000 K, 100 MPa | 0.129 | 6E-14 | 0.485 | 0.015 | 0.364 | 2E-03 | 4E-03 | 1E-05
2000 K, 10 GPa | 0.036 | 6E-17 | 0.035 | 0.296 | 0.581 | 0.011 | 0.039 | 1E-03
6000 K, 100 MPa | 0.217 | 0.248 | 0.511 | 1E-04 | 0.023 | 5E-06 | 3E-04 | 5E-07
6000 K, 10 GPa | 0.129 | 4E-04 | 0.456 | 0.032 | 0.289 | 2E-03 | 0.089 | 3E-03

We can divide the chemical processes that act on high-Z materials into three
distinct stages in the evolution: initial ablation of planetesimals, their
subsequent rainout and the long-term global evolution of the planet. For the
first stage, we use our planetesimal ablation code. Thus, we consider the
values listed in Table 1 to represent the relevant conditions. Cross
referencing these values with Table 2, it is clear that pebbles and small
planetesimals are dissolved in the atmosphere early-on and experience
conditions that drive the chemistry towards methane and water. Large
planetesimals on the other hand might break up due to ram-pressure at
locations where the conditions are more inclined to drive the chemistry
towards less CH4 and more CO (and even CO2), while H2O also diminishes.

The second stage is rainout of high-Z materials towards the core. This process
is included in our growth code for Uranus, however without explicitly
accounting for chemistry, and only for mono-sized large planetesimals. We can
nevertheless use it to approximately estimate the possible range in T-P
conditions in the region of the atmosphere that contains hydrogen, as marked
by the arrows in Figures 6-8. Increase in both temperature and pressure during
rainout may result in the chemistry moving either towards methane or carbon
monoxide. It is hard to determine the outcome without including explicit
feedbacks in the code.

The third stage occurs on long timescales and involves the global thermal
evolution of the entire planet as it cools and contracts. We then expect a
decrease in temperature and increase in pressure, overall. As an example, the
present day pressures in the H+He outer envelope of Uranus and Neptune could
be as high as a few hundred GPa (see Figure 4) which is approximately three
orders of magnitudes more than those obtained with our growth code for Uranus’
early evolution (see Figure 8).

Table 2 generally shows that high temperature drives the chemistry towards
carbon monoxide. However, keeping the same high temperature while increasing
the pressure, always drives the chemistry back in the opposite direction,
towards water and methane. This can be qualitatively understood from Le
Chatelier’s principle, which states that chemistry will move in the direction
that undoes the pressure increase, i.e. in the direction of fewer molecules
(the right hand side of Equation 3). Therefore, if we use our random
statistical code to represent the more evolved state of the planet, we expect
the chemistry to move towards methane.

Given the previous discussion, a full solution to the problem requires a
complex numerical code which tracks the intricate thermal and physical
evolution of the planet from early growth to the present day, while
continually resolving the direction of chemistry both spatially and
temporally. This code should also track the large scale convective motion in
the interior in order to account for mixing or lack thereof. The chemical
calculation should include additional elements and reactions. Such a code far
exceeds the capabilities of any present-day study, and is also beyond the
scope of this paper. While we cannot achieve the ultimate level of detail,
here we use separate codes, showing that chemical methanation appears to be a
likely outcome and is expected to occur. Future efforts must be taken in order
to account for all the various feedbacks we discussed, in a self-consistent
manner.

A last point to consider is the ratio of accreted hydrogen to high-Z material
during the early evolution of the planet. If there is insufficient hydrogen,
the chemical transformation will not go to completion. During the early stage
of core accretion, models predict that the accreted material consists
primarily of high-Z solids, and the relatively little hydrogen is probably
insufficient to drive chemistry. At some point there is however a crossover
where the bulk of the accreted material becomes dominated by H+He. This
crossover depends on the details of the protoplanetary disc and whether
accretion is dominated by km-sized solid bodies or by pebbles. For example, in
a recent pebble accretion growth model (Ormel _et al._ , 2021) this crossover
will occur when a Uranus-mass planet has accreted about 1/3 of its final mass
(however this work considers small semi-major axes). In our Uranus growth
model the crossover point occurs when it has accreted about 2/3 of its mass
(however our growth code does not consider pebble accretion). Thus, the
assumption of full chemical transformation could break down, at least
temporarily. Nevertheless, even if the planet has accreted much of its
hydrogen during its final growth phase, the hydrogen could still interact with
the interior of the planet if there is ongoing long-term mixing. In this case
only the bulk fraction of hydrogen is important. Jupiter and Saturn are so
massive that the bulk mass of H+He gas dwarfs the rest of the high-Z materials
(Helled, 2019). In Uranus and Neptune the H+He mass fraction is up to a few
tens of % (Helled _et al._ , 2020), which is enough to enable a full chemical
transformation.

###  4.3 Interior Composition After Methane Formation

As discussed above, the conditions are likely appropriate for chemistry to
occur in the growing atmosphere of Uranus and Neptune. Demonstrating that
chemical conversion is complete remains a future task for much more advanced
models than currently exist in the literature. We note that even if the
chemical transformation is incomplete, it still applies to a very large
fraction of the accreted solids. Since modeling of partial chemical evolution
is extremely difficult, we instead simplify the problem by assuming, without
full justification, that the chemical reactions we invoke apply to all solids.
The methane fraction resulting from the calculation below may thus be viewed
as an upper limit.

If the accreted material is indeed composed of comet-like solids, as we
assumed, it is possible to estimate the effect of its methanation on the
overall composition of the planet. Let $M_{\mathrm{p}}$ be the total mass of
the planet, and $f_{\mathrm{e}nv}$ be the fraction of the accreted H+He
envelope. Let $r_{\mathrm{h}:he}$ denote the mass ratio of hydrogen to helium.
The envelope mass will then be given by

$M_{\mathrm{e}nv}=f_{\mathrm{e}nv}M_{\mathrm{p}}$ (4)

and the mass of the high-Z materials is

$M_{\mathrm{Z}}=M_{\mathrm{p}}-M_{\mathrm{e}nv}.$ (5)

The masses of hydrogen and helium are, respectively,

$M_{\mathrm{h}}=\frac{r_{\mathrm{h}:he}}{r_{\mathrm{h}:he}+1}M_{\mathrm{e}nv}$
(6)

and

$M_{\mathrm{h}e}=M_{\mathrm{e}nv}-M_{\mathrm{h}}.$ (7)

Let $r_{\mathrm{r}ef:h2o}$ be the mass ratio of refractories to water in the
accreted solids, $r_{\mathrm{h}2o:co}$ the mass ratio of water to CO, and
$r_{\mathrm{f}aya:c}$ the mass ratio of silicates (modeled by fayalite) to
organic refractories (modeled by graphite, i.e. pure carbon). The
refractory/ice mass ratio in the planet’s solid building blocks is then

$r_{\mathrm{r}ef:ice}=r_{\mathrm{r}ef:h2o}\frac{r_{\mathrm{h}2o:co}}{r_{\mathrm{h}2o:co}+1}$
(8)

and the total refractory mass is

$M_{\mathrm{r}ef}=\frac{r_{\mathrm{r}ef:ice}}{r_{\mathrm{r}ef:ice}+1}M_{\mathrm{Z}}.$
(9)

This can be divided between the fayalite (silicate) and graphite (carbon-rich
organic) components, where

$M_{\mathrm{f}aya}=\frac{r_{\mathrm{f}aya:c}}{r_{\mathrm{f}aya:c}+1}M_{\mathrm{r}ef}$
(10)

and

$M_{\mathrm{c}}=M_{\mathrm{r}ef}-M_{\mathrm{f}aya}.$ (11)

Similarly, the ice mass is given by

$M_{\mathrm{i}ce}=\frac{1}{r_{\mathrm{r}ef:ice}+1}M_{\mathrm{Z}},$ (12)

which can be split between water and carbon monoxide via

$M_{\mathrm{h}2o}=\frac{r_{\mathrm{h}2o:co}}{r_{\mathrm{h}2o:co}+1}M_{\mathrm{i}ce}$
(13)

and

$M_{\mathrm{c}o}=M_{\mathrm{i}ce}-M_{\textrm{h2o}}.$ (14)

In what follows, let $N_{i}$ be the number of molecules of species $i$ and
$m_{i}$ be the mass of such a molecule. By invoking chemistry we transform the
above reactants, expecting the following reactions to occur:

Reaction 1: fayalite converts to iron + silica (1) - This is a decomposition +
single displacement type equation, where we assume that all fayalite accreted
onto the planet decomposes into iron and silica, and displaces oxygen to make
water from hydrogen.

The number of fayalite molecules is then

$N_{\mathrm{f}aya}=\frac{M_{\mathrm{f}aya}}{m_{\mathrm{f}aya}}.$ (15)

Reaction 1 thus reduces the total mass of hydrogen by

$\Delta(M_{\mathrm{h}})_{1}=-2m_{\mathrm{h}2}N_{\mathrm{f}aya}$ (16)

and increases the mass of water by

$\Delta(M_{\mathrm{h}2o})_{1}=\frac{m_{\mathrm{o}}}{m_{\mathrm{f}aya}}M_{\mathrm{f}aya}+2m_{\mathrm{h}2}N_{\mathrm{f}aya}$
(17)

The masses of iron and silica masses are given by

$M_{\mathrm{f}e}=\frac{m_{\mathrm{f}e2}}{m_{\mathrm{f}aya}}M_{\mathrm{f}aya}$
(18)

and

$M_{\mathrm{s}io2}=\frac{m_{\mathrm{s}io2}}{m_{\mathrm{f}aya}}M_{\mathrm{f}aya}.$
(19)

Reaction 2: graphite converts to methane (2) -

This is a synthesis type equation, where we assume that all graphite accreted
onto the planet combines with hydrogen to make methane. The number of graphite
molecules formed is

$N_{\mathrm{c}}=\frac{M_{\mathrm{c}}}{m_{\mathrm{c}}}$ (20)

and this decreases the hydrogen mass by

$\Delta(M_{\mathrm{h}})_{2}=-2m_{\mathrm{h}2}N_{\mathrm{c}}.$ (21)

The mass of methane produced is

$M_{\mathrm{c}h4}=M_{\mathrm{c}}+2m_{\mathrm{h}2}N_{\mathrm{c}}.$ (22)

reaction 3: all carbon monoxide converts to methane and water (3) -

This is a double displacement type equation, where we assume that all carbon
monoxide accreted onto the planet reacts with hydrogen, displacing oxygen and
hydrogen to make methane and water. The number of carbon monoxide molecules is

$N_{\mathrm{c}o}=\frac{M_{\mathrm{c}o}}{m_{\mathrm{c}o}}.$ (23)

This reaction reduces the hydrogen mass by

$\Delta(M_{\mathrm{h}})_{3}=-3\times m_{\mathrm{h}2}\times N_{\mathrm{c}o}.$
(24)

The methane mass increases by

$\Delta(M_{\mathrm{c}h4})_{3}=\frac{m_{\mathrm{c}}}{m_{\mathrm{c}o}}M_{\mathrm{c}o}+2m_{\mathrm{h}2}N_{\mathrm{c}o}$
(25)

and the water mass by

$\Delta(M_{\mathrm{h}2o})_{3}=\frac{m_{\mathrm{o}}}{m_{\mathrm{c}o}}M_{\mathrm{c}o}+m_{\mathrm{h}2}N_{\mathrm{c}o}.$
(26)

We can now compare these estimates of the expected chemistry with the range of
compositions determined by random interior modeling, noting that outcomes
depend only on the ratios between the various reactants, and are independent
on the planet mass. We therefore explore a range of values for the reactants,
given the following parameters: (a) the planet’s H+He envelope mass fraction
$f_{\mathrm{e}nv}$. The maximum value is constrained by Fig. 3 to be 0.28 and
0.18 for Uranus and Neptune, respectively. The minimum value is that which
uses all the hydrogen in the atmosphere, enabling full chemical conversion of
the accreted solids. It can be trivially calculated from the molecular weights
of the reactants, giving a different result (in the range 0.1-0.17) for each
combination of the free parameters, as specified in Table 4. There is the
possibility of a still lower fraction, however without full chemistry. We do
not consider this option here for simplicity, hence the resulting methane
fractions we obtain may be viewed as upper limits; (b) the assumed mass ratio
between H and He in the accreted gaseous material $r_{\mathrm{h}:he}$, which
we take to be 3, i.e. close to solar; (c) the assumed mass ratio between
refractories and water in the solid accreted material $r_{\mathrm{r}ef:h2o}$.
Based on the composition of Kuiper belt objects and comets, we take 2 and 4 as
the lower and upper limits, respectively, with a mean of 3 (see Section 1);
(d) the assumed mass ratio between water and carbon monoxide in the initial
accreting ices, $r_{\mathrm{h}2o:co}$. We take 1 for the initially CO-rich end
case, and 100 for the ratio in most contemporary comets, as another end case
(see Section 3.3); and (e) the assumed mass ratio between the original rocky
(fayalite) and organic components (carbon) in the refractory materials
$r_{\mathrm{f}aya:c}$, which we take to be between 1 and 2 (see Section 2).

The results of our simple chemical model are shown in Tables 3 and 4, for the
maximum and minimum H+He fractions, respectively (Table 4 uses up all the
molecular hydrogen in the atmosphere by construction, giving
$f_{\mathrm{h}}=0$). The four left-hand columns depict the various initial
choices for the free parameters. The initial ratio between H and He (solar)
are kept fixed. The other six columns show the results of our chemical model
in terms of the relative mass fraction of each material in the planet (noting
that the results are independent on the planet’s mass itself). Tables 3 and 4
unveil Uranus and Neptune as methane rich planets. Their methane mass fraction
is in the range 20-50%, comparable to and even larger than the mass fraction
of water. However, here we considered full methanation of carbon-bearing
species, which requires sufficiently massive H+He atmospheres, with a minimum
mass fraction between 10-17%.

Table 3: Chemical model results for maximum H+He fractions $f_{\mathrm{e}nv}$ | $r_{\mathrm{r}ef:h2o}$ | $r_{\mathrm{h}2o:co}$ | $r_{\mathrm{f}aya:c}$ | $f_{\mathrm{h}}$ | $f_{\mathrm{h}e}$ | $f_{\mathrm{c}h4}$ | $f_{\mathrm{h}2o}$ | $f_{\mathrm{s}io2}$ | $f_{\mathrm{f}e}$
---|---|---|---|---|---|---|---|---|---
The maximum $f_{\mathrm{e}nv}$ is either 0.28 (Uranus) or 0.18 (Neptune);
$r_{\mathrm{h}:he}$=3 is fixed.
0.28 | 2 | 1 | 1 | 0.1079 | 0.07 | 0.3429 | 0.3275 | 0.0529 | 0.0988
0.18 | 2 | 1 | 1 | 0.0187 | 0.045 | 0.3905 | 0.373 | 0.0603 | 0.1125
0.28 | 4 | 1 | 1 | 0.0996 | 0.07 | 0.3886 | 0.2395 | 0.0706 | 0.1318
0.18 | 4 | 1 | 1 | 0.0092 | 0.045 | 0.4425 | 0.2728 | 0.0804 | 0.1501
0.28 | 2 | 100 | 1 | 0.1251 | 0.07 | 0.3203 | 0.283 | 0.07 | 0.1313
0.18 | 2 | 100 | 1 | 0.0383 | 0.045 | 0.3648 | 0.3223 | 0.0801 | 0.1496
0.28 | 4 | 100 | 1 | 0.1082 | 0.07 | 0.3841 | 0.1954 | 0.0845 | 0.1578
0.18 | 4 | 100 | 1 | 0.0191 | 0.045 | 0.4374 | 0.2225 | 0.0963 | 0.1797
0.28 | 2 | 1 | 2 | 0.1267 | 0.07 | 0.2629 | 0.3381 | 0.0706 | 0.1318
0.18 | 2 | 1 | 2 | 0.0402 | 0.045 | 0.2994 | 0.385 | 0.0804 | 0.15
0.28 | 4 | 1 | 2 | 0.1247 | 0.07 | 0.2819 | 0.2536 | 0.0941 | 0.1757
0.18 | 4 | 1 | 2 | 0.0378 | 0.045 | 0.3211 | 0.2888 | 0.1072 | 0.2001
0.28 | 2 | 100 | 2 | 0.1501 | 0.07 | 0.214 | 0.297 | 0.0938 | 0.1751
0.18 | 2 | 100 | 2 | 0.0668 | 0.045 | 0.2437 | 0.3383 | 0.1068 | 0.1994
0.28 | 4 | 100 | 2 | 0.1383 | 0.07 | 0.2563 | 0.2123 | 0.1127 | 0.2104
0.18 | 4 | 100 | 2 | 0.0533 | 0.045 | 0.2919 | 0.2417 | 0.1284 | 0.2396
Table 4: Chemical model results for minimal H+He fractions $f_{\mathrm{e}nv}$ | $r_{\mathrm{r}ef:h2o}$ | $r_{\mathrm{h}2o:co}$ | $r_{\mathrm{f}aya:c}$ | $f_{\mathrm{h}}$ | $f_{\mathrm{h}e}$ | $f_{\mathrm{c}h4}$ | $f_{\mathrm{h}2o}$ | $f_{\mathrm{s}io2}$ | $f_{\mathrm{f}e}$
---|---|---|---|---|---|---|---|---|---
The minimum $f_{\mathrm{e}nv}$ is calculated to give $f_{\mathrm{h}}=0$ at
each combination; $r_{\mathrm{h}:he}$=3 is fixed.
0.159 | 2 | 1 | 1 | 0 | 0.0397 | 0.4005 | 0.3825 | 0.0618 | 0.1154
0.1698 | 4 | 1 | 1 | 0 | 0.0425 | 0.448 | 0.2762 | 0.0814 | 0.1519
0.1359 | 2 | 100 | 1 | 0 | 0.0338 | 0.3844 | 0.3396 | 0.0844 | 0.1576
0.1586 | 4 | 100 | 1 | 0 | 0.0225 | 0.4854 | 0.2469 | 0.1068 | 0.1994
0.1336 | 2 | 1 | 2 | 0 | 0.0397 | 0.4488 | 0.2283 | 0.0988 | 0.1844
0.1364 | 4 | 1 | 2 | 0 | 0.0341 | 0.3381 | 0.3042 | 0.1129 | 0.2107
0.0999 | 2 | 100 | 2 | 0 | 0.025 | 0.2675 | 0.3713 | 0.1173 | 0.2189
0.1172 | 4 | 100 | 2 | 0 | 0.0293 | 0.3143 | 0.2603 | 0.1382 | 0.258

In Figure 9 all the models in Table 3 (abundant hydrogen) are plotted to show
the scatter in possible interior composition, against the background of
randomly generated models. Using the most recent moment of inertia factor
estimates for Uranus (0.22) and Neptune (0.24) (Neuenschwander and Helled,
2022) each random model is depicted by a single pixel, whose colour
corresponds to the central temperature. The mass fraction of methane in the
planet is shown as a function of the water/rock mass ratio (here rock =
SiO2+Fe). The dashed vertical line denotes equal proportions of water and
rock. The right (left) dotted vertical line indicates a water/rock mass ratio
of 2 (0.5). Superimposed on the same plot are the calculated upper-bound
methane mass fractions based on chemistry, when accounting for the range in
our reactants ratios (red asterisks). The composition derived from our simple
chemical model falls well within the allowed parameter space of internal
structures determined by our random models, independently producing the same
patterns for both planets. In these models with abundant methane we see that
the proportions between rock and water in Uranus and Neptune can shift either
towards rock or water, but since methane itself is an ice, they are strictly
always icy, despite having been initially formed by refractory-dominated
solids. This is better reflected in the ice/rock mass ratio in Figure C1 in
Appendix C. We likewise confirm that iron in our random models is within the
expected range in solar system planets (Figure C2 in Appendix C).

We note that the randomly generated models shown here assume solar composition
for the H+He mixture (H/He$\sim$3), whereas models involving chemistry are
always enriched in He (H/He$<$3). This inconsistency is however ignored for
simplicity, since we cannot produce an infinite number of random models with
every conceivable H/He ratio. We do however consider the end case scenario as
well, where we produce random models that only contain He and no H. In this
case too we find (not shown) that the methane fractions from Table 4 fit the
random models’ scatter as well.

---
(a) Uranus, MOI factor 0.22

---
(b) Neptune, MOI factor 0.24

Figure 9: The internal compositions resulting from chemistry are super-imposed
(red asterisks) on the computer generated phase space of CH4 mass fraction vs.
water/rock mass ratio.

##  5 Summary and Conclusion

Classical models of Uranus and Neptune present them as water-rich, yet their
planetesimal building blocks should have a rock/water mass ratio greater than
2. We have explored the parameter space of planet models using a random planet
generator and find that models fitting the mass, radius, and MOI factor of
Uranus and Neptune can, at best, achieve a rock/water mass ratio somewhat less
than 2, where the only alternative solution is to mix rock with a large
fraction of hydrogen. We therefore suggest a different solution to this
compositional tension problem: a significant fraction of the accreted
planetesimals included highly carbon-rich refractory organic materials,
similar to composition of comets. We show that while the oxygen in rocky
minerals and small amounts of CO ice will react with the hydrogen to form
additional water, the carbon inside refractory organics will form very
significant amounts of methane. Our random model generator shows that such
methane-rich planets can fit the observed properties of Uranus and Neptune.
The rock/water mass ratio in these models now span a considerable range of
values, but since a large fraction of methane is also present, we find that
these models always contain more ice (methane and water) than rock and iron.
Thus, the interior of ’ice giants’ is indeed strictly icy.

Although the process described here would apply universally to the other giant
planets as well, for Jupiter and Saturn the fraction of high-Z material is
negligible compared with that of H+He (Helled, 2019), and, as a result, the
CH4 mass fraction expected will be too small to significantly affect the
resulting models. Additionally, the interior pressures in these giants would
easily dissociate most CH4 into other hydrocarbons or carbon and hydrogen
(Ancilotto _et al._ , 1997; Benedetti _et al._ , 1999; Gao _et al._ , 2010;
Sherman _et al._ , 2012). On the other hand, Uranus and Neptune, as well as
other similar-sized exoplanets, are massive enough to accrete sufficient
hydrogen in order to synthesize ample CH4 but also just small enough to keep
most of this methane from pressure dissociation. In addition, they do not
accrete too much hydrogen and helium so that the heavier materials remain a
significant fraction of the mass.

In addition to the fact that Uranus and Neptune might be methane-rich, there
is another interesting outcome of our proposed chemistry. Since large amounts
of atmospheric hydrogen are required for methanation, the atmospheric H/He
ratio should be significantly sub-solar. While H/He ratio in the bulk is
difficult to measure using current remote techniques which only probe the
uppermost atmosphere (Atreya _et al._ , 2020), a Uranus mission has recently
been recognized as a highest priority target by both NASA and ESA, and is
expected to launch sometime during the next decade (Fletcher _et al._ , 2022;
Mousis _et al._ , 2022; Girija, 2023; Mandt, 2023). We can expect an entry
probe to reveal more high quality data about Uranus’ interior, especially if
there is some mixing in the atmosphere, dredging up material from deeper
within the planet.

##  6 CRediT authorship contribution statement

Uri Malamud: Conceptualization, Methodology, Software, Formal Analysis,
Investigation, Resources, Data curation, Writing - original draft, Writing -
review & editing, Visualization, Project administration, Funding acquisition.
Morris Podolak: Software, Formal analysis, Investigation, Writing - review &
editing, Funding acquisition. Joshua Podolak: Software. Peter Bodenheimer:
Software.

##  7 Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  8 Data availability

The data that has been used is available on request from the corresponding
author.

##  9 Acknowledgements

We wish to thank Marc Brouwers for useful conversations with regard to giant
planet growth and evolution. UM acknowledges support by the Niedersächsisches
Vorab in the framework of the research cooperation between Israel and Lower
Saxony under grant ZN 3630 and grant by MOST-space. MP acknowledges support by
a grant from the Pazy Fund of the Israel Atomic Energy Commission. PB
acknowledges support from NASA grant EW-18-2-0060.

##  Appendix A Are large Kuiper belt objects compatible with comet-like
composition?

In the main text we discuss how comet-like refractory material can be composed
of a large fraction of organic rich material, up to 50% in mass. It needs to
be shown that such organic-rich refractory constituents are also compatible
with the contemporary bulk density of intermediate- and large-sized KBOs.

An underlying assumption in this work, as well as other studies, is that
objects in the outer solar system form roughly in the same region and
therefore have similar compositional characteristics (Kenyon _et al._ , 2008;
Malamud and Prialnik, 2015; Bierson and Nimmo, 2019). KBOs are expected to
incorporate materials similar to those in comets, and Uranus and Neptune are
likewise hypothesized (as the main rationale for this work) to incorporate
similar materials as these two classes of objects - a premise which gave rise
to the composition tension problem this paper addresses. Unlike comets,
however, KBOs are unique in their ability to test this hypothesis, and
therefore we will show that comet-like, organic-rich material, is compatible
with the composition in the Kuiper belt.

We start our discussion with intermediate-sized KBOs, whose internal structure
contains non-negligible residual porosity. We then discuss Pluto, the most
well-characterized object in the Kuiper belt which is both massive and
sufficiently evolved in order to suppress internal porosity.

A decade ago, a trend among KBOs was identified – an increase in the bulk
density of dwarf planets as a function of their mass (Brown, 2012). The
composition, or rock/water mass ratio which was commonly suggested in the past
for Kuiper belt dwarf planets, was based on the casual assumption that
porosity is negligible. If porosity is negligible, then the bulk density
translates directly into composition. If however porosity is non-negligible,
then it implies that the derived rock/water mass ratio is only a lower limit.

A true determination of the composition from the bulk density, must take into
consideration how porosity is internally distributed within the dwarf planet
as a function of the internal (evolving) pressure and temperature profiles, as
well as grain texture, and mixing (if rock and ice are mixed or
differentiated). To this purpose, a differentiation model for icy bodies
(Malamud and Prialnik, 2013) was revised (Malamud and Prialnik, 2015) to
include an equation of state suitable for porous icy bodies with radii of a
few hundred km, based on available empirical studies of ice and rock
compaction. The plausibility of this equation of state was then tested on
three intermediate-sized KBOs of various masses, considering how these icy
bodies evolve due to internal heating. The porosity profile evolves as a
result of internal migration of water, rock/water stratification and
compaction. It was shown that the present-day bulk densities in the model were
quite well matched with the observed trend, without requiring one to vary the
initial composition of those three objects.

Based on this aforementioned work (Malamud and Prialnik, 2015), it was
concluded that the trend is well produced for a rock/water mass ratio of
approximately 3, and a similar conclusion was later echoed in a different
study (Bierson and Nimmo, 2019). The exact rock/water mass ratio predicted by
such models, depends on what one assumes for the rock specific density (i.e.
its grain density). According to the former study (Malamud and Prialnik,
2015), two phases of rocky material were considered. The rock was initially
considered to be anhydrous, as in comets (Wooden _et al._ , 2017). Later
interaction by the flow of liquid water, which was explicitly followed in the
code, transformed the rock and rendered it into the hydrated phase. The
specific density of the hydrated phase is typically about 20% lower than the
anhydrous phase (Malamud and Prialnik, 2013). The anhydrous rock specific
density was a free parameter. Values were considered in the range 3250-3500 kg
m-3. A rock/water mass ratio of 3 matched the Kuiper belt density trend,
corresponding to the 3250 kg m-3 value. Slightly changing the value to 3000 or
3500 kg m-3, only modestly changed these results.

According to the above arguments, for the conclusions to apply, it is merely
necessary to justify their choice of rock specific density. Taking the
mineralogy assumed in the present work, that is, a binary combination of
fayalite and graphite, we can calculate what the specific grain density should
be. At zero porosity, as well as low pressure and temperature which would be
typical of dwarf-planet sized objects, fayalite and graphite have the specific
densities $\varrho_{\mathrm{f}aya}$=4390 kg m-3 and
$\varrho_{\mathrm{c}}$=2260 kg m-3, respectively. Assuming an additive volume
law, the density of their combined refractory mixture $\rho_{\mathrm{r}ef}$ is
given by

$\rho_{\mathrm{r}ef}=\left(\frac{x_{\mathrm{f}aya}}{\varrho_{\mathrm{f}aya}}+\frac{x_{\mathrm{c}}}{\varrho_{\mathrm{c}}}\right)^{-1}=\left(\frac{1-x_{\mathrm{c}}}{\varrho_{\mathrm{f}aya}}+\frac{x_{\mathrm{c}}}{\varrho_{\mathrm{c}}}\right)^{-1},$
(A1)

where $x_{\mathrm{f}aya}$ and $x_{\mathrm{c}}$ are the respective mass
fractions of fayalite and graphite in the mixture, and
$x_{\mathrm{f}aya}+x_{\mathrm{c}}=1$. From Equation A1, a 50:50 ratio leads to
a density of 3000 kg m-3, and a 63:37 ratio leads to 3250 kg m-3. These mix
ratios, where organic fractions are in the range 37-50%, are in excellent
agreement with those proposed in the Section 2 for outer solar system
material. The grain specific density $\varrho_{\mathrm{r}ef}$ of anhydrous
refractories is therefore compatible with previous work (Malamud and Prialnik,
2015), and will lead to the assumed rock-rich intermediate-sized KBOs, having
a rock/water mass ratio of around 3.

Turning to Pluto for additional clues, its mass is much larger than the
previously mentioned intermediate-sized KBOs. The pressure exerted by its own
self-gravity and the higher internal temperatures attained during its
evolution are likely sufficient in order to have rendered its porosity close
to zero (Bierson _et al._ , 2018). It is therefore possible to directly derive
Pluto’s composition without a great deal of porosity-related degeneracy,
adopting a few simplifying assumptions, as follows.

Let us assume that the porosity is either exactly zero, or allow for the
possibility of some small porosity $\psi$ which should be retained primarily
in the icy mantle, where the pressures are estimated in the range 0-200 MPa
between the surface and the core-mantle boundary (Hussmann _et al._ , 2006),
and yield porosities between 0-30% (Malamud and Prialnik, 2015). Let us
further consider a two-layered stratified structure - a rocky core underlying
an icy mantle. For simplicity, water is the only ice accounted for. We also
ignore the fact that differentiation leads to a more intricate structure in
both core and mantle, rather than a simple two-layer scheme (ignoring partial
hydration and dehydration in the core (Malamud _et al._ , 2017)). Finally, let
us assume that the core contains the original refractories which accreted to
form Pluto (i.e. ignoring hydration), and the mantle consists of pure water
ice. In that case Pluto’s density is given by

$\rho_{\mathrm{p}luto}=\left(\frac{x_{\mathrm{r}ef}}{\rho_{\mathrm{r}ef}}+\frac{x_{\mathrm{h}2o}}{\varrho_{\mathrm{h}2o}(1-\psi)}\right)^{-1}=\left(\frac{x_{\mathrm{r}ef}}{\rho_{\mathrm{r}ef}}+\frac{1-x_{\mathrm{r}ef}}{\varrho_{\mathrm{h}2o}(1-\psi)}\right)^{-1},$
(A2)

where $\varrho_{\mathrm{h}2o}$=917 kg m-3 is the specific density of water
ice, $x_{\mathrm{r}ef}$ and $x_{\mathrm{h}2o}$ are the respective mass
fractions of the original refractories and water in the dwarf planet, and
$x_{\mathrm{r}ef}+x_{\mathrm{h}2o}=1$. The residual porosity in the mantle
$\psi$ can be zero, or it can be estimated as some relatively small number. We
take 15% as an average value between 0 and 30% discussed above. We now
substitute $\rho_{\mathrm{r}ef}$, combining Equations A1 and A2, in order to
obtain the value of $x_{\mathrm{c}}$ as a function of the other parameters,
leading to

$x_{\mathrm{c}}=\left(\frac{1}{x_{\mathrm{r}ef}}\left(\frac{1}{\rho_{\mathrm{p}luto}}-\frac{1}{\varrho_{\mathrm{h}2o}(1-\psi)}\right)+\left(\frac{1}{\varrho_{\mathrm{h}2o}(1-\psi)}-\frac{1}{\varrho_{\mathrm{f}aya}}\right)\right)\cdot\left(\frac{1}{\varrho_{\mathrm{c}}}-\frac{1}{\varrho_{\mathrm{f}aya}}\right)^{-1}.$
(A3)

Letting $\rho_{pluto}$=1854 kg m-3 as determined from observations (Stern _et
al._ , 2015), and replacing the refractory fraction $x_{\mathrm{r}ef}$ with
the rock/water mass ratio $r$ using the relation $x_{\mathrm{r}ef}=r/(1+r)$,
we can use the above relation in order to evaluate $r$ as a function of
$x_{\mathrm{c}}$. Figure A1 shows four subplots for different choices of
mantle residual $\psi$, up to 15%.

---
(a) $\psi$=0

---
(b) $\psi$=0.05

---
(c) $\psi$=0.1

---
(d) $\psi$=0.15

Figure A1: Pluto’s rock/water mass ratio as a function of the organic mass
fraction in refractory material, for different mantle porosities $\psi$. The
dotted lines correspond to the plausible range of organics, based on
measurements from comet Halley (see SI).

Although we have made several simplifying assumptions, the well measured bulk
density of Pluto helps to constrain its composition using the analytic
formulation above. The dotted lines in Figure A1 correspond to the expected
range in organic mass fraction. The lower and upper limits on the x-axis are
based on the direct mass-spectrometer measurements obtained during the Giotto
mission to comet Halley (as was discussed in the main text). It can be seen
that Pluto’s rock/water mass ratio falls in the range 2.4-3.6. While there is
some uncertainty in the parameters, these calculations are commensurate with
the rock-rich composition of Pluto, and organic-rich refractories may be
invoked for its constituent building blocks. Both Pluto and intermediate-sized
KBOs support this conclusion.

##  Appendix B Mission Stardust inability to provide bulk carbon content in
comet dust

In the main text we introduced several arguments supporting a large fraction
of carbon-rich organics in cometary dust. Here we add a note about why the
Stardust mission cannot be used in this context, despite being the only comet
sample return mission to date.

The Stardust spacecraft rendezvoused with comet 81P/Wild 2 at a distance of
1.86 AU, and using an aerogel collector, a low-density highly porous silica-
based material, captured grains that were released from the nucleus. Encased
within a capsule, the sample returned to Earth for analysis (Brownlee, 2014).

Although stardust has accomplished an unprecedented achievement so far in the
study of comets, and has delivered a wealth of information, it is less
relevant for uncovering the exact fraction and nature of organics on its host
comet, based on the arguments in Clemett _et al._ (2010) and references
therein.

First, the capture in the aerogel during the flyby occurred at a velocity of
around 6 km s-1. The kinetic energy of a particle impacting aerogel at this
speed is dominated by the linear kinetic energy term. For organic grains in
particular, this energy typically exceeds their vaporization energy by an
order of magnitude. Heating must therefore affect (remove or at least alter)
cometary organics to a considerable degree during capture. Additionally, the
aerogel itself incorporates organic contaminates originating in its
manufacture. Since they are impossible to remove completely, this presents a
problem in the sample analysis. Finally, the grains which are collected by
Stardust spent an estimated 20 minutes to several hours in the coma prior to
collection, which is ample time for an organic grain, or at least a
significant fraction of it, to dissociate by interacting with stellar photons.

Since the heat of capture is most severe at the entry site into the aerogel, a
few Stardust organic particles which piggy-backed behind terminal tracks made
by previously collected large particles, were able to survive to be analyzed
(de Gregorio _et al._ , 2011). Nevertheless, our overall judgment based on the
aforementioned arguments is that the Stardust mission cannot reliably provide
an assessment of the bulk organic fraction.

##  Appendix C Additional information for models with methane

The data shown in Figures 3 and 5 was used in the main text, depicting some
statistical properties related to models where CH4 was used as a second ice.
Here we expand on some additional properties of the random models.

In Figure C1 we show that the ice/rock mass ratio is larger than the
water/rock mass ratio in Figure 3. This is however trivial, since the
numerator (ice) now includes an additional icy material (methane) and is
therefore larger than water alone. The limit ice/rock mass ratio of Uranus and
Neptune is between 0.5-1 (or rock/ice between 1-2, equivalently).

---
(a) Uranus, MOI factor 0.22

---
(b) Uranus, MOI factor 0.23

---
(c) Neptune, MOI factor 0.23

---
(d) Neptune, MOI factor 0.24

Figure C1: Mass fraction of H+He vs. ice/rock mass ratio (logarithmic). Each
point denotes a random interior model, color coded by its central temperature.
The dashed vertical line indicates equal fractions of water and rock. The
right (left) dotted vertical line indicates a ice/rock mass ratio of 2 (0.5).

Figure C2 shows the iron/rock mass ratio on the y-axis and the water/rock mass
ratio on the logarithmic x-axis. We truncate the water/rock mass ratio at a
value of 10. Here we test the feasibility of random compositions in terms of
their iron fraction. In order to judge if the iron fraction is not too high,
we compare the iron/rock mass ratio of each model to that of the known ratio
in solar system planets. Two horizontal dash-dotted lines are drawn. The
bottom line refers to the ratio on Earth, and the top line to Mercury
(Reynolds and Summers, 1969). Mercury is considered an upper limit since it
probably has the highest iron fraction among the planets, perhaps due to a
mantle-striping impact in its history (Benz _et al._ , 1988). Models above the
top line are therefore viewed as non-physical. We however see that the
majority of models fall below the bottom line or between the bottom and top
lines, in broad agreement with expectation, noting that to our knowledge there
is no firm lower bound to the expected iron content.

---
(a) Uranus, MOI factor 0.22

---
(b) Uranus, MOI factor 0.23

---
(c) Neptune, MOI factor 0.23

---
(d) Neptune, MOI factor 0.24

Figure C2: iron/rock mass ratio vs. water/rock mass ratio (logarithmic). Each
point denotes a random interior model, color coded by its central temperature.
The dashed vertical line indicates equal fractions of water and rock. The
right (left) dotted vertical line indicates a water/rock mass ratio of 2
(0.5). The lower (upper) horizontal line indicates the iron/rock mass ratio of
Earth (Mercury).

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