11institutetext:  Institut für Astrophysik, Universität Zürich,
Winterthurerstr. 190, CH-8057 Zürich, Switzerland 22institutetext:
Weltraumforschung und Planetologie, Physikalisches Institut, Universität Bern,
Gesellschaftsstrasse 6, 3012 Bern, Switzerland
22email<EMAIL_ADDRESS>
# Accretion of primordial H-He atmospheres in mini-Neptunes:
the importance of envelope enrichment

M. Mol Lous Accretion of primordial H-He atmospheres in mini-Neptunes: the
importance of envelope enrichmentAccretion of primordial H-He atmospheres in
mini-Neptunes: the importance of envelope enrichmentAccretion of primordial
H-He atmospheres in mini-Neptunes: the importance of envelope
enrichmentAccretion of primordial H-He atmospheres in mini-Neptunes: the
importance of envelope enrichment C. Mordasini Accretion of primordial H-He
atmospheres in mini-Neptunes: the importance of envelope enrichmentAccretion
of primordial H-He atmospheres in mini-Neptunes: the importance of envelope
enrichment R. Helled Accretion of primordial H-He atmospheres in mini-
Neptunes: the importance of envelope enrichmentAccretion of primordial H-He
atmospheres in mini-Neptunes: the importance of envelope enrichment

(Received 12/20/2023; accepted 02/15/2024)

###### Abstract

Context.  Out of the more than 5,000 detected exoplanets a considerable number
belongs to a category called ’mini-Neptunes’. Interior models of these planets
suggest that they have some primordial, H-He dominated atmosphere. As this
type of planet does not occur in the solar system, understanding their
formation is a key challenge in planet formation theory. Unfortunately,
quantifying how much H-He planets have, based on their observed mass and
radius, is impossible due to the degeneracy of interior models.

Aims. Another approach to estimate the range of possible primordial envelope
masses is to use formation theory. As different assumptions in planet
formation can heavily influence the nebular gas accretion rate of small
planets, it is unclear how large the envelope of a protoplanet should be. We
explore the effects that different assumptions on planet formation have on the
nebular gas accretion rate, particularly by exploring the way in which solid
material interacts with the envelope. This allows us to estimate the range of
possible post-formation primordial envelopes. Thereby we demonstrate the
importance of envelope enrichment on the initial primordial envelope which can
be used in evolution models.

Methods. We apply formation models that include different solid accretion rate
prescriptions. Our assumption is that mini-Neptunes form beyond the ice-line
and migrate inward after formation, thus we form planets in-situ at 3 and 5
au. We consider that the envelope can be enriched by the accreted solids in
the form of water. We study how different assumptions and parameters influence
the ratio between the planet’s total mass and the fraction of primordial gas.

Results. The primordial envelope fractions for small- and intermediate-mass
planets (total mass below 15 M⊕) can range from 0.1% to 50%. Envelope
enrichment can lead to higher primordial mass fractions. We find that the
solid accretion rate timescale has the largest influence on the primordial
envelope size.

Conclusions. Primordial gas accretion rates onto small planets can span many
orders of magnitude. Planet formation models need to use a self-consistent gas
accretion prescription.

###### Key Words.:

Planets and satellites: formation – Planets and satellites: atmospheres –
Planets and satellites: composition

##  1 Introduction

Currently, more than 5,000 exoplanets have been detected. Many of these
planets have sizes larger than Earth but smaller than Neptune (Howard et al.,
2012; Fressin et al., 2013; Fulton et al., 2017), and are commonly referred to
as mini-Neptunes. Despite the degeneracy in exoplanetary characterization,
interior models indicate that mini-Neptunes consist of non-negligible hydrogen
and helium (H-He) envelopes (Weiss & Marcy, 2014; Rogers, 2015; Wolfgang &
Lopez, 2015; Jin & Mordasini, 2018; Otegi et al., 2020a; Bean et al., 2021).
These H-He envelopes are thought to be accreted from the protoplanetary disk
during the planetary growth. These atmosphere are then retained despite
evolutionary atmosphere loss processes such as photo-evaporation, and
therefore can be considered as primordial envelopes. Constraining the initial
mass of primordial envelopes of intermediate-mass exoplanets is a key
objective in exoplanet science. For example, constraining the initial mass of
the envelopes could provide a solution to the conundrum of the ’radius
valley’, which is the lack of observed planets with radii between 1.5 R⊕ and 2
R⊕ (Fulton et al., 2017). In addition, planets with primordial envelopes could
be habitable (e.g. Madhusudhan et al., 2021; Mol Lous et al., 2022), but one
of the major concerns is that a planet must accrete a specific amount of a
primordial envelope.
Calculating the primordial envelope mass for a given exoplanet is extremely
challenging. The prevailing exoplanet measuring techniques only yield radii
and masses, through transit measurements and radial velocity detection,
respectively. Solving the interior composition of a planet knowing only the
mean density and irradiation temperature is a highly degenerate problem (Dorn
et al., 2015; Shah et al., 2021; Haldemann et al., 2023). Additionally there
are large errors in the measurements of radii and masses of exoplanets as
these are derived in relation to stellar radii and masses, of which the values
are not always well constrained (Otegi et al., 2020b).
It is likewise difficult to constrain the size of primordial envelopes from
planet formation models. The standard model for planet formation is core
accretion (Mizuno, 1980; Pollack et al., 1996; Alibert et al., 2005; Helled et
al., 2014). In this scenario, planet formation begins with a solid (heavy-
element) core and once this reaches $\sim$0.1 M⊕ the planet starts to accrete
a gaseous envelope. Often, planet formation models predict larger (i.e., more
massive) envelopes than the ones inferred for the observed planetary
population (e.g., Rogers & Owen, 2021). There are several possible
explanations, including the large uncertainty in the opacities of planetary
envelopes (Ormel, 2014; Mordasini, 2014), underestimating the role of
collisions in atmosphere removal (Denman et al., 2020) or a boil-off phase
during disk dispersal (Rogers et al., 2023). Interestingly, three-dimensional
models which include gas-exchange with the surrounding disk predict smaller
accreted envelopes than one-dimensional models at an orbital distance of 0.1
AU around a sun-like star (Ormel et al., 2015; Cimerman et al., 2017;
Moldenhauer et al., 2021) and it is still unknown whether this inefficiency
remains significant at further radial distances such as 3 or 5 au.
Another physical mechanisms that can greatly alter the accretion of primordial
gas in (1D) formation models are the solid-envelope interactions in the
planetary envelope during the planetary growth. As they grow, protoplanets can
accrete solid material and gas simultaneously. Solid material, in the form of
planetesimals or pebbles, travels through the envelope and can fragment or
ablate. This can enrich the envelope in heavy elements instead of simply being
added to the core (Pollack et al., 1986; Podolak et al., 1988). This process
is sometimes referred to as envelope pollution. This heavy-element enrichment
can have two competing consequences on the planetary growth timescale and the
planetary composition. On the one hand, enrichment can increase the envelope’s
opacity, and therefore delay the planetary contraction and inhibiting the
further accretion of nebular gas. On the other hand, heavy-element enrichment
increases the mean molecular weight of the envelope, which enhances the gas
accretion rate. A schematic overview of envelope accretion with and without
the consideration of solid-envelope interactions is shown in Figure 1.
Many previous studies have already demonstrated that envelope enrichment plays
an important role in planet formation. Specifically pebbles are quick to
ablate and fragment (Ormel & Klahr, 2010; Lambrechts et al., 2014; Alibert,
2017; Chambers, 2017; Brouwers et al., 2018; Valletta & Helled, 2019; Brouwers
& Ormel, 2020), but planetesimals have been shown to interact with the
envelope and alter the formation process as well (Stevenson, 1982; Hori &
Ikoma, 2011; Pinhas et al., 2016). Estimates of the maximum core mass that a
planet can grow range from $\sim$ 0.1 M⊕ to $\sim$ 5 M⊕ (Pollack et al., 1986;
Mordasini et al., 2006; Lozovsky et al., 2017; Alibert, 2017; Brouwers et al.,
2018; Steinmeyer et al., 2023). This range is a result of the different
assumptions on planetesimal or pebble sizes, composition, and material
strength.
Valletta & Helled (2020) simulated the formation of Jupiter and Saturn,
accounting for envelope enrichment where the heavy elements were represented
by water. It was found that including envelope enrichment in a self-consistent
way (equation of state and opacity calculation) decreases the growth timescale
of Jupiter and Saturn. This result is in line with previous studies focusing
on giant planet formation (Stevenson, 1982; Hori & Ikoma, 2011; Venturini et
al., 2015, 2016; Venturini & Helled, 2017), but it is not entirely clear if
this can be accepted as a general result. It remains a possibility that in
some cases the planet cannot cool efficiently enough to trigger runaway gas
accretion (see Figure 1). For example, Wang et al. (2023) showed that if icy
pebbles sublimate outside of the accretion radius and enrich the local gas,
this decreases the nebular gas accretion efficiency. Furthermore assumptions
on mixing efficiency, the composition of the accreted solid material and the
strength of the grain opacity can steer the outcome of a one-dimensional
planet formation simulation.
The objective of this work is to investigate how the accretion rates of gas
depend on the model assumptions when envelope enrichment is considered. We
follow Valletta & Helled (2020) and employ a 1-dimensional planet formation
model that considers the ablation and fragmentation of the solid material
(represented by water ice). We focus on the investigation of the envelope’s
composition of the forming planets before they reach runaway gas accretion. We
consider various formation locations, protoplanetary disk properties, and
solid accretion rates. We also investigate the formation timescales to assess
whether the planet is expected to reach the runaway gas accretion and become a
gas giant planet.
Our paper is organized as follows. In Section 2 we present our model setup. In
Section 3 we present our results for the gas accretion rates for different
planets. We distinguish between gas accretion with and without the enrichment
of solid materials. The distribution of possible H-He envelope masses within
the explored parameter space is given. In this section we also demonstrate the
importance of basic assumptions, such as the mixing of supercritical water
with H-He, on our results. In Section 4 we further test assumptions on mixing
and the smoothing of the deposition profile. In this section we also address
the likelihood that planets form with the required amount of H-He to allow for
surface liquid water. The limitations to our model are discussed in Section 5.
Finally in Section 6 we summarize our findings.

Figure 1: A schematic overview of pre-runaway envelope accretion. When the
solid material does not interact with the envelope the gas accretion is
initially determined by the size of the core and the strength of the accretion
luminosity (Phase I). The core stops growing when there is no more solid
material available, so that the envelope accretion rate is determined by the
cooling timescale of the protoplanet (Phase II). If Phase II is efficient the
planet can reach the critical mass within the lifetime of the protoplanetary
disk. It will go into runaway accretion and become a gas giant. Gas accretion
from the nebula is different when the interaction of the solid material with
the envelope is considered. Part of the ice and/or silicates will vaporize
rather than reaching the core in solid form. This increases the envelope
metallicity. The increased metallicity can on the one hand inhibit further gas
accretion by increased opacities in the envelope, which hinders cooling. On
the other hand the increased mean molecular weight increases the mean density
of the envelope which promotes gas accretion.

##  2 Methods

The formation simulations are based on a modified version of the MESA
code111Version 10108 of MESA is used and compiled by the MESA SDK that was
released simultaneously. (Paxton et al., 2011, 2013, 2015, 2018, 2019) which
was properly adapted to simulate planet formation and evolution (Valletta &
Helled, 2020; Müller et al., 2020a, b). The formation model is similar to the
one used in Valletta & Helled (2020), with some modifications as discussed
below.
The initial model has a core mass of 0.1 M⊕ and an envelope of 10-6 M⊕. The
initial envelope metallicity is 0.03, but drops to zero at the beginning of
the evolution when pure H-He is accreted and envelope enrichment is not yet
significant. Three different solid accretion rate prescriptions are considered
to compute the solid accretion rate $\dot{M_{\text{Z}}}$. Based on
planetesimal accretion we use rapid growth (Pollack et al., 1996) and
oligarchic growth (Fortier et al., 2013). We also simulate pebble accretion
(Lambrechts & Johansen, 2014). A summary of these accretion rates are given in
Appendix A. For planetesimals we assume a radius of 100 km and for the pebbles
one of 10 cm.
In the case of planetesimal accretion the simulation starts at 10 kyr. In the
pebble accretion case we also assume that the solid accretion rate starts at
10 kyr, but with a smaller initial model, namely 0.01 M⊕. Integrating the
solid accretion rate of pebbles (see Appendix A.4) from a mass of 0.01 M⊕ at
10 kyr to 0.1 M⊕ gives us the starting times of our planetary embryo. These
starting times ($t_{0,\textrm{ peb}}$) depend on the disk conditions and are
given in Table 1.
In the nominal case we set the lifetime of the disk to 10 Myr. As solar-like
stars should form within 5 - 10 Myr, this is a long but not unlikely formation
time (Pfalzner et al., 2022). We also consider shorter formation in 3 Myr. We
stop our simulations before runaway gas accretion starts, namely when the
crossover mass is reached, where the envelope and core are of equal mass
(Bodenheimer & Pollack, 1986; Pollack et al., 1996).

###  2.1 Boundary conditions and disk assumptions

The outer boundary conditions of our model planetary envelope
($P_{\text{out}}$ and $T_{\text{out}}$) are set equal to the pressure and
temperature in the disk. Following Piso & Youdin (2014) these are given by
scaling relations in distance:

$\displaystyle P_{\text{out}}=0.27\left(\frac{a}{5.2\text{
AU}}\right)^{-45/14}\text{ Ba},$ (1) $\displaystyle
T_{\text{out}}=150\left(\frac{a}{5.2\text{ AU}}\right)^{-3/7}\text{ K},$ (2)

where $a$ is the orbital distance of the protoplanet. The normalization
factors that we apply are higher than the fudicial MMSN (which would be 0.0085
and 60 for pressure and temperature respectively). While this is a significant
increase, we find that it does not influence the envelope masses when they are
above $\sim$ 0.01 M⊕ and saves computation time. These boundary conditions do
play a significant role in the early stages of the protoplanet and could in
theory alter the formation path, for example through the onset of
fragmentation. Similarly, $P_{\text{out}}$ and $T_{\text{out}}$ are assumed to
remain constant in time in this work. More accurate gas accretion simulations
would thus require an improved disk model, especially for the cases presented
with envelope masses below 0.01 M⊕ after formation. In this work, however,
these simplification suffice to demonstrate the importance of envelope
enrichment on gas accretion.
The planetesimal accretion rates scale linearly with the solid surface
density, $\Sigma_{\text{Z}}$, at the location of formation (see Appendix A.1
and A.2). The initial solid surface density $\Sigma_{\text{Z, 0}}$ is given
by:

$\displaystyle\Sigma_{\text{Z, 0}}=C_{1}\left(\frac{a}{5.2\text{
AU}}\right)^{-1}\text{ g cm}^{-2},$ (3)

The solid surface density decreases as solids gets accreted onto the planet.
The pebble accretion rate has a linear dependency on the gas surface density
at 1 au, $\beta$ (see Appendix A.4). $\beta$ is set by an initial gas surface
density $\beta_{0}$ and decreases exponentially in time ($t$):

$\displaystyle\beta=\beta_{0}\exp(-t/\tau_{\textrm{disk}}),$ (4)

where $\tau_{\textrm{disk}}$ is the gas disk lifetime which we fix to 3 Myr.
$C_{1}$ and $\beta_{0}$ are used as free parameters to account for disks of
different masses. We set $C_{1}$ to 5, 7.5 or 10 g cm-2 for a light, medium or
heavy disk, respectively. Values of $\beta_{0}$ for these three disk types are
set to 250, 500 or 750 g cm-2. The corresponding values of $\Sigma_{Z,0}$ and
$\Sigma_{g,0}$ at 3 or 5 au are listed in Table 1.
The inner boundary of the envelope model is the core. The luminosity at the
core-envelope interface is determined by the accretion luminosity:

$L_{\text{acc}}=\frac{GM_{\text{c}}\dot{M_{\text{z}}}}{R_{\text{c}}}\left(1-f_{\textrm{abl}}\right),$
(5)

where $G$ is the gravitational constant and $f_{\textrm{abl}}$ is the fraction
of solid material which is ablated or fragmented in the envelope.
$M_{\text{c}}$ and $R_{\text{c}}$ are the core mass and radius respectively,
where the value of $R_{\text{c}}$ is calculated assuming a constant core
density of 3.2 g cm-3, regardless of the composition of the accreted material.
The significance of this simplification is considered in Appendix C.

|  |  | Rapid accretion
---
Oligarchic accretion
Pebble accretion
|  |  $\Sigma_{\text{Z, 0}}$ (g / cm-2) |  $\Sigma_{\text{g, 0}}$ (g cm-2) |  $t_{0,\textrm{ peb}}$ (kyr)
Heavy disk | 3 au | 17.33 | 250 | 34.4
| 5 au | 10.4 | 150 | 38.7
Medium disk | 3 au | 13 | 167 | 49.2
| 5 au | 7.8 | 100 | 56.5
Light disk | 3 au | 8.67 | 83 | 101
| 5 au | 5.2 | 50 | 119
Table 1: The assumed initial solid surface density and initial gas surface
density in the heavy disk, medium and light disk. $\Sigma_{Z,0}$ is used for
the planetesimal accretion, while $\Sigma_{g,0}$ and t${}_{0,\textrm{ peb}}$
are used for the pebble accretion.

###  2.2 Enrichment from Planetesimals or Pebbles

The interaction between the accreted solids and the envelope is considered via
the fragmentation and/or ablation of the solids (planetesimals or pebbles).
The calculation of the value of $f_{\textrm{abl}}$ is given in Appendix B.
This method also gives the deposition profile $m_{\textrm{dep}}(r)$ at radius
$r$. The amount of water vapor added to the envelope is the product of
$f_{\textrm{abl}}$ and the solid accretion rate.
We consider two deposition methods. The first is direct deposit. In this
method the mass is deposited at the radial locations where ablation and
fragmentation occur. For example: if a planetesimal fragments at radius $r$
and there is no prior ablation, the water mass in layer $m(r)$ is enhanced by
the solid accretion rate. This increases the metallicity.
The second method, homogeneous deposit, is the default in this work. It
assumes that the mass deposition is completely smoothed over the envelope
which has total mass M${}_{\textrm{env}}$. This means that the amount of added
heavy material is distributed over all layers, normalized to the layer’s mass,
where:

$m_{\textrm{dep}}(r)=f_{\textrm{abl}}\times\dot{M_{\text{Z}}}\frac{m(r)}{M_{\textrm{env}}}.$
(6)

As an illustration, Figure 2 shows the difference between direct deposit and
homogeneous deposit specifically for a planet growing by oligarchic growth at
5 au after 47 kyr. The core mass is still the initial 0.1 M⊕ and the envelope
mass is 1.5 $\times 10^{-5}$ M⊕. The envelope is too small to cause
fragmentation of the planetesimals and thus there is only ablation. The
fraction of ablated material increases towards the interior of the envelope.
The homogeneous deposit is completely smoothed over all layers, but the total
deposited material adds up to the same as for direct deposit.

Figure 2: The difference between the two deposit models: direct deposit (blue)
and homogeneous deposit (orange). The x-axis gives the normalized mass
coordinate of the envelope $q$. This figure specifically shows the deposit
models for Oligarchic growth at 47 kyr when the envelope mass is 1.5 $\times
10^{-5}$ M⊕. There is no fragmentation yet. The ablation enriches the envelope
metallicity up to 10% in the most inner region. The homogeneous deposit has
the same total deposited mass, but smoothed. However, the actual enrichment is
not homogeneous, as shown by the orange dashed line, due to some layers
already being saturated. For different formation conditions and at different
times the difference between the deposition and the actual enrichment changes.

While at every timestep the deposition of heavy material is done
homogeneously, this does not necessarily mean that the composition in the
envelope is homogeneous. This is for two reasons. First of all because there
is a gas accretion of pure H-He with zero metallicity added to the outer
layers of the envelope. The inner layers, which are older, will have been
exposed to envelope enrichment for longer and thus have a higher metallicity.
This will create a compositional gradient unless the Ledoux criterium is met,
in which case the convective region will become homogeneously mixed. Secondly,
we consider a maximum metallicity in each layer and if this is already reached
there is no enrichment. An example of this is shown in Figure 2. The orange
solid line shows the deposit profile, while the dashed orange line shows the
actual enrichment. The difference is due to some layers in the envelope
already being saturated, or close to saturation, so that it is not possible to
deposit all the mass without condensation. There are two criteria that could
limit the amount of water that can be deposited in a certain layer.
First, we check the material state of H2O based on the temperature of the
layer and from there calculate the maximum number density of water in layer
$r$:

$n_{\text{H}_{2}\text{O,
max}}(r)=\cases{&}\frac{P_{\text{sat}}(r)}{P(r)+P_{\text{sat}}(r)}\;\;\;\;\;\text{
if }T(r)<\text{T}_{\text{crit}}\\\
&Z_{\textrm{max}}\;\;\;\;\;\;\;\;\;\;\;\;\;\text{if
}T(r)>\text{T}_{\text{crit}}{}$ (7)

$P(r)$ and $T(r)$ are the pressure and temperature at radius $r$.
T${}_{\text{crit}}$ is the critical temperature of 647.096 K. In the cases
where the temperature is below 647 Kelvin, we apply the vapor-liquid phase
boundary from Wagner & Pruß (2002) to calculate the saturation pressure of
water $P_{\text{sat}}$ as follows:

$\frac{P_{\text{sat}}(r)}{\text{P}_{\text{crit}}}=\exp(\frac{\text{T}_{\text{crit}}}{T(r)}\left(a_{1}\nu+a_{2}\nu^{1.5}+a_{3}\nu^{3}+a_{4}\nu^{3.5}+a_{5}\nu^{4}+a_{6}\nu^{7.5}\right)).$
(8)

Here $\text{P}_{\text{crit}}$ is the critical pressure, 220.64 bar.
$\nu=\left(1-\frac{T}{\text{T}_{\text{crit}}}\right)$. The other variables are
a1 = -7.85951783, a2 = 1.84408259, a3 = -11.7866497, a4 = 22.6807411, a5 =
-15.9618719, a6 = 1.80122502.
If the temperature exceeds the supercritical temperature of water we impose a
limit to the water enhancement, $Z_{\textrm{max}}$. Supercritical water and
H-He are expected to be highly miscible (Soubiran & Militzer, 2015),
suggesting that $Z_{\textrm{max}}=1$. However in our nominal model we set this
value lower to 0.9 to ensure that we do not artificially create a loss of
H-He. This would happen if too much H-He inside the envelope is replaced by
water without a sufficiently high primordial gas accretion rate. We also want
to investigate the significance of this miscibility and additionally consider
this maximum metallicity to be to 0.5.
The second criterion for water deposition is that the deposited material can
only be as massive as the shell in which it is deposited. As MESA uses mass
coordinates, this criterion ensures that there is no Rayleigh–Taylor
instability created through the deposition. While this an artificial limit, we
argue that the deposited material we calculate in a certain layer using our
one-dimensional model underestimates the smoothing over different layers.
Thus, allowing this smoothing of the water deposition profile should better
represent the three-dimensional structure.
When it is not possible to deposit part of the heavy material in the envelope,
we transfer the leftover water mass to the core.
Thus the enrichment is only equal to the initial deposition if a layer is not
yet saturated, as is demonstrated in Figure 2. In this specific case for the
inner 10% of envelope mass ($q<0.1$) the critical temperature is exceeded, so
that the maximum metallicity is much higher than for $q>0.1$. Furthermore the
outer envelope ($q>0.4$) contains newer gas which has not been exposed to as
much enrichment, hence the enrichment increases towards the outside of the
envelope. If the envelope is not convective a compositional gradient can be
created.
Finally, we define the metallicity of the envelope at location $r$ as:

$Z(r)=\frac{m_{\text{H}_{2}\text{O}}(r)}{m(r)}.$ (9)

The change in the envelope’s metallicity alters the opacities and the
equation-of-state of the envelope. The total envelope’s metallicity is
referred to as Z${}_{\textrm{env}}$ and is defined by the total water mass
fraction in the envelope:

$Z_{\textrm{env}}=\frac{M_{\textrm{env, H}_{2}\text{O}}}{M_{\textrm{env}}}.$
(10)

The opacities are calculated by adding the molecular opacities from Freedman
et al. (2014) and grain opacities from Alexander & Ferguson (1994), following
(Valencia et al., 2013). We use an equation-of-state that mixes water with
hydrogen and helium taken from Müller et al. (2020b) (see their Appendix A for
details).
Finally the heavy element deposition in layer $r$ has two influence on the
energy. First of all accretion luminosity is added by:

$l_{\text{acc}}(r)=\frac{GM(r)m_{\textrm{dep}}(r)}{r}$ (11)

with $M(r)$ the cumulative mass at radius $r$.
Secondly the vaporization of water decreases the energy by (Pollack et al.,
1986):

$E_{\text{vap}}(r)=-(c_{\text{p}}\times\Delta T+E_{0})\times
m_{\textrm{dep}}(r)$ (12)

where $c_{\text{p}}=4.2\times 10^{7}$ (erg g-1 K-1) is the specific heat of
water, $E_{0}=2.8\times 10^{10}$ (erg g-1) is the latent heat of vaporisation
and $\Delta T$ is the change of temperature to reach vaporization, which we
set to 373 K assuming that the incoming pebble/planetesimal has temperature 0
K.

Figure 3: Core and envelope compositions under the different solid-envelope interaction models presented in Table 2. The dashed black line represents the outer boundary to the envelope and the solid black line the inner boundary. Everything interior to the black solid line is considered as the core. The composition of the core and whether this is mixed is not considered in this work. Rather a constant core density of 3.2 g / cm3 is used. When envelope enrichment is considered this can either create a compositional gradient or there can (a) mixed convective zone(s), as shown by the two halves.  |  | Ablation &
---
fragmentation
of water
| Solid
---
composition
| $Z_{\textrm{max}}$
---
Case-1 | No | 100% silicates | n.a.
Case-2 | Yes | 100% water | 0.9
Case-3 | Yes |  | 50% water
---
50% silicates
0.9
Case-4 | Yes | 100% water | 0.5
Table 2: Solid-envelope interaction models considered in this work. All models
assume that silicates do not react with the envelope and directly reach the
core.

In our simulations we distinguish between five types of solid-envelope
interactions which are presented in Table 2. For simplicity, we neglected the
thermal ablation or fragmentation of silicates and focus only on water.
Therefore, Case-1 is a reference case without any solid-envelope interactions.
All solid material directly reaches the core-envelope boundary and the
envelope never increases in metallicity. We consider the other extreme in
Case-2. We assume that all the solid material is ice and can enrich the
envelope.
With Case-1 and Case-2 the most extreme, we use Case-3 and Case-4 to
investigate other aspects related to our fragmentation and ablation model.
Case-3 is a hybrid between Case-1 and Case-2. Half of the solid material are
icy planetesimals/pebbles which can enrich the envelope. The other 50% of the
solid accretion rate consists of rocky material that directly reaches the core
and does not interact. Finally, in Case-4 we limit the maximum allowed
metallicity $Z_{\textrm{max}}$ (see Equation 7) to 0.5 if the temperature
exceeds the critical temperature. Figure 3 visualizes the effects of these
cases on the planet’s interior and envelope.

###  2.3 Gas accretion

Gas accretion can occur at every timestep. Following Valletta & Helled (2019),
gas is added to the planet until the outer radius of the envelope is within a
factor 1.1 smaller or larger than the accretion radius. We use the accretion
radius as in Lissauer et al. (2009). This formulation is based on the common
assumption that the planet’s accretion radius must be equal to the smallest of
either the Bondi radius or the Hill radius:

$\displaystyle
R_{\text{acc}}=\frac{GM_{\text{p}}}{c_{\text{s}}^{2}/k_{1}+GM_{\text{p}}/\left(k_{2}R_{\text{Hill}}\right)},$
(13)

where $M_{\text{p}}$ is the mass of the protoplanet, $c_{\text{s}}$ is the
speed of sound at the location of formation, $R_{\text{Hill}}$ is the
protoplanet’s Hill radius and $k_{1}$ and $k_{2}$ are reduction factors to
account for the limited availability of gas at the formation location of the
planet. For the small protoplanets considered in this study $k_{1}$ and
$k_{2}$ can be set to 1.
The first 10 kyr are used to relax the envelope mass. The initial model does
not automatically satisfy the criterion that the accretion radius equals the
radius of the initial model. How much these two values deviate depends on the
orbital distance. We smooth this transition by finding $k_{1}$ and $k_{2}$
values such that the initial model radius is close to the accretion radius.
Then we increase $k_{1}$ and $k_{2}$ linearly in time until these are both 1
at 20 kyr.

##  3 Results

We perform a grid of simulations with the following variations: the solid
accretion rate is rapid, oligarchic or pebbles. The formation location is
either 3 or 5 au and the disk is either heavy, medium or light as defined in
Table 1. An overview of all these results is given in Appendix D.

###  3.1 Solid-envelope interaction affecting and H-He gas accretion

This subsection highlights the effect of all four solid-envelope interaction
models on individual formation cases.

####  3.1.1 Rapid Growth

Figure 4: In-situ formation of a planet at 3 au with rapid growth. The
simulations are run until the envelope and core are of equal mass or until 10
Myr. The initial solid surface density is 17.33 g cm-2 (heavy disk). The upper
panels shows the mass of the core and the mass of the envelope over time. The
lower panel shows the same simulations as in the upper panel, but only the
envelope masses over time. Solid lines are the mass from primordial H-He. The
dashed lines are the envelope mass from water vapor. In Case-1 there is no
water vapor as only hydrogen and helium are accreted from the nebula.  Figure
5: The same simulations presented in Figure 4 but showing the heavy element
fraction of the envelope (upper panel) and the primordial envelope mass, from
pure H-He (lower panel). Both are shown as a function of the total mass
(M${}_{\textrm{core}}$ \+ M${}_{\textrm{env}}$). The grey dashes lines
indicate where primordial envelope mass fractions would be 0.1%, 1%, and 10%
of the total mass.

Figure 4 shows the in-situ formation of a planet by rapid growth at 3 au. The
initial solid surface density is 17.33 g cm-2 (heavy disk). The upper panel
shows the masses of the core (solid line) and envelope (dashed line) as time
progresses for the various cases. We find that all the cases include both
Phase I and Phase II of gas accretion, where the transition between them
occurs after $\sim$ 105 yrs at a core mass between 4.4 - 5.2 M⊕. For Case-1
and Case-3 there is still a small increase in core mass during Phase II of gas
accretion. At this stage the planet grows through envelope accretion which
extends the planetary feeding zone and provides more solid material that can
be accreted by the growing planet. In Case-2 and Case-4 on the other hand, the
maximum core mass is reached, as any newly accreted planetesimals fragment and
only add water vapor to the envelope. Another distinction is that Case-2 and
Case-4 reach a crossover mass after 3.81 Myr and 2.5 Myr, respectively, while
Case-1 and Case-3 do not reach crossover mass within 10 Myr. This is because
in the former two cases the ablation-fragmentation transition occurs before
the feeding zone is depleted and solid accretion is high. This promotes the
gas accretion for several reasons. First, the total accretion luminosity is
reduced, as a large fraction of the mass is deposited at larger radii and
meanwhile the evaporation of water decreases energy locally. Second, the mean
molecular weight of the envelope increases so that a more massive envelope can
be bound. Similar to previous work we find that the increased opacities due to
an increased envelope metallicity do not counteract the mechanisms promoting
gas accretion. As such envelope enrichment promotes total envelope accretion.
The lower panel of Figure 4 shows the envelope’s growth, where the
contributions of H-He are separated from the water vapor. Since Case-3 has a
low-metallicity envelope, the total H-He mass in the envelope is similar to
that of Case-1. Note that there would be a larger difference between Case-1
and Case-3 if fragmentation occurred before the solid accretion rate
decreases. Case-2 and Case-4 have significantly more massive H-He envelopes at
a given time. After $\sim$ 105 years it remains a factor 3 higher than Case-1
and Case-3. We find that for a short time the water mass in the envelope
exceeds the H-He mass. This occurs during the transition between Phase I and
Phase II. However, since subsequently mostly H-He is accreted, the envelopes
final atmospheric composition is dominated by H-He.
During Phase II accretion we find that small amounts of envelope mass can be
lost. For Case-1 and Case-3 this concerns small oscillations in the envelope
mass which are a result of an oscillating solid accretion rate. These are in
turn due to the changes in capture radius, which depends on the internal
structure of the envelope (see Appendix A.1). In other words, when the gas
accretion rate is large, the capture radius also increases, promoting a higher
solid accretion rate. However, the increase in luminosity from the gas
accretion and the solid accretion expand the envelope and increase the radius
beyond the accretion radius, which leads to mass loss. While the solid
accretion rate remains small (between 10-8 M⊕ yr-1 and 0) this is sufficient
to influence the envelope. Nonetheless, we do find that smoothing the change
in capture radius during Phase II gas accretion (by only allowing it to change
with 0.1% every timestep) eliminates these oscillations without altering the
final outcome. In Case-2 we find a single instance of mass loss at 1.42 Myr.
Similarly to Case-1 and Case-3 this mass loss proceeds from an increase in the
capture radius. However, in this case the increased capture radius is due to a
change in the internal structure of the envelope, as the size of the
convective zone increases.
The top panel of Figure 5 shows the envelope’s metallicity as a function of
the total planetary mass. For all cases the envelope metallicity peaks when
the feeding zone is depleted. The maximum envelope metallicity in Case-2 and
Case-3 peaks at $\sim$0.8. This is expected from the maximum metallicity in
supercritical states, $Z_{\textrm{max}}$, set to 0.9. Colder outer layers
where water can condense have even lower metallicities which decreases the
total envelope metallicity from $Z_{\textrm{max}}$.
The lower panel of Figure 5 shows H-He envelope mass as a function of the
total planetary mass. Grey dashed lines give the reference fractions of
$f_{\textrm{H-He}}$= 0.1%, 1% and 10%, where $f_{\textrm{H-He}}$ =
$M_{\textrm{env, H-He}}$ / $M_{\textrm{p}}$. When the planet is smaller than
$\sim$ 2 M⊕ the primordial envelope masses of all cases are similar. At higher
masses the solid accretion rate increases and fragmentation occurs, so that
the envelope has a significant amount of water vapor which influences the H-He
accretion. At a total mass of 5 M⊕, Case-2 has a factor 2 higher
M${}_{\textrm{env, H-He}}$ than Case-1 and Case-4 has a factor 5 higher than
Case-1. However, at masses above 6 M⊕ the primordial envelope mass of Case-1,
Case-2 and Case-4 converge, as Phase II of gas accretion sets in.
Interestingly, Case-3 has the transition into Phase II of gas accretion for a
lower mass than the other three cases. Compared to Case-1, Case-3 has a lower
core mass, as there is always a fraction between 0 and 0.5 of solid material
evaporating in the envelope. Also compared to Case-2 and Case-4 the maximum
core mass is smaller. This is because Case-2 and Case-4 are more efficient at
enhancing the envelope and they reach a stage where $f_{\textrm{abl}}$ equals
1 before the feeding zone is depleted. As a result, envelope accretion
accelerates and in this larger envelope planetesimals are captured more easily
(i.e. the capture radius as defined in Appendix A.1 increases). Thus, as the
solid accretion rate increases, the envelope becomes saturated with water
vapor which then allows the core to grow more rapidly as well. This
acceleration of core and envelope formation is not evoked in Case-3 because of
the later onset of fragmentation.

####  3.1.2 Oligarchic Growth

Figure 6: Same as Figure 4 but for a planet forming in-situ at 3 AU by
oligarchic growth in a heavy disk.  Figure 7: Same as Figure 5 but for a
planet forming in-situ growth at 3 AU by oligarchic growth in a heavy disk.

Figures 6 and 7 demonstrate the effect of envelope enrichment on the planetary
mass and bulk composition as well as the formation timescale for oligarchic
growth at 3 au. Similar to the previously presented rapid growth, the initial
solid surface density is 17.33 g cm-2, corresponding to a heavy disk.
The upper panel in Figure 6 shows that in Case-2 the core reaches a maximum of
2.7 M⊕. This is notably lower than the core of the planet formed by rapid
planetesimal accretion at the same location in a heavy disk. The reason for
this difference is that the rapid formation has a high solid accretion rate
with a large accretion luminosity. This makes the total envelope mass smaller
for a given core mass, so that complete fragmentation is reached for a higher
core mass in rapid growth. However, the core mass presented in these results
also contain evaporated water that could not be held in the saturated envelope
layers. Further discussion on the impact of our core model assumptions on the
accretion of H-He is presented in Section C. Case-4 has a larger core
accretion rate than any of the other models in the last 3 Myr. This is because
Case-4 has a more massive envelope and thus a larger capture radius.
Furthermore, because Z${}_{\textrm{max}}$ in Case-4 is lower than those in
Case-2 and Case-3, the envelope becomes saturated earlier.
Since oligarchic growth is much slower compared to rapid growth, it takes
longer before the core is massive enough to accrete an envelope with which the
solids will interact. As shown in the lower panel of Figure 6, the envelopes
in Case-2 and Case-3 become water dominated after 2.7 and 1.8 Myr,
respectively, and this compostion persists during the remaining planetary
growth. The H-He mass is unchanged for Case-1, Case-2, Case-3 until 7 Myr,
while Case-4 always has more H-He.
The upper panel in Figure 7 confirms that the envelope metallicity increases
at a smaller core mass for oligarchic growth compared to rapid growth. The
maximum metallicities also occur at smaller core masses than for rapid growth
because there is not enough time to grow larger cores. The lower panel shows
that Case-1 and Case-2 have very similar H-He envelope fractions until Case-2
reaches its maximum core mass. Also Case-3 and Case-4 have similar H-He
envelope mass fractions. Overall, we find that the H-He mass fractions are
larger in oligarchic growth than in rapid growth, since the envelopes for a
given core mass are larger due to the slower formation.

####  3.1.3 Pebbles

Figure 8: In-situ formation of a planet at 5 au by pebbles. The initial disk
conditions are light. The upper left panel shows how the core mass and
envelope grow in time. The lower left panel shows the envelope mass, with the
separated mass contributions of H-He (solid lines) and H2O (dashed lines).
Figure 9: Same simulations as shown in Figure 8. Dashed grey lines show H-He
envelope mass fractions of 0.1, 0.01 and 0.001.

Since pebble accretion is more efficient than planetesimal accretion, we find
that most of our pebble simulations reach crossover mass before 3 Myr even
when using a later starting time than for the planetesimal accretion. Only in
the case of a light disk at 5 au we find planets in a pre-runaway state after
10 Myr. As a result, this is the formation scenario we highlight in Figures 8
and 9. Due to the small size of pebbles, the value of $f_{\textrm{abl}}$
reaches 1 already at the beginning of the simulation. We use the first 3 kyr
of the simulation to smooth $f_{\textrm{abl}}$ from 0 to 1 linearly in time.
Figure 8 shows that Case-1 and Case-3 do not reach a crossover mass while
Case-2 and Case-3 reach it after 7.7 and 9.3 Myr, respectively. In Case-2 the
maximum core mass is 3.9 M⊕ and for Case-4 it is 5.2 M⊕. Case-1 leads to a
core mass of 6.3 M⊕ and an envelope of 0.88 M⊕ after 10 Myr while Case-3 ends
with a 5.2 M⊕ core and an envelope of 2.6 M⊕.
There are instances of mass loss in Case-2 and Case-3. Contrary to the rapid
cases, this is not linked to the coupling between the solid accretion rate and
the envelope structure. Instead, this is due to the small size of the pebbles
and their immediate fragmentation. In combination with our model set-up, which
allows the envelope to be considered ’full’ and adds additional water to the
core, this can cause the value of $f_{\textrm{abl}}$ to drop when the
metallicity is close to saturation. This allows temporary oscillations in the
accretion luminosity and possibly, mass loss. These changes in
$f_{\textrm{abl}}$ are unphysical and should be modeled more self-consistently
in future work. It must be noted, however, that the interaction between icy
pebbles and nebular gas can already enhance metallicities at distances further
away from the protoplanet than where the gas is bound. In Section 5.3 we
discuss this point and argue that this interaction needs to be well understood
before it can be incorporated in one-dimensional models.
The envelope’s metallicity and primordial envelope mass for the pebble cases
are shown in Figure 9. The metallicities peak at masses of 2 - 6 M⊕. We find
that all the enrichment models follow roughly the same relation between H-He
mass fraction and total mass, as shown in the lower figure. In comparison to
the planetesimal accretion models in Figures 5 and 7 however these are less
smooth. This is because the instant ablation of the pebbles.

###  3.2 H-He mass fractions after formation

Figure 10: H-He mass fraction after 3 Myr (upper panels), 5 Myr (middle
panels), and 10 Myr (lower panels). The left panels show in-situ formation at
3 au and the right panels at 5 au. The colors indicate the heavy-element
interaction models. Case-1, Case-2, Case-3, and Case-4 are given in red, blue,
purple and yellow, respectively. The three different solid accretion rates are
distinguished by different symbols. We also show the light initial disk
results by a transparent marker, the medium disk result with a 0.5 opacity
marker, and the heavy disk result with a full opacity marker. The light,
medium and heavy results of the same model are connected by a line, as we
would expect that in intermediate disk would produce a final H-He fraction
approximately along this line. The total masses in the figure are limited to
below 15 M⊕, focusing on the distribution for mini-Neptune type planets.
Planets which reached the crossover mass (M${}_{\textrm{env}}$ =
M${}_{\textrm{core}}$) are not shown, even if their total mass is below 15 M⊕.

Figure 10 summarizes all the nebular H-He mass fractions for the different
formation scenarios presented in this work. This fraction is indicated by
$f_{\textrm{H-He}}$ which is the H-He mass in the envelope divided by the
total mass of the planet. All panels show $f_{\textrm{H-He}}$ as a function of
the total mass of the planet. Only masses below 15 M⊕ are shown to focus on
mini-Neptune planets and because post-runaway gas accretion was not taken into
account in our simulations. Planets which reached the crossover mass or had a
total mass above 15 M⊕ were also removed from the figure. Dashed vertical
lines reference where $f_{\textrm{H-He}}$ has values of 10%, 1% and 0.1%.
The left panels show $f_{\textrm{H-He}}$ when the planet forms at 3 au. All
the planetesimal cases (rapid and oligarchic) remain pre-runaway up to 10 Myr
with the exception of Case-2 and Case-4 of rapid growth in a heavy disk. Rapid
growth leads to larger masses and larger values for $f_{\textrm{H-He}}$ than
oligarchic growth. This difference in composition between the two formation
models is most visible at 3 Myr. If formation times are longer and there is
strong envelope enrichment (Case-2) then oligarchic growth can create planets
with total masses and H-He mass fractions that overlap those of rapid growth.
Overall rapid growth can create planets of masses 2 - 9 M⊕ with H-He envelope
fractions of 0.03 to 0.5 at 3 au. Oligarchic growth creates planets with total
masses of 0.5 to 4 M⊕ with $f_{\textrm{H-He}}$ values of 5$\times 10^{-4}$ to
0.1.
At 5 au there are not as many datapoints for rapid growth as the planets are
likely to reach the crossover mass quickly. None of the heavy disk cases
remain. The planet forming in a medium disk under Case-1 remains pre-runaway
at 5 Myr and for Case-3 this is past 3 Myr. Planets forming by rapid growth
with Case-2 can only do so in a light disk in 3 Myr. The oligarchic cases all
remain pre-runaway and have smaller H-He mass fractions than at 3 au. This is
because the planets forming at 5 au have smaller accretion radii due to a
smaller Bondi radius. At some point the accretion radius becomes dominated by
the Hill radius, which increases with distance. This will lead to planets at 5
au holding more massive envelopes than those at 3 au for the same total
planetary mass. However for the oligarchic growth cases this transition
happens too late to see reflected in H-He mass fractions at 3, 5 or 10 Myr.
Pebble accretion only forms mini-Neptune planets when there is a light disk,
which is assumed to coincide with a late formation time (relative to a heavier
disk). Furthermore at 3 au a mini-Neptune can only form when Case-3 enrichment
applies if formation lasts longer than 5 Myr. In Case-3 the total mass and
$f_{\textrm{H-He}}$ stay within the same region as the planetesimal accretion
models. Within 3 Myr a 13 M⊕ planet can also form by pebbles with so that
$f_{\textrm{H-He}}$=0.1 assuming Case-1 or Case-4. In Case-2 there are no pre-
runaway planets even after 3 Myr.
At 5 au it is easier to form small planets by pebbles, although still
exclusively for the light disk. This is contrary to planetesimal formation
which favours smaller planets at 3 au. After 3 Myr there is not yet a
distinction between any of the enrichment models for pebble formation as they
all lead to a planet of 2.5 M⊕ and a $f_{\textrm{H-He}}$ of 0.004. After 10
Myr only Case-1 and Case-3 have remained pre-runaway.

###  3.3 Envelope metallicities after formation

Figure 11: The maximum envelope metallicity that is reached during planetary
formation (Z${}_{\textrm{env, max}}$) and the time at which this maximum
metallicity occurs. The horizontal lines indicate the imposed maximum
metallicity for supercritical water (Z${}_{\textrm{max}}$ in Table 2) Figure
12: Same data as in Figure 11 but instead of the time, the total mass of the
protoplanet is shown when the maximum envelope metallicity is reached.

In Figure 11 and 12 the maximum envelope metallicities (Z${}_{\textrm{env,
max}}$) are shown for the same set of models as in Figure 10, with the
exception of Case-1 which by definition evolves to a zero metallicity
envelope. Horizontal lines indicate the maximum imposed metallicity for
supercritical layers, Z${}_{\textrm{max}}$.
The rapid growth always has the maximum metallicity occuring before 400 kyr,
which is significantly shorter than the shortest considered formation time of
3 Myr. It is therefore unlikely that rapid growth at 3 or 5 au can create
mini-Neptune planets with very high metallicity envelopes, i.e. envelope
metallicities that are close to Z${}_{\textrm{max}}$.
On the other hand, oligarchic growth has maximum metallicities occuring very
late, between 4.6 and 10 Myr. These correspond to total masses of 0.5 to 3 M⊕.
The oligarchic cases never reach saturation where the envelope metallicity is
that of Z${}_{\textrm{max}}$.
The pebble cases show a wider spread in times when Z${}_{\textrm{env, max}}$
is reached. For most of the pebble cases, we find that the maximum metallicity
occurs before 3 Myr, but could be delayed to 4 Myr or even 7 Myr if the disk
is light.

##  4 Discussion

###  4.1 The effect of mixing and location of deposit

The results presented above assumed a homogeneous composition of the envelope
due to convective mixing at layers where the Ledoux criterium is met. While it
is expected that there is some mixing in protoplanetary envelopes, it is
unclear how efficient mixing is. Simulating the planetary formation with MESA
allows to include mixing via the mixing length theory (mlt). This, however,
requires the knowledge of a dimensionless parameter $\alpha_{\text{mlt}}$ (see
Section 2.4 in Valletta & Helled (2020)). For the formation of Jupiter
Valletta & Helled (2020) adapted $\alpha_{\text{mlt}}=0.1$ (Vazan et al.,
2015; Müller et al., 2020b) and we use the same mixing in our nominal model.
In this Subsection we investigate the effect of mixing by considering a model
in which mixing is inhibited.
The effect of mixing on the protoplanets core and envelope mass and
composition are shown in Figure 13 for the oligarchic growth at 3 au for
Case-2. The initial solid supply is heavy. When mixing is included this
increases the efficiency of the deposit of heavy elements. The mixing
distributes the H-He through the envelope, so that there are more layers where
the maximum metallicity is not met. This allows for a larger overall deposit
of heavy elements. As a consequence the mixing model reaches a point where
$f_{\textrm{abl}}$ reaches 1 after 7 Myr. The accretion luminosity becomes
zero and gas accretion increases. The final H-He envelop mass is an order of
magnitude larger and the core mass is 0.5 M⊕ smaller.

Figure 13: Oligachic growth at 3 au in a heavy disk. The light blue line
indicates Case-2 without mixing and the dark blue line is for the same
conditions but with mixing ($\alpha_{\text{mlt}}=0.1$). When mixing is
included, it distributes the H-He through the envelope. As our method replaces
H-He with water, mixing increases the efficiency of water deposition. As a
result, in the mixing cases there are cases where all the water can be
deposited in the envelope. This reduces the accretion luminosity which leads
to acceleration of the envelope’s growth.

We also investigate the effect of our assumption of a homogeneous deposition
of heavy elements in the envelope. We compare the homogeneous deposit to the
direct deposit as defined in Section 2.2.
Figures 14 shows the differences between these deposit models for Rapid growth
at 3 au, Case-2. The nominal, homogeneous deposit is the same as shown in
Figure 4. In the case of direct deposit, is takes longer for the envelope to
start growing significantly. This is because initially only ablation occurs,
which means that in the direct deposit case there are only heavy elements in
the lower layers. In the homogeneous deposit case, water is added to the outer
layers as well, so that the density increases and that causes fragmentation to
occur more quickly (see Equation 28 and 29). Due to this fragmentation almost
no solids reach the core so that the accretion luminosity from solid accretion
decreases and gas accretion increases. The model with a direct deposit of
solids reaches fragmentation later. Nevertheless, when both models have
reached fragmentation the gas accretion is more efficient in the direct
deposit. In that case the mass can be deposited high in the envelope and
’trickle down’ to the lower layers. The crossover mass is then reached after
1.39 Myr instead of 3.81 Myr.
A realistic deposition profile of heavy elements should lie somewhere in
between the two extreme cases that we considered in this work. Assessing the
physical importance of mixing on planet formation, in combination with
envelope enrichment, would require the following improvements. First, the one-
dimensional deposit profile needs to be smoothed appropriately to account for
the three-dimensional process (Mordasini et al., 2017). In the case of
planetesimal accretion this initial deposit profile would also need to be
improved upon by using a more realistic size distribution (Kaufmann & Alibert,
2023). Second, the treatment of envelope metallicity should be improved. In
this work, the accreted heavy-element mass was added by increasing the
metallicity and changing the energy in the relevant layers. However, mass in
every layer was conserved during enrichment. Future work should treat mass
deposition and envelope enrichment self-consistently by allowing this process
to directly change the mass of the relevant layers.

Figure 14: Rapid growth at 3 au in a heavy disk. The light blue line shows a
homogeneous deposit of heavy elements, which is the default in our model. The
purple line shows how the results change when the heavy elements are directly
deposited at the relevant radial distance in the envelope.

###  4.2 Primordial envelopes and habitability

Planets with a primordial, H-He dominated envelope have received increased
attention as potentially habitable candidates. The collision-induced
absorption of hydrogen can act as a greenhouse effect and thereby create
temperate surface conditions (Stevenson, 1982; Pierrehumbert & Gaidos, 2011;
Madhusudhan et al., 2021; Mol Lous et al., 2022). Madhusudhan et al. (2021)
coined the term ’Hycean planets’, which host liquid water underneath a
hydrogen-dominated atmosphere. It remains uncertain, however, whether any of
the currently observed transiting exoplanets orbit in what can be considered
the ’Hycean habitable zone’. The role of a runaway greenhouse effect
(Pierrehumbert, 2023; Innes et al., 2023) and atmospheric escape (Wordsworth,
2012; Mol Lous et al., 2022) could move the inner habitable zone boundary in
comparison to an Earth-like planet.
Another open question regarding Hycean planets is whether such planets can
accrete the required amount of H-He to enable these temperate surface
conditions in the first place. In Mol Lous et al. (2022) we showed that
planets of sizes 1 - 10 M⊕ that orbit around a sun-like star could host
temperate conditions if they are beyond 2 au. Their primordial, pure H-He
envelope could be of masses 10-4 \- 10-5 M⊕ at this distance, but more massive
when further out. The results presented in Figure 10 show that H-He envelope
below 10-3 are generally difficult to form, and most of the formed planets
consist of much larger H-He mass fractions. The smallest H-He envelopes are
formed by oligarchic growth at 5 au. After 3 Myr years these planets have
values for M${}_{\textrm{env, H-He}}$ ranging from 2.7 $\times 10^{-5}$ M⊕
($f_{\textrm{H-He}}=10^{-4}$ when the total mass is 0.27 M⊕) up to 2.24
$\times 10^{-4}$ M⊕ ($f_{\textrm{H-He}}=4\times 10^{-4}$ when the total mass
is 0.56 M⊕). These planets are smaller than those considered in Mol Lous et
al. (2022), but still massive enough to hold onto a H-He envelope at 5 au
(Mordasini, 2020).
We therefore conclude that the formation of H-He envelopes which provide
temperature surface conditions is probably rare but possible when the planet
forms beyond the ice-line and the formation timescale is long.

##  5 Caveats

###  5.1 Envelope-core interactions and outgassing

Our model does not include interactions between the envelope and the core. We
assume that all the nebular hydrogen and helium remain in the envelope and
that none is sequestered in the core, to be outgassed at later stages. For
(super-)Earth this can play an important role in the development of the
atmosphere after the gas disk has disappeared (Elkins-Tanton & Seager, 2008;
Schaefer & Fegley, 2010). As silicates in the core are expected to be in the
magma phase there should be a high solubility of hydrogen (Hirschmann et al.,
2012). That hydrogen would over time be outgassed and replenish the envelope
(Chachan & Stevenson, 2018), but that would be accompanied with the
atmospheric escape of mostly hydrogen. There could also be a later increase in
atmospheric hydrogen if metal-rich impactors oxidate (Genda et al., 2017).
Thus, some of the H-He mass calculated in this work could be stored in the
core and released gradually.
The envelope-core interactions for H2O, not considered in this work, should
also be mentioned. While we focus on predicting the mass fraction of H-He
after disk accretion, the treatment of water can be improved, which could lead
to different results. Water can also be stored efficiently in a magma ocean
and outgassed later (e.g., Dorn & Lichtenberg, 2021; Bower et al., 2022; Sossi
et al., 2023), which can increase the envelope’s metallicity after formation.

###  5.2 Ablation and fragmentation of silicates

In this work we only modelled the effect of water enhancement on the envelope.
It is clear that rocky material can also ablate and fragment. This is
especially the case for pebbles (Brouwers et al., 2018; Brouwers & Ormel,
2020; Steinmeyer et al., 2023), but also for planetesimals (Bodenheimer et
al., 2018, e.g.). Similar to water enrichment, the enrichment with silicates
on the one hand increases the mean molecular weight and promotes gas accretion
and on the other hand enhances the opacities in the envelope (Ormel, 2014;
Mordasini, 2014; Menou & Zhang, 2023). The enrichment of silicates alone can
create a composition gradient which inhibits convection (Ormel et al., 2021;
Markham et al., 2022). This silicate enrichment has an effect on the long-term
evolution of mini-Neptunes and not considering this effect can lead to
overpredictions of H-He mass fractions in observed planets (Misener &
Schlichting, 2022; Vazan & Ormel, 2023).
Future work should consider the enrichment of both water and silicates in the
envelopes of protoplanets. However, accounting for both species will introduce
more free parameters concerning the mixture of ice and silicates in the solid
accretion rate.

###  5.3 Limitations of a one-dimensional model

The assumption of spherically symmetric gas and solid accretion that comes
with a one-dimensional model does not accurately reflect the reality and thus
there are some limitations. First of all there is the recycling of gas that
occurs in the outer regions of the accreting envelope. The nebular gas from
the disk has a higher entropy than the already accreted gas in the envelope
and will mix (Ormel et al., 2015). This will delay the cooling and
contraction, thus prolonging Phase II of gas accretion. Recycling is an
important aspect to planet formation. It can significantly delay the formation
timescale which can help to explain the presence of super-Earths and mini-
Neptunes, when one-dimensional models would have predicted a transition into
runaway gas accretion. This could also possibly help with the formation of
Uranus and Neptune (Eriksson et al., 2023). Three-dimensional gas accretion
simulations remain computationally expensive. Recently Bailey & Zhu (2023)
found a more optimistic comparison between three- and one-dimensional models.
They suggest that one-dimensional models can improve their accuracy by
reducing the accretion radius to 0.4 times the Bondi radius and considering
two distinct outer recycling layers.
A second limitation revolves around the deposition profile of heavy elements.
In this work we have considered two extremes. In the nominal case we deposited
the heavy elements homogeneously and alternatively we solved the deposition
profile in the one-dimensional case and deposit the solids accordingly. The
latter is realistic if the solid accretion rate is isentropic and the
timescales of the impacts is shorter than the azimuthal mixing (Mordasini et
al., 2017).

###  5.4 In-situ formation

The effect of migration is neglected in this work although in-situ formation
is rather unrealistic. One origin theory of mini-Neptunes is that they form
around the ice-line and migrate inwards once they reach a critical mass
(Kuchner, 2003; Venturini et al., 2020; Huang & Ormel, 2022; Burn et al.,
2024). The precise value of this critical mass remains uncertain (McNally et
al., 2019; Paardekooper et al., 2023) though derivations of it can be found,
such as in Emsenhuber et al. (2023). This formation scenario would naturally
lead to a diversity in mini-Neptunes (Bean et al., 2021).
In the rapid growth case some planets in our simulations deplete their feeding
zone and enter Phase II gas accretion. We predict that if migration is
included this will lead to larger planets. However, our conclusion that
different treatments of solid-envelope interactions can heavily influence the
outcome of planet formation is robust.

##  6 Summary and Conclusions

We simulated planet formation assuming different solid accretion rates, and
calculated the corresponding gas accretion rate self-consistently. The
planetary formation locations were set to be outside of the ice-line, where
the observed mini-Neptunes could have formed before migrating to shorter
orbital distances. Our study clearly shows that the assumptions used by planet
formation models play a key role in determining the planetary growth history
and therefore also the planetary mass and composition. Our key conclusions can
be summarized as follows:

  * •

The assumptions of the interaction between solids and the planetary envelope
strongly affect the planetary growth and can change the primordial gas mass
fraction by up to a factor of 10. Nevertheless, we have also identified cases
where this interaction has no or only little influence on the forming planet.
In the case of oligarchic growth, the envelope remains small and, although it
is metal-rich, it does not alter the rest of the formation process.

  * •

Forming mini-Neptunes at 3 au is challenging with pebble accretion due to the
high accretion rates. Their formation via pebble accretion becomes more likely
at 5 au when the initial disk is relatively light. However, we place more
caveats to our pebble result compared to the planetesimals because (1) pebble
growth strongly depends on the start of the growth, and (2) pebbles could
sublimate before reaching the accretion radius, which might influence the
results.

  * •

The impact of envelope pollution is complex. The most extreme solid-envelope
interaction cases (Case-1 and Case-2) do not automatically lead to the most
extreme outcomes. On the contrary, we find that our Case-4, which considers
half of the solids to interact, does not necessarily lead to planets in
between the more extreme cases. For example, with Rapid growth at 5 au we find
that Case-4 leads to the smallest planets for a given time.

  * •

Envelopes of protoplanets during Phase I and II of gas accretion can be
dominated by heavy elements.

  * •

Our results are consistent with the observed diversity of exoplanets (e.g.,
Jontof-Hutter, 2019). Variations in the formation location, solid material in
the protoplanetary disk, the composition of the solid material, and the
formation timescale determine the final mass and composition of the forming
planets. We find that $f_{\textrm{abl}}$ can span a range of several order-of-
magnitudes and that the envelope’s metallicity can range from 0 to full
saturation. Our results clearly imply that small- to intermediate- mass
planets should be diverse in terms of mass and composition, depending on the
exact formation conditions and growth history.

We find that gas accretion models that include envelope-solids interactions
can significantly influence planet formation even for small planetary masses.
This has important effects on our understanding of the formation of mini-
Neptunes. We note that assuming pebble or planetesimal accretion exclusively
could lead to an overestimation of cases which enter runaway gas accretion.
Kessler & Alibert (2023) showed that giant planet formation can be suppressed
when both pebbles and planetesimals are considered.
Although the topic is still being investigated, it is often assumed that the
observed mini-Neptunes and super-Earths have a large water mass fraction
(Venturini et al., 2020; Luque & Pallé, 2022). This is further supported by
the observation of volatile-rich planets, e.g. in Kepler 138-c and -d (Piaulet
et al., 2023). Furthermore, there are observations indicating the presence of
atmospheric water vapor (e.g., Mikal-Evans et al., 2023). However, detecting
atmospheric water is challenging due to the possible formation of clouds.
Water signatures can also overlap with those of methane (Bézard et al., 2022).
For K2-18b, which became notorious as the first mini-Neptune detected with
water vapor in the atmosphere (Benneke et al., 2019), JWST data have confirmed
that the measured signature was due to methane, and not water (Madhusudhan et
al., 2023). For highly-radiated planets JWST should be able to better
constrain the volatile abundances (Acuña et al., 2023; Piette et al., 2023).
If future observations can confirm that mini-Neptunes and super-Earths have
water-rich atmosphere, it would support the idea that they have formed beyond
the ice-line and migrated inward, as we assumed here. Other explanations for
water-rich atmospheres of small planets could be a late volatile delivery
(Elkins-Tanton & Seager, 2008) or in-situ water formation (Kite & Schaefer,
2021).
Our results demonstrate that primordial gas accretion rates are not simple.
Assumptions in the solid-envelope interaction, the solid accretion rate and
formation location can greatly influence the fraction of H-He after formation.
These assumptions as well as aspects not considered in this work (migration,
grain opacities) will need to be included in order to explain the observed
mini-Neptune and super-Earth population.

###### Acknowledgements.

We thank Simon Müller for contributing to the re-writing of the original code
used in Valletta & Helled (2020) and for helpful discussions. We also thank
the referee for their valuable comments. This work has been carried out within
the framework of the National Centre of Competence in Research PlanetS
supported by the Swiss National Science Foundation under grants 51NF40_182901
and 51NF40_205606. The authors acknowledge the financial support of the SNSF.

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##  Appendix A Overview of solid accretion rates

###  A.1 Rapid Growth

Here we give an overview of how the solid accretion rates were calculated. In
the case of rapid growth we calculate the solid accretion rate, $\dot{M}_{Z}$,
from Pollack et al. (1996), Eq. 1:

$\dot{M}_{\text{Z}}=F_{\mathrm{g}}\;\Omega_{\mathrm{k}}\pi
R_{\mathrm{cap}}^{2}\Sigma_{\text{Z}}$ (14)

which uses the gravitational focusing factor $F_{\mathrm{g}}$. This is
calculated following Greenzweig & Lissauer (1992) (analytical approximation
from Appendix B). $\Omega_{\mathrm{k}}$ is the orbital frequency.
$\Sigma_{\text{Z}}$ is the solid surface density, described in A.3).
The capture radius ($R_{\mathrm{cap}}$ or ’enhance radius’) is calculated as
in Inaba & Ikoma (2003), by considering the drag of the already accreted
envelope. Using the assumed radius ($r_{\text{pl}}$) and density
($\rho_{\text{pl}}$) of the planetesimals, we find the capture radius
$R_{\mathrm{c}}$ in the envelope where it holds that:

$r_{\mathrm{pl}}=\frac{3}{2}\frac{\rho\left(R_{\mathrm{c}}\right)}{\rho_{\mathrm{pl}}}R_{\mathrm{Hill}},$
(15)

which also uses the Hill radius of the protoplanet ($R_{\mathrm{Hill}}$). When
there is no sufficient envelope accreted, the capture radius is reduced to the
core radius.

###  A.2 Oligarchic growth

Oligarchic growth is based on the idea that the planetary embryo can already
be massive enough to perturb the planetesimals, increasing the temperature and
reducing the solid accretion (Ida & Makino, 1993). It leads to much longer
formation timescales than rapid growth. We adapt the accretion rate from
Fortier et al. (2007), Eg. 10:

$\dot{M}_{Z}=F\frac{\Sigma}{2h}\pi R_{\text{cap}}^{2}v_{\text{rel}},$ (16)

where $F$ is an efficiency factor that needs to compensate the fact that the
accretion rate is underestimated when the planetesimals are assumed to have
eccentricities and inclinations all equal to the rms. In Greenzweig & Lissauer
(1992) it is estimated to be $\approx 3$.
Following Fortier et al. (2007) we approximate the inclinations and
eccentricities as:

$e\approx 2\dot{i}$ (17)

and find $e$ by Equation 10 in Thommes et al. (2003).

###  A.3 Solid surface density

The initial solid surface density, $\Sigma_{\text{Z, 0}}$, is calculated by
Equation 3. The planet accretes solid materials from its feeding zone. The
extend of this zone reaches
$\sqrt{12+(ea_{\mathrm{orb}}/R_{\mathrm{Hill}})^{2}}R_{\text{Hill}}$ in both
the inner and outer direction, following Pollack et al. (1996). Therefore the
inner and outer radii of the feeding zone are given by:

$R_{\mathrm{in}}=a-\sqrt{12+(ea_{\mathrm{orb}}/R_{\mathrm{Hill}})^{2}}\;R_{\text{Hill}},$
(18)
$R_{\mathrm{out}}=a+\sqrt{12+(ea_{\mathrm{orb}}/R_{\mathrm{Hill}})^{2}}\;R_{\text{Hill}},$
(19)

where $e$ is the planetesimal’s eccentricity. In the case of rapid growth this
is calculated by Equation 6 in Pollack et al. (1996).
While the solid surface density is expected to be lower at $R_{\mathrm{in}}$
and higher at $R_{\mathrm{out}}$ compared to $\Sigma_{\text{Z, 0}}$, this will
partly cancel out such that the difference is negligible.
$\Sigma_{\text{Z}}$ does reduce when solid material is accreted onto the
planet. We calculate $\Sigma_{\text{Z}}$ by:

$\Sigma_{\text{Z}}=\Sigma_{\text{Z,
0}}-\frac{M_{\text{acc}}}{\pi(R_{\mathrm{out}}^{2}-R_{\mathrm{in}}^{2})},$
(20)

with $M_{\text{acc}}$ being the already accreted solid material.
Not considered are the scattering of planetesimals by the growing protoplanet
(Zhou et al., 2007; Shiraishi & Ida, 2008) which would decrease the amount of
solid material.

###  A.4 Pebble Accretion

The solid accretion rate of pebbles is adapted from Lambrechts & Johansen
(2014), Eq. 31. It is given by:

$\displaystyle\dot{M}_{\mathrm{Z}}\approx$ $\displaystyle 4.8\times
10^{-6}\left(\frac{M_{\text{c}}}{\mathrm{M}_{\oplus}}\right)^{2/3}\left(\frac{Z_{0}}{0.01}\right)^{25/18}\left(\frac{M_{*}}{\mathrm{M}_{\odot}}\right)^{-11/36}\left(\frac{\beta}{500\mathrm{\leavevmode\nobreak\
g}\mathrm{\leavevmode\nobreak\ cm}^{-2}}\right)$ (21)
$\displaystyle\times\left(\frac{r}{10\mathrm{au}}\right)^{-5/12}\left(\frac{t}{10^{6}\mathrm{yr}}\right)^{-5/18}\left(\frac{\dot{M}_{\mathrm{Z,3D}}}{\dot{M}_{\mathrm{Z,2D}}}\right)\mathrm{M}_{\mathrm{E}}\mathrm{yr}^{-1},$

where $Z_{0}$ is the disk’s metallicity and it is set to 0.010270. $\beta$ is
given in Equation 4 and $\tau_{\mathrm{disk}}=3\times
10^{6}\mathrm{\leavevmode\nobreak\ yrs}$.
Furthermore there is a multiplication with a 3D factor from Venturini & Helled
(2017) which slightly reduces the mass accretion rate. This 3D factor is:

$\frac{\dot{M}_{\mathrm{Z,3D}}}{\dot{M}_{\mathrm{Z,2D}}}=\left(\frac{\pi\left(\tau_{f}/0.1\right)^{1/3}R_{\text{Hill}}}{2\sqrt{2\pi}H_{\mathrm{peb}}}\right),$
(22)

with $H_{\mathrm{peb}}=H_{\mathrm{gas}}\sqrt{\alpha/\tau_{\mathrm{f}}}$.
$\alpha$ is the disk’s viscosity which is set to $\alpha=10^{-4}$.
$H_{\mathrm{gas}}$ is the scale-height of the gas disk
$H_{\mathrm{gas}}=c_{\text{s}}/\Omega$. This is approximated as $(h=H/r)$ with
$h(r)=h_{0}(r/AU)^{1/4}$. In standard case $h_{0}=0.036$. See Venturini &
Helled (2017) Section 2.2 for this. Finally, in Lambrechts & Johansen (2014)
Equation 20:

$\tau_{\mathrm{f}}\approx\frac{\sqrt{3}}{8}\frac{\epsilon_{\mathrm{p}}}{\eta}\frac{\Sigma_{\mathrm{p}}}{\Sigma_{\mathrm{g}}},$
(23)

where we use $\epsilon_{\text{p}}$=0.5 and
$\eta=0.0015\times\left(a/1\text{au}\right)$ Pebble accretion stops when the
pebble isolation mass (M${}_{\textrm{iso}}$) is reached. Based on (Lambrechts
et al., 2014) we use that:

$M_{\textrm{iso}}=20\left(\frac{a}{5\textrm{
au}}\right)^{\frac{3}{4}}\,M_{\oplus}.$ (24)

It is believed that the pebble accretion abruptly halts once the isolation
mass is reached. In this work, however, we smooth the transition by applying a
reduction factor to the pebble accretion rate. This reduction factor is 1
until the total mass is above half the isolation mass, and linearly reduces in
mass, so that it is 0 when the total mass equals the pebble isolation mass.

##  Appendix B Fraction of ablation

The fraction of ablating or fragmenting material ($f_{\textrm{abl}}$) is
calculated every timestep and changes as the size and composition of the
envelope change. We follow the semi-analytical approached derived in Valletta
& Helled (2019), which was based on the mass loss and motion of a planetary
envelope derived in Podolak et al. (1988). The interacting solid material is
assumed to be water. Therefore, the density of the planetesimals or pebbles is
set to be $\rho_{\textrm{pl}}=1\,\textrm{g}\,\textrm{cm}^{-3}$. This is
similar to previous assumptions (Venturini et al., 2016; Valletta & Helled,
2020) that are justified by observation of comets which indicate that water is
the mayor volatile specie in the solar-system. Of course in reality, the
accreted material is unlikely to be made of pure-water. Below we refer to the
impactors as planetesimals although the same method is applied for pebbles.
We integrate the planetesimals’ position, mass, and velocity from the outer
boundary of the envelope model to the inner boundary, or until all the
planetesimal mass is ablated or fragmented. The assumed initial velocity and
location are $v_{\textrm{init}}=$10 km sec-1 and
$r_{\textrm{init}}=2R_{\textrm{pl}}$, respectively, with $R_{\textrm{pl}}$
being the outer boundary of our model. The planetesimal’s velocity at location
$r$ inside the envelope is given by:

$v_{\textrm{pl}}^{2}=v_{\textrm{init}}^{2}+2GM_{\text{p}}\left(\frac{1}{r}-\frac{1}{r_{\textrm{init}}}\right).$
(25)

We neglect the envelope’s mass beyond the location $r$ since the mass is
negligible compared to the core mass and lower part of the envelope. The mass
loss at location $r$ is calculated using Equation 14 in Valletta & Helled
(2019):

$\frac{\text{d}m_{\text{pl}}}{\text{d}r}=-\frac{A}{Q}\left(C_{\text{h}}\frac{\rho(r)v_{\textrm{pl}}^{2}}{2}+\epsilon\sigma
T(r)^{4}\frac{1}{v_{\textrm{pl}}}\right),$ (26)

where $m_{\text{pl}}$ is the mass of the planetesimal, $A$ is the area of the
planetesimal which naturally decreases as the planetesimal loses mass and
$\rho(r)$ is the density at $r$. $C_{\text{h}}$ and $\epsilon$ are efficiency
factors for which the appropriate values are uncertain. $C_{\text{h}}$ is the
fraction of kinetic energy transferred to the planetesimal. $\epsilon$ is the
product of the emissivity of the gas and the planetesimal’s impact
coefficient. We set both to 0.01. In Valletta & Helled (2019), $C_{\text{h}}$
and $\epsilon$ were left as free parameters and while their value affects the
planetary growth, its effect is smaller in comparison to other assumptions
considered in this work, such as the solid accretion rate and envelope mixing.
Unlike in Valletta & Helled (2019), here we simplify the calculation of the
planetesimal’s trajectory by assuming that it moves straight to the core. In
other words, we assume an impact parameter of 0 and no angular contributions
to the velocity. $Q$ is the latent heat caused upon vaporization and is given
by:

$Q=C_{\text{p}}\times T_{\text{f}}+E_{0}.$ (27)

Here $C_{p}$ is the specific heat of water in the liquid phase, set to
4.2$\times 10^{7}$ erg g-1 K-1. $E_{0}=2.8\times 10^{10}$ erg g-1 is the
latent heat of vaporisation in the solid phase. The values are taken from
Pollack et al. (1986) (Table 1). $T_{\text{f}}$ is the difference between the
initial temperature and the present temperature, which is 373 Kelvin.
The planetesimal can be completely destroyed when two conditions are met
(Pollack et al., 1986). First, if the pressure gradient in the envelope
surrounding the planetesimal is larger than the material strength:

$\frac{1}{2}\rho(r)v_{\textrm{pl}}^{2}\geq S$ (28)

with $S$ being the strength of the compressive material, set to 1$\times
10^{6}$ Ba (0.1 Mpa) for ice (Pollack et al., 1979). Second, if the
planetesimal is sufficiently small so that its self-gravity can not prevent
fragmentation.

$r_{\textrm{pl}}<\sqrt{\frac{5v_{\textrm{pl}}^{2}\rho(r)}{8\pi
G\rho_{\textrm{pl}}^{2}}}$ (29)

If both these criteria are met, fragmentation occurs and $f_{\textrm{abl}}$ is
set to 1.

##  Appendix C Importance of the core’s mass-radius (M-R) relation

In our current model the changes in the core’s composition are not considered.
All the material that reaches the core, whether that is ice or solids, adds to
the core mass in the same way. Using the total accreted mass and assuming a
constant core density of 3.2 g cm-3 we calculate the core radius and this core
radius sets the lower boundary in our atmosphere model. A more realistic model
would need to infer the core’s radius with an interior model based on the
assumed accreted material. We evaluate the influence of the core density on
our results by using a gradually decreasing or increasing core density.
We do not consider the interaction of rocky material, and for simplicity the
rock fraction in the core follows directly from the solid accretion rate. The
water/ice mass fraction can have two different sources. First, it was directly
accreted, which happens when there is no fragmentation. Second, there could be
water excess since not all the water could be deposited in the envelope. We
also consider this to be part of the core although in reality this should be
added to the envelope mass at the lower layers, creating an ocean.
The upper panel in Figure 15 shows the core’s composition for oligarchic
growth at 3 au with Case-2. From the assumption that in Case-2 there is no
rock in the solid accretion rate, the rock fraction of the core only
decreases. The lower panel shows the core composition for Case-3 which by
definition always has at least 50% accreted rock directly onto the core. The
dark blue represents ice that is directly accreted to the core. The light blue
is also water added to the core, but water that did not fit in the envelope
and was thus moved to lower layers. For both of these water contributions to
the core it is unknown what their thermodynamic properties would be. While the
directly accreted water would reach the core in the solid phase, a subsequent
impact might still vaporize, therefore also adding water vapor to the
envelope. The water which did not fit in the envelope was either because outer
layers were too cold for water vapor (small envelope) and thus condensed. It
is more common that all layers have reached the maximum metallicity for
supercritical water.

Figure 15: Core composition during oligarchic growth at 3 au with a heavy
initial disk. The upper panel shows Case-2, where only water is accreted. As a
result, the rock mass fraction, which is the initial model, decreases. The
lower panel shows Case-3, where 50% water and 50% rock is accreted. As all the
rock directly reaches the core, but the water can evaporate in the envelop,
the core always has a rock mass fraction above 50%. The water which is added
the core can either directly reach the core (dark blue regions) or be added
because the envelope was saturated (light blue).

Figure 16 shows the M-R relation of the core with a constant density of 3.2 g
cm-3. For comparison, we also show the mass-radius relationships of planets
taken from Zeng et al. (2019). These M-R relations are for planets including
their atmospheres and should thus not be directly compared to the M-R of the
core. Regardless we apply another core density subscription based on a pure
rocky planet, as our aim is merely to determine whether the core’s radius can
affect the results of our formation model. We find that scaling the core
density as $\rho_{\text{c}}$ = 3.4 + $2\sqrt{\text{M}_{\text{c}}/M_{\oplus}}$
gives a similar core radius to the 100% rock planet ( see Increasing $\rho$ in
Figure 16).

Figure 16: The orange, light blue, and dark blue lines are mass-radius
relationships taken from (Zeng et al., 2019), which are (a) 100% made of rock
of Earth-like composition (i.e. 32.5% iron and 67.5% MgSiO3) (b) 50% made of
an Earth-like rocky core and 50 or (c) 100% pure H2O. The purple line shows
the M-R relationship for a constant density of 3.2 g cm-3, the default used in
this work. The Increasing $\rho$ model is a fit to the rocky core composition.
The Decreasing $\rho$ is the minimum decrease in core density which affects
the gas accretion rate. Since the gas accretion rate is only altered when the
core’s density is significantly lower than interior models predict, we
conclude that our assumption of a constant core density does not significantly
influence the results.

Changing the core’s density has two competing effects. On one hand, a higher
core density leads to a smaller core radius and this core radius is used as a
lower boundary in the atmosphere model. Meanwhile the accretion radius stays
the same so that the volume which the envelope occupies is slightly larger,
thus the envelope can be more massive. This effect, however, is very small
since the core radius is about two orders of magnitude smaller than the
accretion radius. On the other hand, a higher core density increases the
accretion luminosity, which decreases the amount of gas which can fit within
the accretion radius.
We apply this increasing core density to the formation with oligarchic growth
at 3 au under Case-1, as this is the case that assumes the solid accretion is
of pure-rock. The final envelope is indeed negligibly larger when using the
higher density core, namely 0.082 M⊕ rather than 0.079 M⊕.
We do a similar study for the water-rich core in Case-2 but using an arbitrary
scaling to the core’s density. Rather than trying to simulate the core’s M-R
realistically, we simply decrease the density until we see a significant
effect in our results. This is achieved when we decrease the density as
$\rho_{\text{c}}$ = 3.4 - $2\left({M_{c}/\text{M}_{\oplus}}\right)^{1/3}$. As
shown in the Figure 16 this leads to core radii that are more than 1 R⊕ larger
than pure water planets. Applying such a low core density to the oligarchic
formation at 3 au with Case-2 leads to a core mass of 3.45 M⊕ rather than 2.71
M⊕ and an envelope of 0.3 M⊕ instead of 1.36 M⊕. The primordial envelope mass,
$M_{\text{env, H-He}}$ also decreases from 0.43 M⊕ to 0.07 M⊕. This occurs
because a low density core eventually leads to a slightly smaller envelope and
the fraction of ablation always remains low because there is not enough
envelope to replace as water. However, the nominal core density model with a
slightly larger envelope does reach a point where most of the solids can
enrich the envelope and this causes a great reduction in the accretion
luminosity, promoting gas accretion for the final 3 Myrs.
The core’s mean density that we have applied to a water-dominated core is
notably lower than interior models predict (see Figure 16 and e.g. (Haldemann
et al., 2020)). We therefore conclude that an extremely low core density can
lead to significantly different primordial envelopes. However, we note that
our variations in the core’s density is merely a parameter study and is not
based on realistic interior models. A more realistic core model could be
implemented in future work, where it would also need to be considered that
rock and water would be mixed in this core (e.g., Vazan et al., 2022). At the
same time, it should be noted that the core’s M-R relationship is likely to be
less important in comparison to the effect of the chemical interactions
between the core and envelope (see Subsection 5.1). This would play an
important role in determining how the primordial gas is distributed within the
planet.

##  Appendix D Result tables

Tables 3, 4, 5 give the composition of the (proto)planet when assuming rapid,
oligarchic and pebble growth respectively. The core mass
(M${}_{\textrm{core}}$), core water mass fraction (x${}_{\textrm{water}}$),
envelope mass (M${}_{\textrm{env}}$) and envelope metallicity
(Z${}_{\textrm{env}}$) are given at 3 Myr and 10 Myr. If the crossover mass is
reached before 3 Myr, then the planetary composition at the crossover mass is
given, rather than at 3 Myr, and no data is given for 10 Myr. If the crossover
mass is reached between 3 and 10 Myr, the composition at the crossover mass is
given rather than the composition at 10 Myr. The value of
x${}_{\textrm{water}}$ strongly depends on the assumed solid accretion type,
as discussed in Appendix C.

Rapid growth | After 3 Myr or at M${}_{\textrm{core}}$ = M${}_{\textrm{env}}$ | After 10 Myr or at M${}_{\textrm{core}}$ = M${}_{\textrm{env}}$
---|---|---
3 au | M${}_{\textrm{core}}$ = M${}_{\textrm{env}}$? | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$ | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$
Heavy | Case-1 | No | 5.64 | 0 | 1.04 | 0 | 6.2 | 0 | 2.74 | 0
Case-2 | 3.8 Myr | 4.89 | 0.98 | 4.57 | 0.31 | 4.89 | 0.98 | 4.89 | 0.31
Case-3 | No | 4.58 | 0.41 | 0.98 | 0.14 | 4.83 | 0.39 | 2.49 | 0.1
| Case-4 | 2.5 Myr | 5.15 | 0.98 | 5.15 | 0.22 | n.a. | n.a. | n.a. | n.a.
Medium | Case-1 | No | 3.59 | 0 | 0.27 | 0 | 3.73 | 0 | 0.72 | 0
Case-2 | No | 3.45 | 0.97 | 0.71 | 0.09 | 3.45 | 0.97 | 2.24 | 0.06
Case-3 | No | 3.38 | 0.46 | 0.43 | 0.11 | 3.45 | 0.45 | 0.82 | 0.09
| Case-4 | No | 3.43 | 0.97 | 0.81 | 0.09 | 3.43 | 0.97 | 2.77 | 0.09
Light | Case-1 | No | 1.97 | 0 | 0.09 | 0 | 2.0 | 0 | 0.16 | 0
Case-2 | No | 1.96 | 0.95 | 0.22 | 0.29 | 1.97 | 0.95 | 0.54 | 0.27
Case-3 | No | 1.93 | 0.47 | 0.1 | 0.11 | 1.95 | 0.46 | 0.17 | 0.1
| Case-4 | No | 1.92 | 0.95 | 0.23 | 0.29 | 1.92 | 0.95 | 0.49 | 0.31
5 au |  | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$ | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$
Heavy | Case-1 | 1.98 Myr | 15.5 | 0 | 15.5 | 0 | n.a. | n.a. | n.a. | n.a.
Case-2 | 0.434 Myr | 10.8 | 0.99 | 10.8 | 0.26 | n.a. | n.a. | n.a. | n.a.
Case-3 | 2.83 Myr | 9.38 | 0.3 | 9.38 | 0.1 | n.a. | n.a. | n.a. | n.a.
| Case-4 | 0.50 Myr | 10.9 | 0.99 | 10.9 | 0.27 | n.a. | n.a. | n.a. | n.a.
Medium | Case-1 | 7.06 Myr | 8.18 | 0 | 2.48 | 0 | 10.1 | 0 | 10.1 | 0
Case-2 | 1.15 Myr | 6.82 | 0.99 | 6.82 | 0.28 | n.a. | n.a. | n.a. | n.a.
Case-3 | 4.73 Myr | 7.13 | 0.42 | 3.63 | 0.11 | 7.63 | 0.39 | 7.63 | 0.08
| Case-4 | 1.27 Myr | 7.27 | 0.99 | 7.27 | 0.24 | n.a. | n.a. | n.a. | n.a.
Light | Case-1 | No | 4.18 | 0 | 0.44 | 0 | 4.41 | 0 | 1.03 | 0
Case-2 | 3.43 Myr | 3.95 | 0.97 | 3.06 | 0.27 | 3.95 | 0.97 | 3.95 | 0.25
Case-3 | No | 3.15 | 0.36 | 0.59 | 0.14 | 3.33 | 0.35 | 1.33 | 0.12
| Case-4 | 5.92 Myr | 3.99 | 0.97 | 1.34 | 0.28 | 4.0 | 0.97 | 4.0 | 0.25
Table 3: Properties of planets grown by rapid growth at 3 or 5 au. Heavy, Medium and Light refer to the disk models presented in Table 1. Oligarchic growth | After 3 Myr or at M${}_{\textrm{core}}$ = M${}_{\textrm{env}}$ | After 10 Myr or at M${}_{\textrm{core}}$ = M${}_{\textrm{env}}$
---|---|---
3 au | M${}_{\textrm{core}}$ = M${}_{\textrm{env}}$? | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$ | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$
Heavy | Case-1 | No | 1.02 | 0 | 2.4 $\times 10^{-3}$ | 0 | 3.53 | 0 | 7.9 $\times 10^{-2}$ | 0
Case-2 | No | 1.03 | 0.9 | 5.35 $\times 10^{-3}$ | 0.56 | 2.71 | 0.96 | 1.37 | 0.68
Case-3 | No | 0.86 | 0.36 | 8.5 $\times 10^{-3}$ | 0.7 | 2.32 | 0.31 | 0.16 | 0.76
| Case-4 | No | 1.02 | 0.9 | 5.7 $\times 10^{-3}$ | 0.38 | 3.7 | 0.97 | 0.32 | 0.45
Medium | Case-1 | No | 0.67 | 0 | 9.2 $\times 10^{-4}$ | 0 | 2.20 | 0 | 2.2 $\times 10^{-2}$ | 0
Case-2 | No | 0.67 | 0.85 | 1.8 $\times 10^{-3}$ | 0.47 | 2.13 | 0.95 | 9.1 $\times 10^{-2}$ | 0.78
Case-3 | No | 0.64 | 0.40 | 1.6 $\times 10^{-3}$ | 0.48 | 1.61 | 0.35 | 3.5 $\times 10^{-2}$ | 0.67
| Case-4 | No | 0.71 | 0.86 | 1.5 $\times 10^{-3}$ | 0.25 | 2.2 | 0.95 | 3.3 $\times 10^{-2}$ | 0.31
Light | Case-1 | No | 0.41 | 0 | 2.9 $\times 10^{-4}$ | 0 | 1.17 | 0 | 4.6 $\times 10^{-3}$ | 0
Case-2 | No | 0.41 | 0.75 | 1.1 $\times 10^{-3}$ | 0.69 | 1.15 | 0.91 | 2.5 $\times 10^{-2}$ | 0.78
Case-3 | No | 0.41 | 0.38 | 2.9 $\times 10^{-4}$ | 0.01 | 0.93 | 0.36 | 1.1 $\times 10^{-2}$ | 0.71
| Case-4 | No | 0.41 | 0.76 | 7.3 $\times 10^{-4}$ | 0.4 | 1.12 | 0.91 | 1.6 $\times 10^{-2}$ | 0.47
5 au |  | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$ | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$
Heavy | Case-1 | No | 0.55 | 0 | 1.8 $\times 10^{-4}$ | 0 | 2.98 | 0 | 2.3 $\times 10^{-2}$ | 0
Case-2 | No | 0.55 | 0.82 | 1.0 $\times 10^{-3}$ | 0.77 | 2.89 | 0.97 | 0.13 | 0.82
Case-3 | No | 0.49 | 0.34 | 1.0 $\times 10^{-3}$ | 0.81 | 1.95 | 0.33 | 0.04 | 0.79
| Case-4 | No | 0.56 | 0.82 | 2.2 $\times 10^{-4}$ | 0.14 | 2.97 | 0.97 | 4.5 $\times 10^{-2}$ | 0.39
Medium | Case-1 | No | 0.39 | 0 | 7 $\times 10^{-5}$ | 0 | 1.77 | 0 | 5.8 $\times 10^{-3}$ | 0
Case-2 | No | 0.39 | 0.74 | 2.1 $\times 10^{-4}$ | 0.66 | 1.73 | 0.94 | 4.4 $\times 10^{-2}$ | 0.83
Case-3 | No | 0.37 | 0.33 | 3.7 $\times 10^{-4}$ | 0.78 | 1.23 | 0.32 | 1.4 $\times 10^{-2}$ | 0.81
| Case-4 | No | 0.38 | 0.74 | 7 $\times 10^{-5}$ | 0.06 | 1.77 | 0.94 | 6.4 $\times 10^{3}$ | 0.21
Light | Case-1 | No | 0.26 | 0 | 3 $\times 10^{-5}$ | 0 | 0.91 | 0 | 1.0 $\times 10^{-3}$ | 0
Case-2 | No | 0.26 | 0.62 | 2.9 $\times 10^{-4}$ | 0.83 | 0.91 | 0.89 | 1.0 $\times 10^{-2}$ | 0.85
Case-3 | No | 0.26 | 0.31 | 1.4 $\times 10^{-4}$ | 0.77 | 0.69 | 0.32 | 4.2 $\times 10^{-3}$ | 0.83
| Case-4 | No | 0.27 | 0.62 | 3$\times 10^{-5}$ | 0.16 | 0.91 | 0.89 | 1.9 $\times 10^{-3}$ | 0.33
Table 4: Same as Table 3, but for oligarchic growth. Pebble growth | After 3 Myr or at M${}_{\textrm{core}}$ = M${}_{\textrm{env}}$ | After 10 Myr or at M${}_{\textrm{core}}$ = M${}_{\textrm{env}}$
---|---|---
3 au | M${}_{\textrm{core}}$ = M${}_{\textrm{env}}$? | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$ | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$
Heavy | Case-1 | 0.763 Myr | 12.4 | 0 | 12.4 | 0 | n.a. | n.a. | n.a. | n.a.
Case-2 | 0.686 Myr | 9.4 | 0.99 | 9.4 | 0.17 | n.a. | n.a. | n.a. | n.a.
Case-3 | 0.793 Myr | 11.6 | 0.48 | 11.6 | 0.04 | n.a. | n.a. | n.a. | n.a.
| Case-4 | 0.84 Myr | 11.8 | 0.99 | 11.8 | 0.09 | n.a. | n.a. | n.a. | n.a.
Medium | Case-1 | 1.74 Myr | 15.5 | 0 | 15.5 | 0 | n.a. | n.a. | n.a. | n.a.
Case-2 | 1.68 Myr | 12.8 | 0.99 | 12.8 | 0.15 | n.a. | n.a. | n.a. | n.a.
Case-3 | 1.69 Myr | 11.9 | 0.38 | 11.9 | 0.22 | n.a. | n.a. | n.a. | n.a.
| Case-4 | 1.17 Myr | 10.4 | 0.99 | 10.4 | 0.07 | n.a. | n.a. | n.a. | n.a.
Light | Case-1 | 3.47 Myr | 10.8 | 0 | 1.64 | 0 | 10.9 | 0 | 10.9 | 0
Case-2 | 2.38 Myr | 7.74 | 0.99 | 7.74 | 0.55 | n.a. | n.a. | n.a. | n.a.
Case-3 | No | 1.54 | 0.01 | 0.01 | 0.33 | 3.18 | 0.01 | 0.35 | 0.33
| Case-4 | 3.72 Myr | 9.22 | 0.99 | 3.12 | 0.39 | 9.28 | 0.99 | 9.28 | 0.15
5 au |  | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$ | M${}_{\textrm{core}}$ (M⊕) | x${}_{\textrm{water}}$ | M${}_{\textrm{env}}$ (M⊕) | Z${}_{\textrm{env}}$
Heavy | Case-1 | 0.91 Myr | 16.6 | 0 | 16.6 | 0 | n.a. | n.a. | n.a. | n.a.
Case-2 | 0.96 Myr | 14.8 | 0.99 | 14.8 | 0.14 | n.a. | n.a. | n.a. | n.a.
Case-3 | 0.89 Myr | 15.0 | 0.46 | 15.0 | 0.07 | n.a. | n.a. | n.a. | n.a.
| Case-4 | 0.85 Myr | 15.4 | 0.99 | 15.4 | 0.1 | n.a. | n.a. | n.a. | n.a.
Medium | Case-1 | 1.74 Myr | 15.5 | 0 | 15.5 | 0 | n.a. | n.a. | n.a. | n.a.
Case-2 | 1.68 Myr | 12.8 | 0.99 | 12.8 | 0.15 | n.a. | n.a. | n.a. | n.a.
Case-3 | 1.69 Myr | 11.9 | 0.38 | 11.9 | 0.22 | n.a. | n.a. | n.a. | n.a.
| Case-4 | 1.49 Myr | 10.7 | 0.99 | 10.7 | 0.20 | n.a. | n.a. | n.a. | n.a.
Light | Case-1 | No | 2.54 | 0 | 8.8 $\times 10^{-3}$ | 0 | 6.35 | 0 | 0.87 | 0
Case-2 | 7.69 Myr | 2.49 | 0.96 | 0.06 | 0.82 | 3.99 | 0.97 | 3.99 | 0.46
Case-3 | No | 2.54 | 0.48 | 0.01 | 0.18 | 5.27 | 0.38 | 2.6 | 0.42
| Case-4 | 9.31 Myr | 2.54 | 0.96 | 1.34 $\times 10^{-2}$ | 0.33 | 5.22 | 0.98 | 5.22 | 0.23
Table 5: Same as Table 3, but for pebble growth.