# Chemical diversity in protoplanetary discs and its impact
on the formation history of giant planets

Elenia Pacetti INAF - Istituto di Astrofisica e Planetologia Spaziali (INAF-
IAPS), Via Fosso del Cavaliere 100, I-00133, Rome, Italy  Dipartimento di
Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, I-00185, Rome,
Italy  Diego Turrini INAF - Istituto di Astrofisica e Planetologia Spaziali
(INAF-IAPS), Via Fosso del Cavaliere 100, I-00133, Rome, Italy  INAF -
Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025, Pino
Torinese (TO), Italy  Eugenio Schisano INAF - Istituto di Astrofisica e
Planetologia Spaziali (INAF-IAPS), Via Fosso del Cavaliere 100, I-00133, Rome,
Italy  Sergio Molinari INAF - Istituto di Astrofisica e Planetologia Spaziali
(INAF-IAPS), Via Fosso del Cavaliere 100, I-00133, Rome, Italy  Sergio Fonte
INAF - Istituto di Astrofisica e Planetologia Spaziali (INAF-IAPS), Via Fosso
del Cavaliere 100, I-00133, Rome, Italy  Romolo Politi INAF - Istituto di
Astrofisica e Planetologia Spaziali (INAF-IAPS), Via Fosso del Cavaliere 100,
I-00133, Rome, Italy  Patrick Hennebelle AIM, CEA, CNRS, Université Paris-
Saclay, Université Paris Diderot, Sorbonne Paris Cité, F-91191, Gif-sur-
Yvette, France  Ralf Klessen Universität Heidelberg, Zentrum für Astronomie,
Institut für Theoretische Astrophysik, Albert-Ueberle-Str. 2, D-69120,
Heidelberg, Germany  Universität Heidelberg, Interdisziplinäres Zentrum für
Wissenschaftliches Rechnen, INF 205, D-69120, Heidelberg, Germany  Leonardo
Testi AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot,
Sorbonne Paris Cité, F-91191, Gif-sur-Yvette, France  ESO - European Southern
Observatory, Karl-Schwarzschild-Strasse 2, D-85748, Garching bei München,
Germany  Ugo Lebreuilly AIM, CEA, CNRS, Université Paris-Saclay, Université
Paris Diderot, Sorbonne Paris Cité, F-91191, Gif-sur-Yvette, France

###### Abstract

Giant planets can interact with multiple and chemically diverse environments
in protoplanetary discs while they form and migrate to their final orbits. The
way this interaction affects the accretion of gas and solids shapes the
chemical composition of the planets and of their atmospheres. Here we
investigate the effects of different chemical structures of the host
protoplanetary disc on the planetary composition. We consider both scenarios
of molecular (inheritance from the pre-stellar cloud) and atomic (complete
chemical reset) initial abundances in the disc. We focus on four elemental
tracers of different volatility: C, O, N, and S. We explore the entire
extension of possible formation regions suggested by observations by coupling
the disc chemical scenarios with N-body simulations of forming and migrating
giant planets. The planet formation process produces giant planets with
chemical compositions significantly deviating from that of the host disc. We
find that the C/N, N/O, and S/N ratios follow monotonic trends with the extent
of migration. The C/O ratio shows a more complex behaviour, dependent on the
planet accretion history and on the chemical structure of the formation
environment. The comparison between S/N* and C/N* (where * indicates
normalisation to the stellar value), constrains the relative contribution of
gas and solids to the total metallicity. Giant planets whose metallicity is
dominated by the contribution of the gas are characterised by
N/O*$>$C/O*$>$C/N* and allow to constrain the disc chemical scenario. When the
planetary metallicity is instead dominated by the contribution of the solids
we find that C/N*$>$C/O*$>$N/O*.

##  1 Introduction

The past decade has witnessed a huge growth in our knowledge and comprehension
of exoplanetary systems, paving the way for a more systematic study of the
initial stages of their evolution. Such growth has been achieved thanks to the
improved resolution of modern observational facilities. For instance,
observations performed with ALMA allowed for the first direct detection of
gaps and rings in the gas and dust of protoplanetary discs, which are thought
to be the signature of forming giant planets (e.g., ALMA Partnership et al.,
2015; Isella et al., 2016; Fedele et al., 2017, 2018; Andrews et al., 2018 and
references therein; Long et al., 2018; Pinte et al., 2018; Currie et al.,
2022). Improvements in the characterisation of the chemical structure of discs
have been made as well, allowing for the first direct comparison between the
volatile inventory of extrasolar systems and the Solar System records
(Drozdovskaya et al., 2019; Bianchi et al., 2019; Öberg & Bergin, 2021). Such
new evidence combines with the information provided by population studies of
exoplanets aimed at investigating the architectures and characteristics of the
more than 4000 exoplanets identified to date. The overall emerging picture is
that the characteristics of planets (their occurrence, formation pathway,
orbital architecture, final mass, and composition) are extremely diverse and
uniquely shaped by the physical and chemical properties of the environment in
which they formed (e.g., Madhusudhan, 2019; Zhu & Dong, 2021 and references
therein).

Among the wide variety of worlds discovered so far, giant planets attract
considerable interest due to the role they play in shaping the architectures
of planetary systems and for their influence on terrestrial planet formation
(Raymond et al., 2014; Sotiriadis et al., 2018; Drazkowska et al., 2022). Our
understanding of such planets will soon take a big step forward, thanks to the
observations by next generation telescopes such as JWST and ELT. Moreover,
giant planets will be the main target of the upcoming Ariel space mission
(Tinetti et al., 2018; Turrini et al., 2018; Edwards et al., 2019).

Comparative studies between giant planets in exoplanetary systems and in the
Solar System revealed an unexpected diversity in their orbital architectures.
Specifically, giant planets are found to cover a wide range of orbital radii,
between 0.01 and 100 au111source: _“The Extrasolar Planets Encyclopaedia”_
http://www.exoplanet.eu. Although the detection of giant planets far from the
star does not raise particular concerns, their existence within 0.1 au poses a
great challenge to understanding their formation history. In particular,
neither the core accretion (Pollack et al., 1996), nor the gravitational
instability (Boss et al., 2000) models, allow a giant planet to form in situ
and very close to the star. Therefore, such planets must have migrated inward
from their original formation location to the current one.

What clearly emerges from this result is that the final orbit is simply the
end point of the formation process of a giant planet. As such, its properties
do not put any constraint on the planet’s birthplace and migration history.
Instead, Öberg et al. (2011) suggested that this information is locked into
the final composition of the planet’s atmosphere. Such idea is motivated by
the fact that giant planets are expected to form by accreting gas and
planetesimals from the surrounding protoplanetary disc. In this regard,
migration allows planets to visit regions of the disc with different chemical
compositions, as set by the thermal structure of the disc itself. As the
chemical abundances in the planet’s atmosphere are expected to reflect the
composition of the accreted material, they can effectively be used as a proxy
into the formation and migration pathways of the planet. A detailed
description of how this can be implemented into models of giant planet
formation can be found in Turrini et al. (2021a), which we refer to as Paper I
hereafter.

In Paper I we simulated the formation and migration of a giant planet in a
protoplanetary disc. By focusing on four elemental tracers in the final
atmosphere of the planet, namely Carbon (C), Oxygen (O), Nitrogen (N), and
Sulphur (S), we built a model to constrain the extent of the migration and the
phase of the accreted material. We employed a stationary protoplanetary disc
that inherited the composition from its parent cloud. However, a complete
picture of the physical and chemical structure of protoplanetary discs is
still elusive. One of the still open questions concerns precisely their
initial chemical setup. Specifically, it is unclear whether discs inherit
their composition from the pre-stellar phase (i.e. from the molecular cloud
out of which the host star formed, as assumed in Paper I), in the so called
inheritance scenario, or experience a complete chemical reset as a result of
ionising irradiation from the protostar, known as reset scenario. To date,
neither the inheritance, nor the reset scenario can completely explain the
Solar System record. In fact, comparative studies of meteorites and comets in
the Solar System with protostellar objects in protoplanetary discs provide
evidence for both of them (e.g., Öberg & Bergin, 2021 and references therein).

In this work, we tested the diagnostic power of our model against different
chemical initial conditions for the protoplanetary disc. To further explore
the parameter space, we also examined the implications of different levels of
ionising radiation reaching the disc midplane.

Section 2 provides a description of our dynamical and compositional model. In
Section 3 we present our results in terms of elemental ratios in the disc and
in the planet envelope. Specifically, we discuss how the elemental ratios can
be used to characterise giant planets in terms of their accretion history
(Section 3.3), migration scenario (Section 3.4), and chemical structure of the
birth environment (Section 3.5). Finally, in Section 4, we draw the
conclusions of our work and summarise the applications of our results.

##  2 Numerical and compositional model

Paper I studied the link between giant planet formation and composition by
coupling a description of the native circumstellar disc, assuming chemical
inheritance by the native cloud, with detailed N-body simulations of the
formation and migration of a giant planet in a dynamically evolving disc of
gas and planetesimals. While migrating through different compositional regions
of the disc, the planet accretes gas and planetesimals, whose mixture sets the
composition of the giant planet and of its extended atmosphere. In this work,
we expand the analysis of Paper I by coupling the outcome of its original
simulations with four compositional models for gas and solids in
protoplanetary discs. Details on the adopted disc model and on the physical
processes and dynamical effects modelled by the simulations are provided in
Sections 2.1 and 2.2. Section 2.3 is dedicated to the modelling of the disc
ionisation environment. The four compositional models of the protoplanetary
disc are described in Section 2.4.

###  2.1 Disc model: gas and planetesimals

Table 1: Parameters of the disc model. Disc
---
Stellar mass |  | $1M_{\odot}$
Inner disc radius |  |  $0.1\,$au
Outer disc radius |  |  $500\,$au
Disc mass |  | $0.053M_{\odot}$
Disc temperature |  |  $280\,$K at $1\,$au
Planetesimals
Radius |  |  $50\,$Km
Density: rock-dominated |  | $2.4\,\text{g cm}^{-3}$
Density: ice-dominated |  | $1\,\text{g cm}^{-3}$
Distribution |  |  $1-150$ au

The protoplanetary disc considered in Paper I and in this work is modelled
over the observed disc HD163296 (Isella et al., 2016; Turrini et al., 2019).
The gas surface density and temperature profiles of HD163296 were rescaled to
match the total disc mass and the estimated temperature profile of the Minimum
Mass Solar Nebula (MMSN; Hayashi, 1981). The physical parameters of the
resulting disc are summarised in Table 1. Our disc model assumes radial
profiles of gas surface density and temperature on the midplane that are
constant in time and parametrised as:

$\displaystyle\Sigma_{gas}(r)=\Sigma_{0}\left(\dfrac{r}{165\,\text{au}}\right)^{-0.8}\exp\left[-\left(\dfrac{r}{165\,\text{au}}\right)^{1.2}\right]$
(1) $\displaystyle
T(r)=T_{0}\left(\dfrac{r}{1\,\text{au}}\right)^{-1/2}\text{K},$ (2)

where $\Sigma_{0}=3.3835\,\text{g/cm}^{2}$ and $T_{0}=280\,\text{K}$.

Regarding the solid component of the disc, we assumed that it is distributed
between 1 and 150 au with a surface density profile:

$\Sigma_{solids}(r)=\left\\{\begin{array}[]{ll}2\,Z_{i}(r)\,\Sigma_{gas}(r)&\quad
x\leq 150\,\rm au\\\ 0&\quad x>150\,\rm au,\end{array}\right.$ (3)

where $Z_{i}(r)$ is the mass fraction of condensed material. The factor 2 was
introduced as a concentration factor to account for the inward drift of dust
and pebbles as a consequence of their dynamical coupling with the gas (see
Paper I and references therein for further discussion).

Following the approach of Paper I, we assumed that the bulk of dust in the
disc is rapidly converted into planetesimals over a timescale of 1 Myr, as
suggested by comparisons between the masses of exoplanetary systems and of the
dust and gas in protoplanetary discs (Manara et al., 2018; Mulders et al.,
2021). The assumption is also consistent with recent observations of dust
temporal evolution in discs (see Testi et al., 2022 and Bernabò et al., 2022
for further discussion) and with meteoritic data from the Solar System (Scott,
2007, and references therein).

The conversion of dust into planetesimals reduces the efficiency of the gas-
grain chemistry, hence slowing down the chemical evolution of the disc itself.
When the disc is severely depleted in dust with respect to the initial stages
of its evolution, its chemical composition can be reasonably approximated as
fixed. We then assumed that the composition of our disc evolves until 1 Myr
and remains fixed thereafter (see Section 2.4 for details on the compositional
model). The implications of physically and chemically evolving discs will be
explored in future works.

The planetesimals in our model are all characterised by a fixed radius of
$r_{p}=50$ km (Klahr & Schreiber, 2016) and are divided in two populations,
depending on whether they are found inside or beyond the water snowline.
Planetesimals inside the water snowline are rock-dominated and characterised
by a density of $\rho_{rock}=2.4$ g cm-3, while those beyond the water
snowline are enriched in ices and characterised by a density of $\rho_{ice}=1$
g cm-3.

The radius and density values are used to compute the effects of gas on the
planetesimal dynamics. All the planetesimals evolve dynamically under the
influence of both the forming giant planet and the disc itself. Specifically,
the planetesimals interact gravitationally with the forming giant planet,
whose collisional cross section determines whether planetesimals are accreted
or scattered by planetary encounters. Moreover, at each location in the disc
the planetesimals are subject to two competing forces: the dynamical
excitation due to the disc self-gravity and the damping effect of the gas
drag. The implementation of the disc self-gravity is based on the analytical
treatment for thin discs by Ward (1981), following the approach of Marzari
(2018) and Nagasawa et al. (2019). The effect of the gas drag on the dynamical
evolution of the planetesimals is modelled following the treatment by Brasser
et al. (2007), with the updated drag coefficients from Nagasawa et al. (2019).
See Paper I and references therein for more details.

The planetesimal disc is simulated by means of a set of dynamical tracers
distributed randomly between 1 and 150 au with a uniform probability
distribution and a spatial density of 2000 tracers/au. Each tracer represents
a swarm of planetesimals and it can be associated with a mass of transported
solids. Such mass can be computed by dividing the total solid mass in an
annular region of the disc (computed by integrating Equation 3) by the number
of dynamical tracers it contains. One can then associate the flux of impacting
tracers recorded by the simulations with a mass flux of accreted planetesimals
on the giant planet (See Paper I for details).

###  2.2 Formation and migration model

The N-body simulations were performed with Mercury-Ar$\chi$es (Turrini et al.,
2019, 2021a), a high-performance implementation of the hybrid symplectic
algorithm of the Mercury 6 software from Chambers (1999). Besides the
improvements in numerical stability and computational efficiency with respect
to Mercury, the algorithm of Mercury-Ar$\chi$es allows to simulate the mass
growth, the radius evolution, and the orbital migration at each stage of giant
planet formation, as well as the effects of the disc self-gravity and of the
gas drag. We refer the reader to Paper I and references therein for more
details on the theoretical treatment of these processes and on their numerical
implementation in the simulations. Here we provide a brief summary of the key
points of the planet formation and migration model.

We simulated a giant planet that forms and migrates on the disc midplane
(i.e., on an orbit with inclination $i=0$) over a timescale of 3 Myr, growing
from a planetary embryo to a Jupiter-like giant planet. We adopted a two-phase
approach to model both the mass growth and the evolution of the physical
radius of the planet, following the growth tracks from Lissauer et al. (2009),
Bitsch et al. (2015), and D’Angelo et al. (2021) using the parametric approach
from Turrini et al. (2011, 2019). In the first 2 Myr, the planet accretes its
core and extended atmosphere and grows from an initial mass of
$M_{0}=0.1\,M_{\oplus}$ (mass of the planetary embryo) to a critical mass of
$M_{c}=30\,M_{\oplus}$, equally shared between the core and the atmosphere. In
this phase, the physical radius of the planet grows following the approach
described by Fortier et al. (2013), which is based on the hydrodynamical
simulations by Lissauer et al. (2009). Over the last 1 Myr, the planet
undergoes its runaway gas accretion phase and its mass grows until the final
value of $M_{F}=317.8\,M_{\oplus}$ (equal to 1 Jovian mass) is reached. At the
onset of the runaway gas accretion, the physical radius of the planet starts
to shrink due to the gravitational infall of the gas, reaching a final value
of $R_{I}=1.15\times 10^{5}$ km (equal to 1.6 Jovian radii).

To model the migration of the giant planet we reproduced the realistic
nonisothermal migration tracks from the population synthesis models by
Mordasini et al. (2015) following a piecewise approach based on the analytical
treatments of Hahn & Malhotra (2005) and Walsh et al. (2011).

The simulations assume the protoplanet being initially located in a
convergence point, i.e. a region of zero net torque due to disc–planet
interactions. During the growth of the planetary core from $M_{0}$ to the
critical value $M_{C}$, the protoplanet follows the inward drift of the
convergence point due to the disc evolution. In this first phase, the
protoplanet proceeds trough a linear regime of slow Type I migration that
accounts for $40\%$ of the total radial displacement of the giant planet.

When the critical mass is reached and the runaway gas accretion phase begins,
the protoplanet exits the convergence point and enters a faster regime of Type
I migration. Once the protoplanet becomes massive enough to open a gap in the
disc, it transitions to the slower Type II migration regime. These two last
phases take the form of a power law migration regime that accounts for the
remaining $60\%$ of the total radial displacement of the giant planet.

The simulations model a total of six migration scenarios, with the protoplanet
starting at 5, 12, 19, 50, 100, and 130 au from the star and ending at 0.4 au.
The initial positions were suggested by the observed architectures of giant
planets in the Solar System ($5-10$ au), in exoplanetary systems (from
fractions of au to 20 au), and in circumstellar discs ($100-150$ au).

###  2.3 Ionisation environment of the disc

As part of our exploration of the parameter space, we investigated how the
final composition of the planet changes as we vary the level of chemical
activity in the disc. To this aim, we took advantage of the results by Eistrup
et al. (2016), who modelled the ionisation environment of the protoplanetary
disc as set by two key sources. Specifically, they considered the ionisation
from the decay of short-lived radionuclides (SLRs) and from cosmic rays (CRs).
For both disc chemical setups they analysed (inheritance and reset, see
Section 2.4), they explored a first case of low ionisation level, in which
SLRs are the only source of ionisation, and a second case of high ionisation
level, in which an additional contribution from CRs of external origin is also
taken into account. This results in a total of four disc chemical scenarios
that we used to derive realistic planetary compositions from the outcomes of
the simulations of Paper I. Future works will address the dependency of our
results on a more detailed treatment of the interaction between the disc and
the sources of ionisation (Padovani et al., 2016; Rodgers-Lee et al., 2020).

In the low ionisation scenario, the dominant contribution to ionisation comes
from the decay products of 26Al, 36Cl, and 60Fe, which have half-lives
$t_{\text{half}}$ of 0.74, 0.30, and 2.6 Myr, respectively (Cleeves et al.,
2014). Eistrup et al. (2016) adopted a simplified version of the analytical
prescription given in Equation 30 by Cleeves et al. (2013b) for the ionisation
rate per H2 molecule at the disc midplane:

$\zeta_{\text{SLR}}(r)=(1.25\times
10^{-19}\,\text{s}^{-1})\,\left(\dfrac{\Sigma(r)}{\text{g}\,\text{cm}^{-2}}\right)^{0.27},$
(4)

where $\Sigma(r)$ is the surface density of the disc as a function of the
radius $r$. Equation 4 does not account for the time decay of the ionisation
rate. Therefore, in the SLRs-dominated environment by Eistrup et al. (2016),
the ionisation rate is constant in time and higher in the inner and denser
regions of the midplane.

For the high ionisation scenario, in addition to SLRs, Eistrup et al. (2016)
included a contribution of cosmic rays originating from outside the system.
They parametrically modelled the CRs’ ionisation rate per H2 molecule as:

$\zeta_{\text{CR}}(r)=(1\times
10^{-17}\,\text{s}^{-1})\,\cdot\exp\left(\dfrac{-\Sigma(r)}{96\,\text{g}\,\text{cm}^{-2}}\right),$
(5)

where the characteristic value of $\zeta_{\text{CR}}\sim
10^{-17}\,\text{s}^{-1}$ is the interstellar rate that is typically assumed in
models of disc chemistry (Webber, 1998; Cleeves et al., 2014). The addition of
CRs allows for higher ionisation in the outer disc, where the surface density
is lower and CRs can more efficiently penetrate into the midplane.

The aim of considering the two scenarios is to investigate the impact of
ionisation-driven chemical activity on the chemical structure of the midplane.
Both the prescriptions given by Eqs 4 and 5 should be interpreted as upper
limits to the rates that are expected to regulate the ionisation environment
of a typical midplane. In particular, the model by Eistrup et al. (2016)
includes only attenuation due to the disc’s surface density. Works by Cleeves
et al. (2013a, b, 2014) revealed that CRs’ fluxes at the disc’s surface may be
strongly attenuated by winds and/ or magnetic fields. In particular,
modulation by stellar winds may reduce the CR flux reaching the midplane by
many orders of magnitude
($\zeta_{\text{CR}}\,\lesssim\,10^{-20}\,\text{s}^{-1}$), leaving SLRs as the
dominant midplane ionisation source (Cleeves et al., 2013a). Nevertheless, as
it was mentioned before, the abundances of SLRs do evolve with time. As
pointed out by Cleeves et al. (2013b), the SLRs ionisation rate scales with
the inverse of $t_{\text{half}}$ and long-lived radionuclides (e.g. 40K) only
produce ionisation rates of the order of $\sim 10^{-22}\,\text{s}^{-1}$ or
even less. Moreover, although there is evidence of an enhanced abundance of
SLRs in the early Solar System, little is still known about their actual
contribution to the total ionisation rate in the midplane of a typical disc
(Cleeves et al., 2013b).

Regarding other sources of ionisation, Cleeves et al. (2013a) showed that
photoionisation from stellar and interstellar UV acts largely on C-bearing
species in the upper atmosphere of the disc, leaving the midplane essentially
unaffected. On the other hand, X-ray photons from the star penetrate deeper
and dominate in the intermediate layers of the disc. Only the scattered
component of stellar X-rays is able to reach the midplane (Ercolano &
Glassgold, 2013). However, for a stellar X-ray luminosity of
$L_{X}=10^{29.5}\,\text{erg s}^{-1}$, the scattered X-rays are expected to
produce ionisation rates falling in the range
$\zeta_{\text{CR}}\sim(1-10)\times 10^{-21}\,\text{s}^{-1}$, far below the
rates associated with SLRs and CRs, with minimal impact on the midplane
chemistry (see Cleeves et al., 2014, and Figures 2 and 3 therein). Concerning
X-ray background fields, as in the case of a protoplanetary disc embedded in a
cluster, recent works revealed that these would have relatively little impact
on disc chemistry (Meijerink et al., 2012; Rab et al., 2018). In particular,
for an ordinary disc in a typical low-mass star-forming region, the ionisation
rate produced by a X-ray background flux of $2\times 10^{-4}\text{erg
cm}^{-2}\,\text{s}^{-1}$ would equal the interstellar $\zeta_{\text{CR}}$ only
in the outer disc, at $r\sim 200$ au from the central star (see Rab et al.,
2018, and Figure 10 therein).

###  2.4 Compositional model of the disc

Figure 1: Radial profiles of molecular abundances of the key volatile C-, O-,
and N-bearing species with respect to total H atoms. The profiles are taken
from Eistrup et al. (2016) and properly rescaled to account for the different
disc and the different compositional model considered in this work. In all
panels, solid and dotted lines indicate the solid and gas phases,
respectively. The top three panels describe the two inheritance scenarios,
while the bottom three ones describe the two reset scenarios. The left-hand
and central panels show the abundances of the key volatile molecules carrying
C and O, including the contribution of refractory organic carbon (Cref).
N-bearing molecules are shown in the two right-hand panels for visual ease.
Here, the low (SLRs only) and high (SLRs and CRs) ionisation levels are shown
in the same plot by means of thick and thin lines, respectively. The grey,
dashed, vertical lines in the plots of the inheritance low scenario indicate
the position of the snowlines of H2O (2.5 au), Cref (3 au), NH3 (9.4 au), CO2
(10.5 au), and CH4 (105 au). CO and N2 condense beyond 130 au in our model and
their snowlines are therefore not seen. Snowlines are omitted in the other
scenarios, as the complexity of the chemistry requires the introduction of
multiple snowlines. Figure 2: Normalised elemental abundances of C, O, N, and
S in the disc for the gas (dashed lines) and the solid (solid lines) phases
(the latter including rocks, organics, and ices). The trends are extracted
from the radial profiles of molecular abundances in the four chemical
scenarios. The grey, dashed, vertical lines in the top four panels indicate,
from left to right, the locations of the snowlines of H2O (2.5 au), refractory
organic carbon (3 au), NH3 (9.4 au), CO2 (10.5 au), and CH4 (105 au). Such
snowlines are unequivocally defined only in the inheritance scenario with low
ionisation level and for that reason they are omitted in the other scenarios.
The black, dashed, horizontal line at 1 indicates stellar value in all plots.

To characterise the chemical environment in which the giant planet forms and
migrates, we used an updated version of the compositional model presented in
Paper I. The model quantifies the composition of the disc over its key
components: rocks, organics, ices, and gas. In this work, we introduce a more
realistic abundance profile for the organics and we revise the treatment of
the volatile component to allow for different evolutionary scenarios and
ionisation levels. Here, we outline the key features of the model and we
provide details on its refinements. We refer the reader to Paper I for a
complete discussion.

We assumed a solar composition for both the star and its hosting
protoplanetary disc. The protosolar elemental abundances, which characterise
the original mixture of gas, were taken from Asplund et al. (2009) and Scott
et al. (2015a, b). Overall, they result in mass fractions $X=0.7148$,
$Y=0.2711$, and $Z=0.0141$, of H, He, and heavy elements, respectively. The
gas composition sets the initial conditions for the condensation sequence
across the disc, which regulates the distribution of the elements among the
different phases (gas and solid) and carriers (rocks, organics, gas, and
ices). As in Paper I, we focused on four tracing elements: C, O, N, and S.
Their partitioning between the three components and the solid and gas phases
was derived from Solar System data and recent results from astrochemical
models, as discussed below.

####  2.4.1 Refractory elements and rocks

Based on meteoritic (Lodders, 2010; Palme et al., 2014) and cometary (Le Roy
et al., 2015; Rubin et al., 2019, 2020; Altwegg et al., 2019) data, we assumed
that rock-forming elements are subtracted from the gas phase and locked into
rocks in meteoritic proportion. In this framework, the comparison between
meteoritic (Lodders, 2010; Palme et al., 2014) and protosolar abundances
(Asplund et al., 2009; Scott et al., 2015a, b) reveals that rock-forming
elements account for a mass fraction $Z_{rock}=6.67\times 10^{-3}$ of the
total gas in the disc. By subtracting the meteoritic abundances from the
protosolar ones, we computed the initial abundances of the elements that
remain in the gas phase as volatiles. Their partition across the different
molecular carriers will be discussed in Section 2.4.2.

Focusing on the four elemental tracers considered in this work, the residual
gas in our model contains $51\%$ of the protosolar O and the majority of C
($91\%$) and N ($97\%$). The fact that almost half of the total O is trapped
into rocks while N remains almost entirely in the gas phase, has important
consequences for the final elemental ratios in giant planet atmospheres. One
of them is that the N/O ratio of giant planets that derived most of their
metallicity from the accretion of gas is always super-stellar (see Section 3
for a more detailed discussion of this and other implications). While the gas
remains substantially enriched in C, O, and N, the totality of S is trapped
into solids as chondritic rocks across the whole disc. In this work, we
assumed rocks to be fully incorporated into planetesimals at the onset of our
simulations, which makes S particularly effective in tracing the planetesimal
accretion (see Section 3.4 for further discussion).

####  2.4.2 Volatiles

The partition of volatiles across the different phases and carriers on the
protoplanetary disc is based on the astrochemical models by Eistrup et al.
(2016) after 1 Myr of evolution of their _full_ chemical network. Eistrup et
al. (2016) simulated four scenarios obtained by varying the chemical initial
conditions and the degree of ionisation of the disc. The first two scenarios
describe a protoplanetary disc that inherited its initial composition from the
original molecular cloud. In these _inheritance_ scenarios, the initial
conditions of the chemistry are set by the abundances of H, He, H2, and eight
key volatile molecules (see Eistrup et al., 2016, and Table. 1 therein). As
anticipated in Section 2.3, Eistrup et al. (2016) considered two cases of
chemical activity, linked to _low_ and _high_ ionisation levels. The other two
scenarios describe a protoplanetary disc whose birth environment was heated by
the protostar, with a consequent reset of the chemistry. Eistrup et al. (2016)
focused on the extreme case in which all the molecular gas of the disc is
assumed to be dissociated into atoms by the time it reaches the midplane,
where planet formation occurs. Therefore, in these _reset_ scenarios, the
chemical network is evolved starting from atomic initial abundances of H, He,
C, O, N, and S, plus molecules of H2. Also in this case, both the _low_ and
_high_ ionisation levels were analysed.

To model the distribution of C, O, and N across their molecular carriers in
the four scenarios discussed above, we employed the radial abundance profiles
by Eistrup et al. (2016) of the main volatile molecules in gas and ice phases,
returned by their full chemical network after 1 Myr of evolution. It is
important to notice that Eistrup et al. (2016) report total initial abundances
that differ from the protosolar ones adopted in this work. Moreover, they do
not take into account the fractions of C, O, and N that condense as rocks. To
build a self-consistent model, we calibrated the radial abundance profiles by
Eistrup et al. (2016) to the adopted protosolar elemental abundances, while
preserving the abundance ratios of the C-, O-, and N-bearing molecules in the
chemical network. Details on the procedure are provided below.

The rescaled molecular abundance profiles used in this work are plotted in
Figure 1 for both the inheritance and the reset scenarios. The most noticeable
effect of varying the level of chemical activity is that the snowlines become
less unequivocally defined, as multiple snowlines for the same molecule can
exist at different locations on the disc. The O2 carries a significant
fraction of O across all scenarios, except for the inheritance low scenario,
where the abundance of O2 drops below $10^{-7}$. Among the C-bearing species,
the dominant contribution comes from organic compounds, whose origin is
discussed below and in more details in Section 2.4.3. The N2 is the main
carrier of N in all scenarios. As its snowline is located beyond 130 au, for
the adopted disc thermal profile, a large fraction of N remains in the gas
phase across most of the disc. The fraction of N that condense as ice is
carried by NH3 in the inheritance scenarios and by NH3, HCN and NO in the
reset ones.

For the two inheritance scenarios, we followed the approach described in Paper
I. Specifically, we scaled up the total abundances of N, NH3, and N2 from
Eistrup et al. (2016) by a common factor of 1.15, to ensure the conservation
of the total mass of N (set by the protosolar abundances). Regarding C and O,
we scaled down the abundance of O by a factor of 0.57 to account for the
fraction of O sequestered into rocks, which was not considered in Eistrup et
al. (2016). The same factor was then used to scale the abundance of C, as the
two elements share a joint chemical network. The C not accounted for by the
network ($\sim 60\%$ of the total protosolar abundance) is associated with
organic compounds, consistently with cometary and interstellar medium (ISM)
data (Bergin et al., 2015). Their condensation behaviour is described in
Section 2.4.3. More details on the balancing process and on what motivated the
adopted procedure can be found in Paper I.

When considering the reset scenarios, two additional N-bearing species, HCN
and NO, reach considerable abundances in most of the disc extension (see
Figure 1). In this case, the formation processes of N-, O-, and C-bearing
species can no longer be treated independently of each other. However,
evidence suggests that both HCN and NO form more rapidly than NH3 and N2.
Specifically, HCN and NO form quickly through gas phase reactions, while the
formation of NH3 is governed by the comparatively slower gas-grain chemistry
processes (Eistrup et al., 2016; Schwarz & Bergin, 2014). The formation
timescale of N2 is intermediate between these two cases (Eistrup et al., 2016;
Schwarz & Bergin, 2014). Therefore, the difference between the formation
timescales of HCN and NO with respect to NH3 and N2 allows to consider the two
networks as separate. We then assumed that the fraction of atomic N that
reacts with C and O to form HCN and NO does not affect the abundances of NH3
and N2 after 1 Myr of chemical evolution. In our model, this translates into
using different scaling factors for the involved species. Since the formation
of HCN and NO is driven by the availability of C and O, we scaled them down by
the same factor applied to O- and C-bearing species, this factor being 0.57.
The leftover N was then used to compute the scaling factor of N2 and NH3 at
each location on the disc.

The focus of our compositional model is on the distribution of the four
tracing elements, C, O, N, and S across the gas and solid phases in the disc,
not on the distribution of the different species they are carried by. When it
comes to volatiles, the relevant quantity to consider is therefore the
abundance of each tracing element, computed as the sum of the abundances of
all the molecular carriers of that specific element. The partition of the
molecular abundance profiles of Figure 1 among C, O, N, and S, is shown in
Figure 2 for both the inheritance and the reset scenarios. The dashed lines
indicate the contribution from the gas phase, while the solid lines indicate
that of solids. Note that the solid phase includes also the contribution from
the rocks (see Section 2.4.1), in addition to the ices and the organics. The
elemental abundances are normalised to their respective stellar value,
indicated by the black, dashed, horizontal line. We adopt the superscript * in
the notation as a general convention to distinguish the normalised abundances
from the absolute ones. In this scale, S is at the stellar level throughout
the disc, being entirely locked in rocks. For the other elements, the trends
reflect the phase changes of the disc material across the multiple snowlines.
The major differences among the four scenarios are in the gaseous O, which
shows higher abundances in the reset scenarios with respect to the inheritance
ones. Such differences have significant impact on the planetary elemental
ratios that are computed from O. For instance, for planets that derived their
metallicity from the accretion of gas in the reset scenarios, the large
fraction of O locked in the gas phase translates into sub-stellar C/O ratios.
See Section 3.3 for further discussion of this and other effects.

####  2.4.3 Refractory Organics

Following Cridland et al. (2019) and previous authors (e.g. Thiabaud et al.,
2014; Bergin et al., 2015; Mordasini et al., 2016), we included in our model
the third C-bearing component in addition to rocks and volatiles suggested by
the comparison between the carbon deficit observed in the Earth (Allègre et
al., 2001) and in meteorites (Wasson & Kallemeyn, 1988; Bergin et al., 2015)
in the Solar System with respect to the ISM and comets (Bergin et al., 2015
and references therein; Bardyn et al., 2017). Such component accounts for
$\sim 60\%$ of the ISM carbon, yet it is unclear whether this carbon is in the
form of graphite, amorphous carbon grains or organics. We adopted the
terminology by Thiabaud et al. (2014) and Isnard et al. (2019), and we refer
to this C reservoir as refractory organic carbon.

For the bulk composition of the inner Solar System to be markedly carbon poor,
refractory organic carbon needs to be destroyed in the inner Solar Nebula. In
particular, following Lee et al. (2010), we assumed that it is completely
destroyed and released into the gas phase within 5 au. Following Mordasini et
al. (2016) and Cridland et al. (2019), we introduced a snowline for refractory
organic carbon with 50$\%$ condensation at 3 au, in both the inheritance and
reset scenarios. The resulting abundance profile for the condensed phase is
zero inside 1 au, then increases linearly up to 5 au, reaching a maximum value
set by the balancing process (see Section 2.4.2) and remains steady
thereafter. The distribution of refractory organic carbon across the disc is
shown Figure 1 by means of light-blue curves.

##  3 Results and Discussion

As introduced in Section 2, in order to explore the implications of the disc
chemistry on planet formation and migration, we coupled the outcome of N-body
simulations from Paper I with the four compositional models of the disc. The
simulations describe a planetary embryo that starts its growth and migration
in different compositional regions of the disc, specifically at 5, 12, 19, 50,
100, and 130 au from the central star, always reaching its final semimajor
axis at 0.4 au. In particular, they trace the accreted masses of both gas and
planetesimals at each location along the migration pathway.

For each simulation and compositional model we proceeded as follows. We first
used the radial abundance profiles of the main molecular carriers of C, O, N,
and S (Figure 1) and the relevant atomic weights to extract the individual
mass contributions of the four tracing elements from the accreted masses of
gas and solids. Under the assumption of a homogeneous mixing of the planetary
envelope (see Paper I for discussion), for each element we computed the total
accreted mass of gas and solids by adding the contributions from each location
along the formation pathway. We then converted the total accreted masses into
total abundances with respect to H atoms. The results were used to compute the
elemental ratios C/O, C/N, N/O, and S/N, in the final atmosphere of the
planet, for both the gas and solid phases. As discussed in Paper I, when
considering both contributions from the accreted gas and solids, the giant
planet is _solid-enriched_. In this case, the planet envelope is characterised
by super-stellar abundances of most if not all the elements. On the other
hand, when considering only the accretion of gas, the giant planet is _gas-
dominated_. For the adopted compositional model of the disc, _gas-dominated_
giant planets are characterised by sub-stellar abundances (see Paper I and
Turrini et al., 2021b for further discussion of alternative scenarios).

The final step in our analysis was to compute the atmospheric elemental ratios
normalised to their stellar counterparts. In this scale, values of the ratios
equal to 1 are reflective of a giant planet whose chemical composition matches
that of its host star. In this sense, normalisation provides a tool to measure
the deviation of the planet’s composition from the stellar one. The results
can then be generalised to systems in which the composition of the host star
differs from the solar one considered in this work. To differentiate between
absolute and normalised ratios, we refer to the latter as C/O*, C/N*, N/O*,
and S/N*.

The results of our analysis are plotted for each of the four scenarios in
Figures 3, 4, 5, and 6, while their trends for each elemental ratios are
compared in Figure 7. Qualitatively, the elemental ratios appear to be
modestly affected by the chemical structure of the protoplanetary disc at the
onset of planet formation. Moreover, the joint use of different elemental
ratios proves to be the key to accurately characterise the formation of giant
planets in terms of:

  * •

the accretion history, i.e. whether it was dominated by the gas or the
planetesimals,

  * •

the extent of the migration,

  * •

the chemical initial conditions of the protoplanetary disc (only for gas-
dominated giant planets).

We will delve into each of these aspects in Sections 3.3, 3.4, and 3.5. Before
doing that, however, it is worth discussing some important concepts to have in
mind when interpreting the information delivered by Figures 3, 4, 5, and 6.

Figure 3: Normalised elemental ratios in the final atmosphere of the planet
(top two panels) and in the disc (bottom four panels) in the inheritance
scenario with low ionisation. The top two panels show results for the six
migration scenarios, corresponding to initial semimajor axes of 5, 12, 19, 50,
100, and 130 au. The increasing size of the markers maps the increasing
distance travelled by the planet in the six migration scenarios. Linking lines
are shown only for visual ease. The plot on the left (with empty markers)
describes a gas-dominated giant planet, while the plot on the right (filled
markers) is for solid-enriched ones. The bottom four panels show the trends of
the elemental ratios on the disc for the gas (dashed lines) and the solid
(solid lines) phases. The grey, dashed, vertical lines, from left to right,
indicate the position of the snowlines of H2O (2.5 au), refractory organic
carbon (3 au), NH3 (9.4 au), CO2 (10.5 au), and CH4 (105 au). The black,
dashed, horizontal line at 1 indicates stellar value in all plots. Figure 4:
Normalised elemental ratios in the final atmosphere of the planet (top two
panels) and in the disc (bottom two panels) in the inheritance scenario with
high ionisation. The top two panels show results for the six migration
scenarios, corresponding to initial semimajor axes of 5, 12, 19, 50, 100, and
130 au. The increasing size of the markers maps the increasing distance
travelled by the planet in the six migration scenarios. Linking lines are
shown only for visual ease. The plot on the left (with empty markers)
describes a gas-dominated giant planet, while the plot on the right (filled
markers) is for solid-enriched ones. The bottom four panels show the trends of
the elemental ratios on the disc for the gas (dashed lines) and the solid
(solid lines) phases. The black, dashed, horizontal line at 1 indicates
stellar value in all plots. Figure 5: Normalised elemental ratios in the final
atmosphere of the planet (top two panels) and in the disc (bottom four panels)
in the reset scenario with low ionisation. The top two panels show results for
the six migration scenarios, corresponding to initial semimajor axes of 5, 12,
19, 50, 100, and 130 au. The increasing size of the markers maps the
increasing distance travelled by the planet in the six migration scenarios.
Linking lines are shown only for visual ease. The plot on the left (with empty
markers) describes a gas-dominated giant planet, while the plot on the right
(filled markers) is for solid-enriched ones. The bottom four panels show the
trends of the elemental ratios on the disc for the gas (dashed lines) and the
solid (solid lines) phases. The black, dashed, horizontal line at 1 indicates
stellar value in all plots. Figure 6: Normalised elemental ratios in the final
atmosphere of the planet (top two panels) and in the disc (bottom four panels)
in the reset scenario with high ionisation. The top two panels show results
for the six migration scenarios, corresponding to initial semimajor axes of 5,
12, 19, 50, 100, and 130 au. The increasing size of the markers maps the
increasing distance travelled by the planet in the six migration scenarios.
Linking lines are shown only for visual ease. The plot on the left (with empty
markers) describes a gas-dominated giant planet, while the plot on the right
(filled markers) is for solid-enriched ones. The bottom four panels show the
trends of the elemental ratios on the disc for the gas (dashed lines) and the
solid (solid lines) phases. The black, dashed, horizontal line at 1 indicates
stellar value in all plots. Figure 7: Elemental ratios C/O*, C/N*, N/O*, and
S/N* normalised to their respective stellar values in the four chemical
scenarios. The data are the same as in the top two panels of Figures 3, 4, 5,
and 6, here being plotted together to be more easily compared with each other.
The blue points refer to the inheritance scenarios, while the red ones refer
to the reset scenarios. Specifically, the light-blue octagons and the light-
red hexagons indicate low ionisation level, while the dark-blue, downward
triangles and the dark-red, upward triangles indicate high ionisation level.
The results are plotted for the six migration scenarios, corresponding to
initial semimajor axes of 5, 12, 19, 50, 100, and 130 au. The increasing size
of the markers maps the increasing distance travelled by the planet in the six
migration scenarios. Trends for both the accretion scenarios are shown: filled
markers for solid-enriched and empty markers for gas-dominated giant planets.
The black, dashed, horizontal line at 1 indicates the reference value for
giant planets whose chemical composition matches that of their host star.

###  3.1 Linking planetary atmospheres to disc structures

The final composition of giant planets is generally expected to reflect the
chemical structure of the protoplanetary disc. We can assess the degree to
which this assumption is reasonable by comparing the elemental ratios in the
two environments. The results of such analysis is illustrated in Figures 3, 4,
5, and 6. Each figure examines one of the four chemical scenarios. The top two
panels show the normalised elemental ratios in the atmosphere of gas-dominated
(on the left) and solid-enriched (on the right) giant planets at the end of
the six simulations. The lines connecting the different scenarios are provided
as visual aid only. The bottom four panels show the radial profiles of the
normalised elemental ratios in the disc, as computed from their counterparts
in Figure 2. Such profiles can be seen as snapshots of the midplane chemical
structure, where solid and dashed lines characterise the solid and gas phase,
respectively.

Overall, these four figures show that the atmosphere of the giant planet
inherits some of the chemical properties of the disc. Specifically, gas-
dominated giant planets follow the same global trends as the disc gas phase:
an elemental ratio that is sub- or super-stellar in the disc will generally
remain so in the planet as well. The same happens with solid-enriched planets,
which globally follow the trends of the disc solid phase.

Despite such disc-planet connection being confirmed in all scenarios, the
results also show that it is not a universal rule. In particular, a careful
examination of the plots reveals that for some migration scenarios the
normalised elemental ratios in the planet and in the disc do not immediately
relate. For instance, although the S/N* is markedly super-stellar throughout
the disc and increases towards the star, its value in the solid-enriched giant
planet that starts forming at 5 au drops to sub-stellar level. The final
composition of the planet envelope depends crucially on how efficiently the
gas and the solids are accreted from each region of the disc along the
migration pathway. For gas-dominated giant planets, the accretion of gas
reaches its maximum during the runaway gas accretion phase (see Section 2.2).
The final atmosphere of the planet will therefore be strongly influenced by
the chemical properties of the region of the disc traversed by the planet
during this phase. In solid-enriched giant planets the accretion of gas and
solids are coupled, but their relative contribution to the planetary
metallicity depends on the extent of the migration. In particular, the larger
the migration the larger the number of planetesimals encountered by the planet
and the larger the contribution of planetesimal accretion to the final
metallicity. As the planet migrates for shorter distances, it encounters and
accretes less planetesimals and the contributions to its metallicity by gas
and solids become comparable. As a consequence, the elemental ratios of solid-
enriched giant planets follow the global trends of the disc solid component
only in large-scale migration scenarios, while diverging from them for limited
migrations.

The decreasing trend of the planetary S/N* ratio from large- to short-scale
migrations can therefore be explained considering how S and N are distributed
on the disc and how they are accreted on the planet. In our disc, S is
entirely locked in the solid phase, while N is almost entirely in the gas
phase as N2, slightly deviating from the stellar value beyond the NH3 snowline
(see Figure 2). As a direct consequence of this partitioning, in the planet
envelope the total S scales linearly with the accreted planetesimals, while
the total N is accreted in stellar proportion almost independently on the
radial migration. Consequently, even though the S/N* ratio of solids in the
inner disc is about 40 times higher than the stellar value, for short-scale
migrations the contribution of the accreted N becomes large enough to bring
the S/N* ratio in the planet to sub-stellar value. Similar arguments can be
used to explain why the N/O*, C/O*, and C/N* ratios of solid-enriched giant
planets approaches stellar values for short-scale migrations.

Our results emphasise how the composition of giant planets and their
atmospheres can be connected to that of their native discs only by coupling
the disc chemical structure with the growth and migration tracks that
characterise planet formation. With this in mind, we can proceed with the
discussion of our results and their implications for planet formation.

###  3.2 The diagnostic power of the C/O ratio

Figure 7 compares the normalised elemental ratios of solid- and gas-dominated
giant planets in the inheritance (blue points) and reset (red points)
scenarios. The results are shown for each of the six simulations, identified
by the location of the planet at the onset of its migration. The C/O* ratio
shows almost flat trends, revealing a rather limited diagnostic power.

For short-scale migrations, the C/O* ratios in all scenarios vary between 1
and 2 times the stellar value. The limited variations between the different
disc chemistry and planet migration scenarios are too small to be
unequivocally resolved with the current accuracy of retrieval methods, which
is of the order of $20\%$ (see Barstow et al. 2020 and Paper I for further
discussion). Therefore, giant planets that formed close to the star in a disc
that inherited its composition from the pre-stellar core cannot be
observationally distinguished from those that formed in a disc that
experienced a complete reset of the chemistry, nor can we resolve whether they
are gas-dominated or solid-enriched.

The degeneracy is partially broken for large-scale migrations, where the C/O*
ratio of solid-enriched and gas-dominated giant planets follows distinct
trends. Specifically, solid-enriched giant planets are characterised by
stellar values of the C/O* ratio, independently on how far the planet started
its migration and on the chemical structure of the birth environment. On the
contrary, gas-dominated giant planets are characterised by a C/O* ratio
significantly deviating from the stellar value. In this case, the C/O* ratio
provides constraints on both the migration and the disc chemical scenario.
Detailed discussion on each of these individual aspects is provided in
Sections 3.3, 3.4, 3.5.

###  3.3 Constraints on the accretion history

As anticipated in Section 3.2, the C/O* ratio provides information on the
accretion history of giant planets only for large-scale migration scenarios.
In this case, stellar values are associated with solid-enriched giant planets,
whereas marked sub- and super-stellar values provide indication of accretion
dominated by the gas. Specifically, sub-stellar C/O* ratios are associated
with gas-dominated giant planets in the reset scenarios, whereas super-stellar
values characterise gas-dominated giant planets in the inheritance scenarios.

To understand the origin of such trends we refer to the disc composition, as
described by the molecular abundance profiles in Figure 1. Beyond 10 au, the
gas phase in the reset scenarios is characterised by a lower abundance of CH4
and higher abundances of CO and O2 with respect to the inheritance scenarios.
Because both CO and O2 typically condense at very low temperatures, they act
as reservoirs of gaseous O in most of the disc extension. Consequently, the
abundance of total O* in the gas phase is higher in the reset scenarios than
in the inheritance ones. In particular, beyond 10 au, the abundance of O*
exceeds that of C*. Such behaviour is shown in Figure 2 and is the reason why
the C/O* ratio of the disc gas phase drops below the stellar value in the
reset scenarios (see Figures 5 and 6). As higher abundances in the gas
correspond to lower abundances in the solids, the opposite trend is observed
for the C/O* ratios of the disc solid phase. Moving from the disc to the
planet, we recall that the final elemental ratios in the planet envelope
depend on the mass accretion rate (see the discussion in Section 3.1). Such
rate is a non-linear function of the orbital distance, and it is highly
influenced by the position on the disc where the runaway gas accretion occurs.
As discussed in Section 2.2, in our model the runaway gas accretion phase
begins once the first 40$\%$ of the total radial displacement of the planet
has been covered. For migrations starting beyond 20 au, in our simulations,
this means that the region traversed by the planet during the runaway gas
accretion phase falls between 10 au and 80 au. Therefore, gas-dominated giant
planets in the reset scenarios accrete most of their gas from the disc region
in which the C/O* ratio is sub-stellar, which is why their final C/O* ratio is
also sub-stellar. When it comes to solid-enriched giant planets, the
contribution of planetesimal accretion to the envelope metallicity increases
with the extent of migration. For large-scale migrations, the metallicity is
dominated by the planetesimals accreted from the disc region in which the C/O*
ratio of the solids is super-stellar, which is why the final C/O* ratio in the
planet envelope is also super-stellar.

We highlight that in the inheritance scenario with high ionisation level, the
C/O* ratio of gas-dominated giant planets slightly decreases with increasing
length of migration. For the accuracy of the current retrieval methods (see
Section 3.2), this introduces a degeneracy on the planet accretion history.
Specifically, for very large-scale migration scenarios, gas-dominated giant
planets in the inheritance high scenario would be indistinguishable from
solid-enriched ones. Such decreasing trend in the inheritance high scenario
originates from the enhanced abundance of O* between 50 au and 100 au in our
disc model (see Figure 2), which in turn results from the peak abundance of O2
in the outer disc (see Figure 1). In this region, the abundance of O* exceeds
that of C*, hence bringing the disc C/O* ratio to sub-stellar values (see
Figure 4). Gas-dominated giant planets that accrete most of their gas from
this region will therefore be characterised by lower C/O* ratios than in the
inheritance low scenario.

The information provided by the C/O* ratio on the planet accretion history can
find independent confirmation in the total metallicity of the planet envelope.
Gas-dominated giant planets are characterised by sub-stellar envelope
metallicities that decrease with migration, as highlighted also in Paper I.
Solid-enriched giant planets are instead characterised by the opposite trend,
i.e., super-stellar envelope metallicities that increase with migration. Note
that here the relevant quantity is the planetary metallicity normalised to
that of the host star, which is reflective of the original composition of the
formation environment. The choice of the right reference frame is key to a
correct interpretation of observational data. The C/O ratio and metallicity of
the Sun are widely used, either explicitly or implicitly, as reference in
interpreting the composition of exoplanetary atmospheres (see e.g. Madhusudhan
(2019) for a recent review). However, both the C/O ratio and the metallicity
of stars in planetary systems can significantly deviate from their respective
values in the Sun (Delgado Mena et al., 2010; Mulders, 2018 and references
therein). Therefore, to avoid introducing biases, one has to look at the
planetary elemental ratios and metallicity in the reference frame of the host
star. For instance, a planetary C/O ratio estimated to be 0.55 (equal to the
solar C/O ratio) would be stellar only if the planet formed around a solar
type star. If, instead, the planet formed around a star characterised by a
sub-solar C/O ratio (e.g. 0.45), the planetary C/O ratio should be correctly
interpreted as super-stellar. Our methodology has been recently applied by
Guilluy et al. (submitted) and Carleo et al. (submitted) for the
interpretation of observational data from the GAPS 2 program.

It is important to notice that when the C/O* ratio can only be estimated as
being either sub- or super-stellar, it is not possible to constrain the planet
accretion history, unless independent measurements of the envelope metallicity
are available.

Our findings reveal that using the C/O* ratio to probe the formation pathways
of giant planets could lead to wrong conclusions. The limited diagnostic power
of the C/O* ratio is essentially due to the high volatility of both C and O,
which makes the ratio unable to unequivocally trace the accretion of gas and
solids. The inclusion of N, one of the most volatile elements, in the set of
elemental tracers allows to compute the C/N* and N/O* ratios and to break the
degeneracies on the accretion history. In particular, Figure 7 shows that for
both the C/N* and N/O* ratios the trends for gas-dominated and solid-enriched
giant planets are well separated and remain above or below the stellar
reference. Such behaviours result from the lower volatility of C and O with
respect to N. Specifically, while the bulk of N in discs remains in the gas
phase as N2, the abundances of gaseous C and O decrease towards the outer disc
(see Figure 2). Variations of the C/N* and N/O* ratios in discs are therefore
driven by variations of C* and O*. As a consequence, the C/N* (N/O*) ratio of
gas-dominated giant planets is globally sub-stellar (super-stellar) and
decreases (increases) with the radial migration. The trends for solid-enriched
giant planets are derived accordingly, following the advices discussed in
Section 3.1.

It is worth noticing that the N/O* ratio of gas-dominated giant planets is
always markedly super-stellar and never drops below 2. Such trend is due to
the fact that in our model about 50$\%$ of O is early sequestered into rocks,
hence subtracted from the gas phase. As a consequence, in short-scale
migration scenarios, even if the other elemental ratios approach the stellar
value, the N/O* ratio does not.

What emerges from our study is that in order to get a complete picture of
planet formation, one has to look at multiple elemental ratios. One of the
advantages of using normalised ratios is that their trends can be more readily
and intuitively compared. A careful examination of the top two panels in
Figures 3, 4, 5, and 6 reveals the existence of fixed relations between the
ratios of gas-dominated and solid-enriched giant planets. In particular,
regardless of the chemical scenario, the N/O* ratio of gas-dominated giant
planets is always larger than the C/O* ratio, which is in turn larger than the
C/N* ratio. The opposite relation holds for solid-enriched giant planets for
moderate and large-scale migrations. Such relations result from the
differences in relative volatility between C, O, and N. More volatile elements
are characterised by lower condensation temperatures. As such, they condense
in solid form farther away from the star and remain in gas form over wider
disc regions. As discussed in Section 2.4, in our compositional model N is
almost entirely present in the gas phase as N2 in all four scenarios.
Therefore, as a result of its high volatility, N will be proportionally more
abundant than C and O in giant planets that accreted only gas, hence leading
to N/O*$>$C/O*$>$C/N*. For solid-enriched planets, which accreted from a solid
phase that is progressively depleted in high-volatile elements, the opposite
is true and C/N*$>$C/O*$>$N/O*. The emerging picture is that a joint
evaluation of normalised elemental ratios allows to immediately and
unequivocally discriminate between planets that derive their metallicity from
the accretion of gas and planets that experienced planetesimal enrichment.
These results expand those of Paper I, showing how the relative values of the
normalised ratios follow the same patterns independently on the underlying
chemical scenario.

###  3.4 How far did the planet start its migration?

Among all the ratios, C/O* appears to be the least robust as a diagnostic tool
for the migration history of the planet. In particular, Figure 7 shows that
the C/O* ratio exhibits limited changes among the migration pathways in almost
all chemical scenarios. Its overall flatness is essentially due to the limited
range of variation of the C/O* ratio in the disc with respect to the other
ratios, as it is shown in the bottom four panels of Figures 3, 4, 5, and 6
(note the different scales on the y axes). The strongest effects on the
planetary C/O* ratio are observed in the two reset scenarios for gas-dominated
giant planets, where the values decrease from 1.6 to 0.5 (i.e. by a factor of
3), for formation regions increasingly farther away from the star. In
particular, the trend decreases steadily up to 50 au and progressively
flattens further out. We discussed the origin of such trend in Section 3.3. On
the contrary, for both gas-dominated and solid-enriched giant planets, the
C/N* and N/O* ratios are significantly more sensitive to the radial migration
than the C/O* ratio. Specifically, they show monotonic trends with deviations
from the stellar value that increase with direct dependency on the migration,
leading to separations up to one order of magnitude between short- and large-
scale migrations.

The S/N* ratio provides additional information for solid-enriched giant-
planets. As it is shown in the top-right panel of Figures 3, 4, 5, and 6,
large-scale migrations of solid-enriched giant planets result in S/N* ratios
systematically above C/N* ratios. Although the difference between the two may
appear marginal, the trend results from the lower volatility of S with respect
to C, which causes a larger fraction of S to be locked in the solid phase than
C. Given that the total budget of S and C is dominated by the accretion of
solids for large-scale migrations, the abundance of S grows faster than that
of C resulting in S/N*$>$C/N*. On the contrary, short-scale migrations of
solid-enriched giant planets result in S/N* ratios markedly lower than C/N*
ratios. Such trend is a consequence of the increasingly dominant contribution
of the gas accretion (see the discussion in Section 3.1) combined with the
fact that the gas is the major carrier of C within the snowline of refractory
organic carbon. Our results confirm that the relation between the S/N* and
C/N* ratios holds for all the disc chemical scenarios in which the bulk of N
remains in gaseous form while the bulk of S is early trapped in refractory
material, so that the planetary S/N* ratio becomes a direct tracer of the
planetesimal accretion.

Given its intrinsic properties, the S/N* ratio is an extremely versatile
diagnostic tool. In particular, the joint evaluation of the S/N* ratio and the
envelope metallicity of the planet allows to discriminate between two
scenarios that are not directly modelled by our simulations. The first one
describes a giant planet that formed in-situ and close to the star, in a hot
environment with $T>700$ K. At such high temperature, all the volatile
elements, including S, are in the gas phase. In cosmochemistry, S is indeed
the most abundant among the moderately volatile elements (Palme et al., 2014),
with condensation temperature $T_{\text{cond}}=672$ K (Wood et al., 2019).
Consequently, the planet would be characterised by stellar values of both the
envelope metallicity and the S/N* ratio. The second scenario is that of a
solid-enriched hot Jupiter that accreted most of its heavy-element budget
beyond the N2 snowline, where even the highly volatile elements (C, O, and N)
are condensed in the solid phase (Öberg & Wordsworth, 2019). At the end of its
migration, when the planet reaches its close-in orbit, its envelope would be
characterised by a markedly super-stellar metallicity and a stellar S/N*
ratio. Therefore, the joint evaluation of the metallicity and the S/N* ratio
allows to distinguish hot Jupiters that formed in-situ and close to the star
from those that underwent extreme migrations.

The metallicity, coupled with the C/N* and the S/N* ratios, opens up the
possibility of identifying and investigating another alternative scenario that
was not directly modelled by our simulations. This is the scenario of a giant
planet that formed in a disc in which the inward drift of dust and pebbles
causes the ices of C, O, and N, to sublimate as they cross their respective
snowlines (Booth & Ilee, 2019; Schneider & Bitsch, 2021). As a direct
consequence of sublimation, the disc gas phase gets enriched in heavy
elements. In such scenario, gas accretion may contribute more than solids to
the total metallicity of the planet, hence mimicking the effect of short-scale
migration in our simulations. Due to the higher volatility of C with respect
to S, sublimation would be more efficient in enriching the gas in C rather
than in S. Therefore, Hot Jupiters that form in such scenario, will be
characterised by super-stellar metallicity and C/N*$>$S/N*.

###  3.5 The chemical structure of the birth environment

As discussed in Sections 3.3 and 3.4, our new analysis confirms and expands
the general conclusion of Paper I about the limited predictive power of the
C/O* ratio. Nevertheless, the C/O* ratio does retain some diagnostic power in
our model. Specifically, the C/O* ratio of gas-dominated giant planets that
underwent moderate and large-scale migrations (beyond 10 au in our model) is
always super-stellar in the inheritance scenarios and sub-stellar in the reset
ones (see the top-left panel in Figure 7). Therefore, provided that it is
possible to independently unveil the origin of the accreted material, e.g. by
looking at the other elemental ratios or through density and metallicity
estimates, the C/O* ratio of gas-dominated giant planets can be used to
constrain the chemical structure of the birth environment. On the other hand,
little can be said about solid-enriched giant planets, whose C/O* ratio does
not deviate appreciably from the stellar value in both scenarios.

For gas-dominated giant planets, the N/O* ratio in the atmosphere provides
additional insights into the chemical characterisation of the disc. As it is
shown in the bottom-left panel of Figure 7, the results for the inheritance
and the reset scenarios follow two distinct trends that diverge with respect
to each other. The separation between the two increases with the radial
migration, resulting in a N/O* ratio systematically higher in the inheritance
scenarios than in the reset ones. Specifically, for large-scale migrations,
the N/O* ratio in the inheritance scenarios can be more than a factor of three
larger than in the reset ones. Such trend is a direct consequence of the
higher abundance of gaseous O in the outer disc in the reset scenarios with
respect to the inheritance ones (see Figure 2). We discussed this feature in
Section 3.3 to explain why the C/O* ratio of gas-dominated giant planets is
sub-stellar in the reset scenarios.

###  3.6 Pairwise comparison of elemental ratios

Figure 8: Comparison between pairs of normalised elemental ratios in the
planet’s atmosphere. Results are shown for the four analysed chemical
scenarios. Each point on the plots represents one of the six analysed
simulations, describing a planet that starts its migration at 5, 12, 19, 50,
100, and 130 au from the star. Colour maps and increasing size of markers are
adopted to distinguish between short-scale (light colours, small markers) and
large-scale (dark colours, big markers) migrations. The blue markers
(hexagons) indicate gas-dominated giant planets, while the red ones (circles)
are for solid-enriched ones. The grey, dashed lines indicate stellar values.
Their intersection identifies four quadrants in which the elemental ratios can
assume sub- or super-stellar values.

To provide a more immediate visualisation of the characteristics of planetary
atmospheres and to facilitate the interpretation of observational data, we
compared pairs of normalised elemental ratios through binary plots. Figure 8
illustrates this comparison for three pairs of ratios, C/N* versus C/O*, N/O*
versus C/O*, and N/O* versus C/N* for both gas-dominated and solid-enriched
giant planets in the four chemical scenarios. The grey dashed lines,
indicating stellar values, delimit four quadrants in which the ratios can
assume sub- or super- stellar values. One of the advantages of this
representation is that the results for different combinations of the
parameters of the system (i.e. the extent of migration, the accretion and
chemical scenarios) occupy distinct quadrants. One can then immediately
identify the conditions that favour one scenario over the others.

For instance, a point in the third quadrant (bottom-left) of the C/N* versus
C/O* plot, i.e. sub-stellar values for both the C/N* and C/O* ratios,
necessarily identifies a planet that started its migration far from the star
(beyond the CO2 snowline) and accreted only gas in a reset scenario.
Conversely, a super-stellar C/O* ratio coupled with a C/N* ratio an order of
magnitude lower than the stellar value indicates that the same formation
process occurred in an inheritance scenario.

Binary plots also show that the comparison between the N/O* and C/N* ratios
allows to constrain the accretion history and the migration of giant planets.
This is clearly visible in the bottom four panels of Figure 8, where the
deviations of the ratios from their respective stellar values are directly
proportional to the radial migration and grow in opposite directions for gas-
dominated and solid-enriched giant planets. In the inheritance scenarios (see
the two bottom-left panels), the N/O* and C/N* ratios of gas-dominated giant
planets can deviate up to one order of magnitude above and below the stellar
value, respectively. For the same planets, moderate deviations of the N/O*
ratio with respect to the C/N* ratio provide strong indication that the
evolution occurred in a reset scenario (see the two bottom-right panels). When
considering solid-enriched giant planets, both the N/O* and C/N* ratios are
less sensitive to migration than in gas-dominated planets and follow the same
trend independently on the chemical scenario. Moreover, no planets populate
the quadrant where both the N/O* and the C/N* ratios are sub-stellar. The
opposite case, i.e. N/O* and C/N* ratios simultaneously super-stellar, is
compatible only with solid-enriched giant planets that experienced very short-
scale migrations, starting within the H2O snowline.

###  3.7 Refractory elements beyond S

One of the main findings of Paper I, confirmed by this study, is the untapped
potential of the S/N* ratio as a diagnostic tool for planet formation. As
discussed in Section 3.4, this is essentially due to the marked volatility
contrast between S and N, which makes their ratio an effective tracer of the
accretion of solids. It is important to notice that any element X more
refractory than O can be used in place of S to perform the same analysis. From
an observational perspective, this means that our set of diagnostic tracers
can be extended to elements that may be more easily detectable than others in
giant planet atmospheres. For instance, for increasing equilibrium
temperatures of the planets the atmospheres get enriched in elements with
increasingly lower volatility. For warm and hot Jupiters, moderately volatile
elements are all good alternatives to S. Such category includes elements that
condense at temperatures between 1250 K and 250 K, such as Na, K, Cl, and F
(Palme et al., 2014; Wood et al., 2019). For ultra-hot Jupiters the list
extends to refractory (e.g Al, Ca, and Ti) and rock-forming (e.g Mg, Si, Ni,
and Fe) elements, which have condensation temperatures in the range 1850-1250
K (Palme et al., 2014; Wood et al., 2019).

In Section 3.4 we stressed that the planetary elemental ratios strongly depend
on the phase of the main carrier of the two involved elements (e.g., the gas
phase for N2, which is the main carrier of N and the solid phase for the rocks
that are the main reservoir of S) and on how efficiently the gas and the
solids contribute to the final envelope metallicity. In particular, we
examined the case of the C/N* and S/N* ratios in solid-enriched giant planets.
Which of the two ratios is larger than the other depends on whether the planet
formation is dominated by the accretion of gas or solids. More generally, the
comparison between two elemental ratios X/N* and Y/N*, where the elements X
and Y have a marked volatility contrast, allows to discriminate between
scenarios in which one of the two elements is predominantly accreted by
different phases.

In Section 3.4 we also highlighted that the S/N* ratio allows to resolve
scenarios of in-situ formation very close to the star, in a hotter environment
than the condensation temperature of S. Specifically, giant planets that
formed by accreting gas from such region will be characterised by stellar
values of both metallicity and S/N* ratio. It is important to notice that the
replacement of S with a more refractory element would push this limit towards
higher temperatures, allowing to resolve formation scenarios even closer to
the star.

In Section 3.4 we also discussed how the comparison between the metallicity
and the C/N* and S/N* ratios allows to identify giant planets that formed by
accreting gas enriched in heavy elements due to the sublimation of inwardly
drifting dust and pebbles. Sublimation is more efficient in enriching the gas
in high volatile elements rather than in refractory ones. In particular, any
refractory element X would produce less enrichment in the gas than C.
Therefore, planets that formed by accreting high-metallicity gas would be
characterised by super-stellar metallicity and C/N*$>$X/N*.

###  3.8 Linking our predictions to observable planetary atmospheres

In the previous sections we discussed how the atmospheric elemental ratios can
be used to trace the formation history of giant planets. From an observational
perspective, such elemental ratios need to be retrieved from the abundances of
the molecular carriers of C, O, N, and S in exoplanetary atmospheres.

The observable chemistry in giant planet atmospheres is shaped by multiple and
diverse processes, such as thermochemical reactions at equilibrium, mixing
processes, photochemistry, and chemical diffusion. Each of them dominates in
different atmospheric layers (e.g., Madhusudhan et al., 2016 and references
therein).

Deep in the atmosphere, the chemistry is at equilibrium and the chemical
composition is a function only of the elemental abundances, the temperature,
and the pressure. For a solar composition, the main molecular carriers of C
and N are CO, CH4, N2, and NH3. Two regimes in the pressure-temperature space
can be identified. At high temperatures and low pressures, C and N are
essentially locked in CO and N2, while at low temperatures and high pressures,
the main reservoirs of C and N are CH4, and NH3. At the pressure of 1 bar, the
transition between the two regimes occurs at the temperatures of 1200 K and
700 K for the C- and N- bearing molecules, respectively. O is largely present
as water vapour, while S is mainly locked into H2S (e.g., Madhusudhan et al.,
2016, Fortney et al., 2021 and references therein). The chemical equilibrium
in atmospheres with super-solar metallicity generally favours the production
of molecules that contain two or more heavy elements. In such atmospheres, CO2
is therefore favoured over CO and N2, which in turn are favoured over CH4 and
NH3 (e.g., Madhusudhan et al., 2016 and references therein).

In the upper layers of the atmosphere, photochemical reactions induced by
stellar UV radiation are proved to alter the chemical composition. Among the
direct and indirect products of photochemistry there are H, HCN, C2H2, OH, O2,
and NO. Regarding S, photochemistry converts H2S in S, HS, S2, SO, and SO2
(e.g., Madhusudhan et al., 2016; Fortney et al., 2021).

Chemistry in observed exoplanetary atmospheres may be more complex than
discussed above, being also influenced by the cooling history of the planet
(Fortney et al., 2020). At intermediate layers, volatile species may be
sequestered into clouds, although these are not expected to form at the high
atmospheric temperatures of hot Jupiters (e.g., Madhusudhan et al., 2016,
Madhusudhan, 2019 and references therein).

Ground-based and high-resolution spectral observations have recently succeeded
in providing the first characterisations of exoplanetary atmospheres. For
instance, observations by Giacobbe et al. (2021) of the hot Jupiter HD209458b,
at the equilibrium temperature of 1500 K, revealed the presence of H2O, CO,
HCN, CH4, NH3, and C2H2 in the atmosphere. Future observations with JWST
(Greene et al., 2016), Twinkle (Edwards et al., 2019), and Ariel (Tinetti et
al., 2018), as well as advances in atmospheric retrieval techniques, will soon
revolutionise the field. The broad spectral range and the high spectral
resolution of JWST will allow for precise determination of the abundances of a
number of species of interest besides H2O, like CO, CO2, CH4, NH3 (Greene et
al., 2016). An even greater number of molecules will be accessible with Ariel,
by the time the telescope starts its operations in 2029 (Tinetti et al.,
2018).

Recent studies have been testing our methodology on real data in preparation
of its use for interpreting the observations of future ground-based and space-
based facilities. Biazzo et al (submitted), propose possible formation
scenarios for giant planets in four planetary systems, around the stars
HAT-P-26, WASP-39, HAT-P-12, and WASP-10. Specifically, they studied the
trends of the normalised planetary metallicity and elemental ratios as
computed from the estimates of metallicity, N/O, and C/O ratios obtained by
Kawashima & Min (2021) through spectral disequilibrium retrieval models. A
similar study was performed independently by Kolecki & Wang (2021) on an
additional set of four giant planets. Guilluy et al. (submitted) and Carleo et
al. (submitted) jointly used the planetary metallicity and C/O* ratio to
constrain the formation histories of WASP-69b and WASP-80b, as well as the
characteristics of their native circumstellar discs (see also Section 3.3). A
more systematic study of the observational implications of our results will be
the focus of dedicated future works.

##  4 Conclusions

In this work, we investigated the link between the final composition of the
atmospheres of giant planets and their formation process, focusing on the
effects of different chemical initial conditions for the host protoplanetary
disc. The study in Paper I was limited to a disc that inherited its
composition from the pre-stellar core and whose chemical evolution was
limitedly affected by cosmic rays and similar energy sources. Here we explored
also the case of a complete reset of the chemistry, as well as the impact of
different levels of the ionisation rate throughout the disc. Specifically, we
considered a first case in which the only source of ionisation is the decay of
short-lived radionuclides (i.e., the same considered in Paper I) and a second
case in which an additional contribution is provided by cosmic rays. In
addition, we introduced a more realistic model of the condensation of
refractory organic carbon.

We analysed the six N-body simulations from Paper I of the growth and
migration of a giant planet in a disc of gas and planetesimals. During the
migration, the planet grows in radius and mass by accreting gas and solids
characterised by different compositions and relative abundances of refractory
and volatile elements. By coupling the outcome of the simulations with the
different compositional models of the disc we derived the composition of the
accreted material, from which we computed the C/O, N/O, C/N, and S/N ratios in
the final atmosphere of the planet. For a better interpretation of the
results, we normalised the elemental ratios to their corresponding stellar
values and we indicated the new values with the superscript *. Our findings
can be summarised as follows:

  * •

The final composition of giant planets is markedly non-stellar due to the
interplay between planetary migration and concurrent accretion of gas and
solids. Therefore, despite being intimately connected, the planetary elemental
ratios are not a direct reflection of those of the disc.

  * •

The diagnostic power of the elemental ratios is maximised when these are
expressed in units of their corresponding stellar value.
First, normalisation brings different elemental ratios to the same scale,
making it easier to compare them and study their mutual relations. As
discussed below, such relations provide unique insights into the formation and
migration histories of giant planets.
Second, normalisation allows to directly measure the deviation of the planet’s
composition from that of its host star, which in turn reflects the original
composition of the formation environment. As a result, we can constrain the
formation history of the planet even in absence of detailed knowledge of the
specific properties of the natal environment. The use of normalised elemental
ratios is therefore key to comparing the formation histories of giant planets
in multi-planet systems and of giant planets orbiting different stars.

  * •

In all the disc chemical scenarios, the joint evaluation of C- and N-based
elemental ratios allows to unequivocally constrain the source of planetary
metallicity. Specifically, gas-dominated giant planets are characterised by
N/O*$>$C/O*$>$C/N*, while solid-enriched giant planets are characterised by
C/N*$>$C/O*$>$N/O*.
Moreover, all the planetary elemental ratios, except C/O*, exhibit monotonic
trends with respect to the starting location of the planet. Specifically, the
deviation of the ratios from their stellar value increases with the extent of
disc-driven migration, independently on the disc chemical scenario and the
ionisation level.

  * •

The global trends of the elemental ratios in the planet composition through
the different migration scenarios are not model-dependent, only their absolute
values are so. Specifically, changing the chemical and thermal profiles of the
disc would only affect the slope of the curves shown in our plots, but not
their overall behaviour. Therefore, our methodology can be directly applied to
compare the formation and migration histories of different giant planets in
the same planetary system.

  * •

For solid-enriched giant planets, the C/O* ratio provides constraints only on
the accretion scenario, as its value changes limitedly with migration. For
gas-dominated ones, it can provide information also on the chemical structure
of the disc, although this is true only for large-scale migration scenarios.

  * •

For gas-dominated giant planets, the joint evaluation of the N/O* and the C/O*
ratios allows to constrain the disc chemical scenario. Specifically, the N/O*
ratio is up to a factor of 3 higher in the inheritance scenarios than in the
reset ones. In parallel, for large-scale migrations the C/O* ratio is super-
stellar in the inheritance scenarios and sub-stellar in the reset ones.
The elemental ratios of solid-enriched giant planets are instead limitedly
affected by whether the disc is characterised by chemical inheritance or
reset, making it hard to discriminate between the two scenarios.

  * •

The S/N* ratio provides additional constraints on the migration history of
solid-enriched giant planets. Specifically, large-scale migrations associated
with substantial accretion of solids result in S/N*$>$C/N*. On the contrary,
short-scale migrations associated with limited supply of solids and increased
contribution of gas accretion result in S/N*$<$C/N*. Such relations holds for
all the disc chemical scenarios in which the bulk of N remains in gaseous form
while the bulk of S is early trapped in refractory material, so that the
planetary S/N* ratio becomes a direct tracer of the planetesimal accretion.

  * •

Given the high volatility contrast between S and N, the joint evaluation of
the planetary S/N* ratio and metallicity allows to distinguish hot Jupiters
that formed in-situ and close to the star from those that underwent extreme
migrations, from beyond the snowline of N2. The first will be characterised by
stellar values of both the metallicity and the S/N* ratio, while the second
will have markedly super-stellar metallicity and stellar S/N* ratio.
The metallicity, combined with the C/N* and the S/N* ratios, allows also to
trace formation histories of giant planets in discs whose gas phase is
enriched in heavy elements due to the sublimation of inwardly drifting dust
and pebbles. Such planets will be characterised by super-stellar envelope
metallicity and C/N*$>$S/N*.

  * •

Pairwise comparison of multiple normalised elemental ratios provides
additional insights into the planet formation history and the chemical initial
conditions of the disc. The associated binary plots allow to immediately
identify the conditions that favour one scenario over the others, hence
providing an effective tool for the interpretation of observational data.

##### Acknowledgements.

The authors wish to thank Aldo Bonomo, Gloria Guilluy and Ilaria Carleo for
discussion and feedback on exoplanetary atmospheric observations. The authors
acknowledge the support of the European Research Council via the Horizon 2020
Framework Programme ERC Synergy “ECOGAL” Project GA-855130, of the Italian
National Institute of Astrophysics (INAF) through the INAF Main Stream project
‘Ariel and the astrochemical link between circumstellar discs and planets’
(CUP: C54I19000700005), and of the Italian Space Agency (ASI) through the ASI-
INAF contract no. 2021-5-HH.0. The authors also acknowledge contribution from
PRIN INAF 2019 through the project “HOT-ATMOS”. The computation resources for
this work were supplied by the Genesis cluster at INAF-IAPS and the technical
support of Scigé John Liu is gratefully acknowledged.

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