11institutetext:  Institute of Physics, University of Bern,
Gesellschaftsstrasse 6, 3012 Bern, Switzerland
11email<EMAIL_ADDRESS>22institutetext: Geneva Observatory, University
of Geneva, Chemin Pegasi 51b, 1290 Versoix, Switzerland

# A framework for the architecture of exoplanetary systems

II. Nature versus nurture: Emergent formation pathways of architecture classes

Lokesh Mishra A framework for the architecture of exoplanetary systemsA
framework for the architecture of exoplanetary systemsA framework for the
architecture of exoplanetary systemsA framework for the architecture of
exoplanetary systems 0000-0002-1256-7261 Yann Alibert A framework for the
architecture of exoplanetary systemsA framework for the architecture of
exoplanetary systems 0000-0002-4644-8818 Stéphane Udry A framework for the
architecture of exoplanetary systemsA framework for the architecture of
exoplanetary systems 0000-0001-7576-6236 Christoph Mordasini A framework for
the architecture of exoplanetary systemsA framework for the architecture of
exoplanetary systems 0000-0002-1013-2811

(Received DD MMM YYYY; accepted DD MMM YYYY)

In the first paper of this series, we proposed a model-independent framework
for characterising the architecture of planetary systems at the system level.
There are four classes of planetary system architecture: similar, mixed, anti-
ordered, and ordered. In this paper, we investigate the formation pathways
leading to these four architecture classes. To understand the role of nature
versus nurture in sculpting the final (mass) architecture of a system, we
apply our architecture framework to synthetic planetary systems — formed via
core-accretion — using the Bern model. General patterns emerge in the
formation pathways of the four architecture classes. Almost all planetary
systems emerging from protoplanetary disks whose initial solid mass was less
than one Jupiter mass are similar. Systems emerging from heavier disks may
become mixed, anti-ordered, or ordered. Increasing dynamical interactions
(planet–planet, planet–disk) tends to shift a system’s architecture from mixed
to anti-ordered to ordered. Our model predicts the existence of a new
metallicity–architecture correlation. Similar systems have very high
occurrence around low-metallicity stars. The occurrence of the anti-ordered
and ordered classes increases with increasing metallicity. The occurrence of
mixed architecture first increases and then decreases with increasing
metallicity. In our synthetic planetary systems, the role of nature is
disentangled from the role of nurture. Nature (or initial conditions) pre-
determines whether the architecture of a system becomes similar; otherwise
nurture influences whether a system becomes mixed, anti-ordered, or ordered.
We propose the ‘Aryabhata formation scenario’ to explain some planetary
systems which host only water-rich worlds. We finish this paper with a
discussion of future observational and theoretical works that may support or
refute the results of this paper.

###### Key Words.:

Planetary systems – Planets and satellites: detection – Planets and
satellites: formation – Planets and satellites: physical evolution

##  1 Introduction

Studying planetary systems as single units of a physical system makes them
amenable to system level examinations. Investigating the ensemble of bound
objects (host star(s), planets, minor bodies) coherently can allow a deeper
and more comprehensive understanding of exoplanetary astrophysics to emerge.
The purview of this multi-body physics covers a breadth of topics including
stability of planetary systems (Gladman 1993; Laskar 1997, 2000; Chambers
1999; Fang & Margot 2013; Pu & Wu 2015; Laskar & Petit 2017; Obertas et al.
2017; Petit et al. 2018; Wang et al. 2019; Yeh et al. 2020; Tamayo et al.
2020; Turrini et al. 2020), stellar host and protoplanetary disk properties
(Petigura et al. 2018; Manara et al. 2019; Mulders et al. 2021), novel
approaches to system-level characterisation (Tremaine 2015; Kipping 2018;
Alibert 2019; Mishra et al. 2019; Gilbert & Fabrycky 2020; Bashi & Zucker
2021; Sandford et al. 2021), and the architecture of planetary systems
(Lissauer et al. 2011; Ciardi et al. 2013; Fabrycky et al. 2014; Weiss et al.
2018; Millholland et al. 2017; Adams 2019; Adams et al. 2020; Mulders et al.
2020; He et al. 2019, 2021; Mishra et al. 2021; Adibekyan et al. 2021;
Millholland & Winn 2021; Winter et al. 2020). Analysing multi-body system
level physics may allow us to understand whether planetary systems are self-
organizing emergent structures – i.e. whether global level patterns are
emerging from local level interactions.

Inspired by the peas in a pod architecture (Weiss et al. 2018; Millholland et
al. 2017; Mishra et al. 2021), we introduced a new framework for studying the
architecture of planetary systems (Mishra et al. 2023, ; hereafter Paper I).
Studying the architecture as a global-system level phenomena, this framework
allows us to characterise, quantify, and compare the architecture of
individual planetary systems. Four classes of planetary system architecture
emerged from this framework. These classes are labelled similar, mixed, anti-
ordered, and ordered depending on the arrangement and distribution of planets
around the host star. The key idea behind this framework is that the
arrangement and distribution of planets contains additional information that
cannot be extracted by studying single planets individually. Hints of the
presence of this additional information were revealed in some works (Tremaine
2015; Laskar & Petit 2017; Kipping 2018; Mishra et al. 2019; Gilbert &
Fabrycky 2020; Sandford et al. 2021).

Explaining the formation, evolution, and final assembly of planetary systems
remains an outstanding theoretical problem. Planet-formation physics spans
astronomical orders of magnitude in mass, size, and time (Udry & Santos 2007;
Armitage 2010). The processes occurring during planet formation convert gases
and micron-sized dust particles from the protoplanetary disk into different
kinds of planets arranged in different architectures over timescales of
millions and billions of years. However, it remains unclear as to how initial
conditions derived from the host star or protoplanetary disk combine with the
formation and evolution processes to give rise to the observed exoplanetary
systems.

We are interested in understanding the role of nature versus nurture in
sculpting the final planetary system and the extent to which the character of
the mature planetary system is influenced by its initial conditions. Kipping
(2018) suggested, using an entropy-like formulation for planetary systems,
that the initial conditions of planet formation could be inferred based on
their present-day architecture. However, the presence of stochastic processes
makes it difficult to connect the initial conditions with the final system. It
is also unclear as to whether or not stochastic physical processes can erase
all memory of initial conditions, or indeed leave their own impressions on the
final architecture. Using ideas from the fields of machine-learning-based
natural language processing, Sandford et al. (2021) showed that planetary
systems are not randomly assembled. While it is clear that planetary systems
are not identical copies of one another, the quest for quantifying the
similarity between planetary systems is a tantalising one.

In this paper, we investigate the formation pathways that lead to the four
architecture classes. Due to the stochastic nature of this problem,
understanding the formation of a single planetary system can be very
complicated. For example, two systems with almost identical initial conditions
may evolve into two completely different planetary systems. Chaos arising from
multi-body gravitational interactions may cause differing formation pathways
for these two systems. However, some patterns are found to emerge when
studying planetary systems as part of an ensemble. These trends, as we show in
this paper, help us to understand the role played by initial conditions and
physical processes in shaping the architecture.

Figure 1: The four classes of planetary system architecture and their emergent
formation pathways.
Top: Reproduced from Paper I – Schematic diagram depicting the Four classes of
planetary system architecture: similar, anti-ordered, mixed, and ordered.
Depending on how a quantity (such as mass or size) varies from one planet to
another, the architecture of a system can be identified. The framework is
model independent.
Bottom: Emergence of formation pathways: Sankey diagram depicting the
emergence of formation pathways of architecture classes. The thickness of the
links and nodes is proportional to the relative number of synthetic systems in
our simulation. This result is derived from synthetic planetary systems around
a solar mass star via the Bern model. Disk gas mass and metallicity are binned
at their median values.

Figure 1 (bottom) summarises the main findings of this paper. We show that the
effects of planet formation and evolution processes are imprinted in the
system-level architecture. Figure 1 shows the formation pathways of the
architecture classes that emerge due to the system-level approach of our
architecture framework (Fig. 1 (top)). This sankey diagram has nodes for
protoplanetary disk gas mass, protoplanetary disk solid mass, metallicity, and
planetary architecture. We find that the formation of similar planetary
systems is dominated by initial conditions. If the initial conditions
disfavour the formation of similar architecture, the other three architectures
may emerge. Whether the final architecture is mixed, ordered, or anti-ordered
seems to depend on the stochastic formation processes. Increasing dynamical
interactions (disk–planet, planet–planet) generally tends to produce mixed,
anti-ordered, and then ordered architectures, respectively.

We first summarise the architecture framework and some results from Paper I in
Sect. 2. We study the role of nature (initial conditions) and nurture
(dynamical processes) in Sects. 3 and 4, respectively. In these sections, we
study the influence of protoplanetary disk mass, metallicity, protoplanetary
disk lifetime, planet–disk interactions, planet–planet interactions, and
N-body interactions on the final architecture of simulated planetary systems.
We summarise our results, suggest possible future studies, and conclude this
paper in Sect. 6.

##  2 Summary of Paper I and the Bern model

###  2.1 Architecture framework

The arrangement of multiple planets and the collective distribution of their
physical properties around the host star(s) characterises the architecture of
a planetary system (Mishra et al. 2021). To quantify the architecture of a
planetary system, we developed a novel model-independent framework Paper I.
Some key aspects of this framework are briefly summarised here, and we refer
the reader to Sect. 3 of Paper I for details.

Conceptually, the framework defines four classes of planetary system
architecture: similar, mixed, anti-ordered, and ordered. Consider a planetary
quantity (such as mass, radius, etc.) as a function of the distance of the
planet to the host star (see Fig. 1). When all planets in a system have
similar values of a planetary quantity, the architecture of such systems is
similar. When the planetary quantity increases with increasing distance, the
system is said to exhibit an ordered architecture. Alternatively, if the
quantity shows an overall decreasing trend with increasing distance, the
architecture is considered to be anti-ordered. Finally, the planetary
quantities could also show variations that are not captured in the three
classes above. A mixed architecture may depict large, increasing, and
decreasing variations with distance. By studying the variation of a planetary
quantity with distance for all planets in the system, our framework captures
the arrangement and distribution of planets in the system.

The architecture of a system is quantified via two coefficients: the
coefficient of similarity, $C_{S}(q_{i})$, and the coefficient of variation,
$C_{V}(q_{i})$. Here, $q_{i}$ represents a planetary quantity (e.g. mass,
radius, eccentricity, density) for the $i^{\text{th}}$ planet. When the
coefficients are calculated using planetary masses, they inform us about the
mass architecture of a system, that is, the arrangement and distribution of
mass in a given system. Likewise, we can study the radius architecture,
density architecture, water-mass-fraction architecture, eccentricity
architecture, and so on. The versatility of our architecture framework lies in
its ability to allow us to study the multifaceted architectures of a planetary
system. In Paper I, we explored the relationship between these different kinds
of architectures. As in Paper I, we identify the architecture of a system by
its bulk mass architecture.

Calibrated on planetary masses, a classification scheme to identify the
architecture class was proposed in Paper I (eq. 8). The $C_{S}$ versus $C_{V}$
plane represents the architecture space for planetary systems (Fig. 3 in Paper
I). This new parameter space was found to be endowed with a curious
mathematical property, namely planetary systems cannot occupy all parts of the
architecture plane, as some regions of this parameter space are mathematically
forbidden.

To understand the implications of this architecture framework, we applied it
on several catalogues in Paper I. These included 41 observed multi-planetary
systems and numerically simulated systems via population synthesis using the
Generation III Bern model (Emsenhuber et al. 2021a, b).

###  2.2 Bern model

For the synthetic planetary systems, as the initial conditions and the
physical processes are known, it is possible (and desirable) to understand how
different architecture classes are formed. As this paper is dedicated to
planet formation and its imprints on architecture, we briefly review the
ingredients of the Bern model here. Readers interested in further details of
this model are referred to the recent NGPPS series of papers (Emsenhuber et
al. 2021a, b; Schlecker et al. 2021a; Burn et al. 2021; Schlecker et al.
2021b; Mishra et al. 2021). The historic development of the Bern model may be
traced through the works of Alibert et al. (2004, 2005); Mordasini et al.
(2009); Alibert et al. (2011); Mordasini et al. (2012a, b); Alibert et al.
(2013); Fortier et al. (2013); Marboeuf et al. (2014b); Thiabaud et al.
(2014); Dittkrist et al. (2014); Jin et al. (2014) and is reviewed in Benz et
al. (2014); Mordasini (2018).

Based on the core-accretion paradigm (Pollack et al. 1996), the Bern model is
a global model of planet formation and evolution. The model studies the growth
of several lunar-mass protoplanetary embryos embedded in protoplanetary disks
(consisting of a gaseous and solid phase) around a solar-type star. The disk
model is based on viscous angular momentum transport (Lynden-Bell & Pringle
1974; Veras & Armitage 2004; Hueso & Guillot 2005). Turbulence is
characterised by the Shakura & Sunyaev (1973) approach. The initial mass of
the solid disk depends on the metallicity of the star and also on the
condensation state of the molecules in the disk (Thiabaud et al. 2014). The
solids in the disk are composed of a swarm of rocky and icy planetesimals. The
solids in the disk evolve via (a) accretion by growing planets, (b)
interaction with gaseous disk, (c) dynamical stirring from planets and other
planetesimals, and so on (Fortier et al. 2013). The 1D geometrically thin disk
evolution is studied up to 1000 au.

This star–disk–embryo numerical system is endowed with several physical
processes, which are occurring simultaneously and in a self-consistently
coupled way. Some of these physical processes are: stellar evolution (Baraffe
et al. 2015), interactions between viscous protoplanetary disk and star
(Lynden-Bell & Pringle 1974; Shakura & Sunyaev 1973; Clarke et al. 2001;
Matsuyama et al. 2003; Veras & Armitage 2004; Nakamoto & Nakagawa 1994; Hueso
& Guillot 2005), condensation of volatile and/or refractory species (Marboeuf
et al. 2014b, a; Thiabaud et al. 2014), planet formation physics (Alibert et
al. 2013; Fortier et al. 2013; Mordasini et al. 2012b), orbital and tidal
migration (Coleman & Nelson 2014; Paardekooper et al. 2011; Dittkrist et al.
2014), gravitational N-body interactions (Chambers 1999; Alibert et al. 2013;
Emsenhuber et al. 2021a, b), atmospheric escape (Jin et al. 2014), bloating
(Sarkis et al. 2021), and so on (see Fig. 1 in Mishra et al. (2019) for a
schematic diagram). In addition, the model also calculates the internal
structure of all planets, assuming them all to be spherically symmetric.

In the synthetic planetary population we use in the present work, some initial
conditions are fixed, namely we use a $1M_{\odot}$ mass star and a disk
viscosity $\alpha=2\times 10^{-3}$, describing the initial shape of the gas
and planetesimal disks via power laws (Veras & Armitage 2004), with a
planetesimal size of $300m$, and a fixed density (rocky 3.2 $\text{g
cm}^{-3}$, icy 1 $\text{g cm}^{-3}$). We add 100 protoplanetary embryos to the
protoplanetary disk. We ensure that no two embryos start within 10 hill radii
of each other (Kokubo & Ida 1998, 2002). This model is then run 1000 times
while varying other initial conditions. We varied the initial gas mass in the
protoplanetary disk, disk lifetime, stellar metallicity, disk inner edge, and
the initial location of the protoplanetary embryos (for details see Emsenhuber
et al. 2021b).

The Bern model includes a significant variety of physics and uses plausible
choices of initial conditions, which are motivated by observations. However,
it is only a simplified low-dimensional approximation of our current
understanding of planet formation. For example, we model planet formation via
core-accretion only and ignore other methods, such as disk instability (Schib
et al. 2021). Among others, we also assume that the dust-to-gas ratio is the
same for both the host star and the disk, and that all dust in the disk is
aggregated into planetesimals. The N-body interactions are tracked for only 20
Myr, which may be inadequate to capture dynamical effects occurring in the
outer parts of the system. The assumptions, choices, and simplifications made
in this model may have a strong impact on the outcome of this paper.
Nevertheless, exploring the implications of our architecture framework using
synthetic populations via the Bern model is a necessary first step. The main
result of this paper is not in understanding the formation of any single
planetary system but to show that, for different architecture classes,
discernible patterns of formation pathways emerge. Future studies could apply
our architecture framework (from Paper I) with other planet formation models.
If the formation pathways for the different architecture classes were found to
remain the same after using different formation models, then our results would
be strengthened and become more robust.

##  3 Nature: Role of star and disk initial conditions

In this section, we study the connection between the initial conditions and
the final architecture of a system. We begin by counting the number of
different architecture classes that emerge from our population synthesis as a
function of the various initial conditions that are varied. The role of
varying disk masses and stellar metallicities is presented in Sect. 3.1, and
that of varying disk lifetimes in Sect. 3.2. For completeness, we measure the
relative count for an architecture class within a bin by dividing the number
of systems of a particular architecture class in a bin by the total number of
systems in that bin. We emphasise that, as in Paper I, the architecture of a
system is identified with its bulk mass architecture. Thus, when we refer to a
similar or ordered system, we are referring to a system whose bulk mass
architecture is similar or ordered, respectively.

###  3.1 Protoplanetary disk: Mass and stellar metallicity

Figure 2: Role of disk mass and the metallicity–architecture correlation. The
top two rows show the binned relative count of each architecture class as a
function of initial disk gas mass (upper left), disk solid mass (upper right),
stellar metallicity in the synthetic population (middle left), and stellar
metallicity in observed systems (middle right). The length of the error bars
corresponds to the total number of systems in each bin as:
${100}/{\sqrt{\mathrm{bin\ counts}}}$. In the bottom panels, each point
corresponds to a single planetary system. The system architecture is indicated
by the colour and shape of the marker. The bottom left panel shows the solid
mass in the disk as a function of the disk gas mass. The two diagonal lines
convey the role of stellar metallicity. The dashed horizontal line indicates
the mass of Jupiter. The bottom right panel shows the total mass in planets as
a function of the solid mass in the protoplanetary disk. The two diagonal
lines indicate the efficiency of converting solids from the disk into planets.
If the planets in a hypothetical system could accrete all the solid mass of
its disk, and these planets had no gaseous atmosphere, then such a system
would lie on the diagonal line corresponding to $100\%$ accretion efficiency.
The dashed vertical line indicates the mass of Jupiter.

Figure 2 (upper left) shows the dependence of the architecture class relative
counts on the initial mass of gas in the protoplanetary disk. Over $96\%$ of
all disks that started with gas masses $\lesssim 0.04M_{\odot}$ give rise to
planetary systems of similar architecture. About $1\%$ of these low-mass disks
lead to each of the other three architecture classes. The relative count of
systems with similar architecture shows a clear decreasing trend with
increasing mass in the disk gas.

The production of the remaining three architecture classes tends to increase
with increasing disk gas mass, but with distinct trends. As the mass in the
gas disk increases, the relative count of mixed architectures increases first,
and then decreases for gas mass $\gtrsim 0.12M_{\odot}$. The relative count
for both anti-ordered and ordered architectures continues to increase with
increasing disk mass. Anti-ordered architectures become the most common
outcome from large disks with gas mass $\gtrsim 0.12M_{\odot}$.

In Fig. 2 (upper right), we see the binned relative count of different
architecture classes as a function of the mass of the solids in the
protoplanetary disk. This plot shows some of the same features that we saw in
Fig. 2 (upper left). About $99\%$ of all disks that have solid masses
$\lesssim 200M_{\oplus}$ give rise to similar planetary systems. The
production of similar architecture decreases as the mass of solids in a disk
is increased.

Before continuing, we note that this is already a result of considerable
importance. The physical processes encoded in the Bern model are the same for
all 1000 planetary systems. The only difference between these synthetic
systems arises from the variations in their initial conditions. We are seeing
that almost all low-mass disks give rise to only one architecture, the similar
class. This occurs despite all the physical processes that can act upon the
system and induce some architectural variation. As we show below, the low mass
of the disk limits some of the physical processes that sculpt a system’s
architecture. We conclude that the production of systems of the similar
architecture class is dominated by initial conditions.

Close to $60\%$ of all observed systems in our multi-planetary systems
catalogue (from Paper I) are similar in their mass architecture (Paper I). For
some of these similar class systems (like Trappist-1, TOI-178, etc), if their
formation is via core-accretion, our work may suggest strong limits on the
initial mass of their protoplanetary disks.

The relative count of the other three architecture classes increases as the
solid mass in the disk increases. The production of mixed architectures peaks
around disks of $\approx 1M_{\text{J}}$ and then decreases. The prevalence of
anti-ordered and ordered architectures continues to increase with increasing
disk mass. For heavy massive disks, anti-ordered architecture is the most
common outcome.

Figure 2 (middle left) shows the relative count of each architecture class in
the synthetic population as a function of stellar metallicity. Figure 2
(middle right) shows the same for the 41 observed multi-planetary systems. The
selection criterion for our observed catalogue is detailed in Paper I. We find
an interesting correlation between the metallicity and the architecture of a
system, hereafter referred to as the metallicity–architecture correlation, and
note the following trends. Over $98\%$ of all systems with Fe/H $<-0.2$ are of
similar type. The relative count of similar architecture decreases as the
metallicity is increased. The relative counts of the other three architecture
classes are below $5\%$ for metallicities $\leq-0.2$. At different rates, the
relative counts of mixed, ordered, and anti-ordered classes increase with
increasing metallicity. Our catalogue of observed planetary systems shows an
encouragingly similar trend.

Our observations catalogue suffers from detection biases and incompleteness.
One way in which these limitations manifest is that we do not find any
observed example of anti-ordered architecture. The qualitative trend for the
relative count of observed system architectures as a function of their stellar
metallicity agrees with our synthetic systems. For example, the relative count
of similar observed systems decreases with increasing metallicity. The
relative count of ordered architectures continues to increase with increasing
metallicity.

To understand the origin of these correlations, we study the relation between
initial disk mass (both in solids and gases), stellar metallicity, and the
final architecture of the systems in our model. In the Bern model, the initial
solid mass of the disk is a fraction of the initial gas mass of the disk. This
fraction is correlated with the dust-to-gas ratio, which also depends on the
gas mass itself because the location of different icelines depend on it. By
simulating systems with varying dust-to-gas ratio ($f_{\rm D/G}$), we simulate
systems around stars with different metallicities. This is due to the
following relation:

$10^{[\mathrm{Fe/H}]}=\frac{f_{\rm D/G}}{f_{\mathrm{D/G,\odot}}},\quad
f_{\mathrm{D/G,\odot}}=0.0149\
\text{\cite[citep]{(\@@bibref{AuthorsPhrase1Year}{Lodders2003}{\@@citephrase{
}}{})}}.$ (1)

The metallicities in our simulations vary from $-0.6$ to $0.5$ following
Santos et al. (2005).

Figure 2 shows the solid disk mass as a function of the gas disk mass (bottom
left) and the total mass in the planets as a function of the solid disk mass
(bottom right). Each point represents one planetary system, and the shape and
colour of the marker shows its final architecture. These two plots help us
understand the correlations discussed above.

The bottom left panel of Fig. 2 shows the relationship between gas disk mass,
solid disk mass, metallicity, and the final architecture of the system.
Generally, when the mass of the solids in a disk is $\gtrsim
1M_{\text{J}}(\approx 318M_{\oplus})$, the production of architectures other
than similar is triggered. We note that up to a certain gas disk mass
($\lesssim 0.02M_{\odot}$), irrespective of the metallicity, all disks lead to
similar architecture. For heavier gas disks ($\gtrsim 0.02M_{\odot}$),
metallicities begin to play a role. If the gas disk mass is high enough, even
low metallicities ($\approx-0.2$) can trigger the production of architectures
other than the similar class. However, for lower gas disk masses, higher
metallicities are required to produce about a $1M_{\text{J}}$ mass in the
solid disk.

It is clear that the mass in the solids of the protoplanetary disk plays an
essential role here. The bottom right panel of Fig. 2 explains the above
statement. The total mass in the planets increases as the mass of solids in
the disk increases. When the mass of solids in the disk is $\sim
1M_{\text{J}}$, the distribution of total mass in planets shows a jump. This
is because massive planets can begin to accrete significant amounts of gas.
For the core-accretion scenario, this plot suggests that similar architectures
occur for low-mass disks because they cannot produce massive giant planets.
Gas giants are very effective in inducing dynamical stirring, which are in
turn responsible for shaping the system architecture. This signifies the role
played by physical processes in producing the mixed, anti-ordered, and ordered
architectures111The architecture framework is not sensitive to the absolute
value of a planetary quantity, such as mass, but only the ratio of the
quantities for adjacent planets. Independent of the architecture framework, we
will present another system-level framework analysing the state of a planetary
system. This other classification framework is sensitive to the absolute mass
of a planet and will address the role of giant planets on system-level
properties. The state classification framework reveals a drastic difference
between systems with and without giant planets (Mishra et al. in prep.)..

###  3.2 Lifetime of the protoplanetary disk

Figure 3: Role of disk lifetime on system architecture. Left: Binned relative
counts of architecture classes as a function of disk lifetime. The length of
error bars corresponds to the total number of systems in each bin, as:
${100}/{\sqrt{\mathrm{bin\ counts}}}$. Right: Scatter plot shows the disk gas
mass as a function of disk lifetime. The solid lines show the binned average
gas disk mass for each architecture class.

In this section, we explore the role of disk lifetime (i.e. the age of a
protoplanetary disk) in defining the final architecture class of a system. The
lifetime of a disk, in the Bern model, is influenced by the external disk
photo-evaporation rate (see Emsenhuber et al. (2021a) for details) and the
mass of the disk.

Figure 3 (left) shows the binned relative count of system architecture as a
function of disk lifetime. About $80\%$ of all disks with lifetimes ranging
from 1 to 5 Myr produce systems of the similar architecture class. The
relative count of similar systems decreases as disk lifetime increases. The
relative count of mixed architecture does not show any significant variation
with disk lifetime. The relative counts of anti-ordered and ordered
architectures vary as the disk lifetime increases. This suggests that the
physical mechanisms by which disks shape the final architectures of systems
play a role in shaping similar, anti-ordered, and ordered architectures.

The trends of the relative counts of architecture classes with disk lifetime
are similar to the distribution of relative counts as functions of disk mass.
We would like to understand whether system architecture is influenced by disk
lifetime directly or via an inherent dependence of disk lifetime on disk mass.
The right panel of Fig. 3 shows the gas disk mass as a function of disk
lifetime. The scatter plot depicting each individual disk shows that,
generally, low-mass disks have short lifetimes. The solid lines depict the
average gas mass for each architecture class for each disk lifetime bin.

The gas mass of the disks that go on to form systems of mixed, anti-ordered,
or ordered architecture shows a weak dependence on disk lifetime. On average,
the more massive disks seem to last longer. For disks that give rise to the
similar architecture class, this trend is clearly visible. If more massive
disks also live longer, this partly explains the relative count distribution
seen in Fig. 3 (left).

However, disks also affect the planetary architecture in other interesting
ways, namely orbital migration and eccentricity, and inclination damping. We
study the effect of these planet–disk interactions in shaping system
architecture in Sect. 4.1.

##  4 Nurture: Role of dynamical stirring

Whether or not the final architecture of a planetary system is pre-determined
by its initial conditions from the host star and the protoplanetary disk
remains unclear. If not, the mechanism by which dynamical processes shape the
architecture of a planetary system remains to be determined. It also remains
unclear as to whether or not dynamical processes remove all traces of initial
conditions from the final system, or whether these stochastic processes leave
their impressions on the final architecture. In this section, we try to answer
these questions. We focus our attention on dynamical interactions between
planets and the protoplanetary disk, and the gravitational multi-body
interactions amongst planets themselves.

While there exist several dynamical mechanisms that shape the final
architecture, we simplify the task before us by concentrating on violent
dynamical instabilities that change a planetary system in a non-trivial
manner. For each synthetic planetary system, we count the number of
planet–planet mergers, planetary ejections, and planets falling into their
host star. We use these counts as a proxy to assess the strength of dynamical
interactions that occur in a system. In the subsequent subsections, we study
planet–disk interactions and planet–planet interactions (mergers, ejections,
stellar accretion). These dynamical effects give rise to stochasticity and are
thereby inherently unpredictable. However, we hope that the underlying
dynamical processes that are sculpting the system architecture emerge as
patterns in the counts of these violent events.

###  4.1 Planet–disk interactions

Figure 4: Effect of planet–disk interactions on architecture. Left: Scatter
plot shows the number of planet–planet mergers and planetary ejections that
occurred in systems as a function of disk lifetime. The solid lines show the
average counts for each disk lifetime bin. Right: Distribution of the total
number of mergers (dashed) and ejections (solid) for the entire synthetic
population. The black line depicts the nominal synthetic population, and the
red line depicts a different synthetic population in which the disk-)planet
interactions were artificially switched off.

Protoplanetary disks interact with planets via several mechanisms. Planets may
experience orbital migration via gravitation interactions with the disk. Low-
mass planets undergo type I migration, which in the Bern model is implemented
following the approaches of Coleman & Nelson (2014); Paardekooper et al.
(2011). Massive planets may open a gap in the disk and undergo type II
migration (Dittkrist et al. 2014). The disk also dampens the eccentricity and
inclination of planets, which is coherently applied within the N-body
integrator. Readers interested in the details of the implementation are
referred to Emsenhuber et al. (2021a, b).

Figure 4 (left) shows the count of mergers and ejections for each planetary
system in our synthetic population as a function of the lifetime of its
protoplanetary disk. For an easier visualisation of any underlying trend, we
also show the average merger and ejection counts for each disk lifetime bin.
The number of planet–planet mergers shows a clear correlation with disk
lifetime. Disks that live longer usually give rise to planetary systems that
undergo more mergers than short-lived disks. We refer to this correlation as
‘migration assisted mergers’. One possible explanation for this correlation
could be that disks allow planets to migrate depending on their mass 222There
could be other scenarios which contribute to the ‘migration assisted mergers’
correlation. For example, migration may allow planets to become more massive
by accreting more material due to increased access to planetesimals (Alibert
et al. 2005). Massive planets may interact more amongst themselves, leading to
more mergers.. Two adjacent planets that are not migrating at the same rate,
perhaps owing to their different masses, can come close enough for a merger to
occur. The number of ejections does not show any clear trend with disk
lifetime. Disks dampen a planet’s eccentricity and inclination. As ejection
requires extremely violent interactions (marked by high eccentricities and
inclinations), disks may essentially inhibit planetary ejections.

To test these ideas, we simulated another population of 1000 planetary
systems. In this population (NG140), planet–disk interactions (gas-driven
migrations, and eccentricity and inclination damping) are artificially
switched off. For all such systems, we count the number of mergers and
ejections and compare them with our nominal population. Figure 4 (right) shows
the distribution of the number of planet–planet mergers and planetary
ejections in the two populations.

As expected, the number of planet–planet mergers decreases (distribution
shifts to the left) when planet–disk interactions are switched off. This
confirms the migration-assisted mergers correlation presented above. The
distribution of ejections, on the other hand, increases significantly when
planet–disk interactions are switched off. When the damping of the planetary
eccentricity and inclination by the disk is switched off, the gravitational
interactions between planets increases, such that many planets are ejected.

We make two observations from the results presented so far. First, counts of
mergers and ejections seem to be a good proxy for the prevalence of dynamical
interactions, as they capture some of the well-established dynamical effects
concerning planet–disk interactions. Second, we observe that disks affect
system architecture in a multitude of ways. While disk mass shows a direct
relation to final architecture, disks also affect system architecture
indirectly by influencing the dynamical interactions that occur therein. Long-
living disks give rise to more mergers and inhibit planetary ejections.
Conversely, systems emerging from short-lived disks experience fewer mergers.

###  4.2 Planet–planet interactions

Figure 5: Effect of planet–planet interactions on system architecture. For
each architecture class, the panels show a histogram of the counts of
planet–planet mergers, ejections, and stellar accretion occurring in the
synthetic population. The y-axis in all panels is scaled to reflect the
percentage of systems in each of the four architecture classes. For example,
$100\%$ of all similar systems lost less than five planets via planetary
ejection.

Above, we show that planet–disk interactions in the Bern model may influence
the dynamical interactions occurring in a system. Now, in this section, we are
interested in understanding how these violent events shape the final
architecture of a system.

Planets interact with each other gravitationally. These multi-body
interactions are tracked via a N-body integrator in the Bern model. The end
result of some of the more violent interactions is that planets are lost via
one of several channels: planet–planet mergers333In our model, when the
distance between two planets becomes smaller than the sum of their radii, a
planet–planet collision is said to occur. We treat such merger events in a
simplified manner: the cores of the target–impactor pair are merged, the
lesser massive body loses its envelope, and the impact energy is added to the
merged new body following Broeg & Benz (2012), which determines what part of
the gaseous envelope is ejected., planetary ejections, accretion by the host
star, and so on. These channels allow a planetary system to fundamentally
alter itself and its architecture.

Figure 5 shows, for each architecture class, the distribution of planet–planet
mergers and the number of planets lost via ejections and stellar accretion. At
first glance, losing planets to the host star may not seem appropriate for
planet–planet interactions. However, many of these planets meet their fate, in
the Bern model, when they are pushed inwards after being captured in mean-
motion resonances with other planets444The model also includes inward
migration of planets as a result of the stellar tides.. Therefore, this
channel of losing planets is included here. We caution the reader that the
absolute number of planets lost via any channel is model-dependent. The
quantity of interest here is the relative difference between the different
architecture classes.

Figure 5 suggests that the similar architecture class is almost completely
shaped by planet–planet mergers. Most similar systems in our simulations have
between 40 and 80 mergers taking place within them, and the median number of
mergers is 63. Violent dynamical interactions that lead to the ejection of
planets seems to be very rare in this architecture type, as $100\%$ of all
similar systems lose less than five planets via planetary ejection (median
ejections is 0). Likewise, similar systems seem to not rely on the stellar
accretion channel for losing planets (median stellar accretions is 0).

Systems with mixed architecture also undergo many planet–planet mergers. The
number of mergers in mixed systems ranges from 50 to 85, and the median of
mergers is 70. In a clear contrast from similar architectures, the ejection
and stellar accretion channels play an important role for mixed systems. The
median number of planets lost via ejections is 7, and via stellar accretions
is 2.

anti-ordered systems utilise all three dynamical channels. The distribution of
mergers in anti-ordered systems is roughly similar to that of mixed systems.
The range is between 50 and 85 and the median number of mergers is 67.
However, anti-ordered systems tend to lose more planets via the ejection
channel. The number of planets lost via dynamical ejection ranges from 0 to 35
with a median value of 14.5. Compared to mixed systems, anti-ordered systems
also tend to lose more planets via stellar accretion (median is 6).

Amongst the four architecture classes, ordered systems seem to undergo the
greatest number of dynamical interactions. The distribution of planet–planet
mergers in ordered systems shows a tail-like feature. The number of mergers
ranges from 55 to 85, with 62 being the median. All ordered systems eject at
least five planets. The number of ejections has a range from 5 to 35, and the
median is 23. The distribution of planets lost via the stellar accretion
channel shows a shift to the right. The number of planets accreted by the star
ranges from 0 to 20 with 8 being the median.

A comprehensive picture of the role of dynamical history in shaping the final
architecture emerges from the four panels in Fig. 5. Similar systems tend to
rely only on the merger channel for shaping their system architecture. As
planetary systems in all four architecture classes undergo a considerable
number of mergers, this channel may not suffice to explain or distinguish the
emergence of the four architecture classes. This is in line with what was
found before, namely that the emergence of the similar class is mostly
governed by the initial conditions.

While initial conditions seem to decide whether a system becomes similar or
one of the other three architectures, there appears to be a trend in the role
of dynamical interactions in shaping mixed, anti-ordered, and ordered
architectures. The distributions of the ejection and accretion channels
distinguish these three architectures. These distributions show a shift to the
right, indicating that more planets are being lost via these two channels as
we move from mixed to anti-ordered and to ordered architectures. Thus, we
conclude that if initial conditions do not allow a system to become similar,
its fate is decided by its dynamical history, among other effects. If the
strength of the dynamical interactions increases in a system, the architecture
of the system changes from mixed to anti-ordered or to ordered.

All systems in the Bern model start with 100 protoplanetary embryos. Above, we
show that systems of different architecture show varying propensity to lose
planets via the different dynamical channels. This suggests that we should
also see an effect of the dynamical history of the four architecture classes
in their multiplicity distribution. We observed this effect in Fig. 6 of Paper
I. We do not have a way to determine the initial number of embryos of the
planetary systems we observe today. Our approach may therefore not be directly
applicable to observed planetary systems. We remind the reader that while the
quantitative aspects we present in this section are probably model dependent,
the qualitative nature of these results is of paramount importance.

##  5 The Aryabhata formation scenario

In this section, we propose a planet-formation scenario to explain a feature
observed by Paper I (Sect. 5.4). We found that many synthetic planetary
systems have a peculiar water-mass-fraction architecture namely that all
planets hosted in these systems are water-rich worlds. We explain this
peculiar feature with the ‘Aryabhata formation scenario’.

The first exoplanets to be discovered were hot Jupiters —giant planets
orbiting their host stars at very short periods (Mayor & Queloz 1995). Orbital
migration was suggested as a possible mechanism to explain these short periods
(Lin et al. 1996; Lin & Ida 1997). Theoretical studies indicate that orbital
migration and planet–star tidal interactions should make many close-in planets
unstable. In the 1990s, Doug Lin described ‘the last of the Mohicans’ scenario
(Garaud 2011). In this scenario, the protoplanetary disk gives rise to
planets, many of which are doomed to fall onto the star. The surviving
observable planets are those that were able to escape annihilation.

For some simulated systems, we noticed a modified version of this scenario.
Protoplanetary disks seem to give rise to planets at different epochs. In the
first epoch, several intermediate-mass planets ($1-100M_{\oplus}$) are formed
within the first 1Myr. Most of these ‘first generation’ planets are
subsequently lost mainly via giant impacts (and a few are lost via orbital or
tidal migration leading to stellar accretion). This purging phase is
catastrophic to all planets that started within the ice line. Over the next
few million years, a second epoch sees the advent of a ‘second generation’ of
planets. Most of these second-generation planets are born outside the ice
line, and are able to migrate inwards during the disk lifetime. After disk
dissipation, migration comes to a halt and many of these planets survive long-
term N-body evolution in our simulations. We call this the Aryabhata formation
scenario. The key difference between the two scenarios is that in the
Aryabhata formation scenario (a) planets (surviving and lost) are born in
different epochs, and (b) most first-generation planets are lost via giant
impacts.

We quantify this scenario with the Aryabhata’s number, $\mu$, which is the
ratio of the surviving planets that started inside the ice line to the total
number of surviving planets:

$\text{Aryabhata's number:}\ \mu=\frac{n(a^{\text{start}}_{\text{embryo}}\leq
a_{\text{ice}})}{n}.$ (2)

At the start of our calculations, all systems have an Aryabhata’s number
$\approx 0.5\pm 0.1$. Figure 12 of Paper I (middle) shows the ice mass
fraction architecture of simulated planetary systems. The colour of each point
shows the Aryabhata’s number.

Most planetary systems with $C_{S}(f_{ice})\approx C_{V}(f_{ice})\approx 0$
have $\mu$ close to zero. This suggests that most (or all) of the surviving
planets in such systems started outside the ice line. The formation path of
these systems falls into the Aryabhata formation scenario. These classes of
systems can be identified by two characteristics: (i) the core water-mass
fraction for different planets in these systems is similar, and (ii) the core
water-mass fraction for most planets is high (owing to their origin outside
the ice line) making them water-rich planets. Approximately, one-fifth of the
simulated systems fall into this scenario. Among these, about half are of
similar class, one-third are anti-ordered, and the remaining systems have
either a mixed or ordered mass architecture.

There exists an almost linear relationship between $C_{V}(f_{ice})$ and $\mu$.
Using scipy’s linear regression module, we obtain a slope of 1.8 and intercept
of 0.18 between these two quantities. The correlation coefficient is $R=0.95$,
indicating a strong correlation between the Aryabhata’s number and the
coefficient of variation of core water mass fraction. This suggests a
possibility to identify observed exoplanetary systems that may have originated
via the Aryabhata formation scenario. By determining the $C_{V}(f_{ice})$ of a
system, the Aryabhata’s number can be estimated. Systems with low $\mu$ values
probably arose from this scenario.

For systems that fall into the default scenario (positive $C_{S}(f_{ice})$,
implying an increasing core water mass fraction inside-out), the Aryabhata’s
number is $\mu>0$. We note that most systems with $\mu\gtrapprox 0.6$ show
similarity in their mass architecture.

Overall, the intra-system core water-mass-fraction architecture of most
planetary systems seems to take one of two forms. (i) Those characterised by
$C_{S}(f_{ice})\approx C_{V}(f_{ice})\approx 0$ and $\mu=0$. These systems are
composed of water-rich planets wherein the core water mass fraction is similar
across the different planets. All surviving planets in these systems started
outside the ice line. The Aryabhata formation scenario explains these systems.
(ii) Those with $C_{S}(f_{ice})>0$ and $\mu>0$. These systems represent the
‘default’ or common outcome of our simulations. The planetary core water-mass
fraction in these systems increases from one planet to another with increasing
distance from the host star. Some of the surviving planets started from inside
the ice line. At the extreme end, systems in which $60\%$ or more surviving
planets started inside the ice line tend to have a similar mass architecture.

##  6 Summary, conclusions, and future work

Paper I of this series introduced a novel, model-independent framework for
characterising the architecture of planetary systems at the system level.
Planetary-system architectures can be separated into four classes: similar,
mixed, anti-ordered, and ordered. This classification is achieved via two
quantities: the coefficient of similarity and the coefficient of variation.
The mathematical $C_{S}$ versus $C_{V}$ architecture space was found to have
forbidden regions – regions in which no planetary system can exist. In Paper
I, the mass architecture classes of observed and synthetic systems were
characterised. The mass architecture of synthetic systems was compared with
their radii architecture, bulk-density architecture, core-mass architecture,
spacing architecture, and water-mass-fraction architecture. As in Paper I, we
identify a system’s architecture with its mass architecture.

In this paper, we explore the core-accretion-based formation pathways —around
a solar-like star— of the four classes of planetary system architecture. We
tried to disentangle the role of nature (initial conditions of planet
formation) from that of nurture (physical processes occurring during planet
formation). Our findings can be summarised as follows:

  1. 1.

System-level analysis: Our findings show that a system-level analysis of
planetary system architecture via our architecture framework (Paper I)
provides an abundance of information. We show that planetary formation and
evolution process leave their imprint on the entire system architecture.

  2. 2.

Solid disk mass: The initial amount of solids in the protoplanetary disk in
our models plays an important role in deciding the architectural fate of a
planetary system. Disks with a solid mass (initial content of planetesimals)
of $\lesssim 1M_{\text{J}}$ almost always give rise to systems with similar
architecture. Mixed architectures arise most often from disks with solid
masses $\approx 1M_{\text{J}}$. Disks with solid mass $\gtrsim 1M_{\text{J}}$
favour the production of anti-ordered and ordered architectures.

  3. 3.

Gas disk mass and metallicity: Initial gas disk mass and stellar metallicity
influences the final architecture of a planetary system by controlling the
initial mass of solids in the disk. Metallicity, in our models, is simply
related to the dust-to-gas ratio, which allows us to convert a fraction of the
initial gas disk mass into initial dust mass (eq. 1). Applying the
architecture framework on the synthetic systems from the Bern model allows us
to predict the existence of a metallicity–architecture correlation. The
observed correlation between metallicity and final architecture is in
qualitative agreement with the Bern model.

  4. 4.

Metallicity–architecture correlation: The architecture of a planetary system
correlates with the metallicity of the host star. Most systems hosted by a
low-metallicity star ($Fe/H<-0.2$) are of similar architecture. As the
metallicity of the star increases, mixed, ordered, and anti-ordered
architectures become increasingly common.

  5. 5.

Disk lifetime: The occurrence of systems of a similar architecture around
short-lived disks is high, and their frequency reduces around long-lived
disks. The frequency of anti-ordered architecture increases as disk lifetime
increases. These correlations are mediated in at least two ways. First, disks
interact with planets, where orbital migration and eccentricity and
inclination damping occur. Due to the ‘migration assisted merger’ correlation,
long-lasting disks allow planetary systems to have, in general, more
planet–planet mergers and inhibit planetary ejections. These dynamical events
shape a system’s final architecture. In addition, in our model, disk lifetimes
are correlated with disk masses, which also strongly influences the system
architecture.

  6. 6.

Dynamical interactions: Planetary systems can significantly alter their
architecture via (at least) three dynamical channels: planet–planet mergers,
planetary ejections, and accretion by the host star. All architecture classes
in our formation model were found to undergo numerous merger events. Similar
systems rely entirely on mergers to shape their final architecture. As the
strength of the dynamical interactions experienced by a system (quantified by
the number of ejections and/or accretions) increases, the architecture of a
system shifts from mixed to anti-ordered to ordered.

  7. 7.

The Aryabhata formation scenario: Systems following this formation scenario
have the following formation pathway. First-generation planets (formed within
1 Myr) are lost mostly via giant impacts. Second-generation planets started
outside the ice line and migrated inwards. The surviving planets are from the
second generation and shape the architecture of the system. This scenario
explains about $20\%$ of simulated systems in which the core water-mass-
fraction architecture is different from the default scenario. Systems
following this formation scenario (i) host only those planets that have a high
core water-mass fraction and (ii) host only those planets that started outside
the ice line. We introduce the Aryabhata’s number to identify those systems
that follow this formation scenario and find that $80\%$ of all anti-ordered
simulated systems are formed via the Aryabhata formation scenario.

  8. 8.

Nature versus nurture: Overall, our model suggests that initial conditions —or
‘nature’— dictate whether a system will have a similar architecture or one of
the other three architecture classes, namely mixed, anti-ordered, or ordered
(via initial disk mass). If nature does not allow a system to have a similar
mass architecture, then the final architecture is controlled by ‘nurture’, or
dynamical interactions, among other possible effects. As the dynamical
interactions increase, the final architecture tends to become mixed, anti-
ordered, and then ordered.

We would like to offer readers some warning when interpreting our results.
Although the architecture framework (from Paper I) is model-independent, the
present results hinge critically on the underlying planet formation model –
the Bern model. There are several assumptions, simplifications, and choices to
be made when simulating synthetic planetary systems using the Bern model. For
example, the treatment of planet–planet merging collisions is relatively
simple (Ali-Dib et al. 2022). We also assume simplified planet-formation
conditions; that is, our star–disk–planet system is isolated enough so that we
may ignore the influence of the stellar neighbourhood, stellar flybys, and so
on (Bate 2012, 2018). The main strength of this study does not lie in
providing an explanation of the formation pathway of any particular system.
Instead, our main result is the observation that when groups of planetary
systems are identified (architecture classes), general trends in formation
pathways emerge. This allowed us to map the roles of nature and nurture in
shaping the final architecture of a planetary system.

The results of this study can be strengthened or challenged in several
observational and theoretical ways. We list some possibilities for future
studies emerging from this work:

  1. 1.

Linking disk mass distribution and architecture occurrence rates: Our model
suggests that there should be a direct relationship between the mass of the
solid disk and the final architecture of a system. While initial disk masses
and the final architecture of the same system will forever remain
unobservable, this relation can be tested statistically. The distribution of
initial disk masses and the distribution of final system architecture can be
linked by formation models. We speculate that in future, when these two
distributions become available, formation models can be used to predict one or
the other. In fact, this problem can also be turned around; we can identify
the right family of models as those that correctly link the observed
distributions of protoplanetary disk masses and architecture occurrence rates.
We believe such tests are crucial for the development and eventual emergence
of a standard model for exoplanetary astrophysics.

  2. 2.

Metallicity–architecture correlation: Our work suggests that the current
architecture of a planetary system should be related to the metallicity of its
host star. As both of these are observable, testing this
metallicity–architecture correlation should be feasible. Here, we used a
catalogue of 41 observed multi-planet systems (from Paper I) to test this
correlation. We find a qualitative agreement between theory and observations.
However, our observational catalogue suffers from incompleteness and low-
number statistics, which prevents us from making any further assertions. More
observational data are required to confirm or reject the proposed metallicity-
architecture correlation. It would also be interesting to estimate the current
architecture occurrence rate based on the known metallicity distributions.

  3. 3.

Confirming formation pathways: Confirming the formation pathways discovered in
the present study with observations is challenging. However, the strength of
our results will increase if different planet-formation models are studied
through the architecture framework. Hence, one possible line of future work
involves repeating the present study using different planet-formation models.

  4. 4.

Extending the architecture framework: So far, we have calibrated our
classification scheme for the mass architectures only. Calibrating the
architecture classification framework on other quantities maybe useful.
Especially for planetary radii, which are observable via transit surveys, the
use of machine learning methods may be necessary.

  5. 5.

Temporal evolution of system architecture: In the nominal Bern model
population studied in this paper, protoplanetary embryos of 100 lunar masses
are initialised in the protoplanetary disk at the start. This necessarily
implies that all planetary systems start as similar type systems. It would be
interesting to inquire whether this is generally true in nature as well. If
this is the case, this implies that the ‘default’ architecture of all
planetary systems is similar and the physical processes playing out in the
system evolve this architecture into other possibilities. Investigating this
may lead to deep insights into the structure of planetary system architecture.
In addition, such studies would be necessary to interpret the observed
architecture occurrences, as observed planetary systems are seldom of the same
age.

  6. 6.

External perturbations: Stellar flybys or multi-planetary systems around
binaries provide excellent theoretical and observational laboratories with
which to study the influence of external perturbations on the architecture of
planetary systems. This problem, when turned around, is also useful in
deducing the perturbed or dynamical (or lack of) history of observed planetary
systems.

This paper presents new insights obtained by analysing planetary systems at
the system-level. We showed that several patterns emerged in the formation
pathways of the four architecture classes. These patterns linked the initial
conditions of planet formation with the final architecture of a system –
bridging the vast temporal gap of several billions of years between the birth
of planets to their final assembly.

###### Acknowledgements.

This work has been carried out within the frame of the National Centre for
Competence in Research PlanetS supported by the Swiss National Science
Foundation. We acknowledge the support of the Swiss National Fund under grant
200020_172746 and 200021_204847 “PlanetsInTime”. LM acknowledges the generous
hospitality of the ”Planet Formation” workshop by the Munich Institute for
Astro-, Particle and BioPhysics (MIAPbP) which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC-2094 – 390783311.
Data: The synthetic planetary populations (NGPPS) used in this work are
available online at http://dace.unige.ch. Software: Python (Van Rossum & Drake
2009), NumPy (Oliphant 2006), Seaborn (Waskom & the seaborn development team
2020), Pandas (pandas development team 2020), Matplotlib (Hunter 2007).

## References

  * Adams (2019) Adams, F. C. 2019, MNRAS, 488, 1446
  * Adams et al. (2020) Adams, F. C., Batygin, K., Bloch, A. M., & Laughlin, G. 2020, Monthly Notices of the Royal Astronomical Society, 493, 5520
  * Adibekyan et al. (2021) Adibekyan, V., Santos, N. C., Demangeon, O. D. S., et al. 2021, Astronomy & Astrophysics, 649, A111
  * Ali-Dib et al. (2022) Ali-Dib, M., Cumming, A., & Lin, D. N. C. 2022, MNRAS, 509, 1413
  * Alibert (2019) Alibert, Y. 2019, Astronomy & Astrophysics, 624, A45
  * Alibert et al. (2013) Alibert, Y., Carron, F., Fortier, A., et al. 2013, Astronomy & Astrophysics, 558, A109
  * Alibert et al. (2004) Alibert, Y., Mordasini, C., & Benz, W. 2004, Astronomy & Astrophysics, 417, L25
  * Alibert et al. (2011) Alibert, Y., Mordasini, C., & Benz, W. 2011, A&A, 526, A63
  * Alibert et al. (2005) Alibert, Y., Mordasini, C., Benz, W., & Winisdoerffer, C. 2005, Astronomy & Astrophysics, 434, 343
  * Armitage (2010) Armitage, P. J. 2010, Astrophysics of Planet Formation
  * Baraffe et al. (2015) Baraffe, I., Homeier, D., Allard, F., & Chabrier, G. 2015, Astronomy & Astrophysics, 577, A42
  * Bashi & Zucker (2021) Bashi, D. & Zucker, S. 2021, A&A, 651, A61
  * Bate (2012) Bate, M. R. 2012, MNRAS, 419, 3115
  * Bate (2018) Bate, M. R. 2018, MNRAS, 475, 5618
  * Benz et al. (2014) Benz, W., Ida, S., Alibert, Y., Lin, D., & Mordasini, C. 2014, in Protostars and Planets VI, ed. H. Beuther, R. Klessen, C. Dullemond, & T. Henning (University of Arizona, Tucson), 691–713
  * Broeg & Benz (2012) Broeg, C. H. & Benz, W. 2012, A&A, 538, A90
  * Burn et al. (2021) Burn, R., Schlecker, M., Mordasini, C., et al. 2021, Astronomy & Astrophysics, 656, A72
  * Chambers (1999) Chambers, J. E. 1999, Monthly Notices of the Royal Astronomical Society, 304, 793
  * Ciardi et al. (2013) Ciardi, D. R., Fabrycky, D. C., Ford, E. B., et al. 2013, The Astrophysical Journal, 763, 41
  * Clarke et al. (2001) Clarke, C. J., Gendrin, A., & Sotomayor, M. 2001, Monthly Notices of the Royal Astronomical Society, 328, 485
  * Coleman & Nelson (2014) Coleman, G. A. & Nelson, R. P. 2014, Monthly Notices of the Royal Astronomical Society, 445, 479
  * Dittkrist et al. (2014) Dittkrist, K. M., Mordasini, C., Klahr, H., Alibert, Y., & Henning, T. 2014, Astronomy & Astrophysics, 567 [arXiv:1402.5969]
  * Emsenhuber et al. (2021a) Emsenhuber, A., Mordasini, C., Burn, R., et al. 2021a, Astronomy & Astrophysics, 656, A69
  * Emsenhuber et al. (2021b) Emsenhuber, A., Mordasini, C., Burn, R., et al. 2021b, Astronomy & Astrophysics, 656, A70
  * Fabrycky et al. (2014) Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2014, The Astrophysical Journal, 790, 146
  * Fang & Margot (2013) Fang, J. & Margot, J.-L. 2013, The Astrophysical Journal, 767, 115
  * Fortier et al. (2013) Fortier, A., Alibert, Y., Carron, F., Benz, W., & Dittkrist, K.-M. 2013, Astronomy & Astrophysics, 549, A44
  * Garaud (2011) Garaud, P. 2011, The Astrophysical Journal Letters, Volume 728, Issue 2, article id. L30, ¡NUMPAGES¿5¡/NUMPAGES¿ pp. (2011)., 728, L30
  * Gilbert & Fabrycky (2020) Gilbert, G. J. & Fabrycky, D. C. 2020, The Astronomical Journal, 159, 281
  * Gladman (1993) Gladman, B. 1993, Icarus, 106, 247
  * He et al. (2019) He, M. Y., Ford, E. B., & Ragozzine, D. 2019, Monthly Notices of the Royal Astronomical Society, 490, 4575
  * He et al. (2021) He, M. Y., Ford, E. B., & Ragozzine, D. 2021, AJ, 161, 16
  * Hueso & Guillot (2005) Hueso, R. & Guillot, T. 2005, Astronomy & Astrophysics, 442, 703
  * Hunter (2007) Hunter, J. D. 2007, Computing in science & engineering, 9, 90
  * Jin et al. (2014) Jin, S., Mordasini, C., Parmentier, V., et al. 2014, ApJ, 795, 65
  * Kipping (2018) Kipping, D. 2018, Monthly Notices of the Royal Astronomical Society, 473, 784
  * Kokubo & Ida (1998) Kokubo, E. & Ida, S. 1998, Icarus, 131, 171
  * Kokubo & Ida (2002) Kokubo, E. & Ida, S. 2002, The Astrophysical Journal, 581, 666
  * Laskar (1997) Laskar, J. 1997, Large scale chaos and the spacing of the inner planets., Tech. rep.
  * Laskar (2000) Laskar, J. 2000, Physical Review Letters, 84, 3240
  * Laskar & Petit (2017) Laskar, J. & Petit, A. C. 2017, Astronomy & Astrophysics, 605, 1
  * Lin et al. (1996) Lin, D. N., Bodenheimer, P., & Richardson, D. C. 1996, Nature, Volume 380, Issue 6575, pp. 606-607 (1996)., 380, 606
  * Lin & Ida (1997) Lin, D. N. C. & Ida, S. 1997, The Astrophysical Journal, Volume 477, Issue 2, pp. 781-791., 477, 781
  * Lissauer et al. (2011) Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, The Astrophysical Journal Supplement Series, 197, 8
  * Lodders (2003) Lodders, K. 2003, The Astrophysical Journal, 591, 1220
  * Lynden-Bell & Pringle (1974) Lynden-Bell, D. & Pringle, J. E. 1974, Monthly Notices of the Royal Astronomical Society, 168, 603
  * Manara et al. (2019) Manara, C. F., Mordasini, C., Testi, L., et al. 2019, Astronomy & Astrophysics, 631, L2
  * Marboeuf et al. (2014a) Marboeuf, U., Thiabaud, A., Alibert, Y., Cabral, N., & Benz, W. 2014a, Astronomy and Astrophysics, 570 [arXiv:1407.7282]
  * Marboeuf et al. (2014b) Marboeuf, U., Thiabaud, A., Alibert, Y., Cabral, N., & Benz, W. 2014b, Astronomy and Astrophysics, 570 [arXiv:1407.7271]
  * Matsuyama et al. (2003) Matsuyama, I., Johnstone, D., & Murray, N. 2003, The Astrophysical Journal, 585, L143
  * Mayor & Queloz (1995) Mayor, M. & Queloz, D. 1995, Nature, 378, 355
  * Millholland et al. (2017) Millholland, S., Wang, S., & Laughlin, G. 2017, The Astrophysical Journal, 849, L33
  * Millholland & Winn (2021) Millholland, S. C. & Winn, J. N. 2021, ApJ, 920, L34
  * Mishra et al. (2021) Mishra, L., Alibert, Y., Leleu, A., et al. 2021, Astronomy & Astrophysics, 656, A74
  * Mishra et al. (2019) Mishra, L., Alibert, Y., & Udry, S. 2019, in EPSC-DPS Joint Meeting 2019, held 15-20 September 2019 in Geneva, Switzerland, id. EPSC-DPS2019-1616, Vol. 2019, EPSC–DPS2019–1616
  * Mishra et al. (2023) Mishra, L., Alibert, Y., Udry, S., & Mordasini, C. 2023, Astronomy & Astrophysics
  * Mordasini (2018) Mordasini, C. 2018, in Handbook of Exoplanets, ed. H. J. Deeg & J. A. Belmonte, 143
  * Mordasini et al. (2009) Mordasini, C., Alibert, Y., & Benz, W. 2009, Astronomy & Astrophysics, 501, 1139
  * Mordasini et al. (2012a) Mordasini, C., Alibert, Y., Georgy, C., et al. 2012a, Astronomy & Astrophysics, 547, A112
  * Mordasini et al. (2012b) Mordasini, C., Alibert, Y., Klahr, H., & Henning, T. 2012b, Astronomy & Astrophysics, 547, A111
  * Mulders et al. (2020) Mulders, G. D., O’brien, D. P., Ciesla, F. J., Apai, D., & Pascucci, I. 2020
  * Mulders et al. (2021) Mulders, G. D., Pascucci, I., Ciesla, F. J., & Fernandes, R. B. 2021 [arXiv:2107.12520]
  * Nakamoto & Nakagawa (1994) Nakamoto, T. & Nakagawa, Y. 1994, The Astrophysical Journal, 421, 640
  * Obertas et al. (2017) Obertas, A., Van Laerhoven, C., & Tamayo, D. 2017, Icarus [arXiv:1703.08426]
  * Oliphant (2006) Oliphant, T. E. 2006, A guide to NumPy, Vol. 1 (Trelgol Publishing USA)
  * Paardekooper et al. (2011) Paardekooper, S. J., Baruteau, C., & Kley, W. 2011, Monthly Notices of the Royal Astronomical Society, 410, 293
  * pandas development team (2020) pandas development team, T. 2020, pandas-dev/pandas: Pandas
  * Petigura et al. (2018) Petigura, E. A., Marcy, G. W., Winn, J. N., et al. 2018, The Astronomical Journal, 155, 89
  * Petit et al. (2018) Petit, A. C., Laskar, J., & Boué, G. 2018, Astronomy & Astrophysics, 617, A93
  * Pollack et al. (1996) Pollack, J. B., Hubickyj, O., Bodenheimer, P., et al. 1996, Icarus, 124, 62
  * Pu & Wu (2015) Pu, B. & Wu, Y. 2015, The Astrophysical Journal, Volume 807, Issue 1, article id. 44, ¡NUMPAGES¿10¡/NUMPAGES¿ pp. (2015)., 807, 44
  * Sandford et al. (2021) Sandford, E., Kipping, D., & Collins, M. 2021, Monthly Notices of the Royal Astronomical Society, Volume 505, Issue 2, pp.2224-2246, 505, 2224
  * Santos et al. (2005) Santos, N. C., Israelian, G., Mayor, M., et al. 2005, Astronomy & Astrophysics, 437, 1127
  * Sarkis et al. (2021) Sarkis, P., Mordasini, C., Henning, T., Marleau, G. D., & Mollière, P. 2021, A&A, 645, A79
  * Schib et al. (2021) Schib, O., Mordasini, C., Wenger, N., Marleau, G. D., & Helled, R. 2021, A&A, 645, A43
  * Schlecker et al. (2021a) Schlecker, M., Mordasini, C., Emsenhuber, A., et al. 2021a, Astronomy and Astrophysics, 656, A71
  * Schlecker et al. (2021b) Schlecker, M., Pham, D., Burn, R., et al. 2021b, Astronomy and Astrophysics, 656, A73
  * Shakura & Sunyaev (1973) Shakura, N. I. & Sunyaev, R. A. 1973, Astronomy & Astrophysics, 24, 337
  * Tamayo et al. (2020) Tamayo, D., Gilbertson, C., & Foreman-Mackey, D. 2020, Stability constrained characterization of multiplanet systems
  * Thiabaud et al. (2014) Thiabaud, A., Marboeuf, U., Alibert, Y., et al. 2014, Astronomy & Astrophysics, 562 [arXiv:1312.3085]
  * Tremaine (2015) Tremaine, S. 2015, Astrophysical Journal, 807, 157
  * Turrini et al. (2020) Turrini, D., Zinzi, A., & Belinchon, J. A. 2020, Astronomy and Astrophysics, 636 [arXiv:2003.05366]
  * Udry & Santos (2007) Udry, S. & Santos, N. C. 2007, Annual Review of Astronomy and Astrophysics, 45, 397
  * Van Rossum & Drake (2009) Van Rossum, G. & Drake, F. L. 2009, Python 3 Reference Manual (Scotts Valley, CA: CreateSpace)
  * Veras & Armitage (2004) Veras, D. & Armitage, P. J. 2004, Monthly Notices of the Royal Astronomical Society, 347, 613
  * Wang et al. (2019) Wang, Y., lin Zhou, J., yao Liu, F., et al. 2019, Monthly Notices of the Royal Astronomical Society, 490, 359
  * Waskom & the seaborn development team (2020) Waskom, M. & the seaborn development team. 2020, mwaskom/seaborn
  * Weiss et al. (2018) Weiss, L. M., Marcy, G. W., Petigura, E. A., et al. 2018, The Astronomical Journal, 155, 48
  * Winter et al. (2020) Winter, A. J., Kruijssen, J. M., Longmore, S. N., & Chevance, M. 2020, Nature, 586, 528
  * Yeh et al. (2020) Yeh, L.-C., Jiang, I.-G., & Gajendran, S. 2020, Astrophysics and Space Science, 365 [arXiv:2012.09431]