1 Introduction

The advent of exoplanet discovery projects, such as the Kepler space telescope [1], the HARPS survey [2 by discussing in more details the concepts of planetary population synthesis followed by the history and different models in Sect. 3. The remainder of this work will focus on the link between the properties of the protoplanetary discs and the diversity of planetary systems and the emergence of four classes of planetary system architectures. The methodology is presented Sect. 4 and the corresponding results in Sect. 5.

2 Principles of planetary population synthesis

The arching goal of planetary population synthesis is to improve our understanding of planet formation by finding the theoretical model that provides the best match of the observed exoplanets from the properties of protoplanetary discs. This also enables to make predictions about the planets that have not been observed (yet). Such an endeavour encompasses many aspects, from theory to observations.

Figure 1 shows how these elements are linked. The starting point are the detailed models of individual processes like disc structure and evolution, planetary solid and gas accretion, orbital migration, planet–planet interaction, and so on. They inform the global end-to-end models which combine the essence of many such detailed models into one big model. This gives end-to-end models the capability to directly predict observable properties of planetary systems based on the initial conditions of the planet formation process, which are the protoplanetary disc properties.

Fig. 1
figure 1

Principle and flow chart of planetary population synthesis method and its links to the overall theory of planet formation and to the construction of astronomical instrumentation. Extended from Mordasini et al. [15]

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From observations it is known that not all protoplanetary discs have the same properties, in terms of essential disc properties like mass, size, or lifetime (e.g. [76, 106, 107].

Planet formation follows the core accretion paradigm [108,109,110]. Initially, a given number of embryos are randomly placed in the disc with a uniform spacing in the logarithm of the distance, so that they are separated by a given amount of Hill radii, as found in N-body simulations of embryo formation [111]. They will accrete planetesimals, which is taken to be in the oligarchic regime [112,113,114]. The gas envelope structure is initially in contact with the surrounding disc and accretion occurs thanks to radiating away the gravitational binding energy released during this process, i.e. on a Kelvin-Helmholtz timescale [115, 116]. The gas accretion rate is thus determined by solving the 1D spherically symmetric internal structure equation of the envelope [117]. The presence of a gaseous envelope also enhances the capture radius of planetesimals, which we compute following the procedure of Inaba and Ikoma [118]. As a protoplanet core grows, its Kelvin-Helmholtz timescale decreases [119]. The gas accretion rate thus increases until it becomes larger than what can be supplied by the disc. When this occurs, the envelope can no longer maintain the contact with the disc and contracts [120]. In this stage, the structure equations are used to compute the planet radius instead, while the gas accretion is set by the supply from the disc [50, 74].

Dynamical interactions between different protoplanets are modelled by means of the mercury N-body package [121]. Also, type I gas-driven migration is computed according to Coleman and Nelson [122] while type II and the transition between the two regimes follow Dittkrist et al. [123]. Migration is applied as additional forces in the N-body.

4.2 Evolution stage

After a fixed time of 100 \(\hbox{Myr}\) (compared to 20 \(\hbox{Myr}\) in [80] to increase the duration of the N-body interactions), the model transitions to the evolution stage, where the planet is individually evolved to 5 \(\hbox{Gyr}\). In addition to the normal thermodynamic evolution (cooling and contraction) of the interior which is found by solving 1D internal structure equations, this stage also tracks XUV-driven atmospheric escape [124, 125] and inward migration due to stellar tides [126, 127].

4.3 Analysis: mass scales

Because of the numerous interaction and feedback (see the many arrows in Fig. 2), global models can produce numerical results of considerable complexity which can be difficult to interpret. To understand what leads to the final shape of the final systems, it is therefore helpful to analytically investigate a number of mass scales of different processes that take place during the formation stage. The purpose is to compare these analytical mass scales to the numerical formation tracks to understand which process is linked to different features.

The first two mass scales are related to the accretion of planetesimals only. Temporally, the first process that occurs is this accretion of planetesimals. Absent radial motion caused by the gas disc (orbital migration) or interactions with other bodies, the embryos can only accrete from nearby planetesimals, which will halt once the local reservoir is depleted. This leads to the planetesimal isolation mass [128],

$$\begin{aligned} M_{\textrm{iso}}=\frac{\left( 2\pi b a^2\Sigma _{\textrm{p}}\right) ^{\frac{3}{2}}}{\sqrt{3M_\star }} \sim {3.6\times 10^{-2}}\,\hbox{M}_\oplus \left( \frac{a}{{1}\,{\hbox{au}}}\right) ^3\left( \frac{\Sigma _{\textrm{p}}}{{10}\,\hbox{g cm}^{-2}}\right) ^{\frac{3}{2}}\left( \frac{M_\star }{{1}\,\hbox{M}_\odot }\right) ^{-\frac{1}{2}}, \end{aligned}$$
(1)

with \(b\sim 10\) the full width of the feeding zone in Hill radii of the planet \(r_{\textrm{H}}=a\left( M/(3 M_\star )\right) ^{1/3}\), a the local position, \(\Sigma _{\textrm{p}}\) the local surface density of planetesimals, and \(M_\star\) the mass of the central star. At large separation, however, the accretion timescales become so slow that the isolation mass cannot be reached in any reasonable time. There, growth is rather stalled at a time comparable to the dispersal of the protoplanetary disc, which halts the dam** of the planetesimals random velocities and causes in turn the accretion rate to strongly decrease. We can use the analytic accretion timescales in the oligarchic regime [112] to estimate a growth mass (see [114], Eq. 14 for the derivation)

$$\begin{aligned} M_{\textrm{grow}}\sim & {} \dot{M}\tau _{\textrm{acc}}= k_{\textrm{grow}}\frac{t_{\textrm{disc}}^3 M_\star ^{1/2} \Sigma _{\textrm{g}}^{6/5} \Sigma _{\textrm{p}}^3}{\rho _m^{4/5} \rho _M a^{3/2} m^{2/5} (H/a)^{6/5}} \end{aligned}$$
(2)
$$\begin{aligned}\sim & {} {6.8}\,\hbox{M}_\oplus \left( \frac{t_{\textrm{disc}}}{{1\times 10^{6}}\,{\hbox{yr}}}\right) ^3\left( \frac{\Sigma _{\textrm{p}}}{{10}\,\hbox{g cm}^{-2}}\right) ^3\left( \frac{a}{{1}\,{\hbox{au}}}\right) ^{-\frac{3}{2}}\left( \frac{M_\star }{{1}\,\hbox{M}_\odot }\right) ^{\frac{1}{2}} \left( \frac{\Sigma _{\textrm{g}}}{{2400}\,\hbox{g cm}^{-2}}\right) ^{6/5}\nonumber \\{} & {} \left( \frac{\rho _m}{{1}\,\hbox{g cm}^{-3}}\right) ^{-4/5} \left( \frac{\rho _M}{{3.2}\,\hbox{g cm}^{-3}}\right) ^{-1} \left( \frac{m}{{1\times 10^{18}}\,{\hbox{g}}}\right) ^{-1/10} \left( \frac{H/a}{0.05}\right) ^{-6/5} \end{aligned}$$
(3)

with a constant prefactor \(k_{\textrm{grow}} = \frac{ \pi ^3 3^{21/15}}{4^{3/5} } \left( \frac{C_D b}{C_e}\right) ^{6/5} G^{3/2} \sim 11.9 \sqrt{G^3}\). Here, G is the Newtonian gravitational constant and reasonable values are for the drag coefficient \(C_D\sim 1\) [129], for the viscous stirring factor \(C_e \sim 40\) [107, 112], and for the width of the feeding zone \(b\sim 10\). Further, \(\Sigma _{\textrm{g}}\) is the local gas surface density, H the disc scale height often combined to the aspect ratio \(H/a \sim 0.1\), \(t_{\textrm{disc}}\) the lifetime of the gaseous disc, \(\rho _{m}\) and \(\rho _{M}\) are the bulk densities of the planetesimals and the protoplanet, and m is the mass of one planetesimal. Note that while \(M_{\textrm{grow}}\) is much larger than the other scales at short distances (within 1 \(\hbox{au}\)), it decreases most rapidly with distance for realistic disc and planetesimal surface density profiles. It should be noted that \(M_{\textrm{grow}}\) only makes sense in the outer parts of the disc (outside of \(\sim 10\) AU) where it holds that \(M_{\textrm{grow}}\le M_{\textrm{iso}}\).

The next two mass scales are related to orbital migration. As the mass of a protoplanet increases, its oligarchic planetesimal accretion rate decreases, whereas its type I migration rate increases. At one point migration becomes faster than growth. In a mass-distance diagram this means that planetary formation track bend from a vertical (upward) direction inward and become more horizontal. This happens at what we call the ‘equality mass’, where the migration timescale is equal to the accretion timescale. It can be estimated by equating the planetesimals accretion timescale from ([130], Eq. 10, or equivalently [114]) with the type I migration timescale neglecting non-linear corotation torques (see [131], Eq. 70, but with non-isothermal torques from [132]). This results in

$$\begin{aligned} M_{\textrm{eq}} = k_{\textrm{eq}}\frac{\Sigma _{\textrm{p}}^{3/4} M_\star ^{5/4} (H/a)^{6/5}}{\Upsilon ^{3/4}\Sigma _{\textrm{g}}^{9/20} \rho _{m}^{1/5} \rho _M^{1/4} a^{3/4} m^{1/10}}, \end{aligned}$$
(4)

where \(k_{\textrm{eq}} \simeq 3^{11/10} \pi ^{3/4} \left( \frac{b C_D}{2 C_e}\right) ^{3/10} \simeq 4.2\), \(\Upsilon \simeq \gamma ^{-1}(2.5 +1.7 \beta _{\textrm{T}} - 0.1 \beta _\Sigma\)) is an order unity prefactor for the Lindblad torque which depends on the adiabatic index of the gas \(\gamma\), the radial slopes of the surface density set to \(-\beta _\Sigma\), and the one of temperature to \(-\beta _{\textrm{T}}\) [132] and m is the mass of an individual planetesimal. Entering typical values, this becomes

$$\begin{aligned}{} & {} M_{\textrm{eq}} = {1.2}\,\hbox{M}_\oplus \left( \frac{\Upsilon }{2.23}\right) ^{-3/4} \left( \frac{M_{\star }}{{M_\odot }}\right) ^{5/4} \left( \frac{\Sigma _{\textrm{p}}}{{10}\,\hbox{g cm}^{-2}}\right) ^{3/4} \left( \frac{m}{{10^{18}}\,{\hbox{g}}}\right) ^{-0.1}\end{aligned}$$
(5)
$$\begin{aligned}{} & {} \left( \frac{H/a}{0.05}\right) ^{6/5} \left( \frac{\Sigma _{\textrm{g}}}{{2400}\,\hbox{g cm}^{-2}}\right) ^{-0.45} \left( \frac{a}{{1}\,{\hbox{au}}}\right) ^{-3/4} \left( \frac{\rho _m}{{1}\,\hbox{g cm}^{-3}}\right) ^{-1/5} \left( \frac{\rho _M}{{3.2}\,\hbox{g cm}^{-3}}\right) ^{-1/4}. \end{aligned}$$
(6)

However, the classical type I Lindblad torques can be counteracted by the co-rotation torque [133, 134]. The co-rotation region exerts a torque on the planet if gradients in specific vorticity and entropy are present. To not balance out those gradients and saturate the corotation torque, the gradients need to be restored by the disc’s viscosity faster than the time it takes for a parcel of gas to perform a full horseshoe orbit [123, 135]. The width of the corotation region is given by

$$\begin{aligned} \Delta a_{\textrm{HS}} = C_{\textrm{HS} }\frac{a}{\gamma ^{1/4}} \sqrt{\frac{M a}{M_\star H}}, \end{aligned}$$
(7)

where \(C_{\textrm{HS}} \approx 1.1\) is a numerical factor determined from numerical experiments [132] and \(\gamma\) is the heat capacity ratio or adiabatic index. Since \(\Delta a_{\textrm{HS}}\) increases with planetary mass, we can determine a critical mass at which the corotation torque saturates by equating the libration and the viscous diffusion timescale across the corotation region [136]

$$\begin{aligned} \frac{8 \pi a}{3 \Omega _{\textrm{K}} \Delta a_{\textrm{HS}}} {\mathop {=}\limits ^{}} \frac{(\Delta a_{\textrm{HS}})^2}{\nu }. \end{aligned}$$
(8)

Solving for the mass results in what we call the saturation mass

$$\begin{aligned} M_{\textrm{sat}} = \left( \frac{8 \pi \alpha }{3}\right) ^{2/3} \sqrt{\frac{\gamma }{C_{\textrm{HS}}^4}} M_\star \left( \frac{H}{a}\right) ^{7/3}. \end{aligned}$$
(9)

This mass scale the upper limit where planets can be trapped between outward and inward migration regions (migration traps). Above this value, planets always migrate towards the star. However, planets below this mass scale are not necessarily trapped into convergence zones, as that depends on the relative strength of the Lindblad and corotation torques. We note that this mass scale depends on H/a, which is distant- and time dependent. This leads to time-dependent predictions, as the aspect ratio decreases as a function of time because of decreasing viscous heating, and stellar evolution (e.g. [137]). The assumption of viscous dissipation as the radial transport method in the disc does also play a major role here, as the inclusion of magnetically driven disc winds combined a lower viscous dissipation would severely reduce the corotation torques (though it may also produce large outward corotation torques under certain conditions, [138]).

The fifth mass scale is linked to giant impacts. This important mass scale does not directly depend on orbital migration but may depend indirectly on it, as migration leads to a radial redistribution of the protoplanets: the final stage of the inner planets’ formation is dominated by giant impacts. This phase can lead to planet masses significantly larger than the planetesimal isolation mass. Assuming that the dam** of the protoplanets’ eccentricities by remaining planetesimals is weak, the resulting mass can be estimated [139]. To obtain this mass scale, we assume that the random velocity are given by mutual gravitational interactions between the planets only. Therefore, the derivation of this mass scale is similar to the isolation mass, but the extent of the feeding zone is modified to include the eccentric excursions from apastron to periastron of the growing planets, and the building blocks are no longer directly the planetesimals, but larger protoplanets with a surface density \(\Sigma _{\textrm{P}}\). The capital ‘P’ here stands in contrast with that of the planetesimals \(\Sigma _{\textrm{p}}\). Here, we assumed for an order-of-magnitude estimate that, locally, all planetesimals have been accreted by the protoplanets, for example bodies that have reached the local planetesimal isolation mass. In detail, this assumption does not hold in regard to the Solar System where there are planetesimal-like bodies that remain until today, but we are here interested at the overall mass budget, which is dominated by the planets. This assumption is also justified a posteriori from the fact that all planetesimals are accreted in the terrestrial region (Sect. 5.3). The radial distribution of these protoplanets may differ from the initial planetesimals depending on whether orbital migration lead to a significant redistribution. Let m be the mass of the planet which is given by

$$\begin{aligned} m = 2 \pi a 2 \Delta a \Sigma _{\textrm{P}} \end{aligned}$$
(10)

for a radial extent of the feeding zone \(\Delta a = ea\) if the orbit is eccentric enough such that the width used for the isolation mass (Eq. (1)) is much smaller than the radial excursion of the planet, that is when \(\Delta a\gg r_{\textrm{H}}\) or \(e\gg \left( M/(3M_\star )\right) ^{1/3}\). Given multiple interacting bodies, the planets’ random velocity is approximately given by the planets’ escape velocity \(v_{\textrm{esc}}=\sqrt{2 G m/R}\) which then leads to an eccentricity of \(e \approx v_{\textrm{esc}}/v_{\textrm{K}}\). Thus,

$$\begin{aligned} m = 4 \pi a^2 \Sigma _{\textrm{P}} \sqrt{\frac{2 G m / R}{G M_\star /a}}, \end{aligned}$$
(11)

where we can assume a constant bulk density \(\rho _{\textrm{p}}\) for each planet and solve for m to get the Goldreich mass

$$\begin{aligned} M_{\textrm{Gold}} = 16 a^3 \Sigma _{\textrm{P}}^{3/2} \left( \frac{2\pi ^7 a^3 \rho _{\textrm{P}}}{3 M_\star ^3}\right) ^{1/4}\,. \end{aligned}$$
(12)

If the effect of a previous orbital migration of the protoplanets can be neglected, we can set \(\Sigma _{\textrm{P}}=\Sigma _{\textrm{p}}\). It should be noted that the Goldreich mass is inherently a mass scale that is not strictly local, as it is obtained via a series of giant impacts. This leads to a radial mass exchange, as can be seen in the simulations below. It is therefore useful to calculate \(M_{\textrm{Gold}}\) not with local quantities only, but take an average over the region where the giant impact phase takes place (like for example inside of the iceline, as it is done in the next section).

Finally, two additional mass scales are important for giant planet formation: the critical (core) mass for gas runaway accretion [108, 109, 140] and the mass where planets switch from faster type I to slower type II migration [141]. Regarding the former, it is important to directly calculate it with the structure equations, as semi-analytical expression can give grossly incorrect results [142].

5 Results

We conduct our main analysis for a synthetic planet population that is based on the nominal population that was presented in Emsenhuber et al. [80] (NG76). However, in the simulation shown here (NG76longshot), the formation stage which includes N-body interactions was extended from 20 to 100 \(\hbox{Myr}\). As a simplification, the model starts with all the solids are in the form of planetesimals, with a steeper radial slope than the gas to account for planetesimal formation simulations (power-law index of \(\beta _{\textrm{s}}=-1.5\) following [143]). The solid component is thus always in the form of planetesimals and does not account for the presence of dust in evolved discs. At the same time, 100 lunar-mass embryos are placed in each of the 1000 simulated planetary systems. This population contains a variety of planetary systems, from terrestrial planet systems to multiple giants, as we will show in this section.

5.1 Four classes of system architectures

To analyse the formation patterns of different kinds of planetary systems and determine how the diversity of protoplanetary discs and environment affect the formation, we start by classifying the final systems into a number of classes of system that show similar properties. We base the classification on mainly the mass-distance diagram plus additionally the bulk composition. For the latter, we distinguish five compositions: Jovian (H/He dominated in mass), Neptunian (volatile (ice)-rich with H/He), water worlds (volatile-rich without H/He), hyterran (silicate-iron core with H/He envelope) and terrestrial (silicate-iron without H/He or ices).

By visually inspecting each of the 1000 synthetic planetary systems in the population, we find that the system architectures emerging in the simulations can be divided into four main classes. These are, ordered in increasing typical mass of formed planets, 1) (compositionally) ordered Earths and ice worlds systems, 2) migrated sub-Neptune systems, 3) mixed systems with low-mass and giant planets, broadly speaking akin to the Solar System, and 4) dynamical active giants. The preponderance of different classes is as follows: 57.6% are part of Class I, 22.1% of the system are found to be part of Class II, 12.3% are part of Class III, and finally 8.0% are found to be part of Class IV. As was already discussed by Emsenhuber et al. [80], most systems contain only low-mass planets (Class I and II) while only about 20.3% of the systems are in classes that contain giant planets (Class III and IV). However, these numbers are sensitive to the choice of initial conditions, which themselves are subject to uncertainties (Sect. 2.3).

We find that the extension of the formation stage to 100 \(\hbox{Myr}\) (compared to 20 \(\hbox{Myr}\) in [80]) affects mostly the Class-I systems. The final architecture of the systems in the other classes is only seldom affected by this change. In the following subsections, we will review the formation pathways of each of the four classes and provide example of system architecture in each of them.

Our classification of planetary systems is based on a human classificator. As becomes clear below, the classes found this way correspond to fundamentally different temporal formation pathways which are governed by different physical processes (like giant impacts, orbital migration, or gas runaway accretion). These processes in turn manifest themselves through different mass scales introduced in Sect. 4.3. Our approach thus stresses the temporal emergence of different system architectures and the (theoretical) physical mechanisms included in the model leading to them. A distinct approach was recently presented by Mishra et al. [6 are about 10 \(\hbox{M}_\oplus\), while that in the bottom right panel of Fig. 4 are slightly more massive. The difference lies in that the planets in the Class II system migrate further than the ones in Class I, and this is related to the disc lifetimes of the respective systems. We shall discuss this effect in greater details in Sect. 5.4.

Fig. 6
figure 6

Final architecture of four systems of Class II. The plot is analogous to Fig. 4. In contrast with Class I, all planets have a volatile-rich composition, and the mass is a decreasing or constant function of orbital distance. One also notes the absence of planets with H/He at small orbital distances. The numerous low-mass planets outside of 10 AU might not have yet settled into their final state after 100 Myr which is the N-body integration time covered in our simulations

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5.1.3 Class III: mixed systems with low-mass and giant planets

Class III systems have giant planets, along with inner less massive planets. They also have outer lower mass planets. Here again, we provide formation tracks of one system in this class in Fig. 7. The initial formation stage is similar to system in Class II, with the embryos inside the iceline accreting in-situ up to the isolation mass, while the embryos beyond the iceline begin to migrate before reaching the isolation mass. The contrast with systems in Class II is that the planets beyond the iceline continue to grow by gas accretion during their inward migration and end up undergoing runaway gas accretion. At this point, the track of the forming giant planet bends strongly upwards. The reason is the higher (equality) mass these proto-giants have (close to 10 \(\hbox{M}_\oplus\), instead of a few Earth masses), as discussed below. Once the planet undergoes runaway gas accretion, the migration rate decreases in part because in the inner disc, the migration rate is limited by the ratio between the planet mass and that of the local gas disc. As a consequence, the giant planets migrate less than the sub-Neptunes in Class II, and leave the inner part of the system free from perturbation. The inner system behaves with features of both Classes I and II. The part resembling to Class I is that there remain terrestrial planets that grow by giant impacts, while the part resembling to Class II is the presence of two Neptunian planets inside of the giant planets that underwent migration.

Fig. 7
figure 7

Example of formation tracks and final architecture for a system that belongs to Class III. In this class and in contrast with Class II, a protoplanet (with typical origin outside the water iceline) manages to grow massive enough to start runaway gas accretion, i.e. it reaches the critical core mass. It then passes into slower Type II migration and grows to a giant planet mass. The final mass of the giant planet is in this system similar to the one of Jupiter, but is found at about 0.7 AU. The system also contains several lower mass planets in- and outside of the giant planet. During its gas accretion phase, the giant accreted a 3 \(M_\oplus\) planet. 8 protoplanets were ejected out of the system and one fell into the star, as visible by grey lines crossing the right and left y-axes, respectively

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The initial disc of the system presented in Fig. 7 is more massive than that in Fig. 5. As a consequence, the isolation mass is also larger while the growth timescale \(\tau _{\textrm{grow}}\) is smaller, meaning faster accretion. This pushes the equality mass to larger values, allowing the planets to reach larger masses for a given amount of migration compared to the Class II system. The larger core masses allow the embryos to bind proportionally even more gas (as the Kelvin-Helmholtz cooling timescales decreases for larger body masses, [119]), allowing them to undergo runaway gas accretion on a time that is less than that of the inward migration.

Four final systems in this class are shown in Fig. 8. As they show, the inner planets can be any combination of rocky or icy bodies. In systems with both rocky and icy inner planets, the same compositional ordering as in Class I is retained. Rocky planets have a formation pattern similar to those in Class I, while the inner icy planets have similar formation patterns as for those in Class II, migrating after having accreted from outside the iceline. This also means that the giant planets are not necessarily the first embryos just beyond the iceline. Outside of the giants, we see groups of several planets with masses of a few  \(\hbox{M}_\oplus\). They consist of mostly ices and some H/He, reminiscent of the Solar System ice giants. They still grow slowly from remaining planetesimals. Over long timescales, they might coalesce to form more massive ice giants.

Fig. 8
figure 8

Final architectures of four systems of Class III, analogous to Fig. 4. The diversity is very large regarding both the orbital configuration, mass, and number of the giant planets but also regarding the composition of the inner low-mass planets which can be of a Neptunian, water world, hyterran or terrestrial type

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One common feature between different systems in this category is that they are dynamically cool. Planet eccentricities remain relatively low, even when multiple giant planets are present. Still, there are low and mid-mass planets in the outer regions that get on wide orbits or ejected (but not the giant planets themselves). However, this is limited to protoplanets outside the position of the giants.

5.1.4 Class IV: dynamically active giants

Systems in Class IV also have giants, like Class III; the difference lies in that they do not contain inner low-mass planets. Overall, they have a small number of remaining objects (sometimes even only one or two), with possible low-mass bodies at large distance. Systems in this class tend to have very massive giant planets (\(\sim 10\) ), although this is not always the case.

As for the other classes, we provide one example of the formation history of a system in this class in Fig. 9. It reveals that the system formed a total of three giant planets, two of which have subsequently been ejected, leaving a single giant on a wide and eccentric orbit. These planets begin their formation similar to the system in Class III discussed above, but the giants dynamically interact and get destabilised. The instability arises if several protoplanets in close-proximity start gas runaway accretion within a short time interval. It should be noted that our model assumption of inserting all embryos at one moment (at \(\hbox{t}=0\)) may artificially lead to such situations. If the early phases of solid growth (from dust via pebbles to planetesimal and embryos) are included, bodies may emerge more gradually [151]. The dynamical instability caused by the giant planets affects the other bodies in the system, starting with close-by bodies, some of which get accreted by the innermost giant. The others are sent on very eccentric orbits, ending up either colliding with the star or ejected.

Fig. 9
figure 9

Formation tracks of a system that belongs to Class IV. The concurrent formation of several (proto)giant planets (bodies reaching the critical core mass for runaway gas accretion) in spatial proximity lead to violent dynamical interactions among these massive planets, strongly disturbing the lower-mass bodies in the system via collisions and ejections. In the end, in this example only one very massive, distant and eccentric giant planet remains

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We also provide the final state of four systems in this class in Fig. 10. While they show a pattern of one or more giant planets, these planets have a diversity of configurations, from compact pairs inside 1 \(\hbox{au}\) (top left panel) to a single body at \({\sim 100}\,{\hbox{au}}\) (top right panel, the same one as in Fig. 9). In addition, other low-mass objects at large distances may remain. These are the remainder of the outer embryos which have not been affected by the instabilities of the inner giant planets. For instance, none remain in system 15 (top right panel) because the planet was sent to an eccentric orbit with large separation while two other giants have been ejected (as revealed by the grey dots at 100 \(\hbox{au}\)); however, in the other systems there are no giant planets either ejected or on wide orbits, leaving the outer region free from perturbation that would destabilise these bodies.

Fig. 10
figure 10

Final architectures of four systems of Class IV, analogous to Fig. 4. From the points next to the left and right y-axis, one notes the significant number of planets that have either been ejected out of these systems or that have collided with the host star. In the system in the bottom left, collisions among giant planets have also occurred

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One of the main differences between systems of Classes III and IV is the dynamical instabilities. The presence of very massive giants does favour the occurrence of planet–planet scatterings, and thus, systems with the most massive giants are more likely to be in Class IV.

5.2 Key mass scales in planetary systems

In Sect. 4.3 we introduced a number of analytical key mass scales which can be used to understand the main mechanisms of planetary formation. We then used them to understand the formation tracks of individual systems. Here, we expand this analysis to the population as a whole. Figure 11 shows the comparison of various mass scales with all the planets in different classes of planetary systems.

Fig. 11
figure 11

Gaussian kernel density estimates of the planet mass function for planets within 5 au. For each line, the same bandwidth of 0.3 dex was chosen. The isolation (Eq. (1)) and Goldreich (Eq. (12)) mass scales are plotted only for initially rocky planets (dotted lines). Most of the planets in the sample were, however, initially icy. For comparison to those, we show with the magenta curve the smaller of the growth \(M_{\textrm{grow}}\) (Eq. (2)) or the equilibrium mass \(M_{\textrm{eq}}\) (Eq. (4)). Due to the lack of statistics (only five planets), we omitted the rocky planets and their mass estimates for Class IV

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It is interesting to see that the typical mass scales can predict to an order of magnitude or better the modelled distribution of solid-dominated masses. For the icy planets, we expect them to be limited by either running out of time to grow or by starting to migrate. The smaller of the growth or equality mass is shown in Fig. 11 as the magenta line. This simple argument predicts a bimodal distribution of ‘failed’ cores at the initial mass of 0.01 \(\hbox{M}_\oplus\) and at a larger mass varying from 1 \(\hbox{M}_\oplus\) to 10 \(\hbox{M}_\oplus\). The actual upper peak shown as the dash-dotted lines are for Class I and II located a factor of \({\sim 4}\) above this simple estimate. From evolution tracks it becomes evident that planets in this category can be trapped at the edges of outward migration regions or undergo giant impacts which enhances their mass.

The fact that collisions between embryos take place is factored into the Goldreich mass, which we, however, only applied to the inner rocky planetary system, where the simulation time is long compared to the growth timescale. The rocky planets in Class I systems are also bimodally distributed. There, we see that the lower peak is located close to the isolation masses while the upper is well reproduced by the Goldreich mass. This can be interpreted as systems or planets which went into a dynamically active stage leading to a giant impact phase forming the upper peak while lower mass planets are also predicted by the simulations remaining at their location without excitation and the possibility to grow from solid material outside their feeding zones.

For Class II systems with significant migration patterns, the low-mass rocky planets remain slightly below the mean isolation masses while the upper peak is reduced. Also, the overall number of initially rocky planets existing at the end of the simulations is reduced, a trend which continues also for the classes with giant planets.

This reduces the statistical significance and meaningfulness of the comparison for the rocky planets. Furthermore, for Classes III and IV, the final planetary systems are dominated by the massive, gas-rich planets which leads to expected deviations from the simple, solid accretion based mass scales. For gas-dominated planets, see Adams and Batygin [152].

5.3 Links between initial conditions, system properties, and architecture classes

Now, we aim to understand the conditions needed for the formation of different kinds of planetary systems. We find that among different initial conditions of our simulations, the initial mass in solids \(M_{\textrm{p,ini}}\) is the one that can be used best to discriminate different classes of systems. However, it is not the only discriminant.

One way of illustrating the impact of the initial mass in solids is shown in Fig. 12. It shows for all 1000 synthetic planetary system as a function of the initial mass of solids (planetesimals) in the natal disc \(M_{\textrm{p,ini}}\) the final mass of the planetary system \(M_{\textrm{sys}}\) (i.e. sum of all planets in a system at 5 \(\hbox{Gyr}\)).

Fig. 12
figure 12

Final mass of the planetary system (i.e. sum of all planets in a system at 5 \(\hbox{Gyr}\)) \(M_{\textrm{sys}}\) as a function of the initial mass of the solids (planetesimals) in the parent disc \(M_{\textrm{p,ini}}\). The colours show the number of planets in each system with a mass of at least 1 \(\hbox{M}_\oplus\). From this colour code, we see how the number of planets as a proxy of system architecture systematically changes with the initial mass in solids. The vertical shaded region ‘MMSN’ shows the estimated initial minimum solid content of the Solar Nebula. The horizontal dotted line ‘Solar System’ is the total system mass today \(({\approx 450}\,\hbox{M}_\oplus\) = essentially the sum of the masses of the four giant planets). Diagonal lines show conversion efficiencies (which can be larger than unity because accreted H/He is included in the system masses, but not in \(M_{\textrm{p,ini}}\))

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For this system mass, one can clearly distinguish the systems without (\(M_{\textrm{sys}}\lesssim {150}\,\hbox{M}_\oplus\)) and with one or several giant planets (\(M_{\textrm{sys}}\gtrsim {150}\,\hbox{M}_\oplus\)). As expected for a formation model based on the core accretion paradigm (e.g. [73]), giant planets only form in discs with sufficiently high \(M_{\textrm{p,ini}}\) of about 150 to 200 \(\hbox{M}_\oplus\). For the systems without giants, there is a clear linear scaling between \(M_{\textrm{p,ini}}\) and \(M_{\textrm{sys}}\) with an efficiency factor of about 30%. For the systems with giants, there is still a correlation, but it is less clear. This is expected, because for the final masses with giant planets the disc gas mass is important and not the solid mass. There are also some special cases with very small \(M_{\textrm{sys}}\) for high \(M_{\textrm{p,ini}}\). They are discussed below.

The colours in Fig. 12 show the number of planets with a mass of at least 1 \(\hbox{M}_\oplus\) in each system (including planets at all semi-major axes). This planetary multiplicity is a proxy for system architecture and exhibits an interesting pattern. At the lowest \(M_{\textrm{p,ini}}\) of about 10 \(\hbox{M}_\oplus\), not even a single planet with a mass of at least 1 \(\hbox{M}_\oplus\) forms, because the planets remain at sub-Earth masses. As we move to higher \(M_{\textrm{p,ini}}\), but still below the value needed for giant planet formation, the number of planets increases. At \(M_{\textrm{p,ini}}\) between 100 and 200 \(\hbox{M}_\oplus\) there are some planetary systems with a very high multiplicity of up to 25. In this region, we see also a vertical stratification where at given \(M_{\textrm{p,ini}}\), systems with a lower \(M_{\textrm{sys}}\) also have a lower number of planets. This is a consequence of another disc property, the disc lifetime, which is influenced mostly by the initial mass of the gas disc and the external photoevaporation rate. At given \(M_{\textrm{p,ini}}\), system with a longer disc lifetime have a lower \(M_{\textrm{sys}}\). The reason is not that these systems form less planets (smaller mass or lower number) in the first place, but that planets have more time to migrate. This has the consequence that more planets fall into the star both by migration in resonant convoys during the presence of the gas disc, and on long timescales after disc dissipation via tides. For the latter, the close-in parking location in systems with significant inward migration makes the difference. This reduces both \(M_{\textrm{sys}}\) and the number of planets. It also influences the classification of the architecture, as we will see soon.

Moving finally to the \(M_{\textrm{p,ini}}\) that are high enough to allow giant planet formation, we see that the number of planets again decreases. This is a consequence of the fact that growing giant planets can destabilise lower mass planets in their vicinity. These planets are then either ejected from the system or accreted by the growing giant planet(s). This is shown in Fig. 7 where the star symbols on the tracks of the growing giants indicate the accretion of lower mass planets during the process of rapid gas accretion. At \(M_{\textrm{p,ini}}\) less than about 400 \(\hbox{M}_\oplus\), there are, however, still many systems with about 10 planets. Violent instabilities among the giants are particularly efficient in removing lower-mass planets. Such cases can be seen at the highest \(M_{\textrm{p,ini}}\). For these systems, the number of planets reaches—as for the lowest \(M_{\textrm{p,ini}}\)—again very low values of just one or two (but now very massive) planets.

5.3.1 Solar system analogues

Figure 12 also shows the location of the minimum-mass Solar nebula (MMSN; with the boundaries discussed in [80]: \(M_{\textrm{p,ini}}\) of 66 to 151 \(\hbox{M}_\oplus\)) and the final Solar System. The Solar System is not straightforward to form in our current model. This is mainly because gas accretion is assumed to remain unaffected by gap formation. Planets undergoing runaway gas accretion therefore will rapidly attain a Jupiter mass or more, leading to systems that are more massive than the Solar System. Obtaining more giant planets whose mass is compatible with that of Jupiter or Saturn would require lower mass accretion rates [153, 154]. Low disc viscosities ([154],and references therein) coupled with the inclusion of gap formation could alleviate this problem. Further, the model does not produce many systems with giant planets for discs whose mass are in the MMSN range. Rather, \(M_{\textrm{p,ini}}\) that are a factor \({\sim 2}\) more massive are need. This is linked to the overall conversion efficiency of the model, where only about a third of the solids are accreted onto the planets, as it can be seen from the many systems without giant planets. In the outer part of the disc, accretion timescale is larger than the disc lifetime (see the pink curve in Figs. 3, 5, and 7) so only a fraction of the mass is converted into planets. On the other hand, the MMSN is a minimum mass indeed, so the actual \(M_{\textrm{p,ini}}\) of the Solar System might had been higher.

In addition, we find that that planets in Solar-system analogues are too close-in (Fig. 8). This is in part due to the difficultly of forming distant planets with planetesimal accretion (e.g. [155]. There are several others aspects in the model which are missing for the specific case of the Solar System. One is that the model does not include the Masset and Snellgrove [156] mechanism that leads to outward migration of a pair of giant planets trapped in a mean-motion resonance and sharing a common gap, which has been invoked for the formation of Jupiter and Saturn (e.g. [157]). Further, we do not take into account that by blocking or at least reducing the pebble flux, giant planet formation can affect the formation of planetesimals, which later form the terrestrial planets [158].

5.3.2 Architecture class

Instead of considering the number of individual planets in a system as a proxy of system architecture, we can also study the four classes directly. We illustrate the effect of different initial disc properties on architecture in Fig. 13, with histograms of the fraction of system class as function of solid mass, metallicity, and gas mass. We find the mass in solids is the quantity that best discriminates different categories, having different classes best confined to certain regions. The effect of the solid disc mass is reflected on the metallicity and gas mass, since the three quantities are related (only two of the three can be freely chosen). The middle panel shows that Class I systems are anti-correlated with metallicity, having the largest fraction of systems at low metallicity and steadily decreasing, while the other three quantity show an initial increase following by a plateau. The gas mass follows a similar pattern as the solid mass, but with more overlap between different classes.

Fig. 13
figure 13

Histograms of the fractions of different system class as function of initial mass in solids (left), metallicity (centre), and initial mass in gas (right). Blue represents Class I, orange represents Class II, green represents Class III, and red represents Class IV. The mass in solids is the quantity that most clearly discriminates the classes

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In Fig. 14 we focus again on the solid mass, by again showing the total mass in planets \(M_{\textrm{sys}}\) in each system is plotted against the initial content of the solids disc \(M_{\textrm{p,ini}}\), but the systems are now colour-coded by the four architecture classes. Here, we note a strong separation between the first two (only low-mass planets) and last two (with giant planets) classes of systems.

Fig. 14
figure 14

System architecture as function of the initial solids mass content in the disc \(M_{\textrm{p,ini}}\) and the total mass (accounting for solids and gas) in the final planets per system \(M_{\textrm{sys}}\) (left) and a histogram of initial solids mass only (right). In the left panel, the three dashed diagonal lines again denote the conversion efficiency from disc solids to planets, with the value given next to them. In the right panel, the small horizontal lines show the extent of each bin

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For the first two categories of systems, most of the planet masses is comprised of solids. Therefore, the masses in planets are generally a given fraction of the disc mass, between 20 and 35%, as mentioned. Usually, the whole inner disc up to 1–2 au is accreted by the protoplanets, some fraction beyond that up to \({\sim 10}\,{\hbox{au}}\), while the outer part remains essentially intact, because the accretion timescales are too long. It should be taken into account that extending the (numerical) integration time (here 100 \(\hbox{Myr}\)) in the simulations could further increase the efficiency for some systems, until ejection of planetesimals will start to limit further planet growth [14]. We also note that there is a trend between Class I and II. The lower-mass discs usually form Class I architectures, while more massive discs tend to result in Class II systems.

As discussed, discs with solids content \(M_{\textrm{p,ini}}\) above 150 to 200 \(\hbox{M}_\oplus\) lead to the formation of giant planets. In this case, planets accrete significant envelopes, which strongly increases the conversion efficiency. However, we do not find a clear separation in the parameter space between Class III and IV architectures, which is interesting. We also searched the other initial conditions, including dust-to-gas ratio and photoevaporation rate, but did not find anything clearer than what is presented in Fig. 14. The right panel shows that Class IV systems have larger disc masses than Class III systems, with median values of nearly 440 \(\hbox{M}_\oplus\) and 563 \(\hbox{M}_\oplus\), respectively, although there is significant overlap. This leaves us to conclude that the distinction between the two classes is not only related to initial disc characteristics and that the chaotic nature of the dynamical interactions between the protoplanets also bears some responsibility. So for similar disc initial conditions, Class III or IV systems can arise, depending on the exact locations and timing of the moments the proto-giant planets in one system start runaway accretion (which depends for example on the exact initial positions of the embryos), and the associated presence or absence of strong gravitational interactions. This (quasi) chaotic nature is different than in a simpler model where only a single embryo per disc is present, whose outcome is easy to predict [83]. It should be noted that our full model used here is also deterministic, i.e. for given model initial conditions it always predicts exactly the same outcome, but small changes of the initial conditions may lead to very different outcomes because of N-body interactions.

Finally, examples of dynamically active Class IV systems leading to the loss of most of the planets are visible in the bottom right corner of Fig. 14’s left panel. Here close encounters lead to the loss of all giant planets and only lower-mass planets remain. This leads to the (rare) points with high \(M_{\textrm{p,ini}}\) but (very) low \(M_{\textrm{sys}}\). If we were to include in the system mass also all planets that were ejected out into interstellar space or that collided with the host star (and not only the planets that actually exist at 5 \(\hbox{Gyr}\)), these points would join the other ‘normal’ points at high \(M_{\textrm{sys}}\).

The classification by Mishra et al. [165, 166]. To check for this effect in our model, we provide in the left panels of Fig. 17 the eccentricities of giant planets as function of metallicity. We first see that overall giant planet eccentricities are rather low in our model, with a mean value of 0.17. Still, we get a slight correlation between stellar metallicity and the mean eccentricity (represented by the black line).

Fig. 17
figure 17

Eccentricity distribution of the giant planets (mass \({>100}\,\hbox{M}_\oplus\)). Left: eccentricity versus initial metallicity. The black line shows the mean value in each interval. Right: cumulative distribution of giant planets eccentricities in Class III and Class IV systems

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In addition, we see a different trend in the eccentricities as function of the system class. The highly eccentric planets (\(e \gtrsim 0.6\)) are mostly found in Class IV systems and the mid-eccentricities (\(0.6 \gtrsim e \gtrsim 0.3\)) are also dominated by Class IV systems, though a higher proportion of planets in Class III systems is also present. Only in the low eccentricity regime (\(0.3 \gtrsim e\)) do planets in Class III systems dominate. To support this, we provide in the right panel of Fig. 17 distinct cumulative functions for the giant planet eccentricities as function of class. There, we see that the eccentricity distribution is different between the two classes, with planets in Class IV systems having overall larger eccentricities.

We also have a different distribution of stellar metallicities between Class III and Class IV systems. We already saw in Fig. 14 that Class IV systems tended to have larger initial disc masses and this is also reflected in the metallicity distribution. The median metallicity of Class III systems is 0.11 while for Class IV systems it is 0.17. Thus, our model can reproduce the general trend of increasing planet eccentricities with stellar metallicities, albeit eccentricities are overall low.

5.6.2 Planet and system kinds

There are observational correlations between kinds of either planets or system architectures and stellar metallicity, both in transit and radial velocity surveys. To assess our model against such trends, we perform two comparisons. First, we compare the metallicity effect of different kind of planets following Petigura et al. [167]. Here we select planets whose orbital period is between 10 and 100 d, apply an selection similar to transit surveys, as described in Emsenhuber et al. [80], and categories the planets according to their radii following the definitions of [167]: super-Earths from 1.0 to 1.7 \(\hbox{R}_\oplus\), mini-Neptunes from 1.7 to 4.0 \(\hbox{R}_\oplus\), sub-Saturns from 4.0 to 8.0 \(\hbox{R}_\oplus\), and Jupiters above 8.0 \(\hbox{R}_\oplus\). The occurrence of small-radii planets, ‘super-Earths’, shows an anti-correlation with stellar metallicity. For comparison, Petigura et al. [167] found that the occurrence of the same planets (panel b of their Fig. 10) has overall no significant correlation with metallicity. However, the occurrence rate of the largest metallicity bin (0.2–0.4) is lower than the others, as in our simulations. Our finding is also consistent with the occurrence of low-mass planets reported by Emsenhuber et al. [80], where systems having Earth-like (mass from 0.5 to 2 \(\hbox{M}_\oplus\)) planets within 1 \(\hbox{au}\) have a median metallicity of \(-0.09\) (the same value for the population presented here is \(-0.11\)). All other categories, from mini-Neptunes to Jupiters, show a positive correlation between occurrence rate and metallicity. For the mini-Neptunes and sub-Saturns, this is consistent with the trend found by Petigura et al. [167], while the authors have been unable to find any trend for Jupiters from the Kepler mission, as there were only few such planets observed. Our model is thus able to reproduce the general trends correlation with metallicity found in the Kepler survey.

87% of super-Earths are found in Class I systems, whose occurrence rate is anticorrelated with metallicities (Fig. 13, centre panel). This, combined with their total absence from Class IV systems, explains the trend of anti-correlation between the occurrence of super-Earths with stellar metallicity. Mini-Neptune planets originate principally from Class II (49%) and Class I (40%) systems. As the occurrence of Class II systems is positively correlated with stellar metallicity, this explains the shift in the metallicity correlation between super-Earths and mini-Neptunes. From this discussion, we also see that Class I systems may produce large planets (both in mass and radii) which enter the mini-Neptune category. Having them originate more from Class II systems would be a way to reinforce the metallicity correlation.

A second comparison can be done on the correlation between inner super-Earths and distant giant planets (or cold Jupiters). Such a correlation was observed [168, 169] and the metallicity effect of these systems was studied by Rosenthal et al. [170]. Schlecker et al. [82] determined there was a correlation between the occurrence of super-Earths and cold Jupiters in the same model as presented in this work (although assuming 20 \(\hbox{Myr}\) for the formation stage), though weaker than in observations. Here we extend the comparison for the trend in metallicity between systems hosting only inner super-Earths and both super-Earths and cold Jupiters in radial velocity survey. For this, we first restrict the synthetic planets to those with a radial velocity semi-amplitude \(K>{2}\,{\hbox{m s}^{-1}}\), as in Schlecker et al. [82]. We use the same definitions as Rosenthal et al. [170], name super-Earths have distances between 0.02 and 1 \(\hbox{au}\) and masses between 2 and 30 \(\hbox{M}_\oplus\) while cold Jupiters have distances between 0.23 and 10 \(\hbox{au}\) and masses between 30 and 6000 \(\hbox{M}_\oplus\). The cumulative distributions of metallicity for system hosting only super-Earths and both super-Earths and cold Jupiters are shown with the blue and green curves in the right panel of Fig. 18, respectively. The p-value of a Kolmogorov–Smirnov (KS) test between the two distributions to assess whether they are drawn from the same sample yields \(3.7\times 10^{-3}\), implying that the two distributions are distinct. Our results are compatible with those Rosenthal et al. [170] in that (1) systems with only super-Earths have lower metallicities than those with both super-Earths and cold Jupiters, (2) a statistical test assesses that the two groups have a distinct underlying distribution, and (3) the metallicity distributions are broadly similar. In Rosenthal et al. [170], the metallicity of systems with only super-Earths spans roughly \(-0.4\) to 0.3 with most of the systems between \(-0.2\) and 0.2, while the metallicity of systems with both super-Earths and cold Jupiters spans roughly 0.05–0.35 (their Fig. 8). In our model, the values are similar although the metallicities of the latter type are shifted towards lower values, with values mainly between \(-0.1\) and 0.3. We find that 77% of the systems with both super-Earths and cold Jupiters have \([\textrm{Fe}/\textrm{H}]>0\), and 41% of them have \([\textrm{Fe}/\textrm{H}]>0.15\). Thus, while the systems containing both kinds of planets are not as strongly biased towards large metallicities in our synthetic population compared to what Rosenthal et al. [170] obtained, we nonetheless obtain a bias.

Fig. 18
figure 18

Metallicity dependence of certain planet and system kinds in the synthetic population, which can be compared to observations. Left: occurrence of planet classes at periods between 10 and 100 \(\hbox{d}\) (including an observability criterion similar to [167]) as function of stellar metallicity (transit detections). Super-Earths are between 1.0 and 1.7 \(\hbox{R}_\oplus\), mini-Neptunes between 1.7 and 4.0 \(\hbox{R}_\oplus\), sub-Saturns between 4.0 and 8.0 \(\hbox{R}_\oplus\), and Jupiters above 8.0 \(\hbox{R}_\oplus\). Right: cumulative distribution of metallicities of systems harbouring an inner Super-Earth (SE; defined as distance between 0.02 and 1 \(\hbox{au}\) and a mass between 2 and 30 \(\hbox{M}_\oplus\)) and with (without) outer giant planet—or cold Jupiter (CJ)—in green (blue), defined as distance between 0.23 and 10 \(\hbox{au}\) and mass between 30 and 6000 \(\hbox{M}_\oplus\). Only planets with a radial velocity semi-amplitude \(K>{2}{\hbox{m s}^{-1}}\) are accounted for

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When comparing these types of systems with our classes, we find that systems with only detectable super-Earths are 77% from Class II, 20% from Class I, and 3% from Class III. Systems with both super-Earths and cold Jupiters are 94% from Class III, with contributions from all others categories. We note that even a system of Class I is included here; this is because that system contains a rocky planet of 18 \(\hbox{M}_\oplus\) at 0.15 \(\hbox{au}\) and an icy planet of 36 \(\hbox{M}_\oplus\) at 0.39 \(\hbox{au}\). Although the planet is massive for their kind (and hence enter the definitions of super-Earth and cold Jupiters of [170]), the formation pattern of the system is similar to that of Class I with only a small effect of migration and was thus categorised as such. Thus, the difference in metallicities between super-Earth systems and with or without cold Jupiters is consistent with Class III systems having overall larger metallicities than Class II systems.

We find different metallicity effects for planets defined as ‘super-Earths’ in transit or radial velocity surveys. The super-Earths according to the definition of Petigura et al. [167] show an anticorrelation, or a negative metallicity effect. Conversely, using the definitions of the radial velocity survey of Rosenthal et al. [170], the median metallicity of systems hosting super-Earths is 0.05, thus showing a positive metallicity effect. The difference is due to 75% of the defined as super-Earths by Rosenthal et al. [170] having radii between 1.7 and 4.0 \(\hbox{R}_\oplus\), the range defined as ‘mini-Neptunes’ by Petigura et al. [167]. We thus caution that there is an inconsistency between the naming of ‘super-Earths’ in transit and radial velocity, and that ‘super-Earths’ in radial velocity are similar to ‘mini-Neptunes’ in transit surveys, which both show a positive metallicity effect.

We further find that the median metallicity of systems where a giant planet has been accreted by the central star is 0.15. This is an indication that our model could reproduce the metallicity effect of the high-eccentricity migration channel of hot-Jupiters [166, 171]. However, we cannot fully confirm this because the model does not account for tidal circularisation.

6 Discussion and conclusion

In this work, we discussed how planetary population synthesis can be used to infer how planetary formation may work although we have at the moment only few direct observations of forming planets. The general idea is to find a theoretical end-to-end model that can reproduce the observed planet population from the diversity of known protoplanetary discs in a quantitative statistical way. The underlying theoretical model has to be global, so that it includes the essence of all (currently known) important governing physical processes and treat their interactions. We can also use such a global model to investigate the physical processes that lead to different classes of architectures of synthetic planetary systems and how the disc and stellar environmental properties affect the outcome. Hence, we classified the 1000 synthetic systems from the nominal NGPPS population of Emsenhuber et al. [80] based on classes of formation pathways seen in the simulations and resulting planetary system architectures. This approach stresses the temporal emergence of system architectures and the underlying physical processes included in the theoretical model. This is in contrast with the approach of Mishra et al. [\(\hbox{M}_\oplus\), which corresponds to the critical core mass, to allow for strong gas accretion while the planets migrate. The distinction between the two classes is whether some strong dynamical instability has taken place between the giants, which lead to the loss of the inner companions. While these two categories together are well separated from the others in the space of initial conditions (Fig. 14), there is no such clear separation in the initial disc properties (like initial mass of solids) between the systems in Class III versus Class IV. Thus, we attribute the discerning factor mainly to the stochasticity of the N-body interactions: for some initial placings of the embryos, the forming giants do undergo dynamical instabilities, for others not. There are nevertheless some trends, with Class IV systems forming preferentially from higher-mass discs (Fig. 14) and where photoevaporation is lower and thus disc lifetimes are longer (Fig. 15).

We further investigate whether the total number of giant planets that ever existed in a system can explain the distinction between Classes III and IV. We find that the loss of a giant planet due to an ejection or a collision with the central star (as the result of a close encounters between two giant planets) will likely result in the destabilisation of the inner system, thereby resulting in a system with only giant planets. However, if the two giant planets collide rather than scatter themselves, this is not sufficient to destabilise the inner system. Thus, the stochastic nature of close encounters plays an important role in the distinction between Class III and IV, rather than only the initial conditions.

The discovery space of extrasolar planet detection methods expands continuously. Several upcoming missions and instruments like GAIA, NIRPS, PLATO, Roman, or Ariel will foster this trend even more, and allow to build observationally a more complete statistical picture of the demographics of extrasolar planets. The times are therefore promising for a method like planetary population synthesis which can take full advantage of the wealth of all these new observational constraints. It will, however, be important to compare the same model with all different constraints and methods simultaneously to uncover the shortcomings in our theoretical understanding, and to avoid building models to reproduce only a certain observation. Furthermore, it is time to conduct comparisons of models and observations in a quantitative, and no longer a qualitative fashion only, given that the observational data are fully on a level to allow for this. Combined with direct observations of ongoing planet formation, this should make it possible to markedly improve our understanding of the planet formation process in the coming decade.