THE ASTEROID BELT AS A RELIC FROM A CHAOTIC EARLY SOLAR SYSTEM

To get a sense of the presence of chaos across the phase space of Jupiter's and Saturn's orbital configuration, we performed about 9000 simulations to build a dynamical map for a wide range of orbital period ratios between Jupiter and Saturn. The MEGNO (Mean Exponential Growth factor of Nearby Orbits; Cincotta & Simo 2000) chaos indicator is a powerful tool used to identify chaos in dynamical systems. Chaotic orbits are characterized by a large MEGNO value ($\langle Y\rangle \gg \,2$) while regular or quasi-periodic orbits are associated with $\langle Y\rangle \to 2$ (e.g., Cincotta & Simo 2000).

Our initial conditions were set as follows. Jupiter was placed at 5.25 au while Saturn's semimajor axis was sampled from about 6.25 to 8.6 au (period ratio between 1.3 and 2.1). Their mutual inclination was sampled randomly from 0° to 05. The eccentricity of Jupiter was randomly selected between 0 and 0.01. Saturn's initial eccentricity ranges from 0 to 0.1. Note that although the initial eccentricity of Jupiter in this set of simulations is smaller than the corresponding initial values in Figures 1 and 2, this quantity does not remain constant over time. Jupiter and Saturn's (secular) interaction leads to eccentricity oscillations that imply that Jupiter's eccentricity may reach values comparable to or even larger than 0.025–0.03. Angular orbital elements of both planets were all sampled randomly between 0° and 360°. Simulations were integrated for 50 Myr using the REBOUND integrator (Rein & Tamayo 2015) computing the MEGNO value of each of these dynamical states.

Figure 6 shows a dynamical map of the behavior of Jupiter's and Saturn's orbits at different orbital separations. The simulations were integrated for fifty million years and the results are color-coded by MEGNO value. Black regions are potentially unstable; they show orbital configurations where Jupiter and Saturn undergo close encounters. Orange regions exhibit chaotic motion (with $\langle Y\rangle \gtrapprox 4$) while blue regions encloses regular motion (with $\langle Y\rangle =2$). Chaotic regions are generally confined to orbital period ratios and are associated with specific MMRs. There is a broad chaotic region near the 2:1 MMR and narrower regions close to the 5:3 (period ratio of 1.66) and 7:4 (period ratio of 1.75) resonances, with some chaos present just exterior to the 3:2 resonance (period ratio of 1.5).

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The dynamical map in Figure 6 shows that chaos is common in the phase space available for Jupiter's and Saturn's orbits. Yet their actual orbits at early times were not chosen at random. Rather, the gas giants' orbital configuration was generated by interactions between the growing planets and the gaseous protoplanetary disk, in particular by a combination of orbital migration (Lin & Papaloizou 1986; Ward 1997) and eccentricity and inclination damping (e.g., Papaloizou & Larwood 2000; Bitsch et al. 2013). The dynamical evolution of Jupiter and Saturn in the gas disk depends on the disk properties (Masset & Snellgrove 2001; Morbidelli & Crida 2007; Zhang & Zhou 2010; Pierens & Raymond 2011). To study chaos in Jupiter's and Saturn's orbits in the context of orbital migration we perform N-body simulations using artificial forces to mimic the effects of the gas and also pure hydrodynamical simulations. We present these results next.

In our simulations of Jupiter and Saturn migrating in a gaseous disk, Jupiter was initially at 5.25 au and Saturn was initially placed exterior to the 2:1 MMR with Jupiter (beyond 8.33 au). Following Baruteau et al. (2014, p. 4293), Jupiter was assumed to migrate in a type-II mode with the migration timescale computed as

$$\begin{eqnarray}&&{t}_{m,\mathrm{Jup}}=\displaystyle \frac{2{a}_{\mathrm{Jup}}^{2}}{3\nu }\times \min \left(1,\displaystyle \frac{{M}_{\mathrm{disk}}}{{m}_{\mathrm{Jup}}}\right),\end{eqnarray} \tag{ 2 }$$

where ${a}_{\mathrm{Jup}}$ and ${m}_{\mathrm{Jup}}$ are Jupiter's semimajor axis and mass, respectively. $\nu$ is the gas viscosity and ${M}_{\mathrm{disk}}$ is the gas disk mass. We modeled the disk viscosity using the standard "alpha" prescription given by $\nu =\alpha {c}_{s}H$ (Shakura & Sunyaev 1973), where ${c}_{s}$ is the isothermal sound speed and H is the disk scale height. In our simulations $\alpha =0.002$ and the disk aspect ratio is h ∼ 0.07. To account for the damping of eccentricity and inclination on Jupiter's orbit we assume the following relationship between migration timescale and eccentricity/inclination damping (Crida et al. 2008):

$$\begin{eqnarray}&&{t}_{e,\mathrm{Jup}}={t}_{i,\mathrm{Jup}}={t}_{m,\mathrm{Jup}}/K.\end{eqnarray} \tag{ 3 }$$

In our simulations, we generally adopted the typical value of K = 10, although we also performed simulations with K = 1 and K = 100.

To mimic the migration of Saturn in the gas disk, for simplicity we use the type-I migration/damping approach (Papaloizou & Larwood 2000; Tanaka et al. 2002; Tanaka & Ward 2004; Cresswell & Nelson 2006, 2008). Since Saturn's gap is not fully open, this is an acceptable approximation. We stress that our goal here is only to have convergent and smooth migration of Jupiter and Saturn such that we can access the plausibility of chaos origin in this kind of simulation. The initial gas surface density at the location of Saturn is ${{\rm{\Sigma }}}_{\mathrm{Sat}}=900{r}_{\mathrm{Sat}}^{-0.5}\,\,{\rm{g}}\,{\mathrm{cm}}^{-2}$. To implement migration, eccentricity, and inclination damping on Saturn, we use the following formulas:

$$\begin{eqnarray}&&{t}_{m,\mathrm{Sat}}=\displaystyle \frac{2}{2.7+1.1\beta }\left(\displaystyle \frac{{M}_{\odot }}{m}\right)\left(\displaystyle \frac{{M}_{\odot }}{{{\rm{\Sigma }}}_{g}{a}^{2}}\right){\left(\displaystyle \frac{h}{r}\right)}^{2}\left(\displaystyle \frac{1+{\left(\tfrac{{er}}{1.3h}\right)}^{5}}{1-{\left(\tfrac{{er}}{1.1h}\right)}^{4}}\right){{\rm{\Omega }}}_{k}^{-1},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}{t}_{e,\mathrm{Sat}} & = & \displaystyle \frac{{t}_{\mathrm{wave}}}{0.780}\left(1-0.14{\left(\displaystyle \frac{e}{h/r}\right)}^{2}+0.06{\left(\displaystyle \frac{e}{h/r}\right)}^{3}\right.\\ & & +\left.0.18\left(\displaystyle \frac{e}{h/r}\right){\left(\displaystyle \frac{i}{h/r}\right)}^{2}\right),\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}{t}_{i,\mathrm{Sat}} & = & \displaystyle \frac{{t}_{\mathrm{wave}}}{0.544}\left(1-0.3{\left(\displaystyle \frac{i}{h/r}\right)}^{2}+0.24{\left(\displaystyle \frac{i}{h/r}\right)}^{3}\right.\\ & & +\left.0.14{\left(\displaystyle \frac{e}{h/r}\right)}^{2}\left(\displaystyle \frac{i}{h/r}\right)\right)\end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray}&&{t}_{\mathrm{wave}}=\left(\displaystyle \frac{{M}_{\odot }}{m}\right)\left(\displaystyle \frac{{M}_{\odot }}{{{\rm{\Sigma }}}_{g}{a}^{2}}\right){\left(\displaystyle \frac{h}{r}\right)}^{4}{{\rm{\Omega }}}_{k}^{-1}\end{eqnarray} \tag{ 7 }$$

and ${M}_{\odot }$, $a$, $m$, $i$, and $e$ are the solar mass, planet's semimajor axis, planet's mass, orbital inclination, and eccentricity, respectively. r is the planet's heliocentric distance. ${{\rm{\Sigma }}}_{g}$ and $\beta$ are the gas disk surface density and gas surface profile index at the planet's location, respectively. In our simulations, in the case of Saturn, $\beta$ was calibrated from hydrodynamical simulations. The synthetic accelerations to account for the effects of the gas on the planet were modeled as

$$\begin{eqnarray}&&{{\boldsymbol{a}}}_{\mathrm{mig}}=-\displaystyle \frac{{\boldsymbol{v}}}{{t}_{m,\mathrm{pla}}}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{{\boldsymbol{a}}}_{e}=-2\displaystyle \frac{({\boldsymbol{v}}.{\boldsymbol{r}}){\boldsymbol{r}}}{{r}^{2}{t}_{e,\mathrm{pla}}}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{{\boldsymbol{a}}}_{i}=-\displaystyle \frac{{v}_{z}}{{t}_{i,\mathrm{pla}}}{\boldsymbol{k}},\end{eqnarray} \tag{ 10 }$$

where ${\boldsymbol{k}}$ is the unit vector in the z-direction.The label "pla," which appears in Equations (8)–(10), takes the form "Jup" or "Sat" to refer to accelerations applied to Jupiter and Saturn, respectively.

To ensure the robustness of this method, we compared selected simulations with true hydrodynamical simulations using similar disk setups and they agree in terms of the final period ratio, eccentricities, and orbital inclination of Jupiter and Saturn.

In simulations in which the ice giants were also included, they were assumed to migrate inward in the type-I mode described above.

We performed 1000 simulations that differed only in the initial surface density of the disk. In each case the gas disk was assumed to dissipate exponentially in 1 Myr (${\tau }_{\mathrm{gas}}$ = 100 kyr). After the gas disk dissipated, simulations were integrated for another 50 Myr. To compute how chaotic the orbits of the giant planets are in each simulation we again used the MEGNO chaos indicator incorporated in the REBOUND integrator (Rein & Tamayo 2015).

Figure 7 shows the results of these simulations. The vertical axis shows the relative initial surface density of the disk and the horizontal axis shows the final orbital period ratio between Jupiter and Saturn. Each final dynamical state of our simulation is represented with a circle whose color shows the MEGNO value of the final dynamical state of Jupiter and Saturn.

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The experiment presented in Figure 7 produced more regular configurations of Jupiter and Saturn than chaotic ones. In agreement with pure hydrodynamical simulations (Zhang & Zhou 2010; Pierens et al. 2014), Jupiter and Saturn typically park in either 3:2 (in high-mass disks) or 2:1 (low-mass disks) MMR. We did not find chaos in simulations where Jupiter and Saturn ends in 3:2. All our instances of chaos were related to Saturn and Jupiter's period ratio being close to 2. We observe chaos in about 1%–25% of these simulations depending on the disk parameters and K-value (a parameter that defines the ratio between the migration timescale and eccentricity/inclination damping; see Equation (3)). We expect that chaotic configurations for Jupiter and Saturn in 3:2 may be more likely for larger eccentricities than our migration and damping prescriptions allow (see Figure 6).

We performed 15 hydrodynamical simulations using the GENESIS code (de Val-Borro 2006). None of the simulations with Jupiter and Saturn migrating in the disk generated chaos in Jupiter's and Saturn's orbits. However, both hydrodynamical simulations and our N-body simulations with synthetic forces are extremely idealized. A number of factors could change the outcome. First, the protoplanetary disks in these simulations are typically laminar and perfectly axisymmetric. Second, the gas disk's dissipation in numerical simulations is in general poorly modeled with an exponential density decay over the entire disk. Third, at the end of the gas disk phase, planets should evolve in a sea of planetesimals and leftover building blocks of their own process of formation. In these simulations we did not include any external perturbation in the system (but see Section 2). Fourth, the migrating ice giants (and/or their building blocks; Izidoro et al. 2015a) represent a further source of perturbations.

In fact, the results shown in Figures 6 and 7 are complementary in nature. The setup of the simulations in Figure 7 favor deep capture in resonance and regular motion between Jupiter and Saturn while that in Figure 6 (because of the random selection of angles) may preferentially put the dynamical state slightly off the libration center and favor mostly chaotic orbits. The existence of many chaotic configurations in Figure 6 demonstrates the prevalence of chaos and even if the gas giants' orbits behaved regularly immediately after the dissipation of the gaseous disk, there are a number of processes that could have rendered them chaotic.

The present-day orbits of the giant planets are thought to have been achieved by an instability in the giant planets' orbits that occurred long after the dissipation of the gases (Tsiganis et al. 2005; Morbidelli et al. 2007; Levison et al. 2011). Our model is entirely consistent with a late (∼500 Myr-later) instability in the giant planets' orbits, whatever the exact configuration of Jupiter and Saturn (see Batygin et al. 2012; Nesvorný & Morbidelli 2012; Pierens et al. 2014). In fact, our model is also consistent with an earlier giant planet instability (Kaib & Chambers 2016), as long as there is a sufficiently long interval during which the belt can be chaotically excited. This interval is roughly longer than 10 Myr in most of the simulations we have run, but can be as short as 2 Myr. To illustrate the possibility of a long-term stability between the giant planets evolving in chaotic orbits, we performed N-body simulations in which Jupiter, Saturn, Uranus, and Neptune migrate in a disk. The prescription for migration used in these simulations is analogous to that explained before in this text. Isothermal type-I migration and damping are also applied to the ice giants. Here, we again used those damping timescales defined in Equations (4)–(7) and accelerations given by Equations (8)–(10). Figure 8 shows a long-term stable dynamical evolution of the gas giants in chaotic orbits. At the end of the gas disk phase, Jupiter and Saturn are locked in 2:1 MMR, Saturn–Uranus in 2:1 MMR, and Uranus–Neptune in 3:2 MMR. The system is dynamically stable for over 500 Myr.

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The "small Mars" problem highlights the fact that a mass deficit is needed from $\sim 1\mbox{--}4\,\mathrm{au}$ to explain Mars' small mass relative to Earth's (Wetherill 1996; Raymond et al. 2009). In the Grand Tack model, this deficit is generated by Jupiter's long-distance orbital migration from several au to 1.5 au, then back out to beyond 5 au (Walsh et al. 2011; Jacobson & Morbidelli 2014; Raymond & Morbidelli 2014; Brasser et al. 2016)

By explaining the asteroid belt's orbital structure, our results revive models in which the asteroid belt was initially very low mass (Izidoro et al. 2015b; Levison et al. 2015). While standard disk models typically invoke a smooth mass distribution within disks (for example, the minimum-mass solar nebula model of Weidenschilling 1977 and Hayashi 1981 generates a smooth disk from discrete planets), it remains unclear whether the planetary building blocks embedded in these gaseous disks should really follow a smooth distribution. In fact, models often find that planetesimals preferentially grow in special locations within the disk, such as at pressure bumps (e.g., Johansen et al. 2014, p. 547). Some models naturally create confined rings of planetesimals within broad disks (Surville et al. 2016). There also exist mechanisms to systematically drain solids from certain areas of the disk, thereby creating localized depletions and enhancements (e.g., Levison et al. 2015; Moriarty & Fischer 2015; Drazkowska et al. 2016).

Our model thus forms the basis of an alternative to the Grand Tack model. Within the context of this model, Mars' small mass can be explained by a broad mass depletion between Earth's and Jupiter's orbits. The asteroid belt, which could not be stirred by the dynamical effects of local embryos, was chaotically excited by Jupiter and Saturn.

The next step is to search for ways to distinguish between the Grand Tack and this new chaotic model. Tests may be based on observations of small bodies in the solar system or geochemical measurements. Alternately, the models may be differentiated by more detailed studies of the underlying physical mechanisms involving planetary orbital migration and pebble accretion.

Like the Grand Tack, our model assumes that Jupiter and Saturn migrated during the gas disk phase into a resonant configuration, mostly likely to 3:2 or 2:1 MMR (Masset & Snellgrove 2001; Morbidelli & Crida 2007; Pierens & Nelson 2008; D'Angelo & Marzari 2012; Pierens et al. 2014). However, the scale of radial migration of Jupiter and Saturn in our scenario could be less specific and much smaller than that in the Grand Tack scenario, where Jupiter and Saturn migrated inward then outward. In our case—since it is quite unlikely that Jupiter and Saturn grew already in resonance—only an inward convergent phase of migration between the giant planets is needed to put them into a resonant configuration.

The Grand Tack and the chaotic excitation models are not contradictory models. In the Grand Tack model, some level of chaotic excitation could have operated between Jupiter's and Saturn's two-phase migration and the Nice model instability. However, since the asteroid belt is already sufficiently dynamically excited after the Grand Tack (Deienno et al. 2016) it would be unnecessary to invoke chaotic excitation. We also note that both the Grand Tack and chaotic excitation mechanisms may operate with Jupiter and Saturn in either the 3:2 or 2:1 MMR (see also Pierens et al. 2014). Of course, the 3:2 resonance has been much more carefully studied for the Grand Tack, and our results suggest that the 2:1 resonance may be favored with regards to chaotic excitation. Yet we caution that Jupiter and Saturn's configuration during the disk phase may not necessarily differentiate between the two models.

It is entirely reasonable to imagine a chaotic young solar system. Several exoplanetary systems with planets on near-resonant orbits are thought to be chaotic, such as 16 Cyg B (Holman et al. 1997), GJ876 (Rivera et al. 2010), and Kepler 36 (Deck et al. 2012). The present-day solar system is well known to be chaotic (Laskar 1989, 2003; Sussman & Wisdom 1992). The orbits of the terrestrial planets undergo chaotic diffusion on a timescale of a few million years (Laskar 1989; Batygin et al. 2015). It is unknown whether the present-day giant planets are in a chaotic configuration; an accurate determination is precluded by uncertainties in our knowledge of the planets' orbital positions (Michtchenko & Ferraz-Mello 2001; Hayes 2008). It is often assumed that the early solar system was characterized by regular motion of the planets (Brasser et al. 2013). Our work suggests that this may not have been the case, and that the structure of the asteroid belt is a signpost that Jupiter's and Saturn's early orbits were in fact chaotic.

We showed that a small nudge could trigger chaos in Jupiter's and Saturn's early orbits (Figure 1). Could another small nudge or accumulated effects of scattering planetesimals make the system regular again? It is important to note that when the scattering of planetesimals takes place, not only damping of eccentricity but also radial migration should be observed. This process would lead to the divergent migration of Jupiter and Saturn (and ice giants) and not necessarily sinking toward the resonant center and consequently regular orbits. Typically, a late dynamical evolution is envisioned in models of solar system evolution (Gomes et al. 2005; Levison et al. 2011) which implies Jupiter and Saturn hanging out in or very near resonances for hundreds of Myr after gas disk phase. What is required for chaotic excitation to work is simply a sufficient time interval in this phase while the giant planets remain in a chaotic state. This also depends on the chaotic configuration of Jupiter and Saturn and the setup over the disk of planetesimals (Nesvorný & Morbidelli 2012).

Obviously, it would be computationally challenging to perform a systematic analysis to identify all dynamical configurations of Jupiter and Saturn that could excite the belt. However, we indeed found that non-resonant or temporary resonant configurations can also excite the belt. We also recognized that it is not clear why previous classical simulations of terrestrial planet formation did not find similar effects in the belt. One possibility is that the chaotic effects on asteroids have been erased or mitigated by the typical presence of large planetary embryos in the belt in classical simulations (Chambers 2001; O'Brien et al. 2006; Raymond et al. 2006; Izidoro et al. 2014, 2015b). Also, these previous simulations have preferentially considered Jupiter and Saturn near their current orbits or in 3:2 MMR (in almost circular orbits; see for example Raymond et al. 2009). In our most successful simulations of belt excitation Jupiter and Saturn have eccentricities of about 0.03–0.05; the latter is consistent with results from hydrodynamical simulations (Pierens et al. 2014).

The Kirkwood gaps in the asteroid population are created by MMRs with Jupiter. The most prominent is the gap created by Jupiter's 3:1 MMR centered at 2.50 au in the present-day belt. The chaotic excitation of the asteroid belt likely took place before a late instability in the giant planets' orbits (the so-called Nice model; Tsiganis et al. 2005; Morbidelli et al. 2007; Levison et al. 2011). If a late instability shifted Jupiter's orbit inward by ∼0.2 au (Tsiganis et al. 2005) then there should exist a fossilized gap in the asteroid belt at about 2.6 au, just exterior to the current Kirkwood gap associated with the 3:1 resonance. No such gap exists (see Section 6.4 of Morbidelli et al. 2010).

We caution that the data set used to study fossilized gaps (or lack thereof) are relatively sparse, containing just 335 large asteroids (with absolute magnitude $H\lt 9.7$) spread across the entire main belt (see Figure 7 in Morbidelli et al. 2010). The vicinity of the 3:1 Kirkwood gap contains only ∼10 asteroids. A careful statistical analysis using a larger data set (potentially extending to smaller bodies) would help quantify the depth and width of the missing Kirkwood gaps.

The belt excitation by the chaotic motion of Jupiter and Saturn seems to require an eccentricity of ∼0.03 or more (although we did not perform a systematic analysis on this issue since it is beyond the scope of this paper). It was proposed by Morbidelli et al. (2010) and Deienno et al. (2016) that the pre-instability giant planets must have had almost perfectly circular orbits. Thus, Jupiter's MMRs were relatively weak and narrow such that the Kirkwood gaps before the instability were virtually nonexistent. Jupiter's eccentricity must remain extremely low (${e}_{J}\lt 0.01$) to avoid clearing the primordial gap. This requires that during the gas disk phase the orbits of the gas giants were very efficiently damped by the gas (Morbidelli et al. 2007; Pierens et al. 2014).

The problem of the fossilized Kirkwood gaps (if it exist al all) appears to have a simple solution. During the Nice model instability, one or two ice giants are often ejected from the solar system after suffering close encounters with Jupiter. To reproduce the solar system's current architecture, one to two additional primordial ice giants may thus have existed before the instability epoch (Nesvorny 2011; Batygin et al. 2012). During the scattering process a doomed ice giant typically spends time with an orbit interior to (but crossing) Jupiter's. The scattered ice giant passes through the asteroid belt, often crossing the main belt. While this interval is short in duration, lasting just a few tens to hundreds of thousands of years, a scattered ice giant perturbs the distribution of asteroids. Brasil et al. (2016) showed that local groupings of asteroids analogous to asteroid families, initially strongly confined in orbital parameter space, are smeared out by the scattered ice giant (see their Figures 4 and 5). This smearing would fill in any fossilized Kirkwood gaps. In the context of this evolution of the giant planets, the present-day Kirkwood gaps must have been created after the Nice model instability.

In summary, we expect that any fossilized Kirkwood gaps would have been erased by the Nice model instability (Brasil et al. 2016). The absence of fossilized gaps cannot be used as a constraint on the giant planets' early orbits.

In this paper we did not address another important characteristic of the asteroid belt: the radial mixing of different taxonomic types of asteroids. The inner part of the main belt is dominated by S-type (water-poor) bodies, while C-type (water-rich) ones are preferentially found in the outer part of the belt, mostly beyond 2.5 au (Gradie & Tedesco 1982; DeMeo & Carry 2014). We demonstrate in an upcoming paper that the belt's chemical dichotomy is a natural, unavoidable outcome of the gas giants' growth in the gaseous protoplanetary disk. During the gas-accretion phase, Jupiter's core (and Saturn's as well) destabilizes the orbits of nearby small bodies and implant a fraction of these bodies in the outer asteroid belt (Izidoro et al. 2016, S. N. Raymond & A. Izidoro 2016, in preparation). These results together with those already presented here will form the basis for a new model to explain the bulk structure of the asteroid belt.

Here we provide more details about the nature of the chaotic excitation mechanism by showing how different chaotically jumping resonances can excite asteroids' eccentricities and/or inclinations.

Figure 9 show examples of asteroids being excited by the perturbation of chaotic Jupiter and Saturn. These asteroids are in the same simulation as those shown in Figure 3, but are simply in different parts of the main belt. Different resonances act to increase the eccentricities and inclinations of bodies in the belt.

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Figures 10–12 show the Fourier analysis of other angles and comparison between the JSREG and JSCHA simulations. The ${\nu }_{16}$ SR is much stronger and wider in the JSCHA simulation than the JSREG simulation (${\nu }_{16}$ appears at ∼2.5 au; see Figures 4, 5, and 11). However, its effects in the JSCHA simulation are much more localized than those of ${\nu }_{6}$ in the sense of its power to affect bodies over the whole belt (if bodies have the same proper inclination and eccentricity). We did not perform a systematic analysis of all resonances that contribute to pump inclination of bodies in the whole belt but high order secular and secondary resonances also play an important role for exciting bodies residing away from ${\nu }_{16}$ in the JSCHA simulation. Perhaps even three body resonances contribute. We stress that the nature of the resonances that contribute to excite bodies in the belt and their strength depends on the chaotic evolution of Jupiter and Saturn.

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In this section we present several examples of dynamical excitation of the belt by different orbital configurations between Jupiter and Saturn. In these simulations we used 8000 test particles (in some cases 2000) uniformly distributed between 1.8 and 4.5 au. We stress that not all our simulations where Jupiter and Saturn had chaotic orbits successfully excited the belt. Rather, we found a spectrum of outcomes. Some simulations were not able to excite parts of the belt while in other cases the perturbations from Jupiter and Saturn were so strong that the belt was destroyed. We did not perform a systematic analysis looking for the optimal resonant or non-resonant chaotic configuration to excite the belt. However, we have used as fiducial case a dynamical configuration where Jupiter and Saturn are in 2:1 MMR.

Figure 13 shows the dynamical evolution of asteroids in a simulation with chaotic Jupiter and Saturn in 2:1 resonance. Figures 14–17 show the dynamical evolution of asteroids in the belt in simulations where Jupiter and Saturn are in different resonant configurations, with period ratios of ∼1.75, ∼1.66, ∼1.66, and ∼1.5, respectively. Simulations were integrated using Symba (Duncan et al. 1998) with a 0.1 year time step. The total duration of each simulation varied between 20 and 125 Myr, and each simulation's duration is indicated in the bottom panel. Figures 13, 14, and 16 show simulations initially with 8000 test particles. Simulations corresponding to Figures 15 and 17 initially contain 2000 test particles.

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Figure 14 shows a simulation in which the whole belt was excited both in eccentricity and inclination. However, some regions of the belt were overly depleted compared with others after 20 Myr of integration.

Figure 15 is a very interesting case. Asteroids in the belt maintained low orbital inclinations and eccentricities for more than 30 Myr. At ∼33 Myr a stochastic jump in the positions of the giant planets (a consequence of their orbits being chaotic) resulted in a very strong perturbation in the belt (see corresponding panel in Figure 15). Within 2 Myr of the jump, the whole belt was excited to the observed levels of the real one. This case shows that excitation of the entire belt may be an extremely fast event. However, typically, the complete belt excitation seems to require about ∼10 Myr or so.

Figure 16 shows a case where the level of eccentricity excitation is consistent with the observed one. However, the region between 2.1 and 2.8 au is under-excited in inclination. In general an under-excited inner belt is less of a problem than an under-excited outer belt, because perturbations from the inner parts of the solar system (e.g., Mars and other remnant planetary embryos) may excite the inner belt but not the outer belt (Izidoro et al. 2015b).

Figure 17 shows a simulation where Jupiter and Saturn are near the 3:2 resonance. In this case, both eccentricities and orbital inclination of bodies in the belt are modestly under-excited when compared with the real belt.

Despite some cases failing to reproduce the dynamical excitation of the belt, we stress that in all these examples the level of dynamical excitation produced is substantially higher than if Jupiter and Saturn had those respective period ratios but regular orbits. When Jupiter and Saturn have regular orbits, only objects near strong SRs (as ${\nu }_{6}$ and ${\nu }_{16}$) and MMRs have their eccentricities and inclinations significantly increased relative to the initial value (almost coplanar and circular orbits; (Raymond et al. 2009; Izidoro et al. 2015b)).

In this section (see Table 1) we provide the initial conditions of our simulations corresponding to Figures 1, 2, 4, and 13–17.