YOUNG SOLAR SYSTEM's FIFTH GIANT PLANET?

Studies of giant planets' interaction with a protoplanetary gas disk show that their orbits radially migrate, and typically achieve a compact configuration, in which the pairs of neighbor planets are locked in orbital resonances (Kley 2000; Masset & Snellgrove 2001). The resonant planetary systems emerging from protoplanetary disks can become dynamically unstable after the gas disappears, leading to a phase when planets scatter off each other. This model can explain the observed resonant exoplanets, commonly large exoplanet eccentricities (Weidenschilling & Marzari 1996), and microlensing data that show evidence for a large number of planets that are free floating in interstellar space (Sumi et al. 2011).

The solar system, with the widely spaced and nearly circular orbits of the giant planets, bears little resemblance to the bulk of known exoplanets. Yet, if our understanding of the physics of planet–gas-disk interaction is correct, it is likely that the young solar system followed the evolutionary path outlined above. Jupiter and Saturn, for example, were most likely trapped in the 3:2 resonance (Masset & Snellgrove 2001; Morbidelli & Crida 2007; Pierens & Nelson 2008), defined as P_Saturn/P_Jupiter = 1.5, where P_Jupiter and P_Saturn are the orbital periods of Jupiter and Saturn (this ratio is 2.49 today).

To stretch to the present state, the outer solar system most likely underwent a violent phase when planets scattered off each other and acquired eccentric orbits (Thommes et al. 1999; Tsiganis et al. 2005). The system was subsequently stabilized by damping the excess orbital energy into the transplanetary disk, whose remains survived to this time in the Kuiper Belt. Finally, as evidenced by dynamical structures observed in the Kuiper Belt, planets radially migrated to their current orbits by scattering planetesimals (Malhotra 1995; Levison et al. 2008).

We conducted computer simulations of the early evolution of the solar system to determine the initial states of planetary orbits (Batygin & Brown 2010). First, we performed hydrodynamic and N-body simulations to identify the resonant configurations that may have occurred among the young solar system's giant planets. Our hydrodynamic simulations used Fargo (Masset 2000) and followed the method described in Morbidelli et al. (2007). As the Fargo simulations are CPU expensive, we used these results as a guide and generated many additional resonant systems with the N-body integrator known as SyMBA (Duncan et al. 1998).

Planets with masses corresponding to those of Jupiter, Saturn, and ice giants were placed in initial orbits with the period ratios slightly larger that those of the selected resonances. The planets were then migrated into resonances with SyMBA modified to include forces that mimic the effects of gas. We considered cases with four and five initial planets, where in the latter case an additional planet was placed into a resonant orbit between Saturn and the ice giants, or beyond the orbit of the outer ice giant. The fifth planet was given a mass between 1/3 and 3 times the mass of Uranus.

Different starting positions of planets, rates of the semi-major axis and eccentricity evolution, and timescales for the gas disk's dispersal produced different results. For Jupiter and Saturn, we confined the scope of this study to the 3:2 and 2:1 resonances, because the former is strongly preferred from previous hydrodynamic studies (Masset & Snellgrove 2001; Morbidelli et al. 2007; Pierens & Nelson 2008). The 2:1 resonance was included for comparison. The eccentricities, inclinations, and resonant amplitudes obtained here were typically e < 0.1, i < 02 and <60°. The inner ice giant had 0.05 < e < 0.1, while the other planets acquired more circular orbits (Jupiter had e ≲ 0.01 in most cases).

The instability of a planetary system can occur after the gas disk's dispersal when the stabilizing effects of gas are removed. Such an instability can be triggered spontaneously (e.g., Weidenschilling & Marzari 1996) or by divergent migration of planets produced by their interaction with planetesimals leaking into the planet-crossing orbits from the transplanetary disk (Thommes et al. 1999). In the latter case, the instability is produced when Jupiter and Saturn cross a major mean motion resonance (Tsiganis et al. 2005; also known as the Nice model).

In the solar system, it is often assumed that the instability occurred at the time of the Late Heavy Bombardment of the Moon some 3.9 Gyr ago, when the lunar basins with known ages formed (Hartmann et al. 2000). If so, the solar system's giant planets would be required to remain on their initial resonant orbits for about 600 Myr. To allow for this possibility, we sifted through the resonant configurations identified above and selected those that were stable over 10⁹ yr, if considered in isolation. Only the stable systems were used for the follow-up simulations, in which we tracked the evolution of planetary orbits through and past the instability.

We included the effects of the transplanetary planetesimal disk in these simulations. The disk was represented by 1000 equal-mass bodies that were placed into orbits with low orbital eccentricities and inclinations, and in radial distances between r_in < r < r_out. The surface density was set to be Σ = 1/r. The outer edge of the disk was placed at r_out = 30 AU, so that the planetesimal-driven migration is expected to park Neptune near its present semi-major axis (Gomes et al. 2004).

We considered cases with r_in = a_N + Δ, where a_N is the semi-major axis of the outer ice giant, N is the number of planets, and Δ = 0.5, 1, and 3.5 AU. The instability was triggered early for Δ = 0.5 AU. To trigger the instability for Δ = 1 and 3.5 AU, we broke the resonant locks by altering the mean anomaly of one of the ice giants. This method was inspired by the recent study of the late instability that showed that planets can slowly exchange orbital energy with a distant planetesimal disk and break from the resonances when the resonant amplitude exceeds certain limits (Levison et al. 2011).

In either case discussed above, the scattering phase between planets starts shortly after the beginning of our simulations, which guarantees a low CPU cost. We considered six different masses of the planetesimal disk, m_disk = 10, 20, 35, 50, 75, and 100 M_Earth, and performed 30 simulations in each case, where different evolution histories were generated by randomly seeding the initial orbit distribution of planetesimals. In total, we completed over 6000 scattering simulations. Each system was followed for 100 Myr with standard SyMBA.

We defined several criteria to measure the overall success of simulations. First, the final planetary system must have four giant planets (criterion A) with orbits that resemble the present ones (criterion B). Note that A means that one planet must be ejected in the five-planet runs, while all four planets need to survive in the four-planet runs. As for B, we claim success if the final mean semi-major axis of each planet is within 20% of its present value and if the final mean eccentricities and mean inclinations are no larger than 0.11 and 2°, respectively. These thresholds were obtained by doubling the current mean eccentricity of Saturn (e_Saturn = 0.054) and mean inclination of Uranus (i_Uranus = 102).

For the successful runs, as defined above, we also checked on the history of encounters between giant planets, evolution of the secular g₅, g₆, and s₆ modes, and secular structure of the final planetary systems. To explain the observed populations of the irregular moons that are roughly similar at each planet (Jewitt & Haghighipour 2007), all planets—including Jupiter—must participate in encounters (Nesvorný et al. 2007). Encounters of Jupiter and/or Saturn with ice giants may also be needed to excite the g₅ mode in the Jupiter's orbit to its current amplitude (e₅₅ = 0.044; Morbidelli et al. 2009).

It turns out that it is generally easy to have encounters between one of the ice giants and Jupiter if planets start in a compact resonant configuration. The amplitudes of the g₆ and s₆ modes also do not pose a problem. It is much harder to excite e₅₅, however. We therefore opt for a criterion in which we require that e₅₅ > 0.022 in the final systems, i.e., at least half of its current value (criterion C). The e₅₅ amplitude was determined by following the final planetary systems for an additional 10 Myr (without planetesimals) and Fourier-analyzing the results (Šidlichovský & Nesvorný 1996).

The evolution of secular modes, mainly g₅, g₆, and s₆, is constrained from their effects on the terrestrial planets and asteroid belt. As outer planets scatter and migrate, these frequencies change. This may become a problem, if g₅ slowly swipes over the g₁ or g₂ modes, because the strong g₁ = g₅ and g₂ = g₅ resonances can produce excessive excitation and instabilities in the terrestrial planet system (Brasser et al. 2009; Agnor & Lin 2011). The behavior of the g₆ and s₆ modes, on the other hand, is important for the asteroid belt (Morbidelli et al. 2010; Minton & Malhotra 2011).

As g₅, g₆, and s₆ are mainly a function of the orbital separation between Jupiter and Saturn, the constraints from the terrestrial planets and asteroid belt can be conveniently defined in terms of P_Saturn/P_Jupiter. This ratio needs to evolve from <2.1 to >2.3 in <1 Myr (criterion D), which can be achieved, for example, if planetary encounters with an ice giant scatter Jupiter inward and Saturn outward. The 1 Myr limit is conservative in that a slower evolution of secular frequencies, which fails to satisfy criterion D, would clearly violate the constraints. Note also that in most simulations that satisfy criterion D, P_Saturn/P_Jupiter evolves from <2.1 to >2.3 in much less than 1 Myr. Most simulations that satisfy D should therefore clearly satisfy the constraints.

In summary, our constraints A and B express the basic requirements of a successful model. Constraint C is more restrictive in that it places a more precise condition on the dynamical structure of the final systems. Constraint D is the least rigorous. It applies only if the instability occurred after the formation of the terrestrial planets, if the amplitudes of the giant planet's secular modes were significant when resonances were crossed, and excluding the low-probability solutions discussed in Brasser et al. (2009).

4.1. Four- and Five-planet Results

We start by discussing the results obtained with four planets that were assumed to be initially locked in the (3:2,3:2,4:3) resonances. The inner ice giant has the largest eccentricity (e₃ = 0.06). According to Levison et al. (2011), the system should therefore be driven toward instability by the energy and momentum exchange between the inner ice giant and planetesimal disk. We set r_in = 15 AU, so that the inner disk edge is well beyond the orbits of the two ice giants (a₃ = 9.6 AU and a₄ = 11.6 AU). This setup should be consistent with the late instability (Levison et al. 2011).

The best results were obtained with M_disk = 35 M_Earth and M_disk = 50 M_Earth. The fraction of simulations producing the final systems with four outer planets is 10% and 13%, respectively (criterion A). Only three of the total of 120 integrations performed for these disk masses satisfied our criterion B. This shows that it is very unlikely that the solar system evolved from these initial states. The results do not improve when M_disk is varied (Section 4.2).

We compare these results with those obtained for the five-planet systems. To start with, the five planets were placed into the (3:2,3:2,4:3,5:4) resonances and fifth planet was given a mass equal to that of Uranus. As in the four-planet case, we fix r_in = 15 AU so that the most eccentric inner ice giant (e₃ = 0.07) has about the same radial distance from the inner edge of the disk.

The best five-planet results were obtained for M_disk = 50 M_Earth with the fraction of systems matching criteria A and B raising to 37% and 23%, respectively (Figure 1). This shows that it is roughly 10 times more likely to obtain a good solar system analog starting from five planets than from four planets, at least in the case discussed above. This result is not unexpected because systems starting from a compact resonant configuration typically suffer a rather violent instability and tend to lose planets (Figure 2).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

4.2. Effects of M_disk

4.3. Effects of r_in

4.4. Mass of the Fifth Planet

4.5. Initial Resonant Configuration

4.6. Planetary Encounters

Only ∼3% of simulations performed for the four-planet case with (3:2,3:2,4:3), r_in = 12.6 AU, and M_disk = 50 M_Earth, satisfy criteria A and B, and show encounters of one of the ice giants with Jupiter. The statistics for other resonances is similarly low. Consequently, in absence of Jupiter encounters, Jupiter and Saturn end up too close to each other if M_disk < 50 M_Earth, and their orbits are too circular.

Larger disk masses produce more plausible final period ratios, as Jupiter and Saturn can slowly separate from each other by scattering planetesimals, but these evolution paths do not satisfy criterion D. Overall, criterion D was satisfied only in ∼1% of the four-planet cases, which is troubling. For a comparison, practically in all five-planet simulations reported here, all planets, including Jupiter, participate in planetary encounters. For example, criterion D was satisfied in 50% of cases that also satisfied A and B for (3:2,3:2,4:3,5:4), M_disk = 50 M_Earth, and r_in = 15 AU (Figure 3), and in over 60% of good cases for (2:1,3:2,3:2,4:3), M_disk = 20 M_Earth, and r_in = 19 AU.

Zoom In Zoom Out Reset image size

4.7. Secular Modes

The formation of Uranus and Neptune is a long-standing problem in planetary science because their accretion at ∼20 and ∼30 AU would require implausibly long timescales (Safronov 1969, Levison & Stewart 2001). The ice giants can form more easily at <15 AU, where densities are higher and dynamical timescales are shorter (e.g., Robinson & Bodenheimer 2010). Our five-planet resonant systems start with all ice giant at <15 AU, if Jupiter and Saturn are in the 3:2 resonance. If Jupiter and Saturn start in the 2:1 resonance, on the other hand, the whole planetary system is more spread out, and the outer ice giant is at ∼18–20 AU, where its formation can be problematic. Future work will need to address these issues.

This work was supported by NLSI and NSF.