The Dynamical Consequences of a Super-Earth in the Solar System

The N-body integrations for the analysis reported here were performed with the Mercury Integrator Package (Chambers 1999), where the methodology was similar to that described by Kane & Raymond (2014) and Kane (2016, 2019). The orbital elements for the major planets were extracted from the Horizons DE431 ephemerides (Folkner et al. 2014); minor planets were not included in the simulation. The simulations used a hybrid symplectic/Bulirsch–Stoer integrator with a Jacobi coordinate system to provide increased accuracy for multiplanet systems (Wisdom & Holman 1991; Wisdom 2006). Given the relatively large semimajor axes involved in our simulations, general relativity was not calculated in the evolution of the planetary orbits. Dynamical simulations conducted by Lissauer et al. (2001) provided a dynamical simulation that added mass to Ceres and several other asteroids, finding dynamically stable configurations for a subset of the simulations. Here, we provide a more general case by systematically exploring the parameter space of planetary mass and semimajor axis of the added body. Specifically, the simulations described here add a planet with mass range 1–10 M_⊕, in steps of 1 M_⊕. The additional planet was placed at various starting locations in a circular orbit, coplanar with Earth's orbit, and with a semimajor axis range 2–4 au, in steps of 0.01 au, as shown in Figure 1. This resulted in several thousand simulations, where each simulation was allowed to run for 10⁷ yr, commencing at the present epoch and an orbital configuration output every 100 simulation years. We used a simulation time resolution of 1 day to provide adequate sampling based on the orbit of the inner planet, Mercury. An initial baseline simulation was conducted without the additional planet, to ensure that the eccentricity variations of the solar system planets were properly reconstructed in agreement with other solar system dynamical models, such as those by Laskar (1988). The baseline simulation provided an important point of comparison from which to evaluate the potential divergence of the main suite of simulations with the additional terrestrial planet. The divergence of the simulation results from the baseline model were primarily tracked via the eccentricity evolution of the solar system planets, where planets are considered removed if they are either ejected from the system or lost to the potential well of the host star.

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As mentioned in Section 3.1, several thousand simulations were conducted, producing a vast variety of dynamical outcomes for the solar system planets. The inner solar system planets are particularly vulnerable to the addition of the super-Earth planet, resulting in numerous regions of substantial system instability. The broad region of 2–4 au contains many locations of MMRs with the inner planets that further amplify the chaotic evolution of the inner solar system. There are also important MMR locations with the outer planets within the 2–4 au region, with potential significant consequences for the ice giants. Here we provide several specific examples of simulation results to highlight the range of possible dynamical outcomes.

Shown in Figure 2 is an example of the eccentricity evolution of the inner planets for the case of an inserted planet with a mass and semimajor axis of 7.0 M_⊕ and 2.00 au, respectively. The eccentricity evolution for the inserted planet is shown in the bottom panel. In addition, we provide the evolution of the semimajor axis for all planets in the panels of Figure 3, shown as the fractional change in the semimajor axis from the initial value. For this example, the orbits of all four inner planets become sufficiently unstable such that they are removed from the system before the conclusion of the 10⁷ yr simulation. The semimajor axis of 2 au places the additional planet in a 2:3 MMR with Mars, leading to a rapid deterioration of the Martian orbit, culminating with the ejection of Mars approximately halfway through the simulation. Mercury is also ejected relatively early in response to interactions with Venus and Earth, whose eccentricities gradually increase and deposit angular momentum into Mercury's orbit. Figures 2 and 3 show the gradual increase in eccentricities for Venus and Earth, along with the increasing semimajor axis of Venus and decreasing semimajor axis of Earth, leading to catastrophic close encounters. Consequently, both Venus and Earth are removed from the system during the time period 8–9 Myr after the commencement of the simulation. For the super-Earth itself, an examination of the periastron longitude relative to those of Jupiter and Saturn reveal a weak connection of the eccentricity evolution to the ν₆ secular resonance with Saturn, and is largely affected by the presence of Jupiter.

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A second example for the inner solar system planets is shown in both Figures 4 and 5, for the case of an inserted planet with a mass and semimajor axis of 8.0 M_⊕ and 3.70 au, respectively. In this case, the orbit of Mars is relatively unaffected by the presence of the additional planet, whose eccentricity undergoes high-frequency oscillations due to interactions with the outer planets. However, the ν₅ secular resonance with Jupiter plays a role in the eccentricity excitations of Venus and Earth. In particular, the initial slight increase in the eccentricities of Venus and Earth, combined with the ν₅ resonance, is sufficient to perturb the orbit of Mercury, resulting in the rapid removal of Mercury from the system. The catastrophic loss of Mercury causes a subsequent injection of that angular momentum into the orbits of Venus and Earth. This results in a substantial periodic evolution of their orbits, with both high- and low-frequency variations in their eccentricities. Figure 5 shows that, although the semimajor axis of Earth and Mars are largely unaltered, the semimajor axis of Venus experiences a slight reduction.

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The effect of the additional planet on the orbital evolution of the outer planets tends to be less severe than for the inner planets, but there are still several cases of significant orbital perturbations. The example shown in Figures 6 and 7 reveals the orbital evolution of the outer planets for the case where the additional planet has a mass and semimajor axis of 7.0 M_⊕ and 3.79 au, respectively. A significant aspect of this orbital configuration is that the additional planet lies at the 8:5 MMR with Jupiter, creating further opportunity for orbital instability within the system. Indeed, the orbits of the planets are largely unaffected for the first several million years. After ∼3.5 Myr, a dramatic change occurs within the system, whereby the orbit of the super-Earth planet is substantially altered, with high eccentricity (0.5–0.7) and with semimajor axis values in the range 10–30 au, as indicated in the bottom panels of Figures 6 and 7. Figure 7 also shows that this change in the super-Earth orbit corresponds with a slight increase in the semimajor axis of Saturn. This period of instability persists for the next several hundred thousand years until the super-Earth is ejected from the system after a total integration time of ∼4.2 Myr. The loss of the super-Earth planet, and subsequent transfer of angular momentum, has a marginal effect on the orbit of Jupiter but a substantial effect on the eccentricities of Saturn, Uranus, and Neptune. In particular, the orbit of Uranus remains in a quasi-perturbed state, and the third and fourth panels of Figure 6 show that Uranus and Neptune thereafter exchange angular momentum with a significantly higher amplitude than their previous baseline interactions.

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A final example of the outer planet eccentricity evolution is shown in Figures 8 and 9. In this case, the additional planet has a mass and semimajor axis of 7.0 M_⊕ and 3.80 au, respectively, so the overall system architecture is only slightly different to the case shown in Figures 6 and 7. Indeed, the super-Earth is similarly influenced by the 8:5 MMR with Jupiter and the 4:1 MMR with Saturn, leading to significant planet–planet interactions after ∼2.5 Myr. These interactions are especially manifest in the increased eccentricity of Jupiter and Saturn, the latter of which also increases in semimajor axis, and the ejection of the super-Earth from the system entirely. These interactions also perturb the orbit of Neptune, which moves through a period of chaotic changes in eccentricity up until ∼4 Myr. The biggest effect of the interactions though is the loss of Uranus after ∼4 Myr, illustrating how the presence of a super-Earth within the solar system can greatly alter the architecture of the outer planets. The substantial difference between the cases shown in Figures 6 and 8 demonstrates the sensitivity of the dynamical outcome to the initial conditions, particularly near locations of MMR. It is also worth noting that, in cases such as these that produce significant interactions in the outer solar system, the orbits of the inner planets generally become unstable also. For example, in the case described here and represented in Figures 8 and 9, Mars is ejected from the solar system after ∼6 Myr.

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As described in Section 3.1, several thousand simulations with the inclusion of a super-Earth planet were executed, in addition to a baseline simulation for the unmodified solar system architecture. The divergence from the baseline scenario was evaluated by dividing the eccentricity range throughout the simulation for each planet with the eccentricity range from the baseline case. This produced a ratio of eccentricity ranges that are generally ≥1, where unity implies that there is relatively little change in the orbital evolution of that planet for a particular simulation. Cases where the ratio of eccentricity ranges is <1 can occur in relatively rare situations where the damping of eccentricity oscillations exceeds the perturbative effects of the other planets.

The results of the above described calculations for all of the simulations are represented as intensity plots, shown in Figures 10, 11, 12, and 13. The calculated eccentricity ranges relative to the baseline model are labeled "Δ Eccentricity" on the color bars. Here, we note some of the major features of these figures, and discuss their implications in Section 4. Figure 10 shows the results for Mercury and Venus, and Figure 11 shows the results for Earth and Mars. These figures demonstrate the sensitivity of the inner planet orbits to the addition of the super-Earth at locations in the range 2.0–2.7 au, particularly for Mars. The orbit of Mercury is also especially vulnerable to a super-Earth location in the range 3.1–4.0 au, resulting in significant eccentricity excursions for Mercury. Venus and Earth appear to have very similar orbital evolution responses to the addition of the super-Earth, regardless of the planetary mass or semimajor axis location.

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Figure 12 shows the results for Jupiter and Saturn, and Figure 13 shows the results for Uranus and Neptune. The features of the eccentricity divergences for the outer planets from the baseline model primarily occur at locations of MMR. Indeed, a location of instability common to both the inner and outer planets is ∼3.3 au which is the 2:1 MMR with Jupiter and a significant location within the Kirkwood gaps (see discussion in Section 2). As expected, the variations in eccentricity are relatively small for Jupiter and Saturn, who remain largely unaffected by the presence of the super-Earth except at resonance locations. However, as demonstrated by the several outer planet examples in Section 3.2, the locations of MMR can have a devastating effect on the orbits of Uranus and Neptune, including the loss of one or both ice giants from the system.

The results provided in Section 3.3, and in Figures 10 and 11, show that there are broad regions of mass and semimajor axis for the additional planet that can have have severe consequences for the inner solar system planets. In general, the inner planets retain their orbital integrity if the additional planet is located within the range 2.7–3.1 au, even for relatively high masses. This semimajor axis region minimizes interactions with the giant planets that would otherwise excite the orbital eccentricity of the additional planet. However, the full range of semimajor axes (2–4 au) for the added super-Earth spans a period range of 1033–2922 days, in which there are numerous MMR locations with the inner planets. For example, the semimajor axis range of 2.0–2.1 au encompasses the 1:3 MMR with Earth, though it is unclear that this MMR plays a major contributing role to the destabilization of the other inner planets. As shown in Figure 11, the stability of Mars is particularly sensitive to the super-Earth being located in the semimajor axis range 2.0–2.7 au. There are various MMR locations throughout this region, including the 1:2 MMR with Mars at 2.42 au and the 4:1 and 3:1 MMRs with Jupiter, located at 2.07 au and 2.50 au, respectively.

Also of note is the evolution of the Mercury orbit into regions of high eccentricity when the additional planet lies within the semimajor axis range of 3.1–4.0 au, occasionally resulting in the ejection of Mercury from the system prior to the conclusion of the simulation. Mercury does not lie near any significant locations of MMR with the full range of semimajor axes for the additional planet. However, there have been numerous studies regarding the chaotic nature of Mercury's orbit (Ward et al. 1976; Correia & Laskar 2004; Lithwick & Wu 2011; Noyelles et al. 2014), including its sensitivity to the eccentricity and inclination of Venus' orbit. Figures 10 and 11 show that the eccentricities of the Venus and Earth orbits are impacted by the 3.1–4.0 au semimajor axis range of the super-Earth, as well, although to a lesser extent than that of Mercury. Additionally, as noted in Section 3.2, the orbits of Venus and Mercury are sensitive to the ν₅ resonance with Jupiter, and the combination of such resonances may be the primary cause of the Venus orbital changes. Consequently, these alterations to Venus's and Earth's orbits may be sufficient to dramatically contribute to the evolution of Mercury's orbit, further demonstrating the complex dynamical interactions that can occur between planets that are relatively widely separated.

As seen from the several simulation results examples described in Section 3.2, the outer planets do not escape the consequences of the super-Earth's presence in the solar system. Section 3.3 mentions that Jupiter and Saturn are largely unperturbed by the presence of the super-Earth. The exceptions to this are locations of MMR, where Jupiter and Saturn experience mild increases in eccentricity. Particularly strong locations of MMR are those associated with Jupiter; specifically, ∼2.06 au (4:1), ∼2.5 au (3:1), ∼3.3 au (2:1), ∼3.58 au (7:4), ∼3.7 au (5:3), and ∼3.97 au (3:2). Many of these locations are represented as Kirkwood gaps in the asteroid belt, as described in Section 2. Note that ∼3.79 au corresponds to the 4:1 MMR with Saturn, but there is little evidence that this MMR plays a significant role in the orbital evolution of the super-Earth. Even at locations of MMR with the super-Earth, the perturbation to the Jupiter and Saturn eccentricities remain relatively small, and there are no simulation outcomes in which either planet are ejected from the solar system.

However, the consequences of the super-Earth's presence near the MMR locations are far more severe for the ice giants, with frequent excitation of their eccentricities, including rare occasions of ejection of the ice giants from the solar system. The elevation of the ice giant eccentricities has two major sources. The first source is via the angular momentum transfer through Jupiter and Saturn, whose inflated eccentricities, though relatively small, deliver substantial dynamical effects to the ice giants. The second source is the result of a dramatic alteration of the super-Earth's orbit, placing it in a highly eccentric chaotic orbit with a semimajor axis beyond the orbit of Saturn. In such cases, the ice giants gravitationally interact directly with the super-Earth and develop their own divergent orbital states. These interactions usually lead to the super-Earth being ejected from the system, and it is unlikely that the ice giant orbits will maintain long-term stability beyond the 10⁷ yr simulation. It is worth noting that these described sources of orbital perturbation for the ice giants are layered on top of complicated interactions between the super-Earth and the overall dynamics of the system, potentially resulting in further sources of instability, particularly related to secular frequencies with Jupiter.

As the known exoplanet population has grown, the study of demographics and their statistical relationship to planetary architectures has become increasingly important. The occurrence rates of terrestrial planets, including super-Earths, have been estimated from radial velocity (e.g., Howard et al. 2010; Bashi et al. 2020; Fulton et al. 2021; Rosenthal et al. 2021; Sabotta et al. 2021) and transit surveys (e.g., Dressing & Charbonneau 2013; Kopparapu 2013; Kunimoto & Matthews 2020; Bryson et al. 2021). The results of these occurrence-rate estimates vary enormously, depending upon observational biases, analysis techniques, and the parameter space being explored. In particular, most of these studies are heavily biased toward short-period planets around M dwarfs, since the radial velocity amplitude and transit probability/depths are maximized within that regime. However, these analyses consistently indicate that terrestrial planets are relatively common, with super-Earth planets occurring in most of the detected planetary systems. The exception to these planetary size distributions occur within a photoevaporation regime of close-in planets, where there is a relative dearth of super-Earths and mini-Neptunes (Lopez & Fortney 2013; Owen & Wu 2013; Fulton et al. 2017).

On the other hand, giant planets are relatively rare, particularly beyond the snow line, with occurrence rates averaging around 10% for solar-type stars, depending on the exact stellar type and semimajor axis range explored (Wittenmyer et al. 2020; Fulton et al. 2021). Giant planets are even scarcer around later-type stars (Zechmeister et al. 2013; Wittenmyer et al. 2016), lending further credence to the hypothesis that the solar system may not be representative of planetary architectures. Indeed, the presence of giant planets can be disruptive to the orbits of dynamically packed systems. Kane et al. (2020a) and Mulders et al. (2021) found that the occurrence of hot super-Earths and cold giant planets are anticorrelated around M dwarfs. These findings are important for the results presented in this work, since a major reason for the regions of instability within the solar system architecture are due to the interaction of the super-Earth with the outer giant planets.

An important feature of exoplanetary systems is their eccentricity distribution, and the important role that eccentricity plays in the evolution and stability of terrestrial planet climates (Williams & Pollard 2002; Dressing et al. 2010; Kane & Gelino 2012; Linsenmeier et al. 2015; Bolmont et al. 2016; Kane & Torres 2017; Way & Georgakarakos 2017; Kane et al. 2020b). As shown in Figure 4, the presence of the super-Earth has significant effects on the orbits for the inner terrestrial planets. These interactions result in large-amplitude oscillation of Venus's and Earth's orbital eccentricities, creating Milankovitch cycles that may potentially influence the long-term climate of these planets (Deitrick et al. 2018a, 2018b; Horner et al. 2020a; Kane et al. 2021b). Given the vast range of planetary system architectures and terrestrial planet climates, it is unclear how the high occurrence rate of super-Earths may contribute to the overall evolution of terrestrial climates in exoplanetary systems (Becker & Adams 2017). The dependence of planetary climates on orbital interactions with super-Earths will require further atmospheric data and modeling to determine if the presence of such planets (or lack thereof) can preferentially lead to eccentricity-driven climate effects.