Outward migration of Jupiter and Saturn in 3:2 or 2:1 resonance in radiative disks: implications for the Grand Tack and Nice models
Embedded in the gaseous protoplanetary disk, Jupiter and Saturn naturally become trapped in 3:2 resonance and migrate outward. This serves as the basis of the Grand Tack model. However, previous hydrodynamical simulations were restricted to isothermal disks, with moderate aspect ratio and viscosity. Here we simulate the orbital evolution of the gas giants in disks with viscous heating and radiative cooling. We find that Jupiter and Saturn migrate outward in 3:2 resonance in modest-mass (, where MMSN is the ”minimum-mass solar nebula”) disks with viscous stress parameter between and . In disks with relatively low-mass () , Jupiter and Saturn get captured in 2:1 resonance and can even migrate outward in low-viscosity disks (). Such disks have a very small aspect ratio () that favors outward migration after capture in 2:1 resonance, as confirmed by isothermal runs which resulted in a similar outcome for and . We also performed N-body runs of the outer Solar System starting from the results of our hydrodynamical simulations and including 2-3 ice giants. After dispersal of the gaseous disk, a Nice model instability starting with Jupiter and Saturn in 2:1 resonance results in good Solar Systems analogs. We conclude that in a cold Solar Nebula, the 2:1 resonance between Jupiter and Saturn can lead to outward migration of the system, and this may represent an alternative scenario for the evolution of the Solar System.
Subject headings:planets and satellites: formation — planet-disk interactions — accretion, accretion disks — hydrodynamics — methods: numerical
Two new models piece together a story of the evolution of the Solar System. They are the Grand Tack and Nice models.
The Grand Tack model addresses the formation of the inner Solar System (Walsh et al 2011; see also Morbidelli et al 2012; Raymond et al 2013; O’Brien et al 2014). It invokes an inward-then-outward migration of Jupiter and Saturn consistent with hydrodynamical simulations (Masset & Snellgrove 2001; Morbidelli & Crida 2007). The change in the direction of migration, or ”tack”, is set to occur when Jupiter is at 1.5 AU, so that the sculpted inner disk of planetary embryos can reproduce the terrestrial planets, in particular the large Earth/Mars mass ratio (Wetherill 1991; Hansen 2009; Raymond et al 2009).
The Nice model proposes that after the gas disk dispersal, the outer Solar System underwent a delayed instability triggered by interactions between the giant planets and an outer disk of planetesimals. In the original version of the model, the giant planets started on an arbitrary but more closely-packed orbital configuration. The instability was triggered by Jupiter and Saturn crossing their mutual 2:1 resonance (Tsiganis et al 2005). In more recent incarnations, the giant planets’ orbital configuration was sculpted by an earlier phase of gas-driven migration (Morbidelli et al 2007), with initial conditions that are consistent with the Grand Tack scenario. Angular momentum exchange between the giant planets and the planetesimal disk can act to extract the giant planets from a resonant chain and trigger the instability (Levison et al 2011).
The Nice model has been invoked to explain the Late Heavy Bombardment (Gomes et al. 2005). It can reproduce the current architecture of the giant planets (Nesvorny 2011; Nesvorny & Morbidelli 2012), the orbital distribution of Jupiter’s Trojan asteroids (Morbidelli et al. 2005; Nesvorny et al. 2013), the irregular satellites of Jupiter (Nesvorny et al. 2014), Saturn, Uranus and Neptune (Nesvorny et al. 2007), and the structure of the Kuiper belt in broad strokes (Levison et al. 2008, Batygin et al. 2012).
Jupiter and Saturn’s gas-driven migration in 3:2 resonance has been confirmed by several studies (Masset & Snellgrove 2001; Morbidelli & Crida 2007; Pierens & Nelson 2008; Pierens & Raymond 2011). Jupiter and Saturn naturally converge into 3:2 resonance and then migrate outward. However, each of these studies used an isothermal disk with a standard value for the aspect ratio .
In this paper we present the outcome of hydrodynamical simulations of the planets embedded in gaseous disks that are subject to viscous heating and radiative cooling. We show that in relatively low-mass (, where MMSN is the minimum-mass solar nebula model of Hayashi 1981) and low-viscosity (viscous stress parameter or ; see Shakura & Sunyaev 1973) disks, Jupiter and Saturn can migrate outward in 2:1 resonance. This arises because for a purely viscously heated disk, a low-disk mass and/or a small viscosity give rise to a very small disk aspect ratio , which tends to favor outward migration. The advantage of having Jupiter and Saturn in 2:1 rather than in 3:2 resonance is that it can increase the success rate of the Nice model instability. Nesvorny & Morbidelli (2012) indeed showed that if Jupiter and Saturn start in a 2:1 resonance, there is a better chance that their period ratio jumps to the correct value.
Our paper is structured as follows. In Sect. 2, we describe the hydrodynamical model and discuss the results of our simulations in Sect. 3, and the effects of stellar heating on these results in Sect. 4. Nice model simulations that study the evolution after disk dispersal are presented in Sect. 5. Finally, we draw our conclusions in Sect. 6.
2. Hydrodynamical model
Simulations were performed using the GENESIS (De Val-Borro et al. 2006) numerical code that solves the equations governing the disk evolution on a polar grid . The code’s energy equation reads:
where is the gas velocity, the thermal energy density, the adiabatic index (set to ). is the pressure which is related to the disk temperature and surface density as , where is the ideal gas constant and the mean molecular weight. is the viscous heating term where viscous stresses are modeled using the standard ’alpha’ prescription for the disk viscosity , where is the sound speed and the disk scale height which is related to the angular velocity and the isothermal sound speed as . is the local radiative cooling term, where is the Stephan-Boltzmann constant and is the effective temperature which is computed using the opacity law of Bell & Lin (1994). In this work, effects resulting from stellar heating are not considered in the energy budget.
We employ radial grid cells uniformly distributed between and AU, and azimuthal grid cells. At the inner edge, we use a viscous outflow boundary condition (see Pierens & Nelson 2008). At the outer edge, we employ a wave-killing zone for to avoid wave reflections.
The initial surface density profile is , where in dimensionless units is the surface density at AU of the MMSN, and is an enhancement factor. The initial temperature profile is such that . Due to the action of source terms in Eq. , however, this initial temperature quickly evolves until an equilibrium state is reached.
Jupiter and Saturn initially evolve on circular orbits with semi-major axes and AU respectively, just exterior to their mutual resonance. To give them sufficient time to open a gap, the planets are held on fixed circular orbits for orbits, and then are released and evolve under the action of disk torques. When calculating the disk torques, we exclude the material contained within a distance from the planets using a Heaviside filter (Crida et al. 2009) , where is the Hill radius.
In our simulations we tested the effect of two parameters: the disk’s surface density and viscosity. We tested , and for each value of we performed four different simulations with , for a total of 16 simulations.
The results of our simulations are illustrated in Fig. 1. There are four qualitatively different outcomes shown with different symbols. Runs that are labelled ”3:2 resonance” (downwards triangles) produced outward migration of Jupiter and Saturn in a 3:2 resonance (Masset & Snellgrove 2001; Morbidelli & Crida 2007). The top panels of Fig. 2 show one such outcome, for the simulation with and . The planets migrate convergently and become trapped in a 2:1 resonance at orbits (see top middle panel of Fig. 2). Due to Saturn’s fast migration, the planets break free from the 2:1 resonance and become trapped in a resonance at orbits. From that time on, Jupiter and Saturn evolve in a common gap and the resulting change in the torque balance, together with the flux of gas from the outer disk across the gap, makes them migrate outward. The 3:2 resonance is maintained throughout (resonant angles shown in the top right panel of Fig. 2).
The middle panels of Fig. 2 show the evolution of a simulation with and . Jupiter and Saturn were captured in 2:1 resonance. This triggered outward migration that was maintained to the end of the simulation. As for the 3:2 resonance, outward migration was maintained by a continuous flux of gas across the planets’ common gap. This flux acts to replenish the inner disk and also to exert a positive corotation torque on the planets. The main difference between the two simulations illustrated in Fig. 2 is that the planets’ eccentricities are much higher migrating outward in 2:1 resonance. This occurs because the gap is wider and deeper in the simulation with the gas giants in 2:1 resonance, resulting in a weaker damping of the planets’ eccentricities.
Jupiter and Saturn should be caught in stable 2:1 resonance when Saturn’s migration is slower than a critical rate. The critical velocity for capture in the 2:1 resonance (D’Angelo & Marzari 2012), where is the Saturn’s angular velocity. Assuming an isothermal type I migration rate for Saturn (Tanaka et al. 2002) and that Jupiter does not migrate, D’Angelo & Marzari (2012) found that Jupiter and Saturn would become trapped in 2:1 resonance if the surface density at AU is . Although we use a non-isothermal disk model here, Fig. 1 shows that our simulations are consistent with this estimate, since all runs with formed a stable 2:1 resonance. An alternative possibility, which is the one that actually happens in the simulations, is that both the disk aspect ratio and viscosity are small enough for Saturn undergo a slow Type II migration (Lin & Papaloizou 1993). In that case, capture in 2:1 resonance is expected provided that the condition for gap opening is satisfied (Crida et al. 2006):
where is the Saturn mass ratio. For a thin disk with , whereas for , capture in the 2:1 resonance should arise for . This estimate also agrees with our simulations since runs with disc masses corresponding to the MMSN and produced a stable 2:1 resonance.
When viscous stresses are the only heating process, a low disk mass (small value of ) and a modest viscosity (small ) produces a thin disk. Figure 3 (top panel) shows the aspect ratio as a function of radius for two disks; a low-mass, low-viscosity disk with , and a high-mass, modest-viscosity disk with , . Here, Jupiter and Saturn are held on circular orbits at and AU, respectively (exterior to 2:1 resonance). In the high-mass disk the aspect ratio is () at the location of Jupiter (Saturn). In contrast, in the low-mass disk () at Jupiter’s (Saturn’s) orbit. For the low-mass, low-viscosity disk, the gaps opened by the planets are wider and deeper than in the high-mass disk (see bottom panel of Fig. 3). In the low-mass, low-viscosity disk the gas near Saturn’s orbit is significantly depleted, imparting a strong positive torque on Jupiter which may favor outward migration (Morbidelli & Crida 2007).
A small aspect ratio thus appears to be the key factor in causing outward migration with Jupiter and Saturn in 2:1 resonance. In principle, outward migration should also occur in isothermal disk models with small aspect ratios. To test this hypothesis, we performed a simulation using an isothermal equation of state with and and which indeed resulted in outward migration with Jupiter and Saturn in a 2:1 resonance, as illustrated in the lower panel of Fig. 2.
Fig. 4 shows the results of radiative simulations that produced stable 2:1 resonances. For a MMSN model, the condition for outward migration is whereas for , outward migration occurs for . This is consistent with the expectation that the aspect ratio in radiative disks should be a function of the product (Bitsch et al. 2014). Outward migration tends to be faster for smaller and smaller . The run with and , which exhibits a smaller outward migration rate than in the case with , is an exception; this probably occurs because the inner edge of Jupiter’s gap is located further away from the planet in that case.
4. Effect of stellar irradiation
Results from the provious section suggest that the 2:1 resonance can lead to outward migration in disks with very small aspect ratios . For a radiative disk subject to viscous heating only, low values of the aspect ratio correspond to small disk masses and/or viscosities and therefore to low accretion rates. In that case, stellar irradiation may contribute significantly to the disk temperature structure and possibly dominate over viscous heating, but this depends strongly on the disk metallicity (Bitsch et al. 2014). Recent hydrodynamical simulations of the structure of disks with constant accretion rate (Bitsch et al. 2014) show that in the late stages of evolution, a metallicity of can lead to an aspect ratio in the inner parts, and it is expected that even smaller values for to be reached for lower metallicities. Moreover, 1D modeling of viscous protoplanetary disks subject to stellar irradiation (Baillié & Charnoz, in prep.) also shows that at very late times, the aspect ratio can be typically at AU. Both of these studies therefore suggest that the conditions for Jupiter and Saturn to migrate outward in 2:1 resonance may be fulfilled under certain conditions. It remains to be seen, however, whether outward migration of Jupiter and Saturn in 2:1 resonance triggered in a part of the disk with a low aspect ratio could be maintained if the planets migrated into an outer, thicker part of the disk. Clearly, more sophisticated hydrodynamical simulations are needed to definitely assess the effect of stellar heating on these results.
5. Evolution after gas disk dispersal
We now turn our attention to the later evolution of the outer Solar System. Our goal is to test whether simulations showing outward migration of Jupiter and Saturn in 2:1 resonance are compatible with the current architecture of the Solar System.
We selected three disk models that produced outward migration in 2:1 resonance at different rates (with , ; , and , ). We restarted each simulation and artificially dissipated the disk by forcing the gas surface density to decay exponentially with an e-folding time yr. We then used the outputs of these runs, rescaled such that the initial semi-major axis of Jupiter was AU, as initial conditions for N-body simulations of the evolution of the outer Solar System after dispersion of the gas disk.
Following Nesvorny (2011) and Nesvorny & Morbidelli (2012), we performed simulations with Uranus and Neptune initially located in a resonant chain with the giant planets. The resonant chain is generated using both hydrodynamic and N-body simulations to identify the resonant configurations that are compatible with Jupiter and Saturn in 2:1 resonance. We also considered a five-planet case where an additional planet with mass comparable to that of Uranus/ Neptune was placed between the orbits of Saturn and Uranus. In each case, the orbit of one planet was purposely shifted by in mean anomaly to artificially disrupt the resonant chain and trigger an instability (see Levison et al 2011). An outer planetesimal disk was included with masses of . For each case, we performed simulations with different, randomly-generated, radial ranges and surface density profiles of the planetesimal disk.
In the four-planet case, the initial states deduced from our three disk models did not produce good Solar System analogs. Low disk masses typically led to final systems with fewer than four planets. For high disk masses the ice giants migrated too far and the giant planets end up on too-circular orbits due to too strong disk-induced eccentricity damping (Nesvorny 2011). We obtained better results in simulations with an extra ice giant. This is broadly consistent with Nesvorny & Morbidelli (2012), who found a much higher success rate in reproducing the current architecture of the Solar System in the five-planet case. This was true for Jupiter and Saturn in both 3:2 and 2:1 resonance.
Figure 5 shows the results of a successful five-planet run with initial configuration from the model with , . The planets were started in a (2:1, 3:2, 3:2, 3:2) resonant chain and the planetesimal disk mass was . The extra ice giant (in blue) was ejected from the system after Myr, and the subsequent migration of the giant planets was marginal. This run satisfies most of the constraints from Nesvorny & Morbidelli (2012). The final semi major axes of the planets are , , and AU and their final eccentricities are , , and . Neptune is a little too close to the Sun and Jupiter’s eccentricity is slightly smaller than its current value, but the lower panel of Fig. 5 shows that the final Jupiter-Saturn period ratio is almost perfect. When the ice giant is ejected at Myr, the Jupiter-Saturn period ratio jumps from to in Myr such that the secular resonance jumps, rather than sweeps, across the inner Solar System (Brasser et al 2009; Morbidelli et al 2010; called constraint D by Nesvorny 2011 and Nesvorny & Morbidelli 2012). In this example the secular amplitude is also close to the current value.
Although we did not perform a broad statistical study as in Nesvorny & Morbidelli (2012), the existence of successful cases such as the one from Fig. 5 shows that the outward migration of Jupiter and Saturn in 2:1 resonance is consistent with the late evolution of the outer Solar System.
6. Discussion and Conclusion
We have shown that a Grand Tack can occur with Jupiter and Saturn in 2:1 resonance for a limited range of the parameter space defined by the disk’s mass and viscosity. Outward migration in 2:1 takes place in relatively low-mass (), low-viscosity () disks that tend to have very small aspect ratios (). Compared with outward migration in 3:2 resonance, the biggest difference in the evolution when Jupiter and Saturn are in 2:1 resonance is that their eccentricities are higher, with 0.05-0.2.
Outward resonance of Jupiter and Saturn in 2:1 resonance is consistent with a later, Nice model instability. We used the simulations in which the gas giants migrated outward in 2:1 resonance as inputs to N-body simulations of the evolution of the outer Solar System after dispersal of the gas disk. Instabilities including an outer planetesimal disk produced good Solar System analogs, matching most of the criteria derived in Nesvorny (2011). Therefore, a scenario in which Jupiter ”tacked” at AU when Saturn caught up and was trapped in a 2:1 resonance may also explain the evolution of both the inner and outer Solar System.
Outward migration in 2:1 resonance is possible in disk models that have a very small aspect ratio, typically at the location of Jupiter. For such a value of , the disk is substantially depleted at Saturn’s orbit, creating a strong positive torque exerted on Jupiter and favoring outward migration. This result was confirmed by locally isothermal runs which resulted in a similar outcome for and . We note that more realistic protoplanetary disks models subject to stellar irradiation are thought to have similar aspect ratios at late stages, especially if their metallicity is small (Bitsch et al. 2014). This suggests that a Grand Tack scenario with Jupiter and Saturn in 2:1 resonance is possible in evolved protoplanetary disks, but this needs to be checked with 3D radiative hydrodynamical simulations.
- affiliation: Univ. Bordeaux, Laboratoire d’Astrophysique de Bordeaux, UMR 5804, F-33270, Floirac, France
- affiliation: CNRS, Laboratoire d’Astrophysique de Bordeaux, UMR 5804, F-33270, Floirac, France
- affiliation: Univ. Bordeaux, Laboratoire d’Astrophysique de Bordeaux, UMR 5804, F-33270, Floirac, France
- affiliation: CNRS, Laboratoire d’Astrophysique de Bordeaux, UMR 5804, F-33270, Floirac, France
- affiliation: Department of Space Studies, Southwest Research Institute, 1050 Walnut Street Suite 300, Boulder, CO 80302, USA
- affiliation: University of Nice-Sophia Antipolis, CNRS, Observatoire de la côte d’Azur, Laboratoire Lagrange, BP4229, 06304 NICE Cedex 4, France
- Baillié, K., & Charnoz, S. 2014, in prep.
- Batygin, K., Brown, M. E., & Betts, H. 2012, ApJ, 744, L3
- Bell, K. R., & Lin, D. N. C. 1994, ApJ, 427, 987
- Bitsch, B., Crida, A., Morbidelli, A., Kley, W., & Dobbs-Dixon, I. 2013, A&A, 549, A124
- Bitsch, B., Morbidelli, A., Lega, E., & Crida, A. 2014, A&A, 564, A135
- Brasser, R., Morbidelli, A., Gomes, R., Tsiganis, K., & Levison, H. F. 2009, A&A, 507, 1053
- Crida, A., Morbidelli, A., & Masset, F. 2006, Icarus, 181, 587
- Crida, A., Baruteau, C., Kley, W., & Masset, F. 2009, A&A, 502, 679
- D’Angelo, G., & Marzari, F. 2012, ApJ, 757, 50
- de Val-Borro, M., Edgar, R. G., Artymowicz, P., et al. 2006, MNRAS, 370, 529
- Gomes, R., Levison, H. F., Tsiganis, K., & Morbidelli, A. 2005, Nature, 435, 466
- Hansen, B. M. S. 2009, ApJ, 703, 1131
- Hayashi, C. 1981, Progress of Theoretical Physics Supplement, 70, 35
- Levison, H. F., Morbidelli, A., Van Laerhoven, C., Gomes, R., & Tsiganis, K. 2008, Icarus, 196, 258
- Levison, H. F., Morbidelli, A., Tsiganis, K., Nesvorný, D., & Gomes, R. 2011, AJ, 142, 152
- Lin, D. N. C., & Papaloizou, J. C. B. 1993, Protostars and Planets III, 749
- Masset, F., & Snellgrove, M. 2001, MNRAS, 320, L55
- Morbidelli, A., Levison, H. F., Tsiganis, K., & Gomes, R. 2005, Nature, 435, 462
- Morbidelli, A., & Crida, A. 2007, Icarus, 191, 158
- Morbidelli, A., Brasser, R., Gomes, R., Levison, H. F., & Tsiganis, K. 2010, AJ, 140, 1391
- Morbidelli, A., Lunine, J. I., O’Brien, D. P., Raymond, S. N., & Walsh, K. J. 2012, Annual Review of Earth and Planetary Sciences, 40, 251
- Nesvorný, D., Vokrouhlický, D., & Morbidelli, A. 2007, AJ, 133, 1962
- Nesvorný, D. 2011, ApJ, 742, L22
- Nesvorný, D., & Morbidelli, A. 2012, AJ, 144, 117
- Nesvorný, D., Vokrouhlický, D., & Morbidelli, A. 2013, ApJ, 768, 45
- Nesvorný, D., Vokrouhlický, D., & Deienno, R. 2014, ApJ, 784, 22
- O’Brien, D. P., Walsh, K. J., Morbidelli, A., Raymond, S. N., & Mandell, A. M. 2014, Icarus, 239, 74
- Pierens, A., & Nelson, R. P. 2008, A&A, 482, 333
- Pierens, A., & Raymond, S. N. 2011, A&A, 533, A131
- Raymond, S. N., Kokubo, E., Morbidelli, A., Morishima, R., & Walsh, K. J. 2013, arXiv:1312.1689
- Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
- Tanaka, H., Takeuchi, T., & Ward, W. R. 2002, ApJ, 565, 1257
- Tsiganis, K., Gomes, R., Morbidelli, A., & Levison, H. F. 2005, Nature, 435, 459
- Walsh, K. J., Morbidelli, A., Raymond, S. N., O’Brien, D. P., & Mandell, A. M. 2011, Nature, 475, 206