On the Formation and Evolution of Giant Planets in Close BinarySystems

Richard P. NelsonAstronomy Unit, School of Mathematical Sciences, Queen Mary, University of London, Mile EndRoad, London, E1, 4NS, U.K.

Abstract. We present the results of hydrodynamic simulations of Jovian mass protoplanets that formin circumbinary discs. The simulations follow the orbital evolution of the binary plus planet systemacting under their mutual forces, and forces exerted by the viscous circumbinary disc. The evolutioninvolves the clearing of the inner circumbinary disc initially, so that the binary plus planet systemorbits within a low density cavity. Interaction between disc and planet causes inward migration ofthe planet towards the inner binary. Subsequent evolution can take three distinct paths: (i) The planetenters the 4:1 mean motion resonance, but is gravitationally scattered through a close encounterwith the binary; (ii) The planet enters the 4:1 mean motion resonance, the resonance breaks, and theplanet remains in a stable orbit outside the resonance; (iii) When the binary has initial eccentricityebin ≥ 0.2, the disc becomes eccentric, leading to a stalling of the planet migration, and apparentstability of the planet plus binary system.

1. Introduction

Extrasolar planets have been observed to exist in binary systems (e.g. γ Cephei, 16Cygni B). Most solar-type stars appear to be members of binary systems (Duquen-noy and Mayor, 1991), and most T Tauri stars, whose discs are thought to bethe sites of planet formation, appear to members of binary systems (e.g. Ghez,Neugebauer, and Matthews, 1993; Leinert et al., 1993). A number of circumbinarydiscs have also been observed (e.g. DQ Tau, AK Sco, UZ Tau, GW Ori). Thereforeit is of interest to explore how stellar multiplicity affects planet formation, andpost-formation planetary orbital evolution, including formation in circumbinarydiscs.

Previous work examined the stability of planetary orbits in binary systems (Dvo-rak, 1986; Holman and Wiegert, 1999). This showed that there is a critical ratio ofplanetary to binary semimajor axis for stability, depending on the binary mass ratio,qbin, and eccentricity ebin. A recent paper (Quintana et al., 2002) explored the latestages of terrestrial planet formation in the α Centauri system, concluding that thebinary companion can help speed up planetary accumumlation.

Recent work (Kley and Burkert, 2000) examined the effect that an externalbinary companion can have on the migration and mass accretion of a giant planetforming in a circumstellar disc. In this work we explore the evolution of Jovianmass protoplanets forming in circumbinary discs. Previous work has shown thata giant protoplanet embedded in a disc around a single star undergoes inwardmigration driven by the viscous evolution of the disc (Lin and Papaloizou, 1986;

PlanetarySystems.tex; 28/12/2006; 12:00; p.181

172 R. P. Nelson

Nelson et al., 2000). Here we are interested in how this process is affected if thecentral star is replaced by a close binary system. In particular we are interested inexploring the orbital stability of planets that migrate towards the central binary. Formultiple planet systems, migration can induce resonant locking (e.g. Snellgrove,Papaloizou, and Nelson, 2001). An issue explored in this article is whether a planetcan become stably locked into resonance with a central binary, with the disc actingas a source of dissipation.

This article is organised as follows. In section 2 we describe the numerical setup.In section 3 we described the results of the simulations. Finally, we discuss theresults and draw conclusions in section 4.

2. Numerical Setup

We consider the interaction between a coplanar binary plus planet system and atwo-dimensional, gaseous, circumbinary disc within which it is supposed the giantplanet formed. The equations of motion are similar to those described in Nelson etal., 2000. Each of the stellar components and the planet experience the gravitationalforce of the other two, as well as that due to the disc. The disc is evolved using thehydrodynamics code NIRVANA (Ziegler and Yorke, 1997). The planet and binaryorbits are evolved using a fifth-order Runge-Kutta scheme (Press et al., 1992)

We adopt a disc model in which the aspect ratio H/r = 0.05, and the viscosityparameter α = 5 × 10−3. The surface density is set up to have an inner cavitywithin which the planet and binary orbit:

�(r) =⎧⎨⎩

0.01�0 if r < rp�0r−1/2 exp [(r − rg)/�] if rp ≤ r ≤ rg�0r−1/2 if r > rg

(1)

where rp is the initial planet orbital radius, rg is the radius where the gap joins themain disc, and � is chosen to ensure that the gap profile correctly joins onto themain disc and the inner cavity. Simulations initiated with no inner cavity show thatone is formed by the action of the binary system and planet clearing gaps in theirlocal neighbourhood. As the planet migrates in towards the central binary thesegaps join to form a single cavity. The disc mass is normalised through the choice of�0 such that a standard disc model with�(r) = �0r−1/2 throughout would containabout 4 Jupiter masses interior to the initial planet radius rp (assumed in physicalunits to be 5 AU). Thus the disc mass interior to the initial planet radius would beabout twice that of a minimum mass solar nebula model. Calculations were alsorun with disc masses a factor of three higher. The total mass of the binary plusplanet system is assumed to be 1 M�. Dimensionless units are used such that thetotal mass of the binary system plus planet Mtot = 1 and the gravitational constantG = 1. The initial binary semimajor axis is abin = 0.4 in all simulations, and the

PlanetarySystems.tex; 28/12/2006; 12:00; p.182

Planet formation in close binary systems 173

Table I. Column 1 gives the run label, column 2 the planet mass (Jupiter masses),column 3 the binary mass ratio, column 4 the binary eccentricity, column 6 thedisc mass (in units described in section 2). Column 7 gives the number of gridcells in the r and φ direction, column 8 describes the mode of evolution which isdefined in the text.

Run label m p (MJ ) qbin ebin md/mmmsn Nr × Nφ ResultA1 1 0.1 0.1 2 160 × 320 Mode 2A2 1 0.1 0.1 2 320 × 640 Mode 2B1 3 0.1 0.1 2 160 × 320 Mode 1B2 3 0.1 0.1 2 320 × 640 Mode 1C1 1 0.1 0.1 6 160 × 320 Mode 2C2 1 0.1 0.1 6 320 × 640 Mode 2D1 3 0.1 0.1 6 160 × 320 Mode 1D2 3 0.1 0.1 6 320 × 640 Mode 1E1 1 0.1 0.05 2 160 × 320 Mode 1E2 1 0.1 0.05 2 320 × 640 Mode 1F1 1 0.1 0.05 6 160 × 320 Mode 1G1 1 0.25 0.1 2 160 × 320 Mode 1G2 1 0.25 0.1 2 320 × 640 Mode 1H1 1 0.25 0.1 6 160 × 320 Mode 1H2 1 0.25 0.1 6 320 × 640 Mode 1I1 3 0.25 0.1 6 160 × 320 Mode 1I2 3 0.25 0.1 6 320 × 640 Mode 1J1 1 0.1 0.2 2 160 × 320 Mode 3K1 1 0.1 0.3 2 160 × 320 Mode 3

initial planet semimajor axis ap = 1.4. The unit of time quoted in the discussionof the simulation results below is the orbital period at R = 1.

3. Numerical Results

The results of the simulations are shown in Table I. They can be divided intothree categories, which are described below, and are most strongly correlated withchanges in the binary mass ratio, qbin, and binary eccentricity ebin. Runs for similarparameters but calculated at different resolutions always gave the same qualitative

PlanetarySystems.tex; 28/12/2006; 12:00; p.183

174 R. P. Nelson

results. In some runs the planet enters the 4:1 mean motion resonance with thebinary. The associated resonant angles are defined by:

ψ1 = 4λs − λp − 3ωs ψ2 = 4λs − λp − 3ωpψ3 = 4λs − λp − 2ωs − ωp ψ4 = 4λs − λp − 2ωp − ωs (2)

where λs , λp are the mean longitudes of the secondary and planet, respectively, andωs , ωp are the longitudes of pericentre of the secondary and planet, respectively.When in resonance ψ3 or ψ4 should librate, or all the angles should librate. Belowwe give a very brief description of the 3 different modes of evolution obtained inthe simulations.Mode 1: The protoplanet migrates inwards and temporarily enters the 4:1 meanmotion resonance with the binary before being gravitationally scattered by thesecondary star. This arises for a binary system with low mass ratio (i.e. qbin = 0.1),when the planet becomes tightly locked into the resonance, because resonant ec-centricity pumping leads to a close encounter between planet and secondary star.Planetary scattering also occurs for central binaries with larger mass ratios (i.e.qbin = 0.25), but via a different evolutionary path. Here, the resonance lockingis weak, and the resonance breaks. The subsequent interaction between planetand binary is strong enough to perturb the planet into a close encounter with thesecondary star.Mode 2: This occurred only for binary mass ratios qbin = 0.1. The planet migratesinwards and enters the 4:1 mean motion resonance. The resonance locking is weak,and the resonance breaks, but the planet does not undergo a close encounter andscattering off the central binary. Instead it migrates outwards slightly through in-teraction with the disc, and becomes ‘parked’ in an orbit beyond the 4:1 resonance,where it remains for the duration of the simulation.Mode 3: This applies when the central binary has ebin ≥ 0.2. The circumbinarydisc is driven eccentric by the binary. The interaction between protoplanet andeccentric disc stalls the inward migration, and the planet remains orbitally stableover long times.

Examples of each of these modes of evolution are described in detail below.Note that similar simulations run at different numerical resolution resulted in thesame qualitative results, as shown in table I.

3.1. PLANETARY SCATTERING – MODE 1

Table I shows that a number of simulations resulted in a close encounter betweenthe planet and binary system, leading to gravitational scattering of the planet tolarger radii, or into an unbound state. These runs are labelled as ‘Mode 1’. Typicallythe initial scattering causes the eccentricity of the planet to grow to values ep � 0.9,and the semimajor axis to increase to ap � 6–8. In runs that were continued forsignificant times after this initial scattering, ejection of the planet could occur aftersubsequent close encounters. We note, however, that the small disc sizes consid-ered in these models preclude us from calculating the post-scattering evolution

PlanetarySystems.tex; 28/12/2006; 12:00; p.184

Planet formation in close binary systems 175

Figure 1. This figure shows the evolution of disc and planet plus binary system for run D2. Time areshown at top right hand corners in orbital periods at R = 1.

accurately, since the planet trajectories take them beyond the outer boundary ofthe disc, usually located at Rout = 4. The eventual ejection of the planets maybe a function of this if continued disc-planet interaction after scattering causeseccentricity damping. Subsequent close encounters may then be prevented.

3.1.1. Low Binary Mass RatiosIn all models with qbin = 0.1 and ebin ≤ 0.1, that resulted in the planet beingscattered, the evolution proceeded as follows. The protoplanet migrates in towardsthe central binary and temporarily enters the 4:1 mean motion resonance with thebinary. The resonant angle ψ3 defined in equation 2 librates with low amplitude, in-dicating that the planet is strongly locked into the resonance. The resonance causesthe eccentricity of the planet to increase, until the planet has a close encounter withthe secondary star, and is scattered out of the resonance into a high eccentricityorbit with significantly larger semimajor axis.

We use the results of model D2 to illustrate the main points discussed above.Figure 1 shows the evolution of the disc and planet plus binary system. The earlymigration stage is shown in the first panel. The second panel shows the system just

PlanetarySystems.tex; 28/12/2006; 12:00; p.185

176 R. P. Nelson

Figure 2. This figure shows the evolution of the semimajor axes and eccentricities of the planet (solidline) and binary system (dashed line) for run D2.

after the initial scattering encounter, with the planet immersed in the main bodyof the circumbinary disc. The third panel shows the planet approaching the binaryduring a subsequent pericentre passage, and the fourth panel shows a time when theplanet orbit takes it out beyond the main body of the disc modeled here. For thosesimulations that resulted in the planet being completely ejected, a circumbinarydisc remains that eventually returns to a state similar to that which would haveexisted had no planet been present.

The orbital evolution of the planet and binary for model D2 is shown in Fig. 2.The upper and middle panels show the semimajor axes versus time. The lowestpanel shows the evolution of the eccentricities versus time. The time evolution of

PlanetarySystems.tex; 28/12/2006; 12:00; p.186

Planet formation in close binary systems 177

Figure 3. This figure shows the evolution of the resonant angles (ψi , i = 1...4) for run D2. Note thelow amplitude libration of ψ3 for t > 300, indicating that planet is strongly locked in resonance.

the resonant angles ψ1, ψ2, ψ3, and ψ4 defined in equation 2 is shown in Fig. 3.The planet enters the 4:1 resonance at t ∼ 300, corresponding to the time whenthe planet eccentricity ep starts to grow steadily (Fig. 2). Figures 2 and 3 showthat the resonance breaks at t ∼ 460, and that ep at this stage has been increasedto ep � 0.4. A close encounter with the secondary star excites ep up to betweenep = 0.8–1. The final state at the end of the simulation has ap � 10 and ep � 0.95.It is likely that a longer integration of this system would result in the planet beingejected from the system, leading to the formation of a ‘free-floating planetary massobject’.

3.1.2. Higher Binary Mass RatiosFor calculations with qbin = 0.25, the evolution differed from that just described,although scattering of the planet still occurred. The planet enters the 4:1 meanmotion resonance, and large amplitude librations of the resonant angle ψ3 occur,accompanied by large oscillations of ep, indicative of weak resonant locking. Theresonance becomes undefined and breaks when ep = 0 during these high amplitudelibrations, and the subsequent interaction between planet and binary causes a closeencounter and scattering of the planet. This is facilitated by the larger value of qbin,

PlanetarySystems.tex; 28/12/2006; 12:00; p.187

178 R. P. Nelson

Figure 4. This figure shows the evolution of the semimajor axes and eccentricities for the planet(solid line) and binary system (dashed line) in run I1.

since a similar run involving weak resonant locking with qbin = 0.1 does not leadto scattering of the planet (see section 3.2 below).

Results for model I1 are shown in Figs. 4 with 5. Figure 4 shows the evolutionof the semimajor axes and eccentricities. Figure 5 shows the time evolution of theresonant angles. Comparing Fig. 4 and Fig. 2 we can see that the eccentricity in thiscase does not increase to such large values before the resonance breaks. Instead theresonance appears to break because ep goes to zero momentarily. After resonancebreaking, the planet-binary interaction is strong enough to perturb the planet intoan orbit that leads to a close encounter with the binary. Comparing Figs. 5 and 3,we see that the resonant angles undergo libration with much greater amplitude inthis case, indicating that the resonant locking is weaker in run I1 than in run D2.

3.2. NEAR-RESONANT PLANET – MODE 2

A mode of evolution was found in some of the simulations with qbin = 0.1 andebin = 0.1 leading to the planet orbiting stably just outside of the 4:1 resonance.These cases are labelled as ‘Mode 2’ in table I. Here, the protoplanet migratesinwards and becomes weakly locked into the 4:1 resonance, with the resonant angleψ3 librating with large amplitude. The evolution in the resonance is similar to thatdescribed for run I1 in section 3.1.2. The resonance is undefined and breaks when

PlanetarySystems.tex; 28/12/2006; 12:00; p.188

Planet formation in close binary systems 179

Figure 5. This figure shows the evolution of the resonant angles (ψi , i = 1...4) for run I1. Note thehigh amplitude libration of ψ3 for t > 300, showing that the planet is weakly locked in resonance.

ep = 0. However, because qbin is smaller, the planet is not perturbed into an orbitthat leads to a close encounter with the binary. Instead it undergoes a period ofoutward migration through interaction with the disc by virtue of the eccentricityhaving reattained values of ep � 0.15–0.2 once the resonance is broken. Calcula-tions by Nelson, 2003 have shown that gap-forming protoplanets orbiting in tidallytruncated discs undergo outward migration if they are given eccentricities of thismagnitude impulsively, due to the sign of the torque exerted by the disc reversingfor large eccentricities. The outward migration moves the planet to a safer distanceaway from the binary, thus avoiding instability.

Once the planet has migrated to just beyond the 4:1 resonance the outward mi-gration halts, since its eccentricity reduces slightly, and the planet remains there forthe duration of the simulation. The system achieves a balance between eccentricitydamping by the disc and eccentricity excitation by the binary, maintaining a meanvalue of ep � 0.12. The torque exerted by the disc on the planet is significantlyweakened by virtue of the finite eccentricity (Nelson, 2003), preventing the planetfrom migrating back towards the binary.

We use calculation C2 to illustrate the points discussed above. The orbital evo-lution of the system is shown in Fig. 6. The upper panel shows the semimajoraxes. The planet initially migrates in towards the binary, and halts as it reaches

PlanetarySystems.tex; 28/12/2006; 12:00; p.189

180 R. P. Nelson

Figure 6. This figure shows the evolution of the semimajor axes and eccentricities of the planet (solidline) and binary system (dashed line) for run C2.

ap � 1. The planet enters the 4:1 resonance and the resonant angle ψ3 undergoeslarge amplitude librations, similar to those observed for run I1 in Fig. 5 and incontrast to those observed in Fig. 3 for run D2. ep also undergoes large amplitudeoscillations, with the planet eccentricity reaching ep = 0 just prior to the breakingof the resonance at t � 500. Once the planet leaves the resonance, ep increases toep � 0.15, and the semimajor axis ap increases. The planet remains outside the 4:1resonance for the duration of the simulation, (i.e. for nearly 3000 planetary orbits)with the eccentricity oscillating between values of 0.05 ≤ ep ≤ 0.18. Continuationof this run in the absence of the disc indicates that the planet remains stable forover 6 × 106 orbits. This is in good agreement with the stability criteria obtainedby (Holman and Wiegert, 1999).

3.3. ECCENTRIC DISC – MODE 3

A mode of evolution was found in which the planetary migration was halted beforethe planet could approach the central binary. This only occurred when the centralbinary had an initial eccentricity of ebin ≥ 0.2. The migration stalls because thecircumbinary disc becomes eccentric. We label runs of this type as ‘Mode 3’ in ta-ble I. Interaction between the protoplanet and the eccentric disc leads to a reductionor even reversal of the time-averaged torque driving the migration. Simulations ofthis type can be run for many thousands of planetary orbits without any significantnet inward migration occurring. Such systems are likely to be stable long after thecircumbinary disc has dispersed, and are probably the best candidates for findingstable circumbinary extrasolar planets.

PlanetarySystems.tex; 28/12/2006; 12:00; p.190

Planet formation in close binary systems 181

Figure 7. This figure shows the evolution of the disc and planet plus binary system for run J1. Notethe formation of the eccentric disc.

Figure 8. This figure shows the evolution of the semimajor axes and eccentricities of the planet (solidline) and binary system (dashed line) for run J1.

PlanetarySystems.tex; 28/12/2006; 12:00; p.191

182 R. P. Nelson

Model J1 is used to illustrate the evolution of the ‘Mode 3’ class of models.The evolution of the disc, binary, and protoplanet are shown in Fig. 7. Early onthe system looks similar to the one shown in Fig. 1. However, the circumbinarydisc eventually becomes eccentric. The orbital evolution of the planet and binary isshown in Fig. 8. The upper panel shows the semimajor axes, and the lower panel theeccentricities. Initially the planet undergoes inward migration. After a time of t �400 the migration reverses. This is the time required for the eccentric disc modeto be established. Over longer time scales the planet oscillates between inward andoutward migration before settling into a non migratory state. The planet remainsorbiting with semimajor axis ap � 1.2, and is unable to migrate in towards thecentral binary. The resulting system is likely to consist of a central binary withebin ≤ 0.2, and a circumbinary planet orbiting with ap � 1.2 and ep � 0.06 afterdispersal of the circumbinary disc.

The disc eccentricity in these models is driven by the central binary, and occursonly when the inner binary has significant eccentricity. Simulations performedwithout a protoplanet included also produced eccentric discs. In these cases, thedisc inner cavity was retained, and those only calculations with ebin ≥ 0.2 gaverise to an eccentric disc. In previous work (Papaloizou, Nelson, and Masset, 2001),an eccentric disc could be excited for companion masses with q ≥ 0.02, corre-sponding to a companion of 20 Jupiter masses orbiting a solar mass star. Thismode of disc-eccentricity driving occurred even for companions on circular orbits,and arose because of nonlinear mode coupling between an initially small m = 1mode in the disc and the m = 2 component of the orbiting companion potential. Tooperate, this requires significant amounts of gas to be present at the 1:3 resonancelocation (which is at r � 2.08 × abin). The initial disc models in this paper haveinner cavities that extend much beyond the 1:3 resonance of the binary, so thiscannot be the cause of the disc eccentricity observed in runs J1 and K1, and thosewithout protoplanets included. Instead, the disc eccentricity is driven by the m = 1component of the eccentric binary potential which excites a global m = 1 mode inthe disc.

4. Conclusions

We have examined the evolution of giant protoplanets orbiting in circumbinarydiscs. For low eccentricity binaries (i.e. ebin ≤ 0.1), when the planet migratesinwards through interaction with the circumbinary disc the resulting interactionwith the binary can lead to the planet being scattered and ejected from the sytem,leading to the formation of a ‘free-floating planet’. However, there is also a finiteprobability that the planet can end up orbiting stably just outside of the 4:1 meanmotion resonance. For binaries with significant eccentricity (i.e. ebin ≥ 0.2), thecircumbinary disc becomes eccentric, and this stalls the inward migration of theplanet, preventing it from approaching the binary and being ejected. Given that

PlanetarySystems.tex; 28/12/2006; 12:00; p.192

Planet formation in close binary systems 183

close binaries often have eccentric orbits, this suggests that circumbinary plan-ets in these systems may be orbiting at safe distances, and could potentially beobservable.

Acknowledgements

The computations reported here were performed using the UK Astrophysical Flu-ids Facility (UKAFF).

References

Duquennoy, A., Mayor, M.: 1991, ‘Multiplicity among solar-type stars in the solar neighbourhood.II – Distribution of the orbital elements in an unbiased sample’, A & A 248, 485.

Dvorak, R.: 1986, ‘Critical orbits in the restricted three-body problem’, A & A 167, 379.Ghez, A. M., Neugebauer, G., Matthews, K.: 1993, ‘The multiplicity of T Tauri stars in the star

forming regions Taurus-Auriga and Ophiuchus-Scorpius: A 2.2 micron speckle imaging survey’,Ap. J. 106, 2005.

Holman, M.J., Wiegert, P.A.: 1999, ‘Long-term stability of planets in binary systems’, Ap. J. 117,621.

Kley, W., Burkert, A.: 2000, ‘Disks and Planets in Binary Systems’, Discs, Planetesimals, andPlanets, ASP Conference Series, eds. F. Garzon, C. Eiroa, D de Winter, T.J. Mahoney.

Leinert, Ch., Zinnecker, H., Weitzel, N., Christou, J., Ridgway, S. T., Jameson, R., Haas, M., Lenzen,R.: 1993, ‘A systematic approach for young binaries in Taurus’, A & A 278, 129.

Lin, D.N.C., Papaloizou, J.C.B.: 1986, ‘On the tidal interaction between protoplanets and theprotoplanetary disk. III – Orbital migration of protoplanets’, Ap. J. 309, 846.

Nelson, R.P., Papaloizou, J.C.B., Masset, F., Kley, W.: 2000, ‘The migration and growth ofprotoplanets in protostellar discs’, MNRAS 318, 18.

Nelson, R.P.: 2003, ‘On the evolution of giant protoplanets forming in circumbinary discs’, MNRAS345, 233.

Papaloizou, J. C. B.; Nelson, R. P.; Masset, F.: 2001, ‘Orbital eccentricity growth through disc-companion tidal interaction’, A & A 366, 263.

Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P.: 1992, ‘Numerical Recipes inFORTRAN’, Cambridge Univ. Press, Cambridge, p. 710.

Quintana, E.V., Lissauer, J.J., Chambers, J.E., Duncan, M.J.: 2002, ‘Terrestrial Planet Formation inthe α Centauri System’, Ap. J. 576, 982.

Snellgrove, M. D., Papaloizou, J. C. B., Nelson, R. P.: 2001, ‘On disc driven inward migration ofresonantly coupled planets with application to the system around GJ876’, A & A 374, 1092

Ziegler, U., Yorke, H.: 1997, ‘ A Nested Grid Refinement Technique for MagnetohydrodynamicalFlows’, Comp. Phys. Comm. 101, 54.

PlanetarySystems.tex; 28/12/2006; 12:00; p.193