The Astrophysical Journal, 678:498Y508, 2008 May 1 # 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.
FORMATION OF HOT PLANETS BY A COMBINATION OF PLANET SCATTERING, TIDAL CIRCULARIZATION, AND THE KOZAI MECHANISM M. Nagasawa, S. Ida, and T. Bessho Tokyo Institute of Technology, 2-12-1, Ookayama, Meguro-ku, Tokyo 152-8550, Japan; [email protected] Received 2007 October 5; accepted 2008 January 8
ABSTRACT We have investigated the formation of close-in extrasolar giant planets through a coupling effect of mutual scattering, the Kozai mechanism, and tidal circularization, by orbital integrations. Close-in gas giants would have been originally formed at several AU beyond the ice lines in protoplanetary disks and migrated close to their host stars. Although type II migration due to planet-disk interactions may be a major channel for the migration, we show that this scattering process would also give a nonnegligible contribution. We carried out orbital integrations of three planets with Jupiter mass, directly including the effect of tidal circularization. We have found that in about 30% of the runs close-in planets are formed, which is much higher than suggested by previous studies. Three-planet orbit crossing usually results in the ejection of one or two planets. Tidal circularization often occurs during three-planet orbit crossing, but previous studies have monitored only the final stage after the ejection, significantly underestimating the formation probability. We have found that the Kozai mechanism in outer planets is responsible for the formation of close-in planets. During three-planet orbital crossing, Kozai excitation is repeated and the eccentricity is often increased secularly to values close enough to unity for tidal circularization to transform the inner planet to a close-in planet. Since a moderate eccentricity can retain for the close-in planet, this mechanism may account for the observed close-in planets with moderate eccentricities and without nearby secondary planets. Since these planets also remain a broad range of orbital inclinations (even retrograde ones), the contribution of this process would be clarified by more observations of Rossiter-McLaughlin effects for transiting planets. Subject headinggs: celestial mechanics — planetary systems: formation — solar system: formation
1. INTRODUCTION account for not only the pile-up of the close-in planets but also planetary pairs locked in mean-motion resonances. The migrating More than 250 planets have been detected around both solar planets may have stalled as they enter the magnetospheric cavity and nonsolar type stars. The recent development of radial velocity in their nascent disks or interact tidally with their host stars. techniques and an accumulation of observations have revealed P : detailed orbital distributions of close-in planets. Figure 1a shows However, several close-in (a 0 1 AU) planets such as HAT-P-2, 1 HD 118203b, and HD 162020b have moderate eccentricities the distribution of semimajor axis and eccentricity for 236 planets (k0.3), but are not accompanied by nearby secondary large discovered around solar type stars by the radial velocity tech- planets. HD 17156b with a ’ 0:15AUmayalsobelongtothis niques. The dotted line shows a pericenter distance q ¼ 0:05 AU. group, since it has very large eccentricity ( 0.67) and q as small At q P 0:05 AU many close-in planets have small eccentricities, ’ as 0.05 AU. It may be difficult for the planet-disk interaction alone which are accounted for by circularization due to the tidal dis- to excite the eccentricities up to these levels. One possible mech- sipation of energy within the planetary envelopes (Rasio & Ford anism for the excitation to moderate eccentricity for close-in 1996). The semimajor axis distribution of discovered extrasolar planets in multiple-planet systems is the passage of secular res- planets shows double peaks around 0.05 and 1 AU (e.g., Marcy onances during the epoch of disk depletion (Nagasawa et al. 2000, et al. 2005; Jones et al. 2004). The location of the outer peak is determined by the observational limits, but the inner peak at 2003; Nagasawa & Lin 2005). Nagasawa & Lin (2005) showed that the relativity effect and the gravity of (distant) secondary around 0.05 AU, in other words, the shortage of planets between planets excite the eccentricity of close-in planets orbiting around 0.06 and 0.8 AU, is likely real. Since the close-in planets have relatively small eccentricities, the peak around 0.05 AU appears a slowly rotating host star. Although a main channel to form close-in planets may be type II as a pile-up at log q/(1 AU) ¼ 1:3, also in the pericenter dis- migration, inward scatterings by other giant planets can be another tribution (Fig. 1b). The close-in planets are also found in the channel, in particular for the close-in planets with moderate ec- multiplanetary systems. Up to now, 23 multiplanet systems have centricities. In relatively massive and/or metal-rich disks, multiple been reported with known planetary eccentricities (Fig. 2). gas giants are likely to form (e.g., Kokubo & Ida 2002; Ida & Lin The discovered close-in planets would have formed at large 2004a, 2004b, 2008). On longer timescales than formation time- distances beyond the ice line and migrated to shorter period orbits scales, the multiple gas giant systems can be orbitally unstable (e.g., Ida & Lin 2004a, 2004b). A promising mechanism for the (Lin & Ida 1997; Marzari & Weidenschilling 2002). In many orbital migration is ‘‘type II’’ migration (e.g., Lin et al. 1996). This cases, a result of orbital crossing is that one of the planets is is the migration with disk accretion due to the gap opening around the planetary orbit caused by gravitational interaction between ejected and the other planets remain in well-separated eccentric orbits (‘‘Jumping Jupiters’’ model; e.g., Rasio & Ford 1996; the planet and the protoplanetary gas disk. This mechanism can Weidenschilling & Marzari 1996; Lin & Ida 1997). Recent studies of the Jumping Jupiter model show fairly good coin- 1 See http://exoplanet.eu/catalog-RV.php. cidence with the observational eccentricity distribution (Marzari 498 FORMATION OF HOT PLANETS 499
Fig. 2.—Semimajor axes of extrasolar planets in multiple planetary systems. The vertical axis is the semimajor axis (a1) of an innermost planet in each system. At the bottom, the solar system is also shown with a1 ¼ aJ. Circle sizes of extra- solar planets are proportional to (m sin i)1=3.
sarily high). Since HD 17156b with e ’ 0:67 is also transiting, the observation of the Rossiter-McLaughlin effect will be important. A more serious problem of the scattering model may be the probability for pericenter distances to become small enough; in other words, for eccentricities to become close enough to unity (e.g., e k 0:98) to allow tidal circularization. In a system with Fig. 1.— (a) Distribution of orbital eccentricity (e) and semimajor axis (a)of only two giant planets, the energy conservation law keeps the observed extrasolar planets. Circle sizes are proportional to (m sin i)1=3. Dotted line shows pericenter distance q ¼ a(1 e) ¼ 0:05 AU. (b) Histogram of peri- ratio of semimajor axes close to the initial value (see x 2.1). In center distance. such systems, the probability for final planets with a large ratio of semimajor axes and very large eccentricity is P3% (Ford et al. 2001). In a system with more giant planets, the situation is slightly & Weidenschilling 2002; Chatterjee et al. 2007; Ford & Rasio improved, especially when planets have nonzero inclinations. The 2007). close-in planets are very rare just after the scattering, but when With the sufficiently small pericenter distance (P0.05 AU), they take into account the longer orbital evolution, the proba- the planetary orbit can be circularized into the close-in orbit by bility of the candidates for close-in planets increases to 10% tidal effects from the star (Rasio & Ford 1996). In this paper, the (Weidenschilling & Marzari 1996; Marzari & Weidenschilling process of planet-planet scattering followed by the tidal circu- 2002; Chatterjee et al. 2007). The contributed planets are planets larization will be referred to as the ‘‘scattering model.’’ To send in the Kozai state (see x 3.5). a planet into the inner orbit, the other planets have to lose their Here we revisit the scattering model through orbital integration orbital energy. Since inward scattering of a lighter planet is more of three giant planets with direct inclusion of the tidal circulari- energy saving, the systems that experienced planet-planet scatter- zation effects and an analytical argument for the Kozai effect. We ing tend to have smaller planets inside. We might see the tendency find that the probability for formation of close-in planets is re- in Figure 2, but it is also true that the radial velocity technique markably increased to 30%. Although the previous studies were tends to detect heavier planets in outer regions. Note that since concerned with orbital states after stabilization by ejection of outer planets have negligible orbital energy, energy conservation some planet(s), we find that the orbital circularization occurs cannot restrict the semimajor axes of the outer planets, and the through ‘‘Kozai migration’’ (Wu 2003; Wu & Murray 2003) outer planets may be located in the regions far beyond the radial caused by outer planets, mostly during three-planet orbital crossing, velocity technique limit (Marzari & Weidenschilling 2002). after a tentative separation of the inner planet and the outer planets. Observation of the Rossiter-McLaughlin effect for five transit- In x 2, we describe orbital instability of a planetary system ing close-in planets, including HAT-P-2 with eccentricity e ’ (x 2.1) and the orbital evolution that becomes important after the 0:52, show that their orbital planes are almost aligned with the system enters a stable state: tidal circularization (x 2.2) and the stellar spin axes (Narita et al. 2007; Winn et al. 2005, 2006, Kozai mechanism (x 2.3). Following a description of its models 2007; Wolf et al. 2007), which may suggest that the scattering and assumptions, we present the results of numerical simulations mechanism is not a main channel for formation of close-in planets, in x 3. We show typical outcomes of the scattering (x 3.2), how because the close scattering may excite orbital inclination, as well the planets are circularized into short-period orbits (x 3.3), and as eccentricity (note that as shown in x 3.4, the orbital inclinations its probability (x 3.5). In x 4 we present analytical arguments. of close-in planets formed by the scattering model are not neces- We summarize our results in x 5. 500 NAGASAWA, IDA, & BESSHO Vol. 678
2. PLANET-PLANET SCATTERING with a few day period can be formed. With e highly excited to a AND THE TIDAL CIRCULARIZATION value close to unity, the pericenter of the planet can approach its host star, since the pericenter distance is given by q a(1 e). In this paper we carry out orbital integration of three equal- ¼ Then, both e and a decrease, while keeping q almost constant by mass (m) giant planets with physical radii R rotatingarounda host star with mass m . Interactions with disk gas is neglected. the tidal dissipation. The tidal dissipation is a strong function of q. As the scattered We consider the case in which the timescale of orbital crossing to planets normally have highly eccentric orbits, we only consider occur is longer than the disk lifetime (see x 2.1). The orbits are integrated until one planet is tidally circularized, two planets are the dynamic tide. In our simulation, we adopt the formula by Ivanov & Papaloizou (2004). They analytically calculated the ejected, or the system becomes stable after one planet is ejected strongest normal modes, l 2 fundamental modes, of the tidal and the others are orbitally well separated. However, we keep ¼ dissipation. It is assumed that the normal modes arisen near the integrating long enough after the orbital stabilization to see the pericenter are fully dissipated before a next pericenter passage. Kozai mechanism that occurs on longer timescales. To understand If the normal modes remain, the angular momentum and energy the numerical results, in this section, we briefly summarize key can either increase or decrease at the new pericenter passage processes for our calculations: orbital instability, tidal circular- (e.g., Mardling 1995a, 1995b). ization, and the Kozai mechanism. 3 Ivanov & Papaloizou (2004) derived the tidally gained an- In the present paper, Jupiter-mass planets (m ¼ mJ ¼10 M ) gular momentum ( Ltide) and energy ( Etide) during a single and a solar-mass host star (m ¼ M ) are adopted. In the following we use a for semimajor axis, e for eccentricity, i for inclination, pericenter passage as ffiffiffi ffiffiffi and ! for argument of pericenter. When we distinguish the planets, p p 32 2 2 ˜ 2 4 2 we use subscripts 1, 2, and 3 from inner to outer planets. A sub- Ltide w˜ 0Q exp wþ 15 3 script asterisk ( ) shows quantities for the host star and a subscript pffiffiffi ‘‘tide’’ is used for values related to dynamic tide. A variable h is 9 4 2 2 2 ; 1 exp ˜ L ; defined as h ¼ (1 e )cos i. The mutual Hill radius of planets k 14 4 pl 1=3 2 (w˜ 0 ) 3 and j is expressed as RH(k; j) ¼½(mk þ mj)/3m (ak þ aj)/2. pffiffiffi pffiffiffi 16 2 4 2 E w˜ 3Q˜ 2 exp w ð2Þ 2.1. Planet-Planet Interactions tide 15 0 3 þ pffiffiffi 2 Timescales for orbital instability to occur (Tcross)verysensi- 3 2 2 tively depend on orbital separation between planets, their eccen- ; 1 þ exp ˜ Epl; ð3Þ 7 2 3 tricities, and masses (Chambers et al. 1996; Chambers & Wetherill 2 (w˜ 0 ) 1998; Ito & Tanikawa 1999, 2001; Yoshinaga et al. 1999). 1=2 2 where Lpl ¼ m(GmR) and Epl ¼ Gm /R are the orbital an- Chambers et al. (1996) found that for systems with equal-mass (m) 3 1=2 3 1=2 gular momentum and orbital energy, ¼ (mq ) (m R ) , planets in initially almost circular orbits of orbital separation bRH, 3 1=2 3 1=2 wþ w˜ 0(Gm/R ) þ r,and˜ 2 r /(Gm/R ) . The value log Tcross ¼ b þ ,where and are constants in terms of b. r is the angular velocity of the planet rotation, w˜ 0 is a dimen- The values and weakly depend on m and the number of ˜ planets. Marzari & Weidenschilling (2002) showed that for three sionless frequency of fundamental mode, and Q is a dimension- Jupiter-mass planets with a ¼ 5 AU, a ¼ a þ bR (1; 2), a ¼ less overlap integral that depends on the planetary interior model. 1 2 1 H 3 The spin axis is assumed to be perpendicular to the orbital plane. a2 þ bRH(2; 3), ’ 2:47, and ’ 4:62. Chatterjee et al. (2007) derived an empirical formula that is not skewed toward We use the same planetary model as Ivanov & Papaloizou (2004). FromtheirFigures6and7weapproximatew˜ 0 ’ 0:53(R/RJ) þ greater timescales: the median of log Tcross is given by 1:07 þ 0:68 and Q˜ ’ 0:12(R/RJ) þ 0:68 for Jovian mass planet, where 0:03 exp (1:10b). Since the resonances modify Tcross, especially in systems with massive planets, a real timescale is sometimes RJ is Jovian radius. much shorter than the above timescale. But these expressions For a nonrotating planet, equations (2) and (3) are simplified to are helpful to know how long we need to continue the simulations. pffiffiffi pffiffiffi Note that the semimajor axis of an final innermost planet 32 2 2 ˜ 2 4 2 Ltide w˜ 0 Q exp w0 Lpl; ð4Þ (aB ) is limited by energy conservation when only mutual scat- 15 3 n;1 pffiffiffi pffiffiffi tering is considered. In an extreme case, in which the other planets 16 2 3 ˜ 2 4 2 are sufficiently far away (aB 3 aB ; k 2; 3;:::), Etide w˜ Q exp w0 Epl: ð5Þ n; k n;1 ¼ 15 0 3 1 XN 1 ’ ; ð1Þ When the tide spins the planet up to its critical rotation, r ¼ B a n;1 k aini;k crit , which gives no additional increase in planetary angular momentum in equation (2), where aini; k is the initial semimajor axis of planet ‘‘k’’ and N is number of planets. This equation implies that aB should be Ltide ¼ 0; ð6Þ n;1 pffiffiffi larger than a /N. That means close-in planets are hardly formed 2 ini;1 1 w˜ 0Q˜ 4 2 as a result of direct scattering of a few planets that originate from Etide pffiffiffi exp w0 Epl: ð7Þ 3 the regions beyond several AU, such as Jupiter and Saturn. 5 2 The values of E /E for a m planet around a M star given 2.2. Tidal Circularization tide pl J in equations (5) and (7) are shown in Figure 3 for R ¼ RJ and As shown above, the planet-planet scattering cannot make 2 RJ (m ¼ mJ). As the figure shows, the tide is only effective in the orbital period of a planet as short as a few days, as long as the vicinity of the star. the energy is conserved. However, if energy is dissipated in the The model is only valid for fully convective planets in highly planetary interior by the tidal force from the host star, a planet eccentric orbits, since it considers only ‘ ¼ 2 f-mode and uses No. 1, 2008 FORMATION OF HOT PLANETS 501
Fig. 4.—Tidal circularization timescales, given as a function of pericenter Fig. 3.—Energy change Etide during one pericenter passage as a function of distance q. The formulae are given by eqs. (8) and (9) with eqs. (4) and (5) for orbital period; Epl is orbital energy. A solar-mass host star and a Jupiter-mass nonrotating planets or eqs. (6) and (7) for planets with the critical rotation, fol- planet are assumed. For larger planetary mass, Etide is larger. Solid lines show lowing Ivanov & Papaloizou (2004). Solid lines are for nonrotating planets with the change for nonrotating planets. Dotted lines are calculated for planets with R ¼ 2 RJ. Dotted lines are for 1 RJ planets rotating with the critical rotation. critical rotation that leads to Ltide ¼ 0. Thick lines correspond to 1 RJ radius Thick lines show a and thin lines show e.Thesemimajoraxisa ¼ 2AUisused planets. Thin lines are for 2 RJ planets. for the estimation. A solar-mass host star and a Jupiter-mass planet are assumed. For larger planetary mass, the timescales are much shorter. impulse approximation. When a planetary orbit is considerably circularized to a level such that the impulse approximation be- inner planet has relatively small z-component of the angular comes invalid and a quasistatic tide becomes important, the above momentum (‘z). This mechanism is called the Kozai mechanism equations are no longer relevant. However, since our purpose of (Kozai 1962). Suppose that the inner planet 1 is perturbed by a this paper is not to study the details of the tidal circularization well-separated outer planet 2. Eccentricity e1, inclination i1,and process itself but to show an available pass for the formation of argument of pericenter of the inner planet evolve conserving its close-in planets, we use the above formulae for dynamic tide time-averaged Hamiltonian and ‘z. In the lowest order of a1 /a2 until the end of the simulations. (a1 and a2 are conserved), the conservation of the time averaged With the estimations dEtide /dt ’ Etide /TKep and dLtide /dt ’ Hamiltonian of the inner planet and that of ‘z give Ltide /TKep (the pericenter passage occurs every TKep), the evo- lution timescale of the semimajor axis and eccentricity are 3h (2 þ 3e2) 1 1 1 e2 1 a Gmm TKep a ¼ ¼ ; ð8Þ 2 h a˙ 2a ( Etide) þ 15e1 1 2 cos (2 ) C ¼ const:; ð10Þ ! 1 e1 rffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 e Gm (1 e )cos i h ¼ const:; ð11Þ ¼ ¼ Gmm T a E þ L ; 1 1 e e˙ Kep tide ae2 tide ð9Þ where is the difference of the argument of the pericenter of the two planets (i.e., ¼ !2 !1). The above equations are inde- 2 2 pendent of a and the parameters of the outer planet, but the where TKep is a Keplerian time and ¼ (1 e )/e ¼ q(2a 1 q)/(a q)2. The damping timescale for a planet scattered into a timescale for the circulation or the libration is in proportion to 3=2 3 1 1 highly eccentric (e 1) orbit at a ¼ 2 AU is shown in Figure 4 (1 e2) (a2 /a1) m2 n2 ,wheren2 is mean motion of planet 2. The conservation of ‘z,i.e.,h ¼ const:, shows that the change [q ¼ a(1 e)]. The stellar and planet masses are 1 M and mJ, in i toward 0 results in an increase in e toward unity, i.e., the respectively. Two limiting cases with strong tide ( r ¼ 0and 1 1 approach of the pericenter to the vicinity of the star. How closely e R ¼ 2RJ) and weak tide ( r ¼ cirt and R ¼ 1 RJ) are plotted. 1 In the strong tide case, the semimajor axis is damped within can approach unity is regulated by equation (10). 1010 yr at q P 0:04 AU. In the weak damping case, q P 0:02 AU. In the following numerical simulations we test both cases. 3. NUMERICAL SIMULATIONS 3.1. Initial Setup 2.3. Kozai Mechanism We study the evolution of systems for three planets with a 3 How closely the pericenter approaches the host star depends Jupiter mass (m ¼ 10 M ) orbiting a solar-mass star in circular on eccentricity. Even after a planet is settled in a stable orbit, its orbits (e1 ¼ e2 ¼ e3 ¼ 0). Their initial semimajor axes are a1 ¼ eccentricity and inclination can oscillate via the secular per- 5:00 AU, a2 ¼ 7:25 AU, and a3 ¼ 9:50 AU, and their incli- turbation from other separated planets, in particular when the nations are i1 ¼ 0:5 , i2 ¼ 1:0 ,andi3 ¼ 1:5 , following the 502 NAGASAWA, IDA, & BESSHO Vol. 678
TABLE 1 Characteristics of Simulations
Set r R/RJ Comments Close-in Planets
Set N ...... No tide ... 2 Set V ...... 0 2 v ¼ 2 Etide 37% Set T1...... 0 2 Eq. (4) 38% Set T2...... 0 1 Eq. (4) 29% Set T3...... crit 2 Eq. (6) 33% Set T4...... crit 1 Eq. (6) 32% simulation setup of Marzari & Weidenschilling (2002). Orbital angles are selected randomly. We repeat the orbital integration with different seeds of random number generation for the initial orbital angles with the same initial a, e, and i. With this choice, the shortest semimajor axis after the scattering is expected to be amin ¼ 2:26 AU (eq. [1]). The tidal damping timescale is a function of mass and radius of planets. Although we fix the planetary mass, we test R ¼ RJ and 2 RJ cases. The latter case corresponds to newborn planets that have not cooled down. Since an averaged orbital separation of the system is Fig. 5.—Final distribution of a and e in the case of set N (without tide). 3.6RH in this choice of semimajor axes, it is expected that the orbital instability starts on timescales of 103 yr (see x 2.1). Oscillation modes are raised in the planetary interior by the 3.2. Outcome of Planet-Planet Scattering: Set N tidal force from the host star in the vicinity of the pericenter. We The result of the planet-planet scatterings without tidal force assume that the energy of the modes is dissipated, and the angular is consistent with previous studies of Marzari & Weidenschilling momentum is transferred to the orbital angular momentum before (2002). Systems ending with two planets are the most common the next pericenter passage. Assuming that the orbital changes are outcome. In 75 cases of set N, one planet is ejected. In 22 cases, small in individual approaches, we change the orbit impulsively at one of the planets hits the host star. This mainly occurs during the pericenter passage as mentioned below. the chaotic phase of planetary interaction, namely, before the first We have performed six sets of simulations (Table 1). In set V, ejection of a planet. The case in which two planets are ejected is we adopt the simplest model, that is, the velocity (v) of the planet rare, as Marzari & Weidenschilling found. We observed such is changed discontinuously to v0 at the pericenter passage as outcome in five runs. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The distribution of semimajor axis and eccentricity of the final 0 2 v v ¼ 2 Etide þ v ; ð12Þ systems is shown in Figure 5. The innermost planets are scattered v into orbits at a ’ 2:5 AU. Since a small difference in the orbital energy causes large difference in the semimajor axis in the outer where E is given by equation (5). In this set we adopt ¼ 0 tide r region, the semimajor axes of outer planets are widely distributed. and R ¼ 2 R . Since we do not change the location of the peri- J The figure includes planets that hit the star; they are clumped center and direction of motion there, the angular momentum var- around e ’ 1. The planets with small pericenter distance q P iation is specified as q v, which is inconsistent with equation (4). 0:05 AU are the star-colliding planets. Since there is no damping In sets T1 and T2, on the other hand, the pericenter distance is also mechanism in set N, the small q planets also have a 2Y3AU. changed, as well as velocity, so as to consistently satisfy both equations (4) and (5) for r ¼ 0, but the direction of the motion does not change. Equations (6) and (7) are satisfied in sets T3 3.3. Orbital Evolution to Hot Planets andT4( r ¼ crit). Planetary radius is R ¼ 2 RJ in sets T1 and The star-approaching planets are circularized to become close- T3, while R ¼ RJ in sets T2 and T4. As we will show later, the in planets when we include tidal force in our simulation. A typical probability for the tidal circularization to occur does not signifi- evolution of semimajor axis, pericenter, and apocenter in the case cantly depend on the models of orbital change by tidal dissipation. of set V is shown in Figure 6. The system enters the chaotic phase For comparison, we also test the case without the tidal circulari- quickly, and originally the outermost planet is scattered inward zation (set N). into a ’ 3 AU through several encounters. The planet is detached Because of the chaotic nature of the scattering processes, we from other planets and becomes marginally stable with a 3AU integrate 100 runs in each set. We integrate orbits for 107Y108 yr. after t 105 yr, until it suffers tidal damping (t k 3:9 ; 106 yr), al- We stop the calculation when a planet hits the surface of the host though the outer two planets still continue orbital crossing. Dur- star with 1 R or when Ltide overcomes the angular momentum ing the tentative isolation period, the eccentricity and inclination that a circularized planet has. The latter condition happens in sets of the innermost planet are mostly varied by secular (distant) per- T1YT4. In set V,a collision against the host star occurs to a tidally turbations from the outer two planets (Fig. 7). Since the perturba- circularized planet. In other cases, we check the stability of the tions are almost secular, the semimajor axis of the isolated planet systems every 106 yr after 107 yr. If only one planet survives and does not change greatly until 3:9 ; 106 yr, but its eccentricity its pericenter is far from the star, or two are left with dynamically randomly varies at the occasions when the middle planet ap- stable orbits with relatively large ‘z, we stop the simulation. We proaches. As a result of one of these repeated close encounters, continue the simulation until 108 yr, as long as the system con- the isolated planet acquires relatively large e and i at t ’ 1:0 ; tains three planets. 106 yr. Its eccentricity oscillates with large amplitude, exchanging No. 1, 2008 FORMATION OF HOT PLANETS 503
Fig. 6.—Typical evolution of the semimajor axes (a) of three planets. Thin Fig. 8.—Evolution of the semimajor axes of three planets. The meaning of lines show evolution of pericenters [a(1 e)] and apocenters [a(1 þ e)]. The ; 6 the lines is the same as Fig. 6. One planet shown by dotted line is ejected from planet indicated by solid line is circularized at 4:13 10 yr. the system at 2:2 ; 106 yr. The planet indicated by solid line is circularized at ’1:1 ; 107 yr. ‘z with the inclination by the Kozai mechanism. Although the amplitude of oscillations of e and i decay after t ’ 1:7 ; 106 yr, ; 6 ; 6 they are pumped up again at t ’ 2:5 10 yr. At 3:7 10 yr, migration.’’ They considered the Kozai mechanism induced by the planet acquires a very large oscillation amplitude of incli- perturbations of a companion star, while we consider that of outer nation and eccentricity. The eccentricity reaches the maximum planets in orbital crossings. If the planet scattered inward has ; 6 value of a Kozai cycle at 3:9 10 yr, at which time the pericenter small h ¼ (1 e2)cos2i, it is subject to tidal circularization. approaches the host star. Then the planet is tidally moved slightly inward, but the damping of the semimajor axis is interrupted when 3.4. Final Orbital Distribution the eccentricity reaches the maximum value of a Kozai cycle and turns to decrease. At 4:1 ; 106 yr, the pericenter distance q can The distribution of final eccentricity of innermost planets is be small enough for tidal circularization during a Kozai cycle. shown in Figure 9. It is divided into three groups: a peak at e > Since q becomes <0.01 AU in this case, the orbit is circularized 0:95, peak at e < 0:05, and a broad distribution between the peaks. on a timescale of 104 yr. The peak at e > 0:95 is composed mainly of planets in set N Another example from set T1 is shown in Figure 8. In this (long-dashed line). This reflects a fact that we have stopped the case, one of the planets enters a hyperbolic orbit at 2:2 ; 106 yr. simulation in set N when the planet hits the surface of the host Two planets are left in stable orbits at 2.3 and 80 AU. However, stars. In other sets these planets are tidally circularized. Almost the innermost planet is in the Kozai state. At the time of isolation, all the circularized planets go to the most prominent peak at e < the eccentricity and inclination of the innermost planet are 0.49 0:05. Excluding these two peaks, the eccentricity is broadly dis- and 1.5 rad, respectively. The eccentricity and inclination oscillate tributed centered at e 0:5, as previous authors found. by the Kozai mechanism. Because the perturbing planet is located The planets that are injected into the inner orbits with moderate much further ( 80 AU) than in the previous case in Figure 6, eccentricities but have not suffered tidal circularization are dis- eccentricity increases more slowly. At 10:5 ; 106 yr, the eccen- tributed at around a 2:3 AU, as the energy conservation law tricity reaches 0.984. Since q reaches ’0.02 AU, the planet’s e requires. The pericenter distribution is shown in Figure 10. The and a are tidally damped. Since the damping timescale of the eccentricity ( e)islongerthanthatofthesemimajoraxis(see Fig. 4), the eccentricity decays after the semimajor axis sig- nificantly decreases. In contrast to set V, q gradually increases during the tidal circularization in set T. Since the damping time- scale is a strongly increasing function of q, the tidal circularization slows down with time in this set. As a result, the eccentricity is not fully damped in 108 yr (e ¼ 0:08 at 108 yr in this example). Wu (2003) and Wu & Murray (2003) also pointed out the mi- gration due to a coupled effect of the Kozai mechanism and tidal circularization for a planet in a binary system and called it ‘‘Kozai
Fig. 9.—Eccentricity histogram of the planets that were scattered into inner Fig. 7.—Evolution of eccentricity e and inclination i of the circularized planet orbits. Long-dashed, solid, dotted, and dash-dotted lines correspond to sets N, V, shown in Fig. 6. Solid and dotted lines indicate e and i, respectively. T1, and T4, respectively. 504 NAGASAWA, IDA, & BESSHO Vol. 678
Fig. 10.—Histogram of pericenter distance. Solid lines represent inner planets, and dotted lines represent outer planets at the end of simulations. distributions at log (q/1 AU) > 0:5havenoremarkablediffer- Fig. 11.—Histogram of inclination of planets. (a) The case without tidal force ences between models. In set V the close-in planets are piled up at (set N). Solid line shows the innermost planets. Dashed lines are all surviving log (q/1 AU) ¼ 2:3. Since we do not take into account the ef- planets. The planets that hit the star are counted as surviving planets. (b) The case fect that the dynamic tide should become less efficient with orbital with tidal force (set T1). The inclination of close-in planets is shown in the shaded circularization, the planet cannot stop the tidal decay once the cir- region. (c) The inclination of formed close-in planets in all simulations (sets V and cularization is started, which leads to the overdensity at the lower- T1-T4). q peak. Such a lower peak does not exist in set T, since the increase in q during the tidal circularization slows down the further cir- the eccentricity is higher. The tidal circularization occurs when cularization,asshowninFigure8.InsetsT3andT4,inwhich the eccentricity is high. Thus, the close-in planet is formed more 2 1=2 Ltide ¼0, the initial angular momentum, ½Gm (1 eini)aini ’ easily when the inclination takes lower values in the Kozai cycle. 1=2 2 1=2 (2Gm qini) , is equal to the final one, ½Gm (1 eBnal)aBnal ’ This effect inhibits overdensity at the highest inclination and (Gm qBnal), so q can increase by a factor 2. The final q of close-in results in a rather broad inclination distribution. Note that we planets are distributed between log (q/1 AU) ¼ 2and 1, like did not include any damping mechanisms for inclination, which the observed planets (see Fig. 1b). The final e are not necessarily generally have very long timescales. Our model shows that the damped fully and distributed in a range of 0Y0.4 in set T, and the nonnegligible fraction of close-in planets have retrograde rotation final a are larger than that in set V. (i > /2) with respect to the equator of the host star, although the The inclination distribution is shown in Figure 11 in the cases retrograde planets can be tidally unstable on longer timescales. of set N (Fig 11a) and set T1 (Fig. 11b). Solid lines show the Since type II migration cannot produce retrograde rotation, if such distribution of only innermost planets. Dashed lines show that retrograde planets are discovered, it would be a proof of some of all remaining planets (inner+outer planets). The overall dis- contribution from the scattering model to the formation of close-in tributions show no difference between models with and without planets. tide. Most planets keep relatively small inclinations as Chatterjee 3.5. Circularization Probability et al. (2007) claimed. However, the close-in planets shown in Figure 11c tend to have a relatively large inclination, because The fraction of runs in which close-in planets are formed is the Kozai migration is effective for planets injected into highly shown in Table 1 for each model. It is 30%, almost indepen- inclined (i /2) orbits, so that close-in planets formed with dent of the detailed model for tidal circularization. This is much this model selectively have highly inclined orbits. In the Kozai higher than the probability ( 10%) of formation of close-in mechanism, however, the inclination takes lower values when planet candidates predicted by Marzari & Weidenschilling (2002). No. 1, 2008 FORMATION OF HOT PLANETS 505
Fig. 14.—Example of evolution of eccentricity of the inner planet. Fig. 12.—Example of evolution of e, i,andh of the inner planet. The evolution of the inclination is shown in secondary axis. Dashed line shows e ¼ 0:98. The bars in the figure show the period for which the planet has small h value, i.e., it is the magnitude of e and that of i shows that the planet is in the in Kozai state. Kozai state. As we will explain in x 4, the maximum eccentricity during the Kozai cycle is determined mainly by the value of h, We found that in 2/3 of runs that form close-in planets, the almost independent of the location and mass of the perturbing planets are produced during three-planet interactions. In the planet. Since there is no energy dissipation in this run (even with residual 1/3 runs, the close-in planets are formed after one of tidal circularization, energy dissipation is practically negligible k the planet is ejected from the system and the system becomes as long as q 0:05 AU), the isolation of the innermost planet is stable. The difference between our result and the prediction by tentative, and it occasionally undergoes a relatively close en- Marzari & Weidenschilling (2002) comes from the fact that counter with the outer planet(s). After the encounter, the planet Marzari & Weidenschilling (2002) were concerned only with the enters a new Kozai state with a changed h. In the case of Figure 12, ; 6 latter cases after stabilization and did not pay attention to the the planet enters the Kozai state at t ’ 1:5 10 yr first. This former cases, which contribute much more. The latter circulari- Kozai cycle is terminated by a relatively close encounter with zation case is 6%Y8% in total simulations, which is consistent an outer planet before eccentricity is fully increased. The next ; 6 with the estimation of Marzari & Weidenschilling (2002). Kozai state starts at t ’ 2:5 10 yr, but h of this cycle is not k P The tidal circularization models that we used could be too small enough to raise e to 0.98 (q 0:05 AU). The third Kozai ; 6 strong, since we use the formula of dynamic tide even after the ec- state starting at t ’ 4:5 10 yr is not a clean Kozai state, because centricity is significantly damped. However, even if a more realistic an outer planet is relatively close. However, thanks to the small k P model is used, the result would not change significantly. We used value of h, e can be 0.99 (q 0:02 AU) in the cycle. If the tidal five different tidal circularization models. Among our models, the damping is included, the orbit would be circularized. Eventually, ; 6 damping force is the weakest in Set T4 (see Fig. 3). In this case, at t ’ 6:5 10 yr, one of the outer planets is ejected, and the only planets with q P 0:02 AU are circularized. Nevertheless, system enters a stable state where the inner planet and the re- the probability for formation of close-in planets is still as high maining outer planet are separated by 70 AU. At the point when as 30%. The relatively high probability is a result of repeated the system enters the stable state, q 0:4 AU. However, since the Kozai mechanisms that are inherent in the scattering of three inner planet is again in the Kozai state with relatively small h,the giant planets. pericenter periodically approaches 0.04 AU, where the tidal dis- Figure 12 shows an example of e, i,andh evolution without sipation is marginally effective in a relatively strong tide model. Since the outer planet is separated by 70 AU, the period of the tidal force. The evolution of the semimajor axis of this example 7 is shown in Figure 13. During the initial phase (t < 1:5 ; 106 yr) Kozai cycle is more than 10 yr. in which all the planets repeat mutual close encounters, their ec- Marzari & Weidenschilling (2002) noticed this long-term or- centricity is chaotically changed. However, after a planet is scat- bital change after the stabilization that can bring the pericenter to teredintoaninnerorbitofa ¼ 2:47 AU, its orbital change is the vicinity of the stellar surface, although they did not refer to the mostly regulated by secular variation. Anticorrelation between Kozai mechanism. But, with their criterion, the third Kozai cycle in which q takes the smallest value in this run is missed. During the three-planet orbit crossings, there are many chances to be in the Kozai state. The Kozai state is quasi-stable, and it often has enough time for the pericenter to stay close to the star and be tidally circularized. Since we also monitored q during the three- planet orbit crossing (in the models with tidal circularization effect, it is automatically monitored), we found many more cases to form close-in planets than previous authors did. Figure 14 gives another example. In this example, the dynam- ical tide is included. After the chaotic phase ends, three planets enter a quasi-stable state. The eccentricity of innermost planet oscillates with a large amplitude, exchanging e and i. The longer period perturbation gradually enhances the mean eccentricity, and the planet’s orbit is circularized at 1:6 ; 107 yr. This is a similar Fig. 13.—Semimajor axis, pericenter, and apocenter evolution in the system example to the one that Marzari & Weidenschilling (2002) found. corresponding to Fig. 12. The close-in planet can be formed even if it is not injected into a 506 NAGASAWA, IDA, & BESSHO Vol. 678
pffiffiffi Fig. 16.—Range of C and phffiffiffiffiffiffiffiffifor given emax.Inh 0:6, the minimum value of C is C ¼ 20 24h þ 12 15h.Inh 0:6, it is given by C ¼ 6h 2. In shaded regions (denoted by ‘‘libration’’ and ‘‘circulation’’), the planet is in an elliptical orbit. In the region denoted by libration, the mutual angle of pericenters (jj ) is restricted between /4 to 3 /4. In the circulation region, jj circulates from 0 to .
that is determined by h, provided that the other planets are well separated and the perturbations from them are secular. The pericenter ( ) liberates when C < 6 h and h Fig. 15.—Hamiltonian map for an innermost planet with h ¼ 0:2 and 0.6. 1 e2 ¼ (16 24h C The contours in upper panel are drawn at every C ¼ 0:5fromC ¼ 4:02 to 8.8 max 36 (inner ones to outer ones)forh ¼ 0:2. The variable C takes the minimum (’ 4) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi at libration center and the maximum at the top contour (=8.8). In lower panel, þ 400 1200h þ 40C þ 576h2 þ 48hC þ C2): contours are drawn at every C ¼ 0:2fromC ¼ 1:6to6.4(lower ones to upper ones)forh ¼ 0:6. The dotted line shows a boundary between libration and ð13Þ circulation. For emax closer to unity, the minimum pericenter distance during the Kozai cycle is smaller. If it is small enough, the tidal cir- good Kozai condition at once, as long as the planetary interaction cularization is efficient, and the formation of a close-in planet is continues. expected. In Figure 16,pffiffiffi contours of emaxpforffiffiffi values close to unity are plotted on the C- h plane. Since h is proportional to an- 4. THE PATH TO KOZAI MIGRATION gular momentum ‘z, this plane is essentiallypffiffiffi the energy-angular In x 3, we have showed that the Kozai mechanism coupled momentum plane. Assuming that C and h is distributed homo- with tidal circularization effectively sends planets to short-period geneously after scattering, although this assumption may not be orbits. In this section we present analytical arguments of the Kozai exactly true, the probability [P(ecrit)] that a planet enters regions mechanism to evaluatethe probability of forming close-in planets of emax > ecrit is 46.8%, 30.8%, and 22.1% for ecrit ¼ 0:95; 0:98; and to explain the numerical results. and 0.99, respectively. In Figure 15, contours of C given by equation (10) In our simulation, in which an initially innermost planet is at (‘‘Hamiltonian map’’) are plotted on the e- plane, where ¼ 5 AU, a typical final semimajor axis of the innermost planet is !2 !1. The topology of the Hamiltonian map depends only on 2 AU (Fig. 5). With the tidal circularization models we adopted, h. The upper and lower panels are the maps with h ¼ 0:2and the circularization is effective on timescales P108 yr, if the peri- 0.6, respectively. When a planet is scattered into an orbit having center distance is smaller than 0.015Y0.03 AU. Then emax k some (C; h), the planet’s orbit moves along the constant C line ecrit ¼ 0:985Y0:993 is required for the circularization. From the No. 1, 2008 FORMATION OF HOT PLANETS 507 probability P(ecrit) estimated above, we expect that formation enhances the probability for the tidal circularization to occur. probability of close-in planets is 20%Y30% for this setting, which Thereby, we have found a much higher probability of formation is consistentpffiffiffi with our numerical results. Realistic distribution ofpffiffiffiC of close-in planets than previously expected. Therefore, scattering and h might be more concentrated in lower C and/or higher h. would contribute to the formation of close-in planets, although the In this case, P(ecrit) may be smaller for one scattering, but the main channel may still be through type II migration. repeated nature of the Kozai mechanism may compensate for A nonnegligible number of close-in planets without nearby the decrease in P(ecrit) to result in the formation probability of secondary planets but with eccentricities k0.3 has been discov- close-in planets at a similar level. Our numerical simulations in ered. It is not easy for the type II migration model to account for x 3 were done for only one initial setting of the planetary semi- the high eccentricities, but the scattering model could account for major axis, but the above estimates suggest that the formation them. As shown in xx 3.3 and 3.4, the tidal circularization can probability of the close-in planets may be similar in other settings. slow down by an increase in q before full circularization, leaving moderate eccentricities. The discovered close-in eccentric planets 5. CONCLUSION AND DISCUSSION are generally massive (having masses larger than Jupiter mass) among the close-in planets (Fig. 1a), so their circularization time- The ‘‘standard’’ model for formation of close-in giant planets scales are much shorter than those shown in Figure 4. Therefore, is that gas giants may originally form at several AU’s beyond the results in our set Tcould be consistent with these discovered ice lines and migrate to the vicinity of the star. The most discussed planets. migration mechanism is type II migration. Here, we have inves- Our results also suggest that the close-in planets have a broadly tigated another channel to move planets to the stellar vicinity, the spread in inclination distribution, including retrograde rotation. ‘‘scattering’’ model (Rasio & Ford 1996), which is a coupled More observations of the Rossiter-McLaughlin effect for transit- process of planet-planet scattering and tidal circularization. If or- ing close-in planets, in particular for those with relatively high bital eccentricity is pumped up to values close to unity, because of eccentricity, such as HD 17156b, will impose constraints on the the proximity of the pericenter, the eccentricity and the semimajor contribution of the scattering model to close-in planets. axis of the planet are damped by the tidal effect from the star, The tidally dissipated energy is GM m/2a in total, where a almost keeping the pericenter distance to form a close-in planet is final semimajor axis of the circularized close-in planet and with relatively small eccentricity. We newly find that the Kozai m is its planet mass. Since this energy is much larger than the mechanism is also a key process in the model. binding energy of the planet, it could be a heat source for in- We have carried out orbital integration of three planets of flated close-in planets such as HD 209458b, OGLE-TR-10, Jupiter mass, including tidal damping in the integration. We fol- OGLE-TR-56, HAT-P-1b, or WASP-1b, if the tidal circulari- low the simulation setup by Marzari & Weidenschilling (2002). zation occurred relatively recently (orbital crossing followed by We found that in 30% runs, close-in planets are formed, which the circularization can start a long time after the formation of is a much higher probability than suggested by previous studies giant planets). (Weidenschilling & Marzari 1996; Marzari & Weidenschilling Note that there are some caveats to the tidal circularization 2002; Chatterjee et al. 2007), because in many cases, the cir- model that we used may not be sufficiently relevant. We made cularization occurs during three-planet scattering, which was not many assumptions for the use of equations (2) and (3). The model monitored by previous studies. Three-planet scattering is usually is only valid for fully convective planets in highly eccentric orbits, terminated by the ejection of one planet and the system entering a since it considers only the ‘ ¼ 2 f-mode and uses an impulse stable state. Previous studies were concerned only with the stable approximation. We apply the model even after the orbits are state. significantly circularized, which may overestimate the effect of Since tidal damping timescale is usually longer than that of tidal dissipation. In multiple encounters with the star, the energy orbital change in the chaotic stage of three-planet interaction, and angular momentum can either increase or decrease (e.g., Press the tidal circularization does not occur during the chaotic stage, & Teukolsky 1977; Mardling 1995a, 1995b), but we include the even if the pericenter distance happens to be small enough. How- tidal effect, as it always gives a decreasing effect. Because of these ever, we have found that when one planet is scattered inward, it simplifications, our simulation may be a limiting case in which the often becomes separated from other outer planets. As long as the tidal force works most effectively, although inclusion of g-modes tidal dissipation is negligible, the isolation is only tentative, but its might strengthen the tidal effect. Our purpose in this paper is not a duration can be long enough for the tidal circularization if the P Y study of the detailed tidal damping process, but to show another pericenter distance is small enough ( 0.02 0.04 AU for Jupiter- available channel for the close-in planets. We also need to take mass planets). We also found that the isolated planet usually enters into account the relativity precession and J2 potential of the central the Kozai state. Even if the eccentricity of the planet is not excited star, which prevent the Kozai mechanism, as well as the long-term enough by the close scattering that injected the planet into an inner tidal disruption process (e.g., Gu et al. 2003). These issues are left orbit, the eccentricity can secularly increase to values close to for further studies. unity during the Kozai cycle, in particular when the planet has an inclined orbit. Although the probability that the eccentricity takes values between 0.98 and 1.0 is quite low just after the scattering, that probability is significantly enhanced to a 30% We would like to thank E. Kokubo, A. Morbidelli, and an anon- level during the Kozai cycle. Even if the Kozai cycle has relatively ymous referee for their helpful suggestions. This work is sup- large angular momentum and the eccentricity does not take such ported by MEXT KAKENHI(18740281) Grant-in-Aid for Young a high value, the outer planets eventually destroy the quasi-stable Scientists (B), Grant-in-Aid for Scientific Research on Priority state and the inner planet enters another quasi-stable state, in other Areas (MEXT-16077202), and MEXT’s program ‘‘Promotion words, another Kozai state. The new Kozai state can have rel- of Environmental Improvement for Independence of Young atively small angular momentum and raise the eccentricity high Researchers’’ under the Special Coordination Funds for Pro- enough for tidal circularization. The repeated Kozai mechanism moting Science and Technology. 508 NAGASAWA, IDA, & BESSHO REFERENCES Chambers, J. E., & Wetherill, G. W. 1998, Icarus, 136, 304 Marcy, G. W., Butler, P. R., Fischer, D. A., & Vogt, S. S. 2004, in ASP Conf. Chambers, J. E., Wetherill, G. W., & Boss, A. 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