I. THE FORMATION OF PROTOPLANETS DR.RUPNATHJI( … · Disk-planet interacti ns, including recent w...

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I. THE FORMATION OF PROTOPLANETS

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We c nsider several pr cesses perating during the late stages f planet f rma-ti n that can affect bserved rbital elements. Disk-planet interacti ns, includingrecent w rk n the fl w induced by embedded pr t planets and gap f rmati n,are reviewed. Recent results n phen mena caused by the tidal interacti n f anrbiting c mpani n with a central star, such as rbital circularizati n and spin syn-

chr nizati n, are reviewed and applied t extras lar planets. Dynamical pr cessesthat c uld pr duce sh rt-peri d planets r planets in highly eccentric rbits, suchas l ng-term rbital instability and the K zai mechanism, are discussed.

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Queen Mary & Westfield College, London

Lick Obser atory

Lick Obser atory

Tokyo Institute of Technology

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ORBITAL EVOLUTION ANDPLANET-STAR TIDAL INTERACTION

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D. N. C. LIN

J. C. B. PAPALOIZOU

C. TERQUEM

G. BRYDEN

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It is generally accepted that the planets in ur s lar system were f rmedin a flattened gase us nebula centered ar und the Sun. In typical star-f rming m lecular cl uds, dense c res are bserved t have specific an-gular m mentum greater than 6 10 cm s (G dman et al. 1993)such that their c llapse leads t r tati nally supp rted disks anal g us tthe prim rdial s lar nebula (Terebey et al. 1984). Between 25 and 75%f the y ung stellar bjects (YSOs) in the Ori n Nebula appear t have

disks (Pr sser et al. 1994; McCaughrean and Stauffer 1994), with typicalmass 10 M , temperature 10 K, and size 40 20 AUD

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1112 D. N. C. LIN ET AL.

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(Beckwith and Sargent 1996; see als the chapters by McCaughrean et al.,Calvet et al., and Wilner and Lay, this v lume). The c mm n existence fpr t stellar disks ar und YSOs, with pr perties similar t th se expectedf r the s lar nebula, suggests that the c nditi ns f r planetary f rmati nmay be generally satisfied.

Recent bservati nal breakthr ughs have led t the disc very fJupiter-mass (M ) planets ar und at least a few percent f nearby s lar-type stars (see chapter by Marcy et al., this v lume). With the presentdata, we can assert that planetary f rmati n is r bust.

In c nventi nal planetary f rmati n m dels, the first stage f pr t -planetary f rmati n is the rapid buildup f s lid c res thr ugh the c ag-ulati n f planetesimals (Safr n v 1969; Hayashi et al. 1985; Lissauerand Stewart 1993). When the c re mass increases ab ve a critical value( 15 M ), quasistatic ev luti n is n l nger p ssible, and a rapid accre-ti n phase begins (Mizun 1980; B denheimer and P llack 1986), leadingt the f rmati n f gase us giant planets (P llack et al. 1996; als see thechapter by Wuchterl et al., this v lume).

The details f this m del are n t yet fully w rked ut, but if we sup-p se that the pr t planet can accrete gas as efficiently as p ssible, it willfirst take in gas in the neighb rh d f its rbit, assumed circular at radius

until it fills its R che radius ( /3) , where / isthe pr t planet central star mass rati , while rbiting in an empty annulus.The mass f the pr t planet is then given by 3[4 ( )/3 ]where ( ) is the characteristic disk mass within radius , with

being the surface density. At 5.2 AU and 200 g cm this gives amass 0 4 M .

Further mass gr wth n w depends n whether the disk has a kine-matic visc sity capable f pr ducing a mass accreti n rate nt thepr t planet. If is finite, then all f the mass fl w thr ugh the uter disksh uld g t the pr t planet. T m del the disk visc sity, we ad pt the pre-scripti n f Shakura and Sunyaev (1973), in which , whereis a dimensi nless c nstant, is the disk semithickness, and is thedisk angular vel city. The m st likely mechanism f r pr viding an effec-tive visc sity in stellar accreti n disks is MHD turbulence (Balbus andHawley 1991), which pr duces 10 in a fully i nized disk. N te,h wever, that may vary thr ugh ut the disk and be much smaller in itsintermediate parts (see the chapter by St ne et al., this v lume).

Observati nally inferred values f the disk accreti n rate10 M yr (Hartmann et al. 1998) are m del dependent and highlyuncertain. N netheless, f r such a fiducial , a pr t planet may attaina mass 10 M within 10 yr. The mass f extras lar planets is

1 M (see chapter by Marcy et al., this v lume). Unless these planetsare preferentially f rmed in l w-mass disks, their gr wth needs t be ter-minated r inhibited such that they are unable t accept all the mass thatfl ws thr ugh the disk. In this paper we f cus n disk-pr t planet tidalinteracti ns as a mechanism f r acc mplishing this.

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A. Protoplanet-Disk Tidal Interactions

ORBITAL EVOLUTION 1113

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A pr t planet exerts tidal perturbati ns that, f r 1 M , may beadequate t induce the f rmati n f a gap in the disk near its rbit andthereby start t limit accreti n fl w nt it (Lin and Papal iz u 1993). Ingeneral, the gravitati nal p tential due t the pr t planet may be F urierdec mp sed in the f rm

c s[ ( )] (1)

where is the azimuthal angle, and the azimuthal m de number areintegers, and ( ) / is the pattern speed, with and

being the angular and epicyclic frequency f the pr t planet, respec-tively. If the rbit f the planet has a small eccentricity , then(G ldreich and Tremaine 1980; Shu 1984). F r circular rbits, nlyneed be c nsidered.

B th utg ing and ing ing density waves are excited in the disk atthe Lindblad res nances, l cated at and where, f r ,

/ (G ldreich and Tremaine 1978). Here is the epicyclic fre-quency f the gas and is the disk angular vel city. In a Keplerian disk,

. The ing ing ( utg ing) waves carry a negative (p sitive) angularm mentum flux measured in their directi n f pr pagati n as they m veaway fr m the pr t planet int the disk interi r (exteri r). The wavesthus carry a p sitive, utward-pr pagating, c nserved angular m men-tum flux r wave acti n In m st cases it is reas nable t assume thatthe waves are dissipated at s me l cati n in the disk, where their angularm mentum density is dep sited (see Lin and Papal iz u 1993 and refer-ences therein). In the limit f a c ld tw -dimensi nal disk, G ldreich andTremaine (1978) f und f r a particular that

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The back reacti n t rque exerted n the disk interi r (exteri r) t the planetis ( ). Thus, the inner disk l ses angular m mentum while theuter disk gains it; hence, the tendency is t f rm a gap. Evaluati n f

the t tal t rque acting n each side f the disk requires the summati n fc ntributi ns fr m all the res nances, which, f r circular rbits, am untst summing ver . As , the l cati n f the res nance appr achesthe rbit.

H wever, f r a n n-self-gravitating disk and large , the waves ares nic in character and thus can exist nly farther than a distance 2 /3fr m the rbit, bey nd which the relative disk fl w is supers nic. Thisresults in a t rque cut ff (G ldreich and Tremaine 1980; Artym wicz1993 ; K rycansky and P llack 1993; Ward 1997) f r / The

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B. Embedded Protoplanets and Gaps

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dynamics in the c rbital regi n 2 /3 are n t wavelike and are c n-sidered bel w. Summing ver all res nances, the t tal angular m men-tum flux carried by the waves is essentially the same as that given byPapal iz u and Lin (1984), namely

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which was btained by direct calculati n f the t rques f r a m del diskin which the pr t planet rbited in a gap f radial width .

Using equati n (3) t estimate tidal t rques and then c nsideringthe c mpetiti n between visc us t rques, which tend t fill a gap, andtidal t rques, which tend t empty it, Lin and Papal iz u (1979 , 1980,1986 , 1993) f und the f ll wing visc us c nditi n f r gap pening:

40 /( ). They pr p sed that gap f rmati n w uld lead t thelimitati n f accreti n, but they were able t c nsider nly empty gaps.With the intr ducti n f p werful numerical finite-difference techniquesand c mputers, it has recently bec me p ssible t study gap f rmati nnumerically m re fully than previ usly (Artym wicz et al. 1998; Brydenet al. 1998; Kley 1998; and see secti n I.B). The first results f this w rkindicate that there is a transiti nal regime in visc sity f r which a gapexists but s me accreti n still ccurs, which is then essentially switchedff f r small en ugh visc sity.

In additi n, Lin and Papal iz u (1993) p inted ut that in a diskwith very small visc sity, t f rm a gap it is necessary that( /3) This thermal c nditi n can be viewed in several ways. It meansthat the pr t planet’s gravity is m re imp rtant than pressure at a distance

fr m the pr t planet; generati n f large en ugh hydr static pressuref rces w uld require gradients that w uld cause a vi lati n f Rayleigh’scriteri n. Fr m K rycansky and Papal iz u (1996), it is apparent thatthe thermal c nditi n is required s that the fl w in the neighb rh d fthe pr t planet is n nlinear en ugh that sh cks are pr duced, which canpr vide the dissipati n ass ciated with gap pening (see bel w). This c n-diti n was f und f r a simple 2D m del disk with a bar tr pic equati nf state.

With m re c mplicated physics and the intr ducti n f three-dimensi nal effects (Lin et al. 1990 ), it may be p ssible t alter wavedissipati n patterns and, thus, angular m mentum dep siti n. H wever,because f the difficulty f clearing material near the pr t planet (it isdifficult t see h w linear waves c uld be dissipated cl ser than severalvertical scale heights), it is likely that gap f rmati n will ccur much lessreadily if the thermal c nditi n is n t satisfied.

The fl w ar und a partially embedded pr t planet has been simulated inthe l w- limit in a shearing-sheet appr ximati n (K rycansky and Pa-pal iz u 1996; Miy shi et al. 1999). These simulati ns pr vide useful

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clues n the fl w pattern near the pr t planet. A gl bal illustrative m delis sh wn in C l r Plate 16, in which we set 10 , / 0 07, and

10 . The fl w pattern can be divided int three regi ns:

1. A circumplanetary disk is f rmed within the R che radius f the pr -t planet, with semithickness . In such a ge metrically thickdisk, the tidal perturbati ns f the h st star induce a tw -arm spiralsh ck wave with an pen pitch angle. Angular m mentum is effi-ciently transferred fr m the disk t the star’s rbit ar und the planet(equivalent t the planet’s rbit ar und the star) s that gas is ac-creted nt the c re f the pr t planet within a few rbital peri ds

2 / .2. An extended arc is f rmed in the c rbital regi n near , with gas

streaming in h rsesh e rbits ar und the L and L p ints. Becausethese are l cal p tential maxima, visc us dissipati n leads t the de-pleti n f this regi n (Lin et al. 1987).

3. In th se regi ns f the disk that have , the pr t planet’stidal perturbati n results in the c nvergence f streamlines t f rmpr n unced trailing wakes, b th inside and utside These high-density ridges have been identified by s me as “stream accreti n,”thr ugh which material fl ws fr m the disk t the pr t planet (Arty-m wicz and Lub w 1996). H wever, the vel city field viewed in aframe r tating with the c mpani n (C l r Plate 16) clearly indicatesthat gas in the p stsh ck regi n al ng the ridge line is m ving awayfr m the pr t planet.

Simulati ns f gap f rmati n need t c nsider m del ev luti n vermany rbital peri ds and have t deal with a large density c ntrast betweenthe gap regi n and ther parts f the disk. Special care is needed t min-imize the tendency f numerical visc sity t pr duce spuri us diffusi nint the gap regi n and thus significantly affect the results. In C l r Plate17, we illustrate the excitati n and pr pagati n f waves and the existencef a clear gap f r a planet (with 10 ) interacting with a disk (with/ 0 04 and 10 ).

In Figure 1, we pl t the gr wth timescale / as a functi n fand / f r pr t planets f mass 1 and 10 M . We c mment that it is n tnecessary f r accreti n t st p entirely f r gap f rmati n t affect the finalmass f a pr t planet. All ne needs is that the mass d ubling time sh uldbec me l nger than either the disk lifetime r the time f r the interi r diskt accrete. In the latter case, the planet w uld appr ach the central starbef re it c uld increase its mass significantly (Ivan v et al. 1998). Fr mFig. 1 we see that, f r 10 and / 0 04, the mass d ubling timef r 1 M is 10 yr. F r 10 M the same results apply f r / 0 07The results in Fig. 1 thus suggest that the tidal truncati n pr cess peratesduring planetary f rmati n and is imp rtant in determining the final massf the planet .

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II. PROTOPLANETARY ORBITAL EVOLUTION

A. Eccentricity Evolution

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Figure 1. The mass d ubling timescale as a functi n f visc sity parame-ter f r ne Jupiter mass with / 0 04 (left) and 10 Jupiter masses with

/ 0 07 (right). As the gas visc sity is decreased, the accreti n rate ntthe pr t planet dr ps ff sharply until reaching a numerical limit f r 10

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Res nant interacti n between the disk and the planet can lead t changes inthe rbital eccentricity (Lin and Papal iz u 1993; Artym wicz 1993 ).The way in which Lindblad t rques cause a small eccentricity t gr whas been reviewed by Lin and Papal iz u (1993). If nly the Lindbladt rques are c nsidered, the disk matter is able t pump eccentricity tidallyin a similar way t a rapidly r tating star (see bel w). H wever, c r tati nres nances in the disk (at which the pattern speed f a F urier c mp nentf the tidal perturbati n c r tates with the disk) als need t be c nsidered

(see secti n I.A). When these res nances exist, a t rque is applied t thedisk at the c r tati n radius , where at a rate given by

( ) (4)2

where /(2 ) is the v rtensity. The r le f c r tati n res nancesdepends str ngly n the distributi n. When vanishes at s me diskedge, r when there is a gap near a l w-mass perturber, G ldreich andTremaine (1980) sh wed that the acti n f t rques due t c r tati n res -nances damps m re effectively than effects due t Lindblad res nancesexcite it. H wever, this result d es n t apply t the situati n where a pr -t planet is embedded in a disk with n edges r gap and where tidal inter-acti n is linear.

Assuming ln / ln 1 in the unperturbed disk, Artym wicz(1993 ) suggested that the eccentricity f l w-mass embedded pr t plan-ets w uld be damped by tidal interacti n. H wever, in the minimum-mass

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s lar nebula m del where , v rtensity is independent f , andthen all c r tati n t rques vanish (Ward 1993), alth ugh the t rque ass -ciated with c rbital Lindblad res nances may persist. F r a planet witha sufficiently large mass t pen a wide gap, b th c r tati n and c rbitalLindblad res nances w uld be severely weakened and c uld then in-crease n a timescale /( / ) ( / ) (Lin and Papal iz u1993) which, f r massive planets, can be sh rter than the inferred diskev luti n timescale.

If the gr wth f pr t planets is limited by the f rmati n f a gap fwidth , tides may cause t increase until / when a p tentialexpansi n valid f r small assumed in the ab ve analysis breaks d wn. Ina c ld disk, the pr t planet’s angular vel city at peri-apheli n w uld thenbe greater/less than ( ), resulting in a t rque reversal. Such a pr -cess c uld pr vide a limit t the gr wth f . H wever, a pr t planet’s in-creased radial excursi n may als cause the gap t widen, in which casemight increase t yet larger values. Analytic treatments and self-c nsistentnumerical simulati ns in the large- limit have n t been carried ut s far.

The characterizati n f extras lar planets’ rbits fr m f rthc mingbservati ns will pr vide s me useful c nstraints n the rigin f eccen-

tricity. The c existence f several planets with similar but very different( r similar but very different ) is n t a natural utc me f the ex-

citati n f eccentricity by disk tidal interacti n, because this w uld tendt pr duce an ( ) relati n (Artym wicz 1992).

Planetary rbital migrati n is induced by the difference between the innerand uter disk t rques, , which react back n the planet, assumed tremain in a circular rbit, such that

2(5)

Based n recent w rk indicating that is alm st always negative, Ward(1997) suggested that embedded planets migrate inward n a timescale

10 ( /M ) yr. If this pr cess ccurs in pr t stellar disks, pr t plan-ets w uld migrate t wards the stellar surface at nce 1 M ,because their gr wth timescale (Lissauer and Stewart 1993) w uld thenbec me l nger than this migrati n timescale. Res luti n f this rapid mi-grati n dilemma may require the c mplete and n nlinear analysis f thedisk resp nse t the pr t planet in the c r tati n regi ns which may bequite c mplex (see C l r Plates 16 and 17). H wever, supp sing that mi-grati n ccurs and can be st pped near the star (see bel w), a scenarileading t the f rmati n f sh rt-peri d planets has been pr p sed (seechapter by Ward and Hahn, this v lume).

After gap f rmati n, there may still be s me residual accreti n de-pending n the value f (see secti n I.B). T rques are still exerted

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III. PLANET-STAR TIDAL INTERACTION


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between disk and pr t planet, and the imbalance due t differing pr per-ties f the disk n either side f the pr t planet causes rbital migrati n(G ldreich and Tremaine 1980; Lin and Papal iz u 1986 ; Takeuchi et al.1996). The estimated effects f advecti n f angular m mentum thr ughthe accreti n fl w n this pr cess are usually f und t be small. F r small

, the pr t planet behaves like a disk particle and s migrates t wardsthe star. F r larger masses, the ev luti n is sl wer. N netheless, the pr -t planet is always expected t reach the star bef re it has time t d ubleits mass (Ivan v et al. 1998).

The ab ve discussi n naturally leads t the suggesti n that sh rt-peri d planets were f rmed at several AU away fr m their h st stars andsubsequently migrated t their present l cati n (Lin et al. 1996). H wever,

f rmati n cann t be c mpletely ruled ut (B denheimer 1997).A pr t planet’s migrati n may be terminated near if (1) the disk

d es n t extend d wn t r (2) the h st star induces angular m mentumtransfer t the pr t planet’s rbit via tidal effects (see secti n III). F rthe first p ssibility, interacti n between the disk and an intense ( 1 kG)stellar magnetic field has been suggested (K nigl 1991) as the cause f rthe m dest bserved r tati n peri d ( 8 days) f classical T Tauristars (B uvier et al. 1993; Ch i and Herbst 1996). Such a magnetic fieldstrength may be c nsistent with recent measurements (Guenther 1997).In this scenari , the stellar field is assumed t induce a cavity ut t themagnet spheric radius ( a few 10 cm) where is equal t the l calKeplerian peri d f the disk. When a pr t planet migrates interi r t the2:1 res nance f the gas at , the pr t planet is dec upled fr m the disk,and its migrati n is stalled. The main uncertainties here are in the bserveddistributi n f (Stassun et al. 1999).

As the planet appr aches , the tides raised n the star r planet by itsc mpani n bec me str ng en ugh that their dissipati n leads t rbitalev luti n. If the distributi n f mass f the perturbed bject has the samesymmetry as the perturbing p tential, n tidal t rque results fr m the in-teracti n. H wever, in general, dissipative pr cesses (e.g., radiative damp-ing r turbulent visc sity) acting n the tides pr duce a lag between theresp nse f the star r planet and the perturbing p tential, enabling me-chanical energy t be l st and angular m mentum t be exchanged be-tween the r tati n f the perturbed bject and the rbital m ti n. Only ifthe system is circular, synchr n us, and c planar (i.e., the spin f the bi-nary c mp nents and that f the rbit are parallel) d es the tidal t rquevanish (Hut 1980). H wever, such an equilibrium state is n t necessarilystable (C unselman 1973; Hut 1980, 1981).

The resp nse f a star r planet t a tidal perturbati n is the sum ftw terms: an equilibrium tide and a dynamical tide. The equilibrium tide

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(Darwin 1879) is the shape that the perturbed bject w uld have if it c uldadjust instantly t the tidal p tential; it is btained by balancing the pres-sure and gravity f rces. The dynamical tide c ntains the scillat ry re-sp nse f the star r planet. It takes int acc unt the fact that gravity rm des can be excited in the c nvectively stable layers f the star r planetand that res nances between the tidal disturbance and the n rmal m desf the star r planet can ccur (C wling 1941).

In massive cl se binaries, which have a c nvective c re and a ra-diative envel pe, the dynamical tide cann t be neglected, because tidalfricti n is caused pred minantly by the radiative damping f the tidallyexcited m des (Zahn 1975, 1977; Sav nije and Papal iz u 1983, 1984,1997; Papal iz u and Sav nije 1985, 1997; G ldreich and Nich ls n1989; Sav nije et al. 1995; Kumar et al. 1995). In that case, m des,which are excited mainly near the c nvective c re b undary, pr pagateut thr ugh the envel pe t the atm sphere, carrying energy and angular

m mentum. They are damped cl se t the surface, where the radiativediffusi n time bec mes c mparable t the f rcing peri d, thus enabling anet tidal t rque t be exerted n the star r planet.

In the case f s lar-type binaries, which have a radiative c re and ac nvective envel pe, it was th ught until recently that tidal fricti n c uldbe well described by the the ry f the equilibrium tide, in which nlyturbulent dissipati n f the equilibrium tide in the c nvective envel pe istaken int acc unt (Zahn 1977). H wever, recent studies have indicatedthat this is n t the case. Even if radiative damping can be ign red, thet rque derived using nly the equilibrium tide is 4–6 times larger than thattaking int acc unt the dynamical tide f r binary peri ds f several days(Terquem et al. 1998 ). The reas n is that the the ry f the equilibriumtide can in principle be applied nly when the characteristic timescales fthe perturbed bject are small c mpared t the f rcing peri d. In a s lar-type star, h wever, the c nvective timescale in the interi r regi n f thec nvective envel pe is as large as a m nth. Furtherm re, because f theuncertainty ver the magnitude f the turbulent visc sity ass ciated withc nvecti n, it is n t clear that the t rque due t turbulent dissipati n actingn the full tide in the c nvective envel pe is m re imp rtant than that due

t radiative damping acting n the m des that pr pagate inwards in theradiative c re (Terquem et al. 1998 ; G dman and Dicks n 1998). Here,we f cus n the case f a planet rbiting ar und a s lar-type star.

Tides raised n the star by the planet can be analyzed in the limitthat the r tati nal frequency f the star is small c mpared t the rbitalfrequency. Then, as a result f tidal fricti n, the star spins up, the rbitdecays (the planet spirals in), and the rbit’s eccentricity, if any, decreases.T calculate the timescales n which this ev luti n ccurs, we need tquantify the dissipative mechanisms that act n the tides. Recent studies(Claret and Cunha 1997; G dman and Oh 1997; Terquem et al. 1998 )have f und that the turbulent visc sity in the c nvective envel pe that is

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required t pr vide the bserved circularizati n rates f main-sequences lar-type binaries (Mathieu 1994) is at least 50 times greater than thatsimply estimated fr m mixing length the ry f r n nr tating stars. This in-dicates either (1) that the bservati ns are questi nable, (2) that s lar-typebinaries are n t circularized thr ugh turbulent visc sity acting n tidalperturbati ns (see Tass ul 1988 and Kumar and G dman 1996 f r thersuggested tidal mechanisms), r (3) that dissipati n in the c nvective en-vel pe f s lar-type stars is significantly m re efficient than is currentlyestimated (see Terquem et al. 1998 f r a m re detailed discussi n fthe uncertainties inv lved). Here we assume that circularizati n f s lar-type binaries d es ccur thr ugh the acti n f turbulent visc sity n thetides, and we then calibrate its magnitude s as t acc unt f r the bservedtimescales. Under these circumstances, when the resp nse f the star is n tin res nance with ne f its gl bal n rmal m des, the tides are dissipatedm re efficiently by turbulent visc sity than by radiative damping. In a res-nance, radiative damping d minates and limits the resp nse f the star at

its surface. Alth ugh the planetary c mpani n may g thr ugh a succes-si n f res nances as it spirals in under the acti n f the tides, f r a fixedspectrum f stellar n rmal m des its migrati n is c ntr lled essentially bythe n nres nant interacti n. F r a n nr tating star, and with the calibrati nmenti ned ab ve, the rbital decay timescale, spinup timescale f the star,and circularizati n timescale, in Gyr, are (Terquem et al. 1998 ):

/ 1Gyr 2 763 10 (6)

/ 1 day

Gyr 1 725 10 (7)1 day

/ 1Gyr 4 605 10 (8)

/ 1 day

where is the rbital peri d. This circularizati n timescale is valid nlyif the initial eccentricity is n t t large. If nly the c nvective envel pe fthe star, where tidal dissipati n ccurs, is spun up during tidal ev luti n,then the spinup timescale has t be multiplied by / , where andare the m ments f inertia f the c nvective envel pe and the entire star,respectively. F r the Sun, / 0 14.

Tides are als raised n the planet by the star. In c ntrast t the giantplanets f ur s lar system, Jupiterlike planets n a cl se rbit are expectedt have an is thermal, and thus radiative (c nvectively stable) envel pe(Guill t et al. 1996; Saum n et al. 1996). F r these planets, b th turbulentdissipati n in the c nvective c re and radiative damping in the envel peact n the tides. S far it is n t clear which mechanism is m re imp rtant.

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We express the timescales ass ciated with turbulent dissipati n f thetides in terms f the parameter , the inverse f which is the effectivetidal dissipati n functi n (MacD nald 1964). Damping f the tides in thec nvective c re f the planet leads t the synchr nizati n f the planet r -tati n with the rbital rev luti n n the f ll wing characteristic timescale,in Gyr (G ldreich and S ter 1966):

1 day 1 dayGyr 4 4 10 (9)

1 day

where is the planet radius, is the semimaj r axis f the rbit, and isthe initial value f the planet r tati nal peri d (when it first underg es tidalinteracti n with the star). Because the r tati nal angular m mentum f theplanet is in general small c mpared t its rbital angular m mentum, syn-chr nizati n f the planet ccurs bef re any significant rbital ev luti ncan take place. Once synchr nizati n is achieved, damping f the tidesraised in the planet always leads t the decay f the rbital eccentricity na characteristic timescale, which is, in Gyr (G ldreich and S ter 1966),

Gyr 2 8 10 (10)1 day

The tides raised n the planet d n t lead t the decay f the rbit ncethe planet is synchr nized. We n te that may depend n the tidal fre-quency as seen by the planet, and theref re n the r tati nal frequency fthe planet. Since is calculated assuming synchr nizati n, the value f

in the ab ve equati n may be different fr m that used f r calculating. Orbital ev luti n and tidal heating f Jupiter’s satellites leads t an

estimate f 10 –10 f r this planet (G ldreich and S ter 1966; Linand Papal iz u 1979 ; Gailitis 1982). H wever, it is n t yet underst dwhere this value f c mes fr m, because turbulent visc sity arisingfr m c nvecti n w uld pr duce 5 10 (G ldreich and Nich ls n1977; see Stevens n 1983 f r an alternative). There is, f c urse, n rea-s n t assume that the values f the extras lar planets are similar tthat f Jupiter. First, s me f these planets rbit very cl se t their parentstars and theref re are much h tter than Jupiter. Als , may depend nthe magnitude and the frequency f the tidal scillati n, in which case itw uld be different f r Jupiter if this planet were synchr n usly r tating na cl ser rbit. Theref re we can nly speculate when applying the ab vef rmulae t extras lar planets.

Radiative damping f the tides in the envel pe, as described ab vef r massive binaries, gives rise t the synchr nizati n and circularizati ntimescales and , respectively. These timescales have been eval-uated (in particular f r 51 Pegasi) by Lub w et al. (1997). H wever, asthey p inted ut, the asympt tic analysis they use is valid nly if the

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initial spin rate f the planet is less than half that f Jupiter. Besides, thisanalysis neglects the effect f r tati n n the tides, which is imp rtant f rnear-synchr n us planets.

In the c ntext f extras lar planets we first c mment n the magni-tude f the perturbed vel city induced by the tides at the stellar surface.Terquem et al. (1998 ) have f und that, in the case f 51 Peg, this vel c-ity is t small t be bserved. This result is insensitive t the magnitude fthe stellar turbulent visc sity and is n t affected by the p ssibility f res -nance. It als h lds f r the ther extras lar planets that have been detecteds far.

As indicated ab ve, the r tati n f planets n a cl se rbit is alm stcertainly synchr n us with the rbital rev luti n. F r the star t be syn-chr nized in less than 5 Gyr (i.e., 5 Gyr), a 1-(3)-Jupiter-mass planetw uld have t be n an rbit with a peri d less than 1.4 (3) day(s). We n tethat in the case f B tis, the r tati n f the star may then have been syn-chr nized as a result f tidal effects. If nly the c nvective envel pe f thestar is spun up, these peri ds have t be multiplied by ab ut a fact r f 2.Theref re, the bservati n fan mal us rapidr tat rsc uldgivefurtherev-idence f the presence f cl se planets and pr vide s me indicati n n h wthe dissipati n f tides affects the r tati n f the star (Marcy et al. 1997).

The circularizati n timescale f r the rbit is such that 1/1/ 1/ 1/ . We n te that, acc rding t the expressi n f thetimescales we have given ab ve, 6 and 10 , s thatcircularizati n can be achieved with ut a significant rbital decay takingplace (in c ntrast t the statement by Rasi et al. 1996). S far it is n tclear which f the timescales , , and is the sh rtest. F r 51 Peg,

requires 2 10 (see als Rasi et al. 1996 and Marcy etal. 1997 f r estimates f ). We p int ut that f r a Jupiter-mass planet,

5 Gyr f r 3 days. Theref re, all the Jupiter-mass planets r-biting with a peri d less than ab ut 3 days sh uld be n a circular rbit.This cut ff peri d is a l wer limit because is an upper limit f thecircularizati n timescale.

Planets f und n eccentric rbits at smaller peri ds are indicative thatthe stellar r tati n frequency (assumed zer up t n w) is large c mparedt the rbital frequency. Indeed, in that case, tidal fricti n in the star pumpsup the rbital eccentricity, pp sing the effect f tidal fricti n in the planet.H wever, the ab ve calculati ns indicate that the planet may have t ben a very cl se rbit f r the tidal fricti n in the star t be able t increase

the rbital eccentricity significantly. Whether the planet can get t such acl se rbit is questi nable if the disk in which the planet f rmed did n textend d wn t .

The timescale n which the rbit decays is . The c nditi n 5Gyr requires 2 days f r a Jupiter-mass planet. If a planet has n t beenable t get t such a cl se rbit (e.g., because the inner parts f the disk

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were truncated), rbital decay may n t ccur as a result f tidal fricti n,and the planet may n t plunge int its parent star. N te t that a giantplanet that spirals in t wards the star may l se mass thr ugh R che l beverfl w (Trilling et al. 1998). Apart fr m reducing the planetary mass,

c nservati n f angular m mentum results in an utward t rque n therbit, which sl ws the inward migrati n. A l w-mass planet in a very smallrbit may result in this way.

H wever, several planets may attain sh rt peri ds. After the firstb rnpr t planet emerges and migrates t a few , subsequently f rmed pr -t planets (beginning farther ut because the f rmati n timescale increaseswith ) may migrate inward, pushing pr t planetary c res ahead f theirinward path, until they bec me trapped at Lindblad res nances. Such res -nant trapping is f und f r the Galilean satellites (e.g., G ldreich and Peale1966; Lin and Papal iz u 1979 ) and it ccurs if the migrati n rate is suf-ficiently sl w. It raises the intriguing p ssibility that planets temp rarilyparked cl se t the star because f absence f disk material c uld be f rcedt plunge int the h st star.

On the ther hand, if the star is a rapid r tat r (with peri d sh rter thanthat f the rbit) and has a weak magnetic field, tidal effects transfer angu-lar m mentum fr m the star t the rbit, with increasing efficiency as therbit is pushed ut, until a tidal barrier is pr duced such that the migrati n

time and inverse rbital decay time balance. Tentatively, using estimatesderived fr m equati n (7), an inward migrati n time f 10 yr c uld bebalanced by tidal effects acting n a 1-M planet nly if the peri d were

0.3 days, which increases t 0.7 days f r a 3-M planet and a migra-ti n time f 10 yr. If planets can survive at such sh rt peri ds (m re likelyf r massive planets and sl w migrati n rates) and if eccentricity gr wthis suppressed by tides within the planet, tidal effects might temp rarilycause the halting f migrati n by transferring angular m mentum fr mthe star t the inner rbit and fr m there t any residual disk material viaLindblad t rques and s n t uter planets (cf. the situati n f r Saturn’srings). Such pr cesses have n t yet been w rked ut in detail. H wever,a s lar-type star is likely t have a small am unt f angular m mentumin c mparis n t the disk/planet system in t tal, and nce the star sl wsd wn sufficiently, the inward migrati n must c ntinue.

As a planet eventually bec mes engulfed by the h st star, it is dis-rupted by tidal breakup, heating, and ram pressure stripping (Sandquist etal. 1998). Supp rting evidence f r such phen mena may be f und in thesupers lar metal abundance f 51 Pegasi (G2 V), 55 Cancri (G8 V), and

B tis (F7 V) (Butler et al. 1997; G nzalez 1997, 1998). The c nvectiveenvel pe f r each f these stars c ntains a few 10 M . The mixing f10–40 M f planetary heavy element material within the stellar envel pe(Zhark v and Gudk va 1991) w uld lead t a significant metal enrichmentthere. Because the depth f the c nvecti n z ne f main-sequence starsdecreases significantly with increasing stellar mass, the r utine engulfing

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IV. LONG-TERM STABILITY OF PLANETARY SYSTEMS


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f planets by their h sts might lead t a tendency f r h tter planetary h stst sh w a general verall metallicity enhancement with respect t c lernes.

In a relatively massive disk, several giant planets may be f rmed with1–3 M and 1 AU (Lin and Ida 1997). Their l ng-term r-

bital stability determines the dynamical ev luti n f the system. After thedepleti n f the disk gas, mutual gravitati nal perturbati n between theplanets may gradually increase their eccentricities until their rbits cr sseach ther n a timescale . Extrap lati ns f existing numerical results(e.g., Franklin et al. 1990; Chambers et al. 1996) give 10 yr f rthe s lar system. H wever, w uld reduce t 10 r 10 yr if allthe gas giants had 1 r 2 M , respectively. M re recent simula-ti ns and cr ssing times f r subsets f planets, c nfigured as in the s larsystem but with a reduced s lar mass, are given in Duncan and Lissauer(1998). These auth rs find that the rbits f the giant planets remain stableunder the expected s lar mass l ss f r up t 10 yr r m re. The m re thecentral mass is reduced, the sh rter is .

Since systems f massive planets f rmed with similar values f andsmall eventually suffer rbit cr ssing (Lin and Ida 1997), massive eccen-tric planets may have acquired their rbital pr perties as a c nsequence frbit cr ssing (Rasi and F rd 1996; Weidenschilling and Marzari 1996;

Lin and Ida 1997; Levis n et al. 1998). Once rbit cr ssing ccurs, cl seenc unters eventually take place. When these inv lve equal-mass plan-ets, they pr duce a vel city perturbati n with magnitude limited by thesurface escape vel city 60( /M ) ( /1 g cm ) km s ,where is the planet’s internal density (Safr n v 1969). This results in

/( ) 2( /M ) ( /1 AU) . Thus, bserved eccentrici-ties can be acc unted f r.

Supp sing that giant planets f rm at 1 AU, massive planets (5 M ) with m derately high 0 3 and m derately small 0 5 AUcan be acc unted f r by rbit cr ssing and merging. Eccentric planets with

1 AU and M can be pr duced by rbit cr ssing f ll wedby ejecti n. Merger, resulting in 0 5, is fav red at small , whereasejecti n ccurs preferentially at large , because the cr ss secti n f r directc llisi ns is independent f whereas that f r cl se enc unters is( ) (Lin and Ida 1997). Lin and Ida (1997) find that the rbitalpr perties f a merged b dy are c nsistent with th se f the planets ineccentric rbits ar und 70 Virginis and HD 114762.

Numerical simulati ns f systems with many planets indicate that al-th ugh s me may be ejected, a residual p pulati n f eccentric planets( 0 5) may remain b und t the central star at large distances (100 AU). Cl se enc unters als excite the relative inclinati n up t /2

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V. EFFECTS OF SECULAR PERTURBATIONSDUE TO A DISTANT COMPANION

A. Kozai Effect


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radians. Detecti n f additi nal c mpani ns ar und stars with an eccentricplanet w uld pr vide tests f these scenari s.

Rasi and F rd (1996) suggested that the sh rt-peri d planets weredynamically scattered int a regi n cl se t the central star. Alth ugh thesh rt-peri d planets may be circularized (see secti n III), numerical sim-ulati ns indicate that the scattering scenari w uld lead t a p pulati n fplanets with high eccentricity at large distances (10–100 AU) fr m theircentral stars. A c mprehensive search f r these planets w uld pr vide auseful test f this scenari as against that f disk-planet interacti n, whichpr duces rbits with smaller eccentricity.

S me extras lar planets are f und in binary systems. F r example, a planetis f und ar und 16 Cyg B that rbits at a distance f 1.7 AU fr m thecentral star, which is kn wn t have a binary c mpani n, 16 Cyg A. Ithas been suggested that the high eccentricity ( 0 67) f 16 Cyg B isexcited by the gyr sc pic perturbati n due t 16 Cyg A (H lman et al.1997). A similar effect may be caused by the gravitati nal perturbati nsfr m ther planetary b dies. Here we discuss the effect f secular perturba-ti ns n a l ng timescale such that a time average may be perf rmed. Therbiting b dies may then be c nsidered as having their mass distributed

c ntinu usly ar und their rbits as in the classical the ries f Laplace andLagrange (see Hagihara 1972 and references therein). Semimaj r axes dn t change under secular perturbati n. H wever, changes t the eccentric-ity and inclinati n may be pr duced.

If rbits are c planar, in general nly m dest changes t eccentricitiesare pr duced if the perturbing b dies are widely separated. An excepti nt this might ccur if secular res nances sweep thr ugh the system be-cause f a changing gravitati nal p tential due t disk dispersal r chang-ing stellar blateness (e.g., Ward 1981). H wever, if rbits are all wed thave high inclinati n, large eccentricity changes can ccur t the rbit fan inner planet perturbed by a distant b dy r binary c mpani n.

T c nsider this gyr sc pic perturbati n effect, let us c nsider the sim-plest case f the m ti n f an inner planet with mass perturbed byan uter c mpani n with mass assumed t be in a circular rbit withsemimaj r axis . The uter c mpani n is assumed t c ntain sufficientangular m mentum that its rbit remains fixed, defining a reference planet which the inner planet rbit has inclinati n . We n te , the l ngitudef the apsidal line measured in the rbital plane fr m the line f n des.

If the distance f fr m is , its speed is , the angle betweenits p siti n vect r and that f is , and nly the d minant quadrup le

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term in the interacti n p tential multip le expansi n is retained, its m ti nis g verned by the Hamilt nian (K zai 1962; Hagihara 1972; Innanen etal. 1997):

13 c s 1 (11)

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The fact that b th and the c mp nent f angular m mentum parallel tthe uter rbital axis, c s (1 ) c s , are c nstant en-ables eliminati n f and . A c mplete s luti n can then be f und fr mthe equati n f r the rate f change f , which can be derived fr m thecan nical equati n / / in the f rm

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with 2 / being the rbital peri d f the planetThe ab ve equati n indicates that can scillate between extremes

ccurring when sin 2 0. There has been particular interest in find-ing c nditi ns under which can start fr m very small values and thenincrease unstably t values cl se t 1. T examine c nditi ns f r this tccur, we use c nst and c s c nst t find and in termsf . Assuming initial values and when , f r in-

finitesimally small but n nzer values f that can be neglected, we find

1 5 c sc s 2 (14)

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Clearly, f r s luti ns f the type we seek, we require c s 2 1 whichrequires that the initial inclinati n exceed a critical value (see Innanen etal. 1997) such that

c s (16)

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If this inequality is satisfied (if it is n t, then cann t be neglected),equati n (13) indicates that gr ws fr m a very small value up until thevalue btained by setting /2 namely

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attained when c s . A range f eccentricities may be generated inthis way. H wever, values cl se t unity may be attained f r initial in-clinati ns cl se t 90 . There are then s luti ns that have large-amplitudescillati ns in eccentricity (K zai effect), and adding the n nsecular terms

back in can lead t cha tic behavi r. The large eccentricity changes ccurindependently f the size f the perturbati n. H wever, the characteristictimescale f r the eccentricity changes t ccur is given fr m equati n (13)as ( )/( ) rbital peri ds f the inner planet, which is l ng f rsmall perturbati ns.

N te, h wever, that the effect requires the m ti n f the apsidal anglebe g verned nly by the perturbati n c nsidered. Other effects may

disrupt this, such as general-relativistic c rrecti ns (e.g., H lman et al.1997) and the blateness f the central star. B th c uld be c nsidered inthe f ll wing discussi n, but here we limit urselves t c nsidering theeffects f relativistic apsidal precessi n, which may lead t c nstraints nthe rbital elements f planets with sh rt peri ds.

T inc rp rate relativistic apsidal precessi n t l west rder, we m difythe Newt nian p tential such that Keplerian elliptic rbits are induced tprecess at the c rrect rate. We m dify the p tential due t the central masssuch that

31 (18)

with being the speed f light. The additi nal term can be added int theHamilt nian and averaged s that equati n (12) bec mes

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2 (1 )

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Here the parameter

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measures the imp rtance f relativistic precessi n relative t the pertur-bati n due t the uter c mpani n. Fr m this, if , the effect is c m-pletely suppressed. When the K zai effect ccurs, the maximum p ssibleeccentricity that can be generated is btained by setting /2 and

90 . This gives

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Here, and den te the rbital peri ds f and , respectively,and / .

Planets with peri ds less than 3–4 days are likely t have their rbitscircularized as a result f tidal effects (see ab ve). Fr m the requirement

, we see that f r inner planets with s mewhat larger peri ds (but stilln the rder f days), binary c mpani ns with mass rati f rder unity

need rbital peri ds sh rter than ab ut 40 yr in rder t pump significanteccentricity. If a massive planet with 10 is c nsidered, it must rbitwith a peri d ten times sh rter, 4 yr. Such an bject sh uld be readilydetectable.

We c mment that f r a given distant binary c mpani n with inclina-ti n cl se t 90 t the planet rbit, and n ther perturbers, there is aplanet with a peri d l ng en ugh such that the K zai effect c uld pr duceeccentricities cl se en ugh t unity that the planet has arbitrarily cl seappr aches t the central star. Then tidal effects may act t wards circular-izati n, s decreasing the distance t ap apse until a circular rbit is pr -duced. The final peri d w uld have t be a few days. Alternatively, sh uldthe cl sest appr ach t the star be bey nd the largest radius at which r-bits can be circularized, the rbit w uld retain a K zai cycle with highmaximum eccentricity.

Here we summarize the vari us pr cesses discussed in this chapter, theirinfluence n the ev luti n f planetary systems, and their bservati nalimplicati ns.

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Acknowledgments

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1. excites p tentially bservable spiraldensity waves and can create clean gaps in the disk, limits mass gr wth,and drives rbital migrati n. Gr wth limitati n by gap f rmati n, t getherwith inward migrati n, may naturally acc unt f r the upper limit in thebserved mass rati distributi n f extras lar planets, pr vided they were

f rmed in disks with 0 1 and 10 . Because the critical massf r gap f rmati n is an rder f magnitude larger than f r dynamical gasaccreti n, a bim dal mass distributi n with a depressi n between thesemasses may be anticipated. After gap f rmati n, massive pr t planetsare expected t migrate with the visc us ev luti n f the disk until theyenc unter a stellar magnet spheric cavity r tidal barrier. The existencef sh rt-peri d planets ar und classical T Tauri stars w uld imply that

the l cal disk visc us ev luti n timescale is sh rter than the typical age( 10 yr). Orbital migrati n may lead t stellar c nsumpti n f pr t plan-ets with surviving planets l cked int c mmensurable rbits. C ntamina-ti n f h st stars (during the latter’s main-sequence ev luti n) by plungingplanets may have led t a relatively high metallicity. Planet-disk interac-ti n is likely t excite nly small 0 3 and lead t a relati n between

and that c uld be bserved. A planet with larger eccentricity w uldverrun the gap and induce n nlinear dissipati n.

2. sh uld rapidly synchr -nize their spin with that f the rbit. Sh rt-peri d planets with Mand 3–4 days (e.g., B ) sh uld have their rbits circularized by

within a few Gyr. In the case f mas-sive planets, the stellar r tati n may als be synchr nized with the rbit.Only very sh rt-peri d planets ( 2 days), h wever, are expected tunderg significant rbital decay.

3. with initiallysmall eccentricity leads t rbit cr ssing n a timescale that is sensitivelydetermined by their masses and initial separati ns. Subsequent cl se en-c unters between c mparable masses can lead t high rbital eccentricityand planets scattered int extended rbits. This mechanism may pr videa supply f planets with a range f sh rt peri ds, s me f which may un-derg tidal circularizati n r plunge int the star, t gether with utlyingacc mplices. The latter c uld be imaged by the next generati n f IR in-terfer meters.

4. , r massive utlying plan-etary, c mpani ns in relatively highly inclined rbits can als excite higheccentricity thr ugh the K zai mechanism. P tentially this pr cess mightpr duce a supply f sh rt-peri d planets ( 2–3 days) that have under-g ne tidal circularizati n. Because f the very high relative inclinati nsrequired, this mechanism is likely t w rk nly in a small number f cases.

This w rk is supp rted by NSF and NASA thr ughgrants AST-9618548, NAG5-4277, and NAG5-7515.

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REFERENCES

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