Obser atoire de la Coˆte d’Azur v Institut fu¨r Astronomie ... · II. INTERIORS OF THE GIANT PLANETS 1082 G. WUCHTERL ET AL. vv the total mass of solar-composition material needed

I. INTRODUCTION

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Giant planet f rmati n is cl sely interrelated with star f rmati n, pr t planetarydisks, the gr wth f dust and s lid planets in th se nebula disks, and finally nebuladispersal. M dels f the interi rs and ev luti n f giant planets in ur s lar systemp int t a bulk enrichment f heavy elements m re than a fact r f 2 ab ve s larc mp siti n and imply heavy element c res up t 17 times the mass f Earth.Detailed m dels f giant planet f rmati n explain the diversity f s lar system andextras lar giant planets by variati ns in the c re gr wth rates caused by a c uplingf the dynamics f planetesimals and the c ntracti n f the massive envel pes int

which they dive, as well as by changes in the hydr dynamical accreti n behavi rf the envel pes resulting fr m differences in nebula density, temperature, andrbital distance.

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Institut fur Astronomie der Uni ersitat Wien

Obser atoire de la Cote d’Azur

NASA-Ames Research Center

GIANT PLANET FORMATION

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¨GUNTHER WUCHTERL

TRISTAN GUILLOT

and

JACK J. LISSAUER

Our f ur giant planets c ntain 99.5% f the angular m mentum f thes lar system but nly 0.13% f its mass. On the ther hand, m re than99.5% f the mass f the planetary system is in th se f ur largest b d-ies. The angular m mentum distributi n can be underst d n the basisf the “nebula hyp thesis” (Kant 1755), which assumes c ncurrent f r-

mati n f a planetary system and a star fr m a centrifugally supp rtedflattened disk f gas and dust with a pressure-supp rted central c nden-sati n (Laplace 1796; Safr n v 1969; Lissauer 1993). The retical m delsf the c llapse f sl wly r tating m lecular cl ud c res have dem n-

strated that such preplanetary nebulae are the c nsequence f the bservedcl ud c re c nditi ns and the hydr dynamics f radiating fl ws, pr videdthere is a macr sc pic angular m mentum transfer pr cess (chapter bySt ne et al., this v lume; Cassen and M sman 1981; M rfill et al. 1985;Laughlin and B denheimer 1994; P d sek and Cassen 1994). Assuming

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II. INTERIORS OF THE GIANT PLANETS

1082 G. WUCHTERL ET AL.

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the total mass of solar-composition material neededto pro ide the obser ed planetary/satellite masses and compositions bycondensation and accumulation

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turbulent visc sity t be that pr cess, dynamical m dels have sh wn h wmass and angular m mentum separate by accreti n thr ugh a visc us disknt a gr wing central pr t star (Tscharnuter 1987; Tscharnuter and B ss

1993; see chapter by St ne et al., this v lume, f r visc sity mechanisms).Th se calculati ns, h wever, d n t yet reach t the ev luti nary state fthe nebula where planet f rmati n is expected. Observati nally inferreddisk sizes and masses are verlapping the retical expectati ns and f rtifythe nebula hyp thesis. High-res luti n bservati ns at millimeter wave-lengths are n w sensitive t disk c nditi ns at rbital distances 50 AU(see, e.g., chapter by Wilner and Lay, this v lume; Dutrey et al. 1998;Guill teau and Dutrey 1998). H wever, bservati ns thus far pr vide lit-tle inf rmati n ab ut the physical c nditi ns in the respective nebulae nscales f 1 t 40 AU, where planet f rmati n is expected t ccur.

Planet f rmati n studies theref re btain plausible values f r disk c n-diti ns fr m nebulae that are rec nstructed fr m the present planetary sys-tem and disk physics. The s btained “minimum rec nstituted nebulamasses,” defined as

, are a few percent f the central b dy,b th f r the s lar nebula and f r the circumplanetary pr t satellite nebulae(Kusaka et al. 1970; Hayashi 1980; Stevens n 1982 ). The t tal angularm menta f the satellite systems, h wever, are nly ab ut 1% f the spinangular m menta f the respective giant planets (P d lak et al. 1993), instr ng c ntrast with the planetary system/Sun rati . Assembling planetsfr m a nebula disk and advecting the angular m mentum due t Keple-rian shear until the present giant planet masses are reached results in t talangular m menta verestimating the present spin angular m menta f the

¨giant planets nly by small fact rs (G tz 1993). Even if giant planets hadkept this angular m mentum, they still w uld n t r tate critically! Giantplanets, unlike stars, theref re d n t have an angular m mentum pr blem.This may justify why m st studies f pr t -giant planets neglect r tati nr treat it as a small perturbati n.

We discuss new results n interi r m dels in secti n II. We reviewrecent w rk n planetesimal f rmati n and gr wth f s lid planets in sec-ti n III. The “nucleated instability hyp thesis” is the nly m del f r thef rmati n f Uranus and Neptune at the m ment, whereas ther m delsals exist f r Jupiter and Saturn; in secti n IV, we review these vari usm dels. We put emphasis n envel pe ev luti n and gas accumulati n us-ing the “nucleated instability” m del in secti n V. We apply the f rmati nthe ries t extras lar planets in secti n VI.

Our kn wledge f the mechanisms that led t the f rmati n f the giantplanets is essentially based n numerical m dels and n the c nstraintspr vided by studies f the internal structure and c mp siti n f Jupiter,

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GIANT PLANET FORMATION 1083

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Saturn, Uranus, and Neptune. This inv lves the calculati n f interi rm dels matching the bserved gravitati nal fields. Each f the f ur giantplanets f ur s lar system are thus believed t c nsist f a central, densec re and a surr unding envel pe c mp sed f hydr gen, helium, and smallam unts f heavy elements. The c res f Jupiter and Saturn are very smallc mpared t the t tal masses f the planets, whereas Uranus and Neptuneare m stly c re and p ssess small (i.e., l w-mass) envel pes.

The giant planets, with the excepti n f Uranus, emit significantlym re energy than received fr m the Sun, a c nsequence f their pr gres-sive c ling and c ntracti n. Tw imp rtant c nsequences can be drawnfr m this:

1. They have inner temperatures f a few th usand kelvins r m re;theref re, their hydr gen-helium envel pes are .

2. They are m stly c nvective (see Hubbard 1968; Stevens n andSalpeter 1977). The c nvective hyp thesis has been challenged(Guill t et al. 1994 ), but the regi ns where c nvecti n c uld besuppressed due t radiative transp rt are limited t a small fracti nf the envel pe, at temperatures between 1500 and 2000 K, r in

l w-temperature regi ns where the abundance f water is small.

These tw c nclusi ns are als expected t h ld f r Uranus f r essentiallytw reas ns: First, it is highly unlikely that its interi r has c led muchm re than that f Neptune (thus, ne can expect that its intrinsic heat fluxis small but larger than zer ; see Marley and McKay 1999); sec nd, itp ssesses a magnetic field f similar strength t that f the ther giantplanets, a sign f c nvective activity in its interi r.

It seems, theref re, l gical t assume that the envel pes f all f urgiant planets are h m gene usly mixed. S me caveats are necessary,h wever:

1. C ndensati n and chemical reacti ns alter chemical c mp siti n(these sh uld be c nfined t the external regi ns).

2. A first- rder phase transiti n (such as the ne between m lecular andmetallic hydr gen) imp ses an abundance disc ntinuity acr ss itself.

3. Hydr gen-helium phase separati n might ccur and lead t a variati nf the abundance f helium in the planet.

4. The envel pes f Uranus and Neptune are small and enriched in heavyelements; it is thus c nceivable that m lecular weight gradients in-hibit c nvecti n and yield n nh m gene us envel pes.

On this basis, a three-layer structure is generally ad pted f r the f urgiant planets (Fig. 1). In the case f the less massive Uranus and Nep-tune, in which hydr gen is believed t remain in m lecular phase, theplanets are divided int a “r ck” c re (a mixture f the m st refract ryelements including silicates and ir n), an “ice” layer (c nsisting f H O,CH , and NH ) and a hydr gen-helium envel pe. The latter is substan-tially enriched in heavier elements, as dem nstrated by the 30 times

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Figure 1. The interi rs f Jupiter, Saturn, Uranus, and Neptune, acc rding t thec nventi nal wisd m. The sizes and blatenesses f the planets are representedt scale. Inside Jupiter and Saturn, hydr gen, which is in m lecular f rm (H )at l w pressures, is th ught t bec me metallic in the 1- t 3-Mbar regi n. Thistransiti n c uld be abrupt r gradual. The equati n f state is very uncertainf r a substantial p rti n f the interi rs f b th planets. Uranus and Neptune,c ntain, in a relative sense, m re heavy elements. There are indicati ns thattheir interi rs are partially mixed (see text).

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s lar enrichment in carb n, in the f rm f CH , spectr sc pically mea-sured in the tr p spheres f Uranus and Neptune (e.g., Fegley et al. 1991;Gautier et al. 1995). Other elements, in particular xygen (m stly in thef rm f H O), are als believed t be substantially enriched c mpared t as lar-c mp siti n mixture but are hidden deep in the atm sphere becausef c ndensati n.

Alth ugh the tw planets share many similarities (mass, magneticfield, atm spheric structure), several fact rs p int t ward s me differencesin their internal structure. Uranus emits scarcely m re energy than re-ceived fr m the Sun, whereas Neptune p ssesses a very significant in-trinsic heat flux, and Uranus’s gravitati nal field indicates that it is m recentrally c ndensed (see Hubbard et al. 1995). Furtherm re, three-layer

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m dels f these planets that assume h m geneity f each layer and adia-batic temperature pr files succeed in repr ducing Neptune’s gravitati nalfield but n t that f Uranus (P d lak et al. 1995). The difficulty is circum-vented by using slightly reduced (by 10%) densities in the ice layer,which is interpreted either as hydr gen mixed t the ice r as higher tem-peratures (superadiabatic temperature gradients). B th explanati ns im-ply that substantial parts f the planetary interi r are n t h m gene uslymixed. The existence f such c mp siti nal gradients c uld als explainthe fact that Uranus’s heat flux is s small: part f its internal heat w uldn t be all wed t escape t space by c nvecti n, but had t escape thr ugha much sl wer diffusive pr cess in the regi ns f high m lecular-weightgradient (P d lak et al. 1991). Such regi ns w uld als be present in Nep-tune, but deeper, thus all wing m re heat t be transp rted utward. Thisc uld als explain the fact that the magnetic fields f these tw planetsp ssess a very significant quadrup lar c mp nent, by all wing a hydr -magnetic dynam t f rm nly in a relatively thin shell rather than in asphere (Ruzmaikin and Starchenk 1991; Hubbard et al. 1995).

The existence f these n nh m gene us regi ns is further c nfirmedby the fact that if hydr gen is supp sed t be c nfined s lely t thehydr gen-helium envel pe, m dels predict ice/r ck rati s f the rder f10 r m re, much larger than the pr t s lar value f 2.5. On the therhand, if we imp se the c nstraint that the ice/r ck rati is pr t s lar, theverall c mp siti n f b th Uranus and Neptune is, by mass, ab ut 25%

r ck, 60–70% ices, and 5–15% hydr gen and helium (Hubbard and Mar-ley 1989; P d lak et al. 1991, 1995; Hubbard et al. 1995). The f rmati nf these n nh m gene us regi ns is certainly c ntemp rane us with the

accreti n f the planets (Hubbard et al. 1995). The imp rtance f st chas-tic pr cesses during that ep ch is sh wn by the 98 bliquity f Uranus,a str ng sign that giant impacts shaped the actual structure f these icegiants (Lissauer and Safr n v 1991; cf., h wever, Tremaine 1991 f r analternative explanati n f giant planet bliquities).

The structure f the much m re massive Jupiter and Saturn, whichare m stly f rmed fr m hydr gen and helium, is c mparatively simpler.M st interi r m dels (Hubbard and Marley 1989; Chabrier et al. 1992;Guill t et al. 1994 ) f these planets assume a three-layer structure: a c re,an inner envel pe where hydr gen is in metallic phase, and an uter newhere hydr gen is m stly in the f rm f H . M re c mplex m dels (e.g.,Zhark v and Gudk va 1991) can be calculated, but these further divisi nsint multiple layers d n t qualitatively affect the main results.

Each layer is assumed t be gl bally h m gene us (i.e., neglectingc ndensati n and chemical reacti ns), a c nsequence f efficient mixingby c nvecti n. Because less helium is bserved in the external layers fJupiter and Saturn than was present in the pr t s lar nebula (v n Zahnet al. 1998; Gautier and Owen 1989), it is believed that the metallic re-gi ns f these planets c ntain m re helium than the m lecular nes. Thedifference is th ught t be due t a first- rder m lecular-metallic phase

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transiti n f hydr gen, a hydr gen-helium phase separati n, r b th (e.g.,Stevens n and Salpeter 1977; Hubbard and Marley 1989). B th phen m-ena are expected t ccur in similar regi ns (e.g., Stevens n 1982 ), andtheref re we d n t differentiate ne fr m the ther. We emphasize, h w-ever, that a first- rder transiti n such as that suggested by Saum n andChabrier (1989), w uld lead t a disc ntinuity f abundance f all chemi-cal elements, whereas a phase w uld principally affect heliumand ther min r species, such as ne n, that tend t be diss lved int heliumdr plets (unless, e.g., water is present in large en ugh abundances and canals separate fr m hydr gen). The lack f ne n measured by thePr be (Niemann et al. 1998) suggests that, in Jupiter, helium phase sepa-rati n has begun (R ulst n and Stevens n 1995), and that, c nsequently,it als ccurs in Saturn, which is c lder.

With these hyp theses, Guill t et al. (1997) and Guill t (1999) cal-culate the ensemble f interi r m dels f Jupiter and Saturn that matchthe gravitati nal m ments within the err r bars f the measurements.Using the inferred mass mixing rati f helium in the pr t s lar nebulaand the presently bserved ne in the atm spheres f Jupiter and Saturn,they retrieve the p ssible abundances f heavy elements in the metal-lic and m lecular regi ns. Their calculati ns include uncertainties in thehydr gen-helium and heavy elements equati ns f state, in the inner tem-perature pr file (c nvective/radiative), and regarding the internal r tati n(s lid/differential).

The resulting c nstraints n the c re mass and t tal mass f heavyelements in Jupiter, Saturn, Uranus and Neptune are summarized in Fig.2. (The cases f Uranus and Neptune are relatively trivial; these planetsc ntain little hydr gen and helium.) A first result is that the gravitati nalfields f Jupiter and Saturn d n t necessarily imply that these planetshave ice/r ck c res. In the case f Jupiter, m dels with ut a c re are b-tained nly in the case f the less fav red interp lated equati n f state fhydr gen, wh se calculati n is n t c mpletely therm dynamically c n-sistent (see Saum n et al. 1995). In the case f Saturn, it is difficult tdistinguish between heavy elements in the c re and th se in the metallicregi n, hence yielding an even larger uncertainty in the c re mass. As aresult, Jupiter has a c re wh se mass lies between 0 and 14 M , and Sat-urn’s c re is between 0 and 22 M . We stress, h wever, that larger c remasses are p ssible if gravitati nal layering ccurs and the c res, p ssiblyer ded by c nvective mixing, extend int the metallic envel pe.

A sec nd result c ncerns the t tal mass f heavy elements. The m delsf Saturn sh w that the planet is significantly enriched in heavy elements,

by a fact r f 10 t 15 c mpared t the s lar value (c rresp nding t 20–30 M , including the c re) and by at least a fact r f 5 when c nsideringnly the envel pe. The c nstraints are much weaker in the case f Jupiter

because f the larger metallic regi n, where the equati n f state is c nsid-erably m re uncertain. Figure 2 sh ws that Jupiter c ntains 10 t 42 M

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Figure 2. Limits n the abundances f heavy elements in the f ur j vian planetsin ur s lar system. F r each planet, the p int n the left represents the t talam unt f high- material, whereas the (l wer) p int n the right sh ws theam unt f heavy elements segregated int the planet’s c re. F r Jupiter andSaturn, the thick lines represent s luti ns with additi nal c nstraints btainedfr m ev luti n m dels. N te the high level f uncertainty, especially regardingthe c re masses f Jupiter and Saturn. M dels f Jupiter with small c res (i.e.,less than 2 M ) require significant enrichments in heavy elements (i.e., m rethan 20 M ).

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f heavy elements, implying that it is m derately t significantly enrichedin heavy elements c mpared t the pr t s lar nebula.

Recent interi r m dels calculated with different assumpti ns (Hub-bard and Marley 1989; Zhark v and Gudk va 1991; Chabrier et al. 1992)generally predict c re masses and heavy-elements abundances that fallwithin the ranges given in Fig. 2. Larger c re masses (10–30 M ) weref und in previ us calculati ns (see Stevens n 1982 f r a review), but thelargest c re masses als yielded helium mass fracti ns well bel w the pr -t s lar value and theref re are unrealistic. The main reas n f r the discrep-ancy with t day’s values is, h wever, that the calculati n f c re masses,especially in the case f Jupiter, is very sensitive t changes in the equati nf state. At present, we can nly h pe that advances in ur understandingf the behavi r f hydr gen and helium at high pressures have led us in

the right directi n. Pr gress in c mpressi n experiments n liquid deu-terium (Weir et al. 1996; C llins et al. 1998) sh uld all w us t check thatasserti n in the near future.

In the case f Jupiter and Saturn, further c nstraints n t day’s in-ternal structure can be s ught fr m ev luti n m dels that acc unt f r thepr gressive sedimentati n f helium (Hubbard et al. 1999; Guill t 1999).M dels with small c res tend t require a m re pr n unced helium dif-ferentiati n and theref re yield l nger c ling times. The time t c lt the present temperature is c nstrained by the age f the s lar system

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III. FORMATION OF PLANETESIMALSAND GROWTH OF SOLID PLANETS

A. Formation of Planetesimals


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(4.56 Gyr). S me static s luti ns are thus ruled ut. Figure 2 sh ws thatan upper limit f 10 M is btained f r Jupiter’s c re, and that Saturn’sc re mass lies between 6 and 17 M .

Finally, bservati ns f the atm spheric c mp siti n give tw imp r-tant clues: First, the C/H rati steadily increases fr m Jupiter t Neptune,a fact that has t be explained by f rmati n m dels (we refer the reader tthe review by P d lak et al. 1993 f r further details). Sec nd, the D/H is -t pic rati s recently measured in Jupiter by the Pr be (Mahaffy etal. 1998), and in Saturn by the (ISO) (Griffinet al. 1996) are, within the err r bars, c nsistent with the pr t s lar valuederived fr m He/ He in the s lar wind, namely 2 1 0 5 10 (Geissand Gl ecker 1998), whereas they are ab ut three times larger in Uranusand Neptune (Feuchtgruber et al. 1999). This has t be c mpared t theD/H values measured in c mets Halley, Hyakutake, and Hale-B pp, which

áre all ab ut 10 times larger than the pr t s lar value (B ckelee-M rvanet al. 1998). If Uranus and Neptune c ntained mixtures f c metlike icesand pr t s lar H that were is t pically h m genized within these plan-ets, their large ice fracti ns w uld have pr duced a m re deuterium-richatm spheric c mp siti n than that bserved. Thus, either a significant is -t pic exchange between vap rized ices and hydr gen t k place in an earlyh t turbulent s lar nebula (Dr uart et al. 1999), r Uranus and Neptunef rmed fr m high-D/H, c metlike ices that had never been fully mixedwith hydr gen in their interi rs. The precise determinati n f D/H in thegiant planets is thus an imp rtant t l f r c nstraining their f rmati n. Weleave a m re th r ugh discussi n f this pr blem t the chapter by Lunineet al., this v lume.

Even a very sl wly r tating m lecular cl ud c re has far t much r ta-ti nal angular m mentum t c llapse d wn t an bject f stellar dimen-si ns, s a significant fracti n f the material in a c llapsing c re fallsnt a r tati nally supp rted disk in rbit ab ut the pressure-supp rted

star. Such a disk has the same elemental c mp siti n as the gr wing star;that is, primarily H and He, with 1–2% heavier elements. Sufficientlyfar fr m the central star, it is c l en ugh f r s me f this material t be ins lid f rm, either remnant interstellar grains r c ndensates f rmed withinthe disk. Dust aggl merates via inelastic c llisi ns and gradually settlest wards the disk midplane as particles gr w large en ugh t be able t driftrelative t the surr unding gas (Weidenschilling and Cuzzi 1993).

Alth ugh t a first appr ximati n the gas in the disk is centrifugallysupp rted in balance with the star’s gravity, negative radial pressure gra-dients pr vide a small, utwardly directed f rce that acts t reduce the

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B. Growth of Solid Planets


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effective gravity, s the gas r tates at slightly less than the Keplerian ve-l city. Small s lid b dies (dust grains) r tate with the gas. Large s lidb dies rbit at the Keplerian vel city, and medium-sized particles m veat a rate intermediate between the gas vel city and the Keplerian vel city;thus, macr sc pic s lid b dies are subjected t a headwind fr m the gas(Adachi et al. 1976). This headwind rem ves angular m mentum fr mthe particles, causing them t spiral inward t wards the central star. Thisinward drift can be very rapid, especially f r particles wh se c uplingtime t the gas is similar t their rbital peri d. Smaller particles drift lessrapidly because the headwind they face is n t as str ng, whereas largeparticles drift less because they have a greater mass-t -surface-area rati .Orbital decay times f r meter-sized particles at 1 AU fr m the Sun havebeen estimated t be nly 100 years (Weidenschilling 1977). The largeradial vel cities f b dies in this size range, relative t b th larger andsmaller particles, implies frequent c llisi ns, s it is p ssible that m sts lid b dies gr w thr ugh the critical size range quickly with ut substan-tial radial drift. H wever, it is als p ssible that a large am unt f s lidplanetary material is l st fr m the disk in this manner.

S lid b dies larger than 1 km in size face a headwind nly slightlyfaster than meter-sized bjects (f r parameters th ught t be representa-tive f the planetary regi n f the s lar nebula), and because f their muchgreater mass-t -surface-area rati they suffer far less rbital decay fr minteracti ns with the gas in their path. The gr wth f s lid b dies fr mthe meter-sized “danger z ne” t the kil meter-sized “safe z ne” c uldccur by c llective gravitati nal instabilities in a thin subdisk f s lids

(Safr n v 1960; G ldreich and Ward 1973) in regi ns f pr t planetarydisks that are n t t turbulent, r (m re likely) via c ntinued binary ac-creti n (Weidenschilling and Cuzzi 1993). Kil meter-sized planetesimalsappear t be reas nably safe fr m l ss (unless they are gr und d wn tsmaller sizes via disruptive c llisi ns) until s me f these planetesimalsgr w int planetary-sized b dies.

The primary perturbati ns n the Keplerian rbits f kil meter-sized andlarger b dies in pr t planetary disks are mutual gravitati nal interacti nsand physical c llisi ns (Safr n v 1969). These interacti ns lead t accre-ti n (and in s me cases er si n and fragmentati n) f planetesimals. Grav-itati nal enc unters are able t stir planetesimal rand m vel cities up tthe escape speed fr m the largest c mm n planetesimals in the swarm(Safr n v 1969). The m st massive planets have the largest gravitati n-ally enhanced c llisi n cr ss secti ns and accrete alm st everything withwhich they c llide. If the rand m vel cities f m st planetesimals remainmuch smaller than the escape speed fr m the largest b dies, then theselarge “planetary embry s” gr w extremely rapidly (Safr n v 1969; cf.Greenzweig and Lissauer 1990, 1992 f r three-b dy gr wth rates). Thesize distributi n f s lid b dies bec mes quite skewed, with a few large

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IV. GAS ACCUMULATION THEORIES


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b dies gr wing much faster than the rest f the swarm in a pr cess kn wnas (Greenberg et al. 1978; Wetherill and Stewart 1989;K kub and Ida 1996). Eventually, planetary embry s accrete m st f the(sl wly m ving) s lids within their gravitati nal reach, and the runawaygr wth phase ends. Planetary embry s can c ntinue t accumulate s lidsrapidly bey nd this limit if they migrate radially relative t planetesimalsas a result f interacti ns with the gase us c mp nent f the disk (Tanakaand Ida 1999).

The eccentricities f planetary embry s in the inner s lar systemwere subsequently pumped up by l ng-range mutual gravitati nal per-turbati ns; c llisi ns between these embry s eventually f rmed the ter-restrial planets (Wetherill 1990; Chambers and Wetherill 1998). H w-ever, timescales f r this type f gr wth in the uter s lar system (at least10 years; Safr n v 1969) are l nger than the lifetime f the gase us disk(cf. Lissauer et al. 1995). M re ver, unless the eccentricities f the gr w-ing embry s are damped substantially, embry s will eject ne an therfr m the star’s rbit (Levis n et al. 1998). Thus, runaway gr wth, p ssi-bly aided by migrati n (Tanaka and Ida 1999), appears t be the way bywhich s lid planets can bec me sufficiently massive t accumulate sub-stantial am unts f gas while the gase us c mp nent f the pr t planetarydisk is still present (Lissauer 1987).

M st m dels f the accumulati n f giant planet atm spheres haveassumed a c nstant accreti n rate f r planetesimals. The m dels f P llacket al. (1996) calculate the planetesimal accreti n rate t gether with that fgas; h wever, these m dels neglect gr wth f c mpeting planetary c res aswell as radial migrati n. M dels f giant planet gr wth will impr ve nceatm spheric accumulati n m dels are c upled t s phisticated m dels fs lid planet gr wth, such as the multiz ned numerical accreti n c de fWeidenschilling et al. (1997), and when radial migrati n f planetesimalsand planets is better underst d and included in the m dels.

The key pr blem in giant planet f rmati n is that preplanetary disks arenly weakly self-gravitating equilibrium structures, supp rted by cen-

trifugal f rces augmented by gas pressure (see chapters by St ne et al.,H llenbach et al., Calvet et al., and Beckwith et al., this v lume). Anyis lated, rbiting bject bel w the R che density is pulled apart by the stel-lar tides. Typical nebula densities are m re than tw rders f magnitudebel w the R che density, s c mpressi n is needed t c nfine a c ndensa-ti n f mass inside its tidal r Hill radius at rbital distance

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A. Nebula Stability


nucleated instability

disk instability

external perturber

fragmentationduring collapse

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A l cal enhancement f self-gravity is needed t verc me the c unter-acting gas pressure. Giant planet f rmati n the ries may be classified byh w they pr vide this enhancement.

1. The m del relies n the extra gravity field f asufficiently large s lid c re (c ndensed material represents a gain ften rders f magnitude in density, and theref re self-gravity, c m-pared t the nebula gas).

2. A may perate n lengthscales between sh rt-scalepressure supp rt and l ng-scale tidal supp rt.

3. An c uld c mpress an therwise stable disk n itsl cal dynamical timescales, e.g., by accreti n f a clump nt the diskr rendezv us with a stellar c mpani n.

If the gravity enhancement is pr vided by a dynamical pr cess as in thelatter tw cases, the resulting nebula perturbati n (say, f a Jupiter mass,M , f material) is c mpressi nally heated, because it is ptically thickunder nebula c nditi ns. Giant planet f rmati n w uld then inv lve a tran-sient phase f tenu us giant gase us pr t planets, which w uld be essen-tially fully c nvective and w uld c ntract n a timescale f 10 yr (seeB denheimer 1985).

An ther mechanism f f rming stellar c mpani ns,, is plausible f r binary stars and p ssibly br wn dwarfs,

but it is unlikely t f rm bjects f planetary masses, because pacity lim-its the pr cess t masses ab ve 10 M (cf. the chapter by B denheimeret al. in this v lume; and B denheimer et al. 1993).

Preplanetary nebulae with minimum rec nstituted mass are stable. Sub-stantially m re massive disks resulting fr m the c llapse f cl ud c resare self-stabilizing by transfer f disk mass t the stabilizing centralpr t star (B denheimer et al. 1993). Nevertheless, a m derate-mass neb-ula disk might be f und that can devel p a disk instability leading t astr ng density perturbati n, especially when f rced with a finite externalperturbati n. Giant gase us pr t planets (GGPPs) might f rm when theinstability has devel ped int a clump (DeCampli and Camer n 1979;B denheimer 1985). B ss (1997, 1998) has c nstructed such an unstabledisk with 0 13 M within 10 AU and btained maximum density enhance-ments (by a fact r 20) with 10 M ab ve the backgr und f r a fewrbital peri ds. (The density enhancement at the surface f a 1-M c re

is between 10 and 10 , f r c mparis n.) These clumps, pr vided theyare stable n a few c ling times, are candidates t bec me pr t -giantplanets via an intermediate state as tenu us GGPPs.

A key issue, as in any the ry inv lving an instability f the disk gas,is then the f rmati n f a c re. Only metals that are presentinitially w uld rain ut t f rm a c re, whereas material added later by

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B. Nucleated Instability


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impacts f small b dies after the GGPP had f rmed w uld be s luble inthe envel pe (Stevens n 1982 ). B ss (1998) utlines h w a c re c rre-sp nding t the s lar-c mp siti n high- material (6 M and 2 M f rJupiter and Saturn mass, respectively) might f rm if the density enhance-ments are l ng-lived, need n m re pressure c nfinement, and ev lve intGGPPs that are . It sh uld be n ted here (see secti n II) thatalth ugh interi r m dels f Saturn d n t rule ut the p ssibility that theplanet has n c re ( r, equivalently a 2-M c re), this is n t the fav reds luti n. Als , GGPP m dels w uld pr bably predict that Jupiter sh uldhave a bigger c re than Saturn, which is nly marginally c nsistent withpresent interi r m dels. Finally, Jupiter and certainly Saturn c ntain a l tf heavy elements (see Fig. 2). T acc unt f r these bulk heavy element

c mp siti ns, planetesimal accreti n must ccur anyway after the GGPPshave f rmed their c res.

If GGPPs need pressure c nfinement, they als require the presencef an (undepleted) nebula and p se a lifetime c nstraint f r the nebula,

namely that nebula dispersal can begin nly after a c ling time, that is10 yr (B denheimer 1985). T determine whether GGPPs are c nvec-

tively stable, s that the n nturbulent c re gr wth scenari can be applied,a detailed calculati n f their thermal structure during c ntracti n is nec-essary. DeCampli and Camer n (1979) f und largely c nvective GGPPs.

One f us (G. W.) checked c nvective stability f GGPPs by a radi-ati n hydr dynamical calculati n. Alexander and Fergus n (1994) paci-ties and time-dependent MLT c nvecti n were used in the descripti n fenergy transfer. The initial c nditi n was a Jeans-critical nebula c nden-sati n f M and a temperature f 10 K. Initially the GGPP had similarpr perties as B ss’s (1998) banana-shaped density enhancements (meandensity 8 10 g cm , central density 3 3 10 g cm ). Acc rdingt the new calculati n, the GGPP needs 1 8 10 yr t c ntract int thetidal radius and is essentially fully c nvective fr m 100 yr t 2 10 yr,when a radiative z ne spreads ut fr m the planet’s center.

Planetesimals in the s lar nebula are small b dies surr unded by gas. Ararefied equilibrium atm sphere f rms ar und such bjects. Early w rk inthe nucleated instability hyp thesis, which assumes that such s lid “c res”trigger giant planet f rmati n, was m tivated by the idea that at a certaincritical c re mass the atm sphere c uld n t be sustained, and is thermal,sh ck-free accreti n (B ndi and H yle 1944; B ndi 1952) w uld set in.Determinati ns f this critical mass were made f r increasingly detaileddescripti n f the envel pes: adiabatic (Perri and Camer n 1974), is ther-mal (Sasaki 1989), is thermal-adiabatic (Harris 1978; Mizun et al. 1978),and with radiative and c nvective energy transfer (Mizun 1980). By then,m deling the f rmati n and ev luti n f a pr t -giant planet had bec meessentially a miniature stellar structure calculati n, with energy dissipa-

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“the rate of gas accretion is determinednot by the rate of deli ery of mass to the planet [as in Bondi accretion]but by the energy losses from the contracting en elope.”

M

Protostars and Planets III

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ti n f impacting planetesimals replacing the nuclear reacti ns as the en-ergy s urce. Present results n the critical mass are reviewed in the nextsecti n. Already, Safr n v and Rusk l (1982) p inted ut that, even afterthe instability at the critical mass,

The planet’s ac-creti n rate is limited by the delivery f mass nly when 100 M .C nsequently, the energy budget f the envel pe has been m deled m recarefully, taking int acc unt the heat generated by gravitati nal c ntrac-ti n (quasihydr static m dels by B denheimer and P llack 1986).

Maj r pr gress since the c nference hasbeen made by a detailed treatment f planetesimal accreti n t calculatethe c re gr wth rate and the capture, diss luti n, and sinking that deter-mines h w much and where in the envel pe the planetesimal kinetic en-ergy is liberated (P llack et al. 1996). That made p ssible the first study fthe c upling between gas accreti n and s lid accreti n. Additi nally, thedescripti n f the mechanics f c ntracti n has been impr ved by hydr -dynamic studies that determine the fl w vel city f the gas by s lving anequati n f m ti n f r the envel pe gas in the framew rk f c nvectiveradiati n-fluid dynamics (e.g., Wuchterl 1993, 1995 , 1999). That all wsthe study f c llapse f the envel pe, accreti n with finite Mach number,and an access t the study f linear adiabatic (Tajima and Nakagawa 1997)and n nlinear, n nadiabatic pulsati nal stability and pulsati ns f the en-vel pe. Furtherm re, the treatment f c nvective energy transfer has beenimpr ved by calculati ns using a time-dependent mixing length the ry fc nvecti n (Wuchterl 1995 , 1996, 1997) in hydr dynamics. The first hy-dr dynamic calculati ns with r tati n in the quasispherical appr ximati n

¨have been undertaken by G tz 1993.M st aspects f early envel pe gr wth, up t 10 M , can be un-

derst d n the basis f a simplified analytical m del given by Stevens n(1982 ) f r a pr t planet with c nstant pacity , c re mass accreti n rate

, and c re density , inside the tidal radius . The key pr pertiesf Stevens n’s m del c me fr m the “radiative zer s luti n” f r spheri-

cal pr t planets with static, fully radiative envel pes, in hydr static andthermal equilibrium. We present here the s luti n relevant t the structuref an envel pe in the gravitati nal p tential f a c nstant mass, f r zer

external temperature and pressure and using a generalized pacity law fthe f rm . The critical mass, defined as the largest mass twhich a c re can gr w while f rced t retain a static envel pe, is thengiven by

3 1 4 3 4(2)

4 1 34 ln /

where / ; and , , and den te the gas c nstant, the grav-itati nal c nstant, and the Stefan-B ltzmann c nstant, respectively. The

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critical mass depends n neither the midplane density , n r n thetemperature f the nebula in which the c re is embedded. The uterradius, , enters nly l garithmically. The str ng dependence f the an-alytic s luti n n m lecular weight, , led Stevens n (1984) t pr p se“superganymedean puffballs” with atm spheres assumed t be enriched inheavy elements. Such bjects w uld have l w critical masses, pr viding away t f rm giant planets rapidly (see als Lissauer et al. 1995). Equati n(2) permits a glimpse f the effect f the run f pacity via the p wer lawexp nents and . Except f r the weak dependences discussed ab ve,a pr t -giant planet essentially has the same gl bal pr perties f r a givenc re wherever it is embedded in a nebula. Even the dependence n isrelatively weak; detailed radiative/c nvective envel pe m dels sh w thata variati n f a fact r f 100 in leads nly t a 2.6 variati n in thecritical c re mass.

This similarity in the static structure f pr t -giant envel pes yieldssimilar dynamical behavi rs characterized by pulsati n-driven mass l ssf r s lar-c mp siti n nebula pacities (see secti n V.B). H wever, therstatic s luti ns are f und f r pr t planets with uter envel pes,which ccur f r s mewhat larger midplane densities than in minimummass nebulae (Wuchterl 1993). These largely c nvective pr t -giant plan-ets have larger envel pes f r a given c re and a reduced critical c re mass.Their pr perties can be illustrated by a simplified analytical s luti n f rfully c nvective, adiabatic envel pes with c nstant first adiabatic exp -nent, :

1(3)

( 1)4

and / . In this case, the critical mass depends n the nebulagas pr perties and theref re the l cati n in the nebula, but it is indepen-dent f the c re accreti n rate. Of c urse, b th the radiative zer and fullyc nvective s luti ns are appr ximate, because they nly r ughly estimateenvel pe gravity, and all detailed calculati ns sh w radiative c nvec-tive regi ns in pr t -giant planets. In Fig. 3 the transiti n fr m “radiative”t “c nvective” pr t planets is sh wn by results fr m detailed static radia-tive/c nvective calculati ns f r 10 M yr (Wuchterl 1993).Nebula c nditi ns are varied fr m l w densities, resulting in radiativeuter envel pes, t enhanced densities that result in largely c nvective

pr t -giant planets. The critical mass can be as l w as 1 M , and sub-critical static envel pes can gr w t 48 M . Calculati ns with updatedpacity and impr ved, mixing-length c nvecti n (Wuchterl 1999) and the

ïnclusi n f r tati nal effects in the quasispherical appr ximati n (G tz1993) sh w a reducti n f the critical c re mass fr m 13 t 7 M f r the“radiative” pr t -giant planets at the l w nebula densities. The new, l wervalues are in better agreement with the new interi r m dels.

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V. DETAILED NUCLEATED INSTABILITY MODELSFOR THE GIANT PLANETS IN THE SOLAR SYSTEM

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Figure 3. Critical masses f static pr t planets as a functi n f nebula mid-plane density. Critical t tal mass and c re mass values are c nnected by a s lidand a dashed curve, respectively. Observe the increased envel pe masses anddecreased c re masses f r the c nvective uter envel pes ccurring at largernebula densities. The c nditi ns in the nebula c rresp nd t Mizun ’s minimum-mass nebula (Mizun 1980); densities at the Neptune, Uranus, Saturn, andJupiter p siti ns are labeled by N, U, S, and J, respectively. They illustrate thec nstancy f the critical mass in the case f radiative uter envel pes. Den-sities t the right f the d tted vertical line are arbitrarily enhanced relativet the minimum-mass values, s that the uter envel pes bec me c nvective(see text). The s lid vertical line gives an estimate f r the critical density fa Jupiter-mass nebula fragment at Jupiter’s p siti n. The value pl tted is themean density f a c ndensati n that is Jeans critical fits int its Hill sphere.


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The early phases f giant planet f rmati n discussed ab ve are d mi-nated by the gr wth f the c re. The envel pes adjust rapidly t the chang-ing size and gravity f the c re. As a result, the envel pes f pr t -giantplanets remain very cl se t static and in equilibrium bel w the criticalmass (Mizun 1980; Wuchterl 1993). This must change when the en-vel pes bec me m re massive and cann t reequilibrate as rapidly as thec res gr w. The nucleated instability was assumed t set in at the criticalmass, riginally as a hydr dynamic instability anal g us t the Jeans insta-bility. With the rec gniti n that energy l sses fr m the pr t -giant planetenvel pes c ntr l the further accreti n f gas, it f ll wed that quasihydr -static c ntracti n f the envel pes w uld play a key r le.

Maj r pr gress has been made since by calcu-lating the gr wth f the c res fr m planetesimal dynamics and the gr wthf the envel pes using hydr dynamics. We review these results bel w.

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A. Quasihydrostatic Models with Detailed Core Accretion



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P llack et al. (1996) c nstructed m dels in which they simulated the c n-current accreti n rates f b th the gase us and s lid c mp nents f giantplanets. P llack et al. (1996) used an ev luti nary m del having three ma-j r c mp nents: a calculati n f the three-b dy accreti n rate f a singled minant-mass pr t planet surr unded by a large number f planetesi-mals, a calculati n f the interacti n f accreted planetesimals with thegase us envel pe f the gr wing giant pr t planet, and a calculati n fthe gas accreti n rate using a sequence f quasihydr static m dels hav-ing a c re/envel pe structure. These three c mp nents f the calculati nwere updated every time step in a self-c nsistent fashi n in which relevantinf rmati n fr m ne c mp nent was used in the ther c mp nents.

The m del f P llack et al. (1996) is very detailed in many respects(c re accreti n rate, planetesimal diss luti n in the envel pe, treatment fenergy l ss via radiati n and c nvecti n, equati n f state), but it includesthe f ll wing simplifying assumpti ns:

1. The planet is assumed t be spherically symmetric.2. Hydr dynamic effects are n t c nsidered in the ev luti n f the en-

vel pe.3. The pacity in the uter envel pe is determined by a s lar mixture

f small grains in m st f the simulati ns. S lar abundances are alsused t calculate the pacity in deeper regi ns f the envel pe, wherem lecular pacities d minate.

4. The equati n f state f r the envel pe is that f r a s lar mixture felements.

5. During the entire peri d f gr wth f a giant planet, it is assumed t bethe s le d minant mass in the regi n f its feeding z ne, i.e., there aren c mpeting embry s, and planetesimal sizes and rand m vel citiesremain small. A c r llary f this assumpti n is that accreti n can bedescribed as a quasic ntinu us pr cess, as pp sed t a disc ntinu usne inv lving the ccasi nal accreti n f a massive planetesimal.

6. Planetesimals are assumed t be well-mixed within the planet’s feed-ing z ne, which gr ws as the planet’s mass increases, but planetes-imals are n t all wed t migrate int r ut f the planet’s feedingz ne as a c nsequence f their wn m ti n. Tidal interacti n betweenthe pr t planet and the disk, r migrati n f the pr t planet (see thechapters by Lub w and Artym wicz, Ward and Hahn, and Lin et al.in this v lume), are n t c nsidered.

It is n t at all bvi us that these vari us assumpti ns are valid, but n well-defined, quantitatively justifiable alternative assumpti ns are available.

The parameters in the calculati ns f P llack et al. (1996) were ad-justed t fit the pr perties f giant planets in the s lar system and bserva-ti ns f disks ar und y ung stars. They judged the applicability f a givensimulati n t planets in ur s lar system using tw basic criteria. One cri-

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teri n is pr vided by the time required t reach the runaway gas accreti nphase. This time interval sh uld be less than the lifetime f the c mp -nent f the s lar nebula, , f r successful m dels f Jupiter and Saturn andgreater than f r successful m dels f Uranus and Neptune. Limited b-servati ns f accreti n disks ar und y ung stars suggest that 10 yr,based n bservati ns f the dust c mp nent (see the chapters by Calvetet al., Natta et al., and Lagrange et al., this v lume). The lifetime f thegas c mp nent is less well c nstrained bservati nally (Str m et al. 1993).See the chapter by Wadhwa and Russell, this v lume, and see als P d sekand Cassen (1994) f r a review f nebula-lifetime estimates.

A sec nd criteri n is pr vided by the am unt f high- mass accreted,. In the case f Jupiter and Saturn, at the end f a successful sim-

ulati n sh uld be c mparable t , but s mewhat smaller than, the currenthigh- masses f these planets, because additi nal accreti n f planetes-imals ccurred between the time they started runaway gas accreti n andthe time they c ntracted t their current dimensi ns and were able t scat-ter planetesimals gravitati nally ut f the s lar system. Updated valuesf the c nstraints n high- material in the j vian planets are discussed in

secti n II f this chapter.In the m dels f P llack et al. (1996), there are three main phases t

the accreti n f Jupiter and Saturn. Phase 1 is characterized by rapidlyvarying rates f planetesimal and gas accreti n. Thr ugh ut phase 1,

/ exceeds the rate f gas accumulati n, / . Initially, thereis a very large difference (many rders f magnitude) between these twrates. H wever, they bec me pr gressively m re c mparable as timeadvances. Over much f phase 1, / increases steeply. After a max-imum at 5 10 years, it declines sharply. Meanwhile, / gr wssteadily fr m its extremely l w initial value. Phase 2 f accreti n is char-acterized by relatively time-invariant values f / and / ,with / / . Finally, phase 3 is defined by rapidly increasingrates f gas and planetesimal accreti n, with / exceeding /by steadily increasing am unts. The accreti n f Uranus and Neptunewas terminated during phase 2, presumably as a result f the dissipati nr dispersal f the gas in the pr t planetary disk.

The m dels f P llack et al. (1996) imply that the cr ss ver mass,at which the s lid and gas c mp nents f the planet are equal in mass,depends alm st exclusively n the surface mass density f s lids and thedistance fr m the Sun. The cr ss ver time is a rapidly decreasing functi nf the initial surface mass density f s lids. A surface mass density f10 g cm at Jupiter yields b th a small en ugh c ndensables mass and

rapid en ugh gas accreti n t be c nsistent with bservati ns f r n minalvalues f ther parameters. G d fits f r Saturn and Uranus are btainedif surface density f s lids dr ps ff with distance fr m the Sun as .C nstraints n the surface density are quite restrictive in the “baseline”case, but a l wer value is all wed if the pacity f the uter envel pe is

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B. Hydrodynamic Accretion beyond the Critical Mass


Protostarsand Planets III

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l w because grains sink, if planetesimal heating is reduced because ac-creted planetesimals diss lve well ab ve the c re and their residue d esn t sink t the c re, r if planetesimal accreti n st ps during phase 2, e.g.,as a result f accreti n by neighb ring embry s (P llack et al. 1996). In-creasing the mean m lecular weight f the envel pe als increases the gasaccreti n rate [cf. equati n (4)], but this parameter variati n was n t m d-eled by P llack et al. (1996). The m del results are relatively insensitive tm derately large changes in the gas density and temperature. Planetesimalsize (which affects the vel city dispersi n) is imp rtant in determining thedurati n f phase 1; f r n minal parameters this has a small effect n theverall gr wth time f r Jupiter, but the accreti n time f Uranus is m re

pr f undly affected by changes in planetesimal size.

The static and quasihydr static m dels discussed s far rely n theassumpti ns that gas accreti n fr m the nebula nt the c re is verysubs nic and that the inertia f the gas and dynamical effects such asdissipati n f kinetic energy d n t play a r le. T check whether hy-dr static equilibrium is achieved and whether it h lds, especially bey ndthe critical mass, hydr dynamical investigati ns are necessary. Tw typesf hydr dynamical investigati ns have been undertaken since

: (1) linear adiabatic dynamical stability analysis f en-vel pes ev lving quasihydr statically (Tajima and Nakagawa 1997) and(2) n nlinear, c nvective radiati n hydr dynamical calculati ns f c re-envel pe pr t -giant planets (Wuchterl 1993, 1995 , 1996, 1997, 1999)that f ll w the ev luti n f a pr t -giant planet with ut assuminghydr static equilibrium and which whether envel pes are hy-dr static, pulsate, r c llapse and at what rates mass fl ws nt the planet.Wuchterl’s m dels s lve the fl w equati ns f r the envel pe gas, essen-tially assuming nly that spherical symmetry h lds. They determine thenet gain and l ss f mass fr m the equati ns f m ti n f r the gas in spher-ical symmetry, whereas quasihydr static calculati ns add mass acc rdingt s me prescripti n and then calculate the structure f r the updated mass,yielding a new equilibrium. Alth ugh the ther assumpti ns made in thehydr dynamic calculati ns agree with th se listed in the previ us secti nf r the quasihydr static m dels, there is a sec nd imp rtant difference:The c re accreti n rate is, f r simplicity, assumed t be either c nstantr calculated acc rding t the particle-in-a-b x appr ximati n (see, e.g.,

Lissauer 1993).The first hydr dynamical calculati n f the nucleated instability

(Wuchterl 1989, 1991 ) started at the static critical mass and br ught asurprise: Instead f c llapsing, the pr t -giant planet envel pe begins tpulsate after a very sh rt c ntracti n phase (see Wuchterl 1990 f r a sim-ple discussi n f the driving mechanism). The pulsati ns f the innerpr t planetary envel pe expanded the uter envel pe, and the utward-

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traveling waves caused by the pulsati ns resulted in a mass l ss fr m theenvel pe int the nebula. The pr cess can be described as a pulsati n-driven wind. After a large fracti n f the envel pe mass has been pushedback int the nebula, the dynamical activity fades, and a new quasiequi-librium state is f und that resembles Uranus and Neptune in c re and en-vel pe mass (Wuchterl 1991 ). The mass l ss pr cess ccurs in a verysimilar way f r nebula c nditi ns at Jupiter t Neptune p siti ns and f rc re mass accreti n rates fr m 10 t 10 M yr . Starting the hydr -dynamics at l w c re mass rather than at the critical mass d es n t changethe eventual mass l ss (Wuchterl 1995 ).

Pulsati ns and mass l ss d n t ccur when “n dust,” zer -metallicitypacities are used; the lack f dust makes c nditi ns m st fav rable f r

energy l ss fr m the envel pe and theref re f r accreti n. It is interest-ing t n te that even f r zer -metallicity pacities the static critical c remass is between 1 5 and 3 M f r 10 t 10 M yr , re-spectively. Envel pe accreti n bec mes independent f c re accreti n atab ut 15 M ; the quasihydr static assumpti ns h ld until infl w vel ci-ties reach a Mach number f 0.01 at ab ut 50 M . At a t tal mass f ab ut100 M the nebula gas influx appr aches the B ndi accreti n rate, and at300 M the envel pe c llapses verall (cf. Wuchterl 1995 ). This resultsh ws that there must be an pacity-dependent transiti n fr m pulsati n-driven winds t efficient gas accreti n at the critical mass.

The main questi n c ncerning the hydr dynamics was then t ask f rc nditi ns that all w gas accreti n (i.e., damp envel pe pulsati ns) f r“realistic” s lar-c mp siti n pacities that include dust. Wuchterl (1993)derived c nditi ns f r the breakd wn f the radiative zer s luti n bydetermining nebula c nditi ns that w uld make the uter envel pe f a“radiative” critical mass pr t -giant planet c nvectively unstable. The re-sulting criteri n gives a minimum nebula density that is necessary f r ac nvective uter envel pe. Pr t planets that gr w under nebula c nditi nsab ve that density have larger envel pes f r a given c re and a reducedcritical mass as described in secti n IV.B. C nvecti n is f great imp r-

¨tance in damping stellar pulsati ns f RR Lyrae and -Cepheıd stars atthe c l, s -called “red” end f the stellar instability strip; similar behav-i r may be expected in pr t -giant planet envel pes. Wuchterl (1995 )calculated the gr wth f giant planets fr m l w c re masses hydr dy-namically f r a set f nebula c nditi ns ranging fr m bel w the criticaldensity t s mewhat ab ve. As the density was increased, the envel pesbecame increasingly m re c nvective at the critical mass but still sh wedthe mass l ss. At a nebula density f 10 g cm (i.e., greater by a fact rf 6 7 than Mizun ’s (1980) minimum rec nstituted mass nebula value),

the dynamical behavi r was different: The pulsati ns were damped, andrapid accreti n f gas set in and pr ceeded t 300 M . Apparently thespreading f c nvecti n in the uter envel pe had damped the pulsati ns,thereby inhibiting the nset f a wind and leading t accreti n. The critical

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VI. FORMATION OF EXTRASOLAR PLANETS


v v v

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c re masses required f r the f rmati n f this class f pr t -giant planetsare significantly smaller than f r the Uranus/Neptune-type (see Wuchterl1993, 1995 ).

M st giantplanet f rmati n studies use zer -entr py-gradient c nvecti n; that is,they set the temperature gradient t the adiabatic value in c nvectivelyunstable layers f the envel pe. That is d ne f r simplicity but can beinaccurate, especially when the ev luti n is rapid and hydr dynamicalwaves are present (see Wuchterl 1991 ). It was, theref re, imp rtant tdevel p a time-dependent the ry f c nvecti n that can be s lved t getherwith the equati ns f radiati n hydr dynamics. Such a time-dependentc nvecti n m del (Kuhfuß 1987) has been ref rmulated f r self-adaptivegrid radiati n hydr dynamics (Wuchterl 1995 ) and applied t giant

¨planet f rmati n (G tz 1993; Wuchterl 1996, 1997). In a ref rmulati nby Wuchterl and Feuchtinger (1998), it cl sely appr ximates standardmixing length the ry in a static l cal limit and accurately describes thes lar c nvecti n z ne and RR Lyrae light curves. In additi n, updatedm lecular pacities (Alexander and Fergus n 1994) are used in a c mpi-

¨lati n by G tz (1993) t impr ve the accuracy f radiative transfer in thepr t -giant planet envel pes. The effect f these impr vements in energytransfer is that the c re mass needed t initiate gas accreti n t a fewhundred Earth masses at vari us rbital radii is reduced t 8.30, 9.48, and9.56 M at 0.052, 5.2, and 17.2 AU, respectively (see Fig. 4), even in aminimum-mass nebula.

M re than a d zen planets have thus far been disc vered t rbit main-sequence stars ther than the Sun; all f these bjects are m re massivethan Saturn, and m st are m re massive than Jupiter (May r and Quel z1995; chapter by Marcy et al., this v lume, and references therein). Theextras lar planets currently kn wn all rbit nearer t their stars than Jupiterd es t the Sun (this is primarily an bservati nal selecti n effect; high-precisi n radial vel city surveys have n t been in perati n l ng en ugh thave bserved a full rbit f m re distant planets). S me f these planetsrbit n highly eccentric paths, suggesting that after they f rmed they were

subjected t cl se enc unters with ther giant planets (Weidenschillingand Marzari 1996; Lin and Ida 1997; Levis n et al. 1998) r, in the casef the c mpani n t 16 Cygni B, secular perturbati ns fr m the star 16 Cyg

A (H lman et al. 1997). S me f the extras lar planets are separated fr mtheir stars by less than 1% f the Jupiter-Sun distance. Guill t et al. (1996)sh wed that giant planets are stable ver the main-sequence lifetime f a1-M star even if they are as cl se as 0.05 AU. M dels inv lving migrati ncaused by disk-planet interacti ns are fav red by many researchers f r thef rmati n f these bjects (e.g., Lin et al. 1996; Trilling et al. 1998; see alsthe chapters by Ward and Hahn and by Lin et al., this v lume). H wever,

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simulati ns als sh w that it may be p ssible t f rm giant planets verycl se t stars, and we review these m dels in this secti n.

B denheimer et al. (2000) have m deled the f rmati n and ev luti n fthe planets recently disc vered in rbit ab ut the stars 51 Pegasi, C r -nae B realis, and 47 Ursae Maj ris, assuming that these planets f rmed inr near their current rbits. They used updated versi ns f the quasihydr -

static c des devel ped by B denheimer and P llack (1986) and P llacket al. (1996). The is lated pr t planet/n migrati n m del f P llack etal. (1996) requires high surface mass density f s lids f r giant planets tf rm cl se t stars within the bserved lifetimes f pr t planetary disks.The primary cause f this restricti n is that the larger Kepler shear near thestar decreases the s lid c re’s is lati n mass unless the am unt f s lids islarge; the increase in temperature cl ser t the star has nly a very smalleffect (Mizun 1980; B denheimer and P llack 1986), and the higher den-sity f gas acts in the pp site sense (Wuchterl 1996). The planet rbiting2.1 AU fr m 47 UMa can f rm in 2 Myr f r a surface density f c n-densed material 90 g cm but requires 18 Myr f r 50 g cm(B denheimer et al. 2000). A value f 90 g cm at 2.1 AU is wellab ve that used by P llack et al. (1996), but still well bel w that requiredf r l cal axisymmetric gravitati nal instabilities (T mre 1964), assuminga s lar-c mp siti n mix.

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The surface mass density f s lids required t f rm giant planets at0.23 AU ( CrB) and 0.05 AU (51 Peg) is pr hibitively large unless rbitaldecay f planetesimals is inc rp rated int the m dels. On the ther hand,Ruzmaikina (1998) has given a m del t pr vide the required am untsf b th gas and s lids. As n well-c nstrained meth d t quantify c re

gr wth cl se t stars is available, B denheimer et al. (2000) made theassumpti n f a c nstant rate f s lid-b dy accreti n f r these inner

planets. M del results f r 51 Peg indicate that if the gr wth rate f the c reis 1 10 M yr , then the planet takes 4 10 years t f rm and has afinal high- mass f 40 M . Using the same definiti n f r the planetaryradius and the same planetesimal accreti n rate as used by B denheimeret al. (2000), Wuchterl btained, in a c mparis n calculati n undertakenf r this w rk, a critical c re mass f ab ut 25 M . The tw gr ups arecurrently attempting t res lve this discrepancy, an eff rt that will includecalculati ns with identical pacities.

A maj r result f the hydr dynamical studies is that pr t -giant planetsmay pulsate and devel p pulsati n-driven mass l ss. Only if the pulsa-ti ns are damped can gas accreti n pr duce Jupiter-mass envel pes. Sinceall extras lar planets disc vered s far have minimum masses 0.5 M ,they pr bably require efficient gas accreti n and sh uld satisfy the c n-vective uter envel pe criteri n (Wuchterl 1993). A glance at Wuchterl’s(1993) Fig. 2 sh ws that pr t -giant planets l cated s mewhat inside fMercury’s rbit in the Hayashi et al. (1985) minimum-mass nebula ful-fill this c nditi n. C nvective radiati n hydr dynamical calculati ns fc re-envel pe gr wth at 0.05 AU, f r particle-in-a-b x c re mass accre-ti n at nebula temperatures f 1250 and 600 K, sh w gas accreti n bey nd300 M at c re masses f 13.5 M and 7.5 M , respectively (Wuchterl1996, 1997).

It is interesting t apply the arguments based n the c nvecti n-c ntr lled bifurcati n in hydr dynamic accreti nal behavi r t an en-semble f preplanetary nebula m dels, t simulate a variety f initialc nditi ns f r planet f rmati n that might have been present ar und therstars. Wuchterl (1993) has sh wn that alm st all nebula c nditi ns, fr ma literature c llecti n f nebula m dels, result in radiative uter envel pesat the critical mass. N nlinear radiati n hydr dynamical calculati ns withzer -entr py gradient c nvecti n sh w that Uranus/Neptune-type giantplanets are pr duced under such circumstances (secti n V.B). Jupiter-mass planets sh uld then be the excepti n. The first calculati ns withtime-dependent mixing length c nvecti n, discussed in secti n V.B, sh wgas accreti n t bey nd a Jupiter mass f r a much wider range f neb-ula c nditi ns. Apparently the impr ved descripti n f c nvecti n (andthe updated pacities) have shifted the instability strip f r pulsati ns andmass l ss at the critical mass. Further calculati ns and a reanalysis f

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VII. CONCLUSIONS


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the c nditi ns f r efficient gas accreti n f r mixing length c nvecti n havet be undertaken bef re an updated expectati n c ncerning the mass dis-tributi n f extras lar planets can be given. An imp rtant requirement f rthat is an extensive the retical and bservati nal study f plausible pre-planetary nebulae.

Jupiter and Saturn are c mp sed primarily f hydr gen and helium, yetthe heavy elements that they c ntain may h ld the key t the pr blem ftheir f rmati n. The density pr files f these planets derived fr m inte-ri r m dels, as well as the c mp siti n f their atm spheres, clearly in-dicate a significantly larger fracti n f heavy elements than was presentin the pr t s lar gas. Were the heavy elements the first t accrete, r didthe enrichment ccur at later stages? Depending n the scenari , Jupiterand Saturn might have received very different am unts f planetesimals,thereby pr viding a way t differentiate a very rapid f rmati n (such asin the nebula instability mechanism) fr m a sl wer ne (such as in thenucleated instability).

Interi r and ev luti n m dels f r Jupiter and Saturn tend t fav r c remasses that lie within the range f acceptable critical c re masses pre-dicted within the nucleated instability hyp thesis. The m dels based nthis hyp thesis als explain why Uranus and Neptune are m stly c re:either because (i) gas accreti n is limited t 1 M by a hydr dynamicinstability that perates under certain nebula c nditi ns, l w gas densitybeing the d minant fact r (Wuchterl 1993, 1995 ); (ii) their c res grewm re sl wly than th se f Jupiter and Saturn because rbital timescalesare l nger farther fr m the Sun, and thus they did n t achieve sufficientmass t accrete large quantities f gas bef re the s lar nebula gas was dis-persed (P llack et al. 1996); r (iii) the gas in the Uranus/Neptune regi nf the nebula was dispersed rapidly via ph t evap rati n, whereas gas re-

mained in the Jupiter/Saturn regi n f r a much l nger peri d f time (Shuet al. 1993).

The nucleated instability hyp thesis thus pr vides a viable m del f rthe f rmati n f the giant planets bserved in ur s lar system and be-y nd. Presently kn wn extras lar planets may have accreted if theirpreplanetary nebulae pr vided sufficient am unts f gas and s lids. Al-ternatively, acc rding t studies f disk-induced migrati n (chapters byWard and Hahn and by Lin et al., this v lume) and gravitati nal enc un-ters with ther planets, they c uld have f rmed elsewhere and m ved intthe present p siti ns. In that case the rbits f m st if n t all planets kn wnt be b und t main-sequence stars suffered substantialrbital ev luti n.

The next few years will be dedicated t the devel pment f a syn-ptic understanding f giant planet f rmati n pr cesses f r a variety f

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REFERENCES

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Adachi, I., Hayashi, C., and Nakazawa, K. 1976. The gas drag effect n the ellipticm ti n f a s lid b dy in the prim rdial s lar nebula.56:1756–1771.

Alexander, D. R., and Fergus n, J. W. 1994. L w-temperature R sseland paci-ties. 437:879–891

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T. G.’s w rk is supp rted by CNRS (UMR 6529)ánd the Pr gramme Nati nal de Planet l gie. T. G. thanks Daniel Gautier

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schaftlichen F rschung (FWF) under pr ject numbers S-7305-AST, S-7307-AST. We thank W. B. Hubbard f r a c nstructive review f thismanuscript.

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