Overcoming migration during giant planet formation

0.34 MB·English

by  E. W. Thommes

ApJL,in press
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
OVERCOMING MIGRATION DURING GIANT PLANET FORMATION
Edward W. Thommes1, Leif Nilsson2 and Norman Murray1
1CanadianInstitute forTheoretical Astrophysics,UniversityofToronto,60St. GeorgeStreet,Toronto, OntarioM5S3H8,Canadaand
2DepartmentofPhysics,UniversityofToronto,60St. GeorgeStreet, Toronto,OntarioM5S1A7,Canada
ApJL, inpress
ABSTRACT
In the coreaccretionmodel, gas giantformationis a race betweengrowthandmigration; for a core
to become a jovian planet, it must accrete its envelope before it spirals into the host star. We use a
multizonenumericalmodeltoextendourpreviousinvestigationofthe“windowofopportunity”forgas
7 giantformationwithin a disk. When the collisioncross-sectionenhancementdue to coreatmospheres
0 is taken into account, we find that a broad range of protoplanetary disks posses such a window.
0
Subject headings: protoplanetary disks, origin: solar system — planets: formation, planets:extrasolar
2

n
a 1. INTRODUCTION tion rate of a protoplanet is
J
Core accretion and direct gravitational instability are dM Σ v2
16 tfohremtw. oTmheaicnorceomacpcerteitniognmpoidcteulsrefoarphpoewargsatsogbiaentfapvloaunreetds dt ≈32HmmπRM2 (cid:18)1+ ver2secl(cid:19)vrel (1)

by both observations (Fischer & Valenti 2005) and the- whereΣ isthesurfacedensityofthe planetesimaldisk,
1 m
ory (Rafikov 2005) but its viability in the face of puta- H is its scale height, R is the protoplanet radius,
v m M
tive rapid orbital decay of protoplanets by planet-disk v istheescapevelocityfromtheprotoplanet’ssurface,
3 esc
4 interactions (Ward 1997; Papaloizou& Larwood 2000; andvrel is the averagerelative velocitybetweenthe pro-
4 Tanaka et al. 2002) has long been a cause of worry. toplanetandaplanetesimal. Theplanetesimalvelocities
1 Thommes & Murray(2006,herafterTM06)showedthat areincreasedthroughgravitationalstirringbytheproto-
0 infact,evenwithrapidmigrationoperating,awindowof planets, and reduced by aerodynamic drag from the gas
7 opportunity for successful core accretion can open later disk;wetakethecorrespondingratesfromIda & Makino
0 in a protostellar disk’s lifetime, once it has dissipated to (1993) and Adachi et al. (1976) respectively. Unlike in
/ thepointthataccretionandmigrationtimescalesbecome TM06, we explicitly calculate the velocity evolution of
h
p comparable. Here, we build on this work, adopting a the plantesimal disk, without assuming equlibrium val-
- moresophisticatedmodelwhichexplicitlytracksthehis- ues for the random velocities of the plantesimals.
o toryofeachindividualprotoplanetallthroughtheeraof We compute the evolution of the plantesimal disk us-
r coreformation. Inthisway,weareabletoassesstherole ingamultizoneapproach,dividingitupintoradialbins.
t
s of protoplanet mergers (as opposed to just planetesimal Each timestep, we compute the accretion zone, as well
a
accretion) in the formation of cores, as well as to make asthe stirredzone, ofeachprotoplanet,noting the plan-
:
v predictions about the actualnumber of potentialcores a etesimal disk annuli it encompasses. The stirred zone
i given system will form. We go on to include the effect has a radial width
X
of gas atmospheres on the protoplanets, which increases
r their accretion cross-sections. In agreement with previ- 4 r 2 1/2
a ∆r ≈4r e2 +i2 +12 M (2)
ousresults,wefindthis greatlyspeedsupaccretiononce stir M 3 m m r
protoplanet masses reach ∼ 10−1 Earth masses (M⊕). " (cid:0) (cid:1) (cid:18) H(cid:19) #
As a result,core accretionis successful, notwithstanding (Ida & Makino 1993) where r is the protoplanet’s or-
M
typeImigration,forawiderangeofdiskparameters. In bital radius, rH ≡ (M/3M∗)1/3rM its Hill radius, and
§2, we describe the model. In §3, we perform a sample M∗ the mass of the central star. The accretion zone
computation using the model. In §4, we add core at- has a width ∆r ≈ ∆r /2. In the oligarchic regime
accr stir
mospheres to the model and then recompute the sample of growth, adjacent protoplanets maintain a separation
case. In §5, we examine core formation likelihood as a of ∼ 10r . We start the protoplanets with this spac-
H
functionofdisk properties. Conclusionsarepresentedin ing, and merge a pair of neighbors whenever they come
§6. within 7r of each other. At early times, such mergers
H
represent the fact that there are winners and losers as
2. THEMODEL neighboring protoplanets compete for building material.
Caution is required in interpreting any mergers which
As in TM06, we adopt for the type I migration rate
occur at late times; we consider this issue in §3 below.
the result obtained by Tanaka et al. (2002). Likewise,
we again use a “particle in a box” estimate of the pro- 3. RESULTS
toplanet accretion rate, assuming it to be in the ”oli-
We begin by re-calculating the “baseline” model of
garchic” regime of growth (Kokubo & Ida 1998, 2000;
TM06 using the above multizone approach. That case
Thommes et al. 2003). Also, we use the simplifying as-
sumption of a uniform planetesimal mass m. The accre- beginswithadiskofmassMd =0.15M⊙aroundaSolar-
mass star. The disk has an α-parameterized viscosity
Electronicaddress: [email protected] with α = 10−2, an initial outer radius of 50 AU, and a
2 Thommes, Nilsson & Murray
25 tion expands their Hill radii, so that subsequent growth
20 proceedsonlybythesweep-upoftheremainingplanetes-
r (AU)1105 imAalns.exceptiontotheorderlysequencesofmergerstakes
placewithinthefirst2×105years,beginningatthesnow
5
line. Here, the wave of mergers starts to go inward with
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 1056.5 time rather than outward, with the result that one pro-
toplanetaccretes allits inner neighbors before falling off
102 theinneredgeofthedisk. Itisabletodothisbecausethe
101 densityjumpatthesnowlineissteepenoughthatwithin
M (Earth masses)1100−01 sfitac,settnehereftgohrraoanwt“ihtpsrteiimndnaeetsorcran”leepigdrheocbtrooepraslsa.ensSeoitnutctoewaathrrides.etTywphheiiscIhsemgtsirgotrwhaes-
10−2
tionrateofabody isproportionaltoits mass,this body
10−3
0 0.5 1 1.5 2 2.5t (yrs) 3 3.5 4 4.5 5 x 1056.5 then overtakes its neighbors, accreting them all one by
Fig. 1.— Protoplanet migration (top) and growth (bottom) in one,andfinallyexitingtheinneredgeofthediskjustasit
anevolvingprotostellardiskwiththeparametersgivenin§3. The attainsM . However,becausethegrowthofthepreda-
vertical line indicates t30AU, the time at which the gas disk mass torbecomrewsydominatedbymergers,andalsobecausethis
cinortehfierisntnreerac3h0esAMUrdwryo,ptshetoru1nMawJauyp.gaFsilalcecdrectiirocnlesmsahsosw. when a is happening in a part of the disk where the majority of
theplanetesimalmasshasalreadybeensweptup,weare
scale height profile H = 0.047(r/1AU)5/4 AU. The gas nolongerinthe true oligarchicgrowthregime. Thus our
disk’s time evolution is calculated analytically using the treatmentofprotoplanetmergers,beingstatisticalinna-
similarity solution of Lynden-Bell & Pringle (1974). At ture, loses its validity. All we can say is that convergent
t = 0, the accompanying solids disk follows the surface migrationof protoplanetstakes place fromthe snow line
densityprofileofthegas,withametallicity[Fe/H]=0.25. inward. Previous N-body work suggests that differential
Itconsistsofplanetesimalswithauniformradiusof1km. migration tends to result in the establishment of mean-
We takethe “snowline” tobe at2.7AUas inthe model motionresonancesamong protoplanets(Thommes 2005;
of Hayashi (1981) (see TM06 for details). As in TM06, McNeil et al. 2005; Terquem & Papaloizou2006)
westopthecalculationwhenthegasmassinteriorto100
AU drops below one Jupiter mass (t = 5.5 Myrs), 4. ATMOSPHERE-ENHANCEDACCRETION
100AU
noting alsothe time atwhichthe gasmassinteriorto 30 In the core-accretion model, a core becomes a gas gi-
AUdropsbelowthisvalue(t =2.4Myrs). Asanew ant when it undergoes runaway accretion from the sur-
30AU
addition,wecalculatethecriticalcoremassM forrun- rounding gas disk, thus obtaining a massive atmosphere
crit
awaygasaccretionasinIkoma et al.(2000),assumingan (Cameron 1978; Mizuno et al. 1978). Well before that
envelopegrainopacityof κ=0.4cm2g−1 (see §4 below). time,however,thecorewillalreadystarttoaccretesome
This expression for M becomes zero when planetesi- gas (Pollack et al. 1996). Planetesimals which intersect
crit
mal accretionceases,so we choose the mass for runaway the atmospherelose energyto aerodynamicdrag;if they
gasaccretiononsettobe Mrwy =min(Mcrit,10M⊕); en- lose enough energy they are captured, even if their orig-
velopegrowthtimefora10M⊕ bodynotaccretingplan- inal trajectory would have caused them to miss the core
etesimals is only ∼105 yrs (Ikoma et al. 2000). (Pollack et al. 1996; Inaba & Ikoma 2003; Inaba et al.
The result of the calculation is shown in Fig. 1. This 2003). Atmospheres thus provide an enhancement to
canbedirectlycomparedtoFig. 6ofTM06. Theoverall the cores’ planetesimal capture cross-sections, which in-
evolutionmatches ourprevioussemianalytic resultquite creaseswiththedensityoftheatmosphereanddecreases
closely: By t , the largest protoplanet mass is just with plantesimal size. As shown by the above authors,
30AU
under 10 M⊕, and by t100AU it is a bit over. No pro- atmospheric capture can substantially boost a proto-
toplanets have reached M by t , while two reach planet’s accretion rate beyond the early stages of its
rwy 30AU
thisthresholdbetweent30AU andt100AU,suggestingthat growth, once it is &10−1 M⊕.
coreaccretiongrowthofgasgiantsdoeshaveachanceto In order to add this effect to our calculation, we use
occurbeforethegasislost. Themultizoneapproachnow the capture model of Chambers (2006) which is in turn
allowsustoseetherolewhichmergersbetweenneighbors a simplified version of the the model of Inaba & Ikoma
playsinthe growthofthe protoplanets. Bothther vs. t (2003). The atmosphere-enhanced capture radius R is
C
andM vs. t panels ofFig. 1 clearlyshowthese mergers. then given by
Because we start the protoplanets with a uniform sepa-
R 3 0.0790µ4cR r M 2 24 dM −1
ration in r , and adopt a uniform separation at which C M H
H =
to merge neighbors, the mergers occur in successive, or- (cid:18)RM(cid:19) κrm (cid:18)M⊙(cid:19) (cid:18)24+5e˜(cid:19)(cid:18) dt (cid:19)
derly generations which propagate outward through the (3)
diskovertime. Fromourchoseninitialprotoplanetmass whereµisthemeanmolecularweightofthegas(takenas
of 10−3 M⊕, three generations of mergers occur, with 2.8, Solar composition), c is the speed of light, rm is the
the lastone taking protoplanetmassesfrom≈0.2 to 0.4 planetesimal radius, κ is the atmospheric opacity, and
M⊕. Thus,protoplanetmergersplayasignificantrolein e˜≡ erM/rH. We substitute RC for RM in our calcula-
growth (for a discussion, see Chambers 2006), but only tion, subject to the condition that R < R < R ,
M C atmos
uptoamass<1M⊕ inthecaseshownhere. Afterthat, where Ratmos is the outer radius of the protoplanet’s at-
the rate at which neighboring protoplanets diverge by mosphere, taken to be the smaller of its Hill and Bondi
type I migration exceeds the rate at which mass accre- radii (R =GM/c2).
B s
Giant planet formation 3
25 [Fe/H]=−0.25 [Fe/H]=0.0 [Fe/H]=+0.25
r (AU)11205050 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Initial disk mass (Msun)10−1510.1 150.1 1 0.1 10−1 1 50.19 51 10.1 10−1 9 9950.151 10.1
x 106
102 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1
Disk viscosity (α)
101
M (Earth masses)111000−−021 fmtaowFgrainiisgtvs,he.tMenh3feds.o—y,mrsαmtoTevarmhietsie;colioflnkoesenrioltgyftytwghipitanaosirf.sagtmtih1ahee0nat5ttteirympo,rnlesaae,nnwteodihtrnsmedmraeoeostwiarsaeltluwifgcnuiainltnsiykc(gcte[iFioloaynenn/tttoHosouf]bwr.iseTni,lsilhitunfeiffiaoMlrlacmdryiegirnsiesnkrt)
10−3 time.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
t (yrs) x 106 slower growth than Alibert et al. (2005), likely because
Fig. 2.—Protoplanetmigration(top)andaccretion(bottom)for their code incorporates weaker protoplanet-planetesimal
the sameparameters as inFig. 1, butincludingthe enhancement
gravitationalstirring. Secondly,theychosearatherlarge
to the planetesimal capture cross-sections due to the growth of
atmospheresontheprotoplanets. initial mass of 0.6 M⊕ for their protoplanets, with the
result that at t = 0, all of them are already migrating
We repeat the calculation of Fig. 1, this time with quite rapidly;throughoutmuchofthe disk, this removes
atmosphere-enhanced capture radii for the protoplan- theopportunityfordiskdispersaltoequalizegrowthand
ets. We use the opacity from the analytic fit of migration timescales.
Inaba & Ikoma (2003), with a grain depletion factor of
f = 0.1 relative to the interstellar value to account for 5. GIANTPLANETFORMATIONASAFUNCTIONOF
graingrowthandsettling(Podolak2003). Forsimplicity, DISKPROPERTIES
we use a single value for the opacity. We conservatively As in TM06, we now build up a global picture of how
choose the highest (i.e. lowest-temperature) value out the likelihood of gas giant formation by core accretion
of the range: κ = 4f cm 2g−1 =0.4 cm2g−1. Fig. 2 changesasafunctionofdiskproperties,thistimeinclud-
showsthe result. Untilprotoplanetmassesreach∼10−1 ingthe effectofcoreatmospheres. We computeagridof
M⊕, accretion proceeds in the same manner as before. models,0.01M⊙ ≤Md ≤0.4M⊙ and10−4 ≤α≤10−1,
However,oncetheatmospheresbecomeeffective,growth and repeat this for metallicities of [Fe/H]=-0.25, 0 and
quickly becomes much faster than in the unenhanced 0.25. In TM06, the largest protoplanet mass at t
100AU
case. Discounting the initial burst of growthat the edge was plotted for each model. Here, we instead plot the
of the snow line (see §3 above), the first protoplanet to length of the time window, t , between the time the
win
M does so at 4.5×105 yrs already. By t = 2.4 first protoplanet reaches M , and t ; we consider
rwy 30AU rwy 30AU
Myrs, a total of eight protoplanets have managed to thistobeabettermeasureofhowlikelyagivensystemis
reachM ,eachoneasitsplanetesimalaccretionceases tosucceedinformingagasgiant. Also,wecutallcompu-
rwy
and M decreases. Since the later ones originate from tationsoffat10Myrs,evenift >10Myrs,sincethis
crit 30AU
furtheroutinthedisk,theysurviveforlongerbeforemi- is the observational upper age limit for gas disks. Fig.
grationremovesthem;thelastfourprotoplanetsreaching 3 shows the results, indicating that t increases with
win
M beforet lastanother∼0.5−1Myrs. Thusthe [Fe/H],whilethe“optimal”valueofα,whichallowssuc-
rwy 30AU
windowofopportunityforacoretobecomeagasgiantis cessful core accretion at the lowest disk mass, decreases
considerablylargerthaninthecalculationwithoutatmo- with[Fe/H].Withintherangeofobservationally-inferred
spheres,§3above. Itshouldbenotedthatthecalculation disk viscosities, 10−3 . α . 10−2 (Hartmann et al.
does not take into account various effects which become 1998), [Fe/H]=0.25 disks as low as ∼ 0.03 M⊙ in ini-
important at large core mass, in particular gap-opening tial mass have t > 105 yrs, and thus stand a chance
win
and the extra mass contributed by the atmosphere, so of accreting a Jovian envelope in time (Hubickyj et al.
any growth which happens at M ≫ 10 M⊕ should not 2005). For disks with [Fe/H]=-0.25, Md & 0.08 M⊙ is
be taken too literally. Likewise, the series of mergers required.
starting inside 15 AU at t ≈ 4 Myrs is subject to the
caveat discussed in §3. 6. CONCLUSIONS
Calculations of concurrent core formation and migra- Wehavepresentedhereamultizonecalculationofcon-
tion were also performed by Alibert et al. (2005), who current planet growth and accretion in an evolving pro-
concluded that type I migration would have to be re- toplanetary disk, focusing on whether giant planet cores
duced by an order of magnitude relative to the result of canformintime. Thefirstpartofourresultsagreeswell
Tanaka et al. (2002) in order for core accretion to suc- withourprevioussemi-analyticmodel,whileelucidating
ceed. The disparity between their results and ours ap- the role played by protoplanet-protoplanet mergers at
pears to stem from two main sources: First, they as- early times. Since the calculation tracks individual pro-
sumedacharacteristicplanetesimalsizeof100kmrather toplanets,wecannowestimatethetotalnumberofbod-
than1km;thisslowsaccretionrelativetoourcalculation ies which reach core mass in a given system. An impor-
by reducing both the damping of planetesimal random tant point is that although migration impedes planetes-
velocities and atmosphere enhancement of protoplanet imal accretion by moving protoplanets to the already-
accretion radii. It is worth noting that in a direct com- depleted inner disk before they can finish their growth,
parison using the same parameters, we actually obtain this actually helps in attaining runaway gas accretion,
4 Thommes, Nilsson & Murray
provided a protoplanet is able to reach & 10 M⊕ before the atmosphere-less calculation of TM06. Now, we find
its planetesimal supply is cut off. that a Solar-metallicity disk of moderate initial mass
When we add to our calculation a simple model for (Md & 0.05 M⊙) stands a reasonable chance of giving
the effect of gas atmospheres on growing protoplanets, birth to a jovian planet. Since we neglect fragmentation
the results change dramatically. In our baseline model, of planetesimals, and the potentially very rapid accre-
the time to grow a body large enough to serve as a tionof fragments atlow relative velocities dominatedby
giant planet core shrinks from over 3 million years to Keplerian shear (Rafikov 2004), which will also further
less than half a million years. Thus in the competi- increasetheroleofatmosphericcapture(Inaba & Ikoma
tionbetweenaccretionandmigration,thebalanceswings 2003), the accretion rates in our calculation are likely
strongly toward the former, relatively speaking. This conservative,and core formation may thus proceed even
makesthewindowofopportunityfortheformationofgas more readily; these effects will be added in future work.
giants by core accretion—the time between the appear-
ance of the first core-sized body, and the disappearance
of the gas disk—much larger for a given set of parame- EWTthanksShigeruIdaandMasahiroIkomaforstim-
ters. As a result, the range of disk properties which can ulating and informative discussions. We also thank the
plausibly produce gas giants, notwithstanding effective referee for valuable comments, in particular for suggest-
type I migration, becomes significantly broader than in ing an improved treatment of critical core masses.
REFERENCES
Adachi, I., Hayashi, C., & Nakazawa, K. 1976, Progress of Papaloizou,J.C.B.,&Larwood,J.D.2000, MNRAS,315,823
Theoretical Physics,56,1756 Podolak,M.2003, Icarus,165,428
Cameron,A.G.W.1978, MoonandPlanets,18,5 Pollack, J. B., Hubickyj, O., Bodenheimer, P., Lissauer, J. J.,
Chambers,J.2006, Icarus,180,496 Podolak,M.,&Greenzweig,Y.1996, Icarus,124,62
Fischer,D.A.,&Valenti,J.2005,ApJ,622,1102 Rafikov,R.R.2004,AJ,128,1348
Hartmann,L.,Calvet,N.,Gullbring,E.,&D’Alessio,P.1998,ApJ, —.2005, ApJ,621,L69
495,385 Tanaka,H.,Takeuchi,T.,&Ward,W.R.2002,ApJ,565,1257
Hayashi,C.1981, Prog.Theor.Phys.,70,35 Terquem,C.,&Papaloizou, J.2006,toappearinApJ
Hubickyj,O.,Bodenheimer,P.,&Lissauer,J.J.2005,Icarus,179, Thommes,E.W.2005,ApJ,626,1033
415 Thommes, E. W., Duncan, M. J., & Levison, H. F. 2003, Icarus,
Ida,S.,&Makino,J.1993,Icarus,106,210 161,431
Inaba,S.,&Ikoma,M.2003,A&A,410,711 Thommes,E.W.,&Murray,N.2006,ApJ,644,1214
Inaba,S.,Wetherill,G.W.,&Ikoma,M.2003,Icarus,166,46 Ward,W.R.1997,Icarus,126,261
Kokubo,E.,&Ida,S.1998, Icarus,131,171 Ikoma,M.,Nakazawa, K.&Emori,H.2006,ApJ,537,1013
—.2000,Icarus,143,15 Alibert,Y.,Mordasini,C.&Benz,W.2005,A&A434,343
Lynden-Bell,D.,&Pringle,J.E.1974, MNRAS,168,603
McNeil,D.,Duncan,M.,&Levison,H.F.2005,AJ,130,2884
Mizuno,H.,Nakazawa,K.,&Hayashi,C.1978,Prog.Theor.Phys.,
60,699