Planetesimal Accretion in Binary Systems: Could Planets Form Around Alpha Centauri B ?

5 MB·English

by  Ji-Wei Xie

Draftversion January15,2010
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
PLANETESIMAL ACCRETION IN BINARY SYSTEMS: COULD PLANETSFORM AROUND α CENTAURI B ?
Ji-Wei Xie1,2, Ji-Lin Zhou1, Jian Ge2
1DepartmentofAstronomy,NanjingUniversity,Nanjing,Jiangsu,210093,Chinaand
2DepartmentofAstronomy,UniversityofFlorida,Gainesville,FL,32611-2055, USA
Draft version January 15, 2010
ABSTRACT
Stellarperturbationsaffectplanet-formationinbinarysystems. Recentstudiesshowthattheplanet-
0
formation stage of mutual accretion of km-sized planetesimals is most sensitive to binary effects. In
1
0 this paper, the condition for planetesimal accretion is investigated around α CenB, which is believed
2 tobeanidealcandidatefordetectionofanEarth-likeplanetinornearitshabitablezone(0.5-0.9AU).
A simplified scaling method is developed to estimate the accretiontimescale of the planetesimals em-
n beddedinaprotoplanetarydisk. Twenty-fourcaseswithdifferentbinaryinclinations(i =0,0.1o 1.0o,
a B
and 10o), gas densities(0.3,1,and 3 times of the Minimum Mass of Solar Nebula, MMSN hereafter),
J
and with and without gas depletion, are simulated. We find: (1) re-phasing of planetesimals orbits
5 is independent of gas depletion in α CenB, and it is significantly reached at 1−2 AU, leading to
1
accretion-favorable conditions after the first ∼ 105 yrs, (2)the planetesimal collision timescale at 1-2

AU is estimated as: TB ∼ (1+100i )×103 yrs, where 0 <i < 10o, (3)disks with gas densities of
] col B B
P 0.3-1.0MMSNandinclinationsof1o-10owithrespecttothebinaryorbit,arefoundtobethefavorable
E conditions in which planetesimals are likely to surviveand growup to planetary embryos,(4)evenfor
. theaccretion-favorableconditions,accretionissignificantlylessefficientascomparedtothesingle-star
h
case,andthetimetakenbyaccretionofkm-sizedplanetesimalsintoplanetaryembryosorcoreswould
p beatleastseveraltimesofTB ,whichisprobablylongerthanthe timescaleofgasdepletioninsucha
- col
o close binary system. In other words, our results suggest that formation of Earth-like planets through
r accretionof km-sized planetesimals is possible in α CenB, while formationof gaseousgiant planets is
t not favorable.
s
a Subject headings: methods: numerical — planetary systems: formation
[

1 1. INTRODUCTION etscanformfromLunar-massprotoplanets,thequestion
v
of whether Lunar-massembryos can form and remain in
4 Planet formation in binary systems is an important
stable orbits is critical towards understanding the like-
1 issue, as a major part of solar-type stars were born
lihood of Earth-like planets forming in binary star sys-
6 in binary or multi-stars systems (Duquennoy & Mayor
tems.
2 1991; Mathieu et al. 2000; Duchˆene et al. 2007) and
. approximately ∼ 25% of detected exoplanets are ex-
1 1.1. Planetesimal Accretion in Binary Systems
pectedtoinhabitbinarystarsystems(Desidera&Barbi-
0
eri2007). Mostofthecurrentdetectedplanet-bearingbi- According to the current standard planet-formation
0
narysystems are wide S-types(Eggenbergeret al. 2004), model, protoplanets or embryos are formed by accretion
1
meaning the companion star acts as a distant satellite, of 1-10 km-sized (in radii) planetesimals whose own ori-
:
v typically orbiting the inner star-planet system over ∼ gin is still a big puzzle to our knowledge (Goldreich &
i 100 AU away. Nevertheless, at least three planets− Ward1973;Weidenschilling&Cuzzi1993;Youdin2008).
X
GJ86b(Queloz et al. 2000), γ Cephei b (Hatzes et al. Theaccretionfromplanetesimalstoembryosusuallyfol-
r 2003), and HD41004b (Zucker et al. 2004)−are found lows a runaway mode in single-star systems(Greenberg
a
in close binary systems with stellar separation of only et al. 1978; Wetherill & Stewart 1989; Barge & Pel-
∼20 AU. In addition, our closest neighbor− α Centauri lat 1993; Kokubo & Ida 1996, 1998, 2000; Rafikov 2003,
AB, of which no planet has yet been detected− is also a 2004). However,thingsareentirelydifferentfortheclose
closebinarysystemwithstellarorbitalsemimajoraxisof binary systems.
23.4 AU and eccentricity of 0.52. The binary proximity Accretion in binary systems may probably not follow
and eccentricity can stir strong gravitational perturba- the normal runaway mode or even be suppressed be-
tions which may pose a threat to planet formation. To causethe binaryperturbationsstirupthe relativeveloc-
address this issue, many researchers(Barbieriet al 2002; ity (△V) among the planetesimals. Recent studies have
Quintana 2004; Quintana & Lissauer 2006; Quintana & foundthatthis issue ismuchmorecomplicatedif the bi-
Lissauer2007;Quintanaetal. 2007;Guedes etal. 2008) nary perturbations are coupled with gas drag forced by
haveperformedaseriesofN-Bodysimulationsandfound the gas disk. Marzari & Scholl (2000) found that the
Earth-mass planets can form in such close binary sys- △V is significantly reduced by the “phasing” of plan-
tems. However, all their simulations have made an as- etesimals’ orbits (“orbital phasing” hereafter) under the
sumption that the planetary disk was initially made of couplingofbinaryperturbationsandgasdrag. However,
proto-planetofatleastLunar-mass. IfEarth-massplan- Th´ebaultetal. (2006,2008,2009)havepointedoutthat
the “orbital phasing” is size-dependent. If assuming a
[email protected] populationofplanetesimalswithasizedistribution,such
2
as a Gaussian distribution with a medium of 5 km and be rewritten as
rangeof1-10km,theaverage△V isstilltoohighforac-
2 a M R
cretionin mostofthe regionofbinarysystems with sep- TS = ×104f−1f−1( )3( A)−1/2( p) yrs. (3)
arationsof ∼ 20AU. Paardekooperetal. (2008)investi- col 3 d ice AU M⊙ km
gated this issue with an evolving-disk model and found
Thisestimationgiveatypicalcollisionaltimescaleof104
the conditions for planetesimalaccretionweremore hos-
yrsroughlyforplanetesimalswith1-10kmsizesat1AU.
tilethaninthenon-evolvingsymmetricaldisk. Torescue
Inbinarysystems,however,neitherEq(1)norEq(2)is
theplanetformationinsuchclosebinarysystems,Xie &
satisfied,andthusthecollisionaltimescaleforbinarysys-
Zhou(2008,2009)havefoundtwoindividualmechanisms
tems (TB ) is not clear. Without any knowledge of TB ,
which may reduce the effects of “size dependence of or- col col
we may misunderstand planetesimal accretion in binary
bitalphasing”andfavortheplanetesimalaccretion. One
systems. For an example, if TB ∼105 yrs or longer, we
mechanismis “orbitalre-phasing”induced by gasdeple- col
may draw wrong conclusions on the conditions of plan-
tion. Taking the γ Cephei system as an example, Xie
etesimal accretion by simulations of a binary system for
& Zhou (2008) have found that “orbital phasing” is sig-
only 104 yrs. The conditions, such as △V, in the later
nificantly restored for all size planetesimals after several
several 104 yrs, may become either hostile for accretion
timescales of gas depletion. This “orbital re-phasing”
byorbitalrandomizationorfavorableforaccretionbyor-
reduces △V to very low values, favoring planetesimal
bital re-phasing. Therefore, an accurate estimate of TB
accretion. Another mechanism is small binary inclina- col
is very crucial to determine whether planetesimal accre-
tion. Taking α CenA as an example, Xie & Zhou (2009)
tion is favored or suppressed in binary systems. In this
foundthatthe△V canbesignificantlyreducedifasmall
paper, we develop a simplified method to estimate TB .
inclination of such as 1o−5o is considered between the col
binary orbit and the gas disk. However, the efficiency
1.2. Planet Formation in α Centauri B
of these mechanisms may be quite diverse in different
binary systems with different stellar mass ratios, eccen- αCentauriAandBaremainsequencestarswithspec-
tricities, inclinations, and separations. For an example, tral types of G2V and K1V, masses of 1.1 M⊙ and 0.93
“orbital re-phasing” mainly depends on the timescale of M⊙, respectively. They are bound together as a binary
gas depletion in γ Cephei systems as shown in Xie & system (or a triple star system if including the M dwarf
Zhou (2008), while it is found to be nearly independent ProximaCentauri,whichorbitstheABpairwithasemi-
of gas depletion in α CenB as shown later in this paper. major axis over 10,000 AU according to Wertheimer &
Inaddition,forhighlyinclinedbinarysystems(i >10o), Laughlin 2006) with relative orbital semimajor axis of
B
Marzariet al. (2009a)found gasdrag has little effect on 23.4 AU and eccentricity of 0.52(Pourbaix et al. 2002).
planetesimals because of their significant vertical excur- As our closest neighbor in space, α Cen AB has been
sionabove the gasdisk plane, and the Kozaimechanism monitoredforoveronehundredyears,andnoplanethas
stronglyinhibits planetesimalaccretionif i largerthan been identified around either α CenA or α CenB by far.
B
40o. It is concluded by Endl et al. (2001) that the mass up-
A crucial problem that has not been addressed in all per limit of a planetary companion is 2.5 M for
Jupiter
previous studies is: what is the timescale for planetesi- α CenA and 3.5 M for α CenB at any orbital ra-
Jupiter
mal accretion in binary systems. This timescale can be dius. Inotherwords,westill cannotrule outpresenceof
roughly treated as the collisional timescale T if most lower mass planets, especially an Earth-like planet. In
col
collisions leadto accretionrather than erosion. In single fact, recent simulations(Quintana et al. 2002; Guedes
star systems, T can be estimated as et al. 2008) have shown Earth-like planets could form
col
around α CenB, and it is the best candidate for search-
1 2a<i> m
TS = = , ing for Earth-like planets. Nevertheless, all their sim-
col n△VπRp2 △V ΣdπRp2 ulations focus on the final stage of planet formation,
and implicitly assume a disk of planetary embryos as
a △V <i> R
=4×105f−1f−1( )5/2( )−1( )( p) yrs, their initial condition. Thus the remaining question is
d ice AU 1ms−1 10−3 km whether these embryoscanform. Th´ebaultet al. (2009)
(1) recently addressed this problem by analyzing the condi-
where n,m,Rp,a,<i> and Σd are the number density, tions for planetesimal accretion. They conclude plane-
mass, radii, semi-major axis, average orbital inclination tary embryos formation through planetesimal accretion
and surface density of planetesimals in the disk. fd and seems impossible aroundα CenB, unless the binary sep-
fice arescalingnumbersofsolidsurfacedensityandsolid aration was wider in its initial stages. However, their
enhancement beyond ice line, respectively. Here, fd = 1 conclusions are limited in the absolutely coplanar case,
correspondstoMMSN,i.e., Σd =10gcm−3 at1AU,and where the inclination between the gas disk and binary
fice =4.2accountsforsolidenhancementbeyondtheice stellar orbit is exactly equal to zero(iB =0)
line, which usually lies in 2-3 AU for solar type stars. In this paper, we numerically reinvestigate the condi-
Considering the relation, tionsforplanetesimalaccretionaroundαCenB.Thema-
jor contributions will focus on: (1)testing the efficiency
△V ≃ <e>V ≃2<i>V
k k ofgas-depletion-induced“orbitalre-phasing”inαCenB,
<i> a M (2)developingascalingmethodtoestimatethetimescale
≃60(10−3 )(AU)−1/2(M⊙A)1/2 ms−1 (2) of planetesimal accretion in binary systems, and (3)ex-
tending the study of Th´ebault et al (2009) by including
whereV isthe localKeplerianvelocityand<e>is the theeffectsofbinaryinclinations. Insection2,wepresent
k
average orbital eccentricity of planetesimals, Eq(1) can the model and results of our simulations. Based on our
3
results, we discuss planet formation in α CenB in sec- Although the collisional timescale, TB, in binary sys-
col
tion 3. Finally, in section 4, we summarize our major tems cannot be derived as Eq (1) and (3) as in single
conclusions. starsystems,wecanestimateTB ifweknowthescaling
col
factor between it and TS . In this paper, subscripts “S”
2. SIMULATIONS and“B”,denotetheprocpoelrtiesofsinglestarsystemsand
2.1. Models and Setup binary systems, respectively.
Generally, collisional timescale is inversely propor-
Using the fourth-order Hermite integrator(Kokubo et
tional to the total collision cross-section of the system,
al. 1998), we follow the evolution of 2×104 test plan-
namely,
etesimals around α CenB, under the coupling of the bi-
TS =C (NR2)−1,
nary’s gravity and gas drag force. Planetesimals with col S P
orafn0do−m5e×cc1en0t−r5ic,itainesdorfan0d−om10r−a4d,iiraonfd1om−i1n0clkinma,tioanres TcBol =CB(NRP2)−1 (4)
initially orbiting around α CenB from 0.5 AU to an as- where N and R are the number and typical radii of
P
sumedouterdisk-edgeof2.5AU.Thisouterdisk-edgeis planetesimals, and C and C are the coefficients for
S B
consistentwiththose values derivedfromArtymowicz & single-star systems and binary systems, respectively. In
Lubow (1994), Holman & Wiegert (1999), and Pichardo arealsystem,N istoolargetocalculate,thusweusually
et al. (2005). Gas drag is modeled by assuming a 3- take a system with less(such as n=103−104) particles
dimensional non-evolving axisymmetric gas disk as that butlargerradii(theinflatedradiusr=10−5−10−4AUin
inour previouswork(Xie & Zhou2009). Such a gas-disk typicalcases)forsimulations. Ithasbeenshown(Brahic,
modelneglectsthereactionofgasdisktothebinaryper- 1976) that a change in the number N of particles or in
turbations,whichcanfurtherincreaseplanetesimals’△V their radius R affects only the collisional rate of the
P
and inhibit their mutual accretion(Paardekooper et al. system,whichisproportionaltothe totalcollisioncross-
2008). Recently,Marzarietal. (2009b)foundgasdiskis section. Therefore, the collisional timescales of simula-
muchless excitedby the binaryperturbationsifthe disk tions t (namely the reverse of impact rate n ) have
col imp
self-gravity is considered. Therefore, our axisymmetric a scaling relation with the collisional timescales of real
gas disk model should be a reliable simplification. systems, such as:
Wetrackalltestplanetesimalcollisionsbysettingeach
body an inflated radius of 5 × 10−5(r /AU)1/2 AU, nS =1/tS =1/(TS NR2/nr2)=C−1nr2,
col imp col col P S
where r is the distance from the collision to α CenB.
col
This choice gives a precision of ∼1.5m s−1 in every im- nB =1/tB =1/(TBNR2/nr2)=C−1nr2. (5)
imp col col P B
pact velocity estimate. These velocity values are then
interpreted in terms of accreting or eroding encounters Combining Eq(4) and Eq(5), the scaling factor between
bycomparingthemtotwothresholdvelocities(V and TS and TB is written as:
low col col
V ,seedetailsinXie&Zhou2009). Thesetwothresh-
olhdigvhelocities are directly related to the specific critical TcBol =TcSol∗(nBimp/nSimp)−1, (6)
energy Q∗ of the planetesimal. Since Q∗ is a broad pa-
where nB /nS is called “normalized impact rate”.
rameterwithout being wellconstrained,we then derived imp imp
V and V from a pessimistic Q∗ and an optimistic SincenB ,nS canbe directly readoutfromoursimu-
low high imp imp
one, respectively(Th´ebault & Augereau 2007). For an lations (runs 0-24), we therefore use Eq(3) and Eq(6) to
example, if △V is greater than Vhigh, then the collision estimate TB .
probably leads to erosion rather than accretion. col
In total, 25 simulations(see the Table 1) were 2.3. Results
performed with the consideration of different initial
2.3.1. radial migration and distribution
conditions such as: gas disk density(M ), binary’s
d
inclination(i ), and gas depletion. These include three Figure 1 shows the planetesimal number distributions
B
differentgasdensitycases: low(0.3MMSN),medium(1.0 at three different epochs for the 24 binary cases. At
MMSN) and high cases (3.0 MMSN), where MMSN de- the beginning, planetesimals orbit the central star with
notestheMinimumMassofSolarNebula(Hayashi1981). a uniform distribution(1000 per 0.1 AU, and 20000 in
For the binary’s inclination, we study four cases (i =0, total) from 0.5 to 2.5 AU. As they are subject to the
B
0.1o, 1o and 10o) since i is probably within 10o for bi- gas drag force, under which they migrate inward, the
B
nary systems closer than 30-40 AU (Hale 1994). The distribution then changes. The major features of these
timescale of gas depletion is the same as that used in distribution changes, as shown in figure 1 are:
Xie & Zhou(2008)asM ∝(t/τ )−1.5 with τ =105 1)For i ≤ 1o, their evolutions of planetesimal radial
d dep dep B
yr. The “run 00” is simulated for a singe-star system, distributions are very similar because their inclinations
fromwhichthe impactrateis normalizedto inthe other aretoosmalltoinducesignificantdifferenceinradialmi-
24 binary cases. The evolving timescale of “run 00” is grationrate. Ontheotherhand,fori =10o (10o ∼0.2
B
104 yrs,whichis longenoughto findenoughimpacts for radian, which is comparable to the binary eccentricity
statistics. For the remaining binary cases, the evolving of 0.52), the planetesimal migration rate are highly in-
timescale is set to 2×105 yrs, since TB is expected to creased, resulting in a significant mass loss near 1 AU.
col
be much longer in inclined binary cases than TS . 2)Since denser gasleadsto strongergasdragandthen
col
faster inward drift, planetesimal inward migrations are
2.2. Estimate of TB: Scaling Method much more significant in high-mass gas disks without
col depletion than in low-mass gas disks with depletion.
4
3)Inninecases(6low-massgasdiskcaseswithi ≤1o, 2.3.3. accretion ratio
B
and 3 medium-mass gas disk cases with depletion and
We define a possible accreting collision if its collision
i ≤ 1o, namely runs 1,7,13, 4,10,16, and 5,11,17), very
B velocity is less than V (see the appendix of Xie &
few planetesimals were lost by inward drift in the habit- high
Zhou, 2009 for details). Figure 4 shows the fraction
ablezone(0.5-0.9AU)andouterregion(suchas1-2AU).
ofpossibleaccretingcollisions(hereafter,accretionratio)
There remains sufficient material for planet formation.
during three episodes for the 24 runs. The majorresults
are summarized as follows.
2.3.2. normalized impact rate: nBimp/nSimp 1)During the first episode (t=0-7×104 yrs), a)for the
Figure2showsthenormalizedimpactrate(nB /nS ) coplanar cases(runs 1-6), the accreting collisions only
imp imp weigh ∼10-20% because planetesimals’ orbital differen-
as a function of radial distance to the central star(α
tial phasing increases their relative velocities(Th´ebault
CenB). The entire 2×105 yrs evolution is divided into
et al. 2009); b)for the small inclined cases(runs 7-18)
3 phases. As shown in figure 2, the major features are with i =0.1o and 1o, the weights of accreting collisions
summarized as follows. increasBeto∼40-50%becausesmallbinaryinclinationre-
Comparing different cases of different binary
duces the relative velocities by separating the orbits of
inclinations(i = 0,0.1o,1.0o,10o, from left to right,
B bodies with different sizes(Xie & Zhou 2009); c)for the
column by column), we find: largeinclinedcases(runs19-24)withi =10o,theaccre-
B
1)Forthecoplanarcase(iB=0),theimpactrateinbi- tionweightsreduceto∼30%becausei =10oissufficient
narysystemsishighlyincreasedtoasmuchas1-2orders B
large to pump up considerable planetesimals’ orbital in-
of magnitude of the value in the single star case. This
clinations, which give an extra contribution to increase
result has also been observed in figure 3 of Xie & Zhou
the relative velocities.
(2009) and is explained as a result of the increase in the 2)Forthefollowingtwoepisodes(7×104<t<20×104
disk volume density inducedby the progressivedamping
yrs), we observe an evident increase in the accretion ra-
of planetesimals’ orbital inclinations.
tio between 1-2 AU in most cases independent of gas
2)Asthebinaryorbitbecomesmoreandmoreinclined,
depletion. Most of these increases can be up to 80%, or
the impact rates decrease progressively. And there ap-
evencloseto100%insomecasesduringthelastepisode:
pearstobeagoodtrend: asiB isincreasedbyanorderof t=14-20×104 yrs. However,for the two exceptions, runs
magnitude, the impact rate is reduced correspondingly.
19 and 22, we have not observed significant increase in
For an example, at 1.5 AU, the normalized impact rates
accretion ratio. Their gas density is only 0.3 MMSN
in the cases of iB =0,0.1o,1.0o,10o are, respectively: and the binary orbital inclination is as large as 10o. In
such conditions, planetesimals are pumped up to highly
nB /nS ∼50, if i =0.0,
imp imp B inclined orbits with little time to experience a consider-
able gas drag force. These cases become close to a gas-
nBimp/nSimp ∼5.0, if iB =0.1o, free case(see some similar cases in Marzari et al 2009a)
where conditions usually become accretion-hostile be-
nB /nS ∼0.5, if i =1.0o, cause of the progressiveorbital randomization(Th´ebault
imp imp B
et al. 2006).
nB /nS ∼0.05, if i =10o. (7)
imp imp B
3. DISCUSSIONS
Focusing on the evolution of the impact rates during
3.1. orbital re-phasing
these 3episodes(t=0−7×104 yrs,t=7−14×104yrs,
and t=14−20×104 yrs ), we find: As shown in figures 3 and 5, the orbital re-phasing is
independent of gas depletion in α CenB, and thus it is
3)In cases such as runs 3, 21, 24, impact rates are sig-
differentfromtheoneidentifiedbyXie&Zhou(2008)in
nificantly reduced with time because of the loss of most
γ Cephei system. To understand the origins and differ-
of their planetesimals caused by the inward drift.
ences of these two kinds of orbital re-phasing,we should
4)Despite similar mass loss by inward drift in runs 3,
review the dynamics of a planetesimal embedded in the
9, 15, impact rates in the later two inclined runs do not
gasdiskinbinarysystems(seedetailsinMazari&Scholl
drop as much as in the coplanar run 3. Furthermore,
2000,Th´ebaultetal. 2006,andXie&Zhou2008). Gen-
in most inclined cases(runs 8,12, 14, 18, for examples),
erally,theplanetesimal’sorbitalelementsfollowdamped
althoughmasslossesaresignificant,impactratesarenot
oscillations. In figure 6, for example, the planetesimal’s
reduced or even increased slightly.
eccentricity follows an oscillation with a frequency of µ
Thereasonforresult4)canbefoundinfigure3,which
(see Eq.(4) of Th´ebault et al. 2006) under the secular
shows evolution of planetesimal orbital elements (i,Ω)
perturbationsofthecompanionstar. Thenthetimescale
in cases of 14 and 17. At the beginning, planetesimals
of secular oscillation, T , can be expressed as
are excited to orbits with different i and Ω, resulting in e
reductioninthevolumedensityoftheplanetesimaldisk.
T =2π/µ
Then, their orbits have a progressive re-phasing process e
duetogasdamping. Thisre-phasingofiandΩincreases
thedisk’svolumedensity,andthenmaintainstheimpact = 4( a )−3/2(aB )3(1−e2)3/2(MA)1/2(MB)−1
rateaslongasthemasslosebygasdragisnottoomuch. 3 AU AU B M⊙ M⊙
For coplanar binary case with i = 0, there is no “re-
B
phasing”ofplanetesimals’ iandΩ,thus no maintenance
M
or enhancement in impact rate. × [1+32(a/a )2(1−e2)−3( B)]−1 yrs. (8)
B B M⊙
5
Because of the gas drag damping, it is also a damped (2008), which matches very well with the dependence of
oscillation with a damping timescale T , which can be a on f as shown in Eq(10).
d c g
expressed as(see the appendix for details)
3.2. mass left for planet formation
R a a (1−e2) M
Td =12fg−1(kmp)(AU)9/4(ABU) eBB (M⊙A)−1/2 yrs. bitOaulrre-rpehsualstisn(gfigoufraelslp3lananetdes5im)aslhsoawretrheaecheeffidcnieenatr 1o.r5-
(9)
AU of α CenB after ∼ 105 yrs, leading to a “possible
Here a and a are the semi-major axes of planetesimals
B planetesimal-accretionzone”there. Thisencouragingre-
and the binary star, M , M are the masses of the pri-
A B sult, however, may be challenged by several problems.
mary and companion, respectively, e is the orbital ec-
B Thefirstoneis,aspointedoutbyTh´ebaultetal. (2009),
centricity of the binary, and f is the scaling number of
g mostplanetesimals in the “possible accretionzone” may
gasdiskdensitywithrespecttoMMSN.Notethatfigure
be removed as a result of inward drift induced by gas
6 is just the solution based on secular perturbation the-
drag if the accretion-favorable episode comes too late.
ory(numericalsolutionofEq(15)andEq(16)inMarzari
Their results shows no objectsmaller than ∼4 km is left
& Scholl 2000). Although this solution ignores the ef- beyond 0.8 AU after 2 × 105 yrs. This issue, in fact,
fects of higher order and short period perturbations, it
is directly related to the size distribution of the initial
basically agrees with the direct N-body calculations as
planetesimal population, which is poorly constrained by
shown in figures 3 and 5. Besides, as this solution only
currentknowledgeontheplanetesimalformation. Inthis
considersthesecularperturbations,itssolutioncurvesin
paper,withtheassumptionofarandomsizedistribution
figure 6 are much more clear than that in figures 3 and
from 1 to 10 km, the inward drift problem is less severe
5, thus it is suitable for us to recognize and understand
or even ignorable in some cases, including runs 1, 4, 5,
orbital re-phasing.
7,10, 11,13,16, and 17. Furthermore,the outer edge of
As shownin figure 6, oscillationis over-dampedin the
planetesimal disk we considered (2.5 AU) is larger than
innerregionwheregasisdenserandthusstrongerdamp-
that(1.5AU) of Th´ebault et al.(2009). Therefore, in the
ing,whereasitisunder-dampedintheouterregion. The
possible accretionregion,say near 1.5 AU, the mass lost
critical distance(a ) to separate these two damped oscil-
c by the inward drift of local planetesimals can be replen-
lations can be roughly estimated by equating T (ignore
e ished by the planetesimals from outer disk(see the left
the high order terms) and T , from which we have
d paneloffigure7). Inaddition,iftheicelineisconsidered
1 R a at2 AU, the supplement ofplanetesimals is very consid-
ac ∼(9)4/15fg4/15(kmp)−4/15(ABU)8/15 erable because of the enhancement of solid mass beyond
the snow line(see the right panel of figure 7). In such a
[e (1−e2)]4/15(MA)4/15 AU. (10) case, gas inward drift does not induce a mass loss prob-
B B M lem but a mass enhancement favoring the planetesimal
B
accretionat1-2AUofαCenB.Fortheseconsiderations,
Although Eq(10) cannot give an accurate value of a , it
c weconclude,masslossbythe inwarddriftis negligibleif
showsthedependencesofac onaB,eB,fg,MA,andMB. M ≤1 MMSN and i <10o.
Generally, under-damped oscillation is more favored in d B
binary systems with high mass ratio(M /M ) and low
B A 3.3. accretion-hostile transition
density of gas disk.
Areliablea canbederivedfromfigure6bymeasuring The second problem is that even when mass is pre-
c
the position where the oscillation begins to over-damp, served in the possible accretion zone, say 1-2 AU, the
namely where the equilibrium eccentricity begins to de- accretion-favorableconditions showninfigure 4 areonly
part from the forced eccentricity. The vertical dashed reached after an accretion-hostile transition duration of
lines in figure 6 denote the location of a ≃ 1.6 AU for at least several 104 yrs. The question that remains, as
c
α CenB and of a ≃ 3.6 AU for γ Cephei. Beyond the pointed out by Th´ebault et al. (2009), is how these
c
a , planetesimals with R ≥ 1 km follow under-damped objects might survive this long erosion period. Dur-
c p
oscillationandeventuallyconvergetheireccentricitiesto ing this period, it is likely most large planetesimals will
e (forced eccentricity). This orbital re-phasing process be fragmented into small debris which will be quickly
f
onlyneedsprogressivedamping,anditisindependentof removed by the inward drift because of their small
gas depletion. Within the a , planetesimals with R ≥1 sizes. To understand the details of this erosion pro-
c p
begin to follow over-damped oscillation and force their cess one needs a N-body code including fragmentation
eccentricitiestodifferentequilibriumsdependingontheir which is out of the scope of this paper. Nevertheless,
radii. In this region, orbital re-phasing will be never we can reach some qualitative conclusions by compar-
reached unless the gas density depletes to some low val- ing the collisional timescale(TB) with the duration of
col
ues. InγCepheicase,over-dampedoscillationdominates the accretion-hostile transition(Taht). The argument is
thewholeplanetesimalsdisk,thereforeorbitalre-phasing the following: if the disk is fragmented into small de-
requiregasdepletionasshowninXie&Zhou(2008). On bris,everyplanetesimalshouldhaveexperiencedatleast
the other hand, in α CenB case, under-damped oscilla- one erosive collision, or in other words, the duration of
tionissignificantbeyond1.5AU,thereforetheorbitalre- theaccretion-hostiletransitionshouldbelongerthanthe
phasingisindependentofgasdepletionasseeninfigures collisional timescale(Taht > TcBol). On the other hand, if
3 and 5. It is worthy to point out that the over-damped TB > T , then the planetesimals are able to survive
col aht
region in γ Cephei would shrink to a ≃2 AU if assum- the accretion-hostile transition . As shown in figure 4,
c
ingalow-massgasdiskofonly1MMSNratherthanthe TB should be at least 105 yrs to survive the accretion-
col
massive gas disk of 10 MMSN adopted by Xie & Zhou hostile transition.
6
According to Eq.(1), the collisional timescale in sin- than their escape velocities. The accretion may proba-
gle star systems is about TS = 104−105 yrs for 1-10 blyfollowatypeIIrunawaymodewhichischaracterized
col
km-sized planetesimals at 1.5 AU where the center of withinitially asloworderlygrowthandthen switchesto
possible accretion zone is located(see figure 4). If we runawaymodewheneverthebiggestbody’sescapeveloc-
take a medium value of TS = 5×104 yrs, then adopt- ityislargerthan△V(Kortenkampetal. 2001;Th´ebault
ingthescalingfactorsofEcqo(l6)andEq(7),thecollisional et al. 2006, 2009; Xie & Zhou 2009). However, even
timescalesat1.5AUfordifferentbinarycasesareroughly for the type II runaway mode, planetesimal accretion,
estimated as: is significantly slowed down compared to the normal ac-
cretion or type I runaway accretion. As shown in figure
TcBol ∼103 yrs, if iB =0.0, 2 of Kortenkamp et al. 2001, the planetesimal accre-
tion timescales are ∼ 105 yrs and ∼ 5×105 yrs at 1.5
TB ∼104 yrs, if i =0.1o,
col B and2 AU,respectively. Althoughthese resultsfromKo-
TB ∼105 yrs, if i =1.0o, rtenkampet al. 2001are basedon the Solar-Jupitersys-
col B tem, they have shown a similar mechanism(the type II
TB ∼106 yrs, if i =10o, (11) runaway mode) which can be applied to α CenB.
col B
Besides, accretion process is further slowed down by
or expressed as a simple formula as: binaryinclination, which stirs the planetesimaldisk to a
larger scale height with lower volume density, and thus
TB ∼(1+100i )×103 yrs, if 0≤i ≤10o. (12)
col B B leading to lower impact rate. For the possible accre-
Therefore, for the cases with iB > 1.0o (runs 13-24), tion favorable cases(iB = 1o − 10o) discussed in the
collisional timescale is long enough for planetesimals to last subsection, the collisional timescale is as long as
survive the accretion-hostile transition. Planetesimals TB ∼ 105 − 106 yrs, i.e., it takes at least 105 − 106
col
are probably able to accrete into planetary embryos for yrs to form planetary embryos or planetary cores. On
the cases with 1o <i <10o, while cases with i ≥10o the other hand, gas disks in close binary systems may
B B
arelessfavorableforplanetesimalaccretion,eventhough deplete veryfast(Ciezaet al. 2009). Forbinarysystems
theirT arelongenoughtosurvivetheaccretion-hostile such as α CenB with a separation of only ∼20 AU, the
col
phase. The reasons are the following: a)the accretion gas depletion timescale can be as short as a few 105 yrs.
timescale is so long(T ≥ 106 yrs if i ≥ 10o) that al- Therefore,in the case of α CenB, although planetesimal
col B
mostalltheplanetesimalswouldberemovedbeforethey accretion is possible, the accretion timescale would be
have accreted into bigger ones(see cases 20, 21, 23, 24 comparable or longer than the gas depletion time scale.
in figure 1), and b) i ≥ 10o would lead systems close In such a case, even if Earth-like planets or planetary
B
to gas-free case, in which conditions become accretion- cores could be formed, there would be no gas left for
hostile(see cases 19, 22 in figure 4). themtoaccreteintogaseousplanet. Thisargumentmay
For the coplanar cases(i = 0), as shown in figure probablyaccountfor the fact that no gas gianthas been
B
4(runs 1-6), more than 80% collisions lead to erosion at found orbiting α CenB(Guedes et al. 2008).
thefirstseveral104yrs. This“erosion-dominatedphase” 4. CONCLUSIONS
is longer than the collisional timescale by 1-2 orders of
We perform a series of simulations to investigate the
magnitude,thusplanetesimaldiskwillbeefficientlyfrag-
conditions for planetesimal accretion from 0.5-2.5 AU
mentedintodebrisbeforeconditionstobecomefavorable
around α CenB. In all of the simulations, we vary
for planetesimal accretion. This resultis consistent with
the gas-disk density(0.3-3 MMSN), binary inclination(0-
the conclusion of Th´ebault et al. (2009)
10o), and consider both the cases with and without
Forthesmallinclinedcases(i =0.1o,runs7-12),dur-
B gas depletion. Three indicators−planetesimal num-
ingthephaseofthefirstseveral104 yrs(1−7×104yrsas
ber(or mass) distribution, planetesimal impact rate(or
in figure 4) the erosion collision still weighs a large part
collisional timescale), and the fraction of accreting
(∼ 60%), but not as dominant as in the coplanar cases.
collisions− are used in our discussions to determine
The chances for accretion and erosion are nearly equal
whether planetesimals can accrete into planetary em-
during this period, and thus the final result of the colli-
bryos or planets around α CenB. Our major conclusions
sionalevolutionduringthisphaseisuncertain. This“un-
are summarized as following:
certain phase” dominates the fate (accretion or erosion)
1)Mostplanetesimals areable to remainin the “possi-
of the planetesimal disk, since its duration (∼ 105 yrs)
bleaccretionzone”(1-2AUaroundαCenB)againstthe
is much longer than the collisional timescale(Tcol ∼ 104 gas drag induced inward drift if the gas density is rela-
yrs,ifiB =0.1o). Forthisreason,thecases(iB=0.1o)are tively small (0.3-1.0 MMSN) and the binary inclination
marginalcases,forwhichwecannotdrawanyconclusion is not too large(i <10o).
B
about whether planetesimals are possible to accrete into 2)For all the cases with and without gas depletion,
planetary embryos or not. within a timescale of ∼105 yrs, we have observedan or-
bitalre-phasingforplanetesimalsat∼1−2AUaroundα
3.4. planetesimals grow slowly
CenB(see figures 3 and 5), leading to an accretionfavor-
As discussed above, planetesimal accretion is possi- ableconditionthere(seefigure4). Thisorbitalre-phasing
ble, most probably at 1-2 AU around α CenB if 1o < is caused by the under-damped oscillation of planetesi-
i <10o. Nevertheless,the accretionprocess in α CenB mals’ eccentricities and inclinations, which is indepen-
B
will be never like the normal accretion around a single dent of gas depletion and thus different from the orbital
star(type I runaway mode), since most relative veloci- re-phasing identified as in γ Cephei (Xie & Zhou 2008).
tiesamongplanetesimalsarepumpeduptovalueslarger 3)We develop a simplified scaling method to estimate
7
the collisionaltimescale in binarysystems, TB . We find if the disk is initially composedof lunar-massplanetary-
col
TB sensitively depends on the binary inclination(i ) as embryos. The possible accretion zone shown in this pa-
col B
TB ∼(1+100i )×103 yrs,if 0≤i ≤10o,at1−2AU per is roughly between 1-2AU, whichmatches wellwith
col B B their planet formation zone(∼0.5-2.0 AU, as shown in
around α CenB. Based on the estimate of TB, we find,
col figures 1 and 2 of Guedes et al. 2008). In addition,
planetesimalsareprobablyabletosurvivethe accretion-
at the time of writing this paper, we note a promising
hostile transition period in the first 105 yrs and accrete
result from Payne et al. (2009) that Earth-like planes
into larger planetary embryos if 1o <i <10o.
B can also form in the habitable zone of α CenB-like bi-
4)Planetesimal accretion into planetary embryos in α
nary systems through outward migration from the inner
CenB,thoughpossible, takea long timescale as muchas
accretion-unperturbered zone(within ∼ 0.7 AU). There-
105 −106 yrs, which is 1-2 orders of magnitude longer
fore, by combining these studies(Guedes et al. 2008;
than that in normal accretion process in single-star sys-
Payne et al. 2009) and our simulations, it is quite pos-
tems. Therefore,the formation of gaseous giant planets,
sible that a habitable Earth-like planet may be hidden
likeJupiterandSaturn,arenotfavoredinαCenB,since
around α CenB.
gasdepletesveryfastinclosebinarysystems(Ciezaetal.
2009) in a timescale as short as 105 yrs.
In summary, although planetesimal accretion in α
We thank Th´ebault, P. for discussions and valuable
CenB is significantly less efficient and slowed-down as
suggestions. This work was supported by the National
compared to single star systems, it is still possible if
Natural Science Foundation of China (Nos.10833001,
gas density is 0.3-1.0 MMSN and binary inclination is
10778603 and 10925313), the National Basic Research
1o < i < 10o. These accretion favorable conditions, in
B Program of China(No.2007CB814800), NSF with grant
fact, aretypicalvaluesfor initial gasdensity(Andrews &
AST-0705139, NASA with grant NNX07AP14G, W.M.
Williams2005)andbinaryinclination(Hale1994,Jensen
Keck Foundation and also University of Florida.
etal. 2004,Moninetal. 2004,2006)fromcurrentobser-
vations. Our results support recent work by Guedes et
APPENDIX
al. (2008), which has shown Earth-mass planets can be
formed near the habitable zone(0.5-0.9 AU) of α CenB
APPENDIX
GAS DAMPING TIMESCALE: T
D
According to Adachi, Hayashi, & Nakazawa (1976), the average decay of planetesimal eccentricity caused by gas
drag can be given as
1 de 1 5 1
( )=− ( e2+ i2+η2)1/2, (A1)
e dt τ 8 2
aero
where η ∼ 0.002(a/AU)2, which describe the sub-Keplerian motion of gas, and e, i are the orbital eccentricity and
inclination of the planetesimal. τ is the characteristic timescale, which is related to the planetesimal size and gas
aero
density. With MMSN modeling the gas disk, τ can be typically expressed as
aero
R a M
τ =15f−1( p)( )13/4( A)−1/2 yrs, (A2)
aero g km AU M⊙
where R and a are the radius and semi-major axis of the planetesimal, M is the mass of primary star, and f is
p A g
the scaling number of gas density with respect to MMSN. In the binary case of our interest(slightly inclined binary
systems), e is much larger than i and η, thus the time scale for damping e is
de 8 τ
T =e/ ∼( )1/2 aero. (A3)
d
dt 5 e
As long as eccentricity is dominated by secular perturbations, we can take e as an average of forced eccentricity (see
Th´ebault et al. 2006) as
5 a e
B
e = , (A4)
forced 4a 1−e2
B B
then we finally get the damping timescale
R a a (1−e2) M
T =12f−1( p)( )9/4( B ) B ( A)−1/2 yrs. (A5)
d g km AU AU eB M⊙
Note this formula will be invalid if e is close to zero, since the evolution is then dominated by others, such as short
B
period perturbations, rather than by secular perturbations.
REFERENCES
Andrews,S.M.,&Williams,J.P.2005,ApJ,631,1134 Barbieri,M.,Marzari,F.,Scholl,H.,2002,A&A,396,219
Artymowicz,P.,&Lubow,S.H.1994,ApJ,421,651 Barge,P.,&Pellat,R.1993,Icarus,104,79
8
Brahic,A.1976,JournalofComputational Physics,22,171 Monin,J.-L.,M´enard,F.,&Peretto, N.2006,A&A,446,201
Cieza,L.A.,etal.2009,ApJ,696,L84 Monin,J.-L.,Clarke,C.J.,Prato,L.,&McCabe,C.2007,
Desidera,S.,&Barbieri,M.2007,A&A,462,345 ProtostarsandPlanets V,395
Duchˆene, G.,DelgadoDonate, E.,Haisch,K.E.,Jr.,Loinard,L., Paardekooper,S.-J.,Th´ebault,P.,&Mellema,G.2008, MNRAS,
Rodrguez,L.F.2007, 379,inProtostarsandPlanetsV,eds.B. 386,973
Reipurth,D.Jewitt,&K.Keil(Tucson: Univ.ArizonaPress) Payne,M.J.,Wyatt, M.C.,&Th´ebault, P.2009,MNRAS,1550
Duquennoy, A.&Mayor,M.1991, A&A,248,485 Pichardo,B.,Sparke,L.S.,&Aguilar,L.A.2005,MNRAS,359,
Eggenberger,A.,Udry,S.,&Mayor,M.2004,A&A,417,353 521
Endl,M.,Ku¨rster,M.,Els,S.,Hatzes,A.P.,&Cochran,W.D. Pourbaix,D.,etal.2002,A&A,386,280
2001, A&A,374,675 Queloz,D.,etal.2000, A&A,354,99
Goldreich,P.,&Ward,W.R.1973,ApJ,183,1051 Quintana,E.V.,2004,Ph.D.thesis,Univ.Michigan
Greenberg,R.,Hartmann,W.K.,Chapman,C.R.,Wacker, J.F. Quintana,E.V.,&Lissauer,J.J.2006,Icarus,185,1
1978, Icarus,35,1 Quintana,E.V.,Adams,F.C.,Lissauer,J.J.,&Chambers,J.E.
Guedes,J.M.,Rivera,E.J.,Davis,E.,Laughlin,G.,Quintana, 2007,ApJ,660,807
E.V.,&Fischer,D.A.2008,ApJ,679,1582 Quintana,E.V.&Lissauer,J.J.2007,inPlanetsinBinaryStar
Hale,A.1994,AJ,107,306 Systems,ed.N.Haghighipour(SpringerPublishingCompany)
Hatzes,A.P.,Cochran,W.D.,Endl,M.,McArthur,B.,Paulson, Rafikov,R.R.2003,AJ,125,942
D.,Walker,G.A.H.,Campbell,B.,&Yang,S.2003,ApJ,599, Rafikov,R.R.2004,AJ,128,1348
1383 Th´ebault,P.,&Augereau,J.-C.2007,A&A,472,169
Hayashi,C.1981, ProgressofTheoretical PhysicsSupplement, Th´ebault,P.,Marzari,F.,&Scholl,H.2006,Icarus,183,193
70,35 −,2008,MNRAS,388,1528
Holman,M.J.,&Wiegert,P.A.1999, AJ,117,621 −,2009,MNRAS,393,L21
Jensen,E.L.N.,Mathieu,R.D.,Donar,A.X.,&Dullighan,A. Weidenschilling,S.J.,&Cuzzi,J.N.1993,inProtostarsand
2004, ApJ,600,789 PlanetsIII, ed.E.H.Levy&J.I.Lunine(Tucson: Univ.
Kokubo,E.&Ida, S.1996,Icarus,123,180 ArizonaPress),1031
−1998, Icarus,131,171 Wertheimer,J.G.,&Laughlin,G.2006, AJ,132,1995
−2000, Icarus,143,15 Wetherill,G.W.,Stewart,G.R.1989,Icarus,77,330
Kokubo,E.,YoshinagaK.,&Makino,J.1998,MNRAS,297,1067 Xie,Ji-Wei,&Zhou,Ji-Lin,2008, ApJ,686,570
Kortenkamp,S.J.,Wetherill,G.W.,&Inaba,S.2001,Science, −,2009,ApJ,698,2066
293,1127 Youdin,A.2008,arXiv:0807.1114
Marzari,F.&Scholl,H.2000,ApJ,543,328 Zucker,S.,Mazeh,T.,Santos,N.C.,Udry,S.,&Mayor,M.2004,
Marzari,F.,Th´ebault,P.,Scholl,H.2009a,A&A,inpress A&A,426,695
Marzari,F.,Scholl,H.,Th´ebault,P.,Baruteau, C.,2009b, A&A,
inpress
Mathieu,R.D.,Ghez, A.M.,Jensen,E.L.N.,Simon,M.2000,
703,inProtostarsandPlanetsIV,eds.Mannings,V.,Boss,
A.P.,Russell,S.S.(Tucson:Univ.ArizonaPress)
9
TABLE 1
InitialConditions for25 Simulations.
Run iB Md τdep
00 - 1 +∞
01 0 0.3 +∞
02 0 1 +∞
03 0 3 +∞
04 0 0.3 105
05 0 1 105
06 0 3 105
07 0.1 0.3 +∞
08 0.1 1 +∞
09 0.1 3 +∞
10 0.1 0.3 105
11 0.1 1 105
12 0.1 3 105
13 1 0.3 +∞
14 1 1 +∞
15 1 3 +∞
16 1 0.3 105
17 1 1 105
18 1 3 105
19 10 0.3 +∞
20 10 1 +∞
21 10 3 +∞
22 10 0.3 105
23 10 1 105
24 10 3 105
Note. —TheunitsofiB,Md andτdep aredeg,MMSN,andyr respectively.
10
Fig.1.— Planetesimal number distribution (N/(0.1 AU)) vs radial distance (AU) to the central star. Trapezoid: t=0, dashed-dot:
t=3×104 yr,dashed: t=10×104 yr,andsolid: t=18×104 yr