On the Likelihood of Planet Formation in Close Binaries

0.28 MB·English

by  Hannah Jang-Condell

ToAppear in The Astrophysical Journal
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
ON THE LIKELIHOOD OF PLANET FORMATION IN CLOSE BINARIES
Hannah Jang-Condell
UniversityofWyoming,DepartmentofPhysics&Astronomy
1000E.University,Dept3905, Laramie,WY82071
To Appear inThe Astrophysical Journal
ABSTRACT
Todate,severalexoplanetshavebeendiscoveredorbitingstarswithclosebinarycompanions(a.30
5
AU).Thefactthatplanetscanforminthesedynamicallychallengingenvironmentsimpliesthatplanet
1
formation must be a robust process. The initial protoplanetary disks in these systems from which
0
planetsmustformshouldbetidallytruncatedtoradiiofafew AU,whichindicatesthattheefficiency
2
of planet formation must be high. Here, we examine the truncation of circumstellar protoplanetary

n disks in close binarysystems, studying how the likelihoodof planet formationis affectedovera range
a of disk parameters. If the semimajor axis of the binary is too small or its eccentricity is too high,
J
the disk will have too little mass for planet formation to occur. However, we find that the stars

4 in the binary systems known to have planets should have once hosted circumstellar disks that were
capable of supporting planet formation despite their truncation. We present a way to characterize

] thefeasibilityofplanetformationbasedonbinaryorbitalparameterssuchasstellarmass,companion
P
mass, eccentricity and semi-major axis. Using this measure, we can quantify the robustness of planet
E
formationinclosebinariesandbetter understandthe overallefficiencyofplanetformationingeneral.
.
h Subject headings: planets: formation — protoplanetary disks — binaries: close
p
-
o 1. INTRODUCTION Dumusque et al. 2012; Hatzes 2013), and HD 41004. In-
r
terestingly,HD41004hassubstellarcompanionsinboth
t Close binary systems are a challenging environment
s A and B components: HD 41004 Ab is a 2.5 M planet
a for planet formation because a binary companion will J
(Zucker et al. 2004) and HD 41004 Bb is a brown dwarf
[ dynamically perturb the protoplanetary disk. However,
(Zucker et al. 2003). The binary orbits and planet char-
severalexoplanets have been found in S-type orbits (cir-
1 acteristicsof the systems discussed here arelisted in Ta-
cumstellar) around stars in binary systems with semi-
v ble 1.
major axes of less than 30 AU (e.g. Raghavan et al.
7 Onewaytostudytherobustnessofplanetformationin
2006). These systems represent interesting test cases
1 binaries is to study the long-term dynamical stability of
for planet formation theories. By comparing the fre-
6 those planets Holman & Wiegert (e.g. 1999). However,
quency of planets in binary systems with that around
0 thefocushereinthisworkisonthe formationofplanets
single stars, we can learn about how robust the planet
0
in the first place. Rather than doing a detailed calcula-
. formation process can be. In this paper, we address the
1 topicofcircumstellarplanets(S-typeorbits),asopposed tionofthe dynamicsofplanetformationasin,forexam-
0 ple, Thebault (2011), we want to understand planet for-
to circumbinary planets (P-type orbits).
5 mationmorebroadlybyconsideringtheeffects ofbinary
HD 188753 is a hierarchical triple system, with a 12.3
1 interactionswiththeinitialprotoplanetarydisk. Thatis,
AUseparationandorbitaleccentrictyof0.5betweenthe
: given that the protoplanetary disk will be truncated by
v A and BC components. Konacki (2005) claimed the de-
a binary companion, what is the likelihood that planet
i tection of planet aroundthe A component. However,we
X formation can take place at all?
foundthatthebinaryparametersforHD188753Adonot
Calculations of disk truncation by binary companions
r allowforinsituplanetformationtooccur(Jang-Condell
a show that stars in binary systems are still able to retain
2007,henceforthPaperI).Indeed,itwaslaterfoundthat
circumstellar disks (Artymowicz & Lubow 1994). This
there was no planet there after all (Eggenberger et al.
is supported by millimeter and submillimeter observa-
2007).
tions of pre-main sequence binaries which show that bi-
On the other hand, γ Cep has a separation of 20
nary stars with separations of tens of AU are able to re-
AU and eccentricity of 0.4, and has a well-established
taintheirdisks(Jensen et al.1996;Osterloh & Beckwith
gas giant planet orbiting the A component at ∼2 AU
1995; Jensen et al. 1994; Beckwith et al. 1990).
(Hatzes et al. 2003; Endl et al. 2011). Although at
Theobjectiveofthispaperistogiveameasureforthe
first glance, its orbit is not very different from that of
difficulty of planet formation across a range of binary
HD 188753, we found that the binary parameters of
starparameters. WeconsiderS-typeorbits,inwhichthe
γ Cep A do allow in situ planet formation to occur
planetorbitsoneofthestarsinthebinarysystem,asop-
(Jang-Condell et al. 2008, henceforth Paper II).
posedtoP-typeorbits,whichrefertocircumbinaryplan-
Circumstellar planets have been discovered in a few
ets. We examine the initial protoplanetary disk around
other binary systems of comparable separations. These
a given star and calculate the degree of truncation of
include HD 196885 (Correia et al. 2008; Fischer et al.
this disk. Rather than determining where exactly plan-
2009;Chauvin et al.2011), αCen(Pourbaix et al.2002;
ets should form in these truncated disks, we focus on
whether planet formation is possible at all.
[email protected]
2 JANG-CONDELL
TABLE 1
Binarysystems analyzedin thispaper.
HD188753A γCepA HD41004A HD41004B HD196885A αCenB
M∗ (M⊙) 1.06 1.40 0.7 0.4 1.31 0.93
Mb (M⊙) 1.63 0.41 0.4 0.7 0.45 1.1
µ 0.39 0.77 0.64 0.36 0.74 0.46
a(AU) 12.3 20.2 20 20 21 23.5
e 0.5 0.41 0.4 0.4 0.42 0.52
mpsini(MJ) — 1.85 2.5 18.4 2.96 0.0034
ap (AU) — 2.1 0.006 7×10−4 2.6 0.04
ep — 0.05 0.7 0.08 0.46 0
reference a b b,c d e f
NM 9 12 12 12 12 12
NDI 0 1 1 1 1 1
NCA 0 8 7 4 8 3
Note. —Orbitalparameterswithoutsubscripts(aande)refertothoseofthebinary. Thesubscript‘p’referstoplanetaryorsubstellar
companions.
References. — (a)Eggenbergeretal.(2007),(b)Endletal.(2011),(c)Zuckeretal.(2004),(d)Zuckeretal.(2003),(e)Correiaetal.
(2008),and(f)Dumusqueetal.(2012)
In this paper, we extend the methods of determining
TABLE 2
thefeasibility ofplanetformationinclosebinariesasde- Stellar parametersasa function of stellar mass
veloped in Papers I and II to a wider range of binary
star configurations. For a given set of binary parame- Stellarmass Teff R∗ L∗†
ters,wedevelopaformalismfordescribingthefeasibility (M⊙) (K) (R⊙) (L⊙)
0.5 3770 2.1 0.8
of planet formation as measured by the number of disk
0.7 4026 2.4 1.4
models that wouldallow foreither coreaccretionordisk
0.9 4209 2.5 1.8
instability to occur.
1.0 4282 2.6 2.0
2. METHODS 1.1 4343 2.7 2.3
1.3 4447 2.9 3.0
2.1. Truncated Disk Models
1.5 4536 3.1 3.7
Webaseourestimatesofthelikelihoodofplanetforma- 1.7 4616 3.3 4.4
tionforagivenbinarysystemontherangeofdiskparam- 1.9 4690 3.4 5.0
2.0 4721 3.5 5.5
etersthat permit enoughmass inthe disk for planetfor-
mation to occur, given that a circumstellar protoplane- †DerivedfromTeff andR∗.
tarydiskwillbetruncatedbyabinarycompanion. These
methods are explained fully in Papers I and II, and are
for a total of 18 disk models for each set of binary pa-
summarized briefly here.
For convenience, we refer to the planet-hosting star rameters (M∗,µ,a,e). The lower accretion rates reflect
values typically observed in T Tauri disks. The highest
as the primary and the binary stellar companion as the
accretionratesaremoretypicalofactiveFUOrionis-like
secondary, regardless of the relative masses between the
objects or very young protoplanetary disks than of pas-
binary components. Accordingly, M∗ and Mb are the sive T Tauri disks. Nevertheless, we include these high
masses of the primary and secondary, respectively. The
values of accretion for completeness.
massratioofthebinaryisdefinedasµ=M∗/(M∗+Mb). A given value of α sets the viscosity, according to
The binaryorbitis definedbythe semimajoraxisa,and
eccentricity e. ν =αc H (3)
s
The primary mass, M∗, affects both the gravitational
potentialofthediskaswellastheamountofstellarirra- where cs is the thermal sound speed and H is the scale
height of the disk (Shakura & Sunyaev 1973). These
diation of the disk since luminosity varies strongly with
stellar mass. We consider a range of masses for the pri- quantities are related by H = csΩ where Ω is the Ke-
mary star from 0.5 to 2.0 M⊙. To calculate the lumi- plerian angular speed, and cs = kT/µ¯ where k is the
nosityoftheprimarystar,neededasaninputparameter Boltzmannconstant,T isthemidpplanetemperature,and
forthediskmodels,weusepre-mainsequencemodelsfor µ¯isthemeanmolecularweightofthegas,whoseassumed
stars of 1 Myr of age (Siess et al. 2000). The values for compositionisthatofmolecularhydrogen. Theviscosity
Teff, R∗ and L∗ for each value of M∗ selected are listed sets the rate of the mass flow through an accretion disk,
in Table 2. according to
The two additional parameters that determine the M˙
properties ofa disk are the mass accretionrate (M˙ ) and νΣ= 3π (1−r/R∗) (4)
viscosityparameter(α). Foreachstellarmass,wecalcu-
where Σ is the surface density, r is the stellocentric ra-
late a suite of disk models by varying α and M˙ , as
dius,andR∗ isthestellarradius(see,e.g.,Pringle1981).
α∈{0.001,0.01,0.1} (1) Ifweassumethattheaccretionrate,M˙ ,isconstantwith
r,thenEq.4givesarelationbetweenT andΣinthedisk.
and
Assuming that the only heating sources in the disk are
M˙ ∈{10−9,10−8,10−7,10−6,10−5,10−4}M⊙yr−1 (2) viscous accretion and stellar illumination, we calculate
ON THE LIKELIHOOD OF PLANET FORMATION IN CLOSE BINARIES 3
a circular orbit is 10 AU, as indicated by a gray line in
Figure 1. However,if the binary companion is eccentric,
the truncation radius varies with disk properties. This
is becausethe Reynoldsnumber is differentin eachdisk,
therefore the balance of tidal and viscous torques occurs
atdifferentradii. The truncationradiifor abinaryorbit
with µ = 0.5 and a = 30 AU with e = 0.3 (0.6) is
shown by hollow (filled) symbols in Figure 1. As the
eccentricity increases, the truncation radii move inward
asexpected,sincetheperiastronoccursatsmallerradius
with increasing eccentricity.
The disk profilesshownin Figure 1 varymore with M˙
and α than with stellar mass. As the stellar mass in-
creases, the disks become slightly more massive overall,
but with less dust and higher values of Q. This is be-
causetheincreasedstellarilluminationheatsthedisksto
higher temperature, which affects both disk sublimation
and pressure support of the gas.
The Toomre Q parameter decreases with radius in
most models of protoplanetary disks as a general rule.
In the disk models presented, the midplane temperature
and suface density typically behave as T ∝ r−0.6 and
c
Fig. 1.— Disk profiles for the least massive (0.5 M⊙, left) Σ ∝ r−0.9, respectively. Since Ω ∝ r−3/2, the overall
and most massive (2.0 M⊙, right) planet host stars examined variation of Q is to decrease with r,
in this work. We show the total disk mass (top), total dust
mass (middle), and Toomre Q parameter (bottom) versus radius
for the disk parameters explored. The accretion rate (M˙) is 2.2. Core Accretion vs. Disk Instability
indicated by line color, while values of the viscosity parameter of
α=0.001,0.01,and0.1areindicatedbysolid,dotted,anddashed Given a set of parameters defining a particular binary
lines. Thetruncationradiusforeachdiskforanequal-massbinary system(M∗, µ, a, ande), wecancounthowmany ofthe
companion (µ= 0.5) at a=30 AU is shown by the vertical gray 18 truncated disk models constructed around the pri-
linefore=0,byhollowsymbolsfore=0.3andfilledsymbolsfor
mary star (1) have sufficient total mass to form planets,
e=0.6.
(2) have sufficient solid material to form a planet core,
or (3) have low enough value of Q for disk instability to
occur. If either condition (2) or (3) is satisfied, then (1)
T(r) and Σ(r) self-consistently. Thus, α and M˙ can be is also satisfied,so we henceforth consider only the more
selected to “tune” the mass of the disk. stringent criteria for planet formation based on core ac-
The truncation radius for each of the 18 disk models cretion and disk instability, rather than the total disk
for a selected primary mass is determined from values mass. We define the number of disks in which core ac-
calculated in Artymowicz & Lubow (1994), who define cretionis possible basedon the mass ofsolidmaterialto
the truncation radius to be where viscous torques bal- beN andthenumberofdisksinwhichdiskinstability
CA
ance the tidal torques. This truncation radius depends is possible based on the value of Q to be N .
DI
bothontheReynoldsnumber,whichiscontrolledbythe In the core accretion process of planet formation, gi-
viscosityν,andthebinaryorbit,representedbythemass antplanetsareformedbyfirstcreatingarockycorefrom
ratio and eccentricity. That is, for each set of M˙ and α the agglomerationof dust particles in the disk. When it
for the disk, and M∗, µ, a, and e for the binary, there is reaches sufficient mass, typically around 10 M⊕, it be-
a unique disk truncation radius. comes massive enough to accrete and retain a gaseous
InFigure1,weshowhowthestructureofthedisksvary atmosphere, and gas giant planet is born. Our criterion
with input parameters. The two sets of plots show disk for determining that a given truncated disk model is ca-
profilesforthe leastmassive(0.5M⊙)andmostmassive pable offorminggiantplanets by coreaccretion,then, is
(2.0 M⊙) stars explored. Each line represents a selected thatitcontainsatleast10M⊕ofsolidmaterials,i.e.dust
valueofM˙ andαforagivendiskmodel,asindicatedby andice. If the totalamountof dust ina giventruncated
colorandlinetypeaccordingtothelegend. Foreachdisk disk model is above 10 M⊕, then we say that that disk
model, we show the enclosed disk mass (top), total dust is capable of supporting planet formation by core ac-
mass (middle), and Toomre Q parameter (bottom) as a cretion. So, NCA is the total number of truncated disk
function ofradius. The ToomreQ parameteris givenby models thatcansupportplanet formationbycoreaccre-
tion for a given set of binary parameters. For example,
c κ
Q= s , (5) NCA for M∗ = 0.5M⊙, µ = 0.5, a = 30 AU, and e = 0
πGΣ can be read off the dust mass plot in Figure 1 (left cen-
where κ is the epicyclic frequency and G is the gravita- ter)bycountingthenumberofprofilesthatintersectthe
tionalconstant. Sincethediskorbitalvelocitiesareclose truncation radius (gray line) at 10 M⊕ or above, giving
to Keplerian, we assume that κ≈Ω. NCA =12. Whentheeccentricityisraisedtoe=0.3,the
If the binary orbit is circular, then all disks will be solid content of the disk model with M˙ =10−6M⊙yr−1
truncated at the same radius. The truncation radius for and α=0.1 drops below 10 M⊕, so NCA =11.
an equal-mass binary companion (µ = 0.5) at 30 AU in The criterion for planet formation by disk instability,
4 JANG-CONDELL
on the other hand, is determined by the value of the refractory species sublimate and those disks have lower
Toomre instability parameter, Q. When Q<1, the disk solid content.
isgravitationallyunstableandcanrapidlyformgasgiant The values of N andN canbe fit to a polynomial
DI CA
planetsthroughfragmentation. Ifagiventruncateddisk functiondependentona,e,µ,andM∗. Thegeneralform
model has Q < 1 at any point in the disk, then we de- of this polynomial is
clare that the disk is capable of forming planets by disk
instability,andNDI isthetotalnumberoftruncateddisk N = c a iejµk M∗ l. (6)
mofobdinelasrythpaatraamreegterrasv.itFaotrioMna∗ll=y u0n.5stMab⊙l,eµfo=r a0.5g,ivaen=s3e0t i,Xj,k,l ijkl(cid:16)1 AU(cid:17) (cid:18)M⊙(cid:19)
AU, and e = 0, N = 4, and when the eccentricity is Since we want to keep the analytic function as simple
DI
raised to e=0.3 N =3. as possible, we do not try to fit the bunched contours,
DI
3. RESULTSANDANALYSIS but rather try to find a smooth fit that will give a rea-
sonable approximation for the true values of N and
CA
InPapersIandII,weusedafixedsetofparametersfor
N . We find best-fitting polynomial coefficients, c ,
DI ijkl
M∗, a, e, and µ and used those parameters to calculate for N , using those points where they have tabulated
CA
NCA, and NDI. Here, we extend the ranges of M∗, a, e, values where 0 < N ≤ 10, because this avoids the
CA
and µ in order to see how N and N vary with each
CA DI plateau in N values at larger separations and low
CA
parameter. InTable1,weshowourcalculationsforN
CA eccentricity. For N , we use tabulated values where
DI
and N for the binary systems listed there.
DI 0<N ≤5. AsshowninFigure2,valuesofN higher
DI DI
Our previous results for HD 188753A(Paper I) and γ than 5 occur only at a&50 AU and at low eccentricity,
CepA (Paper II) are shownin the firsttwo columns, re-
and in a region where N >10.
CA
spectively. We also repeat our calculations for the close
We find that for N we get reasonably good fits for
CA
binary systems HD 41004, HD 196885A, and α Cen B.
a function linear in a and e and quadratic in µ and M∗.
Although HD 41004Bb is a brown dwarf rather than a
ForN , the fitting function is linearin allfour parame-
DI
planet, we include that system in this paper as a com-
ters. The best-fitting coefficients are tabulated in Table
parison.
3. Comparing the directly calculated and tabulated val-
HD188753A,wherethereisnoconfirmedplanet,isthe
uesforN tothepolynomialfitforalltabulatedvalues
CA
onlysystemwithN =0; allothersystemshaveN =
DI DI whereN ≤10,theroot-mean-squaredeviationis0.62.
CA
1. Similarly, HD 188753A is the only system examined
For N ≤5, the rms deviation is 0.44.
DI
here with N = 0. If we use N as a proxy for the
CA CA The values of N and N for the binary systems
CA DI
difficulty of planet formation, then α Cen B should be
consideredin this paper as calculated using this polyno-
the least likely system to harbor a planet, and γ Cep A
mial equation are tabulated in Table 4. Since both N
CA
and HD 196885Athe most likely.
and N cannot be less than 0, negative values should
DI
In Figure 2 we show examples of how N and N
CA DI be interpreted as 0. The resulting values are all within
vary with a and e, holding M∗ and µ fixed. That is, the ±1forN , andwithin±0.6forN ofthe directly cal-
CA DI
masses of the binary system are held fixed and their or-
culated values shown in Table 1. The polynomial values
bits are allowedto vary. One plot shows where γ Cep A
are more discrepant for larger values of N , underesti-
CA
falls in parameter space, while the other shows the loca-
matingthetruevalues. Thisisaresultofthepolynomial
tionofHD 188753A.As eccentricitydecreasesandsemi-
equation smoothing over the bunching of N contours,
CA
majoraxisincreases,N (greensolidcontours)andN
CA DI as discussed above.
(red dashed contours) both increase, as expected. This
The values of N for all binary systems considered
DI
is because as periastron increases, the disk truncation
are less than 2, as shown both in Table 1 and Table 4.
radius also increases, allowing a larger disk to remain
This could be an indication that core accretion is more
around the star. N increases also because Q tends to
DI likely than disk instability in close binaries. As shown
decreasewithincreasingdistancefromthestar,andN
CA in Figure 1, the disk model with the lowestQ values has
increasesbecausemorecolddisk materialremainsinthe thehighestaccretionrate,10−4 M⊙ yr−1 andthe lowest
disk.
viscosity parameter explored. Disks with this high an
The irregularity in the spacing of the green contours
accretionratearetypicallyoutburstsystems,suchasFU
(N ), particularly at large a and small e, results from
CA Orionis objects. Accretion rates this high are typically
the finite sampling of disk models. If we had included
more values of M˙ and α, then there wouldbe more con- jturastns1ienAtU. ,Tshoeidtisiks ceoxntrteaminesly10m0aMssJiveo,rh0e.1ncMe ⊙itwisiththine
tours in this area. Some of the contours bunch up at most gravitationally unstable. Because a steady-state
small separations and low eccentricities, largely because disk with these properties is unlikely, planet formation
the solid content of the disks are not a simple function via disk instability is also unlikely.
of M˙ and α, as shown in the middle panels of Figure 1. If we were to set a minimum threshold for N for
CA
The dust mass versus radius profiles of the disks cross whichplanetformationcanoccurinaclosebinarygiven
each other at small separations, with steeper profiles at the binary systems explored in this paper, we first need
higher accretion rates. This is because holding α con- toconsiderwhichsystemstoinclude. The existenceofα
stant, higher accretion rates give higher disk mass ac- Cen Bb has yet to be definitively confirmed, and it has
cordingtoEquation(4),buttheheatingfromthisaccre- thelowestvalueofN of3,notincludingHD188753A.
CA
tion causes sublimation of the refractory species. These IfαCenBbisconfirmed,thenwemightsetN =3.
CA,min
competingeffects meanthatatlargediskradii,thetotal One might dispute that HD 41004Bb is a brown dwarf
solid content should overall be higher in the more mas- and ought not be considered, also. Then, we might
sive, higher accretion rate disks, but at small radii, the set N = 7, since γ Cep A, HD 41004A, and HD
CA,min
ON THE LIKELIHOOD OF PLANET FORMATION IN CLOSE BINARIES 5
Fig.2.—Planet formationlikelihoodplots for HD188753A (left) andγ Cep A (right). Each plotshows how NCA and NDI varywith
eccentricityandsemi-majoraxisforthebinaryorbitforafixedprimaryandsecondarymass. SolidmagentacontoursshowvaluesforNCA
and dashed blue contours show those for NDI. The positions of HD 188753A (left) and γ Cep A (right) in a-e space are indicated by
crosses.
TABLE 3
Coefficientsfor Equation (6).
i=0,j=0 i=1,j=0 i=0,j=1 i=1,j=1
CoreAccretion
k=0, l=0 −6.7585 0.4308 2.9369 −0.4411
k=1, l=0 −2.0956 1.6243 −9.5050 −1.4578
k=2, l=0 1.8452 −0.5589 16.0528 0.3879
k=0, l=1 1.5486 −0.1609 −1.9967 0.1696
k=1, l=1 −9.3964 0.2216 3.5221 −0.3374
k=2, l=1 7.9117 −0.3244 −5.4470 0.4971
k=0, l=2 −0.2951 0.0364 0.4372 −0.0363
k=1, l=2 2.7651 −0.0993 −1.6003 0.1275
k=2, l=2 −2.2214 0.0920 2.1346 −0.1406
DiskInstability
k=0, l=0 0.3921 0.0521 −0.8329 −0.0431
k=1, l=0 −0.1135 0.1106 1.1926 −0.0975
k=0, l=1 −0.2704 −0.0066 0.3131 0.0012
k=1, l=1 0.2944 −0.0296 −0.7153 0.0327
TABLE 4
Calculatedvaluesof NCA andNDI fromEquation (6).
HD188753 γCep HD41004A HD41004B HD196885 αCenB
NDI 0.54 1.41 1.57 1.25 1.45 1.15
NCA -2.19 7.04 6.87 3.83 7.13 3.45
196885Aallsatisfythis criterion. Setting NCA,min =7is is hyperbolic in a-e space for fixed M∗ and µ.
the morestringentconstraintfor planetformation,since Foragivenbinarypair,giantplanetformationbycore
werequiremorediskstosatisfytherequirementforsuffi- accretion is allowed for greater separations and lower
cient solid mass. Based on these considerations, we pre- eccentricity than the indicated contour, or under each
dict that N & 7 for planet formation to be readily curve. Planet formation also becomes more feasible as
CA
feasible, based on the known planets in binary systems. theprimarymassdecreasesandthemassratioincreases,
For3<N <7,planetformationmaystillbepossible, as shown by the shifting of the contours up and to the
CA
as in α Cen A, but less common. left.
In Figure 3, we show the allowed parameter space for Note that for a = 12 AU, the semi-major axis of the
planet formation in close binaries, with N = 3 HD188753binary,planetformationishighlydisfavored.
CA,min
(left) or N = 7 (right). That is, we set equation Evenconsidering N =3, only stars with relatively
CA,min CA,min
(6)equalto 3 or7, choosefixedvalues ofM∗ andµ, and low-mass companions (µ & 0.7 or Mb/M∗ . 3/7) and
drawthe resultingcurveina-espace. Sinceequation(6) low eccentricity (e . 0.5) might allow planet formation
is linear in a and e, the resulting curve for N =N tooccur. However,fora=20AU,alargerangeofbinary
CA,min
6 JANG-CONDELL
Fig. 3.—Contours ina-espace forNCA =3(left) andNCA =7(right) forvaryingbinarymasses. Eachcontour represents aselected
choice for M∗ and µ, and the axes show allowed values of a and e. The mass of the primarystar is labeled by color, and the mass ratio
ofitsbinarycompanionisindicated bylinestyle: dot-dashed, dashed, dotted, andsolidforµ=0.1,0.3,0.7,and0.9, respectively. Planet
formationbecomes morefeasibletotherightandbeloweachcurve.
properties allow for planet formation to occur, although thatthecircumstellardiskiscoplanarwiththebinaryor-
larger values of µ are favored. bit. It is conceivable that the disk could be misaligned
with the binary, as is the case for the wide pre-main
4. DISCUSSIONANDCONCLUSIONS sequencebinaryHKTau(Jensen & Akeson2014). How-
ever,tidaleffectsfromclosestellarcompanionswith.20
We have calculated the feasibility of giant planet for-
AU separations will probably cause the disk to become
mation in the close binary systems HD 188753, γ Cep,
aligned. The orbits of circumbinary planets discovered
HD 41004, HD 196885, and α Cen. We find that ex-
by the Kepler mission are closely aligned with those of
cept for HD 188753, where there is not a planet, core
the binary, suggesting that disks are likely to be aligned
accretionis the more likely planet formation mechanism
with binary orbits (Doyle et al. 2011; Welsh et al. 2012;
thandiskinstabilitybasedonsimplemodels ofthetrun-
Orosz et al. 2012b,a).
catedprotoplanetarydiskfromwhichtheplanetformed.
We have made the assumption in the core accretion
These truncated disks contain a sufficiency of solid ma-
casethat likelihoodofplanet formationdepends only on
terialfromwhichplanetcorescanform,butdonotmeet
thetotalsolidmassbeing aboveacertainthreshold,and
the criterion for gravitational instability. Based on the
thatitdoesnotdependontheamountabovethatthresh-
results for these systems, we conclude that for binary
old. We do not calculate whether or not these solids can
systems of separations .20 AU and eccentricity &0.4,
come together to form a single core, nor do we consider
circumstellargiantplanetsmayonlyformviacoreaccre-
the possibility of the formation of multiple cores. These
tion.
types of calculation would involve detailed modeling of
More generally, we extend our results to binary sys-
the dynamics of the particles within the disk, and is be-
tems of arbitrary masses, separations, and eccentricities
yond the scope of this paper.
to predict the likelihood of giant planet formationby ei-
Wealsodonotconsiderthegrowthandevolutionofgi-
ther core accretion or disk instability. These likelihoods
antplanets within the disk beyondthe initial formation.
are represented by the quantities N and N , respec-
CA DI Dynamical effects such as tidal torques on planetesimals
tively. We find an empirical fit to N and N as a
CA DI and planet embryos, planet-planet scattering, and disk
functionofbinarymass,massratio,separation. Bycom-
migration will affect the survivability of planets within
paring N and N as predicted in this paper to the
CA DI the system. These effects have been considered in some
observed frequency of planets in binaries, we can gain
individual systems by others (e.g. Thebault 2011).
insight into the planet formation process in general.
We have not considered how metallicity or chemical
The estimates of planet formation likelihood repre-
composition of the disk might affect our results. In-
sented by N and N are not meant to be definitive
CA CI creasedmetallicitywillaffectboththediskopacitiesand
relationsfortheprobabilityofplanetformation. Rather,
the amount of solids in the disks. Qualitatively speak-
thesevaluesshouldbeusedtorepresenttherelativeease
ing, increased opacity will result in slightly higher disk
with which planet formation can take place in binary
temperatures,which will decrease the refractorycontent
systems. As more planets are found in close binary sys-
inthemostseverelytruncateddisks. Ontheotherhand,
tems,those statistics canbe comparedto the likelihoods
highermetallicitywillraisetherefractorycontentoverall,
calculated in this work to yield a better understanding
sothese effects willcompetewith eachother. Thiscould
of how planets can form in the dynamically challenging
result in a lower likelihood of giant planet formation in
environment of a close binary star.
the most tightly bound binary systems, but increased
We have considered only giant planet formation. In
likelihood in wider systems.
principle, the requirement for terrestrial planet forma-
tionislessstringent,sinceless solidmaterialisrequired.
These calculations would be a possible direction for fu- ThanksgotoEricJensenandananonymousrefereefor
ture work. helpfulcommentsthatgreatlyimprovedthispaper. The
Ourresultshavebeencalculatedundertheassumption author acknowledges support from NASA ATP Grant
ON THE LIKELIHOOD OF PLANET FORMATION IN CLOSE BINARIES 7
NNX12AD43G.
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