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ICARUS
ARTICLE NO.
132, 196–203 (1998) IS985900
Habitable Planet Formation in Binary Star Systems
Daniel P. Whitmire, John J. Matese, and Lee Criswell
Department of Physics, University of Southwestern Louisiana, Lafayette, Louisiana 70504 E-mail: whitmire@usl.edu
and Seppo Mikkola
Tuorla Observatory, Turku University, 21500 Piikkio, Finland ¨ Received March 26, 1997; revised December 24, 1997
1. INTRODUCTION Assuming current models of terrestrial planet formation in the Solar System, we num

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ICARUS
132,
196–203 (1998)
ARTICLE NO
. IS985900
Habitable Planet Formation in Binary Star Systems
Daniel P. Whitmire, John J. Matese, and Lee Criswell
Department of Physics, Uni
v
ersity of Southwestern Louisiana, Lafayette, Louisiana 70504
E-mail: whitmire@usl.edu
andSeppo Mikkola
Tuorla Obser
v
atory, Turku Uni
v
ersity, 21500 Piikkio¨ , Finland
Received March 26, 1997; revised December 24, 1997
1. INTRODUCTIONAssuming current models of terrestrial planet formation in
Modeling the accretion of terrestrial planets in the Solar
the Solar System, we numerically investigate the conditions
System (Wetherill and Stewart 1993) and around other
under which the secondary star in a binary system will inhibitplanet growth in the circumstellar habitable zone. Runaway
single stars (Wetherill 1996) suggests that planet formation
accretion is assumed to be precluded if the secondary (1) causes
within a star’s habitable zone may be common. We denote
the planetesimal orbits to cross within the runaway accretion
a terrestrial planet that forms within the circumstellar hab-
time scale and (2) if, during crossing, the relative velocities of
itable zone (Kasting
et al
. 1993; Whitmire and Reynolds
the planetesimals have been accelerated beyond a certain criti-
1996) as a potentially habitable planet, recognizing that
cal value which results in disruption collisions rather than
many other conditions may be necessary for life to actually
accretion. For a two solar mass binary with planetesimals in
evolve on such a planet. Since
Ȃ
2/3 of solar type stars are
circular orbits about one star at 1 AU, and a typical wide binary
known members of multiple star systems it is of interest
eccentricity of 0.5, the minimum binary semimajor axis which
to consider the constraints placed on planet formation due
wouldnotinhibitplanetformation,
a
c
,is32AU.Iftheplanetesi-
to the presence of a secondary star. Two conﬁgurations
mals orbit the center of mass of the binary system,
a
c
0.10AU, which is inside the tidal circularization radius. We obtain
will be considered: the internal-planetesimal geometry in
anempiricalformulagivingthedependenciesof
a
c
onthebinary
whichtheplanetesimalsorbitthe primaryandtheexternal-
eccentricity, secondarymass, planetesimallocation, andcritical
planetesimal geometry in which the planetesimals orbit the
disruption velocity. Based on the distributions of orbital ele-
center of mass of the binary system. We assume that the
ments of a bias-corrected sample of nearby G-dwarfs, we ﬁnd
secondary star formed simultaneously with the primary. If
that
ȂȂ
60% of solar-type binaries cannot be excluded from
this were not the case and the secondary star formed or
having a habitable planet solely on the basis of the perturbative
was captured after the initial stages of planet formation
effect of the secondary star. This conclusion is independent of
were complete, then planet accretion could have pro-
when the secondary star formed, nebula dissipative mecha-
ceeded as expected for a single star. In this case the only
nisms, and the time scale for runaway planetesimal accretion,
restriction would be the dynamical ejection of the planet
and is relatively insensitive to the mass of the secondary star,
itself(GrazianiandBlack1981;PendletonandBlack1983).
thecriticaldisruptionvelocity,andthelocationofplanetesimalswithin the circumstellar habitable zone. An earlier study of
The assumption of simultaneous star formation is conser-
planet formation in binary star systems came to a different
vative in the sense that if it is invalid the probability of
conclusion, namely that planet formation, even at Mercury’s
habitable planet formation is greater than our analysis
distance, is unlikely except in widely separated systems (
ՆՆ
50
will suggest.
AU), or when the secondary has a very low mass and near
The accretion of planetesimals from a turbulent gas and
circular orbit as in the Sun–Jupiter system. The discrepancy
dust nebula is not well understood theoretically even in
with the present numerical study is due in part to the different
the case of our own Solar System. Nonetheless, dust grains
runaway accretion time scales assumed and the neglect in the
managed to collect into planetesimals even in the parent
earlier study of an exact criterion for crossing orbits.
©
1998
asteroid belt where the perturbation due to Jupiter or
Academic Press
proto-Jupiter would have been stronger than will be as-
Key Words:
planet formation; binary stars; life.
sumed in our binary star analysis. More direct evidence
196
0019-1035/98 $25.00Copyright
©
1998 by Academic PressAll rights of reproduction in any form reserved.
HABITABLE PLANETS IN BINARY STAR SYSTEMS
197that planetesimals can form from dust grains in binary star it is possible for some fragments to achieve escape velocityeven when
U
Յ
U
c
, but most mass will remain bound.systems is given by Kalas and Jewitt (1997), who haveobserved a dust disk surrounding the BV5 binary system Depending on the mass of the planetesimals, velocitypumping by the secondary body will be countered by sev-BD
ϩ
31
Њ
643. Based on dust lifetime arguments they con-clude that the dust must have planetesimal sources located eraldissipativeprocesses,whichincludeinelasticcollisions,gas drag, viscous stirring, and dynamical friction. For grow-at 1,000 AU. The observed binary separation is 200 AU.This system is an (external-planetesimal case) example of ing planetesimals of size
Ն
10 km the effects of dissipativeprocesses other than dynamical friction and inelastic colli-a binary in which planetesimals have formed at distancesof ﬁve times the observed stellar separation. If the binary sions are usually assumed to be negligible. We conserva-tively neglect
all
damping mechanisms in our analysis sinceorbit were circular, contrary to expectations, the ratio of planetesimal formation to binary semimajor axis would be theneteffectoftheseprocessesistomakeplanetformationmoreprobable.Velocityaccelerationduetomutualpertur-5:1. This is within a factor of about 2 of the dynamicallimit for planetesimal ejection (of coplanar orbits) from bations betweenlarge planetesimalsand embryosis impor-tant in current models for planet formation in the Solarthe system and much closer than our analysis will require(typical ratio 30:1). Although the entire disk in this system System and other single stars, causing growing embryos tocollide and grow. The additional effect of the secondarymay have more total mass and therefore potentially moredissipation than disks typical of solar-type binaries, the starontheinteractionsbetweenplanetesimalswillbetakeninto account directly in our numerical calculations dis-dust-to-planetesimal accretion distance is 1,000 AU fromthe center of mass of the binary, making it likely that the cussed below.Internal-planetesimal formation in binary star systemssurface density is less than at 1 AU in a solar type disk.The most perturbation sensitive stage in the terrestrial wasﬁrstconsideredinsomedetailbyHeppenheimer(1974;1978, hereafter H78). That investigation was based on theplanet formation process is the runaway planetesimal ac-cretion phase (Wetherill and Stewart 1993, hereafter WS; then current Goldreich–Ward (1973) model of terrestrialplanet formation and on the assumption that planetesimalLissauer 1993). This phase is rapid because of the smallrelative velocities between the planetesimals and the grow- collisions would lead to disruption rather than accretionif the relative velocities between planetesimals exceededing embryo as a result of dynamical friction. The runawayaccretion time scale is
ȁ
2
ϫ
10
4
yr (WS). This phase ends the critical velocity
U
c
Ȃ
100 m s
Ϫ
1
. The time scale forplanetesimal–planetesimal accretion during which thiswhen the relative velocities of the growing embryos in-crease due to their mutual gravitational interactions. Sub- value of
U
c
would be relevant was assumed to be 10
7
yr.The increase in relative velocity due to planetesimal eccen-sequent growth by nondestructive embryo–embryo colli-sions to a ﬁnal planet occurs on a much longer time scale tricity pumping and differential precession was then com-puted from secular perturbation theory. A reduction inof
ȁ
10
7
yr.If the relative velocities at inﬁnity
U
of orbit-crossing eccentricity due to two different models of nebular gasdrag was also considered in the analysis.planetesimals are accelerated beyond a certain criticalvalue
U
c
then runaway accretion and thus planet formation Theconclusion ofHeppenheimer’s studywas thatplanetformation even at Mercury’s distance was not likely exceptwill be precluded since collisions will then cause disruptionrather than accretion. In a binary star system or a system for widely separated binaries (
Ն
50 AU) or when the sec-ondary mass was small and its orbit circular as in the casecontaining a massive planet or brown dwarf the secondarybody may accelerate planetesimals beyond this limit. For of the Sun–Jupiter system. Even for this system it wasassumed necessary to invoke the gravitational damping of planetesimals bound by nongravitational forces alone (ex-pected for planetesimal radii up to
ȁ
100 km) the critical eccentricities by the solar nebula to allow accretion of theterrestrial planets. If accurate, that study would imply thatvelocityisindependentofmassandis
ȁ
100ms
Ϫ
1
accordingto laboratory measurements (Greenberg
et al
. 1977) and
Ȃ
2/3ofsolar-typestarscouldbeexcludedascandidatesforhabitable planets. The present study comes to a differentsimple theoretical arguments (Heppenheimer 1978; Weth-erill 1991). conclusion and we return to the explanation for thisdiscrepancy after presentation of our analysis andIncluding the gravity of the colliding planetesimals willincrease
both
the impact velocity
ϭ
͙
U
2c
ϩ
V
2es
and the results.escape velocity from the merged pair. The additional rela-tive kinetic energy due to free fall will result in the (two)
2. ANALYSIS
collision fragments having nonzero velocities but, begin-ning with a velocity at inﬁnity
Յ
U
c
, these fragments cannot
2.1. Crossing Orbits
have a velocity greater than
V
es
, thus justifying the neglectof gravity in our conservative analysis. In a more realistic We wish to determine an analytic criterion for the cross-ing of two coplanar planetesimal orbits given bycollision with a spectrum of fragment masses and velocities
198
WHITMIRE ET AL.
2.2. Relati
v
e Velocity
1
r
1
ϭ
1
ϩ
e
1
cos (
Ϫ⌬
Ͷ
˜
)
p
1
(1)Given that two elliptical orbits satisfy the above crossingcriterion we determine their relative velocities
U
Ϯ
at thetwo crossingpoints (corresponding tothe twogeneral solu-1
r
2
ϭ
1
ϩ
e
2
cos(
)
p
2
, (2)tions for cos
) fromwhere
r
1
(
r
2
) is the radial position of planetesimal 1 (2),
is their common polar angle at the point(s) of intersection,(
U
Ϯ
)
2
ϭ
ͫ
e
1
sin(
Ϯ
Ϫ⌬
Ͷ
˜
)
͙
p
1
Ϫ
e
2
sin
Ϯ
͙
p
2
ͬ
2
(7)
⌬
Ͷ
˜
ϭ
Ͷ
˜
1
Ϫ
Ͷ
˜
2
is the longitude of periastron (i.e., theangle between the two periastron vectors),
e
1
(
e
2
) is theeccentricity, and
p
1
(
p
2
) is the semilatus rectum of the
ϩ
ͫ
͙
p
1
r
Ϯ
Ϫ
͙
p
2
r
Ϯ
ͬ
2
,ellipse
ϭ
(1
Ϫ
e
21(2)
)
a
1(2)
.At orbit intersectionwhere
U
is divided by the orbital velocity at 1 AU
ϭ
30 km s
Ϫ
1
and
r
,
p
are in AU,
ͩ
1
r
1
Ϫ
1
r
2
ͪ
ϭ
0
ϭ
1
p
1
Ϫ
1
p
2
ϩ
e
1
cos(
Ϫ⌬
Ͷ
˜
)
p
1
Ϫ
e
2
cos
p
2
. (3)
r
Ϯ
ϭ
p
2
1
ϩ
e
2
cos
Ϯ
ϭ
p
1
1
ϩ
e
1
cos (
Ϯ
Ϫ⌬
Ͷ
˜
). (8)The intersection point angles areWe note that even when orbits cross and the relativecos
ϭϪ
AB
Ϯ
C
͙
C
2
ϩ
B
2
Ϫ
A
2
B
2
ϩ
C
2
(4)velocity is greater than 100 m s
Ϫ
1
an actual collision neednot occur in 2
ϫ
10
4
yr since the orbits are not entirelyﬁlledwithplanetesimals.Theassumptionthatadestructivewherecollision will always occur when orbits cross and the rela-tive velocity is greater than
U
c
is therefore conservative.
A
ϭ
p
2
Ϫ
p
1
2.3. Calculation of Orbital E
v
olutionB
ϭ
e
1
p
2
cos
⌬
Ͷ
˜
Ϫ
e
2
p
1
(5)For the internal conﬁguration case where the planetesi-
C
ϭ
e
1
p
2
sin
⌬
Ͷ
˜
.mals orbit the primary star, the relevant secular perturba-tion theory equations for the time dependence of
e
and
Ͷ
˜
Physical solutions (i.e., orbit intersections) occur whenfor planetesimals with initial eccentricities
ϭ
0 are (H78)
C
2
ϩ
B
2
Ϫ
A
2
Ն
0,
e
(
t
)
ϭ
52
aa
B
e
B
1
Ϫ
e
2B
ͯ
sin
ͩ
u
2
t
ͪͯ
(9)yielding the necessary criterion for orbit crossingtan(
Ͷ
˜
t
)
ϭϪ
sin
ut
1
Ϫ
cos
ut
(10)(
e
1
p
2
)
2
ϩ
(
e
2
p
1
)
2
Ϫ
2
e
1
p
2
e
2
p
1
cos
⌬
Ͷ
˜
Ն
(
⌬
p
)
2
, (6)where
⌬
p
ϭ
p
2
Ϫ
p
1
.
u
ϭ
32
ȏ
1(1
Ϫ
e
2B
)
3/2
M m
1/2
a
3/2
a
3B
, (11)Twoisolated planetesimalswill eventuallycollide iftheirorbits come within a critical distance
ϭ
2.4(
Ȑ
1
ϩ
Ȑ
2
)
1/3
,where
Ȑ
is the planetesimal mass in solar units (Gladman Where
a
B
and
e
B
are the semimajor axis and eccentricityof the binary star system and
M
and
m
are the masses of 1993). For planetesimals of density
ϭ
1 g cm
Ϫ
3
, located at1 AU, and having radii in the range 1–1,000 km the critical the primary and secondary star, respectively.The secular perturbation approximation is valid only incollision radius is 4
ϫ
10
Ϫ
6
–4
ϫ
10
Ϫ
3
AU. As discussedbelow, if the separation between planetesimals at 1 AU is the limit (
a
/
a
B
)
Ӷ
1 for the internal-planetesimal conﬁgu-ration. This approximation is also inaccurate in the limitless than
ȁ
10
Ϫ
3
AU the perturbation of the secondary star(not present in Gladman’s analysis) will rapidly, within a
e
B
ϭ
0 since Eq. (9) predicts that the planetesimal eccen-tricity
e
ϭ
0inthis limit.Testcalculationsusing anumericalfew orbits, increase
͉
⌬
a
͉
to values
Ն
10
Ϫ
3
AU. The time inwhich the two nearly circular orbits are within the critical
N
-body code and an integration time of 2
ϫ
10
4
yr resultedinsystematicallylargervaluesofthecriticalbinarysemima-distance of each other is negligibly small for the binaryparameters used in this analysis. The orbits can still cross jor axis
a
c
(beyond which planet accretion would not beinhibited) compared with the secular approximation. Itand a collision can be inferred to have occurred.
HABITABLE PLANETS IN BINARY STAR SYSTEMS
199wasalsofoundthatplanestesimalsemimajoraxisvariations the binary parameter space of interest, values of
⌬
a
o
Յ
0.001AUresultedintheosculating
⌬
a
increasingto
ȁ
0.001were not negligible as assumed in the secular approxima-tion. The secular approximation formulae are obtained by AU or larger within a few binary orbital periods. It wasalso found that values of
⌬
a
o
signiﬁcantly greater thanaveraging over both the planetesimal and binary orbitsand therefore information about variations in the orbital 0.01 AU never produced crossings within 2
ϫ
10
4
yr. Oursimulations were carried out for three values of
⌬
a
o
, 0.001,elements within a single binary orbit are lost. The numeri-cal calculations showed that typically these variations were 0.003,and0.01AU.Itwasfoundthatthemaximumrelativevelocity obtained at orbit intersection was essentially inde-dominant during the short 2
ϫ
10
4
yr integration time of interest.Applyingourconservativetheme,andbecausethe pendent of
⌬
a
o
for
⌬
a
o
Յ
0.01 AU. On the other hand,orbit crossing itself was typically dependent on
⌬
a
o
. Fornumericalresultsaremoreaccurate,weusedthenumericalcalculations to obtain the present results and employed a given set of binary parameters, if crossing and a givenmaximum relative velocity were achieved for
⌬
a
o
ϭ
0.001the secular approximation only as a test of the
N
-bodycode in an appropriate limit. AU, the principal change when the simulation was re-peated with
⌬
a
o
ϭ
0.003 and 0.01 AU was that crossingsThe
N
-body code used for these calculations is basedonthesymplecticmappingmethodofWisdomandHolman occurred for a shorter time interval at comparable maxi-mum relative velocities.(1991). It was developed by S. Mikkola and K. Innanenand has proved itself in earlier works (e.g., Mikkola and The recorded planetesimal orbital elements were in-serted into Eqs. (6) and (7) to determine if the orbitsInnanen 1995). For the present application several addi-tional tests were performed to ensure that the program crossed and, if they did, what the relative velocities wereat the points of intersection. As noted earlier, even whenwas functioning properly, including comparison with theanalytic secular perturbation equations in the limit orbits cross an actual collision need not occur in 2
ϫ
10
4
yr since the orbits are not entirely ﬁlled with planetesi-(
a
/
a
B
)
Ӷ
1, large binary semimajor axis, and moderateeccentricity. In another test we compared the numerical mals. The assumption that a collision will necessarily occuris thus conservative.output with an analytically derived Jacobi constant. Forthis test
e
B
ϭ
0 and the planetesimal was given an arbitrary
3. RESULTS
eccentricity and inclination. TheJacobi constant was foundto be conserved to the expected high accuracy at each
3.1. Internal-Planetesimal Case
computational step.The initial conditions of the simulated 4-body systems Wefocusonplanetesimalaccretioninthehabitablezoneof a 1
M
᭪
primary. The habitable zone for a 1
M
᭪
starwere chosen as follows. Two planetesimals of negligiblemass were started in coplanar circular orbits around the depends on the evolutionary state of the star and the habit-ability time scale of interest (Kasting
et al
. 1993; Whitmireprimary star in the invariant plane of the binary systemand orbiting in the same sense as the binary. The initial and Reynolds 1996): The narrowest habitable zone consid-ered was the 4.5 Gyr continuously habitable zone. Usingpositions of the two planetesimals were
Ϯ
90
Њ
from thebinarysemimajor axisand thesecondary starwasstarted at the most restrictive climatic assumptions this zone ex-tended from 0.95–1.15 AU. Less restrictive climatic as-binary periastron. This conﬁgurationwas chosen because itwas found empirically that it resulted in the largest pertur- sumptions or shorter time scales result in somewhat largerhabitable zones. The presence of a second star of luminos-bation (relative velocities) during crossing and is thereforethe most conservative initial conﬁguration. This is likely ity equal to or less than that of the primary will not signiﬁ-cantly alter the location of the habitable zone for the rangedue to the fact that this conﬁguration results in the greatestinitial differential torque on the planetesimals. The orbital of semimajor axes of relevance for the internal-planetesi-mal case.elements of the planetesimals were recorded only at binaryperiastron since it was found empirically, as expected, that Figures 1–4 show the dependence of the critical binarysemimajor axes
a
c
on the binaryeccentricity
e
B
, the mass of the orbital phase corresponding to the maximum perturba-tion of the planetesimals (as measured by relative velocity) the secondary star
m
, the mean radius of the planetesimals
a
, and the critical disruption velocity
U
c
. Habitable planetwas near periastron. Comparing the orbital elements atﬁxed time intervals (and therefore at random orbit phases) growth will not be inhibited in binaries with semimajoraxes greater than
a
c
. Except when speciﬁed otherwise, thesystematically resulted in smaller relative velocities, andtherefore smaller values of
a
c
, during the ﬁxed integra- secondary star mass
m
ϭ
1
M
᭪
, the binary star systemeccentricity
e
B
ϭ
0.5, the average planetesimal semimajortion time.From Eqs. (6) and (7) it is seen that in general crossing axes
a
ϭ
1.0 AU, and the critical planetesimal disruptionvelocity
U
c
ϭ
100 m s
Ϫ
1
ϭ
0.003
V
Kepler
.andrelativevelocityduringcrossingdependontheosculat-ing values of
⌬
a
,
⌬
e
, and
⌬
Ͷ
˜
. For our initial conditions Fitting the numerical data in Figs. 1–4 gives an empiricalfunction relating
a
c
to the system parameters:both
⌬
e
o
and
⌬
Ͷ
˜
o
ϭ
0. Experimentation showed that, for

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