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Mon. Not. R. Astron. Soc.000, 1–11 (2010) Printed 9 June 2010 (MN LATEX style file v2.2)

Dynamics of Planetesimals due to Gas Drag from an EccentricPrecessing Disk

C. Beauge1⋆, A.M. Leiva1, N. Haghighipour2 and J. Correa Otto11Observatorio Astronomico, Universidad Nacional de Cordoba, Laprida 854, (X5000BGR) Cordoba, Argentina2Institute for Astronomy and NASA Astrobiology Institute, University of Hawaii-Manoa, Honolulu, HI 96822, USA

ABSTRACTWe analyze the dynamics of individual kilometer-size planetesimals in circumstellar orbits ofa tight binary system. We include both the gravitational perturbations of the secondary starand a non-linear gas drag stemming from an eccentric gas diskwith a finite precession rate.We consider several precession rates and eccentricities for the gas, and compare the resultswith a static disk in circular orbit.

The disk precession introduces three main differences withrespect to the classical staticcase: (i) The equilibrium secular solutions generated by the gas drag are no longer fixed pointsin the averaged system, but limit cycles with frequency equal to the precession rate of the gas.The amplitude of the cycle is inversely dependent on the bodysize, reaching negligible valuesfor∼ 50 km size planetesimals. (ii) The maximum final eccentricity attainable by small bodiesis restricted to the interval between the gas eccentricity and the forced eccentricity, and apsidalalignment is no longer guaranteed for planetesimals strongly coupled with the gas. (iii) Thecharacteristic timescales of orbital decay and secular evolution decrease significantly withincreasing precession rates, with values up to two orders ofmagnitude smaller than for staticdisks.

Finally, we apply this analysis to theγ-Cephei system and estimate impact velocities fordifferent size bodies and values of the gas eccentricity. For high disk eccentricities, we findthat the disk precession decreases the velocity dispersionbetween different size planetesimals,thus contributing to accretional collisions in the outer parts of the disk. The opposite occursfor almost circular gas disks, where precession generates an increase in the relative velocities.

Key words: planets and satellites: formation - stars: individual:γ-Cephei.

1 INTRODUCTION

The detection of giant planets in moderately close binary systemshas raised many questions regarding the formation of these objects.For many years simulations of the dynamical evolution of circum-stellar disks suggested that planets may not form around thestars ofa binary as the perturbation of the secondary star may (i) truncatethe disk and remove the material that may be used in the forma-tion of planets, (ii) increase the relative velocities of planetesimals,which may cause their collisions to result in breakage and fragmen-tation, and (iii) destabilize the regions where the building blocks ofthese objects may exist. However, the discovery of planets aroundthe primaries of the binariesγ Cephei (Hatzes et al. 2003, Neuhuseret al. 2007), GL 86 (Els et al. 2001, Lagrange et al. 2006), HD41004 (Zucker et al. 2004; Raghavan et al. 2006), and HD 196885(Correia et al. 2008), where the stellar separation is smaller than20 AU, suggest that planet formation in such systems may be asefficient as around single stars.

According to the core-accretion model (e.g. Safronov 1969,

⋆ email: beauge@oac.uncor.edu

Goldreich and Ward 1973, Pollack et al. 1996, Kokubo and Ida1998, Alibert et al. 2004, 2005), planetary cores are the resultsof accretional collisions between solid planetesimals immersed inthe nebular gas disk. If this process is sufficiently fast to reach afew Earth-masses before the gas removal, gaseous envelops cancollapse onto the embryos and result in giant planets. If growthis slower and/or if the bodies are located inside the ice line, thevolatile component cannot participate in the accretion process,leading to the formation of small rocky terrestrial-type planets.

Accretional collisions require low impact velocities in order toavoid disruption. In single star systems, relative velocities are usu-ally low, particularly during the early stages of planetaryformation,and accretion appears as a natural outcome of most impacts. In bi-nary stellar systems, however, collisions are more complicated, es-pecially if the pericentric distance between stellar components islower than∼ 20 AU. For instance, as the simulations show, it ispossible to form terrestrial-class bodies from large∼ 1000 km-sized protoplanets around a star of close binaries (e.g. Quintana etal. 2002, Haghighipour and Raymond 2007). However, simulationsof the collision and accretion of planetesimals have not been able toexplain how these embryos formed in the first place (see Haghigh-

2 C. Beauge, A.M. Leiva, N. Haghighipour and J. Correa Otto

ipour 2008 for an extensive review). The gravitational perturbationsof the secondary star may affect the motion of bodies by excit-ing their orbits and increasing their eccentricities, which, for smallkilometer-size planetesimals, may lead to encounter velocities wellbeyond the accretional limit (Heppenheimer 1978a, Whitmire etal. 1998). Had it not been for the fact that planets have alreadybeen detected in close binaries (e.g.γ Cephei, GJ 86, HD41004A,HD196885), it would have been tempting to conclude that planetformation in these environments would be extremely unlikely.

A possible mechanism for reducing the relative velocitiesof impacting planetesimals was presented by Marzari and Scholl(2000). After analyzing the effect of the nebular gas on the plan-etesimal dynamics, these authors found that the combined effect ofthe gravitational perturbation of the stellar companion and gas dragresults in the appearance of an attractor in the dynamical system,causing the eccentricities of planetesimals to approach anequilib-rium valueeeq. At this state, the pericenters of planetesimals orbitsalign and the difference between the longitude of the periastron of aplanetesimal and that of the secondary star∆eq for all planetes-imals approaches equilibrium. Since encounter velocitiesdependon the dispersion in both the orbital eccentricity and longitude ofperiastron, at the equilibrium state the relative velocities of plan-etesimals are reduced and collisions occur below the fragmentationlimit even (although the planetesimal swarm would still maintain aneccentric orbit). Subsequent studies by (Thebault et al. 2004, 2008)show that the periastron alignment is size-dependent an is more ef-fective for planetesimals with equal sizes. The latter implies thateven if at the onset of accretion, initial population of planetesimalsconsisted of equal-mass bodies, subsequent collision and growth ofthese objects would result in a mass spectrum which would lead todifferent values ofeeq and∆eq, causing the relative velocities ofplanetesimals to increase and bringing their accretion process to ahalt.

All the above-mentioned studies assume that the circumpri-mary gaseous disk is in circular motion. However, recent hydrody-namical simulations have portrayed a more complex picture.Apartfrom the truncation of the circumprimary disk near mean-motionresonances with the secondary star (Artymowicz and Lubow 1994),and the subsequent alteration of the surface density, Kley et al.(2008) and Paardekooper et al. (2008) have shown that the gaseousdisk develops an eccentricityeg plus a retrograde precession withfrequencygg. The values ofeg andgg appear very sensitive to thedisk parameters (Kley et al. 2008). However, as long as the viscoustime scale is much smaller than the induced precession time,thedisk rotates as a rigid body and all fluid elements precess with thesame rate.

More recently, Marzari et al. (2009) found that self-gravitycan damp the gas eccentricity and precession rate, and lead to analmost circular disk with no secular precession. However, these re-sults were obtained for disks of massmD ∼ 40MJup and thusan order of magnitude more massive than those studied in Kleyetal. (2008) or Paardekooper et al. (2008). Since Toomre’s parameteris inversely proportional tomD, it is not yet clear if self-gravitywould be relevant in disks with surface densities of the order of theMMSN.

Paardekooper et al. (2008) studied the interactions of plan-etesimals in an static (i.e. no precession) eccentric disk in a binarystellar system. They found that, at any given semimajor axis, thevalue of the equilibrium eccentricityeeq depends on the size of theplanetesimal. For small planetesimals (mainly coupled to the gas)this value is close toeg and for larger bodies it is close to the forcedeccentricityef . A similar behavior was also noted for∆eq. Inter-

estingly, ifeg is sufficiently low, it is possible to find a critical semi-major axis for whicheg = ef . In that case, all planetesimals wouldhave the same equilibrium eccentricity independent of their sizes.More importantly, they would also exhibit apsidal alignment, con-stituting a very favorable breeding ground for planetary embryos.

Unfortunately, two issues conspire against this idea. First, atleast for the simulations done for theγ Cephei system, the gaseousdisk acquired too high eccentricity. Second and more importantly,the disk was assumed to be static. As can be seen from the hydrody-namical simulations by Kley and Nelson (2008), a precessionin thedisk appears to cause oscillations in the eccentricities ofthe plan-etesimals, which no longer seem to reach the stationary solutions.

In this paper we aim to perform a detailed analysis of the dy-namics of individual planetesimals in a precessing eccentric gasdisk that is also perturbed by the gravitational force of a secondarystar. We will adopt the classical approach of considering a restrictedplanar three-body system in which the gas effects are mimickedwith a non-linear drag force. Gas self-gravity will not be included,thus our results will only be applicable to light gas disks. Althoughsuch a model is a very simplified version of the real physical prob-lem, it has the advantage of isolating individual dynamicaltraitsand allowing a separate analyze. As we will show, even this toy-model yields dynamical behavior quite distinct from that presentedfor circular gas disks.

In Section 2, we construct the variational equations of the av-eraged secular system due to this drag. Although such expressionsmight have been used in an implicit form in hydrodynamical sim-ulations such as those mentioned above, we were unable to foundexplicit expressions for them in the literature. We presentthem inthis paper, believing they would be beneficial to any reader wishingto undertake similar studies.

Section 3 is devoted to the evolution of individual orbits with-out an external perturber. We search for equilibrium solutions fora precessing disk and compare the results with the case of a staticone. In Section 4, we include the gravitational perturbations of abinary stellar component and discuss its consequences on the dy-namics of the system. We show that the stationary orbits evolvetowards limit cycles, in which bothe and oscillate around equi-librium values with frequency equal togg.

Finally, in Section 5, we apply this simplified analysis to theγCephei binary and estimate encounter velocities for different plan-etesimal pairs, as functions of semimajor axis and gas eccentricity.Conclusions and discussions end this articles in Section 6.

2 GAS DRAG DUE TO AN ECCENTRICCIRCUMPRIMARY DISK

Our binary system consists of two stars with massesmA andmB

wheremA is the primary. We place the origin of the coordinatesystem onmA and assume thatmB has an orbit with a semimajoraxisaB and an eccentricityeB. We also assume that both the gasdisk and the disk of planetesimals orbit the primary star. Our focuswill then be on the dynamics of planetesimals in the gaseous diskaround the primary when subject to gas drag and the gravitationalperturbation of the secondary star.

The dynamics of the gas in its rotation around the primary staris given by Euler’s equation,

dvg(r)

dt= −GmA

r3r− 1

ρg(r)

∂Pg(r)

∂r. (1)

In this equation,r represents the position vector of a gas element,

Dynamics of Planetesimals in an Eccentric Precessing Disk3

Pg(r) is the pressure of the gas,ρg(r) is the gas density,vg(r) isthe velocity of the gas at positionr, andG is Newton’s constant.The negative pressure gradient in equation (1) causes each gas el-ement to orbitmA with a sub-Keplerian velocity. Usually it is as-sumed that the rotation of gas is circular, where the deviation of thegas velocity from Keplerian(Vk(r))circ is equal to

(∆V (r))circ ∼ − 1

ρg(r)

( r2

2GmA

)∂Pg(r)

∂r(Vk(r))circ . (2)

The quantityη ≡ (∆V (r)/Vk(r))circ in such disks has been con-sidered to be between 0.005 and 0.01, suggesting that in a circu-larly rotating disk, the deviation from circular Keplerianveloc-ity due to pressure gradient is no more than 1%. In other words,α ≡ (1 + η) ∼ (Vg/Vk)circ > 0.99. Different authors have useddifferent values forα. While Adachi et al. (1976), Gomes (1995),and Armitage (2010) consideredα = 0.995, Supulver and Lin(2000) mentioned that it is probably larger than∼ 0.99.

In an eccentric disk, however, the situation is more compli-cated. A gas element in such disks rotates around the centralstar inan elliptical motion, and its Keplerian velocity varies with its po-sition vector. In other words, in eccentric gaseous disks weshouldadoptα = α(r). However, for the present simplified model wewill restrict ourselves to a simple case with a constant value ofα = 0.995.

2.1 Gas Drag

The magnitude of gas drag is a function of the size of an objectandits velocity relative to the gasvrel = v − vg. In this equation,vis the velocity of a planetesimal with a position vectorr, andvg

is the velocity of gas at that location. For planetesimals inkm-sizerange, gas drag is a non-linear function of the relative velocity andits magnitude is proportional tov2rel (Adachi et al 1976, Weiden-schilling 1977a, Supulver and Lin 2000, Haghighipour and Boss2003). The acceleration of a planetesimal due to the gas draginthis case is given by

r = −C|vrel|vrel, (3)

where

C =3CD

8sρpρg(r) (4)

In equation (3),s is the radius of the planetesimal andρp is itsvolume density. The quantityCD in this equation is the coefficientof the drag and has different functional forms for differentvalues ofthe gas Reynolds number and planetesimal’s radius. For km-sizedplanetesimals,CD = 0.44 (Adachi et al 1976, Weidenschilling1977a).

2.2 Relative Velocity

We consider a gaseous disk with an eccentricityeg. We also as-sume that all gas elements have the same orbital orientation(i.e.longitude of pericenter g) at any given instant of time. Althoughhydrodynamical simulations (e.g. Marzari et al. 2009) haveshownthat botheg andg can be a function of the radial distance, for oursimplified model we have chosen constant values. This has beendone primarily to avoid complications in the averaging of the vari-ational equations; however, it is fairly straightforward to implementa more realistic representation of the gas orbital dynamics.

Figure 1 shows an schematic view of such a disk foreg = 0.2and an arbitrary value of the longitude of pericenter. The orbits of

-3 -2 -1 0 1 2 3x

-3

-2

-1

0

1

2

3

y

r

ϖf

vg

v

ϖg

Figure 1. Astrocentric Cartesian coordinates showing the elliptic orbit ofa solid planetesimal (black curve) immersed in an eccentricgas disk. Theorbits of reference gas elements are shown by broad orange curves.

reference gas elements are shown by broad orange curves. Theblack ellipse represents the orbit of a solid planetesimal with asemimajor axisa and eccentricitye. The planetesimal’s positionis defined by its distancer from the primary star, its true anomalyf , and its longitude of the pericenter.

To calculate the relative velocity of a planetesimal, we con-sider a two-body system consisting of the primary star and the plan-etesimal, and a Cartesian coordinate system with its originat thelocation of the primary (Figure 1). In this system, the plane-polarcoordinates of the velocity of the planetesimal are given by(Murrayand Dermott 1999)

vr = r =

√

µ

pe sin f (5)

vθ = rf =

√

µ

p(1 + e cos f),

whereµ = GmA andp = a(1 − e2) is the planetesimal’s semi-lactus rectum. The mass of the planetesimal is assumed negligiblewith respect tomA.

As mentioned previously, we assume that the deviation ofthe velocity of a gas element from Keplerian is very small(α =0.995). As a first approximation, this assumption allows us to useequations similar to (5) to calculate the components of the velocityof the gas at the position of the planetesimal. As shown in Figure 1,the true anomaly of the orbit of a gas element at the radial distancer is related tof by

fg = f + −g = f +∆. (6)

The plane-polar components of the velocity of gas at the positionof the planetesimal can then be written as

vgr = α

√

µ

pgeg sin (f +∆) (7)

vgθ = α

√

µ

pg

[

1 + eg cos (f +∆)

]

,

4 C. Beauge, A.M. Leiva, N. Haghighipour and J. Correa Otto

wherepg(r) = ag(r)(1 − e2g). Given that in a two-body systemr = a(1−e2)/(1+e cos f), the semimajor axis of the gas elementcan be written as

ag = a

(

1− e2

1− e2g

)

1 + eg cos (f +∆)

1 + e cos f, (8)

and subsequently

pg(r) = p1 + eg cos (f +∆)

1 + e cos f. (9)

The dependence onr is given implicitly through the true anomalyf .

For the purpose of numerical simulations, it proves useful tocalculate the contribution of the gas drag to the total acceleration ofa planetesimal in Cartesian coordinate system. From equation (3),we can write,

x = −C|vrel|(x− vgx) (10)

y = −C|vrel|(y − vgy).

To calculate the Cartesian components ofvg we note that fromFigure 1,

f + = fg +g = arctan(y/x), (11)

and therefore the Cartesian components of the gas velocity can bewritten as,

vgx = −α

√

µ

pg

[

sin (fg +g) + eg sing

]

(12)

vgy = α

√

µ

pg

[

cos (fg +g) + eg cosg

]

.

In a precessing disk, the longitude of pericenterg is a function oftime and can be included in numerical simulations as

g = ggt+g0, (13)

wheregg is the precessional frequency of the disk andg0 is thevalue of longitude of pericenter at the beginning of the simulations.

2.3 Averaged Variational Equations

Finally, in this section we construct a simple analytical model forthe differential equations averaged over the short period terms. In aperturbed two-body problem, the variational equations for(a, e,)can be obtained from Gauss’ perturbation equations (Roy 2005):

da

dt=

2a2

õp

[

R′ e sin f + T ′ (1 + e cos f)]

de

dt=

√

p

µ

[

R′ sin f + T ′ (cos f + cos u)]

(14)

d

dt=

1

e

√

p

µ

[

−R′ cos f + T ′ (1 + r/p) sin f]

.

Hereu is the eccentric anomaly of the planetesimal,R′ is the radialcomponent of the acceleration due to gas drag, andT ′ is its trans-verse component. From equation (3), these quantities are given by

R′ = −C|vrel|(vr − vrg) (15)

T ′ = −C|vrel|(vθ − vθg).

Substituting for the components of the planetesimal and gasveloc-ities from equations (5) and (7), expanding to the second order ineandeg, retaining only lowest order terms inα and averaging overthe planetesimal’s mean anomaly, equations (14) can be written as

da

dt= −C√µa

[

2(1− α)2 +O(e2, e2g)]

, (16)

for the semimajor axis, and

dk

dt= −C π

4

√

µ

a(k − kg)

√

(k − kg)2 + (h− hg)2 (17)

dh

dt= −C π

4

√

µ

a(h− hg)

√

(k − kg)2 + (h− hg)2 ,

for the secular variables. Here we have adopted the regular vari-ables(k, h) = (e cos, e sin) for the motion of the planetesi-mal, and(kg, hg) = (eg cosg, eg sing) for the gas elements(Paardekooper et al. 2008). Due to its complicated form, thevari-ational equation for the semimajor axis (16) has been written onlyto its lowest order terms. We refer the reader to Gomes (1995)formore a detailed analysis of the complications of deducing anana-lytical expression for valid for any value ofα and eccentricity.

3 DYNAMICS OF THE TWO-BODY PROBLEM

Our first analysis is restricted to the dynamics of the planetesi-mal (around the primary star) without considering the perturbationsfrom the binary companion. The orbital evolution of the planetes-imal will then be governed by the gravitational attraction of theprimary star plus the effects of gas drag. In all the numerical sim-ulations performed in this section we assume a planetesimalwith abulk density ofρp = 3 gr/cm3 and initial orbital elements ofa = 2AU, e = 0.2 and∆ = 120◦. The mass of the primary star ischosen asmA = 1.59M⊙, equal to the mass of the largest compo-nent ofγ-Cephei. For the gas disk we assume a volumetric densityof ρ = 5× 10−10 gr/cm3 (value ata = 2 AU), consistent with thevalues adopted by Paardekooper et al. (2008) for a MMSN. Finally,in our simulations both the gas eccentricityeg and the retrogradeprecession frequencygg are considered free parameters.

Since we expect the motion of the planetesimal to be tightlycoupled to the gas, we define a new set of regular variables(K,H) = (k − kg, h − hg) (Paardekooper et al. 2008) and re-write equations (17) as

dK

dt= −C′K

√

K2 +H2 + gghg (18)

dH

dt= −C′H

√

K2 +H2 − ggkg,

where

C′ = C π4

√

µ

a. (19)

In the case of a static disk wheregg = 0, equations (18) canbe solved analytically to give

K(t) =K0

1 + E0C′ t; H(t) =

H0

1 + E0C′ t. (20)

HereK0 andH0 are the values ofK andH at t = 0, andE20 =

K20 + H2

0 . The circularization time is thus proportional to1/t,unlike the exponential decay time associated with the linear dragregime. The final equilibrium solution is given byK = H = 0 im-plying that, at equilibriume → eg and → g. Thus, the overalldynamical behavior of the planetesimal in a static eccentric disk isvery similar to the circular case, except that the final equilibriumtrajectory is now an ellipse.

Neglecting terms proportional to the eccentricity, the rate ofthe change of the semimajor axis of the planetesimal can be writtenas

Dynamics of Planetesimals in an Eccentric Precessing Disk5

-0.3 -0.15 0 0.15 0.3K = k - k

g

-0.3

-0.15

0

0.15

0.3

H =

h -

hg

-0.1 -0.05 0 0.05 0.1k

-0.1

-0.05

0

0.05

0.1

h

-0.1 -0.05 0 0.05 0.1

K = e cos(∆ϖ)

-0.1

-0.05

0

0.05

0.1

H =

e s

in(∆ϖ

)

Figure 2. Top: Solution of the secular equations (18) for planetesimal withs = 1 km in a retrograde precessing disk with period of1000 yrs.Middle:Orbital evolution, shown in the variables(k, h). Bottom: Same, but for newvariables(K, H) = (e cos∆, e sin∆), where∆ = − g . Inall cases the initial condition is marked with a filled black circle.

dL

dt= −C(1− α)2, (21)

whereL =õa is the Delaunay momentum associated toa. Since

(1 − α) ≪ 1, equation (21) suggests that the orbital decay of theplanetesimal is much slower than the circularization time.In thelimiting case whereα = 1, this equations indicates that no sec-ular change exists in the semimajor axis of the planetesimaloncethe eccentricity and longitude of the pericenter have reached theirequilibrium values.

In the case of a precessing disk, system (18) is much morecomplicated to solve analytically. We can, however, focus on theequilibrium solutions. Figure 2 shows a typical example of the or-bital evolution of a one-kilometer planetesimal in a retrograde pre-cessing disk with period of1000 yrs. The solution was obtainedsolving numerically the secular equations (18). Due to the diskprecession,(K,H) no longer reach a stationary point but exhibitoscillations with frequencygg around a center located near the ori-gin. The same behavior is also noted for the time evolution oftheoriginal variables(k, h) (middle plot). However, the system stillretains an equilibrium solution in a new set of variables defined as(K, H) = (e cos∆, e sin∆), where∆ = − g. Thisbehavior has been noted for all values ofs and as well as for anyinitial condition.

To obtain analytical expressions for the equilibrium solutionsin this case, we note that from the definitions of(k, h) and(kg, hg),it is possible to write

egK = kkg + hhg (22)

egH = hkg − khg .

The corresponding variational equations ofK andH are then givenby

egdK

dt=

dk

dtkg +

dh

dthg − ggegH (23)

egdH

dt=

dh

dtkg − dk

dthg − ggegK .

In an equilibrium state, the values of(K, H) are constant and equa-tions (23) can be simplify to

dk

dtkg +

dh

dthg = ggegH (24)

dh

dtkg − dk

dthg = ggegK .

Equations (24) are a set of algebraic equations that can be solvedanalytically. Substituting for the time derivatives ofk andh fromequations (17), the equilibrium values ofK andH are given by

Keq =e2eqeg

(25)

H2eq = e2eq − K2

eq ,

where

e2eq = e2g +1

2

(

ggC′

)2

− ggC′

√

√

√

√4e2g +

(

gg2C′

)2

. (26)

For a static gas disk (i.e.gg = 0), this equation indicates thatthe planetesimal’s eccentricityeeq → eg, whereas for an eccentricdisk eeq varies according to the ratio between the disk precessionfrequency and the drag coefficient (i.e. object size). Figure 3 showsthe equilibrium values of the eccentricity and∆ for different val-ues of planetesimal radius and three values of thegg. Even smallvalues of the precession frequencies cause significant changes inthe dynamics of the system. Except for very small sizes wherethecoupling to the gas is strong, the planetesimal does not follow thegas exactly. At large sizes, even though the eccentricity still reachesan equilibrium, its value is smaller thaneg, reachingeeq → 0 forlarge bodies or faster precession rates. Similarly, even though thelongitude of pericenter of the planetesimal is still lockedto the pre-cession of the disk and circulates with the same frequency, the ap-sides are no longer aligned. After a transition time,∆ acquiresan equilibrium value of∆ > 0 (for retrograde precessing disks),

6 C. Beauge, A.M. Leiva, N. Haghighipour and J. Correa Otto

0.1 1 10 100planetesimal radius [km]

0

0.02

0.04

0.06

0.08

0.1

e eq

0.1 1 10 100planetesimal radius [km]

0

30

60

90

∆ϖeq

2π/|gg|=4000 yrs

1000

200

2π/|gg|=4000 yrs

gg=0

gg=0

1000

200

Figure 3. Top: Equilibrium eccentricity, as function of the planetesimalradiuss, in a disk with eccentricityeg = 0.1 with different values of theprecession frequencygg. Bottom: Equilibrium value of∆ = − g .We assume a retrograde precession (i.e.gg < 0).

indicating that the planetesimal always is behind the motion of thegas1.

4 SECULAR DYNAMICS IN THE RESTRICTEDTHREE-BODY PROBLEM

4.1 Equations of Motion

We now analyze the dynamics of the planetesimal including thegravitational perturbations from the stellar companionmB . Similarto the previous case, both the gas disk and planetesimal orbit theprimary starmA, and the coordinate system will also be centeredon this body.

Outside mean-motion resonances, the dynamics is dominatedby secular perturbations. Averaged over short-period variables (i.e.,the mean anomalies), the disturbing function can be writteninterms of regular variables (Heppenheimer 1978a),

R =3

8

GmB

(1− e2B)3/2

a2

a3B

[

(k2 + h2)− 5

2

aeBaB(1− e2B)

k

]

(27)

1 Direct precession of the disk implies∆ < 0 (planetesimal trails disk)while retrograde precession implies∆ > 0 (planetesimal precedes disk).

truncated up to second-order in the eccentricity. The complete av-eraged equations of motion for(k, h) are given by

dk

dt=

dk

dt|drag − gh (28)

dh

dt=

dh

dt|drag + g(k − ef )

where the first terms are the contributions of the gas drag (expres-sions (17)) and

g =3

4

mB

mA

√

mA +mB

mA

na3

a3B(1− e2B)

3/2(29)

ef =5

4

aeBaB(1− eB2)

. (30)

In the absence of gas drag, the semimajor axis is constant andthe solution of(k, h) are given by the classical linear Lagrange-Laplace model

k(t) + ih(t) = epEi(gt+φ0) + ef , (31)

where we have used the notationEx = expx. In this equationep and ef are the proper (or free) and forced eccentricities, re-spectively, andφ0 is the initial phase angle. The expression forthe forced eccentricity is given by (30), and the free eccentricityis equal to

ep =√

(k0 − ef )2 + h20. (32)

Herek0, h0 denote the initial values of their corresponding quan-tities att = 0. Note thatg is the secular frequency of the system.We refer the reader to Heppenheimer (1978b), Marzari and Scholl(2000), Thebault et al. (2004, 2006), and Paardekooper et al. (2008)for more details. For any initial values of the quantitiesk, h, andφ,the minimum and maximum values of the eccentricity are equaltoemin = |ef − ep| andemax = ef + ep, respectively. In the case ofinitially circular orbits,emin = 0 andemax = 2ef .

4.2 Combined Effects with a Precessing Disk

The complete dynamical system contains perturbations withtwodistinct frequencies:gg and g. Except for large semimajor axes,we expectg ≪ |gg | and resonances between the two frequen-cies should not to be significant. In order to have an insight onthis complex problem, we show to examples of numerical integra-tions in Figure 4. Left frames correspond to a planetesimal of radiuss = 0.5 km, while the right plots are drawn fors = 50 km. Thefigure caption gives initial conditions for the planetesimals and theparameters of the disk. Red symbols correspond to the solutions ofthe secular system (28), while results from full N-body simulationsare shown in black. The agreement between these two results showthat the analytical equations give a very good representation of thereal dynamics of the system.

At first sight, the orbital evolution of both planetesimals ap-pear significantly different. For the smaller body, neithere nor∆reach zero-amplitude equilibrium values, but evolve towards a limitcycle displaying significant oscillations with frequencygg. Never-theless,∆ librates while the longitude of pericenter of the plan-etesimal circulates. For the larger object the plots seem toindicatean opposite behavior. In this case, oscillates with a small ampli-tude around an equilibrium value, even if this behavior implies acirculation with respect to the gas disk.

Figure 5 shows the solutions of equations (28) in the(k, h)plane, after the transient period is over and the orbit reaches the

Dynamics of Planetesimals in an Eccentric Precessing Disk7

0 5000 100000

0.03

0.06

0.09

0.12

ecce

ntric

ity

0 50000 1e+050

0.05

0.1

0.15

0.2

0 5000 10000-180

-90

0

90

180

ϖ

0 50000 1e+05-200

-100

0

100

200

0 5000 10000time [yrs]

-180

-90

0

90

180

∆ϖ

0 5000 10000time [yrs]

-180

-90

0

90

180

s = 0.5 km s = 50 km

Figure 4. Simulation of the orbital evolution of two planetesimals (s = 0.5km on the left, ands = 50 km on the right) under the combined effectsof gas drag from an eccentric disk (eg = 0.1), precession of the disk(2π/|gg| = 1000 yrs) and the gravitational perturbations from a secondarystar equal to that ofγ-Cephei. Initial conditions for the planetesimals werea = 2 AU, e = 0.1 and∆ = 120◦. Red lines show results from theanalytical differential equations (28) while black corresponds to exact nu-merical integrations. Gas volumetric density at2 AU was chosen equal toρ = 5× 10−10 gr/cm3.

limit cycle. Graphs have been made for different values of the plan-etesimals radiuss. There appears to be a smooth transition betweensmall and large planetesimals with no topological changes in the so-lutions. The only difference is in the relative magnitude between theforced and free eccentricities. For small planetesimals, the limit cy-cle includes the origin of the coordinates and circulates, whereasfor large bodies the opposite occurs and librates.

5 RELATIVE VELOCITIES BETWEENPLANETESIMALS

5.1 Orbits of Different-Size Bodies

In order to determine the effect of the precession of the gaseousdisk on the process of the accretion of planetesimals, we will ana-lyze the relative motions of different size bodies in crossing orbits.Figure 6 shows the evolution of the orbital eccentricity forplan-etesimals withs between0.5 and50 kilometers. In each case we

-0.06 -0.03 0 0.03 0.06k = e cos(ϖ)

-0.06

-0.03

0

0.03

0.06

h =

e s

in(ϖ

)

s=0.5 km

1

2 510

2050

Figure 5. Limit cycles in the(k, h) plane for various planetesimal radii.Initial conditions and gas parameters are the same as in the Figure 4.

1.98e+05 2e+05time [yrs]

0

0.05

0.1

0.15

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ecce

ntric

ity

1.98e+05 2e+05time [yrs]

0

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0

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ntric

ity

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0

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eg = 0.02 ; g

g = 0

eg = 0.2 ; g

g = 0

eg = 0.02 ; 2π/g

g = -1000 yr

eg = 0.2 ; 2π/g

g = -1000 yr

s=0.5km

510

25

50

1

Figure 6. Equilibrium solutions for static (left) and precessing disks (right)for two values of the gas eccentricityeg . Semimajor axis of planetesimalswasa = 2 AU giving a forced eccentricity ofef = 0.057. The gas densityat that point was chosen to beρ = 5× 10−10 gr/cm3 .

have only plotted the orbit after the limit cycle has been achieved.The top panels show results for a quasi-circular gaseous disk witheg = 0.02, while in the bottom graphs we choseeg = 0.2. Forcomparison, on the left-hand side we have plotted the results for astatic disk (gg = 0) while on the right we chose a retrograde diskwith a precessional period of1000 years. In all cases the initialvalue of the semimajor axis of the planetesimal was set toa = 2AU which gives a forced eccentricity equal toef = 0.057.

In a static disk (upper left panel), the equilibrium eccentrici-ties lie in the interval betweeneg andef depending on the size ofthe planetesimal (Paardekooper et al. 2008). Since in the simula-

8 C. Beauge, A.M. Leiva, N. Haghighipour and J. Correa Otto

tions depicted by the top panels the value ofeg is close toef , thereis little dispersion in the eccentricities and all orbits have similartrajectories independent of their sizes. However, for the more ec-centric disk (bottom panels), the spread in eccentricitiesis muchmore pronounced. Such a separation between objects of differentsizes results in a larger relative velocities during physical encoun-ters.

The effect of the disk precession can be seen comparing theright-hand plots. A non-zero value ofgg lowers the equilibrium ec-centricity of a planetesimal to values beloweg, even when the sizeof the planetesimal is small (see also Figure 3). In the case wherethe forced eccentricity is larger thaneg, the precession of the diskincreases the range of the equilibrium eccentricity of a planetesimalcompared to its corresponding value in the static disk. Thisimpliesthat for gg 6= 0 the relative velocities of planetesimals would belarger than in a static disk. Conversely, ifeg > ef , such as depictedin the bottom panels, the precession will induce a smaller range ineccentricity for different size bodies. This can be seen by compar-ing the eccentricities of two planetesimals with sizes of5 km and50 km in both the static and precessing disks.

A second effect of the disk precession, when coupled with thegravitational perturbation from the binary companion(mB), is theappearance of limit cycles, causing the periodic time variations ofplanetesimals’ final eccentricities. Fortunately, however, the phaseangles of the limit cycles are only weakly dependent on the radiusof a planetesimals, which causes these oscillations to appear al-most coherent.

In short, in disks with high values ofeg, the disk precessioncan actually reduce the relative velocities of planetesimals with dif-ferent sizes, compared to the case of a static disk. However,as canbe seen from Figure 5, although the eccentricities tend to groupneareg for a wide range ofs, the pericenters do not show such ageneral alignment. Thus, it is difficult to say at this point whetherthe precession of the disk will help or hinder the accretional pro-cess. A more detailed analysis is thus necessary.

5.2 Disk Density Profile

It is important to note that in all our simulations presentedso far,we have assumeda = 2 AU and a gas volumetric density ofρg = 5× 10−10 gr/cm3. Since the gas drag scales linearly with thegas densityρg, the drag dynamics of a given planetesimal scaleswith s/ρg. For example, the largest limit cycle in Figure 5 corre-sponds tos = 0.5 km for the chosen gas density. However, if thegaseous disk becomes smaller by a factorN , the orbital motionof that planetesimal would become similar to one with a size ofs = 0.5/N km. This suggests that in order to be able to assess theaccretional probability of planetesimals, it is importantto considera realistic gas density profile for the gaseous disk.

As shown by the results of hydrodynamical simulations of agaseous disk around the primary star of theγ-Cephei system byPaardekooper et al. (2008) and Kley and Nelson (2008), one ofthemost important consequences of the secondary star is to truncatethe original circumprimary disk at a point close to5 AU from theprimary and to introduce both an eccentricity and precession rateto the gaseous disk. Although the eccentricity and precession ratesseem to depend on the disk parameters (Kley et al. 2008), the trun-cation radius and final density profile of the disk seem to be morerobust and independent of the initial values. In particular, Kley andNelson (2008) show that after∼ 100 orbital periods of the sec-ondary, the surface density profile acquires a near-equilibrium state

s1 s2 gg = 0 gg 6= 0∆V ∆Vmin 〈∆V 〉

1 2 119 267 2771 5 195 451 4682 5 77 190 1972 10 102 256 2665 10 25 67 695 20 20 95 103

10 20 36 28 3410 40 305 86 12620 40 267 109 122

Table 1.Relative collision velocities (in m/s) for planetesimals with a = 2AU, eg = 0.02 and for the precessing disk2π/|gg| = 1000 yrs. Radii arein kilometers.

which is practically linear withr, reaching a zero value atr ≃ 5AU.

In this paper, we adopt the conclusions by Kley and Nelson(2008), and for all subsequent applications of our simulations totheγ-Cephei system, we employ a gas density law of the form:

Σ(r) = B(ag(out) − ag) (33)

whereag(out) = 5 AU is the outer edge of the disk andB is a con-stant depending on the total mass of the disk. Since the gaseousdisk is expected to be eccentric, its density should be constantalong lines of constant semimajor axisag and not along fixed val-ues of radial distancer. For this reason, we have expressed (33)in terms ofag. However, it must be noted that the transformationΣ(r) → Σ(ag) makes use of our assumption that all the gas ele-ments share the same longitude of pericenter at any given time. Ina more realistic gas disk this may not be the case.

Also, because the density of the gaseous is not very steep nearthe origin, we can approximately express the total mass of the disk(MT ) as:

MT ≃ π

3Ba3

g(out) (34)

The value of the constantB can be explicitly calculated from thisequation. The gas surface densityΣ at1 AU can now be written asΣ0 = B(ag(out) − 1) which implies that the volume densityρg isequal to

ρg(r) =B

2HR

(

ag(out)

ag− 1

)

. (35)

In this equation,HR = 0.05 is the scale height of the disk,taken constant for all values ofag. To specify the total mass ofthe gas disk, we note that the known planet in the binary systemof γ-Cephei has a mass of∼ 1.6MJup. We, therefore, chooseMT = 3MJup. This value is larger than the one used by Kleyand Nelson (2008) and smaller than the one by Paardekooper etal.(2008) and results inB = 1.3× 10−11 g/cm3.

5.3 Encounter Velocities

Tables 1 and 2 show values of encounter velocities between bodiesof different sizes but with the same semimajor axisa = 2 AU. Wecompare two cases: a static disk (gg = 0) and a retrograde precess-ing disk with2π/|gg| = 1000 yrs. Table 1 corresponds to an almostcircular gaseg = 0.02 while in Table 2 we considered a more ec-centric disk:eg = 0.2. The relative velocity at the encounter was

Dynamics of Planetesimals in an Eccentric Precessing Disk9

s1 s2 gg = 0 gg 6= 0∆V ∆Vmin 〈∆V 〉

1 2 877 39 5951 5 1874 212 11572 5 998 317 6522 10 1414 482 9285 10 412 206 3155 20 619 319 481

10 20 211 118 17210 40 315 178 25920 40 105 61 88

Table 2.Relative collision velocities (in m/s) for planetesimals with a = 2AU, eg = 0.2 and for the precessing disk2π/|gg| = 1000 yrs. Radii arein kilometers.

1 2 3 40

250

500

750

1000

∆V [m

/s]

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250

500

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1000

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250

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m/s

]

1 2 3 40

250

500

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1 2 3 4semimajor axis [AU]

0

250

500

750

1000

∆V [m

/s]

1 2 3 4semimajor axis [AU]

0

250

500

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1000

s1=1 , s

2=2 s

1=2 , s

2=5

s1=2 , s

2=10 s

1=5 , s

2=10

s1=5 , s

2=20 s

1=10 , s

2=20

eg=0.05

eg=0.10

eg=0.20

Figure 7. Relative collision velocities, as function of the semimajor axis,for several pairs of different sizes (values given in kilometers). Line colorscorrespond to different gas eccentricities, indicated in the upper left-handplot. The broad horizontal brown line gives the limit for disruption colli-sions (Stewart and Leinhardt 2009).

calculated using the equations of Whitmire et al. (1998). Since theprecessing disk introduces a limit cycle in the orbits, the impact ve-locity ∆V will also oscillate. We can characterize this spread by itsaverage〈∆V 〉 and its minimum value∆Vmin.

As shown in Table 1, encounter velocities are larger in a quasi-circular precessing disk, related to a larger eccentricitydispersionnoted fors < 2 km (see Figure 5). Such high encounter velocitieshinder the accretion of small planetesimals in disks wheregg 6= 0.

In a static and more eccentric disk (Table 2), the range of pos-sible equilibrium eccentricities lies betweenef andeg (Figure 6).This means that, the larger the gas eccentricity, the broader theeccentricity dispersion. The encounter velocities in thiscase arehigher and the collision among planetesimals may results inerosionand fragmentation. In an eccentric precessing disk, on the otherhand, the disk precession reduces planetesimals’ eccentricities tovalues belowef and causes bodies with different sizes to acquiresimilar eccentricities (bottom panels of Figure 6). In thiscase, eventhougheg may be much higher thanef , the disk precession reducesthe relative velocities during encounters even for small size bodies.

As shown in Table 2, even with the eccentricity-damping ef-fect of the disk precession, the impact velocities for objects withradii s < 5 km in an eccentric disk are still larger than the dis-ruption velocity. However, it is important to note that these resultswere obtained for planetesimals ata = 2 AU. Because a truncateddisk will have a much smaller gas density in its outer regions, it ispossible that different semimajor axes will give differentresults.

Figure 7 shows the variation of∆V for six different planetes-imal pairs, as a function of the semimajor axis and for three valuesof eg. In horizontal dashed lines we have also plotted the criticalrelative velocity for a catastrophic disruptionV ∗

RD using the recipedeveloped by Stewart and Leinhardt (2009) for weak aggregates.As shown in this figure,∆V decreases sharply in the outer parts ofthe disk for small planetesimals. Although for collisions betweenvery small bodies the impact velocity is still aboveV ∗

RD (upper leftpanel), we find that∆V < V ∗

RD for planetesimals withs ∼ 2km and semimajor axis beyond3 AU (upper right panel). Colli-sions between larger bodies are even more favorable (middleandlower panels). In all these cases the outcome of a collision seemsto be accretion, at least in a significant portion of the disk.Even ifthe inner disk still appears hostile, it is possible to envision a sce-nario in which accretional collisions between small planetesimalsoccurs preferentially in the outer disk. Then, as these bodies growand reach the inner regions due to orbital decay with the gas,theycould continue their growth closer to the star.

5.4 Timescales for Orbital Decay and Secular Equilibria

In the previous section we analyzed the relative velocity oftwoplanetesimals after they acquired their limit cycles. However, de-pending on the gas density and the radius of the object, the timerequired to reach the equilibrium solution may be longer than thetypical collisional timescale (Paardekooper and Leinhardt 2010).During this time, the planetesimals may also undergo orbital de-cay.

Although the interplay between dynamical and collisionalevolution can only be evaluated with full numerical simulations,here we present an estimate of the characteristic timescales of thesecular equilibria (τe) and semimajor axis decay (τa). Starting withinitial circular orbits anda = 2 AU, we defineτe as the time nec-essary for a planetesimal with radiuss to reach the final limit cyclewith a relative error less than1%. Since there are no equilibria inthe semimajor axis, we defineτa as the time necessary for the bodyto decrease its semimajor axis by∆a = 1 AU. Figure 8 showsτeandτa as functions of the planetesimal radiuss, for different val-ues of the disk precession rate. The case of a non-precessingdiskis identified with black lines, while color curves representdifferentprecession rates (see figure caption for details). In the twotop pan-els, we have assumed an almost circular gas disk (eg = 0.02). Asshown here, even a slow precession rate causes a significant reduc-tion in τe, which, for instance for as = 0.1 km planetesimal, falls

10 C. Beauge, A.M. Leiva, N. Haghighipour and J. Correa Otto

0.1 1 10100

1000

10000

1e+05

1e+06

τ e [yr

s]

0.1 1 10100

1000

10000

1e+05

1e+06

τ a [yr

s]

0.1 1 10s [km]

100

1000

10000

1e+05

1e+06

τ e [yr

s]

0.1 1 10s [km]

100

1000

10000

1e+05

1e+06

τ a [yr

s]

eg = 0.02 e

g = 0.02

eg = 0.2 e

g = 0.2

gg = 0

gg = 0

gg = 0 g

g = 0

gg =/ 0

gg =/ 0

gg =/ 0

gg =/ 0

Figure 8. Characteristic timescales for secular evolution in(k, h) variables(τe) and orbital decay (τa), as function of the planetesimal radiuss, forseveral values of the disk precession rate. Top plots were drawn assuminga quasi-circular gas disk (eg = 0.02) while in the two bottom graphs weadoptedeg = 0.2. In all cases the initial semimajor axes of the planetes-imals were chosen asa = 2 AU, and gas density given by equation (35).Line colors are indicative of disk (retrograde) precessionperiods: 200 years(blue), 1000 years (green) and 4000 years (red). The case of astatic (non-precessing) disk is shown in black.

from ∼ 105 years to∼ 103 years. Hence, a precession in the diskcauses a much faster evolution from the initial conditions towardsthe limit cycle.

Figure 8 also shows a comparison betweenτe and thetimescale for orbital decay. For a static diskτe > τa, implying thatthe decay in semimajor axis occurs faster than the time necessaryfor the planetesimal to reach the secular solution. The opposite,however, occurs for a precessing disk, where nowτe < τa for allvalues ofs. In this case then, it is expected that the body will reachthe limit cycle before suffering any significant orbital decay.

Figure 9 shows the results of two sets of N-body simulations.Planetesimals were initially ata = 2 AU with e = 0 and anadopted radius ofs = 10 km. The gas disk was assumed to have aneccentricity ofeg = 0.2. Black curves correspond to a static disk(no precession) while red curves denote a disk with (retrograde)precession period of1000 years. As expected from Figure 8, thesemimajor axis of a planetesimal falls more rapidly for a precess-ing disk, reaching1 AU in timescales slightly over105 years. Theorbital decay rate increases with time due to the greater gasden-sity near the central star (see equation (35)). The evolution of theeccentricity in this case shows different behaviors. Whilethe plan-etesimal approaches a quasi-circular orbit in the precessing disk,the opposite occurs in the static case, reaching values close to egfor small values ofa.

The two bottom panels of Figure 9 show the eccentricity asa function of the semimajor axis. On the left-hand panels we haveagain plotted the results of the exact N-body simulations. Due to theorbital decay, the orbital evolution occurs from the right to the leftof the graph. On the right, we plot the values of the center of thelimit cycles (continuous lines) as well as the minimum and max-imum values of the eccentricity (dashed lines), each determined

0 1e+05 2e+05time [yrs]

1

1.2

1.4

1.6

1.8

2

sem

imaj

or a

xis

[AU

]

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ntric

ity

1 1.2 1.4 1.6 1.8 2semimajor axis [AU]

0

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ntric

ity1 1.2 1.4 1.6 1.8 2

semimajor axis [AU]

0

0.05

0.1

0.15

ecce

ntric

ity

gg = 0

gg =/ 0

Figure 9. Two orbital simulations of a planetesimal withs = 10 kmin initially circular orbit with a = 2 AU. Black curves correspond to astatic disk while red shows results using a retrograde precessing disk with2π/|gg| = 1000 yrs. In both cases the gas eccentricity waseg = 0.2. Thecontinuous lines in the lower right-hand plot shows the center of the limitcycles as function of the semimajor, while the dashed lines show the max-imum and minimum values of the eccentricity. Notice the excellent agree-ment with the numerical simulation shown in the left-hand plot.

numerically from the averaged model (28) for fixed values of thesemimajor axis.

Except for the first few tenths of AU close to the initial con-dition, the rest of the planetesimal’s evolution occurs very close tothe instantaneous limit cycles for each value of the semimajor axis.In other words, even in the presence of the significant orbital decay,the secular dynamics of the planetesimals is expected to be dictatedby the limit cycles in(k, h). Moreover, except for the first few103

years, it is expected that the encounter velocities of collisions be-tween planetesimals be determined by the equilibrium solutions ofthe secular problem, and not by the transient evolution fromtheinitial conditions to the limit cycles.

6 CONCLUSIONS

In this paper we have presented an initial study of the dynamicsof individual planetesimals in circumstellar orbits of a tight binarysystem. Apart from the gravitational perturbations of the secondarystar, we have also considered the effects of a non-linear gasdragdue to an eccentric precessing gaseous disk.

A non-zero precession frequencygg in the gas disk introducesthree main changes in the secular dynamics of the solid bodies.On one hand, the eccentricity and longitude of pericenter nolongerreach stable fixed points (in the averaged system), but display peri-odic orbits in the(k, h) plane with the same frequency as the disk.This appears to be independent of the semimajor axis of the plan-etesimal, and thus of the secular frequencyg of the gravitationalperturbations due to the secondary star. The amplitude of the limitcycle is inversely proportional to the planetesimal radiuss, reach-ing values close to zero for large bodies.

Another issue is the range of planetesimals’ final eccentrici-

Dynamics of Planetesimals in an Eccentric Precessing Disk11

ties. For large planetesimals, the limite → ef is still observed forall values ofgg, whereef is the forced eccentricity induced by thesecondary star. However, the behavior of small bodies is dependenton gg. In a static disk,e → eg for planetesimals strongly coupledto the gas. However, the final eccentricity is significantly lower ina precessing disk, reaching almost circular orbits for highvalues ofgg.

Finally, we have also noted that the precession also reducesthetime necessary for a given planetesimal to reach the stationary limitcycle, starting from initially circular orbits. Although the orbital de-cay is also accelerated, the characteristics timescale associated withthe semimajor axis is always longer than the secular timescale. Thisseems to imply that, except for a very small time interval immedi-ately following the initial conditions, the full dynamics of the plan-etesimals should be well represented by the equilibrium periodicorbits obtained for each instantaneous value ofa.

Applying these results to theγ-Cephei binary system, and as-suming a gas density profile similar to those obtained from hydro-dynamical simulations, we find that the precession of the disk alsocauses significant differences in the relative velocity of collidingplanetesimals of different radii. For quasi-circular gas disks, a non-zero value ofgg appears to increase the encounter velocities, hin-dering the possibility of accretional collisions among small ∼ 1−5km bodies. However, the opposite effect is observed for values ofeg ∼ 0.2, where a non-zero precession of the gas actually helpsreduce the eccentricity dispersion and reduce the collisional veloc-ities.

Assuming a gas disk of total mass∼ 3MJup we found thataccretional collisions may occur in the outer parts of the disk fors ≥ 2 km. For larger bodies, the location of minimum relative ve-locities appears closer to the central star. However, full N-body sim-ulations are necessary in order to fully evaluate the consequencesof this dynamics on a mutually interacting protoplanetary swarm.

ACKNOWLEDGMENTS

This work has been supported by the Argentinian Research Coun-cil -CONICET- and by the Cordoba National University. A.M.L. isa post-doctoral fellow of SECYT/UNC. NH acknowledges supportfrom the NASA Astrobiology Institute under Cooperative Agree-ment NNA04CC08A at the Institute for Astronomy, UniversityofHawaii, and NASA EXOB grant NNX09AN05G. This project wasinitiated during the program, ”Dynamics of Disks and Planets” thatwas held from August 15 to December 12, 2009 at the Newton’sInstitute of Mathematical Science at the University of Cambridge(UK). C.B. and N.H. would like to thank the organizers of the pro-gram, and acknowledge the warm hospitality of the Newton’s Insti-tute. Finally, we are grateful to F. Marzari for his helpful review ofthis work.

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