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Planetesimal Collisions in Binary Systems

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Planetesimal Collisions in Binary Systems Mon. Not. R. Astron. Soc. 403, L64–L68 (2010) doi:10.1111/j.1745-3933.2010.00816.x Planetesimal collisions in binary systems S.-J. Paardekooper andZ.M.Leinhardt DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA Accepted 2010 January 15. Received 2010 January 15; in original form 2009 December 17 ABSTRACT Downloaded from https://academic.oup.com/mnrasl/article/403/1/L64/1187217 by guest on 24 September 2021 We study the collisional evolution of kilometre-sized planetesimals in tight binary star systems to investigate whether accretion towards protoplanets can proceed despite the strong gravi- tational perturbations from the secondary star. The orbits of planetesimals are numerically integrated in two dimensions under the influence of the two stars and gas drag. The masses and orbits of the planetesimals are allowed to evolve due to collisions with other planetes- imals and accretion of collisional debris. In addition, the mass in debris can evolve due to planetesimal–planetesimal collisions and the creation of new planetesimals. We show that it is possible in principle for kilometre-sized planetesimals to grow by two orders of magnitude in size if the efficiency of planetesimal formation is relatively low. We discuss the limitations of our two-dimensional approach. Key words: planets and satellites: formation – stars: individual: γ Cephei – stars: individual: α Centauri – planetary systems: protoplanetary discs. sizes. Thebault,´ Marzari & Scholl (2006) suggest that the destruc- 1 INTRODUCTION tive collisions prohibit the growth of planetesimals in γ Cephei at We consider the problem of planetesimal evolution in tight binary 2 au, the current location of the planet. star systems such as γ Cephei and α Centauri. To date, several Although planets have yet to be detected in the α Centauri system, hundred extrasolar planets have been detected – 20 per cent of which the proximity of the system and the roughly solar mass of both stellar orbit the primary of a binary or multiple system (Desidera & Barbieri components make the system a prime target for finding an Earth 2007). In most of these cases the stars are widely separated and do mass planet in the habitable zone (Guedes et al. 2008). However, not significantly affect the evolution of planets, but γ Cephei, Gl 86, the α Centauri system is more extreme than γ Cephei with a mass HD 41004A and possibly HD 196885 are examples of tight binary ratio close to 1 (MA = 1.1 M and MB = 0.93 M), eb = 0.52 systems with a semimajor axis, ab, around 20 au, that have Jupiter and ab = 23.4 au. Thebault,´ Marzari & Scholl (2008, 2009) found mass planets. γ Cephei has the most extreme binary parameters that impact speeds between different sized planetesimals would be of the known tight binaries with detected planets. The primary above the disruption threshold due to differential orbital phasing, star is a K giant (MA ∼ 1.4 M) and the secondary is an M and planetesimals could not grow in the habitable zone in either α dwarf (MB ∼ 0.4 M); the binary has a high eccentricity (eb = Centauri A or B. 0.4) and a small ab of 20 au. The planet is 1.6 MJ with an a ∼ In this Letter, we show that planetesimal growth from 1 to 100 km 2 au (Neuhauser¨ et al. 2007). Due to its extreme characteristics, is possible in principle in these tight binaries when the collisional γ Cephei is often considered one of the most stringent tests for evolution of the planetesimal disc is taken into account. Disruptive planet formation models. collisions will prevent differential orbital phasing from being set up Previous work by Thebault´ et al. (2004) and Quintana et al. and, at the same time, create a reservoir of small debris that can be (2007) has shown that planets can form in γ Cephei if planetary accreted on to the leftover planetesimals. embryos can form within the stable region of <3au(Wiegert& Holman 1997). However, work on the earlier stage of planet for- 2 DIFFERENTIAL ORBITAL PHASING mation has shown that it is very difficult to grow planetesimals that are initially in the kilometre size regime because of the secular In the absence of any gas, a tight eccentric binary companion will perturbations from the secondary star on the planetesimals and the excite eccentricities of particles orbiting the primary. For particles gas disc. The coupled effect of perturbations from the binary and starting on circular orbits, their eccentricity will oscillate around the gas drag from the disc causes differential orbital phasing, and the forced eccentricity ef : high-speed destructive impacts between planetesimals of different 5 a e e (a) = b , (1) f − 2 4 ab 1 eb with amplitude 2ef ,wherea is the semimajor axis of the particle E-mail: [email protected] (Heppenheimer 1978). Since there is no dissipative force in the C 2010 The Authors. Journal compilation C 2010 RAS Planetesimal collisions in binary systems L65 0.15 10 for the collision time-scale −2 R 17gcm ρp − 8 τ ≈ 11 750 1, (2) c −3 0.10 km s 3gcm 6 where ρp is the bulk density of the particles. This means that for typ- e ical parameters at 1 au, a kilometre-sized planetesimal will undergo 0.05 4 a collision once every few thousand years. 2 0.00 0.04 0.06 0.08 0.10 0.12 0.14 0.16 3.2 Binary star case a/a b Perturbations due to a coplanar, eccentric binary will increase the Figure 1. Eccentricity versus semimajor axis of collisionless, massless test eccentricity of the planetesimals. Differential orbital phasing can particles in the γ Cephei system after 100 binary orbits in a gas disc with a easily give rise to the velocity dispersion entering the collision − − 9 3 Downloaded from https://academic.oup.com/mnrasl/article/403/1/L64/1187217 by guest on 24 September 2021 constant gas density of ρg = 1.4 × 10 gcm . Colour indicates the size time-scale which will go up by two orders of magnitude compared in km; the black curve with the large oscillations denotes a gas-free case. to the single-star case. Keeping all other parameters the same, a kilometre-sized planetesimal will now undergo a collision every system, the oscillations do not damp and their spatial frequency 10–20 orbits at 1 au. If we take the binary companion to have a increases with time, which eventually leads to orbital crossing of semimajor axis of 20 au, this collision time-scale amounts to 25 per neighbouring particles (Thebault´ et al. 2006). This is illustrated cent of a binary orbit. This should be compared to the time-scale for by the black curve in Fig. 1. As soon as particles at an eccentricity a particle to reach its equilibrium orbit in the binary system, which maximum can collide with particles at an eccentricity minimum, the happens on a secular time-scale (many binary orbits). Therefore, resulting encounter velocities are too high for accretion to occur. physical collisions are of crucial importance for the evolution of the The presence of a gas disc and the associated aerodynamic gas planetesimal population. It is important to stress that not only, as has drag on the particles can effectively damp the oscillations leading been realized before, are collisions destructive when differential or- bital phasing happens, they can actually prevent this size-dependent to a well-defined equilibrium eccentricity distribution e0 = e0(a) (see Paardekooper, Thebault´ & Mellema 2008), with correspond- orbital structure from being set up in the first place. It is therefore ing periastron alignment. The time-scale for this equilibrium to be necessary to take collisions into account when studying the effect reached is the secular time-scale. Therefore, it takes many binary of differential orbital phasing on planetesimal accretion. orbits before planetesimals are on their equilibrium orbits. Once Another consequence of having many possibly catastrophic col- they have reached their equilibrium orbits, particles of the same lisions in the system is a large amount of small debris. This small size have orbits that are in phase and therefore they will not suf- debris can act as a source of new planetesimals or be accreted on to fer from high-speed collisions with each other (Marzari & Scholl remaining larger bodies. All these ingredients need to be taken into 2000). However, since the magnitude of the gas drag force depends account in a consistent model of planetesimal accretion in binary systems. on the particle size, each particle size will have a different e0(a), a phenomenon that is called differential orbital phasing (Thebault´ et al. 2006). In Fig. 1, particles of a single size (and colour) follow 3.3 Two-dimensional disc phased orbits with no oscillations. The residual oscillations that can be seen for the largest particles will eventually be damped com- In order to reduce the computational cost of the models, we will pletely by gas drag. However, for particles of differing sizes there is work in a two-dimensional (2D) geometry. The collision time-scale, for orbits with a random phase, is then given by a considerable spread in eccentricity and, not shown, longitude of periastron. The difference in the eccentricity magnitude can easily 1 2.4 × 10−4 R 2 17gcm−2 ρ τ = = p −1, c,2D −3 exceed 0.01 for particles of 1 and 5 km. This implies that particles 2Rnv¯ e km s 3gcm of different sizes will undergo high-speed collisions, which are usu- (3) ally destructive. As a result, it is difficult to form planetary cores in tight binary systems if the only growth mechanism is accretionary where n¯ is the surface number density and e is a measure of the ≈ × −5 = planetesimal–planetesimal collisions.
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