Regions of Dynamical Stability for Discs and Planets in Binary Stars of the Solar Neighborhood 3

arXiv:1208.2051v2 [astro-ph.EP] 17 Aug 2012 ∗ & Quist 1999; (Söderhjelm 1999; al. astrometry et spatial (Simon 2001), optics Close adaptive 2001), Guilloteau Launhar 2001, 2000, al. sub et long-baselin (Launhardt and ima- interferometry infrared millimeter millimeter submillimeter and 2001a,b), optical 2010), (Quirrenback 2000), al. interferometry al. et et 1999; Borucki (Smith 1997, 2001; ging al. al. et et (Padgett HST Reid imaging the NICMOS) as obser- & such in (WFPC2 instrumentation, advances and recently, techniques activity More vational & observational stimulated. (Mayor 1998), greatly Butler star was & sequence Marcy main extrasola 1995; a first Queloz the around after (Wolszc- discovered particularly pulsar planet and th 1992), a of Frail around discovery & system the zan after et planetary right (Mathieu extrasolar addition, regions first In star-forming 1994). of 1992, variety bi- al. a young on of surveys stars large nary enabled have technology imaging 1994). primary (Mathieu the process formation is star formation the binary of that branch suggests This 1983).

oa egbrod h rcinge pto up the goes in fraction particular the in & and Neighborhood, (Duquennoy 1992), Solar Marcy systems & multiple Fisher or 1991; Mayor binary of members are INTRODUCTION 1 cetdxx eevd;i rgnlform original in ; Received xxx. Accepted o.Nt .Ato.Soc. Astron. R. Not. Mon. us .Jaime G. Luisa Neighborhood in Solar Planets the and of Discs Stars for Binary Stability Dynamical of Regions c 2 3 1 nttt eCeca ulae,UiesddNcoa Aut Nacional Universidad Nucleares, Ciencias de Instituto nttt eAtooá nvria ainlAutónoma Nacional Astronom´ıa, Universidad de Instituto nttt eAtooá nvria ainlAutónoma Nacional Astronom´ıa, Universidad de Instituto -al [email protected](BP) [email protected] E-mail: RAS infiatavne nhg-nua-eouininfrared high-angular-resolution in advances Significant ti nw htms o-asmi-eunestars main-sequence low-mass most that known is It 1 , 000 2 abr Pichardo Barbara , –6( rne 4d gsod 08(NL (MN 2018 de agosto de 24 Printed () 1–16 , nets. hr ic a aeetbihdadpaesformed. words: sy planets Key binary and fo around established limits have planets precise may for defines discs search method each where our the In reg since in stability parameters, (2005,2008). guide orbital calculated know al. a our within provide et lay a to Pichardo planets 116727 intends of the HIP of approximation 67275, position the HIP reported to 4954, par test HIP orbital 10138, a known nume HIP with systems constructed planets: S binary the confirmed we five in Additionally, around stars parameters. binary loops) (invariant 161 orbital for regio known orb study determine stable this with in we produce survive We (2005,2008), could general) systems. al. in stellar discs et (or Pichardo planets where of stability results the Using ABSTRACT ∼ icmtla atr ic iay tr,SlrNihoho,e Neighborhood, Solar stars, binary: – discs matter, circumstellar 8%(Abt % 78 1 o oad ´xc,Ad.psa 053Cua Universitari Ciudad 70-543 postal México, Apdo. ónoma de eMeio po otl87 20 neaa México Ensenada, 22800 877, postal México, Apdo. de eMeio po otl7-6,Cua nvriai,Mé Universitaria, Ciudad 70-264, postal México, Apdo. de dt ∗ e e r - n usAguilar Luis and , nams l ae,tepae risi -yeconfiguratio separations. S-type in orbital orbits of planet the range cases, wide all reside a almost In to with known systems are % binary 10 in about stars, sequence ex- main http://exoplanet.eu of around (see hundreds on far several www.exoplanets.org) of influence so http://planetquest.jpl.nasa.gov, the presence strong confirmed From al. the planets very planets. trasolar cases et a and have these Queloz discs should In 2011; both star 2003). al. companion al. et De- a et bi- Doyle 2010; Hatzes close Weinberger 2007; around & 2000; Barbieri Prato lie & 2011; formed, sidera al. be et to (Wright assumed w naries ad- are discs In planets circumstellar (2010). re discovered al. obser- recently et of several review Muterspaugh dition, a see For techniques 2006). vational Both Muterspaugh al. & etc.). et (Pfahl 1988, Udry cesses in Lyne 2003; formed 1993; of al. Phinney suspected Eg- et & 2005; Sigurdsson 2008; Sigurdsson al. 2004; al. 2002; et al. et Mugrauer Fischer et 2007; 2005; genberger Barbieri Konacki Deeg & 2006; 2008; Desidera Raghavan (Correia 2008; discovered al. been et have systems star ple and stage. systems formation the binary during study of built to physics discs surrounding possibility the the the better now and understand technology have and new we this availa- all work, also to observational Thanks are studies. 2009), binary Rattenbury in ble 2008; 2001 al. al. et et (Alcock Dong-Wook microlensing and 2001), 2000, Lindegren ntercn at eea lnt nbnr rmulti- or binary in planets several past, recent the In A T E tl l v2.2) file style X 3 situ raqie ydnmclpro- dynamical by acquired or , h tberegions, stable the r lrneighborhood olar mtr n with and ameters t rudbinary around its dKpe 6 as 16, Kepler nd ial h discs the rically so dynamical of ns os hsstudy This ions. tm ihwell with stems igecs,the case, single xico ,DF,México D.F., a, xopla- obe to he- ns ; , 2 L. G. Jaime B. Pichardo L. Aguilar

(Dvorak 1986), while the second star acts as a perturber to of time and the original hamiltonian as two extra dimensions the planetary system. A circumbinary planet (P-type or- in phase space. Regular orbits will lie on 3-D manifolds and bit) has recently also been detected in Kepler 16 (Doyle et be multiple periodic with three frequencies, one of which is al. 2011). This has motivated the search for stable perio- given by the binary orbital frequency. If we take snapshots dic orbits around binary systems where planets (and discs at a fixed binary phase, the projections of a regular orbit in general) can settle down in a stable configuration. Most will lie on a 2-D manifold. If the orbit has an additional theoretical studies have focused on binaries in near-circular isolating integral of motion, this projection will now lie on orbits (Hénon 1970; Lubow & Shu 1975; Paczyński 1977; Pa- a 1-D manifold: an ¨ınvariant loop”(Maciejewski & Sparke paloizou & Pringle 1977; Rudak & Paczyński 1981; Bonell & 1997, 2000). Bastien 1992; Bate 1997; Bate & Bonnell 1997). Even very Stable invariant loops represent the generalization to precise analytical methods to approximate periodic orbits in periodically time-varying potentials of stable periodic orbits circular binaries are available (Nagel & Pichardo 2008). in steady potentials. PSA1 and PSA2, implemented a nu- Due to the lack of conservation of the Jacobi integral, merical method to find them. The equations of motion are the case of eccentric binaries is qualitatively more complica- solved in an inertial reference frame with Cartesian coordi- ted. Artymowicz & Lubow (1994) and Pichardo et al. (2005, nates with the origin at the binary barycenter. An ensemble 2008, hereafter PSA1 and PSA2) calculate the extent of zo- of test particles is launched when the binary star is at perias- nes in phase space available for stable, non self-intersecting tron, and from the line that joins both stars at that moment, orbits around each star and around the whole system. In to search for invariant loops. A more detailed explanation of this study we use the results of PSA1 and PSA2, to calcula- the method and a study of the phase space in binary systems te stable regions for planets or discs in binary systems in the is in those references. In this work we employ the formulae Solar Neighborhood with known orbital parameters such as, from PSA1 (Eq. 6) and PSA2 (Eq. 6). These relations pro- mass ratio, eccentricity and semimajor axis. vide the maximum radius of circumstellar stable zones and Although the existence of stable zones, as the ones we the inner radius of the circumbinary stable zone, in terms are calculating here, is a necessary, but not a sufficient con- of the mass ratio (q = m2/(m1 + m2), where m1 and m2 dition to the existence of planets (or discs in general), if any are the masses of the primary and secondary stars, respec- stable material (planets, gas, etc.), exists in a given system, tively), and the eccentricity (e = p1 − b2/a2, where a and irrespective of their formation mechanism, they would be ne- b are the semimajor and semiminor axes of the binary or- cessarily located within the limits of the stable zones. In this bit). The radius of the outer limit of the circumstellar stable direction, a fruitful line of investigation is the intersection zones from PSA1, is given by, between phase space available zones for the long term evolution of planetary systems and habitable regions allowed by × − 1,20 0,07 the binary system (Haghighipour et al. 2007; Haghighipour Ri = Ri,Egg [0,733(1 e) q ] , (1) et al. 2010). and a similar study in PSA2 but for the circumbinary region, We present a table with the compilation of all the bina- gives a relation for the inner radius, ries in the Solar Neighborhood with known orbital parameters from different sources. In the same table are presented 0,32 0,043 the results for our calculated circumstellar and circumbinary RCB (e, q) ≈ 1,93 a (1+1,01e ) [q(1 − q)] . (2) stable zones around each binary system. This paper is organized as follows. In Section 2, we ex- In these relations Ri,Egg is the approximation of Eggleton plain briefly the method employed to calculate regions of to the maximum radius of a circle circumscribed within the stable non self-intersecting orbits around binary stars. The Roche lobe (Eggleton 1983): binary star sample is presented in Section 3. Results, inclu- ding stable regions of orbital stability for circumstellar and 2/3 0,49qi circumbinary planetary discs (and discs in general), and an Ri,Egg/a = 2/3 1/3 , (3) application to observations of five binaries with observed 0,6qi + ln(1 + qi ) planets, are given in Section 4. In Section 5 we present our and conclusions. 1 − q q q1 = m1/m2 = and q2 = m2/m1 = . (4) q 1 − q 2 THE METHOD In Figure 1 we present a schematic figure of the geometry In the simpler case of circular binary orbits, the poten- of the system. We show in this figure some of the variables tial is time-independent in the co-rotating frame and thus involved in the problem. the Jacobi energy is conserved. This allows the existence of We must emphasize that the regions of stable orbits closed orbits that, when stable, spawn the orbital structure we report here, are the regions were these invariant loops of the system (Carpintero & Aguilar 1998). In the gene- exist, and furthermore, we restrict ourselves only to non self- ral case of binaries in eccentric orbits, the problem is more intersecting orbits, where gas could settle and planets may complex, as now the potential is time-dependent. However, form. It is in this sense that the term ”stable region”should we can exploit the fact that the time-dependency is strictly be understood in this work. Using these formulae and res- periodic. triction, we have calculated circumbinary and circumstellar A time-periodic potential in 2-D can be casted as a 3-D radii for our sample with a total of 161 binary systems in system with an autonomous hamiltonian, with the addition the solar neighborhood with known orbital parameters.

c RAS, MNRAS 000, 1–16 Regions of Dynamical Stability for Discs and Planets in Binary Stars of the Solar Neighborhood 3

lar masses. Finally, the last column is a shcematic ﬁgure that gives the relative sizes and positions of the stable zones (see Figure 1). For instance, for BD -8o 4352 (9th entry in the table) the circumstellar regions are symmetric and cover a good fraction of the inner hole of the circumbinary region, this is is due to the low eccentricity of the system and its high q(=0.5). In contrast, the binary called Gamma Vir (6th entry in the table), present circumstellar regions slightly asymmetric, due to the small mass diﬀerence between the stars, and quite narrow, due to its high eccentricity.

4 RESULTS The presence of a stellar companion, particularly in an eccentric orbit, severely curtails the size and shape of the stable zones. While a single star has circular stable orbits at all radii (neglecting finite stellar size and tidal distor- tion effects), the presence of a stellar companion severely curtails the region where stable, non-self intersecting orbits may exist, both in extent and, to a lesser extent, shape. It is unclear at present the way in which these effects impose restrictions in the process of star formation. What is clear is that the effect is in the sense of inhibiting, rather than promote it. From the observational side, the statistics of the obser- Figure 1. Schematic figure of the geometry of the stable zones ved systems suggests that binarity does indeed has en effect in a binary system. The black outer circle represents the inner boundary of the circumbinary region (eq. 2). The orange and on planetary masses and orbits (Eggenberger et al. 2004), green curves are the primary and secondary stars. The gray cir- even restricting terrestrial planet formation for binary pe- cular areas around each star represent the circumstellar stable riastron smaller than 5 AU affecting discs to within ∼1 AU zones (eq. 3). The red dot at the center represents the system of the primary star (Quintana et al. 2007, Quintana & Lis- barycenter. In the upper part we show a schematic rectangle that sauer 2010). we will use in Table 4, to show graphically the relative extent and The goal of this study is to determine the extent of situation of the stable regions, in relation to the stars, for each circumstellar and circumbinary regions of stable, non-self entry in our sample. Dotted lines join the corresponding parts in intersecting orbits, as plausible regions where planets could the upper and lower representations. have formed and may exist. It could also indicate possible regions of planetary formation. Figure 2 shows the binary semimajor axes vs. the or- 3 THE SAMPLE bital eccentricity of our entire sample, split in mass ratio intervals. As it is already known, the region of small semi- The previous equations require, besides the stellar mass major axes and high eccentricites is depleted, i.e. very close ratio, the semimajor axis and orbital eccentricity for each binaries, tidally locked in general, have eccentricities clo- binary system, two parameters that are difficult to deter- se to zero (Duquennoy & Mayor 1991). A large percentage mine observationally. There is a diversity of methods that (about 60 %) of binaries in the sample have low eccentricities have been used to estimate them. Our sample is a compi- (e . 0,5), small semimajor axes (a . 50 AU) and large mass lation from different sources (Jancart et al. 2005; Martin et ratios (q & 0,4), as shown in the histograms in Figure 3. Con- al. 1998; Strigachev & Lampens 2004; Bonavita & Desidera sequently (as seen in Figure 4), circumstellar stable regions 2007; Holman & Wiegert 1999; Mason et al. 1999; Latham in this sample have small radii (. 2 AU), with both stable et al. 2002; Balera et al. 2006; Cakirli et al. 2009; Milone regions (circumprimary and circumsecondary) in most sys- et al. 2005; D´ıaz et al. 2007; Konacki et al. 2010; Desidera tems having similar size. On the other hand, the majority & Barbieri 2007; Doyle et al. 2011). We include all binaries of circumbinary gaps have radii smaller that ∼50 AU. with an estimation of these parameters within 100 pc from the Sun (currently 161 systems). In Table 4 we present our sample. Columns 1 and 2, are the name of the systems in the Hipparchos catalogue and an alternate name, if it exists. Columns 3 to 6 are our input data: the semimajor axis, orbital eccentricity and stellar masses as reported in the references listed in column 10. Columns 7 to 9 list our results: the circumprimary, circumstellar, and circumbinary boundaries of the stable regions. In the circumstellar cases, the value is the radius of the outer boundary. In the circumbinary, is the radius of the inner boundary. All lengths are given in AU and masses in so-

c RAS, MNRAS 000, 1–16 4 L. G. Jaime B. Pichardo L. Aguilar

Disc Radii (Solar Neighborhood Binary Sample)

0 < q 0.1 0.1 < q 0.2 20 0.2 < q 0.3 0.3 < q 0.4 10 0.4 < q 0.5 0 0 2 4 6

0 0 2 4 6

0 0 50 100 150 200

Figure 2. Binary semimajor axes (AU) vs eccentricities of our Figure 4. Histograms (from top to bottom): Circumprimary, cir- sample. The mass ratio is indicated with various colors and sym- cumsecondary and circumbinary radii, for all systems in our sam- bols, as shown in the upper left corner. ple

25 20 15 10 5 0 0 0.2 0.4 0.6 0.8 1

0 0 50 100 150 200 250

0 0 0.1 0.2 0.3 0.4 0.5

Figure 3. Histograms (from top to bottom): Eccentricity, semimajor axis and mass ratios, for all systems in our sample.

c RAS, MNRAS 000, 1–16 Regions of Dynamical Stability for Discs and Planets in Binary Stars of the Solar Neighborhood 5 2 ∗ 1 ∗ Ref Scheme cb R 2 Rce 1 Rce ) (AU) (AU) (AU) ⊙ 2 M ) (M ⊙ 1 a e M 2505 4.58 0.73 0.29 0.29 0.25 0.25 15.93 5 4352 1.35 0.05 0.42 0.42 0.33 0.33 3.41 5 o o Com 12.49 0.5 1.43 1.37 1.45 1.42 41.08CMa 5 Cen 19.89 0.59 2.11 23.57 0.52 1.04 1.12 2.12 0.95 1.54 2.71 66.70 2.52 5 77.86 5 Vir 37.84 0.88 0.94 0.9 0.78 0.77 135.53 5 Equ 4.73 0.42 1.66 1.59 0.66Boo 0.64 15.18 33.14 5 0.51 0.86 0.73 3.85 3.58 109.35 5 Cet 1.57 0.27CrB 1.3 1.3 13.98 0.28 0.28 0.79 0.28 0.78 4.75 2.51 5 2.49 42.43 5 ǫ γ α ǫ δ α α ξ HIP (AU) (M Object Alter. name ––– HD 10361– HD 145958A 52.2– 124 HD 145958B 0.53 124 HD 0.39 146362 0.77– 0.9 0.39 130 0.75– 0.89 0.89 0.76– 5.61 0.9 2.19 18.17– 5.54 18.08 1.12 18.17– 173.15 393.95 18.08 4 6.98 BD-8 4 393.95– 5.14 BD 4 45 – 452.9– 4 – Kpr 37– 99 Her 9 Pup 9.67– 0.15 16.39– 1.2 0.74– 10.00 0.89 0.69 0.89– 0.52 0.98– 2.26 G9-42 0.98 0.87 G62-30 1.97 0.77– 0.67 G165-22 27.23 0.44– 56.97 0.63 5 0.79 0.81– 5 G65-52 34.49 0.1 0.59 0.77– G178-27 5 0.68 0.08 0.04– G15-6 0.48 0.82 0.04 0.02 0.33 G66-65 0.26 0.07 0.1 0.01 0.43 0.62 G141-8 0.73 0.03 0.68 0.03 1.45 0.04 0.61 0.45 0.01 0.05 2.49 0.8 7 0.12 0.9 0.67 0.25 0.06 7 0.04 0.7 0.06 0.58 7 0.02 0.77 1.36 0.13 0.04 1 0.02 0.04 7 0.01 0.11 2.25 0 7 0.02 7 2.05 2.43 7 7

c RAS, MNRAS 000, 1–16 6 L. G. Jaime B. Pichardo L. Aguilar 2 ∗ 1 ∗ Ref Scheme cb R 2 Rce 1 Rce ) (AU) (AU) (AU) ⊙ 2 M ) (M ⊙ 1 a e M HIP (AU) (M Object Alter. name ––– G18-35– V821 Cas473 SV Cam BS Dra 4.48 HD1349 38 0.044 0.371955 0.13 HD 1273 0.75 0.016 2.0462237 HD 0 2070 0.07 1.626 0.0602552 HD 2475 73.01 0.01 0 0.93 0.8622762 0.22 1.25 0.646 0.01 0.32 0.76 1.294 HD 0.54 0.57 3196 0.00452941 0.12 1.276 13.44 0.98 0.74 0.004 7.08 0.333821 0.02 ADS520 0.03 7 1.13 10 0.55 14.41 03850 HD 4614 0.016 0.48 14.24 11 5.13 0.135531 0.11 1.56 HD 4747 215.35 41.32 0.1 0.775842 0.1 3 1.24 9.57 0 11 1.71 HD 0.07 4.18 0.22 72 7693 1.967078 1.14 0.4 0.7 1.66 6.77751 1 1.76 0.25 0.49 HD 9021 0.29 0.77918 0.99 1 0.64 12.87 HD 10360 0.21 11.6 23.4 0.828903 0.51 2 HD 1.87 10307 17.96 5.00 0.04 0.048903 10.02 9.54 HD 0.64 11636 2 0.89 1.87 0.72 75.05 0.73 52.2 HD 11636 0.31 1.17 7.0548904 28.23 0.84 3 7.1 0.53 1.21 0.19 235.07 1.169480 5 5.96 0.77 0.63 4 0.7 21.21 0.4210138 0.29 WDS 0.75 02019+7054 5.81 0.66 0.88 0.8 4 HD1344510321 0.12 22.5 1.86 0.29 57.89 5.61 0.9 0.1411231 HD 13507 0.39 0.09 17.36 1.05 4 2.07 5.54 HD 1.92 1.26 15064 18.4 1.97 8 0.01 1.28 173.15 0.027 1.19 0.4 0.57 4 1 0.01 0.01 0.01 4.3 0.77 3.59 22.13 0.365 2.25 0.64 0.01 0.49 0.14 0.348 2 2.88 0.29 1 2.86 2.37 1 0.007 71.31 1.01 2.33 0.006 2 6 0.05 0.68 0.06 58.53 1.34 0.12 13 9 0.36 0.1 11.18 1.95 4 1

c RAS, MNRAS 000, 1–16 Regions of Dynamical Stability for Discs and Planets in Binary Stars of the Solar Neighborhood 7 2 ∗ 1 ∗ Ref Scheme cb R 2 Rce 1 Rce ) (AU) (AU) (AU) ⊙ 2 M ) (M ⊙ 1 a e M HIP (AU) (M Object Alter. name 1206212114 HD 1586212153 HD 1616012623 HD 16234 2.0412777 HD 16739 15 HD 0.26 1689513769 4.22 0.95 0.7514954 HD 18445 1.27 0.88 0.44 HD19994 0.7618512 249.5 11 0.66 0.4319206 0.09 HD 24916 0.13 1.13 1.06 0.320087 9.41 1.01 1.24 120 1.39 HD 0.09 0.56 27176 0.43 0.39 6.1120935 0.26 0.1 174.55 0.78 0.08 66.5 1.35 50.2622429 HD 0 28394 1 0.09 0.18 0.35 15.1123395 41.09 HD 4 30339 7.05 0.35 0.13 27.34 4.35 684.2523835 WDS 2 05017+2640 9.290 14.83 0.17 0.17 0.99 0.07 4 10.2824419 HD 32923 354.86 2 0.69 1.76 52.36 0.13 0.33 HD 13 3.47 0.24 34101 0.960 0.9525662 37.68 1 1.13 0.25 0.79027913 HD 4 1.66 315.65 35956 2.86 0.64 1.39 1.1129860 HD 0.72 3 39587 1.75 1.26 0.9 0.07 0.58 0.1933451 HD 1.79 43587 20.08 0.08 0.03 1.11 32.1 2.6 0.1934164 HD 51825 1.54 0.9 2 1.03 5.9 0.01 HD 2.95 53424 8 0.62 31.938657 0.21 11.6 0.05 0.36 0.98 0.4539064 HD 1 64468 9.3 0.52 6 0.05 0.8 1.05 0.1839893 HD 4 65430 1.7 0.27 1.06 10.28 0.43 0.14 0.2844248 1.61 4.52 0.27 0.34 0.56 1.0144892 HD 4 0.13 76943 1.09 1.26 4 0.54 HD 0.41 78418 0.26 1 8.58 0.66 1.31 0.81 0.32 18.41 0.32 10.51 4 0.34 1.17 0.14 40.39 4 0.83 0.184 0.1 1.81 0.27 0.13 29.93 4 0.06 0.2 1.53 0.21 5.13 0.06 2 0.92 1.173 0.92 0.95 1.64 1 1.011 0.29 2.69 0.52 0.04 4 11.66 0.4 2.13 0.04 4 28.27 0.31 0.53 2 5.29 12 1

c RAS, MNRAS 000, 1–16 8 L. G. Jaime B. Pichardo L. Aguilar 2 ∗ 1 ∗ Ref Scheme cb R 2 Rce 1 Rce ) (AU) (AU) (AU) ⊙ 2 M ) (M ⊙ 1 a e M HIP (AU) (M Object Alter. name 4557152940 HD 8067154204 HI5294055016 WDS 11053-2718 4.2256809 HD 97907 6.04 HD 2.6 0.51 101177 0.3560129 3.66 1.9363406 HD 0.37 107259 6.87 3.66 1.93 1.12 240.3965026 HD 112914 0.42 0.47 0.05 0.9565343 WDS 0.12 13198+4747 2.62 10.48 1.95 0.47 14.3666077 HD 0.95 0.53 116495 2.32 1.59 0.08 13.92 1.36 0.23 18.91 0.2 6.0166492 0.97 63.9 0.66 0.33 2 6 7.84 39.6566640 0.67 0.92 0.82 54.21 0.5867275 WDS 0.84 605.53 22.05 3.37 13396+1044 0.23 4 2.85 HD120136 0.61 10.767422 3 1.26 2 0.32 2.6867422 HD 0.58 120476 26.45 0.55 51.51 0.18 42.58 HD 245 1.17 1.24 120476 0 2 4.8668682 6 46.59 1.15 1.19 0.91 33.268682 HD 122742 1 0.39 0.61 1.35 140.96 1.09 33.1569226 HD 122742 0.45 0.54 0.4 0.35 3 1.0771094 HD 0.44 0.76 123999 0.39 13.91 5.46 4.38 35.68 0.8371681 0.75 13.24 4.23 2.52 5.3 0.55 HD 0.76 6 93.65 128621 4.3 868.91 3.65 0.12471683 1.11 4.45 0.48 13 3 71729 157.58 HD 0.19 4.27 128620 0.55 0.92 4.27 1.411 3 22.7672659 HD 107.59 129132 0.63 107.08 0.54 1.36872848 HD 0.51 3 131156 13.98 0.45 4 0.026 22.76 0.7 1.1272848 HD 131511 0.025 0.16 18.11 8.28 0.51 0.55 HD 0.89 0.359 131511 1.89 1.12 32.8 2 0.4 17.28 2.67 12 1.16 0.53 0.89 0.51 3.34 2.4 4 3.28 0.52 2.67 0.92 0.51 3.29 75.04 2.62 2.4 0.79 0.51 0.79 1.19 39.6 4 0.93 0.45 3.8 75.04 1.18 0.45 0.07 2 3.54 4 26.4 0.07 0.05 108.17 2 0.05 1.74 4 1.71 1 4

c RAS, MNRAS 000, 1–16 Regions of Dynamical Stability for Discs and Planets in Binary Stars of the Solar Neighborhood 9 2 ∗ 1 ∗ Ref Scheme cb R 2 Rce 1 Rce ) (AU) (AU) (AU) ⊙ 2 M ) (M ⊙ 1 a e M Object Alter. name 7344074392 HD 13362175312 WDS 15123+1947 14.9375379 WDS 15232+3018 1.25 0.25 16.1576382 HD 137502 3.54 0.22 0.26 ADS9716 1.03 1.26 2.5176852 0.15 1.18 0.9177152 HD 2.98 14015978727 0.32 HD 0.68 3.02 2.55 140913 19.15 1.26 0.5979101 0.14 WDS 2.93 44.69 16044+1122 12.4 1.14 0.68 15.6480346 HD 3.56 48.64 6 145389 0.55 0.15 0.71 1.14 0.07 6 1 2 0.9280686 0.54 1.73 0.05 1.17 0.92 2.2481126 HD 147584 1.73 3.12 1.98 0.0482817 HD 0.47 0.94 149630 64.54 2.7 1 3.4782860 0.08 HD 0.94 152771 5 0.12 1.31 2.6983895 HD 0.02 54.18 153597 2.07 6.33 0.06 34.96 0.33 HD 1.67 6 155763 1.05 0.67 1.3884140 0.53 0.21 2 0.5 4 3.04 0.37 0.3384720 HD 0.05 155876 7.23 0.33 1.5 0.13 7.0984949 0.04 HD 0.21 156274 1 1.18 0.5685141 0.02 HD 0 0.18 157482 0.77 5.01 0.5286201 0.38 HD 0.3 0.1 157498 0.55 5.94 91.65 0.75 0.08 HD 0.3 160922 20.89 6.98 0.78 0.38 3.65 1 4.8786221 0.05 0.79 3.47 2 0.37 9.2986400 WDS 0.67 1 2.05 17370+2753 0.96 0.47 1.15 9 2 0.08286722 0.25 HD 0.58 1.64 1360346 1 4.33 0 1.79 2.6286974 HD 0.25 12.86 161198 0.21 3.41 1.7587895 0.29 HD 17.5 2 161797 1.460 0.39 0.64 321.18 0.87 HD 0.42 1.180 163840 2 0.23 4 0.63 3.97 0.02 0.86 16.6 0.72 22 0.94 1.8 0.011 31.22 0.39 2 0.94 2.14 0.143 1.79 0.32 2 0.08 0.34 0.41 12 1.15 26.4 0.06 0.99 0.04 0.13 6 1.15 0.68 0.03 4.92 0.32 14.21 1 1.85 0.27 2 65.17 6.84 4 1 HIP (AU) (M

c RAS, MNRAS 000, 1–16 10 L. G. Jaime B. Pichardo L. Aguilar 2 ∗ 1 ∗ Ref Scheme cb R 2 Rce 1 Rce ) (AU) (AU) (AU) ⊙ 2 M ) (M ⊙ 1 a e M Object Alter. name 8993790355 HD 17015391768 HD 16982292418 HD 173739 1.0592835 HD 174457 0.84 0.41 HP Dra93017 49.51 1.18 0.48 ADS11871 0.5393506 1.9 0.91 0.77 0.3993574 WDS 19026+2953 0.3 0.16 0.23 0.123 13.36 0.3495028 HD 22.96 175986 0.13 1.07 0.12 0.06 0.25 0.295575 HD 5.43 181602 3.35 0.06 1.65 1.102 0.07 HD 2.97 5.1 183255 1.099 1.58 1 0.5295995 9.01 2.71 2.4 0.03 164.2 4.3496302 0.14 0.85 HD 184467 0.39 4 3 0.03 4.26 2.8196471 0.62 5.26 HD 1.89 184759 0.37 68.77 0.31 2.5598001 HD 1.4 184860 0.15 4 1.65 5 1.45 HD188753 11 38.9399965 0.78 1.35 0.5 4.68 0.37 6 HD 193216 0.38 1.27103641 1.4 1.22 0.82 0.15 11.65 0.15 28.62 HD 200077104019 3.34 0.47 0.46 0.1 0.67 2 0.11 WDS 1.3 21044+1951105969 1.24 1.59 0.26 0.77 2.65 12.86 1.74 HD 204613107354 1.11 0.08 0.18 0.17 0.587 0.03 0.39 HD 1 206901 1 108473 0.88 1.48 0.66 0.13 4.53 1.67 0.14 HD 208776 1.38 1.186 0.56 2.06 16.48 2 109176 0.03 1 37.98 0.941 2 0.32 8.24 HD 210027 0.13110893 4.42 2 0.04 0.26 4.2 HD 2.06 1.01 239960 0.31110893 4 0.04 3.26 ADS15972 1.63 1.56 0.49111170 0.27 2.01 0.12 HD 40.74 213429 1 2.6113718 0.52 1.14 9.51 12 0 HD 6 217580 0.38 0.51 9.53 1.53 0.42 5.68 1.25 0.87 1.21 1.74 0.28 0.41 0.8 1 0.61 1.16 25.32 0.27 0.38 0.15 2 12.62 1.08 0.17 0.54 0.03 1.46 4 0.76 0.7 1.51 0.03 1.1 0.18 0.22 1.20 0.28 30.4 0.15 30.41 1 0.23 3 5 0.08 5.5 3.78 1 1 HIP (AU) (M

c RAS, MNRAS 000, 1–16 Regions of Dynamical Stability for Discs and Planets in Binary Stars of the Solar Neighborhood 11 an and 009, (11) 11, (16) 2 ∗ 1 ∗ Ref Scheme cb R 2 Rce 1 Rce ) (AU) (AU) (AU) ) Balega et al. 2006, (9) Diaz et al. 2007, (10) Cakirli et al. 2 d Barbieri 2007, (14) Doyle et al. 2011, (15) Faigler et al. 20 ⊙ 2 ev and Lampens 2004, (4) Bonavita and Desidera 2007, (5) Holm M ) (M Muterspaugh et al. 2010 ⊙ 1 a e M Kepler16K10848064 0.22K08016222 0.049 0.16K09512641 0 0.69K07254760 0.065 0.20K05263749 0.044 0.060 1.2 0.06K04577324 1.1 0 0.042 0.073 0.03 0 0.055 0.086K06370196 0.018 0.63 1.2 0 0.023 0.039 0.005 1.2 0 0.007 14 0.140 0.083 0.061 1.3 0.154 0.021 0.215 0 15 1.2 0.008 0.014 0.266 15 0.105 0.007 0.018 0.241 1.3 0.075 0.009 0.013 15 0.359 0.097 0.006 15 0.020 0.069 15 0.011 15 0.108 15 Milone et al. 2005, (12) Konacki et al. 2010, (13) Desidera an HIP (AU) (M Object Alter. name 116310 HD221673116727 HD222404117666 95 WDS 23517+0637 10.2 18.5 0.322 0.36 0.3 2.0 1.59 2.0 0.6 0.4 15.70 0.58 3.55 15.70 1.77 294.14 1.9 16 1.74 57.04 31.29 13 6 (1) Jancart et al. 2005, (2) Martin et al. . 1998, (3) Strigach Wiegert 1998, (6) Mason et al. 1999, (7) Latham et al. 2002, (8

c RAS, MNRAS 000, 1–16 12 L. G. Jaime B. Pichardo L. Aguilar

4.1 Application to Real Systems The perturbing effect of stellar companions lead to a widespread belief that the presence of planets in binary sys- ap = 0.113 AU tems was very unlikely. Now we know that binaries can have ep = 0.042 planets, and thus should have stable regions around them where planet formation took place. Queloz et al. 2000, and Hatzes et al. 2003, discovered two giant planets in the binary systems GJ 86 and γCephei. Since then, binaries have be- come important targets in the search for extrasolar planets, particularily given their abundance. Until now about 70 binary systems with planets have been discovered (Wright et al. 2011), but only for eight of them, orbital parameters (semimajor axis, eccentricity and mass ratio) are available (Desidera & Barbieri 2007, Doyle et al. 2011, Muterspaugh et al. 2010). Table 4 shows our pre- diction for the extent of the stable orbits regions for these systems (in bold type letter). In all cases, where the semimajor axis of the planet is known, the observed planet is located within the predicted stable zone. We present figures with the calculated stable regions constructed with invariant loops, for the five cases where planets are confirmed, and the values for the semimajor axis of observed planets are known. In particular, notice the case of HD 120136, this is an open binary with a very high eccentricity, usually treated as single star for this reason. The large eccentricity results in a very narrow, circumstellar stable region. Even so, the discovered planet lies in a P-type orbit inside our predicted circumstellar stable region.

4.1.1 HIP 10138 (HD 13445 or GL 86) This binary system is located at a distance of 10.9 ± 0.08 pc. The companion of this object, discovered by Els et al. 2001, has a semimajor axis of 18.4 AU, with an eccentricity of 0.4. The spectral type of the most massive component is K1 with a mass of 0.77 M⊙ and it is a white dwarf (Mugrauer & Neuhauser 2005). The companion has a mass Figure 5. Stable zones around the system HD 13445. The upper panel shows the circumstellar regions (primary to the left) and the of 0.49 M⊙. At the moment, one planet was found in this bottom panel the circumbinary region. Notice the change in scale. J system with a mass of Mp sin i = 4.0 M . It has a semima- The green lines show the stellar orbits, the barycenter is at the jor axis of 0.113 AU and an eccentricity of 0.042 (Bonavita center and the system is shown at periastron. Orbital parameters: & Desidera 2007). For this binary the calculated stable zone M1 = 0.77 M⊙,M2 = 0.49 M⊙, e = 0.4, a = 18.4 AU located around each star is Rce1 = 3.06 AU around the prin- cipal component, Rce2 = 2.49 AU around the companion, in both cases this radii is the outermost radii possible to have an eccentricity of 0.30 and Mp sin i= 1.69 MJ (Desidera & stable orbits, and Rcb = 58.53 AU as the innermost radii Barbieri 2007). available for circumbinary orbits. In Figure 6, circumstellar, circumbinary stable regions, In Figure 5, we present the circumprimary, circumse- and planetary orbit (in red) are shown. Stellar orbits (green) condary and circumbinary regions of orbital stability, for are also indicated. planets in this case, to settle down, calculated at periastron with our method. The stellar orbits are marked in green. The known planet is orbiting the primary star in a very 4.1.3 HIP 67275 (HD 120136 or τ Boo) small orbital radius, indistinguishable in this ﬁgure. This system is located at 15.62 pc, and it has the lar- gest eccentricity for the cases of binary systems with known orbital parameters, which here are semimajor axis 245 AU, 4.1.2 HIP 14954 (HD 19994 or 94 Cet) eccentricity of 0.91 and masses 1.35 M⊙ for the primary and This binary system is located at a distance of 22.6 pc. 0.4 M⊙ for the secondary component. The planet observed Its semimajor axis is 120 AU and it has eccentricity of 0.26. in this system has a semimajor axis of 0.048 AU while our The mass of the primary star is 1.35 M⊙ with a spectral type approach predicts a maximum radii of 4.86 AU, the eccentri- F8 V, the mass of the secondary star is 0.35 M⊙. The planet city planet is 0.023 and Mp sin i is 4.13 (Desidera & Barbieri orbiting this object has a semimajor axis of 1.428 AU with 2007).

c RAS, MNRAS 000, 1–16 Regions of Dynamical Stability for Discs and Planets in Binary Stars of the Solar Neighborhood 13

a p =1.428 AU a p = 0.048 AU

e p = 0.3 e p = 0.023

Figure 6. Stable zones around the system HD 19994 (94 Cet). The upper panel shows the circumstellar regions, primary star Figure 7. Stable zones around the system HD 120136 (τ Boo). to the left secondary to the right. The bottom panel shows the The upper panel shows the circumstellar regions, primary star circumbinary region, notice the change in scale. The green curves to the left secondary to the right. The bottom panel shows the show the stellar orbits, the system is presented at periastron. circumbinary region, notice the change in scale. The green curves Orbital parameters: M1 = 1.35 M⊙, M2 = 0.35 M⊙, e = 0.26, a show the stellar orbits, the system is presented at periastron. = 120 AU Orbital parameters: M1 = 1.35 M⊙, M2 = 0.4 M⊙, e = 0.91, a = 245 AU

In Figure 7, we present the circumprimary, circumsecondary and circumbinary stable regions, for planets in this stable region (gray), and planetary orbit (in red) are shown. case, calculated at periastron. The stellar orbits are marked Stellar orbits (green) are also indicated. in green. The known planet is orbiting the primary star in a very small orbital radius, indistinguishable in this ﬁgure. 4.1.5 Kepler 16 Recently it was found in this system a planet in cir- 4.1.4 HIP 116727 (HD 222404 or Gamma Cep) cumbinary orbit, this the ﬁrst observed of this kind (on the This system is located at a distance of 14.1 pc. The circumbinary disc). The primary star has a mass of 0.69 M⊙, spectral type of the main component is K, and its mass is and the companion has a mass of 0.20 M⊙, the semimajor 1.59M⊙, while the mass of the companion is 0.4 M⊙, the axis is 0.22 AU with an eccentricity of 0.16 (Doyle et al. semimajor axis is 18.5 AU with an eccentricity of 0.36. The 2011). The discovered planet has a semimajor axis of 0.71 planet in this system has a semimajor axis of 2.14 AU, with AU, with an eccentricity of 0.0069 and a mass of 0.33 MJ. an eccentricity 0.12 and Mp sin i 1.77 MJ (Desidera & Bar- In Figure 9, circumstellar stable regions, circumbinary bieri 2007). stable region, and planetary orbit (in red) are shown. Stellar In Figure 8, circumstellar stable regions, circumbinary orbits (green) are also indicated.

c RAS, MNRAS 000, 1–16 14 L. G. Jaime B. Pichardo L. Aguilar

a p = 2.14 AU

e p = 0.12

a p = 0.71 AU ep = 0.0069

Figure 8. Stable zones around the system HD 222404 (Gamma Figure 9. Stable zones around the system Kepler 16. The upper Cep). The upper panel shows the circumstellar regions, primary panel shows the circumstellar regions, primary star to the left star to the left secondary to the right. The bottom panel shows the secondary to the right. The bottom panel shows the circumbi- circumbinary region, notice the change in scale. The green curves nary region, notice the change in scale. The green curves show show the stellar orbits, the system is presented at periastron. In the stellar orbits, the system is presented at periastron. Orbital this case planet orbit is not to close to the star, line darker (red) parameters: M1 = 0.69 M⊙, M2 = 0.20 M⊙, e = 0.16, a = 0.22 around the primary star shows its orbit. Orbital parameters for AU the star: M1 = 1.59 M⊙, M2 = 0.4 M⊙, e = 0.36, a = 18.5 AU

. may constraint discs sizes, however, what we are providing here are the regions where the most important orbits of the 5 CONCLUSIONS binary, i.e., the ones that represent the backbone of the dy- We have compiled a sample of binary stars with known namical system (the ones that are followed for the most of orbital parameters (semimajor axes, eccentricities and ste- the orbits), lay. We have computed the spatial limits of these llar masses) of the Solar neighborhood and present some circumstellar and circumbinary zones for a sample of 161 bi- basic statistics. naries in the Solar neighborhood where orbital data is known We calculate on this binary stars sample the extent of and presented it in the form of a table where all the relevant regions of stable non-self intersecting orbits where planets parameters are provided. may exist. For this purpose, we have applied the concept We compare our results with observations in the 5 cases of “invariant loops” and used the formulae of PSA1 and where planets have been discovered in binary systems, and PSA2. Our approximation is ballistic, thus, the application where semimajor axis for the planets are provided. We ﬁnd is straightforward to debris discs (planets, cometary nucleii, that all the planets lay down within our computed regions of asteroid belts, etc.). In the case of gas discs, further physics stability. In particular, for HD 120136, our predicted region

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