Galactic perturbations on the population of wide binary stars with exoplanets
aa r X i v : . [ a s t r o - ph . E P ] O c t Astronomy & Astrophysicsmanuscript no. wbssynthetic_arxiv c (cid:13)
ESO 2018November 6, 2018
Galactic perturbations on the population of widebinary stars with exoplanets.
J. A. Correa-Otto and R. A. Gil-Hutton
Grupo de Ciencias Planetarias, Dpto. de Geofísica y Astronomía, Facultad de Ciencias Exactas,Físicas y Naturales, Universidad Nacional de San Juan - CONICET, Av. J. I. de la Roza 590 oeste,J5402DCS, Rivadavia, San Juan, Argentinae-mail: [email protected]
Received ; accepted
ABSTRACT
Aims.
The aim of this work is to study the dynamical e ff ects of the Galaxy on binary star systemswith physical and orbital characteristics similar to those of the population of known wide binarystars with exoplanets. As secondary goal we analyse the possible consequences on the stabilityof a hypothetical planetary system orbiting one of the stellar components. Methods.
We numerically reproduced the temporal evolution of a sample of 3 × binary starsystems disturbed by the Galactic potential and passing stars in an environment similar to thesolar neighbourhood. Results.
Our results show that the dynamical evolution of the population of wide binary starswith exoplanets in the solar neighbourhood is modelled by the process of disruption of binarystar systems induced by the Galaxy. We found that this process depends mainly on the separationbetween both stars, whereas it is almost independent of the initial orbital configuration. More-over, our calculations are in agreement with the results of previous works regarding the indirectinfluence of the Galaxy on the stability of planetary systems in wide binary stars. However, thee ff ects on the planetary region show a dependence on the initial configuration of binary stars.Finally, we obtain an indirect test of the impulse approximation model for dynamical studies ofbinary star systems. Key words.
Galaxy: kinematics and dynamics – Stars: binaries – (Galaxy:) solar neighbourhood– Methods: numerical – planets and satellites: dynamical evolution and stability
Use \ titlerunning to supply a shorter title and / or \ authorrunning to supply a shorter list of au-thors. Article number, page 1 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets
1. Introduction
Binary star systems are generally classified as tight binary stars, for which the distance between thecomponents is less than 1000 au, and wide binary stars (WBS), with mutual distance greater than1000 au. Currently, there are 18 known WBS harbouring 28 exoplanets (Roell et al. 2012), whichcomprises ∼
35% of the planets detected in binary stars.Exoplanets in WBS have been ignored in dynamical studies, probably because a distant stellarcompanion produces only very weak e ff ects on the planetary evolution (Rabl & Dvorak 1988;Holman & Wiegert 1999; Andrade-Ines & Michtchenko 2014). However, WBS are not isolatedsystems; they evolve in a Galactic environment under the influence of external perturbations, wherepassing stars and the tidal field of the Milky Way are the most important disturbers. Their e ff ects onthe dynamical evolution of WBS were studied in several works (Heggie 1975; Bahcall et al. 1985;Jiang & Tremaine 2010). Moreover, similar studies have also been applied in external solar systemdynamics research, in particular the Kuiper belt and Oort cloud (Duncan et al. 1987; Brunini 1995;Eggers & Woolfson 1996; Levison & Dones 2001; Fouchard et al. 2006; Kaib et al. 2011).The discovery of structures similar to the Kuiper belt in other planetary systems (Moro-Martinet al. 2015; Kennedy et al. 2015) and the existence of exoplanets in WBS have stimulated inves-tigations of the influence of the Galactic environment on extrasolar planetary systems. Recently,Kaib et al. (2013) have proposed that the apparent more eccentric orbits of exoplanets in WBS canbe explained by such external perturbations.Although the pioneering work of Kaib et al. (2013) has shown interesting results, a completeportrait of the problem remains unknown. The dynamics of planets orbiting one of the WBS com-ponents and the e ff ects of the galactic environment define a complex problem with a large numberof possible initial configurations and physical parameters. In fact, Kaib et al. (2013) only studied2600 possible binary star configurations, in a phase space of six dimension and two parameters (i.e.stellar masses), and considering only a uniform distribution on log a and e .Thus, even though we are mainly interested in studying the evolution of planets orbiting one ofthe stars of a WBS, since the Galaxy a ff ects the planetary system indirectly through perturbationson the stellar companion and the masses of the planets are much lower than those of the stars, wecan start studying the dynamical behaviour of the binary star system and then get information onthe stability of the planetary system using the critical periastron criterion defined by Kaib et al.(2013).Additionally, both Kaib et al. (2013) and other authors have considered a model of impulseapproximation to simulate the stellar passage (Levison & Dones 2001; Zakamska & Tremaine2004; Rickman et al. 2008; Jiang & Tremaine 2010; Kaib et al. 2011; Kaib & Raymond 2014).Although Yabushita et al. (1982), Scholl et al. (1982), Dybczynski (1994), and Eggers & Woolfson(1996) have demonstrated a good agreement of that model with the numerical simulations for thecase of a restricted three-body problem (e.g. Sun, comet, and star), it is unclear if it will work in thegeneral three-body problem. This is an important topic because in a restricted three-body problem Article number, page 2 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets the two massive bodies describe a parabolic orbit and there is no chance for a mutual capturein an elliptical orbit. Instead, in a general three-body problem we initially have a binary systemwhose centre of mass moves in a hyperbolic orbit with the third body, but in the instance of a closeapproach the dynamical evolution becomes chaotic and could occur a stellar reconfiguration withthe formation of a new binary star system or even a triple star system. Despite the increase of thecomputing capacity in recent years, we did not find a test for the impulse approximation model insystems with more than two massive bodies in the literature.The aim of this paper is to study the dynamical evolution of a synthetic sample of WBS underthe e ff ects of the gravitational potential of the Galaxy and passing stars to improve our understand-ing of the dynamic e ff ects of the Galaxy on wide binary stars and to study the indirect e ff ects onplanetary systems. In Section 2 we describe the numerical methods employed and the choice of theinitial configurations. In section 3 we present our results. Finally, conclusions close the paper inSection 4.
2. Numerical methods
Table 1 shows the list of binary star systems harbouring exoplanets with projected separations ( ρ )greater than 800 au (Roell et al. 2012; Kaib et al. 2013). We used these data to make a syntheticpopulation of 10 binary star systems and follow their temporal evolution during a period of 10 Gyr,i.e. the estimated age of the thin disk of the Milky Way. The masses of the primary and secondarystars of each WBS, m and m , are taken at random from Table 1 and we assume that the planetsare always orbiting the primary star.We considered a Cartesian astrocentric coordinate system ( x , y , z ) with origin in the main star( m ). This system is at a distance R g from the Galactic centre and corotates with the Galaxy. The z − axis is perpendicular to the Galactic plane and points towards the south galactic pole, the y − axis points in the direction of the Galactic rotation, and the x − axis points radially outwards from theGalactic centre. In such a system, the secondary star ( m ) evolves around the primary ( m ) in anorbit of size and shape defined by a semi-major axis a and eccentricity e . For the angular orbitalelements we assume an isotropic distribution and the inclination of the orbit is defined with respectto the Galactic plane.The initial distributions of the semi-major axis and eccentricity are di ffi cult to determine be-cause we only know the projected distance ρ (fifth column, Table 1) between the members of theWBS. So, we define three di ff erent combinations of both elements to be assigned to three syntheticpopulation sets. We define a first set of 10 synthetic WBS (called c1 ) with an isotropic distributionof e . Then, assuming that the line between the two stars has a random angle to the line of sight, wecan estimate the distribution of the real separation between the binary components; this distributionwas used together with the eccentricity and mean anomaly to obtain the distribution of semi-majoraxis, taking as lower and upper limits semi-major axes of 1000 au and 100000 au, respectively. Article number, page 3 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets
Fig. 1.
Parameters and initial configuration of the three sets. Top panel: Initial distribution of masses for thethree sets of WBS is shown. The black and open green histograms correspond to the main star ( m ) and thesecondary star ( m ), respectively. The middle and bottom panels show the initial distribution of the semi-majoraxis and eccentricity for the three sets of WBS: c1 , k2, and j3 in black, open red, and open green histograms,respectively. Article number, page 4 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets Table 1.
Physical parameters of the known WBS with exoplanets. The stellar masses and their projectedseparation ( ρ ) are in the third, fourth, and fifth columns, respectively. The index 1 in the masses indicates themember of the binary system harbouring the planetary system. The mass of the secondary stars indicated with‘RD’ (red dwarf) corresponds to the range 0.08 − ⊙ . The second and last columns indicate the numberof exoplanets and semi-major axis ( a p ) of the most external planet orbiting the main star ( m ). WBS N P m m ρ a p Name m ⊙ m ⊙ au auGJ 676 A 1 0.71 −
800 1.82HD 142022 1 0.9 −
820 2.9311 Comae 1 2.04 −
995 1.29HD 11964 2 1.08 0.67 1044 3.3455 Cancri 5 0.905 0.13 1065 5.74HD 80606 1 0.958 − − − − − − synthetic WBS (called k2 ) by keeping the distribution in e ofthe previous set, but considering a uniform distribution in log a , with a ∈ (1000 , j3 ), we considered an initialrandom distribution in e , following Jiang & Tremaine (2010) and Kaib et al. (2013), and foundthe semi-major axis from e and the projected distance as in the c1 set.The initial configuration of the three sets is shown in Fig. 1, where the mass distribution usedis in the top panel, while in the middle and bottom panels are shown the distribution of a and e inblack, open red, and open green histograms for the c1 , k2 , and j3 sets, respectively. The purposeof using these three sets is to take into account the di ff erent initial configurations analysed in theliterature to improve our understanding of the Galactic e ff ects on the population of WBS withexoplanets. The total sample of 3 × WBS is two orders of magnitude larger than that of Kaibet al. (2013).The temporal evolution of the 3 × WBS was solved numerically by integrating the exactequations of motion using a Bulirsch-Stoer code with adopted accuracy of 10 − . During the simu-lation the WBS are a ff ected by the external influence of the tidal field of the Milky Way and passingstars. We considered the solar neighbourhood as the Galactic environment for our simulations be-cause ∼
80 % of the WBS with exoplanets are at distances less than 50 pc from the Sun, but it isimportant to mention that these objects possibly spend much time in denser regions closer to theGalactic centre (Sellwood & Binney 2002; Roskar et al. 2008; Kaib et al. 2011) and the typicalperturbations would be more powerful than we considered in our scenario.
Article number, page 5 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets
Finally, we repeated the simulation of the c1 set considering each perturbation separately toimprove our comprehension about the external perturbations acting, their interactions, and the ef-fects produced on the WBS. Then, we define two subsets that are identical to the c1 set. But for thefirst subset, called c1a , we only considered the e ff ect of the potential of the Galaxy, while for thesecond subset, c1b , we only took the influence of stellar passages into account. We use the Hill approximation (Heggie 2001; Binney & Tremaine 2008) to describe the motionof a WBS in the Galaxy and assume that it is at R g = z =
0, the corresponding equations of motion are¨ x = − µ xr + Ω G ˙ y + Ω G A G x , ˙ y = − µ yr − Ω G ˙ x , ¨ z = − µ zr − ν G z , (1)where, r = p x + y + z , µ = G ( m + m ), G is the Gravitational constant, and x , y , z , ˙ x , ˙ y , ˙ z are the components of the astrocentric position and velocity of the secondary body m around m .Moreover, Ω G , A G , and ν G are the angular speed of the Galaxy, the Oort constant, and the frequencyfor small oscillations in z , respectively. At the approximated distance of the Sun from the Galaxycentre (i.e. R g = Ω G = . × − yr − , A G = . × − yr − ,ν G = . × − yr − . (2)For more details about the deduction of Eq. (1) see Jiang & Tremaine (2010). The e ff ect of a passing star is included as the encounter of a WBS with a third star, which weidentified as m . Usually, this three-body interaction is computed with the model of impulse ap-proximation (Rickman 1976). However, in spite of the high relative velocities between the starson the solar neighbourhood, we solve numerically by a direct integration of a three-body problemwith the additional perturbation of the Galactic potential. Article number, page 6 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets
For a relative velocity of 40 km s − and a number density of stars of 0.05 pc − in the solarneighbourhood, the total number of stellar encounters with impact parameter less than q during thetotal time of integration ( T =
10 Gyr) is defined as (Jiang & Tremaine 2010; Brunini & Fernández1996) N = q . ! . (3)We consider for all the sets a maximum impact parameter of q M ∼ N = × stellar passages of a background star in the vicinity of a WBS, occurring atrandom times during the total time of integration. Each stellar passage was generated following themethod of Rickman et al. (2008) and Kaib & Raymond (2014). The mass of the third star ( m ) isselected from the mass-luminosity function in the solar neighbourhood (Reid et al. 2002; Ninkovic& Trajkovska 2006) and the initial relative velocity for the encounter is taken from the velocitydispersion of nearby stars available in the Hipparcos data (Garcia-Sanchez et al. 2001), which is afunction of the stellar masses.For each one of the WBS on the three sets, the 8 × stellar passages are randomly distributedalong the total time of integration. Then, in our simulations each one of the 3 × WBS has adi ff erent set of stellar encounters.
3. Results
The main influence of the Galaxy on the WBS are twofold: first, the disruption of the binary system,which produces gaps in the orbital elements distributions of the three sets; and second, the changesin the orbital configuration of the surviving WBS, which modify the orbital element distributionsof the population in each set.We considered that a WBS is disrupted when the main star loses its companion or the distancebetween their components is less than 2 stellar radii (Kaib & Raymond 2014; Correa-Otto et al.2017). Then, the disruption conditions are e ≥ a (1 − e ) ≤ R ∗ for each case, respectively.We obtain that ∼
28 % of the WBS in each sample (i.e. c1 , k2, and j3 ) are disrupted by theexternal perturbations. This is an interesting result because it seems to indicate that the rate ofdisruption of WBS in the solar neighbourhood is independent of the initial distribution of the orbitalconfiguration. On the other hand, for the set where we only take into account the influence of stellarpassages ( c1b ) we obtain a 40 % of disrupted WBS. This result indicates that the stellar passageswould have a destabilizing e ff ect on the WBS population, but its intensity would be reduced when itacts in combination with the Galactic potential. Moreover, in the sample c1a all the disrupted WBScorrespond to stellar collisions indicating that the Galactic potential alone is not able to ionizethe WBS. For all the sets the probability of a close encounter between m and m is statisticallynegligible (i.e. ≪ Article number, page 7 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets between m and m or a triple star system, but their percentage of occurrence is very small ( ≪ ∼
25 % of disruptedsystems in a sample with a distribution in a and e similar to our sets k2 and j3 , respectively. This isan interesting result because these authors used the model of impulse approximation for the stellarpassages, while we solved the complete three-body problem. Then, the impulse approximationseems to be su ffi ciently good to study the population of WBS with exoplanets.In order to determine the importance of the orbital configuration of WBS during their dynamicevolution on the Galaxy, we analyse the disruption process as function of the initial distribution ofthe elements that define the extension, form, and orientation of the orbit (i.e. a , e , and I ). We developthis study following the technique of Kaib et al. (2013), which consists in calculating the proportionof disrupted systems along an orbital element. Additionally, this method allow us to compare ourresults with that work. For the semi-major axis we limit our analyses to a ∈ (1000, 30000) au inour sample k2 because this is the range used by Kaib et al. (2013) and also because the other twosets have a small percentage of WBS with a > c1 ), red ( k2 ), and green ( j3 ), and some results of ? are indicated byblue squares. We can see an agreement between our results and that work, which indicates thatthe probability of the rupture of a WBS increases with the separation between the components ofthe binary star. On the other hand, the subset c1b (yellow line) confirms our previous results thatthe dissociation process of binary star systems in the solar neighbourhood is dominated by stellarpassages; however, the combination with the e ff ect of the Galactic potential reduces the number ofdisrupted WBS.The middle and bottom panels of Figure 2 also show the fraction of disrupted WBS as functionsof their initial eccentricity and inclination, where the sets are indicated with the same colours usedin the top graph. The results for the three sets look similar and confirm the independence of thedisruption process with the initial configuration of the WBS. The systems with e > I > ± ◦ ) have a large percentage of disruptions in comparisonwith WBS in low and intermediate inclinations even when the e ff ects of the Galactic tide and stellarpassages are combined, reaching a percentage of disruptions similar to that of the c1b set.Furthermore, our results show that the disruption process is independent of the WBS massesor, in other words, the disruption probability is approximately the same for any value of m or m .Thus, our dynamical study is not a ff ected by the poorly determined values of the secondary starmasses. Additionally, an important consequence of this result is that the evolution of the WBS in Article number, page 8 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets the Galaxy does not change the distribution of their masses (Table 1). Then, the initial and finalmass distribution normalized with the total number of cases considered should be similar.On the other hand, the dynamical influence of the Galactic environment in the surviving syn-thetic population of WBS is analysed considering the final distribution of the orbital elements a , e , and I of each set. Figure 3 shows our results for the samples c1 , c1b , k2, and j3 . As in thedisruption dynamical process, the three populations end with similar orbital characteristics, whichconfirms the independence with respect to the initial configuration of WBS. The final distributionof the semi-major axis is in the top panel, where we plot it in log-scale because of the high per-centage of disrupted WBS with large separations between the stars. We observe for the three sets amaximum at ∼ a (see Fig. 1) and the disruptive process of the e ff ects of the Galaxy for highvalues of a . This maximum at ∼ c1b , which seems to indicate alimit for the external e ff ects on WBS and provides a dynamical explanation for the empirical limitof 1000 au between tight and wide binary star systems.The middle panel of Fig. 3, shows the final distribution of the eccentricity for the three sets.We can see a maximum at high eccentricities, close to a value of 0.8. Since this maximum is notpresent in the c1b subset, it is possible that the evolution of the eccentricity could be a consequenceof the interaction of the two phenomena, the stellar passages, and the Galactic potential, but in anycase this is a marginal result.In any case, the final distribution of e represents an important dynamical result. For the samples c1 and k2 we assume that the binary stars do not have a preferred eccentricity when they are born,but the e ff ects of the Galaxy change the uniform distribution and define an excited or hot popu-lation. For the j3 set the initial distribution of e corresponds to an initially hot population, but thedynamical evolution disrupts WBS with very high values of e , and the final distribution is similarto those of the other two samples. Then, our results show that the influence of the Galaxy changesthe form of the WBS orbits and reaches a final distribution with a high eccentricity maximum withindependence of the initial distribution of e .The final distribution of I (bottom panel of Fig. 3) is modelled by the disruption process. Thereis a gap in the region of ±
90 degrees, which as explained seems to be consequence of the dynam-ical e ff ects of the combined phenomena of Galactic potential and stellar passages. Hence, mostof the population of WBS in the solar neighbourhood would have orbits with low and mediuminclinations. c1 and c1b In order to understand the e ff ects of the Galactic potential and stellar passages it is necessary toperform a better analysis to explain the di ff erences between the results obtained for sets c1 and c1b .Such di ff erences show an important dependence on the separation of the pair and the inclination ofthe orbit. Hence, we have to explain these results. Article number, page 9 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets
For a binary star system disturbed by the Galactic tidal field there is a Jacobi or tidal radius,which is a stationary solution of the Eq. (1), and it is defined by r J = " µ Ω g A g / . (4)Jiang & Tremaine (2010) found that r J defines a characteristic scale for the distribution of binarystars systems with large separations. The existence of a stability limit for the separation of a binarystar disturbed by the Galaxy can give us the wrong impression that we have to expect a higherdisruption rate for the sample c1 , which is the opposite of our numerical result. However, it isimportant to understand that such a limit is defined in the context of a disturbed two-body problem(i.e. considering Galactic perturbations) and we cannot directly compare this limit with the isolatedtwo-body problem. Then, the key to our analysis is that the characteristic scale define by r J isobtained from the Jacobi constant, which is an integral of motion of the problem defined by E J = v − µ r − A g Ω g x + ν g z , (5)where v = ˙ x + ˙ y + ˙ z . The two first terms only depend on the semi-major axis and correspondto the energy of an isolated two-body problem (i.e. − . µ a − ), and the two last terms dependon all orbital elements and correspond to the tidal influence of the Galactic potential. Then, forlarger separations the two tidal terms in eq. (5) become important, and unlike an isolated two-bodyproblem (i.e. set c1b ) the rate of disruption of the WBS in the set c1 depends not only on thesemi-major axis, but also on the other orbital elements.The Jacobi constant can be separated into two parts: first, the terms corresponding to the two-body problem, which represent the energy of the WBS in the sample c1b ; and second, the termsof the Galactic potential, which have influence on the disruption process of the set c1 . In particularthe third term in eq. (5) is negative similar to the energy of the two-body problem, and it can beconsidered as a protection produced by the tide forces against the disruption of the pair by anexternal perturbation, such as a stellar passage. So, it is possible to predict that more energy isnecessary to break up the pair because of this term. On the other side, the last term has the oppositee ff ect because it is positive, which reduces the bond energy of the pair and increases the probabilityof disruption. It is worth noting the dependence of the last term on the inclination of the WBS orbit.Our numerical study (Fig. 2, bottom panel) shows, for the c1b set, that number of disrupted sys-tems does not depend on the inclination, while for the c1 set the probability of disruption reaches amaximum for I = ± ◦ . Such results are in agreement with our analytical predictions and demon-strate that the inclusion of the Galactic potential reduces the random e ff ect of stellar passages. ForWBS with the orbital plane close to the Galactic plane the z -component is close to 0, so the tidee ff ect (third term of eq. (5)) increases the bond energy of the pair and the stellar encounters needto transfer more energy to break up the binary star. Instead, for highly inclined orbits we have the Article number, page 10 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets opposite case; now the x -component is close to 0 and the Galactic potential (fourth term of eq. (5))reduces the energy required to disrupt the stellar pair. ? found numerically for a sample similar to our c1 set that the influence of all the orbitalelements in the disruption process can be represent by the separation of the stellar pair r , scaled by r J . In Figure 4 we plot the distributions of the final separation ( r ) and the final semi-major axis ( a )for all the WBS of each sample, even the disrupted systems, where the histogram for the c1 set isshown in red and that for the c1b set in open blue, while the initial distribution is indicated witha green histogram. In the top panel of Fig. 4 the distribution of r / r J is shown, and for the c1 setour results agree with the prediction of Jiang & Tremaine (2010). There is a second maximum orexterior peak beyond 100 r J that corresponds to the disrupted WBS, and there is a minimum at ∼ r J , which is indicated by a black line in the graph. On the other hand, for the c1b set the secondpeak is higher than the maximum observed for the c1 set but it is below 100 r J , while the minimumis close to ∼ r J . On the bottom panel of Fig. 4 we can see that the distribution of the semi-majoraxis has the same structure of two peaks as the top panel. If we approximate the evolution of aWBS in the Galaxy by a perturbed two-body problem, in a first approximation the semi-major axiscan be considered as indicative of the energy of the system. Therefore, the shift of the exterior peakfor the c1 set seems to confirm that a WBS disturbed by the Galaxy must take more energy fromthe passing stars to be disrupted.In order to show the dependence of the bond energy with the orbital elements for the c1 setwe considered the works of Heisler & Tremaine (1986) and Correa-otto et al. (2017), who foundthat the phase space of a binary star in the Galaxy is parameterized by the semi-major axis and thedimensionless projection of the angular moment ( J = √ − e cos I ). From Figures 9 and 10 ofCorrea-otto et al. (2017) we can deduce that for a WBS with high values of J the extension of thephase space decrease and it can not reach high values of e . So, just as we deduced in our previousanalysis, it is possible to predict that in the case of a sample a ff ected by tidal e ff ects ( c1 ) the stellarpassages have to transfer more energy to break up the pair in comparison with the other sample( c1b ) because the encounters have to change not only the semi-major axis but also the parameter J .Figure 5 shows the disrupted fraction as function of the initial semi-major axis ( a ) and the initialdimensionless projection of the angular moment ( J ) in a grid of 6 by 10 bins, and we can seethat for the c1 set (top panel) the process of rupture depends on the parameter J (i.e. e and I ).This result agrees with our theoretical prediction that when the value of J increases, the disruptionprobability decreases. For small values of J where the phase space extends to e ∼ J < . ff ect of the Galactic potential as a protection mechanism for theWBS against the random disruption e ff ects of the stellar passages. In the dynamic evolution of asample of WBS in the solar neighbourhood, each binary star can gain or lose energy through stellarpassages; successive encounters can reduce the bond energy until the stellar pair breaks up, butthe tidal secular influence of the Galactic potential increases such bond energy in some direction. Article number, page 11 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets
Then, the final result is a lower probability of dissociation for the sample with respect to the casewithout tidal influence.
In our simulations we do not include planets, but it is possible to estimate the disturbing e ff ectsof the Galaxy in the planetary region around m . The main external phenomena considered in thiswork are the tidal field of the Milky Way and passing field stars, which act indirectly in the area ofinterest. The first case corresponds to a long duration e ff ect and takes place in the interval betweenstellar passages. During this stage the planetary region can be disturbed by m if the secondarystars occasionally reaches a small distance of pericentre (Kaib & Raymond 2014; Correa-Otto etal. 2017). In the second case, the disturbance by an encounter is a fast phenomena in which thegravitational influence of m and / or m is defined by a fortuitous close approach of each one ofthese objects to the main star.Most of the known exoplanets have their orbits with a < ? mentioned that we are probably observing the most inner planets ofthe exoplanetary systems due to the methods of detection. Then, if we suppose a planetary systemsimilar to the solar system orbiting the star m of the binary, the extent of the gravitational influenceof the secondary on the planetary region can be estimate by the critical periastron ( q C ) defined inKaib et al. (2013).Thus, for a wide binary star we consider that the hypothetical planetary system is destabilizedwhen in the numerical integration the minimal distance between m and m evolves below thecritical distance q C , at least one time. The results of Kaib et al. (2013) were obtained for a mainstar of one solar mass and more than 70 % of our sets of WBS have primary stars with masses ∼ ⊙ ; this makes it possible to compare our results with those of Kaib et al. (2013).Table 2 shows the percentage of planetary regions perturbed by m for each set and subset. Wecan see a di ff erence between the percentage obtained for the j3 set and the other two sets. Sincethe main di ff erence between these sets is a di ff erent initial distribution of e for j3 , we can deducethat this orbital element is responsible for the planetary dynamics in WBS. On the other hand,the disturbed fraction in the c1a subset is twice that of c1b and is close to that of c1 . This seemsto indicate that the Galactic potential is the main e ff ect that modulates the process of planetarydestabilization.We also reproduce the dynamical map presented in Fig. 2 (a) of Kaib et al. (2013) using oursamples c1 and j3 . We divided each initial sample in a grid of 13 by 5 bins in semi-major axis (inlog-scale) and mass of the companion, and we calculated the total number of WBS in each bin withtheir hypothetical planetary system destabilized and divided this quantity into the total number ofsystems in that bin. Figure 6 shows our results, where we can see values in a range of 5-30 % forthe set with uniform distribution of e and values in a range of 5-55 % for the set with the initial Article number, page 12 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets
Table 2.
Percentage of WBS with the planetary region around m disturbed by m for all the sets. sample disturbedfractionc1 0.16c1a 0.11c1b 0.06k2 0.15j3 0.27distribution of eccentricity similar to that of Kaib et al. (2013) (i.e. e ). The percentages shown forthe set j3 are similar to those of Kaib et al. (2013), which confirm the e ffi ciency of the method ofcritical periastron ( q C ) and the outer influence of the Galaxy as an important source of perturbationfor the planetary systems in WBS.The dynamical structure of our maps is similar to that of the map shown by Kaib et al. (2013)despite the di ff erence in the initial distributions of semi-major axis and secondary mass. Therefore,from our results we can say that the initial distribution of the eccentricity and its subsequent evo-lution, which is a ff ected mainly by the secular e ff ect of the Galactic potential Heisler & Tremaine(1986) and Correa-otto et al. (2017), modulates the planetary stability in WBS.Finally, we analyse the direct e ff ect of the third star on the planetary region. Following Laughlin& Adams (2000), we estimate the change of eccentricity in a hypothetical Neptune, at 30 au with e N = m produces an instant force,we estimate the ∆ e in the orbit of our hypothetical Neptune in the context of a perturbed two-body problem (Murray & Dermott 1999). We found a probability smaller than 0.5 % for a change ∆ e > .
01 on the three sets. Then, for statistical studies we can ignore the direct influence of thethird star in the planetary region.
4. Conclusions
In this paper, we presented a statistical study about the temporal evolution of a synthetic set ofbinary star systems in the solar neighbourhood with orbital and physical characteristic similar tothe population of wide binary stars with exoplanets (Table 1) and taking into account the e ff ectsproduced by the tidal field of the Milky Way and the perturbations produced by passing stars. Wecarry out simulations using three sets of 10 WBS with di ff erent initial distributions of a and e . Thedynamic evolution of the WBS in each set was followed by solving the exact numerical equationsof motion during 10 Gyr, i.e. the approximated age of the thin disk.Our results show that the three sets of WBS have similar final orbital element distributionsregardless their initial configuration. Therefore, we conclude that the external e ff ects modified thesamples and makes them converge to a standard configuration during the temporal evolution. Themost important phenomenon that alters a population of WBS is the disruption of binary pairs, andthis e ff ect is independent of the masses of the pair. Thus, for the population of WBS studied themain characteristics of its final standard configuration are Article number, page 13 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets – There is an accumulation of systems within a ∈ (2000 , – The population is dynamically hot (i.e. high eccentricities). – There is a gap in the distribution of WBS in a direction perpendicular to the Galactic plane (i.e. I ∼ ± ◦ ). – The final mass distribution preserves the original form.A denser galactic environment could be more e ffi cient at generating a standard configuration , forexample by a faster disruption process on the most separated WBS.From our numerical results and the analytical interpretation of these results, we conclude thatthe disruption process is dominated by the cumulative e ff ect of stellar passages with the Galacticpotential acting like a protection mechanism against the break up of the stellar pair. We foundthat the most important characteristics of the WBS that regulates this phenomenon are first, theseparation of the pair ( r ), which has been demonstrated by Jiang & Tremaine (2010); and second,its orientation with respect to the Galactic plane ( I ). This last point predicts a lack of binary systemswith orbits perpendicular to the Galactic plane in the solar neighbourhood, which agree with thepredictions of our analytical study: a stellar pair in an orbit with high inclination has a smaller bondenergy than those in a coplanar orbit.Thus, as the population of WBS with exoplanets reaches a standard configuration as a resultof the Galactic e ff ects, we can get rid of the dynamical problem of the definition of the initialdistributions for the orbital elements. Our results strengthen the statistical study of Kaib et al.(2013) because the 2600 combinations of WBS considered in that work (100 times less than oursample) are enough to reach correct results. However, our results imply a problem for the formationstudies of WBS (e.g. Tokovinin (2017)) because the current distribution of the orbital elementswould not show traces of the initial distribution.Additionally, we found an agreement between our results and those of Kaib et al. (2013) forthe percentage of disrupted WBS. The importance of this agreement is highlighted when weconsider that we solved the complete equations of motion, while they applied the model ofimpulse approximation.
Although this model has been tested in the case of the restricted three-body problem (Yabushita et al. 1982; Scholl et al. 1982; Dybczynski 1994; Eggers & Woolfson1996), there are no studies about its accuracy for a general three-body problem. Thus, we concludethat for a dynamical analysis about the evolution of the population of WBS with exoplanets in theGalaxy, the impulsive hypothesis gives reliable statistical results.Finally, we applied the criteria of critical periastron proposed by Kaib et al. (2013) to estimatethe indirect influence of the Galactic environment on the planetary region around the main starof a WBS. Our results show that the e ff ects of the external perturbations considered in this workare in agreement with that obtained by Kaib et al. (2013). However, we find that di ff erent initialconfigurations of the samples of WBS produce varying levels of dynamic excitation in the plan-etary system, with the initial distribution of eccentricity of the secondary component as the mainmodulator of the problem. Therefore, from our partial analysis (i.e. without planetary systems), Article number, page 14 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets we concluded that more studies about the topic of exoplanets in WBS are needed to improve ourunderstanding of the dynamical influence of the secondary star orbit on the planetary region aroundthe main star.
5. Bibliography – Andrade-Ines, E. & Michtchenko, T. A. 2014, MNRAS, 444, 2167 – Bahcall, J. N., Hut, P., & Tremaine, S. 1985, AJ, 290, 15 – Binney, J. & Tremaine, S. 2008, Galactic Dynamics: Second Edition (Princeton UniversityPress) – Brunini, A. 1995, A&A, 293, 935 – Brunini, A. & Fernández, J. 1996, A&A, 308, 988 – Correa-Otto, J., Calandra, F., & Gil-Hutton, R. 2017, A&A, 600, 59 – Duncan, M., Quinn, T., & Tremaine, S. 1987, AJ, 94, 1330 – Dybczynski, P. A. 1994, CeMDA, 58, 139 – Eggers, S. & Woolfson, M. 1996, MNRAS, 282, 13 – Fouchard, M., Froeschlé, C., Valsecchi, G., & Rickman, H. 2006, CeMDA, 95, 299 – García-Sánchez, J., Weissman, P., Preston, R., & et al. 2001, A&A, 379, 634 – Heggie, D. C. 1975, MNRAS, 173, 729 – Heggie, D. C. 2001, in The Restless Universe, ed. B. A. Steves & A. J. Maciejew- ski, 109–128 – Heisler, J. & Tremaine, S. 1986, ICARUS, 65, 13 – Holman, M. J. & Wiegert, P. A. 1999, AJ, 117, 621 – Jiang, Y. F. & Tremaine, S. 2010, MNRAS, 401, 977 – Kaib, N. A. & Raymond, S. N. 2014, ApJ, 782, 60 – Kaib, N. A., Raymond, S. N., & Duncan, M. 2013, Nature, 493, 381 – Kaib, N. A., Roskar, R., & Quinn, T. 2011, Icarus, 215, 491 – Kennedy, G. M., Matrá, L., Marmier, M., et al. 2015, MNRAS, 449, 3121 – Laughlin, G. & Adams, F. C. 2000, Icarus, 145, 614 – Levison, H. F. & Dones, L. 2001, AJ, 121, 2253 – Moro-Martín, A., Marshall, J. P., Kennedy, G., et al. 2015, ApJ, 801, 143 – Murray, C. D. & Dermott, S. F. 1999, Solar system dynamics (Cambridge Uni- versity Press) – Ninkovic, S. & Trajkovska, V. 2006, Serb. Astron. J., 172, 17 – Rabl, G. & Dvorak, R. 1988, A&A, 191, 385 – Reid, I. N., Gizis, J. E., & Hawley, S. L. 2002, AJ, 124, 2721 – Rickman, H. 1976, BAICz, 27, 92R – Rickman, H., Fouchard, M., Froeschlé, C., & Valsecchi, G. B. 2008, CeMDA, 102, 111 – Roell, T., Neuhauser, R., Seifahrt, A., & Mugrauer, M. 2012, A&A, 542, A92 – Roskar, R., Debattista, V. P., Quinn, T. R., Stinson, G. S., & Wadsley, J. 2008, ApJ, 684, 79 – Scholl, H., Cazenave, A., & Brahici, A. 1982, A&A, 112, 157 – Sellwood, J. A. & Binney, J. J. 2002, MNRAS, 336, 785 – Tokovinin, A. 2017, MNRAS, 000, 000 – Yabushita, S., Hasegawa, I., & Kobayashi, K. 1982, MNRAS, 200, 661 – Zakamska, N. L. & Tremaine, S. 2004, AJ, 128, 869
Article number, page 15 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets
Acknowledgements
The authors gratefully acknowledge financial support by CONICET through PIP 112-201501-00525. The authors are grateful to the anonymous referee for numerous suggestions and correctionson this paper.
Article number, page 16 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets
Fig. 2.
Top panel: Fraction of disrupted WBS in the initial range a ∈ (1000, 30000) au. The blue squaresare results taken from the Figure S8 of Kaib et al. (2013) included for comparison. Middle and bottom panel:Fraction of disrupted WBS as a function of the initial eccentricity and inclination, respectively. In the threegraphs the sets c1 , c1b , k2, and j3 are shown in black, yellow, red, and green, respectively.Article number, page 17 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets Fig. 3.
Orbital distribution for the sets c1 , k2, and j3 after T =
10 Gyr of evolution. The three samples areindicated with histograms in black, red, and green, respectively, while the subset c1b is indicated with ayellow spike histogram. The top, middle, and bottom panels correspond to the orbital elements a , e , and I ,respectively. Article number, page 18 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets Fig. 4.
Final distribution for the sets c1 (red), and c1b (open blue) after T =
10 Gyr of evolution. In greenwe include the histogram of the initial distribution of both samples. The top and bottom panels correspond tothe final separation divided by r J and the final semi-major axis, respectively. The black vertical line in the topgraph corresponds to ∼ r J . Article number, page 19 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets Fig. 5.
Fraction of disrupted WBS as a function of a and J . The top and bottom panels correspond to thesets c1 and c1b , respectively. We can see a dependence of the c1 set on the initial dimensionless projection ofthe angular moment because for small values of J the disruption rate increases.Article number, page 20 of 21. A. Correa-Otto and R. A. Gil-Hutton: Galactic perturbations on wide binary stars with exoplanets Fig. 6.
Percentage of WBS in the sets c1 and j3 with the planetary region around m disturbed by the secondarystar, as a function of the semi-major axis ( a ) and secondary mass ( m ). The maps show a good agreement withthe results of ??