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Edited by: Francesco Marzari, University of Padua, Italy

Reviewed by: Philippe Thebault, Université de Sciences Lettres de Paris, France; Luca Malavolta, University of Padua, Italy

This article was submitted to Exoplanets, a section of the journal Frontiers in Astronomy and Space Sciences

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In order to assess the habitability of planets in binary star systems, not only astrophysical considerations regarding stellar and atmospheric conditions are needed, but orbital dynamics and the architecture of the system also play an important role. Due to the strong gravitational perturbations caused by the presence of the second star, the study of planetary orbits in double star systems requires special attention. In this context, we show the important role of the main gravitational perturbations (resonances) and review our recently developed methods which allow a quick determination of locations of secular resonances (SRs) in binary stars for circumstellar planetary motion where a giant planet has to move exterior to the habitable zone (HZ). These methods provide the basis for our online-tool ShaDoS which allows a quick check of circumstellar HZs regarding secular perturbations. It is important to know the locations of SRs since they can push a dynamically quiet HZ into a high-eccentricity state which will change the conditions for habitability significantly. Applications of SHaDoS to the wide binary star HD106515 AB and the tight system HD41004 AB reveal a quiet HZ for both systems. However, the study of these systems indicates only for the tight binary star a possible change of the HZ's dynamical state if the orbital parameters change due to new observational data.

Since the discovery of 51 Peg b (Mayor and Queloz, ^{1}^{2}

Thus, in this study we considered only systems where the giant planet is exterior to the HZ and investigate the occurrence of strong perturbations in the HZ. Studies of planetary motion in binary stars are of special interest as a high fraction of stars in the solar neighborhood are members of binary and multiple star systems. Observational surveys by Duquennoy et al. (

The ^{3}

Theoretical stability studies of S- and P-type motion have been carried out decades before the detection of exoplanets (see e.g., Harrington, ^{4}

The detections of exoplanets in tight binary stars like

In case of circumstellar motion mainly the outer edge of the disk is influenced (Kley and Nelson,

During the planet-formation stage where planetesimals (km-sized bodies) collide and merge to Moon-sized embryos dynamical perturbations play an important role, as planetesimal accretion requires low encounter velocities. Thébault et al. (

Apart from these problems of planetary embryo formation, the late planet-formation stage where embryo-sized bodies grow to planets can be easily simulated by N-body calculations (Haghighipour and Raymond,

A weak point in all these numerical studies is certainly the treatment of collisions where usually the so-called

The habitability of planets in binary stars is certainly an interesting issue, especially due to the fact that most of the stars in the solar neighborhood build such stellar systems. Considering the different spectral types, especially F to M type stars are of interest for habitability studies, since the life-times of these stars on the main sequence are sufficiently long (see Kasting et al.,

In a binary star system, an important requirement for habitability is certainly the dynamical stability of the HZ. Moreover, the influence of the secondary star will increase the planet's eccentricity which also affect the insolation on a planet in the HZ where the perturbations depend on the distance and the eccentricity of the secondary star. Eggl et al. (

In recent years, different approaches for the determination of HZs in binary stars have been published e.g.,(Haghighipour and Kaltenegger,

In this study, we focus on the gravitational influence of the secondary star on planetary motion in the HZ. We consider binary star systems with a gas giant in an orbit exterior to the HZ and a terrestrial planet in the HZ. The interplay of these bodies causes perturbations like mean motion resonances (MMRs) and secular resonances (SRs) whose locations strongly depend on the architecture of the system. Such resonances can also influence the habitability of planets in the HZ. Thus, it is important to determine the locations of resonances, which is easy for MMRs but not for SRs. However, for the SRs new methods have been developed that allow a quick determination of the SR location (Pilat-Lohinger et al.,

In this paper, we will briefly discuss these methods and show some applications to real binary systems that host a giant planet. The structure of this paper is the following: First we define the dynamical model and describe the gravitational perturbation. Then we introduce the semi analytical method (SAM) to determine the location of an SR. Subsequently, we describe the basics of the recently published combined analytical method (CAM) and its application for the online tool named

In our study, we focus on circumstellar planetary motion where (i) a terrestrial planet is located in the primary stars' HZ, (ii) a gas giant is moving in an orbit exterior to the HZ, and (iii) the secondary star is a far away perturber. First of all, the planets have to move on stable circumstellar orbits around the primary star. The stability of the planetary motion depends strongly on the distance of the two stars, their eccentricity and their masses. To ensure that the outer planet i.e., the giant planet moves inside the stable region of the primary star, it is advisable to apply the investigations by Pilat-Lohinger and Dvorak (

Nevertheless, the presence of a distant secondary star will perturb the giant planet, which might pass the perturbations to the terrestrial planet in the HZ. In our study, we are primarily interested in the orbital behavior of the terrestrial planet in the HZ and how perturbations of the giant planet and the secondary star may affect its motion. In case these perturbations cause high eccentricity motion in the HZ then the planet's habitability would be affected, especially when it leaves the HZ periodically.

It is well-known that in N-body systems consisting of more than two massive bodies variations of the orbital parameters occur due to gravitational interactions between these bodies. Thus, in binary star systems that host a planet such variations occur. These mutual gravitational interaction between celestial bodies can lead to resonances which appear when the ratio of two frequencies _{1} and _{2} can be expressed as a rational number: |_{1}/_{2}| =

^{5}_{1}, _{2} are integers. The location of an MMR can be easily derived from the third Kepler law:

where _{res} is the resonant semi-major axis and

where _{TP} and _{GP} are the proper frequencies of the test-planet and the giant planet, respectively. For ^{6} years are needed. However, the computation time increases with the stellar distance _{B} (as discussed in Bazsó et al.,

To overcome the time-consuming procedure of identifying the proper frequency via Fourier analysis of each orbit, a new semi-analytical approach has been introduced by Pilat-Lohinger et al. (

For the development of our semi-analytical method, the tight binary star HD41004 AB has been used. In this binary star system a giant planet (of 2.5_{Jup}) has been discovered (Zucker et al., _{Sun}) at about 1.64 au. The secondary star (a M dwarf of 0.42 _{Sun}) is at a distance of about 23 au. The eccentricities of the binary and of the giant planet are not well-known. Thus, we used an eccentricity of 0.2 for both. This binary-star–planet configuration shows strong perturbations around 0.4 au that indicate an SR which has been studied in detail by Pilat-Lohinger et al. (

In the study by Pilat-Lohinger et al. (_{TP}) was deduced using the following secular linear approximation (see e.g., Murray and Dermott,

where α_{i} = _{i} for _{1}, _{2}, _{1} and _{2} are the masses of the giant planet and the secondary and _{0} is the mass of the primary star.

Pilat-Lohinger et al. (

Thus, to satisfy Equation (2) only the numerical integration of the giant planet's orbit in the binary star has to be carried out for a computation time which covers the secular period of the binary star. From this numerical simulation, the proper frequency of the giant planet has to be determined via Fourier analysis. Therefore, this semi-analytical method (SAM) represents a quick approach to define the position of SRs for circumstellar planetary motion in binary stars.

The location of the SR is defined by the semi-major axis of the test-planet which proper frequency equals the one of the giant planet.

Shows on the y-axis the proper periods either for test-planets orbiting the host-star between 0.1 and 1.5 au (x-axis) or for the giant planet (horizontal lines). These plots are the result from a study of the HD41004 AB system, where the giant planet orbits the primary star at 1.64 au and the eccentricities of the binary and the giant planet were set to 0.2 in a planar configuration. Curves represent the proper periods of the test-planets in the considered area (x-axis), where the different colors belong to the binary separations of 20 au (red), 30 au (green), and 40 au (blue) in the left panel or various stellar types of the secondary star: M-type (red, 0.4 solar mass), K-type (green, 0.7 solar mass), and G-type (blue 1 solar mass) in the right panel. In both panels, the proper periods of the giant planet are given by the horizontal lines. Intersections of a curve with a horizontal line of the same color define the location of an SRs for a certain configuration. The left panel indicates the dependency of the SR location on the distance of the two stars where the SR location moves toward the host-star when the distance between the two stars increases. The right panel shows the changes due to the secondary star's mass where the SR location moves toward the host-star if the mass of the secondary decreases. These panels are reproduced from Pilat-Lohinger et al. (

The left panel of

In the same way, the right panel of _{Sun}, red line/curve) it can be seen that the SR is closer to the host-star than for a more massive secondary, like a K-type (0.7 _{Sun}, green line/curve) or a G-type star (1 _{Sun}, blue line/curve).

Furthermore, the study by Pilat-Lohinger et al. (_{GP} and found good results for the binary star HD41004 AB.

An application of SAM to binary star systems with stellar separations up to 100 au and a discovered exoplanet in a circumstellar orbit has been published by Bazsó et al. (

Even if the application of SAM saves a lot of computation time when studying numerous binary-star–planet configurations regarding dynamical conditions for habitability. However, the method cannot be used for an internet-tool.

Therefore, further improvements are needed to speed up the procedure which is only feasible with an analytical approach for the giant planet's proper frequency.

Assuming that the dynamical evolution of the giant planet's orbit is dominated by the secular interaction with the perturber (the secondary star), a simple approach to obtain analytical values for the giant planet's proper frequency has been provided by Heppenheimer (

where _{H} is a first order approximation (in masses) for the proper frequency of the giant planet, _{GP} is its mean motion, _{A} and _{B} are the masses of the primary and the secondary star, _{GP} and _{B} are the semi-major axes of the giant planet and the secondary star and _{B} is the eccentricity of the binary.

The Heppenheimer model has been developed for the restricted three body problem, where the planet's mass is negligible compared to the stellar masses. A suitable improvement^{6}

The new approach by Andrade-Ines and Eggl (

where δ_{g} is an empirical correction term. The model expresses the secular precession frequency _{GP} as a polynomial function of (i) the mass-ratio of the binary star, (ii) the semi-major axis ratio _{GP}/_{B}, and (iii) the perturber's eccentricity _{B}. The expression of δ_{g} has usually less than 20 terms. For the exact values we refer to Equation (24) in Andrade-Ines and Eggl (

Equation (5) is applicable to a wide range of binary stars. Regarding the semi-major axes ratio, it covers the range of currently observed exoplanets in binary systems. Only the range of binary's eccentricities does not cover all systems where an exoplanet has been discovered.

Applying Equation (5) for the solution of _{GP} in Equation (2) we obtain a fully analytical approach which is called

Our main interest is to define the dynamical state of circumstellar HZs and to classify

^{7}_{eff} ≤ 7200 K. For some stellar types, namely F, G, K, and M-type main-sequence stars the standard input data is provided by the tool which can be modified via the

_{B} ≤ 0.6 due to limitations in the analytical model.

Flow chart of the online tool

With the input of all parameters the online-tool performs the calculations and provides a 2D plot depending on the selection in Step 4 which can be saved if desired. This plot usually shows a gray-shaded area which defines the parameters that might perturb the host-star's HZ.

Due to the constraints of the methods used in SHaDoS and the requirement that the giant planet has to be exterior to the HZ, the application to real systems is still very limited. In addition, for most binary star systems the eccentricity is not known which also has strong influence on dynamical studies. Even if nowadays the number of suitable circumstellar systems is rare, we expect an increase of such systems in the near future, especially when the PLATO 2.0 mission starts the observations.

To demonstrate the functionality of

HD106515 AB consists of two G-type stars, with masses of 0.91 and 0.88 solar-masses for the primary and the secondary star, respectively. The distance of the two stars is 345 au and the binary's eccentricity is 0.42. Desidera et al. (

The first 2D-plot of

Result of the online-tool

More possible parameter combinations—according to the input in Step 4—are shown in

Same as

Same as

Same as

HD41004 AB consists of a K- and an M-type main-sequence star as primary and secondary, respectively. The masses are 0.7 solar mass for the K star and 0.4 solar mass for the M star. Actually, both stars have a sub-stellar companion, but in our study, the close-in brown dwarf orbiting the M-type star has been ignored.

In the vicinity of the primary (between 1.3 and 1.64 au) a gas giant (of about 2.5 Jupiter-mass) has been detected (Zucker et al.,

_{GP} = 1.64 au and _{B} = 23 au) there are secular perturbations in the vicinity of the HZ of HD41004 A but the HZ itself is not perturbed. The middle panel displays the secondary star's semi-major axis vs. binary eccentricity which indicates that in case of high eccentricity motion of the binary (0.4 < _{B} < 0.9) there would be a high probability that the HZ of the K-type star would be perturbed by an SR. The bottom panel indicates the same but for different masses of the giant planet on the x-axis.

Results for HD41004 AbB. The gray stripe in each panel indicates conditions that cause perturbations in the primary's HZ due to an SR. The top panel shows the result for the semi-major axis of the secondary star (x-axis) and the giant planet (y-axis). In the middle panel the y-axis shows the binary's eccentricity. In the bottom panel, the x-axis shows the giant planet's mass.

Since the eccentricities of the HD41004 AbB system are not well-determined, it is quite probable that the HZ of the K-type star might change into a pHZ if the orbital parameters vary due to new observational data.

In this study, we presented recently developed methods to determine the location of secular perturbations for circumstellar (or S-type) planetary motion in binary stars with a special emphasis to planets in the cirumstellar HZ in such systems. We did not take into account circumbinary (or P-type) planetary motion as we assume stronger stellar perturbations for a planet in the HZ due to interactions of the stars (e.g., colliding stellar winds; Johnstone et al.,

The appearance of secular resonances (SR) in circumstellar HZs was investigated for certain binary-star–planet configurations where the giant planet orbits the host-star exterior to the HZ. Considering planar systems, SRs are caused by the precession of the giant planet's perihelion which results from gravitational interaction with the secondary star. Moreover, limitations of the adopted methods have restricted our study to eccentricities ≤ 0.6 for the planet and the binary star.

In a first step we replaced the fully numerical approach by a

However, a general parameter study to distinguish between systems with either qHZ or pHZ needs a parameter space of at least five variables (i.e., mass-ratio of the binary star, mass of the giant planet, semi-major axes of the secondary star and the giant planet, and the eccentricity of the binary) where a change of each parameter will modify the location of a resonance makes therefore, the application of SAM is still too costly in terms of time. Only a fully analytical approach could realize such a study in a reasonable time.

We strove to replace the numerical part for the determination of the giant planet's proper frequency by an analytical method. In this context, the approach by Andrade-Ines and Eggl (

Instead of doing a huge amount of calculations in order to compile a catalog which states the dynamical behavior of the circumstellar HZ in various binary-star–planet configurations, we decided to develop an online-tool which applies CAM. The online-tool is named ^{8}

Moreover, the plots of the wide binary star indicate that the probability of a change in the dynamical behavior of the HZ is extremely low—only if new observations should yield significantly different orbital parameters. While in case of the tight binary star changes of the orbital parameters could lead to a pHZ.

Thus, the presented methods can be considered as an important contribution to the habitability research of exoplanets as an SR could affect the motion of a planet in the HZ by increasing the eccentricity. Such variations of eccentricity due to SRs usually happen on a long time-scale and could lead to a misestimation of planetary habitability in such systems. Detected planets in the HZ on nearly circular orbits could move onto highly eccentric orbits after some thousands to millions of years. This would certainly influence the conditions of habitability of these planets.

Even if the number of real systems for which SHaDoS can be applied is still very low, we expect that the number of such systems will increase significantly in the near future when large telescopes like ELT or the Rubin Observatory start observing and especially, when the space mission PLATO 2.0 is launched and starts to explore the sky.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors acknowledge the support from the Austrian Science Fund (FWF), project S11608-N16, part of the NFN Pathways to Habitability. EP-L wants to thank F. Marzari for the invitation to this Special Edition on The Effect of Stellar Multiplicity on Exoplanetary Systems. And we want to thank the referees who helped to improve the paper.

^{1}

^{2}

^{3}

^{4}The FLI is a well-known chaos indicator introduced by Froeschlé et al. (

^{5}mean motion=

^{6}Such a correction to the secular precession frequency can be found in previous attempts, e.g., Giuppone et al. (

^{7}Note that the single star HZ corresponds to the averaged HZ in binary stars.

^{8}