To better understand the orbital dynamics of exoplanets around close binary stars, i.e., circumbinary planets (CBPs), we applied techniques from dynamical systems theory to a physically motivated set of solutions in the Circular Restricted Three-Body Problem (CR3BP). We applied Floquet theory to characterize the linear dynamical behavior -- static, oscillatory, or exponential -- surrounding planar circumbinary periodic trajectories (limit cycles). We computed prograde and retrograde limit cycles and analyzed their geometries, stability bifurcations, and dynamical structures. Orbit and stability calculations are exact computations in the CR3BP and reproducible through the open-source Python package pyraa. The periodic trajectories produce a set of non-crossing, dynamically cool circumbinary orbits conducive to planetesimal growth. For mass ratios $μ\in [0.01, 0.50]$ we found recurring features in the prograde families. These features include: (1) an innermost near-circular trajectory, inside which solutions have resonant geometries, (2) an innermost stable trajectory ($a_{cr} \approx 1.61 - 1.85 \, a_\textrm{bin}$) characterized by a tangent bifurcating limit cycle, and (3) a region of dynamical instability ($a \approx 2.1 \ a_\textrm{bin}; Δa \approx 0.1 \ a_\textrm{bin}$), the exclusion zone, bounded by a pair of critically stable trajectories -- bifurcating limit cycles. The exterior boundary of the exclusion zone is consistent with prior determinations of $a_{cr}$ around a circular binary. We validate our analytic results with N-body simulations and apply them to the Pluto-Charon system. The absence of detected CBPs in the inner stable region, between the prograde exclusion zone and $a_{cr}$, suggests that the exclusion zone may inhibit the inward migration of CBPs.
The recent discovery of a number of circumbinary planets lends a new tool to astrophysicists seeking to understand how and where planet formation takes place. Of the increasingly numerous circumbinary systems, Kepler-16 is arguably the most dynamically interesting: it consists of a planet on an almost perfectly circular orbit (e = 0.0069) around a moderately eccentric binary (e = 0.16). We present high-resolution 3D smoothed-particle hydrodynamics simulations of a Kepler-16 analogue embedded in a circumbinary disc, and show that the planet's eccentricity is damped by its interaction with the p…
We investigate the secular dynamics of three-body circumbinary systems under the effect of tides. We use the octupolar non-restricted approximation for the orbital interactions, general relativity corrections, the quadrupolar approximation for the spins, and the viscous linear model for tides. We derive the averaged equations of motion in a simplified vectorial formalism, which is suitable to model the long-term evolution of a wide variety of circumbinary systems in very eccentric and inclined orbits. In particular, this vectorial approach can be used to derive constraints for tidal migration,…
We present numerical simulations of circumbinary accretion onto eccentric and circular binaries using the moving-mesh code AREPO. This is the first set of simulations to tackle the problem of binary accretion using a finite-volume scheme on a freely moving mesh, which allows for accurate measurements of accretion onto individual stars for arbitrary binary eccentricity. While accretion onto a circular binary shows bursts with period of ~5 times the binary period P_b,accretion onto an eccentric binary is predominantly modulated at the period ~1P_b. For an equal-mass circular binary, the accretio…
We examine characteristics of circumbinary orbits in the context of current planet formation scenarios. Analytical perturbation theory predicts the existence of nested circumbinary orbits that are generalizations of circular paths around a single star. These orbits have forced eccentric motion aligned with the binary as well as higher frequency oscillations, yet they do not cross, even in the presence of massive disks and perturbations from large planets. For this reason, dissipative gas and planetesimals can settle onto these "most circular" orbits, facilitating the growth of protoplanets. Ou…
We investigate the resonant rotation of circumbinary bodies in planar quasi-circular orbits. Denoting $n_b$ and $n$ the orbital mean motion of the inner binary and of the circumbinary body, respectively, we show that spin-orbit resonances exist at the frequencies $n\pm kν/2$, where $ν= n_b - n$, and $k$ is an integer. Moreover, when the libration at natural frequency has the same magnitude as $ν$, the resonances overlap and the rotation becomes chaotic. We apply these results to the small satellites in the Pluto-Charon system, and conclude that their rotations are likely chaotic. However, the …
The herein presented analytical framework fully describes the motion of coplanar systems consisting of a stellar binary and a planet orbiting both stars on orbital as well as secular timescales. Perturbations of the Runge-Lenz vector are used to derive short period evolution of the system, while octupole secular theory is applied to describe its long term behaviour. A post Newtonian correction on the stellar orbit is included. The planetary orbit is initially circular and the theory developed here assumes that the planetary eccentricity remains relatively small (e_2<0.2). Our model is tested a…
We investigate the orbital dynamics of circumbinary planetary systems with two planets around a circular or eccentric orbit binary. The orbits of the two planet are initially circular and coplanar to each other, but misaligned with respect to the binary orbital plane. The binary-planet and planet-planet interactions result in complex planet tilt oscillations. We use analytic models and numerical simulations to explore the effects of various values of the planet semi-major axes, binary eccentricity, and initial inclination. Around a circular orbit binary, secular tilt oscillations are driven by…
BEBOP is a radial-velocity survey that monitors a sample of single-lined eclipsing binaries, in search of circumbinary planets by using high-resolution spectrographs. Here, we describe and test the methods we use to identify planetary signals within the BEBOP data, and establish how we quantify our sensitivity to circumbinary planets by producing detection limits. This process is made easier and more robust by using a diffusive nested sampler. In the process of testing our methods, we notice that contrary to popular wisdom, assuming circular orbits in calculating detection limits for a radial …
We study the mutual evolution of the orbital properties of high mass ratio, circular, co-planar binaries and their surrounding discs, using 3D Smoothed Particle Hydrodynamics simulations. We investigate the evolution of binary and disc eccentricity, cavity structure and the formation of orbiting azimuthal over-dense features in the disc. Even with circular initial conditions, all discs with mass ratios $q>0.05$ develop eccentricity. We find that disc eccentricity grows abruptly after a relatively long time-scale ($\sim 400\textrm{--}700$ binary orbits), and is associated with a very small incr…
All circumbinary planets currently detected are in orbits that are almost coplanar to the binary orbit. While misaligned circumbinary planets are more difficult to detect, observations of polar aligned circumbinary gas and debris disks around eccentric binaries suggest that polar planet formation may be possible. A polar aligned planet has a stable orbit that is inclined by 90 degrees to the orbital plane of the binary with an angular momentum vector that is aligned to the binary eccentricity vector. With n- body simulations we model polar terrestrial planet formation using hydrodynamic gas di…
We present sensitive ALMA observations of TWA 3, a nearby, young ($\sim$10 Myr) hierarchical system composed of three pre-main sequence M3--M4.5 stars. For the first time, we detected ${}^{12}$CO and ${}^{13}$CO $J$=2-1 emission from the circumbinary protoplanetary disk around TWA 3A. We jointly fit the protoplanetary disk velocity field, stellar astrometric positions, and stellar radial velocities to infer the architecture of the system. The Aa and Ab stars ($0.29\pm0.01\,M_\odot$ and $0.24\pm0.01\,M_\odot$, respectively) comprising the tight ($P=35$ days) eccentric ($e=0.63\pm0.01$) spectros…
As of today ten circumbinary planets orbiting solar type main sequence stars have been discovered. Nearly all of them orbit around the central binary very closely to the region of instability where it is difficult to form them in situ. It is assumed that they formed further out and migrated to their observed position. We extend previous studies to a more realistic thermal disc structure and determine what parameter influence the final parking location of a planet around a binary star. We perform two-dimensional numerical simulations of viscous accretion discs around a central binary that inclu…
We analyse the evolution of a mildly inclined circumbinary disc that orbits an eccentric orbit binary by means of smoother particle hydrodynamic (SPH) simulations and linear theory. We show that the alignment process of an initially misaligned circumbinary disc around an eccentric orbit binary is significantly different than around a circular orbit binary and involves tilt oscillations. The more eccentric the binary, the larger the tilt oscillations and the longer it takes to damp these oscillations. A circumbinary disc that is only mildly inclined may increase its inclination by a factor of a…
Many unstable periodic orbits of the planar circular restricted 3-body problem (PCRTBP) persist as invariant tori when a periodic forcing is added to the equations of motion. In this study, we compute tori corresponding to exterior Jupiter-Europa and interior Jupiter-Ganymede PCRTBP resonant periodic orbits in a concentric circular restricted 4-body problem (CCR4BP). Motivated by the 2:1 Laplace resonance between Europa and Ganymede's orbits, we then attempt the continuation of a Jupiter-Europa 3:4 resonant orbit from the CCR4BP into the Jupiter-Ganymede PCRTBP. We strongly believe that the re…
With $n$-body simulations we investigate the stability of tilted circumbinary planetary systems consisting of two nonzero mass planets. The planets are initially in circular orbits that are coplanar to each other, as would be expected if they form in a flat but tilted circumbinary gas disc and decouple from the disc within a time difference that is much less than the disc nodal precession period. We constrain the parameters of stable multiple planet circumbinary systems. Both planet-planet and planet-binary interactions can cause complex planet tilt oscillations which can destabilise the orbit…
Three transiting circumbinary planets (Kepler-16 b, Kepler-34 b, and Kepler-35 b) have recently been discovered from photometric data taken by the Kepler spacecraft. Their orbits are significantly non-Keplerian because of the large secondary-to-primary mass ratio and orbital eccentricity of the binaries, as well as the proximity of the planets to the binaries. We present an analytic theory, with the planet treated as a test particle, which shows that the planetary motion can be represented by the superposition of the circular motion of a guiding center, the forced oscillations due to the non-a…
The presence of a body in an orbit around a close eclipsing binary star manifests itself through the light time effect influencing the observed times of eclipses as the close binary and the circumbinary companion both move around the common centre of mass. This fact combined with the periodicity with which the eclipses occur can be used to detect the companion. Given a sufficient precision of the times of eclipses, the eclipse timing can be employed to detect substellar or even planetary mass companions. The main goal of the paper is to investigate the potential of the photometry based eclipse…
When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation as well as rotation numbers; 3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds; 4) regularize the equations of motion; and 5) globalize these manifolds. Our methodology impro…
We investigate the dynamics of a nonzero mass, circular orbit planet around an eccentric orbit binary for various values of the binary eccentricity, binary mass fraction, planet mass, and planet semi--major axis by means of numerical simulations. Previous studies investigated the secular dynamics mainly by approximate analytic methods. In the stationary inclination state, the planet and binary precess together with no change in relative tilt. For both prograde and retrograde planetary orbits, we explore the conditions for planetary orbital libration versus circulation and the conditions for st…
The stability limit for circumbinary planets (CBPs) is not well defined and can depend on initial parameters defining either the planetary orbit or the inner binary orbit. We expand on the work of Holman & Wiegert (1999, AJ 117, 621) to develop numerical tools for quick, easy, and accurate determination of the stability limit. The results of our simulations, as well as our numerical tools, are available to the community through $\texttt{Zenodo}$ and $\texttt{GitHub}$, respectively. We employ a grid interpolation method based on $\sim$150 million full N-body simulations of initially circular, c…
For a body with negligible mass moving in the gravitational field of a star and one planet in a circular orbit (the circular restricted three-body problem) five equilibrium points exist and are known as the Lagrangian points. The positions of the Lagrangian points are not valid for dust particles because in the derivation of the Lagrangian points is assumed that no other forces beside the gravitation act on the body with negligible mass. Here we determined positions of the equilibrium points for the dust particles in the circular restricted three-body problem with radiation. The equilibrium po…
This contribution investigates the properties of a category of orbits around Enceladus. In a previous investigation, a set of heteroclinic connections were designed between halo orbits around the equilibrium points L1 and L2 of the circular restricted three-body problem with Saturn and Enceladus as primaries. The geometrical characteristics of those trajectories makes them good candidates as science orbits for the extended observation of the surface of Enceladus: they are highly inclined, they approach the moon and they are maneuver-free. However, the low heights above the surface and the stro…
We study circumbinary accretion discs in the framework of the restricted three-body problem (R3Bp) and via numerically solving the height-integrated equations of viscous hydrodynamics. Varying the mass ratio of the binary, we find a pronounced change in the behaviour of the disc near mass ratio $q \equiv M_s/M_p \sim 0.04$. For mass ratios above $q=0.04$, solutions for the hydrodynamic flow transition from steady, to strongly-fluctuating; a narrow annular gap in the surface density around the secondary's orbit changes to a hollow central cavity; and a spatial symmetry is lost, resulting in a l…
In recent years, stable and unstable manifolds of invariant objects (such as libration points and periodic orbits) have been increasingly recognized as an efficient tool for designing transfer trajectories in space missions. However, most methods currently used in mission design rely on using eigenvectors of the linearized dynamics as local approximations of the manifolds. Since such approximations are not accurate except very close to the base invariant object, this requires large amounts of numerical integration to globalize the manifolds and locate intersections. In this paper, we study hyp…
The discovery of planets in close orbits around binary stars raises questions about their formation. It is believed that these planets formed in the outer regions of the disc and then migrated through planet-disc interaction to their current location. Considering five different systems (Kepler-16, -34, -35, -38 and -413) we model planet migration through the disc, with special focus on the final orbital elements of the planets. We investigate how the final orbital parameters are influenced by the disc and planet masses. Using 2D, locally isothermal, viscous hydrodynamical simulations we first …
We have studied the possibility that a third circumbinary planet in the Kepler-47 planetary system be the source of the single unexplained transiting event reported during the discovery of these planets. We applied the MEGNO technique to identify regions in the phase space where a third planet can maintain quasi-periodic orbits, and assessed the long-term stability of the three-planet system by integrating the entire 5 bodies (binary + planets) for 10 Myr. We identified several stable regions between the two known planets as well as a region beyond the orbit of Kepler-47c where the orbit of th…
We report the detection of a planet whose orbit surrounds a pair of low-mass stars. Data from the Kepler spacecraft reveal transits of the planet across both stars, in addition to the mutual eclipses of the stars, giving precise constraints on the absolute dimensions of all three bodies. The planet is comparable to Saturn in mass and size, and is on a nearly circular 229-day orbit around its two parent stars. The eclipsing stars are 20% and 69% as massive as the sun, and have an eccentric 41-day orbit. The motions of all three bodies are confined to within 0.5 degree of a single plane, suggest…
We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted TBP, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurc…
We investigate the non-resonant, 3-D (spatial) model of the hierarchical system composed of point-mass stellar (or sub-stellar) binary and a low-mass companion (a circumbinary planet or a brown dwarf). We take into account the leading relativistic corrections to the Newtonian gravity. The secular model of the system relies on the expansion of the perturbing Hamiltonian in terms of the ratio of semi-major axes $α$, averaged over the mean anomalies. We found that the low-mass object in a distant orbit may excite large eccentricity of the inner binary when the mutual inclination of the orbits is …
Within the framework of restricted four-body problem, we study the motion of an infinitesimal mass by assuming that the primaries of the system are radiating-oblate spheroids surrounded by a circular cluster of material points. In our model, we assume that the two masses of the primaries $m_2$ and $m_3$ are equal to $μ$ and the mass $m_1$ is $1-2μ$. By using numerical approach, we have obtained the equilibrium points and examined their linear stability. The effect of potential created by the circular cluster and oblateness coefficients for the more massive primary and the less massive primary,…
The existence of planets orbiting a central binary star system immediately raises questions regarding their formation and dynamical evolution. Recent discoveries of circumbinary planets by the Kepler space telescope has shown that some of these planets reside close to the dynamical stability limit where it is very difficult to form planets in situ. For binary systems with nearly circular orbits, such as Kepler-38, the observed proximity of planetary orbits to the stability limit can be understood by an evolutionary process in which planets form farther out in the disk and migrate inward to the…
In this paper, the near-optimal capture problem is numerically investigated to locate the optimal capture points around Charon using the Pluto-Charon planar circular restricted three-body problem (PCRTBP). Capture orbits are divided into the pre- and post-maneuver portions. In the pre-maneuver portion, the gravitational capture conditions are discussed by backward numerical integration. In the post-maneuver portion, the smaller alignment index (SALI) is applied to numerically investigate the orbital characters of long-term capture orbits around Charon. The initial conditions corresponding to t…
Given the interest in future space missions devoted to the exploration of key moons in the solar system and that may involve libration point orbits, an efficient design strategy for transfers between moons is introduced that leverages the dynamics in these multi-body systems. The moon-to-moon analytical transfer (MMAT) method is introduced, comprised of a general methodology for transfer design between the vicinities of the moons in any given system within the context of the circular restricted three-body problem, useful regardless of the orbital planes in which the moons reside. A simplified …
The regularization of a new problem, namely the three-body problem, using 'similar' coordinate system is proposed. For this purpose we use the relation of 'similarity', which has been introduced as an equivalence relation in a previous paper (see \cite{rom11}). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function, we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to regularize the equations of motion. Explicit formulas are given for the regularization in the…
We study the fourth-order stability of the triangular libration points in the absence of resonance for the three-body problem when the infinitesimal mass is affected not only by gravitation but also by light pressure from both primaries. A comprehensive summary of previous results is given, with some inaccuracies being corrected. The Lie triangle method is used to obtain the fourth-order Birkhoff normal form of the Hamiltonian, and the corresponding complex transformation to pre-normal form is given explicitly. We obtain an explicit expression for the determinant required by the Arnold-Moser t…
In this paper we use the planar circular restricted three-body problem where one of the primary bodies is an oblate spheroid or an emitter of radiation in order to determine the basins of attraction associated with the equilibrium points. The evolution of the position of the five Lagrange points is monitored when the values of the mass ratio $μ$, the oblateness coefficient $A_1$, and the radiation pressure factor $q$ vary in predefined intervals. The regions on the configuration $(x,y)$ plane occupied by the basins of attraction are revealed using the multivariate version of the Newton-Raphson…
The planetary exospheres are poorly known in their outer parts, since the neutral densities are low compared with the instruments detection capabilities. The exospheric models are thus often the main source of information at such high altitudes. We present a new way to take into account analytically the additional effect of the stellar radiation pressure on planetary exospheres. In a series of papers, we present with an Hamiltonian approach the effect of the radiation pressure on dynamical trajectories, density profiles and escaping thermal flux. Our work is a generalization of the study by Bi…
In the present paper, using the first-order approximation of the $n$-body Lagrangian (derived on the basis of the post-Newtonian gravitational theory of Einstein, Infeld, and Hoffman), we explicitly write down the equations of motion for the planar circular restricted three-body problem. Additionally, with some simplified assumptions, we obtain two formulas for estimating the values of the mass/distance and velocity/speed of light ratios appropriate for a given post-Newtonian approximation. We show that the formulas derived in the present study, lead to a good numerical conservation of the Jac…
In this paper, we determine the effect of radiation pressure, Poynting-Robertson drag and solar wind drag on the Sun-(Earth-Moon) restricted three body problem. Here, we take the larger body of the Sun as a larger primary, and Earth+Moon as a smaller primary. With the help of the perturbation technique, we find the Lagrangian points, and see that the collinear points deviate from the axis joining the primaries, whereas the triangular points remain unchanged in their configuration. We also find that Lagrangian points move towards the Sun when radiation pressure increases. We have also analysed …
In the spatial circular restricted three-body problem librational motion around the Lagrangian points $L_4$ and $L_5$ exists up to high inclinations of the massless object. We report the results of numerical investigations on the stability of this librational motion for systematic variations in the inclination of the Trojan and the mass-ratio $μ$ of the two massive bodies. We show that stable motion prevails for inclinations below 60 degrees and mass-ratios $μ< 0.04$ for typical integration times up to $10^6$ revolution periods. At even higher mass-ratios -- beyond the critical mass-ratio for …
The Lidov-Kozai mechanism allows a body to periodically exchange its eccentricity with inclination. It was first discussed in the framework of the quadrupolar secular restricted three-body problem, where the massless particle is the inner body, and later extended to the quadrupolar secular nonrestricted three body problem. In this paper, we propose a different point of view on the problem by looking first at the restricted problem where the massless particle is the outer body. In this situation, equilibria at high mutual inclination appear, which correspond to the population of stable particle…
Rapid GPU-Assisted Search and Parameterization-Based Refinement and Continuation of Connections between Tori in Periodically Perturbed Planar Circular Restricted 3-Body Problems
The dynamics near the Lagrange equilibria $L_1$ and $L_2$ of the Circular Restricted Three-body Problem has gained attention in the last decades due to its relevance in some topics such as the temporary captures of comets and asteroids and the design of trajectories for space missions. In this paper we investigate the temporary captures using the tube manifolds of the horizontal Lyapunov orbits originating at $L_1$ and $L_2$ of the CR3BP at energy values which have not been considered so far. After showing that the radius of convergence of any Hamiltonian normalization at $L_1$ or $L_2$ comput…
Exploration of vertical self-resonant bifurcations from Distant Retrograde Orbits (DROs) in the Earth-Moon Circular Restricted Three-Body Problem (CR3BP)
Planetary, stellar and galactic physics often rely on the general restricted gravitational N-body problem to model the motion of a small-mass object under the influence of much more massive objects. Here, I formulate the general restricted problem entirely and specifically in terms of the commonly-used orbital elements of semimajor axis, eccentricity, inclination, longitude of ascending node, argument of pericentre, and true anomaly, without any assumptions about their magnitudes. I derive the equations of motion in the general, unaveraged case, as well as specific cases, with respect to both …
The halo orbits of the spatial circular restricted three-body problem are largely considered in space-flight dynamics to design low-energy transfers between celestial bodies. A very efficient analytical method for the computation of halo orbits, and the related transfers, has been obtained from the high-order resonant Birkhoff normal forms defined at the Lagrangian points L1-L2. In this paper, by implementing a non-linear Floquet-Birkhoff resonant normal form, we provide the definition of orbits, as well as their manifold tubes, which exist in a large order approximation of the elliptic three-…
We consider the planar circular equilateral restricted four body-problem where a test particle of infinitesimal mass is moving under the gravitational attraction of three primary bodies which move on circular orbits around their common center of gravity, such that their configuration is always an equilateral triangle. The case where all three primaries have equal masses is numerically investigated. A thorough numerical analysis takes place in the configuration $(x,y)$ as well as in the $(x,C)$ space in which we classify initial conditions of orbits into four main categories: (i) bounded regula…
In the circular case of the coplanar Restricted Three-body Problem, we studied how the family of quasi-satellite (QS) periodic orbits allows to define an associated libration center. Using the averaged problem, we highlighted a validity limit of this one: for QS orbits with low eccentricities, the averaged problem does not correspond to the real problem. We do the same procedure to L 3 , L 4 and L 5 emerging periodic orbits families and remarked that for very high eccentricities F L4 and F L5 merge with F L3 which bifurcates to a stable family.
We consider the special case of the restricted circular three-body problem, when the two primaries are of equal mass, while the third body of negligible mass performs oscillations along a straight line perpendicular to the plane of the primaries (so called periodic vertical motions). The main goal of our investigation is to study the stability of these periodic motions in the linear approximation. A special attention is given to the alternation of stability and instability within the family of periodic vertical motions, whenever their amplitude is varied in a continuous monotone manner.
