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We investigate the orbital stability of a putative Jovian planet in a compact binary ν Octantis reported by Ramm et al. We
re-analysed published radial velocity data in terms of a self-consistent Newtonian model and we found stable best-fitting
solutions that obey observational constraints. They correspond to retrograde orbits, in accord with an earlier hypothesis
of Eberle & Cuntz, with apsidal lines anti-aligned with the apses of the binary. The best-fitting solutions are confined to
tiny stable regions of the phase space. These regions have a structure of the Arnold web formed by overlapping low-order mean
motion resonances and their sub-resonances. The presence of a real planet is still questionable, because its formation would
be hindered by strong dynamical perturbations. Our numerical study makes use of a new computational Message Passing Interface
framework mechanic developed to run massive numerical experiments on CPU clusters.

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... It may become a helper tool for a wide range of astronomical applications, particularly focused on processing large data sets, such as dynamical studies of long-term orbital evolution of planetary systems with Monte Carlo methods, dynamical maps or evolutionary algorithms. It has been already applied in numerical experiments conducted for Kepler-11 (Migaszewski et al., 2012) and νOctantis planetary systems (Goździewski et al., 2013). In this paper we describe the basics of the framework, including code listings for the implementation of a sample user's module. ...

... In the field of dynamical astronomy several numerical techniques have been proposed to determine the nature of the phase space of planetary systems. The Monte Carlo methods (e.g., Holman and Wiegert, 1999), evolutionary algorithms (e.g., Goździewski et al., 2008b;Goździewski and Migaszewski, 2009) or dynamical maps (e.g., Froeschlé et al., 2000;Guzzo, 2005;Migaszewski et al., 2012;Goździewski et al., 2013) have become standard research tools for determining possible or permitted configurations, mass ranges or other physical data. These experiments usually require intensive tests of sets of initial conditions, that represent different orbital configurations. ...

... This might be insufficient for large and long-term numerical tests, such as studying the dynamics of planetary systems. In particular, our recent work on Kepler-11 (Migaszewski et al., 2012) and νOctantis (Goździewski et al., 2013) systems required developing a new code framework, the Mechanic, dedicated to conducting massive parallel simulations. It has been turned out into generalpurpose master-worker framework, built on the foundation of the Message Passing Interface (Pacheco, 1996). ...

We introduce the Mechanic, a new open-source code framework. It is designed
to reduce the development effort of scientific applications by providing
unified API (Application Programming Interface) for configuration, data storage
and task management. The communication layer is based on the well-established
Message Passing Interface (MPI) standard, which is widely used on variety of
parallel computers and CPU-clusters. The data storage is performed within the
Hierarchical Data Format (HDF5). The design of the code follows em core-module
approach which allows to reduce the user's codebase and makes it portable for
single- and multi-CPU environments. The framework may be used in a local user's
environment, without administrative access to the cluster, under the PBS or
Slurm job schedulers. It may become a helper tool for a wide range of
astronomical applications, particularly focused on processing large data sets,
such as dynamical studies of long-term orbital evolution of planetary systems
with Monte Carlo methods, dynamical maps or evolutionary algorithms. It has
been already applied in numerical experiments conducted for Kepler-11
(Migaszewski et al., 2012), and nuOctantis planetary systems (Go\'zdziewski et
al., 2013). In this paper we describe the basics of the framework, including
code listings for the implementation of a sample user's module. The code is
illustrated on a model Hamiltonian introduced by (Froeschle et al., 2000)
presenting the Arnold diffusion. The Arnold Web is shown with the help of the
MEGNO (Mean Exponential Growth of Nearby Orbits) fast indicator (Go\'zdziewski
et al., 2008a) applied onto symplectic SABA integrators family (Laskar and
Robutel, 2001).

... Chaotic regions can alter the periodicity of the orbit, as well as the orientation of the orbit relative to some reference direction (i.e., precession of Eulerian angles). Goździewski et al (2013) and Migaszewski et al (2012) demonstrated this principle with a chaos indicator (for details, see section 4.5). ...

... This posed a problem for the confirmation of the exoplanet. However, if one considers an exoplanet in retrograde orbit relative to the orbiting binary, then enlarged regions of stability arise, which would be consistent with the putative exoplanet (Eberle and Cuntz 2010b, Quarles et al 2012a, Goździewski et al 2013. ...

A brief description of the four-body problem and the N-body problems is given. Main differences and similarities between each one of these problems and the three-body problem are presented and discussed, with the purpose of preparing the readers for situations that may require to go beyond the three-body problem.

... Chaotic regions can alter the periodicity of the orbit, as well as the orientation of the orbit relative to some reference direction (i.e., precession of Eulerian angles). Goździewski et al (2013) and Migaszewski et al (2012) demonstrated this principle with a chaos indicator (for details, see section 4.5). ...

... This posed a problem for the confirmation of the exoplanet. However, if one considers an exoplanet in retrograde orbit relative to the orbiting binary, then enlarged regions of stability arise, which would be consistent with the putative exoplanet (Eberle and Cuntz 2010b, Quarles et al 2012a, Goździewski et al 2013. ...

The astronomical applications of the three-body problem are extensive. We summarize the historical and prominent cases where it has been used, including cases from radial velocity and photometric surveys. Moreover, the three-body problem has been applied to a range of contexts, where we focus on those for single star and multiple planets, binary stars with a single planet, or binary systems that could potentially host Trojans in a three-body approximation. We provide a series of examples based upon the techniques outlined in the previous chapters.

... Chaotic regions can alter the periodicity of the orbit, as well as the orientation of the orbit relative to some reference direction (i.e., precession of Eulerian angles). Goździewski et al (2013) and Migaszewski et al (2012) demonstrated this principle with a chaos indicator (for details, see section 4.5). ...

... This posed a problem for the confirmation of the exoplanet. However, if one considers an exoplanet in retrograde orbit relative to the orbiting binary, then enlarged regions of stability arise, which would be consistent with the putative exoplanet (Eberle and Cuntz 2010b, Quarles et al 2012a, Goździewski et al 2013. ...

The most important theoretical developments (historical and recent) in the three-body problem are presented and discussed. The first part of the presentation is devoted to periodic solutions to the equations of motion, the second part to non-periodic solutions, and in the third part a detailed description of different stability criteria is given. Each part contains extensive discussions of the general, circular restricted and elliptical restricted three-body problem, as well as the Hill problem.

... In the (a 1 , a 2 )-plane (bottom panel), the 7:3 MMR between the middle Other strips corresponding to the (1:-5:7) MMR are visible too. Such a complex, fractal-like network of unstable strips in the two-dimensional maps may be interpreted as the Arnold web, as described by Guzzo (2005) for the Solar system, and in the binary ν-Octantis (Goździewski et al. 2013). ...

We study the orbital architecture, physical characteristics of planets, formation and long-term evolution of the Kepler-30 planetary system, detected and announced in 2012 by the Kepler team. We show that the Kepler-30 system belongs to a particular class of very compact and quasi-resonant, yet long-term stable planetary systems. We re-analyse the light curves of the host star spanning Q1-Q17 quarters of the Kepler mission. A huge variability of the Transit Timing Variations exceeding 2 d is induced by a massive Jovian planet located between two Neptune-like companions. The innermost pair is near to the 2:1 mean motion resonance (MMR), and the outermost pair is close to higher order MMRs, such as 17:7 and 7:3. Our re-analysis of photometric data allows us to constrain, better than before, the orbital elements, planets' radii, and masses, which are 9.2 ± 0.1, 536 ± 5, and 23.7 ± 1.3 Earth masses for Kepler-30b, Kepler-30c, and Kepler-30d, respectively. The masses of the inner planets are determined within ~1 per cent uncertainty. We infer the internal structures of the Kepler-30 planets and their bulk densities in a wide range from 0.19 ± 0.01 g·cm⁻³ for Kepler-30d, 0.96 ± 0.15 g·cm⁻³ for Kepler-30b, to 1.71 ± 0.13 g·cm⁻³ for the Jovian planet Kepler-30c. We attempt to explain the origin of this unique planetary system and a deviation of the orbits from exact MMRs through the planetary migration scenario. We anticipate that the Jupiter-like planet plays an important role in determining the present dynamical state of this system.

... ν Oct Ab-Located in the orange region, indicating strong orbital instability. This system has been the subject of discussion in many works (Eberle and Cuntz 2010;Quarles et al. 2012, Goździewski et al. (2013; among others), suggesting that the planet may orbit the central star in a retrograde orbit. Table 3 Forced eccentricities e F and secular frequencies g S calculated for the systems presented on Table 2 with the second-order secular models The secular frequencies were calculated for a fictitious planet at the stationary secular solution (e 1 = e F ) and for the actual value of the planetary eccentricity given by Table 2. ...

We analyse the secular dynamics of planets on S-type coplanar orbits in tight
binary systems, based on first- and second-order analytical models, and compare
their predictions with full N-body simulations. The perturbation parameter
adopted for the development of these models depends on the masses of the stars
and on the semimajor axis ratio between the planet and the binary.
We show that each model has both advantages and limitations. While the
first-order analytical model is algebraically simple and easy to implement, it
is only applicable in regions of the parameter space where the perturbations
are sufficiently small. The second-order model, although more complex, has a
larger range of validity and must be taken into account for dynamical studies
of some real exoplanetary systems such as $\gamma$-Cephei and HD 41004A.
However, in some extreme cases, neither of these analytical models yields
quantitatively correct results, requiring either higher-order theories or
direct numerical simulations.
Finally, we determine the limits of applicability of each analytical model in
the parameter space of the system, giving an important visual aid to decode
which secular theory should be adopted for any given planetary system in a
close binary.

... In such a scenario, test particles would chaotically hop from one resonance to another, without spending any significant period of time trapped in any specific resonance. Such behaviour has been invoked in the past to explain regions of stability for other exoplanetary systems [27]. ...

With improvements in exoplanet detection techniques, the number of multiple
planet systems discovered is increasing, while the detection of potentially
habitable Earth-mass planets remains complicated and thus requires new search
strategies. Dynamical studies of known multiple planet systems are therefore a
vital tool in the search for stable and habitable planet candidates. Here, we
present a dynamical study of the three-planet system HD 204313 to determine
whether it could harbour an Earth-like planet within its habitable zone for a
sufficient time to develop life. We found two semi-stable regions in the
system, but neither prove stable for long enough for a terrestrial planet to
develop life. Our investigations suggest that overlapping weak and high order
resonances may be responsible for these semi-stable regions. This study
established a framework for a larger project that will study the dynamical
stability of the habitable zone of multiple planet systems, providing a list of
interesting targets for future habitable low-mass planet searches.

The single-lined spectroscopic binary ν Octantis provided evidence of the first conjectured circumstellar planet demanding an orbit retrograde to the stellar orbits. The planet-like behaviour is now based on 1437 radial velocities (RVs) acquired from 2001–2013. ν Oct’s semimajor axis is only 2.6 au with the candidate planet orbiting ν Oct A about midway between. These details seriously challenge our understanding of planet formation and our decisive modelling of orbit reconfiguration and stability scenarios, whilst having a host that is equally inconsistent with accepted stellar variability scenarios. Thus, the final solution will be be an important discovery. All non-planetary explanations are less credible based on multiple qualitative and quantitative tests including previous spectroscopic studies of bisectors and line-depth ratios, photometry from Hipparcos and the more recent space missions TESS and GAIA (whose increased parallax classifies ν Oct A closer still to a subgiant, ∼ K1 IV). We conducted the first large survey of ν Oct A’s chromosphere: 198 $\rm Ca\, {\rm \scriptstyle II}$ H-line and 1160 $\rm {H}\, \alpha$ indices using spectra from a previous RV campaign (2009–2013). We also acquired 135 spectra (2018–2020) primarily used for additional line-depth ratios, which are extremely sensitive to the photosphere’s temperature. We found no significant RV-correlated variability. Our line-depth ratios indicate temperature variations of only ±4 K, as achieved previously. Our atypical $\rm Ca\, {\rm \scriptstyle II}$ analysis models the indices in terms of S/N and includes covariance significantly in their errors. The $\rm {H}\, \alpha$ indices have a quasi-periodic variability which we demonstrate is due to telluric lines. Our new evidence provides further multiple arguments realistically only in favor of the planet.

Exoplanets represent a broad new class of astronomical objects, which became accessible for observations and studies only just before the end of the last century. Owing to continually improving techniques of ground-based observations, and especially observations from space, for a little bit more than two decades thousands of planetary systems of other stars have been discovered, and this process is escalating. Exoplanets are of paramount interest for astrophysical, astrochemical, and dynamical studies. Exoplanetary studies have opened up new horizons to gain insights into fundamental problems of stellar-planetary cosmogony and, in particular, into the question of the origin and evolution of the Solar System. Discoveries of Earth-like planets, especially those orbiting in stellar habitable zones favorable to giving rise to and sustaining life, open new prospects for progress in astrobiology.

We present a detailed orbital and stability analysis of the HD~59686 binary-star planet system. HD~59686 is a single-lined moderately close ($a_{B} = 13.6\,$AU) eccentric ($e_{B} = 0.73$) binary, where the primary is an evolved K giant with mass $M = 1.9 M_{\odot}$ and the secondary is a star with a minimum mass of $m_{B} = 0.53 M_{\odot}$. Additionally, on the basis of precise radial velocity (RV) data a Jovian planet with a minimum mass of $m_p = 7 M_{\mathrm{Jup}}$, orbiting the primary on a nearly circular S-type orbit with $e_p = 0.05$ and $a_p = 1.09\,$AU, has recently been announced. We investigate large sets of orbital fits consistent with HD 59686's radial velocity data by applying bootstrap and systematic grid-search techniques coupled with self-consistent dynamical fitting. We perform long-term dynamical integrations of these fits to constrain the permitted orbital configurations. We find that if the binary and the planet in this system have prograde and aligned coplanar orbits, there are narrow regions of stable orbital solutions locked in a secular apsidal alignment with the angle between the periapses, $\Delta \omega$, librating about $0^\circ$. We also test a large number of mutually inclined dynamical models in an attempt to constrain the three-dimensional orbital architecture. We find that for nearly coplanar and retrograde orbits with mutual inclination $145^\circ \lesssim \Delta i \leq 180^\circ$, the system is fully stable for a large range of orbital solutions.

All attempts to solve the three-body problem described in the previous chapters have greatly enriched celestial and classical mechanics. However, the ubiquity of fast computers with modes of operation that allow for parallel computations, numerical solutions have been the driving force in finding and studying possible solutions to the three-body problem. In this chapter, we provide a brief overview of common numerical schemes in terms of the mathematics of the algorithms used, as well as short examples written in the open-source programming language Python. The topics included in our discussion deal with numerical integration of the three-body problem, Fourier analysis, determination of mean motion resonances, and chaos indicators.

We describe the Reversibility Error Method (REM) and its applications to planetary dynamics. REM is based on the time-reversibility analysis of the phase-space trajectories of conservative Hamiltonian systems. The round-off errors break the time reversibility and the displacement from the initial condition, occurring when we integrate it forward and backward for the same time interval, is related to the dynamical character of the trajectory. If the motion is chaotic, in the sense of non-zero maximal Characteristic Lyapunov Exponent (mLCE), then REM increases exponentially with time, as exp λt, while when the motion is regular (quasi-periodic) then REM increases as a power law in time, as tα, where α and λ are real coefficients. We compare the REM with a variant of mLCE, the Mean Exponential Growth factor of Nearby Orbits (MEGNO). The test set includes the restricted three body problem and five resonant planetary systems: HD 37124, Kepler-60, Kepler-36, Kepler-29 and Kepler-26. We found a very good agreement between the outcomes of these algorithms. Moreover, the numerical implementation of REM is astonishing simple, and is based on solid theoretical background. The REM requires only a symplectic and time-reversible (symmetric) integrator of the equations of motion. This method is also CPU efficient. It may be particularly useful for the dynamical analysis of multiple planetary systems in the Kepler sample, characterized by low-eccentricity orbits and relatively weak mutual interactions. As an interesting side-result, we found a possible stable chaos occurrence in the Kepler-29 planetary system.

Context: For over 12 years, we have carried out a precise radial velocity survey of a sample of 373 G and K giant stars using the Hamilton \'Echelle Spectrograph at Lick Observatory. There are, among others, a number of multiple planetary systems in our sample as well as several planetary candidates in stellar binaries. Aims: We aim at detecting and characterizing substellar+stellar companions to the giant star HD 59686 A (HR 2877, HIP 36616). Methods: We obtained high precision radial velocity (RV) measurements of the star HD 59686 A. By fitting a Keplerian model to the periodic changes in the RVs, we can assess the nature of companions in the system. In order to discriminate between RV variations due to non-radial pulsation or stellar spots we used infrared RVs taken with the CRIRES spectrograph at the Very Large Telescope. Additionally, to further characterize the system, we obtain high-resolution images with LMIRCam at the Large Binocular Telescope. Results: We report the likely discovery of a giant planet with a mass of $m_{p}~\sin i=6.92_{-0.24}^{+0.18}~M_{Jup}$ orbiting at $a_{p}=1.0860_{-0.0007}^{+0.0006}$ au from the giant star HD 59686 A. Besides the planetary signal, we discover an eccentric ($e_{B}=0.729_{-0.003}^{+0.004}$) binary companion with a mass of $m_{B}~\sin i=0.5296_{-0.0008}^{+0.0011}~M_{Sun}$ orbiting at a semi-major axis of just $a_{B}=13.56_{-0.14}^{+0.18}$ au. Conclusions: The existence of the planet HD 59686 Ab in a tight eccentric binary system severely challenges standard giant planet formation theories and requires substantial improvements to such theories in tight binaries. Otherwise, alternative planet formation scenarios such as second generation planets or dynamical interactions in an early phase of the system's lifetime should be seriously considered in order to better understand the origin of this enigmatic planet.

In this chapter we discuss in a pedagogical way and from the very beginning the Mean Exponential Growth factor of Nearby Orbits (MEGNO) method, that has proven, in the last ten years, to be efficient to investigate both regular and chaotic components of phase space of a Hamiltonian system. It is a fast indicator that provides a clear picture of the resonance structure, the location of stable and unstable periodic orbits as well as a measure of hyperbolicity in chaotic domains which coincides with that given by the maximum Lyapunov characteristic exponent but in a shorter evolution time. Applications of the MEGNO to simple discrete and continuous dynamical systems are discussed and an overview of the stability studies present in the literature encompassing quite different dynamical systems is provided.

We report 1212 radial-velocity (RV) measurements obtained in the years 2009-2013 using an iodine cell for the spectroscopic
binary ν Octantis (K1III/IV). This system (abin ∼ 2.6 au, P ∼ 1050 days) is conjectured to have a Jovian planet with a semi-major axis half that of the binary host. The extreme geometry
only permits long-term stability if the planet is in a retrograde orbit. Whilst the reality of the planet (P ∼ 415 days) remains uncertain, other scenarios (stellar variability or apsidal motion caused by a yet unobserved third star)
continue to appear substantially less credible based on CCF bisectors, line-depth ratios and many other independent details.
If this evidence is validated but the planet is disproved, the claims of other planets using RVs will be seriously challenged.
We also describe a significant revision to the previously published RVs and the full set of 1437 RVs now encompasses nearly
13 years. The sensitive orbital dynamics allow us to constrain the three-dimensional architecture with a broad prior probability
distribution on the mutual inclination, which with posterior samples obtained from an N-body Markov chain Monte Carlo is found to be 158.4° ± 1.2. None of these samples are dynamically stable beyond 106 years. However, a grid search around the best-fitting solution finds a region that has many models stable for 107 years, and includes one model within 1-sigma that is stable for at least 108 years. The planet's exceptional nature demands robust independent verification and makes the theoretical understanding of
its formation a worthy challenge.

We start by reviewing our previous work on retrograde orbital configurations and on modeling and identifying retrograde resonances. Then, we present new results regarding the enhanced stability of retrograde configurations with respect to prograde configurations in the low mass ratio regime of the planar circular restricted 3-body problem. Motivated by the recent discovery of small bodies which are in retrograde resonance with the Solar System’s giant planets we then explore the case with mass ratio 0.001 and show new stability maps in a grid of semi-major axis versus eccentricity for the 2/1 and 1/2 retrograde resonances. Finally, we explain how the stability borders of the 2/1 and 1/2 retrograde resonances are related to the resonant orbits’ geometry. © 2015, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.

We explore the possibly that either star-spots or pulsations are the cause of a periodic radial velocity (RV) signal (P ∼ 400 d) from the K-giant binary ν Octantis (P ∼ 1050 d, e ∼ 0.25), alternatively conjectured to have a retrograde planet. Our study is based on temperatures derived from 22 line-depth ratios (LDRs) for ν Oct and 20 calibration stars. Empirical evidence and stability modelling provide unexpected support for the planet since other standard explanations (star-spots, pulsations and additional stellar masses) each have credibility problems. However, the proposed system presents formidable challenges to planet formation and stability theories: it has by far the smallest stellar separation of any claimed planet-harbouring binary ($a_{_{\rm bin}} \sim 2.6$ au) and an equally unbelievable separation ratio ($a_{_{\rm pl}}/a_{_{\rm bin}} \sim 0.5$), hence the necessity that the circumstellar orbit be retrograde. The LDR analysis of 215 ν Oct spectra acquired between 2001 and 2007, from which the RV perturbation was first revealed, have no significant periodicity at any frequency. The LDRs recover the original 21 stellar temperatures with an average accuracy of 45 ± 25 K. The 215 ν Oct temperatures have a standard deviation of only 4.2 K. Assuming the host primary is not pulsating, the temperatures converted to magnitude differences strikingly mimic the very stable photometric Hipparcos observations 15 years previously, implying the long-term stability of the star and demonstrating a novel use of LDRs as a photometric gauge. Our results provide substantial new evidence that conventional star-spots and pulsations are unlikely causes of the RV perturbation. The controversial system deserves continued attention, including with higher resolving-power spectra for bisector and LDR analyses.

We study the mid-egress eclipse timing data gathered for the cataclysmic binary HU Aquarii during the years 1993–2014. The
(O−C) residuals were previously attributed to a single ∼7 Jupiter mass companion in ∼5 au orbit or to a stable two-planet
system with an unconstrained outermost orbit. We present 22 new observations gathered between 2011 June and 2014 July with
four instruments around the world. They reveal a systematic deviation of ∼60–120 s from the older ephemeris. We re-analyse
the whole set of the timing data available. Our results provide an erratum to the previous HU Aqr planetary models, indicating
that the hypothesis for a third and fourth body in this system is uncertain. The dynamical stability criterion and a particular
geometry of orbits rule out coplanar two-planet configurations. A putative HU Aqr planetary system may be more complex, e.g.
highly non-coplanar. Indeed, we found examples of three-planet configurations with the middle planet in a retrograde orbit,
which are stable for at least 1 Gyr, and consistent with the observations. The (O−C) may be also driven by oscillations of
the gravitational quadrupole moment of the secondary, as predicted by the Lanza et al. modification of the Applegate mechanism.
Further systematic, long-term monitoring of HU Aqr is required to interpret the (O−C) residuals.

We have explored the possibility of a third circumbinary planet having a
dynamically stable orbit in the Kepler-47 system and producing the single,
unexplained transit event (not associated with either the binary star or the
two known circumbinary planets) reported in the discovery paper (Orosz et al.
2012). We applied the dynamical mapping MEGNO technique to identify regions in
the phase space of the system where this third planet can maintain stable,
quasi-periodic orbits. The long-term, Lagrangian stability of the entire 5-body
configuration (eclipsing binary + three planets) is confirmed by direct
numerical integrations for 10 Myr. We identified several long-term stable
regions between the two confirmed planets, and also an extended region beyond
the orbit of the outer planet Kepler-47c. To further constrain the orbit of the
hypothetical third planet, we compared the synthetic single transit duration it
produces from the ensemble of stable orbits to the measured duration of the
unexplained transit event (~4.15 hours). Due to the rich dynamics of the
system, different stable orbits of such a hypothetical, third circumbinary
planet can produce similar single-transit durations. To remove this degeneracy,
we fixed the planet's orbit as circular and use the observed duration of the
unexplained transit to analytically place an upper limit of 424 days for the
planetary period. Our analysis strongly suggests that, if the yet unexplained
single transit event is indeed due to a planetary object, then the most
probable orbit for this undetected planet will be between Kepler-47b and
Kepler-47c -- a region characterized by low-order mean motion resonances. We
present our methodology in details, and discuss the implication of our results.

The three-body problem, which describes three masses interacting through
Newtonian gravity without any restrictions imposed on the initial positions and
velocities of these masses, has attracted the attention of many scientists for
more than 300 years. In this paper, we present a review of the three-body
problem in the context of both historical and modern developments. We describe
the general and restricted (circular and elliptic) three-body problems,
different analytical and numerical methods of finding solutions, methods for
performing stability analysis, search for periodic orbits and resonances, and
application of the results to some interesting astronomical and space dynamical
settings. We also provide a brief presentation of the general and restricted
relativistic three-body problems, and discuss their astronomical applications.

The planet occurrence rate for multiple stars is important in two
aspects. First, almost half of stellar systems in the solar neighborhood
are multiple systems. Second, the comparison of the planet occurrence
rate for multiple stars to that for single stars sheds light on the
influence of stellar multiplicity on planet formation and evolution. We
developed a method of distinguishing planet occurrence rates for single
and multiple stars. From a sample of 138 bright (KP <13.5) Kepler
multi-planet candidate systems, we compared the stellar multiplicity
rate of these planet host stars to that of field stars. Using dynamical
stability analyses and archival Doppler measurements, we find that the
stellar multiplicity rate of planet host stars is significantly lower
than field stars for semi-major axes less than 20 AU, suggesting that
planet formation and evolution are suppressed by the presence of a
close-in companion star at these separations. The influence of stellar
multiplicity at larger separations is uncertain because of search in-
completeness due to a limited Doppler observation time baseline and a
lack of high resolution imaging observation. We calculated the planet
confidence for the sample of mutlti-planet candidates, and find that the
planet confidences for KOI 82.01, KOI 115.01, KOI 282.01 and KOI 1781.02
are higher than 99.7% and thus validate the planetary nature of these
four planet candidates. This sample of bright Kepler multi-planet
candidates with refined stellar and orbital parameters, planet
confidence estimation, and nearby stellar companion identification
offers a well-characterized sample for future theoretical and
observational study.

We determine the fraction of G-dwarf stars that could host stable planetary
systems based on the observed properties of binaries in the Galactic field, and
in various postulated primordial binary populations, which assume that the
primordial binary fraction is higher than that in the field. We first consider
the frequency of Solar System analogues - planetary systems that form either
around a single G-dwarf star, or a binary containing a G-dwarf where the binary
separation exceeds 100-300au. If the primordial binary fraction and period
distribution is similar to that in the field, then up to 63 per cent of G-dwarf
systems could potentially host a Solar System analogue. However, if the
primordial binary fraction is higher, the fraction of G-dwarf systems that
could host a planetary system like our own is lowered to 38 per cent.
We extend our analysis to consider the fraction of G-dwarf systems (both
single and binary) that can host either circumprimary planets (orbiting the
primary star of the binary) or circumbinary planets (orbiting both stars in the
binary) for fiducial planetary separations between 1 - 100au. Depending on the
assumed binary population, in the circumprimary case between 65 and 95 per cent
of systems can host a planet at 1au, decreasing to between 20 and 65 per cent
of systems that can host a planet at 100au. In the circumbinary case, between 5
and 59 per cent of systems can host a planet at 1au, increasing to between 34
and 75 per cent of systems that can host a planet at 100au.
Our results suggest that the assumed binary fraction is the key parameter in
determining the fraction of potentially stable planetary systems in G-dwarf
systems and that using the present-day value may lead to significant
overestimates if the binary fraction was initially higher.

The aim of our study is to investigate the possibility of habitable moons orbiting the giant planet HD 23079b, a Jupiter-mass planet, which follows a low-eccentricity orbit in the outer region of HD 23079’s habitable zone. We show that HD 23079b is able to host habitable moons in prograde and retrograde orbits, as expected, noting that the outer stability limit for retrograde orbits is increased by nearly 90% compared with that of prograde orbits, a result consistent with previous generalised studies. For the targeted parameter space, it was found that the outer stability limit for habitable moons varies between 0.05236 and 0.06955 AU (prograde orbits) and between 0.1023 and 0.1190 AU (retrograde orbits), depending on the orbital parameters of the Jupiter-type planet if a minimum mass is assumed. These intervals correspond to 0.306 and 0.345 (prograde orbits) and 0.583 and 0.611 (retrograde orbits) of the planet's Hill radius. Larger stability limits are obtained if an increased value for the planetary mass mp
is considered; they are consistent with the theoretically deduced relationship of m
1/3
p
. Finally, we compare our results with the statistical formulae of Domingos, Winter, & Yokoyama, indicating both concurrence and limitations.

20 years after the discovery of the first planets outside our solar system, the current exoplanetary population includes more than 700 confirmed planets around main sequence stars. Approximately 50% belong to multiple-planet systems in very diverse dynamical configurations, from two-planet hierarchical systems to multiple resonances that could only have been attained as the consequence of a smooth large-scale orbital migration. The first part of this paper reviews the main detection techniques employed for the detection and orbital characterization of multiple-planet systems, from the (now) classical radial velocity (RV) method to the use of transit time variations (TTV) for the identification of additional planetary bodies orbiting the same star. In the second part we discuss the dynamical evolution of multi-planet systems due to their mutual gravitational interactions. We analyze possible modes of motion for hierarchical, secular or resonant configurations, and what stability criteria can be defined in each case. In some cases, the dynamics can be well approximated by simple analytical expressions for the Hamiltonian function, while other configurations can only be studied with semi-analytical or numerical tools. In particular, we show how mean-motion resonances can generate complex structures in the phase space where different libration islands and circulation domains are separated by chaotic layers. In all cases we use real exoplanetary systems as working examples.

In this paper we deal with an alternative technique to study global dynamics in Hamiltonian systems, the mean exponential growth factor of nearby orbits (MEGNO), that proves to be efficient to investigate both regular and stochastic components of phase space. It provides a clear picture of resonance structures, location of stable and unstable periodic orbits as well as a measure of hyperbolicity in chaotic domains which coincides with that given by the Lyapunov characteristic number. Here the MEGNO is applied to a rather simple model, the 3D perturbed quartic oscillator, in order to visualize the structure of its phase space and obtain a quite clear picture of its resonance structure. Examples of application to multi-dimensional canonical maps are also included.

The aim of this study is to explore an enigmatic finding about the ν Octantis binary system that indicates the possible existence of a Jupiter-type planet even though the planet seems to be located outside the zone of orbital stability. We perform a detailed analysis of orbital stability based on previous studies that carefully considers the ν Octantis system parameters including their observationally deduced uncertainties. In our analysis, we confront the probability distribution of the parameter space of the system with the requirements of planetary orbital stability. Our results indicate that the suggested planet, if in a prograde orbit with respect to the motion of the binary components, is virtually impossible. However, the estimated probability of existence for a planet in a retrograde orbit is nearly 60%, an estimate that encapsulates the probability distribution of the mass ratio of the stellar components. This estimate increases if a relatively low stellar mass ratio (within the error bars) is assumed. The principal possibility of a planet in a retrograde orbit is also consistent with long-term orbital stability simulations pursued as part of our study. Thus, the existence of the suggested planet in the ν Octantis system constitutes a realistic possibility.

We represent graphically the evolution of the set of resonances of a quasi-integrable dynamical system, the so-called Arnold
web, whose structure is crucial for the stability properties of the system. The basis of our representation is the use of
an original numerical method, whose definition is directly related to the dynamics of orbits, and the careful choice of a
model system. We also show the transition from the Nekhoroshev stability regime to the Chirikov diffusive one, which is a
generic nontrivial phenomenon occurring in many physical processes, such as slow chaotic transport in the asteroid belt and
beam-beam interaction.

A simple question of celestial mechanics is investigated: in what regions of phase space near a binary system can planets persist for long times? The planets are taken to be test particles moving in the field of an eccentric binary system. A range of values of the binary eccentricity and mass ratio is studied, and both the case of planets orbiting close to one of the stars, and that of planets outside the binary orbiting the system's center of mass, are examined. From the results, empirical expressions are developed for both 1) the largest orbit around each of the stars, and 2) the smallest orbit around the binary system as a whole, in which test particles survive the length of the integration (10^4 binary periods). The empirical expressions developed, which are roughly linear in both the mass ratio mu and the binary eccentricity e, are determined for the range 0.0 <= e <= 0.7-0.8 and 0.1 <= mu <= 0.9 in both regions, and can be used to guide searches for planets in binary systems. After considering the case of a single low-mass planet in binary systems, the stability of a mutually-interacting system of planets orbiting one star of a binary system is examined, though in less detail. Comment: 19 pages, 5 figures, 7 tables, accepted by the Astronomical Journal

A planet orbiting around a star in a binary system can be ejected if it lies too far from its host star. We find that instability boundaries first obtained in numerical studies can be explained by overlap between sub-resonances within mean-motion resonances (mostly of the j:1 type). Strong secular forcing from the companion displaces the centroids of different sub-resonances, producing large regions of resonance overlap. Planets lying within these overlapping regions experience chaotic diffusion, which in most cases leads to their eventual ejection. The overlap region extends to shorter-period orbits as either the companion's mass or its eccentricity increase. Our analytical calculations reproduce the instability boundaries observed in numerical studies and yield the following two additional results. Firstly, the instability boundary as a function of eccentricity is jagged; thus, the widest stable orbit could be reduced from previously quoted values by as much as 20%. Secondly, very high order resonances (e.g., 50:3) do not significantly modify the instability boundary despite the fact that these weak resonances can produce slow chaotic diffusion which may be missed by finite-duration numerical integrations. We present some numerical evidence for the first result. More extensive experiments are called for to confirm these conclusions. For the special case of circular binaries, we find that the Hill criterion (based on the critical Jacobi integral) yields an instability boundary that is very similar to that obtained by resonance overlap arguments, making the former both a necessary and a sufficient condition for planet instability. Comment: Accepted for publication in the Astrophyical Journal. Consists of 8 pages, 6 figures

I present an empirical model for predicting a star's radial velocity jitter from its B - V color, activity level, and absolute magnitude. This model is based on observations of ∼450 well-observed stars from Keck Observatory for the California and Carnegie Planet Search Program. The model includes noise from both astrophysical sources and systematic errors, and describes jitter as generally increasing with a star's activity and height above the main sequence. © 2005. The Astronomical Society of the Pacific. All rights reserved.

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As of today over 40 planetary systems have been discovered in
binary star systems.
In all cases the configuration appears to be circumstellar,
where the planets orbit around one of the stars, the secondary acting as
a perturber. The formation of planets in binary star systems is more difficult
than around single stars due to the gravitational action of the companion
on the dynamics of the protoplanetary disk.
In this contribution we first briefly present the relevant observational
evidence for planets in binary systems. Then the dynamical influence that a
secondary companion has on a circumstellar disk will be analyzed through
fully hydrodynamical simulations. We demonstrate that the disk becomes
eccentric and shows a coherent precession around the primary star.
Finally, fully hydrodynamical simulations of evolving protoplanets
embedded in disks in binary star systems are presented.
We investigate how the orbital evolution of protoplanetary embryos
and their mass growth from cores to
massive planets might be affected in this very dynamical environment.
We consider, in particular, the planet
orbiting the primary in the system γ Cephei.

New astrometric–spectroscopic orbital solutions for the single-line K-giant binaries β Reticuli (P≈ 5.2 yr, e= 0.3346 ± 0.0004) and ν Octantis (P≈ 2.9 yr, e= 0.2358 ± 0.0003) have been derived based on high-precision spectroscopic radial velocities (RVs) and the Hipparcos astrometry. For the case of ν Oct, the simultaneous solution is particularly robust and an inclination of i= 70.8 ± 0.9° has been derived. This is one of the most precise inclinations yet calculated based on a spectroscopic solution and the Hipparcos astrometry.
We have also discovered low-amplitude periodic behaviour in the residuals of the orbital solution for ν Oct. This RV perturbation has a semi-amplitude of 50 m s−1 and a 418-d period which is coherent over several years. The RV curve of the perturbation is apparently in resonance with that of the binary: every second maximum of the binary coincides with every fifth minimum of the perturbation, hence the periods have the simple ratio 5:2.
The possible causes of such a perturbation are rotational modulation of surface phenomenon, pulsations or an orbiting body. We have assessed these alternatives in terms of the suspected photometric stability (Hp= 3.8981 ± 0.0004), a lack of evidence of other RV periodicities, no correlation of cross-correlation function bisectors with the residual velocities, no compelling evidence of wavelength dependency for the amplitude or relative phase of the perturbation, our bounds on the rotational period of the primary star and the need for long-lived relatively fixed surface features. The results of these analyses lack consistency with both rotational modulation and pulsations and so imply that a planetary mass is a realistic cause. The planet hypothesis, however, is strongly constrained and challenged by our precise binary orbit. The hypothetical planet would have an orbit (e≈ 0.1, a3≈ 1.2 au) about mid-way between the stars whose periastron distance is only 1.9 au. This orbit, supposedly in resonance with the binary system, appears to be highly unlikely based on current planet formation and orbit-stability expectations.
Without knowing the cause of the perturbation, we cannot be certain if the suspected RV and hence period resonance are merely coincidental or not. Establishing the true cause of the perturbation requires renewed observation of the system, re-assessment of the possible resonance if this is redetected and the acquisition of similar and additional precise diagnostic parameters with respect to each of the possible causative mechanisms.

The role of radial velocity (RV) jitter in extrasolar planet search surveys is discussed. Based on the maximum-likelihood principle, improved statistical algorithms for RV fitting and period search are developed. These algorithms incorporate a built-in jitter determination, so that resulting estimations of planetary parameters account for this jitter automatically. This approach is applied to RV data for several extrasolar planetary systems. It is shown that many RV planet search surveys suffer from periodic systematic errors which increase effective RV jitter and can lead to erroneous conclusions. For instance, the planet candidate HD 74156 d may be a false detection made due to annual systematic errors.

We investigate a non-resonant, three-dimensional (spatial) model of a hierarchical system composed of a point-mass stellar (or substellar) 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 a low-mass object in a distant orbit may excite a large eccentricity of the inner binary when the mutual inclination of the orbits is larger than about 60°. This is related to the strong instability caused by a phenomenon that acts similarly to the Lidov–Kozai resonance (LKR). The secular system may be strongly chaotic and its dynamics unpredictable over long-term time-scales. Our study shows that in the Jupiter- or brown-dwarf-mass regime of the low-mass companion, the restricted model does not properly describe the long-term dynamics in the vicinity of the LKR. The relativistic correction is significant for the parametric structure of a few families of stationary solutions in this problem, in particular for direct orbit configurations (with mutual inclination less than 90°). We found that the dynamics of hierarchical systems with small α∼ 0.01 may be qualitatively different in the realms of Newtonian (classic) and relativistic models. This holds true even for relatively large masses of the secondaries.

We describe numerical tools for the stability analysis of extrasolar planetary systems. In particular, we consider the relative Poincaré variables and symplectic integration of the equations of motion. We apply the tangent map to derive a numerically efficient algorithm of the fast indicator Mean Exponential Growth factor of Nearby Orbits (MEGNO), a measure of the maximal Lyapunov exponent, that helps to distinguish chaotic and regular configurations. The results concerning the three-planet extrasolar system HD 37124 are presented and discussed. The best-fitting solutions found in earlier works are studied more closely. The system involves Jovian planets with similar masses. The orbits have moderate eccentricities, nevertheless the best-fitting solutions are found in dynamically active region of the phase space. The long-term stability of the system is determined by a net of low-order two-body and three-body mean motion resonances. In particular, the three-body resonances may induce strong chaos that leads to self-destruction of the system after Myr of apparently stable and bounded evolution. In such a case, numerically efficient dynamical maps are useful to resolve the fine structure of the phase space and to identify the sources of unstable behaviour.

This is a numerical study of orbits in the elliptic restricted three-body problem concerning the dependence of the critical orbits on the eccentricity of the primaries. They are defined as being the separatrix between stable and unstable single periodic orbits. As our results are adapted to the existence of planetary orbits in double stars we concentrated first on the P-orbits (defined to surround both primaries). Due to the complexity of the elliptic problem there is no analytical approach possible. Using the results of some 300 integrated orbits for 103 to 3. 103 periods of the primaries we established lower and upper bounds for the critical orbits for different values of the eccentricity.

The Kepler-11 planetary system hosts at least six transiting super-Earth planets detected through the precise photometric
observations of the Kepler mission (Lissauer et al.). In this paper, we re-analyse the available Kepler data, using the direct N-body approach rather than an indirect transit timing variation method as employed in the discovery paper. The orbital modelling
in the realm of the direct approach relies on the whole data set, not only on the mid-transits times. Most of the results
in the original paper are confirmed and extended. We constrained the mass of the outermost planet g to less than 30 M⊕. The mutual inclinations between orbits b and c as well as between orbits d and e are determined with a good precision, in
the range of [1°, 5°]. Having several solutions to the four qualitative orbital models of the Kepler-11 system, we analyse
its global dynamics with the help of dynamical maps. They reveal a sophisticated structure of the phase space, with narrow
regions of regular motion. The dynamics are governed by a dense net of three- and four-body mean motion resonances, forming
the Arnold web. Overlapping of these resonances is a main source of instability. We found that the Kepler-11 system may be
long-term stable only in particular multiple resonant configurations with small relative inclinations. The mass–radius data
derived for all companions reveal a clear anticorrelation between the mean density of the planets and their distance from
the star. This may reflect the formation and early evolution history of the system.

We investigate the stability of prograde versus retrograde planets in circular binary systems using numerical simulations.
We show that retrograde planets are stable up to distances closer to the perturber than prograde planets. We develop an analytical
model to compute the prograde and retrograde mean motion resonances’ locations and separatrices. We show that instability
is due to single resonance forcing, or caused by nearby resonances’ overlap. We validate our results regarding the role of
single resonances and resonances’ overlap on orbit stability, by computing surfaces of section of the circular restricted
three-body problem. We conclude that the observed enhanced stability of retrograde planets with respect to prograde planets
is due to essential differences between the phase-space topology of retrograde versus prograde resonances (at p/q mean motion ratio, the prograde resonance is of order p−q while the retrograde resonance is of order p+q).

The motion of the giant planets from Jupiter to Neptune is chaotic with Lyapunov time of approximately 10 Myr. A recent theory explains the presence of this chaos with three-planet mean-motion resonances, i.e. resonances among the orbital periods of at least three planets. We find that the distribution of these resonances with respect to the semi-major axes of all the planets is compatible with orbital instability. In particular, they overlap in a region of 10−3 AU with respect to the variation of the semi-major axes of Uranus and Neptune. Fictitious planetary systems with initial conditions in this region can undergo systematic variations of semi-major axes. The true Solar System is marginally in this region, and Uranus and Neptune undergo very slow systematic variations of semi-major axes with speed of order 10−4 AU/Gyr.

In this paper we numerically detect the web of three-planet resonances (i.e., resonances among mean anomalies, nodes and perihelia of three planets) with respect to the variation of the semi-major axis of Saturn and Jupiter, in a model including the planets from Jupiter to Neptune. The measure confirms the relevance of these resonances in the long-term evolution of the outer Solar System and provides a technique to identify some of the related coefficients. (c) 2004 Elsevier Inc. All rights reserved.

This paper explores the intermediate-time dynamics of newly formed solar systems with a focus on possible mechanisms for planetary migration. We consider two limiting corners of the available parameter space—crowded systems containing = 10 giant planets in the outer solar system and solar systems with = 2 planets that are tidally interacting with a circumstellar disk. Crowded planetary systems can be formed in accumulation scenarios—if the disk is metal rich and has large mass—and through gravitational instabilities. The planetary system adjusts itself toward stability by spreading out, ejecting planets, and sending bodies into the central star. For a given set of initial conditions, dynamical relaxation leads to a well-defined distribution of possible solar systems. For each class of initial conditions, we perform large numbers (hundreds to thousands) of N-body simulations to obtain a statistical description of the possible outcomes. For = 10 planet systems, we consider several different planetary mass distributions; we also perform secondary sets of simulations to explore chaotic behavior and longer term dynamical evolution. For systems with 10 planets initially populating the radial range 5 AU ≤ a ≤ 30 AU, these scattering processes naturally produce planetary orbits with a ∼ 1 AU and the full range of possible eccentricity (0 ≤ ε ≤ 1). Shorter period orbits (smaller a) are difficult to achieve. To account for the observed eccentric giant planets, we also explore a mechanism that combines dynamical scattering and tidal interactions with a circumstellar disk. This combined model naturally produces the observed range of semimajor axis a and eccentricity ε. We discuss the relative merits of the different migration mechanisms for producing the observed eccentric giant planets.

We provide a detailed theoretical study aimed at the observational finding about the ν Octantis binary system that indicates
the possible existence of a Jupiter‐type planet in this system. If a prograde planetary orbit is assumed, it has earlier been
argued that the planet, if existing, should be located outside the zone of orbital stability. However, a previous study by
Eberle & Cuntz concludes that the planet is most likely stable if assumed to be in a retrograde orbit with respect to the
secondary system component. In the present work, we significantly augment this study by taking into account the observationally
deduced uncertainty ranges of the orbital parameters for the stellar components and the suggested planet. Furthermore, our
study employs additional mathematical methods, which include monitoring the Jacobi constant, the zero velocity function and
the maximum Lyapunov exponent. We again find that the suggested planet is indeed possible if assumed to be in a retrograde
orbit, but it is virtually impossible if assumed in a prograde orbit. Its existence is found to be consistent with the deduced
system parameters of the binary components and of the suggested planet, including the associated uncertainty bars given by
observations.

We model the secular evolution of a star’s orbit when it has a nearby binary system. We assume a hierarchical triple system
where the inter-binary distance is small in comparison with the distance to the star. We show that the major secular effect
is precession of the star’s orbit around the binary system’s centre of mass. We explain how we can obtain this precession
rate from the star’s radial velocity data, and thus infer the binary system’s parameters. We show that the secular effect
of a nearby binary system on the star’s radial velocity can sometimes mimic a planet. We analyse the radial velocity data
for ν-Octantis A which has a nearby companion (ν-Octantis B) and we obtain retrograde precession of −0°.86 ± 0°.02 yr−1. We show that if ν-Octantis B was itself a double star, it could mimic a signal with similarities to that previously identified
as a planet of ν-Octantis A. Nevertheless, we need more observations in order to decide in favour of the double-star hypothesis.

We use full available array of radial velocity data, including recently
published HARPS and Keck observatory sets, to characterize the orbital
configuration of the planetary system orbiting GJ876. First, we propose and
describe in detail a fast method to fit perturbed orbital configuration, based
on the integration of the sensitivity equations inferred by the equations of
the original $N$-body problem. Further, we find that it is unsatisfactory to
treat the available radial velocity data for GJ876 in the traditional white
noise model, because the actual noise appears autocorrelated (and demonstrates
non-white frequency spectrum). The time scale of this correlation is about a
few days, and the contribution of the correlated noise is about 2 m/s (i.e.,
similar to the level of internal errors in the Keck data). We propose a
variation of the maximum-likelihood algorithm to estimate the orbital
configuration of the system, taking into account the red noise effects. We
show, in particular, that the non-zero orbital eccentricity of the innermost
planet \emph{d}, obtained in previous studies, is likely a result of
misinterpreted red noise in the data. In addition to offsets in some orbital
parameters, the red noise also makes the fit uncertainties systematically
underestimated (while they are treated in the traditional white noise model).
Also, we show that the orbital eccentricity of the outermost planet is actually
ill-determined, although bounded by $\sim 0.2$. Finally, we investigate
possible orbital non-coplanarity of the system, and limit the mutual
inclination between the planets \emph{b} and \emph{c} orbits by
$5^\circ-15^\circ$, depending on the angular position of the mutual orbital
nodes.

The sudden eccentricity increases discovered by Wisdom (1982) are reproduced in numerical integrations of the planar ecliptic restricted three-body problem, verifying that this phenomenon is real. Mapping derivations are qualitatively reviewed and the maximum Liapunov characteristic exponent and its importance for determining the character of a trajectory are explained. The results of a number of calculations of this exponent using the differential equations for the unaveraged three-body problem are shown and compared to equivalent calculations using a mapping. In all cases the two approaches agree whether the orbits are chaotic or quasiperiodic. The mappings are used to trace out the chaotic zone near the 3/1 commensurability, both in the planar-ecliptic problem and in the three-dimensional elliptic problem. The outer boundary of the chaotic zone coincides with the boundary of the 3/1 Kirkwood gap in the actual distribution of asteroids within the errors of the asteroid orbital elements.

Surfaces of section, plotted in configuration space, have been computed for the motion of the massless particle in the restricted problem of three bodies. Nine mass ratios and a wide variety of Jacobi constants were investigated; over four thousand orbits were computed, about half of which were finally used. The plots of surface of section have been reduced to plots of stability regions, following a method due to Henon (1965a, 1965b, 1966b, 1969). Sample surfaces of section are also given. The complete set of 276 surfaces of section has been published as a report (Jefferys, 1971) and is available upon request from the author.

Mixed-variable symplectic integrators exhibit no long-term accumulation of energy error, beyond that owing to round-off, and
they are substantially faster than conventional N-body algorithms. This makes them the integrator of choice for many problems
in Solar system astronomy. However, in their original formulation, they become inaccurate whenever two bodies approach one
another closely. This occurs because the potential energy term for the pair undergoing the encounter becomes comparable to
the terms representing the unperturbed motion in the Hamiltonian. The problem can be overcome using a hybrid method, in which
the close encounter term is integrated using a conventional integrator, whilst the remaining terms are solved symplectically.
In addition, using a simple separable potential technique, the hybrid scheme can be made symplectic even though it incorporates
a non-symplectic component.

We show that short-term perturbations among massive planets in multiple planet systems can result in radial velocity variations of the central star which dier substantially from velocity variations derived assuming the planets are executing independent Keplerian motions. We discuss two tting methods which can lead to an improved dynamical description of multiple planet systems. In the rst method, the osculating orbital elements are determined via a Levenberg-Marquardt minimization scheme driving an N-body integrator. The second method is an improved analytic model in which orbital elements are allowed to vary according to a simple model for resonant interactions between the planets. Both of these methods can determine the true masses for the planets by eliminating the sin i degeneracy inherent in ts that assume independent Keplerian motions. We apply our tting methods to the GJ 876 radial velocity data (Marcy et al. 2001), and argue that the mass factors for the two planets are likely in the 1.25-2.0 range.

We present an analysis of high precision radial velocity (RV) observations of stars hosting multi-planet systems with Jovian companions. We use dynamical stability constraints and quasi-global methods of optimization. As an illustration, we present new results derived for the RV data of the Sun-like dwarfs HD 155358 and $\tau^1$ Gruis.

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