Modern applications of celestial mechanics include the study of closely packed systems of exoplanets, circumbinary planetary systems, binary-binary interactions in star clusters and the dynamics of ...stars near the Galactic centre. While developments have historically been guided by the architecture of the Solar System, the need for more general formulations with as few restrictions on the parameters as possible is obvious. Here, we present clear and concise generalizations of two classic expansions of the three-body disturbing function, simplifying considerably their original form and making them accessible to the non-specialist.
Governing the interaction between the inner and outer orbits of a hierarchical triple, the disturbing function in its general form is the conduit for energy and angular momentum exchange and as such, governs the secular and resonant evolution of the system and its stability characteristics. Focusing here on coplanar systems, the first expansion is one in the ratio of inner to outer semimajor axes and is valid for all eccentricities, while the second is an expansion in eccentricity and is valid for all semimajor axis ratios, except for systems in which the orbits cross (this restriction also applies to the first expansion). Our generalizations make both formulations valid for arbitrary mass ratios. The classic versions of these appropriate to the restricted three-body problem are known as Kaula's expansion and the literal expansion, respectively. We demonstrate the equivalence of the new expansions, identifying the role of the spherical harmonic order m in both and its physical significance in the three-body problem, and introducing the concept of principal resonances.
Several examples of the accessibility of both expansions are given including resonance widths, and the secular rates of change of the elements. Results in their final form are gathered together at the end of the paper for the reader mainly interested in their application, including a guide for the choice of expansion.
ABSTRACT Explaining the origin and evolution of exoplanetary hot Jupiters remains a significant challenge. One possible mechanism for the production of hot Jupiters is planet-planet interactions, ...which produce them from planets born far from their host stars but near their dynamical stability limits. In the much more likely case of planets born far from their dynamical stability limits, can hot Jupiters be formed in star clusters? Our N-body simulations answer this question in the affirmative, and show that hot Jupiter formation is not a rare event, occurring in ∼1% of star cluster planetary systems. We detail three case studies of the dynamics-induced births of hot Jupiters on highly eccentric orbits that can only occur inside star clusters. The hot Jupiters' orbits bear remarkable similarities to those of some of the most extreme exoplanets known: HAT-P-32b, HAT-P-2b, HD 80606b, and GJ 876d. If stellar perturbations formed these hot Jupiters, then our simulations predict that these very hot inner planets are often accompanied by much more distant gas giants in highly eccentric orbits.
Of the 14 transiting extrasolar planetary systems for which radii have been measured, at least three appear to be considerably larger than theoretical estimates suggest. It has been proposed by ...Bodenheimer, Lin & Mardling that undetected companions acting to excite the orbital eccentricity are responsible for these oversized planets, as they find new equilibrium radii in response to being tidally heated. In the case of HD 209458, this hypothesis has been rejected by some authors because there is no sign of such a companion at the 5 ms−1 level, and because it is difficult to say conclusively that the eccentricity is non-zero. Transit timing analysis as well as a direct transit search has further constrained the existence of very short period companions, especially in resonant orbits. Whether or not a companion is responsible for the large radius of HD 209458b, almost certainly some short-period systems have companions which force their eccentricities to non-zero values. This paper is dedicated to quantifying this effect. The eccentricity of a short-period planet will only be excited as long as its (non-resonant) companion's eccentricity is non-zero. Here, we show that the latter decays on a time-scale which depends on the structure of the interior planet, a time-scale which is often shorter than the lifetime of the system. This includes Earth-mass planets in the habitable zones of some stars. We determine which configurations are capable of sustaining significant eccentricity for at least the age of the system, and show that these include systems with companion masses as low as a fraction of an Earth mass. The orbital parameters of such companions are consistent with recent calculations which show that the migration process can induce the formation of low-mass planets external to the orbits of hot Jupiters. Systems with inflated planets are therefore good targets in the search for terrestrial planets.
The recent discovery of a transiting short-period planet on a slightly non-circular orbit with a massive highly eccentric companion orbiting the star HAT-P-13 offers the possibility of probing the ...structure of the short-period planet. The ability to do this relies on the system being in a quasi-equilibrium state in the sense that the eccentricities are constant on the usual secular time-scale (typically, a few thousand years), and decay on a time-scale which is much longer than the age of the system. Since the equilibrium eccentricity is effectively a function only of observable system parameters and the unknown Love number of the short-period planet, the latter can be determined with accurate measurements of the planet's eccentricity and radius. However, this analysis relies on the assumption that the system is coplanar, a situation which seems unlikely given the high eccentricity of the outer planet. Here we generalize our recent analysis of this fixed-point phenomenon to mutually inclined systems in which the outer body dominates the total angular momentum, and show that (1) the fixed point of coplanar systems is replaced by a limit cycle in eb–η space, where eb is the eccentricity of the inner planet and η is the angle between the periapse lines, with the average value of eb, e(av)b, decreasing and its amplitude of variation increasing with increasing mutual inclination. This behaviour significantly reduces the ability to unambiguously determine the Love number of the short-period planet if the mutual inclination is higher than around 10°. (2) We show that for Q-values less than 106, the HAT-P-13 system cannot have a mutual inclination between 54° and 126° because Kozai oscillations coupled with tidal dissipation would act to quickly move the inclination outside this range, and (3) that the behaviour of retrograde systems is the mirror image of that for prograde systems in the sense that (almost) identical limit cycles exist for a given mutual inclination and π minus this value. (4) We derive a relationship between e(av)b, the equilibrium radius of the short-period planet, its Q-value and its core mass, and show that given current estimates of eb and the planet radius, as well as the lower bound placed on the Q-value by the decay rate of e(av)b, the HAT-P-13 system is likely to be close to prograde coplanar, or have a mutual inclination between 130° and 135°. Lower rather than higher core masses are favoured. (5) An expression for the time-scale for decay of the mutual inclination is derived, revealing that it evolves towards a non-zero value as long as eb > 0 on a time-scale which is much longer than the age of the system. (6) We conclude with a scattering scenario for the origin of the HAT-P-13 system and show that almost identical initial conditions can result in significantly different outer planet eccentricities, stellar obliquities and planet radii. The implications for systems with high stellar obliquities such as HAT-P-7 and WASP-17 are briefly discussed.
ABSTRACT
The dynamical stability of quadruple-star systems has traditionally been treated as a problem involving two ‘nested’ triples which constitute a quadruple. In this novel study, we employed a ...machine learning algorithm, the multilayer perceptron (MLP), to directly classify 2 + 2 and 3 + 1 quadruples based on their stability (or long-term boundedness). The training data sets for the classification, comprised of 5 × 105 quadruples each, were integrated using the highly accurate direct N-body code mstar. We also carried out a limited parameter space study of zero-inclination systems to directly compare quadruples to triples. We found that both our quadruple MLP models perform better than a ‘nested’ triple MLP approach, which is especially significant for 3 + 1 quadruples. The classification accuracies for the 2 + 2 MLP and 3 + 1 MLP models are 94 and 93 per cent, respectively, while the scores for the ‘nested’ triple approach are 88 and 66 per cent, respectively. This is a crucial implication for quadruple population synthesis studies. Our MLP models, which are very simple and almost instantaneous to implement, are available on Github, along with python3 scripts to access them.
ABSTRACT
We present two approaches to determine the dynamical stability of a hierarchical triple-star system. The first is an improvement on the Mardling–Aarseth stability formula from 2001, where we ...introduce a dependence on inner orbital eccentricity and improve the dependence on mutual orbital inclination. The second involves a machine learning approach, where we use a multilayer perceptron (MLP) to classify triple-star systems as ‘stable’ and ‘unstable’. To achieve this, we generate a large training data set of 106 hierarchical triples using the N-body code MSTAR. Both our approaches perform better than previous stability criteria, with the MLP model performing the best. The improved stability formula and the machine learning model have overall classification accuracies of $93{{\ \rm per\ cent}}$ and $95{{\ \rm per\ cent}}$ respectively. Our MLP model, which accurately predicts the stability of any hierarchical triple-star system within the parameter ranges studied with almost no computation required, is publicly available on Github in the form of an easy-to-use python script.
Abstract
The distant scattered disk is a vast population of trans-Neptunian minor bodies that orbit the Sun on highly elongated, long-period orbits. The orbital stability of scattered-disk objects ...(SDOs) is primarily controlled by a single parameter—their perihelion distance. While the existence of a perihelion boundary that separates chaotic and regular motion of long-period orbits is well established through numerical experiments, its theoretical basis as well as its semimajor axis dependence remain poorly understood. In this work, we outline an analytical model for the dynamics of distant trans-Neptunian objects and show that the orbital architecture of the scattered disk is shaped by an infinite chain of exterior 2:
j
resonances with Neptune. The widths of these resonances increase as the perihelion distance approaches Neptune’s semimajor axis, and their overlap drives chaotic motion. Within the context of this theoretical picture, we derive an analytic criterion for instability of long-period orbits, and demonstrate that rapid dynamical chaos ensues when the perihelion drops below a critical value, given by
q
crit
=
a
N
ln
(
(
24
2
/
5
)
(
m
N
/
M
⊙
)
a
/
a
N
5
/
2
)
1
/
2
. This expression constitutes an analytic boundary between the “detached” and actively “scattering” subpopulations of distant trans-Neptunian minor bodies. Additionally, we find that within the stochastic layer, the Lyapunov time of SDOs approaches the orbital period, and show that the semimajor axis diffusion coefficient is approximated by
a
∼
(
8
/
(
5
π
)
)
(
m
N
/
M
⊙
)
M
⊙
a
N
exp
−
q
/
a
N
2
/
2
. We confirm our results with direct
N
-body simulations and highlight the connections between scattered-disk dynamics and the Chirikov Standard Map. Implications of our results for the long-term evolution of minor bodies in the distant solar system are discussed.
Abstract
We introduce self-consistent spin, tidal, and dynamical equations of motion into
REBOUNDx
, a library of additional effects for the popular
N
-body integrator
REBOUND
. The equations of ...motion used are derived from the constant time lag approximation to the equilibrium tide model of tidal friction. These effects will allow the study of a variety of systems of which the full dynamical picture cannot be encapsulated by point particle dynamics. We provide several test cases and benchmark the code’s performance against analytic predictions. The open-source code is available in the most recent release of
REBOUNDx
.
Tidal interactions in star cluster simulations Mardling, Rosemary A.; Aarseth, Sverre J.
Monthly notices of the Royal Astronomical Society,
03/2001, Letnik:
321, Številka:
3
Journal Article
Recenzirano
Odprti dostop
We describe the implementation of tidal circularization of binaries in an N-body code for star cluster simulations. The first part contains the theoretical framework for normal and chaotic tidal ...interactions, including capture from hyperbolic orbits. This formulation yields convenient expressions which are used to modify the binary elements. Stars are represented as polytropes, with a time-dependent effective polytropic index calculated for evolving stars. Stellar evolution is treated using a fast look-up table for stellar masses and radii. This gives a consistent astrophysical description of open clusters containing a significant proportion of primordial binaries with a wide range of masses and periods. An analytic expression for the chaos boundary for arbitrary mass ratio and polytropic indices is presented. We provide detailed correction procedures for tidal circularization and chaotic motion for perturbed binaries which are studied by the classical Kustaanheimo—Stiefel two-body regularization method and also outline a similar treatment for multiple regularization of temporary subsystems involving 3–6 members. Strong interactions in the latter lead to the formation of chaotic binaries and stable hierarchical systems in which the eccentricity of the inner binary may be subject to systematic changes on relatively short time-scales. Finally, we illustrate the effect of tidal circularization by presenting some results of a realistic cluster simulation involving 104 single stars and 500 primordial binaries.
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