Stability Of Planetary Orbits Research Articles

Long‐term stability of planetary orbits between Jupiter and Saturn

AbstractWe extend our two previous studies on the existence of stable orbits in the Solar System by examining the domain between Jupiter and Saturn. We place (1) a massless object, (2) a Moon‐mass object, (3) a Mars‐mass object, (4) an Earth‐mass object, and (5) a Uranus‐mass object in the said region. Note that these objects are considered separately in the framework of our simulations. Our goal is to explore the orbital stability of those objects. We employ the Lie‐integration method, which is fast and well established, allowing us to solve the respective differential equations for the ‐body system. Hence, we consider the celestial bodies spanning from Jupiter to Neptune, including the aforementioned test object, the main focus for our model simulations. The integrations indicate that in some models the test objects placed in the region between Jupiter and Saturn reside in that region for more than 600 Myr. Between 5 and 10 au, mean‐motion resonances (MMRs) take place acting upon the test objects akin to simulations of Paper I and II. Our models indicate relatively small differences for the long‐term stability of the five test objects notwithstanding their vastly different masses. Generally, it is found that between and 7.13 au the orbits become unstable mostly within 5 million years and further out, that is, up to au, the duration of stability lengthens to up to hundreds of millions of years.

Update on WASP-19

Tidal interaction between a star and a close-in massive exoplanet causes the planetary orbit to shrink and eventually leads to tidal disruption. Understanding orbital decay in exoplanetary systems is crucial for advancing our knowledge of planetary formation and evolution. Moreover, it sheds light on the broader question of the long-term stability of planetary orbits and the intricate interplay of gravitational forces within stellar systems. Analyzing Transiting Exoplanet Survey Satellite (TESS) data for the ultra-short period gas giant WASP-19, we aim to measure orbital period variations and constrain the stellar tidal quality parameter. For this, we fitted the TESS observations together with two WASP-19 transits observed using the Las Cumbres Observatory Global Telescope (LCOGT) and searched for orbital decay in combination with previously published transit times. As a result, we find a deviation from the constant orbital period at the 7σ level. The orbital period changes at a rate of P˙=−3.7±0.5msyear−1, which translates into a tidal quality factor of Q★′=(7±1)×105. We additionally modeled WASP-19 b’s phase curve using the new TESS photometry and obtained updated values for the planet’s eclipse depth, dayside temperature, and geometric albedo. We estimate an eclipse depth of 520±60 ppm, which is slightly higher than previous estimates and corresponds to a dayside brightness temperature of 2400±60 K and geometric albedo of 0.20±0.04.

On the stability of planetary orbits in binary star systems I. The S-type orbits

Many exoplanets are discovered in binary star systems in internal or in circumbinary orbits. Whether the planet can be habitable or not depends on the possibility to maintain liquid water on its surface, and therefore on the luminosity of its host stars and on the dynamical properties of the planetary orbit. The trajectory of a planet in a double star system can be determined, approximating stars and planet with point masses, by solving numerically the equations of motion of the classical three-body system. In this study, we analyze a large data set of planetary orbits, made up with high precision long integration at varying: the mass of the planet, its distance from the primary star, the mass ratio for the two stars in the binary system, and the eccentricity of the star motion. To simulate the gravitational dynamics, we use a 15th order integration scheme (IAS15, available within the REBOUND framework), that provides an optimal solution for long-term integration. In our data analysis, we evaluate if an orbit is stable or not and also provide the statistics of different types of instability: collisions with the primary or secondary star and planets ejected away from the binary star system. Concerning the stability, we find a significant number of orbits that are only marginally stable, according to the classification introduced by Musielak et al. (Astron. Astrophys. 434:355, 2005). For planets of negligible mass, we estimate the critical semi-major axis a_{c} as a function of the mass ratio and the eccentricity of the binary, in agreement with the results of Holman and Wiegert (Astron. J. 117:621, 1999). However we find that for very massive planets (Super-Jupiters) the critical semi-major axis decrease in some cases by a few percent, compared to cases in which the mass of the planet is negligible.

Resilient habitability of nearby exoplanet systems

ABSTRACT We investigate the possibility of finding Earth-like planets in the habitable zone of 34 nearby FGK-dwarfs, each known to host one giant planet exterior to their habitable zone detected by RV. First we simulate the dynamics of the planetary systems in their present day configurations and determine the fraction of stable planetary orbits within their habitable zones. Then, we postulate that the eccentricity of the giant planet is a result of an instability in their past during which one or more other planets were ejected from the system. We simulate these scenarios and investigate whether planets orbiting in the habitable zone survive the instability. Explicitly we determine the fraction of test particles, originally found in the habitable zone, which remain in the habitable zone today. We label this fraction the resilient habitability of a system. We find that for most systems the probability of planets existing [or surviving] on stable orbits in the habitable zone becomes significantly smaller when we include a phase of instability in their history. We present a list of candidate systems with high resilient habitability for future observations. These are: HD 95872, HD 154345, HD 102843, HD 25015, GJ 328, HD 6718, and HD 150706. The known planets in the last two systems have large observational uncertainties on their eccentricities, which propagate into large uncertainties on their resilient habitability. Further observational constraints of these two eccentricities will allow us to better constrain the survivability of Earth-like planets in these systems.

Relativistic mass due to a dilatant vacuum leads to a quantum reformulation of the relativistic kinetic energy

Relativistic mass change with speed is considered here as the effect of a viscous, dilatant vacuum, whose apparent viscosity is related to the Lorentz factor. Transient solid-like vacuum due to shear stress is presented as the reason why vacuum prevents the speed of massive objects from being indefinitely increased. Such a vacuum – that in a previous study allowed to exactly calculate the Pioneer anomaly, Mercury’s perihelion precession, and was shown to be compatible with stable planetary orbits – leads here to a quantum formula for the relativistic kinetic energy. A formula which distinguishes between the case of accelerated charges in a vacuum, for which a Stokes–Einstein radius comes into play, and the case of accelerated macroscopic bodies, for which the quantum potential term vanishes. In this way, incidentally, one obtains again correct results for the Pioneer 10, confirming the role of vacuum’s viscous force. This description of a quantum mechanism underlying the relativistic kinetic energy may be also helpful in constructing a theory of quantum relativity and may even tell us more about the interactions of matter with the Higgs field and the dark sector: two issues which can be themselves linked to a dilatant vacuum.

Polar orbits around binary stars

Oks proposes the existence of a new class of stable planetary orbits around binary stars, in the shape of a helix on a conical surface whose axis of symmetry coincides with the interstellar axis, and rotates with the same orbital frequency as the binary pair. We show that this claim relies on the inappropriate use of an effective potential that is only applicable when the stars are held motionless. We also present numerical evidence that the only planetary orbits whose planes are initially orthogonal to the interstellar axis that remain stable on the time scale of the stellar orbit are ordinary polar orbits around one of the stars, and that the perturbations due to the binary companion do not rotate the plane of the orbit to maintain a fixed relationship with the axis.

Fitting Formulae and Constraints for the Existence of S-type and P-type Habitable Zones in Binary Systems

Abstract We derive fitting formulae for the quick determination of the existence of S-type and P-type habitable zones (HZs) in binary systems. Based on previous work, we consider the limits of the climatological HZ in binary systems (which sensitively depend on the system parameters) based on a joint constraint encompassing planetary orbital stability and a habitable region for a possible system planet. Additionally, we employ updated results on planetary climate models obtained by Kopparapu and collaborators. Our results are applied to four P-type systems (Kepler-34, Kepler-35, Kepler-413, and Kepler-1647) and two S-type systems (TrES-2 and KOI-1257). Our method allows us to gauge the existence of climatological HZs for these systems in a straightforward manner with detailed consideration of the observational uncertainties. Further applications may include studies of other existing systems as well as systems to be identified through future observational campaigns.

Dynamics of a Probable Earth-mass Planet in the GJ 832 System

Abstract The stability of planetary orbits around the GJ 832 star system, which contains inner (GJ 832c) and outer (GJ 832b) planets, is investigated numerically and a detailed phase-space analysis is performed. Special attention is given to the existence of stable orbits for a planet less than 15 M ⊕ that is injected between the inner and outer planets. Thus, numerical simulations are performed for three and four bodies in elliptical orbits (or circular for special cases) by using a large number of initial conditions that cover the selected phase-spaces of the planet’s orbital parameters. The results presented in the phase-space maps for GJ 832c indicate the least deviation of eccentricity from its nominal value, which is then used to determine its inclination regime relative to the star–outer planet plane. Also, the injected planet is found to display stable orbital configurations for at least one billion years. Then, the radial velocity curves based on the signature from the Keplerian motion are generated for the injected planets with masses 1 M ⊕ to 15 M ⊕ in order to estimate their semimajor axes and mass limits. The synthetic RV signal suggests that an additional planet of mass ≤15 M ⊕ with a dynamically stable configuration may be residing between 0.25 and 2.0 au from the star. We have provided an estimated number of RV observations for the additional planet that is required for further observational verification.

Replenishment of the comet Oort Cloud during the outward migration of Uranus and Neptune

In the past, the formation process of the distant comet Oort Cloud was simulated assuming the ejecting giant planets in their current orbits and with their current masses. Today, it is however known that the planets formed in a tighter configuration and the comet cloud could be replenished, mostly, in the era of their radial migration. We simulate the Oort-Cloud formation during the planetary migration in course to see if the formation efficiency is larger than that for the stable planetary orbits. As well, we try to answer the question if the ratio of the Oort-Cloud and scattered-disk populations is larger in the case of migration in course to alleviate the problem of too a low ratio in the past simulations. The formation efficiency in the simulations with migrating planets appears to be larger than in that with the planets in stable orbits, but only for the first 100Myr. Later, it practically disappears. The simulations with the migrating planets result in a higher ratio of the Oort-Cloud and scattered-disk populations. However, this enhancement is only moderate and cannot explain the problem of too a low ratio of both populations.

HIGH-PRECISION RADIO AND INFRARED ASTROMETRY OF LSPM J1314+1320AB. I. PARALLAX, PROPER MOTIONS, AND LIMITS ON PLANETS

ABSTRACT We present multi-epoch astrometric radio observations with the Very Long Baseline Array (VLBA) of the young ultracool-dwarf binary LSPM J1314+1320AB. The radio emission comes from the secondary star. Combining the VLBA data with Keck near-infrared adaptive-optics observations of both components, a full astrometric fit of parallax (π abs = 57.975 ± 0.045 mas, corresponding to a distance of d = 17.249 ± 0.013 pc), proper motion (μ αcos δ = −247.99 ± 0.10 mas yr−1, μ δ = −183.58 ± 0.22 mas yr−1), and orbital motion is obtained. Despite the fact that the two components have nearly identical masses to within ±2%, the secondary’s radio emission exceeds that of the primary by a factor of ≳30, suggesting a difference in stellar rotation history, which could result in different magnetic field configurations. Alternatively, the emission could be anisotropic and beamed toward us for the secondary but not for the primary. Using only reflex motion, we exclude planets of mass 0.7–10 M jup with orbital periods of 600–10 days, respectively. Additionally, we use the full orbital solution of the binary to derive an upper limit for the semimajor axis of 0.23 au for stable planetary orbits within this system. These limits cover a parameter space that is inaccessible with, and complementary to, near-infrared radial velocity surveys of ultracool dwarfs. Our absolute astrometry will constitute an important test for the astrometric calibration of Gaia.

STABLE CONIC-HELICAL ORBITS OF PLANETS AROUND BINARY STARS: ANALYTICAL RESULTS

Studies of planets in binary star systems are especially important because it was estimated that about half of binary stars are capable of supporting habitable terrestrial planets within stable orbital ranges. One-planet binary star systems (OBSS) have a limited analogy to objects studied in atomic/molecular physics: one-electron Rydberg quasimolecules (ORQ). Specifically, ORQ, consisting of two fully stripped ions of the nuclear charges Z and Z' plus one highly excited electron, are encountered in various plasmas containing more than one kind of ion. Classical analytical studies of ORQ resulted in the discovery of classical stable electronic orbits with the shape of a helix on the surface of a cone. In the present paper we show that despite several important distinctions between OBSS and ORQ, it is possible for OBSS to have stable planetary orbits in the shape of a helix on a conical surface, whose axis of symmetry coincides with the interstellar axis; the stability is not affected by the rotation of the stars. Further, we demonstrate that the eccentricity of the stars' orbits does not affect the stability of the helical planetary motion if the center of symmetry of the helix is relatively close to the star of the larger mass. We also show that if the center of symmetry of the conic-helical planetary orbit is relatively close to the star of the smaller mass, a sufficiently large eccentricity of stars' orbits can switch the planetary motion to the unstable mode and the planet would escape the system. We demonstrate that such planets are transitable for the overwhelming majority of inclinations of plane of the stars' orbits (i.e., the projections of the planet and the adjacent start on the plane of the sky coincide once in a while). This means that conic-helical planetary orbits at binary stars can be detected photometrically. We consider, as an example, Kepler-16 binary stars to provide illustrative numerical data on the possible parameters and the stability of the conic-helical planetary orbits, as well as on the transitability. Then for the general case, we also show that the power of the gravitational radiation due to this planet can be comparable or even exceed the power of the gravitational radiation due to the stars in the binary. This means that in the future, with a progress of gravitational wave detectors, the presence of a planet in a conic-helical orbit could be revealed by the noticeably enhanced gravitational radiation from the binary star system.

S-TYPE ANDP-TYPE HABITABILITY IN STELLAR BINARY SYSTEMS: A COMPREHENSIVE APPROACH. I. METHOD AND APPLICATIONS

A comprehensive approach is provided to the study of both S-type and P-type habitability in stellar binary systems, which in principle can also be expanded to systems of higher order. P-type orbits occur when the planet orbits both binary components, whereas in case of S-type orbits the planet orbits only one of the binary components with the second component considered a perturbator. The selected approach encapsulates a variety of different aspects, which include: (1) The consideration of a joint constraint including orbital stability and a habitable environment for a putative system planet through the stellar radiative energy fluxes ("radiative habitable zone"; RHZ) needs to be met. (2) The treatment of conservative, general and extended zones of habitability for the various systems as defined for the Solar System and beyond. (3) The providing of a combined formalism for the assessment of both S-type and P-type habitability; in particular, mathematical criteria are presented for which kind of system S-type and P-type habitability is realized. (4) Applications of the attained theoretical approach to standard (theoretical) main-sequence stars. In principle, five different cases of habitability are identified, which are: S-type and P-type habitability provided by the full extent of the RHZs; habitability, where the RHZs are truncated by the additional constraint of planetary orbital stability (referred to as ST and PT-type, respectively); and cases of no habitability at all. Regarding the treatment of planetary orbital stability, we utilize the formulae of Holman & Wiegert (1999) [AJ 117, 621], as also used by previous studies. In this work we focus on binary systems in circular orbits. Future applications will also consider binary systems in elliptical orbits and provide thorough comparisons to other methods and results given in the literature.

Application of the Titius–Bode Rule to the 55 Cancri System: Tentative Prediction of a Possibly Habitable Planet

Abstract Following the notion that the Titius–Bode rule (TBR) may also be applicable to some extrasolar planetary systems, although this number could be relatively small, it is applied to 55 Cancri, which is a G-type main-sequence star currently known to host five planets. Following a concise computational process, we tentatively identified four new hypothetical planetary positions, given as 0.081, 0.41, 1.51, and 2.95 AU from the star. The likelihood that these positions are occupied by real existing planets is significantly enhanced for the positions of 1.51 and 2.95 AU in view of previous simulations on planet formation and planetary orbital stability. For example, Raymond, Barnes, and Gorelick (2008, ApJ, 689, 478) argued that additional planets would be possible between 55 Cnc f and 55 Cnc d, which would include planets situated at 1.51 and 2.95 AU. If two additional planets are assumed to exist between 55 Cnc f and 55 Cnc d, the deduced domains of stability would be given as 1.3–1.6 and 2.2–3.3 AU. The possible planet near 1.5 AU appears to be located at the outskirts of the stellar habitable zone, which is, however, notably affected by the stellar parameters as well as the adopted model of circumstellar habitability. We also computed the distance of the next possible outer planet in the 55 Cnc system, which, if existing, is predicted to be located between 10.9 and 12.2 AU, which is consistent with orbital stability constraints. The inherent statistical significance of the TBR was evaluated following a method by Lynch (2003, MNRAS, 341, 1174). Yet it is up to future planetary search missions to verify or falsify the applicability of the TBR to the 55 Cnc system, and to obtain information on additional planets, if existing.

The instability transition for the restricted 3-body problem

Aims. We establish a criterion for the stability of planetary orbits in stellar binary systems by using Lyapunov exponents and power spectra for the special case of the circular restricted 3-body problem (CR3BP). The criterion augments our earlier results given in the two previous papers of this series where stability criteria have been developed based on the Jacobi constant and the hodograph method.

ON THE REALITY OF THE SUGGESTED PLANET IN THE ν OCTANTIS SYSTEM

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.

Stability of planetary orbits in binary systems

Stability of planetary orbits in binary systems

Stringent Criteria for Stable and Unstable Planetary Orbits in Stellar Binary Systems

The existence of planets in stellar binary (and higher order) systems has now been confirmed by many observations. The stability of planetary orbits in these systems has been extensively studied, but no precise stability criteria have so far been introduced. Therefore, there is an urgent need for developing stringent mathematical criteria that allow us to precisely determine whether a planetary orbit in a binary system is stable or unstable. In this Letter, such criteria are defined using the concept of Jacobi's integral and Jacobi's constant. These criteria are used to contest previous results on planetary orbital stability in binary systems.

A numerical study of the 2:1 planetary resonance

We numerically explore the long-term stability of planetary orbits locked in a 2:1 mean motion resonance for a wide range of planetary mass ratios and orbital parameters. Our major tool is Laskar's frequency map analysis. Regions of low diffusion rate are outlined in a phase space defined by the two planetary eccentricities and the libration amplitude of a critical resonance argument. Resonant systems that are dynamically stable on a long timescale must lie within these regions. The resonance locking between planets in high eccentric orbits may be destroyed by mutual close encounters. We discuss various dynamical protection mechanisms related to the resonant configuration, among which is the well-known apsidal corotation. In the case of moderate-to-low eccentricities, we find that apsidal circulators, little discussed till now, are very common among stable orbits. We also map the different types of resonant behaviour predicted by analytical theories in the phase space.

Stability of planetary orbits in binary systems

Stability of S-type and P-type planetary orbits in binary systems of different mass and separation ratios is inves- tigated. Criteria for stable, marginally stable and unstable planetary orbits are specified. These criteria are used to determine regions of stability of planetary orbits in different binary systems with Jupiter-type planets. The obtained results show that the regions of stability for S-type orbits depend on the distance ratio between the star and planet, and the stellar companions, in the range of 0.22 and 0.46, depending on the mass ratio. For P-type orbits, the regions of stability also depend on that distance ratio, in the range of 1.75 and 2.45, again depending on the the mass ratio. Applications of these results to three observed binary systems with giant planets, namely, τ Boo, HD 195019 and GJ 86, show that the orbits of the giant planets in those systems can be classified as stable, as expected.

Testing Protoplanetary Disk Alignment in Young Binaries

We present K-band (2.2 � m) imaging polarimetry that resolves 19 T Tauri binary and multiple systems in the Taurus-Auriga and Scorpius-Ophiuchus star-forming regions. We observed systems with projected separations 1B5–7B 2( � 200–1000 AU) in order to determine the relative orientation of the circumstellar disks in each binary system. Scattered light from these disks is polarized, allowing us to deduce the position angle of the disk on the sky from the position angle of polarization even though our observations do not resolve the disks themselves. We detected measurable polarization (typically 0.5%–2%, with typical uncertainty 0.1%) from both stars in 14 of the systems observed. In eight of the nine binary systems, the two stars’ polarization position angles are within 30 � of each other, inconsistent with random orientations. In contrast, the five triple and quadruple systems appear to have random disk orientations when comparing the polarization position angles of the widest pair in the system; the close pairs are unresolved in all but one system. Our observations suggest that disks in wide (200–1000 AU) binaries are aligned with each other within P20 � but not perfectly coplanar. However, we cannot conclusively rule out random relative disk orientations if the observed polarizations are significantly contaminated by interstellar polarization. Even in the presence of interstellar polarization our observations securely exclude coplanar disks. These results provide constraints on possible binary formation mechanisms if the observed orientations are primordial. On the other hand, models of disk-binary interactions indicate that the disks may have had time to decrease their relative inclinations since formation. If the common orientation of the disks in these binaries is a tracer of the binary orbital plane, then our results also have significance for the stability of planetary orbits, suggesting that planetary systems in wide binaries should be stable over 10 9 yr timescales. Subject headings: circumstellar matter — planetary systems: formation — planetary systems: protoplanetary disks — polarization — stars: formation — stars: pre–main-sequence