Mutual Inclination Research Articles

Dynamical Structures under Nonrestricted Hierarchical Planetary Systems with Different Mass Ratios

Secular dynamics have been extensively studied in both the inner and outer restricted hierarchical three-body systems. In the inner restricted problem, the quadrupole-order resonance (i.e., the well-known Kozai resonance) causes large coupled oscillations of eccentricity and inclination when the maximum inclination is higher than 39.2°, and the octupole-order resonance leads to the behavior of orbital flips. In the outer restricted problem, the behavior of orbital flips is due to the quadrupole-order resonance. Secular dynamics under the inner and outer restricted systems are distinctly different. The mass ratio of inner and outer bodies could change the ratio of circular orbital angular momenta β, which significantly influences dynamical structures of the system. But this influence is still unclear. In this paper, we focus on nonrestricted hierarchical planetary systems where β > 1 and investigate the secular dynamics by changing mass ratios. Dynamical structures are systematically explored from four aspects: periodic orbits, secular resonances, orbital flips, and chaos detection. We find that (a) it tends to lead to more bifurcations in the host family of prograde periodic orbits associated with Kozai resonance with smaller β; (b) with the decrease of β, fewer orbits inside the octupole-order resonance can realize flip; (c) for given initial conditions, the forbidden region appears in the retrograde region and becomes larger as β decreases, meaning that the mutual inclination cannot reach a very high value if β is small; and (d) chaotic orbits are distributed in the low-eccentricity, high-inclination region when β > 1.

The effects of general relativity on close-in radial-velocity-detected exosystems

Aims. The detection of the first exoplanet around a solar-type star revealed the existence of close-in planets. Several of these close-in planets are part of multi-planet systems. For systems detected via the radial velocity (RV) method, we lack information on the mutual inclination of the orbital planes. The aim of this work is to study the long-term stability of RV-detected two-planet systems with close-in planets and identify possible three-dimensional configurations for these systems that are compatible with observations. To do so, we focused on the protective mechanism of the Lidov-Kozai (LK) secular resonance and studied the effects of general relativity (GR) on long-term evolution. Methods. By means of an analytical study based on a high-order secular Hamiltonian expansion in the eccentricities and inclinations, we first identified ranges of values for the orbital and mutual inclinations that are compatible with the presence of the LK resonance in the purely gravitational case. Then, adding the secular contribution of the relativistic corrections exerted by the central star on the inner planet, namely the advance of its pericenter precession, we analysed the outcomes of the two sets of simulations. We compared our results to analytical estimates to determine the importance of GR effects. Results. We find that for the majority of the systems considered, GR strongly affects the dynamics of the system and, most of the time, voids the LK resonance, as observed for GJ 649, GJ 832, HD 187123, HD 190360, HD 217107, and HD 47186. The long-term stability of these systems is then possible whatever the mutual inclination of the orbits. On the contrary, for GJ 682, HD 11964, HD 147018, and HD 9446, the LK resonant region in the parameter space of the orbital and mutual inclinations is left (almost) unchanged when GR effects are considered, and consequently their long-term stability is only possible if the mutual inclination of the orbits is low or if the systems are in the LK regime with a high mutual inclination.

Kepler Multitransiting System Physical Properties and Impact Parameter Variations

We fit a dynamical model to Kepler systems that contain four or more transiting planets using the analytic method AnalyticLC and obtain physical and orbital parameters for 101 planets in 23 systems, of which 95 are of mass significance better than 3σ, and 46 are without previously reported mass constraints or upper limits. In addition, we compile a list of 71 Kepler objects of interest that display significant transit impact parameter variations (TbVs), complementing our previously published work on two- and three-transiting-planet systems. Together, these works include the detection of significant TbV signals of 130 planets, which is, to our knowledge, the largest catalog of this type to date. The results indicate that the typical detectable TbV rate in the Kepler population is of order 10−2 yr−1 and that rapid TbV rates (≳0.05 yr−1) are observed only in systems that contain a transiting planet with an orbital period less than ∼20 days. The observed TbV rates are only weakly correlated with orbital period within Kepler’s ≲100-day-period planets. If this extends to longer periods, it implies a limit on the utility of the transit technique for long-period planets. The TbVs we find may not be detectable in direct impact parameter measurements, but rather are inferred from the full dynamics of the system, encoded in all types of transit variations. Finally, we find evidence that the mutual inclination distribution is qualitatively consistent with the previously suggested angular momentum deficit model using an independent approach.

Dynamical Architectures of S-type Transiting Planets in Binaries. I. Target Selection Using Hipparcos and Gaia Proper Motion Anomalies* * Also referred to as Hipparcos and Gaia astrometric accelerations.

The effect of stellar multiplicity on planetary architecture and orbital dynamics provides an important context for exoplanet demographics. We present a volume-limited catalog of up to 300 pc of 66 stars hosting planets and planet candidates from Kepler, K2, and TESS with significant Hipparcos-Gaia proper motion anomalies, which indicates the presence of companions. We assess the reliability of each transiting planet candidate using ground-based follow-up observations, and find that the TESS Objects of Interest (TOIs) with significant proper anomalies show nearly four times more false positives due to eclipsing binaries compared to TOIs with marginal proper anomalies. In addition, we find tentative evidence that orbital periods of planets orbiting TOIs with significant proper anomalies are shorter than those orbiting TOIs without significant proper anomalies, consistent with the scenario that stellar companions can truncate planet-forming disks. Furthermore, TOIs with significant proper anomalies exhibit lower Gaia differential velocities in comparison to field stars with significant proper anomalies, suggesting that planets are more likely to form in binary systems with low-mass substellar companions or stellar companions at wider separation. Finally, we characterize the three-dimensional architecture of LTT 1445 ABC using radial velocities, absolute astrometry from Gaia and Hipparcos, and relative astrometry from imaging. Our analysis reveals that LTT 1445 is a nearly flat system, with a mutual inclination of ∼2.°88 between the orbit of BC around A and that of C around B. This coplanarity may explain why multiple planets around LTT 1445 A survive in the dynamically hostile environments of this system.

Searching for low-mass companions at small separations in transition disks with aperture masking interferometry

Context. Transition disks have large central cavities that have been resolved by imaging surveys during recent years. Cavities and other substructures in circumstellar disks are often interpreted as signposts to massive companions. Detecting companions at small angular separations is challenging with coronagraphic imaging observations. Aims. We aim to search for stellar and substellar companions in the central regions of transition disks. Such companions could be responsible for the large dust-depleted cavities. We want to determine if these disks might be circumbinary in their nature, similar to the HD 142527 system. Methods. We observed four systems, HD 100453, HD 100546, HD 135344 B, and PDS 70, with the sparse aperture masking mode of VLT/SPHERE, also leveraging the star-hopping method with the adaptive optics system. We extracted the complex visibilities and bispectra from the H2 and H3 imaging data. A binary model was fit to the closure phases to search for companions and estimate detection limits. For validation, we also analyzed four archival datasets of HD 142527 and inferred the orbital elements and atmospheric parameters of its low-mass stellar companion. Results. We have not detected any significant point sources in the four observed systems. With a contrast sensitivity of ≈0.004, we can rule out stellar companions down to ≈2 au and partially explore the substellar regime at separations ≳3–5 au. The analysis of HD 142527 B revealed that its projected orbit is aligned with dust features in the extended inner disk and that the mutual inclination with the outer disk is close to coplanar for one of the two solutions. Atmospheric modeling confirms the low-gravity and slightly reddened spectral appearance (Teff ≈ 3300 K, log 𝑔 ≈ 3.7, and AV ≈ 0.7). The inferred and derived bulk parameters (log L*/L0 ≈ −0.65, M* ≈ 0.4 M⊙, and R* ≈ 1.46 R⊙) are in agreement with dynamical constraints and evolutionary tracks. Conclusions. In contrast to HD 142527, we find no evidence that a close-in stellar companion is responsible for the resolved disk features of HD 100453, HD 100546, HD 135344 B, and PDS 70. Instead of a dynamical effect by a stellar companion, the formation of giant planets or even low-mass brown dwarfs could be shaping the innermost environment (≲20 au) of these circumstellar disks, as is the case with the planetary system of PDS 70.

Photodynamical Modeling of the Compact, Multiply Eclipsing Systems KIC 5255552, KIC 7668648, KIC 10319590, and EPIC 220204960

We present photodynamical models of four eclipsing binary systems that show strong evidence of being members of higher-order multiple systems via their strong eclipse timing variations and/or via the presence of extra eclipse events. Three of these systems are from the main Kepler mission, and the other is from the K2 mission. We provide some ground-based radial velocities measurements for the three Kepler systems and make use of recent light curves from the TESS mission. Our sample consists of two 2 + 1 systems and two 2 + 2 systems. The first 2 + 1 system, KIC 7668648, consists of an eclipsing binary (Pbin = 27.8 days) with late-type stars (M1=0.8403±0.0090M⊙, R1=1.0066±0.0036R⊙ and M2=0.8000±0.0085M⊙, R2=0.8779±0.0032R⊙) with a low-mass star (M3=0.2750±0.0029M⊙, R3=0.2874±0.0010R⊙) on a roughly coplanar outer orbit (P3=208 days). There are several eclipse events involving the third star that allow for the precise determination of the system parameters. The second 2 + 1 system, KIC 10319590, consists of a binary (Pbin=21.3 days) with late-type stars (M1=1.108±0.043M⊙, R1=1.590±0.019R⊙ and M2=0.743±0.023M⊙, R2=0.7180±0.0086R⊙) that stopped eclipsing about a third of the way into the nominal Kepler mission. We show here that the third star in this system is a Sun-like star (M3=1.049±0.038M⊙, R3=1.39±0.11R⊙) on an inclined outer orbit (P3=456 days). In this case, there are no extra eclipse events. We present the first comprehensive solution for KIC 5255552 and demonstrate that it is a 2 + 2 system consisting of an eclipsing binary (Pbin,1=32.5 days) with late-type stars (M1=0.950±0.018M⊙, R1=0.9284±0.0063R⊙ and M2=0.745±0.014M⊙, R2=0.6891±0.0051R⊙) paired with a non-eclipsing binary (Pbin,2=33.7 days) with somewhat lower-mass stars (M3=0.483±0.010M⊙, R3=0.4640±0.0036R⊙ and M4=0.507±0.010M⊙, R4=0.4749±0.0031R⊙). The two binaries, which have nearly coplanar orbits, orbit their common barycenter on a roughly aligned outer orbit (Pout=878 days). There are extra eclipse events involving the component stars of the non-eclipsing binary, which leads to relatively small uncertainties in the system parameters. The second 2 + 2 system, EPIC 220204960, consists of a pair of eclipsing binaries (Pbin,2=13.3 days, Pbin,2=14.4 days) that both consist of two low-mass stars (M1=0.54M⊙, R1=0.46R⊙, M2=0.46M⊙, R2=0.37R⊙ and M3=0.38M⊙, R3=0.40R⊙, M4=0.38M⊙, R4=0.37R⊙) that orbit their common barycenter on a poorly determined outer orbit. Because of the relatively short time span of the observations (≈80 days for the photometry and ≈70 days for the radial velocity measurements), the masses and radii of the four stars in EPIC 220204960 can only be determined with accuracies of ≈10% and ≈5%, respectively. We show that the most likely period of the outer orbit is 957 days, with a 1σ range of 595 to 1674 days. We can only place weak constraints on the mutual inclinations of the orbital planes, and additional radial velocity measurements and/or additional eclipse observations would allow for much tighter constraints on the properties of the outer orbit.

Planets Across Space and Time (PAST). IV. The Occurrence and Architecture of Kepler Planetary Systems as a Function of Kinematic Age Revealed by the LAMOST–Gaia–Kepler Sample

One of the fundamental questions in astronomy is how planetary systems form and evolve. Measuring the planetary occurrence and architecture as a function of time directly addresses this question. In the fourth paper of the Planets Across Space and Time series, we investigate the occurrence and architecture of Kepler planetary systems as a function of kinematic age by using the LAMOST–Gaia–Kepler sample. To isolate the age effect, other stellar properties (e.g., metallicity) have been controlled. We find the following results. (1) The fraction of stars with Kepler-like planets (F Kep) is about 50% for all stars; no significant trend is found between F Kep and age. (2) The average planet multiplicity () exhibits a decreasing trend (∼2σ significance) with age. It decreases from ∼ 3 for stars younger than 1 Gyr to ∼ 1.8 for stars of about 8 Gyr. (3) The number of planets per star (η = F Kep× ) also shows a decreasing trend (∼2σ–3σ significance). It decreases from η ∼ 1.6–1.7 for young stars to η ∼ 1.0 for old stars. (4) The mutual orbital inclination of the planets (σ i,k ) increases from to as the stars age from 0.5 to 8 Gyr with a best fit of . Interestingly, the solar system also fits such a trend. The fact that F Kep remains relatively constant at approximately ∼ 50% across different ages suggests the robustness of planet formation throughout the history of the Galaxy. The age dependence of and σ i,k demonstrates that the planetary architecture is evolving, and planetary systems generally become dynamically hotter with fewer planets as they age.

Prospects from TESS and Gaia to Constrain the Flatness of Planetary Systems

The mutual inclination between planets orbiting the same star provides key information to understand the formation and evolution of multiplanet systems. In this work, we investigate the potential of Gaia astrometry in detecting and characterizing cold Jupiters in orbits exterior to the currently known Transiting Exoplanet Survey Satellite (TESS) planet candidates. According to our simulations, out of the ∼3350 systems expected to have cold Jupiter companions, Gaia, by its nominal 5 yr mission, should be able to detect ∼200 cold Jupiters and measure the orbital inclinations with a precision of in ∼120 of them. These numbers are estimated under the assumption that the orbital orientations of the CJs follow an isotropic distribution, but these only vary slightly for less broad distributions. We also discuss the prospects from radial velocity follow-ups to better constrain the derived properties and provide a package to do quick forecasts using our Fisher matrix analysis. Overall, our simulations show that Gaia astrometry of cold Jupiters orbiting stars with TESS planets can distinguish dynamically cold (mean mutual inclination ≲5°) from dynamically hot systems (mean mutual inclination ≳20°), placing a new set of constraints on their formation and evolution.

Sweeping secular resonances and giant planet inclinations in transition discs

ABSTRACT The orbits of some warm Jupiters are highly inclined (20°–50°) to those of their exterior companions. Comparable misalignments are inferred between the outer and inner portions of some transition discs. These large inclinations may originate from planet–planet and planet–disc secular resonances that sweep across interplanetary space as parent discs disperse. The maximum factor by which a seed mutual inclination can be amplified is of the order of the square root of the angular momentum ratio of the resonant pair. We identify those giant planet systems (e.g. Kepler-448 and Kepler-693) that may have crossed a secular resonance, and estimate the required planet masses and semimajor axes in transition discs needed to warp their innermost portions (e.g. in CQ Tau). Passage through an inclination secular resonance could also explain the hypothesized large mutual inclinations in apsidally-orthogonal warm Jupiter systems (e.g. HD 147018).

The effect of cant angle on the sound production of mutually inclined rectangular supersonic jets

Rectangular nozzle geometries offer a number of benefits over their axisymmetric counterparts, particularly for military aviation. Among these benefits are altered thermal and noise signatures, ease of airframe integration, and simpler thrust vectoring. Many next-generation tactical aircraft concepts employ two rectangular jets in close proximity. These jets can couple together, producing high-amplitude acoustic tones that are deleterious to aircraft structure. In this work, we take a combined experimental and theoretical approach to examine the effect of cant angle (i.e., mutual inclination angle) on the jets' proclivity to couple. We employ modal decomposition of flow visualization data alongside resonance predictions based on linear stability theory to explore the degree to which mutual inclination can suppress resonance.

Visual Orbits and Alignments of Planet-hosting Binary Systems

Roughly half of Solar-type planet hosts have stellar companions, so understanding how these binary companions affect the formation and evolution of planets is an important component to understanding planetary systems overall. Measuring the dynamical properties of planet host binaries enables a valuable test of planet formation in multistar systems and requires knowledge of the binary orbital parameters. Using high-resolution imaging, we have measured the relative astrometry and visual orbits of 13 binary systems where one of the stars is known to host a transiting exoplanet. Our results indicate that the mutual inclination between the orbits of the binary hosts and the transiting planets are well aligned. Our results for close binary systems (a < 100 au) complement past work for wide planet host binaries from Gaia.

Four new compact triply eclipsing triples found with Gaia and TESS

ABSTRACT This paper presents a comprehensive analysis of four triply eclipsing triple star systems, namely TIC 88206187, TIC 14839347, TIC 298714297, and TIC 66893949. The four systems with third-body eclipses were found in the TESS light curves from among a sample of ∼400 matches between known eclipsing binaries and the Gaia DR3 Non-Single Star solution data base. We combined photometric light curves, eclipse timing variations, archival spectral energy distributions, and theoretical evolution tracks in a robust photodynamical analysis to determine the orbital and system parameters. The triples have outer periods of 52.9 d, 85.5 d, 117 d, and 471 d, respectively. All dozen stars have masses ≲ 2.6 M⊙. The systems are quite flat with mutual inclination angles between the inner and outer orbital planes that are all ≲ 4°. The outer mass ratios (q ≡ M3/Mbin) range from 0.39 to 0.76, consistent with our earlier collection of compact triply eclipsing triples. TIC 88206187 exhibits a fractional radius of the outer tertiary component (rB ≡ RB/aout) exceeding 0.1 (only the third such system known), and we consider its future evolution. Finally, we compare our photodynamical analysis results and the orbital parameters given in the Gaia DR3 NSS solutions, indicating decent agreement, but with the photodynamical results being more accurate.

A 5MJup non-transiting coplanar circumbinary planet around Kepler-1660AB

ABSTRACT Over a dozen transiting circumbinary planets have been discovered around eclipsing binaries. Transit detections are biased towards aligned planet and binary orbits, and indeed all of the known planets have mutual inclinations less than 4.5○. One path to discovering circumbinary planets with misaligned orbits is through eclipse timing variations (ETVs) of non-transiting planets. Borkovits et al. (2016) discovered ETVs on the 18.6 d binary Kepler-1660AB, indicative of a third body on an ≈236 d period, with a misaligned orbit and a potentially planetary mass. Getley et al. (2017) agreed with the planetary hypothesis, arguing for a 7.7MJup circumbinary planet on an orbit that is highly misaligned by 120○ with respect to the binary. In this paper, we obtain the first radial velocities of the binary. We combine these with an analysis of not only the ETVs but also the eclipse depth variations. We confirm the existence of a 239.5 d circumbinary planet, but with a lower mass of 4.87MJup and a coplanar orbit. The misaligned orbits proposed by previous authors are definitively ruled out by a lack of eclipse depth variations. Kepler-1660ABb is the first confirmed circumbinary planet found using ETVs around a main sequence binary.

Long-term Evolution of Warps in Debris Disks—Application to the Gyr-old System HD 202628

We present the results of N-body simulations meant to reproduce the long-term effects of mutually inclined exoplanets on debris disks, using the HD 202628 system as a proxy. HD 202628 is a Gyr-old solar-type star that possesses a directly observable, narrow debris ring with a clearly defined inner edge and nonzero eccentricity, hinting at the existence of a sculpting exoplanet. The eccentric nature of the disk leads us to examine the effect on it over Gyr timescales from an eccentric and inclined planet, placed on its orbit through scattering processes. We find that, in systems with dynamical timescales akin to that of HD 202628, a planetary companion is capable of completely tilting the debris disk. This tilt is preserved over the Gyr age of the system. Simulated observations of our models show that an exoplanet around HD 202628 with an inclination misalignment ≳10° would cause the disk to be observably diffuse and broad, which is inconsistent with Atacama Large Millimeter Array (ALMA) observations. With these observations, we conclude that, if there is an exoplanet shaping this disk, it likely had a mutual inclination of less than 5° with the primordial disk. The conclusions of this work can be applied either to debris disks appearing as narrow rings (e.g., Fomalhaut and HR 4796) or to disks that are vertically thick at ALMA wavelengths (e.g., HD 110058).

Chasing extreme planetary architectures

Context. Planet(s) in binaries are unique architectures for testing predictions of planetary formation and evolution theories in very hostile environments. Their presence in such a highly perturbed region poses a clear challenge from the early phase of planetesimals accretion to the dynamical evolution and stability through a very long period of time (several Gyrs in some case). Aims. The combination of radial velocity, speckle interferometry, high-contrast imaging and high-precision astrometry with interfer-ometry, offers a unique and unprecedented set of observable to push the exploration of the extreme planetary system HD 196885. By dissecting the physical and orbital properties of each component, we aim at shedding light on its global architecture and stability. Methods. We used the IRDIS dual-band imager of SPHERE at VLT, and the speckle interferometric camera HRCAM of SOAR, to acquire high-angular resolution images of HD 196885 AB between 2015 and 2020. Radial velocity observations started in 1983 with CORAVEL at OHP have been extended with a continuous monitoring with CORALIE at La Silla, and ELODIE and SOPHIE at OHP over almost 40 yr extending the radial velocity measurements HD 196885 A and resolving both the binary companion and the inner giant planet HD 196885 Ab. Finally, we took advantage of the exquisite astrometric precision of the dual-field mode of VLTI/GRAVITY (down to 30 µas) to monitor the relative position of HD 196885 A and B to search for the 3.6 yr astrometric wobble of the circumpri-mary planet Ab imprinted on the binary separation. Results. Our observations enable to accurately constrain the orbital properties of the binary HD 196885 AB, seen on an inclined and retrograde orbit (iAB = 120.43 deg) with a semi-major axis of 19.78 au, and an eccentricity of 0.417. The GRAVITY measurements confirm for the first time the nature of the inner planet HD 196885 Ab by rejecting all families of pole-on solutions in the stellar or brown dwarf masses. The most favored island of solutions is associated with a Jupiter-like planet (MAb = 3.39 MJup), with moderate eccentricity (eAaAb = 0.44), and inclination close to 143.04 deg. This results points toward a significant mutual inclination (Ф = 24.36 deg) between the orbital planes (relative to the star) of the binary companion B and the planet Ab. Our dynamical simulations indicate that the system is dynamically stable over time. Eccentricity and mutual inclination variations could be expected for moderate von Zipele Kozai Lidov cycles that may affect the inner planet.

The 14 Her Planetary System: Companion Masses and Architecture from Radial Velocities and Astrometry

We combine Hubble Space Telescope (HST) Fine Guidance Sensor, Hipparcos, and Gaia DR3 astrometric observations of the K0 V star 14 Her with the results of an analysis of extensive ground-based radial velocity data to determine perturbation orbits and masses for two previously known companions, 14 Her b and c. Radial velocities obtained with the Hobby–Eberly Telescope and from the literature now span over 25 years. With these data we obtain improved radial velocity (RV) orbital elements for both the inner companion, 14 Her b and the long-period outer companion, 14 Her c. We also find evidence of an additional RV signal with P ∼ 3789d. We then model astrometry from Hipparcos, HST, and Gaia with RV results to obtain system parallax and proper motion, perturbation periods, inclinations, and sizes due to 14 Her b and c. We find P b = 1767.6 ± 0.2 days, perturbation semimajor axis α b = 1.3 ± 0.1 mas, and inclination i b = 36° ± 3°, P c = 52160 ± 1028 days, perturbation semimajor axis α c = 10.3 ± 0.7 mas, and inclination i c = 82° ± 14°. In agreement with a past investigation, the 14 Her b, c orbits exhibit significant mutual inclination. Assuming a primary mass M * = 0.98 ± 0.04M ☉, we obtain companion masses and .

On the frequencies of circumbinary discs in protostellar systems

ABSTRACT We report the analysis of circumbinary (CB) discs formed in a radiation hydrodynamical simulation of star cluster formation. We consider both pure binary stars and pairs within triple and quadruple systems. The protostellar systems are all young (ages &amp;lt; 105 yrs). We find that the systems that host a CB disc have a median separation of ≈11 au, and the median characteristic radius of the discs is ≈64 au. We find that 89 per cent of pure binaries with semimajor axes a &amp;lt; 1 au have a CB disc, and the occurrence rate of CB discs is bimodal with log-separation in pure binaries with a second peak at a ≈ 50 au. Systems with a &amp;gt; 100 au almost never have a CB disc. The median size of a CB disc is between ≈5 and 6 a depending on the order of the system, with higher order systems having larger discs relative to binary separation. We find the underlying distributions of mutual inclination between CB discs and binary orbits from the simulation are in good agreement with those of observed CB discs.

Method of Circular Gaussian Rings in Perturbation Theory

An extended approach to the model of circular Gaussian rings is developed to study the secular evolution of the orbits of two planets under the influence of mutual gravitational perturbation. The orbits of the planets have a small mutual inclination angle and are represented by circular rings, on which the masses, semimajor axes and inclination angles of the orbits, as well as the orbital angular momenta of the planets are transferred. The role of the perturbation function in the problem is played by the mutual gravitational energy of the rings, which is obtained in integral form and as a series in powers of slope angles. The coefficients of the series are expressed in terms of elliptic integrals. The method for the first time considers the discrepancy between the nodes of the planets’ orbits and has been developed in two versions: (i) with a large and (ii) small angles between the nodes of the orbits. For each of these options, systems of 4 differential equations describing the secular evolution of orbits are compiled and solved in the final analytical form. It is proved that the angle of mutual inclination of the orbits remains constant in the process of evolution in both cases. Option (i) is tested on the example of the Sun–Jupiter–Saturn; option (ii) is tested on the evolution of orbits of exoplanets Kepler-10b and Kepler-10c. For both systems, the precession parameters were calculated and the diagrams were plotted.

The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination

We revisit the secular 3D planetary three-body problem, aiming to provide a unified formalism representing all basic phenomena in the phase space as the mutual inclination between the planetary orbits increases. We propose a ‘book-keeping’ technique allowing to decompose the Hamiltonian as mathcal {H}_textrm{sec}=mathcal {H}_textrm{planar}+mathcal {H}_textrm{space}, with mathcal {H}_textrm{space} collecting all terms depending on the planets mutual inclination i_textrm{mut}. We numerically compare several models obtained by multipole (Legendre) or Laplace–Lagrange expansions of mathcal {H}_textrm{sec}, aiming to define suitable truncation orders for these models. We explore the transition, as i_textrm{mut} increases, from a ‘planar-like’ to a ‘Lidov–Kozai’ regime. Using a numerical example far from hierarchical limits, we find that the structure of the phase portraits of the (integrable) planar case is reproduced to a large extent also in the 3D case. A semi-analytical criterion allows to estimate the level of i_textrm{mut} up to which the dynamics remains nearly integrable. We propose a normal form method to compute the basic periodic orbits (apsidal corotation orbits A and B) in this regime. We explore the sequence of saddle-node and pitchfork bifurcations by which the A and B families are connected to the highly inclined periodic orbits of the Lidov–Kozai regime. Finally, we perform a numerical study of phase portraits for different planetary mass and distance ratios and qualitatively describe the approach to the corresponding hierarchical limits.

Crab crossing in inverse Compton scattering

Inverse Compton scattering is a promising x-ray source, very bright, quasi-monochromatic, and compact. In this paper, we present a generalized theory of Compton backscattering in terms of luminosity, suitable for both classical and quantum regimes. We show that the optimal parameters, which require a certain mutual orientation and inclination of the fronts of the laser and electron beams described by 3D Gaussians, correspond to the crab scheme. This scheme is widely used in particle physics but is not yet used for x-ray sources. The constructed theory not only predicts the optimal geometry for laser and electron beams but also describes the luminosity. Our results reveal the opportunity to sharply increase the luminosity of compact x-ray sources based on Compton/Thomson backscattering.