Perpendicular Orbit Research Articles

Sculpting of Exoplanetary Systems Driven by a Misaligned Disk and Stellar Oblateness: Origin of Perpendicular Orbits in HD 3167

A significant proportion of exoplanets have been detected with highly tilted or even polar orbits relative to their host stars’ equatorial planes. These unusual orbital configurations are often linked to postdisk secular interactions among multiple bodies. However, many aspects remain elusive. In this study, we investigate the role of disk-induced spin–orbit misalignments in shaping the architecture of multiplanet systems, taking into account the combined effect of the host star’s oblateness and the full-space disk potential. We demonstrate that large mutual planetary inclinations can arise from a saddle-center bifurcation occurring during the photoevaporation of the disk. This bifurcation triggers an instant, nonadiabatic transition in the planet’s libration. Following this process, the orbital evolution diverges into several distinct patterns. Notably, in scenarios involving a near-polar primordial misalignment, the orbit, consistently librating about a coplanar equilibrium axis, can be captured by an orthogonal equilibrium during the decay of the stellar oblateness. However, the orbit will be eventually recaptured by the coplanar equilibrium, aligned or antialigned with the orientation of the outer orbit, resulting in either a prograde or retrograde inner–outer orbit configuration. Additionally, general relativity contributes to maintaining eccentricity stability within these dynamic scenarios. Through the proposed mechanism, we can provide a plausible explanation for the unique, near-perpendicular, and likely retrograde orbit architecture observed in the HD 3167 system, enhancing our understanding of exoplanetary system dynamics.

The Rossiter–McLaughlin effect revolutions: an ultra-short period planet and a warm mini-Neptune on perpendicular orbits

Comparisons of the alignment of exoplanets with a common host star and each other can be used to distinguish among concurrent evolution scenarios for the star and the planets. However, multi-planet systems usually host mini-Neptunes and super-Earths, whose sizes make orbital architecture measurements challenging. We introduce the Rossiter-McLaughlin effect Revolutions (RMR) technique, which can access the spin-orbit angle of small exoplanets by exploiting the full extent of information contained in spectral transit time series. We validated the technique through its application to published HARPS-N data of the mini-Neptune HD 3167c (P = 29.8 days), refining its high sky-projected spin-orbit angle (−108.9−5.5+5.4°), and we applied it to new ESPRESSO observations of the super-Earth HD 3167 b (P = 0.96 days), revealing an aligned orbit (−6.6−7.9+6.6°). Surprisingly different variations in the contrast of the stellar lines occulted by the two planets can be reconciled by assuming a latitudinal dependence of the stellar line shape. In this scenario, a joint fit to both datasets constrains the inclination of the star (111.6−3.3+3.1°) and the 3D spin-orbit angles of HD 3167b (29.5−9.4+7.2°) and HD 3167c (107.7−4.9+5.1°). The projected spin-orbit angles do not depend on the model for the line contrast variations, and so, with a mutual inclination of 102.3−8.0+7.4°, we can conclude that the two planets are on perpendicular orbits. This could be explained by HD 3167b being strongly coupled to the star and retaining its primordial alignment, whereas HD 3167c would have been brought to a nearly polar orbit via secular gravitational interactions with an outer companion. Follow-up observations of the system and simulations of its dynamical evolution are required to search for this companion and explore the likelihood of this scenario. HD 3167 b (R = 1.7 REarth) is the smallest exoplanet with a confirmed spectroscopic Rossiter-McLaughlin signal. The RMR technique opens the way to determining the orbital architectures of the super-Earth and Earth-sized planet populations.

A Preponderance of Perpendicular Planets

Abstract Observing the Rossiter–McLaughlin effect during a planetary transit allows the determination of the angle λ between the sky projections of the star’s spin axis and the planet’s orbital axis. Such observations have revealed a large population of well-aligned systems and a smaller population of misaligned systems, with values of λ ranging up to 180°. For a subset of 57 systems, we can now go beyond the sky projection and determine the 3D obliquity ψ by combining the Rossiter–McLaughlin data with constraints on the line-of-sight inclination of the spin axis. Here we show that the misaligned systems do not span the full range of obliquities; they show a preference for nearly perpendicular orbits (ψ = 80°–125°) that seems unlikely to be a statistical fluke. If confirmed by further observations, this pile-up of polar orbits is a clue about the unknown processes of obliquity excitation and evolution.

Intersystem crossing mechanism of thermally activated delayed fluorescence copper(I) thiolate complex: The roles of exchange coupling and magnetic spin interactions

The exchange coupling (J) and magnetic dipole (D) interactions of thermally activated delayed fluorescence copper(I) thiolate complex have been investigated by employing computational chemistry methods. The situation of strict orthogonality of the "hole”–“electron" orbitals with the overlap integral 〈ϕ175α|ϕ175β〉 = 0 leads to small J and excessive triplet T1 population, and is unfavorable to the formation of singlet S1 state via the exchange coupling induced reverse intersystem crossing in the triplet state T1. In magnetic dipole interactions, the mutually perpendicular orbitals provides a significant one-center heavy atomic contribution to the SOC, 〈1φ|hySO|3φ〉 = -41.76 and 〈1φ|hzSO|3φ〉 = −63.14 cm−1, which will enhance the probability of intersystem crossing from the T1 to S1, but system needs to overcome a large intersystem crossing barrier height of 11.05 kcal/mol relative to that of the T1 state at the CAM-B3LYP/def2-TZVP(-f) level. Under magnetic field limit, S1−T0, S1−T+1, and S1−T-1 mixing are all inefficient because the large energy gap between T1 and S1, |2J| » gβB prohibits significant mixing in the triplet state T1, but for the S1/T1 crossing point, the S1−T-1 energy gap is comparable to that of the S1−T0 energy gap, allowing for mixing of S1 with T-1 and T0, |Tz〉 = 56%|1, 0〉 + 20%|1, −1〉 + 16%|0, 0〉.

Formation of discs around super-massive black hole binaries

AbstractWe model numerically the evolution of 104M⊙ turbulent molecular clouds in near-radial infall onto 106M⊙, equal-mass supermassive black hole binaries, using a modified version of the SPH code gadget-3. We investigate the different gas structures formed depending on the relative inclination between the binary and the cloud orbits. Our first results indicate that an aligned orbit produces mini-discs around each black hole, almost aligned with the binary; a perpendicular orbit produces misaligned mini-discs; and a counter-aligned orbit produces a circumbinary, counter-rotating ring.

A study of the high-inclination population in the Kuiper belt – I. The Plutinos

The dynamics of the high-inclination Plutinos is systematically studied. We first present the peculiar features of the 2:3 Neptune mean motion resonance (NMMR) for inclined orbits, especially for the correlation of resonant amplitude Aσ with inclination i. Using the numerical integrations for the age of the Solar system, the dynamical structure of the 2:3 NMMR is mapped out on the plane of semimajor axis versus i for different eccentricities. We have shown that i of stable resonant orbits could be as high as 90°; and the stable region is roughly surrounded by the contours of Aσ = 120°. These new findings allow us to further explore the 2:3 NMMR capture and retention of planetesimals with initial inclinations i0 ≤ 90° in the frame of the planet migration model. We find that the outward transportation of Plutinos is possible for any inclined or even perpendicular orbits. The role of i0 in the formation of Plutinos during Neptune's migration is highlighted and interesting results are obtained: (1) the capture efficiency of the 2:3 NMMR decreases drastically first with the increase of i0, but it then raises instead when i0 exceeds ∼50°; (2) the magnitude of i-variation is limited to less than 5° for any i0, and moreover, for Plutinos with i ≳ 48°, their i are forced to decrease throughout the outward migration; (3) Plutinos with i ≳ 48° are certainly outside the Kozai mechanism, since an inclination increase is prohibited by the migrating 2:3 NMMR; (4) the 7:11 inclination-type NMMR could be responsible for nearly circular Plutinos, and a minimum i0 ∼ 15° is required to intrigue this mechanism.

Semiclassical Calculation of Recurrence Spectra of Rydberg Hydrogen Atom Near a Metal Surface

Using closed orbit theory, we give a clear physical picture description of the Rydberg hydrogen atom near a metal surface and calculate the Fourier transformed recurrence spectra of this system at different scaled energies below ionization threshold. The results show that with the increase of the scaled energy, the number of the closed orbit increases greatly. Some of the orbits are created by the bifurcation of the perpendicular orbit. This case is quite similar to the Rydberg atom in an electric field. When the scaled energy increases furthermore, chaotic orbits appear. This study provides a different perspective on the dynamical behavior of the Rydberg atom near a metal surface.

Polar Ring Galaxies and Warps

Polar ring galaxies, where matter is in equilibrium in perpendicular orbits around spiral galaxies, are ideal objects to probe the 3D shapes of dark matter halos. The conditions to constrain the halos are that the perpendicular system does not strongly perturb the host galaxy, or that it is possible to derive back its initial shape, knowing the formation scenario of the polar ring. The formation mechanisms are reviewed: mergers, tidal accretion, or gas accretion from cosmic filaments. The Tully-Fisher diagram for polar rings reveals that the velocity in the polar plane is higher than in the host plane, which can only be explained if the dark matter is oblate and flattened along the polar plane. Only a few individual systems have been studied in details, and 3D shapes of their haloes determined by several methods. The high frequency of warps could be explained by spontaneous bending instability, if the disks are sufficiently self-gravitating, which can put constraints on the dark matter flattening.

Relativistic electron nonlinear orbit for a combined plasma wave wiggler and an axial guide magnetic field near magnetoresonance

The relativistic electron nonlinear orbit is investigated for a combined electrostatic plasma wave wiggler and an axial guide magnetic field free electron laser near magnetoresonance. Through simplification, the perpendicular orbit equation is reduced to be identical to the wave-driven nonlinear oscillator equation [R. C. Davidson, Phys. Lett. A 126, 21 (1987)] but with different expressions of parameters. It is found that the maximum perpendicular orbit excursion Δxmax is in proportion to (aw/γ0)1/3, which is largely determined by the nonlinear effect, where aw is the wiggler strength and γ0 is the initial relativistic factor. The analytical results are in good agreement with the numerical calculation for small ratio of aw to γ0.

Electron orbit in a combined plasma wave wiggler and an axial guide magnetic field free electron laser near magnetoresonance

The nonlinear electron orbit dynamics is presented for a combined electrostatic plasma wave wiggler and an axial guide magnetic field free electron laser near magnetoresonance. The perpendicular orbit equation is derived and simplified identical to the wave-driven nonlinear oscillator equation but with different expressions of parameters. The nonlinear effect induced by the inclusion of beta /sub perpendicular to //sup 2/ in the relativistic factor gamma is dominant in governing the perpendicular wiggling velocity and the orbit excursion even in the case of beta /sub perpendicular to //sup 2/<< beta /sub ////sup 2/, where gamma =(1- beta /sub 1//sup 2/- beta /sub ////sup 2/)/sup -1/2/. The dependence of the maximum perpendicular wiggling velocity and the orbit excursion on the wiggler constant, the wiggler frequency, and the initial parameters of motion is given analytically.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Weakly relativistic nonlinear orbit dynamics for intense ordinary-mode propagation near electron cyclotron resonance

The nonlinear orbit equations are investigated analytically and numerically for a constant-amplitude electromagnetic wave (ωs, ks) with ordinary-mode polarization propagating perpendicular to a uniform magnetic field B0 êz. Relativisitic electron dynamics are essential in determining the maximum orbit excursion when the incident wave frequency ωs is near the nth harmonic of the electron cyclotron frequency Ωc. Coupled nonlinear equations are obtained for the slow evolution of the amplitude A(т) and phase ø(t) of the perpendicular orbit when ωs ≈ nΩc. The dynamical equations are reduced to quadrature and simplified analytically. At exact resonance (ωs = nΩc), if relativistic effects are (incorrectly) neglected, the maximum orbit excursion Amax approaches the unacceptably large value determined from the first (non-zero) solution to Jn(ns ΩsAmax) = 0, which totally invalidates the non-relativistic approximation. (Here ns Ωs = cks/Ωc.) As a contrasting illustrative example, for n = 1, ωs = Ωc and moderate pump strength, weak relativistic effects limit the maximum orbit excursion to the approximate value , where єs = ns ωs (Vz/c) Vq/c ≪ 1, and Vq is the axial quiver velocity in the applied oscillatory field. Physically, relativistic effects produce a nonlinear frequency shift that dynamically ‘detunes’ the particle motion from exact cyclotron resonance, thereby limiting amplitude growth.

Nonlinear particle trajectories for intense ordinary-mode propagation near electron cyclotron resonance

The nonlinear orbit equations are investigated analytically for a constant-amplitude electromagnetic wave ( ω s , k s ) with ordinary-mode polarization propagating perpendicular to a uniform magnetic field B 0 ê z . Coupled nonlinear equations are obtained for the slow evolution of the amplitude A(τ) and phase π(τ) of the perpendicular orbit when the incident wave frequency ω s, is near the nth harmonic of theelectron cyclotron frequency Ω c . The dynamical equations are reduced to quandrature and simplified analytically. Relativistic electron dynamics are essential in determining the maximum orbital excursion. As an illustrative example, for n = 1 and moderate wave amplitude, weak relativistic effects limit the maximum perpendicular kinetic energy achieved to k max = ( 2 ϵ s) 2 3 mc 2 , where ϵ s=n sΩ s(V z/c)(V q/c) ⪡ 1 , and V q is the axial quiver velocity in the applied oscillatory field.

Did the Moon Form in a Perpendicular Orbit?

Did the Moon Form in a Perpendicular Orbit?