Oblate Planet Research Articles

Evolution of Planetary Obliquity: The Eccentric Kozai–Lidov Mechanism Coupled with Tide

Planetary obliquity plays a significant role in determining the physical properties of planetary surfaces and climate. As direct detection is constrained due to the present observation accuracy, kinetic theories are helpful for predicting the evolution of planetary obliquity. Here the coupling effect between the eccentric Kozai–Lidov effect and the equilibrium tide is extensively investigated; the planetary obliquity is observed to follow two kinds of secular evolution paths, based on the conservation of total angular momentum. The equilibrium timescale of the planetary obliquity t eq varies along with r t , which is defined as the initial timescale ratio of the tidal dissipation and secular perturbation. We numerically derive the linear relationship between t eq and r t with the maximum likelihood method. The spin-axis orientation of S-type terrestrials orbiting M-dwarfs reverses over 90° when r t > 100, then enters the quasi-equilibrium state between 40° and 60°, while the maximum obliquity can reach 130° when r t > 104. Numerical simulations show that the maximum obliquity increases with the semimajor axis ratio a 1/a 2, but is not so sensitive to the eccentricity e 2. The likelihood of an obliquity flip for S-type terrestrials in general systems with a 2 < 45 au is closely related to m 1. The observed potentially oblique S-type planets HD 42936 b, GJ 86 Ab, and τ Boo Ab are explored and found to have a great possibility of rotating head-down over the secular evolution of spin.

Revisiting Universal Variables for Robust, Analytical Orbit Propagation Under the Vinti Potential

To meet the growing complexity and demands of modern spacecraft missions, analytical solutions to initial value problems see continued use, typically supporting global searches of large trajectory design spaces. These efforts often employ universal two-body orbit propagators for their recognized speed and robustness, but many applications, like active space debris removal, would benefit from a comparable propagator with greater accuracy. Vinti propagators, which consider planetary oblateness, may serve this purpose, but existing Vinti solutions possess computational difficulties in certain orbital regimes. To mitigate these deficiencies, the present study develops and validates an analytical, third-order universal Vinti propagator free of computational difficulties by leveraging standard, oblate spheroidal (OS) universal variables and OS equinoctial orbital elements. Accuracy of the third-order approximation is assessed for multiple examples across an array of orbital regimes. Computational runtime is also evaluated, and performance is directly compared to the benchmark Vinti6 algorithm. On average, the Vinti propagator implemented in this work is only slower than a typical universal Kepler propagator by a factor of 4.0 and slower than Vinti6 by a factor of 1.8, but with greater robustness than the benchmark. The new form of the equations of motion also has favorable implications for the associated boundary value problem.

Uranus ring occultation observations: 1977–2006

The Uranian rings were discovered serendipitously on 10 March 1977 during a stellar occultation (Elliot et al., 1977a; Millis et al., 1977), and a rich set of subsequent Earth-based occultations revealed that these narrow and sharp-edged rings were eccentric and inclined, precessing under the gravitational influence of the oblate central planet. Considerable progress has been made in understanding the observed characteristics of narrow rings and sharp edges (Nicholson et al., 2018) and their associated dynamics (Longaretti, 2018), but ever since their discovery, the Uranian rings have posed dynamical puzzles that resist simple explanations. The observational basis to address these questions for the Uranus system rests largely on occultation measurements of the narrow rings spanning nearly 30 years, beginning in 1977 and concluding most recently in 2006. Nearly all of these occultation data sets are available in digital form on NASA’s Planetary Data System (PDS) Ring-Moon Systems node, but many of them have not been previously published or described in detail. This paper serves as a guide to the PDS archive and provides essential information about the observations and the methods used to determine the ring widths, mean optical depths, and occultation event times from individual occultation profiles. Additional detail is provided in the Supplementary Online Material accompanying this publication. In a companion paper (French et al., 2023b), we make use of these observations to determine the Uranus ring orbits, pole direction, and gravity field, and the orbital characteristics and masses of three small Uranian moons – Cressida, Ophelia, and Cordelia – from their forced normal modes on the rings.

Superhabitability of High-obliquity and High-eccentricity Planets

Planetary obliquity and eccentricity influence climate by shaping the spatial and temporal patterns of stellar energy incident at a planet’s surface, affecting both the annual mean climate and magnitude of seasonal variability. Previous work has demonstrated the importance of both planetary obliquity and eccentricity for climate and habitability, but most studies have not explicitly modeled the response of life to these parameters. While exaggerated seasons may be stressful to some types of life, a recent study found an increase in marine biological activity for moderately high obliquities <45° assuming an Earth-like eccentricity. However, it is unclear how life might respond to obliquities >45°, eccentricities much larger than Earth’s, or the combination of both. To address this gap, we use cGENIE-PlaSim, a 3D marine biogeochemical model coupled to an atmospheric general circulation model, to investigate the response of Earth-like marine life to a large range of obliquities (0°–90°) and eccentricities (0–0.4). We find that marine biological activity increases with both increasing obliquity and eccentricity across the parameter space we considered, including the combination of high obliquity and high eccentricity. We discuss these results in the context of remote biosignatures, and we argue that planets with high obliquity and/or eccentricity may be superhabitable worlds that are particularly favorable for exoplanet life detection.

Periodic and Quasi-Periodic Orbits near Close Planetary Moons

Upcoming missions toward remote planetary moons will fly in chaotic dynamic environments that are significantly perturbed by the oblateness of the host planet. Such a dominant perturbation is often neglected when designing spacecraft trajectories in planetary moon systems. This paper introduces a new time-periodic set of equations of motion that is based on the analytical solution of the zonal equatorial problem and better describes the dynamic evolution of a spacecraft subject to the gravitational attraction of a moon and its oblate host planet. Such a system, hereby referred to as the zonal hill problem, remains populated by resonant periodic orbits and families of two-dimensional quasi-periodic invariant tori that are calculated by means of numerical continuation procedures. The resulting periodic and quasi-periodic trajectories are investigated for the trajectory design of future planetary moons explorers.

Measured spin–orbit alignment of ultra-short-period super-Earth 55 Cancri e

A planet's orbital alignment places important constraints on how a planet formed and consequently evolved. The dominant formation pathway of ultra-short period planets ($P<1$ day) is particularly mysterious as such planets most likely formed further out, and it is not well understood what drove their migration inwards to their current positions. Measuring the orbital alignment is difficult for smaller super-Earth/sub-Neptune planets, which give rise to smaller amplitude signals. Here we present radial velocities across two transits of 55 Cancri e, an ultra-short period Super-Earth, observed with the Extreme Precision Spectrograph (EXPRES). Using the classical Rossiter-McLaughlin (RM) method, we measure 55 Cnc e's sky-projected stellar spin-orbit alignment (i.e., the projected angle between the planet's orbital axis and its host star's spin axis) to be $\lambda=10\substack{+17\\ -20}^{\circ}$ with an unprojected angle of $\psi=23\substack{+14\\ -12}^{\circ}$. The best-fit RM model to the EXPRES data has a radial velocity semi-amplitude of just $0.41\substack{+0.09\\ -0.10} m s^{-1}$. The spin-orbit alignment of 55 Cnc e favors dynamically gentle migration theories for ultra-short period planets, namely tidal dissipation through low-eccentricity planet-planet interactions and/or planetary obliquity tides.

THE EFFECT OF SOLAR RADIATION PRESSURE AND PLANET OBLATENESS ON THE LOCATION AND STABILITY OF LAGRANGIAN POINTS IN SOLAR SYSTEM PLANETS

The combined effect of the solar radiation pressure and the oblateness of the planets on the position and stability of the Lagrangian points within the restricted elliptic three-body problem was studied. We found that the solar radiation pressure and planet oblateness slightly changed the location of the Lagrangian points. New formulas for the location of the collinear points using the Lagrange inversion method were introduced. For large values of the force ratio B; the ratio between the solar radiation force and the gravitational attraction force due to the Sun; we found that the collinear point [Formula: see text] is stable.

Inner third-body perturbations including the inclination and eccentricity of the perturbing body

ABSTRACT In the field of the orbital perturbations dealt with an approach based on the mean orbital elements theory, the outer third-body gravitational attraction has been widely investigated. On the contrary, since it represents a less common case in the Solar system, the inner third-body perturbation has only recently been considered. The aim of this paper is to provide a more rigorous formulation of the inner third-body perturbation using a double-averaged analytical model. The disturbing potential function of the inner third-body is expanded in Legendre polynomials up to the second order. Afterwards, it is averaged first with respect to the orbital period of the disturbing body and then with respect to the orbital period of the probe. This procedure eliminates the short periodic motion terms. By applying to the double-averaged disturbing potential, the Lagrange planetary equations, the equations which describe the long-term and the secular variations of the classical orbital elements have been obtained: they present an analogy with those related to the planetary oblateness. Lastly, several cases of inner third-body perturbation in the Solar system are discussed, with the conclusion that this is a disturbance of relevance for binary asteroidal systems.

System Architecture and Planetary Obliquity: Implications for Long-term Habitability

In the search for life beyond our solar system, attention should be focused on those planets that have the potential to maintain habitable conditions over the prolonged periods of time needed for the emergence and expansion of life as we know it. The observable planetary architecture is one of the determinants for long-term habitability as it controls the orbital evolution and ultimately the stellar fluxes received by the planet. With an ensemble of n-body simulations and obliquity models of hypothetical planetary systems, we demonstrate that the amplitude and period of the eccentricity, obliquity, and precession cycles of an Earth-like planet are sensitive to the orbital characteristics of a giant companion planet. A series of transient, ocean-coupled climate simulations show how these characteristics of astronomical cycles are decisive for the evolving surface conditions and long-term fractional habitability relative to the modern Earth. The habitability of Earth-like planets increases with the eccentricity of a Jupiter-like companion, provided that the mean obliquity is sufficiently low to maintain temperate temperatures over large parts of its surface throughout the orbital year. A giant companion closer in results in shorter eccentricity cycles of an Earth-like planet but longer, high-amplitude, obliquity cycles. The period and amplitude of obliquity cycles can be estimated to first order from the orbital pathways calculated by the n-body simulations. In the majority of simulations, the obliquity amplitude relates directly to the orbital inclination whereas the period of the obliquity cycle is a function of the nodal precession and the proximity of the giant companion.

On the Effects of Planetary Oblateness on Exoplanet Studies

When studying transiting exoplanets it is common to assume a spherical planet shape. However, short rotational periods can cause a planet to bulge at its equator, as is the case with Saturn, whose equatorial radius is almost 10% larger than its polar radius. As a new generation of instruments comes online, it is important to continually assess the underlying assumptions of models to ensure robust and accurate inferences. We analyze bulk samples of known transiting planets and calculate their expected signal strength if they were to be oblate. We find that for noise levels below 100 ppm, as many as 100 planets could have detectable oblateness. We also investigate the effects of fitting spherical planet models to synthetic oblate lightcurves. We find that this biases the retrieved parameters by several standard deviations for oblateness values >0.1–0.2. When attempting to fit an oblateness model to both spherical and oblate lightcurves, we find that the sensitivity of such fits is correlated with both the signal-to-noise ratio as well as the time sampling of the data, which can mask the oblateness signal. For typical values of these quantities for Kepler observations, it is difficult to rule out oblateness values less than ∼0.25. This results in an accuracy wall of 10%–15% for the density of planets which may be oblate. Finally, we find that a precessing oblate planet has the ability to mimic the signature of a long-period companion via transit-timing variations, inducing offsets at the level of tens of seconds.

Low spin-axis variations of circumbinary planets

ABSTRACT Having a massive moon has been considered as a primary mechanism for stabilized planetary obliquity, an example of which being our Earth. This is, however, not always consistent with the exoplanetary cases. This article details the discovery of an alternative mechanism, namely that planets orbiting around binary stars tend to have low spin-axis variations. This is because the large quadrupole potential of the stellar binary could speed up the planetary orbital precession, and detune the system out of secular spin-orbit resonances. Consequently, habitable zone planets around the stellar binaries in low inclination orbits hold higher potential for regular seasonal changes comparing to their single star analogues.

Seismic-Perturbed Obliquity Change as a Discrete Trigger Mechanism of El Niño and La Niña Episodes

The global climate disruptor El Niño Southern Oscillation (ENSO) is difficult to forecast some years in advance due to lack of understanding of its root cause. An alternative physical mechanism is hereby proposed to describe the nature and sustaining force, and predict the occurrence of El Niño and La Niña phenomena. This is based on the earthquake-perturbed obliquity change model previously proposed as a major mechanism of the global climate change problem. Massive quakes can impart a very strong oceanic force that can move the moon which in turn pulls the earth’s axis and change the planetary obliquity. Analysis of the annual geomagnetic north-pole shift and global seismic data revealed this previously undiscovered force. Using a higher obliquity and constant greenhouse gas forcing in the global climate model EdGCM showed that the seismic-induced polar motion and associated enhanced obliquity could be the major mechanism governing the mysterious climate anomalies attributed to El Nino and La Nina cycles. The apparent eastward migration of high SST in the Pacific and the warming of the Indian and Atlantic Oceans south of the equator during ENSO years were correctly simulated by the model. The annual time series of global surface temperatures computed by EdGCM was compared with the observed global temperature and the results showed relatively good agreement. In addition, the heat wave that occurred in Europe during the summer of 2003 and the Russian heat wave of 2010 that killed over 55,000 people appeared to have been correctly simulated with higher obliquity. This study can help affected countries in water shortage contingency planning, disaster mitigation and may help prevent adverse economic and commercial impacts due to ENSO.

Moderately High Obliquity Promotes Biospheric Oxygenation

Planetary obliquity is a first-order control on planetary climate and seasonal contrast, which has a number of cascading consequences for life. How moderately high obliquity (obliquities greater than Earth’s current obliquity up to 45°) affects a planet’s surface physically has been studied previously, but we lack an understanding of how marine life will respond to these conditions. We couple the ROCKE-3D general circulation model to the cGENIE 3D biogeochemical model to simulate the ocean biosphere’s response to various planetary obliquities, bioessential nutrient inventories, and biospheric structure. We find that the net rate of photosynthesis increased by 35% and sea-to-air flux of biogenic oxygen doubled between the 0° and 45° obliquity scenarios, which is an equivalent response to doubling bioessential nutrients. Our results suggest that moderately high obliquity planets have higher potential for biospheric oxygenation than their low-obliquity counterparts and that life on moderately high obliquity habitable planets may be easier to detect with next-generation telescopes. These moderately high obliquity habitable planets may also be more conducive to the evolution of complex life.

Nonsingular vectorial reformulation of the short-period corrections in Kozai’s oblateness solution

We derive a new analytical solution for the first-order, short-periodic perturbations due to planetary oblateness and systematically compare our results to the classical Brouwer–Lyddane transformation. Our approach is based on the Milankovitch vectorial elements and is free of all the mathematical singularities. Being a non-canonical set, our derivation follows the scheme used by Kozai in his oblateness solution. We adopt the mean longitude as the fast variable and present a compact power-series solution in eccentricity for its short-periodic perturbations that relies on Hansen’s coefficients. We also use a numerical averaging algorithm based on the fast-Fourier transform to further validate our new mean-to-osculating and inverse transformations. This technique constitutes a new approach for deriving short-periodic corrections and exhibits performance that are comparable to other existing and well-established theories, with the advantage that it can be potentially extended to modeling non-conservative orbit perturbations.

Global searches of frozen orbits around an oblate Earth-like planet

A frozen orbit is beneficial for observation owing to its stationary apsidal line. The traditional gravitational field model of frozen orbits only considers the main zonal harmonic terms J2 and limited high-order terms, which cannot meet the stringent demands of all missions. In this study, the gravitational field is expanded to J15 terms and the Hamiltonian canonical form described by the Delaunay variables is used. The zonal harmonic coefficients of the Earth are chosen as the sample. Short-periodic terms are eliminated based on the Hori–Lie transformation. An algorithm is developed to solve all equilibrium points of the Hamiltonian function. A stable frozen orbit with an argument of perigee that equals neither 90◦ nor 270◦ is first reported in this paper. The local stability and topology of the equilibrium points are obtained from their eigenvalues. The bifurcations of the equilibrium points are presented by drawing their global long-term evolution of frozen orbits and their orbital periods. The relationship between the terms of the gravitational field and number of frozen points is addressed to explain why only limited frozen orbits are found in the low-order term case. The analytical results can be applied to other Earth-like planets and asteroids.

Primordial Giant Planet Obliquity Driven by a Circumplanetary Disk

Abstract Detached circumplanetary disks are unstable to tilting as a result of the stellar tidal potential. We examine how a tilted circumplanetary disk affects the evolution of the spin axis of an oblate planet. The disk is evolved using time-dependent equations for linear wave-like warp evolution, including terms representing the effect of the tidal potential and planetary oblateness. For a disk with a sufficiently large mass, we find that the planet spin quickly aligns to the misaligned disk. The tilt of the planetary spin axis then increases on the same timescale as the disk. This can be an efficient mechanism for generating primordial obliquity in giant planets. We suggest that directly imaged exoplanets at large orbital radii, where the disk mass criterion is more likely to be satisfied, could have significant obliquities due to the tilt instability of their circumplanetary disks.

Stratigraphy and Volumes of the Units Within the Massive Carbon Dioxide Ice Deposits of Mars

AbstractThe south polar layered deposits of Mars contain a massive CO2 ice reservoir formed during periods of low planetary obliquity and partial atmospheric collapse. Within the reservoir are bounding layers (BLs) that are believed to consist entirely of water ice and modify CO2 ice stability during periods of higher obliquity and insolation. Here we use three‐dimensional data from the Shallow Radar instrument to extend and update past mapping efforts of the reservoir in order to evaluate its internal structure and mass distribution. By mapping the entirety of each BL as two distinct reflectors, we provide individual volumetric calculations of the multiple CO2 units for the first time (146, 1,472, and 5,493 km3) and discover a new, smaller unit (12 km3). Our volumetric measurements provide critical parameters to understanding past climatic states and climatological modeling, including the magnitude of atmospheric collapse and rebound over three separate events. We also provide updated thickness and areal extent measurements of BL1 (31 ± 5 m, 1,001 km2), BL2 (32 ± 9 m, 5,415 km2) and BL2+3 (42 ± 12 m, 4,232 km2), with maximum thickness of ∼80 m. By mapping the entirety of the BLs, we find novel radar observations, attainable only in 3D, which support previous interpretations that both BL are composed of water ice. Our novel measurements will support climatological studies of Mars’ atmosphere in the past 500 kyr.

Inverted channel belts and floodplain clays to the East of Tempe Terra, Mars: Implications for persistent fluvial activity on early Mars

The climate on early Mars is one of the major unsolved problems in planetary science. It is unclear whether early Mars was warm and wet or cold and icy. Morphological features on Mars such as sinuous ridges could provide critical constraints to address this issue. Here we investigate several sinuous ridges to the east of Tempe Terra, located at the dichotomy boundary of the highland terrain and lowland plains and find these ridges may have recorded persistent fluvial activity on early Mars. Our analysis indicates that these ridges may represent exhumation of the channel belts and overbank deposits formed from meandering rivers over significant geologic time. Layered smectite-bearing minerals are distributed along the flank of the ridges, and could be detrital or authigenic floodplain clays. Our interpretation of the stratigraphic relationships indicates that the layered smectite-bearing materials lie between channel belt deposits, which has rarely been previously reported on Mars and supports the floodplain interpretation. Our results suggest a persistent warm period, perhaps triggered by volcanism, impacts, and/or variations in planetary obliquity, that persisted for a geologically significant interval (tens of thousands of years) during the Noachian period of Mars.

Circular Rydberg states of helium atoms or helium-like ions in a high-frequency laser field

Abstract In the literature, there were studies of Rydberg states of hydrogenic atoms/ions in a high-frequency laser field. It was shown that the motion of the Rydberg electron is analogous to the motion of a satellite around an oblate planet (for a linearly polarized laser field) or around a (fictitious) prolate planet (for a circularly polarized laser field): it exhibits two kinds of precession – one of them is the precession within the orbital plane and another one is the precession of the orbital plane. In this study, we study a helium atom or a helium-like ion with one of the two electrons in a Rydberg state, the system being under a high-frequency laser field. For obtaining analytical results, we use the generalized method of the effective potentials. We find two primary effects of the high-frequency laser field on circular Rydberg states. The first effect is the precession of the orbital plane of the Rydberg electron. We calculate analytically the precession frequency and show that it differs from the case of a hydrogenic atom/ion. In the radiation spectrum, this precession would manifest as satellites separated from the spectral line at the Kepler frequency by multiples of the precession frequency. The second effect is a shift of the energy of the Rydberg electron, also calculated analytically. We find that the absolute value of the shift increases monotonically as the unperturbed binding energy of the Rydberg electron increases. We also find that the shift has a nonmonotonic dependence on the nuclear charge Z: as Z increases, the absolute value of the shift first increases, then reaches a maximum, and then decreases. The nonmonotonic dependence of the laser field-caused energy shift on the nuclear charge is a counterintuitive result.

Classical Dynamics of Muonic-Electronic Helium and Heliumlike Ions: The Allowance for the Eccentricity of the Muon and Nucleus Orbits

In the previous paper by one of us (hereafter paper I), the author considered Rydberg states of the muonic-electronic helium atom or helium-like ion and used the fact that the muon motion occurs much more rapidly than the electron motion. Assuming that the muon and nucleus orbits are circular, he applied the analytical method based on separating rapid and slow subsystems. He showed that the electron moves in an effective potential that is mathematically equivalent to the potential of a satellite orbiting an oblate planet like the Earth. He also showed that the “unperturbed” elliptical orbit of the electron engages in two precessions simultaneously: the precession of the electron orbit in the plane of the orbit and the precession of the orbital plane of the electron around the axis perpendicular to the plane of the muon and nuclear orbits. The problem remained whether or not the allowance for the ellipticity of the orbit could significantly change the results. In the present paper, we address this problem: we study how the allowance for a relatively low eccentricity ε of the muon and nucleus orbits affects the motion of the electron. We derive an additional, ε-dependent term in the effective potential for the motion of the electron. We show analytically that in the particular case of the planar geometry (where the electron orbit is in the plane of the muon and nucleus orbits), it leads to an additional contribution to the frequency of the precession of the electron orbit. We demonstrate that this additional, ε-depen- dent contribution to the precession frequency of the electron orbit can reach the same order of magnitude as the primary, ε-independent contribution to the precession frequency. Therefore, the results of our paper seem to be important not only qualitatively, but also quantitatively.