The purpose of this work is to study the phenomenon of tidal locking in a pedagogical framework by analyzing the effective gravitational potential of a two-body system with two spinning objects. It is shown that the effective potential of such a system is an example of a fold catastrophe. In fact, the existence of a local minimum and saddle point, corresponding to tidally-locked circular orbits, is regulated by a single dimensionless control parameter which depends on the properties of the two bodies and on the total angular momentum of the system. The method described in this work results in compact expressions for the radius of the circular orbit and the tidally-locked spin/orbital frequency. The limiting case in which one of the two orbiting objects is point-like is studied in detail. An analysis of the effective potential, which in this limit depends on only two parameters, allows one to clearly visualize the properties of the system. The notorious case of the Mars' moon Phobos is presented as an example of a satellite that is past the no return point and, therefore, will not reach a stable or unstable tidally-locked orbit.
The inner moons of Jupiter and Saturn migrate outwards due to tidal energy dissipation within the planets, the details of which remain poorly understood. We demonstrate that resonance locking between moons and internal oscillation modes of the planet can produce rapid tidal migration. Resonance locking arises due to the internal structural evolution of the planet and typically produces an outward migration rate comparable to the age of the solar system. Resonance locking predicts a similar migration timescale but a different effective tidal quality factor $Q$ governing the migration of each mo…
Tidal Venuses: Triggering a Climate Catastrophe via Tidal Heating
传统上，恒星辐射被认为是唯一能够在长时间尺度上决定全球气候的热源。在这里，我们表明，围绕低质量恒星运行的类地系外行星可能会在足够高的水平上被潮汐加热，从而在足够长的时间内引发失控的温室效应，使所有的氢逃逸。如果没有氢，地球上就没有水，无法维持生命。我们称这些行星为“潮汐金星”，这种现象为“潮汐温室”。“潮汐效应也使轨道循环，从而减少潮汐加热。因此，有些行星可能形成时有很大的离心率，伴随着很大的潮汐加热，并失去了它们的水，但最终在宜居带（HZ）中进入接近圆形的轨道（即潮汐加热可以忽略不计）。然而，这些行星不适合居住，因为过去的潮汐加热使它们干燥，因此不应该被列为旨在检测生物特征的详细后续观测的高度。围绕质量小于0.3太阳质量的恒星运行的行星可能会由于潮汐加热而处于干燥的危险之中。我们将这些概念应用于Gl 667 C c，一颗质量约为4.5地球的行星，在0.12天文单位绕一颗质量为0.3太阳质量的恒星运行。我们发现，它可能没有失去它的水，通过潮汐加热的轨道稳定性是不可能的潮汐温室所需的高偏心率。由于HZ的内边缘是由辐射驱动的失控或潮湿的温室的发作所定义的，我们的结果代表了对HZ的非圆形轨道的根本性修订。在附录中，我们回顾了a）潮湿和失控的温室，b）氢逃逸，c）恒星质量-半径和质量-光度关系，d）类地行星质量-半径关系，e）线性潮汐理论。[删节]
A large fraction of known exoplanets have short orbital periods where tidal excitation of gravity waves within the host star causes the planets' orbits to decay. We study the effects of tidal resonance locking, in which the planet locks into resonance with a tidally excited stellar gravity mode. Because a star's gravity mode frequencies typically increase as the star evolves, the planet's orbital frequency increases in lockstep, potentially causing much faster orbital decay than predicted by other tidal theories. Due to nonlinear mode damping, resonance locking in Sun-like stars likely only op…
Potentially habitable planets can orbit close enough to their host star that the differential gravity across their diameters can fix the rotation rate at a specific frequency, a process called tidal locking. Tidally locked planets on circular orbits will rotate synchronously, but those on eccentric orbits will either librate or rotate super-synchronously. I calculate how habitable planets evolve under two commonly-used models and find, for example, that one model predicts that the Earth's rotation rate would have synchronized after 4.5 Gyr if its initial rotation period was 3 days, it had no s…
Tidal effects in planetary systems are the main driver in the orbital migration of natural satellites. They result from physical processes occurring deep inside celestial bodies, whose effects are rarely observable from surface imaging. For giant planet systems, the tidal migration rate is determined by poorly understood dissipative processes in the planet, and standard theories suggest an orbital expansion rate inversely proportional to the power 11/2 in distance, implying little migration for outer moons such as Saturn's largest moon, Titan. Here, we use two independent measurements obtained…
Escaping particle fluxes in the atmospheres of close-in exoplanets. ii. reduced mass loss rates and anisotropic winds
我们建立了一个多流体二维流体力学模型，详细描述了粒子的辐射转移。我们发现，潮汐力提供显着的加速度和各向异性风的结果。潮汐力的一个重要作用是它严重抑制了极地附近粒子的外流。因此，大多数粒子从低纬度地区逃离地球表面。比较潮汐和非潮汐锁定的情况下，我们发现，它们的光学深度是非常不同的，所以流也出现了不同的模式。在非潮汐锁定的情况下，风底的径向速度高于纬向速度。然而，在潮汐锁定的情况下，纬向速度在风底处占主导地位，并且它们可以有效地将质量和能量从昼侧传递到夜侧。此外，我们还发现，风的差异表现出中等程度的大半径。结果表明，由于光学厚度较大，二维辐射转移可以改变风的结构，但这种改变的程度随半径的增大而减小。由于逃逸在极区受到抑制，非潮汐锁定模型预测的质量损失率在$10^{10}$ g s $^{-1}$的数量级上，比一维流体动力学模型预测的质量损失率低2倍。潮汐锁定的结果表明，由于夜面光学厚度大，质量损失率降低了一个数量级，只有4.3 $\times$ $10^{9}$ g s $^{-1}$。此外，还发现氢原子的分布从昼侧到夜侧有明显的变化，因此应利用多维流体动力学模型重新研究Ly $α$中过量吸收的起源。
The objective of this paper is to study the tidally locked 3:2 spin-orbit resonance of Mercury around the Sun. In order to achieve this goal, the effective potential energy that determines the spinning motion of an ellipsoidal planet around its axis is considered. By studying the rotational potential energy of an ellipsoidal planet orbiting a spherical star on an elliptic orbit with fixed eccentricity and semi-major axis, it is shown that the system presents an infinite number of metastable equilibrium configurations. These states correspond to local minima of the rotational potential energy a…
The discovery and characterization of Earth-sized planets that are in, or near, a tidally-locked state are of crucial importance to understanding terrestrial planet evolution, and for which Venus is a clear analog. Exoplanetary science lies at the threshold of characterizing hundreds of terrestrial planetary atmospheres, thereby providing a statistical sample far greater than the limited inventory of terrestrial planetary atmospheres within the Solar System. However, the model-based approach for characterizing exoplanet atmospheres relies on Solar System data, resulting in our limited inventor…
Thermal, orbital, and rotational dynamics of tidally loaded exoplanets are interconnected by intricate feedback. The rheological structure of the planet determines its susceptibility to tidal deformation and, as a consequence, participates in shaping its orbit. The orbital parameters and the spin state, conversely, control the rate of tidal dissipation and may lead to substantial changes of the interior. We investigate the coupled thermal-orbital evolution of differentiated rocky exoplanets governed by the Andrade viscoelastic rheology. The coupled evolution is treated by a semi-analytical mod…
The TRAPPIST-1, Proxima Centauri, and LHS 1140 systems are the most exciting prospects for future follow-up observations of potentially inhabited planets. All orbit nearby M-stars and are likely tidally locked in 1:1 spin-orbit states, which motivates the consideration of the effects that tidal locking might have on planetary habitability. On Earth, periods of global glaciation (snowballs) may have been essential for habitability and remote signs of life (biosignatures) because they are correlated with increases in the complexity of life and in the atmospheric oxygen concentration. In this pap…
Synchronisation of a tidal binary by inward orbital migration. The case of Pluto and Charon
US Naval Observatory（美国海军天文台）；Charles University（查理大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；California Institute of Technology（加州理工学院）；ETH Zürich（苏黎世联邦理工学院）
Since the launch of Kepler and Hubble more than a decade ago, we have come a long way in the quest to find a potentially habitable exoplanet. To date, we have already discovered more than 4000 exoplanets most of which are not suitable for sustaining life. Of all those that can potentially sustain life, A large number has been found rotating synchronously around their parent star, mostly Red dwarf star. Due to their synchronous rotation, these planets receive very uneven stellar heating. Synchronous rotation of these planets causes one side of the planet to permanently face the parent star whil…
Planets in the "Habitable Zones" around M-type stars are important targets for characterization in future observations. Due to tidal-locking in synchronous spin-orbit rotations, the planets tend to have a hot dayside and a cold nightside. On the cold nightside, water vapor transferred from the dayside can be frozen in ("cold trap") or the major atmospheric constituent could also condense ("atmospheric collapse") if the atmosphere is so thin that the heat re-distribution is not efficient, in the case of a single M-type star. Motivated by the abundance of binary star systems, we investigate the …
Exomoons may play an important role in determining the habitability of worlds outside of our solar system. They can stabilize conditions, alter the climate by breaking tidal locking with the parent star, drive tidal heating, and perhaps even host life themselves. However, the ability of an exoplanet to sustain an exomoon depends on complex tidal interactions. Motivated by this, we make use of simplified tidal lag models to follow the evolution of the separations and orbital and rotational periods in planet, star, and moon systems. We apply these models to known exoplanet systems to assess the …
Atmospheric Collapse and Habitability on Tidally-Locked Exoplanets
Earth-Life Science Institute, Institute of Science Tokyo（地球生命研究所，东京科学大学）；Department of Earth and Planetary Sciences, Institute of Science Tokyo（地球行星科学系，东京科学大学）；Laboratoire de Météorologie Dynamique/IPSL, CNRS, Sorbonne Université, Ecole Normale Supérieure, PSL Research University, Ecole Polytechnique（气象动力学实验室/IPSL，法国国家科学研究中心，索邦大学，巴黎高等师范学院，PSL研究大学，巴黎综合理工学院）；Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS（波尔多天体物理实验室，波尔多大学，法国国家科学研究中心）；Univ. Grenoble Alpes, CNRS, IPAG（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，地球与行星物理学研究所）；Institute of Science Tokyo（东京科学大学）
In this paper we propose a new mechanism that could explain the survival of hydrogen atmospheres on some hot super-Earths. We argue that on close-orbiting tidally-locked super-Earths the tidal forces with the orbital and rotational centrifugal forces can partially confine the atmosphere on the nightside. Assuming a super terran body with an atmosphere dominated by volcanic species and a large hydrogen component, the heavier molecules can be shown to be confined within latitudes of $\lesssim 80^{\circ}$ whilst the volatile hydrogen is not. Because of this disparity the hydrogen has to slowly di…
Close-by planets may excite various kinds of oscillations in their host stars through their time-varying tidal potential. Magnetostrophic oscillations with a frequency much smaller than the stellar rotation frequency have recently been proposed to account for the spin-orbit commensurability observed in several planet-hosting stars. In principle, they can be resonantly excited in an isolated slender magnetic flux tube by a Fourier component of the time-varying tidal potential with a very low frequency in the reference frame rotating with the host. However, due to the weakness of such high-order…
At mid-mission perijove 17, NASA's Juno mission has revealed a $7σ$ discrepancy between Jupiter's observed high-degree tidal response and the theoretical equilibrium tidal response, namely the Love number $k_{42}$. Here, we propose an interpretation for this puzzling disagreement based on an interior-orbital resonance between internal gravity waves trapped in Jupiter's dilute core and the orbital motion of Io. We use simple Jupiter models to calculate a fractional correction $Δk_{42}$ to the equilibrium tidal response that comes from the dynamical tidal response of a $g$-mode trapped in Jupite…
A planet's atmospheric constituents (e.g., O$_2$, O$_3$, H$_2$O$_v$, CO$_2$, CH$_4$, N$_2$O) can provide clues to its surface habitability, and may offer biosignature targets for remote life detection efforts. The plethora of rocky exoplanets found by recent transit surveys (e.g., the Kepler mission) indicates that potentially habitable systems orbiting K- and M-dwarf stars may have very different orbital and atmospheric characteristics than Earth. To assess the physical distribution and observational prospects of various biosignatures and habitability indicators, it is important to understand…
The Kepler data show that habitable small planets orbiting Red Dwarf stars (RDs) are abundant, and hence might be promising targets to look at for biomarkers and life. Planets orbiting within the Habitable Zone of RDs are close enough to be tidally locked. Some recent works have cast doubt on the ability of planets orbiting RDs to support life. In contrast, it is shown that temperatures suitable for liquid water and even for organic molecules may exist on tidally locked planets of RDs for a wide range of atmospheres. We chart the surface temperature distribution as a function of the irradiatio…
Despite numerous attempts, no exomoon has firmly been confirmed to date. New missions like CHEOPS aim to characterize previously detected exoplanets, and potentially to discover exomoons. In order to optimize search strategies, we need to determine those planets which are the most likely to host moons. We investigate the tidal evolution of hypothetical moon orbits in systems consisting of a star, one planet and one test moon. We study a few specific cases with ten billion years integration time where the evolution of moon orbits follows one of these three scenarios: (1) "locking", in which the…
