Unless a planet is perfectly spherical and totally rigid (which implies it has no atmosphere), gravitational interactions between host star, other planets and satellites will deform the planet through redistribution of the mass of its fluid fields – as observed in the ocean tides twice a day on Earth. These gravitational tides transfer angular momentum between planet and star and result in a net torque on the planet2. If the torque exhibits a sufficient drag on the rotation of the planet, it will erode its rotation rate, eventually approaching synchronous rotation, commonly know as becoming tidally locked. An effect of this, as observed from the planet, is an eradication of the diurnal cycle. One “day” hemisphere permanently faces the host star, the other “night” side in perpetual darkness. This is the case for our Moon, tidal interactions with Earth have caused its rotation to become synchronised with its orbit and as such, we only ever see one face of the Moon.
For a planet only forced by gravitational tidal interactions, the time taken for a planet to become tidally locked can be given by
τlocking' Q a3 G M (Ω − Γ ) M M∗ R a 6 , (3.1)
where G is the universal gravitational constant, a, M ,Ω, Γ are the planet’s radius, mass, rotation rate and orbital rate, M∗the mass of the star and R the orbital radius (Showman and Guillot 2002). It is worth noting that its dependence on the orbital radius goes like ∼ R6, and as such as we move out from the star the timescale of locking quickly becomes very long, for the case of an Earth-like planet orbiting the Sun τlocking exceeds the star’s lifetime at R ' 0.7 au. The major unknown in this
2A torque is applied to the star too, however due to the mass difference it is proportionally much
3.2 ORBITAL SYNCHRONICITY (TIDAL LOCKING) 39
expression is Q, the a non-dimensional metric of the planet’s tidal dissipation quality factor, a function of its fluidity, rheology, density and frequency of the tidal forcing (Correia 2003). On Earth, the primary mechanism of tidal dissipation is the drag on the ocean at shorelines, giving a relatively low Q∼ 13 and therefore high tidal dissipation (Edson et al. 2011). For a planet with no ocean or no shoreline, this increases by an order of magnitude (Kasting, Whitmire, and Reynolds 1993), and for a gas giant such as Jupiter it can be of order Q∼ 1 × 105(Goldreich and Soter 1966). The tidal forces on terrestrial and gas planets are not limited to gravitational in- teractions. The incoming radiation from the star is differentially absorbed by the planet’s atmosphere and surface (fig. 1.2), creating temperature and pressure gradi- ents, redistributing the mass and producing a thermal tide. There are additionally still torque forces, such as the deformation of the planetary body through surface pressure variations and self-gravitational equilibrium; these are much smaller and we will not address them here – for a comprehensive review, see Correia (2003).
The idea that an additional thermal torque may prevent a planet from becoming truly tidally locked has been posited since the first accurate measurements of Venus’s atmosphere provided a platform for a more complete investigation of its circulation and tidal evolution. Ingersoll and Dobrovolskis (1978) showed that the asynchronous rotation of Venus, which should in the absence of an additional force have been eroded to be tidally locked with its orbit, could be maintained by a thermal tide. Due to thermal inertia of the surface, the thermal tide will not, in general, be aligned with either the direction of the gravitational or thermal forcing vector (Figure 3.3).
At the outset it was assumed that a high-mass atmosphere, such as that of Venus, was necessary to provide sufficient torque to prevent tidal locking, but this has been complicated by studies of hot Jupiters which show that the deep gas atmosphere can both damp (Gu and Ogilvie 2009) and enhance (Arras and Socrates 2010) the thermal torque, changing the likelihood of tidal locking. For terrestrial planets, if we assume that the majority of heating is occurring at the solid lower boundary, the lower bound on the mass of the atmosphere required for a thermal tide to prevent tidal locking has been shown to be as little as 1 bar (Leconte et al. 2015). While somewhat dependent on the treatment of the diffusive quality factor Q, given the radiative
40 EXOPLANETS Ω Star Planet Thermal Tide Gravity Tide Γ
Figure 3.3: Gravitational and thermal tide bulges on an idealised planet.Depending on
the planet, they are not, in general, aligned with each other, nor the vector along which the forcing acts.
emission from the star the balance between gravitational and thermal torques can be shown to give a critical orbital radius beyond which tidal locking is prevented, shown in fig. 3.4.
3.2.1 Eccentricity and obliquity
The seasons on Earth are the result of the 23.5° obliquity in its rotational axis relative to the orbital plane normal, denoted δ in fig. 1.3. The effect of this obliquity, as observed from Earth, is an oscillation in the latitude of the subsolar point over the year peaking at the tropics of Capricorn and Cancer at the two solstice. In the absence of other tidal interactions (for example, the Moon is partially responsible for the stability of Earth’s obliquity) rotational obliquities are very quickly eroded by gravitational interactions (Heller, Leconte, and Barnes 2011).
Eccentricity is another cause of seasonality; for an elliptic orbit the star will not be central in the orbit, as such as the planet circles the star it gets closer and further, varying the total incoming radiation. Earth’s orbit is not fully circular, its periapsis
3.2 ORBITAL SYNCHRONICITY (TIDAL LOCKING) 41 unstable (21). The diversity of equilibria might be
even richer in eccentric systems where these num- bers could change (16, 17). The synchronous spin state is stable. Knowing that Venus, despite such a
rheology, did not end up synchronized tells us that a planet can avoid being trapped in such a stable synchronous state and constrains the his- tory of the Venusian atmosphere (21).
In addition, the number and location of equilib- ria undergo a bifurcation because asynchronous spin states exist only when the amplitude of the thermal tide reaches a threshold. Thus, our results reveal the existence of a critical distance acbeyond
which the planet can be asynchronous, which, using a constant-Q rheology, reads
ac¼ 10p3 ! "1=6 GM*rR2 p k2 q0Q ! "1=3 ð2Þ where k2is the Love number and Q is the tidal
quality factor (21). Both ac(Fig. 3) and the equi-
librium asynchronicity {jw − nj ¼ wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0½ða=acÞ3þ
ða=acÞ6− 1
q
&
} (fig. S3) can be computed forvarious cases by using Table 1. The corollary is that even without any spin-orbit trapping due to a permanent asymmetry of the mantle (triaxiality), planets on circular orbits for which atmospheric tides are too weak should be in exact spin-orbit resonance.
Our results provide a robust framework for the quantitative assessment of the efficiency of thermal tides for different atmospheric masses without having to rely on scaling arguments calibrated on Venus. This is crucial because Venus thermal tides turn out to be relatively weak (Fig. 1B). As can be seen in Fig. 3, Earth- like planets with a 1-bar atmosphere are ex- pected to have a nonsynchronous rotation if they are in the habitable zone of stars more massive than ~0.5 to 0.7M⊙(depending on their location
in the habitable zone). This lower limit de- creases to≲0:3M⊙for a 10-bar atmosphere. These
limits are much less restrictive than the one obtained from our Venus model (Fig. 3, purple line). This realization required full atmospheric modeling.
Atmospheres as massive as 1 bar are a rea- sonable expectation value given existing models and solar system examples. This is especially true in the outer habitable zone, where plan- ets are expected to build massive atmospheres with several bars of CO2(7). So, our results
demonstrate that asynchronism mediated by thermal tides should affect an important frac- tion of planets in the habitable zone of lower- mass stars.
This has many implications. On one hand, the difficulties in sustaining a habitable cli- mate far from the star due to the presence of a permanent cold, night side (9–15) may not be as severe as usually thought. On the other hand, the habitable zone has been recently shown to be more extended near the star for synchronous planets (12). For these objects, if the atmosphere is thick enough, the nonsynchronous rotation that should ensue may thus come to limit the extent of the habitable zone around lower- mass stars.
The thermal tide mechanism presented here does not only affect habitable-zone planets, so many other terrestrial bodies with substantial atmospheres could potentially have asynchronous rotations, depending on their orbital location (Fig. 3). With that in mind, observational methods
634 6 FEBRUARY 2015 • VOL 347 ISSUE 6222 sciencemag.org SCIENCE
ω n ω n a ac Constant Q Constant Q Constant Q a ac a ac a ac Andrade ac a Andrade Andrade Torque Torque Torque 1 0 1 2 3 1 0 1 2 3 0 0 0 a ac
Fig. 2. Equilibrium spin states of the planet. Atmospheric (dashed), gravitational (dotted), and total (solid) torque as a function of spin rate for two tidal models, (A to C) Andrade and (D to F) Constant-Q. Arrows show the sense of spin evolution. (A) and (D) show weak atmospheric torque, only one equilibrium, and synchronous spin state exists (blue circle). (B) and (E) show the bifurcation point (a = ac). In (C) and (F), the atmospheric torque is strong enough to generate four asynchronous equilibrium
spin states, two being unstable (red open circles) and two being stable (blue circles; one is retrograde in the case shown). The synchronous spin state remains stable.The figure is to be compared with figure 6 of (24).
Fig. 3. Spin state of planets in the habit- able zone. The blue region depicts the hab- itable zone (14, 25), and gray dots are detected and candidate exopla- nets. Each solid black line marks the critical orbital distance (ac)
(Eq. 2) separating syn- chronous (left, red arrow) from asyn- chronous planets (right, blue arrow) for ps= 1 and
10 bar (the extrapolation outside the habitable zone is shown with dotted lines). Objects in the gray area are not spun down by tides. The error bar illustrates how limits would shift when varying the dissipation inside the planet (Q ~ 100) (21) within an order of magnitude.
0.10 0.15 0.20 0.30 0.50 0.70 1.00 1.00 0.50 0.20 0.30 0.70
Semi–Major Axis (AU)
S te lla r M as s ( MSun Habitable Zone ps=1 bar ps=1 0bar Scaled Venus limit Asynchronous Rotation Synchronous Rotation no spin down ) RESEARCH | REPORTS on August 8, 2016 http://science.sciencemag.org/ Downloaded from
Figure 3.4: The limit of tidal locking for terrestrial planets. The balance of gravitational
and thermal tidal torques determines whether a planet will be tidally locked. The solid lines within the shaded habitable zone region show the critical orbital radius beyond which thermal tides prevent tidal locking. Grey dots show known exoplanets, as of February 2015. The black error bar shows how the curves would
shift in semi-major axis ifQis changed by an order of magnitude. From Leconte
et al. (2015).
– time of closest approach to the sun – occurs in early January and results in the Southern hemisphere summer receiving slightly more radiation that the Northern hemisphere summer3.
Changes in obliquity and eccentricity have the potential to dramatically change the climate of a planet; the distribution and amplitude of insolation at the planet surface at high obliquity is especially exotic when compared with Earth or Venus (Dobrovolskis 2009). Numerical studies of super-Mercuries — close-in planets most likely stripped of their atmosphere — suggest that the eccentricity and obliquity of
3There are also higher-moments of orbital mechanics — variation in eccentricity and obliquity, for
Earth known as the Milankovitch cycles, and axial precession. These modulate the Earth’s climate over timescales of thousands of years, and are likely to do the same for exoplanets.
42 EXOPLANETS
a planet may be detectable in their thermal phase curve (Selsis et al. 2013). In the exoplanet studies presented in the upcoming chapters of this thesis, we will assume both obliquity and orbital eccentricity are zero. This is primarily a matter of scope and clarity; we do not wish to address the question of seasonal variation in this study, but as the previous work referenced above shows, they can be a potential source of variation in the observed thermal phase curve.