Mention red dwarf habitable zones and tidal lock invariably comes up. If a planet is close enough to a dim red star to maintain temperatures suitable for life, wouldn’t it keep one face turned toward it in perpetuity? But tidal lock, as Ashley Baldwin explains in the essay below, is more complex than we sometimes realize. And while there are ways to produce temperate climate models for such planets, tidal lock itself is a factor in not just M-dwarfs, but K- and even G-class stars like the Sun. Flip a few starting conditions and Earth itself might have been in tidal lock. The indefatigable Dr. Baldwin keeps a close eye on the latest exoplanet research, somehow balancing his astronomical scholarship with a career as consultant psychiatrist at the 5 Boroughs Partnership NHS Trust (Warrington, UK). Read on to learn a great deal about where current thinking stands on a subject critical to the question of red dwarf habitability.

by Ashley Baldwin

“Tidal locking”, “captured rotation” or “spin-orbit locking” etc occurs in most recognised guise when an orbiting astronomical body (be it a moon, planet or even a star) always presents the same face towards the object it is orbiting. In this instance, the orbit of the “satellite” body can be referred to as “synchronous”, whereby the tidally locked body takes as long to rotate around its own axis as to orbit its partner. This occurs due to the primary body’s gravity flexing the orbiting body into an elongated “prolate” shape. This in turn is then exposed to varying gravitational interaction with the central body.

Figure 1: Tidal stresses and tidal locking

As the “orbiter” rotates, its now elongated axis falls out of line with the central mass, which consequently perturbs it as it rotates across its orbit. It thus becomes subject to gravitationally induced torques that can act as a brake — through energy exchange and dissipation, the latter via friction-induced heat loss in the perturbed orbiting body. As M dwarf habitable zones are closer to their central star and their gravitational influence thus greater, it’s easy to see how this dissipated heat can contribute substantially to an exoplanet’s overall energy flux and can even affect its habitability potential – possibly tipping it into a runaway greenhouse scenario. (Kopparapu 2013).

Over millions of years (or more) this process can lead to “orbital synchronisation”. This arises when the orbiting body reaches a state where there is no longer any net exchange of rotation during the course of a completed orbit (Barnes 2010). Leaving a tidal locking state would only be possible with the addition of energy to the system. This might occur should some other massive object (such as a planet, or a star in, say, a binary system) break the equilibrium. If the masses of the two bodies (for instance Pluto & Charon) are similar, they can become tidally locked to each other.

Not all tidal locking involves synchronisation. “Super-synchronisation” occurs where an orbiting body becomes tidally locked to its parent body but rotates at a fixed but quicker rate. A topical example of this is the erstwhile “geosynchronous transfer orbit” (GTO). We see this on launcher specs all the time: “Payload to GTO”. This orbit is external to geosynchronous orbit, where many satellites start their operational lives, but allows for pre-orbital insertion inclination changes — economically expending less propellant prior to final insertion. Alternatively, such orbits can be used as dumping grounds for non-functioning satellites or related debris, so-called “geo-graveyard belts” (Luu 1998). Simulations suggest many exoplanets could exist in variants of such orbital types.

Gravitational interaction with a central star leads to progressive rotational slowing of a smaller planetary body like Mercury via energy exchange and heat dissipation. This is due to subtle but important tidal force variations across the orbiting body (remembering that gravity is inversely proportional to the square of the distance between any two bodies — thus “gravitational gradients” exist across solid bodies, leading to bulges). However, if the initial planetary orbit is significantly eccentric, this effect varies substantially across the orbital period (especially at periapsis — the point of strongest gravitational interaction) and can instead result in a spin-orbit resonance. In Mercury’s case, this is 3:2 (three rotations per two orbits) but other ratios can occur from 2:1 through 5:2 (Mahoney 2013). It’s worth noting that this effect is most pronounced for closer-in planets where the gravitational effects are greatest, so the effect should be even more relevant for the tightly packed exoplanetary architectures (e.g. TRAPPIST-1) that seem to be prevalent.

In extreme cases where the orbiting body’s orbit is nearly circular AND has a minimal or zero axial tilt — such as with the Moon — then the same hemisphere (libration allowing) faces the primary mass.

That said, for simplicity we will now assume that a smaller mass body (exoplanet) is orbiting a very much more massive body (star) — this is the focus of this review, with an unavoidable nod towards habitability.

For reasons of brevity and also pertaining to the exoplanet subject matter of recent posts, we will limit ourselves to the specific case of terrestrial exoplanets and their orbits around smaller main sequence stars.

The time to tidal locking can even be described by the adapted equation :

Tlock ≈ wa6 (0.4 m*R2) / (3 Gmp2 kR5) (Goldreich, Goldreich & Soter 1966); (Peale 1977); (Gladman 1996); (Greenberg 2009)

Where Tlock is “time to tidal locking”, w and k are constants which can be ignored for simplicity, m* is mass of the star, mp is mass of the planet, R is the exoplanet radius and “G” is Newton’s all important gravitational constant.

Tlock is substantially lengthened by “a” — increasing planetary semi-major axis (to the sixth power!). Tidal locking time is also increased by 0.4 X m* in this equation. However it is important to remember the context and just how massive a star, indeed ANY star, is — even an M dwarf star — many times, orders of magnitude even, more massive than a planet. A star thus plays the major role in the tidal locking of its attendant planets.

The gravitational constant G ensures that increasing stellar mass will substantially decrease Tlock. All other things being equal, increasing stellar mass is a major factor in reducing time to tidal locking.

Figure 2: Stellar mass & type versus semi-major axis orange / red graph with superimposed Tsync for 0.1,1 and 10 gigayear times for an Earth mass planet. (Penz 2005)

The concept of synchronisation is relatively new, dating back to Stephen Dole’s seminal Habitable Planets for Man at the beginning of the space age in the early 1960s. The concept was purely theoretical, with somewhat arbitrary parameters at this point, but it implied that tidal lock would be a major impediment to the human-friendly “habitable” exoplanets Dole had in mind for his book. It was here that tidally locked orbits and planets in M-dwarf systems were first linked, in a negative way that to some extent still exists today (before we even get to coronal mass ejections, EUV and stellar flares et al !) Atmospheric collapse due to freezing out on the side of the planet facing away from the star is not the least of these problems.

It was only in 1993 that Kasting et al employed sophisticated 1-D climate modelling as part of describing what constituted habitable planets. Habitable planets essentially now meant planets with conditions that could sustain liquid water on their surfaces. This is a rather lower bar than that set by Dole thirty years earlier, but far more applicable and still a pillar of exoplanet science today. More importantly, Kasting’s team also simulated star/planet gravitational interaction.

They did this by utilising the “Equilibrium Tide” model (ET). Refined variants of this have now become THE staple of all subsequent related studies, as it too has “evolved”. The model essentially assumes that the gravitational force of the tide-raiser (star) produces an elongated shape in the perturbed body (exoplanet) and that its long axis is slightly misaligned with respect to an imaginary line connecting the two centres of mass.

The misalignment is crucial and is due to the dissipating processes within the “deformed” exoplanet, leading to evolution of the orbit and spin angular moments. From this, various equations can be created which map out the orbital and rotational evolutionary history of exoplanets over time (see above). ET was originally derived from the Earth/Moon system by Darwin in 1880 before refinement by Pearle in 1977. Iterations vary in subtle but significant ways and are used as the basis for increasingly sophisticated simulations as computing power increases. Barnes 2017 has carried out a detailed review of synchronising and ET modelling (see below).

Kasting et al showed synchronisation of putative exoplanets orbiting in the habitable zones of M-dwarfs, stars with a mass of up to 0.42 Msun, within 4.5 billion years. They introduced the now familiar term “tidal locking radius”. Though a big step forward, this had the unfortunate consequence of continuing to propagate a pessimistic view of habitable exoplanets orbiting such stars. Importantly, stellar mass was still viewed as the major if not sole cause of synchronisation. The graph below (from Yang et al 2014), though based on sophisticated modelling, still captures this type of thinking. Here various habitable zone model ranges are superimposed on a graph of relative stellar insolation (and star type) versus semi-major axis examples of known exoplanets, adding realistic perspective. You will note also that for a 0.42 Msun star, with a temperature around 3500 K, the 1-D inner habitable range is very close to the value attributed to recently discovered TOI 700d — mid-80s percent.

Figure 3: Temperature of star versus stellar flux graph with superimposed coloured star classes and dashed gray “tidal locking radius” line.

The effects of other factors — such as starting orbital eccentricity (already encountered above with Mercury), baseline rotation rate, the presence of companion bodies (Greenberg, Corriea 2013) thermal tides arising from atmospheres (Leconte et al 2015), and stellar and planetary interiors (Driscoll & Barnes 2015), orbital tilt (Barnes 2017) — were not considered. As can be seen, it has only been over the last five years or so that these things have been added to simulations. Indeed, the results of these studies very much alter the whole tidal locking paradigm with particular relevance to habitable zones, which despite refinement (Kopparapu 2013, Selsis 2007) have only changed slightly, a big compliment to Kasting’s work in 1993.

Taken altogether, habitable zone planets of M,K and G stars all have the potential to become tidally locked. Not just M dwarfs — though their potential remains very much the greatest and especially for < 0.1 Msun stars such as TRAPPIST-1. Even the Earth, had its starting rotation been greater than just three days, according to Barnes 2017, might have become synchronous.

For the sake of brevity, this review has largely focused on stellar mass as a major driver in exoplanetary synchronisation. As can be seen above, as knowledge in this area progresses, other processes come into account. It is also becoming increasingly difficult to tease these out from drivers of exoplanetary habitability. So to this end we must look in more detail at some of the factors named above.

The planet Venus is unusual in many ways, but one in particular stands out: its retrograde and slow rotation rate that is longer than its orbital period. Why? What makes Venus different? One factor is that it is a rocky planet with a substantial atmosphere (92 bar at its surface). We all know about the infamous runaway greenhouse effect this drives, making Venus the hottest planet in the Solar System despite being further from the Sun than (spin/orbit resonant) Mercury. However, does this atmosphere have any other effects?

On Earth, the day/night cycle leads to variations in heat distribution in the atmosphere. It is known that the hottest time of day on Earth does not occur when the Sun is at its zenith and thus nearest to the Earth, but rather several hours later. This is because of thermal inertia. There is a delay between solar heating and thermal response, leading to mass redistribution. As the atmosphere and the Earth’s surface are generally well linked via friction, this will give rise to non-negligible thermal torques.

These torques are akin to the torques arising from the Sun’s uneven gravitational interaction with the Earth described above, though not as potent. On the Earth with its extended 1 AU orbit, they are largely inconsequential, but for 0.3 AU nearer Venus, they become significant. Depending on their direction, they can either slow up OR speed up planetary rotation, but either way they help to resist synchronisation. Over time, torques arising in Venus have acted to slow down its rotation, so much so that it has reversed to the retrograde pattern we see today.

So if this is true of Venus, how about exoplanets? Can these atmospheric torques resist or at least delay synchronisation and tidal locking in vulnerable areas around a star? This has been extensively modelled by Leconte et al 2015 and the answer was a resounding yes, especially for smaller, less luminous stars with close-in habitable zones, and not just for exoplanets with 90 bar atmospheres, either. Even 1 bar Earth-like atmospheres could help resist synchronisation for the habitable zones in stars of 0.5 Mearth – 0.7 Mearth.

Ten bar atmospheres were simulated and shown to resist synchronisation even for habitable zone planets orbiting 0.3 Mearth stars (mid-M dwarfs). These are the high bar “maximum greenhouse” CO2 atmospheres that are postulated to occur in the outer regions of stellar habitable zones. But there are limits. Venus’ 92 bar atmosphere is ironically so thick that most of the incident sunlight that isn’t reflected back into space is either absorbed or scattered before it can reach the planetary surface and exert the driving effect of thermal torques (Leconte et al 2015).

Figure 4: Red arrow synchronous rotation / blue arrow asynchronous rotation graph (Leconte 2015).

Orbital synchronisation and exoplanet habitability remains a contentious theoretical field that is subject to continual debate and constant change. Modern Global Climate Modelling (GCM) has become a sophisticated sub-science. Using an earlier iteration of GCM, Yang et al showed in 2013 that synchronised M-dwarf habitable zone planets would form thick cloud banks above their sub-stellar point. This would then reflect much of the incident stellar flux, thus reducing the energy reaching the surface. In turn, this would reduce the overall energy reaching the planet and so reduce global temperatures. The net effect in theory is to extend the stellar habitable zone inwards. However, the same author collaborated with Wolf and Kopparapu in 2016 to apply an updated 3-D model to the same problem. This showed that a sub-stellar cloud bank could not form, or would form and then move, a result effectively rebutting the 2013 findings and moving the habitable zone back to its original pre 2013 starting point. Expect more of this !

So, all things considered, just how easy is it for an exoplanet to become tidally locked and just how easy can habitable zone planets become tidally locked ? Barnes 2017 attempted to address just this question for exoplanets in circular orbits. He applied two well recognised refined variants (CPL left, CTL right in the graphic below) of the ET to two model populations of exoplanets orbiting differing stellar masses, and ran thousands of giga-year simulations for each (think of the computing power and time!) One population had a starting orbital period of 8 hours and an orbital tilt of 60°. The other had a starting period of ten days and a tilt of 0°. This produced the four outcomes illustrated below. The superimposed grey shading represents the latest habitable zones (Kopparapu 2013) iteration, with the dark grey representing the “conservative” and the light the “optimistic”.

Figure 5: “Four in one” black and white stellar mass vs semi-major axis / superimposed greyscale habzone graphs.

These results are indicative and significantly different from the status quo, which is that tidal locking is only something that applies to exoplanets orbiting in close to M dwarf and smaller K dwarf stars. For one thing, even this older paradigm implies that at least some “Goldilocks” stars are not quite as homely as expected (more Kasting than Dole). The Barnes work hints at potential overlap of the habitable zone for potentially a large fraction of K-class and even many G-class stars, driven by factors beyond simple stellar mass. Clearly planets with a slow initial rotation rate and low orbital tilt are at greater risk, as may prove the case. Opposed to this are non-synchronising factors such as, inter alia, higher baseline orbital eccentricities and the close proximity of other orbiting bodies (moons, planets …thinking TRAPPIST-1 and binary stars/brown dwarfs, as with the recently described Gliese 229Ac system).

What this also shows is the inextricable link between orbital features and planet habitability. No more so demonstrated than by Kepler, and likely even more so with its greater number of short orbital period planets, with any potential habitable zone planetary candidates lying within just tenths or less of an AU from their parent star. This is very much in the “red arrow” synchronous zone in the Leconte graphic above.

There are now over 4000 known exoplanets. The current focus is on their “characterisation” and this is largely about atmospheres and biosignatures. However, it is obvious that we need to know far more about their evolving and historical orbital properties. This is a part of a process of determining habitable planets/zones, which are about so much more than stellar mass.

Most of the exoplanets discovered already by Kepler et al orbit close in to their stars, including those few in the potential tidal lock habitable zone. Ongoing Doppler photometry and TESS will identify thousands more such exoplanets, many of which will be even closer to their latest star given TESS’ shorter 27 day observation runs. TOI 700d and Gliese 229Ac are just for starters. Hopefully the search for habitability will expand to encompass the unavoidable connexion with planetary orbital features.

Know the star to know the planet, but know the orbit to know them both.

Figure 6: Stellar effects/planetary properties/planetary systems (Meadows and Barnes 2018)


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