# Continuous reorientation of synchronous terrestrial planets due to mantle convection

J. Leconte

<figure><img src="image_1.png" /></figure>

## To cite this version:

J. Leconte. Continuous reorientation of synchronous terrestrial planets due to mantle convection. Nature Geoscience, 2018, 11 (3), pp.168 - 172. 10.1038/s41561-018-0071-2. hal-01868924

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# Continuous reorientation of synchronous terrestrial planets due to mantle convection

Jérémy Leconte

Many known rocky exoplanets are thought to have been spun down by tidal interactions to a state of synchronous rotation, in which a planet's period of rotation is equal to that of its orbit around its host star. Investigations into atmospheric and surface processes occurring on such exoplanets thus commonly assume that day and night sides are fixed with respect to the surface over geological timescales. Here we use an analytical model to show that true polar wander—where a planetary body's spin axis shifts relative to its surface because of changes in mass distribution—can continuously reorient a synchronous rocky exoplanet. As occurs on Earth, we find that even weak mantle convection in a rocky exoplanet can produce density heterogeneities within the mantle sufficient to reorient the planet. Moreover, we show that this reorientation is made very efficient by the slower rotation rate of a synchronous planet when compared with Earth, which limits the stabilizing effect of rotational and tidal deformations. Furthermore, a relatively weak lithosphere limits its ability to support remnant loads and stabilize against reorientation. Although uncertainties exist regarding the mantle and lithospheric evolution of these worlds, we suggest that the axes of smallest and largest moment of inertia of synchronous exoplanets with active mantle convection change continuously over time, but remain closely aligned with the star-planet and orbital axes, respectively.

Long before their discovery, it was hypothesized that many exoplanets would be close enough to their stars to undergo tidal synchronization¹. This was supported by the synchronous rotation of all major Solar System satellites, including the Moon.

This has quite dramatic implications for the planetary climate. Because one hemisphere never receives any light from the star, it has been argued that this night side could completely trap volatiles such as water²⁻⁴, carbon dioxide⁵,⁶, or even the whole atmosphere⁷,⁸. However, the amount of volatiles that can be trapped depends crucially on various parameters—land/ocean distribution on the day side, topography¹, geothermal heat flux below ice caps⁶ and so on—that themselves depend on the planetary orientation. Maybe more importantly, the efficiency of the carbonate-silicate cycle⁹, which may control the potential presence of liquid water on numerous temperate-to-cold planets around low-mass stars, strongly depends on the insolation and precipitation over continents. Whether the substellar point lies above a continent or a large ocean¹⁰, and whether it changes over the course of the planet's lifetime, implies major changes for the atmospheric content and its stability¹¹.

To understand the atmospheric and surface processes at play on tidally spun-down planets, it is crucial to know not only whether they are in what we usually call a synchronous spin state or not—we do not discuss here eccentricity or obliquity-driven librations—but also if the orientation of their surface is truly fixed with respect to their stars over geological timescales!

Considering the first question, it has been shown that the processes that keep Mercury and Venus out of synchronicity could be at play on planets with eccentric orbits or having a dense enough atmosphere¹²,¹³. However, this still leaves us expecting plenty of synchronized close-in exoplanets.

Our goal is to address the second question, which can be recast as follows: does a planet that has been tidally synchronized always shows the same face to its star? Indeed, all the studies on the rotation of exoplanets have focused on the evolution of the planetary angular velocity vector (ω). However, as is well known in solid mechanics,

the axis of rotation of a solid body usually changes with respect to this body. In other words, in a frame rotating with this fixed axis of rotation and with an angular velocity ω ≡ ||ω|| (in our case the frame where the star is fixed), the orientation of the solid (in our case the planet) can change over time.

This process, called true polar wander (TPW), has happened and is still happening on Earth, as evidenced by both geological and historical records¹⁴,¹⁵. These records show that Earth's rotation axis has undergone large excursions (possibly up to 90°) with respect to the planet over geological timescales, thus changing continuously which parts of the surface were receiving more (equator) or less (poles) insolation. Although, on Earth, plate motion creates an extra shift of the pole relative to the surface, this is distinct from TPW. Indeed, reorientation is also observed on many Solar System bodies¹⁶. This is crucial, as many exoplanets may not be subjected to plate tectonics. Finally, note that TPW has nothing to do with the precession-nutation of the rotation axis with respect to an inertial reference frame¹⁷ that entails exchange of angular momentum, and only adds its effects to the aforementioned one.

Here, to assess the presence of TPW on synchronous planets, we will apply a simple formalism developed for the Earth which will allow us to identify the key processes and parameters involved. Then we will show how the various stabilization mechanisms ought to be inefficient on known rocky exoplanets. It turns out that even a weak convection in their mantle could create a deformation sufficient to continuously reorient the planet as the convection pattern evolves.

## Dynamics of TPW

Because of heterogeneities in their interior and deformation due to rotation and tides, planets are not truly spherically symmetric bodies. The simplest, general form that the inertia tensor, $\tilde{\mathcal{I}}$, can take is thus:

$$ \tilde{\mathcal{I}} = \begin{pmatrix} A & 0 & 0 \\ 0 & B & 0 \\ 0 & 0 & C \end{pmatrix} \qquad (1) $$
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 1 | Schematic representation of TPW driven by mantle convection on a synchronous planet.** a, Hot, low-density (lighter shading) upwelling plumes rise and cause surface uplift. The net effect is a positive geoid/mass anomaly²² (exaggerated here) that coincides with the axis of the smallest moment of inertia. In contrast, cold downwellings (darker shading) are negative anomalies, where the axis of the largest moment of inertia will lie. b, TPW will tend to align these axes with the star-planet and rotation axes, respectively. If numerous plumes are present instead of the two-cell pattern shown³⁰, the principal axes will be determined by the resulting degree-2 moment. The surface topography follows the reorientation. If plate tectonics occurs, continents will undergo an additional drift with respect to the mantle.</figcaption></figure>

where the three principal moments of inertia satisfy $A<B<C$. The axes diagonalizing the inertia tensor are the axes of the figure that define a frame attached to the solid planet and rotating with it. In this frame, the conservation of angular angular momentum, $L \equiv \hat{\mathcal{I}} \cdot \omega$, yields the Liouville equation

$$ \frac{d}{dt} \hat{\mathcal{I}} \cdot \omega + \omega \times (\hat{\mathcal{I}} \cdot \omega) = N \quad (2) $$

where N is the external torque. This equation shows that even without any torque, if $\omega$ is not collinear to one of the figure axes, the rotation axis will have to wander with time¹⁸.

Solving this equation is not trivial. However, the end state of the torque-free motion of a viscous planet can be determined by simply minimizing the energy

$$ E = \frac{1}{2} \omega \cdot \hat{\mathcal{I}} \cdot \omega \equiv \frac{\|\mathcal{L}\|^2}{2\mathcal{I}} \quad (3) $$

at constant angular momentum. The lower energy state thus corresponds to a rotation about the largest moment of inertia, that is, where the moment of inertia about the instantaneous rotation axis $\mathcal{I} \equiv \omega \cdot \hat{\mathcal{I}} \cdot \omega / \|\omega\|^2$ is equal to C, as observed for all Solar System planets. For a synchronously rotating body, the tidal potential further aligns the axis of smallest moment of inertia, A, with the tide-raising body, as is observed for all major Solar System moons.

However, in determining these axes of largest and smallest moments of inertia, one should remove the instantaneous rotational and tidal deformations¹⁴,¹⁹,²⁰. This means that the elements of the inertia tensor should first be decomposed as follows²⁰:

$$ \mathcal{I}_{i,j} = \mathcal{I}_s \delta_{i,j} + C_{i,j} + \hat{k} \frac{R^5 \omega^2}{G} \left[ \frac{1}{3} \left( m_i m_j - \frac{1}{3} \delta_{i,j} \right) - \left( e_i e_j - \frac{1}{3} \delta_{i,j} \right) \right] \quad (4) $$

where $\mathcal{I}_s$ is the spherically symmetric component of the moment of inertia, $\hat{k}$ the Love number¹⁴, e the unit vector directed toward the tide raising body, $m \equiv \omega/\omega$, R the mean planetary radius, $\delta_{i,j}$

the Kronecker symbol and G the gravitational constant. The last term accounts for induced deformation and only affects the characteristic TPW timescale ($\tau_{\text{TPW}}$)²¹. Thus, the axes of the largest and smallest moments of inertia should be determined by diagonalizing the intrinsic deformation tensor caused by all non-hydrostatic effects ($\hat{\mathcal{C}}$).

## TPW and convective cycles on Earth

Many processes cause intrinsic deformations, but evidence from the gravity field shows that, on Earth, the deformation is dominated by mantle convection²¹,²². Hot rising plumes create negative density anomalies but cause surface uplift, which results in net positive gravity anomalies²²,²³. Cold downwellings have the opposite effect. Interestingly, this convection pattern evolves over time²³ with a characteristic convective timescale ($\tau_c$), and so does $\hat{\mathcal{C}}$.

Therefore, the rotation axis of a planet should follow the axis of largest inertia created by mantle convection, as observed in Earth geological records¹⁵. In particular, major recorded TPW events seem to be due to the formation of hot upwelling plumes²²,²³, which are driven to the equator (Fig. 1). While these convective cycles drive both TPW and the aggregation/dispersal of supercontinents¹⁵,²⁴, the two latter processes can occur independently.

This qualitative behaviour follows well the quantitative theory outlined above. Following ref. ²¹, one can construct a figure of merit, $X_{\text{TPW}}$, estimating the ability of a planet to undergo TPW. Then, the maximum TPW reached for a forcing of period $\tau$ (hereafter equal to $\tau_c$) is:

$$ \text{TPW}_{\text{max}} = \tan^{-1}[\sinh X_{\text{TPW}}] \quad (5) $$

where

$$ X_{\text{TPW}} \equiv \frac{1}{2\pi} \frac{\langle \hat{\mathcal{C}} \rangle}{C-A} \frac{\tau_c}{\tau_R} = \left( \frac{3\gamma}{2\pi k_T^f} \right) \frac{\langle \hat{\mathcal{C}} \rangle}{\mathcal{I}_s} \frac{GM}{R^3 \omega^2} \frac{\tau_c}{\tau_R} \quad (6) $$

$\tau_R = (19\eta)/(2\rho_m gR)$, $M, g, k_T^f$ and $\gamma = \mathcal{I}_s / (MR^2)$ are respectively the effective viscous relaxation timescale, mass, gravity, fluid Love number and inertia factor of the planet, and $\eta$ and $\rho_m$ the effective viscosity and density of its mantle.
<figure><img src="image_5.png" /><figcaption>**Fig. 2** | $X_{\text{TPW}}$ on known rocky exoplanets as a function of their orbital period (dots). The ⊕ symbol shows the value of $X_{\text{TPW}}$ for the Earth. The colour of the dots refers to the equilibrium black-body temperature of the planet ($T_{\text{BB}}$) determined assuming a complete redistribution of the incoming stellar energy (Methods). As expected, cooler planets have longer orbital periods. All synchronous planets with an orbital period above 1 day should undergo TPW very easily. The right-hand ordinate axis shows the timescale on which the pole is able to follow the axis of the largest moment of inertia ($\tau_{\text{TPW}}$; Methods).</figcaption></figure>

$X_{\text{TPW}}$ is the ratio of the characteristic amplitude of the driving non-hydrostatic inertia anomaly ($\langle\hat{C}\rangle$) multiplied by the forcing timescale over the stabilizing hydrostatic rotational bulge ($C-A=I_s(k_T^f/3\gamma)(R^3\omega^2/GM)$) multiplied by the relaxation timescale. When $X_{\text{TPW}} \ll 1$, TPW$_{\text{max}}$ is small ($\sim X_{\text{TPW}}$). When $X_{\text{TPW}} \gtrsim 3$, the pole can shift by 90° during a single convective cycle. On Earth, the convective timescale and amplitude seem to be of the order$^{21}$ of $\tau_c \sim 100$ Myr, and $\langle\hat{C}\rangle / I_s \sim 10^{-5}$. Using the parameters from Table S1 yields $\tau_R \approx 3 \times 10^4$ yr and $X_{\text{TPW}} \approx 1-2$, meaning that large TPW events can arise over timescales of around 100 Myr or more, but rotation drastically filters out shorter events.

## The case of synchronous terrestrial exoplanets

What is the efficiency of TPW on synchronous planets? Of course, very little is known about these objects beyond a few global parameters. Fortunately, our analysis reveals that the most important parameter is the rotation period, which is equal to the orbital one for a synchronous planet.

Indeed, simple boundary layer scaling arguments predict that $\tau_c \sim \delta^2/\kappa$, where $\delta=5(\eta_{\text{conv}}\kappa/\rho_{\text{m}}\alpha\Delta T)^{1/3}$ is the boundary layer depth, $\kappa$ the thermal diffusivity, $\eta_{\text{conv}}$ the effective viscosity for convection, $\alpha$ the thermal expansion coefficient and $\Delta T$ the temperature difference driving the convection. The convective inertia anomaly can be roughly estimated using $\langle\hat{C}\rangle_{\text{conv}} / I_s \sim 0.1\alpha \Delta T \delta / R$, which gives reasonable values for Earth$^{21}$.

Interestingly, combining these scalings yields:

$$X_{\text{TPW}} \approx 0.2 \frac{\eta_{\text{conv}}}{\eta} \frac{GM}{R^3\omega^2} \quad (7)$$

where all internal parameters have disappeared except for the ratio $\eta_{\text{conv}}/\eta \approx 1/30$ for the Earth$^{21}$; Methods). Therefore, TPW efficiency is determined only by measurable quantities along with the viscosity structure. As the absolute viscosity does not appear, global variations in viscosity due to a hotter/colder interiors should not substantially affect our results$^{21,22}$.

Figure 2 shows the TPW efficiency for all known transiting planets with $R < 1.6 R_⊕$ to select terrestrial bodies$^{25}$. As anticipated, rotation rate is the determining factor, planetary density not varying much in the range of radii considered. It demonstrates that planets with an orbital period of about 1 day or more offer very favourable conditions for TPW, similar to Solar System moons$^{16}$. Because the TPW timescale is shorter than the convective one, we can expect the reorientation to follow the convective pattern quasi-statically.

<figure><img src="image_6.png" /><figcaption>**Fig. 3** | Minimal inertia anomaly ($\langle\hat{C}\rangle_{\text{min}}/I_s$) needed to excite significant polar wander as a function of the planetary temperature ($T_{\text{BB}}$). The size of the dot is proportional to the size of the planet. The ⊕ and ⊙ symbols show the convective inertia anomalies for the Earth and Venus. The shaded area illustrates the range of convective contribution predicted by our simple scaling by varying the radius of the planet between 0.5 and 1.6 $R_⊕$ and the convective viscosity within two orders of magnitude. For temperate planets, mantle convection would have to be two to three orders of magnitude less vigorous than on Earth to suppress TPW.</figcaption></figure>

One limitation of our calculation is that our convective contribution to the inertia anomaly is scaled on the Earth, where plate tectonics is strongly coupled to mantle convection$^{24}$. While there would be no continental motions on a planet without plate tectonics, there is no reason that convection could not distort its crust, or simply cause sufficient density heterogeneities within the mantle. To quantify this we computed the minimal inertia anomaly that would have to be created by mantle convection to excite TPW (Methods). Figure 3 shows that the absence of moving plates would have to reduce the distortion by two to three orders of magnitude to suppress TPW, which seems unlikely. Indeed Venus' triaxiality is only about a factor of two smaller than Earth's, despite its lack of plate tectonics.

Another impediment to a truly continuous reorientation would be a lithosphere with a permanent elasticity$^{20,26}$. For Mars, the stabilizing effect of the remnant rotational bulge inherited from the solidification of the lithosphere plays an important role in the TPW event that followed the formation of Tharsis$^{26,27}$. As the lithosphere was strong enough to support part of the Tharsis load once at the equator, the load itself became a stabilizing factor against subsequent TPW events, strongly decoupling the motion of the pole from the convective cycle.

Do we expect an equivalent stabilization by the lithosphere of exoplanets? Importantly, Earth's lithosphere does not seem able to support such loads permanently$^{28}$. This is demonstrated by the absence of a geoid signature of a remnant rotational bulge inherited from a faster, past rotation and by the weakness of the topography to geoid correlation at long wavelengths. This difference is easily explained by estimating the dimensionless rigidity of a spherical lithosphere due to membrane stresses$^{28,29}$

$$\epsilon = \frac{\epsilon d}{g R^2 \rho_*} \quad (8)$$

where $\epsilon$ and $d$ are the Young modulus and thickness of the lithosphere, and $\rho_*$ is a density characterizing the load. Decreasing with planetary mass, the rigidity is sufficient to support massive loads on Mars ($\epsilon \sim 0.5$), but not on Earth ($\epsilon \sim 0.02$)$^{28}$, and even less on larger planets.

The effect of an elastic lithosphere is assessed in Fig. 4. It shows the contribution of a potential remnant rotational/tidal bulge frozen in during the formation of the lithosphere$^{16,20,26,29}$ (Methods), and how it compares with the convective contribution. Except for the hottest planets, the elastic bulge is much weaker than the convective
<figure><img src="image_7.png" /></figure>

**Fig. 4** | Dimensionless contribution of the elastic remnant bulge to the inertia deformation tensor as a function of the planet radius for all known rocky exoplanets ($\langle\langle\hat{C}\rangle\rangle_{III}/\mathcal{I}_s$; Methods). The colour of the dot shows the equilibrium temperature of the planet. For comparison, the dashed black line shows the contribution of convective motions to the inertia tensor expected from our simple scaling ($\langle\langle\hat{C}\rangle\rangle_{\text{conv}}/\mathcal{I}_s$; see text). The shaded area illustrates the uncertainty on this prediction by varying the convective viscosity within two orders of magnitude. For all warm and temperate planets, the deformation is expected to be dominated by convective motions.

one. This results from both the low rotation rate of these objects and their lower rigidity. For the same reason, the stabilizing effect of a topographic load such as Tharsis would be reduced by orders of magnitude.

Our findings collectively suggest that reorientation of warm-
to-temperate rocky exoplanets should be continuous and con-
trolled by mantle convection, as on Earth. Hence, it should
still occur today, except for the smallest bodies devoid of tidal
heating, for which fast interior cooling may have shut down
convection.

Conclusion

There is ample evidence that Earth has undergone several episodes
of large TPW. It is thus sensible to assume that many rocky exo-
planets undergo TPW as well. Our analysis reveals that synchro-
nous planets should exhibit efficient TPW because the stabilization
by their hydrostatic and elastic bulges is weak. Interestingly, if plate
tectonics does occur on these planets, we predict that superconti-
nents should be at either the substellar or antistellar point—depend-
ing on the initial conditions—during their formation. Facilitating
the formation of hot upwellings beneath them<sup>23,24</sup>, supercontinents
are expected to sit on the axis of the smallest moment of inertia<sup>15,22</sup>,
which should align with the star.

Knowing this, it would be interesting to quantify how the par-
titioning of volatiles between the atmosphere, surface and man-
tle varies over time. By simply affecting the day-side ocean/land
fraction, TPW periodically changes, among other things, the heat
redistribution efficiency of the atmosphere and ocean towards the
night side, the weathering efficiency<sup>10,11</sup> and thus the planetary
global greenhouse gas content, temperature and climate. Changes
in the orientation of the large-scale topography and geologically
active regions may play a role as well<sup>4,6</sup>. Generally, when studying
geological processes linked to the orientation of the planet, the
carbonate-silicate cycle being one example<sup>10</sup>, we should consider
that this orientation can evolve as fast as or faster than the pro-
cesses studied.

**Methods**

Methods, including statements of data availability and any asso-
ciated accession codes and references, are available at https://doi.
org/10.1038/s41561-018-0071-2.

Received: 7 August 2017; Accepted: 24 January 2018;
Published online: 26 February 2018

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**Acknowledgements**

This project has received funding from the European Research Council (ERC) under the
European Union's Horizon 2020 research and innovation programme (grant agreement
No. 679030/WHIPLASH). The author thanks E. Couderc, R. Jolivet, L. Londeix and
F. Selsis for comments on the initial manuscript.

**Competing interests**

The author declares no competing interests.

**Additional information**

**Supplementary information** is available for this paper at https://doi.org/10.1038/
s41561-018-0071-2.

Reprints and permissions information is available at www.nature.com/reprints.

Correspondence and requests for materials should be addressed to J.L.

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Methods

**Mass-radius relationship.** The mass of rocky bodies discovered by transit surveys is often difficult to measure. To infer it from a given measured radius, we use the mass-radius relationship from ref. 31, which reads

$$R = (0.3102\text{RMF} + 0.7932) + (0.2337\text{RMF} + 0.4938)\log_{10}M + (0.0592\text{RMF} + 0.0975)(\log_{10}M)^2 \quad (9)$$

where RMF is the rock mass fraction, taken to be 0.67 as for the Earth. Being a degree-2 polynomial in $\log_{10}M$, this equation can be inverted analytically.

**Definition of the black-body temperature scale.** For most known transiting planets, the exoplanets.org catalogue provides the stellar radius, mass and effective temperature, along with the orbital semi-major axis ($R_*$, $M_*$, $T_*$ and $a$, respectively). The total power received by the planet is thus given by

$$L_p = \sigma_{\text{sb}} T_*^4 \left( \frac{R_*}{a} \right)^2 \pi R^2 \quad (10)$$

where $\sigma_{\text{sb}}$ is the Stefan-Boltzmann constant. As we only want a 'flux temperature', we equate this incoming power with the power that would be emitted by the planet if it were a black body with a uniform temperature ($T_{\text{BB}}$)

$$L_p = 4\pi R^2 \sigma_{\text{sb}} T_{\text{BB}}^4 \quad (11)$$

which yields

$$T_{\text{BB}} = \frac{T_*}{4^{1/4}} \sqrt{\frac{R_*}{a}} \quad (12)$$

This gives us a temperature scale, rather than an accurate idea about the real temperature at the surface, which can be affected by both the albedo and the greenhouse effect. With these conventions, the equilibrium temperatures are 279 K and 330 K for the Earth and Venus, respectively. The fact that $T_{\text{BB}}$ depends strongly on the stellar type explains the huge scatter observed in the orbital period-temperature relation (Fig. 2).

**Limit period for synchronization.** We can obtain a rough estimate of the maximal distance at which planets are synchronized by tides over a time $\tau_{\text{tid}}$ by equating the angular momentum to be removed ($\sim I_s \omega_0$) to the integrated tidal torque over that time. Assuming a simple constant phase lag model, one finds

$$\int \mathcal{N}_{\text{tid}} dt \approx \mathcal{N}_{\text{tid}} \tau_{\text{tid}} = \frac{3 k_T}{2} \frac{GM_*^2 R_*^5}{a^6} \tau_{\text{tid}} \quad (13)$$

where $k_T$ and $Q$ are the effective tidal Love number and quality factor. Interestingly, using Kepler's third law, we can get rid of stellar parameters, which yields

$$I_s \omega_0 = \frac{3 k_T R_*^5}{2 Q} \left( \frac{2\pi}{P_{\text{orb}}} \right)^2 \tau_{\text{tid}} \quad (14)$$

where $P_{\text{orb}}$ is the orbital period. This stems from the fact that the tidal and rotational potentials have very similar forms when the rotation rate is replaced by the orbital mean motion. Therefore, we can find the orbital period at which a planet will be synchronized independently of the stellar type. This limit orbital period below which planets are expected to be synchronous is given by

$$\begin{aligned} P_{\text{orb}}^{\text{sync}}(\tau_{\text{tid}}) &= 2\pi \left( \frac{3 k_T R_*}{2 \gamma Q GM_*} \omega_0 \right)^{\frac{1}{4}} \\ &= 245 \text{ days} \left( \frac{13}{Q} \frac{\tau_{\text{tid}}}{4.5 \text{Gyr}} \right)^{\frac{1}{4}} \end{aligned} \quad (15)$$

where the numerical estimate is for a planet the size of Earth starting with a 1 day rotation period. $Q \approx 13$ seems appropriate for Earth$^{32}$. As the oceans are expected to generate about 90% of the dissipation, we can expect $Q \approx 100$ for a dry planet such as Venus, lowering the critical period to about 150 days. Because of the crude dissipation model, and the assumptions needed on the initial spin, this should be regarded as a guide rather than a hard limit. Indeed, Venus has been efficiently spun down despite its approximately 225 day orbital period. This suggests that all the rocky planets shown in Fig. 2 could potentially be synchronous if no other process is at play.

**Effective viscosity and viscosity structure in the mantle.** In our analysis, we differentiate the viscosity that is supposed to be representative of convective

processes ($\eta_{\text{conv}}$) from the effective viscosity entering the relaxation timescale of the mantle ($\eta$). This might be necessary, as viscosity varies over several orders of magnitude throughout the mantle$^{33}$ and has a non-Newtonian behaviour. Different processes, exerting different stresses and being affected differently by the viscosity structure of the mantle, may thus exhibit different effective viscosities.

Empirically, a low convective viscosity of the order of $\eta \sim 10^{21}$ Pa s is needed to recover a convective timescale of the order of 100 Myr for Earth, which is consistent with the viscosities inferred in the upper mantle$^{31}$. However, using this viscosity to compute the viscous relaxation timescale of the mantle yields a timescale of about 100 yr, which is very short compared with the observed timescales for the postglacial rebounds. The difference can be understood by saying that relaxation has occurred only when the slower, more viscous, deep mantle has relaxed, but non-Newtonian effects can be at play as well$^{34}$. Following the authors of ref. 21, we thus use a value of $\eta \sim 30\eta_{\text{conv}}$, keeping in mind that this may vary in a complex manner from one planet to another. Note that reducing the difference between the two values here would only increase the efficiency of polar wander.

**Efficiency of TPW of a Maxwell Earth and limitations.** The ability of a planet to undergo fast TPW can be quantified through different means. In Fig. 2, we chose to represent the dimensionless efficiency $X_{\text{TPW}}$ given by equation (6), which quantifies the maximum polar motion over one excitation period ($\tau_c$) through equation (5). However, TPW can also be seen as a low-pass filter, with all perturbations with a timescale below $\tau_{\text{TPW}}$ damped. This timescale can be defined through

$$X_{\text{TPW}} \equiv \frac{\tau_c}{\tau_{\text{TPW}}} \quad (16)$$

yielding

$$\tau_{\text{TPW}} \equiv 2\pi \frac{C-A}{\langle \hat{C} \rangle} \tau_R \quad (17)$$

This is a useful timescale, as it also tells us at what rate the pole can adapt to a change in the inertia tensor. Particularly relevant here is the fact that the pole will follow the axis of maximum inertia closely if $\tau_c \gg \tau_{\text{TPW}}$ (Fig. 2).

Another metric, used in Fig. 3, is the minimum characteristic amplitude of the inertia deformation needed to cause a significant shift over one convective period. It can be defined as

$$X_{\text{TPW}} \equiv \frac{\langle \hat{C} \rangle}{\langle \hat{C} \rangle_{\text{min}}} \quad (18)$$

which yields

$$\frac{\langle \hat{C} \rangle_{\text{min}}}{I_s} \equiv 2\pi \frac{\tau_R}{\tau_c} \frac{C-A}{I_s} \quad (19)$$

In their analysis, the authors of ref. 21 disregarded a term of the order of $C/(\omega\tau_R(C-A))$, because it is indeed very small for the Earth. For very slow rotation rates, this term starts to be significant. Indeed, it has been shown that for Venus this term is responsible for the observed wobble of the rotation axis of about 0.5° around the axis of maximum moment of inertia$^{35}$. For the planets with the largest orbital periods in our sample (≥100 days) we can thus expect such small deviations from a true alignment.

Another limitation of this analysis is that the effect of the tidal bulge is neglected, as appropriate for an application to the Earth. However we are interested only in the timescale involved and not the precise trajectory followed by the planet during TPW events. To estimate the impact of the tidal bulge on this timescale, it is important to remember that, for a synchronous planet, the magnitudes of the tidal and rotational bulges are about equal. This stems from the fact that the tidal bulge scales as $M_* R^5 / a^3$, which is equal to $\omega^2 R^5 / G$ for a synchronous planet. From equation (4) we can therefore show that the stabilizing bulge is equal to

$$C - \frac{A + B}{2} = \begin{cases} 1/3(\text{rotation only}) \\ 5/6(\text{including tides}) \end{cases} \left\{ \hat{k} \frac{R^5 \omega^2}{G} \right\} \quad (20)$$

We thus expect the tidal bulge to impact the order of magnitude estimates presented here only by a factor of the order of unity, which is acceptable considering the other sources of uncertainty.

**Remnant rotational bulge supported by an elastic lithosphere.** Before a planet has cooled down sufficiently to form an elastic lithosphere, it responds to a static deforming potential hydrostatically and the final deformation is given by the last term in equation (4), where the Love number is taken to be equal to the secular, hydrostatic tidal Love number $k_T^1$, which depends on the density profile alone. For an isodensity planet $k_T^1 = 3/2$. Once a lithosphere forms, the secular tidal Love
number changes to account for its elasticity. This also creates a remnant inertia bulge supported by the lithosphere, which is given by<sup>16,20</sup>

$$ \mathcal{I}_{i,j}^{R} = (k_{T}^{f} - k_{T}) \frac{R^{5} \omega^{2}}{G} \left[ \frac{1}{3} \left( m_{i}^{*} m_{j}^{*} - \frac{1}{3} \delta_{i,j} \right) - \left( e_{i}^{*} e_{j}^{*} - \frac{1}{3} \delta_{i,j} \right) \right] \quad (21) $$

Here *e* and *m* are the unit vectors directed toward the tide-raising body and the rotation pole at the moment of the lithosphere formation.

The difficulty generally lies in the determination of $k_T$, which must account for the rheology of the whole planet. As we are concerned with order of magnitude estimates, we will assume that only the thin lithosphere supports the remnant bulge<sup>26,29</sup>. It can then be shown from equation (1), (2), (3) and (16) of ref. <sup>26</sup> that for the rotational part

$$ \mathcal{I}_{i,j}^{R} = \frac{1-c}{2} \frac{R^5 \omega^2}{G} \left( m_i^* m_j^* - \frac{1}{3} \delta_{i,j} \right) \quad (22) $$

where c is the degree of compensation that measures the resistance of the lithosphere to deformation<sup>26,29</sup>

$$ c \equiv \left[ 1 + \frac{4\epsilon}{(5+\nu)\left(1-\frac{3\rho_{\text{m}}}{\bar{\rho}}\right)} \right]^{-1} \quad (23) $$

In this formula, $\nu$ is the Poisson ratio of the lithosphere, and $\bar{\rho}$ the mean density of the planet. $\epsilon$ is given by equation (8). This formula neglects the effect of bending stresses that have been shown to be unimportant for the degree-2 perturbations considered here<sup>26</sup>. It is however valid for topographic loads as well as tidal and rotational deformations as long as the right definition of $\rho_*$ is taken in $\epsilon$. For topographic loads, $\rho_*$ must be equal to the density difference between the mantle and the load above. To compute the effect of rotation and tidal deformation, one must use<sup>29</sup> $\rho_* = \rho_m$.

By identification, this simply yields

$$ k_T^f - k_T \equiv \frac{3}{2} (1-c) \quad (24) $$

Care should be taken when using this expression, as ref. <sup>29</sup> uses a thin-shell approximation, which is most valid in the case of a homogeneous, incompressible planet. We thus test the validity of this formula by comparing the prediction for the secular Love number of Mars with published models<sup>27,36</sup> for various lithospheric thicknesses. Discrepancies are found to be below the 15% level (see Supplementary Fig. S1).

As a result, the characteristic amplitude of the dimensionless inertia anomaly linked to the remnant stabilizing bulge is given by<sup>26</sup>

$$ \frac{\langle \hat{C} \rangle_{\text{lit}}}{\mathcal{I}_s} = \frac{k_T^f - k_T \omega^2 R^3}{3\gamma GM} = \frac{1-c \omega^2 R^3}{2\gamma GM} \quad (25) $$

where factors of the order of unity due to the geometry of the tidal and rotational bulges have been discarded. This is the quantity shown in Fig. 4. In this context, the values calculated for $k_T^f - k_T$ supersede the value shown in Table S1.

Our estimate is conservative in several ways. First, the results in Fig. 4 assume a constant lithospheric thickness for all planets of 50 km, which is a commonly used value for Earth<sup>26</sup>. It can be argued that this value should be smaller for larger and/ or hotter planets because of the higher temperatures in the crust and mantle due to both the higher irradiation and the larger geothermal flux. Mars, for example, is observed to have a much thicker lithosphere. In addition, higher temperatures should tend to decrease the Young modulus of the lithosphere, further weakening the latter. Hence, for the hottest planets in our sample the increase in the size of the remnant bulge visible in Fig. 4 could be offset by these effects. These rough trends do not hold, however, if the internal heating is sufficient to initiate a heat-pipe type of cooling<sup>27</sup>. In this case, the lithosphere can be larger than predicted for a conductive, stagnant lid planet. The effect of such heat-pipe lithospheres on TPW is however difficult to predict because they tend toward an equilibrium shape as they are recycled over time. Heat-pipe cooling also reduces the convective stress in the mantle<sup>18</sup>. The overall effect of the regime of heat transfer on the properties of the elastic lithosphere and on TPW in general should be further studied using more realistic thermal/rheological models.

**Code availability.** Data processing routines are available on request from J.L.

**Data availability.** Data taken from the exoplanets.org database; retrieved in July 2017.

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