NO PSEUDOSYNCHRONOUS ROTATION FOR TERRESTRIAL PLANETS AND MOONS

The ongoing quest for extraterrestrial life has placed exoplanets and their properties into the forefront of scientific investigation. The trend has provided additional momentum to the development of a broad variety of techniques and approaches employed in the planetary sciences. For example, the recent revival of interest in mechanics of bodily tides is partly due to the importance of the planetary spin for the prospects of finding habitable worlds near other stars.

Well-known examples of dynamical equilibria achieved via tidal coupling include our Moon, which is in a 1:1 spin–orbit resonance with the Earth, and Mercury which makes exactly three sidereal rotations over every two orbital revolutions around the Sun. Similar behavior is expected of the growing number of known super-Earths—especially if their composition happens to be similar to that of the terrestrial planets of the solar system, i.e., if they have massive solid or partially molten mantles of rocky minerals.

Unfortunately, some of the published far-reaching conclusions about specific exoplanets are based on incomplete or ad hoc models which should never be used for solid materials, including those with partial melt. Both these models, introduced by Goldreich (1966) mainly for the ease of analytical treatment, predict quasi-static pseudosynchronous rotation states, with the planet being trapped in a slowly changing equilibrium state at a faster-than-synchronous rotation rate and a vanishing orbit-averaged tidal torque. Except in specific (very narrow) frequency bands, these models are incompatible with the rheological properties of realistic mantles and crusts. Analysis based on actual rheologies demonstrates the impossibility of pseudosynchronous rotation for homogeneous terrestrial objects. Whether this prohibition extends to planets and moons with internal or surface oceans remains an open issue and needs further research.

A consistent linear theory of bodily tides is based on Fourier decomposition of the tide, with subsequent inclusion of the response at each separate mode. The ensuing level of complexity has tempted many to circumvent it by developing simpler approaches. Serving as good illustrations and reflecting some qualitative aspects of the tidal interaction, such models are not necessarily applicable for quantitative purposes (Efroimsky & Lainey 2007) and should certainly be eschewed when fine features of near-resonant dynamics are explored.

2.1. The Essence of the Method

2.2. The Synchronous Spin Case

2.3. The Case of Vanishing Tidal Torque

The second important application of the constant angular lag model is the situation where the orbit-averaged tidal torque vanishes: $\langle {\cal {T}}_z\rangle =0$. Vanishing of the average tidal torque entails a dynamical equilibrium: the tidally disturbed body keeps spinning at a steady rate. A calculation of this rate, borrowed from Goldreich (1966), is presented in Murray & Dermott (1999) and is often cited in the literature. According to that development, the equilibrium is achieved, for a zero obliquity, at the spin rate of

$$\begin{equation} {\bf \dot{\theta }}_{{\rm eq}}\,=\,n\,\left(1\,+\,\frac{19}{2} e^2\,\right), \end{equation} \tag{ 2 }$$

e being the eccentricity. At first glance, the result looks unassailable. Indeed, for $\dot{\theta }=n$, the bulge is lagging behind the central line over one-half of the time period (around the periastron), so the torque accelerates the rotation. Over the other half of the period, the torque decelerates, being weaker due to a larger distance. So the state $\dot{\theta }=n$ looks unstable, as the overall average torque seems to be accelerating.

It is however well known that the Moon is not staying in this pseudosynchronous regime (which would be 3% faster than the synchronous rotation wherein the Moon is presently locked). Murray & Dermott (1999) point at the lunar quadrupole moment as the reason why the Moon is not pseudosynchronous. A deeper reason though lies in the constant geometric lag model being genuinely flawed, and in the entire calculation leading to Equation (2) being invalid.

2.4. A Major Objection Against the Constant-angular-lag Model

As well known, the generic expression for the tidal amendment to the perturbed body's potential is furnished by a Fourier series developed by Kaula (1964). We term it the Darwin–Kaula expansion, as a partial sum of that series was written down by Darwin (1879). Accordingly, the generic expression of the tidal torque also must look as an infinite series. The series remains infinite even if we include into it only the degree-2 terms, i.e., those proportional to the quadrupole Love number k₂. The very fact that the expansion for the torque can be wrapped into a short and neat form (1) is an indicator of some extra, very special assumption being involved.

As was pointed out in Williams & Efroimsky (2012), such an assumption indeed is present in the constant-angular-lag model, though this assumption is never stipulated explicitly. The situation is elucidated in detail in the paper by Efroimsky & Makarov (2013) to which we refer the reader. Here we shall provide only a brief summary.

As explained in Williams & Efroimsky (2012), the concise expression (1) presented above for the torque is equivalent to the full Darwin–Kaula expansion for the potential, only if the following assumptions are made.

• 1.  
  In all terms of the Darwin–Kaula series for the tidal amendment to the potential of the perturbed body, i.e., for all Fourier tidal modes ω_lmpq, the time lags are endowed with the same frequency-independent value Δt.
• 2.  
  The obliquity is set small.
• 3.  
  Only the terms with l = m = 2 are retained, and the Love number k₂ entering these terms is assumed frequency-independent. Here, the degree l and the order m are the first two integers of the four-number set lmpq used to number the Fourier modes showing up in the Darwin–Kaula expansion.

The so-processed Darwin–Kaula series for the tidal potential becomes equivalent to a concise expression (Equation (16b) in Williams & Efroimsky 2012) wherefrom our expression (1) for the torque ensues.

Alternatively, the above three assumptions could be applied directly to the Darwin–Kaula expansion for the tidal torque. Once again, the outcome would be the above expression (1) for the torque. This is demonstrated in Efroimsky & Makarov (2013, Equation (34)).

Under the three assumptions, the instantaneous geometric lag angle turns out to be ²

$$\begin{eqnarray} &&\epsilon _g\,\equiv (\dot{\nu }-\dot{\theta })\,\Delta t, \end{eqnarray} \tag{ 3 }$$

Δt being the frequency-independent time lag, ν being the true anomaly of the perturber, and θ being the sidereal angle of the tidally perturbed body. From this expression, it is straightforward that the validity of formula (3) is incompatible with the geometric lag being constant. Indeed, the road to Equation (3) is paved with the aforementioned three assumptions, one of which being that of a constant Δt. As can be observed from Equation (3), the latter is incompatible with the geometric lag being constant, unless the eccentricity is nil.³

On all these grounds, the constant-geometric-lag (constant-torque) model should be discarded as such.

As distinct from the constant geometric lag approach, the constant time lag model sets the time delay Δt independent of the tidal mode frequency. Pioneered by Darwin (1879), this assumption was a part of numerous works, e.g., Hut (1981) and Eggleton et al. (1998). The assumption was also the base for one of the two models considered by Goldreich & Peale (1966, Equation (23)), the other model addressed in that paper being the constant geometric lag method discarded before.

The constant time lag model is unique, in that it makes the full Darwin–Kaula expansion for the tidal potential (or torque) equivalent to a much shorter and simpler expression. In regard to the tidal potential, this is the equivalence of the full series (18) and a simpler expression (17) in our preceding paper Efroimsky & Makarov (2013). In application to the torque, this is the equivalence of the appropriate full series to the simpler expression (34) in Efroimsky & Makarov (2013).

If we agree to limit our approximation to the lowest degree and order, l = m = 2, the aforementioned simpler expressions read as

$$\begin{eqnarray} U({\boldsymbol r})&=& {-}\frac{3}{4} {\it GM}_{1}\,k_{2} \frac{\,R^{5}\,}{\,r^6\,} \cos (2\,(\dot{\nu }-\dot{\theta })\,\Delta t)\, \nonumber\\ &=& \frac{3}{4} {\it GM}_{1}\,k_{2} \frac{\,R^{5}\,}{\,r^6\,} \cos (2\,|\,\dot{\nu }-\dot{\theta }\,|\,\Delta t), \end{eqnarray} \tag{ 4 }$$

for the potential, and as

$$\begin{eqnarray} {\cal {T}}_z &=& \frac{3}{2} {{\it GM}_{1}^{\,2}}\,k_{2}\;\frac{\,R^{5}\,}{\,r^6\,}\;\sin (2\,(\dot{\nu }-\dot{\theta })\,\Delta t) \nonumber\\ &=& \frac{3}{2} {{\it GM}_{1}^{\,2}}\,k_{2}\;\frac{\,R^{5}\,}{\,r^6\,}\;\sin (2\,|\,\dot{\nu }-\dot{\theta }\,|\,\Delta t) \,\mbox{Sgn}\,(\dot{\nu }-\dot{\theta }) \,,\quad \end{eqnarray} \tag{ 5 }$$

for the polar torque.

A detailed derivation of Equation (4) can be found in Williams & Efroimsky (2012), while derivation of Equation (5) is offered in Efroimsky & Makarov (2013).

To average the torque, it is instrumental to insert into Equation (5) the distance r expressed through the semimajor axis a, eccentricity e, and true anomaly ν, and to integrate over the orbital cycle. The procedure gets simplified greatly for small lags, when one can substitute the sine with its argument. Then the calculation (presented in detail in the Appendix to Williams & Efroimsky 2012) renders

$$\begin{eqnarray} &&\langle \,{\cal {T}}_z \,\rangle \propto \left[ \frac{ 1 + \frac{15}{2} e^2\,+ \frac{45}{8} e^4\,+ \frac{5}{16} e^6 }{(1 - e^2)^{{{6}}}} - \frac{\,\dot{\theta }\,}{\,n\,} \frac{ 1 + 3 e^2\,+ \frac{3}{8} e^4 }{ (1 - e^2)^{{{9/2}}}}\right] \,. \nonumber\\ \end{eqnarray} \tag{ 6 }$$

This expression was obtained by Eggleton et al. (1998), though its equivalent was present in an earlier paper by Hut (1981, Equation (11)). In a somewhat disguised form, this expression can be found in a much earlier work by Goldreich & Peale (1966, Equation (24)).

We find readily that the equilibrium (i.e., vanishing of the average tidal torque) is achieved at

$$\begin{equation} \dot{\theta }_{\rm equ} = n \left[\,1\,+\,6e^2\,+\, \frac{3}{8} e^4\,+\, \frac{173}{8} e^6\,+\,O(e^8)\,\right] \,. \end{equation} \tag{ 7 }$$

Note that the pseudosynchronous rate of rotation depends only on the mean motion and eccentricity. This enables us to solve Equation (7) with respect to e, for a fixed dimensionless spin rate $\dot{\theta }/n$. The outcome will be the equilibrium eccentricity e_equ, i.e., the eccentricity that ensures the vanishing of the average torque at a certain value of $\dot{\theta }/n$. In Figures 1 and 2, the equilibrium eccentricity e_equ is depicted as a function of $\dot{\theta }/n$. For the model leading to expression (6) for the torque, this is a monotonically rising curve.

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The curve divides the plane into two parts corresponding to the two opposite signs of the average polar torque $\langle {\cal {T}}_z\rangle$. While $\langle {\cal {T}}_z\rangle$ is positive (accelerating) everywhere above the curve, it stays negative (decelerating) everywhere below the curve. Indeed, if we fix the eccentricity and make $\dot{\theta }/n$ very large, this will guarantee us that we get into the lower right part of the picture, i.e., below the rising curve. In this situation, i.e., for a fixed eccentricity and a sufficiently swift spin, the second term of the torque (6) must be leading, wherefore the torque must be negative, i.e., despinning. Similarly, by fixing the eccentricity and making the spin rate very small, we ensure getting into the upper left part of the picture, and also ensure that the first term in Equation (6) is leading, so the torque is positive, i.e., accelerating the spin. Since the smoothly rising curve⁴ $e_{{\rm equ}}(\dot{\theta }/n)$ corresponds to a zero $\langle {\cal {T}}_z\rangle$, it is impossible to change the sign of $\langle {\cal {T}}_z\rangle$ without crossing the curve.

Within the constant time lag model, the function $e_{{\rm equ}}(\dot{\theta }/n)$ being a smoothly rising curve explains the emergence of pseudosynchronism. To see this, consider a point on this curve, corresponding to a certain pseudosynchronous state. A small perturbation in $\dot{\theta }/n$ makes the tidally perturbed body rotate either faster or slower than the pseudosyncronous rate, and a nonzero tidal torque emerges. Illustrated by the two counter-directed short arrows on the plot in Figure 1, the tidal torque is restoring, in that its action is opposite to the sign of perturbation. Thus, the tidal torque will return the perturbed rotator to the equilibrium state, i.e., to the initial position on the curve. So the equilibrium is stable. ⁵

The constant time lag model ignores the important contribution of rigidity (Segatz et al. 1988) and inelasticity (Karato & Spetzler 1990) to the tidal response of Earth-like planets. As a result, the model is incapable to account correctly for creep. As will be discussed in the following section, the above derivation of quasi-stable pseudosynchronism, from the linear torque model, is inapplicable to Earth-like planets with rigid mantles. However, in the viscous limit, this model may still be applicable to celestial bodies that do not have appreciable rigidity or inelasticity, such as gaseous planets and stars. Observations of binary stars, especially of short-period active stars on eccentric orbit, hold the best prospect of proving or disproving the linear torque model for this type of object (Ferraz-Mello 2012; Torres et al. 2010).

The spin rate of active stars can be inferred from the characteristic periods of photometric variations caused by the passage of large spots or groups of spots across the visible disk of the star. The orbital period and the eccentricity are determined from spectroscopic radial velocity measurements. We find somewhat conflicting evidence for the existence of pseudosychronism in binary stars. Some stars with considerable eccentricities appear to have pseudosynchronous rotation (Hall 1986; Fekel et al. 1998), which is consistent with the original prediction by Hut (1981). Other stars clearly rotate faster or slower than the predicted rate (Fekel et al. 1993; Strassmeier et al. 2011). Even more puzzling, a significant number of tight binary systems have been found on circularized orbits, albeit spinning clearly asynchronously. This fact comes into contradiction with one of the important predictions of the linear torque theory: that synchronization (or pseudosyncronization) of rotation is achieved much sooner than circularization (P. P. Eggleton 2011, private communication). Thus, the impression created by the current body of observations is that the constant time lag model is, at least, not universally applicable to stars. This should not come as a surprise, because there exist theoretical indications that stars may have magnetic rigidity (Williams 2004, 2005, 2006; Ogilvie 2008; Garaud et al. 2010). ⁶

Finally, it should be mentioned that, contrary to a common belief, the purely viscous model does not render a frequency-independent time lag at all frequencies. Stated differently, the purely viscous model does not imply that the factors k_l(ω_lmpq)sin _l(ω_lmpq) are linear functions of the tidal mode ω_lmpq for all values of the mode. It can be demonstrated that this linearity takes place at low frequencies, but gets violated at frequencies higher than Gρ²R²/η, where G, ρ, R, and η are the Newton gravity constant, mean density, radius, and the mean viscosity of the perturbed body, respectively. We shall address this topic elsewhere.

To build a consistent theory of bodily tides, one has, first, to decompose the tide into a Fourier series and, second, to attribute to each Fourier component its own phase delay and magnitude decrease (the latter being expressed by the Love number appropriate to the said Fourier mode). Development of the decomposition technique was started by Darwin (1879) and accomplished in full by Kaula (1964). Attribution of phase delays and Love number values to the Fourier modes took a much longer time, because of the necessity to explore rheological properties of the mantle at various frequencies. This exploration, by both seismological and geodetic methods, has been going on intensively through the past dozens of years. Merger of the Darwin–Kaula decomposition technique with the results from solid-Earth rheology is explained in Efroimsky (2012a). The paper relied on a combined rheological model (Andrade at higher frequencies, Maxwell at lower frequencies), because of this model's ability to best match laboratory experiments and both seismic and geodetic measurements of dissipation over a range of frequencies in the solid Earth.⁷ The merger of the Darwin–Kaula expansion with the combined rheological model, worked out in Efroimsky (2012a), has been used to predict spin–orbit resonances of a Mercury analogue having a constant eccentricity and a zero obliquity (Makarov 2012) and tidal properties of super-Earths (Efroimsky 2012b).

4.1. Expression for the Tidal Torque

It is explained in Appendix A that the average polar component of the tidal torque can be approximated with the following expression, provided that (1) the obliquity is small, and (2) the perturbed body and the perturber are not too close to one another (so only the terms with degree-2 Love number are important):

$$\begin{eqnarray} \langle \,{\cal {T}}_z\rangle _{{l=2}} &=& \frac{3}{2} \frac{\,{\it GM}_{1}^{\,2}}{a}\,\left(\frac{R}{a}\right)^{5}\sum _{q=-1}^{7}\,G^{\,2}_{{20q}}(e) k_2(\omega _{{220q}}) \nonumber\\ &&\times \sin \epsilon _2(\omega _{{220q}})\,+O(e^8\,\epsilon)+ O(i^2\,\epsilon) \quad \quad \quad \end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray} &&=\frac{3}{2} \frac{\,{\it GM}_{1}^{\,2}}{a}\,\left(\frac{R}{a}\right)^{5}\sum _{q=-1}^{7}\,G^{\,2}_{{20q}}(e) k_2(\omega _{{220q}}) \sin |\,\epsilon _2(\omega _{{220q}})\,| \, \nonumber\\ &&\quad \times \mbox{Sgn}\,(\omega _{220q})+O(e^8\,\epsilon)+O(i^2\,\epsilon). \quad \end{eqnarray} \tag{ 8b }$$

This is the polar (orthogonal to the equator) component of the torque wherewith the tidally perturbed body is acted upon by the perturber. The angular brackets denote orbital averaging, G stands for the Newton gravitational constant, M₁ signifies the mass of the perturber (the star, if the perturbed body is its planet; or the planet, if the perturbed body is a satellite), a is the semimajor axis, while R is the radius of the tidally perturbed body. The degree-2 dynamical Love number k₂ and the phase lag ₂ are functions of the Fourier tidal mode

$$\begin{eqnarray} &&\omega _{{220q}}\,= (2 + q) n - 2 \dot{\theta }{.} \end{eqnarray} \tag{ 9 }$$

While the dynamical Love numbers k₂(ω_220q) are positive definite, the sign of each phase lag ₂(ω_220q) coincides with that of the Fourier mode ω_220q, as can be understood from formulae (A4) and (A5) in Appendix A. It is for this reason that the products k₂(ω_220q) sin ₂(ω_220q) emerging in expression (8a) are rewritten in expression (8b) as k₂(ω_220q) sin | ₂(ω_220q) | Sgn (ω_220q).

A generic expression for the torque implies summation over the four integer indices lmpq serving to number the Fourier tidal modes ω_lmpq entering the spectrum—see Appendix A below. The terms of that series also depend upon the obliquity. As the lmpq term contains a factor (R/a)^{2l + 1}, it is often sufficient to keep only the degree-2 terms (l = 2). In this case, with an extra assumption of small obliquity, it is enough to limit the summation to the terms with m = 2, p = 0. This gives expression (8).

While the full expression for the torque implies summation over all integer values of q, numerical tests demonstrate that, for eccentricities not exceeding ∼0.3, it is enough to take into account the terms up to e⁷, inclusive. This would require summation from q = − 7 through q = 7. However, the values of the numerical factors entering the eccentricity polynomials G_20q(e) are such that in practice it turns out to be sufficient to include only the terms with q varying from − 1 through 7.

4.2. The Tidal Torque and the Equilibrium Eccentricity as Functions of the Spin Rate

Expression (9) makes each product k₂ sin ₂ a function of the planetary spin rate $\stackrel{\bf \centerdot }{\theta }$:

$$\begin{eqnarray} && k_2(\omega _{{220q}})\;\sin |\,\epsilon _2(\omega _{{220q}})\,| \,\mbox{Sgn}\,(\omega _{{220q}}) = k_2(2(n-\dot{\theta })\,+\,q\,n) \nonumber \\ &&\quad\times \sin |\,\epsilon _2(2(n-\dot{\theta })\,+\,q\,n)\,| \mbox{Sgn}\,(2(n-\dot{\theta }) \,+\,q\,n)\,. \quad \end{eqnarray} \tag{ 10 }$$

Consequently, the entire sum (8) can be treated as a function of $\stackrel{\bf \centerdot }{\theta \,}$. The mean motion and eccentricity will play the role of parameters whose evolution is much slower than that of $\stackrel{\bf \centerdot }{\theta}$.

Each product k₂ sin ₂ is an odd function of the tidal mode ω_220 q and has the shape of a kink centered around ω_220 q = 0. When we employ relation (10) to write these products as functions of the spin rate, the new functions will still be kinks, though centered around $\dot{\theta }=n(1+{q}/{2})$. In Figure 3, the dotted line depicts the product⁸

$$\begin{eqnarray} &&k_2(\omega _{{2202}}) \sin \epsilon _2(\omega _{{2202}})\,=\, k_2(4\,n\,-\,2\,\dot{\theta }) \sin \epsilon _2(4\,n\,-\,2\,\dot{\theta }) \nonumber \\ &&= k_2(4\,n\,-\,2\,\dot{\theta }) \sin |\,\epsilon _2(4\,n\,-\,2\,\dot{\theta })\,|\, \mbox{Sgn}\,(4\,n\,-\,2\,\dot{\theta })\,. \nonumber \\ \end{eqnarray} \tag{ 11 }$$

The kink shape of the k₂ sin ₂ products is determined by the rheological properties of the planet and its self-gravitation (see Appendix B for details and references). A kink transcends nil and changes sign continuously as the spin rate goes through the appropriate resonance.

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In the sum (8), the kink-shaped products stand with multipliers G²_20q(e). So the overall tidal torque (8), as a function of $\dot{\theta }$, is a superposition of many kinks having different magnitudes and centered at different resonances (nine kinks, if we sum over q = − 1, . . ., 7). The ensuing curve will cross the horizontal axis in points extremely close to the resonances $\stackrel{\bf \centerdot }{\theta }\,\,=n\,(1+\,{q}/{2})$, but not exactly in these resonances—like the solid line in Figure 3.

4.3. The Physical Meaning of the Kink

The physical forcing frequencies in the mantle, χ_220q, are absolute values of the Fourier modes:

$$\begin{eqnarray} &&\chi _{{220q}}\,= |\,\omega _{{220q}}\,|\,. \end{eqnarray} \tag{ 12 }$$

The dynamical Love number is an even function of the tidal mode ω_220q, while the phase lag is an odd function. For this reason, the product k₂(ω_220q) sin _l(ω_220q) can be rewritten as a function of the physical frequency χ, multiplied by the sign of the appropriate Fourier mode:

$$\begin{eqnarray} k_2(\omega _{{220q}})\,\sin \epsilon _l(\omega _{{220q}})\,&=& k_2(\chi _{{220q}})\,\sin |\epsilon _2(\chi _{{220q}})|\,\nonumber \\ &&\times {\mbox{Sgn}}\,(\omega _{{220q}})\,. \qquad \end{eqnarray} \tag{ 13 }$$

Outside the inter-peak interval (i.e., at frequencies that are not too low), the positive definite quantity ⁹ k₂(χ) sin |₂(χ)| is decreasing monotonically with increase of the frequency χ = χ₂₂₀₂. This happens for two reasons. One, intuitively obvious, is that the dynamical Love number decreases at higher frequencies, because materials are inertial, and it is getting harder for their shape to keep up with the varying stress as the frequency goes up. Less obvious is the circumstance that the sine of the phase lag (i.e., the inverse tidal quality factor), too, decreases as the frequency grows.¹⁰ Supported by a mighty volume of seismological, geodetic, and laboratory data, this behavior may look counterintuitive because this is not what one would expect from a viscous fluid. The fact, however, is that at physically interesting frequencies the mantle behaves not as a viscous or a Kelvin–Vogt body but as an Andrade body dissipation wherein obeys the law sin ∝χ^− α, with α ≈ 0.14–0.4 for most solids (Efroimsky 2012a, 2012b).

Finally, the steep (but still continuous) near-resonant jumps connecting the peaks of the kink are explained by the fact that at those locations self-gravitation "beats" rheology (Efroimsky 2012a, 2012b).

The kink shape of k₂ sin ₂ (generally, of k_l sin _l) entails somewhat counterintuitive consequences for the phase lag and the geometric lag angle. Consider the principal, semidiurnal bulge. Its phase lag and the geometric lag angle are

$$\begin{eqnarray} &&\epsilon_2 (\omega_{{2200}})\,=\,\omega _{{2200}}\,\Delta t_2(\omega_{{2200}})\,=\,2\,(n\,-\,\dot{\theta })\,\Delta t_2(\omega_{{2200}}) \qquad \end{eqnarray} \tag{ 14 }$$

and

$$\begin{eqnarray} &&\delta_2 (\omega_{{2200}}) = \frac{1}{2} |\,\epsilon_2 (\omega_{{2200}})\,|\,=\,|\,n\,-\,\dot{\theta }\,| \Delta t_2(\omega_{{2200}})\,. \end{eqnarray} \tag{ 15 }$$

Were a planet composed of a material with the time lag Δt₂(ω₂₂₀₀) insensitive to the value of the principal tidal mode $\omega _{{2200}}\,=\,2\,(n\,-\,\dot{\theta })$, the geometric lag angle would be larger for a higher value of this frequency. This indeed is what one would, intuitively, expect: the higher the difference between $\dot{\theta }$ and n the larger the angle. However, for a realistic rheology, an increase of the δ₂₂₀₀ angle due to an increase in $2\,|n\,-\,\dot{\theta }|$ will take place only within an extremely close proximity of the 1:1 resonance. Stepping beyond the kink's peak, we shall find that an increase in $2\,|n\,-\,\dot{\theta }|$ will be accompanied by such a decrease in the time lag Δt₂(ω₂₂₀₀) that the product of these two quantities will, overall, be decreasing with growing frequency. So both the phase lag and the geometric lag angle will become smaller.

4.4. Instability of Pseudosynchronous Rotation: Physical Interpretation

4.5. On the Choice of the Values for Physical Parameters

We have determined that a stable spin–orbit equilibrium is achieved at spin rates where the value of the secular polar tidal torque is zero, while the derivative of the equilibrium eccentricity with respect to the spin rate, ${de_{{\rm equ}}}/{d(\dot{\theta }/n)}$, is negative. With the realistic tidal model discussed in Appendix B, this may happen only in the vicinity of spin–orbit resonances because the derivative of the torque is positive elsewhere. The secular torque (8), as well as the angular acceleration caused by it, has "kinks" in the vicinity of spin–orbit commensurabilities $\dot{\theta }/n\,=\,1\,+\,q/2$, with an integer q of either sign. An example thereof is shown in Figure 4 for the super-Earth model, within a segment of the spin rate around $\dot{\theta }/n=5/2$. The kink is comprised by a local maximum below the resonance and a minimum above the resonance.

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To understand the plot in Figure 4, recall that in the vicinity of each resonance q', i.e., for $\dot{\theta }/n$ close to 1 + q'/2, the right-hand side of Equation (8) can be decomposed into two parts. One part is the q = q' term. It is a kink-looking odd function of the tidal mode ω_220q, and it goes through nil at exactly ω_220q = 0. From Equation (9), this term can also be interpreted as a function of the spin rate, antisymmetric with respect to the resonance point $\dot{\theta }/n\,=\,1\,+\,{q^{\prime }}/{2}$, where this term goes through nil. The second part, called bias, is the rest of the sum, i.e., the input of all the q ≠ q' Fourier modes into the values assumed by the torque in the vicinity of the q = q' resonance. The bias can be negative or positive in value, depending on the eccentricity. For not too large eccentricities, it is usually negative. Being a very slowly changing function within the resonance interval, it can, to a good approximation, be assumed constant there.

Having summed up all the terms in Equation (8), and exploring the behavior of this sum near $\dot{\theta }/n\,=\,1\,+\,{q^{\prime }}/{2}$, we see that the resulting curve does not cross the horizontal axis at $\dot{\theta }/n\,=\,1\,+\, {q^{\prime }}/{2}$. One can say that the bias slightly displaces the location of zeros. The zeros are located close to resonances but not exactly in resonances.

In Figure 4(a), the values of the overall torque (in fact, of the total angular acceleration proportional to the torque) are defined mostly by the q = 3 term which, in this vicinity, looks like an antisymmetric kink. However, the curve is shifted down due to the bias which is defined mainly by the right slope of the q = 1 kink located to the left. Since the right slope of the q = 1 kink is negative, the q = 3 kink is shifted down. As a result, the maximum secular torque barely rises above zero, and the curve has two zeros located to the left of the point $\dot{\theta }/n\,=\,5/2$, close to the maximum of the kink. The interval of the resonance is defined by the location of the peaks: $\dot{\theta }/n=2.4998$ and 2.5002.

Figure 4(b) is a blow-up of Figure 4(a) showing in greater detail the area around the maximum of the kink. The root of the function within the resonance interval is at 2.49985 rather than exactly 2.5. For the same reason, the torque value is negative at $\dot{\theta }/n=2.5$.

In the framework of this model, it is reasonable to define capture in resonance as an equilibrium state in which the body's average rotation rate stays between the values corresponding to these two peaks. This definition is adequate because, as we saw above, stable equilibrium is possible only on negative slopes of the angular acceleration (or tidal torque) as a function of the spin rate.

When a triaxial planet is captured in a 5:2 resonance, its time-averaged spin rate is exactly 2.5 n. Similarly, the Moon's spin rate, captured in synchronous rotation, is exactly 1 n. However, we know that the time-averaged tidal torque is nonzero at this spin rate. Why does the Moon not accelerate? For a triaxial body, the nonzero secular tidal torque is compensated by a counteracting triaxiality-caused torque, through a small tilt of the average inertia axis with respect to the line connecting the centers of the bodies. A nonzero time-averaged tilt generates a secular triaxial torque. This mechanism of torque compensation obviously does not work for the rather hypothetical case of axisymmetric body, which would be subject to only tidal forces. Would the Moon be facing the Earth always with the same side if it were completely axisymmetric? First, we have to find out if capture in spin–orbit resonance is at all possible. The answer is yes, as long as the secular torque changes sign in the vicinity of that resonance.¹² Even though the maximum torque in Figure 4 barely rises above zero at spin rates below the resonance, it turns out to be sufficient for capture in 5:2. Figure 5 displays the results of a numerical integration of the evolution of spin rate for the super-Earth model (Table 1) at τ_M = 50 yr with an initial spin rate of $\dot{\theta }(0)= 2.51\,n$. The planet is captured in about 8500 yr, but the equilibrium spin rate at 2.49985 n is clearly below the resonance value. This value is consistent with the root of the secular torque within the resonance interval, Figure 4(b). Thus, the equilibrium resonance state of an axisymmetric body is achieved at the spin rate where the secular tidal torque equals zero, as expected.

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As was demonstrated above, stability of pseudosynchronous spin hinges upon rheology. Being stable for the constant time lag model, the regime is expected to be transient for realistic mantles, insofar as their k₂(χ)sin ₂(χ) has one pronounced peak (not counting the opposite one at the negative value of the tidal mode)—see Figure 3. The situation will have to be re-analyzed for bodies of complex structure (with surface or internal oceans) as well as for bodies of yet unexplored rheologies. Specifically, if it happens that somewhere in the universe there exist bodies with not one but two pronounced peaks of k₂ sin ₂ at positive frequencies, the "ditch" between these peaks may, in principle, lead to the emergence of a pseudosynchronous rotation state. Such a peak may emerge at the boundary of two frequency bands dominated by different friction mechanisms, i.e., when a new mechanism is "turned on" very quickly with the increase of frequency. Although highly hypothetical, such situations should not be written off completely.

The shortness of Earth's day was undoubtedly beneficial for proliferation of biological life, making the daily temperature variation moderate. The situation may be drastically different for the growing class of potentially habitable super-Earth exoplanets. Due to the observational selection effect, the spectroscopically detected super-Earths are found mostly around lower-mass stars, whose habitable zones are inevitably narrower and closer. For such systems, any conclusion about potential habitability of a given exoplanet becomes especially uncertain, and the analysis becomes intricately involved with regard to such parameters as the amount of stellar irradiation, the chemical composition of a hypothetical atmosphere, and the internal heating. The rate of rotation is also a crucial parameter which for now remains unavailable from observation. The most advanced climatic simulations are based on a certain assumption of the spin–orbit state of the planet, e.g., a tidal synchronization (1:1 resonance) is assumed, as in Selsis et al. (2011). A tidally synchronized planet showing the same side to its host star has this side always exposed to plentiful irradiation, the other side staying dark. Such planets can hardly retain an atmosphere and can hardly be habitable. However, a spin–orbit locking into higher commensurabilities (e.g., 3:2, as in the case of Mercury) allows the planet to rotate with respect to the host star, and leaves the planet a possibility of sustaining a stable atmosphere and having water in the liquid form on the surface. Three-dimensional climatic simulations of the potentially habitable super-Earth GJ 581d, by Wordsworth et al. (2010), were performed for a set of possible spin–orbit resonances, including 2:1. As was later explained in Makarov et al. (2012), this resonance is the likeliest state of GJ 581d for a wide range of rheological parameters, assuming a terrestrial composition of its mantle. Wordsworth et al. (2010) drew attention to the fact that a tidally synchronized atmosphere may be short-lived because of the collapse of CO₂ on the night side. A similar phenomenon may occur on a slowly rotating planet. Both the constant angular lag tidal model and the constant time lag model predict that oblate planets with moderate and large eccentricities are captured in stable pseudosynchronous rotations, in which case their spin rate only slightly exceeds their orbital mean motion. This would make an entire class of detected exoplanets unsuitable for biological life. In this paper, we prove that the prediction of pseudosynchronism is germane to the abovementioned simplified models of tidal interactions, models inapplicable to solid planets or moons. Super-Earth exoplanets of Earth-like composition on eccentric orbits are likely to be captured into spin–orbit resonances higher than 1:1, but there is no such thing as pseudosynchronous rotation for these objects.

The authors thank the referee, Stanton Peale, who provided several incisive comments and criticisms, and who urged the authors to present physical interpretation of the results obtained in the paper. One of the authors (M.E.) is indebted to Sylvio Ferraz Mello and James G. Williams for numerous stimulating discussions on the topic of this work.

This research has made use of NASA's Astrophysics Data System.

As explained in Efroimsky (2012a, 2012b), the products k_l(ω_lmpq)sin _l(ω_lmpq) can be expressed through the mass and radius of the planet, and the real and imaginary parts of the complex compliance of its mantle. Thereby, the shape of the functional dependence of k_lsin _l upon ω_lmpq is defined by both the self-gravitation of the planet and its rheological properties. The functions turn out to be odd. They go continuously through nil, changing their sign, when the argument ω_lmpq goes through nil, i.e., when a commensurability is crossed.

These odd functions can then be written down as k_l(ω_lmpq)sin | _l(ω_lmpq) | Sgn (ω_lmpq). Here the product k_l(ω_lmpq)sin | _l(ω_lmpq) | is an even function of the tidal mode and can thus be treated as a function not of the tidal mode ω_lmpq but of its absolute value χ_lmpq = | ω_lmpq | , which is the actual frequency of the oscillating stress in the mantle:

$$\begin{eqnarray} \nonumber k_l(\omega _{{lmpq}}) \sin \epsilon _l(\omega _{{lmpq}})&=&k_l(\omega _{{lmpq}}) \sin |\,\epsilon _l (\omega _{{lmpq}})\,| \,\mbox{Sgn}\,(\omega _{{lmpq}})\\ &&\nonumber \\ &=&k_l(\chi _{{lmpq}}) \sin |\,\epsilon _l (\chi _{{lmpq}})\,| \,\mbox{Sgn}\,(\omega _{{lmpq}})\,. \nonumber\\ \end{eqnarray} \tag{ B1 }$$

The development in Efroimsky (2012a, 2012b) results in the following frequency dependence:

$$\begin{eqnarray} && k_l(\omega _{{lmpq}})\;\sin \epsilon _l(\omega _{{lmpq}})\,=\;\frac{3}{2\,({\it l}\,-\,1)}\;\,\nonumber\\ &&\quad \times\frac{{-}A_l\;J\;{\cal {I}}{\it {m}}[\bar{J}(\chi)]}{ ({\cal {R}}{\it {e}}[\bar{J}(\chi)]+A_l\;J)^2+({\cal {I}}{\it {m}}[\bar{J}(\chi)]\;)^2} \,\mbox{Sgn}\,(\omega _{{lmpq}}), \nonumber\\ \end{eqnarray} \tag{ B2 }$$

where χ is a shortened notation for the frequency χ_lmpq, while coefficients A_l are given by

$$\begin{eqnarray} &&A_{\it l}\, \equiv \;\frac{{(2\,{\it {l}}^{\,2}\,+\,4\,{\it {l}}\,+\,3)\,{\mu }}}{{{\it {l}}\,\mbox{g}\, \rho \,R}}=\frac{{3\;(2\,{\it {l}}^{\,2}\,+\,4\,{\it {l}}\,+\,3)\,{\mu }}}{{4\;{\it {l}} \,\pi \,G\,\rho ^2\,R^2}}{.} \end{eqnarray} \tag{ B3 }$$

with G being the Newton gravitational constant, and R, ρ, μ, and g being the radius, mean density, unrelaxed rigidity, and surface gravity of the planet. The functions

$$\begin{eqnarray} &&{\cal R}{\it e} [ \bar{J}(\chi)]=J+J\,(\chi \tau _{A})^{-\alpha }\;\cos \left(\frac{\alpha \,\pi }{2}\,\right) \;\Gamma (\alpha \,+\,1) \end{eqnarray} \tag{ B4 }$$

and

$$\begin{eqnarray} &&{\cal I}{\it m} [ \bar{J}(\chi)]={-}J (\chi \tau _{M})^{-1}-J\,(\chi \tau _{A})^{-\alpha }\;\sin \left(\frac{\alpha \,\pi }{2}\,\right)\;\Gamma (\alpha \,+\,1) \nonumber\\ \end{eqnarray} \tag{ B5 }$$

are the real and imaginary parts of the complex compliance $\bar{J}(\chi)$ of the mantle. Here, α is the Andrade parameter assuming values of about 0.3 for solid silicates and about 0.14–0.2 for partial melts. In our computations, we used α = 0.2. The quantity J is the unrelaxed compliance of the mantle, which is the inverse of the mantle's unrelaxed rigidity μ. The parameters τ_M and τ_A are typical timescales characterizing the mantle's viscoelastic and inelastic response, correspondingly.

The Maxwell time τ_M is the ratio of the mantle's viscosity η to its rigidity μ. While for the Earth's mantle it has a value of about 500 years, it may be much shorter for warmer planets and moons due to the exponential temperature dependence of the viscosity.

The inelastic (Andrade) time τ_A is expected to be of the same order as or lower than τ_M, over the frequencies higher than some threshold. For these frequencies then, the inelastic (containing τ_A) terms in (B4–B5) will be comparable to or larger than the viscoelastic (containing τ_M) terms. However, at frequencies below the threshold, inelasticity ceases to play a major role in the internal friction, giving way to viscous friction which becomes dominant. Therefore at very low frequencies the mantle's behavior approaches that of a Maxwell body. Mathematically, this implies that below the threshold the parameter τ_A increases rapidly as the frequency goes down (Efroimsky 2012a, 2012b). So only the first term in (B4) and the first term in (B5) are important, and we arrive at the complex compliance of a Maxwell material. The location of the frequency threshold may vary considerably for different planets. For the Earth, it is of the order of 1 yr⁻¹ (Karato & Spetzler 1990).

Numerical computations show that the probabilities of capture into resonances are not very sensitive to the value of τ_A, nor to the location of the threshold, nor to how quickly inelasticity yields to viscoelasticity with the decrease of frequency. In our numerics, we treat τ_A in the same way as in Makarov et al. (2012) and Makarov (2012). We set the threshold to be the same as for the solid Earth, 1 yr⁻¹. We then kept τ_A = τ_M over the frequencies above the threshold. For frequencies lower than the threshold, we set τ_A to grow exponentially with the decrease of the frequency. This way, at low frequencies the rheological model approaches the Maxwell one exponentially. For details, see Makarov et al. (2012) and Makarov (2012).

Writing a code, it is easier to divide both the numerator and denominator of (B2) by J²:

$$\begin{eqnarray} k_l(\omega _{{lmpq}})\sin \epsilon _l(\omega _{{lmpq}})&=&\frac{3}{2\,({\it l}\,-\,1)}\;\,\frac{{-}A_l\;{\cal {I}}}{({\cal {R}} +A_l)^2+{\cal {I}}^{2}} \nonumber\\ &&\times \mbox{Sgn}\,(\omega _{{lmpq}}), \end{eqnarray} \tag{ B6 }$$

where ${\cal R}$ and ${\cal I}$ are the dimensionless real and imaginary parts of the complex compliance:

$$\begin{eqnarray} &&{\cal R}=1+(\chi \tau _{A})^{-\alpha }\;\cos \left(\frac{\alpha \,\pi }{2}\,\right) \;\Gamma (\alpha \,+\,1), \end{eqnarray} \tag{ B7 }$$

$$\begin{eqnarray} &&{\cal I}={-}(\chi \tau _{M})^{-1}-(\chi \tau _{A})^{-\alpha }\;\sin \left(\frac{\alpha \,\pi }{2}\,\right)\;\Gamma (\alpha \,+\,1), \quad \quad \quad \quad \quad \end{eqnarray} \tag{ B8 }$$

χ being a short notation for the physical forcing frequency χ_lmpq ≡ | ω_lmpq |.