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Dynamics of rotation of super-Earths

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We numerically investigate the dynamics of rotation of several close-in terrestrial exoplanet candidates. In our model, the rotation of the planet is disturbed by the torque of the central star due to the asymmetric equilibrium figure of the planet. We model the shape of the planet by a Jeans spheroid. We use surfaces of section and spectral analysis to explore numerically the rotation phase space of the systems adopting different sets of parameters and initial conditions close to the main spin–orbit resonant states. One of the parameters, the orbital eccentricity, is critically discussed here within the domain of validity of orbital circularization timescales given by tidal models. We show that, depending on some parameters of the system like the radius and mass of the planet, eccentricity etc., the rotation can be strongly perturbed and a chaotic layer around the synchronous state may occupy a significant region of the phase space. 55 Cnc e is an example.

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  1. In this work only the prograde direction of both, orbital and rotational motion, are considered. However, exoplanets in retrograde orbits with respect to the rotation axis of the star have been detected (e.g. Hébrard et al. 2011, and references therein). Retrograde rotation is also a possible equilibrium state (Correia et al. 2008). The study of spin–orbit resonances in retrograde motion would be interesting.

  2. Note however that the timescale for orbital circularization decreases with the stellar mass (e.g. Rodríguez and Ferraz-Mello 2010). Thus, some single-planets orbiting small stars may have large circularization timescales (for example, GJ 1214b, among others).

  3. In this hypothesis we are neglecting the dynamic tidal torque raised by the central star which also affects the planet rotation. However, several planets which we study here are so close to their star that tidal effects may be important, mainly in the case of eccentric orbits. Appendix 2 presents one case in which the tidal torque is taken into account and applied to CoRoT-7b super-Earth planet.

  4. The dependence on \(\left( \frac{\varOmega }{n}\right) ^2\) in (3) is in fact weak. For instance, in the case of 55 Cnc e, \(\epsilon \sim 2.8\times 10^{-2}\). If we neglect the rotation term and consider \(\varOmega =0\), we have \(\epsilon \sim 2.75\times 10^{-2}\). These values are smaller than when the uncertainty of the radius of the planet is considered in the calculation of \(\epsilon \). For instance, in the case of 55 Cnc e, Gillon et al. (2012) give \(R=2.17\pm 0.10\) \(R_E\), which results the maximum values: \(\epsilon \sim 3.21\times 10^{-2}\) (\(\varOmega =n\)), \(\epsilon \sim 3.14\times 10^{-2}\) (\(\varOmega =0\)).

  5. As we will see in details in Sect. 3, for \(\epsilon >0.01\) the rotation may be highly disturbed by the torque of the central star in the case of eccentric orbits. However, this division between “small” and “large” prolateness is arbitrary and will be used in this study only for didactic purposes.

  6. We remark that very large values of \(\epsilon \) close to the unit are not valid within the first-order approximation in \(\epsilon \).

  7. We use the code of Everhart (1985).

  8. In the Sect. 4 we list the results for some planets with \(7.5<P<33\) days.

  9. Following classical astrodynamics texts (e.g. Danby 1988), we use the term physical librations to refer to motions associated with the non-spherical shape of the secondary body. The often called optical librations do not require an asymmetry of the secondary since they are not associated with a disturbed rotator. For instance, the optical libration in longitude exists due to differences between a (constant) velocity of rotation and the instantaneous orbital velocity in an eccentric orbit. In the disturbed (non-spherical) case, the optical libration is superposed with the forced component.

  10. Note that \(\omega _0\) is a linear function of \(n\) with linear coefficient depending on the prolateness and \(H(p,e)\). In the case of 1:1 resonance, at first order of eccentricity, the relation between \(\omega _0\) and \(n\) does not vary with the value of eccentricity since \(H(1,e)=1\). This does not occur for large eccentricity for other low-order resonances like 3:2, where \(H(3/2,e)=\frac{7}{2}e\) at first order.

  11. Utilizing \(\omega _0^2=3n^2|H(p,e)|\frac{B-A}{C}\) we can obtain the value 9.783 days for the period of free libration, which agrees with the period of the long-term oscillation given in the result of the numerical simulation of the full equations shown in Fig. 4a. Note also that \(\frac{2\pi }{\omega _0}\sim 28.5\frac{2\pi }{n}\), and the factor 28.5 is comparable to the number of short-term oscillations seen over one period of the free libration.

  12. Our numerical experiments show that in the cases of strong perturbation, the domain of the 1:2 resonance is isolated from the chaotic regions located close to it. Wisdom et al. (1984), in their study on the rotation of the satellite Hyperion, discuss this property of the 1:2 resonance. In fact, the averaged torques associated with the 1:1 and 1:2 spin–orbit resonances show that they are the strongest ones among all low-order resonances (e.g. Goldreich and Peale 1966).

  13. According to Valencia et al. (2007a), supposing a solid-like structure for 55 Cnc e, its radius cannot be larger than \(\simeq \!1.8 R_E\). Since the estimated radius of 55 Cnc e is \(\simeq \!2.17 R_E\) it cannot be considered a pure solid-like planet, admitting a non-negligible fraction of other non-solid components in its composition. However, recent calculations with models of interior structure of 55 Cnc e show the possibility of its solid-like nature (Madhusudhan et al. 2012). Moreover, aiming to improve our model, it could be interesting to consider the perturbations in the rotation of the planet adopting different layers (e.g. van Hoolst et al. 2008).

  14. Wisdom (2004) shows that, in the case of regular satellites of the Solar System, a capture in such rotational state may enhance the tidal heating in the satellite interior by several orders of magnitude. The same would occur for close-in super-Earths.

  15. Moreover, evolution through resonances in the case of strong perturbation can result in the chaotic tumbling of the rotation axis of the planets, similarly to the case of some Solar System bodies (Wisdom et al. 1984; Wisdom 1987).

  16. For comparison it is necessary to consider the relationship between \(J_{22}\) (the equatorial ellipticity) and \(B-A\), that is, \(B-A=4C_{22}mR^2\) (see Beutler 2005).


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Appendix 1: Orbital circularization due to tidal effect

We consider an interacting pair composed of a slow-rotating star and a close-in planet. The aim is to analyze the timescale for orbital circularization due to tidal interaction. We refer to Ferraz-Mello et al. (2008) and Rodríguez and Ferraz-Mello (2010) for assumptions, definitions of quantities and further details.

The average variation of the eccentricity due to the combined effects of stellar and planetary tides is given by

$$\begin{aligned} \langle \dot{e}\rangle =-\frac{1}{3}nea^{-5}(18\hat{s}+7\hat{p}), \end{aligned}$$


$$\begin{aligned} \hat{s}=\frac{9}{4}\frac{k_{20}}{Q_{0}}\frac{m}{m_{0}}R_{0}^5 ; \qquad \hat{p}=\frac{9}{2}\frac{k}{Q}\frac{m_{0}}{m}R^5, \end{aligned}$$

are two parameters which stand for stellar and planetary tides, respectively. The symbol \(0\) refers to the star, \(k_2\) is the second degree Love number, \(Q\) is the dissipation function or quality factor. It can be shown that the orbital circularization can be accounted by planetary tides only. Indeed, \(\hat{s}\) is proportional to \((m/m_0)(1/Q_0)\) which becomes a small quantity for small mass planets and typical stellar \(Q_0\) values. Thus, the contribution of stellar tides can be safely neglected in our analysis (see Rodríguez and Ferraz-Mello 2010).

The timescale for orbital circularization can be defined by \(\tau _e\equiv e/|\langle \dot{e}\rangle |\), or

$$\begin{aligned} \tau _{e}=\frac{3n^{-1}a^{5}}{7\hat{p}}. \end{aligned}$$

Writing Eq. (11) as a function of the semi-major axis, we obtain

$$\begin{aligned} \tau _{e}=A\,a^{13/2}, \end{aligned}$$

where \(A=\frac{3\hat{p}^{-1}}{7\sqrt{Gm_{0}}}\). Alternatively, we can express the result as a function of the orbital period \(P\) as follows

$$\begin{aligned} \tau _{e}=BP^{13/3}, \end{aligned}$$

where \(B=A\Big (\frac{Gm_{0}}{4\pi ^2}\Big )^{13/6}\). Figure 8 shows the plot of Eq. (13) for a planet with the properties of CoRoT-7b, a super-Earth planet with \(m=8.5m_E\) (Ferraz-Mello et al. 2011), \(R=1.68R_E\), assuming the values \(Q=100\), \(Q=500\), \(Q=1000\), and \(k_2=0.35\).

We clearly see that the circularization timescale decreases for short-period planets, as Eq. (13) indicates. Moreover, noting that \(\tau _e\propto Q\), the orbital circularization would become faster in the case of small \(Q\) values (i.e., large dissipation). As an example, for \(P=4\) days and \(Q=100\) we have \(\tau _{e}\simeq 181\) Myr.

It is important to note that the parameter \(B\) is linearly dependent on \(Q/k_2\), which is a quantity poorly known for extrasolar planets. Hence, the plot shown in Fig. 8 can be strongly modified if other values of the planet dissipation are considered.

Fig. 8
figure 8

Plot of Eq. (13) for CoRoT-7b super-Earth planet. The mass of the host star is \(m_{0}=0.93m_{ SUN}\)

The result (13) can be useful for quantifying the efficiency of tides to produce orbital circularization of close-in planets. Note that, in some cases, \(\tau _e\) can be compared with the age of the system, indicating that the orbit of the close-in planet should be circularized during the planet lifetime. However, for large \(P\), \(\tau _e\) can be even larger than the age of a typical planetary system, in which case a non-circular orbit should be expected due to tidal interaction.

Appendix 2: Capture in spin–orbit resonance

The numerical exploration of the planet rotation, which is subject to the gravitational torque of the star, has shown different behaviors, including the oscillation around spin–orbit resonances. When a dissipative effect like the tidal torque is included, the rotation can be captured in a resonant motion. The specific capture depends on the eccentricity, and also on \(Q\) and \(\epsilon \). Hence, as \(e\) is tidally damped, the capture should become unstable and the rotation can achieve another resonant state, which, at the same time, should result in a temporary trapping. When the orbit completes the circularization due to the tidal torque, the final evolution results in synchronization of the orbital and rotational periods (i.e., the 1:1 spin–orbit resonance). The reader is referred to Goldreich and Peale (1966) and Rodríguez et al. (2012) for further details on the spin–orbit evolution of close-in planets.

Let us first consider the torque due to the prolateness or permanent equatorial deformation (i.e., \(\text{ a }\ne \text{ b }\)) on a rotating body of mass \(m\) and radius \(R\). The maximum torque, averaged over an orbital period, is given by

$$\begin{aligned} \langle N\rangle =-\frac{3}{2}n^2(B-A)H(p,e) \end{aligned}$$

(see Goldreich and Peale 1966 and Eq. 1), where \(H(p,e)\) are power series in \(e\) and \(p=\varOmega /n\), with \(\varOmega \) the angular velocity of rotation of the deformed body (Goldreich and Peale 1966). Equation (14) assumes that there is commensurability between \(\varOmega \) and \(n\), indicating that \(p=\ldots -1,-1/2,1,1/2\cdots \)

In addition to the above torque, we also consider the tidal torque driven by the central body of mass \(m_0\). The average tidal torque reads

$$\begin{aligned} \langle T\rangle =-\frac{3k_2Gm_0^2R^5}{8a^6}[4\varepsilon _0+e^2 (-20\varepsilon _0+49\varepsilon _1+\varepsilon _2)] \end{aligned}$$

(see Ferraz-Mello et al. 2008), where \(k_2\) is the second degree Love number, whereas \(\varepsilon _i\) are the phase lags of tidal waves with frequency \(\nu _i\). The phase lags account for the internal viscosity, which introduces a delay between the action of the tidal force and the corresponding deformation.

Several tidal models can be used to fix the dependence between phase lags and frequencies, that is, the function \(\varepsilon _i(\nu _i)\). We first consider what is usually referred to the linear model, where \(\varepsilon _i=\nu _i\Delta t\), where \(\Delta t\) is known as time lag and is considered constant in the linear model (Mignard 1979). The frequencies associated with the phase lags appearing in Eq. (15) are \(\nu _0=2\varOmega -2n\), \(\nu _1=2\varOmega -3n\) and \(\nu _2=2\varOmega -n\) (see Ferraz-Mello et al. 2008). Replacing these values in Eq. (15) and applying the linear model, we obtain

$$\begin{aligned} \langle T\rangle =-\frac{3k_2\Delta tGm_0^2R^5n}{2a^6}[2p-2+(15p-27)e^2]. \end{aligned}$$

The time lag can be related to the most used quantity \(Q\), the dissipation function. Since \(\varepsilon _i\simeq Q_i^{-1}\), it follows, under the assumption of a linear model, \(Q_0^{-1}\simeq 2n\Delta t(p-1)\), \(Q_1^{-1}\simeq n\Delta t(2p-3)\), \(Q_2^{-1}\simeq n\Delta t(2p-1)\). We note that singularities in \(Q_i\) are associated with spin–orbit commensurability with \(p=1\), \(p=3/2\) and \(p=1/2\). Since we restrict our investigation to the cases of \(1:1\) and \(3:2\) spin–orbit resonances, we call \(Q=Q_2\) in order to avoid \(Q_i\)-singularities. Hence, the relationship between \(Q\) and \(\Delta t\) is given by \(Q=1/[n\Delta t(2p-1)]\).

Fig. 9
figure 9

Critical value of the equatorial ellipticity as a function of orbital eccentricity for two spin–orbit resonances. The CoRoT-7+7b system illustrates the example

1.1 Stationary solutions

By definition, the stationary solutions of the rotation are those which satisfy

$$\begin{aligned} \langle N\rangle +\langle T \rangle =0. \end{aligned}$$

Because we are interested in those solutions for which there exists commensurability between spin and orbital revolutions, we can determine a critical value of \(\epsilon \) (we call it \(\epsilon ^{*}\)) which allows a spin–orbit resonance motion to be maintained when the planet’s rotation is under simultaneous action of two torques. Thus, using Eqs. (14, 16, 17) we obtain

$$\begin{aligned} \epsilon ^{*}=-\frac{1}{\xi }\frac{k_2}{Q}\frac{m_0}{m}\left( \frac{R}{a}\right) ^3\frac{[2p-2+(15p-27)e^2]}{(2p-1)H(p,e)}. \end{aligned}$$

where we have used the third Kepler law and \((B-A)\simeq C\epsilon =\xi mR^2\epsilon \), where \(C\) is the moment of inertia about the rotation axis and \(\xi =\frac{C}{mR^2}\), \(0<\xi \le 2/5\). (\(\epsilon ^{*}\) must not be confused with that given in Eq. (8)).

The condition \(\epsilon >\epsilon ^{*}\) is usually known as stability condition of the \(p\)-resonance (e.g. Goldreich and Peale 1966). In fact, the stability condition requires that \(\langle T\rangle \) not exceed the maximum restoring torque \(\langle N\rangle \) and, for that reason, \(\epsilon ^{*}\) should be considered as a critical value.

We note that the case \(p=1\) is in agreement with the result found in Ferraz-Mello et al. (2008) for the synchronous motion.Footnote 16

Figure 9 shows the variation of \(\epsilon ^{*}\) with the orbital eccentricity, taking the planet CoRoT-7b as an example. We adopt \(k_2=0.35\), \(\xi =0.4\) and \(Q=100\). The cases of \(1:1\) (synchronous rotation) and \(3:2\) spin–orbit resonances are shown. For second order in eccentricity, we have \(H(1,e)=1-5e^2/2\) and \(H(3/2,e)=7e/2\). The dashed horizontal line indicates the value of \(\epsilon \) used in the simulations for CoRoT-7b (\(\epsilon =0.00992\); Fig. 2a, b). We note that \(\epsilon >\epsilon ^{*}\) in both cases, indicates that the resonant motion should be stable for the range of considered eccentricity. However, \(\epsilon ^{*}\) can reach high values for very small \(e\), as Eq. (18) indicates. On the other hand, we have seen in the numerical simulations (assuming no tides, Fig. 2b) that the domain of the \(3:2\) spin–orbit resonance is very small for almost-circular orbits.

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Callegari, N., Rodríguez, Á. Dynamics of rotation of super-Earths. Celest Mech Dyn Astr 116, 389–416 (2013).

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