Impact of a moon on the evolution of a planet’s rotation axis: a non-resonant case

Abstract

We investigate how the temporal evolution of the rotation axis of a hypothetical exo-Earth is affected by the presence of a satellite, an exo-Moon. Namely, we study analytically and numerically how the range of the nutation angle of an exo-Earth changes if an exo-Moon is added to a system comprised of an exo-Sun, the exo-Earth and exoplanets. We say that the impact of an exo-Moon is stabilising if upon including the exo-Moon the range of the nutation angle decreases, and destabilising otherwise. The problem is considered in a general set-up. The exo-Earth is supposed to be rigid, axially symmetric and almost spherical, the difference between the largest and the smallest principal moments of inertia being a small parameter of the problem. Assuming the orbits of the celestial bodies to be quasi-periodic, we apply time averaging over fast variables associated with order one frequencies to study rotation of the exo-Earth at times large relative the respective periods. Non-resonant frequencies are assumed. For a system comprised of the exo-Sun and exoplanets in the absence of small orbital frequencies, the system is integrable, which allows to calculate the range of the nutation angle as a function of initial conditions. Using these expressions, we identify a class of systems for which we prove analytically that the impact of the exo-Moon is stabilising and a class where it is destabilising. Namely, if the orbits of the planets are circular and their orbital planes coincide then the impact is destabilising. The impact is stabilising if the angle between orbital planes of the exo-Moon and the exo-Earth vanishes. We also investigate numerically how the impact of the exo-Moon in a particular system comprised of a star and two planets varies on modifying parameters of the orbits of the exo-Moon and the second planet, and the initial nutation angle.

Appendices

Calculation of $\Delta$ in a planetary system, comprised of the exo-Sun, exo-Earth and exoplanets

Here we give a detailed derivation of approximations (28) for the range of the nutation angle from the Eqs. (18), (26), (27) for the temporal evolution of the angles I and h.

Steady states of system (18) satisfy $\,\mathrm{d}\,I/\,\mathrm{d}\,t=\,\mathrm{d}\,h/\,\mathrm{d}\,t=0$. Therefore, due to (26) they can be found from the equations

$$\begin{aligned} \begin{array}{l} \sin 2I(-D+\alpha \cos 2h')+\beta \cos 2I\cos (h'+\gamma )=0,\\ -\sin ^2I\alpha \sin 2h'-\beta \sin 2I\sin (h'+\gamma )=0. \end{array} \end{aligned}$$

(37)

Since the mass of the exo-Sun is much larger than the masses of planets, due to (15), (16) and (27) we have that $D\gg \max (|\alpha |,|\beta |)$. Therefore, the first equation in (37) implies $\sin 2I\approx 0$. Hence, the steady states are

$$\begin{aligned} I\approx 0;\quad I\approx \pi ;\quad I\approx \pi /2,\ h'\approx 0,\pi /2,\pi ,3\pi /2, \end{aligned}$$

where $(I\approx \pi /2,h'\approx 0)$ and $(I\approx \pi /2,h'\approx \pi )$ are saddles and the other ones are centres. The saddles are connected by heteroclinic trajectories that divide the celestial sphere into four regions, each comprised of a centre and closed trajectories around it, see Fig. 2a.

Since $\,\mathrm{d}\,I/\,\mathrm{d}\,h'=\,\mathrm{d}\,I/\,\mathrm{d}\,t(\,\mathrm{d}\,h'/\,\mathrm{d}\,t)^{-1}$, the extrema of $I(h')$ take place at

$$\begin{aligned} -\sin ^2I\alpha \sin 2h'-\beta \sin 2I\sin (h'+\gamma )=0. \end{aligned}$$

(38)

Below we evaluate $\Delta (I_0,h_0)$ defined by (1), where $h_0=\pi /2$ (shown by grey line in Fig. 2a) and consider $0\le I_0\le \pi /2$. Since D is large compared with $\alpha$ and $\beta$, the nutation angle $I(h')$ is close to $I_0$ and we can write $I(h')=I_0+I_1(h')$. Therefore,

$$\begin{aligned} \sin ^2I\approx \sin ^2I_0+I_1\sin 2I_0+I_1^2\cos 2I_0,\quad \sin 2I\approx \sin 2I_0+2I_1\cos 2I_0. \end{aligned}$$

Since ${\tilde{{{\mathcal {G}}}}}(I,h')$ is a constant on trajectories, substituting the above expressions into (26) we obtain a quadratic equation on $I_1(h')$

$$\begin{aligned} \begin{array}{l} I_1^2(-D\cos 2I_0)+I_1(-D\sin 2I_0+2\beta \cos 2I_0\cos (h'+\gamma ))+\\ \alpha \sin ^2I_0(\cos 2h'-1)+\beta \sin 2I_0(\cos (h'+\gamma )+\sin \gamma )=0. \end{array} \end{aligned}$$

(39)

The range $\Delta$ is a continuous function of $I_0$ and $h_0$ inside a region and is discontinuous at a boundary. For a heteroclinic trajectory through $(\pi /2,0)$, we have

$$\begin{aligned} {\tilde{{{\mathcal {G}}}}}(I,h')=(-D+\alpha ), \end{aligned}$$

hence trajectories with the initial conditions $(I_0,\pi /2)$ such that

$$\begin{aligned} \sin ^2I_0(-D-\alpha )+\beta \sin 2I_0\sin \gamma <-D+\alpha \end{aligned}$$

(40)

belong to the equatorial regions, while the other ones to the polar regions. Since $D\gg \max (\alpha ,\beta )$, the inequality (40) may be simplified to

$$\begin{aligned} |I_0-{\pi \over 2}|<\biggl ({2\alpha \over D}\biggr )^{1/2}, \end{aligned}$$

(41)

i.e. the trajectories through $(I_0,\pi /2)$ belong to an equatorial region if $\pi /2-\delta _{het}<I_0<\pi /2+\delta _{het}$, where $\delta _{het}=(2\alpha /D)^{1/2}$, and to a polar one otherwise.

To solve Eq. (39) we consider three possibilities for $I_0$ if the initial condition $(I_0,h_0)$ belongs to the polar region:

• (i) $D^{1/2}|\sin I_0|<\beta ^{1/2}$, $|\sin I_0|<|\cos I_0|$;

• (ii) $D^{1/2}\sin ^2I_0>\max (\alpha ,\beta ^{1/2}|\sin I_0|)$;

• (iii) $D^{1/2}\sin ^2I_0<\alpha ^{1/2}$, $|\sin I_0|>|\cos I_0|$, $|I_0-{\pi \over 2}|>(2\alpha )^{1/2}D^{-1/2}$.

and the equatorial region:

• (iv) $|I_0-{\pi \over 2}|<(2\alpha )^{1/2}D^{-1/2}$.

Since $D\gg \beta$, in case (i) we have that $I_0$ is close to 0 or $\pi$. Therefore, from (38) the extrema of $I(h')$ take place at $\sin (h'+\gamma )\approx 0$. Substituting $h'+\gamma =0$ and $\pi$ into (39), solving the quadratic equation and subtracting the root at $h'+\gamma =0$ from the one at $h'+\gamma =\pi$ we find that

$$\begin{aligned} \Delta \approx {2\beta \over D}. \end{aligned}$$

(42)

In case (ii) in Eq. (39), the quadratic term can be neglected and the remaining linear equation can be easily solved for any value of $h'$. We cannot derive from (38) the particular value of $h'$ where the maxima and minima take place; hence, we give upper and lower bound for $\Delta$ (they differ less than a factor 2):

$$\begin{aligned} {\max (|\alpha |\sin I_0,2|\beta |\cos I_0)\over D\cos I_0}<\Delta \le {|\alpha |\sin I_0+2|\beta |\cos I_0\over D\cos I_0}. \end{aligned}$$

(43)

Alternatively, we introduce the function

$$\begin{aligned} \chi (I_0,\alpha ,\beta ,\gamma )= \max _{0\le h\le 2\pi }f(I_0,\alpha ,\beta ,\gamma ,h)- \min _{0\le h\le 2\pi }f(I_0,\alpha ,\beta ,\gamma ,h) \end{aligned}$$

(44)

where

$$\begin{aligned} f(I_0,\alpha ,\beta ,\gamma ,h)=\alpha \sin ^2I_0\cos 2h+\beta \sin 2I_0\cos (h+\gamma ). \end{aligned}$$

(45)

Then

$$\begin{aligned} \Delta (I_0,h_0)\approx {\chi (I_0,\alpha ,\beta ,\gamma )\over D\cos I_0}. \end{aligned}$$

(46)

As we noted $D\gg \alpha$, therefore in case (iii) we have that $I_0\approx \pi /2$. Hence (see (38)), the minima of $I(h')$ take place at $h'=0$ and $\pi$ and the maxima at $h'=\pi /2$ and $3\pi /2$. Solving (39) we obtain that

$$\begin{aligned} \Delta \approx {D\sin 2I_0-\beta \cos 2I_0\cos \gamma + ((-D\sin 2I_0+\beta \cos 2I_0\cos \gamma )^2+8D\alpha )^{1/2}\over 2D}. \end{aligned}$$

(47)

In case (iv) a trajectory twice intersects the meridian $h'=\pi /2$, at the intersection points I takes the maximum and minimum values, $I_{\max }$ and $I_{\min }$, for this particular trajectory (see (38)). Moreover, (39) implies that $I_{\max }-\pi /2\approx \pi /2-I_{\min }$. Hence,

$$\begin{aligned} \Delta \approx 2|\pi /2-I_0|. \end{aligned}$$

(48)

Overall, for $0\le I_0\le \pi /2$ we have

$$\begin{aligned} \begin{array}{lll} \hbox { case } &{} I_0 &{} \Delta \\ \hline (i) &{} D^{1/2}|\sin I_0|<\beta ^{1/2} &{} \hbox { equation }(42) \\ (ii) &{} D^{1/2}\sin ^2I_0>\max (\alpha ,\beta ^{1/2}|\sin I_0|) &{} \hbox { inequality }(43) \\ &{}&{}\hbox { or equation }(46) \\ (iii) &{} D^{1/2}\sin ^2I_0<\alpha ^{1/2},\ |I_0-{\pi \over 2}|>(2\alpha )^{1/2}(D')^{-1/2} &{} \hbox { equation }(47) \\ (iv) &{} |I_0-{\pi \over 2}|<(2\alpha )^{1/2}D^{-1/2} &{} \hbox { equation} (48) \end{array} \end{aligned}$$

(49)

Remark 2

We have calculated $\Delta (I_0,h_0)$ for $h_0=\pi /2$ only. Unless $I_0$ is close to $\pi /2$, the dependence of $\Delta$ on $h_0$ is marginal, as illustrated by Fig. 2a. By contrast, near $I_0=\pi /2$ the range essentially depends on $h_0$. For example, for an initial condition corresponding to a saddle steady state the range vanishes, while for a slightly perturbed initial condition one can get $\Delta$’s close to $2\delta _{het}$ (inside the equatorial region) or $\delta _{het}$ (inside the polar region), see in Fig. 2a. Investigation of the dependence of $\Delta (I_0,h_0)$ on $h_0$ for $I_0$ near $\pi /2$, which can be carried out similarly, is left for future studies.

The canonical change of variables employed in Sect. 4.3

Let the functions $D_j(t)$ in the averaged Eqs. (18), (19) be represented as the sums

$$\begin{aligned} D_j=D_j^*+D^{(2)}_j,\ \quad D_j^*=D_j^{(1)}+\sum _{k=3}^n D_j^{(k)}, \end{aligned}$$

where $D_j^{(1)}$ (21) result from the torque from the Sun, $D^{(2)}_j$ (24) from the Moon and $D_j^{(3)},\ldots ,D_j^{(n)}$ from the planets. As discussed in Sect. 2, the terms $D_j^*$ are time-independent.

The mean Hamiltonian takes the form

$$\begin{aligned} \begin{array}{l} {{\overline{{{\mathcal {H}}}}}}_1={\displaystyle -{3\varepsilon \over 2}(C_1-A_1)[ \sin ^2I(-(D_1^*+D_1^{(2)})\sin ^2h-(D_2^*+D_2^{(2)})\cos ^2h}\\ +(D_3^*+D_3^{(2)})+(D_4^*+D_4^{(2)})\sin (2h))- \sin 2I((D_5^*+D_5^{(2)})\sin h-(D_6^*+D_6^{(2)})\cos h)]. \end{array} \end{aligned}$$

(50)

Substitution of (24) and (29) into (50) followed by a series of algebraic transformations yield

$$\begin{aligned} \begin{array}{l} {{\overline{{{\mathcal {H}}}}}}_1= {\displaystyle -{3\varepsilon \over 2}(C_1-A_1)[ (-D_1^*\sin ^2h-D_2^*\cos ^2h+D_3^*+D_4^*\sin (2h)+}\\ {\displaystyle \Xi ({1\over 2}\sin ^2i-\cos ^2i)+{\Xi \over 2}\sin ^2i\cos (2\Omega -2h)) \sin ^2I-}\\ {\displaystyle (D_5^*\sin h-D_6^*\cos h +{\Xi \over 2}\sin 2i\cos (\Omega -h))\sin 2I]}. \end{array} \end{aligned}$$

(51)

Using the generating function $F_2(h,H')=H'(h-\Omega _0-\sigma _\mathrm{n}t)$, we obtain that in the canonical coordinates $(H',h')=(H,h-\Omega _0-\sigma _\mathrm{n}t)$ the Hamiltonian (51) takes the form

$$\begin{aligned} {{\mathcal {H}}}'={{\overline{{{\mathcal {H}}}}}}_1+{\partial F_2\over \partial t}= {{\overline{{{\mathcal {H}}}}}}_1-\sigma _\mathrm{n}H'. \end{aligned}$$

Therefore, (10), (11) imply that

$$\begin{aligned} {\,\mathrm{d}\,h'\over \,\mathrm{d}\,t}={\rho \over \sin I} {\partial {{\mathcal {G}}}'\over \partial I},\qquad {\,\mathrm{d}\,I\over \,\mathrm{d}\,t}=-{\rho \over \sin I} {\partial {{\mathcal {G}}}'\over \partial h'}, \end{aligned}$$

(52)

where

$$\begin{aligned} \begin{array}{l}{\displaystyle {{\mathcal {G}}}'(I,h')=\sin ^2I (-D_1^*\sin ^2h'-D_2^*\cos ^2h'+D_3^*+D_4^*\sin (2h')+}\\ {\displaystyle \Xi ({1\over 2}\sin ^2i-\cos ^2i)+{\Xi \over 2}\sin ^2i\cos 2h')+}\\ {\displaystyle \sin 2I(D_5^*\sin h'-D_6^*\cos h'+ {\Xi \over 2}\sin 2i\cos h')+{\sigma _\mathrm{n}\over \rho }\cos I} \end{array} \end{aligned}$$

(53)

and $\rho =3\varepsilon (C_1-A_1)/2G$. In the new variables $(H,h')$, we have that ${{\mathcal {G}}}'=const$ along the trajectories. Note that the Eq. (53) is invariant under the symmetry $h'\rightarrow -h'$

$$\begin{aligned} {{\mathcal {G}}}'(I,h')={{\mathcal {G}}}'(I,-h'). \end{aligned}$$

(54)

The equation is also invariant after the transformation$(\sigma _\mathrm{n},I,h')\rightarrow (-\sigma _\mathrm{n},\pi -I,h'+\pi )$, therefore without the loss of generality we consider non-negative $\sigma _\mathrm{n}$’s only.