Olga Podvigina

We investigate the temporal evolution of the rotation axis of a planet in a system comprised of the planet (which we call an exo-Earth), a star (an exo-Sun) and a satellite (an exo-Moon). The planet is assumed to be rigid and almost spherical, the difference between the largest and the smallest principal moments of inertia being a small parameter of the problem. The orbit of the planet around the star is a Keplerian ellipse. The orbit of the satellite is a Keplerian ellipse with a constant inclination to the ecliptic, involved in two types of slow precessional motion, nodal and apsidal. Applying time averaging over the fast variables associated with the frequencies of the motion of exo-Earth and exo-Moon, we obtain Hamilton’s equations for the evolution of the angular momentum axis of the exo-Earth. Using a canonical change of variables, we show that the equations are integrable. Assuming that the exo-Earth is axially symmetric and its symmetry and rotation axes coincide, we identify possible types of motions of the vector of angular momentum on the celestial sphere. Also, we calculate the range of the nutation angle as a function of the initial conditions. (By the range of the nutation angle we mean the difference between its maximal and minimal values.)

The complementary subspaces (CS) method is aimed at numerically solving equations of the form $P_{\cal U}^{}(Av-f)=0$ emerging when evolutionary equations are integrated by spectral methods. Here $v$ must belong to the finite-dimensional space ${\cal V}$ comprised of functions satisfying the prescribed boundary conditions and $P_{\cal U}^{}$ is an orthogonal projection on a finite-dimensional space ${\cal U}$. The idea of the CS method is to find the solution to the original problem by solving a~modified problem, $P_{\cal G}^{}(Aw-\widetilde f)=0$, $w\in{\cal W}$, and either adding the correction $\widetilde f-f$ before solving the modified problem, or computing the correction $v-w$ afterwards. The spaces ${\cal W}$ and ${\cal G}$ are chosen in such a way that solving the modified problem requires less operations than the original one. The method was introduced in [23]; we propose now its modification allowing more freedom in choosing the spaces for the modified problem and in computing the correction. The algorithm is discussed in the general form and in the case of the spaces spanned by linear combinations of Chebyshev polynomials. The method is applied for numerical investigation of convective flows in a plane horizontal layer heated from below and rotating about an inclined axis with no-slip horizontal boundaries. For the employed values of control parameters the temporal behavior is comprised of repeating events and is possibly related to heteroclinic connections between unstable steady states.